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{"url":"http:\/\/math.stackexchange.com\/questions\/29297\/solvable-group-of-order-pqr2","text":"# solvable group of order $pqr^2$\n\n$|G|=pqr^2$ where $3\\leq p<q<r$ prime. Show that if $r>\\frac{1}{2}(pq-1)$ then $G$ is solvable.\n\nI took $H\\leq G$ $r$-sylow subgroup of $G$, there is a theorem claiming that there exists a homomorphism from $G$ to $S_{pq}$ ($pq$ is the index of $H$).\n\nIf the homomorphism is injective then $G$ is a subgroup of $S_{pq}$, that means $pqr^2|(pq)!$ so $r^2|(pq-1)!$ but we know $r^2>2r>pq-1$ from what follows that the homomorphism can't be injective, that means there is a non trivial kernel which is a maximal subgroup of $H$ and a normal subgroup of $G$, it can be of order $r$ or $r^2$, lets name it K.\n\nIf $|K|=r^2$ then the sequence $\\{e\\}\\triangleleft K\\triangleleft G$ has a factor of order $r^2$ which is abelian as a square of a prime and a factor of order $pq$.\n\nThe other option is $|K|=r$, then the factors are of order: $r$ and $pqr$.\n\nI am a bit stuck from here.\n\n-\nI don't know if it matters, but if |G| = pqrr with 2 \u2264 p < q < r, then G is solvable. Squaring the largest prime factor doesn't help to get nonsolvable groups. You need to square the smaller ones. \u2013\u00a0 Jack Schmidt Mar 27 '11 at 16:03\n\nGroups of order $r$ are solvable. Groups of order $pqr$ are solvable. So your group is an extension of two solvable groups, so solvable. In a group of order $pqr$, the Sylow $r$-subgroup is normal, and in the quotient of order $pq$, the Sylow $q$-subgroup is normal.\n\n-\ndid you mention a group of order $pqr$ is solvable, once they are all prime and regardless of duplicity? May I ask how to get that? \u2013\u00a0 Honghao Aug 30 '12 at 16:45\n\nYou are left to show that a group of order $pqr$ is solvable, and this can be done exactly the same way you began the $pqr^2$ case.\n\n-\nWhen p=3, q=5, r=11, Sym(pq) does not have order prime to r. The order just is not divisible by r*r. \u2013\u00a0 Jack Schmidt Mar 27 '11 at 15:59\noops, sorry I thought we had $r>pq$ \u2013\u00a0 Plop Mar 27 '11 at 16:00\n\nThere are two cases to consider: either the Sylow $r$-group is maximal, or it is normal in $G$. Both cases lead to a solvable group. No conditions on $r$, further than $p < q< r$, are necessary.\n\n-","date":"2014-12-23 03:48:04","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\": 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\": 0.9339215159416199, \"perplexity\": 187.85563148225557}, \"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-2014-52\/segments\/1418802778068.138\/warc\/CC-MAIN-20141217075258-00137-ip-10-231-17-201.ec2.internal.warc.gz\"}"} | null | null |
\section{Introduction}
Supersymmetry (SUSY) stands out as one of the most interesting
alternatives beyond the Standard model (SM) of elementary particles. The Minimal Supersymmetric
extension of the Standard Model (MSSM)
necessarily contains two Higgs doublets, which, on electroweak symmetry
breaking (EWSB), leads to 5 physical Higgs bosons, namely, two CP-even neutral
scalars ($h, H$), one CP-odd Higgs ($A$) and two mutually conjugates
charged scalars ($H^\pm$). In such a framework, the Higgs phenomenology is
obviously richer than in the SM \cite{Djouadi:2005gi,AguilarSaavedra:2005pw}. With the Large Hadron Collider (LHC)
all set to take off, the hunt for this yet undiscovered scalar sector
has assumed a spacial significance, concurrently with the search for SUSY.
The phenomenology of the Higgs sector in the MSSM is considerably
enriched by the interaction of the various scalar (Higgs) states
with SUSY particles \cite{Haber:1984rc,HHG,Gunion:1984yn}. Notwithstanding the prevailing emphasis on
Higgs production in processes driven by SM interactions, the prospect
of extracting information from SUSY channels and cascades should
therefore be always kept within sight. As for example, viable Higgs
signals in associated production of superparticle have been suggested
in recent studies, for cases where SM channels fail due to the
effects of CP-violating phases \cite{Bandyopadhyay:2007cp}. In this paper, we suggest the
utilisation of Higgs production in SUSY cascades in probing the
chargino-neutralino sector of the MSSM. In particular, we show that
the relative rates of $h$-and $H^{\pm}$-production in cascades can
provide insight on whether $M_1$ and $M_2$, the $U(1)$ and $SU(2)$ gaugino
masses, respectively, are related by a high-scale universality condition.
The significance of SUSY cascades as the source of the MSSM
Higgs bosons has been discussed in detail in the recent
past \cite{Datta:2003iz} within an MSSM
framework, but keeping $M_1$ and $M_2$ constrained by universality.
The central idea in such a study has been to exploit the huge
production cross sections for the strongly interacting SUSY
particles the squarks and the gluino
at a hadron collider like the LHC.
These sparticles, once produced, may undergo long cascade-decays that
ultimately lead to stable SM particles (like leptons and quarks (jets))
along with the LSP's which escape detection if R-parity (defined
as $R~=~(-)^{3B+L+2J}$) is conserved. It was pointed out in
\cite{Datta:2003iz,Baer:1992ef}that one or
more MSSM Higgs bosons can be produced at different stages of these
cascades. It was also shown \cite{Datta:2001qs} that the suppressions
resulting from different decay
branching fractions under SUSY cascades are more than compensated for by
the huge production cross-section of the strongly interacting particles.
It is to be noted that such cascades, in order to be instrumental in
Higgs production, necessarily require SUSY particles widely separated
in mass. A high energy machine like LHC is an
ideal hunting ground for MSSM Higgs bosons under such cascades.
As has been already pointed out in the earlier works
\cite{Datta:2001qs}, cascades can be very efficient sources of MSSM
Higgs bosons in certain regions of MSSM parameter space (viz., with
intermediate $\tan\beta$ values) where usual modes cease to deliver.
Here we make use of these sources, with the $M_1 - M_2$ universality
conditions relaxed, and discover some rather spectacular consequences
on the overall rates.
In canonical SUSY scenarios, the neutralinos and the charginos (which are
the mass eigenstates and mixtures of electroweak gauginos and the higgsinos)
can be much lighter compared to the strongly interacting sparticles like
squarks and gluinos. Thus, charginos and neutralinos may take control of the
proceedings at an early stage of the cascade. Further, their
compositions (in terms of the gaugino and Higgsino contents), which play a
crucial role. The compositions in turn are determined by the soft
masses of the electroweak
gauginos, namely, $M_1$ and $M_2$, $\mu$,
the so-called higgsino mass parameter, and
$\tan\beta$, the ratio of the vacuum expectation values of the two Higgs
doublets. In particular, the relative magnitudes of $M_1$,
$M_2$ and $\mu$ largely
determine their physical states.
There is no {\it a priori} justification as to why gaugino
universality, albeit highly predictive and hence, popular,
would necessarily hold at a high scale. In fact, it has its root
in the trivial nature of the so-called gauge kinetic function from
which the common gaugino mass arises at a high scale as SUSY breaks
in the hidden sector. More specifically, such a universality
arises when the gauge kinetic function involves a
combination of the hidden sector fields which is singlet
under the underlying gauge group of SUSY Grand Unified Theory(GUT). It has been
shown [\cite{Martin}-\cite{Cremmer}] that involving higher GUT representations for the purpose
would in general trigger nonuniversality among the gaugino soft masses at a
high scale itself. Such a nonuniversality inevitably distorts the
weak-scale gaugino spectrum thus modifying the compositions of the
charginos and the neutralinos and their masses vis-a-vis the
gluino mass. From a purely phenomenological point of view one can thus
think of a completely uncorrelated gaugino sector at the weak scale.
This can have profound implications
in collider data [\cite{Bt1}-\cite{Huitu:2007vw}]. It is
interesting to note that imprints of such nonuniversality can be recognised
even in SUSY-Higgs searches at the LHC. Also, unlike in earlier works
\cite{Datta:2003iz,Datta:2001qs} we keep sleptons light enough so that
they have a nontrivial role to play.
The main consequence of relaxing the universality condition
on the $SU(3)$ gaugino mass ($M_3$) \cite{Datta:2003iz,Datta:2001qs} is that it gives a free hold to the
gluino mass (and, in schemes of scalar mass evolution, the squark
masses). This affects the rates of cascades through gluino and
squark decay branching ratios only. By relaxing $M_1 - M_2$
universality, on the other hand, one opens up additional
possibilities, as far as the cascade branching ratios of the
charginos and neutralinos themselves are concerned. In addition,
the lack of correlation between $M_1$ and $M_2$ affects the coupling
strengths of a charged or neutral Higgs to a chargino-neutralino
pair. Since such effects have not been studied systematically so far,
we present an analysis here, in the context of the LHC.
In section 2 we outline the Higgs production process in cascades and
the factors that control them. In section 3 we the production rates of the
charged $H^\pm$ and the lightest neutral Higgs
($h$) bosons and contrast them systematically. We demonstrate how
such a knowledge could reflect on the nonuniversality of gaugino-masses
We conlclude in section 4.
\section{Higgs production in SUSY cascades}
The squarks and the gluinos, once produced at LHC, would first undergo
strong two-body decays like $\tilde{q} \to q \tilde{g}$ (for $m_{\tilde{q}} > m_{\tilde{g}}$)
or $\tilde{g} \to q \tilde{q}$ (for $m_{\tilde{g}} > m_{\tilde{q}}$). Beyond this point, the
cascade decays are electroweak in nature where Higgs bosons could appear
\footnote{A possible exception could be when all squarks except the
ones from the third
generation ($\sstop1$ or $\sbot1$) are heavier than the gluino. In such scenarios,
a cascade of strong decays of squarks and gluinos might end up with $\sstop1$
or $\sbot1$ whose electroweak decays would lead to the Higgs bosons.}.
Higgs production under such cascades mainly involves the
charginos and neutralinos in the intermediate stages. With gluinos
initiating a cascade, this is inevitable, since gluinos do not couple
directly to the Higgs bosons at the tree level.
For squarks, couplings to Higgs bosons are proportional to the corresponding quark
masses, and are thus significant only for the squarks of the third family.
Since the generic yield of such squarks is smaller in comparison to those
of the first
two families, most Higgs production processes in cascades involve
the charginos and neutralinos in the intermediate stages.
Schematically, the chains
of cascades leading to the Higgs bosons
are as follows:
\begin{equation}
pp \to \tilde{g} \tilde{g},\; \tilde{q} \tilde{q} , \; \tilde{q} \tilde{q}^*, \; \tilde{q} \tilde{g}
\longrightarrow \; \chpm2, \; \ntrl3, \; \ntrl4 \, + \, X \longrightarrow \;
\chpm1, \; \ntrl2, \; \ntrl1 \, + \, H^\pm, h, H, A \, + \, X
\end{equation}
\vskip -15pt
\begin{equation}
pp \to \tilde{g} \tilde{g},\; \tilde{q} \tilde{q}, \; \tilde{q} \tilde{q}^*, \; \tilde{q} \tilde{g}
\longrightarrow \; \chpm1, \; \ntrl2 \, + \, X \longrightarrow \;
\ntrl1, \, + \, H^\pm, \, h, H, A, H^\pm \, + \, X
\end{equation}
The first decay chain above is a longer one as it involves direct
decays of squark and gluinos to the heavier chargino/neuralinos followed by
subsequent decays of the latter ones to lighter gauginos
and the Higgs bosons.
On the other hand, the second chain is shorter since it exploits
direct decays of squarks and gluinos to the lighter
chargino/neutralinos which then decay to Higgs bosons
and the LSP. In the literature \cite{Datta:2001qs} the first scheme was
called the `big cascade' while the latter one was dubbed as the
`little cascade'. For convenience, we
adopt the same terminology in this work.
It is thus expected that the final yield of Higgs bosons under such SUSY-cascades crucially
depends upon the branching fractions of the relevant decay
processes. Thus on a complicated, though comprehensible, interplay of
different SUSY parameters in the form of various couplings and masses. Out of these,
the couplings of the Higgs bosons with the charginos/neutralinos play
the most important role. It is well known \cite{Gunion:1984yn,Datta:2001qs}
that the Higgs bosons couple
favourably to charginos and neutralinos when the latter are mixtures of gauginos and
Higgsinos while for gauge bosons the couplings are maximal when the charginos and the
neutralinos are Higgsino-dominated.
Naturally, then, the compositions of the charginos and neutralinos would play a crucial role
in our study \footnote{In scenarios with a universal gaugino mass at a high scale
(like the GUT scale), $M_1$ and $M_2$ gets related at the weak scale by the simple
relation
$M_2\simeq 2 M_1$. Thus, in that case, one only talks about 3 input parameters that govern
the chargino-neutralino sector. In contrast, the present work addresses the issue of
nonuniversality of gaugino masses in a particular context. Hence, $M_1$ and $M_2$
are taken to be two free parameters.}. Out of the determining parameters, the values
of $M_1$, $M_2$ and $\mu$ have the most crucial bearings on the masses and the
contents of the charginos and the neutralinos. For $\mu >> M_1, M_2$ , one is in the
so-called `gaugino region' where the lighter neutralinos and chargino
($\ntrl1,\ntrl2,\chpm1$) are
gaugino-dominated with $\mntrl1 \simeq min(M_1, M_2)$ and
$\mntrl2, \mchpm1 \simeq max( M_1,M_2)$
while the heavier ones ($\ntrl3, \ntrl4, \chpm2$) are mostly Higgsinos with
$\mntrl3, \mntrl4, \mchpm2 \simeq \mu$. On the other hand,
for $\mu << M_1, M_2$, we are in the `Higgsino region' for which the lighter neutralinos
and the chargino are predominantly Higgsinos with
$\mntrl1, \mntrl2, \mchpm1 \simeq \mu$ while the heavier ones are dominated by
gauginos with
$\ntrl3 \simeq min(M_1, M_2) $ and $\ntrl4, \mchpm2 \simeq max(M_1, M_2)$.
As expected, for different charginos and neutralinos,
the masses and the contents have one-to-one correspondences in such `pure' regions of the
SUSY parameter space. For $M_1,M_2 \simeq \mu$, the charginos and the neutralinos become maximally
mixed in gauginos and Higgsinos with their masses showing no particular pattern, albeit
restricted within a range determined by the values of $M_1, M_2$ and $\mu$.
In a nutshell, the `big cascades' are favoured in regions where, between
($\chi_2^{\pm}, \chi_3^0, \chi_4^0$) and ($\chi_1^{\pm}, \chi_1^0, \chi_2^0$),
one set is gaugino-dominated, and the other, Higgsino-dominated. In
situations where they are kinematically allowed, little cascades are
on the hand possible when the members of the second set above
comparable gaugino and Higgsino components.
We investigate charged as well as neutral Higgs production rates in cascades.
For a ready comparison, we closely follow the earlier analyses
\cite{Datta:2003iz,Datta:2001qs}. Explicit
expressions for most cross-sections and decay widths of
relevance are found in the above references.
\section{The results of nonuniversality: numerical results}
We are looking at `effective cross-sections' of Higgs production of
various kinds, which essentially means
$$
pp \longrightarrow Higgs + X
$$
\noindent
where the cross-sections of all possible (from $2 \to 2$ strong
productions) cascades are added up, so
long as there is at least one Higgs of any kind in the final state.
As cross-checks of the calculation, we have reproduced the results
in \cite{Datta:2003iz,Datta:2001qs} in the appropriate limits. We have used PYTHIA
\cite{Sjostrand:2006za} for our analysis and CTEQ3L \cite{CTEQ}
as parton distributions interfaced via LHAPDF \cite{CTEQ1}. The factorization/renormalization scale
set at the average of the masses of the particles (squarks and/or
gluinos) produced in the hard scattering. the analysis is based on
leading order production only. Also, following
\cite{Datta:2001qs}, the set of relevant SM and SUSY inputs (at the weak scale) chosen
for the analysis (unless otherwise specified) is:
\[ m_{top}=175 \ {\mathrm {GeV}} \quad \tan\beta=10 \quad
m_A=162.1 (237.5) \ \mathrm{GeV} \]
\[ m_{\tilde{g}}=900 \ \mathrm{GeV} \quad m_{\tilde{q}}=800 \
\mathrm{GeV} \quad A_f=0 \]
%
This resulted in the following masses for the different Higgs bosons:
\[ m_h=109 (110) \ \mathrm{GeV} \quad m_H=164 (238) \ \mathrm{GeV}
\quad m_{H^\pm}=180 (250) \
\mathrm{GeV} \]
which are in close agreement with those used earlier in the literature
\cite{Datta:2003iz,Datta:2001qs}. As for the sleptons, the only way they may signficantly contribute is by affecting the decay modes of
the gauginos. As indicated in section 1, we demonstrated the role of
light sleptons, taking them to be degenerate at 400 GeV \footnote{We have checked that the use of more recent parton distribution
functions, the variations in rates with renormalization/factorization
scale and the use of an updated top-quark mass do not alter the
basic findings of the present work.}.
The parameter-dependence of Higgs production rates under SUSY cascades is investigated in two ways: (i) variation with
$M_2$ and (ii) variation with $\mu$.
\subsection{Variation with $M_2$}
\begin{figure}[hbt]
\begin{center}
{\epsfig{file=sl400U.ps,width=4.7 cm,height=5.2cm,angle=-90.0}}
{\epsfig{file=sl400n.ps,width=4.7cm,height=5.2cm,angle=-90.0}}
{\epsfig{file=sl400m1200.ps,width=4.7cm,height=5.2cm,angle=-90.0}}
\caption{Effective cross-sections for universal ($M_1 = M_2/2$, (a)) and
nonuniversal (with $M_1=100$ GeV, (b) and with $M_1=200$ GeV (c)) for
$\mu=150$ GeV and $m_{H^{\pm}}=180$ GeV.}
\end{center}
\label{fig11}
\vspace*{-1.0cm}
\end{figure}
\begin{figure}[hbt]
\begin{center}
{\epsfig{file=sl400H+250u.ps,width=4.7 cm,height=5.2cm,angle=-90.0}}
{\epsfig{file=sl400H+250m100.ps,width=4.7cm,height=5.2cm,angle=-90.0}}
{\epsfig{file=sl400H+250m1200.ps,width=4.7cm,height=5.2cm,angle=-90.0}}
\caption{Effective cross-sections for universal ($M_1 =
M_2/2$, (a)) and nonuniversal (with $M_1=100$ GeV, (b), with
$M_1=200$ GeV (c)) for $\mu=150$ GeV and $m_{H^{\pm}}=250$ GeV.}
\end{center}
\label{fig11}
\end{figure}
\vspace*{0.8cm}
\begin{figure}[hbt]
\begin{center}
{\epsfig{file=sl400mu700U.ps,width=4.7 cm,height=5.2cm,angle=-90.0}}
{\epsfig{file=sl400mu700.ps,width=4.7cm,height=5.2cm,angle=-90.0}}
{\epsfig{file=sl400mu700m1200.ps,width=4.7cm,height=5.2cm,angle=-90.0}}
\caption{Effective cross-sections for universal ($M_1 = M_2/2$,
(a)), nonuniversal (with $M_1=100$ GeV (b) and $M_1=200$ GeV (c)) for $\mu=700$ GeV and $m_{H^{\pm}}$=180 GeV.}
\end{center}
\label{fig11}
\vspace*{-0.5cm}
\end{figure}
\begin{figure}[hbt]
\vspace*{-1.2cm}
\begin{center}
{\epsfig{file=H+250mu700U.ps,width=4.7 cm,height=5.2cm,angle=-90.0}}
{\epsfig{file=slH+250mu700.ps,width=4.7cm,height=5.2cm,angle=-90.0}}
{\epsfig{file=H+250mu700m1200.ps,width=4.7cm,height=5.2cm,angle=-90.0}}
\caption{Effective cross-sections for universal ($M_1 = M_2/2$,
(a)), nonuniversal (with $M_1=100$ GeV (b) and $M_1=200$ GeV (c))
for $\mu=700$ GeV and $m_{H^{\pm}}$=250 GeV.}
\end{center}
\label{fig11}
\hspace*{-0.5cm}
\end{figure}
Since $m_{\tilde{\ell}}$ is taken to be 400 GeV, the
sleptons have a significant role in the cascades.
Thus, the natural expectation is that once these sleptonic
decay modes of the charginos/neutralinos open up, cascades to Higgs would get
suppressed. Hence the overall rates presented are of a conservative nature.
This is clear from the set of Figures 1 to 4. We discuss below some
generic features present in these figures and the information we get
from them.
Plot (a)s depict the universal scenario while plot (b)s and plot (c)s
are for the nonuniversal scenario. Figure 1 and Figure 2 are for $\mu=150$ GeV
while Figure 3 and Figure 4 are for $\mu=700$ GeV. For
$\mu=150$ GeV the lighter gauginos are too closely degenerate
(for both universal and nonuniversal scenarios) for the `little cascades'
to open up. Hence, the entire cascade contribution to Higgs production
comes from the `big cascade'. The sudden rises in some curves at
specific $M_2$ values in the universal scenario indicate attaining
the right mass-splitting between $\ntrl3$ and the LSP such that
$\ntrl3$ decaying to the lightest neutral (charged) Higgs boson and
the LSP (the lighter chargino) becomes possible. This feature is not
there in the nonuniversal cases (where $M_1$ is set to 100 and 200
GeV respectively) as $\mntrl3\sim \mu$.
One should note the different pattern of $M_2$-dependence of the
rates for $\mu=150$ GeV (Figures 1 \& 2) and 700 GeV (Figures 3 \& 4), respectively. This is because the
former situation allows Higgs production mostly through `big
cascades'. The latter case, where larger separation amongst the low-lying
states is possible, `little cascade' more abundantly, thus
making the variation of Higgs production rates with $M_2$ look
different. In particular, slepton masses of the order of 400 GeV affect
`big cascades' less for larger value of $M_2$ throught the enhancement
of the effective coupling for Higgs production. Little cascades are
affected much more in such a case, thus causing difference in the way
the rates fall with increasing $M_2$.
In Figure 2 we illustrate a case similar to Figure 1 except for
$m_{H^\pm}=250$ GeV. This needs a substantial increment in the mass of
$A$ as input and results in a larger $H$ compared to those for Figure
1. Thus, in the universal case (Figure 2(a)) $\ntrl3$ needs to be heavier such that the heavier
Higgs bosons may be produced in the decay of $\ntrl3$ along with
$\ntrl1$ or $\chpm1$. Note that increasing $m_{H^\pm}$ does not affect the
rate for the lightest higgs ($h$) significantly, when compared to the
corresponding plot in Figure 1 since $m_h$ remains almost unaffected by
such an increase in $m_{H^\pm}$. Thus, for $m_{H^\pm}=250$ GeV, there is no
cross-over between the curves for $h$ and $H^\pm$ in the universal case
unlike $m_H^{\pm}$=180 GeV.
The general observation is that with increasing $m_{H^\pm}$ the threshold
value of $M_2$ shifts naturally to the right leading to more massive
heavier charginos and neutralinos such that the `big cascades' may
take place. This eventually pushes the cross-over point (of the
rate-curves for the lightest and charged Higgs bosons) to larger
values of $M_2$. Also, the rates for heavier Higgs bosons are
smaller for $m_{H^\pm}=250$ GeV as compared to $m_{H^\pm}=180$ GeV. There are
two reasons for this. First, the heavier Higgs bosons now become more
massive whose rates suffer a phase-space suppression for similar
chargino and neutralino masses. Second, the charginos and neutralinos
whose decay results in the Higgs bosons (with increased masses) have to
become heavier as well. Thus the production
rates for the latter also get affected. Figures 2(b) and 2(c) illustrating
the nonuniversal cases with $M_1=100$ GeV and 200 GeV are to be
contrasted with the corresponding ones in Figure 1. They only differ
by the generic features as described above.
Figure 4 illustrates a situation similar to Figure 3 except for $m_{H^\pm}=250$
GeV. In the universal case (Figure 4(a)), the peak in the $H^\pm$ rate
disappears when compared to Figure 3(a). One should note that this peak is due
predominantly to a `little cascade' like $\chpm1 \to \ntrl1 H^\pm$.
As indicated earlier, with growing $m_{H^\pm}$ the mass splitting between
the gaugino states involved above are not enough to accommodate the
above cascade. The situation could be a little different for a
nonuniversal case with a lower value of $M_1$ (100 GeV) as shown in
Figure 4(b). The smaller value of $M_1$ now ensures a lower mass for the
$\ntrl1$ thus help regaining the required splitting when the peak in
the $H^\pm$ is back (at around 400 GeV). In Figure 4(c), $M_1$
is 200 GeV and this again blocks the above decay mode at around 400
GeV. Of course, with increasing value of $M_2$ ($\geq 500$ GeV) the
Higgs productions under such cascades open up again. But this time,
Br[$\chpm1 \to \ntrl1 H^\pm$] starts getting suppressed as the two-body sleptonic decay modes
of $\chpm1$ take off for our choice of slepton mass (400 GeV).
\subsection{Variation with $\mu$}
\begin{figure}[hbt]
\begin{center}
\vspace*{-0.5cm}
\hskip -65pt
{\epsfig{file=MuM2400u.ps,width=6.0 cm,height=6.7cm,angle=-90.0}}
{\epsfig{file=MuM1100M2400.ps,width=6.0 cm,height=6.7cm,angle=-90.0}}
\caption{Effective cross-section as a function of $\mu$ for universal
(with $M_1=M_2/2$, (a)) and nonuniversal (with $M_1=100$ GeV, (b))
scenarios with $M_2=400$ GeV and $m_{H^{\pm}}=180$ GeV.}
\end{center}
\label{fig11}
\end{figure}
\begin{figure}[hbt]
\begin{center}
\vspace*{-0.5cm}
\hskip -65pt
{\epsfig{file=MuH+250u.ps,width=6.0 cm,height=6.7cm,angle=-90.0}}
{\epsfig{file=MuH+250M100.ps,width=6.0cm,height=6.7cm,angle=-90.0}}
\caption{Effective cross-section as a function of $\mu$ for universal
(with $M_1=M_2/2$, (a)) and nonuniversal (with $M_1=100$ GeV, (b))
scenarios with $M_2=400$ GeV and $m_{H^{\pm}}=250$ GeV.}
\end{center}
\label{fig11}
\vspace*{-1.0cm}
\end{figure}
In Figures 5 \& 6, we illustrate the variation of the rates for Higgs production under
SUSY cascades with $\mu$. Plot (a)s represent the universal scenario
while plot (b)s illustrate the same in a nonuniversal situation. In both
cases, for small values of $\mu$, the lighter charginos and
neutralinos are Higgsino-like and their masses are of the order of
$\sim \mu$ with a definite split from their heavier mates governed by
the value of $M_2$ chosen. With increasing $\mu$, the ``mixed region'' is approached
and the mass-splittings among the heavier and lighter charginos
decrease. This gradually eliminates the possible sources of the
heavier Higgs bosons, especially those of $H^\pm$. Further, with
increasing $\mu$, channels like $\chpm2 \to \ntrl1 H^\pm$ again opens
up followed by $\chpm1 \to \ntrl1 H^\pm$ at about $\mu=525$ GeV in the
universal case. A further mild rise in the $H^\pm$-rate is observed at
around $\mu=550$ GeV when $\ntrl3 \to \chpm1 H^\pm$ opens up. However,
in the nonuniversal case, due to a fixed low value of $M_1$ the LSP mass
remains almost the same ($\sim M_1$). Hence channels like $\chpm1 \to
\ntrl1 H^\pm$ opens up as early as $\mu=300$ GeV. The features observed
in these plots are generic as long as the relative splittings of
gaugino masses and $\mu$ used here remain similar.
In Figure 6 we illustrate the case similar to Figure 5 except for
$m_{H^\pm}=250$ GeV. In the universal case, when compared to Figure 5, we find
that the rise in the rate for $H^\pm$ at around $\mu=500$ GeV is missing.
This is because of the lack of enough splitting between the lighter
chargino and the LSP masses that might lead to $H^\pm$ unlike what happened in
the universal case when $m_{H^\pm}=180$ GeV. For $M_1=100$ GeV, $\chpm1
\to \ntrl1 H^\pm$ opens up at a later stage compared to $m_H^{\pm}$=180
GeV case. For higher neutral Higgses similar things happen.
\subsection{Comparison between $H^\pm$ and $h$ production rates}
Figures 1, 3 and 5 ($m_{H^{\pm}}$=180 GeV) along with the discussions
in sections 3.1 and 3.2 reveal a rather complicated interplay of
several masses and couplings. It is undoubtedly a difficult task
to extract useful information from these processes in a systematic
way. However, a closer look at these reveal that we have a somewhat
clean feature in the variations of rates for $H^\pm$ and $h$ which we
can use to our benefit. These rates behave in a distinctly
complementary fashion as functions of $M_2$ (Figures 1 and 3) and
$\mu$ (Figure 5) and show up as multiple cross-over points. For
universal and nonuniversal scenarios, these
cross-overs take place at different values of $M_2$ and $\mu$. This can be
understood in a straight-forward manner following the discussions in sections 3.1 and 3.2.
The observations prompt us to look out for a suitable illustration of
the situation. In Figure 7 we present the scattered plots of the
two-rates compared in the $M_2-\mu$ plane for both universal and
nonuniversal cases. Over the grey regions the rates for the charged
Higgs bosons is greater than that for the lightest neutral Higgs boson and
the reverse is the case over the darker regions. At suitable points on
these plots, direct correspondences can be made with Figures 1, 3 and
5. There are quite a few distinctive patches on these two plots. If
such relative rates can be known from the LHC experiments, these
plots could be used to have a preliminary understanding of the gaugino
mass pattern.
In Figure 8 we present a similar set of scattered plots but for
$m_H^{\pm}$=250 GeV. In the universal case, all over the $M_2-\mu$ space
the rate for lightest Higgs is greater than that for the charged Higgses.
For the nonuniversal case, with increasing $M_1$ the region where the
rate for the charged Higgses is larger than that for the lightest Higgs
gradually shrinks. Again, all these can be read off from the plots in Figures
1 to 6. Thus, we see that $m_{H^\pm}$ and $M_1$ play some significant roles
in characterizing the $M_2-\mu$ plane.
We have checked that the results presented here are robust against
wide variation of squark and gluino masses, even when their ordering
is reversed. Increasing the slepton masses would only make life
simpler in the sense that they get decoupled from the cascades and we
checked that this does not bring out any new feature. We also
checked that the effect of $\tan\beta$ is anything but significant.
This validates one of the very basic expectations for studying the
Higgs production under SUSY cascades and hence its power
of probing the Higgs sector for intermediate values of $\tan\beta$.
It is useful to mention here that the confirmation of gaugino
nonuniversality in Higgs production presupposes some knowledge of
the SUSY spectrum, as obtained from the heaviness or
missing-${E_T}$ distribution of the SUSY signals. As the scatter plots
in figures 7 and 8 indicate, one is able to infer conclusively about
universality (or the lack of it) from the relative magnitudes of the
$h$- and $H^{\pm}$-production rates, provided that two quantities out
of $M_1$, $M_2$ and $\mu$ are known. However, even that does not rob
the present study of its merits, since any unambiguous conclusion
about the particle spectrum of a new physics scenario requires a
multichannel analysis. In that sense, the inclusive Higgs production
processes provide a channel of supreme importance in understanding the
electroweak gaugino-Higgsino sector.
\vspace*{0.5cm}
\begin{figure}[hbt]
\begin{center}
\vspace*{-0.5cm}
\hskip 0pt
{\epsfig{file=sim1.ps,width=5.0 cm,height=5.2cm,angle=-90.0}}
\hskip 2pt
{\epsfig{file=sim2.ps,width=5.0 cm,height=5.2cm,angle=-90.0}}
\hskip 2pt
{\epsfig{file=simM1200.ps,width=5.0 cm,height=5.2cm,angle=-90.0}}
\caption{The pink (dark) region is the region where the rate for H$^{\pm}$
is less than that for $h$ production, for the sky-blue region (grey)
it is the opposite and white regions indicate zero production for both.
We take $M_1=M_2/2$ (a), $M_1=100$ GeV (b) and $M_1=200$ GeV
(c). $m_{H^{\pm}}$ is set to 180 GeV.}
\end{center}
\label{fig11}
\end{figure}
\begin{figure}[hbt]
\begin{center}
\vspace*{-0.5cm}
\hskip 0pt
{\epsfig{file=simH+250U.ps,width=5.0 cm,height=5.2cm,angle=-90.0}}
\hskip 2pt
{\epsfig{file=simH+250m1100.ps,width=5.0 cm,height=5.2cm,angle=-90.0}}
\hskip 2pt
{\epsfig{file=simH+250m1200.ps,width=5.0 cm,height=5.2cm,angle=-90.0}}
\caption{The pink (dark) region is the region where the rate for H$^{\pm}$
is less than that for $h$ production, for the sky-blue region (grey)
it is the opposite and white regions indicate zero production for both.
We take $M_1=M_2/2$ (a), $M_1=100$ GeV (b) and $M_1=200$ GeV
(c). $m_{H^{\pm}}$ is set to 250 GeV.}
\end{center}
\label{fig11}
\end{figure}
\section{Conclusions}
We have investigated the rates for production of different physical
Higgs states in SUSY cascades at the LHC. The new inputs are
(a) relaxation of the universality condition relating electroweak
gaugino masses and (b) allowing sleptons to be light enough to be
produced on-shell in cascades. As we show through the dependence of
production rates on various parameters, both these inputs can
significantly affect the
phenomenology at the LHC. The most important observation is that the
presence or absence of gaugino universality is reflected in a rather
interesting complementarity between the relative production rates of
$H^{\pm}$ and $h$. Therefore, one should aim at a careful
identification of charged as well as neutral Higgs signals in
cascades to extract useful information about the underlying SUSY
scenario. A detailed discussion of such signals {\it all} the physical
Higgs states in SUSY and methods of
suppressing their backgrounds will be presented in a separated study.
\vskip 20pt
{\bf Acknowledgments:}
We acknowledge S. Mrenna for resolving and clarifying some issues in
Pythia and Peter Skands for useful discussions.
Computational work for this study was partially carried out in the cluster
computing facility at Harish-Chandra Research Institute
(http://cluster.mri.ernet.in). This work is partially supported by
Regional Center for Accelerator-based Particle Physics, Harish-Chandra
Research Institute and funded by the Department of Atomic Energy, Government of India under the XIth 5-year Plan.
\section{Introduction}
Supersymmetry (SUSY) stands out as one of the most interesting
alternatives beyond the Standard model (SM) of elementary particles. The Minimal Supersymmetric
extension of the Standard Model (MSSM)
necessarily contains two Higgs doublets, which, on electroweak symmetry
breaking (EWSB), leads to 5 physical Higgs bosons, namely, two CP-even neutral
scalars ($h, H$), one CP-odd Higgs ($A$) and two mutually conjugates
charged scalars ($H^\pm$). In such a framework, the Higgs phenomenology is
obviously richer than in the SM \cite{Djouadi:2005gi,AguilarSaavedra:2005pw}. With the Large Hadron Collider (LHC)
all set to take off, the hunt for this yet undiscovered scalar sector
has assumed a spacial significance, concurrently with the search for SUSY.
The phenomenology of the Higgs sector in the MSSM is considerably
enriched by the interaction of the various scalar (Higgs) states
with SUSY particles \cite{Haber:1984rc,HHG,Gunion:1984yn}. Notwithstanding the prevailing emphasis on
Higgs production in processes driven by SM interactions, the prospect
of extracting information from SUSY channels and cascades should
therefore be always kept within sight. As for example, viable Higgs
signals in associated production of superparticle have been suggested
in recent studies, for cases where SM channels fail due to the
effects of CP-violating phases \cite{Bandyopadhyay:2007cp}. In this paper, we suggest the
utilisation of Higgs production in SUSY cascades in probing the
chargino-neutralino sector of the MSSM. In particular, we show that
the relative rates of $h$-and $H^{\pm}$-production in cascades can
provide insight on whether $M_1$ and $M_2$, the $U(1)$ and $SU(2)$ gaugino
masses, respectively, are related by a high-scale universality condition.
The significance of SUSY cascades as the source of the MSSM
Higgs bosons has been discussed in detail in the recent
past \cite{Datta:2003iz} within an MSSM
framework, but keeping $M_1$ and $M_2$ constrained by universality.
The central idea in such a study has been to exploit the huge
production cross sections for the strongly interacting SUSY
particles the squarks and the gluino
at a hadron collider like the LHC.
These sparticles, once produced, may undergo long cascade-decays that
ultimately lead to stable SM particles (like leptons and quarks (jets))
along with the LSP's which escape detection if R-parity (defined
as $R~=~(-)^{3B+L+2J}$) is conserved. It was pointed out in
\cite{Datta:2003iz,Baer:1992ef}that one or
more MSSM Higgs bosons can be produced at different stages of these
cascades. It was also shown \cite{Datta:2001qs} that the suppressions
resulting from different decay
branching fractions under SUSY cascades are more than compensated for by
the huge production cross-section of the strongly interacting particles.
It is to be noted that such cascades, in order to be instrumental in
Higgs production, necessarily require SUSY particles widely separated
in mass. A high energy machine like LHC is an
ideal hunting ground for MSSM Higgs bosons under such cascades.
As has been already pointed out in the earlier works
\cite{Datta:2001qs}, cascades can be very efficient sources of MSSM
Higgs bosons in certain regions of MSSM parameter space (viz., with
intermediate $\tan\beta$ values) where usual modes cease to deliver.
Here we make use of these sources, with the $M_1 - M_2$ universality
conditions relaxed, and discover some rather spectacular consequences
on the overall rates.
In canonical SUSY scenarios, the neutralinos and the charginos (which are
the mass eigenstates and mixtures of electroweak gauginos and the higgsinos)
can be much lighter compared to the strongly interacting sparticles like
squarks and gluinos. Thus, charginos and neutralinos may take control of the
proceedings at an early stage of the cascade. Further, their
compositions (in terms of the gaugino and Higgsino contents), which play a
crucial role. The compositions in turn are determined by the soft
masses of the electroweak
gauginos, namely, $M_1$ and $M_2$, $\mu$,
the so-called higgsino mass parameter, and
$\tan\beta$, the ratio of the vacuum expectation values of the two Higgs
doublets. In particular, the relative magnitudes of $M_1$,
$M_2$ and $\mu$ largely
determine their physical states.
There is no {\it a priori} justification as to why gaugino
universality, albeit highly predictive and hence, popular,
would necessarily hold at a high scale. In fact, it has its root
in the trivial nature of the so-called gauge kinetic function from
which the common gaugino mass arises at a high scale as SUSY breaks
in the hidden sector. More specifically, such a universality
arises when the gauge kinetic function involves a
combination of the hidden sector fields which is singlet
under the underlying gauge group of SUSY Grand Unified Theory(GUT). It has been
shown [\cite{Martin}-\cite{Cremmer}] that involving higher GUT representations for the purpose
would in general trigger nonuniversality among the gaugino soft masses at a
high scale itself. Such a nonuniversality inevitably distorts the
weak-scale gaugino spectrum thus modifying the compositions of the
charginos and the neutralinos and their masses vis-a-vis the
gluino mass. From a purely phenomenological point of view one can thus
think of a completely uncorrelated gaugino sector at the weak scale.
This can have profound implications
in collider data \cite{Bt1,Bt2,Bhattacharya:2007dr,Huitu:2005wh,Huitu:2007vw}. It is
interesting to note that imprints of such nonuniversality can be recognised
even in SUSY-Higgs searches at the LHC. Also, unlike in earlier works
\cite{Datta:2003iz,Datta:2001qs} we keep sleptons light enough so that
they have a nontrivial role to play.
The main consequence of relaxing the universality condition
on the $SU(3)$ gaugino mass ($M_3$) \cite{Datta:2003iz,Datta:2001qs} is that it gives a free hold to the
gluino mass (and, in schemes of scalar mass evolution, the squark
masses). This affects the rates of cascades through gluino and
squark decay branching ratios only. By relaxing $M_1 - M_2$
universality, on the other hand, one opens up additional
possibilities, as far as the cascade branching ratios of the
charginos and neutralinos themselves are concerned. In addition,
the lack of correlation between $M_1$ and $M_2$ affects the coupling
strengths of a charged or neutral Higgs to a chargino-neutralino
pair. Since such effects have not been studied systematically so far,
we present an analysis here, in the context of the LHC.
In section 2 we outline the Higgs production process in cascades and
the factors that control them. In section 3 we the production rates of the
charged $H^\pm$ and the lightest neutral Higgs
($h$) bosons and contrast them systematically. We demonstrate how
such a knowledge could reflect on the nonuniversality of gaugino-masses
We conlclude in section 4.
\section{Higgs production in SUSY cascades}
The squarks and the gluinos, once produced at LHC, would first undergo
strong two-body decays like $\tilde{q} \to q \tilde{g}$ (for $m_{\tilde{q}} > m_{\tilde{g}}$)
or $\tilde{g} \to q \tilde{q}$ (for $m_{\tilde{g}} > m_{\tilde{q}}$). Beyond this point, the
cascade decays are electroweak in nature where Higgs bosons could appear
\footnote{A possible exception could be when all squarks except the
ones from the third
generation ($\sstop1$ or $\sbot1$) are heavier than the gluino. In such scenarios,
a cascade of strong decays of squarks and gluinos might end up with $\sstop1$
or $\sbot1$ whose electroweak decays would lead to the Higgs bosons.}.
Higgs production under such cascades mainly involves the
charginos and neutralinos in the intermediate stages. With gluinos
initiating a cascade, this is inevitable, since gluinos do not couple
directly to the Higgs bosons at the tree level.
For squarks, couplings to Higgs bosons are proportional to the corresponding quark
masses, and are thus significant only for the squarks of the third family.
Since the generic yield of such squarks is smaller in comparison to those
of the first
two families, most Higgs production processes in cascades involve
the charginos and neutralinos in the intermediate stages.
Schematically, the chains
of cascades leading to the Higgs bosons
are as follows:
\begin{equation}
pp \to \tilde{g} \tilde{g},\; \tilde{q} \tilde{q} , \; \tilde{q} \tilde{q}^*, \; \tilde{q} \tilde{g}
\longrightarrow \; \chpm2, \; \ntrl3, \; \ntrl4 \, + \, X \longrightarrow \;
\chpm1, \; \ntrl2, \; \ntrl1 \, + \, H^\pm, h, H, A \, + \, X
\end{equation}
\vskip -15pt
\begin{equation}
pp \to \tilde{g} \tilde{g},\; \tilde{q} \tilde{q}, \; \tilde{q} \tilde{q}^*, \; \tilde{q} \tilde{g}
\longrightarrow \; \chpm1, \; \ntrl2 \, + \, X \longrightarrow \;
\ntrl1, \, + \, H^\pm, \, h, H, A, H^\pm \, + \, X
\end{equation}
The first decay chain above is a longer one as it involves direct
decays of squark and gluinos to the heavier chargino/neuralinos followed by
subsequent decays of the latter ones to lighter gauginos
and the Higgs bosons.
On the other hand, the second chain is shorter since it exploits
direct decays of squarks and gluinos to the lighter
chargino/neutralinos which then decay to Higgs bosons
and the LSP. In the literature \cite{Datta:2001qs} the first scheme was
called the `big cascade' while the latter one was dubbed as the
`little cascade'. For convenience, we
adopt the same terminology in this work.
It is thus expected that the final yield of Higgs bosons under such SUSY-cascades crucially
depends upon the branching fractions of the relevant decay
processes. Thus on a complicated, though comprehensible, interplay of
different SUSY parameters in the form of various couplings and masses. Out of these,
the couplings of the Higgs bosons with the charginos/neutralinos play
the most important role. It is well known \cite{Gunion:1984yn,Datta:2001qs}
that the Higgs bosons couple
favourably to charginos and neutralinos when the latter are mixtures of gauginos and
Higgsinos while for gauge bosons the couplings are maximal when the charginos and the
neutralinos are Higgsino-dominated.
Naturally, then, the compositions of the charginos and neutralinos would play a crucial role
in our study \footnote{In scenarios with a universal gaugino mass at a high scale
(like the GUT scale), $M_1$ and $M_2$ gets related at the weak scale by the simple
relation
$M_2\simeq 2 M_1$. Thus, in that case, one only talks about 3 input parameters that govern
the chargino-neutralino sector. In contrast, the present work addresses the issue of
nonuniversality of gaugino masses in a particular context. Hence, $M_1$ and $M_2$
are taken to be two free parameters.}. Out of the determining parameters, the values
of $M_1$, $M_2$ and $\mu$ have the most crucial bearings on the masses and the
contents of the charginos and the neutralinos. For $\mu >> M_1, M_2$ , one is in the
so-called `gaugino region' where the lighter neutralinos and chargino
($\ntrl1,\ntrl2,\chpm1$) are
gaugino-dominated with $\mntrl1 \simeq min(M_1, M_2)$ and
$\mntrl2, \mchpm1 \simeq max( M_1,M_2)$
while the heavier ones ($\ntrl3, \ntrl4, \chpm2$) are mostly Higgsinos with
$\mntrl3, \mntrl4, \mchpm2 \simeq \mu$. On the other hand,
for $\mu << M_1, M_2$, we are in the `Higgsino region' for which the lighter neutralinos
and the chargino are predominantly Higgsinos with
$\mntrl1, \mntrl2, \mchpm1 \simeq \mu$ while the heavier ones are dominated by
gauginos with
$\ntrl3 \simeq min(M_1, M_2) $ and $\ntrl4, \mchpm2 \simeq max(M_1, M_2)$.
As expected, for different charginos and neutralinos,
the masses and the contents have one-to-one correspondences in such `pure' regions of the
SUSY parameter space. For $M_1,M_2 \simeq \mu$, the charginos and the neutralinos become maximally
mixed in gauginos and Higgsinos with their masses showing no particular pattern, albeit
restricted within a range determined by the values of $M_1, M_2$ and $\mu$.
In a nutshell, the `big cascades' are favoured in regions where, between
($\chi_2^{\pm}, \chi_3^0, \chi_4^0$) and ($\chi_1^{\pm}, \chi_1^0, \chi_2^0$),
one set is gaugino-dominated, and the other, Higgsino-dominated. In
situations where they are kinematically allowed, little cascades are
on the hand possible when the members of the second set above
comparable gaugino and Higgsino components.
We investigate charged as well as neutral Higgs production rates in cascades.
For a ready comparison, we closely follow the earlier analyses
\cite{Datta:2003iz,Datta:2001qs}. Explicit
expressions for most cross-sections and decay widths of
relevance are found in the above references.
\section{The results of nonuniversality: numerical results}
We are looking at `effective cross-sections' of Higgs production of
various kinds, which essentially means
$$
pp \longrightarrow Higgs + X
$$
\noindent
where the cross-sections of all possible (from $2 \to 2$ strong
productions) cascades are added up, so
long as there is at least one Higgs of any kind in the final state.
As cross-checks of the calculation, we have reproduced the results
in \cite{Datta:2003iz,Datta:2001qs} in the appropriate limits. We have used PYTHIA
\cite{Sjostrand:2006za} for our analysis and CTEQ3L \cite{CTEQ}
as parton distributions interfaced via LHAPDF \cite{CTEQ1}. The factorization/renormalization scale
set at the average of the masses of the particles (squarks and/or
gluinos) produced in the hard scattering. the analysis is based on
leading order production only. Also, following
\cite{Datta:2001qs}, the set of relevant SM and SUSY inputs (at the weak scale) chosen
for the analysis (unless otherwise specified) is:
\[ m_{top}=175 \ {\mathrm {GeV}} \quad \tan\beta=10 \quad
m_A=162.1 (237.5) \ \mathrm{GeV} \]
\[ m_{\tilde{g}}=900 \ \mathrm{GeV} \quad m_{\tilde{q}}=800 \
\mathrm{GeV} \quad A_f=0 \]
%
This resulted in the following masses for the different Higgs bosons:
\[ m_h=109 (110) \ \mathrm{GeV} \quad m_H=164 (238) \ \mathrm{GeV}
\quad m_{H^\pm}=180 (250) \
\mathrm{GeV} \]
which are in close agreement with those used earlier in the literature
\cite{Datta:2003iz,Datta:2001qs}. As for the sleptons, the only way they may signficantly contribute is by affecting the decay modes of
the gauginos. As indicated in section 1, we demonstrated the role of
light sleptons, taking them to be degenerate at 400 GeV \footnote{We have checked that the use of more recent parton distribution
functions, the variations in rates with renormalization/factorization
scale and the use of an updated top-quark mass do not alter the
basic findings of the present work.}.
The parameter-dependence of Higgs production rates under SUSY cascades is investigated in two ways: (i) variation with
$M_2$ and (ii) variation with $\mu$.
\subsection{Variation with $M_2$}
\begin{figure}[hbt]
\begin{center}
{\epsfig{file=sl400U.ps,width=4.7 cm,height=5.2cm,angle=-90.0}}
{\epsfig{file=sl400n.ps,width=4.7cm,height=5.2cm,angle=-90.0}}
{\epsfig{file=sl400m1200.ps,width=4.7cm,height=5.2cm,angle=-90.0}}
\caption{Effective cross-sections for universal ($M_1 = M_2/2$, (a)) and
nonuniversal (with $M_1=100$ GeV, (b) and with $M_1=200$ GeV (c)) for
$\mu=150$ GeV and $m_{H^{\pm}}=180$ GeV.}
\end{center}
\label{fig11}
\vspace*{-1.0cm}
\end{figure}
\begin{figure}[hbt]
\begin{center}
{\epsfig{file=sl400H+250u.ps,width=4.7 cm,height=5.2cm,angle=-90.0}}
{\epsfig{file=sl400H+250m100.ps,width=4.7cm,height=5.2cm,angle=-90.0}}
{\epsfig{file=sl400H+250m1200.ps,width=4.7cm,height=5.2cm,angle=-90.0}}
\caption{Effective cross-sections for universal ($M_1 =
M_2/2$, (a)) and nonuniversal (with $M_1=100$ GeV, (b), with
$M_1=200$ GeV (c)) for $\mu=150$ GeV and $m_{H^{\pm}}=250$ GeV.}
\end{center}
\label{fig11}
\end{figure}
\vspace*{0.8cm}
\begin{figure}[hbt]
\begin{center}
{\epsfig{file=sl400mu700U.ps,width=4.7 cm,height=5.2cm,angle=-90.0}}
{\epsfig{file=sl400mu700.ps,width=4.7cm,height=5.2cm,angle=-90.0}}
{\epsfig{file=sl400mu700m1200.ps,width=4.7cm,height=5.2cm,angle=-90.0}}
\caption{Effective cross-sections for universal ($M_1 = M_2/2$,
(a)), nonuniversal (with $M_1=100$ GeV (b) and $M_1=200$ GeV (c)) for $\mu=700$ GeV and $m_{H^{\pm}}$=180 GeV.}
\end{center}
\label{fig11}
\vspace*{-0.5cm}
\end{figure}
\begin{figure}[hbt]
\vspace*{-1.2cm}
\begin{center}
{\epsfig{file=H+250mu700U.ps,width=4.7 cm,height=5.2cm,angle=-90.0}}
{\epsfig{file=slH+250mu700.ps,width=4.7cm,height=5.2cm,angle=-90.0}}
{\epsfig{file=H+250mu700m1200.ps,width=4.7cm,height=5.2cm,angle=-90.0}}
\caption{Effective cross-sections for universal ($M_1 = M_2/2$,
(a)), nonuniversal (with $M_1=100$ GeV (b) and $M_1=200$ GeV (c))
for $\mu=700$ GeV and $m_{H^{\pm}}$=250 GeV.}
\end{center}
\label{fig11}
\hspace*{-0.5cm}
\end{figure}
Since $m_{\tilde{\ell}}$ is taken to be 400 GeV, the
sleptons have a significant role in the cascades.
Thus, the natural expectation is that once these sleptonic
decay modes of the charginos/neutralinos open up, cascades to Higgs would get
suppressed. Hence the overall rates presented are of a conservative nature.
This is clear from the set of Figures 1 to 4. We discuss below some
generic features present in these figures and the information we get
from them.
Plot (a)s depict the universal scenario while plot (b)s and plot (c)s
are for the nonuniversal scenario. Figure 1 and Figure 2 are for $\mu=150$ GeV
while Figure 3 and Figure 4 are for $\mu=700$ GeV. For
$\mu=150$ GeV the lighter gauginos are too closely degenerate
(for both universal and nonuniversal scenarios) for the `little cascades'
to open up. Hence, the entire cascade contribution to Higgs production
comes from the `big cascade'. The sudden rises in some curves at
specific $M_2$ values in the universal scenario indicate attaining
the right mass-splitting between $\ntrl3$ and the LSP such that
$\ntrl3$ decaying to the lightest neutral (charged) Higgs boson and
the LSP (the lighter chargino) becomes possible. This feature is not
there in the nonuniversal cases (where $M_1$ is set to 100 and 200
GeV respectively) as $\mntrl3\sim \mu$.
One should note the different pattern of $M_2$-dependence of the
rates for $\mu=150$ GeV (Figures 1 \& 2) and 700 GeV (Figures 3 \& 4), respectively. This is because the
former situation allows Higgs production mostly through `big
cascades'. The latter case, where larger separation amongst the low-lying
states is possible, `little cascade' more abundantly, thus
making the variation of Higgs production rates with $M_2$ look
different. In particular, slepton masses of the order of 400 GeV affect
`big cascades' less for larger value of $M_2$ throught the enhancement
of the effective coupling for Higgs production. Little cascades are
affected much more in such a case, thus causing difference in the way
the rates fall with increasing $M_2$.
In Figure 2 we illustrate a case similar to Figure 1 except for
$m_{H^\pm}=250$ GeV. This needs a substantial increment in the mass of
$A$ as input and results in a larger $H$ compared to those for Figure
1. Thus, in the universal case (Figure 2(a)) $\ntrl3$ needs to be heavier such that the heavier
Higgs bosons may be produced in the decay of $\ntrl3$ along with
$\ntrl1$ or $\chpm1$. Note that increasing $m_{H^\pm}$ does not affect the
rate for the lightest higgs ($h$) significantly, when compared to the
corresponding plot in Figure 1 since $m_h$ remains almost unaffected by
such an increase in $m_{H^\pm}$. Thus, for $m_{H^\pm}=250$ GeV, there is no
cross-over between the curves for $h$ and $H^\pm$ in the universal case
unlike $m_H^{\pm}$=180 GeV.
The general observation is that with increasing $m_{H^\pm}$ the threshold
value of $M_2$ shifts naturally to the right leading to more massive
heavier charginos and neutralinos such that the `big cascades' may
take place. This eventually pushes the cross-over point (of the
rate-curves for the lightest and charged Higgs bosons) to larger
values of $M_2$. Also, the rates for heavier Higgs bosons are
smaller for $m_{H^\pm}=250$ GeV as compared to $m_{H^\pm}=180$ GeV. There are
two reasons for this. First, the heavier Higgs bosons now become more
massive whose rates suffer a phase-space suppression for similar
chargino and neutralino masses. Second, the charginos and neutralinos
whose decay results in the Higgs bosons (with increased masses) have to
become heavier as well. Thus the production
rates for the latter also get affected. Figures 2(b) and 2(c) illustrating
the nonuniversal cases with $M_1=100$ GeV and 200 GeV are to be
contrasted with the corresponding ones in Figure 1. They only differ
by the generic features as described above.
Figure 4 illustrates a situation similar to Figure 3 except for $m_{H^\pm}=250$
GeV. In the universal case (Figure 4(a)), the peak in the $H^\pm$ rate
disappears when compared to Figure 3(a). One should note that this peak is due
predominantly to a `little cascade' like $\chpm1 \to \ntrl1 H^\pm$.
As indicated earlier, with growing $m_{H^\pm}$ the mass splitting between
the gaugino states involved above are not enough to accommodate the
above cascade. The situation could be a little different for a
nonuniversal case with a lower value of $M_1$ (100 GeV) as shown in
Figure 4(b). The smaller value of $M_1$ now ensures a lower mass for the
$\ntrl1$ thus help regaining the required splitting when the peak in
the $H^\pm$ is back (at around 400 GeV). In Figure 4(c), $M_1$
is 200 GeV and this again blocks the above decay mode at around 400
GeV. Of course, with increasing value of $M_2$ ($\geq 500$ GeV) the
Higgs productions under such cascades open up again. But this time,
Br[$\chpm1 \to \ntrl1 H^\pm$] starts getting suppressed as the two-body sleptonic decay modes
of $\chpm1$ take off for our choice of slepton mass (400 GeV).
\subsection{Variation with $\mu$}
\begin{figure}[hbt]
\begin{center}
\vspace*{-0.5cm}
\hskip -65pt
{\epsfig{file=MuM2400u.ps,width=6.0 cm,height=6.7cm,angle=-90.0}}
{\epsfig{file=MuM1100M2400.ps,width=6.0 cm,height=6.7cm,angle=-90.0}}
\caption{Effective cross-section as a function of $\mu$ for universal
(with $M_1=M_2/2$, (a)) and nonuniversal (with $M_1=100$ GeV, (b))
scenarios with $M_2=400$ GeV and $m_{H^{\pm}}=180$ GeV.}
\end{center}
\label{fig11}
\end{figure}
\begin{figure}[hbt]
\begin{center}
\vspace*{-0.5cm}
\hskip -65pt
{\epsfig{file=MuH+250u.ps,width=6.0 cm,height=6.7cm,angle=-90.0}}
{\epsfig{file=MuH+250M100.ps,width=6.0cm,height=6.7cm,angle=-90.0}}
\caption{Effective cross-section as a function of $\mu$ for universal
(with $M_1=M_2/2$, (a)) and nonuniversal (with $M_1=100$ GeV, (b))
scenarios with $M_2=400$ GeV and $m_{H^{\pm}}=250$ GeV.}
\end{center}
\label{fig11}
\vspace*{-1.0cm}
\end{figure}
In Figures 5 \& 6, we illustrate the variation of the rates for Higgs production under
SUSY cascades with $\mu$. Plot (a)s represent the universal scenario
while plot (b)s illustrate the same in a nonuniversal situation. In both
cases, for small values of $\mu$, the lighter charginos and
neutralinos are Higgsino-like and their masses are of the order of
$\sim \mu$ with a definite split from their heavier mates governed by
the value of $M_2$ chosen. With increasing $\mu$, the ``mixed region'' is approached
and the mass-splittings among the heavier and lighter charginos
decrease. This gradually eliminates the possible sources of the
heavier Higgs bosons, especially those of $H^\pm$. Further, with
increasing $\mu$, channels like $\chpm2 \to \ntrl1 H^\pm$ again opens
up followed by $\chpm1 \to \ntrl1 H^\pm$ at about $\mu=525$ GeV in the
universal case. A further mild rise in the $H^\pm$-rate is observed at
around $\mu=550$ GeV when $\ntrl3 \to \chpm1 H^\pm$ opens up. However,
in the nonuniversal case, due to a fixed low value of $M_1$ the LSP mass
remains almost the same ($\sim M_1$). Hence channels like $\chpm1 \to
\ntrl1 H^\pm$ opens up as early as $\mu=300$ GeV. The features observed
in these plots are generic as long as the relative splittings of
gaugino masses and $\mu$ used here remain similar.
In Figure 6 we illustrate the case similar to Figure 5 except for
$m_{H^\pm}=250$ GeV. In the universal case, when compared to Figure 5, we find
that the rise in the rate for $H^\pm$ at around $\mu=500$ GeV is missing.
This is because of the lack of enough splitting between the lighter
chargino and the LSP masses that might lead to $H^\pm$ unlike what happened in
the universal case when $m_{H^\pm}=180$ GeV. For $M_1=100$ GeV, $\chpm1
\to \ntrl1 H^\pm$ opens up at a later stage compared to $m_H^{\pm}$=180
GeV case. For higher neutral Higgses similar things happen.
\subsection{Comparison between $H^\pm$ and $h$ production rates}
Figures 1, 3 and 5 ($m_{H^{\pm}}$=180 GeV) along with the discussions
in sections 3.1 and 3.2 reveal a rather complicated interplay of
several masses and couplings. It is undoubtedly a difficult task
to extract useful information from these processes in a systematic
way. However, a closer look at these reveal that we have a somewhat
clean feature in the variations of rates for $H^\pm$ and $h$ which we
can use to our benefit. These rates behave in a distinctly
complementary fashion as functions of $M_2$ (Figures 1 and 3) and
$\mu$ (Figure 5) and show up as multiple cross-over points. For
universal and nonuniversal scenarios, these
cross-overs take place at different values of $M_2$ and $\mu$. This can be
understood in a straight-forward manner following the discussions in sections 3.1 and 3.2.
The observations prompt us to look out for a suitable illustration of
the situation. In Figure 7 we present the scattered plots of the
two-rates compared in the $M_2-\mu$ plane for both universal and
nonuniversal cases. Over the grey regions the rates for the charged
Higgs bosons is greater than that for the lightest neutral Higgs boson and
the reverse is the case over the darker regions. At suitable points on
these plots, direct correspondences can be made with Figures 1, 3 and
5. There are quite a few distinctive patches on these two plots. If
such relative rates can be known from the LHC experiments, these
plots could be used to have a preliminary understanding of the gaugino
mass pattern.
In Figure 8 we present a similar set of scattered plots but for
$m_H^{\pm}$=250 GeV. In the universal case, all over the $M_2-\mu$ space
the rate for lightest Higgs is greater than that for the charged Higgses.
For the nonuniversal case, with increasing $M_1$ the region where the
rate for the charged Higgses is larger than that for the lightest Higgs
gradually shrinks. Again, all these can be read off from the plots in Figures
1 to 6. Thus, we see that $m_{H^\pm}$ and $M_1$ play some significant roles
in characterizing the $M_2-\mu$ plane.
We have checked that the results presented here are robust against
wide variation of squark and gluino masses, even when their ordering
is reversed. Increasing the slepton masses would only make life
simpler in the sense that they get decoupled from the cascades and we
checked that this does not bring out any new feature. We also
checked that the effect of $\tan\beta$ is anything but significant.
This validates one of the very basic expectations for studying the
Higgs production under SUSY cascades and hence its power
of probing the Higgs sector for intermediate values of $\tan\beta$.
It is useful to mention here that the confirmation of gaugino
nonuniversality in Higgs production presupposes some knowledge of
the SUSY spectrum, as obtained from the heaviness or
missing-${E_T}$ distribution of the SUSY signals. As the scatter plots
in figures 7 and 8 indicate, one is able to infer conclusively about
universality (or the lack of it) from the relative magnitudes of the
$h$- and $H^{\pm}$-production rates, provided that two quantities out
of $M_1$, $M_2$ and $\mu$ are known. However, even that does not rob
the present study of its merits, since any unambiguous conclusion
about the particle spectrum of a new physics scenario requires a
multichannel analysis. In that sense, the inclusive Higgs production
processes provide a channel of supreme importance in understanding the
electroweak gaugino-Higgsino sector.
\vspace*{0.5cm}
\begin{figure}[hbt]
\begin{center}
\vspace*{-0.5cm}
\hskip 0pt
{\epsfig{file=sim1.ps,width=5.0 cm,height=5.2cm,angle=-90.0}}
\hskip 2pt
{\epsfig{file=sim2.ps,width=5.0 cm,height=5.2cm,angle=-90.0}}
\hskip 2pt
{\epsfig{file=simM1200.ps,width=5.0 cm,height=5.2cm,angle=-90.0}}
\caption{The pink (dark) region is the region where the rate for H$^{\pm}$
is less than that for $h$ production, for the sky-blue region (grey)
it is the opposite and white regions indicate zero production for both.
We take $M_1=M_2/2$ (a), $M_1=100$ GeV (b) and $M_1=200$ GeV
(c). $m_{H^{\pm}}$ is set to 180 GeV.}
\end{center}
\label{fig11}
\end{figure}
\begin{figure}[hbt]
\begin{center}
\vspace*{-0.5cm}
\hskip 0pt
{\epsfig{file=simH+250U.ps,width=5.0 cm,height=5.2cm,angle=-90.0}}
\hskip 2pt
{\epsfig{file=simH+250m1100.ps,width=5.0 cm,height=5.2cm,angle=-90.0}}
\hskip 2pt
{\epsfig{file=simH+250m1200.ps,width=5.0 cm,height=5.2cm,angle=-90.0}}
\caption{The pink (dark) region is the region where the rate for H$^{\pm}$
is less than that for $h$ production, for the sky-blue region (grey)
it is the opposite and white regions indicate zero production for both.
We take $M_1=M_2/2$ (a), $M_1=100$ GeV (b) and $M_1=200$ GeV
(c). $m_{H^{\pm}}$ is set to 250 GeV.}
\end{center}
\label{fig11}
\end{figure}
\section{Conclusions}
We have investigated the rates for production of different physical
Higgs states in SUSY cascades at the LHC. The new inputs are
(a) relaxation of the universality condition relating electroweak
gaugino masses and (b) allowing sleptons to be light enough to be
produced on-shell in cascades. As we show through the dependence of
production rates on various parameters, both these inputs can
significantly affect the
phenomenology at the LHC. The most important observation is that the
presence or absence of gaugino universality is reflected in a rather
interesting complementarity between the relative production rates of
$H^{\pm}$ and $h$. Therefore, one should aim at a careful
identification of charged as well as neutral Higgs signals in
cascades to extract useful information about the underlying SUSY
scenario. A detailed discussion of such signals {\it all} the physical
Higgs states in SUSY and methods of
suppressing their backgrounds will be presented in a separated study.
\vskip 20pt
{\bf Acknowledgments:}
We acknowledge S. Mrenna for resolving and clarifying some issues in
Pythia and Peter Skands for useful discussions.
Computational work for this study was partially carried out in the cluster
computing facility at Harish-Chandra Research Institute
(http://cluster.mri.ernet.in). This work is partially supported by
Regional Center for Accelerator-based Particle Physics, Harish-Chandra
Research Institute and funded by the Department of Atomic Energy, Government of India under the XIth 5-year Plan.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,596 |
Q: How do I integrate $\sqrt{\frac{1+x}{1-x}}$? How do I integrate $\sqrt{\frac{1+x}{1-x}}$ using standard calculus techniques?
I tried trig substitution but it doesn't seems to work. is that some kind of u-substitution?
A: To start, rationalize the numerator:
$$\begin{align}
\int\sqrt{{1+x\over 1-x}}\,dx & =\int\frac{1+x}{\sqrt{1-x^2}}\,dx\\
& =\int\frac{dx}{\sqrt{1-x^2}}+\int\frac x{\sqrt{1-x^2}}\,dx.
\end{align}$$
On the last line, you can use trig substitution on the first integral and $u$-substitution on the second.
A: To find $ \int \sqrt{\frac{1+x}{1-x}}dx $
You begin by multiplying $\frac{\sqrt{1+x}}{\sqrt{1+x}}$ to the integral.
then you get
$ \int \sqrt{\frac{(1+x)^2}{1-x^2}}dx $
now use trig substitution of
$x=sin \theta$
$dx = cos \theta d\theta$
we get $ \int \sqrt{\frac{(1+sin \theta)2}{1-sin^2 \theta}}cos \theta \, d\theta = \int 1+sin \theta \, d\theta = \theta - cos\theta + C = arcsin x - \sqrt{1-x^2} + C$
A: $$\sqrt{\frac{1+\cos \theta}{1-\cos \theta}}=|\cot\frac{\theta}{2}|$$
let $$x=\cos \theta$$
$$dx=-\sin \theta d\theta$$
$$\int \sqrt{\frac{1+ x}{1- x}}dx= \sqrt{\frac{1+\cos \theta}{1-\cos \theta}}(-\sin \theta d\theta)=\int |\cot\frac{\theta}{2}|(-\sin \theta d\theta)$$
$$\int |\frac{1+\cos \theta}{\sin \theta}|(-\sin \theta d\theta)=-\int(|1+\cos \theta|)d\theta$$
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,942 |
Crocanthes diula is a moth in the family Lecithoceridae. It was described by Edward Meyrick in 1904. It is found in Australia, where it has been recorded from Queensland.
The wingspan is . The forewings are bronzy fuscous, irrorated (sprinkled) with dark fuscous and with an oblique whitish-ochreous mark on the costa before the middle. There is a narrow wedge-shaped whitish-ochreous mark along the costa beyond three-fourths. The hindwings are dark bronzy-fuscous.
References
Moths described in 1904
Crocanthes | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,443 |
Tony Wilding ha battuto in finale Frank Fisher 6-0 6-4 6-4.
Tabellone
Fase finale
Collegamenti esterni
Australasian Championships 1906 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,664 |
You may remember me borrowing this dress from Jessica of Midwest Muse for a Flock Together post a while ago. I was sad to part with it--I mean, look at those pretty colors! I'll admit, I started obsessing over it for a while. Luckily, I happened to see they were having a Minuet flash sale on Ideeli a couple of months ago, and I ended up getting my own for about $40 (Score!).
I wore it for my birthday dinner with my friends and styled it with my new My Flat in London bangle and bow clutch, black "Ampersand" heels, and my silver bunny necklace from Katrina. I also swiped on a bright, matte orange-red lipcolor (Stila's 'Stay All Day' in "Tesoro") to match the red stripes. I seriously can't wait to wear this dress again! It's still available on ModCloth if you're in love with it too. It's the most joyful piece of clothing I own! | {
"redpajama_set_name": "RedPajamaC4"
} | 9,517 |
Q: Selected Check box value display in URL with parameter using Php Javascript I need to get selected check box value in browser url with parameter name.
My HTML code:
<div style="width:200px; float:left">
Price
<div>
<input type="checkbox" name="price" value="0-1000" class="filters" />0-1000<br />
<input type="checkbox" name="price" value="1000-2000" class="filters" />1000-2000<br />
<input type="checkbox" name="price" value="3000-4000" class="filters" />2000-3000<br />
</div>
Colors
<div>
<input type="checkbox" name="colors" value="Red" class="filters" />RED<br />
<input type="checkbox" name="colors" value="Green" class="filters" />GREEN<br />
<input type="checkbox" name="colors" value="Blue" class="filters" />BLUE<br />
</div>
</div>
My Javascript Code
<script type="text/javascript">
$('.filters').on('change',function(){
var price = new Array();
$('input[name=price]:checked').each(function(){
price.push($(this).val());
});
var colors = new Array();
$('input[name=colors]:checked').each(function(){
colors.push($(this).val());
});
location.href = 'http://localhost/test/javascript.php?price='+price;
});
</script>
I need Browser Url like below, after selecting two prices and one color
http://localhost/test/javascript.php?prices=0-1000,1000-2000&colors=Red
A: You are looking for .join()
The join() method joins the elements of an array into a string, and
returns the string.
The elements will be separated by a specified separator. The default
separator is comma (,).
location.href = 'http://localhost/test/javascript.php?price=' + price.join(',') + '&colors=' + colors.join(',');
Even though the comma seems to be the default separator , I would add it anyway just to be sure.
var url = 'http://localhost/test/javascript.php?';
if(price.length) {
url += '&price=' + price.join(',');
}
if(colors.length){
url += '&colors=' + colors.join(',');
}
location.href = url; // or url.replace('?&','?');
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,338 |
\section{Introduction}
\section{Two unsolved problems}
\label{twoproblems}
The problem section of the June 1986 issue of \emph{SIAM Review} lists the following,
labeled \emph{Problem 86-6$^{*}$}~\cite{SRproblem866}:
\begin{quote}
In many audit populations items may have partial errors. Suppose each item in the population has an error size known
to be in the interval $[0,1]$. Suppose the mean population error is $m$ where $0<m<1$. A simple random sample of size
$n$ is drawn with replacement from that population. Let $S_n$ be the random variable representing the sum of the error
sizes of the $n$ sampled items. Given a constant $t <mn$ how should the error sizes be distributed in the population
to maximize \prob{S_n \leq t}?
It is conjectured that for each $m$ and $t$, there is a population with just two error sizes, one of which is $0$ or
$1$, such that \prob{S_n \leq t} is maximized. Prove or disprove.
\end{quote}
It is added that if this conjecture is true, then it will be possible to determine simple bounds on upper confidence
limits for some audit sampling problems.
The problem section of \emph{Statistica Neerlandica}, Vol.~47, no.~1, lists the following as
\emph{Problem 294}~\cite{SNproblem294}:
\begin{quote}
Consider i.i.d.~random variables $X_1, \ldots, X_n$ with $0 \leq X_i \leq 1$ and $\expec{X_i}=m$ given.
Let $S_n=X_1+ \cdots +X_n$. Consider the following statement:
\[ p= \prob{S_n \leq t} \quad \text{is maximal if} \quad \prob{X_1=1}=1-\prob{X_1=0}=m. \]
Show that this statement holds for all $t$ such that $p \leq p_0$ for some $p_0<1$ and find such a value for $p_0$.
\end{quote}
It appears that no solutions to these problems have been published.
This paper addresses them and presents a (partial) solution by considering:
\begin{quote}
Let $0<m<1$ and let \ensuremath{\mathcal{D}_{m}}\ be the set of probability measures on $[0,1]$ with mean $m$. Let $X_1, \ldots, X_n$ be
i.i.d.\ $\mu \in \ensuremath{\mathcal{D}_{m}}$ and $S_n=X_1+\cdots+X_n$. Determine $p(m,t) = \sup_{\mu \in \ensuremath{\mathcal{D}_{m}}}
\pmu{S_n \leq t}$ and, if possible, (all) $\mu$ attaining the maximum.
\end{quote}
Note
that $p(m,t)=1$ for $mn \leq t$: set $X_i = m$, all $i$;
for $mn \leq t$ the question of maximizing \prob{S_n \geq t} would be more natural.
However, since $mn < t \leq n$ implies $n-t<n(1-m)$ and
$\prob{S_n \geq t} = \prob{n-S_n \leq n-t} \leq p(1-m,n-t)$, this case is
included in the problem statement; $t=mn$ once again is the trivial case.
So henceforth, $0 \leq t < mn$ is assumed.
In the sequel, when emphasis on the dependence on~$\mu$ is required,
$\pmu{\cdot}$ will be used.
The supremum $p(m,t)$ is indeed attained by an element of \ensuremath{\mathcal{D}_{m}}, by Weierstra\ss' theorem,
because the set \ensuremath{\mathcal{D}_{m}}\ is weak*-compact and $\mu \mapsto \pmu{S_n \leq t}$ is weak*-continuous.
A $\mu \in \ensuremath{\mathcal{D}_{m}}$ is called \emph{extremal} (for certain $n$, $m$ and $t$) if $\pmu{S_n \leq t}=p(m,t)$.
\medskip
The problem at hand satisfies a common rule:
$n=1$ is trivial, $n=2$ can be solved with a reasonable amount of work, and $n \geq 3$ is hard.
After the $n=1$ case, we start with some general observations.
After that, some relevant results from the literature are discussed, which show that part of the $n=2$ case
follows from a paper by Hoeffding and Shrikande~\cite{hoeffdingshrikande} from~1955. We embark on a different
approach, applying a Lagrange multiplier technique from Mattner~\cite{mattner}, in Section~\ref{mattnersapproach},
for arbitrary $n$.
The resulting Lagrange conditions provide a characterization of extremal distributions.
For the $n=2$ case, this allowed us to show that supports of extremal distributions necessarily look like
$\{s, t-s\}$ or $\{s, t-s, 1\}$, for some $s$.
After this reduction, shown in Section~\ref{threesupportpoints}, the search for extremal distributions may be
restricted to this more manageable class and a complete analysis is carried out.
As it turns out, the conjectures stated in the problems above
can be refuted based on these results (Section~\ref{conjectures}).
The first conjectured solution, however, seems ``almost true'' and its exception is understandable,
so in Section~\ref{revised} a revised and sharpened conjecture is formulated, including the specification, for each
$n$, of a collection of distributions of which the extremal one is conjectured to be a member.
\subsection{The case $n=1$}
Markov's inequality implies that
\begin{equation}
\label{markov}
\prob{X \leq t} = \prob{1-X \geq 1-t} \leq \frac{1 - \expec{X}}{1-t} = \frac{1-m}{1-t}
\end{equation}
and this upper bound is attained by the following two-point distribution:
$\prob{X=t}=(1-m)/(1-t)$ and $\prob{X=1}=(m-t)/(1-t)$.
\subsection{Some results from the literature}
Hoeffding and Shrikande~\cite{hoeffdingshrikande} obtained results on
the supremum of the distribution function of the
i.i.d.~sum of \emph{two} random variables, given $k$ moment conditions and a restricted range.
They showed that the supremum over all such distributions is the same as that over all discrete distributions with
at most $2k+2$ support points. In addition, they provide the following bound for \emph{nonnegative} i.i.d.~$X_1$
and $X_2$ with $\expec{X_1}=\gamma$:
\begin{equation}
\label{hsbound}
\prob{X_1+X_2 \geq c \gamma} \leq
\begin{cases}
1 & \text{if $c \leq 2$;}\\
4/c^2 & \text{if $2 \leq c \leq \frac52$;} \\
2/c- 1/c^2 & \text{if $\frac{5}{2} \leq c$.}
\end{cases}
\end{equation}
For i.i.d.~$X_1$ and $X_2$ on $[0,1]$, with $\expec{X_1}=m$,
one may translate the above result to one on the \emph{left} tail,
by switching to the complements with respect to 1:
\begin{equation*}
\prob{X_1+X_2 \leq t}
= \prob{{1-X_1}+{1-X_2} \geq 2-t},
\end{equation*}
and (\ref{hsbound}) applies with $\gamma= 1-m$ and $c=(2-t)/(1-m)$.
The distributions attaining the resulting bound have as their support, respectively, $\{m\}$, $\{t/2,1\}$, and $\{t-1,1\}$.
The first applies to $c\leq 2$, or $t\geq 2m$, the trivial case;
the second to $2 \leq c \leq 5/2$, or $t/2 \leq m \leq (2t+1)/5$;
the third to $5/2 \leq c$, or: $t \geq 1$ and $(2t+1)/5 \leq m \leq 1$.
These results resolve the $n=2$ case for a \emph{subset} of $(m,t)$-values.
Furthermore, Hoeffding and Shrikande~\cite{hoeffdingshrikande}
did not address the question of uniqueness of the extremal distributions.
\medskip
Hoeffding's well known inequality (see \cite{hoeffding}) bounds
deviations from the expected value for the average of independent, not-necessarily identically distributed, random
variables. The author states that the bound is not optimal, but the best bound that can be obtained via his
method, based on the moment generating function. Applying Theorem~1~\cite{hoeffding}, one obtains:
\begin{equation}
\label{ineqHoeffding}
\prob{S_n \leq t} \leq \left ( \frac{1-m}{1-t/n} \right )^{n-t} \, \left (\frac{m}{t/n} \right)^t.
\end{equation}
For $n=1$ it is clear that (\ref{markov}) is sharper, since $m>t$.
\subsection{Some general observations}
\label{generalobs}
\paragraph{One may assume $0$ or $1$ to be in the support.}\label{zeroandone}
Recall the definition of the \emph{support} of a measure $\mu$: the smallest closed set with measure 1; we denote it
by \supp{\mu}.
Suppose, for some extremal $\mu$ one has $\supp{\mu} \subset [a,b]$ with $0<a<b<1$.
Let $X_i$ have distribution $\mu$ and $Z_i = m + \alpha (X_i -m)$, $i=1,\ldots, n$.
Then $\expec{Z_i}=m$ and there exist $\alpha>1$ such that $\supp{Z_i} \subset [0,1]$.
Writing $S^*_n= Z_1+\cdots+Z_n=\alpha S_n-(\alpha-1)\,m n$ one has for $t <m n$:
\[ \prob{S^*_n \leq t}=\prob{S_n \leq \tfrac1\alpha\, t + \tfrac{\alpha-1}{\alpha} \,m n} \geq \prob{S_n \leq t}. \]
Thus, if $\mu$ is extremal, a measure $\mu^*$ can be found that is extremal as well, and if the largest $\alpha$ is
chosen that satisfies $\supp{\mu^*} \subset [0,1]$, then \supp{\mu^*} will contain $0$ or $1$.
\paragraph{The supremum $p(m,t)$ is non-increasing in $m$.}
Fix $n$ and $t$. Suppose $m_1 < m_2$ and $\mu_2$ attains the maximum value $p(m_2,t)$.
Suppose $\mu_2$ has distribution function $G$. Define, for $0 \leq r \leq 1$, the distribution
function $F_r$ by $F_r(x)=\max(r, G(x))$.
As $r$ goes from $0$ to $1$ the expectation of the corresponding distribution decreases from $m_2$ to $0$,
continuously, so for some $r$ the corresponding distribution $\mu_1$ has expectation~$m_1$.
This measure is stochastically smaller than $\mu_2$.
Hence, $p(m_1,t) \geq \psub{\mu_1}{S_n \leq t} \geq \psub{\mu_2}{S_n \leq t} = p(m_2,t)$.
\paragraph{Subprobability measures with expectation at least $m$.}
Instead of taking the supremum over \ensuremath{\mathcal{D}_{m}}\ one could take the supremum over
the set of all \emph{sub}probability measures on $[0,1]$ with mean \emph{at least} $m$,
and the same $p(m,t)$ would result.
In order to show this, suppose $\nu$ is a subprobability measure with defect $\rho=1-\nu([0,1]) \geq 0$ and
$\int_0^1 x \, \nu(\mathrm{d} x) = r \geq m$. Then
\[
\pnu{S_n \leq t} \leq \psub{\nu+ \rho \cdot \delta_0}{S_n \leq t} \leq p(r,t) \leq p(m,t),
\]
where the first inequality states that putting the mass-defect in $0$ will lead to an improvement, and the last
inequality follows from the non-increasing\-ness proved above. Considering that $d>0$ or $r>m$ will make at least one
of the inequalities strict, it is clear that $p(m,t)$ can only be attained with $d=0$ and $r=m$.
\paragraph{An upper confidence bound on $m$.}
Using $S_n$ as test statistic one may define a non-parametric confidence bound on $m$, as follows.
Let $m_u$ be solution of $p_n(m,t)=\alpha$. By the non-increasing\-ness proved above,
this implies that $p_n(m,t) \leq \alpha$ for $m \geq m_u$. So, for
$m \geq m_u$ it follows that $\prob{S_n \leq t} \leq p_n(m,t) \leq p_n(m_u,t)=\alpha$.
\section{Mattner's Lagrange approach}
\label{mattnersapproach}
Mattner~\cite{mattner} developed a general method for treating extremal problems for probability distributions.
His main theorem is stated below and subsequently applied to the problem:
\begin{theorem}
Let $Z$ be a Banach space,
$\varphi_i : Z \rightarrow \mathbb{R}$, $i=0, \ldots, k$ and
$\psi_j : Z \rightarrow \mathbb{R}$, $j=1, \ldots, l$, continuously Fr\'echet-differentiable,
and $C$ a convex cone in~$Z$.
Define the Lagrange functional
\[ \ensuremath{\mathcal{L}}(z) := \lambda_0 \varphi_0(z)+ \sum_{i=1}^k \lambda_i \varphi_i(z) + \sum_{j=1}^l \alpha_j \psi_j(z), \]
and let $\partial\ensuremath{\mathcal{L}}(z;w)$ denote the Fr\'echet-derivative of $\ensuremath{\mathcal{L}}(z)$ in direction $w$.
If $z \in Z$ minimizes $\varphi_0$ subject to
\begin{align*}
\varphi_i(z) &\leq 0, \quad i=1, \ldots, k,\\
\psi_j(z) &= 0, \quad j=1, \ldots, l,\\
z & \in C,
\end{align*}
then there exist $\lambda_0, \lambda_1, \ldots, \lambda_k, \alpha_1, \ldots, \alpha_l \in \mathbb{R}$ with
\begin{tabbing}
(i)\quad \= not all $\lambda_i$ and $\alpha_j$ vanish,\\
(ii) \> $\lambda_i \geq 0$, $i=0, \ldots, k$, \\
(iii) \> $\partial\ensuremath{\mathcal{L}}(z;w) \geq 0$, $w \in C$ \\
(iv) \> $\partial\ensuremath{\mathcal{L}}(z;z) = 0$.
\end{tabbing}
\end{theorem}
\paragraph{Application to the problem.}
Let \ensuremath{\mathcal{M}}\ be the set of signed Borel measures on $[0,1]$, with
norm $\norm{\mu}=\int |\mu(\mathrm{d} x) |$ (total variation). The pair $(\ensuremath{\mathcal{M}}, \norm{\,\cdot\,})$ is a
Banach space and probability measures are contained in the positive
cone~$C=\{\mu \in \ensuremath{\mathcal{M}}: \mu(B) \geq 0 \: \text{for every Borel set $B$}\}$.
Define
\begin{equation*}
\varphi_0(\mu) = -\prob{S_n \leq t}= - \int_A \mu(\mathrm{d} x_1) \cdots \mu(\mathrm{d} x_n),
\end{equation*}
where $A=\{(x_1,x_2, \ldots, x_n): x_1+\cdots+x_n \leq t\}$.
The (clearly continuous) Fr\'echet-derivative of $\varphi_0$ is given by
\begin{equation*}
\partial\varphi_0(\mu;\nu)=-n \, \int_A \mu(\mathrm{d} x_1) \cdots \mu(\mathrm{d} x_{n-1}) \nu(\mathrm{d} x_n) =
-n \int_0^1 \prob{S_{n-1}\leq t-x} \, \nu(\mathrm{d} x).
\end{equation*}
Indeed, this follows from the next formula, which is established by binomial expansion of the $n$-fold
product of the measure $\mu+\nu$:
\begin{equation*}
\norm{\varphi_0(\mu+\nu)-\varphi_0(\mu)-\partial\varphi_0(\mu;\nu)} = O\left(\norm{\nu}^2\right).
\end{equation*}
For the constraints define
\begin{equation*}
\varphi_1(\mu) = \mu([0,1]) -1\quad\text{and}\quad \varphi_2(\mu) =m - \int_0^1 x \, \mu(\mathrm{d} x),
\end{equation*}
whose (continuous) Fr\'echet-derivatives are given by
\begin{equation*}
\partial\varphi_1(\mu;\nu) =\nu([0,1]) \quad\text{and}\quad \partial\varphi_2(\mu;\nu) = - \int_0^1 x \, \nu(\mathrm{d} x).
\end{equation*}
Now, define the Lagrange functional:
$\ensuremath{\mathcal{L}}(\mu) = \lambda_0\, \varphi_0(\mu) + \lambda_1\, \varphi_1(\mu) + \lambda_2\, \varphi_2(\mu)$.
From Mattner's theorem one concludes:
if $\mu$ minimizes $\varphi_0$ subject to $\varphi_1(\mu) \leq 0$,
$\varphi_2(\mu) \leq 0$, $\mu \geq 0$, then there exist nonnegative $\lambda_0$, $\lambda_1$, and $\lambda_2$, not all
zero, such that
\begin{align}
\label{Lcond1}
\partial\ensuremath{\mathcal{L}}(\mu;\nu) & \geq 0, \quad \text{for $\nu \geq 0$,}\\
\label{Lcond2}
\partial\ensuremath{\mathcal{L}}(\mu;\mu) & =0,
\end{align}
where $\partial\ensuremath{\mathcal{L}}(\mu;\nu)$ is the Fr\'echet-derivative of $\ensuremath{\mathcal{L}}$ at $\mu$ in direction $\nu$
and given by
\begin{equation*}
\partial\ensuremath{\mathcal{L}}(\mu;\nu) = \int_0^1 \ell(x) \, \nu(\mathrm{d} x)
\end{equation*}
with
\begin{equation*}
\ell(x)= -n \, \lambda_0 \, \prob{S_{n-1} \leq t-x} + \lambda_1 -\lambda_2 \, x.
\end{equation*}
Note that $\ell(x)$ is continuous from the left and that jump-discontinuities (if any) are upwards.
\subsection{The Lagrange conditions}
\label{Lagrange}
From Mattner's theorem some properties of extremal distributions can be derived, as well as an expression of
\prob{S_n\leq t} in terms of the Lagrange multipliers. First, the redundant Lagrange multiplier
$\lambda_0$ is removed.
From Lagrange condition (\ref{Lcond1}), by substituting $\nu = \delta_x$ (point-mass at $x$), one may conclude
$\ell(x) \geq 0$, for $0 \leq x \leq 1$. Combining this with the second Lagrange condition (\ref{Lcond2}) results in:
\begin{equation}
\label{lzeroae}
\ell(x)=0 \quad \text{for $\mu$-a.e.~$x$}.
\end{equation}
It is first argued that $\lambda_0$ cannot be zero.
If $\lambda_0=0$, then $\ell(x)=\lambda_1 -\lambda_2 \, x$ should be nonnegative for
$0 \leq x \leq 1$, whence $\lambda_1 \geq \lambda_2 \geq 0$ and, necessarily, $\lambda_1 >0$, for they cannot all
three be zero. However, $\ell(x)=0$ must
have at least one solution, or else $\supp{\mu}=\emptyset$. This leaves $\lambda_1=\lambda_2$ as sole possibility,
implying that $\mu=\delta_1$, which contradicts the assumption $\expec{X}=m <1$.
Therefore, $\lambda_0>0$ and without loss of generality it is henceforth assumed that $n \lambda_0=1$.
Lagrange condition~(\ref{Lcond2}), $\varphi_1(\mu)\leq 0$, and $\varphi_2(\mu)\leq 0$, imply
\begin{equation}
\label{probbound}
\prob{S_n\leq t} = \lambda_1 \mu([0,1])-\lambda_2 \int_0^1 x \, \mu(\mathrm{d} x).
\end{equation}
The Lagrange conditions can be restated as
\begin{align}
\label{L1}
\prob{S_{n-1} \leq t-x } & \leq \lambda_1 -\lambda_2 \, x, \quad 0 \leq x\leq 1,\quad
\text{and}\\
\label{L2}
\prob{S_{n-1} \leq t-x } & = \lambda_1 -\lambda_2 \, x, \quad\text{for $x \in \supp{\mu}$}.
\end{align}
The following lemma shows
that the last statement follows from (\ref{lzeroae}):
\begin{lemma}
\label{elliszero}
Let $\mu$ be extremal. Then $\ell(x)=0$ for $x \in \supp{\mu}$.
\end{lemma}
\textbf{Proof.}
Let $x \in \supp{\mu}$ and suppose a Borel set $A \subset [0,1]$ satisfies $\mu(A)=1$ and $\ell(y)=0$ for $y \in A$.
If $x$ is an atom of $\mu$ then $x \in A$ and $\ell(x)=0$ follows.
Otherwise, if $x$ is an interior point or a right boundary point
of \supp{\mu}, a sequence $(x_k)$ can be found within $A$ such that $x_k \uparrow x$,
whence $\ell(x)=0$, by left-continuity of $\ell$. If $x$ is a left boundary point,
one can find within $A$ a sequence $x_k \downarrow x$, whence $0\leq \ell(x)\leq \ell(x+)=\lim
\ell(x_k)=0$, since jumps cannot go down.~\ensuremath{\quad\square}
\medskip
It is shown that $\lambda_2>0$ must hold.
Let $s=\min \supp{\mu}$ and $u=\max \supp{\mu}$,
then $s \leq m \leq u$ and gaps in \supp{S_{n-1}} cannot exceed $u-s$ in length.
Lagrange conditions (\ref{L1}) and (\ref{L2}) imply $\prob{t-u < S_{n-1} \leq t} \leq \lambda_2 \, u$.
The probability, however, must be positive:
$\expec{S_{n-1}}= (n-1)\,m > t -m \geq t-u$ implies that \prob{t-u < S_{n-1}} must be positive;
$\prob{t < S_{n-1}}=1$ cannot be the case, or else $\prob{S_{n}\leq t}=0$ and $\mu$ is not extremal.
\paragraph{Support conditions.} The Lagrange conditions imply several properties for the support of $S_{n-1}$
and $S_{n}$. An immediate consequence of the next lemma is that $t \in \supp{S_n}$.
\begin{lemma}
\label{support}
Let $\mu$ be extremal, $x \neq 1$.
If $x \in \supp{\mu}$ then $t-x \in \supp{S_{n-1}}$.
\end{lemma}
\textbf{Proof.} By contraposition.
Suppose $t-x \not\in \supp{S_{n-1}}$, for some $0 < x < 1$.
Let $B_\epsilon(x)=(x-\epsilon,x+\epsilon)$.
Then $\prob{S_{n-1} \in B_\epsilon(t-x)}=0$ for some $\epsilon>0$,
implying that $\prob{S_{n-1}\leq t-y}$ is constant for $y \in B_\epsilon(x)$
and that $\ell(y) \geq 0$ is linearly decreasing on this set,
which implies $\ell(x)> 0$.
For $x=0$, this reasoning shows that $\ell(y)$ is linearly decreasing for $y \in [0,\epsilon)$,
with $\ell(0)>0$ as conclusion.
For $x=1$, nothing about the positivity of $\ell(1)$ can be concluded from the fact that $\ell(y)$
is linearly decreasing and positive for $y \in (1-\epsilon,1]$; $\ell(1)=0$ is still possible.
So, for $0\leq x <1$, $t-x \not\in \supp{S_{n-1}}$ implies $\ell(x)>0$,
which by Lemma~\ref{elliszero} implies $x \not \in \supp{\mu}$.~\ensuremath{\quad\square}
\medskip
The Lagrange conditions in this section provide necessary conditions (\ref{L1}), (\ref{L2}), and
Lemma~\ref{support}, that should be satisfied by extremal distributions.
It is not difficult, for general $n$, to identify a number of distributions that satisfy them
(see Section~\ref{revised}).
However, unless \emph{all} the solutions are identified, there are no guarantees that the best of
the solutions found indeed attains the supremum $p(m,t)$.
\section{The case $n=2$}
\label{threesupportpoints}
Lemma~\ref{support} yields an especially strong result for $n=2$, because $S_{n-1}=X_1$ and the lemma
characterizes the support of (candidate) extremal distributions.
Below, certain two and three point solutions to the Lagrange conditions will be identified.
Other solutions (if any) cannot be extremal:
it will be shown that one can always find a distribution of the two or three point type that has a strictly
larger \prob{S_n \leq t}-value. Hence, all extremal distributions belong to this special class.
\medskip
Suppose $\mu$ satisfies the Lagrange conditions and $s = \min \supp{\mu}$.
Lemma~\ref{support} implies that $t-s \in \supp{\mu}$, and if this is not the largest support point,
then $\max \supp{\mu}=1$. Therefore, two cases are to be considered.
\medskip
First, assume that $t-s=\max \supp{\mu}$. Note that, necessarily, $0 \leq s \leq t-s \leq 1$ and $t-s
>m$ (or else $\int_0^1 x \mu (\mathrm{d} x) < m$), which imply $s <t/2 <t-s$ by $m >t/2$.
Note that $m<t$ must hold, or no such $\mu$ exist.
Let $F$ be the distribution function corresponding to $\mu$.
Lagrange condition~(\ref{L2}) requires that nonnegative $\lambda_1$ and $\lambda_2$ exist such that:
\begin{equation*}
F(t-s) = \lambda_1 -\lambda_2 \, s \quad \text{and} \quad
F(s) = \lambda_1 -\lambda_2 \, (t-s).
\end{equation*}
From the monotonicity and nonnegativity of $F$:
\begin{equation}
\label{intFx}
\int_0^1 F(x) \mathrm{d} x \geq (t-2s) \, F(s) + (1-t+s) \, F(t-s),
\end{equation}
where equality holds (if and) only if $s$ and $t-s$ are the only support points.
Since $F(t-s)=F(1)$, one may write $\lambda_1=F(1)+\lambda_2 \, s$ and $F(s)=F(1)-\lambda_2 \, (t-2s)$.
Combining things, one obtains:
\begin{equation}
\int_0^1 x \mu \, (\mathrm{d} x) = F(1) - \int_0^1 F(x) \, \mathrm{d} x
\leq s \, F(1) + \lambda_2 \, (t-2s)^2
\end{equation}
whence $\lambda_2 \geq (m-s)/(t-2s)^2$.
Starting from~(\ref{probbound}), this results in:
\begin{equation}
\label{Pbound2}
\pmu{S_2 \leq t} \leq \lambda_1 - \lambda_2 \, m = F(1) - \lambda_2 (m-s)
\leq 1 - \left ( \frac{m-s}{t-2s} \right )^2.
\end{equation}
Note that $0 < (m-s)/(t-2s) < 1$ since $s <t/2<m$ and $m <t-s$. Let~$\pi$ be the probability
measure $\pi$ on $\{s,t-s\}$ defined by $\pi_{t-s}= (m-s)/(t-2s)=1-\pi_s$, where $\pi_x:=\pi(\{x\})$.
Then $\int_0^1 x \, \pi(\mathrm{d} x)=m$ and $\ppi{S_2\leq t}$ equals the right
hand side of~(\ref{Pbound2}).
The upper bound on \pmu{S_2 \leq t} is strict, unless $F(1)=1$, $\int_0^1 x \, \mu(\mathrm{d} x)=m$, and equality
holds in~(\ref{intFx}). These conditions, however, uniquely identify $\pi$, showing that $\mu$ can only be extremal
if $\mu=\pi$.
\bigskip
Next, consider the situation where $t-s<\max \supp{\mu}=1$. Necessarily, $0 \leq s \leq t-s < 1$ must hold.
Lagrange condition~(\ref{L2}) specifies for the support points $s$, $t-s$ and $1$, respectively:
\begin{equation*}
F(t-s) = \lambda_1 -\lambda_2 \, s, \quad
F(s) = \lambda_1 -\lambda_2 \, (t-s), \quad \text{and} \quad
F(t-1) = \lambda_1 -\lambda_2.
\end{equation*}
Since $t-1<s=\min \supp{\mu}$, $F(t-1)=0$ and so $\lambda_1=\lambda_2$, which is used to eliminate $\lambda_1$.
Note that $F(t-s) < F(1)$. Further, that $F(1)=1$ must hold, or
the defect could be added as an atom in $0$, which would strictly enlarge $\pmu{S_2 \leq t}$.
If $F(1)=1$ and $\int_0^1 x \mu(\mathrm{d} x) > m$ then a small mass $\epsilon>0$ could be moved from~$1$
to $0$, still keeping the mean above $m$. This would increase \pmu{S_2 \leq t} by at least $\epsilon^2$.
Hence, if $\mu$ is to be extremal, then $F(1)=1$ and $\int_0^1 x \mu(\mathrm{d} x) = m$ must hold.
Combining (\ref{intFx}) with $F(t-s) = \lambda_2 \, (1-s)$ and $F(s) = \lambda_2 \, (1-t+s)$, one obtains
\begin{equation}
\label{intFx2}
1-m = \int_0^1 F(x) \, \mathrm{d} x \geq \lambda_2 \, (1-t+s) (1+t-3s),
\end{equation}
where equality holds (if and) only if $s$, $t-s$ and $1$ are the only support points. Apparently,
\begin{equation*}
\lambda_2 \leq \lambda_2^+ := \frac{1-m}{(1-t+s) (1+t-3s)}.
\end{equation*}
Define the measure $\pi$ on $\{s,t-s,1\}$ by
\begin{equation}
\label{threepointsol}
\pi_s=\lambda_2^+ (1-t+s), \quad
\pi_{t-s}=\lambda_2^+ (t-2s), \quad
\pi_1=1-\lambda_2^+ (1-s).
\end{equation}
(Note that $s=t-s$ leaves a valid probability measure on the set $\{t/2,1\}$.)
If $\lambda_2^+ (1-s) <1$, then $\pi$ is a probability measure with mean $m$:
the probabilities are nonnegative and sum to 1,
$s \, \pi_s + (t-s) \, \pi_{t-s} +\pi_1= 1 - \lambda_2^+ \, (1-t+s) (1+t-3s)=m$.
Furthermore, $\pi$ satisfies the Lagrange conditions (for $\lambda_2^+$) and so
\begin{equation*}
\pmu{S_2 \leq t} =
\lambda_2 \, (1-m) \leq
\lambda_2^+ \, (1-m) =
\ppi{S_2 \leq t} = \frac{(1-m)^2}{(1-t+s) (1+t-3s)}.
\end{equation*}
The inequality is strict unless $\mu=\pi$; this can be seen from~(\ref{intFx2}).
If $\lambda_2^+ (1-s) \geq 1$, then
\begin{equation*}
(1-m)(1-s) \geq (1-t+s) (1+t-3s) = (1-s)^2 -(t-2s)^2,
\end{equation*}
which is equivalent to
\begin{equation}
\label{twopointonly}
(t-2s)^2 \geq (1-s)(m-s).
\end{equation}
Since $\lambda_2 \, (1-s) <1$ by assumption, starting from~(\ref{probbound}) (with $\lambda_1=\lambda_2$),
\begin{equation}
\label{Pbound3}
\pmu{S_2 \leq t} = \lambda_2 \, (1-m) < \frac{1-m}{1-s} = 1- \frac{m-s}{1-s}
\leq 1 - \left ( \frac{m-s}{t-2s} \right )^2,
\end{equation}
where the last inequality follows from~(\ref{twopointonly}).
That inequality also implies $t-2s>m-s$, which combined with $m>s$ guarantees the existence of the
probability measure with support $\{s, t-s\}$ and mean $m$. As was shown before, this measure attains
the value on the right hand side of (\ref{Pbound3}).
In all cases it has now been shown that if $\mu$ satisfies the Lagrange conditions it equals a discrete measure on
$\{s, t-s\}$ or $\{s,t-s,1\}$, for some $s$, or else $\pmu{S_2 \leq t} < \ppi{S-2 \leq t}$ for some $\pi$ from this
class.
The last steps of the solution consist of optimizing within the class of two and three point support distributions just
identified.
\bigskip
\textbf{Remark: an alternative approach?}\label{alternative}
What follows is a sketch of a proof that would work if one could show that
extremal measures cannot have a singular component. The Lagrange conditions imply that if the support contains an
interval, say $A$, then $\mu$ has density equal to $\lambda_2$ on that interval. Lemma~\ref{support} implies that the
same holds for $t-A$, from which it easily follows that $\mu$ can be improved upon by moving the mass to the center
of the intervals. If $\mu$ is purely atomic, a similar argument that exploits the symmetry of the support can be used
to show that $[0,t/2)$ cannot contain more than one atom. After this, four possible support points remain: (some)
$s$, $t/2$, $t-s$, and $1$. A simple mass transfer argument shows that the first three cannot occur together. This
leaves one with the same possibilities as in the current line of reasoning.
\subsection{$\{s,t-s\}$-solutions}
Recall that the bound from~(\ref{Pbound2}) and (\ref{Pbound3}) can be attained by the probability
measure $\pi$ on $\{s,t-s\}$ defined by $\pi_{t-s}= (m-s)/(t-2s)=1-\pi_s$.
The largest \ppi{S_2 \leq t}-value is attained for the smallest feasible $s$, as
$\pi_{t-s}$ is increasing in $s$. For $t \leq 1$ the maximum is at $s=0$, for $t \geq 1$ at $s=t-1$. Thus, the best
solutions of this type are as follows.
For $m < t \leq 1$: $\pi_0=1-m/t$, $\pi_t=m/t$ and
\begin{equation*}
\label{case2a}
\ppi{S_2 \leq t} = 1 - \left ( \frac{m}{t} \right)^2.
\end{equation*}
For $t\geq 1$: $\pi_{t-1}=(1-m)/(2-t)$, $\pi_1=(1+m-t)/(2-t)$ and
\begin{equation*}
\label{case2b}
\ppi{S_2 \leq t} = 1 - \left ( \frac{1+m-t}{2-t} \right)^2.
\end{equation*}
\subsection{$\{s,t-s,1\}$-solutions}
Candidate extremal distributions are the probability measures with support $\{s,t-s,1\}$ and probabilities given
by~(\ref{threepointsol}), provided $0 \leq s \leq t/2$, $s \geq t-1$ and $\lambda_2^+ (1-s) \leq 1$, or:
\begin{equation}
\label{existence3}
(1-m) (1-s) \leq (1-t+s) (1+t -3s).
\end{equation}
For the sake of a simpler exposition two small additions were made to the class considered:
equality in the previous formula corresponds to boundary cases with $\pi_1=0$, which, just as the case $s=t-1$ that
was added, leads to distributions already considered.
Recall that $\ppi{S_2 \leq t} = (1-m)^2/(1-t+s)(1+t-3s)$ for $\pi$ as in~(\ref{threepointsol}). What
remains is maximize over feasible $s$.
Define $p(s)=(1-t+s) (1+t -3s)$. Since $p$ is a concave function, the solutions
to~(\ref{existence3}) constitute an interval; call the left end point $s_0$. Since $p(s)=(1-s)^2
-(t-2s)^2$, (\ref{existence3}) is equivalent to $(t-2s)^2 \leq (1-s)(m-s)$, which shows that $s=t/2$ is always
feasible, and only feasible $s$ between $s_0$ and $t/2$ need to be considered.
Consider the maximization problem: since $p$ is concave, the maximum of $(1-m)^2/p(s)$ is attained at an
end point of the feasible range, i.e., $t/2$ or the left end point. Note, however, that as $s \downarrow s_0$, also $\pi_1
\downarrow 0$, and what results is a distribution on $\{s_0, t-s_0\}$, already considered in the previous section.
This means that if $s_0 \geq 0$ and $s_0 \geq t-1$, the entire range $s_0 \leq s \leq t/2$ corresponds to feasible
solutions, at the left end dominated by solutions already considered. Then, the distribution corresponding
to the right end point is the only new (candidate) extremal distribution. It is given by: $\pi_{t/2}=(1-m)/(1-t/2)$ and
$\pi_1=(m-t/2)/(1-t/2)$ with
\begin{equation*}
\label{case2c}
\ppi{S_2 \leq t} = \left ( \frac{1-m}{1-t/2} \right)^2.
\end{equation*}
Note that this solution exists for all $(m,t)$-pairs under consideration.
What remains now, is to determine whether the maximum can be attained for an intermediate value $s_0<s<t/2$, which
would correspond to a \emph{true} three point distribution.
\medskip
First, a closer look at $p(s)$ is warranted. It has zeros at $t-1$ and $(t+1)/3$, a maximum value of $(2-t)^2/3$ attained at
$(2t-1)/3$; $p(0)=1-t^2$ and $p((5t-4)/6)=p(t/2)=(1-t/2)^2$. These points are ordered in the following manner:
\begin{equation*}
t-1 < \frac{5t-4}6 < \frac{2t-1}3 <\frac{t}2 < \frac{t+1}3,
\end{equation*}
where zero can be anywhere to the left of $t/2$, depending on $t$.
\medskip
From equality in (\ref{existence3}) one sees that $p(s_0)=(1-m)(1-s_0)>0$, and since $t-1$ is the left zero of $p$,
this implies that $s_0>t-1$. Therefore, for $t\geq 1$, the whole range $s_0 \leq s \leq t/2$ corresponds to
feasible solutions dominated by one of the two point solutions corresponding to the end points, i.e.,
with support $\{s_0,t-s_0\}$ or $\{t/2,1\}$.
Next, consider $t <1$. A~true three point solution occurs if $s_0<0$, which happens if inequality
(\ref{existence3}) is strict
for $s=0$, which happens if $t < \sqrt{m}$. However, if $t<4/5$, then $(5t-4)/6<0$ and $p(0)>p((5t-4)/6)=p(t/2)$, whence
$(1-m)^2/p(s)$ attains a higher value at $s=t/2$ than at $s=0$. In summary, this shows that the best solution is
obtained at $s=0$ only for $4/5 \leq t < \sqrt{m}$. It is the extremal probability measure $\pi$ on $\{0,t,1\}$ defined by:
$\pi_0 ={(1-m)(1-t)}/(1-t^2)$, $\pi_t ={(1-m)\,t}/(1-t^2)$, and $\pi_1 =(m-t^2)/(1-t^2)$, with
\begin{equation}
\label{case3}
\ppi{S_2 \leq t} = \frac{(1-m)^2}{1-t^2}.
\end{equation}
The $(m,t)$-range where this distribution dominates all others has just been determined.
On the complement of this range several two point solutions may exist together and therefore
need to be compared.
\subsection{Some comparisons}
In the region $m \leq t \leq 1$, both the $\{0,t\}$ and the $\{t/2,1\}$-solution exist. The second is the best when
\begin{equation}
\label{compareCandA}
1 - \left ( \frac{m}{t} \right )^2 < \left ( \frac{1-m}{1-t/2} \right )^2.
\end{equation}
Substituting $m=a t$, the equivalent inequality $4(1-at)^2 -(1-a^2)(2-t)^2 >0$ is obtained,
which in turn simplifies to
$\left( 5\,{t}^{2}-4\,t+4 \right) {a}^{2}-8\,at-{t}^{2}+4\,t>0$.
The discriminant of this quadratic in $a$ is $20\,{t}^{4}-96\,{t}^{3}+144\,{t}^{2}-64\,t$ which factors as
$4\, t\, ( 5\,t-4 )\, (2-t) ^{2}$. This shows that the inequality~(\ref{compareCandA}) is valid for $0<t<4/5$,
as the discriminant is negative for these values.
For $4/5 \leq t \leq 1$, the boundary curve of the inequality~(\ref{compareCandA}) is given by
\begin{equation}
\label{monet}
m_1(t)= \frac {4\,{t}^{2}- \left( 2-t \right) t\sqrt {t \left( 5\,t-4
\right) }}{5\,{t}^{2}-4\,t+4}.
\end{equation}
For $m<m_1(t)$ the $\{t/2,1\}$ solution is superior; for larger $m$ the $\{0,t\}$ solution is.
\medskip
To determine for which $m$ and $t$ the $\{t/2,1\}$-solution is best
for $t \geq 1$ one needs to solve
\begin{equation*}
1-\left(\frac{m+1-t}{2-t}\right)^2 < \left ( \frac{1-m}{1-t/2} \right )^2.
\end{equation*}
Setting $a=(1-m)/(2-t)$, this becomes $4a^2>1-(1-a)^2$, resulting in $a>2/5$, or $5m<1+2t$.
\bigskip
Summarizing everything, one obtains the following table.
The function $m_1(t)$ on the second line is given in equation~(\ref{monet}).
Figure~\ref{supportpoints} shows the regions with the support of the respective extremal distributions.
\begin{center}
\begin{tabular}{lcl}
support & $(t,m)$-region & \prob{S_2 \leq t}\\[.5ex] \hline
$\{0, t\}$ & $\tfrac45 \leq t \leq 1,\, m_1(t) \leq m \leq t^2$ & $1 - \left ( \frac{m}{t} \right )^2$,
\\[.5ex]
$\{0, t, 1\}$ & $\tfrac45 \leq t < \sqrt{m}$ & $\frac{(1-m)^2}{1-t^2}$, \\[.5ex]
$\{t-1, 1\}$ & $1 \leq t \leq 2$,\, $5m>2t+1$ & $1-\left(\frac{m+1-t}{2-t}\right)^2$,
\\[.5ex]
$\{t/2, 1\}$ & everywhere else & $\left ( \frac{1-m}{1-t/2} \right )^2$.
\end{tabular}
\end{center}
\begin{figure}
\input plotnistwo.pic
\caption{Support points of extremal distributions, for $n=2$.}
\label{supportpoints}
\end{figure}
The extremal measures are unique except on the boundary between the $\{t/2,1\}$-solution and the others, where two distinct
solutions yield the same \prob{S_2\leq t}-value; on the other boundaries, the two \emph{solutions} coincide.
Figure~\ref{versusHoeffding} shows a contourplot of the ratio of the supremum $p_2(m,t)$ and
the Hoeffding bound~(\ref{ineqHoeffding}); the bound is sharp at the boundary $m=t/2$ and
progressively looser as $m$ increases.
\begin{figure}
\includegraphics[width=\textwidth]{versusHoeffding3.pdf}
\caption{Ratio of $p_2(m,t)$ and the Hoeffding bound~(\ref{ineqHoeffding});
contour lines correspond to $0.1$, $0.2$, \ldots, $0.9$, going from top to bottom.}
\label{versusHoeffding}
\end{figure}
\section{The conjectures: one refuted, one revised}
\label{conjectures}
Strictly speaking, the results for $n=2$ suffice to disprove \emph{both} conjectures.
Whereas the \emph{Statistica Neerlandica} conjecture can be utterly disproved,
the \emph{SIAM Review} conjecture is only disproved by a small $(m,t)$-region where
the extremal distribution has three support points (including $0$ and $1$!).
In our view the \emph{SIAM Review} conjecture is close to what may be true and
therefore a revised conjecture is formulated below.
\paragraph{Statistica Neerlandica.}
It seems that what was meant is ``If $p_n(m,t) = \sup \prob{S_n \leq t}$ is small (enough), then the Bernoulli with
mean $m$ is the best distribution.''
Looking at the $n=2$ results, the Bernoulli only appears as extremal distribution for $t=0$ and
for $t=1$, $3/5 \leq m \leq 1$.
Let $b(n,p,x)$ denote the probability that a binomial random variable with parameters $n$ and $p$ attains a value less
than or equal to $x$.
The following is the logical negation of the \emph{Statistica Neerlandica} conjecture:
\begin{lemma}
For any $0 < p_0 <1$ there exist $n$, $t$, and $m$, such that $b(n,m,t) < p_n(m,t) \leq p_0$.
\end{lemma}
\textbf{Proof.} Set $n=2$ and choose any $t$ in $(0,1)$ or $(1,2)$.
Then for $m >t/2$: $b(n,m,t) < p_n(m,t)$ and as $m \uparrow 1$, $p_n(m,t) \to 0$.~\ensuremath{\quad\square}
\subsection{The \emph{SIAM Review} conjecture revised}
\label{revised}
In order to maximize \prob{S_n \leq t}, it seems that as much probability mass as possible should be on or near the
boundary $S_n=t$; the support condition from Lemma~\ref{support} illustrates this. Furthermore, the
fewer support points $\mu$ has, the more mass can contribute to the event $S_n=t$; an illustration
of this can be seen in the remark on page~\pageref{alternative}, where shrinking a continuous
portion of the distribution to one point doubles the contribution to \prob{S_n \leq t}.
Sometimes, however, putting some probability mass at $1$ may enable a redistribution of mass on lower support points that
results in an increase of \prob{S_n \leq t}. This (we think) is the intuitive explanation for the $\{0, t, 1\}$
solution. It is also the reason we think that the number of support points required is no larger than three.
\begin{conjecture}
For any $n \geq 2$, $0<m<1$, and $0 \leq t < m n$, all distributions attaining the supremum $p_n(m,t)$
belong to the collections described below.
\end{conjecture}
\paragraph{Conjectured extremal binary solutions.}
A collection of at most $n$ distributions with two support points is identified below. They may not all satisfy all
of the Lagrange conditions.
However, it is conjectured that if an extremal distribution is binary, it \emph{must} be one of these.
Suppose the support is $\{a, b\}$, with $0 \leq a <m<b\leq 1$. Lemma~\ref{support} implies $t-a \in \supp{S_{n-1}}$,
which means that $t-a = j \, b + (n-1-j) \, a$ for some integer $j=0, 1, \ldots, n-1$. From $\expec{X}=m$ follows that $\pi
:= \prob{X=b} = (m-a)/(b-a)= (m-a) \, j /(t - n \, a)$ (where $b-a = (t -n \, a)/j$ is used) and so
$\prob{S_n \leq t} =b(n, \pi, j)$.
Since $\pi$ is increasing in $a$ and $b(n,\pi,j)$ decreasing in $\pi$, one should minimize $a$.
From $0 \leq a <m$ it follows that $m - (m\,n-t)/j < b \leq t/j$, so for $0 \leq j \leq t$ the constraint $b \leq
1$ becomes active as $a \downarrow 0$. Hence, for these $j$, the solution is $b_j=1$, $a_j=(t-j)/(n-j)$ (from
$a=(t-j\,b)/(n-j)$) and $\pi_j=1- (1-m) (n-j)/(n-t)$.
Considering that $m<b\leq 1$ implies $(t-j)/(n-j) \leq a < (t- j \, m)/(n-j)$, one sees that for $t<j<t/m$ one
should set $a_j=0$, $b_j=t/j$, and $\pi_j=j\,m/t$.
This results in a collection of at most $n$ potential extremal distributions, from which the best is
selected by comparing the values $b(n, \pi_j, j)$, for $j=0, 1, \ldots \ceil{t/m}-1$.
\paragraph{Conjectured extremal ternary solutions.}
It was shown on page~\pageref{zeroandone} that the supremum $p_n(m,t)$ is attained by a distribution with $0$ or $1$
in the support. The intuitive argument given above suggests that an extremal \emph{ternary} distribution will have \emph{both}
$0$ and $1$ in the support.
Using this as an assumption, a collection of (at most $\binom{n}{2}$) possible three point supports can be identified.
In order to precisely specify the distributions, the Lagrange linearity condition~(\ref{L2}) is needed as well.
Suppose the support is $\{0, a, 1\}$ with $0<a<1$. Lemma~\ref{support} implies $\{t-a, t\} \subset \supp{S_{n-1}}$,
whence integers $k\geq 1$ and $l\geq 0$ should exist, such that $k+l\leq n-1$ and $t=k \, a+ l$. Solving the last
equation for $a$, define $a_{k,l} = (t-l)/k$, which is between $0$ and $1$ if $0 \leq l <t < l+k \leq n-1$.
In contrast with the binary solutions above, the requirement that $\expec{X}=m$ is insufficient to fix the
probabilities and as an additional equation one should use the Lagrange linearity condition: \prob{S_{n-1} \leq t-x}
is linear for $x \in \{0, a, 1\}$. This results in
\begin{equation*}
(1-a) \, \prob{t-a < S_{n-1} \leq t} = a \, \prob{t-1 < S_{n-1} \leq t-a}.
\end{equation*}
Since $S_{n-1}$ is distributed as $a\,N_a+N_1$, where $(N_0,N_a,N_1)$ have a trinomial distribution with parameters
$n-1$, $p=\prob{X=1}$, $q=\prob{X=a}$ and $r=\prob{X=0}$,
this last requirement is a polynomial equation in $p$, $q$, and $r$.
The requirements $a\,q+p=m$ and $p+q+r=1$ can be used to eliminate $q$ and~$r$, leaving a polynomial equations of
order $n-1$ in $p$.
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,388 |
\section{Introduction}
A good knowledge of the spectrum and properties of hadrons is one of
the key issues for understanding the strong interaction at low
and intermediate energies. The conventional quark model implies that
quark-antiquark states are produced as nonets, which consist of
mesons with strange and non-strange quarks. Therefore, an
accurate identification of mesons with one strange quark can
help to establish nonet members in the isoscalar sector, where the
situation is more complicated. This is due to a potential mixing between octet and
singlet states as well as possible mixing with glueball states.
The identification of meson radial excitations also helps in the
understanding of quark-antiquark interaction at intermediate energies.
Quark potential models~\cite{Godfrey:1985xj} predict that the squared masses of
radial excitations depend on the excitation number quadratically.
However, in the analysis of proton-antiproton annihilation in flight,
it was found that this dependence is close to the linear one similar
to the Regge trajectories~\cite{Anisovich:2000kxa}.
If correct, this behavior has the potential to reveal a new symmetry of
the quark-antiquark interaction
\cite{Afonin:2013npa,Afonin:2014nya}. Therefore, the experimental confirmation
(or disproof) of this behavior is an important task
in experimental hadron physics.
$J/\psi$ decays are ideal for the study of meson spectra and
the determination of meson properties. They can
provide important information about meson states with masses up to
3~GeV/$c^2$ and partial-wave analysis is facilitated due to
the well-known
quantum numbers of the initial state. Moreover, the $J/\psi$
radiative decay is favored for the production of glueball states
which makes it a perfect tool to search for and study such exotics \cite{Klempt:2007cp}.
In this paper we report the results of a partial-wave analysis
(PWA) of the decay $J/\psi \to K^+K^-\pi^0$.
This decay channel has been previously studied by the
MARK \cite{Vannucci:1976qg}, MARK-II \cite{Franklin:1983ve},
MARK-III \cite{Coffman:1988ve},
DM2 \cite{Jousset:1988ni}, BESII \cite{Ablikim:2006hp}, and BABAR \cite{Aubert:2007ym,Lees:2017ndy} Collaborations, but only two recent
publications report PWA results.
In the first of these~\cite{Ablikim:2006hp}, BESII analyzes
58~million $J/\psi$ decays and observes
a very broad exotic resonance $X(1575)$
with pole position $\left[ ( 1576^{+49}_{-55}\,^{+98}_{-91}) -
i(409^{+11}_{-12}\,^{+32}_{-67}) \right] \,{\rm MeV}/c^2$
and branching fraction $B(J/\psi \to X(1575)\pi^0 \to K^+K^-\pi^0)=
\left(8.5\pm0.6_{-3.6}^{+2.7}\right)\times10^{-4}$. In the second
analysis~\cite{Lees:2017ndy}, BABAR reports a PWA solution
based on a smaller data set of 2102 events, which
consists of $K^*(892)^\pm$, $K^*(1410)^\pm$ and $K_2^*(1430)^\pm$ states in the
$K^\pm\pi^0$ channels, while the enhancement at low $K^+K^-$ invariant masses is attributed to the $\rho(1450)$.
The analysis presented in this paper is based on a data set of 182,972 event candidates selected from $(223.7\pm1.4)\times10^6$ $J/\psi$
decays \cite{Ablikim:2016fal} collected by
the BESIII experiment in 2009.
The high statistics and good data quality allow us to reveal
signals from states that have not been observed before and
precisely determine properties of intermediate states.
Moreover, the obtained PWA
solution can be used for the simulation of the
irreducible background from this channel to the $J/\psi \to \gamma K^+K^-$
decay, which is one of the key channels to be studied in the search for a
low-mass glueball.
\section{BESIII experimental facility}
The BESIII detector is a magnetic
spectrometer~\cite{Ablikim:2009aa} located at the Beijing Electron
Positron Collider (BEPCII)~\cite{Yu:IPAC2016-TUYA01}. The
cylindrical core of the BESIII detector consists of a helium-based
multilayer drift chamber (MDC), a plastic scintillator time-of-flight
system (TOF), and a CsI(Tl) electromagnetic calorimeter (EMC),
which are all enclosed in a superconducting solenoidal magnet
providing a 1.0~T magnetic field. The solenoid is supported by an
octagonal flux-return yoke with resistive plate counter muon
identifier modules interleaved with steel. The geometrical acceptance of
charged particles and photons is 93\% over the $4\pi$ solid angle. The
charged-particle momentum resolution at $1~{\rm GeV}/c$ is
$0.5\%$, and the $dE/dx$ resolution is $6\%$ for electrons
from Bhabha scattering. The EMC measures photon energies with a
resolution of $2.5\%$ ($5\%$) at $1$~GeV in the barrel (end cap)
region. The time resolution of the TOF barrel part is 68~ps, while
that of the end-cap part is 110~ps.
The {\sc geant4}-based simulation software BOOST \cite{BOOST}
is used to simulate the detector response.
An inclusive $J/\psi$ Monte Carlo (MC) sample is used to estimate
the background. In this sample the production of the $J/\psi$ resonance
is simulated by the MC event generator KKMC \cite{Jadach:1999sf, Jadach:2000ir}
and decays are generated by {\sc evtgen} \cite{Lange:2001uf, Ping:2008zz}.
The branching fractions of known decay modes are set to the
Particle Data Group (PDG) \cite{Amsler:2008zzb} world-average values and
the remaining unknown decays are generated according to the Lund-Charm
model \cite{Chen:2000tv}.
\section{Event selection}
The $K^+K^-\pi^0$ candidate events are required to have two charged tracks
with zero net charge and at least two good photons.
Charged tracks must be reconstructed within the geometrical acceptance
of the detector ($|\cos\theta|<0.93$, where $\theta$ is the
angle with respect to the beam axis)
and originate from the interaction point
($|z|<10$~cm and $R<1$~cm, where $z$ and $R$ are minimal distances from
a track to the run-averaged interaction point along the beam direction and
in the transverse plane, respectively).
An event is rejected if the transverse momentum of at least one charged track
is too low ($p_T<120$ MeV/$c$).
Particle identification (PID) is performed using
TOF and MDC $dE/dx$ information. Their measurements are combined
to form particle identification confidence levels (C.L.) for $\pi$, $K$,
and $p$ hypotheses, and the particle type with the highest C.L. is assigned
to the track. Both tracks are required to be identified as kaons.
Signal clusters in the EMC within the acceptance region, which are not associated with charged tracks and
possess energy $E>25$~MeV in the barrel part of the detector
and $E>50$~MeV in the end caps,
are treated as photon candidates.
To exclude showers from association with charged particles,
the angle between the shower direction and the
charged tracks extrapolated to the EMC must be greater
than 10 degrees.
The requirement on the EMC cluster time with respect to the start of
the event ($0\text{ ns} \le t \le 700$~ns)
is used to reject
electronic noise and energy deposits not related to the analyzed event.
Consistency between the detector response and a final state hypothesis
(for the signal and specific background decays)
is evaluated by a four-momentum constrained (4C) kinematic fit.
Firstly, the accepted pair of charged tracks and each pair of
the selected photon candidates with invariant mass
$M_{\gamma\gamma} < 300$ MeV/$c^2$ are fitted under
the $\gamma\gamma K^+K^-$ hypothesis.
A combination with the lowest value of $\chi^2_{(4C)\gamma\gamma K^+K^-}$
is selected and an event is retained if $\chi^2_{(4C)\gamma\gamma K^+K^-} < 60$.
Secondly, the $\chi^2_{(4C)\gamma\gamma K^+K^-}$ is compared to
the corresponding value obtained in the best fits under the main
background hypotheses:
$\gamma\gamma\pi^+\pi^-$, $\gamma K^+K^-$, and, in the cases more than two good photon candidates are selected, $\gamma\gamma\gamma K^+K^-$. If
any of the background hypotheses results in a lower $\chi^2$ value,
the event is rejected.
Finally, the $\pi^0$ candidates are reconstructed requiring the two-photon mass
of the selected pair to be within a
$110\text{ MeV/}c^2 < M_{\gamma\gamma} < 150$~MeV/$c^2$
interval.
For the partial-wave analysis, we use particle momenta after the
five-constrained (5C) kinematic fit, which also constrains
the invariant mass of the selected photon pair to the nominal
$\pi^0$ mass.
A total of 182,972 candidates satisfy the selection criteria.
The corresponding number of background events is estimated from
the inclusive MC: $N_{bg}=565\mpm24$ (or 0.3\%).
The largest background contributions come from the decay channels
$J/\psi \to \gamma \eta_c, \, \eta_c \to
K^+K^-\pi^0$ and $J/\psi \to \gamma K^+K^-$.
The continuum background, i.e. that due to the $e^+e^-\to \gamma^* \to K^+K^-\pi^0$
process, is estimated from the analysis of a data sample
of approximately 280~nb$^{-1}$ collected from $e^+e^-$
collisions at $3.08$~GeV.
It gives $N_{continuum}=855\pm499$,
where the uncertainty is statistical.
The background treatment in the PWA will be described in
the next section.
The Dalitz plot for the selected data is shown in Fig.~\ref{fig:dalitz}(a).
Its most striking feature is a clear $K^*(892)^\pm$ signal.
In the internal region of the plot a clear signal from $K_2^*(1430)^\pm$
is seen as well as structures at $M^2(K^\pm\pi^0) \approx$ 4~GeV$^2$/$c^4$. These structures are likely to be the result of positive interference of resonances in
the $K^\pm\pi^0$ channels. In the $K^+K^-$ channel there are indications
for a resonance signal at 1.6~--~1.7~GeV/$c^2$ and a signal at
higher masses.
\begin{figure}[pt]
\begin{center}
\includegraphics[width=0.45\textwidth]{fig/Dalitz_full_data.pdf}
\includegraphics[width=0.45\textwidth]{fig/Dalitz_full_660r.pdf}
\includegraphics[width=0.45\textwidth]{fig/Dalitz_full_660s.pdf}
\caption{Dalitz plots for the selected data (a), the PWA solution~I (b) and
the PWA solution~II (c).}
\label{fig:dalitz}
\end{center}
\end{figure}
\section{Partial-wave analysis}
We use the isobar model to describe the $J/\psi$ decay into $K^+K^-\pi^0$. The amplitude is
parameterized as a sum of sequential quasi two-body decay processes in this approach.
The subprocess described by intermediate state production and the subsequent
decay to a specific
pair of the final state mesons is referred to as
the decay kinematic channel.
The angular-dependent parts of the partial-wave
amplitudes are calculated in the framework of the covariant tensor
approach as described in detail in Ref.~\cite{Zou:2002ar}.
Note that in our case the conservation of $P$- and $C$-parities
restricts the number of allowed partial waves for
production and decay of any resonance to one.
To account for the finite size of a hadron
each decay vertex also includes Blatt-Weisskopf form factors,
which depend on the Blatt-Wesskopf radius $r$.
The Breit-Wigner term
for the resonance $a$ in the kinematic channel $m$
(labeled by the number of the spectator particle)
is
\begin{eqnarray}
\nonumber
A^{BW}_{m,a}\!=\!\!
\frac{1}{M_a^2\!-\!s_m\!-\!iM_a\Gamma(s_m, J_a)}.
\nonumber
\end{eqnarray}
Here $M_a$, $J_a$ and $s_m$ are the resonance mass, spin and
the invariant mass squared of its daughter particles, respectively.
The width of the $K^*(892)^\pm$ state is defined by its decay
to $K\pi$ and is parameterized as:
\begin{eqnarray}
\nonumber
\begin{split}
\Gamma(s_m, J_a)=\frac{\rho_J(s_m)}{\rho_J(M_a^2)}\Gamma_a, \\
\rho_J(s_m)=\frac{2q}{\sqrt{s_m}}\frac{q^{2J}}{F^2(q^2,r,J)}.
\label{amp}
\end{split}
\end{eqnarray}
Here, $\Gamma_a$ is the resonance width, $q$ is the
relative momentum of the daughter particles
calculated in the resonance rest frame and $F(q^2,r,J)$
is the above-mentioned Blatt-Weisskopf form factor.
The same parameterization is used for the width of
the $K_2^*(1430)^\pm$ resonance, whose decay branching fraction
to $K\pi$ is about 0.5.
For other states we use a constant
width $\Gamma(s_m, J_a) = \Gamma_a$ due to the absence of
reliable information about their branching fractions.
The masses, widths, decay radii (for the $J/\psi$, $K^*(892)^\pm$
and $K_2^*(1430)^\pm$) of resonances as well as
the product of their production and decay couplings
(complex numbers in general case) are initially free parameters
of our fit. We find fit results weakly sensitive to the $J/\psi$
decay radius. Hence, we set this parameter to be $0.7$~fm,
as is obtained in Ref.~\cite{Bugg:1995jq}.
The analysis is performed within the framework of the
event-by-event maximum likelihood method, which allows us to take
into account all correlations in the multidimensional phase space.
The negative log-likelihood function NLL is expressed as
\begin{eqnarray}
\text{NLL}=-\sum\limits_i \ln \frac{\omega_i\epsilon_i}{\int \epsilon \omega d\Phi} =
-\sum\limits_i \ln \frac{\omega_i}{\int \epsilon \omega d\Phi}+const
\nonumber
\end{eqnarray}
and is minimized. Here index $i$ runs over the selected data events,
$\omega_i$ is the decay-amplitude squared, summed over
transverse $J/\psi$ polarizations and evaluated from the four-momenta
of final particles in the event $i$.
The detector and event selection efficiency for the measured
four-momenta is denoted by $\epsilon_i$,
the denominator is a normalization integral over the phase space ($\Phi$),
and the $const$ term is independent of the fit parameters.
The normalization integral is calculated
using phase-space distributed MC events that
pass the detector simulation and the event reconstruction.
To take the background into account we estimate its
contribution to the NLL function and subtract it.
This is done by the evaluation of the NLL function over properly
normalized data samples that have a kinematic distribution
similar to that of the background. We consider two types of
background channels: those producing a peak
at the $\pi^0$ mass in the two-photon invariant-mass distribution
(``peaking'' background) and those exhibiting a smooth shape
below the peak (``non-peaking'' background).
The former is estimated from
$J/\psi \to \gamma \eta_c$, $\eta_c \to \gamma K^+K^-\pi^0$ events selected
under criteria similar to ones of the main event selection,
and the latter is estimated from the
$\pi^0$ mass cut sideband: $190\text{ MeV/}c^2 <M_{\gamma\gamma} < 230$~MeV/$c^2$.
This approach neglects the detector resolution,
which is a good approximation for all resonances except for
the $K^*(892)^\pm$. The MC simulation shows that estimated
bias to the measured width of $K^*(892)^\pm$ is much larger than the
corresponding systematic uncertainty estimated from other sources.
At the same time, this bias is much smaller than the $K^*(892)^\pm$ width,
which allows us to use the approximation proposed in
Ref.~\cite{Denisenko:2015yoa} to take into account the detector resolution.
Due to the significant computation time, this method is used
only to correct the final PWA results.
The quality and consistency of the obtained solution is
evaluated by the comparison of the mass and angular
distributions of the experimental data and reconstructed
phase-space generated MC events weighted according to the
PWA solution.
The conservation of $P$- and $C$-parities strongly restricts the
allowed quantum numbers of intermediate states.
In the $K^\pm\pi^0$ channels only resonances with
quantum numbers $I=1/2$, $J^{P}\!=\!1^-, 2^+,3^-,4^+\ldots$ can be
produced.
The reaction is dominated by $K^*(892)^\pm$
production.
There are two other established vector states which are in the accessible
mass region: $K^*(1410)$ and $K^*(1680)$ \cite{Tanabashi:2018oca}. In the
$2^+$, $3^-$ and $4^+$ partial waves three states are well established:
$K_2^*(1430)$, $K_3^*(1780)$ and $K_4^*(2045)$.
Possible contributions must also be considered from two observations reported by the LASS Collaboration: a $2^+$ state at 1980 MeV/$c^2$ \cite{Aston:1986jb}
(also claimed to be seen by SPEC~\cite{Tikhomirov:2003gg})
and a $5^-$ state at 2380 MeV/$c^2$ \cite{Aston:1986rm},
which needs confirmation. As for the $K^+K^-$ channel,
the produced resonances are restricted to quantum
numbers $J^{PC}=J^{--}$, where $J=1,3,5\ldots$.
For the strong decays of the $J/\psi$
isospin and $G$-parity conservation requires
$I^G=1^+$. There are two well known isovector resonances in the
$J^{PC}=1^{--}$ sector: the $\rho(1450)$ and $\rho(1700)$, and a set of
observations that needs confirmation: the $\rho(1570)$, $\rho(1900)$
and $\rho(2150)$ (see Ref.~\cite{Tanabashi:2018oca}). In the isovector
$J^{PC}=3^{--}$ sector one can expect the production of the
well known and relatively narrow $\rho_3(1690)$ state. At higher energies
there have been observations of two $J^{PC}=3^{--}$ states: the $\rho_3(1990)$ and
$\rho_3(2250)$. The first isovector $J^{PC}=5^{--}$ state is expected
to have a mass of around 2350 MeV/$c^2$. Such a resonance is observed
in the analysis of the
GAMS2 data for the reaction $\pi^-p\to \omega\pi^0n$~\cite{Alde:1994jm}
and in the analyses of proton-antiproton annihilation
in flight into different meson final states (e.g. see
Ref.~\cite{Anisovich:2002su}).
The decay of the $J/\psi$ through a virtual photon does not forbid but
even favors the production of $I^G=0^-$ resonances.
The $J/\psi \to \phi\pi^0$ decay is strongly suppressed \cite{Ablikim:2015mua},
hence the production of excited $\phi$ mesons is expected
to be negligible assuming the absence of strong mixing
of excited $\phi$ and $\omega$ states.
However, the production of excited $\omega$ resonances is possible. The
isovector and isoscalar states can be distinguished in
a combined analysis of the decay under consideration and the $J/\psi$
decay to $K^\pm K^0\pi^\mp$.
\subsection{Fit to the data}
The masses and widths of all states included in the solution
(with the sole exception of the $\rho(770)$) are
initially free fit parameters.
For the well-established $K\pi$ resonances we use results of the LASS
fits to the elastic $K\pi$ scattering amplitudes \cite{Aston:1987ir} as
reference values. The masses and widths of these states are allowed to vary
within $\pm\sigma$ of the LASS measurements
(here $\sigma$ stands for the LASS uncertainty).
If no NLL minimum is found for the mass or width within this
range or the minimum is unstable (with
respect to variations of the PWA solution used for estimation
of systematic errors), the parameter is set to the central
value of the LASS results.
Motivated by the claim of an observation of the $K_2^*(1980)^\pm$ by LASS \cite{Aston:1986jb}
and by Regge trajectories predicting a state at approximately
1.8~GeV/$c^2$ we introduce a second $J^P=2^+$ contribution
with a mass allowed to vary within the 1.75~GeV/$c^2$~--~2.1~GeV/$c^2$ interval.
Two clear resonance-like $K^+K^-$ signals are found
to significantly contribute to the data description in all
fits. The first contribution has a mass of around 1.65~GeV/$c^2$ and
is likely a manifestation of the $\rho(1700)$ or $\omega(1650)$, or interference between the two.
Note that the parameters of both these states remain highly uncertain.
For the $\rho(1700)$, the PDG quotes the results with the mass
varying roughly from 1540~MeV/$c^2$ to 1860~MeV/$c^2$, which may
indicate the presence of two states. Quark potential models \cite{Godfrey:1985xj}
suggest two resonances close to this mass range: $1^3D_1$ and $3^3S_1$.
This possibility is implied in the interpretation of the fit results.
The second contribution has a mass of around 2.0~GeV/$c^2$~--~2.1~GeV/$c^2$,
close to the mass of the $\rho(2150)$. No limitations on their parameters
are imposed in the fits. For the $\rho(1450)$ the
mass range from 1.3~GeV/$c^2$ up to 1.5~GeV/$c^2$ is studied,
but no NLL minima are found, and so its mass and width
are fixed to the PDG estimates \cite{Tanabashi:2018oca}.
In the analysis we find that the PWA solution can not be saturated
with well-known states included as Breit-Wigner resonances
and constant contributions in the lowest
partial waves. At the same time, the ``missing part'' of the
PWA solution can not be reliably attributed to a single
resonance and mainly manifests itself as a slow changing
background in the $J^P=3^-$ partial wave of the $K^\pm\pi^0$
pairs at high $K^\pm\pi^0$ masses. Below we provide
two solutions constructed with and without the smooth
contribution in this partial wave to demonstrate that the
conclusions of this analysis are not strongly affected by
assumptions on the ``missing part'' of the PWA solution.
\subsection{Solution I}
The results for the best fit based on the well-established
resonances and constant contributions in the lowest partial waves
are given in Table~\ref{tab:fbestPDG}.
Only contributions improving the NLL by more than 17
are included to the fit (corresponding to a statistical significance
of $5\sigma$ for 4 degrees of freedom).
The data description as a Dalitz plot are shown in Fig.~\ref{fig:dalitz}(b).
Fig.~\ref{fig:kin_distributions_full}
and Fig.~\ref{fig:kin_distributions} show the corresponding
invariant mass spectra and angular distributions.
The kinematic distributions in Fig.~\ref{fig:kin_distributions}
are restricted to the inner part of the Dalitz plot
($M(K^\pm\pi^0)>1.05$~GeV/$c^2$) to exclude the huge
peaks from the $K^*(892)^\pm$.
The dominant contribution stems from the $K^*(892)^\pm$ and $K_2^*(1430)^\pm$
resonances in the $K^\pm\pi^0$ kinematic channels.
The first decay is well-known and contributes about 90\% to the
total decay rate. The interference term between the $K^{*}(892)^+K^-$
and $K^{*}(892)^-K^+$ intermediate states contributes about 10\%.
The mass and the width of the $K^*(892)^\pm$ are determined with high
statistical precision.
The Blatt-Weisskopf radius of the resonance is found to be
$r=0.25\pm0.02$~fm. The second largest contribution,
with a decay fraction of about 10\%, is the
$K^*_2(1430)^\pm$, which also can be clearly seen
in Fig.~\ref{fig:dalitz}.
The mass and width of this state are also determined with
high precision. Its Blatt-Weisskopf radius can not be
reliably determined from the fit and is set
to $0.4$~fm, which is the meson-interaction radius
used in Ref.~\cite{Aston:1986rm}.
The contribution of the
$K^*_2(1430)^\pm K^\mp$ channel to the reaction is approximately 10
times smaller than the contribution from the $K^*(892)^\pm K^\mp$
channel. Taking into account this result and
using a branching fraction of 49.9\% for the $K_2^*(1430)^\pm$ decay to
$K\pi$ \cite{Tanabashi:2018oca}, we find that the
$J/\psi$ decay to $K^*_2(1430)^\pm K^\mp$ is suppressed
by an approximate factor of 5 compared to the decay to
$K^*(892)^\pm K^\mp$.
For $J^P=1^-$, the inclusion of the $K^*(1680)^\pm$
provides a significant improvement in the data description,
but no NLL minima consistent with its mass and width are found.
The $J^P=2^+$ partial wave requires another $2^+$ state
with a relative contribution of approximately 0.4\%.
Its mass and width are found to be $1817\pm11$ MeV/$c^2$ and $312\pm28$ MeV/$c^2$,
respectively. This mass is much lower than the mass of the $K_2^*(1980)^\pm$
observed by LASS. The $K_3^*(1780)^\pm$ state
provides a significant improvement in the log-likelihood, but
no NLL minima consistent with its measured parameters
are found. Finally, there is a small , but
very distinct and stable
contribution of $(0.18\pm0.02)$\% from the $K_4^*(2045)^\pm$. Its fitted mass is lower
than that obtained in other measurements~\cite{Tanabashi:2018oca}, which
can be attributed to the uncertainties of the PWA solution
(see solution~II).
In the $K^+K^-$ kinematic channel, the first stable contribution has
$J^{PC}=1^{--}$, a mass of $1643\pm3$~MeV/$c^2$,
a width of $167\pm12$~MeV/$c^2$ and a decay fraction of 1\%.
It can also be clearly seen in the
Dalitz plot. As mentioned above, this contribution can be attributed to the
$\rho(1700)$. The structure is also reasonably consistent
with the $\omega(1650)$ (the mass is consistent with the PDG estimate,
and the width is well within the spread of the results quoted by PDG)
or an interference between these states.
The second contribution that can be reliably determined from the data
is a $J^{PC}=1^{--}$ resonance with a mass of $2078\pm6$~MeV/$c^2$ and
width of $149\pm21$ MeV/$c^2$. The largest relative contribution of $(1.8\pm0.2)$\%
comes from the tail of the $\rho(770)$. Since the mass of this state is significantly below
the $K^+K^-$ production threshold,
no reliable claim can be made about its observation.
The $\rho_3(1690)$ and $\rho(1450)$ provide NLL
improvement by 144 and 27, but no NLL
minimum consistent with the parameters of each state is found.
The smooth contribution in the $J^{PC}=1^{--}$ $K^+K^-$
partial wave is also found to be significant.
Additionally, we try to set the mass and the width
of the $J^{PC}=1^{--}$ $K^+K^-$ contribution at
1.65~GeV/$c^2$ to the PDG mean values for the $\rho(1700)$
averaged from $\eta\rho(770)$ and $\pi^+\pi^-$ modes.
In this case, the NLL worsens by 42, and so one
may consider including the $\omega(1420)$ and $\omega(1650)$ in
the fit. In these fits we set their masses and width
to the mean values of the PDG estimates. If the $\omega(1420)$ ($\omega(1650)$)
is included, the NLL is still worse by 14 (7) compared
to the result of solution~I.
If the $\rho(1450)$ is substituted by the $X(1575)$, instead of
adding a resonance, the NLL improves by 28, but
remains worse by 14 than the result for solution~I.
Adding further well-established resonances with the nominal
PDG parameters does not improve log-likelihood by more than 17 units.
Despite this, the solution is not saturated: if additional
contributions (parametrized as Breit-Wigner resonances with
parameters not required to correspond to a physical state)
are added, they can improve NLL
by up to 95 in a single partial wave, which is much
larger than the contribution of other resonances included
to the solution. The only notable additional contribution
indicating resonance behavior is in the $J^P=1^-$ $K\pi$
partial wave with a mass of around 2.4 GeV/$c^2$, but there
is lack of qualitative evidence to report a new state.
The largest improvement
in the NLL function comes from contributions
that tend to be broad and cannot be interpreted as resonances.
These conclusions are not surprising if one considers the measured two-particle
$K\pi$ scattering amplitudes obtained by the LASS Collaboration \cite{Aston:1987ir}.
Here the $F$-wave intensity, apart from the $K^*_3(1780)$ peak,
has a strong contribution from nontrivial structures,
which are not resolved in the LASS analysis.
The inability to provide a consistent data description
for this solution prevents us from making a reliable
estimation of systematic uncertainties.
\begin{figure}[t]
\centerline{\includegraphics[width=0.48\textwidth]{fig/global_upd_LASS_full.pdf}}
\caption{Kinematical distributions for the data (dots),
the PWA solution~I (shaded histograms) and the PWA solution~II (solid line).
The notation $K$ without any specified charge indicates the sum of the
$K^+$ and $K^-$ distributions.
(a-b) Invariant mass of the $K^+K^-$ and $K^\pm\pi^0$ systems.
(c-d) Distributions of the final state particles polar angle ($\theta_{\pi^0}$, $\theta_{K}$)
with respect to the beam axis in the $J/\psi$ rest frame.
(e-f) Polar angle distributions ($\theta_{KK}$, $\theta_{\pi K}$) for $K^+$ in the $K^+K^-$
helicity frame (e) and for $\pi^0$ in the $K\pi^0$
helicity frame (f). The uncertainties are statistical and are within the size of the dots.
}
\label{fig:kin_distributions_full}
\end{figure}
\begin{figure}[t]
\centerline{\includegraphics[width=0.48\textwidth]{fig/global_upd_LASS.pdf}}
\caption{Kinematical distributions for the data (dots),
PWA solution I (shaded histograms) and PWA solution II (solid line)
in the inner region of the Dalitz
plot ($M(K^\pm\pi^0)>1.05$ GeV/$c^2$).
The notation $K$ without any specified charge indicates the sum of the
$K^+$ and $K^-$ distributions.
(a-b) Invariant mass of the $K^+K^-$ and $K^\pm\pi^0$ systems.
(c-d) Distributions of the final-state state particles polar angle ($\theta_{\pi^0}$, $\theta_{K}$)
with respect to the beam axis in the $J/\psi$ rest frame.
(e-f) Polar angle distributions ($\theta_{KK}$, $\theta_{\pi K}$) for $K^+$ in the $K^+K^-$
helicity frame (e) and for the $\pi^0$ in the $K\pi^0$
helicity frame (f).
The error bars represent the statistical uncertainties.
}
\label{fig:kin_distributions}
\end{figure}
\begin{table*}[htb]
\footnotesize
\renewcommand{\arraystretch}{1.6}
\caption{
List of contributions for solution~I, showing for each contribution the mass, width, decay fraction and increase in negative log-likelihood for the removal of the state.
In the $K\pi$ channel $b$ stands for decay fraction through
both charged conjugated modes and $b^{+(-)}$ gives the contribution
of one charged mode, which allows their interference to be determined.
The uncertainties are statistical. Parameters marked with $^\star$ are fixed.
}
\label{tab:fbestPDG}
\center
\setlength{\tabcolsep}{11pt}
\begin{tabular}{ >{$}l<{$} >{$}l<{$} >{$}r<{$} >{$}r<{$} >{$}r<{$} >{$}r<{$} >{$}r<{$} } \hline\hline
\multicolumn{7}{ c }{\textbf{$K^\pm\pi^0$ channels}} \\
J^{PC}\!\! & \text{PDG} & M(\text{MeV/}c^2) &\Gamma (\text{MeV/}c^2) & b(\%) & b^{+(-)}(\%) & \Delta\text{NLL} \\
\hline\hline
\;1^{-}\!\! & K^*(892)^\pm & 894.1 \mpm0.1 & 46.7 \mpm0.2 & 89.2\mpm0.8 & 41.0\mpm0.2 & - \\ \hline
\;1^{-}\!\! & K^*(1680)^\pm & 1677^{\star } & 205^{\star } & 0.59\mpm0.04 & 0.25\mpm0.02 & 398 \\ \hline
\;2^{+}\!\! & K_2^*(1430)^\pm & 1431.4\mpm0.8 & 100.3\mpm1.6 & 9.2 \mpm0.1 & 4.1 \mpm0.1 & - \\ \hline
\;2^{+}\!\! & K_2^*(1980)^\pm & 1817\mpm11 & 312\mpm28 & 0.44\mpm0.05 & 0.17\mpm0.02 & 238 \\ \hline
\;3^{-}\!\! & K_3^*(1780)^\pm & 1781^{\star } & 203^{\star } & 0.08\mpm0.01 & 0.04\mpm0.01 & 83 \\ \hline
\;4^{+}\!\! & K_4^*(2045)^\pm & 2015\mpm7 & 183\mpm17 & 0.16\mpm0.02 & 0.07\mpm0.01 & 192 \\ \hline
\multicolumn{7}{ c }{\textbf{$K^+K^-$ channel}} \\
\quad\;\, J^{PC}\!\! & \text{PDG} & M(\text{MeV/c}^2) &\Gamma (\text{MeV/c}^2) & \multicolumn{2}{ c }{b(\%)} & \Delta\ln L \\ \hline
\quad\;\, 1^{--}\!\! &\rho(770) & 771^{\star} & 150^\star & \multicolumn{2}{ c }{$1.8 \mpm0.2 $} & 220 \\ \hline
\quad\;\, 1^{--}\!\! &\rho(1450) & 1465^\star & 400^\star & \multicolumn{2}{ c }{$1.2 \mpm0.2 $} & 27 \\ \hline
\quad\;\, 1^{--}\!\! & & 1643 \mpm3 & 167 \mpm12 & \multicolumn{2}{ c }{$1.1 \mpm0.1 $} & 281 \\ \hline
\quad\;\, 1^{--}\!\! & & 2078\mpm6 & 149\mpm21 & \multicolumn{2}{ c }{$0.15\mpm0.03 $} & 73 \\ \hline
\quad\;\, 1^{--}\!\! & \text{non-resonant} & -- & -- & \multicolumn{2}{ c }{$1.2 \mpm0.2 $} & 34 \\ \hline
\quad\;\, 3^{--}\!\! &\rho_3(1690) & 1696^\star & 204^\star & \multicolumn{2}{ c }{$0.14\mpm0.01 $} & 144 \\
\hline\hline
\end{tabular}
\end{table*}
\subsection{Solution II}
We find that the largest improvement to the NLL
of the solution~I comes from the inclusion of a smooth contribution
in the $J^P=3^-$ partial wave, which we parametrize with a broad Breit-Wigner shape.
Its mass is found to be close the maximal allowed
invariant mass of the $K^\pm\pi^0$ system.
The width can vary in the approximate interval of 0.5~GeV/$c^2$~--~1.2~GeV/$c^2$,
depending on small variations of the PWA solution, and its
value only slightly affects other components in the fit.
Such a mass and width does not allow an interpretation of
this contribution as a single resonance.
The solution where this broad component is added
and the significance of the resonances is reevaluated is shown in Table~\ref{tab:fbestres}.
For this solution, we use the more conservative resonance
significance criteria: the minimum NLL improvement is
required to be 40.
We ensure that no other allowed resonance
contributions improve the NLL value above
this number, considering possibilities with spins up
to $J=5$, which is the maximum spin of previously reported
states allowed in this decay.
Those contributions which give the most significant
NLL improvement are used to estimate systematic
uncertainties. The NLL value for
this solution is better by 116 than that of solution~I.
The systematic uncertainties listed in Table~\ref{tab:fbestres}
will be discussed later. The Dalitz plot for the solution~II
is shown in Fig.~\ref{fig:dalitz}(c).
Mass and angular distributions are given in Fig.~\ref{fig:kin_distributions_full}
and Fig.~\ref{fig:kin_distributions} for the data and for the two models.
The two descriptions are very similar, but solution~II is superior in specific kinematic regions.
Solution~II has the same set of well-defined contributions
as solution~I. The fitted mass and
width for the $K^*(892)^\pm$ and $K_2^*(1430)^\pm$ are
almost the same. The mass, width and Blatt-Weisskopf radius
of the $K^*(892)$ are found to be
$M=893.6 \mpm0.1 _{-0.3 }^{+0.2}$~MeV/$c^2$,
$\Gamma=46.7 \mpm0.2 _{-0.2 }^{+0.1 }$~MeV/$c^2$ and
$r=0.20\pm0.02_{-0.04}^{+0.14}$~fm, respectively, where here
and subsequently the first uncertainty is statistical, and the second systematic.
The mass lies between
the PDG averages for measurements performed
where the $K^*(892)^\pm$ is produced
in hadronic collisions and those were it is produced in $\tau$ decays~\cite{Tanabashi:2018oca}.
The fitted width is consistent with the $\tau$-decay results~\cite{Epifanov:2007rf}.
For the $K_2^*(1430)^\pm$ we fix
the Blatt-Weisskopf radius to 0.4~fm. The $2^+$ partial amplitude
in the $K^\pm\pi^0$ kinematic channels also requires a second
contribution with a mass higher than that of the previous solution
with large systematic uncertainties for both the mass and width: $M=1868\pm8_{-57}^{+40}$ MeV/$c^2$
and $\Gamma=272\pm24_{-15}^{+50}$ MeV/$c^2$. The mass is approximately
100~MeV/$c^2$ below the LASS measurement for the $K_2^*(1980)$ \cite{Aston:1986jb}, but
both the mass and the width are compatible with the PDG averages within $2.2$
standard deviations.
As in solution~I, there is a very clear contribution to the $J^P=4^+$ partial wave
with $M=2090\pm9^{+11}_{-29}$~MeV/$c^2$ and $\Gamma=201\pm19_{-17}^{+57}$~MeV/$c^2$,
which is consistent with the parameters of the $K_4^*(2045)^\pm$ \cite{Tanabashi:2018oca}.
For the $K^*(1410)$, which is required in this solution, the
$K^*(1680)^\pm$ and the $K_3^*(1780)^\pm$, no NLL minima
consistent with parameters of these resonances are found. In the
$K^+K^-$ kinematic channel we see again two
stable contributions at 1.65~GeV/$c^2$ and 2.05 GeV/$c^2$. The contributions
from the $\rho(1450)$, $\rho_3(1690)$ and $\rho(770)$ are marginal.
A striking feature of solution~II is the presence
of a non-resonance component in the $J^P=3^-$ $K^\pm\pi^0$ partial
waves, which can not be clearly interpreted as an
interference between Breit-Wigner states. A possible interpretation is that this component
is the manifestation of non-resolved contributions present in the
$F$-wave $K\pi$ scattering amplitude \cite{Aston:1987ir}.
This may include the presence of several resonances, non-resonant production and
final-state particle rescattering effects.
The stability of the found NLL minimum with respect to
the parameters of the reported resonances is demonstrated
in Fig.~\ref{fig:mass_scan}.
\begin{figure*}
\centerline{
\includegraphics[width=0.95\textwidth]{fig/MG-scans.pdf}
}
\caption{\label{fig:mass_scan} Mass and width scans for the $K^*_2(1430)$,
$K_2^*(1980)$, $K^*_4(2045)$ and $1^{--}$ structures at 1650 MeV/$c^2$
and 2050 MeV/$c^2$ for solution~II.}
\end{figure*}
The systematic errors due to the uncertainty of the PWA solution
are assigned to be the largest deviations for the following
variations of the solution:
\begin{itemize}
\item variation of the masses and widths for the $K^\pm\pi^0$ resonances with
the parameters fixed in the fit, and varied by one standard
deviation of the LASS results \cite{Aston:1987ir};
\item variation of the Blatt-Weisskopf radius of the $K_2^*(1430)^\pm$ by $\pm0.2$~fm;
\item inclusion of contributions that strongly improve
the log-likelihood below the acceptance criteria
($J^P=1^{-}$ $(K\pi)$ at approximately 2.5 GeV/$c^2$
and $J^{PC}=1^{--}$ $(K^+K^-)$ at
$M(K^+K^-)\approx 2.3$ GeV/$c^2$);
\item reparametrization of the broad background part of partial waves.
\end{itemize}
To evaluate the latter variation, broad contributions in the $1^-$, $2^+$ ($K\pi$)
amplitudes and $1^{--}$ ($K^+K^-$) partial wave parametrized with
$\rho(770)^0$ and $\rho(1450)^0$ are studied. In all these fits the states
$K^*(892)^\pm$, $K_2^*(1430)^\pm$, $K_4^*(2045)^\pm$ and the structures at
1.65~GeV/$c^2$ and 2.05~GeV/$c^2$ in the $K^+K^-$ channels remain stable.
The high-mass
broad $K^\pm\pi^0$ $3^-$ contribution always remains significant, but its relative
fraction varies to much smaller values in some fits.
The $1^-$ additional contribution mostly manifests
resonant behavior. No stable contribution can be associated with the
$\rho(1450)$, but its relative decay fraction at the level of 1\%
does not contradict the data.
The total systematic uncertainties for the masses, widths and decay fraction given in Table~\ref{tab:fbestres} are calculated as a quadratic sum of:
\begin{itemize}
\item the variation in results due to the uncertainty of the PWA solution;
\item the bias introduced by imperfections of the detector simulation and
the event reconstruction;
\item the uncertainties due to the differences in kaon tracking and PID
efficiencies between data and the MC simulation.
\end{itemize}
The differences
in kaon tracking and PID
efficiencies between data and the MC simulation
are studied with a high-purity
control sample of $J/\psi \to K_SK^\pm\pi^\mp$ decays as
a function of kaon transverse momentum $p_T$ and are
found to be within $1\%$ per track both for the tracking
and the PID. The effect on the
PWA result is estimated by varying the selection efficiency
difference for data and MC in $p_T$ bins within
these errors. Uncertainties on the fit parameters due to the efficiency
variation in each bin are summed quadratically.
The background uncertainty, estimated by varying the
subtracted NLL contribution
by 50\%, is found to be negligible.
\begin{table*}[htb]
\footnotesize
\renewcommand{\arraystretch}{1.6}
\caption{
List of components for solution~II.
For the reported states
in the $K\pi$ channel ($K^*(892)^\pm$, $K_2^*(1430)^\pm$, $K_2^*(1980)^\pm$ and $K_4^*(2045)^\pm$) and the reported signals in the $K^+K^-$ channel ($J^{PC}=1^{--}$
signals with masses around 1650 MeV/$c^2$ and 2050 MeV/$c^2$) the first uncertainty is statistical and the second is systematic.
In the $K\pi$ channel the decay fraction is given for both charged conjugated
modes ($b$) and for the contribution of one charged mode ($b^{+(-)}$), so that
their interference can be determined.
As the $K^*(1410)^\pm$, $K^*(1680)^\pm$ and $K_3^*(1780)^\pm$ contributions are not
reliably identified (see main text), their masses and widths are fixed (marked with $^\star$)
and only statistical uncertainties are given for their decay fractions.
}
\label{tab:fbestres}
\center
\setlength{\tabcolsep}{11pt}
\begin{tabular}{ >{$}l<{$} >{$}l<{$} >{$}r<{$} >{$}r<{$} >{$}r<{$} >{$}r<{$} >{$}r<{$} } \hline\hline
\multicolumn{7}{ c }{\textbf{$K^\pm\pi^0$ channels}} \\
J^{PC}\!\! & \text{PDG} & M(\text{MeV/}c^2) &\Gamma (\text{MeV/}c^2) & b(\%) & b^{+(-)}(\%) & \Delta\text{NLL} \\
\hline\hline
\;1^{-}\!\! & K^*(892)^\pm & 893.6 \mpm0.1 _{-0.3 }^{+0.2 } & 46.7 \mpm0.2 _{-0.2 }^{+0.1 } & 93.4\mpm0.4_{-5.8 }^{+1.8 } & 42.5\mpm0.1_{-1.7 }^{+0.5 } & - \\ \hline
\;1^{-}\!\! & K^*(1410)^\pm & 1380^{\star } & 176^{\star } & 0.26\mpm0.04 & 0.11\mpm0.02 & 80 \\ \hline
\;1^{-}\!\! & K^*(1680)^\pm & 1677^{\star } & 205^{\star } & 0.20\mpm0.03 & 0.08\mpm0.01 & 56 \\ \hline
\;2^{+}\!\! & K_2^*(1430)^\pm & 1432.7\mpm0.7_{-2.3}^{+2.2} & 102.5\mpm1.6_{-2.8}^{+3.1} & 9.4 \mpm0.1 _{-0.5 }^{+0.8 } & 4.2 \mpm0.1 _{-0.2 }^{+0.3 } & - \\ \hline
\;2^{+}\!\! & K_2^*(1980)^\pm & 1868\mpm8_{-57}^{+40} & 272\mpm24_{-15}^{+50} & 0.38\mpm0.04_{-0.05}^{+0.22} & 0.15\mpm0.02_{-0.02}^{+0.08} & 192 \\ \hline
\;3^{-}\!\! & K_3^*(1780)^\pm & 1781^{\star } & 203^{\star } & 0.16\mpm0.02 & 0.07\mpm0.01 & 105 \\ \hline
\;4^{+}\!\! & K_4^*(2045)^\pm & 2090\mpm9_{-29}^{+11} & 201\mpm19_{-17}^{+57} & 0.21\mpm0.02_{-0.05}^{+0.10} & 0.09\mpm0.01_{-0.02}^{+0.04} & 212 \\ \hline
\;3^{-}\!\! & \text{non-resonant} & -- & -- & \sim 1.5\% & \sim0.6\% & 629 \\
\hline
\multicolumn{7}{ c }{\textbf{$K^+K^-$ channel}} \\
\quad\;\, J^{PC}\!\! & \text{PDG} & M(\text{MeV/c}^2) &\Gamma (\text{MeV/c}^2) & \multicolumn{2}{ c }{b(\%)} & \Delta\ln L \\ \hline
\quad\;\, 1^{--}\!\! & & 1651\mpm3_{-6}^{+16} & 194\mpm8_{ -7}^{+15} & \multicolumn{2}{ c }{$1.83\mpm0.11_{-0.17}^{+0.19}$} & 796 \\ \hline
\quad\;\, 1^{--}\!\! & & 2039\mpm8 _{-18}^{+36} & 193\mpm23_{-27}^{+25} & \multicolumn{2}{ c }{$0.23\mpm0.04_{-0.06}^{+0.07}$} & 102 \\
\hline\hline
\end{tabular}
\end{table*}
\subsection{Summary on PWA}
Our analysis shows that there is a set of states in the PWA
solutions that remains stable for both considered cases:
when contributions corresponding to well-known
resonances are considered or when broad contributions are introduced
to parameterize the missing part of the partial amplitudes.
In the $K^\pm\pi^0$ channels this set of resonances includes
the $K^*(892)^\pm$, $K_2^*(1430)^\pm$, and $K_4^*(2045)^\pm$.
The second $J^P=2^+$ state, labeled here as $K_2^*(1980)^\pm$, has a mass
much lower than that observed by the LASS Collaboration \cite{Aston:1986jb}.
However, given the large systematic uncertainties on
this quantity, our result is compatible within 2.2~standard deviations.
The first stable structure in the $K^+K^-$ channel has a mass of about 1.65~GeV/$c^2$
and a decay fraction of 1.0\%~--~1.5\%. The absence of a distinct
contribution from the first radial excitation of the $\rho(770)$ favors its interpretation
as a $^3D_1$ $\rho$-resonance. At the same time such a small decay fraction is
consistent with $\omega(1650)$ production in $J/\psi$ decay through
a virtual photon.
Its mass is consistent with the PDG estimate for the $\omega(1650)$ and its
width is well within the spread of experimental results quoted by the PDG.
It could also be the result of interference between these isovector and isoscalar states.
The second stable contribution has a mass of about 2.05~GeV/$c^2$~--~2.10~GeV/$c^2$ and
decay fraction of 0.1\%~--~0.2\%. Given the large systematic uncertainties
it could be interpreted as either the $\rho(2150)$ or as another
isovector-vector state observed in
proton-antiproton annihilation in flight \cite{Anisovich:2000ut}.
Clarification of the nature of these excited vector mesons requires
further investigation.
\section{Branching fractions}
The $J/\psi \to K^+K^-\pi^0$ branching fraction is determined as
$B(J/\psi \to K^+K^-\pi^0) =
\frac{N_{sel}-N_{bg}-N_{continuum}}{\epsilon N_{J/\psi} B(\pi^0 \to
\gamma\gamma)}$. Here $N_{sel}$, $N_{bg}$ and $N_{continuum}$ are
the number of selected events, the estimated background from the $J/\psi$
decays, and the continuum production, respectively. The number of $J/\psi$ events
$N_{J/\psi} = (223.7\pm 1.4(syst.)) \times 10^6$ is taken from Ref.~\cite{Ablikim:2016fal}, and $B(\pi^0 \to \gamma\gamma) = (98.823 \pm 0.034) \times 10^{-2}$
is taken from the PDG~\cite{Tanabashi:2018oca}. The selection efficiency $\epsilon$
is obtained using the PWA solution~II and the detector performance simulation.
The dominant contribution to the statistical uncertainty comes from
$N_{sel}$.
The systematic uncertainty on the branching fraction is estimated
from the sources listed in Table~\ref{tab:brsyst_summ}.
The background uncertainty is estimated by
varying $N_{bg}$ by $\pm50\%$.
The uncertainty associated with the subtraction of the continuum background is assigned to be
the statistical error on
$N_{continuum}$.
The charged track reconstruction
efficiency and the PID efficiency uncertainties are 1\% each per
track as is discussed above. The photon detection efficiency is
studied with the decays $\psi(3686) \to \pi^+\pi^- J/\psi$, $J/\psi \to
\rho^0\pi^0$ and photon conversion control
samples \cite{Ablikim:2010zn, Ablikim:2011kv}. In this analysis,
an uncertainty of 1\% per photon is assigned.
The uncertainty introduced by the cut on
$\chi^2_{K^+K^-\gamma\gamma}$ is estimated using a control sample.
This is selected using similar selection criteria,
with the kinematic-fit cut replaced
by the requirement that at least one particle out of three
($K^+$, $K^-$, $\pi^0$) has a mass hypothesis consistent
with the recoil mass calculated using the other two particles.
Such a procedure accepts a signal event even if one of the
particles is badly reconstructed.
This gives
$B(J/\psi \to K^+K^-\pi^0)=(2.88\pm0.01\pm0.12)\times10^{-3}$.
\begin{table}[h]
\centering
\caption{Summary of systematic uncertainties for $B(J/\psi \to K^+K^-\pi^0)$.}
\label{tab:brsyst_summ}
\begin{tabular}{ l c }
\hline\hline
Source & Uncertainty (\%) \\ \hline\hline
$N_{bg}$ & 0.2 \\
$N_{continuum}$ & 0.3 \\
Track reconstruction efficiency & 2.0 \\
PID efficiency & 2.0 \\
Photon reconstruction efficiency & 2.0 \\
Kinematic fit cut efficiency & 2.4 \\
$N_{J/\psi}$\cite{Ablikim:2016fal} & 0.6 \\
\hline
Total & 4.3 \\
\hline\hline
\end{tabular}
\end{table}
Knowing the $J/\psi \to K^+K^-\pi^0$ branching fraction and the
decay fractions for the individual components from the PWA, we determine branching
fractions for the decay via individual resonances. Results
for solution~II are summarized in Table~\ref{tab:resonance_br}.
The branching fraction $B(J/\psi \to K^+K^-\pi^0)$ and the branching fractions
for the decay via the $K^*(892)^\pm$ that are obtained in solution~II
are compared to the results
from previous experiments in Table~\ref{tab:br_compar}.
Our result for $B(J/\psi \to K^+K^-\pi^0)$
is up to now the most precise measurement. It differs
from the PDG value~\cite{Tanabashi:2018oca}, obtained indirectly from Ref.~\cite{Aubert:2007ym},
by about 2.8 standard deviation.
The systematic uncertainty of our results for decays through the $K^*(892)^\pm$
is somewhat larger than that of Ref.~\cite{Aubert:2007ym},
which can be attributed to the uncertainties present in the PWA model.
\begin{table*}[h]
\small
\renewcommand{\arraystretch}{1.6}
\centering
\caption{Branching fractions for decays via reliably identified intermediate states (solution~II).
$R_{K\pi}$ and $R_{KK}$ denotes $K^\pm\pi^0$ and $K^+K^-$ resonances, respectively, and
$R_{K\pi}^\pm K^\mp$ denotes for one possible charged combination. The first uncertainty
is statistical and the second one is systematic.}
\label{tab:resonance_br}
\begin{tabular}{ >{$}l<{$} >{$}r<{$} >{$}r<{$} }
\hline\hline
\multicolumn{3}{ c }{\textbf{Intermediate resonance in the $K\pi$ system}} \\ \hline
\text{$R_{K\pi}$} & B(J/\psi \to R_{K\pi}^\pm K^\mp \to K^+K^-\pi^0) & B(J/\psi \to R_{K\pi}^+ K^- + c.c. \to K^+K^-\pi^0) \\
\hline
K^*(892) & (1.22\mpm0.01 \er{0.07}{0.05}) \times10^{-3} & (2.69\mpm0.01 \er{0.20}{0.13}) \times10^{-3} \\ \hline
K_2^*(1430) & (1.21\mpm0.02 \er{0.08}{0.10}) \times 10^{-4} & (2.69\mpm0.04 \er{0.19}{0.25}) \times10^{-4} \\ \hline
K_2^*(1980) & (4.3\mpm0.5 \er{0.6}{2.3}) \times10^{-6} & (1.1\mpm0.1 \er{0.1}{0.6}) \times10^{-5} \\ \hline
K_4^*(2045) & (2.6\mpm0.3 \er{0.6}{1.1}) \times10^{-6} & (6.2\mpm0.7 \er{1.4}{2.8}) \times10^{-6} \\ \hline
\multicolumn{3}{ c }{\textbf{Intermediate resonance in the $K^+K^-$ system}} \\ \hline
\text{$R_{KK}$} & \multicolumn{2}{ c }{$B(J/\psi \to R_{KK}\pi^0 \to K^+K^-\pi^0)$} \\\hline
1^{--}(1650\text{ MeV/}c^2) & \multicolumn{2}{ c }{$(5.3\mpm0.3 \er{0.5}{0.6}) \times10^{-5} $} \\ \hline
1^{--}(2050\text{ MeV/}c^2) & \multicolumn{2}{ c }{$(6.7\mpm1.1 \er{1.8}{2.2}) \times10^{-6} $} \\
\hline\hline
\end{tabular}
\end{table*}
\begin{table*}[h]
\renewcommand{\arraystretch}{1.6}
\centering
\scriptsize
\caption{Comparison between this work and previous measurements.
For $B(J/\psi\to K^{*+}K^-\!+\!c.c. \to K^+K^-\pi^0)$ and
$B(J/\psi\to K^{*+}K^-\!+\!c.c.)$ we give two numbers for
solution~II: the first one
is a sum of branching fractions through $K^{*+}$ and $K^{*-}$ and
the second number (in parenthesis) accounts for their interference.
Results marked with ``$^\dagger$'' are obtained by averaging the $K_SK^\pm\pi^\mp$ and
$K^+K^-\pi^0$ final states. Results recalculated by us using numbers from this work
are marked with ``$^{\dagger\dagger}$''.}
\begin{tabular}{ p{.27\textwidth} r r r r r r}
\hline\hline
Channel &\multicolumn{5}{c }{$B(\times10^{-3})$} \\ \cline{2-6}
&This work &BABAR\cite{Aubert:2007ym} & DM2\cite{Jousset:1988ni} & MARK-III\cite{Coffman:1988ve} & MARK-II\cite{Franklin:1983ve} \\ \hline
$B(J/\psi\to K^+K^-\pi^0)$ &$2.88\!\pm\!0.01\!\pm\!0.12 $ & --- & --- & --- & $2.8\!\pm\!0.8$ \\ \hline
$B(J/\psi\to K^{*+}K^-\!+\!c.c.\!\to\!K^+K^-\pi^0)$ &$2.45\!\pm\!0.01_{-0.14}^{+0.10}(2.69\!\pm\!0.01_{-0.20}^{+0.13}) $ &$1.97\!\pm0.16\!\pm0.13$ & $1.50\!\pm\!0.23\!\pm\!0.27^{\dagger\dagger}$ & $1.87\!\pm\!0.04\!\pm\!0.28^{\dagger\dagger}$ & $2.6\!\pm\!0.8$ \\ \hline
$B(J/\psi\to K^{*+}K^-\!+\!c.c.)$ &$7.34\!\pm\!0.03_{-0.43}^{+0.33}(8.07\!\pm\!0.04_{-0.61}^{+0.38}) $ &$5.2\!\pm\!0.3\!\pm\!0.2^\dagger$ & $4.57\!\pm\!0.17\!\pm\!0.70^\dagger$ & $5.26\!\pm\!0.13\!\pm\!0.53^\dagger$ & $7.8\!\pm\!2.4^{\dagger\dagger}$ \\
\hline\hline
\end{tabular}
\label{tab:br_compar}
\end{table*}
\section{Conclusion}
A partial-wave analysis of the decay $J/\psi \to K^+K^-\pi^0$
using a data sample of $(223.7\pm1.4)\times10^6$ $J/\psi$
events collected by the BESIII reveals a set of resonances that
have not been observed by previous experiments.
In the $K^\pm\pi^0$ channels our analysis reveals signals from
$K_2^*(1980)^\pm$ and $K_4^*(2045)^\pm$ resonances. This is the first
observation of these states in $J/\psi$ decays. The mass
of the former state is determined with
a central value around 100 MeV/$c^2$ lower than that
reported by the LASS Collaboration~\cite{Aston:1986rm}. This lower value is in
better agreement with the expectation from the linear Regge trajectory of
radial excitations with the standard slope \cite{Anisovich:2001ig}.
As for the known decays through $K\pi$ resonances, we
determine the parameters, decay ratios, and branching fractions
for the $K^*(892)^\pm$ and $K_2^*(1430)^\pm$ with improved precision
compared to previous measurements.
In the $K^+K^-$ channel we observe a clear $J^{PC}=1^{--}$ resonance structure with a
mass of 1.65~GeV/$c^2$ and another $J^{PC}=1^{--}$ contribution at 2.05~GeV/$c^2$~--~2.10~GeV/$c^2$. The
first structure may be interpreted as the ground $^3D_1$ isovector state.
At the same time its mass, width and small relative contribution
to the decay are reasonably consistent
with the production of the $\omega(1650)$
in $J/\psi$ decays through a virtual photon.
The second state can be interpreted as the $\rho(2150)$ or as another
isovector-vector state that has been observed in
proton-antiproton annihilation in flight \cite{Anisovich:2000ut}.
The precise identification of these two states requires
further analysis of more channels, such as $J/\psi\to K_SK^\pm\pi^\mp$
and $J/\psi\to K^+K^-\eta$.
Our PWA solutions have notable differences from those
presented in Ref.~\cite{Ablikim:2006hp} and
more recently in Ref.~\cite{Lees:2017ndy}.
We also report the most precise measurement of the branching fraction $B(J/\psi \to K^+K^-\pi^0)$.
\section{Acknowledgements}
\input{acknowledgement-2019-04-19.tex}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 621 |
Today, too many job listings for therapists and counselors are vague about pay, if they mention it at all. It's part of a culture in mental health that keeps salaries low and professionals feeling disempowered. When employers #PostThePay — even as a range — both employers and applicants benefit.
That's why we're launching a social campaign encouraging employers to do exactly that.
Applicants have a clear sense of what job opportunities are worth their time pursuing. Many will happily apply for positions where the pay is lower, if they can see that they would be working in support of a compelling mission, or there are other benefits of working in that job.
Employers avoid wasting time on applicants who wouldn't accept their pay scale. They protect themselves against possible concerns over wage discrimination. And they can demonstrate that they are truly interested in attracting the best-qualified applicants, either through pay or through other compelling benefits (great supervision, being part of a unique mission, etc.).
We want to make waves with this. It's time for a cultural shift in how therapists and counselors are recruited into their jobs. We think employers stand to benefit from this shift every bit as much as their employees. We hope you'll join us. #PostThePay in support of fair wage and employment practices for counselors and therapists. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,480 |
package com.lexicalscope.jewel.cli.specification;
public interface SpecificationMultiplicity<T> {
T expectedNoneGotSome();
T expectedOneGotNone();
T expectedOneGotSome();
T expectedExactGotTooFew(int exactly, int valueCount);
T expectedExactGotTooMany(int exactly, int valueCount);
T expectedMinimumGotTooFew(int minimum, int valueCount);
T expectedMaximumGotTooMany(int maximum, int valueCount);
T allowed();
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,902 |
{"url":"https:\/\/socratic.org\/questions\/how-do-you-find-vertical-horizontal-and-oblique-asymptotes-for-f-x-x-1-3","text":"How do you find vertical, horizontal and oblique asymptotes for f(x)= x^(1\/3)?\n\nThis function does not have any asymptotes. It has some nice rotational symmetry, because it is an \"odd\" function. It is the inverse of $f \\left(x\\right) = {x}^{3}$ which means that its graph would be a reflection across the line y = x. If you wish to investigate asymptotes for functions, perhaps take a look at \"Rational\" functions.","date":"2019-03-20 13:34:48","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 1, \"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\": 0.7483901381492615, \"perplexity\": 175.8669507539702}, \"config\": {\"markdown_headings\": false, \"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-2019-13\/segments\/1552912202347.13\/warc\/CC-MAIN-20190320125919-20190320151919-00170.warc.gz\"}"} | null | null |
\section{Introduction}
The action reaction principle (ARP) is as firmly established as the conservation of energy.
Theoretical models where one or the other is not fulfilled have most pro\-ba\-bly incomplete
phase spaces, and additional variables in an extended phase space will restore them. The axiom of measurement
in Quantum Mechanics (QM) violates the ARP in simple examples. This paradox
would not appear if the phase space of standard QM were incomplete. The
wave function collapse, projection of state, generates many paradoxes in the interpretation of QM,
and a possible incompleteness of the theory was already considered in the seminal paper \cite{EPR}.
Alternative formulations and interpretations incessantly appear although the scientific success of QM is overwhelming;
see e.g. \cite{biblioguide} for a detailed bibliographic list up to 2004. No--go mathematical results \cite{Bell} \cite {GHSZ}
rule out alternative formulations with hidden variables fulfilling the hypothesis of the theorems. On the other hand,
alternatives as Bohmian mechanics \cite{Bohm} are explicit examples of consistent models with hidden variables,
obviously not fulfilling the hypothesis of those no--go theorems.
With regards to Bell's theorems, there has been a wide research activity around which of the hypothesis is (or are) not fulfilled by the laws of Physics. Logic, axioms of mathematical probability, even realism, have each open more or less fruitful lines of study in different fields, although none seems to be relevant in fundamental Physics.
Non locality is acknowledged as the
most probable property of the real world that explains the violation of Bell's inequalities
in experiments \cite{Aspect}. But that seems to be incompatible with relativity.
The non local interaction between measurements of an entangled pair is very different from
known fundamental interactions (electromagnetic, gravitational, \ldots). It acts between events either spatially or causally separated, it does not decay with distance and it is specifically directed towards the entangled pair. No theoretical model of its mechanisms (as, e.g., gauge theories) has been proposed, neither an isolated effect of an hypothetical signal over an apparatus has ever been detected.
Standard Quantum Mechanics is non local through the
projection of state, the same axiom that crashes with the ARP,
unitary evolution, etc. It seems reasonable to look at the established theory in order to find some other ingredient of the mathematical formalism that could be the cause of violation of Bell's inequalities, instead of a non local projection of state.
Born's rule, a distribution of probability obtained from a distribution of amplitude, is not considered in the hypothesis of Bell's theorems; e.g., typical interference phenomena as in the two slit experiment can not be reproduced with a distribution of probability
\footnote{In a minimum of the diffraction pattern there is no way to get a near vanishing total probability by addition of two positive independent ones, which is a kind of trivial Bell's type inequality.}.
A window is open to assign the violation of Bell's inequalities in Nature to the description of physical states through distributions of amplitude, instead of non locality of the wave function collapse.
Violation of the ARP in the projection of state suggest to extend the phase space. An extended Hilbert space will necessarily contain
elementary states with joint precise values of non commuting magnitudes.
It seems that such states would violate the uncertainty principle. The point is if the uncertainty principle applies to the description of states or it just determines a fundamental limit of accuracy to the knowledge (through measurement) of conjugate variables.
Commutation relations apply to
evolution in Hamiltonian dynamics. In Classical Mechanics, functions on the phase space (e.g., conjugate variables) commute under the usual product, and do not generically commute under the Poisson bracket operation \footnote{This fact necessarily limits accuracy of classical joint measurements, although there is no Plank's constant as fundamental lower bound.}. The existence of quantum
states with joint precise (hidden) values of non commuting magnitudes could be consistent with an uncertainty principle for joint measurement of these magnitudes.
In \cite{mio}, I presented a mathematical framework with an extended phase space for spin variables,
consistent with the standard theory. The model is inspired in the
path integral formalism, where I consider the formal hypothesis that
there are families of virtual paths characterized by joint precise final values of non
commuting physical magnitudes, as position and momentum $(x,p)$, or spin in two or more directions $(s_1,s_2, \ldots )$.
Sum of $exp(i S/\hbar )$, $S$ the path integral, for all paths in the family determines
a distribution of amplitude $\Psi (x,p)$ or
$\Psi (s_1,s_2, \ldots )$ in an extended phase space.
The orthodox state in the standard phase space is then obtained by addition of amplitudes,
i.e., computation of marginal amplitudes, for all
fa\-mi\-lies with common value of the considered magnitude, e.g., $\Psi (x) = \int dp\,\, \Psi (x,p)$.
Bell's type inequalities do not apply; the fundamental ingredient is,
as in the usual formulation, a distribution of amplitude (here in an extended phase space with hidden variables),
and not a distribution of probability,
The main physical hypothesis of the proposed theory, introduced to restate the ARP, are the existence of an accompanying
subsystem, the de Broglie wave \cite{deBroglie}, and additional variables for the corpuscular (sub)system.
Elementary states of a quantum system are characterized by a family of precise joint values
of all physical magnitudes representing
the particle (corpuscle) behavior, and a distribution of amplitude (ray in a Hilbert space)
representing its wave like behavior. Bell's experiments
with spin variables on a sample of entangled pairs \cite{Bell} (and Aspect's for polarization variables \cite{Aspect})
appear in this
theory as local interference phenomena, analogous to the two slit experiment.
The aim of this paper is to apply the former hypothesis to a generic quantum system and arbitrary family of magnitudes.
Section $2$ reviews the physical arguments, the need to extend the phase space in order to preserve the ARP.
In Section $3$ the mathematical framework of the theory is presented, and its correspondence with standard QM
established.
Section $4$ develops the description of composite systems and entanglement. Interference terms in the marginal amplitudes give account of the
apparently non local correlation between measurements.
Particular cases, the two slit experiment and the phase space of spin variables, are presented in Section $5$.
\section{Quantum measurement and the action reaction principle}
The ARP represents, in its most generic terms, a basic hypothesis of dynamical theo\-ries.
An isolated composite system evolves, with regards to some coarse--grained
variables, as a free elementary system. Internal variables must be correlated:
departure from the free dynamics of a subsystem
is accompanied by a corresponding departure of the other. Third Newton's law
is necessary for the consistency of Classical Mechanics; otherwise, the first law
would not have been established, because elementary classical systems
are composite at a microscopic level. The ARP is independent of the particular type
of interaction between subsystems; it belongs to the foundations of Physics and its generic framework
(ideal isolated systems, free dynamics, \ldots).
Friction forces apparently violate the conservation of energy; obviously, there are additional
(microscopic) variables giving account of the lost energy in the balance, and all known fundamental interactions are
conservative. In Brownian motion, the pollen grain receives impulses
from unobservable systems, the fluid molecules; once incorporated the corresponding variables in an extended phase space,
an opposite impulse balances the momentum equation.
The ARP and the axiom of quantum measurement are contradictory. One or the other must be rejected
in its present formulation. Let $|a_1>$ be the state of a quantum system, eigenstate of a physical magnitude $A$.
When $A$ is measured, the result of measurement is $a_1$ with certainty, and the final state of the system is
again $|a_1>$, with trivial projection of state. The pointer of the measurement apparatus has changed of state, from a neutral position to the result
``$a_1$''. ARP is violated.
In a theory with additional variables in an extended phase space, elementary states of the quantum system should be
described by joint precise values of non commuting magnitudes, because maximal families of compatible magnitudes are already
considered in the standard formulation. In the orthodox Hilbert space $\cal H$
there are not common eigenstates for incompatible magnitudes; an extended Hilbert space ${\cal H}_{ext}$, with different representation of physical magnitudes, will be the first ingredient of the proposed framework.
Let $c_1|a_1>$ $+$ $c_2|a_2>$ be the initial state of a quantum system, an elementary particle,
and suppose wave packets of the two components are spatially separated
(e.g., with a Stern Gerlach apparatus). A particle detector is located
in one of the ``virtual'' paths, $|a_1>$, and the result of measurement is negative
\footnote{It will happen with relative frequency
$|c_2|^2$.}. The new, projected state of the system is $|a_2>$. The detector has not, apparently, changed of state, and the ARP is violated in this indirect measurement.
Obviously, the detector is designed to show an observable response when interacting
with a particle located in the surrounding spatial region.
Perhaps some type of non local interaction between particle at $|a_2>$ region and detector at $|a_1>$ happens,
an interaction that does not generate an observable response in the detector.
I will consider an alternative hypothesis: the detector interacts with a system spatially located in its neighborhood.
This hypothetical system is not the particle, which is certainly located away. An hypothetical wave like system, which will be denoted the de Broglie wave, does not generate the same reaction than a corpuscular system, and another type of detector should be designed to observe a response. An accompanying wave of an isolated particle is a wave in vacuum, the vacuum
becoming a relevant physical ingredient in non relativistic QM, as it is relevant in
the opposite length scales of Quantum Field Theory (e.g., Casimir energy) and Cosmology (dark energy).
Let us suppose that an elementary particle, and in general a quantum system,
is a composite of a corpuscular subsystem and a wave like subsystem. The corres\-ponding phase space
of the composite system must describe states of both subsystems. If the standard representation
$c_1|a_1>$ $+$ $c_2|a_2>$ in $\cal H$, or the co\-rres\-pon\-ding vector in an
extended Hilbert space ${\cal H}_{ext}$, is associated to the de Broglie wave
(as seems to suggest interaction of the detector with the $|a_1>$ wave component and not
with the particle),
additional variables of state for the corpuscular component
(known to be located at the spatial region of the other
wave packet $|a_2>$) must be considered
for a total phase space ${\cal H}_{ext}^{wave} \times {\cal P}^{corp}$.
\section{Extended phase space}
We must confront the standard phase space of a quantum system, the Hilbert space
$\cal H$, with an extended phase space ${\cal H}_{ext}^{wave} \times {\cal P}^{corp}$
describing both components, wave like and corpuscular subsystems. First, a formal path integral formalism
is considered. Then, the definition of an extended phase space for a generic quantum system
$\cal S$ and
arbitrary family of physical magnitudes $\cal F$ is given. A correspondence between states in both formalism
determines equivalence of the theories, with regards to phase space description.
\subsection{Abstract path integral formalism}
$\cal S$ denotes a quantum system, ${\cal F} = \{ A, B, C, \ldots \}$ a generic family of $N$ physical magnitudes (observables) of
$\cal S$, $M_A = \{ a_i \}$ the set of possible values of magnitude $A$,
$M_B = \{ b_j \}$ of $B$, etc. Additional magnitudes can be incorporated if needed; in many
cases some magnitudes are ignored, when they are not correlated to those of our interest \footnote{Magnitudes in $\cal F$ are not necessarily functionally independent, as would be a coordinate description of classical phase space. Angular momentum, for example, is a vector
but components in three independent directions
do not exhaust the quantum information, because operators in additional directions do not commute with the selected three.}.
An element $\lambda \in M_{\cal F} \equiv $ $M_A \times M_B \times M_C \cdots $,
$\lambda = (a_i,b_j,c_k, \ldots )$, is a $N$--tuple of values of all magnitudes in
$\cal F$.
Let us consider an abstract, formal set of virtual paths, such that
we can assign to each of them an action integral $S(path)$ and a final value
$\lambda$ \footnote{A specific description of the set of paths and rules of assignment should be given if no alternative way to determine states is available.}. $[path](\lambda )$ denotes the subset of paths with final value $\lambda = (a_i,b_j, \ldots )$. We formally define the amplitude
\begin{equation}
Z(\lambda ) = \sum _{[path](\lambda )} e ^{\frac {i}{\hbar}S(path)}
\end{equation}
It is possible to fix the values of a subset of magnitudes
${\cal F}_1 \subset {\cal F}$,
and define $Z(\lambda _1 )$, $\lambda _1 \in M_{{\cal F}_1}$, by addition of elementary terms
$exp(i S/\hbar )$ for all paths with common values $\lambda _1$.
The amplitude $Z$ associated to a family of paths union of disjoint subfamilies with corresponding amplitudes $Z_1$, $Z_2$, \ldots,
is $Z= Z_1 + Z_2 + \cdots$. For example,
$Z(\lambda _1 ) = \sum _{\pi _1 (\lambda ) = \lambda _1} Z(\lambda )$ in the previous case
$\lambda _1 \in M_{{\cal F}_1}$,
with $\pi _1$ the natural projection $\pi _1 : M_ {\cal F} \to M_{{\cal F}_1}$.
In the standard
treatment \cite{Feynman}, the family ${\cal F}_1$ considered is one made of compatible magnitudes, with vanishing commutation relations, e.g. final position coor\-dinates.
A quantum state of $\cal S$ is completely determined through a distribution of amplitude $Z(\lambda _1)$,
which defines a vector $|S> = \sum _{\lambda _1} Z(\lambda _1) |\lambda _1>$ in the co\-rres\-pon\-ding Hilbert space $\cal H$,
generated by elementary states $|\lambda _1> = |(a_i,b_j,\ldots )>$, common
eigenvectors of the family of self adjoint operators representing the magnitudes in ${\cal F}_1$.
Another family of magnitudes ${\cal F}_2$ (compatible among themselves, but not with all of
${\cal F}_1$)
has its corresponding basis of eigenvectors, and
the relations between bases of eigenvectors $|\lambda _1>$ and $|\lambda _2>$ is consistent with the commutation relations
(quantization rules) of operators. Given the change of bases the new distribution $Z(\lambda _2)$ is obtained from
$Z(\lambda _1)$,
$Z(\lambda _2) =$ $\sum _{\lambda _1}$ $<\lambda _2|\lambda _1>$ $Z(\lambda _1)$.
In the abstract path integral formalism, we can express the amplitude $Z(\lambda _2)$
as
\begin{equation}
Z(\lambda _2) = \sum _{\lambda _1}
\sum _{[path](\lambda _1)\cap [path](\lambda _2)}
e ^{\frac {i}{\hbar}S(path)}
\end{equation}
where the set of paths $[path](\lambda _2)$ has been decomposed into disjoint subsets
$[path]$ $(\lambda _1)$ $\cap$ $[path](\lambda _2)$.
Denoting $Z(\lambda _1,$ $\lambda _2)$ the amplitude associated to the family of paths
$[path](\lambda _1)$ $\cap$ $[path](\lambda _2)$,
\begin{equation}
Z(\lambda _1,\lambda _2) = \sum _{[path](\lambda _1)\cap [path](\lambda _2)}
e ^{\frac {i}{\hbar}S(path)}
\end{equation}
we can obtain both $Z(\lambda _1)$ and $Z(\lambda _2)$ as marginal amplitudes
\begin{equation}
Z(\lambda _1) = \sum _{\lambda _2} Z(\lambda _1, \lambda _2)
\quad
Z(\lambda _2) = \sum _{\lambda _1} Z(\lambda _1, \lambda _2)
\end{equation}
In this way, we can get any orthodox representation of a state (associated to a family of compatible magnitudes)
from a
representation in an extended phase space (associated to a larger family of non commuting magnitudes)
\footnote{Families ${\cal F}_1$ and ${\cal F}_2$ can contain
a common subfamily ${\cal F}_c = {\cal F}_1 \cap {\cal F}_2$ of magnitudes;
the set of paths characterized by joint parameters $\lambda _1$ and
$\lambda _2$ which do not share the same values in ${\cal F}_c$ is void. Both
$\lambda _1$ and $\lambda _2$ must project onto a common $\lambda _c$.}.
We consider next the formulation of a non relativistic Quantum Mechanics
inspired in the former abstract path integral formalism, making the hypothesis that a
distribution of amplitude defining a vector (ray) in an extended Hilbert space
represents a state of the quantum system.
Being inspired in the path integral formalism, a relativistic quantum theory
in an extended phase space
could be formulated when a Lorentz invariant action were considered. Notice that
virtual paths arriving to a space time event are contained inside its past
light cone, i.e., acausal interactions do not appear through the amplitudes
obtained in this way.
\subsection{Extended phase space}
If the phase space of the quantum system is to be extended, the family of magnitudes considered
in $\cal F$ must contain non commuting operators. In the former abstract path integral, a distribution of amplitude $Z(\lambda )$
determines the quantum state, where $\lambda \in M_{\cal F} $ is a $N$--tuple of
values of magnitudes in $\cal F$.
We define a Hilbert space ${\cal H}_{ext}$, generated by vectors $\|\lambda> =$
$\|(a_i,b_j,$ $c_k,$ $\ldots )>$,
orthonormal vectors representing elementary states of $\cal S$ with joint precise values of all magnitudes in $\cal F$.
Operators representing the magnitudes (denoted with the same symbols) $A$, $B$, etc.,
have the basis $\|\lambda >$ of common eigenvectors, with eigenvalues $a_i$, $b_j$, \ldots
$\|S> = \sum _{\lambda } Z(\lambda ) \|\lambda >$ is a generic vector of state in
${\cal H}_{ext}$.
By construction, $A$, $B$, etc., commute. This represents the fact that elementary states of the system have joint precise values of all considered magnitudes.
Together with operators $A$, $B$, etc. introduced in the
definition of the phase space ${\cal H}_{ext}$, we will need additional operators $A^d$,
$B^d$, \ldots, acting on ${\cal H}_{ext}$ to represent the
dynamical role of magnitudes, e.g., when they appear in the Hamiltonian. The ``phase space''
representation $A$, $B$, etc., preserves
the commutation property of usual multiplication of functions, while
``dynamical'' representation
$A^d$, $B^d$, \ldots,
would fulfill the usual quantization rules. Dynamics is not considered in this article,
only an alternative phase space representation of quantum states is analyzed.
Together with the extended Hilbert space ${\cal H}_{ext}$, we introduce a label
$\lambda _0 \in M_{\cal F}$ characterizing the physical state of the corpuscular component of $\cal S$.
Both subsystems, de Broglie wave and particle, are described by $(\|S>, \lambda _0)$
in a complete phase space ${\cal H}_{ext} \times M_{\cal F}$.
$\lambda _0$ determines the value of an arbitrary measurement on state $(\|S>, \lambda _0)$;
the result of an $A$ measurement is $\pi _A (\lambda _0)$ (some $a_i$), where
$\pi _A : M_{\cal F} \to M_A$ is the natural projection.
$\lambda _0$ is hidden, we can at most know precise values for a maximal family of compatible magnitudes,
a subfamily of $\cal F$. The observable relative frequencies for an arbitrary measurement on an ensemble $\|S>$
of states $(\|S>, \lambda _0)$, i.e., with common component $\|S>$ and all allowed values of
$\lambda _0$,
is obtained from the distribution of amplitude through Born's rule.
Born's rule \cite{Born} at ${\cal H}_{ext}$ is defined in two steps as follows. If we want to obtain a (perhaps formal)
distribution of probability $P(\lambda _1)$ for a subfamily ${\cal F}_1 \subset {\cal F}$,
${\cal F}_1 =$ $\{ D,E,G, \ldots \}$,
$\lambda _1 \in M_{{\cal F}_1}$,
we first project $\|S>$ onto the Hilbert space generated by vectors $\|\lambda _1>$,
$\tau _1 : {\cal H}_{ext} \to {\cal H}_1$,
$\tau _1(\|\lambda>)$ $=$ $\|\pi _1(\lambda )>$ $=$ $\|\lambda _1>$, i.e.,
\begin{equation}
\tau _1(\|(a_i,b_j,c_k,d_l, e_m, g_n, \ldots)>)
= \|(d_l, e_m, g_n, \ldots) >
\end{equation}
with $\lambda = (a_i,b_j,c_k,d_l, e_m, g_n, \ldots) \in M_{\cal F}$ and $\lambda _1 =$
$(d_l, e_m, g_n, \ldots) \in M_{{\cal F}_1}$.
For $\|S> = \sum _{\lambda } Z(\lambda ) \|\lambda >$, the projection $\|S_1>$ is
\begin{equation}
\|S_1> = \tau _1 (\|S>) = \sum _{\lambda _1} \left( \sum _{\pi _1(\lambda ) = \lambda _1} Z(\lambda )
\right) \|\lambda _1>
\end{equation}
The coefficient $Z(\lambda _1)$ of an elementary state
$\|\lambda _1>$ in ${\cal H}_1$ is the marginal amplitude obtained from $Z(\lambda )$,
\begin{equation}
Z(\lambda _1) = \sum _{\pi _1(\lambda ) = \lambda _1} Z(\lambda )
\end{equation}
In the abstract path integral, this is the union of disjoint families of virtual paths
characterized by different $\lambda$ in a larger family characterized by a common $\lambda _1$.
The second step is the standard $P(\lambda _1) = |Z(\lambda _1)|^2$. Obviously,
in general $|Z(\lambda _1)|^2 \neq$ $\sum _{\pi _1(\lambda ) = \lambda _1}$
$|Z(\lambda )|^2$, that is,
the probability distribution $P(\lambda _1)$ does not match the
marginal probability distribution $P'(\lambda _1) =$ $\sum _{\pi _1(\lambda ) = \lambda _1}$ $P(\lambda )$
obtained from the formal probability distribution $P(\lambda )$ $=|Z(\lambda )|^2$ associated to the amplitude distribution $Z(\lambda )$.
The former Born's rule can be interpreted as
a representation of the correlation (or interaction) between the corpuscular and wave like subsystems. Contextuality of QM
is in this formulation a consequence of interference, with different inter\-fe\-rence
results for different marginal amplitudes.
The interference that appears in marginal amplitudes can not be generically reproduced with marginals of a distribution of probability in $M_{\cal F}$, as Bell's theorems show.
The formal $P(\lambda)$ is not observable, no joint measurements of all magnitudes can be consistently performed. The unobservable $P(\lambda )$ does not reproduce observable probabilities through marginal probabilities. For example, in the two slit experiment the diffraction pattern is not obtained from a sum of probability distributions for each individual slit; interference, or sum of wave like degrees of freedom, is a more adequate analogy. Generically, an observable $P(\lambda _1)$ distribution is calculated through the sum of wave like degrees of freedom in the marginal amplitude and Born's rule. According to the path integral point of view, all amplitudes are calculated in a strictly causal way, through
integral along virtual paths inside the past light cone.
\subsection{Correspondence}
For a quantum system $\cal S$, let ${\cal F}$ be a family of physical magnitudes,
and ${\cal H}_{ext}$ the associated extended Hilbert space, generated by the basis
$\{ \|\lambda > \}$. ${\cal H}_{QM}$ will denote the standard Hilbert space,
orthodox quantum phase space of $\cal S$. For each
orthonormal basis $\{ |\lambda _1> \}$ in ${\cal H}_{QM}$ of common eigenvectors of a
maximal family of compatible operators in a subfamily ${\cal F}_1 \subset {\cal F }$, we describe next the mathematical conditions for
a co\-rres\-pon\-dence between vectors $\|S> \in {\cal H}_{ext}$ and
$|S> \in {\cal H}_{QM}$ representing the same physical state (or ensemble) of $\cal S$.
The most direct (and strongest) conditions are simply the equations
($\pi _1(\lambda ) = \lambda _1$)
\begin{equation}
\tau _1 (\|\lambda >) \equiv \|\lambda _1> = |\lambda _1>
\quad
\tau _1 (\|S>) = |S>
\end{equation}
Marginal amplitudes of $Z(\lambda )$ will match $z(\lambda _1)$,
for $\|S> = \sum _{\lambda } Z(\lambda ) \|\lambda >$,
$|S> = \sum _{\lambda _1} z(\lambda _1) |\lambda _1>$, i.e.,
$\sum _{\pi _1(\lambda ) = \lambda _1} Z(\lambda ) = z(\lambda _1)$.
However, the observable properties of the physical state are determined by the ray
$[|S>] = \{ c |S> \}$ ($c$ arbitrary complex numbers), so that it is enough that rays
$[\tau _1(\|S>)]$ and $[|S>]$
coincide. The corresponding equations will be projective ones, which I will denote
with the symbol $::$. Maintaining the identification $\|\lambda _1> = |\lambda _1>$
we have
\begin{equation}
\sum _{\pi _1(\lambda ) = \lambda _1} Z(\lambda ) :: z(\lambda _1)
\end{equation}
The mildest way of the correspondence takes into account the freedom in phase
that exists in the definition of a basis of eigenvectors,
even after normalisation. We could state, for each subfamily ${\cal F}_1$
and each one dimensional eigenspace $[|\lambda _1>]$,
a relation
\begin{equation}
\|\lambda _1 > = e^ {i\theta (\lambda _1)}|\lambda _1>
\end{equation}
which determines the projective equation
\begin{equation}
|\sum _{\pi _1(\lambda ) = \lambda _1} Z(\lambda )| :: |z(\lambda _1)|
\end{equation}
These equations should be solved for the $Z(\lambda)$ (defining $\|S>$),
given the $z(\lambda _1)$, $z(\lambda _2)$, $z(\lambda _3)$, \ldots of $|S>$ for all
maximal subfamilies ${\cal F}_1$, ${\cal F}_2$,${\cal F}_3$, \ldots $\subset {\cal F}$ of compatible magnitudes \footnote{Equivalently, given
$z(\lambda _1)$ and the
change of bases of eigenvectors in ${\cal H}_{QM}$
for all maximal compatible subfamilies ${\cal F}_2$, ${\cal F}_3$, \ldots of the considered physical magnitudes ${\cal F}$.}.
The abstract path integral suggest there could be solution in a generic physical
case; I do not have proof of that. Equations are linear (projective),
and obviously no positivity condition appears, amplitudes are not even real numbers
\footnote{Wigner's quasiprobability distribution \cite{Wigner} in the phase space of a point particle is an example of partial solution for the analogous problem of existence of a distribution of probability in an extended phase space, with hidden variables. The positivity requirement for probabilities can not be generically fulfilled.}.
A correspondence for the dynamical operators $A^d$ would also be needed.
In particular, an extended Hamiltonian in ${\cal H}_{ext}$ should
determine a dynamics $\|S>(t)$ compatible with the standard $|S>(t)$
through the previous correspondence of states.
There is not a correspondence for the corpuscular variables
$\lambda _0$, which do not exist in the orthodox theory.
We could incorporate hidden $\pi _1(\lambda _0)$ labels (for each
set ${\cal F}_1$ of compatible magnitudes) to the interpretation of the
standard theory, without observable consequences.
An interesting property of this formulation is
the classical limit. Classical limit for an elementary particle
has no physical relevance, Plank's constant can not be driven to $0$,
and quantum effects from the wave component are unavoidably relevant. The relevant classical limit applies to complex systems, with many components. For them,
only coarse--grained, global, macroscopic variables are observable. Classical
additive state variables of the corpuscular component survive in the classical limit, while
wave variables become irrelevant and we can apply $\hbar \to 0$. In particular, only corpuscular variables are observable in the interaction with a macroscopic system as a measurement apparatus.
The quantum Hamiltonian $H_0$ for $\lambda _0(t)$ evolution can be
rewritten as $H_0 = H_{class} + V_{QM}$, with $H_{class}$ the classical Hamiltonian.
$V_{QM}$ is the (possibly stochastic) interaction term (quantum potential in Bohm's formulation \cite{Bohm})
between corpuscular and wave subsystems,
and also disappears in the classical limit.
Quantum interaction between particle and vacuum wave is out of reach of observation, we can not
analyse its deterministic or genuinely probabilistic character. Therefore, the observable evolution must be described in a probabilistic formulation, that incorporates wave like
effects through amplitudes and Born's rule. In the classical limit, once these effects become negligible for
global state variables, we get a deterministic theory.
An observable
ingredient that does not exist in the orthodox theory is the de Broglie wave subsystem;
recall it was introduced in order to preserve the ARP in indirect measurements, when particle and
detector in different spatial regions can not interact, if we maintain the hypothesis of locality.
\section{Composite system}
Let $\cal S$ be a composite quantum system, with components (subsystems) ${\cal S}_I$ and
${\cal S}_{II}$, $\cal F$ a family of physical magnitudes of $\cal S$,
${\cal F}_I$ magnitudes specific of subsystem ${\cal S}_I$, ${\cal F}_{II}$ of
${\cal S}_{II}$,
and ${\cal F}_{int}$ magnitudes, as a potential of interaction, defined on the composite.
As before, a Hilbert space ${\cal H}_{ext}$ is defined through
an orthonormal basis of vectors $\|\lambda> =$ $\|\lambda _I>$ $\|\lambda _{II}>$
$\|\lambda _{int}>$, $\lambda _I \in M_{{\cal F}_I}$, etc.
For example, two independent systems with vectors of state
\begin{equation}
\|S_I> = \sum _{\lambda _I} Z_I(\lambda _I) \|\lambda _I>
\quad
\|S_{II}> = \sum _{\lambda _{II}} Z_{II}(\lambda _{II}) \|\lambda _{II}>
\end{equation}
define a state of the composite
\begin{equation}
\|S> = \sum _{\lambda = (\lambda _I, \lambda _{II}}
Z_I(\lambda _I) Z_{II}(\lambda _{II}) \|\lambda _I> \|\lambda _{II}>
\end{equation}
In general, when both subsystems interact the state of the composite will not be a direct product, but some
\begin{equation}
\|S> = \sum _{\lambda } Z(\lambda ) \|\lambda>
\end{equation}
in which additional magnitudes of interaction could be taken into account, and appear in
$\lambda$.
If $A_I$ and $B_{II}$ are a maximal family of compatible operators in ${\cal F}_I$ and ${\cal F}_{II}$ respectively
(magnitudes of a system trivially commute with magnitudes of the other), the projection
of $\|S>$ onto ${\cal H}_{A_I} \times {\cal H}_{B_{II}}$ is
\begin{eqnarray}
\tau _{A_IB_{II}}(\|S>) =
\sum _{i,j} \left(
\sum _{\pi _{A_I}(\lambda ) = a_{Ii}, \pi _{B_{II}}(\lambda ) = b_{IIj}}
\right)
\|a_{Ii}>\|b_{IIj}> = \\
\sum _{i,j} Z(a_{Ii},b_{IIj}) \|a_{Ii}>\|b_{IIj}>
\end{eqnarray}
with $Z(a_{Ii},b_{IIj})$ the corresponding marginal amplitudes.
Correspondence with the standard formalism will be fulfilled if the previous state
is an equivalent representation of the orthodox $|S> = \sum _{i,j}$ $c_{ij}$ $|a_{Ii}>$
$|b_{IIj}>$.
A state is said to be entangled if $c_{ij}$ does not factorize, or in the
extended formalism, if $Z(\lambda )$ does not factorize.
Relative frequencies are obtained as usual, $P(a_{Ii},b_{IIj}) = |c_{ij}|^2$ in the standard
formulation when $|S>$ is normalized, and
$P(a_{Ii},b_{IIj})$ $::$ $|Z(a_{Ii},b_{IIj})|^2$ in the alternative formulation.
Obviously, $P(a_{Ii})$ is obtained as a marginal probability
$P(a_{Ii}) = \sum _j P(a_{Ii},b_{IIj})$, and it is independent of the
chosen magnitude in the second system, e.g.
$P(a_{Ii}) = \sum _k P(a_{Ii},c_{IIk})$ for a magnitude (more precisely, maximal family)
$C_{II}$.
The (extended) state of system $\cal S$ is determined by some vector $\|S>$ and
label $\lambda _0 = $ $(\lambda _{I0}, \lambda _{II0})$, if we ignore for simplicity
magnitudes ${\cal F}_{int}$, for example because after interaction both subsystems are far apart. $\lambda _0$ determines the result of an arbitrary measurement, or pair of independent measurements on each subsystem. When measuring system ${\cal S}_I$ its
new label $\lambda _{I0}'$ will change (magnitudes non commuting with the one measured will
evolve along the measurement interaction), but this does not modify
the label $\lambda _{II0}$, because all magnitudes in ${\cal F}_{II}$ commute with the
measured magnitude in ${\cal F}_I$. With regards to the
relative frequencies, as we said $P(b_{IIj})$ is independent of
magnitude $A_I$ measured, although there will be in general a correlation
in $P(a_{Ii},b_{IIj})$.
If we understand the amplitudes $Z(\lambda )$ as result of a relativistic path integral calculation,
virtual paths characterized by $\lambda = (\lambda _I, \lambda _{II})$
are contained in the past light cones of events corresponding to $\lambda _I$
and $\lambda _{II}$, and in the shared past region
of both there will be contributions of interaction terms
in the
action integral. This is the origin of correlation. After both subsystems
separate, do not interact, the additional contributions along each individual path
will not modify the established correlation, although obviously they can
generate individual evolution of each subsystem. That is, all correlation information in $Z(\lambda)$ has a causal origin in the path integral formalism.
Given a magnitude $A_I$ of ${\cal S}_I$ and two non commuting magnitudes
$B_{II}$ and $C_{II}$ of ${\cal S}_{II}$ we can calculate
the marginal $Z(a_{Ii},$ $b_{IIj},$ $c_{IIk})$, and a formal
$P(a_{Ii},$ $b_{IIj},$ $c_{IIk})$. Again, the marginal $P'(a_{Ii},b_{IIj})$
$= \sum _k P(a_{Ii},b_{IIj},c_{IIk})$ will not match generically
the observable $P(a_{Ii},b_{IIj}$, because of interference in the marginal amplitude
$Z(a_{Ii},b_{IIj}) =$ $\sum _k$ $Z(a_{Ii},b_{IIj},c_{IIk})$.
Let us consider a correlated magnitude $C_T = C_I + C_{II}$, with value $c_T$, as a consequence of a
past interaction. The label $\lambda _0$ (corpuscular degrees of freedom) will fulfil the equation
\begin{equation}
\pi _{C_T}(\lambda _0) = c_T = \pi _{C_I}(\lambda _0) +
\pi _{C_{II}}(\lambda _0) = c_{Ik} + c_{IIk}
\end{equation}
so that perfect correlation appears in a measurement of
magnitudes $C_I$ and $C_{II}$ on the entangled pair.
A subset $M_{corr} \subset M_{\cal F}$ determines the correlated values.
Similarly, the vector of state (wave degrees of free\-dom) fulfils
$C_T\|S>$ $=$ $c_T\|S>$. If $\{C_I,$ $C_{II}\}$ is a maximal family of
compatible magnitudes, the projected state
\begin{equation}
\tau _{C_IC_{II}}(\|S>) =
\sum _{k,l} \left(\sum _{\pi _{C_I}(\lambda ) = c_{Ik},
\pi _{C_{II}}(\lambda ) = c_{IIl}}
Z(\lambda )\right)
\|c_{Ik}>\|c_{IIl}>
\end{equation}
defines, as usual, the marginal amplitudes $Z(c_{Ik},c_{IIl})$, which vanish of
$c_{Ik} + c_{IIl} \neq c_T$.
However,
not all terms $Z(\lambda )$ with $\lambda \in M_{\cal F}/M_{corr}$ necessarily vanish, it is enough that
interference in the marginal $Z(c_{Ik},c_{IIl})$ is destructive. That is, $\lambda _0 \in
M_{corr}$, but parameter $\lambda$ in the sum defining $\|S>$ belongs to
$M_{\cal F}$.
As an example, in the two slit experiment wave amplitudes
coming from both slits do not vanish in a zero of the diffraction pattern,
and a destructive interference of both determines the null probability density there.
A correspondence with the standard state $|S> =
\sum _k z_k |c_{Ik}>|c_{IIk}>$ ($c_{Ik} + c_{IIk} = c_T$) can now be established,
$|z_k| :: |Z(c_{Ik},c_{IIk})|$.
With another correlated magnitude $D_T = D_I + D_{II}$, with value $d_T$, we can
define the marginals
\begin{equation}
Z(c_{Ik},c_{IIl},d_{Im},d_{IIn}) \equiv Z_{klmn}
\end{equation}
as usual, and formal, unobservable, probability distributions $P(c_{Ik},c_{IIl},d_{Im},$ $d_{IIn})$ whose marginal probabilities do not match those obtained through Born's rule.
As pointed out before, $Z_{klmn}$ does not vanish generically
for $c_{Ik} + c_{IIl} \neq c_T$ or $d_{Im} + d_{IIm} \neq d_T$. Only the marginals, used in Born's rule to obtain observable re\-la\-tive frequencies,
$Z(c_{Ik},c_{IIl})$ $=Z_{kl} =$ $\sum _{mn} Z_{klmn} $ will vanish
for $c_{Ik} + c_{IIl} \neq c_T$. Therefore, in a marginal
$Z(c_{Ik},d_{IIn}) =$ $\sum _{lm}Z_{klmn}$ there will be contributions of
terms $Z_{klmn}$ where $c_{Ik} + c_{IIl} \neq c_T$. We can compare this generic
case with the two slit experiment, corpuscular variables $\lambda _0$
have a definite, but hidden, value (e.g., left or right slit) while
wave degrees of freedom, the amplitudes, have ge\-ne\-ri\-ca\-lly all possible components, although
they can interfere destructively in some marginals. In next section a particularly
relevant case, pairs of spin $1/2$ entangled particles with total null spin,
is analysed.
The double role of amplitudes, as source of wave like effects and in Born's rule for
observable relative frequencies, is behind the quantumness of QM in the proposed formulation, instead of a non local wave function collapse. There is no essential
distinction between entanglement and the two slit experiment.
\section{Applications}
\subsection{Two slit experiment}
The relevant variables in the two slit experiment are the slit variable $S$, with value
$L$ or $R$, and the position at the final screen $\bf R$, with values ${\bf r}_i$.
Slit and final position do not commute. Paths can be grouped in families
\footnote{Magnitudes associated to a path
are not necessarily values at its final point. Virtual paths going through $L$ and $R$ slits
are disjoint families.}
$[path](L,{\bf r}_i)$ and $[path](R,{\bf r}_i)$.
The extended Hilbert space is generated by elementary states $\|L,{\bf r}_i>$
and $\|R,{\bf r}_i>$. The vector of state is
\begin{equation}
\|S> = \sum _{i} Z(L, {\bf r}_i) \|L,{\bf r}_i> +
\sum _{i} Z(R, {\bf r}_i) \|R,{\bf r}_i>
\end{equation}
with projections
\begin{equation}
\tau _S (\|S>) = \frac {1}{\sqrt{2}} (\|L> + \|R>)
\end{equation}
and
\begin{equation}
\tau _{\bf R} (\|S>) = \sum _i (Z(L, {\bf r}_i) + Z(R, {\bf r}_i))\|{\bf r}_i>
\end{equation}
This one is also the orthodox vector of state in the
position coordinates representation at the final screen.
The formal distribution of probability
\begin{equation}
P(L,{\bf r}_i)=|Z(L,{\bf r}_i)|^2 \quad
P(R,{\bf r}_i)=|Z(R,{\bf r}_i)|^2
\end{equation}
is not observable. The marginal
\begin{equation}
Z({\bf r}_i)=Z(L, {\bf r}_i) + Z(R, {\bf r}_i)
\end{equation}
determines the observable diffraction
pattern $P({\bf r}_i)$ $=$ $|Z({\bf r}_i)|^2$.
Let us suppose that an external system interacts with our system of interest (particle plus de Broglie wave) at the $R$ slit. For simplicity,
let us consider that the new amplitude $Z'(R, {\bf r}_i)$ is similar in modulus to
the unperturbed $Z(R, {\bf r}_i)$. However, there will be some phase shift
$Z'(R, {\bf r}_i, \phi) \simeq e^{i\phi} Z(R, {\bf r}_i)$. If we use, for example, a beam of coherent photons, and the additional system is a phase plate of known phase shift, we will obtain a
displaced diffraction pattern.
If the additional system is a (more or less complex) measurement apparatus, the phase shift will be unknown. A statistical average on the phase (e.g., with uniform distribution)
determines a total distribution of relative frequencies
\begin{equation}
P'({\bf r}_i) = \frac {1}{2\pi}\int _{-\pi}^{\pi}d\phi
|Z'(R, {\bf r}_i, \phi) + Z(L, {\bf r}_i)|^2 \simeq
|Z(R, {\bf r}_i)|^2 + |Z(L, {\bf r}_i)|^2
\end{equation}
and the diffraction pattern disappears. We can observe now,
by correlation with the positive/negative result of measurement in slit $R$,
the marginal distributions $P'(R,{\bf r}_i)$ and $P(L,{\bf r}_i)$.
In practise, we can apply the projection rule according to the result of measurement
at $R$ slit.
A stochastic phase shift at measurement interactions determines
the projection of state as a practical
rule.
\subsection{Spin variables}
Let us consider a spin $1/2$ particle. The most general family of spin variables
will contain spin (up or down) in an arbitrary number of directions
${\bf n}_j$, unit vectors in space. ${\cal F} = \{ S_1, S_2, \ldots \}$, with
$M_{S_j} = \{ +, - \}$ and $M_{\cal F} = \{ (s_1, s_2, \ldots ) \}$; $s_j$ is $+$ for spin up and $-$ for spin down in direction ${\bf n}_j$ \footnote{Direction $-{\bf n}_j$ is redundant, because $S_{-{\bf n}_j} $ $= - S_{{\bf n}_j} $.}.
The extended Hilbert space ${\cal H}_{ext}$ is generated by
elementary states $\|\lambda >$, $\lambda = (s_1, s_2, \ldots ) \in M_{\cal F}$. Orthodox Hilbert space is a two dimensional one ${\cal H}_{QM}$, where we can use each $\{|+_j>,|-_j>\}$ basis of spin up/down states in direction ${\bf n}_j$.
Let us associate a fix ``path'' amplitude to each spin value, $s_j e^{i\theta _j}$ for coplanar directions
or $s_j N_j$ for directions in space, with $N_j$ a quaternion number (with null real part) $N_j = n_j^x I + n_j^y J + n_j^z K$ associated to the
vector ${\bf n}_j = n_j^x {\bf i} + n_j^y {\bf j} + n_j^z {\bf k}$ \cite{mio}.
The amplitude associated to an elementary state $\|\lambda>$ will be
\begin{equation}
Z(\lambda ) = \sum _j s_j N_j
\end{equation}
A states with known spin $s_1$ in direction ${\bf n}_1$ is
\begin{equation}
\|s_1> = \sum _{\pi _1(\lambda ) = s_1} Z(\lambda ) \|\lambda >
\end{equation}
Projection over the two dimensional Hilbert space for operator $S_1$, when using the basis $\{\|+_1>, \|-_1>\}$,
trivially reproduces the orthodox $|s_1>$: in the marginal amplitude, terms $s_j N_j$
and $-s_j N_j$ for $j>1$ cancel out, and a global factor can be ignored.
Projection over the Hilbert space for operator $S_2$, using the basis $\{\|+_2>, \|-_2>\}$, gives
\begin{equation}
\tau _2 (\|s_1>) = {\cal N} \left( (s_1 N_1 + N_2) \|+_2> +
(s_1 N_1 - N_2) \|-_2> \right)
\end{equation}
with $\cal N$ a global factor. There is a correspondence, when appropriate
phases are introduced in the correspondence of vectors,
between $\tau _2 (\|s_1>)$ in the two dimensional quaternion Hilbert space
${\cal H}_{S_2} = < \|+_2>, \|-_2> >$
and the standard expression
in the orthodox complex Hilbert space.
All orthodox spin states of an individual particle are eigenstates of the spin operator
in some spatial direction. The former correspondence allows to
represent them in the extended phase space. On the other hand, the label
$\lambda _0$ in the state $(\|s_1>, \lambda _0)$ will project onto a label $s_{1}$ (known for state $\|s_1>$)
or $s_{2}$ (unknown at the ensemble state $(s_1 N_1 + N_2) \|+_2> +$
$(s_1 N_1 - N_2) \|-_2>$). The (hidden) label determines the value of spin measurement
in arbitrary direction ${\bf n}_j$, $s_j = \pi _j(\lambda _0)$.
Born's rule in two steps for the extended formalism reproduces the standard results.
Operators $S_j$ commute. For dynamical purposes, additional operators
$S_j^d$ fulfilling the standard $[S_x^d, S_y^d]$ $= i \hbar S_z^d$ should be considered.
Let us consider a composite system of two spin $1/2$ particles
${\cal S}^a$ and ${\cal S}^b$ in a total null spin state.
The label $\lambda _0$ will fulfil $\pi _j(\lambda _{a0})$ $+$ $\pi _j(\lambda _{b0}) = 0$,
i.e., perfect correlation $s_j^a + s_j^b = 0$, determining a subset $M_{corr} \subset M_{\cal F}$;
the total family of operators is ${\cal F} = {\cal F}^a \cup {\cal F}^b$,
${\cal F}^a = \{ S^a_1, S^a_2, \ldots \}$ and ${\cal F}^b = \{ S^b_1, S^b_2, \ldots \}$.
The vector of state for the composite will have the form
\begin{equation}
\|S> = \sum _{\lambda \in M_{\cal F}} Z_T(\lambda ) \|\lambda ^a>\|\lambda ^b>
\end{equation}
Notice that $\lambda$ is not restricted to $M_{corr}$, all we have to
impose is that marginals $Z(s_1^a,s_1^b=s_1^a) = 0$, through destructive interference.
A solution for the amplitude distribution is
\begin{equation}
Z_T(\lambda ) = Z_T(\lambda ^a, \lambda ^b ) =
Z(\lambda ^a) - Z(\lambda ^b)
\end{equation}
where $Z(\lambda )$ is as before $Z(\lambda ) = \sum _j s_j N_j$.
Let us consider two directions ${\bf n}_1$ and ${\bf n}_2$, with variables $(s^a_1, s^a_2, s^b_1, s^b_2)$.
The marginal amplitude becomes
\begin{equation}
Z_{(1,2)^a,(1,2)^b}(s^a_1, s^a_2, s^b_1, s^b_2) =
(s^a_1-s^b_1) N_1 + (s^a_2-s^b_2)N_2
\end{equation}
up to a total factor; all coefficients of $N_j$ for $j>2$ vanish.
Therefore, $Z_{(1,2)^a,(1,2)^b}$ $(s^a_1, s^a_2, s^b_1 = s^a_1, s^b_2)$ $=$
$(s^a_2-s^b_2)N_2$ $\neq 0$. However, the observable amplitudes
of measurement in direction ${\bf n}_1$ for both particles is
\begin{equation}
Z_{1^a,1^b}(s^a_1,s^b_1) =
(s^a_1-s^b_1) N_1
\end{equation}
We get $Z_{1^a,1^b}(s^a_1,s^b_1= s^a_1) = 0$, $Z_{1^a,1^b}(s^a_1,s^b_1= -s^a_1)$
$= 2s^a_1 N_1$,
and a probability distribution
$P(s_1,s_1) = 0$, $P(s_1,-s_1) = 1/2$.
Similarly, the observable amplitude $Z_{1^a,2^b}$ for measurement $S_1$ of particle $a$ and $S_2$ of particle $b$ is
\begin{equation}
Z_{1^a,2^b}(s^a_1,s^b_2) = s^a_1 N_1 - s^b_2 N_2
\end{equation}
with the quantum associated probability distribution
$P(s^a_1,s^b_2) ::$ $|s^a_1 N_1 - s^b_2 N_2|^2$ $=$ $2(1 - s^a_1 s^b_2\,\, {\bf n}_1 \cdot {\bf n}_2)$
\footnote{$N_1^* = - N_1$
for quaternions without real part, and $N_1^* N_2 =$
$- {\bf n}_1 \cdot {\bf n}_2$ $ - {\bf n}_1 \times {\bf n}_2$;
in the former expression, the dot product is the real part and the cross product
the imaginary part of a quaternion.}.
If we calculate the marginal amplitude for direction $S_1$ of particle $a$ and
directions $S_2$ and $S_3$ of particle $b$
\begin{equation}
Z_{1^a,(2,3)^b}(s^a_1,(s_2,s_3)^b) = s^a_1 N_1 - (s^b_2 N_2 + s^b_3 N_3)
\end{equation}
we can define a formal distribution of probability
\begin{equation}
P(s^a_1,(s_2,s_3)^b) =
|Z_{1^a,(2,3)^b}(s^a_1,(s_2,s_3)^b)|^2
\end{equation}
The marginal probability
$P'(s_1,s_2)$ $=$ $P(s^a_1,(s_2,+_3)^b)$ $+$ $P(s^a_1,(s_2,-_3)^b)$ does not match the former one, because of interference
when applying Born's rule in two steps,
\begin{equation}
Z_{1^a,2^b}(s^a_1,s^b_2) = Z_{1^a,(2,3)^b}(s^a_1,(s_2,+_3)^b) +
Z_{1^a,(2,3)^b}(s^a_1,(s_2,-_3)^b)
\end{equation}
and
\begin{equation}
P(s^a_1,s^b_2) :: |Z_{1^a,(2,3)^b}(s^a_1,(s_2,+_3)^b) +
Z_{1^a,(2,3)^b}(s^a_1,(s_2,-_3)^b)|^2
\end{equation}
differs, even projectively, from
\begin{equation}
|Z_{1^a,(2,3)^b}(s^a_1,(s_2,+_3)^b)|^2 + |Z_{1^a,(2,3)^b}(s^a_1,(s_2,-_3)^b)|^2
\end{equation}
There is a full analogy with interference in the two slit experiment,
for exam\-ple if we identify the third magnitude $s_3 \in \{+, -\}$ with the slit $\{L,R\}$. Bell's inequalities
prove that there is no distribution of probability in a space with hidden variables
whose marginal probabilities reproduce the quantum result. What we have here is a distribution of amplitude in the extended space of hidden varia\-bles; through the
defined correspondence (marginals, interference, Born's rule) it reproduces the orthodox quantum phase space
representation.
Corpuscular
variables $\lambda _0$ fulfil the perfect correlation condition, and determine the result of arbitrary measurements in a correlated pair of particles
$(\|S>,$ $(\lambda _{a0},$ $\lambda _{b0}))$. The distribution
of probability for observable magnitudes depends on the wave degrees of freedom $\|S>$
(common for an ensemble), the distribution of
amplitudes. Born's rule in two steps gives way to interference, wave like phenomenon that does not appear in a marginal probability.
The distribution $Z(\lambda)$ of the entangled system, as well as the corpuscular
variable $\lambda _0$, are fixed from the generation event of the two entangled particles.
No non local phenomenon is invoked
in the former mathematical description.
\section{Summary and Outlook}
A formalism of non relativistic Quantum Mechanics in which elementary states
have joint precise values of non commuting magnitudes has been developed in order to
restore the action reaction principle, which is
contradictory with the projection of state.
An abstract path integral formalism, with families of virtual paths more res\-tric\-tive than the
standard one, suggest that a correspondence with all predictions of the orthodox theory could
exist. The two slit experiment and the phase space of spin variables have been formulated in the extended phase space.
Contextuality and non locality of Quantum Mechanics appear here
as interference properties of wave like degrees of freedom. The mysterious
double role of amplitudes, at interference and determining probabilities,
is the source of these non classical properties of quantum systems.
An accompanying, wave like subsystem is predicted
when particle and detector are spatially separated in indirect measurements, in order to preserve
locality of interactions and the action reaction principle. An appropriately designed detector
could show observable reactions; they would be assigned either to a non local interaction
between the corpuscular component and the detector or to
a local interaction between a de Broglie wave component and detector.
\section{Acknowledgements}
Financial support from research project MAT2011-22719 is acknow\-ledged. I also
kindly ack\-now\-led\-ge helpful comments from members of the audience in both
seminars at Zaragoza and Valladolid Universities, where I presented some
conclusions of this research in April and June 2015 respectively.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,379 |
{"url":"http:\/\/riofriospacetime.blogspot.com\/2008\/05\/asteroid-beckons.html","text":"## Tuesday, May 06, 2008\n\n### An Asteroid Beckons\n\nFrom the UK Guardian May 7: NASA engineers have identified asteroid 2000SG344 as a potential landing site for astronauts. A study to be released next month suggests a 3-6 month mission with 2 weeks spent on the surface. When this 40 meter object was discovered in 2000, it was briefly considered a danger to Earth. 2000SG344's orbit close to Earth makes it a tempting target, one of hundreds of new worlds.\n\nBode's Law suggests that another planet should orbit between Mars and Jupiter. When Ceres, the largest asteroid, was discovered it was first thought to be the missing planet. In subsequent years hundreds more asteroids have been found. Ceres was long classified as an asteroid but recently has been promoted to minor planet. The DAWN spacecraft will rendezvous with asteroid Vesta in 2011 and Ceres in 2015.\n\nWe are fortunate to have found meteorites from Vesta. The Camel Donga meteorite seen below now sits in the Hall of Meteorites and New York's Museum of Natural History. Science fiction writers have long dreamed about mining the asteroids for metal. Some theories suggest that Ceres is largely made of water, and could contain even more water than Earth. More than just big rocks, the asteroids are new worlds that could even harbour life.\n\nObservations suggest that Ceres is differentiated into core and mantle, which would mean that it was melted early in its history. Ultraviolet observations have found water vapour near Ceres' North Pole. How such a small body could be heated is a complete mystery. The asteroid's 10^{21} kg mass could easily hide a small Black Hole.\n\nOne can imagine the adventure of landing on this new world. An Orion spacecraft launched on an Ares V booster would carry along a small hab module. Since 2000SG344 very likely rotates, the Orion would have to line up with the asteroid's spin axis and match rotation. This maneuver would be similiar to the Orion docking with a space station in 2001: A SPACE ODYSSEY. Since the asteroid's gravitational field is negligible, the ship would fire pitons to contact the surface. The final touchdown might be accomplished by reeling the spacecraft in.\n\nNew presidents like to impose their own Vision, and an asteroid mission would be a lasting legacy. After a return to the Moon, an asteroid mission will be far simpler than reaching Mars. In addition to the adventure of landing on a new world, the mission could easily be justified. Much needs to be learned if we are to prevent another big impact. If we do not venture into the asteroids, one day they will venture into us.\n\nLabels: ,\n\nYoyo said...\n\nLouise wrote: \"Ceres was long classified as an asteroid but recently has been promoted to minor planet. (It's bigger than Pluto.)\"\n\nNot by a long shot - Pluto's mass is ~10^(22) kg, Ceres' mass is ~9x10^(20) kg.\n\nPerhaps you were thinking of Eris?\n\n8:27 PM\nL. Riofrio said...\n\nFixed that in proofreading, thanks. You earlier had a question about metrics, (-1, -1, -1, -1) vs. other metrics. Einstein used this metric to get (ds)^2 = - (dX1)^2 - (dX2)^2 - (dX3)^2 - (dX4)^2 when dX4 = ict. This \"imaginary\" factor ict allowed him to reach his conclusions about Relativity, but has been abandoned by modern books. You would enjoy Einstein's books, which are surprisingly easy to read.\n\n4:42 AM\nYoyo said...\n\nI have no problem with using $x^4 = ict$ as a mathematical convenience to make the metric look Euclidean, but this doesn't change the signature to $(-1, -1, -1, -1)$. The signature is still Lorentzian. The notion of \"signature\" is only defined for metrics on real vector spaces. It's undefined for complex spaces, since there you can make the \"signature\" anything you like by inserting factors of $i$ in the basis vectors.\n\nI can show that this metric is Lorentzian when restricted to real vectors by choosing the real basis ${\\partial_k, i \\partial_4 }$, then $g(\\partial_k, \\partial_k) = -1$ but $g(i \\partial_4, i\\partial_4) = +1$.\n\n1:45 PM\nYoyo said...\n\nWhile we're on the subject, I'd still like to know what metric components $g_{ij}$ you used to get the RHS of eq (4) here (the metric you pointed me to here doesn't appear to be the right one, since it doesn't have a $z$ or an $A$ term).\n\nWithout knowing what the original $g_{ij}$ was, having $r(t)$ is not very useful. How can I know what role $r(t)$ is supposed to play in the metric? It's not even possible do simple things like evaluate it on a pair of vectors $X$ and $Y$ ie. $g_{ij}X^iY^j$, or raise and lower indices, or calculate a geodesic without knowing what $g_{ij}$ is and where $r(t)$ appears in it.\n\nSo, could you please write down the form of the metric that is used to get the RHS of eq (4). What I'm after is something like\n\n$ds^2 = g_{ij} dx^i dx^j$\n\nwhere the terms $r$, $z$ and $A$ from eq (4) appear in their proper places.\n\n2:07 PM\nKea said...\n\nyoyo, in a cosmology that uses an hbar hierarchy, why would you expect classical GR to be applied universally?\n\n2:26 PM\nYoyo said...\n\nKea\n\nI don't expect GR to apply universally. I'm quite happy to contemplate the idea that it's an approximation to the \"true\" theory of gravity, quantum or otherwise, even with a varying $c$. It's just that in Louise's article below there are lots of terms left undefined, and I'm trying to establish what they are.\n\nMy question relates to Louise's derivation eqs (3)-(9) of the Einstein-Friedman equations. This appears to be using totally standard GR, no modifications inserted. What one usually does to derive these is\n\n1) Propose a simple form (ansatz) for the metric $g_{ij}dx^idx^j$ containing some unknown functions (such as $r$).\n\n2) Propose a form for the energy-momentum tensor (Louise uses pressure-free dust, so $T^{ij} = \\rho U^iU^j$).\n\n3) Apply the Einstein equations $R_{ij} - 1\/2Rg_{ij} = 8\\pi GT_{ij}$, yielding an equation for the unknown functions that appear in $g_{ij}^dx^idx^j$.\n\nFor the FRW-type metrics, it's usual to start with an ansatz for the most general metric for a homogeneous, isotropic spacetime. Louise doesn't give an ansatz for the $g_{ij} dx^i dy^j$ she's starting from (this undefined metric appears in the LHS of eq (4) here), so I don't see how to obtain the RHS from the LHS, since the RHS contains terms like $r$, $z$ and $A$ which are not defined. Now, when I compare her derivation to \"textbook\" derivations I can make an educated guess as to what she means by $r$, $z$ and $A$, but rather than putting words in her mouth I'd rather she tell me exactly what $g_{ij} dx^i dy^j$ she started with. Then I can see where $r = ct$ needs to be inserted to get the metric of her proposed solution.\n\n4:30 PM\nL. Riofrio said...\n\nThank you again, Kea.\n\nAnsatz?! Since my German isn't that great, consult somebody named Einstein. Equations (4) through (8) come from his book \"The Meaning of Relativity\" pages 117-118. I added equation (9) myself.\n\nEinstein would have said: those who insist on doing things the way they have been done before will never come up with anything original.\n\n6:35 PM\nYoyo said...\n\nThis comment has been removed by the author.\n\n11:52 PM\nYoyo said...\n\nThanks for that, I will look it up. Ansatz means something like \"educated guess\". Google Scholar returns 708,000 results for \"ansatz\". It's a perfectly cromulent word :)\n\nLouise wrote:\n\"Einstein would have said: those who insist on doing things the way they have been done before will never come up with anything original.\"\n\nHe might have. Since I can't read the minds of dead people I will respond with a quote from Richard Feynman: \"If you're doing an experiment, you should report everything that you think might make it invalid \u2014 not only what you think is right about it... Details that could throw doubt on your interpretation must be given, if you know them.\"\n\n11:54 PM\nStephen said...\n\nCeres was recently promoted to Dwarf Planet. It was demoted to Minor Planet in something like 1850. IMO, it's round, it orbits a star, it does not orbit some other planet, so it's a planet. But, i recognize that i'm not the IAU. I'm not even a memeber. But i don't like the IAU's definition much, and even less the process they used to get it. By the letter of the definition, we have no planets, since none have 'cleared their orbital region' (whatever that means). By the best explanation i've heard, we have one planet, Jupiter, which is the solar system's gravitational bully (other than the Sun). Everything is in orbital resonance with Jupiter. The Earth clearly needs a minor planet number. I suggest zero.\n\n10:00 AM\nStephen said...\n\nApophis briefly seemed to have a one in 45 chance of hitting Earth. 2000SG344 isn't Apophis. The Guardian article isn't clear about this. It's just sloppy.\n\nThe Guardian article also doesn't say if the Orion capsule will be radiation shielded, in case a CME comes their way. We had one come to Earth last week, and we're at solar minimum. And, the Sun has been doing practically NOTHING for months. Hardly any sunspots. Hardly any flares. No reason to buy an h-alpha solar scope comes to mind. If we can have CME induced aurora now, expect to encounter one on a 3-4 month long trip. It WILL happen. Even if i were 80, i would not sign up for an unshielded 3-4 month mission outside the Earth's magnetic bubble. I MIGHT sign up to go to the Moon, if after three days, i'd get into a good shelter there. If i were 80. I might get lucky.\n\n10:16 AM\nYoyo said...\n\nThis comment has been removed by the author.\n\n3:12 AM\nYoyo said...\n\nThis comment has been removed by the author.\n\n3:15 AM\nYoyo said...\n\nLouise\n\nI've looked up your page references for \"The Meaning of Relativity\" and would like to confirm the metric with you. The coordinates are $(x_1, x_2, x_3, x_4)$ where $x_4 = c t$ (note, not $ict$ this time - Einstein uses a real time coordinate in this section). In the form given in Eq (2) on pg 120\n\n$ds^2 = d{x_4}^2 - G^2(x_4)A^2(r)(d{x_1}^2 + d{x_2}^2 + d{x_3}^2)$\n\nwhere $A = (1+z r^2\/4)^{-1}$ for $r^2 = {x_1}^2 + {x_2}^2 + {x_3}^2$. The $G(x_4)$ corresponds to the $r(t)$ or $R(t)$ you use in your article. I run into a bit of a problem here, because $G$ was stated to be a function of $x_4$ (pg 117 just below eq (2a)), not $t$, whereas you explicitly use $t$ in your formula $R = ct$ and $GM = tc^3$. I think the only sensible way to interpret $R(x_4)$ is to use the fact that it should have the same value regardless of whether the coordinates are $(x_k, x_4)$ or $(x_k, c t)$, so I am assuming that $R(x_4) = x_4$ since $x_4 = ct$ is the way you would interpret that - let me know if this is not what you mean. So, replacing $G$ by $R$ and putting in your condition $z = 0$ which gives $A = 1$, your metric would be:\n\n$ds^2 = d{x_4}^2 - {x_4}^2 (d{x_1}^2 + d{x_2}^2 + d{x_3}^2)$\n\nIs this what you mean? If not, please tell me what it should be.\n\nAnother alternative would be to start from\n\n$ds^2 = c^2(t) d{t}^2 - R^2(t) (d{x}^2 + d{y}^2 + d{z}^2)$\n\nand just plug in your formulae directly to get\n\n$ds^2 = (GM)^{2\/3}t^{-2\/3}d{t}^2 - (GM)^{2\/3}t^{4\/3}(d{x}^2 + d{y}^2 + d{z}^2)$\n\n3:16 AM\nL. Riofrio said...\n\nFor yy: Not what I mean. Note that the G on p. 117-118 is different from the Newtonian gravitaional constant in GM=tc^3. Einstein states elsewhere (like in p. 80 of the same book) that x4 = ict. I recommend reading p. 80 for discussion of the difference between (-1, -1, -1, 1) and (-1, -1, -1, -1) metrics.\n\n9:18 AM\nYoyo said...\n\nThis comment has been removed by the author.\n\n1:14 PM\nYoyo said...\n\nLouise\n\nI realise the $G(x_4)$ on p117 is not Newton's constant. It has exactly the same role in the Einstein-Friedmann equations as your $R(t)$ does, but it is stated as being a function of $x_4$ rather than $t$. I only mentioned $GM = tc^3$ to emphasise the fact that you are using $t$ as a coordinate while Einstein was using $x_4 = ct$ as a coordinate.\n\nI also realise that the coordinate $x_4$ is used to mean $ict$ in most places, but in this section specifically Einstein switches to the real coordinate $x_4 = ct$. You can see this from the metric, since the minus sign is in there. The metric as given on pg120 is\n\n$ds^2 = d{x_4}^2 - G^2(x_4)A^2(r)(d{x_1}^2 + d{x_2}^2 + d{x_3}^2)$\n\nSee the minus sign? This metric is given explicitly with signature $(-1,-1,-1,+1)$. If you want to insist that $x_4 = ict$ then fine, but the metric won't have Lorentzian signature and it won't be possible (for example) to have real null (lightlike) vectors.\n\nOk, so if what I said above is not the correct metric, then what is? Once more, could you please state it in whatever coordinates you prefer?\n\n1:15 PM\nL. Riofrio said...\n\nOffice hour nearly over. Read Einstein and see me next week.\n\n5:37 PM\nYoyo said...\n\nThis comment has been removed by the author.\n\n8:01 PM\nYoyo said...\n\nLouise\n\nOn your blog you've repeatedly complained about being ignored by the physics community. Other people have berated your critics like Ethan for not making an effort to understand your work.\n\nI've put many hours of time into trying to follow your derivation and relate it to the references you've given. This is made very difficult by the fact that you use different symbols and coordinates than your sources, and you avoid answering simple questions about what those symbols mean. You've ignored multiple requests to simply state your metric, something which ought to be very straightforward.\n\nI've made a couple of guesses at what your metric is and you tell me they're wrong. OK, for the fourth time, I'm asking you to write down your metric. It's one line of mathematics I'm asking for here. You've expressed a lot of confidence in your theory - put your money where your mouth is.\n\n8:05 PM\nL. Riofrio said...\n\nEinstein also said that the definition of insanity is trying the same thing repeatedly and expecting a different result.\n\n7:11 PM","date":"2015-03-02 16:29:43","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\": 0.6444219350814819, \"perplexity\": 899.6597478229959}, \"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-2015-11\/segments\/1424936462898.92\/warc\/CC-MAIN-20150226074102-00169-ip-10-28-5-156.ec2.internal.warc.gz\"}"} | null | null |
The LazyCtrl Project
=================================
*A hybrid control plane design for large-scale cloud data centers.*
* Version: 0.21
* Authors: [Baohua Yang](mailto:baohyang@cn.ibm.com), [Kai Zheng](mailto:zhengkai@cn.ibm.com)
* Homepage: <https://github.com/yeasy/lazyctrl>
## Download
SSH:
```
git clone git@github.com:yeasy/lazyctrl.git
```
Https:
```
git clone https://github.com/yeasy/lazyctrl.git
```
## Organization
### Central Control Modules
The directory ```ccm``` contains essential centralized control modules that are responsible for handling flow-based centralized control, as well as local control group management, for LazyCtrl.
* Central controller: Our implementation of the central controller in LazyCtrl is based on the [Floodight](http://www.projectfloodlight.org/floodlight) project, which we call ```floodlight-lc```. We extend the basic [OpenFlow 1.0](http://archive.openflow.org/documents/openflow-spec-v1.0.0.pdf) protocol in ```floodlight-lc``` by introducing a function to support installing a new action *Encap* for packet encapsulation and forwarding at edge switches. Besides, the central controller in LazyCtrl is only in charge of inter-group traffic flows.
* Switch grouping management: Some daemons are provided to maintain switch grouping and updating in the network, as well as providing communication channels between the central controller and local control groups.
* The grouping process is based on [METIS](http://glaros.dtc.umn.edu/gkhome/metis/metis/overview) project (version 5.0.2) and the gpmetis binary is suggested to be compiled on your own platform.
### Local Control Modules
All the modules for local control in switch groups are contained under directory ```lcm```. Our implementation of flow switch (called ```openvswitch-lc```) is based on the [Open vSwitch](http://openvswitch.org) project and we extend Open vSwitch by implementing the *Encap* action. Besides, the *ovsd* module in ```openvswitch-lc``` will take the charge of maintaining L-FIB and G-FIB, while the *datapath* kernel module is also modified to provide essential forwarding functions.
We also implement some agent daemons at every flow switch to report its state to the central controller.
###Others
A test platform to test the performance of the grouping algorithm, workload and packet forwarding delay in LazyCtrl.
##Installation
###Basic Requirements:
* Servers supporting [OpenvSwitch](http://openvswitch.org), e.g., most Linux based servers.
* Servers supporting [Floodlight Controller](http://www.projectfloodlight.org/floodlight), e.g., most Linux based servers.
* Servers supporting [METIS](http://glaros.dtc.umn.edu/gkhome/metis/metis/overview), e.g., most Linux based servers.
* Physical switches that support IP multi-cast.
* Server (as controller) can login into the servers (as edge switches, with at least 2 separate network interfaces) running OpenvSwtich via ssh without authorization (Need to put the public authorization key previously).
###Sample Testbed:
* Three edge switch instances (in two local control groups) are connected through a physical switch (IP network), and are connected to the central controller.
* Each edge switch has connections to two subnets: control layer subnet and the datapath layer subnet.
* Edge switches can be installed at Linux based servers.
* Edge switches can login to the central controller via ssh without manual authorization.
* Deploy the local control modules at every edge switches following the [Installation Documents](lcm/openvswitch-lc/INSTALL) inside.
* Modify the address information as necessary. For example, the default control plane IP of edge switch #1 is ```192.168.57.10```, while for edge switch #2, the IP is ```192.168.57.11```. The datapath plane IP addresses are set to ```10.0.0.0/8``` subnet.
* Start the agent.sh daemon at each edge switch.
* Setup the central control modules at the central controller.
* Start floodlight-lc in the central controller to receive PACKET_IN msg, see [Floodlight Getting Started](http://www.projectfloodlight.org/getting-started/). Run ```ant; java -jar target/floodlight.jar``` if all dependencies are already satisfied.
* Start the collect_agent.sh daemon to collect upward statistics.
* Start the *groupManager* daemon to update the grouping.
* Test the deployment
* Capture IP multicast message at each edge switch, confirming that they are broadcasting the LCG location/statistics information.
* Ping between two hosts in the same LCG, it will be handled by the G-FIB, hence no openflow message is generated and forwarded to the controller. The latency will be similar to direct transfer.
* Ping between two hosts in different LCGs, openflow messages are generated and forwarded to the controller and the the central controller will handle it using the C-LIB. The latency would be much higher as the edge switch has to talk with the central controller. Besides, the *Encap* message will be sent to the edge switch.
## How does it work?
Basically, the ```floodlight-lc``` + ```openvswitch-lc``` cooperate as the basic control-datapath model in SDN. Based on them, we enhanced ```openvswitch-lc``` + agent to behave as local control groups, while ```floodlight-lc``` + daemons work as the central controller. More details, e.g., the group maintain algorithm and the implementation technical issues, are discussed in the paper.
##Documentation
Besides the readme files, you can easily make the API document with python/java/c doc tools. Plenty comments in the code would also be helpful.
##Support
Feel free to email to the authors. The project is planned to be community-supported soon.
##Contributing
There are still numbers of components available to optimize. For example, to build more robust message switching system.
Everyone is encouraged to download the code, examine it, modify it, and submit bug reports, bug fixes, feature requests, new features and other issues and pull requests. Thanks to everyone who has contributed to the project.
| {
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from .prod import * # NOQA
# Hosts/domain names that are valid for this site; required if DEBUG is False
# See https://docs.djangoproject.com/en/1.7/ref/settings/#allowed-hosts
ALLOWED_HOSTS = ['*']
USE_X_FORWARDED_HOST = False
DATABASES = {
'default': {
# Add 'postgresql_psycopg2', 'mysql', 'sqlite3' or 'oracle'.
# 'ENGINE': 'django.db.backends.postgresql_psycopg2',
'ENGINE': 'django.contrib.gis.db.backends.postgis',
# Or path to database file if using sqlite3.
'NAME': 'sunlumo_dev',
# The following settings are not used with sqlite3:
'USER': 'dodobas',
'PASSWORD': '',
# Empty for localhost through domain sockets or '127.0.0.1' for
# localhost through TCP.
'HOST': 'postgis',
# Set to empty string for default.
'PORT': '',
}
}
LOGGING = {
'version': 1,
'disable_existing_loggers': False,
'formatters': {
# define output formats
'verbose': {
'format': (
'%(levelname)s %(name)s %(asctime)s %(module)s %(process)d '
'%(thread)d %(message)s')
},
'simple': {
'format': (
'%(name)s %(levelname)s %(filename)s L%(lineno)s: '
'%(message)s')
},
},
'handlers': {
# console output
'console': {
'class': 'logging.StreamHandler',
'formatter': 'simple',
'level': 'DEBUG',
},
'logfile': {
'class': 'logging.FileHandler',
'filename': 'app-dev.log',
'formatter': 'simple',
'level': 'DEBUG',
}
},
'loggers': {
'django.db.backends': {
'handlers': ['logfile'],
'level': 'INFO', # switch to DEBUG to show actual SQL
},
'django': {
'handlers': ['logfile'],
'level': 'DEBUG',
'propagate': True
}
# example app logger
# 'app.module': {
# 'level': 'INFO',
# 'handlers': ['logfile'],
# # propagate is True by default, which proppagates logs upstream
# 'propagate': False
# }
},
# root logger
# non handled logs will propagate to the root logger
'root': {
'handlers': ['logfile'],
'level': 'DEBUG'
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,284 |
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Gold Bullion Worth $1 Billion To Be "Repatriated" From NY Fed To New Texas Bullion Depository
by The Daily Coin · June 16, 2015
by Mark O'Byrne, Gold Core
– Texas creates state gold depository – bringing gold home from New York Fed
– Move to remove gold from Federal Reserve highlights distrust
– Follows repatriation moves by Germany, Netherlands, Austria and others
– Legislation will prevent Federal government from confiscating gold
– Includes provisions that may lead to return to using gold as currency in the U.S.
– New gold electronic payments system protect from "national financial or currency crisis"
– European, UK and Irish governments could learn from prudent monetary move
The state of Texas has just passed legislation to build its own gold bullion depository, to repatriate $1 billion dollars worth of gold currently stored by the Federal Reserve in New York and to create a new gold electronic payments system to protect from "national financial or currency crisis".
New York Federal Reserve Employees Auditing Gold?
The move is being widely perceived as a vote of no confidence in the privately owned, bank owned central bank and the federal government.
Governor Abbott said that establishing this Depository means Texas will be, "increasing the security and stability of our gold reserves and keeping taxpayer funds from leaving Texas to pay for frees to store gold in facilities outside our state." (see Governor Abbott Signs Legislation To Establish State Bullion Depository )
The law will go into effect immediately and the new Texas Bullion Depository – soon to be built- will cater to businesses, state agencies and citizens.
Representative Giovanni Capriglione who introduced the bill was reported by the Star-Telegram as saying:
"People have this image of Texas as big and powerful … so for a lot of people, this is exactly where they would want to go with their gold," leading some commentators to puzzle over whether New York was not "big and powerful".
Heretofore, it has been Venezuela and European countries that have been repatriating gold – Germany, the Netherlands and Austria have sought to bring their sovereign gold home from New York amid fears that the Fed – whose gold stocks have not been publicly audited since 1953 – may not be in possession of the gold it claims to hold.
It is highly significant therefore that a powerful state from within the U.S., such as Texas, should display such apparent distrust of the Federal Reserve.
Two of Texas' largest public pension funds from the University of Texas (UoT) and the Teacher Retirement System (TRS) showed similar concerns in 2011 when they took delivery of $1 billion worth of gold bars out of storage with HSBC in New York.
The asset had been held in an ETF but the pension funds were apparently uneasy about not actually owning the physical gold. ETFs track the price of gold and ETF speculators or investors are merely creditors of the ETF and therefore, are very vulnerable to counter-party risk and exposure to the many banks, who are custodians and indeed sub custodians.
The legislation even seeks to protect the state's gold from confiscation by the Federal government. Section A2116.023 of the bill states: "A purported confiscation, requisition, seizure, or other attempt to control the ownership … is void ab initio and of no force or effect."
Under the Tenth Amendment the rights of the state trumps any order from the federal government.
Also significant is a provision which may lead to a return to sound money as proposed by Article 1, section 10 the constitution, i.e. gold and silver. In one section the bill states: "depository account holder may transfer any portion of the balance of the holder's depository account by check, draft, or digital electronic instruction to another depository account holder or [to a person who at the time the transfer is initiated is not a depository account holder.]" [underline]
The man who initially drafted this legislation is Rick Cunningham of the Texas Center for Economics, Law, and Policy. Mr. Cunningham is respected and is the Executive Director of the Center, but he is also a magna cum laude graduate of Texas A&M with a degree in Economics, as well as a graduate of the University of Chicago Law School, where he served as associate editor of the Law Review journal.
According to Mr. Cunningham
"this proposal consists of two parts – the "depository" part and the "system" part. The "depository" part … provides simply for hedging the state's investment risk by allocating a recommended portion of state and local investment assets to physical gold and other precious metals, and housing those metals in a state-operated facility…"
"But the truly game-changing aspect of this proposal … lies in the "system" part. This would be an advanced, state-owned and operated system of electronic payments and settlements, denominated in ounces of precious metals, barred from engaging in lending, leasing, speculative or derivative transactions, and always maintaining a 100% ratio of bullion reserves to account balances. At full scale, not only could it sustain state and local government operations, it could potentially sustain large swaths of the Texas economy, even in the face of a national financial or currency crisis."
If gold and silver were to become widely circulating currencies in Texas, the Federal Reserve issued and continuously devaluing dollar may slowly fall out of favour. Not maintaining a currency monopoly could ultimately lead to a return to using gold and silver as currency in the U.S.
In the coming months and years it is likely that the Federal Reserve and the Federal government of the U.S. will come under increasing pressure from Russia and China to back the dollar with something of intrinsic value rather than simply increasingly empty promises.
If either of those two countries chose to back their currencies with gold bullion, of which they have been accumulating vast volumes in recent years, the U.S. would be forced to follow suit in order to prevent sharp falls in the value of the dollar and in an attempt to preserve reserve currency status.
Now, it would seem, the Federal Reserve note – the dollar bill as issued by a private central bank in defiance of the Constitution – may face pressure from within the U.S. as 'Lone Star' state Texas begins to bring gold back into the monetary system.
The days of paper and electronic currencies – backed by little more than faith in governments that the public increasingly do not trust – are numbered. Texas is preparing for this as are European nations such as Germany, Austria and the Netherlands.
Preserve your wealth by acquiring an allocation to history's only enduring money – gold.
Must Read Guide: 7 Key Gold Must Haves
Today's AM LBMA Gold Price was USD 1,182.10, EUR 1,050.06 and GBP 759.36 per ounce.
Yesterday's AM LBMA Gold Price was USD 1,178.25, EUR 1,049.57 and GBP 760.01 per ounce.
Gold climbed $5.20 or 0.44 percent yesterday to $1,186.20 an ounce. Silver rose $0.17 or 1.07 percent to $16.11 an ounce.
Gold in USD – 1 Week
Gold in Singapore for immediate delivery was marginally lower at $1,185.6 an ounce towards the end of the day, while bullion in Switzerland fell a dollar.
Gold inched down this morning but stayed in lock down in a very narrow range. The yellow metal looks well supported at these levels with safe haven bids increasing due to the unresolved Greek debt crises as the time runs out before the deadline at the end of the month.
Some investors wait for more guidance from the U.S. Federal Reserve during its meeting that begins today. Fed Chair Jane Yellen's comments and wording of the Fed policy statement tomorrow will be closely watched.
U.S. economic data is still weak. Yesterday's data from industrial production was poor underlining concerns about the U.S. economy.
The Bank of China has joined the ICE Benchmark Administration (IBA) gold price benchmarking process. This increases the number of participants to eight including – JPMorgan Chase Bank, Scotiabank, HSBC, Société Générale, UBS, Barclays and Goldman Sachs in the LBMA Gold Price, which formally replaced the London Gold Fix this spring.
In late morning European trading gold is down 0.16 percent at $1,184.87 an ounce. Silver is off 0.38 percent at $16.01 an ounce and platinum is also down 0.15 percent at $1,086.56 an ounce.
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Contact The Daily Coin | {
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} | 3,914 |
The U.S. Senate has voted in favor of overturning privacy rules instituted by the Federal Communications Commission in 2016, which —if the measure survives a second vote in the House of Representatives —will allow internet service providers to sell customer data to advertisers without people having a say.
The Senate vote was split along party lines, with 50 Republicans in favor and 48 Democrats opposing, Reuters reported. Two Republicans were absent.
It's unknown when the House of Representatives will decide on the matter.
The prior FCC rules required ISPs to get consent before sharing data on location, health, children, finances, or browser history, whether for advertisers or internal marketing. They were opposed not only by the ISPs but by political supporters like Senate Majority Leader Mitch McConnell (R-KY), who after the Senate vote argued they created "an uneven playing field," discouraging "competition, innovation, and infrastructure investment."
Republicans on the FCC —including new chairman Ajit Pai —have suggested the rules give Web companies like Google and Facebook the ability to collect more data than ISPs, and hence an unfair edge in ads.
According to Vocativ, citing the Center for Responsive Politics, the 22 Republican senators behind the move to overturn have received over $1.7 million in donations from trade organizations and ISPs like AT&T and Comcast since 2012. Some of the bigger individual recipients include Texas senators John Cornyn and Ted Cruz, Florida's Marco Rubio, and South Dakota's John Thune.
The biggest corporate donors were AT&T, Comcast, and Verizon, which gave $357,000, $309,000, and $273,000, respectively.
Apple has typically limited the amount of demographic data it shares with outside parties, in keeping with a privacy focus favored by CEO Tim Cook. This has created problems in some cases, for instance hurting the appeal of selling digital magazines and newspapers through the App Store, since publishers have less data to attract advertisers. It was infamously a factor in tanking iAd, Apple's in-house attempt at mobile advertising. | {
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} | 1,496 |
Don't forget to rest and enjoy amongst responsibilities.
As a child, one has the freedom to genuinely care free. You can trust that your parents will provide for you, and responsibilities are never placed on your shoulders.
Little by little, we find ourselves gaining more freedom to make our own decisions, yet less freedom in the sense that there is far more expected of us.
Paying rent, making good grades, performing well, cleaning the house, working hard, providing, and caring — all of these are expected more and more as you age.
In a sense, one's character is strengthened by serving others and removing yourself from the center of your life.
However, it has become clear to me that the stress of balancing and completing these responsibilities robs you of an overall quality of life.
If someone is able to balance adult life without stress, then damn, props to you. However, from what I've seen, the psychological effects of stress drain the spirit and ability to enjoy what is in front of us.
Is there a solution? If you denounce these responsibilities, you will most definitely fail to provide for your family or be financially stable. On the other hand, if we let stress take us over, we lose sight of what really matters.
Balance is key. But even more so, it is important to catch yourself and notice when you are stressing to an extent that is detrimental to yourself and those around you. By having a proactive awareness like this, we are able to control our stress rather than allow it to control us.
In the end, with freedom and luxuries, does come responsibility.
As adults we may not experience the carefree lifestyle we once did, however, we have the power to allow ourselves the grace to relax and enjoy amongst our responsibilities. | {
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Ielena Danilova (), née le , est une triathlète russe, double championne de Russie de triathlon (2013 et 2017).
Biographie
Palmarès
Le tableau présente les résultats les plus significatifs (podium) obtenus sur le circuit national et international de triathlon et aquathlon depuis 2012.
Notes et références
Voir aussi
Article connexe
Aquathlon
Coupe du monde de triathlon
Liens externes
Triathlète russe
Naissance en février 1991
Naissance en RSFS de Russie | {
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Candona foviolata är en kräftdjursart som beskrevs av Dobbin 1941. Candona foviolata ingår i släktet Candona och familjen Candonidae. Inga underarter finns listade.
Källor
Musselkräftor
foviolata | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 101 |
Tarot, Tibetan bowls and a marathon of desires: what esotericism do Russians now believe in
January 18, 2023 adminadmin 0 Comments
Esotericism has firmly established itself in our lives and does not intend to give up positions yet: according to the Levada Center, from a third to a half of Russians believe in fortune telling, horoscopes and amulets. VTsIOM also cites similar figures – 59% of Russians trust predictors, 36% trust astrological forecasts.
The last surge of interest in mysticism was understandably in the 90s: the higher the level of uncertainty, the more people turn to magic and the occult, psychologists say. Tourism has also become completely unpredictable: borders are closed, flights are canceled, tours are canceled – how can one not turn to the supernatural in such a situation? The five most fashionable irrational trends of our difficult times are in the review of "Subtleties".
1. Tarot cards
Tarot cards are breaking popularity records: in 2022, in Russia, the demand for esoteric books as a whole increased by 53%, and for card reading manuals – almost 6 times, Wildberries calculated. Cards and their interpretations are sold in bookstores, artists draw their own versions of the decks, they are laid out at secular parties, trying to get an answer to the question that is more relevant than ever – what will happen tomorrow?
"Since 2020, everything has become so unstable that people have lost their bearings. Now the turbulence is only increasing. People have nothing to rely on, so they began to turn to tarot in an attempt to somehow predict their lives, "tarot reader Nina Yudova explains obvious things. >
Los Muertos Crew/3p/kf/3pkfqruxth4wgk4wkc8kow4s0.jpg
2. Divination in Tiktok
Modern magicians and clairvoyants are using the Internet to promote their services with might and main, and Tiktok has become the most popular platform for predicting the future. Videos with the hashtag "fortune teller" collect hundreds of millions of views, and especially advanced fortune tellers have up to 300,000 subscribers. Using all the same tarot cards and other improvised means, tiktok fortune-tellers live answer simple questions: loves – does not love, will the husband return to the family and will the child finally pass the exam. In return, they only ask for a like and a subscription, but a deeper analysis costs money – for example, a personal forecast for a year will cost 500-1500 RUB. According to statistics, young people from 18 to 24 years old are most interested in esotericism in Russia. In the West, the age limits are wider: practical magic is a hobby not only for zoomers, but also for millennials.
/2s/07/2s07vue7k6m88ogc4sk8ko804.jpg
3. Astrology
The ancient teaching that conquered Russia back in the 90s has returned updated and fully armed: astrologers of the 21st century maintain YouTube blogs and Telegram channels, equate themselves with psychologists and believe that not today or tomorrow their work will be recognized as a full-fledged science. Skeptics, of course, remind that even a light bulb affects a person more than retrograde Mercury, but they listen to them little and reluctantly: stellar forecasts help to gain confidence in themselves and the future, because astrologers give people what they are looking for – ready-made advice in difficult situations. By the way, the first astrological column was published in the Sunday Express newspaper during the Great Depression in August 1930.
Today, 16 % of employers pay attention to the astrological sign of the applicant, and three-quarters of Russians admitted that they always read horoscopes.
/45/lk/45lkxx32h6yo88cwsg088sgg4.jpg
4. Marathons of desires
If clairvoyants and fortune-tellers only reveal to a person the cards of his fate, then energetic gurus and coaches teach to control their life and change it. One of the most common techniques today is the wishing marathon. After registration, its participants receive homework: make a list of wishes, set dates for their implementation and, of course, visualize dreams. There are also paid options: for money, the mentor will check the assignments, give useful advice, and even talk in person. At the end of the training, the wish list is tied to a balloon and released into the sky. So you will see such a ball with a note – do not be surprised.
Marathon organizers claim that miracles happen in the process and unrealizable things come true, but the participants of special races themselves, as a rule, assess what is happening quite prosaically: only those wishes are fulfilled that they worked hard to fulfill.
/6t/3x/6t3xntxg2yw4ksgw4w44g0so4.jpg
What else to read on topic
Top 13 places of power in Russia
In a black-and-black city: top 5 horror stories from pioneer camps
Is blood type discrimination in Japan a myth or a truth?
5. Tibetan bowls
Alternative medicine sometimes uses the most unexpected devices – for example, Tibetan singing bowls. This ancient musical instrument was widely used in Asia during religious rituals. Today, bowls serve as an aid to meditation, yoga, massage and other medical practices. Adherents of this technique believe that the vibrating sounds extracted from the bowl with the help of a wooden stick pass through the body and improve the functioning of the organs, allow you to get rid of stress and achieve, if not nirvana, then at least peace. And at the same time – success in business and personal life, this is really a trifle for cups.
Los Muertos Crew/bl/98/bl980b7za7ks8ks4cgoowog44.jpg
Orlando Reveals 23 New Experiences for Visitors in 2023
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Moving to Istanbul: housing, prices, jobs, people
"My husband and I have been thinking about moving for a long time, but we didn't really prepare for it.[...]
Direct Travel Launches Leisure Air, Hotel Booking Tool for Advisors
Travel agents save their clients both time and money. (photo via Yuri_Arcurs / iStock / Getty Images Plus) Travel management[...] | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,390 |
Q: INSERT in the query part of BigQuery, using google-api-php-client I wanted to insert thousands of records in the database using INSERT command in php script , as it will easier to access, but https://developers.google.com/bigquery/docs/query-reference , this doesn't show the INSERT command that can be used while querying in php , like ,
$quert->setQuery("Insert into ... values...") , Have tried this in the Query Table of BigQuery web console, but it doesn't seem to work , Is there anyway to use setQuery() with some other command, to insert data ?
A: BigQuery doesn't support the INSERT command. You would need to create a load job. See https://developers.google.com/bigquery/docs/import#localimport for more information.
A: In addition to Jordan's answer, here's a snippet of code that should get you started using the Google BigQuery API and the Google API PHP client library for loading your own data into BigQuery programmatically. Note, this snippet simply spits out the raw API response of the load job, including the job Id to the screen - you'll have to add your own polling logic to check on the load job status.
(We will be adding additional documentation about loading your own data, as well as more PHP samples soon).
<?php
require_once "google-api-php-client/src/Google_Client.php";
require_once "google-api-php-client/src/contrib/Google_BigqueryService.php";
session_start();
$client = new Google_Client();
// Visit https://code.google.com/apis/console to generate your
// oauth2_client_id, oauth2_client_secret, and to register your oauth2_redirect_uri.
$client->setScopes(array('https://www.googleapis.com/auth/bigquery'));
$client->setClientId('XXXXXXXXX.apps.googleusercontent.com');
$client->setClientSecret('XXXXXXXXX');
$client->setRedirectUri('http://YOURAPPLICATION/index.php');
// Instantiate a new BigQuery Client
$bigqueryService = new Google_BigqueryService($client);
if (isset($_GET['code'])) {
$client->authenticate();
$_SESSION['token'] = $client->getAccessToken();
header('Location: http://' . $_SERVER['HTTP_HOST'] . $_SERVER['PHP_SELF']);
}
?>
<!doctype html>
<html>
<head>
<title>BigQuery API Sample</title>
</head>
<body>
<div id='container'>
<div id='top'><h1>BigQuery API Sample</h1></div>
<div id='main'>
<?php
if (isset($_GET['logout'])) {
unset($_SESSION['token']);
}
if (isset($_GET['code'])) {
$client->authenticate($_GET['code']);
$_SESSION['token'] = $client->getAccessToken();
header('Location: http://' . $_SERVER['HTTP_HOST'] . $_SERVER['PHP_SELF']);
}
if (isset($_SESSION['token'])) {
$client->setAccessToken($_SESSION['token']);
}
if ($client->getAccessToken()) {
// Your project number, from the developers.google.com/console project you created
// when signing up for BigQuery
$project_number = 'XXXXXXXXXXXXXX';
// Information about the destination table
$destination_table = new Google_TableReference();
$destination_table->setProjectId($project_number);
$destination_table->setDatasetId('php_test');
$destination_table->setTableId('my_new_table');
// Information about the schema for your new table
$schema_fields = array();
$schema_fields[0] = new Google_TableFieldSchema();
$schema_fields[0]->setName('first');
$schema_fields[0]->setType('string');
$schema_fields[1] = new Google_TableFieldSchema();
$schema_fields[1]->setName('last');
$schema_fields[1]->setType('string');
$destination_table_schema = new Google_TableSchema();
$destination_table_schema->setFields($schema_fields);
// Set the load configuration, including source file(s) and schema
$load_configuration = new Google_JobConfigurationLoad();
$load_configuration->setSourceUris(array('gs://YOUR_GOOGLE_CLOUD_STORAGE_BUCKET/file.csv'));
$load_configuration->setDestinationTable($destination_table);
$load_configuration->setSchema($destination_table_schema);
$job_configuration = new Google_JobConfiguration();
$job_configuration->setLoad($load_configuration);
$load_job = new Google_Job();
$load_job->setKind('load');
$load_job->setConfiguration($job_configuration);
$jobs = $bigqueryService->jobs;
$response = $jobs->insert($project_number, $load_job);
echo '<pre>';
print_r($response);
echo '</pre>';
$_SESSION['token'] = $client->getAccessToken();
} else {
$authUrl = $client->createAuthUrl();
print "<a class='login' href='$authUrl'>Authorize Access to the BigQuery API</a>";
}
?>
</div>
</div>
</body>
</html>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 119 |
module Seymour
module Channels
class Feed < Base
def deliver(activity)
actor_id = activity.load_instance(:actor).id.to_s
recipients = recipients(activity)
return unless recipients
begin
Mongoid.unit_of_work(disable: :all) do
if recipients.kind_of?(Mongoid::Criteria)
# We can only work with Criteria in batches
batch_size = ::Seymour::Config.feed_batch_size
0.step(recipients.count, batch_size) do |offset|
recipients.limit(batch_size).skip(offset).each do |recipient|
recipient.add_activity_to_feed!(activity) unless recipient.id.to_s == actor_id
end
end
else
# Anything else we get for recipients must respond to each
recipients.each do |recipient|
recipient.add_activity_to_feed!(activity) unless recipient.id.to_s == actor_id
end
end
end
rescue ArgumentError
# We're using a non-patched version of Mongoid without a way to disable identity map.
if recipients.kind_of?(Mongoid::Criteria)
# We can only work with Criteria in batches
batch_size = ::Seymour::Config.feed_batch_size
0.step(recipients.count, batch_size) do |offset|
recipients.limit(batch_size).skip(offset).each do |recipient|
recipient.add_activity_to_feed!(activity) unless recipient.id.to_s == actor_id
end
end
else
# Anything else we get for recipients must respond to each
recipients.each do |recipient|
recipient.add_activity_to_feed!(activity) unless recipient.id.to_s == actor_id
end
end
end
end
end
end
end | {
"redpajama_set_name": "RedPajamaGithub"
} | 6,113 |
Play Gym is open during weekday daytimes for all from 0 to 10 years. You and your child have the run of the whole gym (except for the high trampolines). As well as our amazing facilities, we have bouncy castles, soft play and a whole host of other play gym equipment. Come and bring your little ones and play in the foam pits, the sunken trampoline, the fast-track, beams and much, much more.
In the holidays we also allow older siblings up to 10 years along too, so long as they are careful as there are lots of babies crawling around. We ask that every parent is responsible for their own child. Come and play whenever you like!
Additional Adults will be £1.00.
The Big Trampolines and the corner trampoline are out of bounds as is the rings over the pit.
No Drinks or biscuits will provided during these sessions however we do have a water fountain.
Play corner is not open during the holidays.
Timetable remains the same through the holidays. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,694 |
package com.example.bootweb.httpclient.quick.example;
import java.io.IOException;
import java.io.InputStream;
import org.apache.http.HttpEntity;
import org.apache.http.client.methods.CloseableHttpResponse;
import org.apache.http.client.methods.HttpGet;
import org.apache.http.impl.client.CloseableHttpClient;
import org.apache.http.impl.client.HttpClients;
import com.example.bootweb.httpclient.quick.Constants;
/**
* This example demonstrates the recommended way of using API to make sure the
* underlying connection gets released back to the connection manager.
*
* This example demonstrates how to ensure the release of the underlying HTTP
* connection back to the connection manager in case of a manual processing of
* HTTP responses.
*/
public class ClientConnectionRelease {
public final static void main(String[] args) throws Exception {
CloseableHttpClient httpclient = HttpClients.createDefault();
try {
HttpGet httpget = new HttpGet(Constants.URL_ABOUT);
System.out.println("Executing request " + httpget.getRequestLine());
CloseableHttpResponse response = httpclient.execute(httpget);
try {
System.out.println("----------------------------------------");
System.out.println(response.getStatusLine());
// Get hold of the response entity
HttpEntity entity = response.getEntity();
// If the response does not enclose an entity, there is no need
// to bother about connection release
if (entity != null) {
InputStream instream = entity.getContent();
try {
instream.read();
// do something useful with the response
} catch (IOException ex) {
// In case of an IOException the connection will be released
// back to the connection manager automatically
throw ex;
} finally {
// Closing the input stream will trigger connection release
instream.close();
}
}
} finally {
response.close();
}
} finally {
httpclient.close();
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,973 |
Q: IMAP Access to BCC field I'm currently using IMAP and trying to access my gmail mailbox with it. I'm fetching email details such as Subject, Email Body, CC email Addresses etc.
My issue is that i need to access the BCC email addresses as well. I'm not sure whether the BCC field could be accessed due to the fact that..
1 - I'm sending this email which could be seen in the screenshot below
(it has a BCC email address added which is highlighted)
2 - This is my C# Code, which shows the BCC email count to be 0
So basically my question is that
*
*Can the BCC email addressed can be fetched using IMAP? Any sort of help/hint would really be appreciated.
A: The BCC field is available in the IMAP ENVELOPE structure, along with from, to, cc, sender and in-reply-to. It's usually empty since most software removes it when sending the message, but if you can see it in the gmail interface you'll also be able to see it via IMAP, in the same way as your existing code accesses from/to/etc.
See RFC 3501 page 77 for details, or look for uid fetch ... envelope in your IMAP library.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,902 |
Today is Bastille Day, a sort of French Independence Day. While not the same as our 4th of July, the two are often compared and, for our purposes, that'll do just fine.
So, break out the champagne and spread un peu d'amour. And maybe try some of these delicious French recipes.
Remember Mama's Quiche Lorraine? Tried and true, Quiche Lorraine is as French as the day is long. Serve with a green salad and a gentle Sancerre. C'est si bon!
What's more French than soup a l'onion? Serve with a baguette and a Cote du Rhone. Magnifique!
And what about crepes? We Americans love crepes! You can fill these yummy pastry shells with just about anything.
So, here's to liberte, egalite, fraternite! Bon Appetit! | {
"redpajama_set_name": "RedPajamaC4"
} | 7,150 |
html, body, md-tabs {
overflow-x: hidden;
}
#section-main {
overflow-y: hidden;
}
.app-optimized {
-webkit-transform: translateZ(0);
-moz-transform: translateZ(0);
-ms-transform: translateZ(0);
-o-transform: translateZ(0);
transform: translateZ(0);
-webkit-backface-visibility: hidden;
-moz-backface-visibility: hidden;
backface-visibility: hidden;
-webkit-perspective: 1000px;
-moz-perspective: 1000px;
perspective: 1000px;
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,898 |
import uuid
import requests
class Book(object):
"""An object representation of a book.
Returns a librarian.Book type"""
def __init__(self):
"""Initial value.
No return."""
self.id = uuid.uuid4()
self.title = "Untitled"
self.author = ["Unknown"]
self.length = 0
self.content = "Blank"
def assign_id(self):
"""Assigns a book a unique id.
Returns a uuid."""
self.id = uuid.uuid4()
return self.id
def assign_title(self, title):
"""Assigns a book a title.
Returns a string of the book title."""
self.title = str(title)
return self.title
def assign_author(self, author):
"""Assign a book one more author.
Returns a list of authors."""
self.author = [str(x) for x in self.author if x != "Unknown"]
self.author.append(str(author))
return self.author
def assign_length(self, pageCount):
"""Assigns a book a length.
Returns an int of book length."""
self.length = int(pageCount)
return self.length
def assign_content(self, bookURL="https://raw.githubusercontent.com/shakna-israel/NecromancersApprentice/master/docs/000-Chapter-Zero.md"):
"""Assigns a book some content from a given URL.
Returns a string of content."""
self.content = requests.get(str(bookURL)).text
return self.content
class Series(object):
"""An object representation of a group of books.
Returns a librarian.Series type."""
def __init__(self):
"""Inital values.
No return."""
self.books = {}
def add_book(self, book):
"""Add a book to a series.
Returns a single key/value pair of uuid to title."""
self.books[book.id] = book.title
return self.books[book.id]
def get_titles(self):
"""Get the titles of the books in the Series.
Returns a list of titles."""
return self.books.items()
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,388 |
\section{Introduction}\label{intro}
With the growing interest in sensor networks and multi-agent
systems, the problem of estimating the state of a dynamical system
whose measured outputs are distributed across a network has been
under study in one form or another
for a number of years
\cite{KhanAli2011ACC,shamma,carli,xxx,saber2,bullo.observe,sanfelice}.
Despite this, only quite recently have provably correct distributed state estimators begun to
emerge which solve this problem under reasonably non-restrictive assumptions
\cite{martins,TAC.17,MitraPurdue2016,
Kim2016CDC,trent,trent2,CDC17.1,ACC19}.
In its simplest form, the discrete-time version of the distributed
state estimation
problem
starts with
a network of $m>1$ agents labeled $1,2,\ldots,m$ which are able to
receive information from their neighbors. Neighbor relations are
characterized by a directed graph $\mathbb{N}$, which may or may not
depend on time, whose
vertices correspond to agents and whose arcs depict neighbor relations.
Each agent $i$ senses a
signal $y_i\in{\rm I\!R}^{s_i},\;i\in\mathbf{m} = \{1,2,\ldots,m\}$ generated by a discrete-time system of the form
$x(\tau+1)=Ax(\tau),\;y_i(\tau) = C_ix(\tau),\;i\in \mathbf{m}$ and
$x\in{\rm I\!R}^n$. It is typically assumed that $\mathbb{N}$ is strongly
connected and that the system is jointly observable. It is
invariably assumed that each agent receives certain real-time
signals from its
neighbors although what is received can vary from one problem formulation to the next.
In all formulations, the goal is to devise local estimators, one for each agent, whose outputs
are all asymptotically correct estimates of $x$.
The local estimator dynamics for agent $i$ is typically assumed to depend only on the pair $(C_i,A)$
and certain properties of $\mathbb{N}$.
The problem is basically the same in continuous time, except that
rather than the discrete-time model just described, the
continuous-time model
$\dot{x}=Ax,\;y_i(t) = C_ix, \; i\in \mathbf{m}$ is considered
instead.
One way to try to address the estimation problem is to recast it as
a discrete-time
classical decentralized control
problem \cite{corfmat} as was done in \cite{martins}. Following this
approach, it is possible to devise a provable correct procedure for
crafting a distributed linear filter with a prescribed spectrum
which solves
the continuous-time version of the problem assuming
$\mathbb{N}$ is a constant strongly connected graph \cite{TAC.17}; the same procedure is easily modified to
deal with the discrete-time version of the problem.
Prompted by work in \cite{Kim2016CDC}, an entirely different and simpler approach
to the continuous-time version of the estimation problem was developed in
\cite{trent}. The same approach was simplified still further in \cite{ACC19} by exploiting certain
well-known properties of
invariant subspaces. There are however two distinct limitations of the types of estimators
discussed in \cite{Kim2016CDC,trent,ACC19}. First, as they stand these estimators cannot deal with
time-varying neighbor graphs. Second, there does not appear to be a way to easily modify these estimators to
address the discrete-time state estimation problem; this is because the continuous-time estimators rely on a
``high gain'' concept for which there is no discrete-time counterpart.
Despite these limitations, there is a very useful idea in these papers, stemming from the work in
\cite{Kim2016CDC}, which can be used to advantage in developing a discrete-time solution to the problem.
Roughly speaking, the idea is to using the invariance of the
unobservable spaces of the the pairs $(C_i,A)$ to ``split'' the
estimators into two parts - one for which conventional spectrum
assignment tools can be used to control convergence rate and the
other for which convergence rate can be controlled by switching and
averaging.
This paper is organized as follows. Certain basic properties of
invariant subspaces are reviewed in
\S\ref{in}. The specific problem to be addressed is then formulated in \S\ref{form}. In \S\ref{obs}
the observer which solves this problem is described. The error
model needed to analyze the observer is developed in \S\ref{errors}. Finally in \S\ref{switch},
several techniques are outlined for picking the number of switches required between ``event times''
in order to achieve a prescribed convergence rate.
\subsection{Invariant Subspaces}\label{in}
Throughout this paper certain basic and well-known algebraic
properties of invariant subspaces will be exploited. To understand
what they are,
let $A$ be any square matrix, and suppose $\scr{V}$ is an $A$-invariant subspace. Let $Q$
be any full row rank matrix whose kernel is $\scr{V}$ and suppose that $V$ is any ``basis matrix''
for $\scr{V}$; i.e., a matrix whose columns form a basis for $\scr{V}$.
Then the linear equations $$QA =\bar{A}_{V}Q\;\;\;\;\;
\text{and}\;\;\;\;\; AV=VA_{V}$$
have unique solutions $\bar{A}_V$ and $A_V$ respectively. Let $V^{-1}$ be any left inverse of $V$ and let
$Q^{-1}$ be that right inverse of $Q$ for which $V^{-1}Q^{-1} =0 $.
Then
$$A = H^{-1}\matt{\bar{A}_V & 0\\ \widehat{A}_V & A_{V}}H$$
where
$$H = \matt{Q \\ \\ V^{-1}}$$
and
$\widehat{A}_V = V^{-1}AQ^{-1}$.
Use will be made of these simple algebraic facts in the sequel.
\section{Problem}\label{form}
We are interested in a time-varying network
of $m>1$ agents labeled $1,2,\ldots,m$ which are able to receive
information from their neighbors where by a {\em neighbor} of agent
$i$ is meant any agent in agent $i$'s reception range.
We write $\scr{N}_i(t)$ for the set of labels of agent $i$'s neighbors at real time $t$ and
take agent $i$
to be a neighbor of itself for all $t$. Relations between neighbors are characterized by a directed graph
$\mathbb{N}(t)$ with $m$ vertices and a set of arcs defined so that there
is an arc from vertex $j$ to vertex $i$ whenever
agent $j$ is a neighbor of agent $i$.
Each agent $i$ can sense a discrete-time
signal $y_i(\tau)\in{\rm I\!R}^{s_i}$ at {\em event times} $\tau T$, $\tau = 0, 1,2,\ldots $
where $T$ is a positive constant;
for $i\in\mathbf{m} \stackrel{\Delta}{=} \{1,2,\ldots,m\}$ and $\tau=0,1,2,\ldots $
\eq{y_i(\tau) = C_ix(\tau) ,\;\;\;\;\;x(\tau +1) =Ax(\tau )
\label{syss}} and $x\in{\rm I\!R}^n$. We assume throughout that
$\mathbb{N}(t)$ is strongly connected and that the system defined by
\rep{syss} is {\em jointly observable}; i.e., with $C =
\begin{bmatrix}C_1' &C_2' & \cdots & C_m'\end{bmatrix}'$, the matrix
pair $(C,A)$ is observable. Joint observability is equivalent to the
requirement that
$$\bigcap_{i\in\mathbf{m}}\scr{V}_i = 0$
where $\scr{V}_i$ is the {\em unobservable space} of $(C_i,A)$; i.e.
$\scr{V}_i = \ker \begin{bmatrix}C_i' &(C_iA)' & \cdots
&(C_iA^{n-1})'\end{bmatrix}'$. As is well known, $\scr{V}_i$ is
the largest $A$-invariant subspace contained in the
kernel of $C_i$.
Each agent $i$ is to estimate $x$ using a dynamical system whose
output $x_i(\tau)\in{\rm I\!R}^n$
is to be an asymptotically correct estimate of $x(\tau)$ in the sense that the estimation error
$x_i(\tau)-x(\tau)$ converges to zero as $\tau\rightarrow \infty$
as fast as $\lambda^{\tau} $ does,
where $\lambda$ is an
arbitrarily chosen but fixed
positive number\footnote{For the type of observer to be developed, finite-time convergence is not possible.} less than $1$. To accomplish this it is assumed that
the information
agent $i$ can receive from neighbor $j$ at event time $\tau T$ is
$x_j(\tau)$. It is further
assumed that agent $i$
can also receive certain additional information from its neighbors
at a finite number of
times between each successive pair of event times; what this information is will be specified below.
\section{The Observer}\label{obs} In this paper it will be assumed that each agent's neighbors do not
change between event times. In other words, for $i\in\mathbf{m}$, $$\scr{N}_i(t) =
\scr{N}_i(\tau T),\;\;\;\; t\in [\tau T,(\tau+1)T),\;\;\;\;\tau = 0,1,2,\ldots $$
With this assumption, the observer to be considered
consists of $m$ private estimators, one for each agent.
The estimator for agent $i$ is of the form
\eq{ x_i(\tau+1) =
(A+K_iC_i)\bar{x}_i(\tau) -K_iy_i(\tau)\label{est1}}
where $\bar{x}_i(\tau)$ is an ``averaged state'' computed
recursively during the real time interval $[\tau T,\; (\tau+1)T)$
using the update equations
\begin{eqnarray}
z_i(0,\tau) &=&x_i(\tau)\label{p1}\\
z_i(k,\tau) &=& (I-P_i)z_i(k-1,\tau)\nonumber\\ &+& \frac{1}{m_i(\tau)}P_i\sum_{j\in\scr{N}_i(\tau T)}z_j(k-1,\tau),\;\;k\in \mathbf{q}\\
\bar{x}_i(t) &= &z_i(q,\tau)\label{p3}\end{eqnarray} Here
$m_i(\tau)$ is the number of labels in $\scr{N}_i(\tau T)$, $q$ is a
suitably defined positive integer,
$\mathbf{q}\stackrel{\Delta}{=}\{1,2,\ldots, q\}$, and $P_i$ is the orthogonal projection on the unobservable space of $(C_i,A)$.
Each matrix $K_i$ is defined as follows.
For fixed $i\in\mathbf{m}$, write $Q_i$ for any full rank matrix whose kernel
is the unobservable space of $(C_i,A)$, and let $\bar{C}_i$ and $\bar{A}_i$ be the unique solutions to
$\bar{C}_iQ_i = C_i$ and $Q_iA=\bar{A}_iQ_i$ respectively. Then the matrix pair $(\bar{C}_i,\bar{A}_i)$ is
observable. Thus by using a standard spectrum assignment algorithm,
a matrix $\bar{K}_i$ can be chosen to ensure that the convergence
of $(\bar{A}_i + \bar{K}_i\bar{C}_i)^{\tau}$ to zero as $\tau\rightarrow \infty$
is as fast as the convergence to zero
of $\lambda ^{\tau}$ is.
Having chosen such $\bar{K}_i$, $K_i$
is then defined to be $K_i = Q_i^{-1}\bar{K}_i$ where $Q_i^{-1}$ is a right inverse for $Q_i$.
The definition implies that $Q_i(A+K_iC_i) =
(\bar{A}_i+\bar{K}_i\bar{C}_i)Q_i$ and that
$(A+K_iC_i)\scr{V}_i\subset \scr{V}_i$. The latter, in turn, implies
that there is a unique matrix $A_i$ which satisfies $(A+K_iC_i)V_i
= V_iA_i$ where $V_i$ is a basis matrix\footnote{For simplicity, we
assume that the columns of $V_i$ constitute an orthonormal basic for $\scr{V}_i$
in which case $P_i = V_iV_i'$.} for $\scr{V}_i$.
To explain what needs to
be considered in choosing
$q$ it is necessary to describe the structure of the ``error model'' of the overall observer. This will be done next.
\section{The Error Model}\label{errors}
For $i\in\mathbf{m}$, write $e_i(\tau)$ for the {\em state
estimation error} $e_i(\tau) = x_i(\tau)-x(\tau )$. In view of
\rep{est1},
$$e_i(\tau+1) = (A+K_iC_i)\bar{e}_i(\tau)$$
where $\bar{e}_i(\tau) =\bar{x}_i(\tau)-x(\tau)$. Moreover if
$\epsilon_i(k,\tau) \stackrel{\Delta}{=} z_i(k,\tau)-x(\tau),\;k\in\{0,1,\ldots, q\}$
then
\begin{eqnarray}
\epsilon_i(0,\tau) & =& e_i(\tau)\\
\epsilon_i(k,\tau) &=& (I-P_i)\epsilon_i(k-1,\tau)\nonumber\\
& +&\frac{1}{m_i(\tau)}P_i\sum_{j\in\scr{N}_i(\tau T)}\epsilon_j(k-1,\tau),\;\;k\in \mathbf{q}\\
\bar{e}_i(\tau) &= &\epsilon_i(q,\tau)\end{eqnarray} because of
\rep{p1} -- \rep{p3}. It is possible to combine these $m$
subsystems into a single system. For this let
$e = $ column $\{e_1,e_2,\ldots, e_m\}$, define
$\bar{A} = $ block diagonal $\{A+K_1C_1
,A+K_2C_2,\ldots,A+K_mC_m\}$,
$P = $ block diagonal $\{P_1 ,P_2,\ldots, P_m\}$ and write $S(\tau)$ for the stochastic matrix
$S(\tau) = D_{\mathbb{N}(\tau T)}^{-1}A'_{\mathbb{N}(\tau T)}$ where
$A_{\mathbb{N}(\tau T)}$ is
the adjacency matrix of $\mathbb{N}(\tau T)$ and $ D_{\mathbb{N}(\tau T)}$
is the diagonal matrix whose $i$th diagonal entry is the in-degree
of $\mathbb{N}(\tau T)$'s $i$th vertex. Note that $\mathbb{N}(\tau T )$ is the graph\footnote{The {\em graph} of an $n\times n$ matrix $M$ is that directed graph
on $n$ vertices possessing
a directed arc from vertex $i$ to vertex $j$ if $m_{ij}\neq 0$ \{p. 357, \cite{horn1}.\}}
of $S'(\tau)$ and that the diagonal entries of $S'(\tau)$
are all positive because each agent is a neighbor of itself.
Let $\bar{e}(\tau) = $ column
$\{\bar{e}_1(\tau),\bar{e}_2(\tau),\ldots, \bar{e}_m(\tau)\}$ and
$\epsilon(k,\tau) =
\text{column}\{\epsilon_1(k,\tau),\epsilon_2(k,\tau),\ldots
,\epsilon_m(k,\tau)\}$. Then
$$e(\tau +1) = \bar{A}\bar{e}(\tau)$$
and
\begin{eqnarray*}
\epsilon(0,\tau) &=&e(\tau)\\
\epsilon(k,\tau) &=& (I_{mn}-P(I_{mn}-\bar{S}(\tau)))\epsilon(k-1,\tau),\;k\in\mathbf{q}\\
\bar{e}(\tau) &= &\epsilon(q,\tau)\end{eqnarray*} where
$\bar{S}(\tau ) = S(\tau)\otimes I_n$; here $\otimes $ denotes
Kronecker product, and $I_n$ and $I_{mn}$ are the $n\times n$ and
$mn\times mn$ identity matrices respectively. Clearly
$$\bar{e}(\tau) = (I_{mn}-P(I_{mn}-\bar{S}(\tau)))^qe(\tau) $$
so \eq{e(\tau+1) = \bar{A}(I_{mn}-P(I_{mn}-\bar{S}(\tau)))^qe(\tau)
\label{error}}
Our aim is to explain why for $q$ sufficiently large, the
time-varying matrix
$\bar{A}(I_{mn}-P(I_{mn}-\bar{S}(\tau)))^q$ appearing in \rep{error} is a discrete-time stability matrix
for which the product \eq{\Phi(\tau) = \prod_{s =
1}^{\tau}\bar{A}(I_{mn}-P(I_{mn}-\bar{S}(s)))^q\label{phi}}
converges to zero as $\tau\rightarrow\infty $ as fast as $\lambda
^{\tau}$ does. As a first step towards this end, note that the
subspace $\scr{V} = \scr{V}_1\oplus\scr{V}_2\oplus \cdots \oplus
\scr{V}_m$
is
$\bar{A}$ - invariant because $(A+K_iC_i)\scr{V}_i\subset
\scr{V}_i,\;i\in\mathbf{m}$. Next, let $Q = $ block diagonal $
\{Q_1, Q_2,\ldots ,Q_m\}$ and $V = $ block diagonal $ \{V_1,
V_2,\ldots ,V_m\}$ in which case $Q$ is a full rank matrix whose
kernel is $\scr{V}$ and $V$ is a basis matrix for $\scr{V}$ whose
columns form an orthonormal set.
It follows that $P = VV'$, and that
\begin{eqnarray}
Q\bar{A}& = &\bar{A}_VQ\label{prr1}\\
\bar{A}V & = & V\tilde{A}\label{prr2}
\end{eqnarray}
where
\eq{\bar{A}_V = \text{block diagonal}\;
\{\bar{A}_1+\bar{K}_1\bar{C}_1 ,\ldots,
\bar{A}_m+\bar{K}_m\bar{C}_m\}\label{sunday}} and
$$ \tilde{A} =\text{block diagonal}\;\{A_1,A_2,\ldots ,A_m\};$$
as before, $(A+K_iC_i)V_i=V_iA_i$. Moreover
\begin{eqnarray}Q(I_{mn}-P(I_{mn}-\bar{S}(\tau)))^q &=& Q\label{pr1}\\
(I_{mn}-P(I_{mn}-\bar{S}(\tau)))^qV& =& V(
V'\bar{S}(\tau)V)^q\label{pr2}\end{eqnarray} Note that \rep{pr1}
holds because $QP = 0$. To understand why \rep{pr2} is true, note
first that $(I_{mn}-P(I_{mn}-\bar{S}(\tau)))V = V(I_{\bar{n}} -
V'(I_{mn} - \bar{S}(\tau))V)$ because $P =VV'$; here $\bar{n} =
\dim(\scr{V})$. But $I_{\bar{n}} - V'(I_{mn} - \bar{S}(\tau))V =
V'\bar{S}(\tau)V$ because
$V'V = I_{\bar{n}}$. Thus \rep{pr2} holds for $q=1$; it follows by induction that \rep{pr2}
holds for any positive integer $q$.
Using \rep{prr1} -- \rep{pr2}, one obtains the equations
\begin{eqnarray}Q\bar{A}(I_{mn}-P(I_{mn}-\bar{S}(\tau)))^q &=& \bar{A}_VQ\label{xpr1}\\
\bar{A}(I_{mn}-P(I_{mn}-\bar{S}(\tau)))^qV& =& VA_V(\tau)
\label{xpr2}\end{eqnarray} where \eq{A_V(\tau) =
\tilde{A}(V'\bar{S}(\tau)V)^q\label{ench}} These equations imply
that
\eq{\bar{A}(I_{mn}-P(I_{mn}-\bar{S}(\tau)))^q = H^{-1}\matt{\bar{A}_V & 0\\
\hat{A}_V(\tau) & A_V(\tau)}H\label{as}} where $$H =\matt{Q \\ \\
V^{-1}}$$ and $\widehat{A}_V(\tau) =
V^{-1}\bar{A}(I_{mn}-P(I_{mn}-\bar{S}(\tau)))^qQ^{-1}$.
Since the spectrum of each
$\bar{A}_i+\bar{K}_i\bar{C}_i,\;i\in\mathbf{m}$, is assignable with
$\bar{K}_i$, and $\widehat{A}_V(\tau)$ is a bounded matrix, to show
that for suitably defined $\bar{K}_i$ and $q$ sufficiently large,
the matrix $\Phi(\tau )$ defined in \rep{phi}
converges to zero as fast as $\lambda^{\tau}$ does, it is sufficient to show that for $q$ sufficiently large,
$A_{V}(\tau )$ is a discrete-time stability matrix whose state-transition matrix converges to zero
as fast as $\lambda^{\tau}$ does. To accomplish this, use will be
made of the following results.
\begin{lemma} Let $M$ be an $m\times m$ row stochastic matrix whose transpose has a strongly connected graph.
There exists a diagonal matrix $\Pi_M$ whose diagonal entries are
positive for which the matrix $L_M= \Pi_M- M'\Pi_M M $ is positive
semi-definite; moreover $L_M\mathbf{1} = 0$ where
$\mathbf{1}$ is the $m$-vector of $1$s. If, in addition,
the diagonal entries of $M$ are all positive, then
the kernel of $L_M$ is one-dimensional.
\label{brian} \end{lemma}
\noindent{\bf Proof of Lemma \ref{brian}:} Since $M$ is a stochastic
matrix, it must have
a spectral radius of $1$ and an eigenvalue at $1$ as must $M'$. Moreover, since
the graph of $M'$ is strongly
connected, $M'$ is
irreducible \{Theorem 6.2.24, \cite{horn1}\}. Thus by the Perron-Frobenius Theorem there
must be a positive vector $\pi$
such that $M'\pi= \pi$. Without loss of generality, assume
$\pi$ is normalized so that the sum of its entries equals $1$; i.e., $\pi$ is a probability vector.
Let $\Pi_M $ be that diagonal matrix
whose diagonal entries are the entries of $\pi$. Then $\Pi_M\mathbf{1} = \pi$.
Since $M\mathbf{1} = \mathbf{1}$, $\Pi_M \mathbf{1} = \pi$, and
$M'\pi = \pi$, it must be true that $M'\Pi_M M\mathbf{1} = \pi$
and thus that $L_M\mathbf{1} = 0$. To show that $L_M$ is
positive-semidefinite note first that
$L_M$ can also be written as $L_M=D-\hat{A}$ where $D$ is a
diagonal matrix whose diagonal entries are the diagonal
entries of $L_M$
and $\hat{A}$ is the nonnegative matrix $\hat{A} = D- L_M$.
As such, $L_M$ is the generalized Laplacian \cite{graph}
of that
simple undirected graph $\mathbb{G}$ whose adjacency matrix is the matrix which results when the
nonzero entries $a_{ij}$
in $\hat{A}$ are replaced by ones. Since $L_M$ can also be written as
$$L_M=\sum_{(i,j)\in\scr{E}} a_{ij}(e_i-e_j)(e_i-e_j)'$$ where $e_i$ is the $i$th unit vector
and $\scr{E}$ is the edge
set of $\mathbb{G}$, $L_M$ is
positive semi-definite as claimed.
Now suppose that the diagonal entries of $M$ are all positive. Then
the diagonal entries of $M'\Pi_M $ must also all be positive. It
follows that
every arc in the graph of $M'$ must be an arc in the graph of $M'\Pi_M M$ so the graph of $M'\Pi_M M$ must
be strongly connected. Since $I-\Pi_M$ is a nonnegative matrix, the graph of $M'\Pi_M M$ must be
a spanning subgraph
of the graph of $I-\Pi_M +M'\Pi_M M$.
Since $I-L_M = I-\Pi_M +M'\Pi_M M$ and the graph of $M'\Pi M$ is strongly connected, the graph of $I-L_M$ must
be strongly connected as well.
But $I-L_M$ is a nonnegative matrix so it must be irreducible. In
addition, since
$(I-L_M)\mathbf{1} = \mathbf{1}$, the row sums of $(I-L_M)$ all equal one. Therefore the infinity norm of
$I-L_M$ is one so its spectral radius is no greater than $1$. Moreover $1$ is an eigenvalue of $I-L_M$.
Thus
by the Perron-Frobenius Theorem, the geometric multiplicity of this
eigenvalue is one. It follows that the geometric multiplicity of the
eigenvalue of $L_M$ at $0$ is also one;
ie, the dimension of the kernel of $L_M$ is one as claimed. \rule{.1in}{.1in}
\begin{proposition}
For each fixed value of $\tau$, \eq{(V'\bar{S}(\tau)V)'
R(\tau)(V'\bar{S}(\tau)V) - R(\tau) < 0\label{ly}} where $R(\tau )$
is the positive definite matrix, $R(\tau) = V'(\Pi_{S(\tau)}
\otimes I_n)V$.
\label{mp} \end{proposition}
Note that \rep{ly} shows that for each fixed $\tau$, $x'R(\tau)x$ a
discrete-time Lyapunov function for the equation $w(k+1)=V'\bar
S(\tau)Vw(k)$. Thus for fixed $\tau$,
$V'\bar{S}(\tau)V$ is a discrete-time stability matrix.
\noindent{\bf Proof of Proposition \ref{mp}:} Fix $\tau $ and write
$S$ for $S(\tau)$ and $\bar{S}$ for $\bar{S}(\tau)$. Note that the
graph of $S'$, namely $\mathbb{N}$, is strongly connected.
In view of Lemma \ref{brian}, the matrix $L = \Pi_S-S'\Pi_S S$
is positive semi-definite and $L\mathbf{1} = 0$. Moreover, since
the diagonal entries of $S$ and thus $S'$ are all positive, the
kernel of $L$ is one-dimensional.
Write $R$ for $R(\tau )$.
To prove the proposition it is enough to show that the matrix \eq{
Q = R- (V'\bar{S}'V)R(V'\bar{S}V)\label{1}} is positive definite.
To proceed, set $\bar{L} = L\otimes I_n$ in
which case $\bar{L}$ is positive semi-definite because $L$ is. Moreover,
$\bar{L} = \bar{\Pi}- \bar{S}'\bar{\Pi }\bar{S}$ where $\bar{\Pi} = \Pi_{S}\otimes I_n$.
Note that that $VRV' = P\bar{\Pi}P$ where $P$ is the orthogonal
projection matrix $P=VV'$. Clearly $VRV' =
P\bar{\Pi}^{\frac{1}{2}}\bar{\Pi}^{\frac{1}{2}}P$. Note that both
$P$ and $\bar{\Pi}^{\frac{1}{2}}$ are block diagonal matrices with
corresponding diagonal blocks of the same size. Because of this and
the
fact that each diagonal block in $\bar{\Pi}^{\frac{1}{2}}$
is a scalar times and identity matrix, it must be true that $P$ and $\bar{\Pi}^{\frac{1}{2}}$ commute; thus
$P\bar{\Pi}^{\frac{1}{2}} = \bar{\Pi}^{\frac{1}{2}}P$. From this and
the fact that $P$ is
idempotent, it follows that
$VRV' =\bar{\Pi}^{\frac{1}{2}}P\bar{\Pi}^{\frac{1}{2}}$.
Clearly $\bar{\Pi}^{\frac{1}{2}}P\bar{\Pi}^{\frac{1}{2}}\leq
\bar{\Pi}^{\frac{1}{2}}\bar{\Pi}^{\frac{1}{2}}$ so $VRV' \leq
\bar{\Pi}$. It follows using \eqref{1} that $Q\geq
R-V'\bar{S}'\bar{\Pi}\bar{S}V = R+ V'\bar{L}V - V'\bar{\Pi}V $.
Therefore \eq{Q\geq V'\bar{L}V\label{water}} In view of this, to
complete the proof it is enough to show that $V'\bar{L}V$ is
positive definite.
Since $\bar{L}$ is positive semi-definite, so is $V'\bar{L}V$. To
show that $V'\bar{L}V$ is positive definite, let $z = \text{column}
\{z_1,z_2,\ldots, z_m\}$ be any vector such that $z'V'\bar{L}Vz=0$.
Then $\bar{L}Vz = 0$. Since the kernel of $L$ is spanned
$\mathbf{1}$, the kernel of $\bar{L}$ must be spanned by
$\mathbf{1}\otimes I_n$. It follows that
$V_iz_i=V_jz_j,\;i,j\in\mathbf{m}$. But because of joint
observability, $\bigcap_{i\in\mathbf{m}} \scr{V}_i = 0$ so $V_iz_i =
0,\;i\in\mathbf{m}$. Thus $z_i = 0,\;i\in\mathbf{m}$
so $z=0$.
Therefore $V'\bar{L}V$ is positive definite. Therefore $Q$ is
positive definite because of \rep{water}. From this and \rep{1}
it follows that \rep{ly} is true.
\rule{.1in}{.1in}
\section{Choosing $q$}\label{switch}
In what follows it will be assumed that each $\bar{K}_i$ has been
selected so that the the matrix
$\bar{A}_V$ defined by \rep{sunday}, is such that $\bar{A}_V^{\tau}$ converges
to zero as $\tau\rightarrow \infty$ as fast as $\lambda^{\tau}$ does.
This can be done using standard
spectrum assignment techniques to make the spectral radius of
$\bar{A}_V$ at least as small as $\lambda $.
In view of \rep{as}, it is clear that to assign the convergence rate of the state transition matrix of
$\bar{A}(I_{mn}-P(I_{mn}-\bar{S}(\tau)))^q$ it is necessary and
sufficient to control the convergence rate
of the state transition matrix of $A_{V}(\tau)$. This can be accomplished by choosing $q$ sufficiently large.
There are two different ways to do this, each utilizing a different
matrix norm. Both approaches
will be explained next
using the abbreviated notation $B(\tau) = V'\bar{S}(\tau)V$; note
that with this simplification, $A_V(\tau) = \tilde{A}B^q(\tau)$
because of \rep{ench}.
\subsection{Weighted Two-Norm}
For each fixed $\tau$ and each appropriately-sized matrix $M$, write
$\|M\|_{R(\tau)}$ for the matrix norm induced by the vector norm
$\|Mx\|_{R(\tau)} \stackrel{\Delta}{=} \sqrt{x'R(\tau )x}$. Note that
$\|M\|_{R(\tau)}$ is the largest singular value of
$R^{\frac{1}{2}}(\tau)MR^{-\frac{1}{2}}(\tau)$.
Note in addition that
$$ (R^{\frac{1}{2}}(\tau)B(\tau)R^{-\frac{1}{2}}(\tau))'
(R^{\frac{1}{2}}(\tau)B(\tau) R^{-\frac{1}{2}}(\tau)) < I
$$
because of \rep{ly}. This shows that the largest singular value of
$R^{\frac{1}{2}}(\tau)B(\tau) R^{-\frac{1}{2}}(\tau)$ is less than
one. Therefore \eq{\|B(\tau)\|_{R(\tau)}<1\label{ino}}
\subsubsection{$\mathbb{N}$ is constant}
In this case both $B(\tau)$ and $R(\tau)$ are constant so it is
sufficient so choose
choose $q$ so that
$\|\tilde{A}B^q(\tau)\|_{R(\tau)}\leq \lambda $. Since
$\|\cdot\|_{R(\tau )}$ is submultiplicative, this can be done by
choosing $q$ so that
$$\|B(\tau)\|^q_{R(\tau)} \leq \frac{\lambda}{\;\;\;\;\;\|\tilde{A}\|_{R(\tau)}}$$
This can always be accomplished because of \rep{ino}.
\subsubsection{$\mathbb{N}$ changes with time}
In this case it is not possible to use the weighted two-norm
$\|\cdot\|_{R(\tau )}$ because it is time-dependent. A simple fix,
but perhaps not the most efficient one, would be to use
the standard two-norm $\|\cdot\|_2$ instead since it does not depend on time.
Using this approach, the first step would be to
first choose, for each fixed $\tau$,
an integer $p_1(\tau )
$ large enough so that $\|B^{p_1(\tau)}(\tau )\|<1$. Such values of
$p_1(\tau)$ must exist because each $B(\tau)$ is
a discrete-time stability
matrix or equivalently, a matrix with a spectral radius less than $1$.
Computing such a value amounts to looking at the
largest singular value of $B^{p_1(\tau)}(\tau)$ for successively largest values of $p_1(\tau)$
until that singular value is less than $1$. Having accomplished this, a number $p$ can easily be computed
so that $\|B^p(\tau )\|<1\;\forall \tau$ since there are only a finite number of
distinct strongly connected graphs on $m$ vertices and consequently only a finite number of distinct
matrices $B(\tau)$ in the set $\scr{B} = \{B(\tau ):\tau\geq 0\}$.
Choosing $p$ to be the maximum of $p_1(\tau)$ with respect to $\tau
$ is thus a finite computation. The next step would be to compute
an integer $\bar{p}$
large enough so that each $\|\tilde{A}(B^{p}(\tau))^{\bar{p}}\|_2\leq \lambda$.
A value of $q$ with the
required property would then be $q = p\bar{p}$.
\subsection{Mixed Matrix Norm }
There is a different way to choose $q$ which does
not make use of either Lemma \ref{brian} or Proposition \ref{mp}. The approach exploits the
``mixed matrix norm'' introduced in \cite{lineareqn}. To define this norm requires several steps. \;
To begin, let $\|\cdot\|_{\infty}$
denote the standard induced
infinity norm
and write
${\rm I\!R}^{mn\times mn}$ for the vector space of all $m\times m$ block matrices $M = \matt{M_{ij}}$
whose $ij$th entry is a matrix $M_{ij}\in{\rm I\!R}^{n\times n}$. With
$n_i = \dim \scr{V}_i,\;i \in\mathbf{m},$ and $\bar{n} =
n_1+n_2+\cdots n_m$,
write ${\rm I\!R}^{mn\times \bar{n}}$
for the vector space of all $m\times m$ block matrices $M =
\matt{M_{ij}}$ whose $ij$th entry is a matrix
$M_{ij}\in{\rm I\!R}^{n\times n_j}$. Similarly write ${\rm I\!R}^{ \bar{n}\times
mn}$ for the vector space of all $m\times m$ block matrices $M =
\matt{M_{ij}}$ whose $ij$th entry is a matrix
$M_{ij}\in{\rm I\!R}^{n_i\times n}$. Finally write
${\rm I\!R}^{\bar{n}\times \bar{n}}$ for the vector space of all $m\times m$ block matrices $M = \matt{M_{ij}}$
whose $ij$th entry is a matrix $M_{ij}\in{\rm I\!R}^{n_i\times n_j}$.
Note that $B\in {\rm I\!R}^{mn\times mn}$, $\tilde{A}\in {\rm I\!R}^{\bar{n}\times
\bar{n}}$, $V\in {\rm I\!R}^{mn\times \bar{n}}$, and $V'\in{\rm I\!R}^{\bar{n}\times
mn}$.
For $M$ in any one of these four spaces, the {\em mixed matrix norm } \cite{lineareqn} of $M$, written $\|M\|$, is
\eq{\|M\| = \|\langle M\rangle \|_{\infty}\label{mmn}}
where $\langle M\rangle $ is the matrix in ${\rm I\!R}^{m\times m}$ whose $ij$th entry is $\|M_{ij}\|_2$.
It is very easy to verify that $\|\cdot\|$ is in fact a norm. It is
even sub-multiplicative whenever matrix multiplication is defined.
Note in addition that $\|V\| = 1$ and $\|V'\| = 1$ because the
columns of each $V_i$ form an orthonormal set.
Recall that $P = VV'$ is an orthogonal projection matrix. Using
this, the definition of $B(\tau)$ and the fact that $PV = V$, it is
easy to see that for any integer $p>0$
$$B^{p}(\tau) =V'(P\bar{S}(\tau)P)^pV $$
Thus
$$ \|B^p(\tau)\|\leq \|(P\bar{S}(\tau)P)^{p}\|$$
Using this and the fact that the graph of $S'$ is strongly
connected, one can conclude that
$$\|(P\bar{S}(\tau)P)^{p}\|<1,\;\;\;p\geq (m-1)^2$$
This is a direct consequence of Proposition
2 of \cite{lineareqn}. Thus
\eq{\|B^p(\tau)\| <1 ,\;\;\;p\geq (m-1)^2\label{rajit}}
\subsubsection{$\mathbb{N}$ is constant}
In this case $B(\tau)$ is constant so it is sufficient to choose
$q$ so that
$\|\tilde{A}B^q(\tau)\|\leq \lambda $.
This can be done by choosing $q = p\bar{p}$ where $p\geq (m-1)^2$
and $\bar{p}$ is such that
\eq{\|B^p(\tau)\|^{\bar{p}} \leq \frac{\lambda}{\|\tilde{A}\|}\label{rajit2}}
This can always be accomplished because of \rep{rajit}.
\subsubsection{$\mathbb{N}$ changes with time}
Note that \rep{rajit} holds for all $\tau $. Assuming $p$ is chosen
so that $p\geq (m-1)^2$ it is thus possible to find, for each
$\tau$, a positive integer $\bar{p}(\tau)$, for which
\eq{\|B^p(\tau)\|^{\bar{p}(\tau)} \leq
\frac{\lambda}{\|\tilde{A}\|}\label{rajit3}}
Having accomplished this, a number $\bar{p}$ can easily be computed
so that
\eq{\|B^p(\tau)\|^{\bar{p}} \leq
\frac{\lambda}{\|\tilde{A}\|}\label{rajit3}} holds for all $\tau$,
since there there are only a finite number of
distinct strongly connected graphs on $m$ vertices and consequently only a finite number of distinct
matrices $B(\tau)$ in the set $\scr{B} $ defined earlier.
Choosing $\bar{p}$ to be the maximum of
$\bar{p}(\tau)$ with respect to $\tau $ is thus a finite
computation.
A value of $q$ with the
required property would then be $q = p\bar{p}$.
\section{Concluding Remarks}
The state estimator developed in this paper relies on an especially
useful observation about distributed observer structure
first noted in \cite{Kim2016CDC} and subsequently exploited in \cite{trent} and \cite{ACC19}.
Just how much further this idea
can be advanced remains to be seen. For sure, the synchronous switching upon which the local
estimators in this paper depend, can be relaxed by judicious application
of the mixed matrix norm discussed here. This generalization will be addressed in a future paper.
\bibliographystyle{unsrt}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,802 |
See Previously Unreleased Photos from 'The Last Waltz'
By newadmin / Published on Wednesday, 25 Nov 2020 13:35 PM / No Comments / 30 views
On Thanksgiving Day 1976, the Band and an array of special guests famously gathered at the Winterland Ballroom in San Francisco to play their final show, The Last Waltz. Forty-four years later, on the same day the concert took place — and one day before Thanksgiving 2020 — the Morrison Hotel Gallery is launching a new online exhibition featuring never-before-seen photographs from that concert and other moments in the Band's history.
The exhibit launches Wednesday, November 25th, and features photos by Elliot Landy, Ed Perlstein, Neil Preston, Ken Regan, Norman Seeff, and Barrie Wentzell. While the focus is on The Last Waltz, the exhibit provides a broader photographic history of the Band and the years leading up to that famous show, with shots from past tours, rehearsals, and recording sessions.
A handful of select photographs from the exhibition are available to peruse exclusively on Rolling Stone, starting with an Elliot Landy infrared film shot of the Band in 1969 at Richard Manuel and Garth Hudson's house above the Ashokan reservoir in Woodstock, New York.
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Lily Allen: I considered taking heroin during Miley Cyrus tour | Ents & Arts News
January 17, 2021, No Comments on Lily Allen: I considered taking heroin during Miley Cyrus tour | Ents & Arts News | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,194 |
Q: Bottom Navigation Bar not shown in activity I have an activity with a title, a fragment with a recyclerview full of cards and a bottom navigaton bar.
I cannot manage to show all 3 together without problems, the closest I have been is showing all 3, but with the bar taking space from the cards space.
Right now this is the XML which I thought should be working:
<LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:app="http://schemas.android.com/apk/res-auto"
xmlns:tools="http://schemas.android.com/tools"
android:layout_width="match_parent"
android:layout_height="match_parent"
android:orientation="vertical"
tools:context=".habits.HabitsActivity">
<ImageView
android:id="@+id/imageView"
android:layout_width="313dp"
android:layout_height="166dp"
android:layout_gravity="center_horizontal"
app:srcCompat="@drawable/goodreasons" />
<android.support.design.widget.CoordinatorLayout
android:id="@+id/coordinatorLayout"
android:layout_width="match_parent"
android:layout_height="match_parent">
<FrameLayout
android:id="@+id/contentFrame"
android:layout_width="match_parent"
android:layout_height="match_parent"></FrameLayout>
</android.support.design.widget.CoordinatorLayout>
<FrameLayout
android:layout_width="match_parent"
android:layout_height="match_parent"
android:layout_gravity="bottom">
<com.aurelhubert.ahbottomnavigation.AHBottomNavigation
android:id="@+id/bottom_navigation"
android:layout_width="match_parent"
android:layout_height="wrap_content"
android:layout_gravity="bottom" />
</FrameLayout>
</LinearLayout>
But this gives this output:
This is the xml of the fragment, in case you think it is relevant too.
<?xml version="1.0" encoding="utf-8"?>
<LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:app="http://schemas.android.com/apk/res-auto"
xmlns:tools="http://schemas.android.com/tools"
android:layout_width="match_parent"
android:layout_height="match_parent"
android:layout_gravity="center"
android:orientation="vertical"
android:padding="5dp"
>
<android.support.v7.widget.RecyclerView
android:id="@+id/rv_list"
android:layout_width="wrap_content"
android:layout_height="match_parent"
app:layout_constraintBottom_toBottomOf="parent"
app:layout_constraintEnd_toEndOf="parent"
app:layout_constraintHorizontal_bias="0.0"
app:layout_constraintStart_toStartOf="parent"
app:layout_constraintTop_toTopOf="parent"
app:layout_constraintVertical_bias="0.0"
android:layout_gravity="center"
/>
</LinearLayout>
A: Here's your layout, stripped down:
<LinearLayout
android:layout_width="match_parent"
android:layout_height="match_parent"
android:orientation="vertical">
<ImageView
android:layout_width="313dp"
android:layout_height="166dp"/>
<android.support.design.widget.CoordinatorLayout
android:layout_width="match_parent"
android:layout_height="match_parent"/>
<FrameLayout
android:layout_width="match_parent"
android:layout_height="match_parent"/>
</LinearLayout>
Your problem is that the CoordinatorLayout has a match_parent height, so it will take up all of the space that the ImageView doesn't. That leaves no space at all for the FrameLayout, so you won't be able to see anything inside of it.
Probably the best thing to do would be to use layout_weight on the CoordinatorLayout to make it take up as much space as it can while still leaving room for other views. That would look something like this:
<LinearLayout
android:layout_width="match_parent"
android:layout_height="match_parent"
android:orientation="vertical">
<ImageView
android:layout_width="313dp"
android:layout_height="166dp"/>
<android.support.design.widget.CoordinatorLayout
android:layout_width="match_parent"
android:layout_height="0dp"
android:layout_weight="1"/>
<FrameLayout
android:layout_width="match_parent"
android:layout_height="wrap_content"/>
</LinearLayout>
The only changes I made were to set the CoordinatorLayout's height to 0dp and the FrameLayout's height to wrap_content, and to add the weight attribute to the CoordinatorLayout.
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Q: How to display specific cell values while using a dictionary in vba? I am slightly new to dictionary usage so appreciate any help extended in this area
I have employee related details in an excel sheet. i.e Col A to Col BD. The key created is a combination of Name, Dept and Grade.
I need to display various attributes of the employee such as DOJ, Training undertaken etc when user selects the employee displayed in the combo box of the Form.
I have managed to create the key in the dictionary. If i read a particular row(no) as Value and Key as the above combination how do i then display specific attributes of the employee selected. Listbox displays the same combination as the key created i.e Name, Dept and Grade
Any ideas suggestions on this is appreciated
I have done this using arrays but when the size of the range increases array search slows down the overall performance Hence was looking at using dictionary
varArray_TP is the array which contains all the data. A Listbox displays Name, Grade and Dept of the employee. This combination is stored in the array with | as the separator. When user selects an option from the listbox this is checked against the key in the array to display other attributes from the array.
' loading the data into the array
For i = 1 To TotalRows_TP
For j = 1 To TotalCols_TP
varArray_TP(i, j) = Sheets("TP").Cells(i + 1, j).Value
Next j
Next i
' generating the key for each row in the array
For i = 1 To TotalRows_TP
varArray_TP(i, TotalCols_TP + 1) =
WorksheetFunction.Concat( _
ThisWorkbook.Sheets("TP").Range("C" & i + 1).Value, "|", _
ThisWorkbook.Sheets("TP").Range("D" & i + 1).Value, "|", _
ThisWorkbook.Sheets("TP").Range("AP" & i + 1).Value _
)
Next i
' find the data based on the selected key
Sub Profile_Details()
Dim i As Integer
Dim j As Integer
With UserForm1
If .vLB_SearchValues.ListIndex > -1 And .vRb_ByName.Value = True Then
For i = 1 To TotalRows_TP
If varArray_TP(i, TotalCols_TP + 1) = WorksheetFunction.Concat( _
.vLB_SearchValues.List(.vLB_SearchValues.ListIndex, 0), "|", _
.vLB_SearchValues.List(.vLB_SearchValues.ListIndex, 1), "|", _
.vLB_SearchValues.List(.vLB_SearchValues.ListIndex, 2) _
) Then
.vLB_Grade.Caption = varArray_TP(i, 4)
.vLB_PositionTitle.Caption = varArray_TP(i, 33)
.vLB_Dateentered_JobGrade.Caption = varArray_TP(i, 7)
.vLB_PersonnelArea.Caption = varArray_TP(i, 42)
.vLB_Dateentered_Position.Caption = varArray_TP(i, 6)
.vLB_BizRating.Caption = varArray_TP(i, 25)
.vLB_PplRating.Caption = varArray_TP(i, 26)
.vLB_UltimatePotential.Caption = varArray_TP(i, 10)
.vLB_Name.Caption = varArray_TP(i, 3)
UserForm2.vLB_TalentNaForm2.Caption = "Talent Name : " & .vLB_Name.Caption
End If
Next i
End If
End With
End Sub
Pasted below is the sample data table (tab separated):
User ID Name - LFM Job Grade Comp Grade Date Entered Position Date Entered Job Grade Personnel Area
T22MXIQ Pillay Avinash 8 07 7-1-2015 7-1-2015 PU03 - JD India Technology Center
B19CXIV Baishali 08 07 7-1-2015 7-1-2015 PU03 - JD India Technology Center
B22MXIR Bhargavi 08 07 7-1-2015 7-1-2015 PU03 - JD India Technology Center
T22MXIG Ghosh Anurag 09 07 7-1-2015 7-1-2015 PU03 - JD India Technology Center
PU00177 Kadam Ashok 08 07 7-1-2015 7-1-2015 PU03 - JD India Technology Center
PU00178 Kadam Sushma 03 07 7-1-2015 7-1-2015 PU03 - JD India Technology Center
PU00180 Kadam Sushma 08 07 7-1-2015 7-1-2015 PU03 - JD India Technology Center
M22MXIL Mugdha 08 07 7-1-2015 7-1-2015 PU03 - JD India Technology Center
PU00179 Pandey Shobha 06 07 7-1-2015 7-1-2015 PU03 - JD India Technology Center
T22MXIS Sanjyot 03 07 7-1-2014 7-1-2018 PU02 - JD China Technology Center
T22MXIP Shah Pradnesh 07 07 7-1-2015 7-1-2015 PU03 - JD India Technology Center
T22MXPS Shah Pradnesh 06 07 7-1-2015 7-1-2015 PU03 - JD India Technology Center
T22MXUS Sushma 03 07 7-1-2015 7-1-2015 PU03 - JD India Technology Center
T22MXVI Vrushali 08 08 7-1-2015 7-1-2015 PU02 - JD China Technology Center
I need to show only the list of employees that match the name typed in the textbox using a string search so I used the below code. In case you have a better approach it would be welcome as I am still looping through the keys so I am guessing if there is an optimum search ability where i do not loop through the keys multiple times
str = UserForm1.vTB_SearchString.Text
UserForm1.vLB_SearchValues.Clear
With UserForm1
If .vRb_ByName.Value = True Then
UserForm1.vLB_SearchValues.ColumnWidths = "2.2 in; 0.5 in; 2.8 in"
For Each EmpKey In EmployeeDict.Keys
EmpParts = Split(EmpKey, "|")
If InStr(1, EmpParts(0), str, vbTextCompare) Then
.vLB_SearchValues.AddItem
.vLB_SearchValues.List(.vLB_SearchValues.ListCount - 1, 0) = EmpParts(0)
.vLB_SearchValues.List(.vLB_SearchValues.ListCount - 1, 1) = EmpParts(1)
.vLB_SearchValues.List(.vLB_SearchValues.ListCount - 1, 2) = EmpParts(2)
End If
Next
A: You can certainly use a Dictionary here, and I would even go so far as to say you should. Not for performance (although a Dictionary is more performant), but because it's simpler by far to get back a value from a Dictionary than to loop over an array searching for a matching key.
But I would suggest the main cause of the performance issues is this: when you use Excel objects (worksheets, ranges, worksheet functions) and their properties and methods, it's generally orders of magnitude slower than using pure VBA alternatives. Specifically:
*
*Instead of loading the data by looping over the cells with varArray_TP(i, j) = Sheets("TP").Cells(i + 1, j).Value in a loop, the Excel Range object supports creating a 2-dimensional array directly:
Dim varArray_TP As Variant
varArray_TP = Sheets("TP").Range("R1C1:R" & TotalRows_TP & "C" & TotalCols_TP)
Admittedly, the dimensions of this array are fixed based on the range size, and we won't have an additional column to store the key in the array; but if we use a Dictionary we shouldn't have to.
*Don't use WorksheetFunction.Concat. Instead, use the VBA string concatenation operator -- &. For example to generate the key for a given row number in the array:
key = varArray_TP(row, 3) & "|" & varArray_TP(row, 4) & "|" & varArray_TP(row, 42)
The following code shows how to use a Dictionary. Add a reference to Microsoft Scripting Runtime from Tools -> References.... Then, define your dictionary in the same place you've defined the array:
Dim dict As New Scripting.Dictionary
Then:
Dim wks As Worksheet
Set wks = Sheets("TP")
' load data into array
Dim varArray_TP As Variant
varArray_TP = wks.Range( _
wks.Cells(1, 1), _
wks.Cells(TotalRows, TotalCols) _
)
' fill the dictionary
' the key for each entry will be the Name, Grade and Dept, separated by |
' the corresponding value will be the first index of that person's data within the array
Dim rows as Long
rows = UBound(varArray_TP, 1)
Dim row As Long, key As String
For row = 1 To rows
key = varArray_TP(row, 3) & "|" & varArray_TP(row, 4) & "|" & varArray_TP(row, 42)
dict(key) = row
Next
There's still a piece missing: how do you populate the listbox with the keys? You might do something like this:
Dim key As Variant
For Each key In dict.Keys
Dim parts() As String
parts = Split(key, "|")
' parts is now a 1-dimensional array, with the Name, Dept and Grade
' TODO fill the listbox with the individual parts
Next
Then, when you want to retrieve the details, you could do something like this:
' find the data based on the selected key
Sub Profile_Details()
With UserForm1
If .vLB_SearchValues.ListIndex > -1 And .vRb_ByName.Value = True Then
Dim row As Long
row = dict( _
.vLB_SearchValues.List(.vLB_SearchValues.ListIndex, 0) & "|" & _
.vLB_SearchValues.List(.vLB_SearchValues.ListIndex, 1) & "|" & _
.vLB_SearchValues.List(.vLB_SearchValues.ListIndex, 2) _
)
.vLB_Grade.Caption = varArray_TP(row, 4)
.vLB_PositionTitle.Caption = varArray_TP(row, 33)
.vLB_Dateentered_JobGrade.Caption = varArray_TP(row, 7)
.vLB_PersonnelArea.Caption = varArray_TP(row, 42)
.vLB_Dateentered_Position.Caption = varArray_TP(row, 6)
.vLB_BizRating.Caption = varArray_TP(row, 25)
.vLB_PplRating.Caption = varArray_TP(row, 26)
.vLB_UltimatePotential.Caption = varArray_TP(row, 10)
.vLB_Name.Caption = varArray_TP(row, 3)
UserForm2.vLB_TalentNaForm2.Caption = "Talent Name : " & .vLB_Name.Caption
End If
End With
End Sub
After having written all this, I realized you can do something simpler: you could store the row number in the listbox itself, as a hidden column, and use that number to extract data from the array. This would mean you don't need a dictionary.
For more information about Scripting.Dictionary, see here.
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\section{Introduction}
\label{sect:introduction}
According to quantum information theory, quantum information is
represented by states of a number of qudits, which are idealized
$d$-level quantum systems. Here, we consider finite quantum
information and do not explicitly refer to the underlying
tensor-product structure of the state space of more than one
qudit. Thus, we consider quantum information represented by the states
of an ideal system $\mathsf{I}$ whose state space is determined by a
Hilbert space ${\cal H}_{\mathsf{I}}$ of finite dimension $N$. In order to
realize quantum information in a physical system $\mathsf{P}$, two
problems need to be solved. The first is to determine the ways in
which the state space of $\mathsf{P}$ can usefully encode the desired
quantum information, the second is to determine which among these ways
can be best protected from decoherence due to the dynamics of
$\mathsf{P}$ and its interactions. If quantum information is intended
for use in quantum algorithms, a third problem is to ensure that the
dynamics of the system and its interactions can be used to implement
the desired quantum gates. Here we consider the first two
problems. The mathematical conventions for objects and notations are
explained at the end of the Introduction.
\subsection{Encoding Quantum Information and the Subsystems Principle}
\label{sect:encoding}
A general method for realizing or encoding quantum information is as a
subsystem of ${\cal H}_{\mathsf{P}}$~\cite{viola:qc2000c}. This method
involves a decomposition of ${\cal H}_{\mathsf{P}}$ as
\begin{equation}
{\cal H}_{\mathsf{P}} = \left({\cal H}_{\mathsf{I'}}\tensor{\cal H}_{\mathsf{S}}\right)
\oplus
{\cal H}_{\mathsf{R}},
\label{eq:subenc}
\end{equation}
where $\oplus$ denotes the orthogonal sum of Hilbert spaces. This
decomposition identifies $\mathsf{I'}$ as a subsystem of $\mathsf{P}$,
written as $\mathsf{I'}\hookrightarrow\mathsf{P}$. We use ``primes''
(as in $\mathsf{I'}$) for systems with identical state spaces. With
this convention, Eq.~(\ref{eq:subenc}) yields a representation of the
states of the ideal quantum information system $\mathsf{I}$ in a
subsystem of ${\cal H}_{\mathsf{P}}$. We say that $\mathsf{I}$ is
``encoded'' in $\mathsf{P}$ and call the decomposition of
Eq.~(\ref{eq:subenc}) a ``subsystem encoding''. For lack of a better word,
we refer to $\mathsf{S}$ as the ``cosubsystem'' of $\mathsf{I'}$ in
$\mathsf{P}$. $\mathsf{R}$ is the ``remainder system''. An example
of such an encoding is that of a vibrational qubit of a single ion
trapped in a one-dimensional harmonic potential. In this case the state
space is spanned by the internal and vibrational levels of the
ion. The space ${\cal H}_{\mathsf{I'}}\tensor {\cal H}_{\mathsf{S}}$ is formed
from the first two vibrational levels and the internal levels of the
ion, respectively. The space ${\cal H}_{\mathsf{R}}$ consists of states
with more than two vibrational quanta irrespective of the internal
state of the ion. The familiar cases of such encodings have the
property that ${\cal H}_{\mathsf{I'}}$ belongs to a physically meaningful
degree of freedom. However, for the purpose of protecting against
errors, it is usually necessary to use ``entangled'' encodings. An
example is the noiseless qubit encoded in three spin-$1/2$ particles
subject to collective decoherence~\cite{knill:qc1999b}. A feature of
subsystem encoding is that the states of $\mathsf{I}$ are not uniquely
encoded as states of $\mathsf{P}$. This is because any change of state
of the cosubsystem $\mathsf{S}$ does not affect the states of
$\mathsf{I'}$.
Subsystem encodings of $\mathsf{I}$ in $\mathsf{P}$ are equivalent to
$\dagger$-preserving isomorphic embeddings of $B({\cal H}_{\mathsf{I}})$
into $B({\cal H}_{\mathsf{P}})$~\cite{knill:qc1999b,zanardi:qc1999c}. Here
$B({\cal H})$ denotes the set of (bounded) operators on ${\cal H}$. In
particular, given the subsystem encoding of Eq.~(\ref{eq:subenc}),
$B({\cal H}_{\mathsf{I}})$ is isomorphic to the algebra of operators of the
form $A\tensor \mathbf{I} \oplus \mathbf{0}$ with $A$ acting on
${\cal H}_{\mathsf{I'}}$, $\mathbf{I}$ on ${\cal H}_{\mathsf{S}}$ and $\mathbf{0}$ on
${\cal H}_{\mathsf{R}}$, where the operators are transported to
${\cal H}_{\mathsf{P}}$ via the isomorphism implicit in the subsystem
decomposition as needed. Conversely, if ${\cal A}$ is a subalgebra of
bounded operators on ${\cal H}_{\mathsf{P}}$ and ${\cal A}$ is
$\dagger$-isomorphic to $B({\cal H}_\mathsf{I})$, then there is a unique
subsystem encoding such that the operators of ${\cal A}$ are the operators
of the form $A\tensor\mathbf{I} \oplus\mathbf{0}$ as above.
Are there ways of encoding quantum information that do not involve a
subsystem encoding? In an attempt to answer this question, it is
worth considering other prescriptions for encoding quantum
information. There are two operationally defined ways of
characterizing encoded information. The first is by a traditional
encoding operation that isometrically embeds ${\cal H}_{\mathsf{I}}$ into
${\cal H}_{\mathsf{P}}$. (Isometries are linear maps preserving the inner
product.) This is the prescription used in the traditional theory of
quantum error correction and corresponds to a subsystem encoding with
trivial cosubsystem. In this case, the subspace
${\cal H}_{\mathsf{I'}}\tensor{\cal H}_{\mathsf{S}} = {\cal H}_{\mathsf{I'}}$ of
${\cal H}_{\mathsf{P}}$ is known as a ``quantum code''. That this is
inadequate is apparent when one considers enlarging $\mathsf{P}$ by
other relevant degrees of freedom. Subspaces also fail to capture the
location of quantum information in realistic error-control settings,
in particular fault-tolerant quantum computing. This is because in
practice, error control never results in restoration of the encoded
quantum information to any fixed quantum code. Assuming that this is a
requirement leads to the conclusion that fault tolerance is not
possible~\cite{alicki:qc2004b}.
The second operational realization of quantum information involves
specifying an ideal decoding procedure. Such a decoding procedure
adjoins $\mathsf{I}$ and (possibly) an ancilla system $\mathsf{A}$ to
$\mathsf{P}$, where $\mathsf{I}$ and $\mathsf{A}$ are in specified
initial states $\kets{0}{I}$ and $\kets{0}{A}$. The total state space
is determined by
${\cal H}_{\mathsf{P}}\tensor{\cal H}_{\mathsf{I}}\tensor{\cal H}_{\mathsf{A}}$. The
decoding operation is a unitary operator on this state space. After it
is applied, the desired quantum information resides in $\mathsf{I}$.
The decoding-based view of quantum information has been used
successfully in analyses of fault-tolerant quantum architectures (see,
for example,~\cite{aliferis:qc2005b}). To connect the decoding-based
realization of quantum information to subsystems, note that the
decoding operation is, in effect, an isometry from an extended space
${\cal H}_{\mathsf{P_e}}\oplus{\cal H}_{\mathsf{T}}$ to
${\cal H}_{\mathsf{I}}\tensor{\cal H}_{\mathsf{U}}$. Here we have identified
${\cal H}_{\mathsf{P_e}}$ with
${\cal H}_{\mathsf{P}}\tensor\kets{0}{I}\tensor\kets{0}{A}$ and
${\cal H}_{\mathsf{U}}$ with ${\cal H}_{\mathsf{A}}\tensor{\cal H}_{\mathsf{P}}$.
Decoding-based realization is therefore equivalent to subsystem
encoding in an extension of the physical state space, where the
extension need not be physically meaningful. One can consider
generalizing decoding-based realizations by means of isometries
that provide the identification
\begin{equation}
{\cal H}_{\mathsf{P}}\tensor{\cal H}_{\mathsf{I}}\oplus{\cal H}_{\mathsf{T}}
={\cal H}_{\mathsf{I'}}\tensor{\cal H}_{\mathsf{S}}\oplus{\cal H}_{\mathsf{R}},
\end{equation}
which is obtained if the decoding operation also involves additional
physical systems in unspecified initial states. Although this is more
general than subsystem encoding, most such isomorphisms do not result
in quantum information that can be considered to be faithfully encoded
in system $\mathsf{P}$. We resolve this problem at the end of
Section~\ref{sect:faithful} by pairing decoding and encoding
operations.
A third approach to encoding of quantum information uses operators
to characterize quantum information and makes it possible to give a
reasonable definition of ``faithful encoding of $\mathsf{I}$ in
$\mathsf{P}$''. We give such a definition at the beginning of
Sect.~\ref{sect:faithful} and prove that every such encoding is
associated with a subsystem. The intuition is that the states of
$\mathsf{I}$ are characterized by the expectations of a linearly
closed set of observables ${\cal O}$. To ensure the correct dynamics of
these states, the complex multiples of the observables should form an
algebra $\dagger$-isomorphic to $B({\cal H})$. Thus, one expects that
faithful encoding of quantum information requires identifying a
$\dagger$-closed subalgebra ${\cal A}$ of $B({\cal H}_{\mathsf{P}})$ that is
isomorphic to $B({\cal H}_{\mathsf{I}})$. If this has been done, then the
representation theory of $\dagger$-closed algebras uniquely identifies
a decomposition ${\cal H}_{\mathsf{P}}=
{\cal H}_{\mathsf{I'}}\tensor{\cal H}_{\mathsf{S}}\oplus{\cal H}_{\mathsf{R}}$ such
that ${\cal A}$ consists of all the operators acting only on
${\cal H}_{\mathsf{I'}}$. The results of Sect.~\ref{sect:faithful} provide
support for the ``subsystems principle'' for realizing quantum
information:
\noindent\textbf{The subsystems principle:}
\emph{
Any faithful representation of quantum information in
a physical system requires that at every point in time
there are identifiable subsystems encoding the desired
quantum information.
}
The subsystem principle is powerful, but it is worth noting that it is
sometimes convenient to use realizations of quantum information that
do not satisfy this principle perfectly. For example, in optical
quantum computing with ``cat'' states, it is convenient to represent
the logical states of qubits by non-orthogonal coherent
states~\cite{ralph:qc2001b,ralph:qc2003a}. Another example is the
study of ``initialization-free'' decoherence-free subsystems, where
the probability amplitude of the encoded information may be less than
$1$ and the nature of the remaining amplitude must be taken into
account~\cite{shabani:qc2005a}.
\subsection{Protecting Quantum Information}
\label{sect:protect}
In most physical settings there are sources of errors that can affect
encoded quantum information. Ideally, we would like exact knowledge of
the error behavior of a physical system under all circumstances in
which it is used. Since this knowledge is usually unavailable, one of
a number of idealizing assumptions can be made. In the context of
quantum channels, or when unwanted interactions are expected to have
weak temporal correlations, we assume errors to be due to a known
markovian process (in the continuous time setting) or a known quantum
operation (in the discrete time setting). Both may be described by
a collection of possible error events ${\cal E}=\{E_i\}$. In general,
the goal of quantum error control is to find quantum information
subsystems for which the effects of the markovian process or quantum
operation can be suppressed to the largest extent possible. Because
the exact nature of the errors is usually not known, this goal is
typically difficult to pursue. To make the task more tractable, we can
consider only those errors that are expected to be likely and look for
subsystems that allow for ``good'' protection against such errors. We
can then bound the effect of other errors by making estimates of their
maximum probability (or amplitude) of occurrence.
In this paper, we focus on subsystems that enable perfect protection
against a fixed set of errors $\{E_i\}_i$, with or without active
intervention. Because of the linearity of quantum mechanics, perfect
protection against the $E_i$ implies perfect protection against any
error in the linear span ${\cal E}$ of the $E_i$. Subsystems whose states
are unaffected by the errors are known as ``noiseless'' or
``decoherence free'' subsystems~\cite{onns1} and were introduced
in~\cite{knill:qc1999b,zanardi:qc1999c} in the context of
$\dagger$-closed ${\cal E}$ (or the $\dagger$-closure of a
non-$\dagger$-closed ${\cal E}$), in which case they can be characterized
by irreducible representations of the commutant of ${\cal E}$,
which is the set of operators that commute with all members of
${\cal E}$. In general, noiseless subsystems are not as easily
characterized. In~\cite{kempe:qc2001a} an explicit characterization of
noiseless subsystems for any ${\cal E}$ is obtained. This characterization
is readily seen to be equivalent to the statement that the subsystem
$\mathsf{I'}$ of the decomposition ${\cal H}_{\mathsf{P}} =
{\cal H}_{\mathsf{I'}}\tensor{\cal H}_{\mathsf{S}}\oplus{\cal H}_{\mathsf{R}}$ is
noiseless if and only if the restriction of the $E_i$ to the subspace
${\cal H}_{\mathsf{I'}}\tensor{\cal H}_{\mathsf{S}}$ acts as
$\slb{\mathbf{I}}{I'}\tensor \slb{E'_i}{S}$. Several equivalent
characterizations for when ${\cal E}$ is the span of the operation elements
of a specific quantum operation were obtained
in~\cite{kribs:qc2004a,kribs:qc2005a}. These characterizations do not
directly address the question of how one can computationally search
for noiseless subsystems. A strategy for this search was offered
in~\cite{choi:qc2005b}. This strategy requires finding
${\cal E}$-invariant subspaces and decomposing them into the canonical
subsystems associated with the irreducible representations of a fixed-point algebra for a
quantum operation whose operation elements span ${\cal E}$. In
Sect.~\ref{sect:ns} we develop this strategy into an algorithm that
does not require explicit constructions of algebras other than that
generated by ${\cal E}$. The mathematical structure of algebras over the
complex numbers plays a crucial role. Interestingly, if there exists a
quantum operation whose operational elements span ${\cal E}$, then the
algorithm simplifies substantially and is efficient in the dimension
of the Hilbert space. Note that there is no a priori requirement that
the likely errors included in ${\cal E}$ be derived from a quantum
operation. However, in most cases ${\cal E}$ does satisfy this condition.
To ensure that this condition holds, one can add $\mathbf{I}-\lambda\sum_i
E_i^\dagger E_i$ for a sufficiently small $\lambda$, although the
choice of spanning set $E_i$ and $\lambda$ may affect the availability
of large-dimensional noiseless subsystems.
When no noiseless subsystem of sufficiently large dimension can be
found, it is necessary to use active intervention to protect encoded
quantum information. The idealized setting for active intervention
involves alternating steps consisting of error events $E_i$ and a
quantum operation ${\cal R}$ that ensures that the errors do not affect the
encoded information. An operation ${\cal R}$ with this property is known
either as a ``recovery'' or as an ``initialization'' operation,
depending on context. According to the subsystems principle, there
must be two subsystems, one in which quantum information resides after
error events but before ${\cal R}$ is applied, and another after ${\cal R}$ is
applied. We call the first a ``protectable'' subsystem. The second is
known as an ``error-correcting'' subsystem. Provided the encoded
quantum information has been successfully protected, both subsystems
are noiseless. The first is noiseless for the products $E_iR_j$,
where the $R_j$ are the operation elements of ${\cal R}$, whereas the
second is noiseless for the operators $R_jE_i$. Protectable
subsystems are defined (but not named) in~\cite{knill:qc1999b}, where
it was shown how to determine the protectable subsystem in the case
where the error-correcting subsystem is a quantum code, that is, the
cosubsystem is one-dimensional. Error-correcting subsystems are the
main feature of operator quantum error
correction~\cite{kribs:qc2004a,kribs:qc2005a} and directly generalize
traditional error-correcting codes.
Knowledge of the protectable subsystem and the error-correcting
subsystem associated with a recovery/initialization operation and the
relationship between the two helps us to understand how quantum
information is stored at all times. An advantage of the protectable
subsystem is that in many cases it is a simple extension of the
error-correcting subsystem. That is, the former's cosubsystem is a
consistent extension of the latter's cosubsystem. As a result, the
observables associated with the protectable subsystem induce the
correct observables on the error-correcting subsystem. This implies
that for the purpose of identifying the current value of the stored
quantum information, it suffices to know the protectable subsystem,
regardless of whether the last event was a recovery operation or an
error. Examples of this situation are stabilizer codes with decoding
algorithms based on syndrome extraction. It is readily verified that
the associated protectable subsystem contains the stabilizer code as a
subspace where the cosubsystem is in a particular state. In
particular, this property holds for the stabilizer-based
error-correcting subsystems identified in~\cite{poulin:qc2005a} and
used to simplify Shor's 9-qubit one-error-correcting
code~\cite{bacon:qc2005b}, except that the error-correcting subsystem
is defined by a subspace of the protectable subsystem's
cosubsystem. In general, this relationship between protectable and
error-correcting subsystems always holds if $\mathbf{I}\in{\cal E}$. It becomes
particularly useful in the context of fault-tolerant quantum
computation, where the recovery operation and error events can no
longer be easily separated. In this case the ideal error-correcting
codes or subsystems associated with a scheme are typically not where
quantum information resides. It resides in the protectable subsystems
of the scheme. Note that in this setting it is usually the case that
the subsystems containing quantum information vary in time. This
happens, for example, when teleportation is used for error correction,
when quantum information is stored in memory versus being actively
manipulated, and in cluster-state-based schemes as part of the
model~\cite{raussendorf:qc2001a}.
An advantage of error-correcting subsystems is that there are simple
criteria and algorithms for determining whether there exists an
associated recovery operation for which it becomes
noiseless~\cite{kribs:qc2004a,kribs:qc2005a,nielsen:qc2005b}. Not
having to specify the recovery operation simplifies the search for
subsystems suitable for protecting quantum information and makes it
natural to talk about error-correcting subsystems without specifying
the recovery operation. The same cannot be said for protectable
subsystems. In Sect.~\ref{sect:initialization} we partially remedy
this situation by reducing the problem of determining whether a
subsystem is protectable to a number of other problems not involving
the existence of a quantum operation.
\subsection{Conventions}
Capital letters in sans-serif font such as $\mathsf{A,...,H,...,P}$
are used to label quantum systems. The state space of a system
$\mathsf{A}$ is determined by a Hilbert space, denoted by
${\cal H}_{\mathsf{A}}$. We label states according to the quantum system
they belong to. For example, $\kets{\psi}{A}$ is a pure state of
$\mathsf{A}$ and $\slb{\rho}{A}$ is a density matrix for
$\mathsf{A}$. The tensor product symbol $\tensor$ may be omitted in
tensor products of labeled states and operators. We frequently
consider instances of identical state spaces realized by and in
different systems. We use primes to distinguish the different systems
with identical state spaces. Thus, $\mathsf{I}$, $\mathsf{I'}$ and
$\mathsf{I''}$ are systems whose state spaces are identified via
implicit isometries, which are inner-product-reserving linear maps.
In particular, a state $\kets{\psi}{I}$ of $\mathsf{I}$ is identified
with the states $\kets{\psi}{I'}$ and $\kets{\psi}{I''}$ of systems
$\mathsf{I'}$ and $\mathsf{I''}$. One way to interpret this is to
consider $\psi$ as a symbol labeling a vector in an appropriate
Hilbert space ${\cal H}$ and $\ket{\psi}\mapsto\kets{\psi}{I}$ as the
isometry identifying ${\cal H}$ and ${\cal H}_{\mathsf{I}}$. We use the
equality symbol ``$=$'' not just to denote strict mathematical
equality but also for identifying objects which are equal via an
isomorphism. The isomorphisms involved are defined only implicitly,
provided the meaning is clear. For a Hilbert space ${\cal H}$, $B({\cal H})$
denotes the algebra of operators of ${\cal H}$. $U({\cal H})$ denotes the group
of unitary operators of ${\cal H}$. In this work, all state spaces are
finite dimensional.
\section{Faithful Encodings of Quantum Information}
\label{sect:faithful}
To formalize the idea of ``faithful encoding'' we consider more
general ways of encoding quantum information. A faithful encoding of
$\mathsf{I}$ in $\mathsf{P}$ is a map $D$ from density operators $\rho$ on
${\cal H}_{\mathsf{I}}$ to non-empty sets of density operators on
${\cal H}_{\mathsf{P}}$ together with a map $O$ from observables (hermitian
operators) $A$ of ${\cal H}_{\mathsf{I}}$ to non-empty sets of observables of
an extension ${\cal H}_{\mathsf{Q}}$ of ${\cal H}_{\mathsf{P}}$ that satisfies three
faithfulness requirements:
\begin{itemize}
\item[1.] Statics: For all $\sigma\in D(\rho)$ and $X\in O(A)$,
\begin{equation}
\label{eq:statics}
\mbox{tr}(\sigma X)=\mbox{tr}(\rho A).
\end{equation}
This requirement ensures that we can identify the expectation
values of faithfully encoded states.
\item[2.] Unitary dynamics:
For all $\sigma\in D(\rho)$ and $X\in O(A)$,
\begin{equation}
\label{eq:udynamics}
e^{-iX}\sigma e^{iX} \in D(e^{-iA}\rho e^{iA}).
\end{equation}
With this requirement satisfied, we can evolve the states using
conventional quantum control so that the evolved states are consistent
with the first requirement.
\end{itemize}
For the next requirement, extend the domain of $D$ to all positive
semidefinite operators by defining $D(\mathbf{0}) = \{\mathbf{0}\}$ and for
$\rho\not=\mathbf{0}$, $D(\rho) = \mbox{tr}(\rho)D(\rho/\mbox{tr}(\rho))$. For
an operator $Z$, let $\Pi(Z,\lambda)$ be the projector onto the
$\lambda$-eigenspace of $Z$, or, equivalently, the projector onto the
null space of $Z-\lambda$. For $\lambda$ not in the spectrum of $Z$,
the projector is $\mathbf{0}$.
\begin{itemize}
\item[3.] Measurement dynamics: For all $\sigma\in D(\rho)$ and $X\in
O(A)$ and $\lambda$ real,
\begin{equation}
\label{eq:mdynamics}
\Pi(X,\lambda)\sigma\Pi(X,\lambda)\in
D(\Pi(A,\lambda)\rho\Pi(A,\lambda)).
\end{equation}
Faithful measurement dynamics ensures that projective
measurements can be implemented correctly.
\end{itemize}
The support of a positive semidefinite Hermitian operator $\rho$ is
the span of its non-zero eigenvalue eigenvectors and is denoted by
$\mbox{Supp}(\rho)$. For a set of such operators $D$, $\mbox{Supp}(D)$ is the
span of the supports of the members of $D$.
\begin{Theorem}
\label{thm:qi=subs}
If $D$ and $O$ are a faithful encoding of $\mathsf{I}$ in
$\mathsf{P}$, then one can identify a subsystem encoding
${\cal H}_{\mathsf{P}} = {\cal H}_{\mathsf{I'}}\tensor{\cal H}_{\mathsf{S}}\oplus
{\cal H}_{\mathsf{R}}$ such that for all $\rho$, $D(\rho)$ has support in
${\cal H}_{\mathsf{I'}}\tensor{\cal H}_{\mathsf{S}}$, and for all $A$,
${\cal H}_{\mathsf{I'}}\tensor{\cal H}_{\mathsf{S}}$ and ${\cal H}_{\mathsf{R}}$ are
invariant subspaces of $O(A)$, and $O(A)$ acts as $A'\tensor\mathbf{I}$ on
${\cal H}_{\mathsf{I'}}\tensor{\cal H}_{\mathsf{S}}$.
\end{Theorem}
The conclusion of the theorem does not hold if we assume only faithful
statics and faithful unitary dynamics. For example, any irreducible
representation of $U({\cal H}_{\mathsf{I}})$ leads to an encoding
satisfying these two faithfulness requirements, and such
representations that have dimension larger than $N$ exist. For
example, if $\mathsf{I}$ is a qubit, then any spin $>1/2$
representation of $SU(2)$ yields an encoding that lacks faithful
measurement dynamics. An other example is ensemble quantum computing
with pure or pseudo-pure
states~\cite{cory:qc1996a,chuang:qc1997a,knill:qc1998c}. In the case
of pseudo-pure states, faithful statics is only satisfied up to a
scale. Nevertheless, quantum information is still associated with
subsystems. It may be interesting to determine the nature of
encodings satisfying only faithful statics (perhaps weakened to allow
for an overall scale factor) and faithful unitary dynamics. Are they
always equivalent to a sum of subsystems transforming under distinct
irreducible representations of $U({\cal H}_{\mathsf{I}})$? On the other
hand, we conjecture that faithful statics and measurement dynamics
imply faithful unitary dynamics. However, our proof of the theorem
requires all three faithfulness properties.
An apparently more general faithfulness property, ``faithful
interactions'', requires that the encoding of $\mathsf{I}$ in
$\mathsf{P}$ behaves correctly in interactions with other idealized
systems. Faithful interactions are needed if we use the encoded
quantum information in a quantum information processing setting with
multiple physical systems, each encoding quantum information in some
way. Faithful measurement dynamics can be seen to be a special case of
faithful interactions, and, according to the theorem, it implies
faithful interactions in general.
\proofof{~\ref{thm:qi=subs}} Let ${\cal V}$ be the linear sum of the
supports of operators in $D(\rho)$ for all $\rho$. Let ${\cal V}^\perp$ be
its orthogonal complement. By assumption, ${\cal V}\subseteq
{\cal H}_{\mathsf{P}}$. The proof proceeds in three stages. In the first,
we show that the operators of $O(A)$ have ${\cal V}$ and ${\cal V}^\perp$ as
invariant subspaces. We can then redefine $O(A)$ by restricting its
operators to ${\cal V}$. ${\cal H}_{\mathsf{R}}$ is identified as
${\cal V}^\perp\cap{\cal H}_{\mathsf{P}}$. We then show that $O(A)$ consists of
exactly one operator and deduce that $O$ extends to an algebra
isomorphism when restricted to commuting subsets of observables. The
underlying reason for this involves showing that the eigenspaces of
$O(A)$ may be faithfully identified with eigenspaces of $A$. The
first two stages of the proof do not require faithful unitary
dynamics. The last stage involves analyzing $SU(2)$ subgroups of
$U({\cal H}_{\mathsf{I}})$ and corresponding subgroups of $U({\cal V})$ induced
by $O$. Their action on eigenspaces of operators in the range of $O$
implies the desired subsystem encoding.
For an operator $X$, let $\mbox{Null}(X)$ denote the null space of $X$.
\begin{Lemma}
\label{lemma:eigendec}
Let $X\in O(A)$. Then $\mbox{Null}(X-\lambda)\cap{\cal V}$ is non-empty if and
only if $\lambda$ is in the spectrum of $A$. Furthermore,
${\cal V}=\sum_{\lambda}\left(\mbox{Null}(X-\lambda)\cap{\cal V}\right)$, and
$\mbox{Null}(X-\lambda)\cap{\cal V}$ is the linear span of the supports of
$\rho\in D(\sigma)$ with $\mbox{Supp}(\sigma)\subseteq\mbox{Null}(A-\lambda)$.
\end{Lemma}
\paragraph*{Proof.} Suppose that $\lambda$ is in the spectrum of $A$, and consider
any $\rho\in D(\Pi(A,\lambda))$. By faithfulness of measurement
dynamics, $\Pi(X,\lambda)\rho\Pi(X,\lambda)\in D(\Pi(A,\lambda))$. By
faithfulness of statics, $\Pi(X,\lambda)\rho\Pi(X,\lambda)$ is not
zero. Since the support of $\Pi(X,\lambda)\rho\Pi(X,\lambda)$ is
contained in $\mbox{Null}(X-\lambda)\cap{\cal V}$, this intersection is
non-zero. Conversely, suppose that $\mbox{Null}(X-\lambda)\cap{\cal V}$ is
non-empty. Then there exist $\sigma$ and $\rho\in D(\sigma)$ such that
the support of $\rho$ is not orthogonal to $\mbox{Null}(X-\lambda)$. Thus
$\Pi(X,\lambda)\rho\Pi(X,\lambda)$ is not zero and is a member of
$D(\Pi(A,\lambda)\sigma\Pi(A,\lambda))$. Because
$D(\mathbf{0})=\{\mathbf{0}\}$, $\Pi(A,\lambda)\sigma\Pi(A,\lambda)$ is not
zero. Hence $\mbox{Null}(A-\lambda)$ is non-zero, so that $\lambda$ is in
the spectrum of $A$.
To prove that ${\cal V}$ is spanned by the subspaces $\mbox{Null}(X-\lambda)\cap{\cal V}$,
we use the following sequence of inclusions:
\begin{eqnarray}
{\cal V} &\supseteq&
\sum_{\lambda}\left(\mbox{Null}(X-\lambda)\cap{\cal V}\right) \nonumber\\
&\supseteq&
\sum_{\lambda}\sum_\sigma\sum_{\rho:\rho\in D(\sigma)}
\left(\mbox{Supp}(\Pi(X,\lambda)\rho\Pi(X,\lambda))\cap{\cal V}\right)\nonumber\\
&=&
\sum_\sigma\sum_{\rho:\rho\in D(\sigma)}\sum_{\lambda}
\left(\mbox{Supp}(\Pi(X,\lambda)\rho\Pi(X,\lambda))\cap{\cal V}\right)\nonumber\\
&\supseteq&
\sum_\sigma\sum_{\rho:\rho\in D(\sigma)}
\mbox{Supp}(\rho)\\
&=& {\cal V},
\label{eq:incseq}
\end{eqnarray}
where in each expression, $\lambda$ ranges over the spectrum of $A$.
The critical step in the sequence requires the inclusion
\begin{equation}
\sum_{\lambda}
\left(\mbox{Supp}(\Pi(X,\lambda)\rho\Pi(X,\lambda))\cap{\cal V}\right)
\supseteq \mbox{Supp}(\rho).
\end{equation}
To prove this inclusion, observe that
$\mbox{Supp}(\Pi(X,\lambda)\rho\Pi(X,\lambda))\subseteq{\cal V}$ because
$\Pi(X,\lambda)\rho\Pi(X,\lambda)\in
D(\Pi(A,\lambda)\rho\Pi(A,\lambda)$. Faithfulness of measurement
dynamics implies that $\Pi(X,\lambda)\rho\Pi(X,\lambda)=\mathbf{0}$ for
$\lambda$ not in the spectrum of $A$. It then suffices to recall that
for a complete set of orthogonal projectors $P_i$,
$\sum_i\mbox{Supp}(P_i\rho P_i)=\mbox{Supp}(\sum_i P_i\rho P_i)\supseteq
\mbox{Supp}(\rho)$. \ignore{ This is not entirely obvious, but I think,
well-known. One way to prove it is to show that if $\ket{x}$ is
orthogonal to the support of $\sum_i P_i\rho P_i$, then it is
orthogonal to the support of $\rho$. To prove this, it helps to
decompose $\rho=\sum_j\ketbra{y_j}$. Using positivity, one can see
that the first orthogonality statement implies that
$\sum_i\bra{x}P_i\ketbra{y_j}P_i\ket{x}=0$, from which one gets
$\bra{x}P_i\ketbra{y_j}P_i\ket{x}=0$ for each $i$, or $\bra{x}
P_i\ket{y_j}=0$ and because $\sum_iP_i=\mathbf{I}$, $\braket{x}{y_j} = 0$.
}
For the last claim of the lemma, let $W$ be the set of density
operators with support in $\mbox{Null}(A-\lambda)$. If $\sigma\in W$, then
$\Pi(A,\lambda)\sigma\Pi(A,\lambda)=\sigma$ and for all
$\lambda'\not=\lambda$, $\Pi(A,\lambda')\sigma\Pi(A,\lambda')=\mathbf{0}$.
Faithful measurement dynamics imply that for $\rho\in D(\sigma)$,
$\mbox{Supp}(\rho)\subseteq \mbox{Null}(X-\lambda)$. Thus, $\sum_{\sigma\in
W}\sum_{\rho\in D(\sigma)}\mbox{Supp}(\rho) \subseteq
\mbox{Null}(X-\lambda)\cap{\cal V}$. The following sequence of relationships
proves the reverse inclusion:
\begin{eqnarray}
\mbox{Null}(X-\lambda)\cap{\cal V} &=&
\mbox{Null}(X-\lambda)\cap
\left(\sum_\sigma\sum_{\rho:\rho\in D(\sigma)} \mbox{Supp}(\rho)\right)
\nonumber\\
&\subseteq&
\Pi(X,\lambda)
\left(\sum_\sigma\sum_{\rho:\rho\in D(\sigma)} \mbox{Supp}(\rho)\right) \nonumber \\
&=&\sum_\sigma\sum_{\rho:\rho\in D(\sigma)}
\mbox{Supp}(\Pi(X,\lambda)\rho\Pi(X,\lambda))\nonumber\\
&\subseteq&
\sum_\sigma\sum_{\rho:\rho\in D(\Pi(A,\lambda)\sigma\Pi(A,\lambda))}
\mbox{Supp}(\rho)\nonumber\\
&=&
\sum_{\sigma\in W}\sum_{\rho:\rho\in D(\sigma)}
\mbox{Supp}(\rho),
\end{eqnarray}
where we have used the fact that
for a projector $\Pi$ and a positive semidefinite hermitian operator
$\rho$, $\Pi\mbox{Supp}(\rho) = \mbox{Supp}(\Pi\rho\Pi)$.
\qed
\begin{Corollary}
\label{cor:invariant}
Let $X\in O(A)$. Then ${\cal V}$ and ${\cal V}^\perp$ are invariant
subspaces of $X$.
\end{Corollary}
\paragraph*{Proof.}
Because the eigenspaces of $X$ are orthogonal,
Lemma~\ref{lemma:eigendec} implies that $X$ can be block diagonalized
with respect to an orthonormal basis whose first members
span ${\cal V}$.
\qed
Lemma~\ref{lemma:eigendec} and Cor.~\ref{cor:invariant} imply that
without loss of generality, we can assume that for all $X\in O(A)$,
$X$ restricted to ${\cal V}^\perp$ is $\mathbf{0}$. If not, replace every
member of $O(A)$ with its restriction to ${\cal V}$. This does not affect
any of the faithfulness requirements.
The last statement in Lemma~\ref{lemma:eigendec} together with the
assumption that $X\in O(A)$ has trivial action on ${\cal V}^\perp$ implies
that $X$'s eigenspaces and eigenvalues are determined by $A$ and the
map $D$. It follows that $O(A)$ consists of exactly one
operator. Thus, without loss of generality, we now take $O(A)$ to be a
function from observables of ${\cal H}_{\mathsf{I}}$ to observables of
${\cal H}_{\mathsf{P}}$. Lemma~\ref{lemma:eigendec} also implies that
inclusion relationships between eigenspaces of observables of
${\cal H}_{\mathsf{I}}$ are preserved by $O$.
\begin{Corollary}
\label{cor:subcommute}
Suppose that $\mbox{Null}(A-\lambda_1)\subseteq \mbox{Null}(B-\lambda_2)$.
Then $\mbox{Null}(O(A)-\lambda_1)\subseteq \mbox{Null}(O(B)-\lambda_2)$.
\end{Corollary}
If observables $A$ and $B$ commute, we can construct an observable $C$
whose eigenspaces are the maximal common eigenspaces of $A$ and $B$. By
using the eigenspaces of $O(C)$ to derive the eigenspaces of $O(A)$ and
of $O(B)$, we can see that $O(AB) = O(A)O(B)$ and $O(\alpha A+\beta B) =
O(\alpha A)+O(\beta B)$, so that $O$ preserves the algebraic structure
of commuting sets of observables. Similarly,
for any eigenbasis $\ket{\lambda_i}$ of $A$, we can use
an operator $C$ with non-degenerate eigenvalues having the same
eigenbasis to see that the spaces
$\mbox{Supp}(D(\ket{\lambda_i}\bra{\lambda_i}))$
are a complete orthogonal decomposition of ${\cal V}$ into eigenspaces
of $A$. From this it follows that $O(A)$ is determined
by the values of $D$ on pure states.
For the last stage of the proof of Thm.~\ref{thm:qi=subs}, we fix an
orthonormal basis $\kets{i}{I}$ of ${\cal H}_{\mathsf{I}}$. Let $e_{ij} =
\kets{i}{I}\bras{j}{I}$, $X_{ij} = e_{ij}+e_{ji}$, $Y_{ij} = -i
e_{ij}+ie_{ji}$, $C=\sum_{i} i e_{ii}$ and ${\cal V}_i = \mbox{Null}(O(C)-i)$.
Note that $ie^{-i X_{ij}\pi/2} \kets{i}{I} = e^{-i
Y_{ij}\pi/2}\kets{i}{I}= \kets{j}{I}$. According to faithfulness of
unitary dynamics, $e^{-iO(A)}D(\sigma)e^{iO(A)} \subseteq
D(e^{-iA}\sigma e^{iA})$. (For sets $D$ and operators $U$, $UD =
\{Ux\mbox{$\mathbf{:}$} x\in D\}$.) The inclusion is an equality because we
also have $D(e^{-iA}\sigma e^{iA}) =
e^{-iO(A)}e^{iO(A)}D(e^{-iA}\sigma e^{iA})e^{-iO(A)} e^{iO(A)}
\subseteq e^{-iO(A)} D(\sigma ) e^{iO(A)}$. This and the earlier
results imply that $e^{-iO(X_{ij})\pi/2} {\cal V}_i = {\cal V}_j$. By using
Cor.~\ref{cor:subcommute} with the eigenspaces of $X_{ij}$, $Y_{ij}$
and $e_{ii}+e_{jj}$, we can see that the non-zero eigenspaces of
$O(X_{ij})$ and $O(Y_{ij})$ are contained in ${\cal V}_i\oplus {\cal V}_j$.
Because of the algebraic properties of $O$ mentioned above and
$X_{ij}^2=e_{ii}+e_{jj}$, $e^{-iO(X_{ij})\pi/2} = -iO(X_{ij})$. We
can therefore fix an orthonormal basis $\ket{il}$ of ${\cal V}_i$ such that
$O(X_{0i})\ket{0l} = \ket{il}$ and $O(X_{0i})\ket{il}=\ket{0l}$. The
goal is to show that we can identify $\ket{il}$ with
$\kets{i}{I'}\kets{l}{S}$ such that $O(A)$ acts as the identity on the
cosubsystem $\mathsf{S}$. Note that the operators $X_{0j}$ and
$e_{jj}$ generate the Lie algebra of $U({\cal H}_{\mathsf{I}})$. Thus
compositions of exponentials of the form $e^{-i X_{0j}t}$ or $e^{-i
e_{jj} s}$ act transitively on the pure states of
${\cal H}_{\mathsf{I}}$. It follows that for any $\kets{\psi}{I}$,
$\mbox{Supp}(D(\kets{\psi}{I}\bras{\psi}{I}))$ is an image of corresponding
compositions of exponentials of the form $e^{-i O(X_{0i}) t}$ or
$e^{-i e_{jj} s}$ acting on ${\cal V}_0$. Such compositions are completely
determined by the basis $\ket{il}$. Now $O(A)$ is determined by
$\mbox{Supp}(D(\kets{\psi}{I}\bras{\psi}{I}))$, with $\kets{\psi}{I}$
ranging over eigenvectors of $O(A)$. Since for fixed $l$, $O(X_{0i})$
and $O(e_{jj})$ act as they should on the states $\ket{kl}$, we have
that $O(A)$ necessarily satisfies $\bra{kl}O(A)\ket{kl} =
\bras{k}{I}A\kets{k}{I}$, as desired. \qed
We return to the issue of the relationship between decoding operations
and faithful encodings. Decoding as defined in the introduction is the
traditional way of identifying quantum information and generalizes
recovery operations. A general form of the situation addressed by
decoding involves one or more encoding isometries
$C_i:{\cal H}_{\mathsf{I}}\rightarrow{\cal H}_{\mathsf{P}}$, one or more
possible events $E_i$ that are operators on ${\cal H}_{\mathsf{P}}$ that
may occur before we decode, and a decoding operation $D$ that (after
purification, if necessary) isometrically maps ${\cal H}_{\mathsf{P}}$ into
${\cal H}_{\mathsf{I}}\tensor{\cal H}_{\mathsf{A}}$ for some possibly composite
system $\mathsf{A}$. We say that $\{C_i\}, \{E_j\}, D$ preserve
quantum information if for all $i,j$, $DE_jC_i\kets{\psi}{I} =
\kets{\psi}{I}\kets{\phi_{ij}}{A}$ for some unnormalized vector
$\kets{\phi_{ij}}{A}$ that does not depend on $\kets{\psi}{I}$. Here,
which event $E_i$ occurred is assumed to be unknown. We could
consider the case where the decoding operation is chosen after the
events and depends on partial knowledge of the events. However, by
conditioning on the knowledge, we return to the situation just
described. The only difference is that the subsystem associated with
the situation may depend on the partial knowledge. To capture the
case where quantum information is stored in error-correcting
subsystems, let $C_{i}$ be given by the isometries identifying
${\cal H}_{\mathsf{I}}$ with ${\cal H}_{\mathsf{I'}}\tensor\kets{i}{S}$, where
the $\kets{i}{S}$ range over any spanning set of ${\cal H}_{\mathsf{S}}$.
In order to justify the subsystems principle, we prove the next theorem.
\begin{Theorem}
\label{thm:ced}
If $\{C_i\}, \{E_j\}, D$ preserve quantum information, then there
exists a subsystem encoding
${\cal H}_{\mathsf{P}}={\cal H}_{\mathsf{I'}}\tensor{\cal H}_{\mathsf{S'}}\oplus{\cal H}_{\mathsf{R}}$
such that for all $i,j$ and $\kets{\psi}{I}$, $E_jC_i\kets{\psi}{I} =
\kets{\psi}{I'}\kets{\phi'_{ij}}{S'}\in{\cal H}_{\mathsf{P}}$ and
$D\kets{\psi}{I'}\kets{\phi'_{ij}}{S'}=\kets{\psi}{I}\kets{\phi_{ij}}{A}$,
where the $\kets{\phi'_{ij}}{S'}$ and $\kets{\phi_{ij}}{A}$ do not
depend on $\kets{\psi}{I}$.
\end{Theorem}
\paragraph*{Proof.} This follows from the fact that there exists a protectable
subsystem associated with any quantum error-correcting code and
associated recovery operation, which was proven
in~\cite{knill:qc1995e,knill:qc1999b}. Alternatively, we could prove
the theorem from Thm.~\ref{thm:qi=subs} by defining $D(\kets{\psi}{I})
= \{E_jC_i\kets{\psi}{I}\}$ and $O(A)$ by pullback of the appropriate
operators via the decoding operator $D$. Here we give a direct proof.
Let the $\kets{\phi_{ij}}{A}$ be as required according to the
definition of preserving quantum information. Let ${\cal S}$ be the set of
vectors $\kets{\phi}{A}$ such that
${\cal H}_{\mathsf{I}}\tensor\kets{\phi}{A}$ is contained in the range of
$D$. Then ${\cal S}$ contains the $\kets{\phi_{ij}}{A}$. Because the range
of $D$ is linearly closed, so is ${\cal S}$. Define
${\cal H}_{\mathsf{S}}={\cal S}$. Using the isometric properties of $D$, we can
define a subsystem encoding
${\cal H}_{\mathsf{P}}={\cal H}_{\mathsf{I'}}\tensor{\cal H}_{S'}\oplus{\cal H}_{\mathsf{R}}$
such that $D(\kets{\psi}{I'}\kets{\phi}{S'}) =
\kets{\psi}{I}\kets{\phi}{S}$. This subsystem encoding has the desired
properties. \qed
\section{Finding Noiseless Subsystems}
\label{sect:ns}
If ${\cal A}$ is a $\dagger$-closed subalgebra of $B({\cal H}_{\mathsf{P}})$,
the canonical decomposition of ${\cal H}_{\mathsf{P}}$ is
\begin{equation}
{\cal H}_{\mathsf{P}} = \sum_i
{\cal H}_{\mathsf{I_i}}\tensor{\cal H}_{\mathsf{S_i}}\oplus{\cal H}_{\mathsf{R}},
\end{equation}
where operators $A\in{\cal A}$ act as $\sum_i
\slb{\mathbf{I}}{I_i}\tensor\slb{S_i(A)}{S_i} + \slb{\mathbf{0}}{R}$. For every
operator of the form $\sum_i \slb{\mathbf{I}}{I_i}\tensor\slb{B_i}{S_i} +
\slb{\mathbf{0}}{R}$, there exists an $A\in{\cal A}$ with $S_i(A)=B_i$. The
${\cal H}_{\mathsf{I_i}}$ are noiseless subsystems for ${\cal A}$. We also
consider ${\cal H}_{\mathsf{R}}$ to be noiseless for ${\cal A}$, but note that
error operators in ${\cal A}$ have probability zero for states in this
subspace. The tensor products and direct sums in the decomposition
must be consistent with the Hilbert space's inner product. This is
implicit in the construction and the identification via an isometry.
Let ${\cal E}$ be a linearly closed set of error operators in
$B({\cal H}_{\mathsf{P}})$. For now, we do not assume that ${\cal E}$ is the
span of the operation elements of a quantum operation. Let
${\cal H}_{\mathsf{I'}}\tensor{\cal H}_{\mathsf{S}}\oplus{\cal H}_{\mathsf{R}}$
define a subsystem encoding of $\mathsf{I}$ in $\mathsf{P}$. Let
$\Pi$ be the projector onto the support of $\mathsf{I'}$,
${\cal H}_{\mathsf{I'}}\tensor{\cal H}_{\mathsf{S}}\subseteq{\cal H}_{\mathsf{P}}$.
The subsystem is noiseless for ${\cal E}$ if and only if for all $E\in{\cal E}$,
the restriction of $E$ to ${\cal H}_{\mathsf{I'}}\tensor{\cal H}_{\mathsf{S}}$
acts as the identity on ${\cal H}_{\mathsf{I}}$. Equivalently, for all
$E\in{\cal E}$, $E\Pi = \slb{\mathbf{I}}{I}\tensor \slb{S(E)}{S}$. It is
straightforward to verify that if the subsystem is noiseless, then
$\Pi$ projects onto an invariant subspace of ${\cal E}$ and ${\cal E}\Pi$
generates a $\dagger$-closed subalgebra of operators acting on the
support of $\Pi$ whose canonical decomposition contains noiseless
subsystems with state space dimension at least $N$, the dimension of
${\cal H}_{\mathsf{I}}$. Such noiseless subsystems are also noiseless for
${\cal E}$. This leads to a strategy for finding noiseless subsystems with
maximum dimensional ${\cal H}_{\mathsf{I'}}$ that is equivalent to the
strategy proposed in~\cite{choi:qc2005b}: 1. Pick an invariant
subspace of ${\cal E}$ and let $\Pi$ be its projector. 2. Determine the
canonical decomposition of the $\dagger$-closed algebra generated by
$\Pi{\cal E}$. The noiseless subsystems of this algebra are candidate
noiseless subsystems for ${\cal E}$. Our goal is to provide an explicit
algorithm for finding suitable $\Pi$ and associated subsystems. The
algorithm involves the decomposition of a matrix algebra, for which
efficient algorithms are known, as we explain below. Note that in
addition to the noiseless subsystems identified in this way, one can
construct other noiseless subsystems as subsystems of already obtained
noiseless subsystems, or by combining cosubsystems of identical
dimensional noiseless subsystems with orthogonal supports. These
constructions cannot yield larger dimensional noiseless subsystems,
but they may generate ones with greater error tolerance or more
efficiently controllable states.
Let ${\cal A}$ be the algebra generated by ${\cal E}$. Any noiseless
subsystem for ${\cal E}$ is a noiseless subsystem for ${\cal A}$. ${\cal A}$ is not
necessarily $\dagger$-closed. As a result, ${\cal A}$ does not have a
canonical decomposition of ${\cal H}_{\mathsf{P}}$ as a direct sum of
tensor products of Hilbert spaces. Nevertheless, we can identify a
special subspace within which a similar decomposition is possible and
where maximum dimensional noiseless subsystems may be found. This
subspace is the span ${\cal S}$ of the irreducible subspaces of ${\cal A}$. A
subspace ${\cal V}$ of ${\cal H}_{\mathsf{P}}$ is ``irreducible'' for ${\cal A}$ if
it is ${\cal A}$-invariant, ${\cal A}{\cal V}\not=0$ and there is no non-zero
${\cal A}$-invariant proper subspace of ${\cal V}$. Let ${\cal Z}$ be the null space
of ${\cal A}$. Both ${\cal S}$ and ${\cal Z}$ are invariant.
\begin{Lemma}
\label{lemma:maxnsloc}
A maximum dimensional noiseless subsystem for ${\cal A}$ can
be found in ${\cal S}$ or in ${\cal Z}$.
\end{Lemma}
Note that ${\cal Z}$ is itself a noiseless subsystem. This subsystem is
trivial in the sense that the probability of ${\cal E}$-errors is zero for
any state in ${\cal Z}$. This means that in a realistic setting, there must
be operators acting on the system not included in ${\cal E}$, and for ${\cal Z}$
to be at least approximately noiseless, they need to act as
operators close to the identity when restricted to ${\cal Z}$.
\proofofx{Lemma~\ref{lemma:maxnsloc}} Suppose that ${\cal H}_{\mathsf{I'}}$
is a noiseless subsystem of ${\cal H}_{\mathsf{P}}$ with cosubsystem
${\cal H}_{\mathsf{S}}$. Then
${\cal V}={\cal H}_{\mathsf{I'}}\tensor{\cal H}_{\mathsf{S}}$ is invariant under ${\cal A}$
and for $A\in{\cal A}$, $A$ acts as $\slb{\mathbf{I}}{I}\tensor \slb{S(A)}{S}$ on
${\cal V}$. If for all $A\in{\cal A}$, $\slb{S(A)}{S}=0$, then ${\cal V}\subseteq{\cal Z}$
and we are done. If not, then there exists a nontrivial irreducible
subspace ${\cal S}_i$ of ${\cal H}_{\mathsf{S}}$ under the action of
$\{S(A)\mbox{$\mathbf{:}$} A\in{\cal A}\}$. For each state $\kets{\psi}{I'}$,
$\kets{\psi}{I'}\tensor{\cal S}_i$ is an irreducible representation for ${\cal A}$. In particular,
${\cal H}_{\mathsf{I'}}\tensor{\cal S}_i\subseteq {\cal S}$. Since
${\cal H}_{\mathsf{I'}}\tensor{\cal S}_i$ is also a noiseless subsystem, the
proof is complete. \qed
According to the theory of $R$-modules, ${\cal S}$ is a module for
${\cal A}$. The definition implies that it is semisimple, from which it
follows that ${\cal S}=\sum_i {\cal S}_i$ where the ${\cal S}_i$ are irreducible and
the sum is over independent subspaces, see~\cite{hungerford:qc1980a},
Chapter 9. ${\cal S}_i$ and ${\cal S}_j$ are isomorphic with respect to the
action of ${\cal A}$ if there exists an invertible linear map $U_{ij}$ from
${\cal S}_i$ to ${\cal S}_j$ such that for $\ket{x}\in{\cal S}_i$, $AU_{ij}\ket{x} =
U_{ij}A\ket{x}$. The map $U_{ij}$ is said to ``intertwine'' ${\cal S}_i$
and ${\cal S}_j$. We can relabel the ${\cal S}_i$ to form sets $\{{\cal S}_{ik}\}_i$ of
isomorphic irreducible representations. For each $k$, let ${\cal V}_k$ be the span of the
${\cal S}_{ik}$ and let $U^{(k)}_{0j}$ be an intertwiner from ${\cal S}_{0k}$ to
${\cal S}_{jk}$. Choose a basis $\ket{i0k}$ of ${\cal S}_{0k}$ and define
$\ket{ijk} = U^{(k)}_{0j}\ket{i0k}$. Note that these vectors need not
be orthogonal or normalized. Nevertheless, they define invertible
linear maps from tensor products ${\cal J}_k\tensor{\cal S}_{0k}$ to ${\cal V}_k$ via
the linear extension of $\ket{j}\tensor\ket{i0k}\mapsto\ket{ijk}$.
The action of $A\in{\cal A}$ with respect to this factorization is on
${\cal S}_{0k}$ only.
\begin{Lemma}
\label{lemma:maxnslock}
A maximum dimensional noiseless subsystem for ${\cal A}$ in ${\cal S}$
can be found in one of the ${\cal V}_k$.
\end{Lemma}
\paragraph*{Proof.} This follows from the argument given in the proof of
Lemma~\ref{lemma:maxnsloc}. It suffices to observe that the
irreducible representations $\kets{\psi}{I}\tensor{\cal S}_i$ are
isomorphic for different $\kets{\psi}{I}$. \qed
The main remaining problem in narrowing the search space for maximum
dimensional noiseless subsystems is that the factorization of the
${\cal V}_k$ may fail to preserve the inner product. To simplify the
notation, fix $k$ and let ${\cal V}={\cal V}_k$, ${\cal S}_0={\cal S}_{0k}$ and
${\cal J}={\cal J}_k$. Let $U$ be an invertible linear map from
${\cal J}\tensor{\cal S}_0$ to ${\cal V}$ that implements the above-mentioned
factorization of ${\cal V}$. Thus, for $A\in{\cal A}$ and
$\ket{x}\in{\cal J}\tensor{\cal S}_0$, $AU\ket{x} = U(\mathbf{I}\tensor R(A))\ket{x}$,
where $R$ is a well-defined, irreducible representation of ${\cal A}$ on
${\cal S}_0$. Note that an irreducible representation of ${\cal A}$ on ${\cal S}_0$
is onto $B({\cal S}_0)$ (Burnside's theorem). This implies that any
noiseless subsystem of ${\cal V}$ must be associated with a subspace ${\cal J}'$
of ${\cal J}$ such that the restriction of $U$ to ${\cal J}'\tensor{\cal S}_0$ has
the property that there are linear operators $W$ on ${\cal J}'$ and $V$ on
${\cal S}_0$ such that $U (W\tensor V)$ is an isometry. Fortunately, in
cases where ${\cal A}$ is generated by the operational elements of a
quantum operation, we do not need to search for such subspaces.
\begin{Lemma}
\label{lemma:nsunit}
If ${\cal A}$ is generated by the operational elements of a quantum
operation, then there exist linear operators $W$ on ${\cal J}$ and $V$ on
${\cal S}_0$ such that $U(W\tensor V)$ is unitary.
\end{Lemma}
\paragraph*{Proof.} Let $\{E_i\}_i$ generate ${\cal A}$, where the $E_i$ are the
operational elements of a quantum operation ${\cal O}$. By composing ${\cal O}$
with itself sufficiently many times, it is possible to obtain a
quantum operation ${\cal O}'$ such that its operational elements span
${\cal A}$. Thus, without loss of generality, assume that the $E_i$ span
${\cal A}$ and $\sum_i E_i^\dagger E_i = \mathbf{I}$. We have $E_i =
U(\mathbf{I}\tensor R(E_i))U^{-1}$. In order to continue, assume, without
loss of generality, that ${\cal J}\tensor{\cal S}_0 ={\cal V}$. This can be done by
means of any isometry between ${\cal V}$ and ${\cal J}\tensor{\cal S}_0$. This
implies that $U$ is an invertible but not necessarily unitary linear
map from ${\cal J}\tensor{\cal S}_0$ to itself. We have
\begin{equation}
\sum_i {U^{-1}}^\dagger (\mathbf{I} \tensor R(E_i)^\dagger) U^\dagger U
(\mathbf{I}\tensor R(E_i)) U^{-1} = \mathbf{I},
\end{equation}
or, equivalently,
\begin{equation}
\sum_i (\mathbf{I}\tensor R(E_i)^\dagger ) U^\dagger U
(\mathbf{I}\tensor R(E_i)) = U^\dagger U.
\end{equation}
This implies that for all positive semidefinite $\sigma$
on ${\cal J}$,
\begin{equation}
\label{eq:fixed}
\sum_i R(E_i)^\dagger \mbox{tr}_{\cal J}((\sigma\tensor\mathbf{I})U^\dagger U (\sigma\tensor\mathbf{I})) R(E_i)
= \mbox{tr}_{\cal J}((\sigma\tensor\mathbf{I})U^\dagger U (\sigma\tensor\mathbf{I})),
\end{equation}
where $\mbox{tr}_{\cal J}$ is the partial trace over ${\cal J}$. Let $\mathbf{R}$
be the operation defined by $\mathbf{R}(X) = \sum_i R(E_i)^\dagger X
R(E_i)$. According to Eq.~\ref{eq:fixed}, for all positive
semidefinite $\sigma$, $\mbox{tr}_{\cal J}((\sigma\tensor\mathbf{I})U^\dagger
U(\sigma\tensor\mathbf{I}))$ is a positive semidefinite fixed point of $R$.
The spanning assumption on the $E_i$ and irreducibility of ${\cal S}_0$
under $R({\cal A})$ imply that the $R(E_i)$ span $B({\cal S}_0)$. It follows
that if $\rho\not=\mathbf{0}$ is positive semidefinite and
$\mathbf{R}(\rho) = \rho$, then the support of $\rho$ is ${\cal S}_0$. It
also implies that $\mathbf{R}$ has at most one positive
fixed point (up to positive multiples): If $\rho'$ is
another one, then so is $\rho-\epsilon\rho'$ for all $\epsilon$. Let
$\epsilon$ be the largest such that $\rho-\epsilon\rho'$ is positive
semidefinite. Then $\rho-\epsilon\rho'$ is a fixed point with
non-maximal support, which implies that it is $\mathbf{0}$. Let $\rho$ be
the unique trace $1$ positive fixed point of $R$. Then, for all
positive semidefinite $\sigma$,
$\mbox{tr}_{\cal J}((\sigma\tensor\mathbf{I})U^\dagger U (\sigma\tensor\mathbf{I}))$ is a
multiple of $\rho$. We can now deduce that $U^\dagger U =
\rho'\tensor\rho$ for some strictly positive $\rho'$. \ignore{ This
is not entirely trivial. But we can see that for all $X$ orthogonal to
$\rho$ in the trace inner product,
$\mbox{tr}((\sigma\tensor\mathbf{I})U^\dagger U (\sigma\tensor\mathbf{I})
(\mathbf{I}\tensor X)) = \mbox{tr}(\rho X) = 0$ The left hand side is $\mbox{tr}(
U^\dagger U (\sigma^2\tensor X))$, by cyclicity of trace. The result
then follows because the operators $\sigma^2$ for positive
semidefinite $\sigma$ linearly span all operators on ${\cal J}$. } Defining
$V=\rho^{-1/2}$ and $W=\rho'^{-1/2}$ yields the lemma. \qed
The above suggests the following strategy for finding
maximum-dimensional noiseless subsystems: 1. Determine the span ${\cal S}$
of the irreducible subspaces of ${\cal A}$. 2. Decompose ${\cal S}$ into a
direct sum $\bigoplus_i {\cal I}_i$ of subspaces spanned by isomorphic
irreducible subspaces. 3. For each ${\cal I}_i$, let ${\cal A}_i$ be the
restriction of ${\cal A}$ to ${\cal I}_i$ and find the canonical decomposition
for the $\dagger$-closed algebra generated by ${\cal A}_i$. This strategy
will find maximum-dimensional noiseless subsystems provided that ${\cal A}$
is generated by the operational elements of a quantum operation.
There are efficient algorithms for each step of this strategy; for a
review, see~\cite{struble:qc2000a}. For completeness, we outline an
algorithm that implements the strategy.
To find ${\cal S}$, consider the structure of ${\cal A}$ in more detail. If
${\cal A}$ does not contain $\mathbf{I}$, replace ${\cal A}$ by ${\cal A}+\mathbb{C}\mathbf{I}$. By
doing so, the action of ${\cal A}$ on ${\cal Z}$ is no longer zero, but ${\cal Z}$ is
still distinguishable from the other irreducible subspaces. Every
one-dimensional subspace of ${\cal Z}$ is irreducible and not isomorphic to
the irreducible subspaces of ${\cal S}$. There exists a maximal chain of
invariant subspaces
$0={\cal V}_0\subset{\cal V}_1\subset\ldots\subset{\cal V}_n={\cal H}_{\mathsf{P}}$ such
that the action of ${\cal A}$ induced on the quotients ${\cal V}_{k+1}/{\cal V}_k$ is
irreducible or zero. In a basis $\ket{e_{kj}}$ of ${\cal H}_{\mathsf{P}}$
where $\ket{e_{(k+1)j}}\in{\cal V}_{k+1}\setminus{\cal V}_{k}$ ($\setminus$
denotes set difference), the operators of ${\cal A}$ are block upper
triangular. Let ${\cal J}$ be the members of ${\cal A}$ that act as $\mathbf{0}$ on
each of these quotients. ${\cal J}$ is known as the Jacobson radical of
${\cal A}$. Let ${\cal N}$ be the null space of ${\cal J}$, which is the set of
vectors in the intersection of the null spaces of operators of
${\cal J}$. Then ${\cal N}$ is invariant (because ${\cal J}$ is a two-sided ideal)
and ${\cal S}\subseteq {\cal N}$ (because ${\cal S}$ is invariant and the span of
irreducible subspaces). A fundamental property of ${\cal J}$ is that
${\cal A}/{\cal J}$ is a semisimple algebra. Let ${\cal A}_{\cal N}$ be the restriction of
${\cal A}$ to ${\cal N}$. Then ${\cal A}_{\cal N}$ is isomorphic to a quotient of
${\cal A}/{\cal J}$, which implies that ${\cal A}_{\cal N}$ is semisimple. According to
the representation theory of semisimple algebras, ${\cal N}$ is a
semisimple ${\cal A}_{\cal N}$-module, which implies that ${\cal N}={\cal S}$. Thus, to
determine ${\cal S}$, we can use an efficient algorithm for finding the
Jacobson radical and then compute its null space. Decomposing ${\cal N}$
into independent irreducible subspaces can be done by means of an
efficient algorithm for the decomposition of semisimple algebras over
the complex numbers. A randomized algorithm can be based on the
observation that ${\cal A}_{\cal N}$ is isomorphic to a direct sum of complete
matrix algebras ${\cal A}_k$ on the ${\cal S}_{0k}$, acting canonically on the
(non-unitary) decomposition of ${\cal S}$ into a direct sum of products
${\cal I}_k={\cal J}_k\tensor{\cal S}_{0k}$. It follows that a random matrix in
${\cal A}_{\cal N}$ (with respect to a suitably chosen probability distribution)
typically has generalized eigenspaces that generate (by
multiplication by members of ${\cal A}_{\cal N}$) exactly one of the invariant
subspaces ${\cal I}_k={\cal J}_k\tensor{\cal S}_{0k}$. This yields the desired matrix
algebras ${\cal A}_k$. For each ${\cal A}_k$, let ${\cal A}^*_k$ be the
$\dagger$-closed algebra generated by ${\cal A}_k$. The canonical
factorization of ${\cal I}_k$ with respect to ${\cal A}^*_k$ can also be
obtained by a randomized algorithm. By construction,
${\cal I}_k={\cal H}_1\tensor{\cal H}_2$ (isometrically), with ${\cal A}^*_k$ acting only
on ${\cal H}_2$. The eigenspaces of a randomly chosen Hermitian operator
$H_2$ in ${\cal A}^*_k$ are typically of the form ${\cal H}_{1i}\tensor\ket{i}$
for an orthonormal basis of ${\cal H}_2$, where ${\cal H}_{1i}={\cal H}_1$, but with
the isometry for making this identification not yet known. With high
probability, these isometries can be determined from an independently
chosen second $H'_2$ by expressing $H'_2$ in an orthonormal basis
whose $i$'th block of vectors is a basis of
${\cal H}_{1i}\tensor\ket{i}$. Because $H'_2$ is a Kronecker product with
identity action on ${\cal H}_1$, the $i,j$ block of $H'_2$ must define an
isometry between ${\cal H}_{1i}$ and ${\cal H}_{1j}$ (if it is nonzero). These
isometries must be consistent and induce the desired tensor product
structure.
Components of the algorithm of the previous paragraph not given
explicitly include the generation of an algebra from a set of matrices
(this comes up in generating ${\cal A}$ from an error set and generating
$\dagger$-closed algebras from a given one) and various standard matrix
manipulations such as matrix multiplication, eigenvalue and eigenspace
determination, etc. We do not discuss the latter here. To generate
the matrix algebra from a set of operators $\{E_i\}$, assume without
loss of generality that the $E_i$ are independent. Then iteratively,
choose $i,j$ and determine whether $E_i E_j$ is in the span of the
$E_i$. If not, adjoin it to the set. Stop when for all $i,j$, $E_i
E_j$ is in the span of the $E_i$.
\section{Protectable Subsystems}
\label{sect:initialization}
As above, let $\{E_i\}_i$ be a set of error operators on
${\cal H}_{\mathsf{P}}$. Let ${\cal H}_{\mathsf{P}} =
{\cal H}_{\mathsf{I'}}\tensor{\cal H}_{\mathsf{S}}\oplus{\cal H}_{\mathsf{R}}$ be a
subsystem encoding. The subsystem $\mathsf{I'}$ is ``initialization
protectable'' (or ``protectable'' for short), if there exists a
quantum operation with operation elements $\{R_i\}_i$ such that
$\mathsf{I'}$ is noiseless for $\{E_iR_j\}_{i,j}$. The goal of this
section is to reduce the problem of determining whether a given
subsystem is protectable to the problem of searching for certain
extremal error-detecting codes. We then reduce this problem to several
linear algebra problems.
Let $\kets{i}{S}$ be an orthonormal basis of ${\cal H}_{\mathsf{S}}$. For
any state $\ket{\psi}$ of
${\cal H}_{\mathsf{I'}}\tensor{\cal H}_{\mathsf{S}}\subseteq{\cal H}_{\mathsf{P}}$, we
define $\bras{i}{S}\ket{\psi}\in{\cal H}_{\mathsf{I'}}$ by the identity
$\sum_i(\bras{i}{S}\ket{\psi})\tensor\kets{i}{S}=\ket{\psi}$. Let
${\cal V}$ be the intersection of the inverse images of
${\cal H}_{\mathsf{I'}}\tensor{\cal H}_{\mathsf{S}}$ under the errors $E_i$.
\begin{Lemma}
\label{lemma:protiso}
With the definitions of the previous paragraphs,
$\mathsf{I'}$ is protectable if and only if there exists a
subspace ${\cal D}\subseteq{\cal V}$ with the property that
the maps $F_{ij}:\ket{\psi}\mapsto \bras{j}{S}E_i\ket{\psi}$
are proportional to a single isometry from ${\cal D}$ to ${\cal H}_{\mathsf{I'}}$.
\end{Lemma}
\paragraph*{Proof.} For the ``if'' part of the lemma, we show that ${\cal D}$ is an
error-correcting code for $\{E_i\}_i$. We can reconstruct $E_i$ on
${\cal D}$ from the $F_{ij}$ by the identity $E_i\ket{\psi} = \sum_j
(F_{ij}\ket{\psi})\tensor\kets{j}{S}$. Let $U$ be the isometry such
that $\bras{j}{S}E_i\ket{\psi} = \alpha_{ij} U\ket{\psi}$. Then
$E_i\ket{\psi} = (U\ket{\psi})\tensor \sum_j\alpha_{ij}\kets{j}{S}$.
That ${\cal D}$ is an error-correcting code follows immediately. The
operators $R_i$ are given by $U^{-1}\ketbras{i}{i}{S}$.
For the converse, we can use the subsystems principle (more
specifically, Thm.~\ref{thm:ced}), according to which there must be a
subsystem decomposition ${\cal H}_{\mathsf{P}} =
{\cal H}_{\mathsf{I''}}\tensor{\cal H}_{\mathsf{T}}\oplus {\cal H}_{\mathsf{Q}}$ such
that $R_i(\kets{\psi}{I'}\tensor\kets{j}{S}) =
\kets{\psi}{I''}\tensor\kets{\phi_{ij}}{T}$ and
$E_i\kets{\psi}{I''}\kets{j}{T}=\kets{\psi}{I'}\kets{\varphi_{ij}}{S}$.
The desired subspace is given by
${\cal H}_{\mathsf{I''}}\tensor\kets{0}{T}$ for any base state $\kets{0}{T}$
of $\mathsf{T}$. Note that the desired isometry is implicity defined
via the two subsystem decompositions. \qed
The maps $F_{ij}$ defined in the statement of
Lemma~\ref{lemma:protiso} are well defined from ${\cal V}$ to
${\cal H}_{\mathsf{I}}$. Let $M,N$ be the dimensions of ${\cal V}$ and
${\cal H}_{\mathsf{I}}$, respectively. By choosing orthonormal bases
$\{\kets{i}{V}\}_i$ of ${\cal V}$ and $\{\kets{j}{I'}\}_j$ of
${\cal H}_{\mathsf{I'}}$, the $F_{ij}$ are expressible as $N\times M$
matrices (also denoted by $F_{ij}$) with entries
$(F_{ij})_{kl}=\bras{k}{I'}F_{ij}\kets{l}{V}$. Without loss of
generality, $M\geq N$, for otherwise the subsystem $\mathsf{I'}$ is
clearly not protectable. The condition in Lemma~\ref{lemma:protiso}
can be seen to be equivalent to the requirement that there exists a
unitary matrix $V$ such that $F_{ij} V$ contains a multiple of the
$N\times N$ identity matrix as its first $N\times N$ block. The code
${\cal D}$ is spanned by the first $N$ columns of $V$. This requirement is
reminiscent of the familiar condition on the existence of an
$N$-dimensional error-detecting quantum code, according to which there
must exist a unitary matrix $W$ such that $W E_i W^\dagger$ has a
multiple of an $N\times N$ identity matrix as its first diagonal
subblock. The protectability requirement can indeed be reduced to the
existence of an error-detecting code. In particular, $\mathsf{I'}$ is
protectable if and only if there exists an $N$-dimensional
error-detecting code for the operators $\{F_{ij}^\dagger F_{kl}\}$.
Note that this is equivalent to requiring the existence of an
$N$-dimensional error-correcting code for the operators $F'_{ij}$,
where $F'_{ij}$ is the square matrix obtained from $F_{ij}$ by
expanding with rows of zeros. However, we do not have to consider all
operators $F_{ij}^\dagger F_{kl}$. It suffices to find an
$N$-dimensional error-detecting code for operators of the form
$F_{ij}^\dagger F_{ij}$ and $F_{ij}^\dagger F_{\pi(ij)}$, where $\pi$
is a cyclic permutation of the index pairs.
We call subspaces ${\cal D}$ satisfying the condition in
Lemma~\ref{lemma:protiso} ``protecting'' codes (for $\mathsf{I'}$).
There are several procedures that can be used to reduce the difficulty
of the search for protecting codes.
\begin{Lemma}
\label{lemma:protnull}
All protecting codes are contained in the null space of the $F$ in
the linear span of $\{F_{ij}\}_{ij}$ whose rank is strictly less than $N$.
\end{Lemma}
\paragraph*{Proof.} Let $V$ be as specified in the paragraph before the statement
of the lemma. If the rank of $F$ is less than $N$, then the first
$N\times N$ block of $FV$ must be zero, from which the result follows.
\qed
Let $G_1,\ldots, G_k$ be $N\times M$ matrices. We say that
$\{G_1,\ldots G_k\}$ has maximal row rank if the span of the rows of
the $G_i$ has dimension $kN$. The next lemma generalizes
Lemma~\ref{lemma:protnull}.
\begin{Lemma}
\label{lemma:protextnull}
Let $G_1,\ldots,G_k$ be in the span of the $F_{ij}$ such that
$\{G_1,\ldots, G_k\}$ does not have maximal row rank, but for every
$k-1$ independent $G'_1,\ldots G'_{k-1}$ in the span of the $G_l$,
$\{G'_1,\ldots G'_{k-1}\}$ has maximal row rank. Then any protecting
codes are contained in the intersection of the null spaces of the
$G_i$.
\end{Lemma}
\paragraph*{Proof.} Let $V$ be such that $G_iV$ has an initial block proportional
to the $N\times N$ identity matrix and ${\cal D}$ is spanned by the first
$N$ columns of $V$. Suppose that ${\cal D}$ is not contained in the
null space of some $G_i$. Then $G_iV$'s initial $N\times N$ block is
not zero. The space ${\cal G}$ of matrices $G$ in the span of the $G_j$
such that $GV$ has an initial $N\times N$ zero block is
$k-1$-dimensional. Because the row span of $G_i$ is independent of
the linear span ${\cal R}$ of the rows of the matrices in ${\cal G}$, the
dimension of ${\cal R}$ is strictly less than $(k-1)N$, contradicting the
assumption of the lemma. \qed
Lemma~\ref{lemma:protextnull} means that in principle, the problem of
finding ${\cal D}$ can be reduced to the case where each $F_{ij}$ has full
rank and its row space is independent of the space spanned by the rows
of the other $F_{kl}$. In this case there are at most $M/N$
independent matrices $F_{ij}$. Unfortunately, we do not know of an
efficient algorithm for checking the condition of
Lemma~\ref{lemma:protextnull} that would enable reducing the problem
to this case. Nevertheless, we can show that one can reduce to the
case where there are at most $M-1$ independent $F_{ij}$.
\begin{Lemma}
\label{lemma:protextM}
For $N>1$, if there are $M$ or more independent $F_{ij}$,
then there exists a nonzero $G$ in the span of the $F_{ij}$ such
that $G$ does not have full rank.
\end{Lemma}
\paragraph*{Proof.} Let $\{G_i\}_{i=1}^{l}$ be a basis of the linear span of the
$F_{ij}$. Let $g^j_{i}$ be the $j$'th row of $G_i$. If one of the
$g^j_{i}$ is zero, we are done. Suppose $l>M$. Then the $g^1_{i}$ are
dependent, so there is a non-trivial linear combination of the $G_i$
with zero first row. Suppose $l=M$. Consider the matrices $A_j$ whose
$i$'th rows are the $g^j_{i}$. Then there exists a non-zero linear
combination $\alpha A_1+\beta A_2$ with determinant zero. Let
$x\not=\mathbf{0}$ be in the null space of $(\alpha A_1+\beta A_2)^T$.
Then $G=\sum_ix_iG_i$ is not zero and the row vector
$y=(\alpha,\beta,0\ldots)$ satisfies $yG=0$, so that $G$ does not have
full rank. \qed
Note that the proof of the lemma contains an efficient algorithm
for finding a non-full rank $G$.
Let $\slb{\rho}{AB}$ be a density matrix on systems $A$ and $B$. What
states $\sigma$ of $\mathsf{A}$ can be obtained by projecting
$\mathsf{B}$ onto a pure state $\kets{\psi}{B}$? The ``pure
$\sigma$-projection problem'' for $\slb{\rho}{AB}$ is to determine a
state $\kets{\psi}{B}$ such that
$\bras{\psi}{B}\slb{\rho}{AB}\kets{\psi}{B} = p\sigma$ for some $p\not=0$,
if such a state exists.
\begin{Theorem}
\label{thm:projection}
If the span of the rows of the $F_{ij}$ is $M$-dimensional, the
problem of determining whether subsystem $\mathsf{I'}$ is protectable
is efficiently reducible to a pure $\mathbf{I}$-projection problem.
\end{Theorem}
If the rows of the $F_{ij}$ do not span the full space, then the
protectability problem may be reduced to a generalization of the pure
$\mathbf{I}$-projection problem. However, in situations where the original
error operators are associated with quantum operations, the $F_{ij}$'s
do not have a common null space, even after the restrictions of the
previous lemmas have been applied. Otherwise there would be states for
which all $E_i$ have zero probability.
\paragraph*{Proof.} Let $G_1,\ldots, G_k$ be a basis for the linear space spanned
by the $F_{ij}$. We can choose an orthonormal basis of ${\cal V}$
such that in this basis, the matrices $G_i$ have a block form
$[G_{i1},G_{i2},\ldots,G_{ii},\mathbf{0},\ldots,\mathbf{0}]$, where the $G_{ij}$
are $N\times N_j$ matrices of full rank. We attempt to find the
desired subspace ${\cal D}$ by choosing an orthonormal basis for ${\cal D}$. Let
$X$ be the matrix whose columns are members of this basis. We wish to
solve the $k$ identities $\alpha_i\mathbf{I} = G_i X$ for $X$ and
$\alpha=(\alpha_i)_i$. We can write $X$ in block form,
$X=[X_1;\ldots;X_k]$, where $X_i$ is $N_j\times N$ and the $X_i$ are
placed one above the other. The desired identities can be expanded as
\begin{equation}
\label{eq:alphaone}
\alpha_i\mathbf{I} = \sum_{j=1}^i G_{ij}X_j.
\end{equation}
The $X_j$ can be eliminated by solving the equations in order. That
is, from $\alpha_1\mathbf{I} = G_{11}X_1$ we obtain $\alpha_1=0$ and $X_1=0$
if $N_1\not=N$, and $X_1=\alpha_1G_{11}^{-1}$ otherwise. We write this
as a linear constraint $L_1\cdot\alpha=0$ and an identity
$X_1=\alpha_1\tilde G_{11}$, where $L_1$ may be ``empty'' (if $N_1=N$)
and we set $\tilde G_{11}$ to be any left inverse of $G_{11}$. Once
we have obtained $X_{j}=\sum_m\alpha_m \tilde G_{jm}$ and linear
constraints $L_j\alpha=0$ for $j<i$, we can solve for $X_i$ by
substituing in Eq.~\ref{eq:alphaone}:
\begin{equation}
G_{ii}X_i =
\alpha_i\mathbf{I}-\sum_{j=1}^{i-1}\sum_{m=1}^{j} \alpha_m G_{ij}\tilde G_{jm}.
\end{equation}
The right hand side of this identity is a matrix $H_i$ that depends
linearly on $\alpha$. $G_{ii}X_i = H_i$ can be solved if and only if
the columns of $H_i$ are in the column span of $G_{ii}$. This
condition yields a set of linear constraints $L_i\alpha = 0$. If the
constraints are satisfied, then we can compute $X_i = G_{ii}' H_i$,
where $G_{ii}'$ is a left inverse of $G_{ii}$. We can therefore
define $\tilde G_{im}$ by the identity $X_i=\sum_{m=1}^i
\alpha_j\tilde G_{im}$. At the end of this process, the only free
variables remaining are the $\alpha_j$, which must be chosen to
satisfy the orthonormality constraint on $X$, $\sum_i X_i^\dagger X_i
= \mathbf{I}$. Expanding, we get
\begin{equation}
\sum_i\sum_{jk}\bar\alpha_j\alpha_k\tilde G_{ij}^\dagger\tilde G_{ik} =
\mathbf{I},
\end{equation}
subject to $L_i\alpha = 0$ for all $i$. If the linear
constraints cannot be solved, we are done.
Define $\slb{\rho}{AB}$ by
\begin{equation}
\slb{\rho}{AB} =
t \sum_i\sum_{jk} \slb{\tilde G_{ij}}{A}{}^\dagger\slb{\tilde G_{ik}}{A}
\ketbras{j}{k}{B},
\end{equation}
where $t$ is chosen so that $\mbox{tr}(\slb{\rho}{AB})=1$. Any state
$\kets{\psi}{B}$ in the subspace defined by $L_i\kets{\psi}{B} = 0$
(with $L_i$ defined with respect to the basis consisting of the
$\kets{j}{B}$) that solves the pure $\mathbf{I}$-projection problem yields a
solution for $\alpha$ by letting $\alpha_j$ be a suitably scaled
multiple of the coefficient of $\kets{j}{B}$ in the solution. It
follows that to complete the proof, it suffices to project
$\slb{\rho}{AB}$ onto the subspace of $\mathsf{B}$ satisfying the
linear constraints $L_i$ and renormalize the resulting positive
semidefinite operator. This operator is a density matrix for which the
pure $\mathbf{I}$-projection problem is equivalent to the problem of whether
$\mathsf{I'}$ is protectable. \qed
The pure $\mathbf{I}$-projection problem may be reduced to a problem
of finding special matrices in a linear space of matrices.
\begin{Theorem}
The pure $\mathbf{I}$-projection problem is polynomially equivalent to
the problem of finding a matrix with orthonormal columns in a linear
space of matrices.
\end{Theorem}
\paragraph*{Proof.}
Consider the pure $\mathbf{I}$-projection problem for $\slb{\rho}{AB}$. By
purifing $\slb{\rho}{AB}$ with the addition of an environment $E$, we
obtain a pure state $\kets{\psi}{ABE}$ whose reduced density matrix on
$\mathsf{AB}$ is $\slb{\rho}{AB}$. The pure $\mathbf{I}$-projection problem
is now equivalent to the problem of finding $\kets{\phi}{B}$ such that
$\bras{\phi}{B}\kets{\psi}{ABE}$ is proportional to a maximally
entangled state between $\mathsf{A}$ and $\mathsf{E}$. Note that
without loss of generality, the dimension of $\mathsf{E}$ is greater
than that of $\mathsf{A}$. Otherwise, the problem has no solution. We
can expand everything in a basis for the different systems' Hilbert
spaces: $\kets{\phi}{B} = \sum_i \alpha_i\kets{i}{B}$,
$\kets{\psi}{ABE} =
\sum_{ijk}m_{ijk}\kets{i}{A}\kets{j}{B}\kets{k}{E}$. Let $M_j$ be the
matrix with coefficients $(M_j)_{ki}=m_{ijk}$. The property that
$\bras{\phi}{B}\kets{\psi}{ABE}$ is maximally entangled is equivalent
to the property that $\sum_j\alpha_j M_j$ has orthonormal columns.
Given any set of matrices $M_j'$ we can reverse the reduction of the
previous paragraph by setting $M_j = t M_j'$ with $t=1/(\sum_j
\mbox{tr}(M_j'^\dagger M_j'))$ to obtain a state such that its
pure $\mathbf{I}$-projection problem is equivalent to the problem of finding
a matrix with orthonormal columns in the span of the $M_j'$.
\qed
Whether there is an efficient algorithm for finding a matrix with
orthonormal columns in a linear space of matrices is an open question.
\begin{acknowledgments}
Thanks to S.~Glancy and D.~Leibfried for helpful comments and
assistance in preparing this manuscript. This paper is a contribution
of the National Institute of Standards and Technology, an agency of
the U.S. government, and is not subject to U.S. copyright.
\end{acknowledgments}
\raggedright
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,155 |
\section{Introduction.}
Let $S:\{f=0\}$ be a germ of an irreducible analytic plane curve. An important analytic invariant of $S$ is the Tjurina number $\tau(S)=\dim_{\mathbb{C}}\frac{\mathbb{C}\{x,y\}}{(f)+J(f)}$ where $J(f)$ denotes the Jacobian ideal of $f$.\\
In general, the computation of $\tau(S)$ is not easy. For instance,
we can obtain it consider a Gr\"obner basis for the ideal
$(f)+J(f)$, or alternatively, it is possible to compute $\tau$ by
the dimension of $\frac{J(f):\left ( f\right )}{J(f)}$ (see Theorem
1 in \cite{zbMATH01549534}) that is related with the
$\mathbb{C}\{x,y\}$-module $\Omega^1(S)$ of all germs of
$1$-holomorphic forms $\omega\in\mathbb{C}\{x,y\}\textup{d}
x+\mathbb{C}\{x,y\}\textup{d} y$ such that $f$ divides $\omega\wedge\textup{d} f$.
More precisely, according to K. Saito \cite{MR586450}, $\Omega^1(S)$
is freely generated by two elements $\{\omega_1,\omega_2\}$. It will
be shown that $\tau(S)$ can be expressed from, among other
invariants, the codimension of the ideal $(g_1,g_2)$ where
$\omega_i\wedge \textup{d} f=g_if\textup{d} x\wedge\textup{d} y$.\\
If $L$ denotes a topological class of a plane curve - for instance,
given by the characteristic exponents - then the Milnor number
$\mu=\dim_{\mathbb{C}}\frac{\mathbb{C}\{x,y\}}{J(f)}$ is constant
for any $S:\{f=0\}\in L$ and $\tau_{\min}\leq \tau(S)\leq \mu$.
Generically, an element $S\in L$ is such that $\tau(S)=\tau_{\min}$,
so $\tau_{\min}$ can be express using the topological data that
characterizes $L$. Delorme in \cite{Delorme1978}, presented a
formula to compute the generic dimension $d(\beta_0,\beta_1)$ of the
moduli space for an irreducible plane curve with characteristic
exponents $(\beta_0,\beta_1)$. As $d(2,\beta_1)=0$ and
$d(\beta_0,\beta_1)=\frac{(\beta_0-3)(\beta_1-3)}{2}+\left [
\frac{\beta_0}{\beta_1}\right ]-1-\mu+\tau_{\min}$ (see
\cite{laudal-martin-pfister-below}) we can compute the minimal
Tjurina number for this topological class. On the other hand,
Peraire in \cite{Peraire} developed an algorithm to compute
$\tau_{\min}$ by means of a flag of $J(f)$.\\
In this paper we present a way to express the difference $\mu-\tau$ for a singular irreducible plane curve $S$ when $\Omega^1\left(S\right)$ admits a basis $\{\omega_1,\omega_2\}$ of special kind, that we call a \emph{good} Saito basis (see Definition \ref{goodbasis}).\\
More specifically, we present a formula (see Theorem \ref{lem:Formula}) to compute the difference between $\mu(S)-\tau(S)$ and $\mu(\widetilde{S})-\tau(\widetilde{S})$ where $\widetilde{S}$ denotes the strict transform of $S$.\\
If $S$ is generic in $L$, then, according to \cite{YoyoBMS}, $S$ admits a good basis and this fact allows us to obtain a formula to compute $\tau_{\min}$ in $L$ by the sole topological data: the sequence of multiplicities in the canonical resolution or the characteristic exponents for instance. In particular, for irreducible plane curves, we are able to present a
lower bound for the minimum Tjurina number in $L$ in terms of the Milnor number that allow us to give an
affirmative answer to a question of Dimca and Greuel \cite{Dimca-Greuel} about the inequality $4\tau >3\mu$ and obtained simultaneously by Alberich-Carrami\~nana {\it et al.} in \cite{dimca-conj} published in ArXiv a few days before the first version of this paper.\\
The paper is organized as follows. In the section 2 we present some general properties of a Saito basis. The concept of a good Saito basis is introduced in the section 3 and its properties as well. The section 4 is devoted to the formula for the minimal Tjurina number, a lower bound for the Tjurina number using the Milnor number and consequently an answer to the Dimca-Greuel question.
\section{The Saito basis.}
Let $S:\{f=0\}$ be a germ of an analytic plane curve and consider
the $\mathbb{C}\{x,y\}$-module $\Omega^1(S)$ of all germs of
$1$-holomorphic forms $\omega\in\mathbb{C}\{x,y\}\textup{d}
x+\mathbb{C}\{x,y\}\textup{d} y$ such that $f$ divides $\omega\wedge\textup{d} f$.
It is equivalent to require that the foliation induced by $\omega$
lets invariant $S$. Saito in \cite{MR586450} shows that
$\Omega^1(S)$ is a free module of rank $2$ and a basis of
$\Omega^1(S)$ is called a Saito basis.\\
It is not trivial to obtain a Saito basis, but there is a simple
criterion to verify if $\left\{ \omega_{1},\omega_{2}\right\}$ is a
basis for $\Omega^1(S)$ (see Theorem, page 270 in \cite{MR586450}).
\begin{thm*}[Saito criterion] The set $\left\{
\omega_{1},\omega_{2}\right\}$ is a Saito basis for $S:\{f=0\}$ if
and only if $\omega_1\wedge\omega_2=uf\textup{d} x\wedge\textup{d} y$, where $u$
is a unit in $\mathbb{C}\{x,y\}$.
\end{thm*}
This criterion can be interpreted as follows : $\left\{
\omega_{1},\omega_{2}\right\}$ is a basis for $\Omega^1(S)$ if the tangency locus between the two forms reduces to $S$.\\
Below, we present some examples of Saito basis for $S:\{f=0\}$. All of them will illustrate, in the sequel, various sensitivities of the Saito basis with respect to small perturbations of the curve $S$. In the whole article, we will keep the same numbering of the examples for the convenience of the reader.
\begin{ex*}[\textbf{1}]
The simplest case is when $f=y^p-x^q$, that is $S_1:\{f=0\}$ is
quasi-homogeneous. In fact, if $\omega_1=qy\textup{d} x- px\textup{d} y$ and
$\omega_2=\textup{d} f$, then $$\omega_1\wedge\omega_2=pqf\textup{d} x\wedge\textup{d}
y$$ and $\{\omega_1,\omega_2\}$ is a basis for $\Omega^1(S_1)$.
\end{ex*}
\begin{ex*}[\textbf{2}]
If $f=y^5-x^6+x^4y^3$ then $S_2:\{f=0\}$ is topologically
quasi-homogeneous, that is, $S_2$ presents characteristic exponents
$(5,6)$, but not analytically equivalent to $y^5-x^6=0$. One can
show that the set $\{\omega_1,\omega_2\}$ where
\begin{eqnarray*}
\omega_1&=& \left ( -6xy+\frac{16}{15}x^3y^2-\frac{8}{5}xy^5\right
)\textup{d} x+\left ( 5x^2+\frac{4}{3}y^3+\frac{4}{5}x^2y^4\right )\textup{d} y\\
\omega_2&=&\left ( -6y^2+\frac{8}{5}x^4-\frac{12}{5}x^2y^3\right )\textup{d}
x+\left ( 5xy+\frac{6}{5}x^3y^2 \right )\textup{d} y
\end{eqnarray*}
satisfy $\omega_1\wedge\omega_2=8f\textup{d} x\wedge\textup{d} y$, so
$\{\omega_1,\omega_2\}$ is a Saito basis for $\Omega^1(S_2)$.
\end{ex*}
\begin{ex*}[\textbf{3}]
The curve $S_3:\{f=0\}$ with $f=y^5-x^{11}+x^6y^3$ is topologically
equivalent to the any curve with characteristic exponents $(5,11)$
and its strict transform is $S_2$. The set $\{\omega_1,\omega_2\}$
where
\begin{eqnarray*}
\omega_1&=& \left ( 605y^2+198xy^3-88x^6\right
)\textup{d} x-\left ( 275xy+66x^2y^2\right )\textup{d} y\\
\omega_2&=&\left ( 605x^4y+150x^5y^2\right )\textup{d} x-\left (
40y^3+275x^5+90x^6y\right )\textup{d} y
\end{eqnarray*}
satisfy $\omega_1\wedge\omega_2=(-24200-7920xy)f\textup{d} x\wedge\textup{d} y$,
so $\{\omega_1,\omega_2\}$ is a Saito basis for $\Omega^1(S_3)$.
\end{ex*}
\begin{ex*}[\textbf{4}]
The class of curve with characteristic exponents the form $(n,n+1)$
has been extensively studied by Zariski \cite{zariski}. The curve
$S_4$ given by $$f=y^7-x^8-7x^6y^2-\frac{147}{8}x^4y^4$$ that,
belongs to the latter class, will be shown of a peculiar interest.
The forms
\begin{eqnarray*}
\omega_1&=& \left(
8x^2y-\frac{147}{8}x^4-\frac{3087}{4}x^2y^2-\frac{21609}{16}y^4\right
)\textup{d} x+\\
&& +\left(
-7x^3+\frac{7}{4}xy^2+\frac{64827}{64}xy^3+\frac{5145}{8}x^3y\right
)\textup{d} y\\
\omega_2&=&\left ( 8xy^2+\frac{1029}{8}x^3y\right )\textup{d} x+\left (-7x^2y+\frac{7}{4}y^3-\frac{1029}{8}x^4 \right )\textup{d} y.
\end{eqnarray*}
produce a Saito basis for $\Omega^1(S_4)$ because
$\omega_1\wedge\omega_2=-\frac{151263}{64}f\textup{d} x\wedge\textup{d} y$.
\end{ex*}
Given a $1$-form $\omega=A\textup{d} x+B\textup{d} y$ we denote by
$\nu(\omega)=\min\{\nu(A),\nu(B)\}$ its algebraic multiplicity,
where $\nu(H)$ indicates the multiplicity of $H\in\mathbb{C}\{x,y\}$
at $(0,0)\in\mathbb{C}^2$.\\
Among all the possible basis $\left\{ \omega_{1},\omega_{2}\right\}$
for $\Omega^1(S)$ we choose some that maximizes the sum
$\nu(\omega_1)+\nu(\omega_2)$ that, following the Saito criterion,
cannot be bigger than $\nu=\nu(f)=\nu\left(S\right)$. For such basis
we denote
\[
\nu_{1}:=\nu\left(\omega_{1}\right)\qquad\nu_{2}:=\nu\left(\omega_{2}\right).
\]
The following result is immediate and identify a new analytical
invariant of $S$.
\begin{prop}
The couple $\left(\nu_{1},\nu_{2}\right)$, up to order, is an
analytical invariant of $S$.
\end{prop}
Remark that the pair $\left(\nu_{1},\nu_{2}\right)$ is not a
topological invariant. For instance, following the examples above, for
$S_1$ with $p=5$ and $q=6$ we have $\left(\nu_{1},\nu_{2}\right)=(1,4)$.
But the curve $S_3$ which is topological equivalent to
$S_1$ has corresponding
pair of multiplicities $(2,2)$.\\
From now on, we consider $S:\{f=0\}$ singular and irreducible (a
plane branch) with a Saito basis $\left\{
\omega_{1},\omega_{2}\right\}$ such that $\omega_{i} =A_{i}\textup{d}
x+B_{i}\textup{d} y$ In particular, we have
\begin{equation}\label{eq:01}
A_{1}B_{2}-A_{2}B_{1} = uf\ \ \ \ \mbox{and}\ \ \ \
A_{i}\frac{\partial f}{\partial y}-B_{i}\frac{\partial f}{\partial
x} = g_{i}f
\end{equation}
where $u(0,0)\neq 0$ and $g_{i}\in\mathbb{C}\{x,y\}$ is called the
cofactor of $\omega_{i}$.\\
Applying a generic linear change of coordinates if necessary, we can
suppose that for $i=1,2$, one has
$\nu\left(A_{i}\right)=\nu\left(B_{i}\right)=\nu_{i} $ and in this
coordinates $\left(x,y\right)$ the tangent cone of $f$, i.e. its
$\nu$-jet, is $f^{(\nu)}=\left(y+\epsilon x\right)^{\nu}$.
\begin{ex*}[\textbf{1}] Consider the irreducible curve $S_1$. Suppose by symmetry that $p<q$, we have
$\nu(A_1)=\nu(B_1)=\nu_1=1$ but $q-1=\nu(A_2)>p-1=\nu(B_2)=\nu_2$.
Consider the change of coordinates $T(x,y)=(x,y-\epsilon x)$ with
$\epsilon\neq 0$ we obtain $f_1=T^*(f)=(y-\epsilon x)^p-x^q$ and the
Saito basis $\eta_1=T^*(\omega_1)$ and $\textup{d} f_1$
\begin{eqnarray*}
\eta_1&=&(q(y-\epsilon x)+\epsilon
px)\textup{d} x-px\textup{d} y\\
\textup{d} f_1&=&(-\epsilon p(y-\epsilon
x)^{p-1}-qx^{q-1})\textup{d} x+p(y-\epsilon x)^{p-1}\textup{d} y
\end{eqnarray*}
satisfying the above
condition. In addition, $\eta_1\wedge\textup{d} f_1=pqf_1\textup{d} x\wedge\textup{d} y$,
that is, $g_1=pq$ and $g_1=0$.
\end{ex*}
\begin{ex*}[\textbf{2}]
For the curve $S_2$, we have
\[ \omega_1\wedge\textup{d} f=(-30x-8xy^4)f\textup{d} x\wedge\textup{d} y
\ \ \mbox{and}\ \ \omega_2\wedge\textup{d} f=(-30y-12x^2y^2)f\textup{d} x\wedge\textup{d}
y,
\]
that is, $g_1=-30x-8xy^4$ and $g_2=-30y-12x^2y^2$.
\end{ex*}
\begin{ex*}[\textbf{3}]
Considering the curve $S_3$, we have $\nu(A_1)=\nu(B_1)=2$ but
$5=\nu(A_2)>\nu(B_2)=3$. By the change of coordinates
$T(x,y)=(x,x+y)$ we obtain $f_1=T^*(f)=(y+x)^5-x^{11}+x^6(y+x)^3$
and $\eta_i = T^*(\omega_i) = \left ( A_i+B_i\right )\textup{d} x+ B_i\textup{d}
y$ with $\nu(A_1+B_1)=\nu(B_1)=2$ and $\nu(A_2+B_2)=\nu(B_2)=3$. In
addition,
\[ \eta_1\wedge\textup{d} f=(3025(x+y)+990x(y+x)^2)f_1\textup{d} x\wedge\textup{d} y
\]
\[ \eta_2\wedge\textup{d} f=(3025x^4+990x^5(y+x))f_1\textup{d} x\wedge\textup{d}
y,
\]
consequently, $g_1=3025(x+y)+990x(y+x)^2$ and
$g_2=3025x^4+990x^5(y+x)$.
\end{ex*}
\begin{ex*}[\textbf{4}]
Finally, for $S_4$ we find
\begin{eqnarray*}
\omega_1\wedge\textup{d} f&=&\left (56x^2-\frac{151263}{16}y^3-\frac{21609}{4}x^2y\right )f\textup{d} x\wedge\textup{d} y\\
\omega_2\wedge\textup{d} f&=&(56xy+1029x^3)f\textup{d} x\wedge\textup{d} y.
\end{eqnarray*}
\end{ex*}
Notice that any generator $\omega_i$ in a Saito basis
$\{\omega_1,\omega_2\}$ has an isolated
singularity, that is, $\gcd(A_i,B_i)=1$. In addition, by
(\ref{eq:01}), we have that $\nu(g_i)\geq \nu_i-1$.
\section{Good Saito basis and the Tjurina number for $S$.}
As we mentioned before, given a Saito basis
$\{\omega_1,\omega_2\}$ for $\Omega^1(S)$ we get $\nu_1+\nu_2\leq
\nu$. In \cite{YoyoBMS}, the first author shows the following theorem:
\begin{thm*}[Generic Basis Theorem] In a fixed
topological class $L$, generically any curve $S$ admits a Saito basis
satisfying
\begin{center}
\begin{tabular}{rl}
$\nu_1=\nu_2=\frac{\nu}{2}$ & if $\nu=\nu(S)$ is even \\ \\
$\nu_1=\nu_2-1=\frac{\nu-1}{2}$ & if $\nu=\nu(S)$ is odd.
\end{tabular}
\end{center}
\end{thm*}
Notice that, generically $\nu_1+\nu_2$ is maximum. Of course,
Example $1$ shows that we can obtain $\nu_1+\nu_2=\nu$
in other cases. This motives the following definition.
\begin{defn}\label{goodbasis}
We say that $S$ (or $\Omega^1(S)$) admits a \emph{good basis} if $\nu_{1}+\nu_{2}=\nu$.
\end{defn}
This section is devoted to present some properties of a good basis.
One of them is related with the index $\mathfrak{i}\left(S\right)$
we introduce in the sequel.\\
Let $E$ be the standard blowing-up of the origin in $\mathbb{C}^{2}$
with coordinates $\left(x,y\right)$ and suppose that, in the chart
$\left(x_{1},y_{1}\right)$ such that
$E\left(x_{1},y_{1}\right)=\left(x_{1},x_{1}y_{1}\right)$, the strict
transform of $S$ goes through $(0,y_{1})$.
\begin{defn} For any $\omega=A\textup{d} x+B\textup{d} y\in\Omega^{1}\left(S\right)$,
we denote by $i\left(\omega\right)\in\mathbb{N}\cup\left\{ \infty\right\}$
the valuation given by
\[
i\left(\omega\right)=\nu_{y_{1}=-\epsilon}\left(A^{\left(\nu\left(\omega\right)\right)}\left(1,y_{1}\right)+y_{1}B^{\left(\nu\left(\omega\right)\right)}\left(1,y_{1}\right)\right)
\]
where $\nu_{y_{1}=-\epsilon}(G)$ denotes de multiplicity of $G\in\mathbb{C}\{y_1\}$ at $-\epsilon\in\mathbb{C}$.\\
Moreover, we denote by $\mathfrak{i}\left(S\right)\in\mathbb{N}$
the integer
\[
\mathfrak{i}\left(S\right)=\min_{\omega\in\Omega^{1}\left(S\right)}i\left(\omega\right).
\]
\end{defn}
The value $i\left(\omega\right)$ is nothing but the index $\rm{Ind}(\mathcal{F},C,0)$ introduced in \cite{camacho} for a germ of foliation $\mathcal{F}$ having $C$ as a smooth invariant curve.\\
Notice that for a given $\omega$, the index $i\left(\omega\right)$
is infinite if and only if $\omega$ is dicritical, that is,
$
A^{\nu\left(\omega\right)}\left(1,y_{1}\right)+y_{1}B^{\nu\left(\omega\right)}\left(1,y_{1}\right)=0.
$\\
However, for any curve $\mathfrak{i}\left(S\right)$ is finite. Indeed,
if $f$ is a reduced equation for $S$ then $\textup{d} f$ belongs to $\Omega^{1}\left(S\right)$
and it is not dicritical, thus $\mathfrak{i}\left(S\right)\leq i\left(\textup{d} f\right)<\infty$. In particular, if $\omega\in\Omega^1(S)$ is non dicritical, then $i(\omega)\leq \nu(\omega)+1$.
\begin{ex*}[\textbf{1}]
For $S_1$ the considered Saito basis is a good basis. Moreover,
$i\left(\omega_1\right)=1$ and $i\left(\omega_2\right)=p$.
\end{ex*}
\begin{ex*}[\textbf{2}]
Having a good basis is a property sensitive to perturbation. Indeed, for instance, the basis of $S_3$ computed in the example is not good, and actually $S_3$ does not admit any good basis. Besides that, we have $i\left(\omega_1\right)=1$ and
$i\left(\omega_2\right)=2$.
\end{ex*}
\begin{ex*}[\textbf{3}]
Good basis is not preserved by blowing-up. In fact, $S_3$ has a good
basis, but its strict transform is analytically equivalent to $S_2$
that does not admit good basis. For $S_3$ we have
$i\left(\omega_1\right)=2$ and $i\left(\omega_2\right)=4$.
\end{ex*}
\begin{ex*}[\textbf{4}]
Finally, $S_4$ does not have a good basis. We find
$i\left(\omega_1\right)=1$ and $i\left(\omega_2\right)=2$.
\end{ex*}
The next result shows that if $S$ admits a good basis, the index
$\mathfrak{i}\left(S\right)$ is achieved for one of its elements.
\begin{prop}
If $S$ admits a good basis $\left\{ \omega_{1},\omega_{2}\right\}$
then
\[
\mathfrak{i}\left(S\right)=\min\left\{ i\left(\omega_{1}\right),i\left(\omega_{2}\right)\right\} .
\]
\end{prop}
\begin{proof}
By Saito criterion, one has $\omega_{1}\wedge\omega_{2}=uf$ with
$u(0,0)\neq 0$. Since $\nu_{1}+\nu_{2}=\nu,$ one has $
\omega_{1}^{\left(\nu_{1}\right)}\wedge\omega_{2}^{\left(\nu_{2}\right)}\neq0,
$
where $\omega_{i}^{\left(\nu_{i}\right)}=A_{i}^{\nu_{i}}\textup{d} x+B_{i}^{\nu_{i}}\textup{d} y$. In particular, both forms $\omega_1$ and $\omega_2$ cannot
be dicritical and therefore $\min\left\{ i\left(\omega_{1}\right),i\left(\omega_{2}\right)\right\} <\infty.$\\
Now, consider any form $\omega=P_{1}\omega_{1}+P_{2}\omega_{2}\in\Omega^{1}\left(S\right)$ with $P_i\in\mathbb{C}\{x,y\}$ and $m_{i}=\nu\left(P_{i}\right)$. Since $P_{1}^{\left(m_{1}\right)}\omega_{1}^{\left(\nu_{1}\right)}+P_{2}^{\left(m_{2}\right)}\omega_{2}^{\left(\nu_{1}\right)}$
cannot identically vanish, it is the homogeneous part of smallest
degree of $\omega.$ Therefore
\begin{align*}
i\left(\omega\right) & =\nu_{y_{1}=-\epsilon}\left(P_{1}^{\left(m_{1}\right)}\left(1,y_{1}\right)\left(A_{1}^{\nu_{1}}\left(1,y_{1}\right)+y_{1}B_{1}^{\nu_{1}}\left(1,y_{1}\right)\right)\right.\\
& \qquad\qquad\left.+P_{2}^{\left(m_{2}\right)}\left(1,y_{1}\right)\left(A_{2}^{\nu_{2}}\left(1,y_{1}\right)+y_{1}B_{2}^{\nu_{2}}\left(1,y_{1}\right)\right)\right)\\
& \geq\min\left\{ i\left(\omega_{1}\right),i\left(\omega_{2}\right)\right\} .
\end{align*}
\end{proof}
In the previous section, we remark that for an element $\omega_i$ in
a Saito basis we get $\nu(g_i)\geq \nu_i-1$ and
$i(\omega_i)\leq\nu(\omega_i)+1$. For good basis it is possible to
obtain the following result.
\begin{lem}\label{i-v1}
Given a good basis $\{\omega_1,\omega_2\}$ for $S$, if
$\nu\left(g_{i}\right)\geq\nu_{i}$ then
$i\left(\omega_{i}\right)=\nu_{i}+1.$
\end{lem}
\begin{proof}
By symmetry let us consider $i=1$ and suppose that
$\nu\left(g_{1}\right)\geq\nu_{1}$. The $(\nu_{1}-1+\nu)$-jet of
\[
A_{1}\frac{\partial f}{\partial y}-B_{1}\frac{\partial f}{\partial
x}=g_{1}f \ \ \ \ \mbox{is}\ \ \ \
A_{1}^{\left(\nu_{1}\right)}\nu\left(y+\epsilon
x\right)^{\nu-1}-B_{1}^{\left(\nu_{1}\right)}\nu\epsilon\left(y+\epsilon
x\right)^{\nu-1}=0,
\]
thus $A_{1}^{\left(\nu_{1}\right)}=\epsilon
B_{1}^{\left(\nu_{1}\right)}$. On the other hand the $\nu$-jet of
$A_{1}B_{2}-A_{2}B_{1}=uf
$
where $u(0,0)\neq 0$ reduces to
$$
A_{1}^{\left(\nu_{1}\right)}B_{2}^{\left(\nu_{2}\right)}-
A_{2}^{\left(\nu_{2}\right)}B_{1}^{\left(\nu_{1}\right)}=
B_{1}^{\left(\nu_{1}\right)}\left(\epsilon
B_{2}^{\left(\nu_{2}\right)}-A_{2}^{\left(\nu_{2}\right)}\right)
=u(0,0)\left(y+\epsilon x\right)^{\nu}.
$$
Thus, there exists some constant $c\neq 0$ such that $
B_{1}^{\left(\nu_{1}\right)}=c\left(y+\epsilon x\right)^{\nu_{1}}$.
Therefore, $\omega_{1}$ can be written
\[
\omega_{1}=\frac{c}{\nu_{1}+1}\textup{d}\left(\left(y+\epsilon
x\right)^{\nu_{1}+1}\right)+ \textup{h.o.t.}
\]
thus $i\left(\omega_{1}\right)=\nu_{1}+1.$
\end{proof}
Notice that the above proof ensures that the inequality
$\nu\left(g_{i}\right)\geq\nu_{i}$ cannot hold for both elements in
a good basis. Moreover, given a good basis for $\Omega^1(S)$ we can
always get a good basis with some nice properties. To do this we
present the following lemmas.
\begin{lem}\label{v-1}
If $\Omega^1(S)$ admits a good basis $\left\{
\omega_{1},\omega_{2}\right\}$, then we can suppose that
$$i\left(\omega_{1}\right)=\mathfrak{i}\left(S\right)\ \ \ \
\mbox{and}\ \ \ \nu\left(g_{1}\right)=\nu_{1}-1.$$
\end{lem}
\begin{proof}
By symmetry we can suppose that
$i\left(\omega_{1}\right)=\mathfrak{i}\left(S\right).$\\
{\bf Case 1.} If
$i\left(\omega_{2}\right)=i\left(\omega_{1}\right)$, then, as
mentioned above, for $i=1$ or $2,$ one has
$\nu\left(g_{i}\right)=\nu_{i}-1$. Switching maybe the two forms, we
can suppose that $\omega_{1}$ satisfies the conclusion of the lemma.\\
{\bf Case 2.} Suppose now that
$i\left(\omega_{1}\right)<i\left(\omega_{2}\right).$\\
{\bf Subcase 2.a} if $\nu_{1}\leq\nu_{2}$, we consider, the family
$\left\{ \omega_{1},\overline{\omega_{2}}\right \}$, where
$\overline{\omega_{2}}=\omega_{2}+c x^{\nu_{2}-\nu_{1}}\omega_{1}$
and $c\in\mathbb{C}$. For a generic value of $c$, we still have a
good basis for $S$. Moreover, the $\nu_{2}$-jet of
$\overline{\omega_{2}}$ is
\[
\left(A_{2}^{\left(\nu_{2}\right)}+c
x^{\nu_{2}-\nu_{1}}A_{1}^{\left(\nu_{1}\right)}\right)\textup{d}
x+\left(B_{2}^{\left(\nu_{2}\right)}+c
x^{\nu_{2}-\nu_{1}}B_{1}^{\left(\nu_{1}\right)}\right)\textup{d} y.
\]
Thus, to evaluate its index, one writes
{\small
\begin{align*}
i\left(\overline{\omega_{2}}\right) & =\nu_{y=-\epsilon}\left(A_{2}^{\left(\nu_{2}\right)}\left(1,y\right)+c A_{1}^{\left(\nu_{1}\right)}\left(1,y\right)+y\left(B_{2}^{\left(\nu_{2}\right)}\left(1,y\right)+c B_{1}^{\left(\nu_{1}\right)}\left(1,y\right)\right)\right) & \\
& =\nu_{y=-\epsilon}\left(A_{2}^{\left(\nu_{2}\right)}\left(1,y\right)+yB_{2}^{\left(\nu_{2}\right)}\left(1,y\right)+c\left(A_{1}^{\left(\nu_{1}\right)}\left(1,y\right)+yB_{1}^{\left(\nu_{1}\right)}\left(1,y\right)\right)\right) & =i\left(\omega_{1}\right).
\end{align*}
} Thus we are led to the previous case $(1)$.\\
{\bf Subcase 2.b.} Finally, if $\nu_{1}>\nu_{2}$, suppose that
$\nu\left(g_{1}\right)\geq\nu_{1}$, then by Lemma \ref{i-v1} we have
$i\left(\omega_{1}\right)=\nu_{1}+1$. Consequently
$i\left(\omega_{1}\right)>\nu_{2}+1$ and then
$i\left(\omega_{2}\right)>\nu_{2}+1$. If $\omega_2$ is not
dicritical, the inequality above leads to a contradiction, thus
$\omega_{2}$ is dicritical. Therefore, it can be seen that
$\nu\left(g_2\right)=\nu_2-1$. Let us consider now
$\overline{\omega}_1=\omega_1+ x^{\nu_1-\nu_2}\omega_2$. Then, the
family $\left\{\overline{\omega}_1,\omega_2\right\}$ is still a good
basis and one has
\begin{eqnarray*}
\overline{\omega}_1\wedge\textup{d} f&=&\overline{g}_{1}f\textup{d} x\wedge\textup{d} y\qquad \textup{with }\ \nu(\overline{g}_{1})=\nu_1-1 \\
i\left(\overline{\omega}_1\right)&=&i\left(\omega_1\right)=\mathfrak{i}\left(S\right).
\end{eqnarray*}
\end{proof}
In addition, from a basis for $\Omega^1(S)$ we can get a
basis satisfying the following lemma.
\begin{lem}
\label{lem:We-can-suppose}
Given a basis
$\{\omega_1,\omega_2\}$ for $\Omega^1(S)$ with $i(\omega_1)\leq i(\omega_2)$ we can suppose that
\[ \gcd\left (B_{i},\frac{\partial f}{\partial y}\right )=1,\ \ \ \mbox{for}\ i=1,2.
\]
\end{lem}
\begin{proof}
Suppose that $H=\gcd\left (B_{1},B_{2},\frac{\partial f}{\partial
y}\right )$. Since by (\ref{eq:01})
$
A_{1}B_{2}-A_{2}B_{1}=uf,
$
$H$ would divide $f$. As $\frac{\partial f}{\partial y}$ and $f$ are
relatively prime, we get
\begin{equation}
\gcd\left(B_{1},B_{2},\frac{\partial f}{\partial
y}\right)=1.\label{eq:gcd}
\end{equation}
Now consider the family $\left\{
\overline{\omega_{1}}=\omega_{1}+P_{1}\omega_{2},\overline{\omega_{2}}=\omega_{2}+P_{2}\omega_{1}\right\}
$ where $P_{i}\in\mathbb{C}\left\{ x,y\right\} $ with
$\nu\left(P_{i}\right)\gg1$. Note that for $P_{i}$ of algebraic
multiplicity big enough, the forms
\begin{align*}
\overline{\omega_{1}} & =\left(A_{1}+P_{1}A_{2}\right)\textup{d} x+\left(B_{1}+P_{1}B_{2}\right)\textup{d} y=\overline{A_{1}}\textup{d} x+\overline{B_{1}}\textup{d} y\\
\overline{\omega_{2}} & =\left(A_{2}+P_{2}A_{1}\right)\textup{d}
x+\left(B_{2}+P_{2}B_{1}\right)\textup{d} y=\overline{A_{2}}\textup{d}
x+\overline{B_{2}}\textup{d} y
\end{align*}
satisfy $
\nu\left(\overline{\omega_{i}}\right)=\nu\left(\overline{A_{i}}\right)=\nu\left(\overline{B_{i}}\right)=\nu_{i},
$ and $i(\omega_1)=i(\overline{\omega_{1}})\leq
i(\overline{\omega_{2}})$.\\
Moreover, $\{\overline{\omega_1},\overline{\omega_2}\}$ is a basis
for $\Omega^1(S)$.
Now the relation (\ref{eq:gcd}) ensures that for a generic choice of
the $P_{i}'s, i=1,2$ - in the sense of Krull -, one has
\[
\gcd\left(\overline{B_{i}},\frac{\partial f}{\partial y}\right)=1.
\]
\end{proof}
As a consequence we obtain the following.
\begin{cor}\label{cor1} For any basis $\{\omega_1,\omega_2\}$ for $\Omega^1(S)$ satisfying the previous Lemma
we have
\[ \gcd(B_i,g_i)=\gcd\left ( \frac{\partial f}{\partial
y},g_i\right )=1.
\]
\end{cor}
\begin{proof} As $A_i\frac{\partial f}{\partial y}-B_i\frac{\partial f}{\partial x}=g_if$,
if $1\neq H=\gcd(B_i,g_i)$ then $H$ must divide $A_i\frac{\partial
f}{\partial y}$. By the previous lemma, $\gcd(B_i,\frac{\partial
f}{\partial y})=1$ so $H$ divides $A_i$, a contradiction because
$\omega_i$ has an isolated singularity.\\
Suppose $H'=\gcd\left ( \frac{\partial f}{\partial y},g_i\right )$,
so $H'$ divides $B_i\frac{\partial f}{\partial x}$. As $\gcd\left (
\frac{\partial f}{\partial y},\frac{\partial f}{\partial x}\right
)=\gcd(B_i,g_i)=1$, we must have $H'=1$.
\end{proof}
In particular, the above lemma allow us to consider a good Saito basis $\{\omega_1,\omega_2\}$ with $\mathfrak{i}\left(S\right)=i(\omega_1)$ and $\gcd\left(B_{i},\frac{\partial f}{\partial y}\right)=\gcd(B_i,g_i)=\gcd\left ( \frac{\partial f}{\partial
y},g_i\right )=1$.
\begin{lem}
\label{lem:The-intersection-of} If $S:\{f=0\}$ admits a good basis satisfying the previous conditions, then the intersection of the tangent cone
of
\begin{enumerate}
\item $g_{1}$ and $g_{2}$,
\item $B_{i}$ and $g_{i}$, for $i=1,2$,
\item $B_{i}$ and $\frac{\partial f}{\partial y}$, for $i=1,2$
\end{enumerate}
is empty or equal to $y+\epsilon x=0$.
\end{lem}
\begin{proof}
The $\nu$-jet of (\ref{eq:01}) is
\begin{equation}
A_{1}^{\left(\nu_{1}\right)}B_{2}^{\left(\nu_{2}\right)}-A_{2}^{\left(\nu_{2}\right)}B_{1}^{\left(\nu_{1}\right)}=c\left(y+\epsilon
x\right)^{\nu}.\label{eq:1}
\end{equation}
where $c\neq0$ and $\epsilon\in\mathbb{C}$. Now, for $i=1,2$, both
following relations $ A_{i}^{\left(\nu_{i}\right)}-\epsilon
B_{i}^{\left(\nu_{i}\right)}=0$ cannot be true all together since it
would yield a contradiction with the relation (\ref{eq:1}). Suppose
the relation above is not true for at least $i=1$, then the cofactor
relations ensures that
\[
A_{1}^{\left(\nu_{1}\right)}-\epsilon
B_{1}^{\left(\nu_{1}\right)}=\frac{1}{\nu}g_{1}^{\left(\nu\left(g_{1}\right)\right)}\left(y+\epsilon
x\right).
\]
Combining the above relations yields
$$
g_{1}^{\left(\nu\left(g_{1}\right)\right)}B_{2}^{\left(\nu_{2}\right)}-
g_{2}^{\left(\nu\left(g_{2}\right)\right)}B_{1}^{\left(\nu_{1}\right)}
=c\nu\left(y+\epsilon x\right)^{\nu-1},\ \ \ \mbox{or}\ \ \
g_{1}^{\left(\nu\left(g_{1}\right)\right)}B_{2}^{\left(\nu_{2}\right)}
=c\nu\left(y+\epsilon x\right)^{\nu-1} $$ from which is derived
$\left(1\right)$ and $\left(2\right)$. The point $\left(3\right)$
follows from the fact that the tangent cone of $\frac{\partial
f}{\partial y}$ and $f$ are the same.
\end{proof}
In what follows we denote by $I_P(G,H)$ the intersection
multiplicity of $G,H\in\mathbb{C}\{x,y\}$ at the point
$P\in\mathbb{C}^2$. If $P=(0,0)$ then we write $I(G,H):=I_P(G,H)$,
that is, $I(G,H)=\dim_{\mathbb{C}}\frac{\mathbb{C}\{x,y\}}{(G,H)}$.\\
An important topological invariant for $S:\{f=0\}$ is the Milnor
number $\mu$ which can be computed by
\begin{equation}\label{milnor}
\mu:=I\left(\frac{\partial f}{\partial
y},\frac{\partial f}{\partial x}\right)=\sum_{i=1}^{N}\nu_{(i)}(\nu_{(i)}-1)
\end{equation}
where $\nu_{(i)};\ i=1,\ldots ,N$ denote the sequence of multiplicities in
the canonical resolution of $S$. In addition, by Zariski (see (2.4)
in \cite{zariski}), we have
\begin{equation}\label{derivative}
I\left(\frac{\partial
f}{\partial y},f\right)=\mu+\nu-1.
\end{equation}
Combining the Lemma \ref{v-1} and the above result we can obtain
an expression for $I(g_1,g_2)$.
\begin{lem} If $g_1$ and $g_1$ are the cofactors for a good basis
for $\Omega^1(S)$, then $I(g_1,g_2)$ is finite and
\[
I\left(g_{1},g_{2}\right)=I\left(\frac{\partial f}{\partial
y},B_{1}\right)-I\left(B_{1},g_{1}\right)-\nu+1.
\]
\end{lem}
\begin{proof}
By Lemma \ref{v-1} we have $\nu(g_1)=\nu_1-1<\nu$. As $f$ is
irreducible it follows that $\gcd(f,g_1)=1$ and $I\left
(f\frac{\partial f}{\partial y},g_1\right )<\infty$. So, from
(\ref{eq:01}) that
{\small$$ I\left(f\frac{\partial f}{\partial
y},g_{1}\right) =I\left(A_1B_2\frac{\partial f}{\partial
y}-A_{2}B_1\frac{\partial f}{\partial y},g_1\right)
=I\left ( B_1B_2\frac{\partial f}{\partial x}-A_{2}B_1\frac{\partial f}{\partial y},g_1\right )
=I\left(B_1g_2f,g_1\right).
$$} Hence,
\begin{equation}\label{aux1}
I(g_1,g_2)=I\left(\frac{\partial f}{\partial
y},g_{1}\right)-I(B_1,g_1).
\end{equation}
The Corollary \ref{cor1} insures that $\frac{\partial f}{\partial
y}$ and $g_{1}$ are coprime. So, by (\ref{aux1}) and using
(\ref{derivative}) we obtain
\begin{align*}
I\left(g_{1},g_{2}\right) & =I\left(\frac{\partial f}{\partial y},g_{1}\right)
+I\left(\frac{\partial f}{\partial y},f\right) -I\left(\frac{\partial f}{\partial y},f\right) -I\left(B_{1},g_{1}\right)\\
& =I\left(\frac{\partial f}{\partial y},g_{1}f\right)-I\left(\frac{\partial f}{\partial y},f\right)-I\left(B_{1},g_{1}\right)\\
& =I\left(\frac{\partial f}{\partial y},A_{1}\frac{\partial f}{\partial y}-B_{1}\frac{\partial f}{\partial x}\right)-\left(\mu+\nu-1\right)-I\left(B_{1},g_{1}\right)\\
& =I\left(\frac{\partial f}{\partial
y},B_{1}\right)-\nu+1-I\left(B_{1},g_{1}\right).
\end{align*}
\end{proof}
Let us consider the Tjurina number $\tau$ of a plane curve $S:\{f=0\}$, that is,
\[ \tau:=\dim_{\mathbb{C}}\frac{\mathbb{C}\{x,y\}}{\left (f,\frac{\partial f}{\partial
y},\frac{\partial f}{\partial
x} \right )}.
\]
Zariski (see Theorem 1 in \cite{zbMATH03232554}) considered the
torsion submodule $T\Omega^1_{\mathcal{O}/\mathbb{C}}$ of the
K\"ahler differential module $\Omega^1_{\mathcal{O}/\mathbb{C}}$
over $\mathcal{O}=\frac{\mathbb{C}\{x,y\}}{(f)}$ and he showed that
$\tau=\dim_{\mathbb{C}}T\Omega^1_{\mathcal{O}/\mathbb{C}}.
$\\
On the other hand, Michler (Theorem 1 in \cite{zbMATH01549534}) proved that
$T\Omega^1_{\mathcal{O}/\mathbb{C}}$ is isomorphic as
$\mathcal{O}$-module, to $\frac{\left ( \frac{\partial f}{\partial
y},\frac{\partial f}{\partial x}\right ):\left ( f\right )}{\left (
\frac{\partial f}{\partial y},\frac{\partial f}{\partial x}\right )}$.
As $\left ( \frac{\partial f}{\partial y},\frac{\partial f}{\partial x}\right ):(f)$ is precisely the cofactor ideal of $S$, that is, $(g_1,g_2)$, one has
\[\tau=\dim_{\mathbb{C}}\frac{(g_1,g_2)}{\left (
\frac{\partial f}{\partial y},\frac{\partial f}{\partial x}\right )}
=\dim_{\mathbb{C}}\frac{\mathbb{C}\{x,y\}}{\left ( \frac{\partial f}{\partial
y},\frac{\partial f}{\partial x}\right
)}-\dim_{\mathbb{C}}\frac{\mathbb{C}\{x,y\}}{(g_1,g_2)}=\mu-I(g_1,g_2),
\]
that is,
\[\mu-\tau=I(g_1,g_2).\]
Denoting $\widetilde{\mu}$ the Milnor number
of $\widetilde{S}$ we provide in the next theorem a precise relation between $\mu-\tau$ and
$\widetilde{\mu}-\widetilde{\tau}$ by means of the analytic invariants
we have introduced previously for curves that admit a good basis.
\begin{thm}
\label{lem:Formula} If $S$ admits a good basis, then
\[
\mu-\tau=\widetilde{\mu}-\widetilde{\tau}
+\left(\nu_{1}-1\right)\left(\nu_{2}-1\right)+\mathfrak{i}\left(S\right)-1.
\]
\end{thm}
\begin{proof}
By symmetry, one can suppose $
\mathfrak{i}\left(S\right)=\min\left\{
i\left(\omega_{1}\right),i\left(\omega_{2}\right)\right\}
=i\left(\omega_{1}\right).$ By Lemma \ref{lem:The-intersection-of}
and the Max-Noether formula one has,
\[
\mu-\tau=I(g_1,g_2)=I_{(0,-\epsilon)}\left(\tilde{g}_{1},\tilde{g}_{2}\right)+\nu\left(g_{1}\right)\nu\left(g_{2}\right),
\]
where $\widetilde{H}:=E^*(H)$ and $E$ denotes the standard
blowing-up of the origin in $\mathbb{C}^2$.\\
In addition, the previous lemma and Lemma
\ref{lem:The-intersection-of}, yield {\small
\begin{align*}
I(g_1,g_2)
& =I\left(\frac{\partial f}{\partial y},B_{1}\right)-I\left(B_{1},g_{1}\right)-\nu+1\\
& =I_{(0,-\epsilon)}\left(\tilde{\frac{\partial f}{\partial y}},\tilde{B}_{1}\right)-
I_{(0,-\epsilon)}\left(\tilde{B}_{1},\tilde{g}_{1}\right)+\nu\left(\frac{\partial f}{\partial y}\right)\nu\left(B_{1}\right)-\nu\left(B_{1}\right)\nu\left(g_{1}\right)-\nu+1.
\end{align*}
} If $\tilde{\omega}_{i}=\frac{E^{*}\omega_{i}}{x^{\nu_{i}}}$, then
the Saito criterion yields $ x^{\nu_{1}}\tilde{\omega}_{1}\wedge
x^{\nu_{2}}\tilde{\omega}_{2}=\widetilde{u}x^{\nu}\tilde{f}x\textup{d}
x\wedge\textup{d} y. $ Since we have a good basis, that is,
$\nu_{1}+\nu_{2}=\nu,$ one has $
\tilde{\omega}_{1}\wedge\tilde{\omega}_{2}=u\tilde{f}x\textup{d} x\wedge\textup{d}
y. $ Locally around $(0,-\epsilon)$ for $i=1,2$ we have
$$
\tilde{\omega}_{i} =\left(A_{i}^{\nu_{i}}\left(1,y\right)+yB_{i}^{\nu_{i}}\left(1,y\right)+x\left(\cdots\right)\right)\textup{d} x+x\left(B_{i}^{\nu_{i}}\left(1,y\right)+\left(\cdots\right)\right)\textup{d} y\\
$$
We notice that the form
\[
\overline{\omega}_{2}=\frac{1}{x}\left(\tilde{\omega}_{2}-\frac{A_{2}^{\nu_{2}}\left(1,y\right)+yB_{2}^{\nu_{2}}\left(1,y\right)}{A_{1}^{\nu_{1}}\left(1,y\right)+yB_{1}^{\nu_{1}}\left(1,y\right)}\tilde{\omega}_{1}\right)
\]
is holomorphic at $\left(0,-\epsilon\right)$ and $\left\{
\tilde{\omega}_{1},\overline{\omega}_{2}\right\} $ is a Saito basis
for $\widetilde{S}:\{\tilde{f}=0\}$. A computation shows that the
cofactor associated to $\tilde{\omega}_{1}$ is written $
g_{1}^{'}=\tilde{g}_{1}+\nu\tilde{B}_{1}. $ Moreover, one has $
\tilde{\omega}_{1}=\left(\tilde{A}_{1}+y\tilde{B}_{1}\right)\textup{d}
x+x\tilde{B}_{1}\textup{d} y=A'\textup{d} x+B'\textup{d} y.
$
Now,
\[
\left(A_{1}^{\nu_{1}}\left(1,y\right)+yB_{1}^{\nu_{1}}\left(1,y\right)+x\left(\cdots\right)\right)\frac{\partial\tilde{f}}{\partial y}-x\tilde{B}_{1}\frac{\partial\tilde{f}}{\partial x}=g_{1}^{'}\tilde{f}.
\]
If $x$ divides $g_{1}^{'}$ then $\tilde{\omega}_{1}$ would be
dicritical and this is not possible. Therefore,
\begin{align*}
I_{(0,-\epsilon)}\left(x,g_{1}^{'}\tilde{f}\right) &
=I_{(0-\epsilon)}\left(A_{1}^{\nu_{1}}\left(1,y\right)+yB_{1}^{\nu_{1}}\left(1,y\right),x\right)+I_{(0,-\epsilon)}\left(x,\frac{\partial\tilde{f}}{\partial
y}\right).
\end{align*}
and, by Corollary \ref{cor1}, $
I_{(0,-\epsilon)}\left(x,g_{1}^{'}\right)=i\left(\omega_{1}\right)-1=\mathfrak{i}\left(S\right)-1.
$\\
Notice that $\tilde{B}_{1}$ and $g_{1}^{'}$ cannot have a common
divisor, since it would be a common divisor of $\tilde{g}_{1}$ and
$\tilde{B}_{1}$ that is impossible by Lemma
\ref{lem:We-can-suppose}. So,
$$
I_{(0,-\epsilon)}\left(\tilde{B}_{1},\tilde{g}_{1}\right)
=I_{(0,-\epsilon)}\left(x\tilde{B}_{1},g_{1}^{'}\right)-\mathfrak{i}\left(S\right)+1
=I_{(0,-\epsilon)}\left(B_{1}^{'},g_{1}^{'}\right)-\mathfrak{i}\left(S\right)+1.
$$
Moreover,
\begin{align*}
I_{(0,-\epsilon)}\left(\frac{\partial \tilde{f}}{\partial y},\tilde{B}_{1}\right) & =I_{(0,-\epsilon)}\left(\frac{\partial\tilde{f}}{\partial y},B_{1}^{'}\right)-I_{(0,-\epsilon)}\left(\frac{\partial\tilde{f}}{\partial y},x\right)\\
& =I_{(0,-\epsilon)}\left(\frac{\partial\tilde{f}}{\partial
y},B_{1}^{'}\right)-I_{(0,-\epsilon)}\left(\tilde{f},x\right)+1.
\end{align*}
So, as $\frac{\partial \tilde{f}}{\partial y}=\tilde{\frac{\partial f}{\partial y}}$ and combining all the above relation yields
\begin{align*}
\mu-\tau & =I_{(0,-\epsilon)}\left(\frac{\partial\tilde{f}}{\partial y},B_{1}^{'}\right)-I_{(0,-\epsilon)}\left(\tilde{f},x\right)+1-\left(I_{y=-\epsilon}\left(B_{1}^{'},g_{1}^{'}\right)-\mathfrak{i}\left(S\right)+1\right)\\
& \qquad+\nu\left(\frac{\partial f}{\partial y}\right)\nu\left(B_{1}\right)-\nu\left(B_{1}\right)\nu\left(g_{1}\right)-\nu+1\\
& =I_{(0,-\epsilon)}\left(g_{1}^{'},g_{2}^{'}\right)+\left(\nu-1\right)\nu_{1}-\nu_{1}\nu\left(g_{1}\right)-\nu+\mathfrak{i}\left(S\right).
\end{align*}
As
$I_{(0,-\epsilon)}\left(g_{1}^{'},g_{2}^{'}\right)=\widetilde{\mu}-\widetilde{\tau}$
and $\nu\left(g_{1}\right)=\nu_1-1$, we obtain finally
\[
\mu-\tau=\widetilde{\mu}-\widetilde{\tau}+\left(\nu_{1}-1\right)\left(\nu_{2}-1\right)+\mathfrak{i}\left(S\right)-1.
\]
\end{proof}
Let us analyze the examples previously considered.
\begin{ex*}[\textbf{1}]
For $S_1$ we have a good basis with $\nu_1=1$, $\nu_2=p-1$ and
$\mathfrak{i}\left(S_1\right)=1$, then $\mu-\tau=0$ as classically
known.
\end{ex*}
\begin{ex*}[\textbf{2}]
Notice that for $S_2$ we have
$\mathfrak{i}\left(S_2\right)=i\left(\omega_1\right)=1$,
$\nu_1=\nu_2=2$ and $\widetilde{S}_3$ is regular, so
$\widetilde{\mu}-\widetilde{\tau}=0$. In this way,
\[1=I(g_1,g_2)=\mu-\tau=0+(2-1)(2-1)+1-1.
\]
So, the formula in the previous theorem holds although $S_2$ does
not admit any good basis.
\end{ex*}
\begin{ex*}[\textbf{3}]
For $S_3$ we get
$\mathfrak{i}\left(S_4\right)=i\left(\omega_1\right)=2$, $\nu_1=2,
\nu_2=3$ and $\widetilde{S}_3$ is analytically equivalent to $S_2$,
so $\widetilde{\mu}-\widetilde{\tau}=1$. In this way,
\[4=I(g_1,g_2)=\mu-\tau=1+(2-1)(3-1)+2-1.
\]
\end{ex*}
\begin{ex*}[\textbf{4}]
As we presented above, $S_4$ does not have a good basis. We have
$\mathfrak{i}\left(S_4\right)=i\left(\omega_1\right)=1$, $\nu_1=\nu_2=3$ and $\widetilde{\mu}-\widetilde{\tau}=0$, but in this case,
\[5=I(g_1,g_2)=\mu-\tau\neq 4=0+(3-1)(3-1)+1-1.\]
A more detailed analysis shows that Lemma \ref{lem:The-intersection-of}
is not valid in this case because the intersection of the tangent cone
of $g_1$ and $g_1$ is $x=0$ that is distinct to the tangent cone $y=0$ of $S_4$.
\end{ex*}
\section{The minimal Tjurina number and the Dimca-Greuel question for plane branches. }
Given a curve $S$, we denote by $L=L(S)$ its topological class. Although the Milnor number is constant in $L$, the same is not true for the Tjurina number $\tau(S)$. On the other hand, as $\tau(S)$ is upper semicontinuous, the minimum value $\tau_{\min}$ for curves in $L$ is achieved generically and it should be computed by the sole topological data (see Chapitre III, Appendice of \cite{zariski} by Teissier).\\
For a topological class $L$ given by characteristic exponents $(\beta_0,\beta_1)$, Delorme in \cite{Delorme1978} presented a formula for the dimension of the generic component of the Moduli space that allow us to compute $\tau_{\min}$. For an arbitrary topological class, Peraire (see \cite{Peraire}) presented an algorithm to compute the $\tau_{\min}$ using flag of the Jacobian ideal.\\
In this section, using the last theorem and results of \cite{YoyoBMS}, we give an alternative method to compute $\tau_{\min}$ in a fixed topological class $L$ and as a bonus we are able to answer a question of Dimca-Greuel for the irreducible plane curves.\\
If $S$ admits a good basis we can not insure that the same is valid
for $\widetilde{S}$ (see Example (3)). However, this property is
true generically.
\begin{thm}\label{theo-tau-min}
Let $L$ the topological class of plane branch given by the characteristic exponents $(\beta_0,\beta_1,\ldots ,\beta_s)$, $\tau_{\min}$
the minimal Tjurina number in $L$ and $\widetilde{\tau}_{\min}$
the minimal Tjurina number in $\widetilde{L}$. If $S$ is generic in $L$, then
\begin{equation}\label{formula}
\mu-\tau_{\min}=\widetilde{\mu}-\widetilde{\tau}_{\min}
+\left ( \left [\frac{\beta_0}{2}\right ]-1\right )\left (\beta_0-\left [\frac{\beta_0}{2}\right ]-1\right )+\mathfrak{i}\left(S\right)-1.
\end{equation}
Moreover, if $n=\left\lceil \frac{\beta_{1}}{\beta_{1}-\beta_0}\right\rceil $, then $\mathfrak{i}\left(S\right)=\left [ \frac{\beta_0}{2}\right ]+1-p_1(S)$, where $p_1(S)$ can be computed in the following
way:
\[
\bullet\ \mbox{if}\ \beta_0\ \mbox{is even then}\ p_{1}\left(S\right)=\left\{ \begin{aligned}1 & \ \textup{ if }n=2\\
1 & \ \textup{ if \ensuremath{\beta_{1}} is even}\\
\frac{n-1}{2} & \ \textup{ if \ensuremath{\beta_{1}} is odd and \ensuremath{n} odd}\\
\frac{n-2}{2} & \ \textup{ if \ensuremath{\beta_{1}} is odd and \ensuremath{n} is even}
\end{aligned}
\right.
\]
\[
\bullet\ \mbox{if}\ \beta_0\ \mbox{is odd then}\ p_{1}\left(S\right)=\left\{ \begin{aligned}0 & \ \textup{ if }n=2\\
1 & \ \textup{ if \ensuremath{\beta_{1}} is odd}\\
\frac{n-3}{2} & \ \textup{ if \ensuremath{\beta_{1}} is even and \ensuremath{n} odd}\\
\frac{n-2}{2} & \ \textup{ if \ensuremath{\beta_{1}} is even and \ensuremath{n} is even.}
\end{aligned}
\right.
\]
\end{thm}
\begin{proof} Suppose that $\beta_0=\nu(S)$ is even. According to the Generic Basis Theorem, $S$ admits a good basis $\{\omega'_1,\omega'_2\}$ with $\nu(\omega'_{1})=\nu(\omega'_{2})=\frac{\beta_0}{2}$. For generic $\alpha_1,\alpha_2\in \mathbb{C}$
$
\left\{\omega_1=\omega'_1+\alpha_2\omega'_2,\omega_2=\omega'_2+\alpha_1\omega'_1\right\}
$ remain a good basis with $\nu_1=\nu_2=\frac{\beta_0}{2}$ and
$i(\omega_1)=i(\omega_2)$.\\
Now, according to \cite{YoyoBMS} - using the notations of the
mentioned paper, it refers to the case $\delta_{1}=0$ and
$\delta_{2}=1$ - we obtain
$\nu_1+1=\frac{\beta_0}{2}+1=\sum_{q\in\mathbb{P}^1}{\rm
Ind}(\widetilde{\mathcal{F}},C,q)=i(\omega_1)+p_1(S),$ that is,
\[\mathfrak{i}\left(S\right)=i\left(\omega_1 \right)=\nu_1+1-p_1\left(S\right)=\frac{\beta_0}{2}+1-p_1\left(S\right)=\left [\frac{\beta_0}{2}\right ]+1-p_1(S)\]
where $p_{1}\left(S\right)$ is described in \cite{YoyoBMS}.\\
Now, suppose $\beta_0$ is odd and let $\left\{
\omega'_{1},\omega'_{2}\right\}$ be a Saito basis for $S\cup l$ with
$l$ a generic line that without loss of generality can be considered
$x=0$. As $\nu(S\cup l)$ is even, by the same above argument, we can
suppose that
\[\nu(\omega'_1)=\nu(\omega'_2)=\frac{\beta_0+1}{2}=\left
[\frac{\beta_0}{2}\right ]+1\ \ \mbox{and}\ \ \mathfrak{i}(S\cup
l)=i\left(\omega'_1\right)=i\left(\omega'_2\right).\] Denoting
$\omega'_{i}
=\left(a_{i}\left(y\right)+x\left(\cdots\right)\right)\textup{d}x+x\left(\cdots\right)\textup{d}y
$ and considering generic $\alpha_1,\alpha_2\in \mathbb{C}$ we
obtain a good Saito basis
$\{\omega_1=\omega'_{1}+\alpha_2\omega'_{2},\omega_2=\omega'_{2}+\alpha_1\omega'_{1}\}$
such that
$\nu\left(a_{1}(y)+\alpha_2a_2(y)\right)=\nu\left(a_{2}(y)+\alpha_1a_1(y)\right)$,
\[i\left(\omega_{1}\right)=i\left(\omega'_{1}\right)=i\left(\omega'_{2}\right)=i\left(\omega_{2}\right)
\ \mbox{and}\
\nu(\omega_1)=\nu(\omega'_1)=\nu(\omega'_2)=\nu(\omega_2).\] Now the
family
\[
\left\{
\omega_{1},\frac{1}{x}\left(\omega_{2}-\frac{a_{2}(y)+\alpha_1a_1(y)}{a_{1}(y)+\alpha_2a_2(y)}\omega_{1}\right)\right\}
\]
is a good Saito basis for $S$. Finally, since
$i\left(\frac{1}{x}\left(\omega_{2}-\frac{a_{2}(y)+\alpha_1a_1(y)}{a_{1}(y)+\alpha_2a_2(y)}\omega_{1}\right)\right)\geq
i\left(\omega_{1}\right)$, one has $
\mathfrak{i}\left(S\right)=i\left(\omega_{1}\right). $
By the description of $p_1(S\cup l)$ given in \cite{YoyoBMS} - using
the notations of the article, it refers to the case $\delta_{1}=1$
and $\delta_{2}=1$ - we get
\[\mathfrak{i}\left(S\right)=i\left(\omega_1 \right)=\frac{\nu\left(S\right)+1}{2}-p_1\left(S\right)=
\left [\frac{\beta_0}{2}\right ]+1-p_1\left(S\right).\]
Thus, the proof of the formula is a consequence of Theorem \ref{lem:Formula} noticing that by the Generic Basis Theorem we have $\nu_1=\left [\frac{\beta_0}{2}\right ]$ and $\nu_2=\beta_0-\left [\frac{\beta_0}{2}\right ]$.
\end{proof}
\begin{ex*}[\textbf{5}]
In \cite{Peraire}, Peraire computed the minimum Tjurina number for
the topological class whose characteritic exponents are $(9,12,17)$.
After five blowing-ups, we obtain a curve with multiplicity $2$. The
corresponding characteristics exponents of the sequence of blown-up
curves are $(3,14),\ (3,11),\ (3,8),\ (3,5),\ (2,3)$. Applying
inductively the formula (\ref{formula}), one accumulates
contribution to the difference $\mu -\tau_{\textup{min}}$. Actually,
it can be seen that the respective contributions are $15,\ 1,\ 1,\
1,\ 0,\ 0$. Thus $\tau_{\textup{min}}=\mu-18=98-18=80$ which
coincides with the computation of Peraire.
\end{ex*}
The last theorem allow us obtain a formula to compute the minimum Tjurina number in a topological class using the multiplicity sequence.
\begin{cor} Let $L$ a topological class of a singular plane branch determined by the multiplicity sequence $\nu_{(1)},\nu_{(2)},\ldots ,\nu_{(N)},\nu_{(N+1)}=1,\ldots$. The minimal Tjurina number achieved in $L$ is \[\tau_{\min}=\sum_{i=1}^{N} \left (\nu_{(i)}^2+\left [ \frac{\nu_{(i)}}{2}\right ]\left ( \left [\frac{\nu_{(i)}}{2} \right ]-\nu_{(i)}-1\right )-1+p_1(S_{(i)}) \right )
\]
where $S_{(i)}$ denote the curve with multiplicity $\nu_{(i)}$ in the canonical resolution process for a generic curve in $L$.
\end{cor}
\begin{proof}
Applying inductively the formula presented in the last theorem and using that $\mathfrak{i}\left(S_{(i)}\right )=\left [ \frac{\nu_{(i)}}{2}\right ]+1-p_1(S_{(i)})$ yields
\[
\tau_{\min} = \mu-\sum_{i=1}^{N} \left (\left (\left [ \frac{\nu_{(i)}}{2}\right ]-1\right )\left ( \nu_{(i)}-\left [\frac{\nu_{(i)}}{2} \right ]-1\right )-(\mathfrak{i}\left(S_{(i)}\right )-1) \right )
\]
\begin{equation}\label{genericformula}
\hspace{0.5cm} = \mu +\sum_{i=1}^{N} \left (\left [ \frac{\nu_{(i)}}{2}\right ]\left (\left [\frac{\nu_{(i)}}{2} \right ]-\nu_{(i)}-1\right )+\nu_{(i)}-1+p_{1}(S_{(i)}) \right ).
\end{equation}
As $\mu=\sum_{i=1}^{N}\nu_{(i)}\left ( \nu_{(i)}-1\right )$, we get the proof.
\end{proof}
In \cite{Dimca-Greuel}, Dimca
and Greuel present an interesting question about the Tjurina number for curves in a given topological class $L$. More specifically, they ask if $4\tau(S) >3\mu(S)$ for any curve in $L$.\\
As the Tjurina number is semicontinuous in $L$ and we have obtained a formula for the $\tau_{\min}$, we are able to given a lower bound for the Tjurina number in terms of the Milnor number and it answered positively the previous question for the irreducible case.
\begin{cor}\label{dimca}
Let $S$ be a singular irreducible plane curve. Then
\[
\tau\left(S\right)\geq \frac{3}{4}\mu\left(S\right)+\frac{\sqrt{1+4\mu(S)}-1}{8}.
\] In particular, $4\tau(S) >3\mu(S)$.
\end{cor}
\begin{proof}
We denote $\mu=\mu(S)$.
It is sufficient to show the inequality for the $\tau_{\min}$.\\
By (\ref{genericformula}), the relation below holds
\[ 4\tau_{\min}-3\mu= \mu +4\sum_{i=1}^{N} \left (\left [ \frac{\nu_{(i)}}{2}\right ]\left (\left [\frac{\nu_{(i)}}{2} \right ]-\nu_{(i)}-1\right )+\nu_{(i)}-1+p_{1}(\nu_{(i)}) \right ).
\]
Now, using that $\mu=\sum_{i=1}^{N}\nu_{(i)}\left (
\nu_{(i)}-1\right )$ and $4\left [ \frac{\nu_{(i)}}{2}\right ]\left
(\left [\frac{\nu_{(i)}}{2} \right ]-\nu_{(i)}-1\right
)=-\nu_{(i)}^2-2\nu_{(i)}+\delta_i
$
with $\delta_i =0$ if $\nu_{(i)}$ is even and $\delta_i=3$ if
$\nu_{(i)}$ is odd, we obtain
\[ 4\tau_{\min}-3\mu= \sum_{i=1}^{N} \left ( \nu_{(i)}+\delta_i +4(p_{1}(S_{(i)})-1)\right ).
\]
Now, by Theorem \ref{theo-tau-min} we have that:
\begin{itemize}
\item if $\nu_{(i)}$ is even, then $p_1(S)\geq 1$ and $\nu_{(i)}+0 +4(p_{1}(\nu_{(i)})-1)\geq \nu_{(i)}$,
\item if $\nu_{(i)}$ is odd, then $p_1(S)\geq 0$ and $\nu_{(i)}+3 +4(p_{1}(\nu_{(i)})-1)\geq \nu_{(i)}-1.$
\end{itemize}
So, the following inequality follows
\begin{equation}\label{inequality}
4\tau_{\min}-3\mu\geq \sum_{i=1}^{N}(\nu_{(i)}-1).
\end{equation}
As $\mu=\sum_{i=1}^{N}(\nu_{(i)}-1)^2+\sum_{i=1}^{N}\left (
\nu_{(i)}-1\right )$ we get $4\tau_{\min}-3\mu\geq
\mu-\sum_{i=1}^{N}(\nu_{(i)}-1)^2.$ Using (\ref{inequality}), that
is, $-(4\tau_{\min}-3\mu)^2\leq -\left (
\sum_{i=1}^{N}(\nu_{(i)}-1)\right )^2\leq
-\sum_{i=1}^{N}(\nu_{(i)}-1)^2$, we obtain $4\tau_{\min}-3\mu\geq
\mu-(4\tau_{\min}-3\mu)^2$ and consequently
\[
\tau\left(S\right)\geq\tau_{\min}\geq \frac{3}{4}\mu+\left (\frac{-1+\sqrt{1+4\mu}}{8} \right ).
\]
\end{proof}
\begin{ex*}[\textbf{6}] Let us consider the topological
class $L$ determined by the characteristic exponents $(141,142)$.
The Milnor number for any curve in $L$ is
$\mu=(141-1)(142-1)=19740$. Using the lower bound presented in the
las result we obtain $\tau_{\min}\geq 14840$. For this topological
class it follows by the Delorme result ({\it cf.}
\cite{Delorme1978}) that $\tau_{\min}=14910$.
\end{ex*}
While we submit the first version of this paper to Arxiv, we discover that, at the same time, a positive answer for the Dimca-Greuel question was obtained by Alberich-Carrami\~nana {\it et al.} and published in Arxiv \cite{dimca-conj} a few days before. Although the methods are a bit different, the key ingredient is still the formula for the generic dimension of the moduli space obtained in \cite{YoyoBMS}.
\bibliographystyle{plain}
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{"url":"http:\/\/ethos.bl.uk\/OrderDetails.do?uin=uk.bl.ethos.556955","text":"Use this URL to cite or link to this record in EThOS: http:\/\/ethos.bl.uk\/OrderDetails.do?uin=uk.bl.ethos.556955\nTitle: On perhexiline and its application to myocardial protection during cardiac surgery\nAuthor: Drury, Nigel Edward\nAwarding Body: University of Birmingham\nCurrent Institution: University of Birmingham\nDate of Award: 2012\nAvailability of Full Text:\n Access through EThOS: Access through Institution:\nAbstract:\nPerhexiline is an anti-anginal drug that is thought to shift myocardial metabolism from $$\\beta$$-oxidation of fatty acids to glucose utilisation. An associated improvement in energy efficiency may be beneficial in ischaemia-reperfusion as an adjunct to established techniques for myocardial protection during cardiac surgery. In this thesis, I conduct a prospective double-blind randomised placebo-controlled trial of oral perhexiline in patients undergoing coronary artery surgery, obtaining samples of serum, right atrium and left ventricle. I measure the concentration of perhexiline using high performance liquid chromatography and find that although highly concentrated in the heart, it may not have reached steady-state in the ventricular myocardium. I perform enzymatic colourimetry and ultra-high resolution mass spectrometry to detect changes in carbohydrate and lipid metabolism; however, the myocardial metabolic profiles of patients on perhexiline are indistinguishable from controls. On analysing the results of the clinical trial, I find no improvement in the primary endpoint, the incidence of a low cardiac output episode, or any secondary outcomes. I conclude that preoperative oral perhexiline does not improve clinical markers of myocardial protection and despite significant accumulation in the myocardium, it has no significant effect on the measurable metabolic profile of the heart at the time of surgery.\nSupervisor: Not available Sponsor: Not available\nQualification Name: Thesis (Ph.D.) Qualification Level: Doctoral\nEThOS ID: uk.bl.ethos.556955\u00a0 DOI: Not available\nKeywords: QH301 Biology ; QM Human anatomy ; QP Physiology ; R Medicine (General) ; RZ Other systems of medicine\nShare:","date":"2017-08-18 15:38:59","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\": 0.1928662657737732, \"perplexity\": 11837.953474249685}, \"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-2017-34\/segments\/1502886104681.22\/warc\/CC-MAIN-20170818140908-20170818160908-00068.warc.gz\"}"} | null | null |
Most of our moshes are now running postgres 9.4, whereas before the jessie upgrade they were running postgres 9.1.
Upon reboot, it appears that the old postgres 9.1 cluster is not started and the new postgres 9.4 is started, but using a non-default port.
That means that any postgres databases are not running, pending an upgrade to 9.4.
It also means we are getting errors when backups are run because the backup can't dump the postgres databases.
At this point we don't really have anything to upgrade, but it doesn't hurt and if you do the upgrade process it will ensure that the newest version of postgres is running on the default port.
And... there are still more moshes that need to be checked.
During this when attempting to upgrade the databases on rose I failed to realize that in this case someone had already configured version 9.4 to run on port 5432 and the previous postgres version on port 5433. I dumped the database and imported in the wrong direction and subsequently dropped the wrong cluster. I believe I have provoked data loss. I have attempted to recover files from backups but require more assistance.
The problem on rose turned out to be fine - the database dropped was no longer needed.
This is done now. Postgres has been upgraded to 9.4 on all moshes and all older versions have been removed. | {
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Missouri's black players to boycott football over university's handling of racial harassment
Filed under News at Nov 2015
Clay Morton
Connect with Clay Morton
COLUMBIA, Mo. — Missouri football players announced Saturday night on Twitter that they will not participate in team activities until the university president is removed from office.
The statement from the athletes of color was tweeted out Saturday by several members of the football team, including starting running back Russell Hanbrough.
The move aligns the team with campus groups that have been protesting the way university President Tim Wolfe has dealt with issues of racial harassment during the school year. Jonathan Butler, a black graduate student, is in the sixth day of a hunger strike to call attention to the issue.
The statement from the players included a photograph of 32 black men, including Butler. In the photo, two players have their arms linked through his.
The message reads: "The athletes of color on the University of Missouri football team truly believe "Injustice Anywhere is a threat to Justice Everywhere" We will no longer participate in any football related activities until President Tim Wolfe resigns or is removed due to his negligence toward marginalized students' experience. WE ARE UNITED!!!!!"
Wolfe met with Butler and student groups on Friday to discuss the university's handling of racial harassment cases.
Butler also has expressed concern about other student issues, including the removal of graduate student health care subsidies and an end to university contracts with a Planned Parenthood clinic.
Student groups have been critical over how Wolfe has handled issues of race and discrimination, and blocked his car at homecoming in an attempt to speak with him. They have complained that the use of racial slurs is prevalent. In April, a swastika was found in a Missouri dormitory, and officials said last month that they were investigating another act of vandalism that targeted minorities.
In a text message statement to The Associated Press, the Missouri athletic department said it was aware of the statement by "many of our student-athletes."
"We all must come together with leaders from across our campus to tackle these challenging issues and we support our student-athletes right to do so."
The campus is about 120 miles west of Ferguson, the St. Louis suburb where tensions erupted following the police shooting death of 18-year Michael Brown. Brown, who was black, was unarmed when he was shot by a white police officer during a confrontation in a street in August 2014. His death sparked protests and helped spawn the national "Black Lives Matter" movement rebuking police treatment of minorities.
The campus of 35,448 is mostly white. Approximately 17 percent of the student population identifies with a minority group, according to the university website.
Missouri last played Thursday night at home, losing to Mississippi State to fall to 4-5 on the season. The Tigers have three games left, including next Saturday at home against BYU. | {
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} | 248 |
Q: @angular/forms/FormsModule missing I am trying to import the FormsModule from @angular/forms@0.3.0 inside Angular 2 RC5. Should be simple right?
import { FormsModule } from '@angular/forms';
However this is not part of the @angular/forms library which I installed using
npm install @angular/forms
Some have suggested using:
import { ReactiveFormsModule } from '@angular/forms';
However this doesn't work either.
I've looked into the forms package and it appears that it's in the form_providers typing file (form_providers.d.ts) and it does mention FormsModule. In the forms.js it mentiones __export(require('./form_providers')); so this means it should be available right?
Any ideas how I fix this problem?
Many thanks
JT
app.module.ts
import {NgModule} from '@angular/core';
import {FormsModule} from '@angular/forms';
import {BrowserModule} from '@angular/platform-browser';
import {platformBrowserDynamic} from '@angular/platform-browser-dynamic';
import {MeteorModule} from 'angular2-meteor';
import {AppComponent} from './app.component';
@NgModule({
imports: [BrowserModule, MeteorModule, FormsModule],
declarations: [AppComponent],
bootstrap: [AppComponent]
})
export class AppModule { }
platformBrowserDynamic().bootstrapModule(AppModule,[ disableDeprecatedForms(),provideForms()]));
Package.json (that is being used by "npm install")
{
"name": "angular2-meteor-base",
"private": true,
"scripts": {
"start": "meteor run",
"test": "meteor test --driver-package practicalmeteor:mocha",
"test:ci": "meteor test --once --driver-package dispatch:mocha-phantomjs"
},
"devDependencies": {
"chai": "3.5.0",
"chai-spies": "0.7.1"
},
"dependencies": {
"@angular/common": "^2.0.0-rc.5",
"@angular/compiler": "^2.0.0-rc.5",
"@angular/core": "^2.0.0-rc.5",
"@angular/forms": "latest",
"@angular/platform-browser": "^2.0.0-rc.5",
"@angular/platform-browser-dynamic": "^2.0.0-rc.5",
"@angular/router": "^3.0.0-rc.1",
"angular2-meteor": "latest",
"angular2-meteor-auto-bootstrap": "latest",
"angular2-meteor-polyfills": "latest",
"angular2-meteor-tests-polyfills": "latest",
"es6-shim": "0.35.1",
"lodash": "^4.15.0",
"meteor-node-stubs": "0.2.3",
"reflect-metadata": "0.1.3",
"rxjs": "5.0.0-beta.6",
"zone.js": "0.6.12"
}
}
app.component.ts
import {Component} from '@angular/core';
@Component({
selector: 'app',
template: 'Hello World'
})
export class AppComponent {
constructor() {
}
}
A: I had a similar issue and had to update my package.json to specify the version of forms to the following:
"@angular/forms": "^0.3.0",
After that running an npm update.
A: make sure you added FormsModule to the app module or module you are working on and restart the server.
A: I solved it just using the following steps:
*
*write import {} from '@angular/forms';
*after that, write FormsModule in the {}
*finally write FormsModule in the imports
It worked.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,299 |
\section{Introduction}
\label{sec:introduction}
Since the detection of the first exoplanet orbiting a main sequence star, 51~Peg, a large population of these objects has been discovered covering a wide range of masses and orbital periods mostly with indirect methods such as radial velocities measurements and transits (see e.g. \citealt{santos2008} for a review). Although mainly sensitive to planets with period of less than 10~years, radial velocities surveys have found stars which start to show long term trends indicating possible low mass companions orbiting at large orbital separations \citep{wittenmyer2007}. The wide use of adaptive optics (AO) systems and coronagraphy in large telescopes instrumentation for high-contrast imaging has allowed to start probing the vicinity of nearby stars for low mass companions at large orbital separations. Over the last decade, a handful of objects close to the planetary mass regime have been imaged with existing instruments, such as 2M~1207~b \citep{chauvin2005a}, DH~Tau~B \citep{itoh2005}, GQ~Lup~b \citep{neuhauser2005}, AB~Pic~b \citep{chauvin2005b}, CHXR~73~B \citep{luhman2006}, and more recently Fomalhaut~b \citep{kalas2008}, 1RXS~J1609~b \citep{lafreniere2008}, $\beta$~Pic~b \citep{lagrange2008} and the triple system around HR~8799 \citep{marois2008b}. However, the large uncertainty on the mass of these objects may place some of them in the sub-stellar rather than the planetary mass regime.
The next generation of planet finding instruments currently being built will combine: (i) extreme AO systems with a large number of actuators \citep{angel1994,stahl1995,langlois2001} to reach very high corrections in the near-infrared \citep{fusco2006,allercarpentier2008}; and (ii) high-efficiency coronagraphs such as the apodized pupil Lyot coronagraph (\citealt{soummer2005} and references therein) or the achromatic 4-quadrant phase mask \citep{rouan2000,mawet2006} to obtain optimal star extinction. GPI (Gemini Planet Imager) for Gemini South \citep{macintosh2006} and SPHERE (Spectro-Polarimetric High-contrast Exoplanet REsearch) for the ESO-VLT \citep{beuzit2006} are the two leading instruments of that category. These will both start operation in 2011, along with HiCIAO (High-contrast Coronagraphic Imager for Adaptive Optics) for Subaru \citep{hodapp2008}. They will aim at detecting exoplanets down to the Jupiter mass (\MJup) around nearby young stars by reaching contrast values of 15 to 17.5~mag ($10^{-6}$ to $10^{-7}$) at angular separations of $\sim$0.1\as. Both GPI and SPHERE will incorporate diffraction limited integral field spectrographs (IFS) in the near infrared, allowing to obtain images simultaneously at several wavelengths. SPHERE will also incorporate a differential imager named IRDIS (InfraRed Dual Imaging Spectrograph, \citealt{dohlen2008a}) that will provide simultaneous images at two close wavelengths in either one of its 5 different filter pairs over the Y to Ks bands.
These instruments will allow to use different observing strategies such as Spectral Differential Imaging (SDI, \citealt{racine1999}) or Angular Differential Imaging (ADI, \citealt{marois2006a}) to obtain data which will be analyzed using advanced methods such as Spectral Deconvolution \citep{sparks2002} or Localy Optimized Combination of Images (LOCI, \citealt{lafreniere2007}) for IRDIS data. Specific methods of signal extraction have also been developed within the SPHERE consortium \citep{mugnier2008,smith2009} to be used in the data reduction pipeline of IRDIS. Data analysis methods are of extreme importance when it comes to the detection of faint objects in coronagraphic images dominated by speckles. In particular, the precise estimation of the object flux after applying these methods is critical for calibration of planetary mass objects model atmospheres (\citealt{allard2001,allard2003,allard2007}, and in preparation; \citealt{burrows2006,ackerman2001,tsuji2005}) based on effective temperature (\teff) and surface gravity (\logg), and the corresponding evolutionary models \citep{burrows1997,chabrier2000,baraffe2003,fortney2005}. The only currently available method for mass estimation, when no dynamical mass estimations are known for the object, consists in incorporating photometric measurements and their error assessment into these physical models. Although spectroscopy is the method of choice for characterization, it may not be possible to obtain high-quality spectra of extremely faint sources \citep{vigan2008,janson2010}.
In this paper we investigate the photometric limitations in high-contrast data obtained at different wavelengths with a dual-band imager like IRDIS, and we study how it translates in terms of characterization of planetary mass objects. After reminding in Sect.~\ref{sec:limitations_high_contrast_imaging} the origin of the speckle noise which is the fundamental limitation in high-contrast imaging and the methods to overcome it, we present in Sect.~\ref{sec:end_to_end_simulations} the end-to-end simulations of IRDIS which were performed to obtain a realistic observing sequence. Section~\ref{sec:photometric_accuracy} briefly describes the detection limits obtained with ADI and SDI+ADI data analysis methods before studying the performances in terms of photometric accuracy. Finally, in Sect.~\ref{sec:photometric_characterization} we advocate a filter pair procedure for IRDIS and present an analysis of the characterizations which will be possible from aperture photometry.
\section{Limitations in high-contrast imaging}
\label{sec:limitations_high_contrast_imaging}
Detecting very faint planetary objects requires to obtain diffraction limited images with high-order AO systems in order to overcome the large contrast ratio between the star and the planet with coronagraphy. In high-Strehl ratio coronagraphic images, the factor that limits the accessible dynamical range is the speckle noise \citep{soummer2007} induced by atmospheric phase residuals and instrumental quasi-static aberrations not corrected by the AO system. The quasi-static speckles are caused by the instrumental aberrations that slowly change during a long exposure. Telescope orientation, temperature variations or rotating optical elements cause small mechanical variations in the optical elements which make the speckle pattern evolve. Long time exposures are typically decomposed in a series of short exposure images of a few seconds during which the atmospheric residuals are averaged out, forming a smooth halo over which the quasi-static speckles are superimposed, because their coherence time is much longer than the atmospheric residuals \citep{langlois1998,macintosh2005,hinkley2007}. To optimize AO performances and speckle rejection, observations can be performed in pupil stabilized mode, leading to a very high stability of the star Point Spread Function (PSF) and a slow rotation of the field of view during the observations, at a rate which depends on the star position in the sky.
The speckle noise can be reduced by subtracting a reference PSF from each science frame in order to remove the star halo and speckles, and possibly reveal a faint planetary object. This reference PSF can be obtained by observing a reference star taken in the same observing conditions (parallactic angle and atmospheric conditions if possible) as the original target to reproduce a similar pattern of quasi-static speckles. This is a very time consuming task because the time spent on the reference star is equal to the one spent on the target to precisely match both PSF, and the aberrations between the two stars cannot be exactly reproduced. The reference can also be built from the science frames by using either spectral or angular information.
The SDI method has first been proposed by \citet{racine1999} for faint companion detection. It has been extensively studied \citep{marois2000}, and subsequently tested on sky with TRIDENT on the CFHT \citep{marois2005} and with NACO on the VLT \citep{lenzen2004}. The technique relies on the fact that planetary objects have large intrinsic molecular features in their spectrum, while the host star has a relatively flat spectrum. By taking simultaneously two images of a system at two close wavelengths located around one of these sharp features and subtracting them, the star contribution can be partially eliminated, and the planet signal revealed. SDI is most effective when used for detecting cool companions that show deep molecular absorption bands caused by H$_2$O, CH$_4$ and NH$_3$ at low \teff according to state of the art planetary mass objects atmosphere models. With carefully selected filter pairs, a contrast of several magnitudes on the planet flux between the two filters can be obtained. However, the presence of the molecular features expected for the detection of cool planetary companions should not be taken for granted since recent atmosphere models \citep{fortney2008}, as well as observations of 2M1207b \citep{mohanty2007,patience2010} and the HR8799 planets \citep{marois2008b,metchev2009,janson2010}, seem to show that non-equilibrium CO/CH$_4$ chemistry could play an important role in young low surface gravity objects. In particular, the CH$_4$ band head near 1.6~\mic could appear at much lower \teff than predicted by current atmosphere models. The SDI method is straightforward to implement: the images taken at $\lambda_1$ need to be spatially rescaled to account for the spectral dependence of the PSF and subtracted from the images at $\lambda_0$, with a possible amplitude correction factor to minimize the residual speckle noise. The main advantage of SDI is to significantly reduce the seeing halo, but it is intrinsically limited by speckle chromaticity and differential aberrations when going through two separate optical paths.
The ADI method proposed by \citet{marois2006a} requires observations made in pupil-stabilized mode. It uses the field rotation to build an optimized reference PSF that contains very little signal from the planet. For each image $I_i$, a reference PSF is calculated using images taken before and images taken after, and for which a field rotation of at least 1.5~$\lambda/D$ has occurred in between. These images are then combined to eliminate the planet signal and produce a reference PSF that is subtracted from the image $I_i$. These operations are performed for all images in annuli of increasing radius. A thorough description of the complete procedure can be found in Sect.~5.2 of \citet{marois2006a}. This technique is essentially limited by the temporal evolution of the speckles which cannot be controled. The global efficiency of the ADI method is controled by the rotation rate of the field of view, which depends on the star declination, and by the angular separation, which constrains the actual motion on the detector. At the latitude of ESO-Paranal observatory (-24\degre~3\am~38\as) and for a star at declination $\delta = -45$\degre, the field rotation varies between 0.006~deg~s$^{-1}$ at an hour angle of $\pm2$h and 0.011~deg~s$^{-1}$ at an hour angle of 0h. This defines a strong constraint on the telescope time necessary to calibrate the speckles.
Finally, the SDI and ADI methods can be efficiently combined to further reduce the speckle noise. SDI is first performed on short exposure images acquired simultaneously to remove the fast varying atmospheric residuals that have not been averaged out. ADI is subsequently applied on this set of data to combine the images with different angular positions of the field of view.
\section{End-to-end simulations}
\label{sec:end_to_end_simulations}
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{fig1}
\caption{1-$\sigma$ noise levels after applying ADI and SDI+ADI data analysis methods in H band for high mass (left, F0 at 10~pc, V~=~2.7) and low mass (right, M0 at 10~pc, V~=~8.8) stars during a 4~h exposure time. The hatched area below 0.1\as is covered by the opaque coronagraphic mask. The coronagraphic profile is calculated by the average of the coronagraphic image in annuli of increasing radius, and the different noise levels by the standard deviation of the residuals in the same annuli. All curves are normalized to the maximum of the PSF without coronagraph.}
\label{fig:detection_limits_H2H3}
\end{figure*}
\begin{table}
\caption{List of IRDIS filter pairs.}
\label{tab:irdis_filter_pairs}
\centering
\begin{tabular}{ccccc}
\hline
Pair name & \multicolumn{2}{c}{Filter 0} & \multicolumn{2}{c}{Filter 1 } \\
& $\lambda_0$ & $R_0$ & $\lambda_1$ & $R_1$ \\
& (\mic) & & (\mic) & \\
\hline
Y2Y3 & 1.020 & 20 & 1.073 & 20 \\
\hline
J2J3 & 1.190 & 25 & 1.270 & 25 \\
\hline
H2H3 & 1.587 & 30 & 1.667 & 30 \\
H3H4 & 1.667 & 30 & 1.731 & 30 \\
\hline
K1K2 & 2.100 & 20 & 2.244 & 20 \\
\hline
\end{tabular}
\end{table}
\begin{table}
\caption{Atmosphere models included in our library.}
\label{tab:models_library}
\centering
\begin{tabular}{lcc}
\hline
Model & \teff & \logg \\
& (K) & (dex) \\
\hline
AMES-COND$^{\mathrm{a}}$ & 350 -- 1300 & 2.5 -- 6.0 \\
BT-SETTL$^{\mathrm{b}}$ & 1100 -- 2300 & 4.5 -- 5.5 \\
AMES-DUSTY$^{\mathrm{c}}$ & 1700 -- 2500 & 2.5 -- 6.0 \\
BSH$^{\mathrm{d}}$ & 700 -- 2000 & 4.5 -- 5.5 \\
\hline
\end{tabular}
\begin{list}{}{}
\item[$^{\mathrm{a}}$] \citet{allard2003}
\item[$^{\mathrm{b}}$] \citet{allard2007}
\item[$^{\mathrm{c}}$] \citet{allard2001}
\item[$^{\mathrm{d}}$] \citet{burrows2006}
\end{list}
\end{table}
A complete end-to-end model of SPHERE has been developed to test the instrument performances and different data analysis methods. This model is a diffractive code written in IDL (Interactive Data Language) based on the CAOS (Code for Adaptive Optics System) problem solving environment \citep{carbillet2004} with a specific package developed for the SPHERE project \citep{carbillet2008}.
Realistic data cubes have been simulated to represent typical 4 hours exposure
with IRDIS and an apodized Lyot coronagraph at different wavelengths where the star goes from -2 to +2 hour angle. Every data cube is composed of 144 images representing a cumulative 100 seconds esposure each, and several parameters are modified in the course of the simulation to take into account the variations of optical aberrations on a long timescale. The seeing and wind speed have been varied on ranges typical for the ESO-Paranal observatory, respectively $0.85 \pm 0.15\as$ and $14.2 \pm 4.6$~m~s$^{-1}$. For each individual images, the AO-corrected atmosphere was simulated by a set of 100 decorrelated phase screens. The typical millisecond timescale of the uncorrected atmospheric residuals is not considered here: we assume that on a 100-second timescale, these residuals are averaged out and produce a smooth halo; only the correlated residuals with timescales longer than a few hundred seconds will remain. From the instrumental point of view, variations of the beam shift as well as rotation of the entrance window, atmospheric dispersion corrector (ADC) and derotator have been translated into wavefront error. Chromatic shifts associated to the ADC have been calculated from its optical design. Slow achromatic drifts, such as defocus and tilt, associated with temperature changes have been added. Finally, differential aberrations between the two filters in the differential imager have been taken into account: considering the prototypes of IRDIS DBI filters \citep{dohlen2008b}, 7.55~nm~RMS of differential aberrations have been introduced.
In these end-to-end simulations, the Fresnel propagation of the wavefront is not considered. However, the overall impact of Fresnel propagation has been evaluated in separate simulations (not detailed here) where pre-coronagraphic and post-coronagraphic propagation effects have been simulated. The main result is that while the region beyond AO cut-off (0.8\as in H band) is mostly dominated by the pre-coronagraphic propagation effects, resulting in a loss of up to a factor 2 in contrast, the inner region is affected by a mix of both effects, resulting in a loss of at most 1.5.
Four complete data cubes have been simulated, corresponding to the filter pairs Y2Y3, J2J3, H2H3 and K1K2 of IRDIS (Table~\ref{tab:irdis_filter_pairs}). The output of the diffractive code is a series of normalized coronagraphic and non-coronagraphic images of the star at the two wavelengths of the considered filter pair. A second code was used to create data cubes representing realistic planetary systems. For each star, 3 series of planets separated by 120\degre have been simulated at 0.2\as, 0.5\as, 1.0\as, 1.5\as and 2.0\as, taking into account the slow field rotation which is a function of the star elevation. The star was chosen at a declination of -45\degre and an hour angle of -2~h at the beginning of the simulated observation, representing a total field rotation of $\sim$120\degre.
To calculate realistic photometry we used standard Kurucz models \citep{kurucz1979,castelli2003} for stars with spectral types regularly distributed from F0 to M0 at a distance of 10~pc (V~=~2.7 to 8.8). For the planets we constituted a library of $\sim$220 synthetic spectra including the AMES-Dusty models of \citet{allard2001}, the BT-Settl models of \citet{allard2007}, the AMES-Cond models of \citet{allard2003} and the models of \citet{burrows2006} with effective temperature ranging from \teff~=~350~K to \teff~=~2500~K, and surface gravity ranging from \logg~=~2.5 to \logg~=~6.0. The steps in the grids of models are of 100~K in \teff and 0.5 in \logg. We assume these models are complementary, and Table~\ref{tab:models_library} gives a list of the models with the \teff and \logg ranges over which they are considered. For each filter pair we generated 66 data cubes with different combination of star and planet models to cover contrast values from 5~mag to 16.5~mag ($\sim$$2 \times 10^{-7}$ to $\sim$$10^{-2}$).
In each data cube, sky contribution has been added to match typical values for ESO-Paranal observatory. Thermal background from the instrument was also included: the value is low in I, J and H bands ($<$~2~photon~sec$^{-1}$~pixel$^{-1}$), while it becomes significant in K band (60 to 220~photon~sec$^{-1}$~pixel$^{-1}$). The code also accounts for the global throughput of the instrument and the atmospheric transmission, but does not consider OH lines variability. Finally, a realistic amount of noise for the IRDIS detector was included in the images: photon noise, flat field noise (0.1\%) and read-out noise (10 e$^-$/read).
The final output of the photometric code represents a 4~h observation with IRDIS after standard cosmetic correction and calibrations (dark, sky background and thermal background subtraction, flat field division, bad pixels correction). Due to the large number of parameters taken into account and the important computing time required for the simulation, only one data set representing a standard case has been produced. This means that all generated data cubes present the same speckle pattern, and only differ by the photometric and noise values.
ADI and SDI+ADI data analysis methods were then applied on all our simulated data cubes to attenuate the speckle noise. The ADI data analysis was implemented in IDL following the algorithm described by \citet{marois2006a}: frames separated by 2.0~$\lambda/D$ were selected and combined in 5 annuli covering our simulated planets to produce 2 final images for data taken at $\lambda_0$ and at $\lambda_1$. The SDI data analysis was implemented in IDL using a custom routine for the precise spatial rescaling (L. Mugnier, private communication) based on zero-padding in both real and Fourier spaces. ADI was then applied on the subtraction of data taken at $\lambda_0$ and at $\lambda_1$ to produce the final SDI+ADI image.
\begin{figure*}
\centering
\includegraphics[width=0.49\textwidth]{fig2_a}
\hfill
\includegraphics[width=0.49\textwidth]{fig2_b}
\caption{Magnitude difference between the star and the planet for which the photometric precision is better than 0.2~mag, as a function of wavelength and angular separation using ADI (left) and SDI+ADI (right) data analysis methods. The oblique dashed-line shows the AO control radius limit.}
\label{fig:flux_error_summary}
\end{figure*}
\section{Photometric accuracy}
\label{sec:photometric_accuracy}
\subsection{Noise level with ADI and SDI+ADI}
\label{sec:noise_level}
1-$\sigma$ noise level were evaluated from the data products of the ADI and SDI+ADI data analysis methods by measuring the standard deviation of the residual speckle noise in annuli of increasing radius, normalized to the maximum of the PSF without coronagraph. Although it is known from \citet{goodman1968}, \citet{aime2004}, \citet{fitzgerald2006} and \citet{smith2009} that the speckle noise statistics in AO-corrected images with and without coronagraph is not Gaussian. \citet{marois2008a} have shown that residual noise after applying ADI on $\sim$20 or more images is quasi-Gaussian. ADI was applied on our data cubes with more than 20 images in every case, so we will consider the residual noise as Gaussian and we will use standard 5-$\sigma$ level for detection limits
Fig.~\ref{fig:detection_limits_H2H3} shows the 1-$\sigma$ noise level for H2H3 filter pair in two different regimes: a high flux case corresponding to a high mass star (F0 star at 10~pc, V~=~2.7) and a low flux case corresponding to a lower mass star (M0 star at 10~pc, V~=~8.8). The ADI noise level in filter H3 is not shown because it is at the same level as in filter H2. In high flux, the speckle noise attenuation is almost constant from 0.1\as (inner-working angle of the coronagraph) to 2.5\as, both with ADI and SDI+ADI, allowing to reach a contrast of $\sim$16~mag at 0.2\as and more than 20~mag at 2.5\as. In low flux, the level of the background noise (sky and instrumental thermal emission, read-out noise) becomes limiting, and the attenuation reaches an almost constant level in SDI+ADI at 1.5\as. The noise levels are similar in Y, J and H bands, but in K band, where the thermal emission is 10 to 15 times higher than in J or H band, the accessible contrast in low flux regime is limited at 15.5~mag.
If we compare these 1-$\sigma$ noise level to the ones derived for IRDIS Long Slit Spectroscopy (LSS) mode \citep{vigan2008}, we see that they are comparable between ADI and LSS. However, SDI+ADI clearly brings an improvement of 2 to 3 magnitudes compared to LSS, justifying the need to quantify characterization capabilities in DBI mode for planets that are not detectable with LSS.
\subsection{Planet flux estimation}
\label{sec:planet_flux_estilation}
We have estimated the signal of all planets detected at more than 5-$\sigma$ with aperture photometry in a 2.44~$\lambda/D$ radius aperture. The aperture is calculated to take into account the effect of using a round aperture on square pixels. The measured signal has been converted to a flux in phot~s$^{-1}$~m$^{-2}$ with the formula:
\begin{equation}
\label{eq:flux_norm}
f_i = \frac{S_i}{E_e~S_{\mathrm{Tel}}~T_r~t_i},
\end{equation}
\noindent where $S_i$ is the measured signal on image $i$, $E_e$ is the encircled energy in the aperture, $S_{\mathrm{Tel}}$ is the telescope collecting surface, $T_r$ is the transmission of the atmosphere, telescope and instrument, and $t_i$ is the exposure time for image $i$ (100~s in our case).
We consider that we are in a case where we know the value of the encircled energy $E_e$ in the aperture. This value varies mostly with seeing conditions because the AO correction will concentrate more energy in the PSF core when seeing improves. Moreover, we consider that the planet position is known exactly to center the aperture on the planet PSF and avoid photometric error bias induced by inaccurate centering. Finally we also take into account the error induced by the field rotation. When the field rotates, it will slowly smear the planet PSF, especially at large angular separations. The effect in our case is significant at separations larger than 1.0\as because we simulated long exposures for individual images (100~s). In practice exposures for individual images will typically last a few seconds to avoid detector saturation, reducing the effect of PSF smearing up to a few arcseconds.
\subsection{Photometric accuracy in ADI}
\label{sec:photometric_accuracy_adi}
For our simulated test case, the planet flux has been evaluated in all data cubes after using ADI data analysis method. In each filter pair and for each simulated planet, 2 independent values are obtained at $\lambda_0$ and $\lambda_1$. They are compared to the flux value introduced at the beginning of the simulation to evaluate the photometric error. Figure~\ref{fig:flux_error_summary}, left, illustrates the photometric performance as a function of wavelength and angular separation. The contours indicate the contrast value between the star and planet below which the photometric precision is better than 0.2~mag. Such a precision is necessary in order to be able to disentangle between different planet masses when comparing actual measurements to evolution nary models. We see two major effects: (i) the photometric performance clearly depends on wavelength, and (ii) there are two different regimes depending on the position compared to the AO control radius. The first effect is directly related to the chromaticity of the PSF: in speckle-limited regime the noise attenuation is almost constant with angular separation compared to the coronagraphic profile, and the level of the coronagraphic profile linearly depends on wavelength. The second effect is related to the AO correction inside the control radius. Inside that region we see a stabilization of the performance: 0.2~mag photometric precision can be reached up to contrast of 10 to 11~mag ($10^{-4}$ to $4 \times 10^{-4}$) from 0.2\as to the AO control radius, which extends from 0.5\as in Y band to 1.0\as in K band. Outside of the AO control radius, the photometric performance increases almost linearly with angular separation at all wavelengths to reach contrast values of 14 to 15~mag ($2.5 \times 10^{-6}$ to $10^{-6}$) around 2.0\as. These numbers are given in the context of our simulated test case, but the general effects should be similar for any data obtained with high contrast coronagraphic imagers.
\subsection{Photometric accuracy in SDI+ADI}
\label{sec:photometric_accuracy_sdi_adi}
Similarly to the noise level, using the SDI+ADI data analysis method improves the photometric accuracy. However, using SDI+ADI will only provide an estimation of the differential flux of the planet between the 2 filters, contrary to ADI which provides an absolute measurement. To preserve the planet differential flux, the amplitude correction factor usually applied for SDI in the subtraction is taken equal to a fixed value of 1. The photometric error estimated with SDI+ADI follows the same variations as for ADI, but at higher contrast values. Figure~\ref{fig:flux_error_summary}, right, illustrates the photometric performance as a function of wavelength and angular separation in SDI+ADI. The trends are similar to ADI alone, but the chromatic effect is less significant because the PSF chromaticity has been mitigated by the SDI part of the analysis. Compared to ADI alone, the contrast values at which a 0.2~mag photometric error is reached are 1.5 to 2.5~mag higher. At shorter wavelengths, in Y2Y3 filters, performances at separations larger than 1.0\as decrease. This effect is related to the size of the aperture for photometry which is very small in Y band (4~pixels in diameter), and to the field rotation which has a strong effect on encircled energy at separations larger than 1.0\as in Y band. Considering shorter exposures for individual images where the field rotation is negligible would decrease the photometric errors in that particular case.
\subsection{Empirical photometric accuracy}
\label{sec:empirical_photometric_accuracy}
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{fig3}
\caption{Empirical photometric error as a function of contrast in IRDIS filter pairs using either ADI or SDI+ADI data analysis methods. Errors bars have been represented only for a small set of data points. Their amplitude is defined by the optimal and pessimistic error curves described in the text.}
\label{fig:error_curves_phot_all}
\end{figure*}
We hereafter combine the photometric accuracy obtained in ADI and SDI+ADI to define empirical photometric error curves for each filter pair as a function of contrast. The photometric error curves as a function of contrast at each angular separation have been fitted with the empirically defined function:
\begin{equation}
\label{eq:empirical_phot_err}
\mathrm{phot_{err}} = \frac{p_1}{c^{p_2}}+p_3
\end{equation}
\noindent where $\mathrm{phot_{err}}$ is the photometric error, $c$ the contrast and $(p_1, p_2, p_3)$ the fitted parameters. This function approaches the measured points with a precision of $\sim$1\%. The fitting has been performed for ADI and SDI+ADI. To take into account the scattering of the error with the planet position in the images, different cases have been considered at each separation, corresponding to the 3 different simulated planet positions: a standard case with an average photometric error, an optimal case corresponding to the lowest estimation of the error and a pessimistic case corresponding to the upper estimation of the error. These empirical photometric errors are plotted in Fig.~\ref{fig:error_curves_phot_all} for the 4 simulated filter pairs. The amplitude of the error bars is defined by the optimal and pessimistic error curves described above. We assume that the photometric error in ADI is the same in the two filters of a pair, which is legitimate given the amplitude of the error bars. These empirical error curves lie in the same range as the expected photometric accuracy of other data analysis methods developed within the SPHERE consortium by \citet{mugnier2008} and \citet{smith2009}.
\begin{table}
\caption{Contrast limit over which the differential photometric error in SDI+ADI becomes smaller than the photometric error in ADI.}
\label{tab:methods_limits}
\centering
\begin{tabular}{ccccc}
\hline
Separation & \multicolumn{4}{c}{Filter pair} \\
& Y2Y3 & J2J3 & H2H3 & K1K2 \\
(arcsec) & (mag) & (mag) & (mag) & (mag) \\
\hline
0.2\as & 6.8 & 6.8 & 6.5 & 6.8 \\
0.5\as & 6.0 & 6.8 & 8.3 & 8.0 \\
1.0\as & 10.8 & 9.3 & 8.0 & 7.2 \\
1.5\as & 12.8 & 11.0 & 11.0 & 10.0 \\
2.0\as & & 11.5 & 12.2 & 12.2 \\
\hline
\end{tabular}
\end{table}
Table~\ref{tab:methods_limits} gives for each filter pair and each angular separation the contrast value at which the photometric error in ADI becomes lower than the differential photometric error in SDI+ADI. These values give the contrast at which it becomes more interesting in terms of photometric error to obtain a differential flux estimation. As explained in Sect.~\ref{sec:photometric_accuracy_sdi_adi}, aperture photometry in Y band is extremely sensitive to errors introduced by the position of the aperture or the field rotation because the aperture is very small. This is why in Y2Y3 pair at 2.0\as there is no contrast limit between ADI and SDI+ADI: for that particular case the flux estimation error is slightly better in ADI than SDI+ADI.
\section{Photometric characterization}
\label{sec:photometric_characterization}
In this section we evaluate characterization capabilities of IRDIS in imaging mode, i.e. how well the physical parameters \teff and \logg of the planets can be estimated from photometric measurements in different spectral bands.
\subsection{Characterization simulation}
\label{sec:characterization_simulation}
To estimate the characterization capabilities of IRDIS, we performed a new simulation using as input the 5-$\sigma$ detection limits obtained from Sect.~\ref{sec:noise_level} and the empirical error curves obtained in Sect.~\ref{sec:empirical_photometric_accuracy}. The goal of the simulation was to test the efficiency of all filter pair sequences for characterization at different stellar magnitudes and for a large number of planetary atmosphere models. These simulations are based on current state of the art atmosphere models. Although these models will clearly evolve with new detections in the future, they allow to test the expected performances of IRDIS, as well as to estimate the intrinsic errors of our signal extraction and comparison to models. It has been performed for all stellar types and atmosphere models included in our library (see Table~\ref{tab:models_library}).
For the simulation we assume that a same planetary system is observed with different filter pairs in a given order. For each possible combination of parameters (filter pairs sequence; star magnitude; angular separation; planet atmosphere model) we proceed as follow: the star and planet fluxes are calculated in the filters of the first pair; if the planet is not detectable (considering the 5-$\sigma$ detection limit), simulation for that combination of parameters is stopped; if it is indeed detectable, a photometric measurement is obtained. Depending on the contrast between the planet and the star in each filter, different informations are obtained: 2 direct photometric measurements if the planet is detectable with ADI in both filters, a differential flux measurement if the planet is only detectable with SDI+ADI or a direct and a differential measurement if the planet is only detectable in ADI in one of the filters. Once the flux measurements are obtained, the photometric error is determined from the empirical error curves and added to the measured values to obtain lower and upper limits to the planet flux. Models that can correspond to these limits are then searched in our models library. If only one model corresponds, we stop iterations considering that the planet has been fully characterized within the limits of the grid of atmosphere models that is used. If many models match those limits, we switch to the next filter pair in the sequence and the same process is started again. In a given sequence, each filter will bring some additional information that will help to find the appropriate atmosphere model and constrain the values of \teff and \logg. At the end of the filter pairs sequence, 4 distinct outcomes are possible:
\begin{itemize}
\item \emph{Non-Detection (ND):} the planet is not detected in the first filter pair; the sequence is stopped. \\
\item \emph{Non-unique Characterization (NC):} the planet is detected at least in the first filter pair of the sequence; at the end of the sequence many models match the flux measurements and they have different values of \teff and \logg. \\
\item \emph{\teff Characterization (TC):} the planet is detected at least in the first filter pair of the sequence; at the end of the sequence many models match the flux measurements and they all share the same value for \teff but not for \logg. \\
\item \emph{Full Characterization (FC):} the planet is detected at least in the first filter pair of the sequence; at the end of the sequence only one model matches the flux measurements, which means that the \teff and \logg values have been determined.
\end{itemize}
The TC and FC are considered within the limits of the atmosphere models grid, which is 100~K in \teff and 0.5 in \logg, i.e. the \emph{full characterization} corresponds to the determination of \teff and \logg with an error equal to the limits of the models grid. Another important point is that in practice the photometric error will have to be estimated from the science data itself. Although, we consider here an ideal case where the photometric error is known, other data analysis methods such as the one proposed by \citet{smith2009} will allow direct estimation of the error from the data, with a precision that it compatible with the results presented here.
\subsection{Filter pair sequence analysis}
\label{sec:filter_pair_sequence_analysis}
\begin{table*}
\centering
\caption{Analysis of the filter pairs sequences.}
\label{tab:sequences_std}
\begin{tabular}{c|ccccc|c|c|c|c|c}
\hline
Spectral type & P1$^{\mathrm{a}}$ & P2$^{\mathrm{a}}$ & P3$^{\mathrm{a}}$ & P4$^{\mathrm{a}}$ & P5$^{\mathrm{a}}$ & ND$^{\mathrm{b}}$ & NC$^{\mathrm{c}}$ & TC$^{\mathrm{d}}$ & FC$^{\mathrm{e}}$ & TC+FC \\
& & & & & & (\%) & (\%) & (\%) & (\%) & (\%) \\
\hline
M0 & H2H3 & & & & & 3 & $20_{-1}^{+1}$ & 0 & $77_{-1}^{+1}$ & $77_{-1}^{+1}$\rule{0cm}{0.4cm} \\
V = 8.8 & H2H3 & Y2Y3 & & & & 3 & $ 2_{-1}^{+1}$ & 0 & $95_{-1}^{+1}$ & $95_{-1}^{+1}$\rule{0cm}{0.35cm} \\
H = 5.3 & H2H3 & Y2Y3 & J2J3 & & & 3 & $ 1_{-1}^{+0}$ & 0 & $96_{-1}^{+1}$ & $96_{-0}^{+1}$\rule{0cm}{0.35cm} \\
& H2H3 & Y2Y3 & J2J3 & H3H4 & & 3 & $ 1_{-1}^{+0}$ & 0 & $96_{-1}^{+1}$ & $96_{-0}^{+1}$\rule{0cm}{0.35cm} \\
& H2H3 & Y2Y3 & J2J3 & H3H4 & K1K2 & 3 & $ 1_{-1}^{+0}$ & 0 & $96_{-1}^{+1}$ & $96_{-0}^{+1}$\rule{0cm}{0.35cm} \\
\hline
K0 & H2H3 & & & & & 6 & $35_{-8}^{+1}$ & 0 & $59_{-1}^{+8}$ & $59_{-1}^{+8}$\rule{0cm}{0.4cm} \\
V = 5.9 & H2H3 & Y2Y3 & & & & 6 & $ 4_{-1}^{+2}$ & $2_{-1}^{+1}$ & $88_{-2}^{+3}$ & $90_{-1}^{+2}$\rule{0cm}{0.35cm} \\
H = 4.0 & H2H3 & Y2Y3 & J2J3 & & & 6 & $ 2_{-1}^{+1}$ & $1_{-1}^{+0}$ & $92_{-1}^{+1}$ & $92_{-1}^{+1}$\rule{0cm}{0.35cm} \\
& H2H3 & Y2Y3 & J2J3 & H3H4 & & 6 & $ 2_{-1}^{+1}$ & $1_{-0}^{+0}$ & $92_{-1}^{+1}$ & $92_{-1}^{+1}$\rule{0cm}{0.35cm} \\
& H2H3 & Y2Y3 & J2J3 & H3H4 & K1K2 & 6 & $ 2_{-1}^{+1}$ & $1_{-0}^{+0}$ & $92_{-1}^{+1}$ & $92_{-1}^{+1}$\rule{0cm}{0.35cm} \\
\hline
G0 & H2H3 & & & & & 7 & $43_{-7}^{+3}$ & 0 & $50_{-3}^{+7}$ & $50_{-3}^{+7}$\rule{0cm}{0.4cm} \\
V = 4.4 & H2H3 & Y2Y3 & & & & 7 & $ 7_{-2}^{+4}$ & $3_{-1}^{+1}$ & $83_{-3}^{+5}$ & $86_{-2}^{+4}$\rule{0cm}{0.35cm} \\
H = 3.0 & H2H3 & Y2Y3 & J2J3 & & & 7 & $ 3_{-1}^{+1}$ & $2_{-1}^{+1}$ & $88_{-2}^{+2}$ & $90_{-1}^{+1}$\rule{0cm}{0.35cm} \\
& H2H3 & Y2Y3 & J2J3 & H3H4 & & 7 & $ 3_{-1}^{+1}$ & $1_{-1}^{+1}$ & $88_{-2}^{+2}$ & $90_{-1}^{+1}$\rule{0cm}{0.35cm} \\
& H2H3 & Y2Y3 & J2J3 & H3H4 & K1K2 & 7 & $ 3_{-1}^{+1}$ & $1_{-1}^{+1}$ & $88_{-2}^{+2}$ & $90_{-1}^{+1}$\rule{0cm}{0.35cm} \\
\hline
F0 & H2H3 & & & & & 10 & $55_{-7}^{+3}$ & 0 & $35_{-3}^{+7}$ & $35_{-3}^{+7}$\rule{0cm}{0.4cm} \\
V = 2.7 & H2H3 & J2J3 & & & & 10 & $11_{-2}^{+2}$ & $3_{-1}^{+0}$ & $76_{-3}^{+2}$ & $79_{-2}^{+2}$\rule{0cm}{0.35cm} \\
H = 1.5 & H2H3 & J2J3 & Y2Y3 & & & 10 & $ 7_{-3}^{+3}$ & $3_{-1}^{+1}$ & $80_{-4}^{+4}$ & $83_{-3}^{+3}$\rule{0cm}{0.35cm} \\
& H2H3 & J2J3 & Y2Y3 & H3H4 & & 10 & $ 6_{-2}^{+2}$ & $2_{-1}^{+1}$ & $82_{-3}^{+4}$ & $84_{-2}^{+2}$\rule{0cm}{0.35cm} \\
& H2H3 & J2J3 & Y2Y3 & H3H4 & K1K2 & 10 & $ 6_{-2}^{+2}$ & $2_{-1}^{+1}$ & $82_{-3}^{+4}$ & $84_{-2}^{+2}$\rule{0cm}{0.35cm} \\
\hline
\end{tabular}
\begin{list}{}{}
\item[$^{\mathrm{a}}$] P1 to P5 designate the filter pairs
\item[$^{\mathrm{b}}$] Not detected
\item[$^{\mathrm{c}}$] No characterization
\item[$^{\mathrm{d}}$] \teff characterization
\item[$^{\mathrm{e}}$] Full characterization
\end{list}
\end{table*}
The output of this simulation allows to determine the most significant filter pairs sequence for characterization, i.e. the sequence which maximizes the number of characterizations. The strategy is to progressively build an \emph{optimal sequence} by adding each time the filter pair that increases the most the number of characterizations. All possible filter pair sequences have been systematically tested to find the one that maximizes the number of \teff characterizations and full characterizations (TC+FC), as a function of stellar magnitude. The simulation shows the following important results (see Table~\ref{tab:sequences_std} for details):
\begin{itemize}
\item H2H3 is the filter pair which minimizes the number of non-detections, reflecting the fact that CH$_4$ absorption band near 1.6~\mic in cool substellar objects spectra is the optimal feature for their detection. However it should be reminded that this specific spectral feature might not always be present in lower mass objects as we mentionned in Sect.~\ref{sec:limitations_high_contrast_imaging}. \\
\item When adding a new filter pair to a sequence, adding Y2Y3 or J2J3 will increase the number of characterized models more than H3H4 or K1K2. This result does not depend on the stellar magnitude. \\
\item The scattering of the flux error between the pessimistic optimal photometric error curves has no major influence on the number of characterized models when more than one filter pair is used. In particular, we see the error bars of the different flux cases do not significantly overlap, confirming that the chosen sequence is appropriate for all cases of flux.
\end{itemize}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{fig4}
\caption{Proportion of models in our library characterized by adding filter pairs from the optimal filter pair sequence for 4 stellar magnitudes. The error bars are given by the optimal and pessimistic photometric error curves.}
\label{fig:filter_pair_sequences}
\end{figure}
This last result is particularly important as it advocates for a given sequence of filter pairs and allows to set general priorities on the filter pairs for characterization. When there are no \emph{a priori} assumptions on the nature of the objects, the priorities are defined from highest to lowest as follow:
\begin{enumerate}
\item[0.] H2H3 \\
\item[1.] Y2Y3~/~J2J3 \\
\item[2.] H3H4~/~K1K2
\end{enumerate}
\noindent The H2H3~--~Y2Y3~--~J2J3~--~H3H4~--~K1K2 will be referred to as the \emph{optimal sequence} from now on, and we will only consider the standard empirical errors since the differences with the other error curves are small. Assuming the use of the optimal sequence, Fig.~\ref{fig:filter_pair_sequences} represents the proportion of characterized models (TC+FC) from our library when new filter pairs are added. The trends are identical for the 4 different stellar magnitudes. When using only H2H3 the proportion of characterized models is comprised between 30\ and 80\%, and the error bars are of $\sim$10\% for bright stars. Adding a second filter pair greatly improves the proportion of characterized models which is above 70\% for all stellar magnitudes. Adding more filter pairs confirms this trend and tends to reduce the error bars to less than 5\% for all magnitudes. The main conclusion is that most of the information for characterizing any given model is obtained using 2 filter pairs around low mass stars when the contrast is favorable, and 3 filter pairs around high mass stars for which the contrast is more challenging.
\subsection[Lowest estimations of T$_{eff}$]{Lowest estimations of \teff}
\label{sec:lowest_estimations_teff}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{fig5}
\caption{Smallest value of \teff which can be characterized as a function of
star magnitude (y-axis) and angular separation (x-axis) when using 1, 2 or 3
filters from the optimal sequence.}
\label{fig:best_sequence_teff_limit}
\end{figure}
In the previous section we set priorities for characterization on the different filter pairs of IRDIS. We will now detail the lowest values of \teff that IRDIS will be able to characterize as a function of stellar magnitude and angular separation. Figure~\ref{fig:best_sequence_teff_limit} gives the smallest values of \teff which have been characterized when using 1 to 3 filter pairs from the optimal sequence. Colder planets can be detected, but we were not able to find the appropriate values of \teff and \logg. When using only H2H3, planets with \teff down to 900~K should be characterized at an angular separation of 0.2\as from high mass bright stars and 700~K from lower mass stars. Adding a second filter pair considerably improves these results by 200~K, while adding a third pair confirms these limiting values.
With the considered data analysis methods and according to the evolutionary models from \citet{baraffe2003} for the COND atmosphere models, we can estimate that in a very young system of 10~Myr, we should be able to characterize a planet of 1~\MJup with H2H3 at separations larger than 0.5\as around a low mass star (M0 at 10~pc) where the star-planet contrast is favorable, but only further than 2.0\as around a high mass star (F0 at 10~pc) where the contrast difference is larger. With two filter pairs, the limit would be 0.2\as around a low mass star and 1.0\as around a high mass star. For older systems, only planets of a few masses of Jupiter could be characterized. At 100~Myr, a Jupiter mass planet would remain out of reach for characterization with H2H3 filters around a high mass star, and only at separations larger than 1.5\as around a low mass star. At this age, the \teff limits of 700~K and 500~K which can be reached at small angular separation around high and low mass stars would respectively correspond to planets with masses of $\sim$6.5~\MJup and $\sim$3~\MJup. Using improved signal extraction methods providing more accurate photometry of the companion would certainly push down those limits.
\subsection{Study of the non-unique characterizations}
\label{sec:study_non_unique_characterizations}
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{fig6}
\caption{Distribution of the errors on \teff and \logg for non-unique characterizations with H2H3 filter pair around a high mass star (F0 at 10~pc, plain line) and a low mass star (M0 at 10~pc, dashed line) cases.}
\label{fig:best_sequence_nochar}
\end{figure*}
NCs are the cases where several models correspond to the flux measurements in all filter pairs with which they are detected. From these remaining models, it is possible to determine if a combination of (\teff; \logg) is more represented than others, making this combination the most probable values of \teff and \logg. If several combinations are counted an equal number of times, an average value and an error can be determined for the values of \teff and \logg. In any case, the error is at least equal to the steps in the grids of models. The estimation of the most probable values for \teff and \logg has been performed for all non-uniquely characterized models at all simulated angular separations and magnitudes.
Figure~\ref{fig:best_sequence_nochar} shows an histogram of the errors on \teff and \logg when using H2H3 filter pair for high mass (F0 at 10~pc) and low mass (M0 at 10~pc) stars. NCs are mostly dominated by errors on the determination of \logg. In particular we see that around a low mass star where the contrast is more favorable, the proportion of models with no error on \teff increases by $\sim$20\%, while the errors on \logg keep the same distribution. Adding other filter pairs improves the determination of \teff for the non-unique charaterizations, in particular around high mass (brighter) stars, reaching more than 95\% for all stellar magnitudes. The determination of \logg is also improved, but even when using 3 filter pairs the number of cases where the error is less or equal to 0.5 never reaches more than 85\% around a high mass star or 95\% around a low mass star.
\subsection{Impact of errors on \teff and \logg}
\label{sec:impact_error_teff_logg}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{fig7}
\caption{Isochrones for the COND planetary atmosphere models covering an age of $40 \pm 30$~Myr used for the determination of the mass of hypothetical 2~\MJup planets orbiting at 5~A.U. from M0 and F0 stars at 10~pc. The error boxes defined by the possible values for \teff and \logg of both planets are respectively represented by dotted and dashed rectangles. The planet mass derived from the models and error box is displayed at the bottom left corner of each error box (see text for an explanation on how the planet masses are derived). The position of the planet predicted by the evolutionary models is represented by the star symbol, and the error box defined by the atmosphere models grid precision is given for reference around that position by a plain rectangle.}
\label{fig:char_error_impact}
\end{figure}
The influence of errors on the determination of \teff and \logg on the determination of the planet mass can be studied using evolutionnary models such as those published by \citet{baraffe2003} for the COND atmosphere models. The preliminary version of the SPHERE target list (S. Desidera, private communication) was used to define a standard young test case based on age considerations. The average age for targets younger than 100~Myr is $44 \pm 20$~Myr and the average error on the determation of the target age is 30~Myr. Using these values we can define two test cases considering a planet of 2~\MJup aged of $44 \pm 30$~Myr orbiting at 5~A.U. from M0 and F0 stars at 10~pc. According to the evolutionary models from \citet{baraffe2003}, such planets should have \teff~=~516~K and \logg~=~3.54~dex, resulting in a contrast of 11.9~mag and 15.6~mag in H band respectively around the M0 and F0 stars. Considering the results from Sect.~\ref{sec:lowest_estimations_teff} and \ref{sec:study_non_unique_characterizations}, the planet around the F0 star cannot be characterized with IRDIS using one filter pair, while the planet around the M0 star is close to the measured limit.
The expected spectra of these planets were introduced in our simulation to test the accuracy of extracting flux information and inversely deriving physical parameters, and to estimate their mass using evolutionary models. The results are presented in Fig.~\ref{fig:char_error_impact}: the areas covered by the values of \teff and \logg are shown as rectangular boxes on the predicted isochrones for both planetary systems. The bin size of the atmosphere model grid is also plotted for reference. For each case, the planet mass is estimated by selecting all the isochrones of the masses that cross the error box and weighting them by the integral of the isochrone inside the box. The isochrone that has the largest intersection with the error box is supposed to be the most likely (or the average if several isochrones have the same integral). The upper and lower limits of the mass are given by the highest and lowest mass isochrones that cross the box.
Around a low mass star, the parameters \teff and \logg are estimated with an accuracy close to the one given by the atmosphere model grid ($1.9^{+1.3}_{-0.7}$~\MJup), leading to an estimation of $1.9^{+1.2}_{-1.0}$~\MJup. Around a high mass star, the planet is very close to the detection limit at 0.5\as, resulting in a poor estimation of both \teff and \logg: the important photometric error in H2H3 leads to a very large uncertainty on \logg ($4.33 \pm 1.23$). The mass of the planet is then estimated to $1.1^{+2.6}_{-0.5}$~\MJup. In this case, the planet mass is stronly underestimated, and the large uncertainties on \teff and \logg lead to a large upper limit for the mass estimation. The origin of the large offset on the determination of \logg is still uncertain, and further simulations are still needed to investigate thoroughly the complete parameter space. In particular we see in this case that the age uncertainty unquestionably increases the uncertainty on the mass estimation by increasing the number of isochrones crossing the possible values of \teff and \logg. Such simulations would greatly benefit from updated homogeneous atmosphere models grids covering a large span of \teff, \logg and age.
\section{Conclusions}
\label{sec:conclusions}
Next generation instruments for ground based exoplanet direct imaging such as SPHERE and GPI will provide data intrinsically limited by the speckle noise. This noise needs to be attenuated using \emph{a posteriori} data analysis methods, such as spectral and angular differential imaging. In this paper, we have quantified the exoplanet characterization capabilities of IRDIS, the differential imager of SPHERE, using photometric and differential photometric information.
The photometric performances have been evaluated with aperture photometry on the detectable planets as a function of contrast and wavelength for a standard test case. In particular we have shown that the photometric performance strongly depends on wavelength because of the PSF chromaticity and on the position with respect to the AO control radius. With ADI, a photometric accuracy of 0.2~mag is reached inside the AO control radius for contrast values of 10 to 11~mag between the star and planet, while at larger radius the precision can be reached for contrasts up to 15~mag. With SDI+ADI, the overall photometric performance is improved, increasing by 1.5 to 2.5~mag the contrast at which a 0.2~mag precision is reached.
We have defined empirical photometric accuracies for IRDIS in its different filter pairs, which were used to test the characterization capabilities of IRDIS with all possible combinations of filter pairs. Priorities for characterization have been set on the different filter pairs by finding the pairs that maximize the number of possible characterizations in various conditions. We showed that when there is no \emph{a priori} knowledge on the planet, the filter pairs Y2Y3 and J2J3 allow a larger number of characterizations than the pairs H3H4 and K1K2. Then we showed that using filter pair H2H3, it will be possible to characterize planets with \teff~$\simeq$~900~K around high mass stars at small angular separation and \teff~$\simeq$~700~K around lower mass stars. Adding Y2Y3 and J2J3 filter pairs allows to decrease the characterizable \teff by 200~K at all separations and for all stellar magnitudes, while considerably decreasing the number of non-characterizations for warmer planets.
Finally, we showed that non-unique characterizations, i.e. planets for which the \teff and \logg values could not be determined exactly (within the limits of the grid of atmosphere models) are mostly dominated by errors on the determination of \logg. In particular, we showed that around low mass stars, where the contrast is more favorable, the determination of \teff is largely improved, while the errors on \logg remain identical around a high mass star. Considering evolutionary models, and including typical ages from the
future SPHERE target list, we showed that such errors on the determination of a low mass planet (2~\MJup) results in a large uncertainty around a high mass bright star, but is very close to the limits fixed by the models grid around a fainter low mass star.
With this work we showed that IRDIS, the dual-band imager of SPHERE, should be able to fulfill the goal set for a high-contrast imager, that is the ability to detect and characterize planetary companions down to the Jupiter mass around nearby young stars. Similar developments could also be performed for an Integral Field Spectrograph in the future, allowing to quantify precisely the performances of SPHERE in the near-infrared, and to work on the aspect of characterization strategy for the detected objects.
\section*{Acknowledgments}
We wish to thank Silvano Desidera (INAF) for generously providing a preliminary version of the SPHERE target list with all associated physical parameters, and Laurent Mugnier (ONERA) for providing an optimized procedure for spatial rescaling of the images.
SPHERE is an instrument designed and built by a consortium consisting of LAOG, MPIA, LAM, LESIA, Laboratoire Fizeau, INAF, Observatoire de Gen\`eve, ETH, NOVA, ONERA and ASTRON in collaboration with ESO.
\bibliographystyle{mn2e}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,071 |
Q: How to clean up after Sitecore template "Shared" setting was "packaged" and "installed" and items using this field are unaware we faced very specific scenario in our Sitecore enviroment. In our Sitecore we have a item, lets call it "Promotion". Promotion was using "End date" field that was shared.
On our dev instance we "unshared" the field. Which naturally triggers the background process that changes the items to use field in unshared mode.
Similar process is described here: http://sitecoreblog.alexshyba.com/2011/10/changing-field-sharing-settings-in.html
So then we packaged and installed change of "unsharing field" on production "master" database. As I assume during installation the background process of "updating the items" has not been triggered. Which now behaves in the way, that "unshared" field on our production master database cannot be saved. Cahnges of value after clicking save are "vanishing". I am sure they are now being saved in some language agnostic mode.
Of course simple fix for that is to "share" it back and "unshare" it again. However when we tried to do this experiment on copy of our enviroment and we noticed all the values were lost. As the items from mentioned template are heavily used, we cannot really afford loosing those values.
Any ideas?
A: I would go "database digging". Sitecore stores these field values in their respective databases inside the "SharedFields", "VersionedFields" and "UnversionedFields" tables.
Assuming you shut off your Sitecore instances (this is important), you should be able to SELECT the data out of the wrong table, and INSERT it into the correct one.
(you need to look for items where FieldId matches the field you are having trouble with)
From what you've described, I don't believe Sitecore has removed any data on your production environment (yet).
A: So the solution we came up to, was to use Sitecore Rocks tool. We exported all the Items containing the fields before changing the field to "Share". The query was more or less like that:
SELECT @@ID, @#Start Date#, @#End Date# FROM //*[@@templateid='{993DC54F-6724-46C3-B8D2-3EE13F15366A}']
It gave us proper values at that point, even though to items were pointing to the SharedFields table. We just simply converted the result of this query (around 9000 rows) in Excel to Sitecore Rocks update query -
UPDATE SET @#Start Date#='20120531T000000',@#End Date#='20120614T000000' FROM //* [@@ID='{E3FD9819-3DBD-4FAA-8DEF-FEF2A6272723}'];
After prepared this migrations script, we shared the appropriate field and apply the script of 9000 updates queries through Sitecore Rocks. We need to to exactly the same on Live database. Everything went pretty smooth.
The same approach could be easily done with the database I believe, however this solution was better for us, because of non-technical reasons (security policies etc.). Anyway Sitecore Rocks rocks!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,375 |
{"url":"https:\/\/hilbertthm90.wordpress.com\/2012\/05\/11\/galois-deformations-2-representability\/","text":"# Galois Deformations 2: Representability\n\nToday we\u2019ll describe what is meant by a deformation of a Galois representation. Since our motivation was Taniyama-Shimura we\u2019ll quickly recall the type of Galois representations that came up there. There we technically had what are called ${\\ell}$-adic representations, because we considered ${\\rho_X: \\text{Gal}(\\overline{\\mathbb{Q}}\/\\mathbb{Q})\\rightarrow GL_n(\\mathbb{Q}_\\ell)}$. Our only caution was that we could have problems at places of bad reduction or ramification, so we will build that into the representations we consider.\n\nLet ${S}$ be a finite set of primes. Define ${G_S=\\text{Gal}(\\overline{\\mathbb{Q}}_S\/\\mathbb{Q})}$ to be the Galois group of the maximal algebraic extension of ${\\mathbb{Q}}$ unramified outside ${S}$. Note that ${G_S}$ will always have the profinite topology. The term \u201cGalois representation\u201d from now on will mean a continuous representation ${\\rho: G_S\\rightarrow GL_n(A)}$ where ${A}$ is a topological ring. Maybe this is a little loose because when we write ${GL_n(A)}$ we really mean ${Aut(A)}$, but we\u2019ve chosen a basis to get actually matrices. Thus we really only want to consider two representations as different if they can\u2019t be conjugated to one another. This is standard in representation theory, so we won\u2019t dwell on it.\n\nThe idea of deformations of Galois representations is roughly to extrapolate information when ${A=\\mathbb{Q}_p}$ or better $\\mathbb{Z}_p$ by using information about representations when ${A=\\mathbb{F}_p}$. If we think to the last post we can almost see how the deformation functor formalism can play a role here. We will set ${k=\\mathbb{F}_p}$ in which case ${\\Lambda=\\mathbb{Z}_p}$ is a local Noetherian ${\\Lambda}$-algebra with augmentation to ${k}$. In fact, suppose ${A\\in \\ _\\Lambda Noeth_k}$, then the augmentation gives a natural map ${GL_n(A)\\rightarrow GL_n(k)}$, so if we fix some ${\\overline{\\rho}: G_S\\rightarrow GL_n(k)}$ called a residual representation a deformation of ${\\rho}$ should be a continuous representation ${\\rho: G_S\\rightarrow GL_n(A)}$ which is (up to equivalence) ${\\overline{\\rho}}$ when composed with this map. The functor ${Def_{\\overline{\\rho}}}$ is now defined in the same way to be the set of all deformations of ${\\overline{\\rho}}$.\n\nNow as was pointed out last time, in order to define our functor there is ambiguity about whether to define it on the completed category or on the full subcategory of Artin rings. It turns out that since our functor is continuous, for the purposes of representability we can check it on this full subcategory. In order to prevent a large amount of tedium and space there is a lot being brushed over here. I highly recommend Gouv\u00eaa\u2019s great article on Galois Deformations in the book Arithmetic Algebraic Geometry for a more precise discussion of these points.\n\nThe punchline is that ${Def_{\\overline{\\rho}}}$ is actually a deformation functor. Moreover if ${\\overline{\\rho}}$ is absolutely irreducible, then Mazur showed that the functor satisfies Schlessinger\u2019s criterion and hence is prorepresentable (a far more general case was actually considered). Let\u2019s unravel why this is important. What this says is that there exists a universal deformation ring (in the completed category), ${R}$, so that given any ${A\\in Art_k}$ we have ${Def_{\\overline{\\rho}}(A)=Hom(R, A)}$. Even stronger we know there is a universal deformation ${\\psi: G_S\\rightarrow GL_n(R)}$ so that the correspondence ${Def_{\\overline{\\rho}}(A)=Hom(R,A)}$ is actually given by ${\\phi:R\\rightarrow A}$ goes to the deformation given by composing ${G_S\\stackrel{\\psi}{\\rightarrow} GL_n(R)\\rightarrow GL_n(A)}$. This is wonderful. If we can somehow get our hands on this universal ring and universal deformation it will completely control all deformations.\n\nSchlessinger unfortunately only tells us it exists, but subsequent work does tell us these things. That will be the subject of the next post.","date":"2017-04-30 12:55:33","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\": 35, \"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\": 0.9240821003913879, \"perplexity\": 180.31691929495165}, \"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-2017-17\/segments\/1492917125532.90\/warc\/CC-MAIN-20170423031205-00552-ip-10-145-167-34.ec2.internal.warc.gz\"}"} | null | null |
Home News New Zealand News NZAF Acknowledges Support From the Community
NZAF Acknowledges Support From the Community
Since 2010 more than $62,000 have been raised for the New Zealand Aids Foundation(NZAF) by local community members Paul Heard and Campbell Orr.
With the most recent donation of over $9,700 being received by the NZAF at an event at Orr's bar Caluzzi in Auckland.
Together since 2010, Paul and Campbell organised the bars and dance tents at the annual Big Gay Out in Auckland's Point Chevalier Park and over the years have donated the proceeds to NZAF.
Speaking to Eikon, Campbell Orr explained how the idea came about over a few beverages at Surrender Dorothy's one Sunday afternoon.
"We thought it would be fun to swap roles at our bars which is how Bear Drag came about and has been a main fixture at Caluzzi for the last 6 years," Orr explained.
"From that we came up with the idea to work the beer garden at the Big Gay Out with the intention of raising as much money as we could for the NZAF, and so the amazing army of volunteers and supporters was born."
Paul Heard and Campbell Orr receiving their certificate of appreciation from the NZAF
Executive Director of NZAF, Shaun Robinson said, "It is awesome to have friends, people that like you, believe in you and help you out. NZAF has some of the best of friends in Paul Heard of Urge and Campbell Orr of Caluzzi."
"We thanked them in the kiwi tradition of a barbi on Sunday 29th March in Auckland's traditional hot sun and occasional rain.
"As Paul and Campbell were quick to point out then, it's not just about them (great as they are); it's about the dozens of volunteers and the whole community of people who Paul and Campbell have been able to call on to get this done.
"This is a caring supportive bunch of guys who enjoy being together, enjoy looking out for each other and by helping the NZAF to respond to HIV and help the whole community,"
The gentleman put their fundraising motivation down to a sense of wanting to help and contribute to the wider community.
"It's about giving back and assisting organisations that often struggle with funding. And yes just showing the love & support," Heard explained.
Both Paul and Campbell feel very strongly about the prevention of HIV and also support for people living with HIV and Aids.
"I lost a few good friends over the years and have always felt HIV & AIDS is such a misconstrued disease," said Heard
Orr explained about his experience that led him to assist the NZAF.
"The first drag queen I ever met was Miss Courtney Cartier who also happened to have HIV so was also the first person I had met that was open about living with the virus," Orr explained.
"Watching Courtney in her last days was very difficult and I learnt a lot about compassion and love from everyone around Courtney, especially from Mother Karen Ritchie who later set up the Cartier Bereavement Trust in Courtney's name."
Paul Heard of Urge with the latest cheque for the NZAF
The extremely humble pair say that they have only been able to raise the amount of money they have raised because of the team of volunteers who have worked with them.
"There are literally hundreds of people to thank, basically everyone and anyone who has been a part of the beer garden or dance tent over the last six years deserves a huge thanks," said Orr.
"The suppliers and sponsors that gave us product and other support.
"The community itself who came along and bought a drink all contributed to the $60,000 we have raised.
"The NZAF also had a huge hand in making the events as successful as they are."
With so many positive moments working together over the years raising money for the NZAF, Paul Heard and Campbell Orr say they are humbled and filled with pride.
Both men intend to continue their support for the NZAF in the future.
More information on the New Zealand Aids Foundation can be found in our Community Section.
Last Updated on Apr 10, 2015
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Cause and effects of air pollution essays
Magnetic field and electromagnet
Richard sennett flexibility
Home why do we write essays Politis and sport
Politis and sport
Staying starstruck is an indulgence of our arrested developments, even for jaded middle-aged reporters—sweaty ones, in this case.
Be entertained. The U. For nearly three decades, he was the Olympics reporter for the Chicago Tribune. What if he had been welcomed, rather than repeatedly shunned, by the N.
A need was also seen to reorient sportsmen towards professionalism, discipline and rules and regulations. Olympians Tommie Smith and John Carlos raised their fists in solidarity with the black power movement—to the Munich Olympics—where 11 Israeli athletes were kidnapped and killed by a Palestinian terrorist group—politics have shone through the veil of non-political competition that is often touted by the International Olympic Committee. Stepping back, we take a look at how Lance Armstrong became so powerful when he started out as just a bike rider. The innovations that followed largely involved tone—at some point, in the seventies, it became acceptable to belittle, rather than humor, an out-of-control caller. All the other stuff in the show was a table setting for this actual life-and-death incident, where no amount of posturing could help. For good and bad, the sports world is bigger and more powerful than ever, with athletes wielding more and more influence over our culture and our politics. The football team threatened a boycott, joining student activists who were concerned about a string of racist incidents on campus. Why was the United States government the one to finally step in to say, enough? But, within a few years, such stations dominated the coveted twenty- and thirtysomething male demographic, and beginning in the nineties WFAN became one of the top-billing stations in the country. When LeBron James and his Miami Heat teammates wore black hoodies to show their support for slain Florida teenager Trayvon Martin, it spurred countless dinner time conversations about whether athletes should even involve themselves in serious issues. Further, "Of the remaining 24 countries that recognize Taiwan, four are in the Caribbean and two of these play cricket. Why do sports offer such an enticing venue for countries to show off? The players are often interchangeable, an assembly line of heroes and villains, symbols of overachievement or wasted talent. A belief that the well of cable-subscription dollars would never run dry in turn affected the professional leagues themselves, whose fortunes depend on lucrative broadcast deals.
Stepping back, we take a look at how Lance Armstrong became so powerful when he started out as just a bike rider. The country made a similar protest for the Bandy World Championship because of the Warsaw Pact invasion of Czechoslovakia that year, handing over the hosting of the event to Sweden.
For example, the popularity of cricket in Indiabrought to the country by British sailors during the British Empire, which has continued to be the most popular sport in India, long after independence from Great Britain.
Rated 10/10 based on 109 review
Politics and Sports: A Long and Complicated Relationship | {
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Fatmire "Lira" Alushi , nacida Bajramaij, (Istok, Yugoslavia, hoy Kosovo; 1 de abril de 1988) es una futbolista alemana de etnia albana nacida en Yugoslavia. Desde junio del año 2014 juega para el equipo Paris Saint-Germain FC.
Su familia huyó de Kosovo en 1992 a raíz de las protestas anti-burocráticas en esa región yugoslava. Llegaron a Alemania como refugiados y se radicaron en la ciudad de Remscheid. En el año 2001 todos los miembros de su familia obtuvieron la ciudadanía alemana. Desde el 1 de julio de 2007 es miembro del grupo deportivo de la Bundeswehr.
En octubre de 2009 publicó su autobiografía titulada "Mein Tor ins Leben – Vom Flüchtling zur Weltmeisterin" (Mi gol en la vida - De refugiada a campeona del mundo).
El 10 de diciembre de 2013, Fatmire Bajrama contrajo matrimonio con el futbolista internacional kosovar Enis Alushi y adoptó su apellido.
Trayectoria deportiva
Bajramaj se unió al equipo femenino FCR 2001 Duisburg en 2004 y casi inmediatamente se convirtió en jugadora del primer equipo. Su equipo obtuvo tres años consecutivos el subcampeonato de la Bundesliga femenina. En el año 2009 el FCR 2001 Duisburg ganó la Liga de Campeones de la UEFA femenina.
En la temporada 2009/2010 fue transferida el equipo 1. FFC Turbine Potsdam con el que logró el campeonato alemán, y por segundo año consecutivo, el título de la Liga de Campeones de la UEFA femenina. El 18 de abril de 2011 se anunció su trasferencia al 1. FFC Fráncfort, los archirrivales del FFC Turbine Potsdasm, lo que causó mucho malestar entre la fanaticada en Potsdam.
Selección nacional
El 20 de octubre de 2005 hizo su debut con la selección femenina de fútbol de Alemania de mayores en un juego amistoso contra Escocia. En el año 2006 ganó el Campeonato Europeo Femenino Sub-19 de la UEFA con la selección juvenil alemana. El 30 de septiembre de 2007 se coronó campeona del mundo en la Copa Mundial Femenina de Fútbol de 2007 celebrada en China.
Obtuvo la medalla de bronce en los Juegos Olímpicos de Pekín 2008, donde anotó los dos goles alemanes que le dieron la victoria sobre Japón. El año siguiente se coronó campeona europea en la Eurocopa Femenina 2009.
Referencias
Enlaces externos
Ficha en FIFA.com
Futbolistas femeninas de Alemania
Futbolistas de la Copa Mundial Femenina de Fútbol de 2011
Futbolistas de Alemania en la Copa Mundial Femenina de Fútbol de 2007 | {
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using Ivvy.API.Json;
using Newtonsoft.Json;
namespace Ivvy.API.Venue.Bookings
{
public class ConfirmBookingRoomReservationResult : ISerializable
{
public enum Errors
{
Unknown = 0,
AlreadyConfirmed = 1,
IsCancelled = 2,
NoRooms = 3,
GroupCannotConfirmAvailability = 4,
GroupNotEnoughRooms = 5,
UnknownChannelError1 = 6,
IndividualRoomsError = 7,
UnknownChannelError2 = 8,
CanBeEditedFlag = 9,
}
/// <summary>
/// Indicates success or failure of the api action.
/// </summary>
[JsonProperty("success")]
public bool Success
{
get; set;
}
/// <summary>
/// The error code on failure.
/// </summary>
[JsonProperty("errorType")]
public Errors? ErrorType
{
get; set;
}
}
} | {
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} | 7,450 |
\section{Adopted Strategies and Software Solutions}
\label{sec:strategies}
In this section, we summarize the solutions most commonly adopted by teams to address each of the basic functionalities involved in the tests.
It is important to remark that none of the extremes in the abstraction layer is addressed.
On the high-level of abstraction side, all solutions differ, reflecting individual approaches of each team.
On the other side of the spectrum, low-level solutions are in most cases either unreported or often vendor-specific, without the influence or means to alter them by the teams.
\input{pages/stss-frameworks}
\input{pages/stss-asr-and-nlp}
\input{pages/stss-manipulation}
\input{pages/stss-navigation}
\input{pages/stss-object-recognition}
\input{pages/stss-people-recognition}
\input{pages/stss-people-tracking}
\section{\athome}
\label{sec:athome}
The \athome league was created in 2006 with the goal of developing robots capable of realizing all domestic chores and bringing them from the labs into people's homes.
As stated in its website\footnotemark,
\enquote{the \athome league aims to develop service and assistive robot technology with high relevance for future personal domestic applications. It is the largest international annual competition for autonomous service robots and is part of the RoboCup initiative}.
\footnotetext{\label{footnote:robocup-website}Source: \url{http://www.robocupathome.org/} Retrieved: Jan 1st, 2018.}
\subsection{Organization}
\label{sec:athome-organization}
The competition takes place in a test arena.
This arena is configured to look like a typical apartment of the hosting country.
However, the specifications and appearance of the arena are kept undisclosed to participants until they arrive.
The competition normally lasts 6 days.
During the first day and a half, period known as setup, teams prepare their robots and familiarize them with the arena; while the last day is reserved for the final demonstration and the award ceremony.
In the competition, which is divided in two stages, each robot has to solve a set of tests.
In most tests, the robots have to solve a household-related task while their abilities and performance are evaluated.
Some tests may take place outside the competition arena.
However, the competition scenarios are never specified beforehand.
Finally, the best 50\% of the participants in Stage~I advance to the Stage~II, from which only the very best would advance to the Finals.
\subsection{Brief History of \athome}
\label{sec:athome-history}
During the first two years, the tests were scored with boolean criteria.
A team would receive points only if the robot successfully accomplished the given task.
This had a lot of setbacks, but the most important one was that it was barely impossible to analyze the robot's performance in each ability from the scores.
Therefore, a new scoring system was introduced in 2008 that is formally introduced and analyzed in~\cite{Wisspeintner2009}.
From this year on, tests were split in a sequential set of goals.
In this schema, a robot can't advance to the next step unless (successfully) completing the current one.
This allowed to delimit the degree of uncertainty, making possible to estimate performance by measuring the contribution of each ability to each reached goal; as performed by \citeauthor{Iocchi2015} in their analysis in~\cite{Iocchi2015}.
In addition, this schema considers increasing the difficulty every third year, fine-tuning only in between.
This provides teams with enough time to tune their algorithms for the newly introduced challenges.
However, by 2013 the \tc had noticed a generalized decrease in performance (see~\Cref{fig:yearly-performance}).
Many robots failed in tasks that were considered more or less solved.
At the same time, it wasn't rare that robots couldn't even try\footnotemark.
For this reason, a test was specially designed to measure performance in basic skills for 2014.
Unfortunately, most of the robots did not come off well despite the relative simplicity of the benchmarking tasks.
Based on these results, the \tc decided to modify the test scheme proposed by~\citeauthor{Wisspeintner2009}~\cite{Wisspeintner2009} in Stage~I.
\footnotetext{Since 2010, the standard procedure involved waiting for the door to be opened, enter the arena, and retrieve a command.
Failing to detect when the door was opened, or misunderstanding the given command would make it impossible for the robot to advance any further.
Loose USB connectors and ambient noise promoted these problems even more.
This caused many teams with robust manipulation, people recognition, task planning, etc., to leave the arena without showing at what they excelled.}
\begin{figure*}[t]
\centering
\includegraphics[width=0.9\textwidth]{img/yearly-performance.png}
\caption{Performance of teams in the Top5 (2009 -- 2017)}
\label{fig:yearly-performance}
{\centering\footnotesize Obtained score with respect to the maximum attainable excluding special bonuses}
\end{figure*}
Therefore, by 2015 tests with a sequential execution scheme were replaced with tests focused on measuring performance.
The former had the intention of attracting new participants by easing the Stage~I while the difficulty level of the Stage~II continued increasingft.
This new schema, in addition to include semi-isolated benchmarking of relevant abilities, also tried to tackle the luck factor by requiring each ability test to be run three times and considering only the average of the best two runs.
The tested abilities were
\begin{enumerate*}[label=\arabic*)]
\item navigation, localization, and mapping,
\item people recognition,
\item people tracking,
\item object recognition and manipulation,
\item speech recognition and sound-source localization,
and
\item ability integration.
\end{enumerate*}
The introduced changes quickly paid off.
Besides providing benchmarking data to the \tc, the changes made evident to teams where their own weaknesses were.
This led to an immediate increase in performance in 2016 as~\Cref{fig:yearly-performance} shows.
Consequently, the \tc decided to raise once more the difficulty for 2017.
The gathered data was used to analyze strengths and weaknesses, and push in those areas that required more attention.
Notwithstanding, this new schema came also with several setbacks.
It widened the breach between stages, led to a performance decrease in tests relying on integration, and fostered score chasing.
Since 2015 a vast number of new challenges were introduced (many as bait) in the Stage~II with little success.
These new challenges include describing untrained objects, guiding people, avoiding tiny and light-reflecting objects, pouring, etc.
However, the newly introduced challenges were barely addressed.
Despite tests being sequential, the rules now allowed robots to receive help or skip a goal that wasn't a logic requirement to accomplish the next one.
As a consequence, participants went after easy to solve goals, completely ignoring the newly introduced challenges.
Increasing the reward didn't work either.
Nonetheless, \athome became more appealing to the audience since robots were allowed to show something, and team's frustration became almost inexistent.
Finally, 2017 introduced several important changes. \athome was split in three leagues:
\begin{enumerate*}[label=\arabic*)]
\item the legacy \opl with no hardware restriction,
\item the \dspl akin to the \opl, but using standardized hardware,
and
\item the \sspl that focuses in high-level \hri.
\end{enumerate*}
This division caused several veteran teams to migrate to the \spl{}, opening new spaces for new participants.
The former, in addition with the increase in the difficulty of several tasks led to the lowest performances in the history of \athome.
\subsection{Gaining People's Interest and Trust}
\label{sec:challenges-people}
Acquiring and maintaining the sympathy and interest of the general public is critic for research in service robotics.
Without people's trust, developing service robots would be pointless since there wouldn't be a market for them.
Furthermore, the active participation of people is fundamental in several related areas.
\subsubsection*{Engaging People}
\label{sec:challenges-people-engage}
It seems impossible to advance in natural \hri and \nlu without eventually taking advantage of people's innate skills.
For instance, latest advances in \asr and \mt were possible thanks to the huge amount of data for training that became available with the extended use of smart-phones.
However, the data acquired might be useless in \hri because it is very unlikely that people interact with robots in the same way they interact with their smart-phones.
Although there may be similarities, the difference in interaction becomes critical when it comes to data sets for deep learning.
In addition, while in a smart-phone a faulty transcription or an inaccurate translation are slight inconveniences, a robot that fails to correctly understand the given instructions becomes useless.
Moreover, having a good understanding of the operator's intentions is crucial in tasks that heavily depend on \hri.
Therefore, it is clear that instructing a robot must require significantly less time and effort than carry out the task.
The above suggests the need to build a corpus of interactions and, in particular, of spoken commands as a first step.
Steps towards that direction have already been made \citep{Bastianelli2014} and the produced corpus is being used in the \erl, although not yet in \athome\footnotemark.
The creation of corpora of that nature is not a trivial task.
First, people with experience in robotics, \asr, or \nlp cannot be involved in the process.
We have found that the expertise of people with a strong background in any of these fields acts as a biasing factor during the interaction process.
The expertise of experts makes them prone to alter pitch, speed, pauses, etc., when addressing robots.
What is worse, is that the biasing occurs at a subconscious level, making it extremely difficult for a person to prevent such behavior.
Therefore, non-experts shall be involved, and preferably volunteers with no experience in robotics and related fields.
The only requirement, at least in the beginning, is that the operator should have a friendly attitude towards the robot to ease the interaction.
\footnotetext{The generation and use of a corpus for \hri in \athome has been proposed and is discussed in \citep{Matamoros2018rc}.}
From these insights, the first logical step would be to attract the audience's attention.
However, this is not as straightforward as it may seem.
Unlike soccer, domestic chores are not exciting from most people's perspective.
In addition, soccer playing robots benefit from the familiarity of the audience who knows what is happening.
That's something we haven't achieved yet in \athome.
Most domestic chores are much more complex than any sport from a robotic's perspective due to the broadness of involved application domains of a domestic environment.
In consequence, robots perform their tasks slowly and fail more often, making the league even less attractive.
Therefore, new strategies are needed to attract audience's attention to \athome, getting them actively involved to produce the aforementioned corpora.
\subsubsection*{Regaining Trust}
\label{sec:challenges-people-trust}
\athome is directly affected by the \textit{Frankenstein Complex}.
While soccer and rescue robots are designed to operate in specific environments and circumstances, it is expected that service robots blend with people and use the same appliances.
This will be possible only if people consider them safe, even when using potentially harmful tools such as hammers, scissors, knives, and any other utensil of common use.
Media and entertainment industry are playing an important role in this matter.
For instance, most successful blockbusters and series related to robotics feature anthropomorphic machines that are hostile to humans.
Similarly, press and news constantly remind people that robots might not only turn evil, but also remark the high number of jobs that would be in jeopardy once robots are ready to be deployed.
Nonetheless, quite often informative services also make good press to the robots.
Moreover, sometimes their optimism harms what they are trying to exalt, raising expectations beyond what robots can actually do today.
We have witnessed on countless occasions the disenchantment of people who came to the competition to watch the prodigious robot of the news failing to grasp a simple object.
In both cases, competitions like \athome set the perfect opportunity to vindicate robots as the useful and innocuous servants they are.
It is during competitions, when people can effectively learn that robotics is a still-growing research area and why their participation is important to foster robot's development.
Furthermore, is in this kind of events where transdisciplinary networks can be established.
For instance, getting involved lawyers, sociologists, and philosophers can lead to the aperture of forums and research lines in topics of interest such as
licensing and patents,
liability in accidents caused by intelligent robots,
role of robots as \enquote{affective artifacts},
mid- and long-term impact of the integration of robots in human societies,
when a machine can be considered conscious,
and
rights and freedom of conscious robots.
Besides presenting the potential aperture of new research line in their fields, such forums can also contribute to engaging general public with the \athome community, preparing the way for the future.
We believe, in addition, that it's on our best interest to start discussing such matters sooner than later.
\subsubsection*{Conclusion}
\label{sec:challenges-people-conclusion}
The League's organizers are aware of these challenges and are working to overcome them.
One proposed solution is to have an expert moderating tests and explaining what is going on and why it is difficult.
In our opinion, another possible solution could be to show to the audience the goal and challenges of the running test to make it more appealing and understandable.
To conclude, more interdisciplinary research is needed to find solutions to integrate robots into human society.
It is also necessary to find ways to promote people's collaboration with the league to foster research in \nlu and \hri.
Often, hard questions come out, such as why should one acquire a robot while human assistants are much more efficient, cheaper, and hiring them helps creating jobs.
Therefore, it is important to find ways to vindicate robots as the useful and innocuous servants they are, and not present them as a threat to humanity.
\subsection{Available Resources and Safety Constraints}
\label{sec:challenges-resources}
In robotics exists a close relationship between the available resources and a reliable safe execution.
Today, with the internet becoming available everywhere, it becomes hard to resist to connect robots to the cloud.
However, such decisions must be addressed with care since, unlike with other technologies, a faulty execution can lead to a human being harmed.
\subsubsection*{Available Resources}
As the difficulty of the tests increases, it can be expected that the computational power required to solve them grew proportionally, especially when introduced changes are small.
It is true that it is impossible to tell what \textit{proportionally} means, especially in computer sciences and particularly in robotics.
In this regard, even the slightest changes, neglectable from a human perspective, can increase the complexity of a task in several orders of magnitude for the robot.
Nonetheless, in recent years we have detected a disproportional increase in the amount of computation needed to solve barely modified tasks.
Moreover, we have found that solving similar tasks requires more resources today than it required half a decade ago.
We believe this disproportion is partially due to the popularization of \ann and deep-learning-based solutions.
Although it is true that these new approaches are often more robust and normally easier to deploy, it is also true that they require a huge amount of computational power.
Besides, the enormous leap they caused in some disciplines have not fully reached the competition yet.
Therefore, we are expecting to witness some remarkable advances in the open demonstrations\footnotemark in coming years.
\footnotetext{The open demonstrations differ from regular tests in that they do not follow a script nor is there a predefined task to complete. Instead, teams can show whatever they want.}
Notwithstanding, experimenting with these technologies requires a computational power that laptops can't offer.
In addition, robots must become what smartphones are today: affordable devices providing reliable long-term operation with very limited resources.
This latter premise makes inadvisable to have robots performing heavy calculations all the time.
In conclusion, robot's computers might be unfit for most \dnn.
In consequence, roboticists look at cloud services.
\subsubsection*{Safety Constraints}
Cloud services offer several advantages besides reducing the amount of process a robot has to perform and increase the duration of its battery.
They also process information remarkably faster while taking advantage of the data provided by their users to enhance.
This, however, comes with some risks.
First, robots might experience a performance decrease due to lag and bandwidth problems during Internet rush-hours.
This might make them slow and inefficient in the best case but, in worst-case scenarios, they might endanger people's lives while waiting for the server's answer.
This becomes critical when performing nursing or baby-sitting tasks, in which a delay of a couple of seconds can cause severe harm.
Moreover, during emergencies the situation worsen.
Having all robots becoming inoperative due to a disconnection caused by a power shortage could be just an annoyance.
However, if the power shortage occurs during an earthquake or a fire, the mere presence of robots could hinder the evacuation instead of provide assistance.
Finally, and also a matter of security, but in a different sense, robots permanently connected to the internet have the potential risk of opening a window through which a malicious user could see and hear.
Nonetheless, implications on this matter won't be analyzed here any further.
\subsubsection*{Conclusions}
We believe that the robot's primary function should be helping people to have a better life.
This necessarily implies that robots must never compromise people's security and integrity.
Furthermore, the presence of a robot should help people to feel safe.
In consequence, minimum performance standards must be established to ensure a reliable and safe operation.
\athome has already taken the first steps by allowing cloud services and external computers on the condition that robots must remain operational and have full obstacle avoidance.
\section{Extrinsic Challenges}
\label{sec:challenges-extrinsic}
The challenges described in this section are considered to be extrinsic to the league.
Although they operate from outside, the influence they exert is big enough to be taken into account.
The discussed challenges are:
\begin{enumerate*}[label=\arabic*)]
\item how to gain the general public interest and trust (\Cref{sec:challenges-people}),
and
\item how to manage available resources and safety constraints (\Cref{sec:challenges-resources}).
\end{enumerate*}
\input{pages/cext-people}
\input{pages/cext-resources-and-safety}
\section{Intrinsic Challenges}
\label{sec:challenges-intrinsic}
The challenges described in this section are considered to be intrinsic to the league.
Most of them largely correspond to the role of the league in directing the research and development in robotics.
First, we start discussing each of the most relevant abilities:
navigation~(\Cref{sec:challenges-nav}), people recognition~(\Cref{sec:challenges-pr}), people tracking~(\Cref{sec:challenges-pt}), object recognition~(\Cref{sec:challenges-or}),
manipulation~(\Cref{sec:challenges-man}), and \nlu~(\Cref{sec:challenges-asr}).
Finally, we close by addressing the the necessity of planning tasks to be tested (\Cref{sec:challenges-roadmaps}).
\input{pages/cint-navigation}
\input{pages/cint-people-recognition}
\input{pages/cint-people-tracking}
\input{pages/cint-object-recognition}
\input{pages/cint-manipulation}
\input{pages/cint-asr-and-nlp}
\input{pages/cint-roadmaps}
\subsection{\asr[x] and \nlp[x]}
\label{sec:challenges-asr}
Speech is a key aspect to people's communication, hence it is fundamental in \hri.
Besides, it is widely acknowledged that language proficiency is related to intelligence.
Therefore, it is expected that robots can understand people's orders expressed naturally, in contrast to the current state where people learn how to talk to a robot.
Regarding spoken interaction, we have identified three key aspects that require attention and are introduced below.
\subsubsection*{Noise}
Noise is one of the most problematic aspects in spoken communication and, in fact, also in competitions like \athome.
The ambient noise produced by several hundred people greatly exceeds the noise levels of an average apartment.
However, so far filtering is delegated to the \asr engine.
Furthermore, eventually robots will be operating in airports, shopping malls, and other noisy environments.
Thus, we think it is best to deal with this issue in an early stage.
\subsubsection*{Non-Native Biased Operators}
Operators in \athome have two fundamental characteristics:
they are robot-friendly and they all have a strong background in robotics.
This means that, unconsciously, operators in \athome are trying to help the robot to succeed.
In other words, they are biased.
The use of biased operators helps robots to perform better and to give a better impression to the audience.
However, it has the disadvantage of voiding the purpose of the competition of providing real-case scenarios for testing.
Furthermore, few operators are native English speakers, which reduces the chances of a robot to deal with the richness of an unconstrained speech production.
\subsubsection*{Generators}
Despite being a powerful tool during the competitions, the use of generators greatly limits the interactions a robot can experience.
In drastical terms, reading verbatim sentences produced by command generators delves \hri into some sort of computer-computer interaction.
\subsubsection*{Next Steps}
A solution to the aforementioned problems has already been proposed.
The next steps consider the use of unbiased people to instruct the robot.
Generators would still be used, but only to select the task, having the operator to command the robot using their own words.
Besides, to address fairness, the interactions should be recorded beforehand, leaving robots to \textit{listen} the recording, but always having the operator at hand for further interactions.
This way, research would be pushed towards free-speech \asr engines, while at the same time, would enforce the integration of the latest techniques in \nlp.
But not only that, this process would lead to the creation of big annotated corpora of spoken \hri for training.
Nonetheless, carefully planned restrictions shall be imposed to the interactions in the first year.
For instance, while in the first years only imperative sentences are used, later on these can be replaced by declarative sentences that depict the desired goal.
Further on, procedures are explained to the robot, reducing the amount of detail with every passing year until the interaction reaches the level of a fluent dialog.
Last, but not least, noise can be introduced gradually by superposing recordings of noisy environments to the ones containing the commands.
This way, all robots would experiment the same initial conditions for the interaction, easing the benchmarking of components with higher levels of abstraction.
\subsection{Manipulation}
\label{sec:challenges-man}
Manipulation is perhaps the oldest and most mature research area in robotics, especially when it comes to industrial robots.
With this background, it would be reasonable to expect a similar degree of precision and speed from service robots.
However, mounting a manipulator in a mobile base and limiting available power and computational resources drastically increases the complexity of even the simplest tasks.
\subsubsection*{People's Expectations and Requirements}
In our polls people are explicitly asking for robots able to
clean the toilet, wipe windows, do the dishes (by hand), wash, iron, and fold clothes, open flasks and jars, brush and wash the dog, and take out the garbage
to name some examples.
Also of importance, but not even considered by potential customers are
abilities to open doors, move furniture, operate switches, and to operate the control panel of all electric and electronic appliances.
From these mandatory skills, opening doors has been addressed since 2006 as an optional challenge, but to the date has not been made compulsory in any test.
Nonetheless, door opening was impressively solved by team eR@sers in 2016 with the proprietary robot now used in the \dspl[x].
Notwithstanding, it seems there is still a long way to go before this skill can be considered solved in \opl and \sspl.
\subsubsection*{Advances and Current State}
Unfortunately, the advances in manipulation seem to have stalled~\Citep{Matamoros2018}.
Although challenges like pouring, stacking, transporting a tray, and grasping small objects have been introduced, there have been no serious attempt to solve any of these during regular tests since 2014.
What is worse, in 2017 during the storing groceries test, practically no team attempted to move any object.
Furthermore, none of the aforementioned challenges are new in \athome.
In fact, most of them were already demonstrated several times by the three-times-champion team Nimbro between 2010 and 2014 (see \Cref{fig:challenge-man-feats}), and repeated later on by other teams during final demonstrations.
\begin{figure}[t]
\centering
\begin{subfigure}[t]{0.32\columnwidth}
\centering
\includegraphics[width=\textwidth,height=3.5cm,keepaspectratio]{img/man-pour-cosero.png}
\caption{}
\label{fig:challenge-man-feats-pour}
\end{subfigure}
~
\begin{subfigure}[t]{0.58\columnwidth}
\centering
\includegraphics[width=\textwidth,height=3.5cm,keepaspectratio]{img/man-table-cosero.png}
\caption{}
\label{fig:challenge-man-feats-table}
\end{subfigure}
\caption{Team Nimbro's final demo in 2011~\Citep{Stuckler2012}. Robot Cosero (a) pouring pancake mixture into a pan, and (b) helping to move a table.}
\label{fig:challenge-man-feats}
\end{figure}
In addition and compliant with \Cref{sec:challenges-or}, the competition considers mostly moving regular-shaped, small, non-fragile lightweight objects.
Apples, small cereal boxes, and water bottles are some examples of typical objects a robot can be requested to move.
In numbers, most of the objects' weight ranges between 75 and 950 grams.
Dimensions are between 5 and 25 centimeters, with a typical maximum volume of 2 liters.
Also in Stage~I, all robots in the \dspl and the \opl are required to open the door of a cupboard.
This optional task was skipped by all \opl participants in 2017.
More complex tests in Stage~II involve pouring, scrub spots from a table, handling a tray, and moving tableware (bowls, dishes, and cutlery).
Once more, these tasks are often skipped or bypassed, as stated in~\Cref{sec:strategies-man}.
\subsubsection*{Next Steps}
The manipulation capabilities of the robots must be expanded in several directions, including reach, maneuverability, strength, and precision.
Below we describe each of this directions in more detail.
\textbf{Reach:} As of today, robots are required to handle objects in heights ranging from $30cm$ to $1.8m$, but eventually they should be able to reach the floor.
More important is depth, since today robots are only required to grasp objects at most $5cm$ from the table border.
Therefore, this distance should be gradually increased until robots can reach objects within an average human grasping distance.
\textbf{Maneuverability:} Nowadays robot's movement is quite clumsy.
Picking objects from a bag or box and stacking them all together in a narrow space is a trivial task for most people, but difficult even for industrial robots.
At the same time, twisting, uncapping, shaking, folding, levering, and turning are features yet to be tested in \athome.
\textbf{Strength:} In order to assist most people, a robot should be capable of carrying a 20 liters water bottle, hold a big dog by the leash, and uncap a flask of mayonnaise.
Moreover, some applications require enough strength to gently lift a fallen elder.
In consequence, weights and loads should start to gradually increase as soon as possible.
\textbf{Precision:} Today robots are very imprecise.
To effectively operate in human environments, precision in movement and applied strength is required.
Like a human, a robot should be able to store a needle and carefully take an egg without crushing it, but also being able to crack it in the right place.
The same applies to acceleration and speed, either for applying an insulin injection or whipping cream.
Summarizing, some tasks that must be addressed soon within the reach of currently used manipulators include
manipulating a switch,
taking out the bag from the trash bin,
using a ladle to serve soup,
mopping the floor,
dialing a phone number,
picking groceries out of a box,
grasping a towel from a hanger,
and
arranging cutlery into the drawer\footnotemark.
\footnotetext{Several of these tasks have already been proposed for \tc discussion for the next rulebook.}
\subsubsection*{Conclusion}
To foster development in manipulation, several tests were modified for 2018.
Nonetheless, there is still a long way to go before robots can effectively perform in domestic environments.
Obviously, the \athome league would enormously benefit from a manipulation road map with which teams could plan hardware improvements with sufficient time.
For now, it seems sufficient to start by gradually incrementing the weight of most objects, locate them farther away from the corner, and request the robot to perform useful tasks using tools.
\subsection{Navigation}
\label{sec:challenges-nav}
In general terms, navigation can be split into two categories: indoor and outdoor navigation.
Although closely related, the categories need to be approached with disjoint ability sets.
Therefore, they are discussed separately.
Likewise and as in~\Cref{sec:strategies-nav}, navigation also involves obstacle avoidance, localization, and mapping.
\subsubsection*{Indoor-Navigation}
\label{sec:challenge-nav-in}
Despite the continuous improvements, nowadays robots won't make it in most homes.
Steps, wet floors, rough carpets, and in general uneven surfaces are challenges to overcome that have not been addressed yet, let alone houses with staircases.
Another minor detail important to consumers is quietness.
We have found that people prefers robots that move silently and without damaging the floor, specially if they are meant to clean the house during the night.
Other than stairs, big houses and facilities present new challenges, specially when it comes to localization.
The perception range of most robots is typically limited to about 4 meters, a distance short enough to impede localization in wide spaces and long corridors.
Furthermore, the map of the environment is often a given condition (e.g. during startup) that uses a remarkable amount of geometrical data.
Without semantic localization, robots might have it hard to localize themselves in buildings such as apartment towers and hospitals.
Thus, without a pre-existing map, both, mapping and localization, can take advantage of robot's ability to read.
This might prove to be specially useful in the long-term should a room or person be temporarily relocated.
Other important abilities that have been addressed in the past but haven't been solved\footnotemark~include
using an elevator, navigate in narrow corridors, and move furniture around.
The latter is of special importance when cleaning and tidying up rooms.
A robot should be able to move unattended objects that are blocking its path.
\footnotetext{Here it is important to remark that the aforementioned abilities haven't been solved in \athome. That does not necessarily imply that they are not being extensively tested by research groups. For example, \citeauthor{utaustinvilla2017} claim in~\citep{utaustinvilla2017} that their robot can localize itself after using an elevator.}
Last but not least, is functional touching.
People often rely on their body to move or stop objects when moving around.
For instance, it is common to see a person, carrying a lot of thing without a free hand, pushing a door with their hips, or holding a door with their foot.
In \athome, we call this kind of interaction \textit{functional touching} to differentiate it from collisions and intended manipulations.
Although allowed in the rulebook, functional touching hasn't been addressed yet by any team.
\subsubsection*{Outdoor-Navigation}
\label{sec:challenge-nav-out}
Either to take out the garbage, or when going to buy groceries, eventually robots will be required to go outside.
Urban environments come with a whole new set of challenges to overcome other than dealing with the elements.
The first one affects directly the robot's ability to move.
If in interiors the presence of unevenness in floors presents challenges, the terrain diversity outdoors seems more adverse.
Conducted polls reveal that there is a potential market for robots in pet owners, specially when it comes to exercise the animals.
Implementing such feature would require robots to be capable of moving in rough terrain like grass, sand, or gravel.
However, not only the unevenness increases outside.
Distances, and the number of stimuli to process, increases by a huge amount.
To localize themselves in open spaces, robots would require not only the ability to see objects at 20 meters and beyond, but it would be also necessary to deal with occlusions, and correlate the perceived information in real time.
As with autonomous cars, localization using point-clouds would be unreliable but, at the same time, the size difference and low movement speed would make it extremely hard to rely exclusively on GPS data.
Therefore, we foresee many situations in which a robot will not only need to recognize signs, but also read street and shop names in order to reach its destination.
In addition, in crowded streets a robot could encounter hundreds of pedestrians, including children, elders, dogs, and other robots, which it would have to evade.
Furthermore, speed is another important factor to take into account.
While indoors, a robot can take its time to solve a task, outside, the world imposes its own schedule.
Streetlights, public transport, and automatic doors are three examples of elements that require a quick response.
Therefore, the reaction time or robots must be improved, even in conditions of insufficient information.
\subsubsection*{Conclusions}
\label{sec:challenge-nav-conclusions}
Although \athome currently features wheeled-robots only, it is evident that this design is not suitable for most human environments.
Therefore, we will have to choose between making our environments robot-friendly, or provide robots with better means of movement.
Otherwise, robots will be restricted to those areas in which they can safely operate.
On the other hand, people can immediately recognize semantic information in their surroundings, having an innate ability to correlate the current environment with those previously known.
Eventually, robots will need to integrate semantic mapping (or an equivalent ability) capable of mimicking such human abilities in order to efficiently integrate in human environments.
\subsection{Object Detection and Recognition}
\label{sec:challenges-or}
Mankind has transformed its environment giving priority to the sense of sight.
Therefore, in order to effectively blend in human society, robots must be able to decode the visual stimuli of the world we have built around us.
Furthermore, and from a totally utilitarian point of view, perceive objects is necessary for manipulation, a key feature to perform any chore.
\subsubsection*{Advances and Current State}
\label{sec:challenges-or-advances}
Despite the remarkable advances in computer vision in the recent years, anyone who has witnessed recent competitions could safely argue that robots are shortsighted.
On the one hand, is very unlikely that a robot can detect an object more than 4 meters away.
On the other hand, robots continuously fail to identify objects lying in direct line of sight.
Both situations would deem a person as visually impaired and unable to perform in most environments without the proper aids.
The former is mostly due to the short range and limited resolution of used RGB-D sensors.
Nonetheless, this allows robots to do much more than years ago when sensors were limited to VGA cameras and Time-of-Flight sensors with a resolution of only a few thousands of pixels.
Limitations aside, successful detections are also used to update the grid of an occupancy map.
This is an important step further from when visual information wasn't used for obstacle avoidance.
In the case of recognition, features from several sensors are extracted and stored.
Later on, the features of each detected candidate are analyzed (e.g.~as in color and shape detection) or evaluated against the features of all trained objects, depending on the task requirements.
However, objects in the arena are always placed sparsely, and with partial occlusions at most.
Also, the set of objects the robot should know is capped at 25, a very small quantity considering the number of objects people use to deal with at home.
Furthermore, although robots have to identify objects \textit{of a kind} (e.g.~an apple, regardless the color, size and texture of the fruit), such recognition is still on very early stages.
\subsubsection*{Next Steps}
\label{sec:challenges-or-next-steps}
Virtually all reported strategies process the input stream frame-by-frame, computing in a feed-forward manner.
In other words, to track an object, a robot has to identify it and calculate its position in each frame with the possible assistance of a filter.
Furthermore, the extracted visual information once used is discarded (i.e.~not used for reinforcement).
In contrast, advances in neurosciences suggest that the human brain uses contextual information to build scene representations, and it can choose to rely on memory search instead of visual search~\Citep{Oliva2004,Oliva2007}.
Having robots using context to perform object detection and recognition, is an interesting area to explore.
For instance, a robot could analyze a scene from a different perspective, calculating the positions of those objects that have been already identified and removing all their clusters from the frame, focusing only in those with a low recognition confidence.
Following the line of recognition as instance-class matching, there are more problems to address, like
increasing the number of objects to hundreds and beyond, perform recognition of stacked objects, discriminate two identical objects based on its relative position,
recognize translucent and transparent objects, and recognize objects in odd lightning conditions like direct sunlight and in the dark.
In a broader perspective, detection and recognition can be taken further on.
Identifying the orientation of an object prior to grasping, as well as the best location for placement, or even infer the weight of an objects are requirements yet to be addressed.
Moreover, quite often robots will need to deal with occlusions and objects behind crystals, identify dirt, dust and spots on the floor, and objects that change over time (e.g.~food).
\subsubsection*{Conclusions}
\label{sec:challenges-or-conslusions}
Object recognition applied to robotics is in a very early stage.
Similar to other abilities, a roadmap is necessary to direct advances on object recognition.
However, recognizing things, more than characterizing features, is an A.I.-complete problem.
The first logical steps would be those that help with manipulation, like identifying placing surfaces, grasping orientations, and containers.
Other useful features that can be addressed first include occlusions and transparent objects.
Finally, it is necessary to start integrating object recognition with high-level action planning and memory management so abstract constructs like spatial and temporal relationships can be also recognized.
\subsection{People Detection \& Recognition}
\label{sec:challenges-pr}
Detecting and recognizing people is fundamental for \hri[x] (\hri), therefore, it should be extensively tested in \athome.
Moreover, science-fiction has raised people's expectations, depicting robots as entities with an infallible, endless memory.
In consequence, most users expect robots capable of remember or find anyone.
\subsubsection*{Advances and Current State}
Although people detection is being extensively tested since it is involved in all types of \hri, people recognition is not.
Robots need to find a person to retrieve a command, deliver an object, answer a question, and detect a gesture, to name some examples.
However, most of the time the detection occurs with the person standing.
As of 2017, finding people lying or sitting was still problematic for most competitors.
In contrast, recognizing people hasn't bee fully addressed.
For our analysis, this feature is split in
\textit{people recalling}, when the robot needs to recall a person's name or order,
and \textit{people identification}, when the robot needs to identify a person's specific features.
Recalling, as stated in~\Cref{sec:strategies-pr}, usually consists in pairing the person's face with a name and, in some cases, some additional information like an order.
This ability has been tested since the very first competitions, but always with a reduced number of people (less than five).
In addition, although there shouldn't be a problem keeping the information longer, the face-name pairing lasts no longer than the test.
On the other hand, people identification is relatively new.
Robots can be requested both, to describe a group of people, or find a person matching the given description (relevant features are described in \Cref{sec:strategies-pr}).
However, by 2017 the tests addressing this ability either randomized the feature selection, or left its selection to the robot.
From our perspective, while most robots succeeded in finding their target, provided descriptions weren't very accurate, leading to the assumption that apart from relying in identification, heuristics were used to maximize the robot's chance of success.
In consequence, it is impossible to know precise information regarding general performance.
Nonetheless, it must be acknowledged that most robots perform good at counting people grouped for a photo.
\subsubsection*{Next Steps}
Considering the important number of features introduced in recent years, we believe the competition should focus in extending their usage.
Moreover, features like estimated age, gender, and relative position can be used to test awareness, decision making and planning.
Notwithstanding, focusing exclusively on detection and recognition, like with objects, the detection range needs to be increased, as well as the number of people in the memory of the robot.
On the features to be introduced in later stages and of importance in \hri we consider:
emotions and moods, activities, health and vital signs (inebriation, fatigue, sickness, sleep, etc.), skin and hair color, clothing names and styles, and identification by voice.
Also important, and shared with object recognition is addressing occlusions and translucent surfaces.
\subsubsection*{Conclusion}
Much has been achieved in the past eleven years.
However, \hri hasn't been tested as extensively as intended.
Furthermore, several tests require to be redesigned to ensure benchmarking data is available for analysis.
Finally, we find that visual and auditory information are equally important when addressing people.
Therefore, the integration of audio features in people detection and recognition has to be carefully planned.
\subsection{People Tracking}
\label{sec:challenges-pt}
People tracking is one of the first functionalities tested in \athome.
Although not necessarily useful when carrying out chores, it becomes relevant in daily life social events (e.g.~parties) and during shopping.
Its importance grows in non-domestic environments such as museums, hospitals, and restaurants.
\subsubsection*{Advances and Current State}
People following has been greatly improved since 2006.
Every year robots require less training time and are less prone to loose their target.
In addition, they can robustly handle people crossing or standing in between.
Moreover, back in 2014 robots were able to follow a person to an elevator, get inside, and continue following after leaving the elevator.
And not only that, should the target go through a compact crowd blocking the sight, robots were able to go around it and meet their target.
Whether robots are still capable of doing it is unclear, since elevators and crowds are not part of the tests since 2015.
Guiding, on the other hand, was first tested in 2006 with little success, leading to its removal by 2008.
Since its reintroduction in 2015, the league has achieved good performance.
Furthermore, this ability is tested along with navigation.
After reaching a previously unknown location, the robot has to go back, find someone, and guide them to the location it just learned.
\subsubsection*{Next Steps}
To humans, it is unnecessary to study someone in order to follow them.
Moreover, it takes us only a quick glance to get all information required to track or even chase an object, let alone people.
Therefore, one of the first improvements in this basic functionality would be removing training time, so it's important to start reducing it.
Another important aspect is speed, which must start to be gradually increased.
Nowadays, robots follow and guide only \textit{professional walkers}, who walk at constant pace to keep tests fair.
In addition, professional walkers are robot-friendly, instructed to follow the instructions of the robot and even slow down when it has lost track.
However, people usually walk a lot faster than professional walkers, and are more likely to leave the robot behind if it can not follow their pace.
Also, when following and guiding, people have no inconvenience in sliding through crowds or narrow spaces, temporary losing contact to each other.
In consequence, it is important that robots can keep track of people walking fast in crowded environments and be able to predict or estimate local rendezvous points after optimizing trajectories.
Also of relevance in people tracking is side-by-side walking.
When two people walk side by side, it is hard to tell who is guiding the march.
It is one of the most common activities performed by humans.
Thus, it is expected that robots learn how to walk with humans and not in front or behind them.
In direct relationship, walking holding hands is a natural way of interaction.
Moreover, it is of special importance when interacting with children.
Notwithstanding, physical contact is strictly forbidden in \athome for security reasons.
Therefore, this also needs to be addressed, but with caution.
\subsubsection*{Conclusion}
Although unrelated to most domestic chores, people following and guiding are necessary abilities for many other applications.
Yet, both abilities can be analyzed as a hybrid problem, namely people tracking and navigation.
Moreover, tracking can be made extensive to animals and any kind of visual objects.
Thus, tracking gets into the domain of object recognition, for which it becomes important to find the pertinent features that make it possible for robots to track something.
Nonetheless, other problems like reducing training time and increasing speed need to be addressed with a carefully designed strategy.
\subsection{Roadmaps and milestones}
\label{sec:challenges-roadmaps}
We believe the \athome league has the responsibility of carefully planning what needs to be tested and when to introduce changes by establishing milestones for the competition.
Said in other works, the league should have a set of roadmaps and milestones to asses the \tc when adapting and designing tests.
This is not an unfounded belief.
In fact, this need arises from the teams and the members of the \tc[x] themselves.
To the former, knowing in advance the challenges that will come in future competitions allows to prepare and direct their research.
Moreover, some teams have even stated that they could have made a better use of their resources, should they have known in advance the direction of the league.
With regard to the latter, the \tc renews every year with candidates elected by the community.
In consequence, sometimes it happens that most people are replaced and the direction taken by former members is overridden.
This situation worsens when most of the new members are relatively new in the league (e.g.~less than three years), thus lacking of important undocumented empirical experience.
This lack of direction has led in several occasions to brainstorms in which valuable ideas appear.
These ideas require to be analyzed, evaluated against the robot's capabilities, and condensed in milestones and test drafts that can be retaken in latter years.
Unfortunately, this has not occured until now.
In response, we have condensed not only ideas taken from the teams, but also the people's needs and, considering the current capabilities and limitations of the robots, present them as future steps or milestones in former sections.
Nonetheless, this is just a first step, since roadmaps need to be still prepared and adopted by the league which, in the end, is peer-maintained.
\section{Conclusions and Future Work}
\label{sec:conclusions}
In this paper we conduct a thorough summary of the software solutions and strategies used by participating teams in \athome to address the most important abilities required in the tasks of the competition's tests.
Further on, we present an overview of the attained achievements since the league's foundation based on our experience as long-time participants, contributors, and referees in the league.
Finally, also organized per ability and along with the overview, we discuss these achievements while addressing what is expected by potential consumers, what needs to be done, and would be the next logical steps based on the robot's current capabilities.
This study result in two contributions.
First, we believe the presented summary can serve as quick reference guide for new competitors, or for experienced ones looking for alternatives to their current solutions.
Second, our work sets the basis to build road maps that can help the \athome league, as well as other competitions aiming at service robotics, to plan the direction of the competition towards its goal.
Moreover, we believe that road maps are very important and that planning features to be tested can help teams to prepare in advance, set mid and log-term goals, and have a smarter resource management.
In fact, based on this and previous work, we have designed a roadmap for spoken \hri[x] in \athome which will be presented to the league in Montreal 2018 and in the RoboCup Symposium.
However, there is still work to be done.
Not only the roadmaps have to be designed.
This work has also allowed us to identify several important flaws that need to be addressed.
For instance, the presence of certain rules might be undermining the development of certain features.
At the same time, we have found that many successful approaches and strategies are never reported in the \tdps.
This has two important setbacks.
First, it makes it much harder for the scientific community to compare the performance of the different approaches when the best performers are missing.
Second, it leads to an eventual loss of knowledge that worsens as the lifetime of a good team shortens.
These insights are left to the competition organizers to analyze as part of future work, for which we trust this manuscript can come handy.
\section{Summary of Hardware Solutions}
\label{sec:hardware}
In this section we present a summary of the hardware configurations most used in the \opl.
We focus on five key aspects:
\begin{enumerate*}[label=\arabic*\rpar]
\item RGB-D sensor model,
\item drive mechanism of the base or locomotion type,
\item number of \dof of the \textit{head}
\item number of manipulators and their \dof,
and
\item number of \dof of the \textit{torso}.
\end{enumerate*}
The aspects have been chosen for their potential influence in the robot's performance, either by expanding its perception or interaction with the environment.
\subsection{RGB-D Sensor}
\label{sec:hardware-rgbd}
As of 2017, all teams report the use of at least one RGB-D sensor.
From these, the preferred one seems to be the Microsoft Kinect 2 due to its incorporated Time of Flight sensor and better resolution, leaving the Asus Xtion in second place (see \Cref{fig:hardware-rgbd}).
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{img/trends-rgbd.png}
\caption{Most used RGB-D sensors (2017)}
\label{fig:hardware-rgbd}
\end{figure}
\subsection{Base}
\label{sec:hardware-base}
To the date, all robots in \athome use wheels to move around.
Therefore, it can be considered that all robots have a base in which the driving mechanism lies.
Furthermore, there is no evidence in the \tdps pointing at a possible change of paradigm in the preferred locomotion type.
From all possible configurations, the most used is the differential pair followed by omni-drive in either 3-wheeled or 4-wheeled configuration. There is only one reported use of \textit{Swerve drive}, a special type of omni-directional configuration in which all four wheels rotate independently. This information is summarized in \Cref{fig:hardware-base}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{img/trends-base.png}
\caption{Reported locomotion types (2017)}
\label{fig:hardware-base}
\end{figure}
\subsection{Head}
\label{sec:hardware-head}
To humans, identify the head of something in nature is normally easy, intuitive, and straightforward.
With robots, however, things change.
An unfamiliarized operator can get easily confused when speaking to a bicephalous robot (\Cref{fig:hardware-head-exia}), or a robot featuring a face located several centimeters away from its sensors (\Cref{fig:hardware-head-lisa}).
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.48\columnwidth}
\includegraphics[width=\textwidth]{img/head-exia.png}
\caption{Exi@, Hibikino-Musashi}
\label{fig:hardware-head-exia}
\end{subfigure}
~
\begin{subfigure}[b]{0.48\columnwidth}
\includegraphics[width=\textwidth]{img/head-lisa.png}
\caption{Lisa, homer@Uni-Koblenz}
\label{fig:hardware-head-lisa}
\end{subfigure}
\caption{Addressing the right head may be deceiving.}
\label{fig:hardware-head}
\end{figure}
Hence, for the purpose of this summary we propose the following definition:
\begin{center}
\begin{tabularx}{0.9\columnwidth}{ l X }
\bfseries Robot head: & Unit comprising at least a camera and a microphone mounted on a pan-tilt unit. \\
\end{tabularx}
\end{center}
Based on that definition, we are confident to declare that 67\% (10 out 15) of of the participant robots in Nagoya 2017 in the \opl featured a head.
The established standard in the league are heads with 2 DoF.
\subsection{Manipulator}
\label{sec:hardware-manipulator}
Few people conceive a domestic service robot with no arms.
In fact, most of the activities considered important by potential customers involve object handling.
Moreover, most homes are designed to optimize spaces, while still being accessible to their inhabitants.
Therefore, it would make sense that most robots had anthropomorphic configurations.
However, to handle objects, most team use either home-made or proprietary low-cost hardware~\Citep{tobi2017,uchile2017}.
Although professional arms might be seen as the best option due to their strength and precision, their size makes them unfit for domestic narrow spaces.
Therefore, they are rarely used in \athome, with none of these present in Nagoya 2017.
In contrast, home-made manipulators are usually anthropomorphic and much cheaper, although they lack the precision and strength of the former ones.
In Nagoya 2017 the number of \dof[x] for manipulators ranged from 4 to 7 with mode in 5 as \Cref{fig:hardware-manipulator} depicts.
Regarding the final effector strength, it is usually about $1.25kg$, insufficient to lift a $1.5L$ bottle of soda or cutting food.
Finally, since there is no mandatory task requiring two-handed manipulation, only few robots have more than one (see \Cref{fig:hardware-manipulator}).
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{img/trends-manipulator.png}
\caption{Adopted hardware solutions for manipulation}
\label{fig:hardware-manipulator}
\end{figure}
\subsection{Torso}
\label{sec:hardware-torso}
Given an anthropomorphic configuration, a \textit{torso} would provide a robot with a panning and variable elevation for its head and upper limbs.
However, this is rarely the case.
In Nagoya 2017, only 9 out of 15 teams (60\%) reported a torso and, in all cases, it consisted of the elevator only.
However, in some cases, the elevator supported exclusively the manipulator.
Therefore, it is necessary to clarify that, in strict terms, units providing an additional \dof to the manipulator shouldn't be considered torsos.
Nonetheless, due to the lack of schematics, we decided to give our vote of confidence to teams, reporting the devices as alleged.
\section{Introduction}
\label{sec:introduction}
In the eleven years since its foundation in 2006, the \athome league has played an important role fostering knowledge exchange and research in service robotics.
Moreover, nowadays the competition can influence --and sometimes direct-- the course of research in the area of domestic service robotics.
Having such impact is not a minor thing.
In consequence, the \athome league has the responsibility of planning carefully what needs to be tested and when to introduce changes by establishing milestones for the competition.
This, of course, can only be done after analyzing the grounds in which the league is standing.
In response, in this paper we present two main contributions.
First, we present a survey of the approaches and technical solutions reported by teams in each of the different basic functionalities or abilities to accomplish a task.
The conducted overview serves as a basis for our second contribution since it reveals to us the capabilities of the competing robots.
To achieve this, we tap on several information sources, including:
\begin{enumerate*}[label=\alph*)]
\item claims made in the \tdps,
\item relevant publications,
\item rulebooks,
\item multimedia material available on-line,
and
\item our cumulative experience as participants and referees in RoboCup@Home since 2009.
\end{enumerate*}
Second, we discuss the challenges yet to be overcome that we have identified throughout these first eleven years of \athome.
This discussion considers not only the robot's current capabilities, but the feats attained since the foundation of the league.
In addition, we also consider information from conducted polls targeting potential customers necessities all over the globe but specifically from people living in Germany, Japan, Mexico, and The Netherlands.
Consequently, we propose several sets of features, tasks, or applications that have to be addressed to achieve the goal of \athome, but making special emphasis in those relevant in the short term.
Nonetheless, it is important to point out that the discussion focuses exclusively on service robots (referred hereinafter as robots for simplicity), with particular application for domestic environments.
In the same sense, we are focusing only on \athome without ignoring the existence of related robotics competitions like the \erl and \wrs.
\subsection{Audio, Speech, and Natural Language Processing}
\label{sec:strategies-speech}
The most broadly adopted solution to deal with speech consists in a pipeline.
In this pipeline,
a filtered audio signal feeds an \asr[f] engine to get a text-transcript for further processing.
Then, the transcript is sent to a natural language processor that extracts and conceals relevant information.
Finally, the acquired information is consumed by a high-level task planner that triggers the pertinent behaviors.
Although processing raw audio signals is technically possible \cite{Roy2002,Dominey2003}, this is still unexplored in \athome.
Typically, no external filters are used, leaving filtering to the microphone and the \asr engine~\cite{Doostdar2008}.
When reported, the most recurrent solution is HARK\footnotemark~\cite{aisltut2017,happymini2017,hibikinoOPL2017},
although it is intended for sound-source localization and separation.
\footnotetext{HARK (\textbf{H}onda Research Institute Japan \textbf{A}udition for \textbf{R}obots with \textbf{K}yoto University) is an open-source robot audition software that includes modules for \asr and sound-source localization and sound separation. Source: \url{https://www.hark.jp/}}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{img/trends-asr.png}
\caption{Trends in \asr[x] (2017)}
\label{fig:asr-trends}
\end{figure}
Regarding \asr, the most commonly adopted off-line solutions include Julius~\cite{duckers2017}, the Microsoft Speech API~\cite{pumas2017}, and CMU Sphinx~\cite{kamerideropl2017,tobi2017} being the most popular solution as~\Cref{fig:asr-trends} shows.
Due to the limited computing power of their robots, most teams in the \spl used cloud services, where Google speech API is the most popular approach~\cite{aisltut2017,jsk2017}.
However, network connectivity is often unreliable, a reason for which all teams using cloud services also have offline solutions as backup.
Moving forward to \nlp[x], despite the remarkable advances achieved in this area, little has been exploited in \athome.
To the date, we have found many teams still rely in keyword spotting and pattern matching to trigger the execution of a state machine~\cite{tinker2017,kamerideropl2017,Seib2015}, specially in simple tests.
In fact, \nlp and \nlu are mentioned in less than 50\% of the \tdps.
Nonetheless, robust A.I. solutions have always been in play and are now gaining strength.
Among the approaches for processing language, we found
probabilistic semantic parsers~\cite{utaustinvilla2017},
Multimodal Residual \dnn~\cite{wrighteagle2017},
ontology-based parsers over inference engines~\cite{pumas2017},
and
probabilistic parsers for syntax-tree extraction along with lambda calculus for semantic parsing~\cite{aupair2017}.
Of special mention are the Stanford Parser~\cite{stanford2011corenlponline}, the most broadly adopted solution for POS-tagging and syntactic tree extraction, and LU4R~\cite{Bastianelli2016}, a Spoken Language Understanding Chain for HRI developed in La Sapienza~\cite{spqrel2017} by participants of \athome which is also being used by several teams.
\subsection{Frameworks and Middlewares}
\label{sec:strategies-frameworks}
Frameworks and middlewares can operate at many levels of abstraction.
This section refers exclusively to solutions for the intercommunication of the multiple modules that operate a service robot.
In addition, there is no further reference to the platforms imposed by vendors such as NaoQui.
Nowadays, ROS has become a tacit standard in robotics with all teams declaring its use in their \tdps for Nagoya 2017 and Montreal 2018~\Citep{Matamoros2018}.
Nonetheless, older frameworks like Orocos~\Citep{tue2017} are still in use.
Moreover, some teams still make good use of their own solutions~\Citep{pumas2017}.
That being said, it is important to remark that, although it has been discussed in the past, the \tc of the \athome league has discarded the idea of making ROS a compulsory standard.
\subsection{Manipulation}
\label{sec:strategies-man}
How teams address manipulation heavily depends on the platform.
In the \sspl[x], due to limitations of the robot, manipulation is barely addressed and often skipped.
In direct opposition, the robot chosen for the \dspl[x] features a very precise and versatile manipulator that even incorporates a suction tip for lifting lightweight objects.
Last, but not least, manipulators in the \opl range from home-made to professional ones.
Implementation solutions for manipulation are often based on direct-inverse kinematic models with a closed-loop control and camera feedback as an alternative to the ROS manipulation stack.
However, nowadays many teams are migrating to \textit{MoveIt!}
Out of 32 participant teams in Nagoya 2017, manipulation was reported in only 18 (56\%) of the \tdps.
From these, 61\% are using \textit{MoveIt!}, while the rest relies on custom solutions that include
super-ellipsoid fitting~\Citep{tritonsdspl2017},
multiple deep auto-encoders fed with raw images and audio from the sensors~\Citep{erasers2017},
and
the built-in software solution provided~\Citep{spqrel2017}.
Other than picking and placing, robots are required to
open doors, pour liquids, scrub spots from a table, handle a tray, and move tableware.
In the \dspl, handling a tray is particularly problematic since the robot has only one manipulator, while opening doors doesn't seem to be a problem for the robot, and the suction tip comes handy for moving tableware.
In contrast, tray handling is bypassed in the \opl by mounting a custom tray in the robot, while the rest are usually skipped.
Unfortunately, no \tdp addresses the solution of any of these particular challenges.
Finally, it seems to be a growing trend in the use of deep-learning-based methods for manipulation, especially in planning.
In contrast with the traditional methods long-time used in industry, deep-learning-based approaches are computationally much more expensive, but also can be remarkably faster when a good-enough, non-optimal solution is acceptable.
In addition, their supporters claim they require much less expertise and effort to code.
Nonetheless, it is too early to know if these new methods will be a better solution for service robots.
\subsection{Navigation}
\label{sec:strategies-nav}
The basic functionality referred to as navigation in \athome involves four research areas, namely
\begin{enumerate*}[label=\arabic*\rpar]
\item path planning,
\item obstacle avoidance,
\item localization,
and
\item mapping.
\end{enumerate*}
Navigation is fundamental in the competition.
It is assumed that all competing robots can safely navigate inside the arena.
Therefore, path planning, and localization in known environments (i.e. inside the arena) are considered to be solved, while obstacle avoidance is not.
Finally, on-line mapping becomes relevant only when robots are outside the arena, and nowadays is being extensively tested.
With a couple of exceptions, it can be said that all teams rely on the ROS navigation stack (see~\Cref{tbl:strategies-nav}).
However, it must be clarified that, in virtually all cases, this solution is adapted to the robot and the particular necessities of each team.
In this regard, the most broadly adopted solution sums up OpenSlam's Gmapping and \amcl with an $A^*$ path planner~\Citep{tue2017,uchile2017,tobi2017}.
Also a recurrent solution involves incorporating a Kalman filter to Gmapping~\Citep{tinker2017,walkingmachine2017}.
\Cref{tbl:strategies-nav} summarize these trends showing the reported ratio, and the most adopted solution with the percentage of reported use cases.
\begin{table}[t]
\caption{Trends in solutions for navigation. The first column, Report Ratio (R.R.) represents the percentage of \tdp in which the solution used was specified. The second column shows the most adopted solution and its use percentage with respect to the number of reports.}
\centering
\begin{tabularx}{\columnwidth}{ X c l }
\toprule
& \bfseries R.R. & \bfseries Mode (cases)
\\ \midrule
Path planning & 63\% & $A^*$ (40\%)\\
Obstacle avoidance & 56\% & ROS (40\%)\\
& & Occupancy grid (28\%)\\
Localization & 63\% & \amcl (60\%)\\
Mapping & 78\% & GMapping (72\%)\\
\bottomrule
\end{tabularx}
\label{tbl:strategies-nav}
\end{table}
\subsubsection*{Path Planning and Obstacle Avoidance}
Other than $A^*$, several teams just report the use of ROS. In these cases, as well as when nothing is said, we assume teams use ROS' default behavior.
Other employed solutions include
randomized path planners~\Citep{aisltut2017},
and
wave-propagation algorithms based on the Fast Marching Method~\Citep{northeastern2017}.
Regarding obstacle avoidance, the most reported strategy is the use of occupancy grids.
To build the occupancy map, several teams take information from both, the laser range finder and the RGB-D camera.
\subsubsection*{Localization and Mapping}
These two abilities are often reported together.
Regarding localization alone, there are two reported solution differing from the robot's built-in localization and \amcl. These solutions are addressed by different teams and implement \slam using the \icp algorithm~\Citep{airobots2017,unsw2017}.
Both methods aim for accuracy and speed with limited resources.
On the mapping side, reported solutions other than GMapping include
hector \slam\Citep{duckers2017}, \mrpt and \icp\Citep{oittrial2017}, and Omnimapper\Citep{tritonsdspl2017}.
\subsection{Object Detection and Recognition}
\label{sec:strategies-or}
Object detection and object recognition are closely related, although they are often used separately.
Object detection is much faster than object recognition, so a common strategy while looking for objects involves performing a continuous detection until a potential area of interest is found.
A recurrent approach consists in taking the point-cloud of the RGB-D sensor and remove background, floor, and other surfaces (e.g. using Vectorial Quantization~\Citep{pumas2017} or \ransac~\Citep{northeastern2017}).
Later on, multiple color-depth images are extracted from the original RGB-D cloud using the detected clusters that can be further analyzed by the object recognizer.
This approach is also a computationally inexpensive alternative to deep-learning-based approaches that consume raw data from the sensors.
Most teams rely in more than one software solution to recognize objects, either running them in parallel and using some consensus algorithm, or implementing a process pipeline.
In addition, depth information is used for recognition based in contour and shape~\Citep{pumas2017,tritonsdspl2017,wrighteagle2017}.
However, there is no clear tendency on how the related modules are coupled.
For instance, some proposed solutions include
\yolo + \sift + \brief~\Citep{aisltut2017},
\surf + Continuous Hough-space voting + \ism~\Citep{homer2017},
color, size and shape histograms + SIFT~\Citep{pumas2017},
\pcl + \ransac + \yolo~\Citep{northeastern2017},
and
contours using LINEMOD + HSV color histograms + \surf considering joint models for occlusion~\Citep{wrighteagle2017}.
Detections and recognitions can take place either continuously or on demand.
For simplicity, and considering the default implementations of most used packages, it is more likely that both processes are running all the time.
However, unless the A.I. is designed to keep updating the world-model or knowledge-base of the robot with every new stimulus, acquired information will be simply discarded by the task planner.
In this regard, very few teams have reported the mechanisms to take advantage of continuous detection and recognition.
\begin{table}[t]
\caption{\small Adopted software solutions for object recognition. Presented results are based on the 81\% of reported strategies.}
\label{tbl:strategies-or}
\centering
\begin{tabularx}{0.8\columnwidth}{ X c }
\toprule
\bfseries Solution & \bfseries Reported uses
\\ \midrule
\yolo & 23\%\\
\sift & 19\%\\
\surf & 15\%\\
OpenCV Caffe & 12\%\\
Tensorflow & 12\%\\ \bottomrule
\end{tabularx}
\end{table}
It is important to point out, however, that object recognition in \athome is not only about correctly labeling instances of previously trained objects.
Recognizing an object shape and orientation is, in many cases, fundamental for grasping, let alone stacking and storing.
Equally important are color, size, and relative position since
one can always refer to an object by description.
Therefore, robots must be able to identify features of untrained objects and correctly categorize them by likeliness.
Furthermore, since 2015 robots can be requested to describe objects they have never seen before.
Unfortunately, the strategies used to address the aforementioned challenges are not documented in the \tdps.
\subsection{People Detection \& Recognition}
\label{sec:strategies-pr}
People detection, recognition, and tracking are closely related but, at the same time, their approaches differ broadly.
In \athome, people detection and recognition means
localizing a relatively static target, while tracking people
involves a moving target (see~\Cref{sec:strategies-pt}).
This section relates only to detection and recognition
of people using exclusively visual information.
Regarding people detection, combining face and skeleton detection is the most popular approach since it reduces false positives and considers only people within range.
In addition, hybrid techniques like combining 3D object recognition with face detection (e.g. OpenFace), or analysis of thermal images~\Citep{Iocchi2015,uchile2015,tue2017} are also being used.
Focusing exclusively in face detection, all reported strategies are also recognition-capable, from which the more popular are
Openface~\Citep{tue2017},
Viola-Jones algorithm~\Citep{golem2017},
and
Haar-based algorithms~\Citep{happymini2017}.
\begin{table}[t]
\caption{Adopted software solutions for face detection and recognition. Presented results are based on the 67\% of available \tdps.}
\label{tbl:strategies-or}
\centering
\begin{tabularx}{0.8\columnwidth}{ X c }
\toprule
\bfseries Solution & \bfseries Reported uses
\\ \midrule
Openface & 23\%\\
Haar-based algorithms & 14\%\\
Viola-Jones algorithm & 14\%\\
Caffe & ~9\%\\
Microsoft Face & ~9\%\\
OpenCV & ~9\%\\ \bottomrule
\end{tabularx}
\end{table}
On the other hand, all tests requiring to recognize a specific person either provide a description (see below), or provide the means to the robot to memorize the person's features and name.
Typically, getting to know a person for later recognition involves a facial recognition.
The solutions used by teams to achieve this are as diverse as the teams themselves.
Yet, some worth mentioning include
\hog descriptors with \svm classifiers~\Citep{alle2017},
strands perception people~\Citep{tobi2017},
Viola-Jones and eigenfaces~\Citep{golem2017},
Siamese \cnn{}s~\Citep{aupair2017},
and
Haar Cascades with either EigenFaces~\Citep{pumas2017} or \dnn~\Citep{happymini2017}.
Nonetheless, sometimes texture and color segmentation are also used as backup information.
It is, nevertheless, important to mention that cloud services are gaining popularity because of their robustness and ease of implementation.
However, they are also often unreliable due to connectivity problems so many teams prefer their own offline solutions.
When a description is provided, it may include height, gender\footnote{To avoid gambling, incorrect labeling is penalized, for which acceptable solutions are male, female and unknown.}, age\footnote{For simplicity, age estimation in \athome is fuzzy and maps to only three categories, child, adult, and elder.}, pose, relative position, and clothing.
To overcome these challenges, most teams rely on \dnn{}-based libraries and cloud services~\Citep{hibikinodspl2017,homer2017}.
Finally, it should be pointed out that some tasks may consist in requesting the robot to provide an accurate description of a person at the specified location.
\subsection{People Tracking}
\label{sec:strategies-pt}
People tracking is directly linked to two abilities: following and guiding people.
This sets another fundamental difference with object recognition: the robot can't see the face of the people being tracked.
Nonetheless, people tracking is always preceded by detection.
The robot has to memorize the person to be followed or guided, and indicate when that person can start moving.
For Nagoya 2017, many teams relied on the same approaches that were implemented back in 2006, like leg detection using the robot's \laser, and color segmentation with a probabilistic tracker~\Citep{Matamoros2018,Hans2008}.
Today, with better sensors and more powerful computers available, these methods have been improved and combined with other techniques.
Examples of these approaches include
leg-like clusterization with \svm and \ukf~\Citep{aisltut2017},
2D/3D leg detector fusing ellipse-fitting and 3D-windows obtained using a \dnn~\Citep{tritonsdspl2017},
and
\laser-based leg-tracker with an upper-body detector~\Citep{walkingmachine2017}.
Other approaches take advantage of the skeleton detection offered by RGB-D cameras~\Citep{unsw2017}.
However, this latter approach often produces unsatisfactory results due to vibrating sensors and people not fitting in the sensor's FoV.
On the other hand, interesting approaches involve
contour matching~\Citep{tue2017},
head tracking using a monocular camera and a Single Shot MultiBox Detector \cnn~\Citep{rtlions2017},
and
the \tld algorithm that learns from the difference between the estimated position of the target and the position detected in the analyzed frame~\Citep{airobots2017,tinker2017}.
Unfortunately, people tracking is one of the least reported functionalities in \tdps, with a frequency of 34\%.
Finally, at this point in time, there is no important difference between following and guiding.
For guiding, teams resort on the same techniques used for following, but using information from sensors looking backwards.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,562 |
Q: How to rename multiple files in several folders? I'd like to rename all files in several folders with filename containing '*file*' by '*doc*'. I've tried
find . -name "*file*" -exec mv {} `echo {} | sed "s/file/doc/"` \;
but got an error (see below).
~$ ls
my_file_1.txt my_file_2.txt my_file_3.txt
~$ find . -name "*file*"
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./my_file_2.txt
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my_doc_1.txt
~$ find . -name "*file*" -exec echo {} \;
./my_file_1.txt
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~$ find . -name "*file*" -exec mv {} `echo {} | sed "s/file/doc/"` \;
mv: './my_file_1.txt' and './my_file_1.txt' are the same file
mv: './my_file_3.txt' and './my_file_3.txt' are the same file
mv: './my_file_2.txt' and './my_file_2.txt' are the same file
Many thanks for your help!
A: There are a thousand ways to do it, I'd do it with Perl, something like this will work:
find files -type f -name "file*" | perl -ne 'chomp; $f=$_; $f=~s/\/file/\/doc/; `mv $_ $f`;'
*
*-ne process as inline script for each line input
*chomp clean a newline
*$f is new filename, same as old filename
*s/\/file/\/doc/ replace "/file" with "/doc" in the new filename
*mv $_ $f rename the file by running an OS command with back ticks
A: The problem with your solution is that the echo {} | sed "s/file/doc/" is executed before the rest of the find command. I tried to make a command demonstrating this:
find . -name "." -exec date \; -exec echo `date; sleep 5` \;
When the date commands aare executed from left to right, the dates would be equal. However the second date and the sleep are executed before find starts the first date.
Result:
Wed Aug 25 22:33:43 XXX 2021
Wed Aug 25 22:33:38 XXX 2021
The following solution is using print0 and xargs -0 for filenames with newlines. xargs will echo the mv command with two additional slashes.
The slashes will be found by the sed command, changing the target filename.
The result of sed is parsed by a new bash shell.
find . -name "*file1*" -print0 2>/dev/null |
xargs -0 -I {} echo mv '"{}"' //'"{}"' |
sed -r 's#//(.*)file(.*)#\1doc\2#' |
bash
A: See if you have rename command. If it is perl based:
# -n is for testing, remove it for actual renaming
find -name '*file*' -exec rename -n 's/file/doc/' {} +
If it is not perl based, see if this works:
# remove --no-act --verbose for actual renaming
find -name '*file*' -exec rename --no-act --verbose 'file' 'doc' {} +
| {
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} | 5,206 |
Q: message_handler не работает в python начал писать телеграм бота на пайтоне и столкнулся с проблемой.
Я установил pytelegrambotapi, но интерпретатор ругается, что не находит такого атрибута как "message_handler". По некоторым советам из интернета я удалил пакет telebot, но после этого модуль telebot перестал работать. Получается замкнутый круг.
Помогите пожалуйста или дайте совет
import telebot
bot = telebot.TeleBot("YOUR_BOT_TOKEN")
@bot.message_handler(commands=['start', 'help'])
def send_welcome(message):
bot.reply_to(message, "Howdy, how are you doing?")
@bot.message_handler(func=lambda message: True)
def echo_all(message):
bot.reply_to(message, message.text)
bot.infinity_polling()
| {
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} | 7,237 |
{"url":"https:\/\/cs.stackexchange.com\/questions\/111274\/regular-expression-with-kleene-closure?noredirect=1","text":"# regular expression with kleene closure [duplicate]\n\n\u2022 Essentially $1^*$ means \"any word consisting of any (finite) number of ones\". If R is a RE and L(R) is its language, then R^* is \"any word consisting of the concatenation of any (finite) number of words in L(R)\". For example if $R = a+b$ then $R^*$ contains the word \"abba\" as it is the concatenation of \"a\", \"b\", \"b\", and \"a\", all of which are in $L(R) = \\{a, b\\}$. Going back to your case, you have $R = 1^*$ so $R^* = (1^*)^*$ is \"any word obtained by concatenating any (finite) number of words containing only ones\"... a convoluted way to say \"any word containing only ones\". \u2013\u00a0Steven Jun 29 at 0:21","date":"2019-11-22 05:39:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.4068889915943146, \"perplexity\": 497.7449938711978}, \"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\/1573496671239.99\/warc\/CC-MAIN-20191122042047-20191122070047-00007.warc.gz\"}"} | null | null |
Copeland, Nicholas April 25, 1835.
This letter is part of the collection entitled: Texas History Collection and was provided by Other to The Portal to Texas History, a digital repository hosted by the UNT Libraries. It has been viewed 438 times . More information about this letter can be viewed below.
1835 letter of Col. Nicholas Copeland to his son-in-law Martin Bridgman of Arkansas, enticing him to move to Texas. The letter discusses the price of land and cattle, as well as the profitability of crops such as cotton and corn. Copeland adds a note for Harry Currin, a free African-American, stating that Texas is a safe place to settle. His land grant (settlement & fortification) described in the letter was 25 miles west of the Trinity River just before crossing the Navasota River. This letter was written at Robbins' Ferry on the Old San Antonio Road at the crossing of the Trinity River (letters went east from there to be carried & put in the US mail system).
from the personal collection of Sandra Burns of Dallas.
Feb. 15, 2007, 6:30 p.m.
Sept. 19, 2007, 8:46 a.m.
Copeland, Nicholas. [Col. Nicholas Copeland letter to Martin Bridgman, April 25, 1835], letter, April 25, 1835; (https://texashistory.unt.edu/ark:/67531/metapth21088/: accessed April 19, 2019), University of North Texas Libraries, The Portal to Texas History, https://texashistory.unt.edu; crediting Other. | {
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// Copyright (c) 2016-2020 The Bitcoin Core developers
// Distributed under the MIT software license, see the accompanying
// file COPYING or http://www.opensource.org/licenses/mit-license.php.
#ifndef SYSCOIN_WALLET_TEST_WALLET_TEST_FIXTURE_H
#define SYSCOIN_WALLET_TEST_WALLET_TEST_FIXTURE_H
#include <test/util/setup_common.h>
#include <interfaces/chain.h>
#include <interfaces/wallet.h>
#include <node/context.h>
#include <util/check.h>
#include <wallet/wallet.h>
#include <memory>
/** Testing setup and teardown for wallet.
*/
struct WalletTestingSetup : public TestingSetup {
explicit WalletTestingSetup(const std::string& chainName = CBaseChainParams::MAIN);
~WalletTestingSetup();
std::unique_ptr<interfaces::WalletClient> m_wallet_client = interfaces::MakeWalletClient(*m_node.chain, *Assert(m_node.args));
CWallet m_wallet;
std::unique_ptr<interfaces::Handler> m_chain_notifications_handler;
};
#endif // SYSCOIN_WALLET_TEST_WALLET_TEST_FIXTURE_H
| {
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Eclairage Ambiance Led. plafond tendu lumi re led ambiance clairage youtube. les productions multi show clairage d 39 ambiance aux led youtube. 5m mood lighting led under bed settee bedroom ideas living room lights strip ebay. long cabinet pulls kitchen contemporary with bamboo contemporary lighting flush. the difference between dimmable and non dimmable led fixtures modern place. f10 5 series ambient lighting interior ambient lighting and exterior lights page 2. clairage led rgb plafond youtube. nautical design gloster ambient teak led lantern. mkv 08 interior light swap. modern indirect ceiling lighting ideas for a pleasant ambience home decor trends home decor. [Yarlodki.com]. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,390 |
A young woman fights to save her salvage business from a murderous competitor.
"I once complained to Mr. DeMille that he never used me in his pictures. This was after I had become a columnist. 'I didn't think you wanted to act any longer,' he said.
"So he gave me a small part in Reap the Wild Wind. I had to faint in it. I had never done that either off stage or on. I took lessons from Ethel Barrymore and Laura Hope Crews. I was fainting all over Hollywood until the day I was to do the scene.
Reap the Wild Wind cost $2,038,210.91 and grossed $4,000,000. | {
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} | 9,546 |
Q: How to evaluate limit of a user-defined function Limit[(k/(k - 1)), k -> Infinity]
Gives correctly:
1
But,
L[k_Integer] := (k/(k - 1))
Limit[L[k], k -> Infinity]
leaves the limit unevaluated.
How can I tell Mathematica to evaluate this limit?
A: The input restriction means it is basically not amenable to analysis by Limit or even DiscreteLimit. If instead it is defined piecewise then the sequence limit can be found.
lL[k_] :=
Piecewise[{{(k/(k - 1)), Element[k, Integers]}, {Indeterminate,
True}}]
DiscreteLimit[lL[k], k -> Infinity]
(* Out[113]= 1 *)
| {
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} | 5,884 |
\subsection{Success Probability Analysis}
We explore the success of the scheme in terms of the free parameter $P$, the raw probability the photon would be found in the right half of the setup, determined by the setting of HWP1.
The raw conditional probabilities of detection in each of the detectors given Bob was Blocking (B) and Not Blocking (NB), for the infinite inner cycle version of the protocol, are presented in Table~\ref{tab:rawprobs}.
\begin{table}[h!]
\begin{center}
\caption{Raw conditional probabilities}
\label{tab:rawprobs}
\begin{tabular}{l|c|c}
\textbf{} & \textbf{Blocking} & \textbf{Not Blocking}\\
\hline
$D_0$ & $(1-P)$ & $(1-P)$\\
$D_1$ & $P$ & $0$\\
$D_3$ (lost) & $0$ & $P$\\
\end{tabular}
\end{center}
\end{table}
Consider for now the limit in which the probability of losing the photon to Bob's blocking apparatus vanishes.
Since Bob's strategy for not blocking only succeeds with probability $P($$D_0$$|NB) = (1-P)$, Alice must perform these tests many times to get a successfully postselected event, so the raw probability that Bob was Blocking is greater than a half $P_B\geq 1/2$. This leads to apparent communication in the postselected data since the conditional probability of Blocking given detection events at $D_0$ decreases.
We assume that on average Bob successfully sends as many zeros as he does ones. We have $\av{X}=1/2$, leading to the normalized postselected probability for not blocking $\tilde P_B = P_B/N = 1/2$, where the normalization factor is $N = P_B+(1-P_B)(1-P)$.
We then find that the probabilities of the postselected detection events are given by those in Table~\ref{tab:postselectfinal}, and the total probability of loss is given by $P_L = P/(2-P)$. This corresponds to a postselected probability of correct outcome of $P_c =(1+P)/2$.
\begin{table}[h!]
\begin{center}
\caption{Normalized Postselected Probabilities}
\label{tab:postselectfinal}
\begin{tabular}{l|c|c}
\textbf{} & \textbf{Blocking} & \textbf{Not Blocking}\\
\hline
$D_0$ & $(1-P)/2$ & $1/2$\\
$D_1$ & $P/2$ & $0$\\
\end{tabular}
\end{center}
\end{table}
We see that in the limit $P\rightarrow 1$ the protocol becomes deterministic, however the probability of postselection vanishes.
In Figure~\ref{fig:CounterFactualData} we plot the overall probability of successful postselected outcome and postselection probability, and for different values of the probability $P$. Notably, for $P=1/2$ postselection succeeds with $2/3$ probability, and is correct with $3/4$ probability. Increasing $P$ to $2/3$, the likelihood of successful postselection drops to $1/2$ whilst the probability of being correct increases to $5/6$.
Finally, let's illustrate our findings using a somewhat amusing scenario. Imagine an outcome-obsessed lab director in charge of this experiment, who is quite happy firing Alice and Bob if a single run of the experiment fails, replacing them with a fresh pair of experimentalists, to start all over, also nicknamed Alice and Bob. The task for Alice and Bob is to communicate a 10-bit message, one bit at a time. Assume the experiment is set up such that the chance of any given run failing is one half. Therefore, in order to successfully communicate a 10-bit message, the lab director has to, on average, go through just over a thousand pairs of experimentalists---which the director in fact secretly enjoys. Each new pair of experimentalists is provided with a new message. Eventually, a lucky Alice and Bob manage to communicate their message. (Bit accuracy will be above eighty percent on average.) Now the question for the successful pair is: Has any of Alice's photons been to Bob? The answer, as we have shown, is an emphatic no.
In summary, we have shown that, given post-selection, sending a message without exchanging any physical particles is allowed by the laws of physics. What carries information, however, remains an open question.
While preparing this manuscript, the posting of a relevant manuscript was brought to our attention, arXiv:1805.10634 \cite{Aharonov2018}. We will comment on its approach and results, which are distinct from ours, in due course.
\begin{acknowledgments}
{\it Acknowledgments.---}
HS thanks Li Yu-Huai for useful feedback on the basic concept, received in autumn 2016. The authors thank David Lowndes for the components used during the experiment, Henry Webb and Holly Caskie for help with 3D drawing of the experimental setup, and Huxley Sessa for help with experimental prototyping. This work was supported by the UK's Engineering and Physical Sciences Research Council (EP/PS10269/1 and EP/L024020/1).
\end{acknowledgments}
\begin{figure}[h]
\centering
\includegraphics[width=\linewidth]{CounterFactualModel.pdf}
\caption{\label{fig:CounterFactualData}The probability of postselected correct outcome, $P_c$ (Blue), postselected correct outcome for when a photon is detected at detector $D_0$ (green), and total postselection success probability (orange), for a given bit sent from Bob to Alice in the infinite-cycle case of the protocol ---plotted against $P$, the probability of the photon entering inner interferometer.}
\end{figure}
\bibliographystyle{unsrt}
| {
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} | 4,863 |
The Pay It Forward Challenge
Sophia Asare-Nkansah multiplies her $1,000, making a difference in the lives of several people in her Naperville, Illinois, community. She starts with Roshanda, a single mother expecting her third child. Roshanda, who has sickle cell anemia, has been forced from her apartment by a fire. Sophia buys her necessities for the family and some gifts for the new baby. An anonymous donor gives another $1,000 for Roshanda!
The next recipient is Guardian Angel Community Services, which provides shelter for victims of domestic abuse. Sophia buys toiletries and gas cards for the residents and helps one woman visit her family for the holidays. Body Energetics spa donates free massages and facials to the residents, and a local car dealership agrees to sell Sophia a used car for one of the women for only $50!
Through friends, Sophia meets Katrina survivors Jill and Phil, who relocated to Illinois, leaving many of their belongings in Louisiana. The couple is disabled and living on a fixed income, so a trip to Wal-Mart for new clothes is greatly appreciated. Finally, Sophia helps Erica, a single mother who is unable to work due to an illness. Sophia helps Erica with her rent and buys some things for her young daughter, including cold medicine and a winter coat.
"To work with all these different strangers that I have never met really changed my life," Sophia says. "It has given me a whole perspective of how life can be and how life is. I would love to do it again and again and again."
Subscribe to the live your best life newsletter Sign up for the oprah.com live your best life newsletter Get more stories like this delivered to your inbox Get updates on your favorite shows, the latest from Oprah's world and more! Get more inspiration like this delivered to your inbox
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The Jockey to Look Out for at the Kentucky Derby
Oprah: The Two Questions You Should Ask Yourself Each Day
17 Quotes That Point Your Life Forward
The Oprah Challenge: 3 Questions That Will Help You Find Your True Calling
Thought for Today - Renewal
Thought for Today - Fatherhood
Thought for Today - Motherhood
9 Almost Impossible Dreams You Can Never Give Up On | {
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} | 4,127 |
Wearing boots with dresses is the hottest trend in fashion right now. And while this trend may have been started on high fashion runways, it has taken country girl fashion by storm. In fact, pairing boots with dresses is one of our favorite looks because you can't really mess it up. And who doesn't want fashion you can't mess up?
One advantage of this trend is that it doesn't matter what type of boot you choose. You can wear a tall boot, a short boot or even a cowboy boot (our personal favorite) for this look. Simply pull out your tights or leggings and rock the boots and dresses trend through the fall and winter.
1. Watch the length of your hemline.
You want to see a few inches of your legs between the hem of your dress and the top of your boot. This is more flattering and sophisticated.
Even with a cute boot sock peeking out of the top, she has plenty of space between the top of her Justin boot and the hem of her dress.
2. Make sure your boot style and your dress style match.
This is pretty much a no-brainer, but you don't want to wear that edgy moto boot with a formal dress.
3. Remember that a little color goes a long way.
If you wear a statement boot in a bold color, keep your dress or skirt toned down. Let those boots be the star of your outfit. This turquoise boot by Justin is sure to be a show-stopper!
4. Bare legs or tights?
This question is nothing more than personal preference and good old common sense. If the weather permits, a bare leg is perfect. But if it's chilly, a tight may be your best option.
This neutral Justin boot looks great with a bare leg, but she could easily add tights if it is too cold.
5. Wearing boots with dresses is more versatile that you think.
You can make this look as casual or as dressy as you want. We love that you can wear the look from a day at work straight to an evening out on the town.
Boots with dresses is one of the hottest trends in fashion. Find the perfect pair of Justin boots to wear with your skirts and dresses this fall. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,640 |
const state = {
0: 'Invitation pending',
1: 'Accepted',
2: 'Declined',
3: 'Left group',
}
export default state;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,515 |
Don't miss any of Janet Dailey's bestsellers:
Always with Love
Texas Tough
Texas True
Bannon Brothers: Triumph
Bannon Brothers: Honor
Bannon Brothers: Trust
American Destiny
American Dreams
Because of You
Calder Storm
Can't Say Goodbye
Close to You
Crazy in Love
Dance with Me
Everything
Forever
Green Calder Grass
Lone Calder Star
Lover Man
Masquerade
Something More
Stealing Kisses
Tangled Vines
Texas Kiss
That Loving Feeling
When You Kiss Me
Yes, I Do
You're Still the One
Heiress
Rivals
Calder Promise
Shifting Calder Wind
Going My Way
It Takes Two
Happily Ever After
Try to Resist Me
Bring the Ring
Ranch Dressing
With This Kiss
Wearing White
Man of Mine
TEXAS TALL
JANET DAILEY
KENSINGTON BOOKS
http://www.kensingtonbooks.com
All copyrighted material within is Attributor Protected.
KENSINGTON BOOKS are published by
Kensington Publishing Corp.
119 West 40th Street
New York, NY 10018
Copyright © 2016 by Janet Dailey
All rights reserved. No part of this book may be reproduced in any form or by any means without the prior written consent of the Publisher, excepting brief quotes used in reviews.
Kensington and the K logo Reg. U.S. Pat. & TM Off.
Library of Congress Card Catalogue Number: 2015933898
ISBN: 978-0-7582-9401-2
ISBN-10: 0-7582-9401-8
First Kensington Hardcover Edition: September 2016
eISBN-13: 978-0-7582-9402-9
eISBN-10: 0-7582-9402-6
First Kensington Electronic Edition: September 2016
Table of Contents
Don't miss any of Janet Dailey's bestsellers:
Title Page
Copyright Page
Dedication
CHAPTER 1
CHAPTER 2
CHAPTER 3
CHAPTER 4
CHAPTER 5
CHAPTER 6
CHAPTER 7
CHAPTER 8
CHAPTER 9
CHAPTER 10
CHAPTER 11
CHAPTER 12
CHAPTER 13
CHAPTER 14
CHAPTER 15
CHAPTER 16
CHAPTER 17
CHAPTER 18
CHAPTER 19
CHAPTER 20
EPILOGUE I
EPILOGUE II
_Christmas On My Mind_ Teaser
About the Author
With gratitude to Elizabeth Lane, without whom this book would not have been written.
CHAPTER 1
The first November chill had painted the Rimrock Ranch with a golden brush. From the glint of sunrise on the high escarpment to the sweep of yellow grass across the plain, from the fading willows along the creek to the bursts of saffron, where the cottonwoods grew, the land was the color of Spanish gold—the gold that, legend whispered, lay hidden in a canyon on the border of the ranch.
By day, flocks of migrating birds swept southward. Meadowlarks settled on the pastures and brightened the air with their calls. Ducks, geese, and sandhill cranes traced elegant V formations across the autumn sky. By night the stars were diamond sharp, the harvest moon ripe and mellow above the horizon.
A man with an easy mind would have savored the season's rich beauty. But Will Tyler's mind was far from easy. Wherever he looked, what he saw was not so much beauty as trouble.
The grass was weak, still recovering from the summer drought and the raging wildfire that had charred the lower pastures. Even if the winter turned out to be mild, would there be enough to feed the calves and breeding stock he needed to sustain the ranch? Or, with finances strained to the breaking point, would he be forced to buy more hay at a cutthroat price for the cattle he'd kept after selling off most of his herd?
Will had counted on the auction of Sky Fletcher's superbly trained colts to shore up the ranch's funds. But the drought-impoverished Texas ranchers, who would have scrambled to buy the lot, were, like Will, too cash-strapped to pay. Only a few young horses had been sold, and those at cut-rate prices.
Worries gnawed at Will as he drove his twelve-year-old daughter, Erin, home to the ranch for the weekend. While school was in session, the girl lived in Blanco Springs with her mother, Will's ex-wife, Tori. But her weekends and summers were Will's. These days Erin, who loved the ranch, was the only bright spot in his life.
Will drove his pickup carefully, with the headlights on high beam. It was late, almost eleven, and the narrow, two-lane road from town was unfenced here, with a meager shoulder and steep barrow pits on either side. Deer, coyotes, even cattle and horses, had been known to wander onto the asphalt at night and cause serious accidents. He would've brought Erin home before dark, but she'd gone to a friend's house for a birthday party. She'd phoned him when it was over and he'd picked her up there.
Erin had turned on the pickup's radio. The local country music station added a twanging underbeat to the thrum of the truck's engine.
"So what's new with you?" Will asked, breaking the relaxed silence between them. "Anything happening at school? Got a boyfriend yet?"
The boyfriend question was an ongoing joke between them. At two months shy of thirteen, Erin was more interested in horses than in boys. And Will would've run off any boy who got within a dozen yards of his daughter.
"Not yet. But Mom might have one. She's got a date tonight."
A knot jerked tight in the pit of Will's stomach. He and Tori had been divorced for eight years, but some part of him still claimed her. The marriage may have been a disaster, but they'd made Erin together—the best thing they'd ever done.
"You don't say?" He feigned a casualness he didn't feel. "Who's the lucky man?"
"His name's Drew Middleton. He's the new principal at the high school."
The knot pulled tighter. "What's he like?"
"He's okay. Seems nice enough."
"Think it's serious?"
"Maybe. Mom hasn't dated anybody in a long time."
"Uh-huh." Will swallowed hard. If Tori had found a man she wanted to date, that was her business. She deserved to be happy. But damn it all, he didn't have to like it.
With Erin at the ranch, Tori would have the house to herself for two nights. The thought of what could be going on there was enough to make Will grind his teeth. Drew Middleton. The name tasted like sawdust in his mouth. He'd never met the man, but he already wanted to punch him.
Will had paid scant attention to the radio, but when the signal for an emergency news bulletin came on, it caught his attention. "Turn that up," he told Erin.
The voice came through a crackle of static: "The sheriff 's office is asking for your help in tracking down a man who held up the convenience store in Blanco Springs, took cash, and shot a clerk. The suspect fled on a motorcycle going north. He's described as a white male in his thirties, wearing a black leather jacket and a black motorcycle helmet. He is armed and dangerous. If you see him, call nine-one-one."
"Hey!" Erin exclaimed as the music came back on. "What if that guy's out here, on this road? What would we do if we saw him?"
"I'd keep driving and let you make the call," Will said. "Somebody like that, with a gun, I wouldn't take a chance on playing hero, especially with you along."
"But what if—" Her words ended in a yelp as the pickup's right front wheel slammed into something solid and stopped dead. Only her seat belt kept her from flying into the dashboard.
Will switched off the key, cursing under his breath. He wasn't sure what he'd hit. It hadn't felt like an animal, but just in case, he pulled his loaded .38 Smith and Wesson revolver from under the driver's seat. If he'd struck some unlucky creature, he'd want to put it out of its misery—or defend himself if it had any fight left.
He found a flashlight in the console. "Stay put," he told Erin as he opened his door. "Whatever happens, don't get out."
Climbing to the ground, he closed the door behind him and turned on the flashlight. The night air was chilly through his denim jacket, the full moon veiled by drifting clouds. The distant wail of a coyote echoed across the sage flats as Will walked around to the passenger side of the truck.
The pickup had come to rest at a cockeyed angle, probably blown a tire, which he'd need to change. In the beam of the flashlight, he could see what he'd hit. It was the engine block for some kind of vehicle, most likely fallen off the back of a flatbed because the fool driver hadn't bothered to tie it down. Heavy and solid, its edges were sharp enough to puncture a tire, which was just what had happened. If he hadn't been distracted by the announcement on the radio, he might have seen it in time to stop.
Erin rolled down the window. "What is it?" she asked. "Is it an animal?"
"No, just a big, nasty chunk of metal. But I'll have to change the tire."
"Can I help? I can hold the light for you."
"No, just stay put. I'll be fine."
He'd stuck the .38 in his belt and was walking around to get the spare and the jack when he saw it—a single headlight approaching fast down the long, straight road from the direction of town, maybe half a mile away. It looked like a motorcycle, sounded like one, too.
Will turned off the flashlight and laid it on the ground. One hand drew the weapon out of his belt. "Close the window, lock the doors, and get down," he ordered Erin. He caught the flash of her frightened eyes as she obeyed. He'd probably scared her for nothing, but he couldn't take any chances.
The motorcycle was slowing down. Maybe the rider was just some Good Samaritan wanting to help. But Will couldn't lay odds on that. He might be safer inside the truck, but that could expose Erin to more danger. Right now, his daughter's safety was the only thing that mattered.
A few yards ahead of the truck, the motorcycle pulled onto the shoulder and stopped. The rider swung off his machine. He wore a black leather jacket with a dark helmet, the visor pulled down to obscure his face. His right hand held a small pistol with the look of a cheap Saturday-night special. He had to be the robber. Will waited in the shadows, gripping the .38, as the man approached and spoke.
"What the hell happened here? We were supposed to meet down by the crossroad." His whiny-pitched voice sounded vaguely Eastern, and strangely familiar. "Never mind, I got the package on the bike. Show me the money, and we're good."
Will stepped into the moonlight, his pistol leveled at the man's chest. "Hands where I can see them, mister. Now, nice and slow, drop your weapon. Then kick it over here toward me."
"Shit, you're not—" The motorcyclist froze in surprise. He dropped the gun on the ground. As he kicked it toward Will, his hand flashed. Suddenly there was a knife in it. As his arm flexed for the throw, Will pulled the trigger. The .38 roared, striking the man squarely in the chest. He toppled backward, dead by the time he hit the ground.
Will stared down at the bleeding body, cursing out loud. He'd never meant to kill the stupid jackass, but he'd had little choice, especially with Erin to protect.
"Daddy?" Erin had rolled down the window partway. Her voice sounded thin and scared. "Are you all right?"
"I'm fine, honey. Close the window and stay in the truck. Don't try to look. I'll come around." Leaving the body where it lay, he circled behind the truck to the driver's side. Once inside, he reached across the console and gathered his daughter in his arms. She clung to him, trembling.
Will felt shaken, too, when he thought about what he'd just done. His younger brother, Beau, who'd been an army sniper in Iraq, had never revealed how many kills he'd made. But for Will, this was a first. He'd never taken a human life before. Now, even though he'd killed a criminal in self-defense, the thought sickened him.
Erin pulled away as he released her. She'd be all right, Will told himself. She was strong, like Tori. "You need to call the sheriff, Daddy," she said.
"I know. But first I'm going to call your mother."
Will reached for the cell phone in his pocket. As an afterthought, he climbed out of the truck again and closed the door. His conversation with Tori could easily get emotional. It might be better not to have Erin listening.
His legs felt unsteady. Leaning against the side of the cab, he scrolled to Tori's number and pressed the call button. The phone rang, once, then again. Maybe she'd turned it off, the better to enjoy her new boyfriend. But no, it wouldn't be like Tori to do that, not even on a date. She had clients who needed her. More important, she had a daughter.
Will was waiting for his ex-wife's voice message to come on when she picked up. "Will?" He could sense her tension. "What is it? Is Erin all right?"
"Erin's fine," Will said. "But I need you to come and get her. There was an . . . incident on the way to the ranch. The truck's stuck on the road with a blown tire, and I can't leave."
"An incident, you say? What happened, Will? What are you not telling me?"
"I'll explain later. Erin said you had a date. Hell, bring him along if you want. I don't care. Whatever you're doing, just drop it and get here. Now."
"I'm on my way. Tell Erin I'm coming." She ended the call without saying good-bye.
Will waited a couple of minutes, then called 911. The night dispatcher who answered his call was a woman whose voice he recognized.
"Carly, this is Will Tyler," he said. "Tell the sheriff he can stop looking for that convenience store robber on the motorcycle. I just shot him."
"Is he dead?"
"As a doornail. I need somebody to come, pronto."
"Where are you?" There was an odd note to her voice.
"About ten miles out of town, on the road to my ranch. My pickup's got a blown tire."
"I hear you. Stay in your truck and don't touch anything. Somebody will be right there."
She hesitated, as if weighing her next words. "There's something you need to know, Will. Whoever you shot, it wasn't the robber. That man was picked up a few minutes ago, headed for the freeway on his motorcycle. The cash was on him, along with the gun and some cigarettes he took. They're bringing him in now."
* * *
"I can drive you if you want, Tori." Drew Middleton straightened his tie. Tall and slender, with hazel eyes and light brown hair, he was appealing enough to make Tori wonder if there was a chance for something real between them.
"Not a good idea." She dropped the cell phone back in her purse and reached for the door latch of Drew's Honda Accord. "You've never met my ex, but I can guarantee you that tonight wouldn't be the best time."
"Is everything all right?"
"Erin's all right, evidently. Whatever else is going on, that's Will's problem."
"Hang on. I can at least open your door and walk you to the porch."
Tori waited while he climbed out of the car and came around to the passenger side. Drew had taken her to a seafood dinner and a show in Lubbock. They'd made it back to her driveway for a few minutes of pleasurable front-seat necking. Just as she'd begun to wonder whether she should invite him in, her phone had rung.
Stiletto heels clicking, she strode up the front walk and turned for a quick kiss on the porch. "Gotta go," she said. "Thanks for a very nice evening."
"I'll call you, okay?"
"Sure." She let herself in and raced to grab her fleece-lined ranch jacket out of the hall closet. No time to change out of her dressy black sheath. If Erin needed her, she would get there as fast as she could.
Minutes later, she was in her station wagon, roaring down Main Street, past the Shop Mart, the drugstore, and the Blue Coyote bar, on her way out of town.
* * *
Standing outside the pickup, Will gazed down the road at the distant headlights. He hoped to hell it was Tori. He wanted Erin gone before the law showed up. The girl had witnessed next to nothing, and she didn't need to be upset by the ugliness of a crime scene investigation. If the sheriff needed to question her, he could do it tomorrow, at home.
The dispatcher had warned him against touching anything. But Will had taken the liberty of dragging an old blanket out of the truck's backseat and laying it over the dead man's corpse. If he'd just contaminated the evidence, too bad. He didn't need Erin seeing the body, or Tori, either.
He'd thought about phoning Beau, or even Sky, at the ranch, then changed his mind. His brothers had enough on their plates without his adding to the pile. This was his own mess. He would clean it up by himself.
But one question continued to chew on him. If the bastard he'd killed wasn't the robber, who the devil was he?
The headlights were coming closer—a low vehicle going like a bat out of hell. That would be Tori, all right. The truck's hazard lights were blinking. She should be able to see them. But just to make sure, he turned on the flashlight and waved it. Seconds later, the station wagon screeched to a gravel-spitting halt behind the truck.
Tori piled out of the driver's side. Under her open canvas coat, Will could see that she was dressed to kill—tight black dress, dangling earrings, honey-blond hair flowing loose, the way he'd always liked it. And she was wearing those high heels that made her long legs look extra sexy.
All for another man.
But he couldn't let that get to him now. What mattered was that she'd come as soon as she could.
"What's going on? Where's Erin?" Her gaze surveyed the tilted truck. At least, from the driver's side of the cab, she couldn't see the body.
"We hit something and punctured a tire. Erin can tell you more in the car." He opened the driver's-side door of the truck and swung his daughter to the ground. "Just get her out of here, Tori. You can take her to the ranch or home with you. I want her gone by the time the sheriff gets here."
"I'll take her home." Tori caught sight of the motorcycle parked ahead of the truck. "Whose bike is that? Why's the sheriff coming?"
"I'd just gotten out of the truck when that motorcycle pulled up. The man had a pistol and a knife. I had to shoot him."
"Thank God you had a gun." Wide-eyed, Tori clutched her daughter close. After taking a moment to compose herself, she spoke. "Are you in trouble, Will? Do you need me to stay as your lawyer?"
"I'll be fine. There's no way it could've been anything but self-defense." Will glanced down the road and saw the flicker of approaching red lights. "Just take Erin and go. She doesn't need to be part of this. I'll call you later."
Tori needed no more urging. She raced with Erin to her car, backed away from the truck and turned for home. Will stood watching her taillights as she drove past the oncoming sheriff 's vehicle and the ambulance. No one tried to stop her.
Moments later, Blanco County sheriff Abner Sweeney pulled up in his tan SUV. A deputy rode beside him in the passenger seat. The ambulance parked behind him.
Sweeney, a short, pugnacious redhead whose manner had become even cockier since winning the recent election, climbed out of the vehicle. Trailed by his deputy, he stalked up to where Will stood.
"So where's the body, Tyler?" he demanded.
"Around there." Will nodded to the other side of the truck. "I covered him out of common decency, but I didn't touch anything else. I admit to shooting the man, but I fired in self-defense."
"How can I be sure of that?" Sweeny's chin jutted as he glowered up at Will's imposing height. His hand rested on the butt of his pistol.
"You'll see his gun on the ground and the knife still in his right hand," Will said. "And you won't find my prints on anything."
"Did you check his pulse to make sure he was dead?"
"Didn't have to. A thirty-eight blows a mighty big hole, especially at close range. And I know dead when I see it."
Two more deputies had come out of the ambulance with an evidence kit, a stretcher, and a folded body bag. After donning latex gloves, they peeled back the blanket to reveal the dead man sprawled in the headlights of Will's truck. His helmet was still in place. Blood from an ugly chest wound had soaked the shirt beneath his leather jacket. One deputy began taking photos of the scene with a small camera, the flash making little bursts of light. Another checked the motorcycle, pulling a packet of white powder from one of the panniers.
Sheriff Sweeney frowned at the body, then turned back to Will. "He's deceased, all right. Suppose you tell me what happened."
Will related the story to the best of his recollection. He hadn't wanted to mention that Erin was with him, but realized that it might come to light later. Better to come clean now than be caught in a lie.
"So your daughter was a witness. Where is she?" Sweeney demanded.
"I called her mother to come get her. And she wasn't really a witness. I ordered her down on the floor when the bastard showed up. She didn't see anything."
"So why would you send her away? Is there some reason you don't want me to question her?" The sheriff 's eyes narrowed, as if he suspected some dark, hidden secret.
"You have children, Sweeney. Would you put your young daughter through something like this? If you need to talk to Erin, you can do it tomorrow—with her mother present."
"I'll do that." Sweeny ruminated a moment, maybe remembering that Tori was a lawyer. Abruptly he changed his tack. "You say you'd gotten out to change a flat tire. So what were you doing with a gun?"
The little man seemed determined to prove some kind of wrongdoing. Will's nerves were screaming, but he forced himself to answer calmly. "I already told you. I'd hit something on the road, and I took the gun because I thought it might be an animal. It wasn't, but when the motorcyclist showed up with weapons, I used that gun to protect my daughter."
"And you thought the man was the robber we were after?"
"Yes, until I called the dispatcher. By then, he was already dead."
"Did you look at his face? Maybe raise that visor on his helmet?"
"I told you, I knew better than to touch him."
"Then what do you say we have a look? Maybe somebody here will recognize him." Sweeny turned toward the dead man. By now, the deputies were gathering up the evidence, preparing to bag the body and lift it onto the stretcher. One of them had already taken Will's .38.
The sheriff wasn't wearing gloves. He motioned for one of the deputies to remove the helmet.
As the visor came up and the helmet was lifted free, Will's pulse lurched. He exhaled, his breath whistling through his teeth.
The sheriff 's shoulders sagged as if he'd been gut kicked. "God and Jesus," he muttered.
There could be no mistaking the swarthy features and the shaved head with its black Maori tattoos. The man Will had shot dead was Nick Tomescu, the brother of Stella Rawlins, who owned the Blue Coyote.
CHAPTER 2
Slumped on a stool in the darkened bar, Stella Rawlins crushed the butt of her last Marlboro in the overflowing ashtray. Her head ached, and her feet throbbed in their cherry-red high-heeled cowgirl boots. Beneath the black silk blouse she wore, her 38DD bra had chafed a raw line around her ribs. It was well after midnight and the Blue Coyote had been closed for an hour. But she didn't want to leave until her brother Nick showed up—and he was seriously overdue.
Worry chewed at her. What if something had gone wrong? What if he'd screwed up and gotten himself arrested?
True, Nicky wasn't the smartest rooster in the coop. But even he should've been able to carry out the simple errand she'd sent him on—drive to the spot where the road cut off to the burned-out Prescott place, look for a dark blue pickup truck, give the driver the package, take the money, and bring it back to her at the bar. It was a no-brainer. So what could've happened to him?
She ran a nervous hand through her dyed red hair. If something had gone wrong, Stella knew she'd blame herself. She'd looked after her younger half brother since he was a toddler. While their pretty, alcoholic mother had flitted from man to man, Stella had always been there for him. Last year, when he'd fled New Jersey after informing on the Romanian mob to beat a drug charge, she'd given him shelter here in Blanco Springs and hired him as her bartender and bouncer. Surprisingly, he'd been good at his job.
Maybe she shouldn't have risked him tonight. Nicky had never been quite right in the head. Behind his tough biker façade was a shy, almost childlike man, who became flustered if things didn't go as expected. She didn't dare trust him with anything more complicated than running a few drugs, maybe not even that.
What she needed was a new ally who could think on his feet, somebody she could count on to follow her orders while she kept her own hands clean. But such people tended not to last. Former sheriff Hoyt Axelrod, Slade Haskell, and Sky Fletcher's young cousin, Lute, were all dead, tripped up by their own failings. The last and smartest candidate for the job, Lute's sister, Marie, had betrayed her, killed the hit man Stella had hired to take her out, and vanished without a trace. Good riddance, Stella thought. Still, it made her nervous, knowing the woman was out there somewhere, itching for revenge.
Finding the right person would take time. And right now, Nicky, her only living kin, was all she had.
A light rap on the bar's front door broke into her musings. For a split second she hoped it might be Nicky. But she would've heard his bike as he rode up, and he would've come in the back. This was somebody else.
"We're closed," she shouted, not bothering to get up.
"Stella, it's Abner." The familiar, nasal voice came through the thin wooden door. "I saw your car outside. Let me in, I need to talk to you."
Stella got up to unlock the door. The sheriff was a friend. They'd done each other a few favors, but she didn't own him like she'd owned Hoyt Axelrod. Abner valued his job too much to break the law by taking bribes. And, although Stella hadn't lost her powers of seduction, Abner was faithful to his wife, a dumpy woman who seemed to be perpetually pregnant.
Still, if he'd arrested her brother tonight, chances were she could talk him into letting Nicky off.
Wincing with each step, she made her way to the door and opened it. Abner plodded into the bar, moving as if his feet were weighted with cement. At the table nearest the door, he stopped and pulled out a chair. "Sit down," he said. "Trust me, you don't want to hear this standing up."
Stella closed the door and took a seat. Talons of cold dread clutched her heart. Even before she heard the news, Abner's grave expression intimated what it would be. But when the words came, she still wasn't prepared to hear them.
"Your brother's dead, Stella," he said. "We got the call about eleven-thirty and found his body, along with his bike, about ten miles up the north road."
She went cold. Nicky was the one person on this miserable earth she truly cared about. She tried to tell herself that Abner's news was a mistake. But she knew better. Now the only thing she could do was extricate herself from the mess. Even as grief and shock slammed her, Stella's survival instincts kicked in. As far as the law was concerned, she knew nothing about her brother's activities.
She forced herself to respond. "The north road? But what was Nicky doing out there, so late at night?"
"We found a packet of cocaine on the bike," Abner said. "I'm guessing he was on his way to make a drug deal."
"Oh, Lord, no!" Stella shook her head in mock disbelief. "I warned him to stay away from dealing drugs. If only he'd listened to me, he could still be alive." She dabbed at her eyes. "Do you know how it happened?"
She waited for Abner to collect his thoughts. Maybe the drug deal had gone bad and the customer had pulled a gun. Or maybe Nicky's bike had been hit by some fool drunken cowboy. However Nicky had died, she'd have to face the truth and deal with it. "Tell me," she said.
"He was shot," Abner said. "A thirty-eight bullet through the heart at close range. At least you can go forward knowing he didn't suffer."
Stella's jaw tightened, holding back a cry of rage. Whoever had pulled that trigger was going to pay. "Who did it?" she demanded. "Who murdered my brother?"
"It was Will Tyler."
"Will Tyler." Stella uttered the name like a curse. Of all the families in Blanco County, she hated the Tylers most. It was as if they held themselves above ordinary people, like damned royalty. And now, she had even more reason to hate them. The head of the clan had murdered her darling Nicky.
"Tyler claimed it was self-defense," the sheriff said. "According to him, his pickup blew a tire. While he was outside the truck, the motorcycle came up the road and pulled off. The rider had a helmet on, visor down, and he was packing a pistol. Tyler assumed he was the biker who'd robbed the convenience store. He drew his thirty-eight and ordered the man to drop his gun. Your brother did, but then he pulled a knife. That was when Tyler shot him. He swore he didn't know it was Nick, not till we showed up and took the helmet off."
"But Tyler did admit to shooting Nicky."
Abner nodded. "No doubt about that."
"And you believe his story?" Stella felt the anger boiling up in her. She glared at the sheriff, her eyes narrowing to catlike slits.
"No reason not to. Nick's gun was on the ground. The knife was still in his hand. And Tyler said he'd had his little girl in the truck. Protecting her would've made his trigger finger extra jumpy."
"So you haven't arrested him?"
"He hasn't been charged. There'll be an inquest. But unless some new bit of evidence turns up—"
"I see." Stella could imagine, now, what had happened. Nicky had been told to look for a dark blue pickup. On the way he'd spotted Will Tyler's dark vehicle with a flat tire and assumed it was his buyer. He'd stopped to make contact, and Tyler had drawn his pistol. When poor Nicky panicked, Tyler had killed him.
And the Tylers, every last one of them, were going to pay.
Stella's hand flashed across the table and seized the sheriff's wrist. Her red-lacquered nails dug into his flesh.
"Listen to me, Abner," she hissed. "I know you want to keep your job. You may not have broken the law, but you've skated the edge a few times, and I know enough to hurt you. I want Will Tyler prosecuted, hear? If you can't find a reason to bring him in, invent one. Plant evidence if you have to—whatever it takes. The bastard murdered my brother. He's going to pay—in blood!"
* * *
It was barely dawn when Will gave up on sleep. Gritty-eyed and restless, he dragged on his clothes, started the coffeemaker in the kitchen, and wandered out onto the front porch of the rambling stone ranch house. It was still dark, the air chilly, the clouds tinged with pewter above the rolling prairie to the east. The high escarpment, which backed the ranch on the west, lay deep in shadow, its craggy buttes and turrets still awaiting the first touch of light.
The windmill next to the barn creaked as it turned in the faint breeze. There was no other sound at this hour, not even the chorus of birdcalls that would signal the start of a new day. Everything was quiet. Too damned quiet, Will thought. He wanted to shatter the silence with the foulest curses his mouth could form. But it wouldn't help him feel any better. And it sure as hell wouldn't change what had happened last night.
His black pickup was parked next to the porch, where he'd left it, with the ruined tire in the bed and the spare on the right front wheel. By the time he'd finished being fingerprinted, checked for gunshot residue, and grilled by Abner Sweeney, it had been almost midnight. After the lawmen had left, he'd changed the tire and driven home to a silent house, with nobody awake to meet him.
Drawn by the smell of fresh coffee, he returned to the kitchen, poured himself a cup, and took it back outside. As he stood at the porch rail, sipping and trying to focus his thoughts, a voice from behind startled him.
"Say, Will, that coffee smells mighty good. I could use a cup, myself."
Jasper Platt, the Rimrock's retired foreman, had come up onto the porch in Will's absence. He sat in one of the chairs, with the ranch's black-and-white Border collie at his feet. White-haired now, and too arthritic to ride, Jasper shared a duplex with horse boss, Sky Fletcher, behind the main house.
"Sure." Will strode back into the house and was back a moment later with a second cup of steaming black coffee.
"Thanks." Jasper reached for the cup, blew away the steam, and took a careful sip. "Heard you come in last night. You got in mighty late. Did you find yourself a lady friend in town?"
"Lord, I wish I had." Will pulled up another chair and sat next to the old man. Maybe talking things out would make him feel better. "I drove into Blanco to pick up Erin," he said.
"Mighty late for that little gal to be out. Is she here?"
"No, she's with Tori. Something . . . happened on the way back."
Jasper glanced off the side of the porch, which was high enough to give him a view into the truck bed. "Judgin' from the shape of that tire, what happened was bad," he observed.
"The tire was the least of it." Will drew a painful breath. "I had to shoot a man last night, Jasper. I killed him."
The old man listened, frowning and nodding, as Will related the night's events. "So who the hell was it?" he asked as Will neared the end of the story.
"Somebody I would never have shot if I'd known who he was. It was Stella Rawlins's brother, Nick."
"The dude with the tattooed head? Lord almighty!" Jasper swore. "I wouldn't want to be you when Stella hears about that. The woman will be out for your blood."
Will gave him a grim nod. "There's that. And then there's my conscience. What if I killed an innocent man, Jasper? I've seen Nick in town. The man looked mean enough, but I always had the feeling he was scared of his own shadow. If I'd known it was him, I'd have figured the knife was a bluff and talked him into putting it down. Damn it, I can't say I ever wanted to find out how it felt to take a human life."
"You had Erin in the truck. You'd have done anything to keep her safe." Jasper scratched the dog's ears. "I never killed a body myself, but I was along when your dad shot a couple of rustlers that were makin' off with his cows. Bull was a dead shot. Plugged 'em both, right out of the saddle and left 'em in the dust."
"And knowing my father, I'm sure he wouldn't have batted an eye over it," Will said. "All my life I wanted to be just like him. As a kid I learned to walk like him, talk like him. Later on, I made the decisions I thought he'd make. But I could never be half as tough. I don't think any man could be as tough as Bull Tyler was."
"It wasn't like Bull to mention it, but he was always proud of you," Jasper said. "He would've been proud of you last night, doin' what it took to protect your little girl. You didn't ask my advice, but I'll offer it, anyway. You did what you had to. So put this ugly business behind you and move on."
Will set his cup on the porch, rose, and walked to the rail. The Rimrock was stirring to life, the aromas of coffee and bacon wafting from the bunkhouse kitchen. Soon the hands would be setting out for morning chores. Sky Fletcher's steel-blue pickup was already parked outside the horse barn. These days he was spending most of his nights in Blanco with his fiancée, Lauren Prescott. But Sky, Bull's secret son by a Comanche woman, could always be counted on to show up early for work.
In some ways—all the good ways—Sky was almost as much like Bull as Will was. But Will's younger brother, Beau, was cut from a different bolt of cloth. His clashes with Bull had driven him away from the ranch for eleven years. Now, as foreman, he was perpetually butting heads with his brother. Some things, Will mused, never changed.
By now, the blinding edge of the sun had risen above the eastern horizon. The yellowed grasslands glimmered with early frost that would melt away as the day warmed. A raven rose from a tall cedar, flapped its wings and soared into the dawn.
A quarter mile to the southeast, four red brick bungalows, built to house the married hands and their families, were set back from the road in a neat line. Beau, with his pregnant bride, Natalie, had moved into one of them while they waited for their new home to be finished. The newlyweds could've easily stayed in the ranch house with Will. There was plenty of room, and Will would have welcomed them. But someone had warned Natalie that moving into the big house would be a sure way to doom their marriage.
Will had no doubt that someone was Tori.
He and Tori had been wildly in love when they'd married. But here on the ranch, their marriage had degenerated into a tug-of-war between Tori and Bull, with Will caught in the middle. Things had gone from bad to worse until the awful week that had torn them apart for good.
Had Tori been trying to save her friend Natalie from the same fate?
As if the thought could summon her, Will saw Tori's brown station wagon approaching up the long gravel drive. His spirits lightened as he glimpsed Erin in the passenger seat. He was glad Tori had brought her. Having his daughter here would go a long way toward salvaging the weekend.
As the station wagon pulled up to the house and stopped, Erin climbed out the door and bounded up the steps to give Will a quick hug. Will fought the urge just to hold her tight and not let go. He'd been so scared for her last night. But it wouldn't help to let her know that. Just let her be happy. Let everything be normal.
"Can I go see Tesoro?" she asked.
"Sure. Sky should be in the barn. Make sure he's close by." Erin's beloved palomino foal was almost six months old. By now, he'd reached 85 percent of his mature height, but only half his mature weight. He was all legs and spunk. Erin had worked with Sky to train and gentle him, but the spirited young colt was big enough to be rambunctious. Will worried about his precious daughter being hurt by an accidental kick or shove.
He watched her race across the yard to the barn, all sunshine. But as she stepped out of the vehicle, her mother's expression cast a shadow over the morning. Tori wasn't happy, and Will was about to find out why.
She looked up at him, where he stood on the porch. "We need to talk, Will," she said.
Jasper took his cue. "I think I hear Bernice in the kitchen," he said, pushing to his feet. "Maybe I'll go see what she's fixin' for breakfast."
As he headed inside, the dog at his heels, Tori came up onto the porch. She was dressed in jeans and a dark green sweater, her long, honey-blond hair brushed back from a face that was freshly washed and bare of the makeup she usually wore. Seeing her like that reminded Will of mornings on the ranch, waking up and seeing that beautiful, unadorned face on the pillow beside him. But why was he thinking about that now—especially since the new man in her life might already be seeing her in the same way.
"Want some coffee?" he asked her, determined to be civil.
"No thanks." She took the chair where Jasper had been sitting.
"Thanks for bringing Erin," he said.
"She wanted to see her foal. Then I'm taking her back to town with me."
"You can't do that." Will's nerves, already raw, caused him to snap at her. "I get her on weekends, Tori. We signed the damned papers."
"I know." She was maddeningly calm. "But Erin's welfare comes first. And after what happened last night, I don't think she's ready to be out here with you."
"What's the problem?" Will glanced toward the barn, where Erin had already disappeared. "I saw her. I talked with her. She seemed right as rain."
Tori's jaw tightened. "Will, you killed a man last night. Erin saw it happen."
The shock hit Will like a lightning bolt. His first reaction was denial. "No. She was down on the floor of the truck. Honest to God, Tori, I made her get down so he wouldn't see her."
"Erin's twelve years old. How could she not look? She told me what she saw."
"But she seemed fine this morning. Was she upset?"
"Last night? Very. She's putting on an act for you today."
"Lord." Will raked a hand through his hair. "To have her see me pull that trigger, see that man die—I wouldn't have done that to Erin for the world!"
"There's more." Tori was sitting straight on the edge of her chair, hands clasped on her knees. "Abner Sweeney called me after we got home last night. He wants to question her today. According to him, you swore she didn't see anything."
"That's right. What did you tell him?"
"Nothing. I'm guessing her account will back up yours. We certainly can't ask Erin to lie. But you sent her away last night before the sheriff got there. And you claimed she hadn't seen the shooting. By the wrong people, that could be interpreted as obstruction of justice."
"Justice!" Will exploded out of his chair. "What justice? I killed the man in self-defense!"
A thought line deepened between Tori's eyebrows. "That's one way of looking at it. But you shot a man who'd just surrendered his gun, a man who hadn't yet attacked you with his knife. In a different light, that could be construed as manslaughter, or worse."
The implication made Will's gut clench. He paced to the top of the porch steps and turned back to face his ex-wife. "Did Abner tell you the man was somebody we knew?"
Tori shook her head as if to say, How could this mess get any worse?
"It was Stella Rawlins's brother, Nick."
"The bartender." It wasn't a question. Tori's face had gone pale. "Will, you know what that woman's capable of. The rumor is, she's got mob connections. And even without them, she could hurt you. Worse, she could hurt Erin."
"Or even you—anything to make me suffer for killing her brother." Will allowed himself a deep breath. He'd taken enough hits this morning. It was time to take charge of the situation. "For now, this is what we're going to do. Erin will be safer here on the ranch than in town with you. We can call the teacher and arrange for her to do her schoolwork on the computer. If you're concerned about leaving her, you can stay here, too. Beau's old room's available, and except when you need to be in court, you can do most of your work from the ranch office."
She started to protest. "Blast it, Will, you can't just dictate—"
He cut her off. "Why not? If you think Erin needs you, what's wrong with your staying here? Are you afraid it might interfere with your love life?"
Tori's blistering glare told Will he'd overstepped. But at least the issue was in the open now. He braced himself as she rose, quivering with fury. "You don't own me anymore, Will Tyler," she said. "I've tried to keep our relationship civil because of Erin, but my so-called love life is none of your business!"
"Erin says he's the high school principal." Will had nothing to lose by pushing her a little further.
"That's right!" she snapped. "He's attractive, smart, decent, and, unlike you, he doesn't try to run my life. That's all I'm going to tell you. And don't you dare grill Erin about him! I won't have you putting her in the middle!"
"Agreed," Will said. "But speaking of Erin, there's one thing I need you to do. Call Abner and tell him that if he wants to talk to her, he can do it here, with both of us present."
"Fine." Turning away from him, she took her cell phone out of her purse and exchanged a few terse words before ending the call. "Abner will be here in an hour," she said in her crisp, neutral lawyer voice. "He'll want to talk with you as well."
"No problem, I don't have a thing to hide." Will tried to sound more confident than he felt. "Can I have my attorney present?"
"You're shameless." Tori shook her head, but Will knew there was no way she wouldn't be involved.
"How about some breakfast, you two?" Bernice, Jasper's widowed sister, had been the Tylers' cook and housekeeper since Will's boyhood. With his family gone from the house, Will did for himself most mornings. But when Erin was here, or when Beau or Sky dropped by, she enjoyed whipping up a feast of bacon, eggs, potatoes, and pancakes, with coffee for the grown-ups and cocoa for Erin.
From the back porch came the sound of Jasper ringing the iron triangle to call Erin from the barn. Maybe Sky would come, too, though he showed up for breakfast less often now that he was engaged and building a home for his bride on the 100 prime acres Bull had left him. The half-Comanche horse trainer was a very private man. He hadn't known he was Bull's son until Jasper had told him, and he still kept it quiet. He wasn't even aware that Will had guessed the secret. Will planned to let him know when the time was right, maybe on his wedding day.
They sat around the cozy kitchen table—Jasper, Will, Tori, and Erin, with one chair saved for Bernice and another, with the place set, for anybody who happened by. Bernice was just setting the platters of food on the table when the back door opened and Beau stepped in.
"Smells good." He was grinning, his face ruddy from the morning chill. "Hope you saved me a place."
"Right there." Will nodded toward the empty chair. "How's Natalie this morning?"
"Still asleep, I hope." Beau pulled out the chair, sat down, and began filling his plate. "She was up past midnight, tending a sick mare over at the Johnson place. With the baby on the way, she needs more rest than she's getting." He glanced across the table at Will. "Hey, I saw you on the news this morning, brother. It seems you've become a local celebrity, gunning down a drug dealer in the night. Maybe they'll send that hot Mindi Thacker out from the TV station to interview you."
Will groaned. It was typical of his younger brother to turn a crisis into a joke. Today it rankled him, even though he knew that whatever happened, Beau would have his back. "It's not funny," he growled. "Tori just filled me in on the legal implications. And Abner Sweeney's on his way out here now to ask more questions. You're welcome to sit in."
"Maybe I'll do that. Especially if Abner still sees me as a cross between James Bond and Elliot Ness." Beau's past career as a DEA agent had impressed the sheriff, who still called him to discuss the occasional drug case.
"I'm guessing I don't have to tell you anything else." Will's subtle nod toward Erin was a signal that he didn't want to say more in front of his daughter.
"It was on the news." Beau speared a pancake with his fork. "Bernice, I swear your breakfasts just keep getting better. It's lucky for us some gentleman hasn't come courting and stolen you away from us."
Accustomed to his banter, Bernice shook her head. "Beau Tyler, you could sweet-talk a skunk, and I'd pay to see you try it."
Laughter drifted around the breakfast table. In the past Will had treasured mornings like this, with the people he cared about gathered in the warmth of the kitchen. It was sad how rarely it happened these days. In no time at all, with Erin growing up, Bernice and Jasper getting old, Tori making new choices, and Beau, as well as Sky, involved in starting new families, these times would be gone forever.
Where would he be by then?
If the worst happened and he ended up on trial, he could be looking at the world through prison bars.
CHAPTER 3
"You say you couldn't tell who the man was, Erin?" Abner Sweeney checked his antiquated cassette recorder, to make sure it was still working, and put it back on the coffee table. A bead of sweat trickled down the side of his neck. He was under orders from Stella to find something—anything—that might incriminate Will Tyler in her brother's death. So far, this poised twelve-year-old girl wasn't giving it to him.
"He was wearing a helmet that covered his eyes," she said. "I could sort of see his mouth, but not really because it was dark. I thought he was the robber we heard about on the radio. I'm pretty sure my dad did, too."
"Did the man say anything?"
"I couldn't hear. The window was closed."
"I see." Abner nodded, vaguely aware that he had to pee. It tended to happen when he was nervous, and he was nervous now. Not so much because of the girl, but because of her lawyer mother, sitting to one side, watching him the way a cougar would watch a sheep, ready to pounce at the first misstep. At least he'd managed to keep Will out of the room by insisting he had to question him and his daughter separately.
"Let me ask you something else, Erin. Did either of your parents tell you how to answer my questions?"
"Yes. They told me to tell the truth."
"Then tell me the truth now. Did you see the knife?"
"Yes."
"Where was it?" Abner tried to ignore the urges of his bladder. Maybe he needed to have his prostate checked. He was getting to that age.
"Where was the knife, Erin?" he asked again.
"In the man's hand."
"What did he do with it? Here, show me with this." He handed her the ballpoint pen from his pocket.
"He put his arm back like this." She demonstrated, bringing the pen up and back, as if about to throw it.
"Did he throw it, or even start to?"
"No. That was when my dad shot him."
"You're sure of that?"
"Yes."
"Thank you, Erin." And that, Abner told himself as he switched off the recorder, was as good as he was going to get.
* * *
Tori gave Erin a hug and sent her out of the den, where the interview had taken place. "Go see if Bernice needs any help," she said. "I'll be right here if you need me."
Tori thought that Will would've been proud of their daughter, but one thing troubled her. When Erin indicated that the dead man hadn't moved the knife forward to throw it, the sheriff 's bland expression had undergone a subtle change—a narrowing of the eyes, a tightening of the lips. As a courtroom lawyer, she'd learned to read people, and she didn't like what she'd seen.
Should she tell Will, or would that just worry him? She put the question aside as Will walked into the den, so tall and strong, and so totally in command that his presence seemed to fill the room. She didn't have to be here, Tori reminded herself. They'd been divorced for eight years, and she was doing her best to move on. Meeting Drew had given her hope that she really could move on.
Will's domineering ways had always made her a little crazy. Today was just one more reminder of that. But Will had gotten into this mess protecting their daughter. For that, she owed him.
The sheriff had excused himself to rush down the hall to the guest bathroom, giving Tori a moment alone with Will. He walked over to the armchair, where she sat perched on the edge; his broad-shouldered frame loomed above her. "How did Erin do?" he asked.
"She did us proud. Calm and cool, spoke right up—more than a match for the likes of Abner Sweeney."
A smile twitched at one corner of his grimly set mouth. "At least we did one thing right, didn't we?"
"We did." And we did most other things wrong—my open defiance, Will's siding with his father, and the last thing, the darkest thing, when he accused me of something that didn't happen. Will never apologized; and I never forgave him.
"Will, let's get started." The sheriff bustled back into the den, took his seat, and turned on the recorder. Will sat down at the end of the sectional leather sofa. "I hope you won't mind if I record your testimony. There's going to be an inquest, and I want to make sure your version of what happened is accurate."
Will shrugged. "Fine. It won't be any different from what you heard last night. Tell me when to start."
Tori listened while Will related the same story he'd told her. Abner stopped him from time to time to ask questions. Tori could tell the sheriff was probing for any detail that might conflict with what Erin had said. It was almost as if he was trying to build a case against Will. What she didn't understand was why. There was no bad blood between the sheriff and the Tylers. And Abner was no longer running for the election he'd just won. What was driving him?
Partway through the session, Beau wandered into the den, took a seat, and leaned forward to listen. Only when Will had finished his story, which matched Erin's, did he speak up.
"Sheriff, according to Will, the man said something about a package and money. The newscast I saw mentioned that the deputies found cocaine on his bike. Can we assume that Nick Tomescu was on his way to a drug deal, and that he mistook Will for his customer?"
Abner looked flustered. "We can't assume anything," he said.
"Did you question his sister?"
"I did. Stella was grieving, of course. She said she didn't know anything about her brother's activities. I'm inclined to believe her."
"Of course." Beau rolled his eyes in Tori's direction. She responded with a subtle shake of her head. Stella Rawlins would never admit to being involved in anything. And she appeared to have the sheriff wrapped around her little finger.
Restless as a bull in the bucking chute, Will rose to his feet. "Are we finished, Sheriff? I need to get to work."
"Just one more thing, Will," the sheriff said. "Last night you told me you'd sent your daughter away because she hadn't seen anything. That turned out to be untrue. Do you have anything to add in defense of your statement?"
Tori's eyes were on Will—his tightened jaw, his narrowed eyes. She knew that look all too well. It was the look of a man who'd had enough. As he drew in his breath, she braced for the explosion.
Will exhaled, holding himself in check. "I'm done here," he said. "Ask my lawyer."
With that, he strode out of the room. Tori heard the closing of the front door—not quite a slam—and the roar of his pickup as he sped away. She guessed he'd be headed somewhere out of reach, maybe up to the summer pastures on the caprock to check the grass and mend the fences. That had been his way when they were married—in any kind of emotional crisis, Will would simply walk away and disappear into his work.
Abner was staring after him, slack-jawed. Tori rose, speaking into the silence. "Sheriff, I believe you already know the answer to your question. Will didn't know that Erin had seen the shooting until I told him this morning. You've no call to read anything else into the situation. Agreed?"
"Well, yes, I suppose so." Abner stood, turned off the recorder, and stuffed it into his imitation-leather briefcase. "I guess we really are done here."
"You said there'd be an inquest." Beau had risen as well. "Can you give us some idea what's involved and how long it'll take?"
"My best guess is ten days to several weeks. We'll need to get the coroner's report and schedule the judge. If the judge wants to impanel a jury, that'll take more time."
"A jury? Why, for God's sake?" Beau demanded.
"To decide whether Will should be charged and tried—most likely for manslaughter."
With that exit line, Abner picked up his briefcase and keys, and headed out the front door.
* * *
That evening, after a long, restless afternoon, Tori stood at the porch rail and watched the last rays of the setting sun fade behind the escarpment. The canyons lay deep in purple shadow, the high buttes above them still bathed in velvety mauve light. Quail called from the cedars along the foothills. Horses, their nostrils testing the wind, nickered and snorted in the paddock. The evening breeze carried the smell of dust and an ominous chill, a warning, perhaps, that the first norther of the season was already sweeping down the distant plain.
She glanced at the luminous dial of her watch. Drew had planned to pick her up two hours from now for an eight o'clock movie and late-night pizza. Either she would need to go home, change, and be there when he arrived, or call now and cancel their date.
The question was, should she stay here with Erin? Her daughter had seemed fine today, but last night she'd had trouble sleeping. Tori had ended up putting an old Disney movie in the DVD player and watching with her until her head drooped and her eyes closed. If Erin was still traumatized tonight, Tori didn't want to leave her, not even with Will. At times like this, a child needed her mother.
And Will—did he need her, too?
But that couldn't be allowed to matter. Will was his own man, and she was no longer his wife. It made sense that she'd agreed to be his lawyer. But that was where she had to draw the line. Sympathy wasn't part of the bargain.
So, if she canceled her date and stayed here, would that be sending Will the wrong message?
A brisk November wind whistled across the porch. Tori shivered beneath her light wool sweater. Either way, it was time to make a decision.
She was about to go back inside when she felt a warm weight settle on her shoulders. As the smells of sage, wood smoke, and horses enfolded her, she recognized Will's fleece-lined range coat and the strong hands that had wrapped it around her.
"Can't have you freezing out here, can we?" Will's husky baritone rumbled in her ear. He stood behind her, his hands on her shoulders, his breath stirring her hair. A long-forgotten thread of heat uncurled in the depths of her body, recalling the sensual passion that had created Erin and the other baby, the one she'd lost. In those early years their lovemaking had been good. More than good—until the end, when even love hadn't been enough.
Tori closed her eyes, an ache rising in her throat. "You weren't here for supper," she said. "Where've you been?"
"Working. Riding. Thinking. Whatever the hell a man does at a time like this. By the time Abner finished with me, I wasn't fit company for you or anybody else." His hands lingered on her shoulders. "How's Erin?"
"Fine. When I last checked, she was doing her homework in the dining room." The old memory tugged at her, standing in the ranch kitchen, feeling the brush of his stubbled whiskers on the back of her neck . . . She felt the heat rising, her body warming. Even after eight years apart, she wasn't immune to Will's raw masculinity.
But there was only one sensible choice here, and Tori forced herself to make it. "I hope you can keep Erin company awhile," she said. "I was about to leave. I have a date tonight."
His hands dropped from her shoulders. He took a step backward, widening the space between them. "You had a date last night," he said.
"Yes, I know." Turning, she slipped off his coat and thrust it toward him. "My plan is to drive back here afterward, with a bag packed for a night or two. How long I stay here will depend on how Erin is doing. If we agree that she's better off on the ranch for now, I'll call the school on Monday and arrange for her lessons. But you can't keep me here, Will. I'm not a prisoner. I have a life."
Stony-faced, Will took the coat. "But will that life be safe? Damn it, Tori—"
"I'll be just fine. And for heaven's sake, don't wait up for me. I'm not sixteen anymore." Sweeping past him, Tori strode into the house to get her purse and say good night to Erin. Was Will more concerned about her safety or about her being with another man? Either way, she couldn't let his problems dictate her life. She was going on a date with Drew Middleton, and, by heaven, she was going to have a good time.
* * *
The grandfather clock in the front hall struck the hour of twelve. Will counted the chimes from his bed, where he lay on his back, staring up into the darkness. Midnight, and Tori still hadn't come back from her so-called date. She was out there somewhere with some goody-two-shoes bastard who held the power to change all their lives—Tori's, Erin's, and his own. Will had never met Drew Middleton. If he ever did—and it was bound to happen sooner or later—it would take all his restraint to keep from punching the man in the face.
Damn it, Middleton didn't belong in the picture. He didn't have a clue about Tori, didn't even know her. He hadn't watched her grow up, changing from a coltish youngster who tore around the ranch with Beau and Natalie to a stunning woman, returning home with a brand-new law degree. Her beauty had knocked Will's socks off back then. It still did.
It wasn't Drew Middleton who'd driven Tori to the Lubbock hospital in a blue norther the night Erin was born. It wasn't Middleton who'd walked the floor with Erin when she had the croup. And it sure as hell wasn't Middleton who'd held Tori in his arms while she sobbed over the loss of their second baby, five months into her pregnancy.
The man was an outsider. He didn't belong in Tori's world or in Erin's. Why couldn't Tori see that?
Twelve-fifteen. Was Tori in Middleton's arms right now, or maybe even in his bed?
Stop it! Will forced the image from his mind. It was time he quit agonizing over his ex-wife and opened his eyes to the reality that was staring him in the face. He had every right to be concerned about Tori's safety. But her romantic life was her own business. The two of them shared a much-loved child. For Erin's sake, he and Tori kept their connection, talking and meeting often, even sharing Sunday dinners on the ranch. But that didn't make her his. Whether he liked it or not, she hadn't been his in a very long time.
A faint sound shocked him to full alertness—a smooth engine gearing down as it approached the house. Will sat up, ears straining in the darkness. Tori's aging wagon had a distinctive rumble and a squeak in the chassis. He would know the sound of it anywhere. But this vehicle was almost silent, more like a late-model high-end sedan.
Tires crunched on gravel as it pulled up to the porch. Will was already grabbing for his clothes, yanking them on, shoving his bare feet into his boots. Was it a highway patrol car, its driver coming to tell him that Tori had been in some horrible accident? Or could it be one of Stella's minions sneaking up to the house to do some damage?
Heart pumping adrenaline, he opened the top drawer in the nightstand and took out the pistol he kept there. By the time he reached the living room, the sound of the engine had stopped. Headlights were shining through the front window. Whoever it was, at least they weren't trying to sneak up on the place. But this could still mean bad news.
Now Will could hear footsteps and voices—one of them a man's, one of them Tori's. At least she sounded all right—more than all right. She was laughing. He stepped back into the shadows of the hallway—not wanting to be seen, but too curious to turn away and go back to bed. Why would Middleton—if that's who it was—be bringing her here? Why hadn't she driven herself? Was he about to meet his ex-wife's new boyfriend?
The parked car's headlights shone blindingly bright through the plate glass window. As Will's eyes adjusted to the glare, he could make out a silhouette on the porch—two people, one taller, locked in a passionate kiss. His pulse slammed.
Don't look, you damnfool idiot! Go back to bed! Will chastised himself. But he was rooted to the spot, fighting emotions he had no right to feel as the silhouette separated and became two people, the taller one leaving. An instant later, Tori's key turned in the lock. She stepped into the darkened living room.
Will backed into the shadows, but not soon enough. Tori must've heard him, or sensed he was there.
"Will?" Clutching her overnight bag, she stood outlined in the open doorway. Her hair fluttered in the night breeze. "Is that you?"
He stepped out of the shadows. Her breath caught in a low gasp. "Good grief, don't tell me that's a gun in your hand! Who were you planning to shoot?"
"This isn't anything to joke about, Tori. I heard a strange car. I thought it might be a prowler."
She closed the door and locked it behind her. "I told you not to wait up for me. If you'd been asleep, you wouldn't have heard the car."
"I'm not exactly sleeping well lately," Will growled. "Where's your wagon?"
"Dead in my driveway. Bad starter, I think. Drew drove me here."
"Too bad he didn't stick around for an introduction. After seeing how he said good night to you, I wouldn't have minded meeting him."
Her chin went up in defiance. "Stop badgering me, Will. I'm tired. I'm going to bed."
He stood his ground, his silence asking the unspoken question that hung between them.
Tori's patience snapped. "For your information, I haven't slept with the man. If I decide to—which I might—that will be none of your business, Will Tyler. Now get out of my way. I want to check on Erin. Then I'm going to sleep."
Ignoring the knot in his gut, Will stepped aside to let her get by. Part of him ached to crush her in his arms, sweep her off to his bed, and stake his claim on her all over again. But the time when that might have happened was long past. Tori would never be his again.
As they passed in the entrance to the hallway, something awakened and cried out in him. He reached out and caught her cheek with his hand, lightly cupping her face, lifting it to the pale light that filtered from the front window. The pupils of her eyes were large and dark, her lips moistly swollen. He ached to bend close, to brush those lips with his, but she pulled back with a sharp little breath, shook her head, and fled down the hall toward Erin's room.
* * *
The next day was Sunday. Although the Tylers weren't big on church attendance, Sunday dinner on the Rimrock was an honored tradition. It was a time when the family, and those who counted as family, gathered around the dining-room table in relative peace to celebrate their blessings.
Lauren Prescott raised her bowed head after Jasper droned the usual grace over the food. She'd been included since the past summer, when her old family home had gone up in flames and Sky had asked her to marry him. The Tylers always welcomed her, but even after more than three months of being engaged to Sky, she still felt like an interloper.
As family and friends helped themselves to roast beef, potatoes and gravy, salad, and fresh hot rolls, Lauren's gaze drifted around the table. Beau, seated across from her with his wife, had been her friend and champion since last spring when he'd hired her accounting skills for the ranch. But Natalie, petite and dark-eyed, her pregnancy beginning to show, was so busy with her veterinary practice and her new marriage that Lauren had scarcely gotten to know her.
Tori had become Lauren's friend after helping her find an apartment in town. But Tori, in her own way, was also an outsider here. Today the tension hung heavy between her and Will. Something was going on between them. It showed in the way they avoided each other's eyes, the way they spoke not to each other but to their daughter.
Flanked by her parents, Erin basked in the love of everyone at the table. She might not realize it, but Will's daughter was the glue that held the ranch family together, the bond that brought them here and made them—for this brief time—one.
Bernice sat closest to the kitchen. She'd always been kind and friendly. But her brother, Jasper, had never warmed to Lauren—and Lauren understood why. For three generations the Tylers and the Prescotts had been bitter enemies. Jasper remembered every wrong, every misdeed, every dispute, from the beginning. Trust a skunk before you trust a Prescott, he was known to say, though not to Lauren's face.
Bull Tyler and Ferguson Prescott, Lauren's grandfather, had started the feud. Both men were dead now, but the animosity remained. When Lauren's father, the late congressman Garn Prescott, had died this summer, Jasper had refused to attend the burial service. Though he'd been at the graveside, Will, too, had had his own issues with the congressman. Now, as the only living descendant of Ferg Prescott and his son, Garn, Lauren carried a heavy burden of past family sins. Only time would tell what that burden would cost her.
She felt the light press of a hand on her knee. Seated next to her, Sky gave her his secret smile. She reached under the edge of the tablecloth and brushed the back of his hand in a furtive caress. Her history and Sky's were intertwined in ways neither of them could have imagined when they'd first met. That hidden bond made her love him all the more. She could hardly wait to become his wife. Maybe then they could start anew and put the old family scandals to rest.
"How's the new house coming along, Sky?" It was Beau who asked the question. "I've meant to ride over and take a look now that the fall roundup's done."
"The outside's finished," Sky said, "as well as the plumbing, heating, and wiring. Once the Sheetrock's up and prepped, I can turn Lauren loose on the inside."
"I'll be in decorator heaven!" Lauren said. "We'll have a big housewarming when it's done!"
Sky had wanted to build his bride an entire house with his own hands, but his responsibilities on the Rimrock had made that impractical. Under his supervision, the crew he'd hired to put in a well and septic tank, run the power line and construct the log house, with its broad, covered front porch, was doing a fine job. The place wouldn't be big and sprawling like the Tyler home, but with Lauren dipping into her inheritance to decorate the rustic interior, it would be beautifully finished and comfortably furnished.
Will had been uncharacteristically quiet throughout the meal. Sky had told Lauren about the shooting of Stella Rawlins's brother and the possible consequences. No wonder Will looked so troubled. Lauren could sympathize with him. Stella, she suspected, had ruined her father's reputation and contributed to his death. As always there was no proof against her, but if rumors were true, the woman was as dangerous as a coiled rattlesnake. And now she'd be out to avenge her brother by hurting Will any way she could—starting with the law.
* * *
Will walked into the room that served as the Rimrock office and closed the door behind him. He usually looked forward to Sunday dinners, but today's meal had been an ordeal of silence and small talk, with everyone avoiding the one topic that was on their minds—the shooting and what was going to happen next.
Will had excused himself at the end of the meal, muttering something about the need to use the office computer; but the truth was, he'd just wanted to be alone and think things out. Until yesterday he'd felt certain that he'd acted in self-defense, and any case against him would be dismissed. But yesterday's senseless grilling from Abner had changed his mind.
For whatever reason, the sneaky little toad was out to get him.
Sitting, he switched on the computer and brought up a search engine. He spent the next half hour reading up on Texas law, the inquest process, and the precedents for charges that could stem from an incident like the one he'd been involved in. What he found wasn't encouraging. By pulling the trigger a split second too soon, he might have left himself vulnerable.
With a muttered curse he switched off the machine. Most of what he'd read, he could've learned from Tori. But now that she had a new man in her life, he couldn't expect her to drop everything and come running whenever he needed her. And unless her help involved protecting Erin, he had too much pride to ask.
A new man in her life.
The thought deepened the dark hollow Will felt inside. He and Tori were past history, but even now, the thought of Drew Middleton, or any man, taking her away was like having the earth slide out from under his feet. Tori was a beautiful woman, and sooner or later, this was bound to happen. But why the hell did it have to happen now?
Swiveling the chair, he gazed up at the leather-framed sepia-toned photograph on the wall. Bull Tyler had refused to sit for a painted portrait, like the one of his wife that hung above the sideboard in the dining room. But years ago, when he'd been featured in a magazine article about Texas ranching, he'd agreed to be photographed.
The picture, taken when Bull was fifty, showed a handsome, vigorous man in his prime. Dressed in a corduroy jacket, plaid shirt, and leather bolo, and sporting a well-trimmed moustache, he emanated authority. His piercing gaze, from under thick, dark eyebrows, challenged any comer to take him on.
In the last years of his life, after the riding accident that paralyzed his legs, Bull had ruled the ranch family from his wheelchair. But there was no trace of any weakness in this photograph. This was the way Will had chosen to remember his father—powerful, dynamic, and always in charge.
That memory would haunt every decision Will had ever made.
What would you do in my place, Dad? Will gazed up at the blunt, chiseled features as if waiting for an answer. But why ask when he knew what the answer would be? Bull Tyler would have told everybody to go to hell, turned his back, and then walked away.
Maybe for Bull, that would've worked, but not these days. The law had too much power. Will would fight the possible charges with every resource he had. But he'd be a fool not to see the cold reality that was staring him in the face. He'd killed a man—the wrong man. If things went badly, he could find himself spending time behind bars.
Starting now, he needed to get his priorities in order—beginning with his family and the ranch.
Abner had told Beau that scheduling and carrying out the inquest would take several weeks. After that, there'd be a trial—or not, depending on the outcome. Either way, Will would have some free time before any decision was made—time to tie up loose ends and put some things right.
One task nagged him every time he looked at his father's proud face. It was the land—the precious canyon parcel with the spring and the rumored Spanish gold—that Bull had sold to his hated neighbor, Ferg Prescott, for the sum of $1.
Except for that small piece of land, less than an acre, no part of the Rimrock had ever been sold. For the sake of family pride, if nothing else, Will knew he had to get it back.
Last spring he'd made Garn Prescott, Ferg's son, a generous offer for it. The congressman had refused to sell, blaming some deathbed promise to his father. But things had changed since then. Garn was dead.
Now the land belonged to Lauren.
CHAPTER 4
Will stepped out of the office and walked down the hall to the living room. He'd hoped Lauren and Sky would still be here; but except for the hum of the dishwasher in the kitchen and the steady ticktock of the grandfather clock in the entry, the house was quiet. Nobody appeared to be around, not even Erin.
Mildly puzzled, he walked out onto the porch. He discovered Jasper in his customary chair, a Corona in his hand and the dog sprawled at his feet.
"Awfully quiet in there," Will said. "Where is everybody?"
Jasper took a swig of his beer, flecks of foam clinging to his upper lip. "Bernice is napping. Beau and Natalie went home. The others piled into Sky's pickup, and he drove 'em over to see the new house."
"Did they say when they'd be back?" Will glanced off the porch and saw Lauren's vintage black Corvette parked on the gravel. At least she hadn't left for town.
"Don't reckon they'll be long. Not that much to see." His wise, pale eyes studied Will. "So you're thinking about getting that canyon parcel back, are you?"
"Am I that easy to read?"
The old man chuckled. "I've known you since you were in diapers, Will Tyler. You come out here looking all wrought up, you ask where everybody is, and then you check for the Prescott girl's car. Doesn't take much to figure that one out."
"It's been on my mind since Garn died," Will said. "But with Lauren still mourning her father, it didn't seem fitting to ask her about it. Now . . ." Will gazed into the blue distance, where two vultures circled on the updrafts. Some people believed that the ugly black birds were a portent of evil. Will had never held with that old superstition, and he wasn't about to start believing it now. "I'm trying to get some things done before the inquest wraps up. Hopefully, it'll come to nothing, but you never know. If I have to go away for a while . . ."
"Don't even talk like that," Jasper said. "When the girl marries Sky, the land will at least be back in the family."
"But not the way it should be. I want a signed, recorded deed giving that land back to the Rimrock. And I won't settle for less."
"It might not be that easy. For all you know, that land could've gone with the rest of the Prescott Ranch when Garn sold out to the syndicate."
"No, I checked with the county recorder," Will said. "The old deed's still valid, made out to Ferguson Prescott and his heirs in perpetuity. The property's Lauren's to sell."
"If she's willing to sell it. She's a Prescott, after all. Stubborn devils, Ferg and his boy. Garn's daughter won't be no different." Jasper took another swig of his Corona. Something in the old man's look told Will he knew more than he was telling. But Jasper was full of secrets, most of which he would probably take to his grave.
For now, there was no time to pry any more out of him. Sky's steel-blue pickup had come over the last rise, trailing a plume of dust as it bounced across the burned-over flatland toward the house. Will waited as the truck pulled up to the porch; he was pondering what he could say to influence Lauren. How much does she know about the land? he wondered. How much had her father told her—and Sky?
Tori and Erin climbed out of the rear seat. Instead of coming up onto the porch, Erin was tugging her mother toward the paddock, where the colts were romping in the afternoon sunlight. Even from a distance Tesoro's hide gleamed like gold, making it easy to spot him among his darker-coated playmates. In a way it was too bad Erin was so smitten with her young colt. The sale of such an animal would give the ranch a much-needed influx of cash. But no amount of money was worth breaking his daughter's heart.
Lauren waited for Sky to come around the truck and open the passenger door before she climbed to the ground. Sky's fiancée was a stunner, with a model's rangy figure, coppery eyes, and an unruly mane of auburn hair. Reared with wealth, she was accustomed to the best. For Sky, a man of secure but modest means, keeping her happy would be a challenge. But the two of them did seem deeply in love. Will envied them that.
Will came down the steps to meet her and invite her inside for a talk. He meant to offer her a fair price for the land, but there was always the chance she wouldn't agree to sell. If she dug in her heels and refused, he might have a fight on his hands, with Sky siding against him. But there was no way he was backing down. One way or another, he would make the stolen land—and there was no other way to think of it—part of the Rimrock once more.
* * *
Sky settled himself in the leather armchair by the fireplace, a safe distance from the sofa, where Will had invited Lauren to sit. He knew exactly what was on Will's mind. The surprise was that his half brother had waited this long to bring it up.
Sky had shown Lauren the disputed land early on and told her what little he knew about it. He understood how much Will wanted that small parcel back where it belonged. But the decision to sell, or not to sell, would be Lauren's, and he would support her choice. Knowing what was coming, he'd already made that clear to her. Right now, he was nothing more than an interested observer.
"Something to drink?" Will was still on his feet. "A beer? Some wine?"
"No, thank you," Lauren declined, as did Sky.
Will lowered his tall frame to the edge of the sofa, looking ill at ease as he turned toward Lauren. "Something tells me you already know what I want to talk to you about," he said.
"Yes, and I'm familiar with the circumstances," Lauren replied. Sky had to admire her quiet poise.
Will cleared his throat. "I offered Garn a fair price for that parcel of land when he was running for reelection. But he told me he'd promised his father not to sell it. I take it you aren't bound by the same promise."
"I don't even remember my grandfather. I was a toddler when my parents divorced, and my mother took me back to Maryland. So, no. I'm not bound by anything," Lauren said. "The syndicate owns the original ranch, including the land where the house was before it burned. But that little parcel wasn't included. According to my dad's lawyer, it's mine now." She gave Will a knowing smile. "But something tells me you're already aware of that."
Will shifted on the sofa. "Then let's get right down to business," he said. "I'm prepared to offer you the same price as I offered your father. I can give you the check today, and you can sign the deed over in town tomorrow." He waited, the expectant silence broken only by the ticking clock, as Lauren took her time. At last she spoke.
"I don't need your money, Will. As a U.S. congressman my father had excellent life insurance, as well as insurance on the house and my grandfather's antique-car collection. I'm the sole beneficiary."
An expression of cold astonishment flashed across Will's face. Sky had nothing but respect for his secret half brother, but it tickled him to see Will put down so handily by a woman. My woman, he thought.
"I've never owned a piece of land before," Lauren said. "I'm not saying I won't sell it eventually. But I want to get to know it first—to explore it and learn more about its history. Maybe then—"
"You know that story about the hidden Spanish gold is nothing but bunk!" Will snapped.
"I know that my grandfather searched every inch of the land and never found it. But this isn't about the gold." Lauren gave Will a few seconds to stew over what she'd said. "I don't want to sell it yet, but here's what I will do. I'll free up the spring so your cattle will have water in the bigger canyon below. And when I sell it to you—if I sell it, which I won't promise—the price will be exactly what my grandfather paid Bull for it. One dollar."
Right then, Sky would have given anything for a camera to photograph Will's face. He looked as if he'd been smacked with a wet fish. But it didn't take long for Will to recover.
"You can stop grinning now, Sky," he growled. "You're the one who'll be living with this woman. Think about that!" Rising, he extended his hand to Lauren. He'd gained some concessions, but he still didn't look happy. "Given a choice, I'd rather pay the money and buy that land now," he said. "I'll try to be patient. But my father won't rest easy in his grave till this is settled."
"I'll keep that in mind," Lauren said. "I'm sorry I never knew your father. From what I've heard about him, he must've been quite a man."
"He was," Will said. "He was more than a man. He was a force."
He still is. Sky kept his silence. But he knew Lauren had meant those words for him, as well as for Will.
* * *
Stella's feet, in their high-heeled red cowgirl boots, throbbed after a night of tending bar in the Blue Coyote. When Will Tyler had fired a bullet through Nicky's innocent heart, she'd not only lost a brother, she'd lost a damned good bartender. She would mourn him for a long time to come. But for now, she'd channeled her grief into rage. Tyler would pay for what he'd done. Before she was finished with them, his whole family would pay.
So far, she'd left his punishment to Abner and the law. But she couldn't depend on the legal system to give her justice, let alone vengeance. She could always use her Dallas connection to call in a hit on the man. But that would be expensive. It would also be too fast and too easy to give her the satisfaction she craved. She wanted to see Will Tyler squirm. She wanted to see him suffer.
Stella had planned to close the Blue Coyote at ten, as she usually did on Sunday nights, but the sad-eyed cowboy in the corner booth, nursing his can of Dos Equis beer, showed no inclination to leave.
She might have given him a gentle nudge out the door, but Stella had recognized the lanky young man. She'd seen him come in a few times with the crew from the Rimrock. Last spring, early on, he'd given Lute Fletcher a few rides to town in his old rust bucket of a pickup. The kid didn't look like much, but it wouldn't hurt to learn more about him.
What was his name? She searched her memory and found it. Ralph, that's what one of the men had called him. She'd make an effort to remember and use it.
Slipping an old Hank Williams CD in the boom box, she turned the volume down low. Then she popped the tab on a fresh beer, sidled over to the booth, and took the seat across from him.
"That beer of yours must be getting stale, Ralph," she said, smiling. "Here, have a cold one on the house."
"Thanks." He accepted the can with a shy smile. He looked young, barely twenty-one, Stella guessed. His eyes were light brown, and his mud-colored hair wanted cutting. The hand that clasped the beer can was nicked and calloused, the fingernails streaked with embedded dirt.
"You look sadder than a hound dog pup, cowboy," she said in her folksiest manner. "If there's anything you need to get off your chest, I'm a good listener."
The melancholy strains of "Your Cheatin' Heart" drifted through the darkened bar. The young cowhand sipped the cold beer, maybe weighing the wisdom of sharing his troubles. After a long moment's hesitation, he sighed. "It's my wife, Vonda," he said. "We had to get married this summer on account of she was in a family way. Mostly it's okay, bein' married. My boss, Will Tyler, let us move into one of the little family houses on the ranch. It's nice enough, and the rent's a lot cheaper than livin' in town, but . . ."
He tipped the can to his lips, his Adam's apple quivering as he swallowed. "It's always about the damned money!" The words exploded out of him. "Will pays as good as most ranchers around here, and we got insurance for when the baby comes, thank God. But that ain't enough for Vonda. She wants to move to town, where she can hang out with her friends. She wants fancy clothes and her own cell phone and her own car to cat around in. She wants a big-screen TV and all kinds of furniture and gadgets for the baby."
He raked his hand through his unruly hair. "Lord, I work my ass off, but cowboyin' don't pay all that much. Tonight, when I tried to tell her how it was, Vonda threw me out. She says I'm not gettin' any you-know-what till I can figure out a way for her to have what she wants."
"How old is Vonda?" Stella asked.
"Sixteen. Just a kid. If I hadn't married her when she got pregnant, her folks woulda thrown me in jail, her daddy bein' sheriff and all. Even then, they threw her out and won't have nothin'to do with us."
Something clicked in Stella's head. So this downtrodden cowboy was Abner's son-in-law. Interesting. She gave him a sympathetic look. "I'm guessing you haven't done any bartending."
"Nope. But I'm right sorry about you losing your brother, Miss Stella. Damn shame what happened." He brightened. "Say, maybe you could teach me bartendin'."
"I'm afraid I need someone with experience." Stella rose and smoothed out her tight denim skirt. "But if you'd like to earn a little extra money, I might could use some help around the place—cleaning up, fixing things, maybe running a few errands."
"Heck, I can do all that stuff!" He was grinning now, as eager as a puppy. "Just let me know what you need."
"I'll think on it, Ralph. Check back with me the next time you're in town. Right now, it's time to finish your beer and go home to that little wife of yours. With luck, she'll be feeling lonesome by now."
As she closed the bar, Stella watched the taillights on the rattletrap truck fade toward the highway. Ralph might not be the sharpest tack in the barrel, but he was desperate for cash, eager to please, and in the right place to be of use. Given time and a little coaching, he could turn out to be helpful.
How loyal to the Tylers was he? But that wouldn't matter. Get him hooked on the money, get him to cross the line, and then threaten him with exposure. With a wife, and a baby on the way, the kid would do anything she asked him to do.
But Ralph couldn't put Will Tyler in prison. She couldn't even count on Abner to do that. Fortunately, a couple of months ago, an ace had fallen into her hands—an ace she would put into play first thing tomorrow.
* * *
After her late-night encounter with Will, Tori hadn't looked forward to spending Sunday night at the ranch. She could've ridden back to town with Lauren or borrowed a spare vehicle from the ranch, but she was still worried about leaving her daughter. Erin had appeared calm and cheerful all weekend, doing her homework and spending time with her beloved colt. But she'd witnessed something no child should have to see. During Abner Sweeney's interrogation she'd seemed almost too composed, her recollection of the shooting almost too clear. Tori suspected Erin was keeping her emotions bottled up inside, where they could fester if not given a chance to heal.
So here Tori was, curled on her side in Beau's former bed, fervently willing sleep to come. But it wasn't happening—not while her memory kept replaying last night's explosive clash with her ex-husband—the accusations, the anger . . .
And the hunger in Will's eyes when he cupped my cheek with his hand.
For Erin's sake, she and Will had maintained a truce over the years, masking their raw wounds with a layer of polite tolerance. Last night had stripped that layer away.
With a sigh of frustration, Tori turned over and punched her pillow into shape. Why did this have to happen now, when she'd finally met a man who could promise her the secure, stable life she'd always wanted?
Drew was kind, romantic, and thoughtful. More important, he gave her respect and treated her as an equal. With Will, the sex had been amazing. But out of bed he'd treated her more like a possession than a companion. Worse, he'd backed his father, who'd insisted that she abandon her law practice to stay home, mind the house, and breed a tribe of little Tylers. She'd tried that. But the miscarriage and the hemorrhaging that followed had come so close to killing her that it had been necessary for the doctor to perform a partial hysterectomy. She'd given Will one perfect daughter. But Bull had never forgiven her for not having sons.
The grandfather clock, brought here from Savannah by Will's mother, chimed one. With an inward groan Tori shifted in the bed and closed her eyes. She was finally beginning to drift when a cry shattered the darkness.
The sound had come from Erin's room.
Tori bolted out of bed, stumbling over her shoes, where she'd left them on the rug. Still in her silk nightgown, she plunged down the hall. Erin's door was open, the darkness inside broken by a shaft of moonlight falling through the window, lending enough light for Tori to find her way.
"Erin?" She could see her daughter now, sitting up in a nest of covers. Sinking onto the bed beside her, Tori gathered her close. Erin was trembling. Her breath came in little hiccupping sobs.
"It's all right. I'm here, sweetheart." Tori stroked the tangled silk of her hair. "What is it? Did something frighten you?"
"B-bad dream," Erin stammered. "So awful."
"Erin?" Will had turned on the hall light. Clad in the old Indian-blanket patterned flannel robe Tori had given him for their first Christmas together, he stood in the doorway. "What's the matter, honey? Are you all right?"
"She's fine." Tori clutched her child closer. "Just a nightmare, that's all."
Will walked into the room and sat down on Erin's opposite side. "We're right here, girl." His throat was still thick from sleep. "You're safe."
Still shaking, Erin freed her arms to wrap around both her parents. They held each other, the three of them, in a tight, awkward circle. Tori could feel Will's warmth, feel the tension in his clasp. Whatever forces had separated them, they would unite in a heartbeat to protect their precious daughter.
"Talking might make you feel better, Erin," Tori said. "Tell us about your dream."
Erin swallowed hard. "There was this man—a man in a motorcycle helmet. He came in the front door with a gun. I heard him and came out of my room. I had a gun, too, right in my hand. I shot him. He fell down . . ." She sucked in air, as if struggling to breathe. "His helmet fell off, and it was you, Daddy. It was you I shot!" She broke into fresh sobs.
"No, don't cry, honey." Will's arms tightened around her. "It was just a dream. I'm fine. We're all fine."
Erin pulled free, gazing up at Will with big, frightened eyes. "Daddy, are they going to put you in jail for shooting that man?"
The partial light from the hall etched black lines of anguish on Will's face. When he spoke, his manner was confident. "Not much chance of that. I fired in self-defense, and I've got the best lawyer in Texas to help me prove it. So go back to sleep, and don't worry your pretty head about it. Hear?"
She hesitated, then nodded. "Okay. I'll try. But could you and Mom stay here for just a little while, till I fall asleep?"
"Sure." Will's eyes flickered toward Tori. "We'll stay as long as you need us." As Erin snuggled into the covers once more, he pulled a chair close to the bedside and sat down.
Tori rose, walked around to the other side and slipped under the coverlet next to her daughter. One arm lay across Erin's shoulders, cradling her close. "Go to sleep," she whispered. "It's all right. You're safe. We're here with you."
Tori closed her eyes. But she could only pretend to sleep. She was sharply aware of Will's presence next to the bed, the sound of his breathing, the shifting of his weight on the chair. Her ex-husband was tough like his father, with the stubborn resolve and mental stamina to see him through any crisis. If he was worried about the outcome of the shooting—as he must be—he would keep it to himself and soldier on. But what about his tenderhearted young daughter? Erin's nightmare had shown just how deeply affected she was. How would she cope if Will was put on trial, or was even convicted?
As Will's attorney, Tori realized, it would be up to her to save her ex-husband—and in doing so, to save their daughter. Whatever else was on her agenda, it would have to be set aside, including her personal life. Only this case could be allowed to matter.
* * *
The dimmed light from the hall cast Will's long shadow across the bed. Nestled in that shadow lay his slumbering daughter and the woman who'd walked out on him eight years ago, changing his life forever.
When they were married, he'd loved watching Tori sleep. But those days were long over. They'd ended when she'd returned from her father's funeral in Florida, madder than a wet wildcat and ready to serve him with divorce papers. She'd stayed in the guest room long enough to find a place in town, and never shared his bed again.
His eyes traced the contours of Tori's face—the chiseled bones, the creamy, golden skin. He could tell by the tautness of her breathing that she was only pretending to sleep. But seeing her with her eyes closed, one arm cradling their daughter, deepened the empty space inside him.
He'd wanted to be a good husband. But so many pressures—the ranch, his father, her need for a career, and finally his own jealousy—had driven a wedge between them. That wedge was still in place, and Tori's new love interest was driving it even deeper.
As a man with a man's needs, Will hadn't remained celibate since the divorce. He'd had brief relationships, a few one-night stands and a few so-called arrangements with women who didn't expect more than an occasional romp between the sheets. None of the women had lasted. None of them had been Tori. But maybe that was just as well. As Tori herself had pointed out, he was married to his ranch—too much competition for any woman to handle, even her.
Still, she looked so desirable with her eyes closed, her soft lips parted, and her hair flowing over the pillow like spilled honey. Overcome by a sudden impulse, he reached out and brushed a fingertip along a silky tendril. He wouldn't do more. The river of hurt between them was too wide and too deep to be bridged by a touch.
Her eyes flew open. She gazed up at him in the darkness, her expression guarded.
"Erin's asleep," Will whispered. "Go back to bed and get some decent rest. I'll stay here awhile."
She eased away from her daughter and sat up, looking uncertain.
"Come on." Will walked around the bed and offered his hand. Sliding her feet to the floor, she took it, allowing him to pull her up without disturbing Erin. Her palm was warm and soft, but he could sense the tension in her fingers before he released them. Her mauve silk nightgown clung to her slender curves, outlining her small, perfect breasts and shadowing the V at the top of her thighs.
Despite his resolve Will felt the heat surge through his body. His sex rose to a jutting erection beneath the old flannel robe he'd thrown on when he heard Erin. He tugged the ends of the sash to make sure the robe wouldn't fall open and humiliate him. He slept in the raw, something Tori would doubtless remember. She might also remember how easily he became aroused. Right now, the urge to have her—to sweep her up and carry her down the hall to his room, fling her on the bed and lose himself between those long, silky legs—was driving him crazy. But the timing was all wrong. The last thing he wanted was to make a fool of himself with the one woman who could bring him down.
Side by side they moved past the bed and out into the dimly lit hall, where they could talk without waking their daughter. Will forced himself to keep his eyes on her face. Tori was no fool. If his gaze wandered to her body in that sexy nightgown, she would be aware of it.
"It's all right," she said. "I can stay with her until morning."
"No, get some rest," he told her. "You've got a busy day ahead, and I'm not sleeping worth a dang, anyway. If you need to go back to town in the morning, you can borrow that spare red truck. But for now, I hope you'll leave Erin here. After what happened tonight, I don't think she's ready to go back to school."
"For once, I agree with you," she said. "I'll talk to the school tomorrow. Whatever happens, Erin's welfare comes first. Agreed?"
"Agreed. Whatever happens." Dark possibilities played like a slide show through Will's mind. How will it feel, he wondered, hearing prison doors slam shut, knowing I'll be an old man by the time they open again for me?
Tori must've sensed his anxiety. She turned toward him, a softness in her eyes. One hand reached up to brush the collar of his worn flannel bathrobe. "I can't believe you're still wearing this old thing," she said. "It even smells the way I remember."
Will felt his chest constrict. She was so close to him, like a butterfly that would take wing if he so much as breathed. He forced a smile. "So how does it smell to you?"
"Like a man. Sweaty and tired from an honest day's work. Like you."
"Maybe I should toss it in the laundry more often." Will mouthed the words, scarcely aware of what he was saying. She smelled like the gardenia-scented bath soap she'd always favored, the same aroma that used to swim in his senses when he buried his face between her breasts. Right now, he wanted to drown himself in her and never come up for air.
Her hand lingered on the collar of his robe. Was it an invitation? A tease? Or just a gesture she had to know how much he wanted her. Did she want him, too?
Will ached to kiss her, to clasp her in his arms and let his hungering hands feel every curve and hollow of her through the thin silk. But that wasn't going to happen. He and Tori had built a cautious trust over the years. They'd made unspoken rules, drawn lines that were not to be crossed. To cross those lines, to shatter that trust now, when he needed her help, could be the worst mistake of his life.
Summoning the last of his resolve, he lifted her hand from his robe and brushed a kiss across her palm. "Get some rest," he whispered. "Good night, Tori."
Releasing her hand, he turned and walked back into Erin's room.
CHAPTER 5
Blanco County prosecutor Clay Drummond was a man at the top of his game. He'd run unopposed in the recent election, standing on his record of toughness, high conviction rate, and absolute incorruptibility. Now at fifty-three, stocky and muscular as a bulldog, with iron-gray hair and a face chiseled in determination, he was setting his sights on higher office—maybe Texas attorney general, if the party would back him. Meanwhile, he had a job to do; and his future depended on his doing it well.
Abner Sweeney's report was waiting on his desk when he arrived Tuesday morning, after a three-day weekend of bird hunting at a friend's cabin. Preoccupied with other concerns, he barely gave the two-page typed report a glance—until two names jumped out at him. The first was Nikolas Tomescu. The second was Will Tyler.
Drummond scanned the report, then read it again, his pulse pounding like a prizefighter's before a title match. News of the shooting must've been all over the media, but he hadn't read a paper or glanced at TV all weekend. Until now, he'd been unaware of what had happened. But whatever had gone down, he needed to take charge of it—ASAP.
This wouldn't be the first time he'd dealt with the Tylers. Last spring he'd constructed an ironclad case for first-degree murder against the second Tyler brother, Beau. He'd assumed the conviction would be a slam dunk. But then, before the trial, the real killer had been exposed. Beau Tyler had gone free, cleared of all charges—and Drummond had been left with a pile of useless evidence and egg on his face.
This time it was Will Tyler, the respected head of the family, who'd run afoul of the law. There'd been no charges filed and no arrest made, pending the inquest. But Abner seemed to think he had enough on Tyler to charge the boss of the Rimrock with manslaughter, or even second-degree murder.
Drummond had no special quarrel with the Tylers. As far as he knew, neither did Abner. But he liked to win. And the press from a high-profile case like this one could jump-start a man's political rise. Both he and the sheriff had personal reasons to find Will Tyler guilty.
As for the victim, Nikolas Tomescu . . .
In the silence Drummond became aware that beneath his fresh white shirt, his body had broken out in a cold sweat. There was a lot more at stake here than just winning. It was as if everything that truly mattered was about to be laid on the line.
In this small community he was admired and envied. He had plenty of money, thanks to his wife's inheritance. He had a perfect family, a respected career, and a promising future. But two months ago, in a weak moment, he'd made one stupid mistake—a mistake that could cost him his marriage, his children, his career, and even his freedom.
Drummond glanced at the list of missed calls on his office phone. One number appeared three times. No messages, but he didn't need any.
It was time to give the devil her due.
He reached for his desk phone, then changed his mind. He took his cell out of his pocket and punched in the number.
"What the hell took you so long?" The husky female voice was unmistakable. "I called you three times, and you never called back."
"I was off the grid," he said. "Got home from a hunt late last night. I just heard about your brother."
"You could at least say you were sorry. Nicky was all the family I had. All I want now is to see Will Tyler pay for what he did—behind bars."
"So do I. That's why I called you. I'd like to make a deal."
"A deal?" She gave a derisive snort. "You're not exactly the one holding the cards, Mr. Prosecutor."
"I know. And I plan to do my job. But I'll work even harder for you if there's something in it for me."
"I'm listening."
A bead of sweat trickled down Drummond's temple. "I want that surveillance tape, Stella, and your promise that there are no copies. I want this whole mess over and done with."
She had the gall to laugh. "How about this deal? If you don't put Will Tyler away, I'll turn the tape over to the press. When you get out of jail, you'll be lucky to get a job cleaning toilets."
Drummond had tried to remain calm and cool, but his anger now boiled over. "If I go down, I'll take you with me. Procuring an underage girl. How's that for a charge?"
"Oh, but there was no procuring. I hired the girl as a waitress. And I had no idea she was underage. She even showed me a fake driver's license. When she fessed up later that she was just sixteen, I showed her the door. I'm guessing she left town. Taking her upstairs was your idea, not mine. Even if it can't be proved she was underage, that surveillance camera caught you with your tidy whities down. Either way, I'd say you were in big trouble."
Drummond could imagine the smirk on her painted face. He swore under his breath. If he thought he could get away with it, he'd be tempted to find the woman and strangle her with his bare hands.
"I'm counting on you. Keep me posted." She ended the call on a maddeningly cheerful note.
Drummond could feel a headache coming on. The pounding grew worse as he recalled how he'd gotten into this mess. His wife, Louise, could be a real bitch when she was in a bad mood. Last August, after one of their nastiest late-night fights, he'd driven to the Blue Coyote for a drink. By the time he'd downed enough Scotch to cloud his judgment, the perky little waitress had begun looking pretty good to him. Friendly and willing, she'd met him out back on her break, took his money, and led him up the stairs. It was only afterward, as he was pulling up his pants, he'd noticed the surveillance camera mounted in a high corner, well out of reach.
Stella had probably paid the girl to target him. Whether she had or not, he was at her mercy now.
The voice of Glenda, the receptionist, startled him out of his thoughts. "Mr. Drummond, Ms. Tyler is here to see you."
Drummond's shoulders sagged. At any other time he'd have been happy to see his former law partner. But Tori Tyler couldn't have picked a more awkward time to show up—especially if she was here to talk about her ex-husband's case.
"Send her in." Drummond straightened his bolo tie and arranged his features in a welcoming smile. Maybe he could at least learn a thing or two from her.
He heard the familiar click of her high heels on the tile floor. An instant later, the door opened and Tori strode into his office. Dressed in tailored slacks, a white silk blouse, and a suede blazer, she took his breath away. She'd always had this effect on him. But even back when she was his junior law partner, and going through a divorce, he'd known better than to lay a hand on her.
The fact that he'd been half in love with the lady for years made his present situation even more painful.
"Hello, Clay." She gave him a friendly smile, but she looked frayed. Drummond knew her well enough to sense that she was worried. And he could pretty well guess why. Will Tyler was her daughter's father—and anything that affected her little girl affected Tori.
He rose, extending his hand. "How've you been, Tori?"
Her handshake was cool and cautious. She'd be representing her ex, of course. She and Drummond had faced each other in court countless times over the years, but this time it would be personal.
"I've been better," she said. "Family crisis, as you're no doubt aware."
"I just read Abner's report," Drummond said. "Sit down. Tell me what's on your mind."
She took the straight-backed chair that faced his desk and pulled it closer. "It's just . . ." She hesitated, very unlike the confident Tori he knew. "It's Abner," she said. "Will shot that man in self-defense. No question. But when Abner came out to the ranch on Saturday, to interrogate Erin and Will, he seemed to have a personal agenda. He was slanting his questions, making it look like Will had shot a man who was no threat to him."
"Nick Tomescu. He was a tough-looking brute, all right. But Abner's report says he had no history of violence."
"I'm aware of that. But it was late at night. The man had a helmet on. And Will had just heard the radio alert on the biker who'd robbed the convenience store. He thought he was facing a criminal who'd already shot one person." She shook her head, more emotional than Drummond had ever seen her. "What if it had been the robber? He could've killed Will and found Erin in the truck. Anything could've happened."
"According to this report, the man had surrendered his gun."
"But he had a knife, raised to throw. Clay, this case should never go to trial. It was self-defense, pure and simple."
Drummond exhaled, feeling for her but mindful of his own dilemma. "For now, that will be up to the judge at the inquest. If the ruling is self-defense, Will's trouble will be over." But mine will just be starting, he thought.
"I heard there might be a jury," Tori said.
"So did I. But evidently that's not going to happen."
"You've probably guessed that I'll be representing Will." Tori had pulled herself together, speaking calmly now. "I'm doing it because he's Erin's father, and she needs him. But even more important, Will is innocent."
Drummond nodded his understanding. "What can I really do for you, Tori? The last thing I want is to hurt you and your daughter."
"You can find out what's driving Abner and why he'd be so determined to punish a man for doing what any father would."
Maybe Stella's got something on Abner, too, Drummond thought. But I'll be damned if I'm going to ask him. "I'll look into it," he lied. "If I learn anything, I'll call you."
"Thanks." She stood. "You've always been a friend, Clay. I know you have a job to do. But I hope you'll at least keep me informed—and, of course, share any new evidence with me as the attorney for the defense."
He watched her walk out, admiring her leggy stride and the way her slacks clung to her shapely hips. Tori was a magnificent woman and a longtime friend. She'd made a good case for her ex-husband's innocence. But given what Drummond was facing, none of that could be allowed to matter. Whatever happened, he had to save himself from ruin. He had to make sure Will Tyler went to prison.
* * *
A cold wind almost blasted Tori off her feet as she stepped out of the county building. Autumn-bright leaves were flying off their branches in a storm of reds and golds. To the northwest, muddy-looking clouds were roiling in over the caprock. The forecasted norther was moving in fast.
Clutching her blazer around her, Tori raced across the parking lot to the old red pickup she'd borrowed from the ranch while her wagon was in the shop. Flinging herself into the driver's seat, she grabbed for the door, which the wind had blown open, and yanked it shut.
For a moment she sat still, catching her breath and thinking about her meeting with Clay Drummond. In the nearly six years she'd worked as Clay's law partner, he'd never been anything but honest and fair with her. She knew his wife, Louise, and his three children—one in college now, the other two in high school. She'd even had dinner in their home. Even though they were on opposite sides of the legal process now, she'd always believed she could trust the man. But today he'd seemed uncomfortable, as if he couldn't wait for her to leave. When she'd argued in favor of Will's innocence, she could've sworn she'd seen the man squirm. What was even more disquieting, he'd kept breaking eye contact while she was talking to him, which wasn't like Clay at all.
Something wasn't right. And she owed it to her client—to Will—to find out what it was.
Will.
How many times had she relived that encounter in the hall outside Erin's room? If she'd made one more move, where would she be right now? She'd seen the hunger in Will's eyes and felt the heat rising between them. The urge to reach down and touch the sash on his old bathrobe had been almost overpowering. One tug at the loose knot would have been enough to push them over the edge. But Will had saved them both. He had kissed her hand and walked away, leaving her weak-kneed and quivering in her silk nightgown.
Time to put the whole incident out of her mind and get on with her day. It wouldn't happen again. She wouldn't give it a chance. Neither, evidently, would Will.
She was fishing her keys out of her purse when her cell phone rang. It was Drew. "Hi," she said, welcoming the diversion. "Aren't you supposed to be in school, Mr. Middleton?"
He chuckled. "Even the principal needs a recess break. How about you? Are you working?"
"Sort of. I've been laying some groundwork for Will's defense, in case he needs it."
"Can't he get himself another lawyer for that?"
"Not one who'll work for free. Since he's Erin's father, and since she was a witness, I really do have to be involved. It's family business."
There was a beat of silence. "All right," he said. "But I confess I'd feel more secure if you weren't so chummy with your ex."
"We're not chummy. We have a daughter, who means the world to both of us. If it weren't for Erin, I'd probably never speak to him again."
"Ouch!" Drew's laugh sounded forced. "Sorry, that's not why I called. Just wondering if you were free for a movie tonight."
She hesitated, checking her mental to-do list.
"I need to see you, Tori," he said. "Besides, you could use a fun chick flick with popcorn and some good old-fashioned back-row snuggling."
"Isn't the back row where your students sit to make out?"
"At least they won't be sitting behind us. Is that a yes?"
She relented. "Sure. My house, seven-thirty?"
"Let's make it seven. We can get pizza before the movie."
"Fine. Seven. See you then." She ended the call. At least she'd have something to take her mind off Will's troubles tonight. Drew had a way of relaxing her, making her laugh. He never confronted her or made unreasonable demands the way Will had done when they were married. And Will would never offer to watch a chick flick just because he thought she'd enjoy it.
Drew is a gem, Tori told herself. If she didn't grab him fast, some other woman would.
As she started the truck, a gust blasted a shower of leaves onto the windshield. Tori turned on the wipers to clear them away. She'd planned to drive to the ranch tonight to brief Will on the case and check on Erin. But given her date with Drew and the chance of a storm moving in, it made more sense to spend the night in town—which brought up the question of what would happen when Drew drove her home.
He'd said he needed to see her. Did that mean he wanted to take their relationship to a new level? Was she ready for that? Tori wasn't a prude. The one brief fling she'd had when she was still reeling from the divorce had been doomed from the start. And she'd never been one for casual sleeping around. She'd come to believe that love, or at least emotional intimacy, should be there before sex happened. Had she reached that point with Drew?
What would he do if she hadn't?
During her musings she'd let the truck idle too low and killed the engine. Shifting down, she started it again and pulled out of the parking spot. Her station wagon was still waiting for the arrival of a new starter. Maybe it was time she began shopping for a new vehicle, something that wouldn't strand her somewhere at night or break down in bad weather. The next time she saw Will, maybe she'd ask him for some suggestions.
Will again. Damn.
Muttering under her breath, Tori ground the pickup's aging gears and roared out of the parking lot.
* * *
After lunch—a beef sandwich eaten off the kitchen counter—Will chose a sturdy paint gelding from the long barn, saddled it, and took the trail up to the back pastures. The windy weather wasn't the best for riding, but he'd wanted to check the stock and the fences before the coming storm. At least that was his excuse.
He didn't really need to go. He'd put Beau in charge of readying the pastures and the cattle for bad weather, and, much as Will was tempted, he'd learned better than to show up and try to supervise. Beau knew his job, and any interference from his big brother would only rile his temper. Will had resolved to keep his distance, but he wanted to get out of the house and see things for himself.
The events of the past few days had left him shaken and out of sorts. He felt the need to ride the ranch alone, to see the land and see himself as part of it. With so much uncertainty hanging over him, he needed a reminder of who he was, why he was here, and what he was fighting for.
Collar raised against the wind, Stetson jammed on and tied under his chin, he rode across the fire-scarred flat and up toward the edge of the foothills. The stiff breeze whipped waves across the yellow grassland and battered his sheepskin coat. A pair of ravens soared on the windy swells, tumbling as if in play.
In the pastures red-coated Hereford cattle clustered with their backs to the wind. After the summer drought and the fire that followed, Will had sold off most of his steers at a loss. The animals that remained were breeding stock—prime cows and bulls and last spring's half-grown calves—his best hope for the next season. If he could keep them fed and healthy over the winter, he'd have a good start on next summer's herd. But if the coming winter turned harsh, the price of extra hay and the calorie-rich cottonseed cake known as "cow candy" could bankrupt him.
In the distance he could see Beau's crew with the flatbed truck, setting up stacks of baled hay to serve as extra wind breaks for the cattle. Two generations ago, when Bull's father, Williston Tyler, had cleared the land for pasture, he'd had the foresight to leave clumps of cedar growing in place. Last summer's wildfire had destroyed many of the scrubby evergreens. A few stands had been spared, but if the storm turned out to be a bad one, the trees wouldn't be enough. Cold would be the worst danger. The cattle were still growing their long winter coats. They'd been given extra feed to strengthen their resistance, and heaters had been installed to keep their water tanks from freezing over. But the worry wouldn't ease till this early storm had passed.
Last summer, after the drought and the fire, he and Beau had taken out a hundred-thousand-dollar short-term bank loan, secured by some acreage, to tide the ranch over for a few months, pending the sale of the steers and Sky's colts. But the cattle had sold low; and with other Texas ranches in as much trouble as the Rimrock, few of their owners had cash to spend on new horses.
At the first of the year, the loan, along with the interest, would be due. If they could talk the bank into an extension, they had a chance of pulling through. Otherwise, they'd have no choice except to lose the land or sell it—a solution that would make Bull Tyler turn over in his grave.
As if spurred by the thought, he headed the horse uphill toward the escarpment. A forty-minute ride brought him to the mouth of a narrow box canyon with high, red sandstone walls. Sheltered from the wind, it was a mystical place. Soft red sand covered the floor. On the side where a sheer cliff rose straight up, a panorama of Native American petroglyphs—wild animals, warriors, mythic spirits, and many, many horses—paraded across the sandstone face, telling silent stories of a past that would never live again.
Will dismounted, tethered the horse, and walked up the canyon, enjoying the peace of the place. But someone had been here recently. For the space of a breath, Will felt the warning prickle at the back of his neck. Then he relaxed as he recognized the prints of Sky's worn soles and Lauren's narrow designer boots. This, he knew, was one of their favorite places.
Near the spot where Will stood, mesquite bushes screened a small, steep side canyon—the disputed canyon that his father had sold to Ferg Prescott years ago for a dollar. The last time Will had been here, the stream in its bed had been dammed at the top. Barbed wire had blocked the entrance with a sign reading, PROPERTY OF PRESCOTT RANCH. But as Will pushed his way through the brush, he realized something had changed. The barbed wire and the sign were gone. Water trickled down the rocks, the sound of it music to a rancher's ears.
Lauren had kept her word. But the parcel was still in Prescott hands, and she had nothing to gain by selling it. Will was doing his best to be patient, but with the threat of jail hanging over him, he needed to get the matter settled. Whatever happened next, he owed it to his father's memory to make the Rimrock Ranch whole again.
* * *
Will returned to the ranch house, hung up his coat and, hearing voices, found Jasper and Erin at the kitchen table, drinking cocoa with marshmallows. "You look like you could use some thawin' out," Jasper said. "Pan's still hot on the stove. Help yourself to what's left."
"Thanks." Will emptied the steaming cocoa into a mug, skipping the marshmallows, which were too sweet for his taste.
"Daddy, can I go out and see Tesoro?" Erin asked. "Sky's out there. I just saw him drive up."
"Have you finished your schoolwork?"
She grinned. "All done."
"Fine, then. But put on a coat. It's brisk out there."
Erin raced to get her coat. The front door opened and closed as she left the house. Will took a cautious sip of hot cocoa and settled back in his chair. He'd hoped to catch the old man alone for a quiet talk.
"I rode out to the petroglyph canyon today," he said. "Lauren promised me Sunday that she'd free up the water in that little side canyon. It's been done. The fence and the sign are gone, too."
Jasper's gaze narrowed beneath his grizzled brows. "But the gal hasn't budged on selling you back that land, has she?"
"She asked for more time. I'm trying to be patient and give her some rope." Will studied the man who'd been more of a father to him than Bull Tyler ever had. "You don't like her much, do you?"
Jasper's scowl deepened. "She seems nice enough, all right. And she makes Sky smile, which takes some doin'. But she's Garn Prescott's daughter and Ol' Ferg's granddaughter, and they was both rotten, no-good skunks! I'll never trust a Prescott as long as I live!"
Will shook his head. "Well, I hope you change your mind, Jasper. When Lauren marries Sky, she'll be family."
"She'll still be a Prescott. I'll wait to pass judgment."
"Speaking of Old Ferg," Will said, changing the subject, "I've always wondered why my dad sold him that little canyon—and for just a dollar. You've been with our family longer than anybody on the ranch, even me. I know there are stories Bull never wanted told. But he's gone, and I need to know. Are you ready to tell me?"
"Maybe." Jasper's mouth tightened as if holding back the secret. Will waited, giving the old man a moment to ponder. When Jasper cleared his throat, Will braced for what he was about to hear.
"This was after your mother was killed in that wreck, you understand," Jasper began. "Bull loved his wife. He mourned her till the day he died. But there was another woman he loved, too. He sold the land because of her—and to protect you and Beau."
Will nodded, knowing better than to speak.
"I'm telling secrets I swore not to tell," Jasper said. "But since I might not be long for this world, maybe it's time you heard. Bull got the woman pregnant. She knew she couldn't expect him to marry her, so she went home to her people in Oklahoma. She left a letter meant for Bull, but Ferg Prescott got his thievin' hands on it first. There were things in that letter that could've dirtied Bull's reputation, if they came out, and hurt his children down the line.
"The blackmailing bastard offered Bull the letter in exchange for selling him that piece of land." Jasper pushed to his feet, a signal that the story was done. "So now you know. That's just one reason why I don't trust the Prescotts, and there are plenty of others."
"What about the woman?" Will asked, already guessing the answer.
"Bull sent a man to find her and give her money for the baby. But he never saw her again."
"She was Sky's mother, wasn't she? Does Sky know?"
"He does. I told him. And I reckon he's told Lauren." Jasper hobbled toward the kitchen door.
"One more thing," Will said. "What about the Spanish gold? Is there anything to that old rumor?"
Pausing in the open doorway, Jasper turned and gave Will a dark glance. "I've told you enough," he said. "That's a story for another time—if I ever choose to tell it."
* * *
Being with Drew was just the diversion Tori needed. Tonight's date—pizza, cokes at the Burger Shack, and a silly romantic movie—had made her feel seventeen again. They walked out of the theater arm in arm.
Now what?
Wind blasted them as they walked down the block to his sleek gray Honda. "You've got school tomorrow. It's probably past your bedtime." Tori managed a nervous laugh. "I sound like I'm talking to Erin, don't I?"
He ushered her to his car and opened the door, the perfect gentleman. "Actually, I'm not quite ready to turn in. How about a nice, grown-up beer at the Blue Coyote? We can wind down and talk a little. Sound good?"
"Sure," Tori answered, hesitant but wanting to please him. She liked Drew, liked him a lot. But was she ready for what he might be leading up to?
They drove the few blocks to the last corner in town, where the cheap neon sign cast a blueish glow over the customers who wandered in and out. Late on a weeknight, the place wasn't crowded. The big-screen TV above the bar was turned off, the classic country music muted and mellow with a throbbing underbeat.
Drew guided Tori to a quiet corner booth, his hand warm and possessive on the small of her back. They took their seats and ordered two Coronas from the tired-looking blond waitress. The girl came right back with their drinks. She looked too young to be working in a bar, but nobody seemed to care.
Tori studied him across the table. He was a handsome man, with regular features and light brown hair that almost matched his good-natured eyes. He wore a brown cashmere sweater under his fleece-lined wool jacket. Flawless conservative style.
"I had fun tonight," she said. "Thanks for talking me into this."
He reached across the table and captured her hand. "I'm hoping the night might get even better," he said. "I'm falling hard for you, Tori. But before I crash and burn, I need to know where we're headed. Are we ready for the 'My place or yours' question?"
Tori had sensed what was coming, but his words had still caught her off balance. Scrambling for a reply that would put him off, without driving him away, she averted her gaze for a moment and glanced around the room.
In the shadowed space behind the bar, a figure stood, holding a glass and a towel—a woman with a buxom figure and flame-red hair. Everybody knew who she was, of course. But what startled Tori was that Stella Rawlins was looking straight at her, those green eyes blazing with pure, murderous hatred.
CHAPTER 6
Chilled by Stella's look, Tori shifted position, turning inward in the booth. Seated at an angle, she could no longer see the woman behind the bar. But she still felt the prickling awareness of those eyes, like a spider crawling up her back.
For the first time, she realized how wise Will had been to insist on keeping Erin safe at the ranch. Stella Rawlins was capable of anything, and now her hatred was focused on the Tylers.
As Will's ex-wife and his lawyer, was she in danger, too? Maybe. But she refused to let that make a difference. She had her life and her work, and nobody, not even Stella, was going to intimidate her.
"Tori?" Drew's polite voice broke into her thoughts. "Did I say the wrong thing?"
She pulled her attention back to him, choosing her words with care. "No, you said the perfect thing, Drew. I appreciate your honesty. I like you a lot—more than a lot. I like the way you make me feel and the way you always seem to be here for me." She licked her lips, a nervous gesture. "When you kiss me, I feel all the right tingles. But I hope you'll be patient a little longer. Right now, while I'm under so much stress, isn't a good time to be swept into something new—not even if it's something wonderful."
Had she been tactful enough? He looked disappointed. But he managed a smile. "If you're under stress, I can think of a great remedy," he joked. "But I understand—at least I'm trying to. Since I happen to think you're a woman worth waiting for, I'll try to be patient. But don't expect me to wait forever, Tori."
"I know better than that. Just a little more time, that's all I'm asking." She nodded toward her half-emptied glass. "I think I'd better leave the rest. Too much beer has an unflattering effect on me."
"Ready to go, then?" Fishing out his wallet, he left a couple of bills on the table. Then he rose, gave Tori his hand, and ushered her outside. The wind had risen to a howl. It whipped Tori's coat around her body as they walked to the car. The air carried the earthy scent of a coming storm. She filled her lungs with it, breathing away the stale, smoky odors of the Blue Coyote and the memory of those hate-filled eyes, watching her from the shadows.
They drove back to her house, saying little. It isn't too late to reverse course and invite him in, Tori reminded herself. She had little doubt that Drew would be a good lover, tender and considerate. But tonight she would be jittery, nervous, and torn by doubts. No, this wasn't a good time. When it happened—if it happened—she wanted to be ready.
He pulled the car into her driveway, walked her to the front door, and gave her a lingering kiss. "Think about what I said," he murmured as she unlocked the door. "Call me if you change your mind."
From the open doorway, she watched his big, sleek car glide out of the driveway. As the taillights vanished down the street, she closed the door behind her and switched on a lamp. The house was quiet. Safe, she thought, unless one of Stella's minions was hiding in a closet, ready to jump out at her. Maybe she should have invited Drew inside. At least she wouldn't have been here alone.
Laughing at her own fear, she walked through the split-level house, turning on the lights. Nothing. She was being silly. All the same, she was glad Erin was safe on the Rimrock with Will.
She paused, thinking of her daughter. When she'd brought Erin to the ranch on Saturday, she hadn't planned on leaving her there. Most of the clothes Erin liked, along with spare underthings, sanitary pads, schoolbooks, and other necessities, were here. Tori had already packed a suitcase for her and planned to take it when she drove to the ranch tomorrow. But with a storm moving in, the roads might be better tonight. She glanced at her watch. It was barely ten-thirty, not too late to change clothes, make the twenty-mile drive to the ranch, and stay the night in Beau's old room.
Fifteen minutes later, dressed in her jeans and her warm ranch coat, she was driving down Main Street, headed out of town. After she passed the last streetlight, the night was pitch-black, darkened by the clouds that had poured in over the caprock to fill the sky. Wind battered the old pickup, threatening to blow it off the road. Tori's fingers cramped on the wheel as she struggled to hold it steady.
She'd been driving about ten minutes when the storm broke in full fury. Lightning streaked across the sky. A fusillade of marble-sized hailstones blasted the vehicle with a clattering roar, covering the road in an instant. Worried but calm, Tori geared down and turned on the wipers. She'd driven in bad storms before. She'd be fine.
The windshield had fogged over. Remembering too late that the truck had no air-conditioning to clear the glass, she punched the defroster button and cleared a spot with her hand. Her headlights showed nothing but white. She was driving blind. But she'd traveled this road hundreds of times over the years. The ranch turnoff couldn't be more than a few minutes ahead. She didn't dare pull off the road and wait. The storm could get even worse, stranding her. She had to get to the house.
The truck crept forward through the swirling whiteness. Hailstorms tended to pass with the storm front, giving way to rain or sleet. Surely, this one would stop in the next few minutes. If anyone in the house had left a light on, she'd be able to see it in the distance and find the gravel lane that turned off the main road. Maybe she should call. But her purse, with her phone in it, was out of easy reach. If she braked to find it, she could kill the engine or skid on the ice-slicked road.
She pushed on, minute after tension-fraught minute, inching forward with the defroster on full blast. By now, she knew she'd missed the turnoff to the lane. But the road's steep edges gave her no room to turn around without the risk of sliding off and getting stuck. What she needed was a wide spot or, better yet, a side road to a ranch or farm where she could drive in, back out, and make the turn.
The hail had given way to a driving, icy sleet that froze on the surface of the road. As the whiteout cleared to a dark gray, Tori could make out the road's shoulder in the headlights. Just ahead, a rutted lane cut off to the right, probably leading to a farm—just what she'd been looking for. Tapping the brake, she eased the wheel into a careful right turn. So far, so good. But she'd only gone a few feet down the cutoff when she realized her mistake. The farm road sloped at a sharp angle from the high shoulder of the road. Under these icy conditions, its steep surface could be too slippery for the truck to back out.
She stopped the vehicle, pulled the hand brake, and shifted into neutral, with the engine still idling. Surely, the old pickup would have four-wheel drive. Tori searched on and under the dashboard, but couldn't find any way to switch it over. All she could do was try to back out.
With a muttered prayer she shifted into reverse, released the hand brake, and floored the gas pedal. The engine roared. The worn tires spun on the slick ice. But the truck didn't move.
She slumped over the wheel, collecting her thoughts. The only option left was to keep going down the farm road and hope it led to someplace where she could find shelter.
Taking a deep breath, she drove slowly forward. Beyond the reach of the truck's headlights, the road descended into a black fog. Between the storm and the clouded windshield, Tori was driving almost blind. She didn't see the electric wire fence and the bulky forms of cattle on the far side of it until she was about to crash into it.
Swallowing a scream, she slammed her foot on the brake. The truck fishtailed and skidded to a stop, inches from the fence.
The engine had died. Shaking, Tori turned off the ignition and pulled the hand brake. She couldn't go forward; she couldn't back up; she'd be a fool to get out of the truck in the storm. She wasn't going anywhere. It was time to find her phone and call for help.
She'd tossed her purse into the backseat with Erin's suitcase. Hooking the strap with her finger, she dragged the purse into the front and fished out her cell phone. The display screen showed a low-battery signal. Tori muttered an unladylike curse. She usually plugged in the phone when she went to bed, but it was too late for that now, and she'd left her car charger in her station wagon.
If I could just reach one person . . .
Mentally crossing her fingers, she scrolled to Will's number and pressed call. Her heart sank as she counted the rings. No answer. When his voice message came on, she spoke rapidly. "Will, I'm stuck off the road in the storm, somewhere past—"
She broke off in midsentence. Her phone had gone dark.
* * *
Will had gone to bed early in the hope of getting some needed rest. But between the storm outside and the worry demons in his head, sleep was impossible. Around ten-thirty, he rolled out of bed, dressed, and took a moment to look in on Erin. Then, shrugging into his sheepskin coat, he went out to his truck.
Icy sleet spattered the pickup as he drove the back roads of the Rimrock, using his powerful spotlight to check every fence, every pasture where the cattle were gathered. Not that he could do much if any of them were in trouble. That would have to wait for daylight. But every animal was precious. In terms of hard cash, the death of any cow, calf, or steer would mean a two-thousand-dollar loss to the ranch.
The crews had done all they could to protect the cattle against cold and wind. But in an open pasture, there wasn't much that could be done about lightning. As a boy Will had seen what one lightning strike could do to a closely packed herd. The memory of those charred, swollen bodies would haunt his nightmares for the rest of his life.
There'd been lightning strikes, dangerously close, as the storm front moved through. Dawn would tell if the lightning had done any damage. Tonight there was nothing he could do.
He was turning around to go back to the house when the spotlight caught a movement along the fence. Driving closer, Will saw that one of the spring calves, probably panicked by the storm, had run headlong into the fence and become caught in the wire. Unless it was cut loose, the young animal wouldn't last till morning. Turning up his collar, Will climbed out in the icy downpour and hauled his toolbox out of the back of the truck. He got a rope as well. If the six-hundred-pound calf tried to fight him, he'd need a way to control it. Maybe he ought to call the bunkhouse for some help. But he remembered then that he'd left his cell phone on the nightstand by the bed. He was on his own.
Fortunately for him, the calf had worn itself out struggling and didn't put up much resistance. Still, it took Will a good twenty minutes, working in the glare of the headlights, to cut through the tangle of wire and free the calf, which loped off bawling for its mother. By then, his hands were numb inside his soaked, half-frozen leather gloves. His teeth were chattering, his clothes clammy against his chilled skin.
He took time to close the hole in the fence and put away his tools. Then he piled into the truck, turned up the heater, and headed back to the house.
At least he wouldn't have to worry about Tori tonight. She'd phoned Erin that afternoon, saying she had a date that night and planned to drive out to the ranch in the morning. She was probably snuggled in a warm bed with that fancy new man of hers right now. Well, why the hell not? Tori was a free woman. She could damn well sleep with anybody she wanted. What was it to him? Right now, all he wanted was a hot shower and a few hours of decent rest before first light.
He'd made it to his bedroom and was peeling his wet clothes off his shivering body when he remembered his cell phone. Reflexively, he reached for it. He'd been out of the house for more than an hour. Tired as he was, on a night like this, he needed to check for messages.
There was only one. Will's throat jerked tight as he heard Tori's frightened voice, cutting off before she could tell him where she was, if she even knew. The fool woman must've decided to come tonight, after all. And she'd been caught in the storm, driving blind in that old truck with its worn tires. Lord, she could be anywhere. He checked the time on the message. She must've called soon after he'd left the house. Wherever she was, she'd been there for at least an hour.
Will grabbed for dry clothes and pulled them on in urgent haste. Somewhere out there, in the storm, lost and cold and scared, Tori was waiting for him to find her.
* * *
The cab of the rusty pickup was frigid inside. Shivering beneath her midweight coat, Tori searched the backseat for some kind of blanket or even an old spare jacket. But the truck had been left clean. She found nothing.
She was tempted to start the engine and turn on the heater, but the gas tank was almost empty—and in this old vehicle, there was the worry of an exhaust leak filling the cab with deadly carbon monoxide. Likewise, if she left the lights on, the truck might be easier to spot. But if no one came by, she could run the battery so low that the truck wouldn't start.
She checked the luminous dial on her watch. It was after midnight. She'd been stuck here more than an hour. There was no way to know if Will had gotten her message, or if anybody was out looking for her.
Maybe she should have called Drew instead. Sensible fellow that he was, he would probably have called the highway patrol. The troopers would have found her by now. She'd be safe and warm somewhere.
But Will? If she'd reached him at all, the man would be out driving the roads in the storm, growing more frustrated and annoyed by the minute. If he found her, she could expect an angry chewing-out all the way back to the ranch for putting herself in danger. She imagined his Bull Tyler voice, as she'd always called it, dressing her down as if she were a misbehaving child.
But even that would be better than not being found.
Teeth chattering, she pulled her coat tighter. If she'd accepted Drew's polite proposition, the evening would have ended very differently. Maybe she'd been wrong to refuse. She liked Drew a lot, and he was great husband material—good-looking, kind, stable, and great with children. She knew several attractive women in town who'd likely jump at the chance to sleep with him. Was she a fool to risk losing a man who could give her a happy life because she wasn't ready to do the same?
A layer of ice had formed on the outside of the truck. Tori could no longer see through the windows. If Will had missed her phone call, nobody would be looking for her. She could be here all night.
How much cold could a body stand before hypothermia set in? she wondered. Was it possible to freeze inside a closed vehicle like this one?
Drained by cold and fatigue, she yawned. What she wouldn't give right now for a warm bed—with or without Drew Middleton in it. Drew wouldn't have had much luck tonight. All she'd want to do was sleep.
Tori's eyelids were drooping. Her head sagged, then jerked up again. She mustn't sleep. She needed to stay alert, to move, to stay warm. But she was so tired, too tired to keep herself from drifting. She slumped over the steering wheel.
Find me . . . Please find me, Will . . .
She jerked awake with a startled gasp. Something—or someone—was banging hard on the outside of the truck. Ice shattered as the heavy hammer broke through, splintering the safety glass on the side window. Through the fog in her mind, a voice, hoarse with strain, shouted her name. Will's voice.
Seconds later, he'd freed the door and yanked it open. In the glare of headlights, he looked like a wild man, red-eyed and unshaven, his woolen cap askew on his head, his coat crusted with ice. As she stirred and sat up, he lowered his arms and, for a moment, simply stared at her.
"What the hell, Tori?" he said.
Tori didn't even try to respond. She tried to climb down from the driver's seat, but her cramped legs buckled beneath her. She fell out of the truck into his arms. He was cold, his bare hands icy, his stubbled chin rough against her forehead. His arms held her painfully tight, their strength almost crushing her.
"Fool woman!" he muttered. "Come on!"
Scooping her up, he carried her to his pickup, which was parked on the asphalt road with its lights on. The engine was idling. She could feel the heater's blessed warmth as he shoved her onto the seat. "Erin's suitcase . . ." she muttered. "My purse. Get them."
Slamming the door, he vanished down the slope, into the dark. In a moment he was back, climbing into the driver's seat and tossing her things, along with the hammer, into the space behind. From somewhere, he pulled out a moth-eaten blanket and thrust it toward her. It was dusty and smelled like the dog, but it was warm. Tori laid it over her legs as he geared down. The truck roared up the road. Within a quarter mile was a farm gate with a wide, level area to turn around. Only when they were headed back toward the ranch did he speak again.
"Damn it, Tori, you could've died out there! You missed the turnoff to the ranch lane by a couple of miles. What were you thinking?"
"I couldn't see. I was lost."
"At least you could've let somebody know you were on your way—even Erin or Bernice."
"It was late."
"Then maybe you should've waited till morning. Three hundred head of cattle to worry about, and I spend half the night chasing all over creation after one mule-headed woman! Do you know how long it took me to find you?"
"Stop browbeating me, Will. We aren't married anymore."
"Then why didn't you call your fancy new boyfriend to come and find you?"
"Right about now, I'm asking myself the same question." Tori glanced sideways at his angry profile, square jaw set, strong hands clamped on the steering wheel. Will would always be Will—stubborn, hard-charging, and determined to be right. He was the most maddening man she'd ever known. Yet, when she'd found herself in danger, he was the one she'd called.
He drove in brooding silence now, turning the truck up the long gravel lane to the house. Sad, Tori thought, how things can change. Fourteen years ago, when she became Will's bride, she thought she'd found heaven on earth. What a naïve child she'd been. She hadn't stood a chance against Bull's domination, Will's duty to the ranch, and, finally, his senseless jealousy over an older man's attentions—a man she could barely abide. That jealousy had struck the final blow to their crumbling marriage.
But all those things were in the past. Now it was only their daughter who kept them tied into some semblance of a family.
"How's Erin?" she asked as he pulled up to the house.
"Fine. She was asleep when I left."
"I saw Stella Rawlins tonight, in the Blue Coyote," Tori said. "The way she looked at me—it gave me the shivers. I realized then that Erin needed to be here with you, out of harm's way."
He reached behind the seat to get Erin's suitcase and hand Tori her purse. "I don't want you messing with the woman. Don't even go into that bar."
"I was safe enough. Drew and I stopped by there for a beer. We didn't stay long."
Tori's legs were still unsteady, the ground slick with ice. She gripped Will's arm as he helped her up the steps, across the porch, and into the dark entryway of the house. He was like a rock beside her, solid and cold.
Releasing her, he closed the door behind them. "Can Drew—" He spoke the name contemptuously. "Can he protect you? Does he carry a gun?"
"I don't know. I never thought to ask."
"Well, you're going to carry one, at least till this nasty business is over. I have a nine-millimeter Kel-Tec that's small enough to fit in your purse, but mean enough to blow a hole in anybody who threatens you. I'll get it for you in the morning." He set Erin's suitcase on the floor, shed his coat, then tossed it over the rack in the hall. "Who knows, maybe Drew could use some protecting, too. According to Erin, he's a mild-mannered type."
Something in Tori snapped. With a sharp intake of breath, she spun to face him. "How . . . dare . . . you?" She kept her voice low, but every word was charged with fury. "How dare you discuss my personal life with our daughter? What I do is none of your business, Will Tyler!"
"Anything that affects Erin is my business. And that includes the men you bring into her life."
His arrogance shoved Tori over the brink. Her hand flashed upward. He made no move to stop her as she slapped the side of his face—so hard that the sound of it cracked like a pistol shot in the room. The impact stung her palm and hurt her wrist. Pain brought tears to her eyes.
Will stood like stone. Only his eyes reacted to her blow, narrowing, darkening. Then his hands moved up to rest on her shoulders, their weight anchoring her in place. His gaze drilled into hers.
"Damn it to hell, woman, I should've left you in that truck to freeze!" he muttered.
In a swift, sure movement, he bent and captured her mouth with his.
Will's crushing kiss went through Tori like a lightning bolt—a flash of heat that melded all the hurt, all the anger, all the loneliness of the past eight years, into one burning rush of need. For the space of a heartbeat, she resisted. Then, with a whimper, her lips parted. Her body softened against his hard planes. Her fingers raked his thick, damp hair, pulling him down to deepen the kiss. He groaned, his hands sliding down over her curves in an act of pure possession, pulling her in closer.
"We . . . mustn't do this . . ." Tori's faint murmur of protest vanished into darkness as if the words had never been spoken. She was shivering with cold. So was he. They clung together, craving warmth, craving intimacy, both of them aware they were careening toward disaster, and knowing that they'd already gone too far to stop.
He swept her down the hall, pausing for the barest instant at Erin's door to make sure their daughter was asleep. Then, in the next moment, they were in his room, ripping off clothes, leaving garments where they fell on the rug, before they tumbled, naked and shivering, into each other's arms and into his bed.
"You're cold." He reached for the down comforter and pulled it over them.
"So are you." She ran her hands over his big, rugged body, remembering every line and hollow, every nick and scar. Only one scar was new—the short, deep gash along his outer thigh where he'd been bitten by a huge rattler last spring and nearly died. That was part of him now, and part of her memory.
Even the way his erection curved slightly to the left was as she remembered, as was the low growl, from deep in his throat, as he mounted between her willing legs and pushed deep, filling the dark, needing place inside her like a man coming home after a long time gone. No foreplay was needed. She'd been ready for him from the moment of that first soul-shattering kiss.
They made love like two dance partners, separated by years, who still recalled the steps. But the music had changed to a throbbing, hunger-driven beat, pounding in its urgency, savage in its demands. Tori stifled a cry against his shoulder as she climaxed, clenching around him in spasms that rocked her to the core. An instant later, he moaned and shuddered, filling her with the warm flood of his release.
For a moment he lay still, his breath easing out in a long exhalation. Then he moved off her, rolled over, and, without a word, sank into exhausted sleep.
That, too, was very much as Tori remembered. Some things never changed.
She slid out of bed and pattered into the bathroom. Will's old flannel robe hung on a hook behind the door. Tori wrapped it around her and walked back to stand beside the bed, gazing down at the man who lay sprawled in sleep like a tired child. Overcome by tenderness and dismay, she shook her head.
Heaven save her, what had she done?
CHAPTER 7
Will woke to silence at 4:15 a.m. Tori was gone from his bedroom, along with her clothes. No surprise there. She probably hadn't wanted to face waking up next to him. And she definitely wouldn't have wanted Erin to discover her in his bed.
He'd needed her last night. Something told him she'd needed him, too—for the first time in eight long years. But he'd be a fool to think their wild encounter had been anything more than a one-night stand. Knowing Tori, he believed she was already beating herself up with regrets. Unless he missed his guess, today would be back to business as usual, with both of them pretending nothing had happened.
Put it aside, he told himself. Right now, he had more urgent concerns than his ex-wife. The morning stillness told him the norther had passed, leaving bitter cold in its wake. There was nothing to do but get up and deal with the damage.
He rolled out of bed and flipped the light switch. Nothing happened. There were probably lines down between here and Blanco, which meant no heat, no coffee, even, till the power crews got out this way. There was nothing to do but get dressed in the dark, go outside, and face the dawn.
He pulled on layers of clothing—thermal underwear, a wool shirt, and a down vest to wear under his coat. Thick wool socks went under his winter boots. In the living room he took a moment to light the fire that was already laid in the fireplace and check the wood box for more logs and kindling. That done, he added his coat, his thick wool cap, and his leather gloves.
He was about to step outside when Tori walked in from the hall. She was wrapped in Will's old flannel robe, her hair tousled from sleep. The memory of her ripe mouth and eager body rose in his mind. He forced it away.
"It's early," she said. "Is everything all right?"
"That's what I'm about to find out. You might as well get some sleep while the place warms up." He turned to go, but her voice stopped him.
"Will, about last night. We need to forget it ever happened."
He'd expected this from her. Still, it stung. "It's already forgotten," he said. "And don't worry, I'm not going to say anything to your new boyfriend."
Before she could respond, he walked out the door and closed it behind him. On the porch what met his eyes confirmed his worst fears.
In the east the sky was paling to gray. The grim dawn cast enough light to reveal the ice-glazed nightmare the storm had left behind. Frozen sleet coated the roads and buildings. Its weight had bowed the willows to the ground and broken branches off the tall cottonwoods. Worst of all, Will knew from experience, the frozen pastureland would offer no forage for the cold, hungry cattle.
The bunkhouse was already stirring. No lights there, either, but smoke was curling from the chimney. All hands would be needed to get hay to the pastures, to de-ice and refill the watering tanks, and aid the distressed cattle. Will could see where Sky had parked his truck with the headlights on in the open doorway of the long barn. Once he'd made sure the horses were all right, he would join the crews in the pastures.
Beau's Jeep was coming down from the east pasture, its familiar headlights bouncing along the rough road. Will watched as it came nearer, apprehension a dark coil in the pit of his stomach. The news would be bad, his instincts told him—as if any news this morning could be good. He braced his emotions as the Jeep rounded the last curve and rocketed into the yard.
Beau braked the jeep to a halt and climbed out. Red-eyed and unshaven, he looked as if he'd barely slept. Will came down the icy steps to meet him. "Bad?" he asked, meeting his brother's eyes.
Beau nodded, his mouth pressed into a tight line. When he spoke, his voice cracked like an old man's. "More than bad. Lightning strike. I counted seventeen dead around the burnt spot in the pasture. Hope to God there aren't more, but we won't know for sure till the sun's up."
Will's knees had gone weak. He braced a supporting hand on the Jeep's warm hood. "Damn," he muttered. "That's all we need to push us over the edge."
Behind him, the front door opened and closed. Tori had come out onto the front porch. Her gaze took in the frozen landscape and the stricken faces of the two men at the bottom of the steps. "What is it?" she asked. "What's happened?"
Beau gave her the news. She'd been a ranch wife long enough to know what it meant. No dramatics, just bear up and move on. She shook her head. "I'm sorry. If there's anything I can do—"
"Just make sure Erin's all right, and keep her inside today." Squaring his shoulders, Will turned back to Beau. "Let's get the men together. We've got a herd to save."
The two brothers climbed into Beau's jeep Jeep and headed toward the bunkhouse, tires crunching on the icy ground.
* * *
Heartsick, Tori watched them go. The death of that many prime cattle would mean disaster for the future of the ranch. The cows and heifers, many of them pregnant, were the backbone of next year's herd, the spring calves a promise of profit next fall. And the two pedigreed Hereford stud bulls, if either was lost, would cost a small fortune to replace.
Will was tough, like his father. He hid his emotions behind a stoic mask. But Tori knew he was devastated. Last night's losses, coupled with the summer's drought and fire, would put the ranch's survival in serious peril. Couple that with the legal charges hanging over him, and Will would be staggering under his invisible burdens.
Until the moment she'd stepped outside this morning, Tori had been preoccupied with what had happened last night in Will's bed. How could she have dropped her guard that way? What, if anything, would Will expect going forward? And how would it affect her growing relationship with Drew?
Now, compared to the morning's disaster, last night was no more than a pebble in her shoe, to be cast aside and forgotten. Like the storm had done, it had come and gone. There was nothing to do but put it behind her and move on.
But Will's hidden anguish tore at her heart. There was nothing she could do about the problems with the ranch. But as his lawyer it was up to her to see that he didn't pay for killing Nikolas Tomescu. Whatever it took, she couldn't let him down. She would question Erin, question Abner and his deputies, inspect the crime scene, scour every legal book she could find for a precedent. She would fight for Will's innocence with everything she had. He had killed in defense of their daughter, and she wouldn't give up until he was cleared of all blame.
* * *
Ralph Jackson slumped on a barstool in the Blue Coyote, so tired he could barely drink the free Tecate that Stella had shoved in front of him. At ten on a Thursday night, most of the customers had cleared out. The others would soon be gone, too. Nobody was paying any heed to the scruffy cowhand hunched over his beer.
"Cowboy, you look like you just got drug through a manure pit behind a mule." Stella studied him across the bar. Her silk blouse was so tight over her ample bosom that Ralph could see the outline of her nipples. He averted his gaze, reminding himself that the woman was old enough to be his mother.
"Been workin' my ass off all week for those damn Tylers," Ralph said. "Diggin' trenches with the backhoe and shovin' in those stinkin' dead cows. Hell, I oughta get double pay for a dirty job like that."
"But you don't, do you?" Stella clucked sympathetically. "How many cattle did they lose?"
"Nigh onto twenty, most of 'em hit by lightning. And I was on the crew that got to bury 'em."
"Poor boy."
For some reason she looked pleased. But that's natural, she thought, remembering that Will Tyler had gunned down her brother.
A week had passed since the storm. Now, as was typical for Texas, the weather was warming again, and the ice had melted. The work of keeping the cattle fed had eased off some. But taking care of cows was dirty work. When he tried to get close to Vonda, she complained that he smelled like a corral. And there was always the money, which never seemed to be enough.
"How's your wife?" Stella asked.
Ralph sighed. "Vonda's mad at me again. She wants to go to the beauty shop in town and get herself some of them fake fingernails. When I told her we didn't have the money, she threw a hissy fit. Locked me out of the bedroom and told me not to come back till I had it."
"Does she know you've been working for me?"
"Yeah. She's all for it, as long as I'm bringin' in extra cash." He looked up at her. "So, have you got any cleanin' up or fixin' to do around here? I don't need much, just enough for Vonda's nails."
"Couldn't her family give you any help? I'd think her father's sheriff job would pay well enough."
"Hell no!" Ralph's fist clenched around the cold can. "Vonda's folks kicked her out when she got pregnant. They won't have nothin' to do with us. I know Abner Sweeney was voted sheriff, but not by me. I can't stand the little turd."
Stella ran a towel over a damp spot on the bar. The last customer had left. Now Ralph was alone with her. "The work around here's pretty well been done," she said. "But how'd you like to make two hundred dollars?"
"I'd like that a lot." Ralph was already counting the money in his head. He'd give Vonda fifty for the beauty shop and keep the rest for himself. He'd been wanting a new pair of boots, but if he bought them, Vonda would know he'd kept money back. Maybe he could just save it up for something big later on, like a new four-wheel ATV or the down payment on a better truck. "What do I have to do for that?" he asked.
"Not much. Just deliver a package, collect the cash from the customer, and bring it back to me."
Ralph wasn't too dumb to figure out what would be in the package. But as long as he didn't know for sure, and as long as nobody got hurt, what was the harm in it? "Sure," he said. "No problem."
"Fine. Come into the back office. I'll give you some directions—and a few rules. We'll see how this goes."
Ralph followed her, noticing how she limped, as if her red high-heeled boots were hurting her feet. He remembered how a friend of his, Lute Fletcher, had done some work for Stella. Lute had become greedy, gotten in too deep, and ended up dead. But Ralph wasn't like Lute. He knew the limits. Just a little job here and there, when he needed spare cash. That's all he'd do. He could walk away anytime he wanted.
* * *
Clay Drummond didn't bother to get up when the sheriff walked into his office. He had scant respect for the annoying little man whose visits always left him in a bad mood. And this morning, Clay was in a bad mood already. Stella had just given him another of her so-called reminder calls, hinting at what could happen if he failed to put Will Tyler away for shooting her brother. Now, as if the day could only get worse, here was Abner in his face.
"You got the notice about the inquest, right?" Abner took a seat opposite Clay's desk.
"I did," Clay said. "It'll be just you, me, the judge, the coroner, and any witnesses we want to call in."
"What about Tori?"
"She can be there if she wants, but only to listen. And Will won't be there at all. The inquest isn't a trial. Its purpose is to examine the evidence and, based on that, determine whether a suspect should be charged and tried. You'll be a witness, of course, and maybe one or two of your deputies."
Abner quite possibly knew all this, Clay thought. But he enjoyed treating the little man as if he were an ignorant bumpkin.
"What about the little girl?" Abner asked. "She saw the whole thing."
"I spoke with Tori on the phone. She doesn't want her daughter put through having to testify. We agreed that, for now, the interview you taped will be enough."
"Well, I want you to know, Drummond, that I plan to do my job. And I expect you to do yours. Those Tylers have always thought themselves a cut above everybody else. It does a body good to see one of 'em go down and face justice like us ordinary folks." Abner took a tissue from a box on the desk and blew his nose. "Do you think we can get Will for murder?"
Abner was like an attack dog straining at the leash. Earlier, Clay had wondered whether Stella was pulling the sheriff's strings, as well as his own. Now he was convinced of it.
"Murder?" Clay shook his head. "Not likely. We'd have to prove malice, and there's no evidence of that. The inquest will be looking at self-defense versus manslaughter, which carries a sentence of two to twenty years in Texas."
Abner smirked. "Even the minimum would take Will Tyler down a peg. What've we got to prove?"
Clay leaned back in his chair. "Tomescu had already surrendered his gun when Will shot him. As I see it, the case hinges on the knife, and whether a reasonable man would see it as a threat. If so, that would argue for self-defense."
"It wasn't much of a knife," Abner said. "Just a little switchblade. Even if Tomescu had thrown it, it wouldn't have done much damage."
"But throwing it could've wounded Will or maybe distracted him long enough for Tomescu to grab his gun back and use it. That's what the defense will argue if this goes to trial. Like I say, it's a fine line."
Abner's face lit. "What if Tomescu hadn't tried to throw the knife at all? What if Will put it in his hand after the man was shot dead?"
"Wouldn't the knife have Will's prints on it if he'd done that?"
"Not if he'd wiped it clean and used a handkerchief or something to put it in the dead man's hand."
Clay frowned. Evidence tampering was a crime, but if Abner was willing to try, it was no skin off his nose. "Could have happened," he said. "What else can you think of?"
"Will's a cool-headed shot. He could've wounded the man instead of blasting him through the chest. Ever see what a thirty-eight can do to a body at point-blank range?" Abner stood. "Will Tyler deserves to pay for what he done. And it's up to us to see that he does."
With that parting line, the sheriff marched out of Clay's office and closed the door with a click. Clay opened his desk drawer, took out a bottle of Lortab, and gulped one down with the last of his morning coffee. Abner Sweeney was a jackass, but at least they were on the same side. And discussing Will Tyler's case with him had clarified Clay's own concerns about the upcoming inquest. Will Tyler was one of Blanco County's leading citizens. He was respected, even liked by most of the people who knew him. Given the evidence, and the mitigating circumstances, there was a good chance the judge would rule against pressing charges.
Will would walk free, and Stella Rawlins would be out for blood.
Clay liked being county attorney, especially with the prospect of moving on to something bigger. He liked being a respected member of the community and having the kind of family life people admired. If Stella released that security footage, everything he'd worked for would be gone—his job, his marriage, his children, and his future. He'd be lucky to stay out of jail. One way or another, he needed to get that tape and destroy it. Until then, he'd have no choice but to do what she wanted.
And what she wanted was for Will Tyler to go to prison.
The physical evidence alone wouldn't be enough to send the case to trial. Neither would the coroner's findings nor even the testimony of the witnesses. That left the judge.
Apart from the juvenile court, there were just three judges in Blanco County. Clay knew them all—decent men, but human, with human failings. They had their weaknesses, and Clay knew how to use them—a small favor with implied repayment, a concession in some unrelated matter, or just a damned good argument. It was something he did well.
And it wasn't as if an inquest was a life-or-death matter. Any room for doubt would be enough to justify sending a case to trial—a trial that could be delayed by weeks, even months, buying him more time to deal with Stella.
Feeling better, Clay picked up the phone and buzzed the receptionist at the front desk. "Glenda, could you find out which judge is on the Tyler inquest and get him on the phone for me? Thanks."
* * *
Lauren had never been an early riser. But sharing a bed with Sky was changing that. When he spent nights with her in town, he was usually gone by first light. If she wanted any morning time with him, she had to get up, too. Now that she was getting used to it, she'd come to enjoy the peace of early dawn and the beauty of the sunrise that came with it. But waking to full alertness at such an ungodly hour was still a challenge.
This morning, ten days after the terrible ice storm, she woke to the aromas of bacon and fresh coffee. Flinging aside the covers, she pulled on her quilted silk robe and pattered into her apartment-sized kitchen. Sky, dressed and ready for the day, was standing at the stove, scrambling eggs. He glanced around with a heart-melting grin. "Good morning, sleepyhead," he said.
"You're fixing me breakfast?"
"I'm fixing us breakfast. Sit down."
She sank onto a chair, blinking herself awake as he passed her a cup of steaming coffee, bitter and black, the way he liked it. Lauren added cream and sugar before tasting hers. Through the kitchen window she could see the barest glint of morning. The weather had cleared and warmed in the past week, but the autumn colors were gone, the grass brown, the trees bare and broken.
"How can I learn to be a good ranch wife if you spoil me like this?" she joked.
"There'll be plenty of time for that." He set two loaded plates on the table and popped two slices of bread out of the toaster. "Eat up," he said.
"If I eat all this, I'll get fat."
"All the more for me to love."
Laughing, she filled her fork. This was the Sky Fletcher few people knew—laughing, teasing, affectionate, and happy. Opening himself to her as he did was the best gift she could have asked for.
"Aren't you going to work this morning?" she asked.
"Soon. There's plenty to do, and I want to be there for Will, in case he needs anything. The inquest is scheduled for this morning. He's putting on a brave face, but if the decision is to charge him, he's going to take it hard."
"How soon will he know?"
"Tori will be at the inquest. She'll call him as soon as it's over. Lord, I hope it's good news. The ranch has enough trouble as it is. We don't need a trial. We need Will."
"How's Erin taking all this?"
"She's one brave kid—doesn't want her dad to know how scared she is. But I think Will's even more worried for her than he is for himself."
"He would be." Lauren sipped her coffee. "Lately I've been thinking about that piece of canyon land Will wants to buy from me. I know it would please him to get it back. It might even take his mind off his troubles for a little while."
"You'd sell it to him for that reason? It's a nice idea, Lauren, but I thought you wanted time to explore the place while it's yours."
"I do," Lauren said. "So why not do it soon? The weather's supposed to be mild for the next couple of weeks. You and I could take Erin with us and spend some time exploring. Or if you can't get off work, I could just take Erin. We could pack a picnic lunch, make it fun for her. When we've explored to our heart's content, then I'll sell the land to Will for a dollar, as I promised I would. That land has been a sore spot between our families since before you were born. It's time we put an end to it and made peace."
Sky reached across the table and clasped her hand. "That's a great idea, and I love you for thinking of it. Do you want me to bring it up to Will, or would you rather wait and do it yourself?"
"Let me do it," Lauren said. "I could use a few points with the ranch family—especially with Jasper."
Finished with his breakfast, Sky rose. "Don't worry about Jasper. He's a prickly old bird, but he'll come around. Nobody could resist you for long."
"Don't count on his coming around anytime soon. Not as long as my last name's Prescott."
"Don't worry, I've got a plan to fix that." He strode to her side of the table, lifted her to her feet, and gathered her close for a lingering kiss. As her body molded to his through the silk robe, Lauren felt the warm stirrings of desire. It would be tempting to coax him back to bed and make him late for work. But Sky wasn't a man to be coaxed into shirking his duties, not even by a warm and willing woman. Maybe that was one of the reasons she loved him so much.
"I'll call you when I know about the inquest," he said.
"Thanks. I'll be waiting to hear."
She kissed him at the door; then she walked to the window to watch his truck drive away. She was far luckier than she deserved to be, Lauren thought. She had her health, her career as an accountant, all the money she needed, and the love of a wonderful man. It didn't seem right that someone as fair, honest, and good as Will Tyler should be surrounded by problems—a ranch that was sinking into a financial quagmire, a possible prison sentence, and a failed marriage to a woman he clearly still loved.
What had happened to drive Will and Tori apart? Will had never talked about it in Lauren's hearing. Tori, although she was Lauren's friend, had never shared the story of her marriage and how it had ended. Lauren told herself that it was a private matter—as such things should be.
Still, she couldn't help wanting to know—in part because she wanted to avoid similar mistakes, but mostly because she cared deeply about these two people and their lovely young daughter. They'd become part of her life—her future family. She wanted to understand them.
Lauren turned away from the window and began clearing away the breakfast dishes. She would give anything to help Sky's half brother and his loved ones through their troubles. But for now, there was nothing she could do but hope and wait.
* * *
Beau found Will in the stallion barn, cleaning out stalls with a shovel and a broom. He was going at it with a fury that Beau understood all too well. Last spring, Beau had been the one facing trial. Things had turned out all right, but he knew how his brother must be feeling.
"Hey." Beau leaned against a partition, trying to look casual. "We pay teenage boys to do that job. You're the boss, not a stable hand."
Will gave him a glance, then went back to shoveling. "What am I supposed to do, sit on the porch with Jasper and wait for the call? Hell, I might as well make myself useful."
"You'll get through this, Will," Beau said. "I know that sounds like a stupid thing to say, but I've been in your shoes. You're a tough man—even tougher than you think you are."
"Don't be so sure of that." Will straightened, bracing the point of the shovel blade on the floor of the barn. "Wait till you have your own child. Then you'll understand. I'm not afraid of prison. I could survive a few years if it came to that. And I know you'd take care of the ranch. But the idea of leaving Erin, having her grow up without me, with the shame of a father behind bars—that's the worst. And if her mother marries that prissy school principal, knowing he'd be raising Erin in my place . . . Lord, that's what I can't even stand to think about."
"But that's not going to happen," Beau said. "You're innocent. The bastard had a knife up to throw at you. You killed him in self-defense."
Will muttered a curse. "Didn't you learn anything in the DEA? It's not about guilt or innocence, damn it, it's about politics! Both Abner and Clay Drummond are counting on a lot of press from this case. Throw Stella into the mix, and anything could happen. They'd see me hang if they could get away with it!"
The jangle of a cell phone startled both men into silence. Mouth tightening, Will reached for the phone in his vest pocket.
"Relax, it's not yours. It's mine." Beau pulled out his phone and took the call. Will tensed, like a man about to receive a blow, as his brother answered, then turned away.
"Yes," Beau was saying. "He's right here, Tori. Yes, I'll tell him." He ended the call.
"She called you?" Will faced him, bristling.
"She thought it might be easier for me to tell you face-to-face," Beau said. "The case is going to trial. Abner's on his way out here to arrest you."
"Call Abner." Will was stone-faced. "Tell him I'll be driving into town. I won't have my daughter seeing me led away in handcuffs."
"I'll call him," Beau said. "But I'll be driving you."
"No need for that," Will said. "This is my problem, not yours."
Beau put a hand on his brother's shoulder. "It's our family's problem. And you're not going through it alone."
CHAPTER 8
Tori waited in the rear entryway to the county building, which housed the sheriff 's department and the jail. Flanked by an armed deputy, Abner Sweeney stood beside her. His freckled face wore a self-satisfied smirk. A set of handcuffs dangled from his plump fist. He swung them back and forth, humming under his breath, a maddening sound.
"I can't imagine you're going to need those cuffs," Tori said. "Will's coming in on his own. He's not going to run away or attack you. You'd be safer putting them on me."
She was only half joking. Right now, it was all she could do to keep from punching the obnoxious little man in the mouth.
He chuckled. "I always did like your sense of humor, Tori. But the cuffs are part of the arrest process. They let the suspect know who's in charge."
The suspect? Will? Tori was still struggling to accept the unthinkable. The judge's decision had left her reeling, her confidence shaken to the core. How could this be happening?
She steeled herself as Beau's Jeep pulled up to the curb. Inside, she might be falling apart, but she couldn't let Abner know that—or Will. Especially Will.
The inquest had raised her hopes at first. As the evidence was presented, she'd felt sure that the conclusion would be self-defense. But at the last moment, the judge, a man Tori had known and trusted for years, had expressed his doubts and called for a trial. Stunned, Tori had looked around for Clay as the room emptied, hoping he might give her some explanation. But the county prosecutor had already left.
Will climbed out of the Jeep. Without waiting for Beau, who'd driven him, he strode up to the jail entrance and through the automatic doorway. Head high, face expressionless, he appeared proud and unafraid. But Tori knew what he must be feeling. His eyes didn't even flicker toward her as the sheriff cuffed his wrists and read him his rights. That done, the deputy led him back to booking, to be searched, fingerprinted, photographed, and humiliated. The sheriff followed, leaving Tori in the entryway.
Beau had come in through the outside door. As Tori turned and saw him—the friend who'd been there since her childhood—something broke inside her. A sob caught in her throat. She began to shake, as if the earth were breaking apart under her feet.
Beau reached her in two long strides and gathered her close. Holding her gently but firmly, he spoke. "It'll be all right. We can get him out tomorrow, after his bail hearing."
"I'm his lawyer. I know that." Her voice wavered. "But I'm scared, Beau. What if I can't do this? What if I let him down and he goes to prison? Maybe Will should hire somebody else."
"You won't let him down." He eased her away, holding her at arm's length. "When it comes to protecting people you care about, you're a tigress, Tori. Will may not be your husband any longer, but he's Erin's father. You won't just be fighting for him—you'll be fighting for her, for our whole family!"
"But what if I fail? What if I lose? The judge in there had every reason to rule in Will's favor. But in the end he went the other way. Something like that could happen again, and it would kill Will. It would kill Erin."
"That's why we have juries," he said. "Will shot that man in the belief that he was saving himself and Erin. Your job is to help those jurors see the truth." His grip tightened on her shoulders. "I've known you most of my life, Tori. You always had your eye on the prize. I've never known you to doubt yourself."
He was getting to her, as only Beau knew how. But the uncertainty was still there. "I've never had so much depending on me—or so much to lose," she said.
"You still love him, don't you?"
Beau's question caught Tori off guard, but she managed a quick recovery. "That's a low blow if I ever heard one," she said.
"You didn't answer my question. Do you still love him?"
The night of the storm flashed through her mind—Will's crushing arms, his kisses, their bodies seeking each other in desperate need. But that hadn't been love. It hadn't even been close.
"I care for him, of course," she said. "He's Erin's father, after all. But right now, that can't be allowed to matter. Will is my client. That's how I need to think of him."
He let her go with a quick hug. "You'll do us all proud, lady," he said. "And when Will's cleared, we'll have a big celebration. For now, let's both get back to the ranch. This'll be a tough day for Erin. She's going to need us."
Leaving her, he headed out to his Jeep. Tori watched him drive away. Beau had given her a good pep talk. But she'd known him since kindergarten and she'd recognized the look in his eyes.
Beau was as scared as she was.
* * *
Abner strolled down the hall to the row of holding cells, where Will Tyler had been taken after the booking procedure. He'd phoned Stella right after the inquest. She'd been pleased as punch. Abner was pleased, too. Putting a Tyler behind bars was no small accomplishment.
Abner and Will Tyler went back a long way. In school Will had been everything Abner wasn't—popular, smart, admired, and rich, at least by Blanco standards. He'd held class offices, gotten the best grades, and dated the prettiest girls, while Abner, a pudgy nobody, had been ignored. Will had never been unkind to the lonely boy. Like the other popular students, he'd simply treated Abner as if he didn't exist.
Abner had always envied the Tyler men—their power, their self-confidence, their brazen masculinity. Over the years that envy had fermented to hatred. He'd watched from behind the one-way glass as Will was strip-searched, dressed in an orange jumpsuit, and photographed face-on and in profile. It was satisfying to see a proud man like Will brought down to the level of a common criminal. It would be even more satisfying to see him behind bars.
The cells were walled on three sides, with bars open to the hallway. Each cell was designed to hold two men, but today most of them were empty. Will would be alone.
As Abner neared the cell, a nervous prickle stole up his spine. Watching Will from behind mirrored glass was one thing. Facing him, even through iron bars, was another.
Stopping next to the wall, just short of the cell, he took a careful peek around the corner. Will was stretched out on the lower bunk, his long legs crossed, his arms supporting the back of his head. His eyes were closed.
Was he asleep? Not likely, Abner surmised, stepping in front of the bars. He was faking it, as if to show his captors how little this humiliating process had affected him.
Even in the ill-fitting orange jumpsuit, which was inches too short in the legs, Will made an impressive figure—like a sleeping lion, relaxed but alert, and still dangerous.
Maybe this was a bad idea, coming here without a deputy along. Abner inched back toward the wall, intent on leaving. But just then, Will opened his eyes. His left eyebrow slid upward. For the space of a long breath, nothing else moved. Then he spoke.
"Do you need something, Sheriff, or did you just come by to gloat?"
Abner drew himself up. "You've no call to say that, Will. It was the judge who put you in here, not me. I'm just doing my job."
"Well, do it somewhere else. I may have to be here, but I don't have to listen to you whine and make excuses. If you've got anything to say that's worth hearing, call my lawyer." Will rolled over in the bunk, giving Abner a view of his orange-clad back.
Seething, Abner stalked back up the hall, toward the booking area. With a few well-chosen words, Will had cut him down yet again, making him feel like a small, powerless nobody. And the arrogant bastard had done it lying down in a jail cell.
Abner's prostate was acting up again today—or maybe it was just stress. He stopped by the men's room to relieve himself. A glance in the mirror confirmed what he knew: He was fat and homely, with a dowdy wife, a house full of kids, and a sixteen-year-old daughter who was about to make him a grandfather at forty. He was undereducated, underpaid, and would likely never advance beyond his present job. The confrontation with Will had brought it all home.
But this fight wasn't over. Will wouldn't be so high-and-mighty once he'd been locked up in the state prison for a few years; and Abner was determined to put him there. Whatever it took, whatever he had to do, the sheriff vowed, he would break Will Tyler and bring him to his knees.
* * *
Clay got the call from Stella as he was leaving work. Her timing was so spot-on that he suspected the woman was somewhere in the parking lot, watching him from her car. "I take it you've heard the news," he said.
"More or less. But I wouldn't mind hearing your take on the story."
"We lucked out with the judge. But getting a jury to convict him won't be that easy. The evidence that Will Tyler shot your brother in self-defense was pretty persuasive, especially the taped interview with his daughter."
"Well, now, that's your problem, isn't it, Mr. Prosecutor?" Her tone made Clay want to wrap his hands around her neck and shake her till it snapped. "Tell me about the evidence," she said.
"No surprises. There was the knife, the two guns involved, Will's flashlight, Nick's helmet, and the packet of cocaine that the deputy found on the bike. The fact that Nick was probably dealing won't help our side any."
She sighed. "Oh, Lordy, I told him to stay away from that awful stuff. If only he'd listened to me."
Her performance was an act, Clay knew. Stella had her fingers in plenty of dirty pies, including illegal drugs. But trying to prove it would be like slitting his own throat.
"What about the witnesses?" she asked.
"Again, no surprises. Abner, the coroner, one deputy, the tape of the girl, and parts of Will's taped interview."
"But nothing that would cast doubt on his story?"
"Not really. That's why everybody was surprised by the judge's decision."
Not quite everybody, Clay knew. Convincing the judge that justice would be best served by a trial had involved some advance persuasion on his part, along with a bottle of very expensive Scotch. A bit unethical? Maybe, but it was how smart lawyers worked the system.
"Well, Clay, it sounds to me like you've got homework to do."
He imagined her licking her chops like a hungry cat. "Any suggestions?"
"You're a smart man. You'll figure something out. You'd better." She let the implication hang.
"I want that tape when this is over, Stella. You'll owe me that much if I win."
She chuckled. "We'll see about that. Ask me again when Will Tyler's on his way to prison."
She ended the call, leaving Clay standing by his car, cursing silently at his cell phone. The day was brisk, but he could smell the sweat under his suit jacket.
The trial date wouldn't be set until the bail hearing tomorrow. But the court's docket wasn't crowded. A manslaughter case shouldn't take more than a few weeks, a month at most, to schedule and prepare. Meanwhile, as Stella had said, he had homework to do.
When it came to threats, the woman wasn't bluffing. If rumors were to be believed, she'd already taken down one powerful man who'd failed to deliver—the late congressman Garn Prescott. If Will Tyler went free, Clay knew she wouldn't hesitate to do the same to him.
He started the car and pulled out of the parking lot, thinking as he drove. Every defense, even a solid one, had its weak spots, and Clay prided himself on being able to find them. This time he would need to be at his sharpest and most ruthless. His career, his family, and perhaps his freedom would be hanging in the balance.
The evidence was straightforward and had been seen by everyone involved. Not much room for manipulation there. He'd have some leeway with jury selection, but Tori would have to approve any juror he chose. Regarding the witnesses, most of them appeared to be favorable to the defense—except for Abner.
Clay remembered their meeting in his office before the inquest. The sheriff had seemed almost as anxious as Clay was to get a conviction. Either Stella had something on him, or he just plain hated Will Tyler. Maybe both.
Abner could be the key to winning this case, especially if he could be manipulated into twisting a few facts. Much as Clay disliked the pompous little toad, maybe it was time to give some thought to an alliance.
* * *
The next morning Will, dressed in a suit Tori had delivered to the jail, appeared before the judge. The proceedings took no more than a few minutes. Bail was set at $15,000, the trial scheduled for early next month. Beau posted the bond with the clerk, and Will was released to go home.
Beau drove him back to the Rimrock in the Jeep, with Tori following in her station wagon. Will had been gone less than twenty-four hours. But the man returning was no longer the man who'd left the ranch yesterday. Will had experienced rage, shame, and humiliation in a way he'd never known before. And he'd been slapped with the cold possibility of losing all he held dear.
As the Jeep turned off the highway and up the long, straight road to the ranch, he gazed out the window at the autumn landscape. The ice storm had drained the rich gold from the grass and stripped the leaves from the cottonwoods and willows. But there was a stark beauty in the pale sweep of the plain, with the russet cliffs of the escarpment jutting against the November sky. Where the creek ran, the leafless willows hung deep bloodred, a slash of crimson against the ecru skin of the land.
Off to the right, the alkali lake, where Jasper liked to hunt wild turkey, had dried to a glittering white smear. Patches of blackened earth and the skeletons of burnt cedar trees marked where the worst of last summer's fire had burned. But the land was already healing. Next spring the grass would grow and the fire-scarred cottonwoods would leaf out. Bright patches of Indian-blanket gaillardias, Tahoka daisies, blue dayflowers, and blooming cacti would dot the prairie with color, and life would go on, as it always did. Beau and Natalie's son would be born. Sky and Lauren would marry and begin their family. Erin would grow into graceful young womanhood.
Would he be here to see it? But with his fate in the balance, Will knew better than to think that far ahead. For the next few weeks, he would live for each day. He would take Jasper bird hunting. He would ride and play chess with Erin, work the stock with Beau and Sky. He would fill his eyes with the sight of Tori and his ears with the sound of her voice. But he would not forget where he'd been or what lay ahead.
Like ammunition for a coming war, he would store up his anger, his outrage, and his hatred of the corrupt justice system that had allowed this to happen. If the trial went the wrong way, he would need it all to fuel his strength.
Erin was waiting on the front porch when the jeep pulled up and stopped. As Will climbed out of the passenger side, she flew down the steps and flung herself into his arms. She was growing long and lean like her mother. Her hair smelled of hay and horses. She didn't speak, but he could tell from the slight jerk of her breathing that she was holding back sobs. With her involvement as a witness in his case, there'd been no way to shield her from what was going on. She knew what her family was facing, and she was handling it with remarkable courage. Will couldn't have been more proud of her.
"I love you, Daddy," she whispered.
A lump rose in Will's throat. As he hugged his daughter close, it was as if he could feel the slow breaking of his heart.
* * *
Parked next to the Jeep, Tori watched as Erin greeted her father and led him into the house, followed by Beau. This was an emotional time that had little to do with her. She would give them a few minutes before she went inside to join them for lunch.
Will was her client now. The man she'd loved, married, and divorced, the man who'd given her Erin, the man who'd swept her away in an explosion of need on the night of the storm—that man couldn't be allowed to matter now. Emotion would only cloud her ability to defend him.
Once more, applying cold logic, she asked herself the forbidden question—the one she'd been asking, answering, and rethinking all the way on the drive from town to the ranch.
If the unthinkable happened and Will went to prison, what would be best for Erin?
Will's daughter loved being on the ranch, especially her time with Tesoro. She loved the house, loved her room, and loved spending time with Bernice and Jasper. To take that away from her would be cruel. But with Will gone, there'd be no reason to keep the big house open. Bernice could retire and move into Sky's half of the duplex, next to Jasper. Beau and Natalie would have their own home. So would Sky and Lauren. Erin would be welcome to visit, but nothing would be the same. The Sunday dinners around the dining-room table, the long talks with Jasper on the porch, playing in the paddock with Tesoro, watching sports on the big-screen TV with the men of the family, and helping Bernice in the kitchen—all these things would be gone for her.
Erin wouldn't just be losing her father's presence. She'd be losing her whole happy, secure world.
But enough, why agonize over something that hadn't happened—and wasn't going to happen? She was going to win Will's case and set him free, Tori told herself. Failure wasn't an option. The stakes were too high for that.
She was getting out of the car to go into the house when her cell phone rang. She glanced at the display. The caller was Drew.
"Hi," she said, "I was just thinking about you—good thoughts."
He laughed. "Nice to know. I was thinking about you, too. Extremely good thoughts. What are you up to today?"
"I'm at the ranch. Some business with Will's case."
She sensed the slight hesitation. "Oh? How's that going?"
"I can't discuss the details. Lawyer-client privilege. But the trial's four weeks off, and it's shaping up to be a fight. I'll have my work cut out for me."
"I take it that means you're going to be busy."
"I am. I'm counting on you to help keep me sane."
She meant what she'd just said. If she let him, Drew would provide an oasis of calm amid the craziness of the upcoming trial. Besides, she needed to give him some encouragement. Otherwise, he could be gone. Distracted as she was right now, she wasn't ready to lose him. When her life slowed down enough to make future plans, she wanted him there.
"I can do more than that, but not over the phone. Are you free for steak and lobster in Lubbock tonight?"
Was she? Tori had work to do, but she'd already come to the decision to make time for him. "That sounds wonderful," she said.
"Seven o'clock? I'll pick you up. And I loved that black dress on you the last time."
"I'll wear it again, just for you. And seven is fine. Gotta go." Tori ended the call, thinking about the need to structure her life into separate compartments—Will's trial, Will's family, Erin as her daughter, Erin as a witness, and her time out with Drew. If she tried to deal with everything at once, she'd be on the fast track to a meltdown.
In the house she found Will, Beau, Bernice, Jasper, and Erin seated around the kitchen table, eating a lunch of grilled cheese sandwiches, baked beans, and potato salad. Tori grabbed a Diet Coke from the fridge and slipped into the empty chair. As if by mutual agreement, they were laughing and exchanging small talk, putting Will's trouble on hold for now.
"Lauren's taking me on a treasure hunt tomorrow," Erin announced. "We're packing a picnic and going up the canyon to look for the Spanish gold."
"Well, don't get your hopes up about findin' it," Jasper said. "That tale about the Spaniards hidin' their treasure before the Comanches killed 'em all is nothin' but a made-up story."
"Well, who made it up?" Erin demanded. "That's what I'd like to know."
"Can't say for sure," Jasper said. "The story's been goin' around since before my time. But I know that Lauren's granddad Ferg Prescott searched every inch of that little canyon—dug it all up by his self and never found a thing."
"Is that why he bought the canyon from Grandpa Bull? Just so he could look for the treasure?"
"You'd have to ask Old Ferg that question. And he's long gone to his grave."
"Well, maybe he didn't look hard enough or dig deep enough. If the treasure's there, we're going to find it!"
Tori listened, enjoying the exchange. It was a relief to be talking about something besides Will's case. Even Will was getting involved in the conversation.
"You know the canyon belongs to Lauren now," he said. "Anything you find will be hers."
"No," Erin protested. "Lauren promised me if we find the treasure, we'll share it with everybody, even Jasper and Bernice."
"Now that's right nice of her," Jasper said. " 'Specially for a Prescott. But I'm not holdin' my breath till you two make us all rich."
"It's not the treasure that matters, Erin," Tori said. "The important thing is that you and Lauren have a fun adventure. I hope you thanked her for inviting you."
"I did." Erin helped herself to another sandwich. "Sky was going to come, too, but he's got work, so it'll be just us girls. Why don't you come with us, Mom? I know it would be fine with Lauren."
"I'm afraid it'll be a workday for me, too," Tori said. "Go and have a wonderful time."
Disappointment shadowed Erin's face. Then she brightened. "The Vegas rodeo finals are on TV tonight. You could stay and watch with Dad and me."
Guilt was like a cold stab between Tori's ribs. "Sorry, sweetheart, but I've got work in town this afternoon. And tonight I've got plans."
Tori's gaze shifted toward Will. He didn't speak, but his expression had darkened. No doubt he'd guessed what those plans were. Tori could imagine what he was thinking. Here he was, facing a life-changing ordeal, and she was running off to go on a date with another man.
Tori looked away, refusing to acknowledge his silent message. As Will's lawyer, she would give her all to win his case and save him. But she wasn't his wife anymore—and the fact that she'd slept with him in a weak moment didn't give him ownership. If she let him, Will would drown her with his need to be in control. That had happened in the past, but it wasn't going to happen again. She had a private life, and tonight she needed a break.
"Say, Will." Beau came to her rescue. "I could use your advice on where to move the cattle next. Maybe this afternoon we could saddle up and ride out to check the grass in the empty pastures."
"Sounds fine to me," Will said.
"Can I come, too?" Erin asked.
Will gave her a fatherly frown. "Is your schoolwork done?"
"I finished it before you got here."
"Okay, you can ride along on old Belle. And tonight we can make popcorn and watch the rodeo finals, just you and me, unless Jasper wants to join us." The look he gave Tori said it all. They were still a family; and at a time like this, it was wrong for her to be away.
But Tori wasn't about to let Will push her guilt buttons. She was going out with Drew tonight and, by damn, she was going to have a good time.
CHAPTER 9
Tori and Drew lingered over their dessert wine, enjoying the panorama of the city, the glow of candles, and the tinkle of piano blues from the adjoining bar. The steak and lobster had been well-prepared, the atmosphere romantic, the conversation easy.
Tori gazed at the man across the table, grateful that, without being asked, he'd avoided all mention of the upcoming trial. He was, at all times, tactful, soft-spoken, and kind. And he was handsome enough to break hearts in his tweed blazer, dark shirt, and tie—not a bolo, but a real silk tie. With the candlelight sharpening the planes of his face, he could have stepped out of a PBS Edwardian drama.
She found herself imagining Will in this upscale dinner club—his big, rugged presence overpowering the intimate space, his restless nature demanding that they finish the meal and get back to the ranch to check on the cattle. He'd be more at home at a barbecue, where he could fill his plate and grab a beer, eat at his own pace, socialize or not, and leave when he was good and ready, with no apologies.
But why was she thinking about Will tonight?
"A penny for your thoughts." Drew's hand slid across the table to capture hers.
"I was thinking how much I needed this break tonight." It wasn't quite true, but close enough. "Thank you, Drew."
"My pleasure. I know how much you must have on your mind."
That's the right thing to say, at the right time, she thought. "What I've got on my mind right now is you," she said.
"I hope you mean that." His hand tightened on hers, smooth palm, no calluses; but then, he was an educator, not a rancher. "I know I agreed to keep things platonic till the trial's over. But if I'm going to wait, I need to know what I'm waiting for." He cleared his throat. "Not to get too personal, but for a woman who's been divorced eight years, you're pretty involved with your ex."
"It's because of Erin. She loves her father and she loves the ranch. My staying involved gives her a sense of family."
"So what if things were to come together for us, and I was in the picture full-time? Would that mean a sort of ménage à trois, with you running back and forth between two men?"
It wasn't so much his question that surprised Tori as his timing. "Certainly not! Things would have to change. Erin's growing up. She could handle a different arrangement."
"But could Will? Why hasn't he remarried? Is he still in love with you?"
His question rocked her, but only for the instant it took to come up with an answer. "Will is married. He's married to his ranch. No woman on earth could compete with that."
He gave her hand a reassuring squeeze. "I think I'm beginning to get the picture. Not that I want to pry, but I could be more understanding if I knew what split you apart."
Tori gave him the Reader's Digest condensed version of her story—how Will had put his father and the ranch ahead of his marriage, how Bull had disapproved of her working, and how things had gone from bad to worse after her miscarriage and partial hysterectomy.
"So you can't have more children?"
"Not the regular way." Will that be a problem for him? Tori wondered. However Drew might feel about having a family, it couldn't be helped. At least she wouldn't have to worry about telling him later. "My father-in-law wanted a houseful of strapping grandsons. He never forgave me for not being able to produce them."
"And Will?"
"Will adored Erin from the first moment he saw her. He'd wanted more children, of course, but when they didn't come, he lavished all his love on her. She was enough." Tori realized her voice had gone tender.
"So what finally happened between you?" Drew asked.
"That's another story. We could be here awhile."
"I'm listening." His thumb stroked the back of her hand, raising a tingle of awareness.
"First off, you have to know that I grew up in Blanco Springs. My father was a county judge."
"So the law's in your blood." He gave her a smile.
"Something like that. While I was away at law school, he retired and moved with my mother to Florida. I came home to Blanco and married Will. Five years later, with the marriage already crumbling, my father died of a heart attack.
"When I got the word, I took Erin and caught the next flight to Florida. Will was too busy with his father and the ranch to leave right then, but he promised to be there in time for the funeral." Tori paused. She'd told him enough. The rest was better left alone.
"Let me guess," he said. "Will didn't show up for your father's funeral."
"That's right. I stayed two weeks to settle things and be with my mother. He never showed up, and I never lived with him again."
And never slept with him—until the night of the blizzard. But that lapse in judgment was best buried and forgotten.
There was more to the story—like the reason why Will hadn't come to Florida, or so much as called her during the two weeks she was away. But even after eight years, that memory was a raw wound. And the truth—that the deathblow to the marriage had been struck by Will's jealousy and distrust—was too painful to share.
She gave Drew an easy laugh. "Next time we'll talk about you," she said.
"There's not much to talk about." He signed the bill the waiter had left on the table and rose to help her with her chair. "I grew up in Omaha, one older sister. Graduated from college with a master's in educational administration. Worked here and there. Came to Blanco for a change of scene—or maybe to find the right woman. What do you think? Have I found her?"
"That remains to be seen." She snuggled into her coat as he settled it around her shoulders and offered her his arm as they walked to the car. She was tired and she'd drunk a little too much wine. But she had to credit Drew for a pleasant, relaxing date.
It was almost eleven when he pulled up to her house and walked her to her door and gave her a long, slow good-night kiss. "I won't invite myself to come in," he said. "But not because I don't want to. You must look lovely in your sleep."
"Thanks for understanding," she said. "I need to start work on Will's case tomorrow, and I'm going to need a clear head. Besides, you've got school, Mr. Middleton."
"Sleep tight, Tori." He feathered a second kiss across her lips and turned away. As she opened her door, he trotted out to his car and climbed inside.
Tori watched him drive away. Their date had been rewarding, she thought. They'd gotten to know each other without crossing any lines. Maybe she'd revealed too much of herself, but Drew had been a good listener and a perfect gentleman. She wanted the chance to know the man better. But preparing for Will's trial had to come first. For the next few weeks, nothing else could be allowed to matter.
She could only hope Drew would have the patience to wait.
* * *
By ten the next morning, the weather had warmed enough to feel comfortable. Wearing hats and jackets and packing a lunch, a flashlight, and a small shovel, Lauren and Erin mounted up and took the trail that wound through the foothills into the deep canyons. The turquoise sky was winter-bright, dazzling above the time-sculpted towers of the escarpment. A quail, perched atop a stunted cedar, scolded them as they rode past. The earth smelled rich and clean.
They rode side by side, laughing and chatting. Lauren was glad she'd decided to take the girl exploring today. After Will's night in jail, Erin had become more and more worried about her father. A day of treasure hunting in the canyon would provide a welcome diversion.
Lauren, too, had felt the need to explore the sliver of land that had caused so much contention between the Prescotts and the Tylers. She'd been there with Sky when they'd freed up the spring and taken out the barbed-wire fence. But he'd been in a hurry that day so they hadn't stayed long. Today she wasn't expecting to find Spanish treasure; but before selling the land back to the Rimrock, she wanted to at least take a closer look. With winter coming, this brief warm spell might be her last chance.
By the time they reached the petroglyph canyon, they were both hungry. They tethered the horses near the canyon mouth, where the animals could graze and drink from the spring. Then they spread a cloth on the sand at the foot of the decorated cliff and feasted on Bernice's homemade bacon sandwiches, topped off with oatmeal cookies and canned root beer.
"Jasper says we won't find any treasure because it's just a story," Erin said. "But what if it's really there? Do you think it might be?"
"My grandfather never found it," Lauren said. "But what do you think?"
Erin nibbled her oatmeal cookie. "Sky always tells me the best way to handle a horse is to think like the horse. Maybe if we want to find the Spanish gold, we should think like the Spaniards. You know, pretend we're trapped in the canyon and the Comanches are closing in, and we have to hide our treasure fast."
"That's a very clever idea," Lauren said. "What do you say we clean up our picnic and try it? You're in charge."
They packed the remains of their lunch, put on work gloves, and gathered up the tools they'd brought. Erin led the way up the box canyon's narrow, steep entrance to the small clearing on higher, more level ground.
Putting down their tools, they surveyed the spot. On the left, a wall of broken rock rose fifty feet above their heads. Fallen boulders and scree lay thick along its base. Willows, on their right, overhung the spring that trickled down into the lower canyon; behind the spring a high bank of crumbling earth sloped upward, then crested and dipped toward pastureland now owned by the syndicate that had bought out the Prescott Ranch.
The ground under their feet was hard-packed sand and gravel, dotted with tussocks of weedy grass. Almost thirty years had passed since Ferg Prescott had dug for buried treasure here and left empty-handed. The canyon looked as if it hadn't been touched since.
Glancing around her, Lauren experienced a strange unease. She'd never known herself to have psychic gifts, but instinct whispered that something dark had happened here—something evil, best left alone.
Erin, however, was all grins and excitement. Asking her to leave, based on a whim of imagination, would be cruel, Lauren decided. For now, she would play along. But she'd keep an eye out for the first sign of trouble.
Erin's gaze traveled up the crumbling cliff. "This is cool!" she exclaimed. "If there's treasure here, maybe the two of us together will be smart enough to find it! Now let's think. We're Spaniards, we've got a chest full of treasure, and the Comanches are coming after us. What do we do?"
"We need to get away." Lauren joined in the game. "But gold is heavy and hard to carry. We'll have a better chance of escape if we hide it now and come back for it later."
"Right." Erin glanced around. "So what do we do with it?"
"We could bury it," Lauren suggested.
Erin thought a moment, then shook her head. "We're in a hurry. Digging a hole in hard ground takes time. And we might not even have a shovel. Besides, we'd have to camouflage the hole when we were finished. That would take time, too. What else could we do?"
"Drop it into the spring?"
"Too easy to see. And if the treasure's in a wooden chest or a leather bag of some kind, the water could ruin it and scatter the gold. What else?"
"Let's look around." Lauren was enjoying the harmless fun. Maybe the unease she'd felt earlier had been nothing more than her imagination. "If the Spaniards were here at all, it would've been two or three hundred years ago, maybe even longer. Any hiding place would have changed on the surface."
"Maybe." Erin made a slow visual circle of the clearing, muttering half to herself. "We're on the run. We have to hide the treasure someplace fast, someplace safe and dry, where nobody will see it."
A curious raven launched itself from the cliff top, circled, and glided down to perch on a nearby boulder. Unafraid, it studied the visitors with intelligent black eyes. Erin froze, staring at it. "The rocks," she said. "We'd hide the treasure in the rocks. It's the only place that makes sense."
The idea made sense to Lauren, too. But over the years tons of rock would have broken loose and crashed down the cliff to shatter at the bottom. The layer of scree piled along the cliff base was at least five feet thick. If any treasure had been buried in the rocks, it would be buried deep.
"Can you imagine how much rock must've fallen down here since the time of the Spaniards?" Lauren asked. "You and I could never dig through it all. If there's more loose rock up on that high wall, it could even be dangerous. Maybe we should just take Jasper at his word and go home."
"Not yet." As the raven flapped away, Erin stood her ground, gazing at the fallen scree. "I read this in a detective book once. When a lot of rocks fall by themselves, they usually land together—you can see it if you know what to look for. And when you look up, the rocks should match where they came from, and match each other. If not, that's a sign the rocks have been moved. I'm looking for a place where the rocks have been moved."
Lauren suppressed the urge to argue. She'd never seen this stubborn side of Erin before. The girl was definitely her father's daughter. And she was so excited, so determined to find what she was looking for. As long as it made her happy, it wouldn't hurt to play the game a little longer.
"There." Erin pointed. "Right there. Look."
Not far from the steep trail that led down into the petroglyph canyon, rocks were piled against the foot of the cliff. Lauren wouldn't have noticed on her own, but the rocks did indeed look as if they'd been moved there. The way they were stacked wasn't how they'd have landed if they'd fallen naturally. And not only did they not match each other, they didn't match the cliff face above them, where they would have broken off.
Erin was beside herself. "There's got to be something behind those rocks! Come on! Let's get them out of the way!"
"Wait!" Lauren held her back. "Let me make sure it's safe." She scanned the cliff face where it rose above the rock pile. It looked solid enough. To make doubly sure, she flung a fist-sized rock at the cliff. The rock bounced off and clattered to the ground, but nothing else moved. "All right, we'll try it," she said, wishing they'd brought hard hats. "But if you hear something break loose, jump back fast."
The sandstone rocks were heavy, but not massive. None of them would have been too large for a strong man to lift into place. Lauren and Erin, however, had to struggle. They started at the top of the pile, loosening the rocks and rolling them off to one side or the other. It was slow going, but Erin's enthusiasm kept them at it.
"I told you we'd find it," she said. "The treasure's got to be here. It's just got to be!"
Lauren kept her silence. By now, she was certain the girl would be disappointed. If these rocks had been piled up centuries ago by treasure-hiding Spaniards, the exposed surfaces would be uniformly weathered, dotted with lichens and overgrown by native plants. These rocks were still clean, meaning they probably hadn't been here more than a few decades. Whatever lay behind them would likely have been hidden, not by Spaniards, but by Lauren's own grandfather Ferguson Prescott.
* * *
Beau got the phone call as he and Will were riding back to the ranch in Will's pickup. They'd spent much of the morning checking the pasturelands. The past summer's fire had burned most of the grass to the ground. By early fall new grass had sprouted, painting the land with promise. But the ice storm had left it brittle, brown, and stunted. There was no way the cattle would survive the winter without extra feed.
After lunch they'd driven up onto the caprock to buy hay from a farmer they knew. The man was fair and honest, but the price of hay had skyrocketed all over Texas. Will's stomach had clenched as he heard the final amount. But, knowing it had to be done, he'd taken out his pen and scrawled the check. The huge, rolled hay bales would be delivered two days from now.
They were headed back down the winding road to the ranch when Beau's phone rang. Mired in his own gloom, Will didn't pay the call much attention at first. Only when he heard Beau arguing did he start to listen.
"This can't be your final decision," Beau was saying. "Look, we've always paid our bills. I can bring in some collateral. Just a few more months, that's all we're asking—"
He broke off with a curse as the call ended. "The bank's declined to extend our loan," he said. "I tried, but they know you'll be going to trial, and they don't want to take a chance. Damn them!"
Will felt surprisingly calm, but maybe he was just numb. "I was expecting something like this," he said. "Getting arrested doesn't exactly raise a man's standing with the bank."
"So what can we do about it?" Beau demanded.
"For now, not much. We've already cut expenses to the bone. When the loan comes due, if we can't pay, we'll have no choice except to lose the land. The only alternative would be to sell it first. The syndicate's got money, and I'm guessing they'd be happy to add some Rimrock land to their spread, especially if they could pin us to the wall and get it for a bargain."
"The syndicate!" Beau swore. "They'd gobble up the whole damned county if they could. Dad would turn over in his grave."
"He'd turn a lot faster if we went under and lost the ranch. This is about survival—especially if I end up doing prison time."
"Don't even think about that," Beau said.
"I have to think about it. If it happens, I want to leave the Rimrock in manageable shape."
"You think I can't handle the ranch alone?"
"Let's not even go there."
They drove in tense silence for a mile before Beau spoke again. "Hey, maybe Erin and Lauren will find the Spanish treasure and share enough money to bail us out. Wouldn't that be something?"
"Dream on, little brother." With a bitter chuckle Will rounded the last bend in the road and set a straight course for home.
* * *
Erin and Lauren had cleared about three feet off the top of the rock pile. Lauren was getting tired. Her back and shoulders ached. She was going to be sore for a week.
She'd paused a moment to massage the small of her back, when Erin gave a shout.
"There's a hole back here! It looks like . . ." She shoved more rocks off the pile, then scrambled up the remaining heap for a closer look. "Lauren! It's a cave! Bring me the flashlight!"
"Get down!" Lauren caught her waist and dragged her back. "There could be anything in there!"
"Anything?" Erin was grinning. "Like Spanish gold?"
Lauren found the flashlight in the pack and switched it on. "Stay here while I check it out," she said.
Lauren crept up the pile of rocks. The prickling danger sense she'd felt earlier was screaming now. She'd never thought of herself as brave, but if anything came flying out of that cave, she'd want it to get past her before it got to Erin.
Peering over the top of the pile, she shone the faint light into the cave. The walls were uneven rock, the ceiling about five feet above the stone-littered floor, which appeared to drop off near the back of the cave. There was no sign of life, not even bats or spiderwebs. But something about the place gave Lauren cold chills—maybe it was the distinct, musky odor that lingered in the cave's stale air.
"What's in there?" Erin demanded.
"Nothing much, but go ahead and look." Lauren moved back and handed her the flashlight. "Remember, I said look. I didn't say climb over. You're not going in there."
Erin clambered up to the opening and shone the light into the cave. "Wow!" she muttered. "I can see all the way to that hole in the back. Maybe that's where they hid the treasure."
"Don't even think about it," Lauren said.
"Okay, for now. But let's move more rocks away. Then it'll be easier to see what's there."
They shoved the rocks to the sides until the pile was no higher than their knees. Lauren was hurting, but she couldn't help being curious. If they hadn't found Spanish gold, they'd at least discovered something. But her danger sense was still tingling. With most of the rock barrier gone, daylight lit the cave. They could see clearly all the way back to the dark opening in the floor. It was irregular in shape, like a jagged crack where the ancient rock had split and settled. At its widest point it was about two feet across.
Erin had picked up the flashlight. "Let's go see what's down there!"
"Wait!" Lauren handed her a rock. "Throw this down first. If we can hear it land—"
"Got it." Erin gave the stone an easy toss. It bounced off the edge and vanished down the dark crack. They could hear the faint clatter as it ricocheted off the sides, then silence. They glanced at each other. The hole was deep.
"Don't worry, I'm only going to look." Erin started forward.
"No—listen!" Lauren gripped her arm. "Listen!"
For the first few seconds, they could barely hear it—a faint, rattling buzz—one, then a second, then a chorus of sound that seemed to echo off the walls of the cave.
"Rattlesnakes," Erin said. "I know that sound. They've got a winter den down there."
Lauren's knees went weak. She'd always been terrified of snakes. "Let's get out of here!" she said, tugging Erin back.
Erin resisted. "We'll be all right. The snakes are down there to hibernate. As long as we don't bother them, they won't bother us."
"That's enough, Erin. We're going."
"The treasure could still be down there. I just want to look—"
"Not now! If you want to see what's down there, come back with Sky—"
"Did I hear my name mentioned?"
Sky had just come up the trail from the lower canyon. As he stepped into the clearing, Lauren's first impulse was to fling herself into his arms. But it was Erin who needed attention.
"What's going on?" As always, Sky was a man of few words.
"We found this secret cave!" Erin spoke up, talking fast. "There's a hole in the back, and I think the treasure might be down there, but Lauren won't let me look."
"For heaven's sake, Erin, it's a rattlesnake den!" Lauren said.
"What would you say to me taking a look?" Sky asked. "Don't worry, I know about snakes. I'll be careful."
Lauren remembered him telling her how Will had been bitten on the thigh by a Texas diamondback last spring and nearly died. "I wish you wouldn't," she said, knowing that Sky would do what he wanted.
He squeezed her shoulder. "It'll be fine. All right, Erin?"
"All right," Erin said, stepping aside for him. "But promise you'll tell me what you find."
"I promise." He pulled leather gloves from the pockets of his thick denim jacket and tugged them on. Erin handed him the flashlight. He frowned at the cheap dime-store item. "Don't you have anything stronger?"
"This was all we could find," Lauren said. "We didn't plan on needing it."
"Guess it'll have to do." He glanced around, picked up a stout piece of a broken branch, and, turning on the flashlight, ducked under the low ceiling of the cave.
Lauren kept her eyes on him. Sky had lived all his adult life on the Rimrock. Surely, he'd be experienced with snakes. But terror gripped her as he dropped to a crouch beside the hole and shone the light down into the opening. For a moment he moved the light around, probably making sure there were no snakes close enough to strike. Then he bent forward, peering into the darkness as if straining to see by the poor light. Lauren's heart crept into her throat as he lingered, looking down for what seemed like an eternity. At last he rose partway and, ducking to clear his head beneath the ceiling, stepped outside.
One look at his grim face told her something wasn't right.
"What did you see?" Erin was all eagerness.
"Mostly just snakes. Plenty of those. But no treasure. I could see all the way to the bottom, and it wasn't there. Sorry, Erin."
"Me too." Her face fell. Her shoulders slumped. "I was really hoping I could help my dad with the ranch. I've heard him and Uncle Beau talking. I know they're having money troubles."
"Oh, honey!" Lauren hugged the girl, her eyes misting. Glancing up, she met Sky's gaze. His jaw was set, his eyes guarded. She knew her man. He was troubled. And something told her it wasn't just about the ranch.
"You saw something else, didn't you?" she guessed. "What was it?"
He handed her the flashlight and pulled off his gloves. His gaze flickered toward Erin. He hesitated, as if weighing the wisdom of telling her, then decided to go ahead. "I saw bones," he said. "They looked like human bones."
CHAPTER 10
The door to Clay Drummond's office burst open. Tori strode in like an Amazon in full battle gear. Even dressed in jeans and a baggy sweatshirt, with her reading glasses perched atop her rumpled blond mane, she was spectacular, Clay thought. But he knew she hadn't barged in here to be admired.
Ignoring his invitation to take a seat, she loomed over his desk. One hand clutched a sheaf of papers, which she shoved in his face. "Take a look!" she said. "I've spent the past two days researching precedents for Will's case. Here's what I found—five similar cases in Texas alone to support a verdict of self-defense. Read them! This trial is a farce—a waste of time and taxpayer money!"
Clay took a deep breath, forcing himself to stay perfectly calm. "It wasn't my decision to try this case, Tori. It was the judge's. Will took an innocent man's life. My job is to prosecute him to the full extent of the law. That's what I intend to do."
Standing, arms akimbo, she braced her fists on her lovely hips. "How long did we work together, Clay? How many times since then have we faced each other in court? I know how you like to win. But, as far as I'm aware, you've always done it honestly. This case feels different, almost as if somebody's got it in for Will. What's going on here?"
"As I told you, Tori, I'm just doing my job." He straightened the papers she'd flung at him. "By the way, there's been a new development. You'll get the official word in a couple of days, but as long as you're here, I may as well give you a heads-up. Abner and I have been reviewing the evidence, both physical and circumstantial. We're in agreement that, along with the present charges against Will, we should add obstruction of justice."
Tori went rigid. "Obstruction? Good Lord, Clay, how did the two of you come up with that?"
"Think about it," Clay said. "First, Will contaminated the crime scene by covering the dead man with a blanket from his truck. Since his DNA, and who knows what else, was already on the blanket, there'd be no way to tell if he'd touched the body."
"Will wouldn't have been thinking about that," Tori said. "His only intention was to keep Erin from seeing the man."
"Second," Clay continued, "Will sent a key eyewitness away from the crime scene before the sheriff arrived, giving her time to think about her testimony, maybe even change it, before her interview."
"Oh, good grief! Why don't you charge me, too?" Tori snapped. "I was the one who picked her up and drove her home. For heaven's sake, Clay, Erin's a child. She was scared and upset. Besides, Will had told her to get on the floor. He didn't know she'd witnessed anything until I told him the next day."
"I'm aware that Erin's a child, Tori. I also know the girl would say anything to help her father." He rose behind his desk. The feeling that he had this beautiful, powerful woman at his mercy was strangely heady, almost erotic. "One more thing," he said. "Just so it won't be a surprise, we'll be calling your daughter as a witness—for the prosecution."
* * *
The afternoon sun was sinking toward the caprock by the time Sky returned with Erin and Lauren from their canyon adventure. Tired and hungry, they unsaddled their horses, rubbed them down, and put them away.
He glanced at Erin as they left the barn, wanting to make certain she was all right. She'd been quiet on the ride home—but then, none of them had felt like talking much. The discovery of the bones in the rattlesnake den had sobered them all. The flashlight had been too weak for a clear look. But Sky had known what he was seeing.
Maybe he should have kept quiet about the bones. They appeared to have been there a long time. And there could be no question of retrieving them from that deep, narrow space, especially with the snakes denned up for the winter. Leave the dead to lie—that would be sound advice. But there was something Sky had kept to himself—something that would compel him to go back to the cave with a stronger light for another look.
The sight of those fragile bones had touched him deeply. He wouldn't rest easy, Sky knew, until he'd learned more about how they'd come to be there. Light and distance may have fooled his eyes, but every instinct told him he'd been right.
He had looked down into that awful darkness and seen the remains of a child.
"Yum! I smell cinnamon rolls!" Erin's piping voice broke into his thoughts as they neared the house. "Bernice told me this morning she was going to make a big batch. Come on in and have some!"
"That sounds wonderful! I'm famished!" Lauren tugged Sky toward the kitchen, where Bernice had just finished icing a big pan of spicy, fresh-baked cinnamon rolls. Jasper sat at the table, waiting for his share.
Bernice greeted the three of them with a smile. "Something told me you'd come back hungry. You're just in time. Wash up and have a seat."
After a quick cleanup, they joined Jasper at the table. Armed with saucers and forks, they dug into the pan of warm, delicious rolls. Bernice poured glasses of cold milk, then sat down to eat with them. "So, did you find the Spanish gold?" she asked.
Lauren shook her head.
"Told ya, ya wouldn't," Jasper said. "See, it's just a story."
"We didn't find gold." Erin spoke between bites. "But we found a cave with a snake den at the bottom of a hole. Sky looked down there and saw some bones, didn't you, Sky?"
Jasper's fork dropped from his hand and clattered to his plate. " 'Scuse me," he muttered, picking it up again. "Gettin' butterfingers in my old age."
"Do you know anything about a cave in that canyon, Jasper?" Lauren asked.
Jasper frowned, looking down at the table. "Seems like I recollect something about a cave. But I never went up that little canyon. Had no call to, 'specially after Bull sold it to Old Ferg."
"So you never heard anything about the bones?" Lauren persisted.
"Nope. Most likely just some old-time Injun that fell down there and died, rest his heathen soul."
Sky thought that Jasper looked distinctly uncomfortable. He'd bet his best saddle that the old man knew a lot more than he was saying. Maybe later, when they were alone, Jasper would tell him the truth. But before asking, Sky wanted to go back to the cave with a strong light and take another look at those bones.
* * *
"You goin' out again tonight, Ralphie?" Vonda looked up from dabbing black polish on her toenails, which she could barely reach over her bulging belly. Her voice, lately, had taken on a whine that grated on Ralph's nerves like the sound of a mewling cat. He could hardly wait to get out the door.
"Gotta go to work, baby," he said. "Stella don't pay much, but with a kid on the way, we can't make it without me workin' two jobs."
"But we don't have any fun since you started that job—not even in bed."
"That kind of stuff could be bad for the baby." Lord, he'd tried. But sex with Vonda these days was like pumping a beach ball. Some men claimed pregnancy made their wives sexier, but Ralph didn't buy it—especially now that he was getting some on the side.
"You're tired all the time. And I'm cooped up here in the house, just gettin' bigger and doin' nothin'! You haven't even took me to a movie since you started workin' for that woman," Vonda whined. "I thought bein' married would be fun, like a date that doesn't end. But it sucks! And bein' pregnant sucks worse! You got me this way, and you owe me better'n this!"
"Hell, I bought you those damn fake fingernails and gave you money for lunch with your friends. I even bought you a TV to watch while I'm gone."
"That old TV is crap! It's not even a flat screen!"
"Well, too bad. I'm doin' the best I can. Just deal with it, Vonda." Ralph walked out and slammed the door behind him.
As he drove his old pickup into town, he dragged on a cigarette to calm his nerves. Stella was expecting him at eleven for a delivery run, and she wouldn't stand for any slipups. He didn't want to risk riling her and getting fired. The money was too good for that.
Ralph thought about how he had stashed away $7,000 in a secret bank account. It was enough to buy a half-decent used truck. But if he showed up with it at home, Vonda would know he'd been holding out on her, and all hell would break loose. She might even be mad enough to phone her daddy and tell him about her husband's part-time job.
For now, he'd be smart just to leave the money in the bank and keep adding to it. He'd be damned if he was going to be stuck with whiny Vonda and a bawling brat for the rest of his life. When the time came to split, he could give her a little money, take the rest, and leave Blanco County for a new start somewhere. Vonda would be okay. Once her folks got a look at the kid, it would be just like in the movies. They'd take her back for sure.
Stella was just closing up when he pulled into the parking lot of the Blue Coyote. He went in the back way, as the last customers were leaving. Angie, the blond young waitress, was just wiping off the tables. She gave him a tired smile, and he knew he'd be welcome in her room when he finished his run. He'd have to give her some money, of course. But what the hell, Stella always paid him in cash, and he'd have plenty to spare.
With the front door locked, and Angie on her way upstairs, Stella ushered him down the hall to the storage room that doubled as her office. She looked tired, but then she always did, after a night of tending bar on her feet. Her lipstick was gone, and her black eyeliner had smudged into the creases around her eyes. His mother would be about her age, if she hadn't died of breast cancer, Ralph thought. But his mother had been a churchgoing woman, nothing like Stella.
"You were Lute Fletcher's friend, weren't you, Ralph?" The question came out of nowhere.
"Can't say we were real good friends. We worked together for the Tylers, and I gave him a few rides to town before he quit the ranch. Never saw much of him after that."
"You know what happened to him, don't you?"
"I know he's dead. Shot by that crooked sheriff."
"I had nothing to do with his getting shot," Stella said. "But Lute was in trouble before it happened. He got greedy. I gave him my trust, and he tried to steal from me. As for his sister, Marie—" She broke off, as if she'd tasted something bitter. "Did you know her? Now, that woman was a real she-devil."
"I never met Lute's sister," Ralph said, feeling a tad nervous. "But I'm not like Lute, ma'am. All I want is to earn whatever you pay me."
"Now that's what I like to hear." She gave him a feline smile. "I've got plans for you—plans that involve a lot more money than you're making now. But first you need to show me I can count on you. Understand?"
Ralph's mouth had gone dry. He nodded.
"Good. We'll talk more later. For now, here's your package. Instructions are in the bag. Now get going."
Ten minutes later, Ralph was on the road, with two packets of cocaine on the seat beside him, hidden under a wrapped cheeseburger in a take-out bag from the Burger Shack. By now he knew the drill. Drive to an isolated spot on Blanco County's network of backroads. Wait for the customer to show up, turn over the package, collect the cash, and take it back to the Blue Coyote for Stella. After she'd given him his cut, he'd be free to go.
He never asked who the customers were, never even looked at their faces, if he could help it. Most of them, he suspected, were local users or small-time dealers who worked nearby cities like Lubbock and Wichita Falls. The less he knew about them, the better.
The source of Stella's drug supply remained a mystery as well. He'd heard rumors she had connections with a Mexican cartel and a powerful Dallas crime family. But these were only rumors. Stella Rawlins played her cards close to her ample chest. Nobody was in a position to accuse her of any crimes—including Ralph himself.
Ralph was startled from his musings by the flash of red and blue lights in his rearview mirror. His heart dropped like a buckshot quail as he pulled over to the side of the road, braked, and rolled down the window, praying he could bluff his way out of the situation.
The sheriff 's vehicle parked behind him. The door opened and the officer climbed out. Lord help me, it's Abner.
"Hey, Ralph." The sheriff was just tall enough to peer in the window of the truck. His headlights illuminated the space behind him. "I recognized your old truck. D'you know you've got a taillight out?"
"No." Stomach clenching, Ralph forced himself to look his father-in-law in the eye. "Thanks. I'll get it fixed tomorrow."
Abner didn't budge. "What're you doing out here at this hour, anyway? Is Vonda all right?"
"She's fine. Just touchy, with the baby so close and all. We had a spat tonight. Nothing serious, just this and that. I'm taking a drive to cool down and clear my head."
"Vonda was always a feisty one." Abner sounded as if he missed his daughter. "But you'd better be getting home to her. She could go into labor, and her all alone in that little house without a car."
"Yes, sir, I'll do that." Ralph started the engine.
"Oh, and be careful on these back roads at night," Abner added. "We've had some reports of illegal drug trade out this way. Run into those scum balls, and they'd just as soon shoot you as look at you."
"Thanks, I'll be careful. I'm going now." Ralph shifted into low and pressed the gas pedal.
"That's a good boy! Go home to your wife!" Abner slapped the fender of the truck as Ralph drove away. Ralph was shaking like a junkie in need of a fix. Running deliveries for Stella was such easy money and paid so well that he tended to forget how risky it could be. Get caught by the law, and you'd wind up in prison. A deal gone bad, or even a case of mistaken identity, and you could wind up dead, like Stella's bartender brother, who'd been shot by Will Tyler. Anything could go wrong out here.
But he didn't plan to stay in this business long, Ralph reminded himself. All he wanted was enough money to buy a decent vehicle, leave Blanco Springs—and Vonda—in his rearview mirror, and never look back. With the new, better-paying jobs Stella had mentioned, he should be able get there even faster than he'd hoped.
Glancing in his side mirror, he saw the taillights of Abner's big, tan sheriff 's vehicle vanishing down the dark road. He'd handled that encounter like a pro, Ralph told himself. Everything had gone fine. And if he played his cards right, things were bound to get even better. Nobody was going to suspect the sheriff 's son-in-law of carrying drugs in his old rust bucket of a truck—not even the sheriff himself.
"Hello, Stella . . ."
The razor-sharp voice pulled Stella out of a deep sleep. The woman bending over her bed was tall, with ropy muscles and long black hair, which hung in strings over her ragged gray T-shirt. An ugly white scar slashed the left side of her lean Comanche face from her temple to the corner of her mouth. In her right hand she gripped a huge, gleaming kitchen knife. Laughing like a witch, she raised the knife high and brought it down in a swooping arc . . .
Stella woke with a gasp. Her heart was pounding, her body drenched in cold sweat. Jerking bolt upright in bed, she stared into the darkness. It's all right, she told herself. I'm safe. She'd been dreaming again, that was all.
Shaking, she glanced at the bedside clock. The luminous digits read three-fifteen. Too early to get up and make coffee. But how could she go back to sleep after that god-awful nightmare? She should've known better than to mention Lute's sister, Marie Fletcher, to young Ralph. It was as if speaking the name had been enough to trigger the dream that had plagued her for months.
Stella had survived and thrived because of her ability to control people. But she'd never been able to control Marie. In fact, it almost had been the other way around. Using her married name, Marie Johnson, the woman had hired on as a waitress. But her real agenda had been to avenge her brother's death and take over Stella's operation. After Stella hired a Dallas hit man to take Marie out, Marie had fled on her motorcycle, leaving the gunman to burn to death in his blazing car.
Marie was still out there somewhere, and Stella had no doubt that someday she'd be back for revenge.
Now that Nicky was dead, Stella had lost her only protector. She'd changed the locks on the Blue Coyote, bought extra fastenings for the doors and windows of her apartment, had an alarm installed on her Buick, and kept a gun within reach, even in the bathroom. But nothing could lock out her fear, or those blood-chilling dreams.
The bedroom was cool. Stella swung her feet to the floor; she reached for her Chinese silk robe and pulled it around her. At this hour there'd be nothing on the living-room TV but infomercials, shopping shows, and religious rants. But anything would be better than going back to sleep and waking up in the nightmare again, with Marie looming over her bed.
In the kitchen she took a cold beer from the fridge, popped the tab, carried it to the sofa, and switched on the TV. The pitch woman on the shopping channel was selling fake Navajo turquoise jewelry that was probably made in a Shanghai sweatshop. Stella stared blankly at the screen, her thoughts elsewhere. Maybe it was time to pull up stakes and leave the country. There were quiet places in Mexico where Marie would never find her. She had useful contacts there and enough money to last her for years. She'd be fine.
But she had unfinished business here in Blanco Springs. Will Tyler had murdered her brother, and she couldn't walk away until she'd seen the high-and-mighty son of a bitch pay for what he'd done. She'd been counting on the law to put him away, but the process was taking far too long. The trial was still two weeks away. Meanwhile, Will Tyler was out on bail and sitting pretty. She'd wanted him to suffer, and he was doing far too little of that. She was getting impatient. She wanted some action. Maybe it was time she took matters into her own hands.
With Slade Haskell and Hoyt Axelrod both dead, she was short-handed when it came to taking vengeance. All she had was a friendly sheriff, a county prosecutor who was scared to death of her, and a willing but inexperienced young flunky. But she'd managed with less. There had to be something she could do.
It would have to look like an accident—one that couldn't be traced back to her. And it would have to be devastating, something that would strike at the very heart of the Tylers' ranch.
Stella lit a cigarette, inhaled, and blew a smoke ring up into the darkness. Whatever her plan turned out to be, she'd enjoy thinking about it for the rest of the night.
* * *
Tori drove up to the house, parked, and stepped out of the station wagon with her briefcase. Looking across the yard, she could see Will and Erin standing by the paddock fence, watching the spring-born foals romp in the late-November sunshine.
Something tightened around Tori's heart as her gaze took in the two of them. Erin was pointing toward her young palomino, exclaiming about something. Will was nodding, listening to every word, as if memorizing the sound of her voice.
With Thanksgiving three days away, and the trial scheduled for the following Wednesday, these precious days were all about family. No one at the ranch had expressed any doubt that Will would be acquitted; but everyone, including Will, seemed to be quietly preparing for the worst.
Days ago Tori had notarized a document giving Beau power of attorney to sell Rimrock land without the need for Will's signature. Either way the trial went, with the hundred-thousand-dollar bank loan due with interest by January 1, they'd have to find a buyer for the land or lose it to the bank. There was no other way to pay the money off. These were dark days for the Rimrock. But at least this year, the ranch family could celebrate Thanksgiving Day together.
Tori was crossing the yard to join Will and Erin at the fence when her cell phone rang. It was Drew. She stopped to take the call.
"Hi," he said. "I was hoping I could catch you before I left for my sister's."
"Aren't you supposed to be in school today?" she asked.
"No, I decided to play hooky. Since I'll be driving to Omaha and back for the big day, I thought I'd take the whole week off. The assistant principal can ride herd on the students for a few days."
"Well, here's wishing you a great trip." Tori had neglected Drew for the past week. She hadn't meant to do that. But the coming trial and Will's other problems had drained her time and energy until there was nothing left for her so-called social life. "Sorry I've been so unavailable," she said. "I hope you'll let me make it up to you later."
"I plan to give you plenty of chances," he said. "But I want to leave you with one thought. I've stood back and been patient while you've dealt with the trial and the issues related to it. But once that trial is over, no matter how it turns out, you'll need to decide between me and Will. I'm falling in love with you, Tori, but I won't play second fiddle to another man. If you and I are going to make this work, you'll need to be on board with me a hundred percent. Understand?"
"I understand, Drew," Tori said. "You've been a saint through all this, and I promise I won't keep you waiting much longer."
"That's what I wanted to hear," he said. "I'll call you when I'm back, and I'll see you again after the trial."
"Be safe," Tori said, and ended the call. Drew was the perfect gentleman. She really did care for him and wanted to see where their relationship might go. But why did he have to add to the pressure on her now, at a time when she was already a bundle of anxiety?
Slipping her phone back into her purse, she reached the fence and took her place next to her daughter. Will acknowledged her presence with a nod and a flicker of a smile. He was putting on a brave face, but Tori knew the thought of what lay ahead was tearing at his heart.
The foals, all legs at this stage, were romping in the paddock, pushing, rearing, nipping, and galloping. It was serious play, strengthening their young bodies and building their survival skills, as well as their social skills with other horses. It was possible to see which animals, even at this young age, would be dominant later on.
"Look, Mom! Look at Tesoro!" Erin pointed to her golden foal, racing ahead of the others. "See how fast he is! And you can already tell he's the one in charge."
Tori took a moment to admire the beautiful young colt. His palomino coat gleamed in the sunlight. His creamy mane and tail fluttered in the breeze as he ran. Tesoro was going to be a magnificent stallion. But Tori still worried about her daughter owning and raising such an animal. She'd urged Will to have Tesoro gelded for Erin's safety, but it was a lost argument. Not to breed such a valuable creature would be unthinkable, and Erin, who'd been promised the foal before he was born, would have no other.
"Maybe I didn't find the Spanish gold," Erin said, "but we have a golden treasure right here, don't we, Daddy?"
"We certainly do." Will squeezed her shoulders. Lauren had told Tori about Erin's wanting the Spanish gold to help save the ranch. Tori had resolved to wait and tell Will about that later. Right now, his emotions were raw enough.
Will glanced down at the briefcase in Tori's hand. "I see you've come on business," he said.
"I'm afraid so. I was hoping to go over some trial notes. If you're busy with Erin, it can wait."
"No, it's fine." Tori could tell he was weary of the whole business. "Let's get it over with. Erin, don't you have schoolwork to do?"
"A little. If I finish in time, can I go with you to check the herd?"
"We'll see."
They crossed the yard to the house, the three of them walking together with Erin in the middle. Once inside, Erin went to her room. Tori and Will settled themselves in the den with the paperwork on the coffee table between them. From the open ranch office across the hall, they could hear the sound of a ringing phone and Beau's muffled voice as he picked up the call. The closing door cut off whatever they might have heard of his conversation.
Tori found her reading glasses in her purse and slipped them on. She looked up to find Will watching her from his chair. His impossibly blue eyes were ringed in shadow. In all the years she'd known him, she'd never seen him look so tired.
"Are you all right, Will?" she asked gently.
"I'm fine, considering." He shifted in his chair, one hand reaching up to massage the back of his neck. "I'm just so damned weary of this whole mess, Tori. All I want is to have it over and done with. The crazy thing is, if I had that night to do over, I wouldn't have done any different. Whoever the bastard turned out to be, I couldn't take a chance on letting him get to Erin."
"And I'd have done the same in your place," Tori said. "Now let's get to this witness list the prosecution sent me, along with the names I've added, like the nine-one-one dispatcher on duty that night."
"Carly will vouch for me," Will said. "When I called in, I was sure I'd killed the robber."
"She should have a recording of the call. If I can play that, it'll be even more powerful than her testimony." Tori started at the top of the list. "Abner?"
Will shook his head. "I can't figure him out. I've known him since grade school and, in all that time, we've never exchanged an unfriendly word. But I could swear he's out to get me. If we can find out why and use it somehow on the stand—"
"Will." Beau had left the office and walked into the den. "I just got a phone call you need to know about."
Will sighed. "I don't suppose it was the bank calling to say they've changed their mind."
"No. It was Bob Stevens, the new syndicate manager for the Prescott Ranch. He had an interesting proposal."
"He wants to buy some land?"
"Not land. It seems Bob has a wealthy friend in San Antonio with a passion for palomino horses. They were talking, and Bob happened to mention that we had a palomino foal with a lot of promise. The man wants to fly out and look at Tesoro. He said that if the foal's everything he expects, he'll offer you seventy-five thousand for him. Throw in his mother, and he'll make it an even hundred thousand. That would be enough to pay off the bank loan and save the ranch."
Tori sat silent, her eyes on Will. She could imagine what was going through his mind. This was a way out, a way to keep the Rimrock intact and honor the family tradition that no part of it should be sold.
All he had to do was break his daughter's heart.
CHAPTER 11
Mouth set in a flat line, Will rose from the chair, walked out of the room and down the hall. He knew Beau was waiting for some kind of response, but he had none to give him. Right now, what drove him was the need to be alone.
Entering the ranch office, he closed the door behind him, raised the Venetian blind that shaded the wide window, and stood looking out across the ranch yard toward the barns, then to the paddock, where the foals were still romping. Even at a distance Tesoro's coat shone like a polished gold coin in the sunlight.
Beyond the paddock the rolling hills of the pastureland rose to meet the crags of the escarpment. In the pitiless blue sky, a lone vulture circled on outstretched wings.
Will had lived his entire life for the Rimrock. From the time he was old enough to shovel his first forkful of hay, he had taken care of the land and the animals. He had labored till his hands bled, sacrificed a college education and any chance he might've had to see the world—and he'd lost the only woman he'd ever loved when he'd been forced to choose between her and this ranch.
Now it had fallen to him again, the duty to keep the Rimrock whole—but not without another agonizing choice.
Selling Erin's beloved foal and the mare, plus some interest they could pay out of pocket, would clear them with the bank, keep their credit in good standing, and save the acreage that would otherwise be lost. But Erin would be heartbroken. She would never trust him again. Worse, she would learn from this that honor was an illusion, and any promise could be broken on a whim.
Turning away from the window, he studied his father's picture, where it hung on the wall. Even in the black-and-white photograph, Bull had the look of a man who never gave in, never gave up, and never stepped aside.
What would you have done, Dad? Will asked silently, as he often did when faced with a tough decision. How would you have handled this in my place?
But this time, Will realized, he already knew the answer. To the Bull Tyler whom Will remembered, the land had been more important than love and family, more important than life itself. A horse and a child's tears would have meant nothing to him. For all Will knew, Bull would have sold his entire family down the river for the sake of this ranch. In a way he almost had.
Gazing at that face, Will felt a sudden flash of understanding. All his life he'd tried to measure up to his father and had failed. And now he knew why.
I'm not you, Dad, he said, continuing the silent conversation. I may look like you, maybe even talk and act like you sometimes. But I'm not you, and I'm not going to make the decision you would have made. I'm not going to destroy my daughter's happiness for a piece of earth that has no mind, no heart, and no memory. If you don't like it, fine. When I get there, we'll settle our differences in hell.
Decision made, he opened the office door and walked back into the den. Beau was standing by the bar. He'd opened a beer from the miniature fridge. His grip tightened around the can as Will appeared. "Well?" he asked.
"Tesoro isn't mine to sell," Will said. "I promised him to Erin, and we're keeping him. So you can tell Bob's rich friend that the answer is no."
The can came down on the bar with a thud, splattering beer on the mahogany surface. "Are you crazy?" Beau demanded. "That foal is ranch property, and we can sell him if we have to. Erin can always choose another foal to raise and fuss over. But that palomino is the key to saving our land."
Will shook his head. "I can't believe this. You sound like Dad."
Beau's eyes flashed a startled look, but he swiftly recovered. "Well, you know what Dad would say if he was here."
"I do, and for once I don't agree with him. I won't betray Erin and see her hurt. You're about to have a child of your own. When you do, you'll understand."
"I can't believe this!" Beau stalked out of the den, crossed the hall into the office, and slammed the door behind him.
Tori had been so quiet that Will almost had forgotten she was there. Now she rose from the couch and walked toward him. Without a word she reached out and pulled him to her in a hug that couldn't be mistaken for anything but gratitude. "Thank you," she whispered. "For Erin and for me."
Will waited until she'd stepped back. "I need you to do something for me. As you know, if I go to prison, Beau will have power of attorney to sell ranch property. I need a separate document drawn up declaring Erin sole owner of her foal."
"I'll have it for you to sign tomorrow. But do you really think that's necessary?"
"I hope it won't be." Will didn't like going against his brother, but this needed to be done. "Beau means well. But his idea of what's right doesn't always match mine."
Tori nodded. "How much do you think we should tell Erin about this?"
"No more than we have to. The idea that Beau would've sold Tesoro to pay off the bank would just upset her. She's already got enough grief on her young shoulders. I just wish to God I could spare her this mess."
Will turned away, but she seized his wrist. Her grip was surprisingly strong as she yanked him back around to face her. "Listen to me, Will Tyler!" Her eyes blazed, passionate and falcon fierce. Her voice rasped with emotion. "We're going to beat this, do you hear? You're innocent of any wrongdoing. I'm going to prove it to the world, if I have to take on the whole damned justice system. So help me, I won't let you lose your freedom! I'll fight this to the last breath in my body!"
Her honest gaze burned into him, its heat pure enough to sear all pretense from his soul. He'd prided himself on having the strength to stand alone. But he'd never needed anybody the way he needed this woman.
Resistance crumbling, Will jerked her against him. His kiss was hard, hungry, and demanding. He felt the barest flicker of resistance before she caught fire in his arms. Her mouth went molten, tongue dancing tip to tip with his. Her arms clasped his neck. Her frenzied fingers tangled in his hair. Pressing tight against her, his arousal ached for release. Will cursed silently. Under different conditions he would've taken her on the sofa, on the floor, on the bar, anyplace he could get her under him. All he wanted was to push inside her and thrust until the tension and anxiety burst in one sweet explosion.
But there were people all around them. Beau was in the office. Erin was in her room. Bernice was probably in the kitchen. Any one of them could walk in without a warning.
Reluctantly he released her. They were both breathing hard. Tori's blouse was rumpled, her lips wet and swollen. Will's erection was still straining his jeans.
"Damn it, Tori," he muttered.
She shoved him away from her—shoved him hard. "Go," she said. "Just go."
Forcing himself to turn away, Will strode out of the den and headed for the front door.
* * *
Knees shaking, Tori tucked in her blouse, smoothed her hair, and slicked on some tinted lip balm. Will's kiss, and her own fevered response, had left her weak. What a time to rediscover that, under the tension, the bad memories, and the coldly controlled anger that kept them civil for Erin's sake, their old chemistry still sizzled.
The timing couldn't have been worse. Serving as Will's lawyer called for calm detachment and total focus on his case. It wasn't going to work if the two of them couldn't keep their hands off each other.
Tori knew better than to think this sudden compulsion to rip each other's clothes off was leading anywhere. Chalk it up to stress and hormones, nothing more. Will had been impossible when they were married, expecting her to kowtow to his father and be as dedicated to the ranch as he was. And a career woman with a mind of her own was probably the last thing Will had wanted in a wife. They were definitely not going to do that number again.
People didn't change—that was the bitter lesson Tori's failed marriage had taught her. Even so, today, when Will had refused to sell Erin's foal, she'd glimpsed the man she'd fallen in love with—the man she'd lost when he'd tried to turn himself into Bull Tyler.
And she could almost—under different conditions—have fallen in love with him all over again.
* * *
For most of the past week, Sky had planned to go back to the cave in the canyon. But work with the colts, running them through their training to keep them sharp, had kept him too busy to take time off. Still, he hadn't stopped thinking about what he'd found there. The memory of that small skeleton, barely glimpsed in the dark pit, had haunted him day and night. It was as if those lonely bones were calling to him, demanding . . . what? Recognition? Justice?
Today he'd made an effort to finish early. It was midafternoon when he saddled Quicksilver, the gray gelding that had become his favorite, and headed for the foothills. He'd thought about asking Lauren along, but that would have taken extra time, and she was busy with work in town.
Not that she'd have been eager to come. She'd fallen silent, visibly affected, after he'd told her about the bones. No sense exposing her, or Erin, to that dark place again.
He reached the petroglyph canyon with plenty of daylight to spare. Leaving the horse to graze by the spring, he climbed the steep, narrow trail to the clearing and the cave.
Everything was as he'd last seen it, the cave open, with rocks heaped on both sides of the entrance. Sky checked around for rattlesnakes, but, as expected, he didn't find any. By now, the frigid nights would have driven them underground to hibernate till spring. Most snake dens had multiple entrances and passageways. Even with the rocks blocking the cave, they'd have had no trouble coming and going.
Sky had worn gloves, armed himself with a pistol, and brought along the high-powered spotlight he kept in his truck. He didn't plan to kill any snakes unless they threatened him. This was their territory, after all, and he was the intruder. But he planned to be extremely careful.
The back of the cave lay deep in shadow. He switched on the spotlight and inspected the floor, walls, and ceiling. Nothing. Sky's Comanche grandfather had taught him a snake song. He sang it under his breath as he crouched on the edge of the hole—not for the snakes, which, as he'd since learned, couldn't hear, but to steady his own nerves.
The light shining down into the hole revealed nothing near the top, but Sky could see movement a dozen feet down, where a huge Texas diamondback slithered along a ledge. Was it close enough to strike him? He'd have to take that chance if he wanted to see the bones, which were a good twenty-five feet lower and could only be viewed by leaning in at an angle.
The snake wasn't coiled and didn't appear to be bothered by the light. Deciding to go ahead, Sky stretched out on his belly to anchor his weight and pushed his head and shoulders out over the opening. Gripping the spotlight with one gloved hand, he slanted the light toward the bottom of the hole.
At last he could see the bones. He'd been right about their size. They were small, definitely the bones of a child, maybe eight or nine years old. But that wasn't all. As Sky peered downward, he caught a glimpse of color. It looked like a fragment of red-plaid cloth—maybe a neckerchief or what was left of a collar, circling the neck bones. Moving the light lower, he saw something else that made him gasp. Lying across the small pelvic bone was what looked like a leather belt with a brass buckle. This was no old-time Indian, as Jasper had claimed. These bones were the remains of a young boy, dressed in the clothes of a modern-day white child.
The huge rattler raised its head and hissed. Startled, Sky jerked backward, dropping the spotlight. It fell, crashing against the sides of the pit to lie dark and broken somewhere below.
With a muttered curse, he scrambled to his feet. He was through looking. But he knew what he'd seen. Earlier, when he and Lauren had questioned Jasper about the bones, the old man had appeared nervous, as if he might be hiding something. When he got home, Sky was going to find him and demand the truth.
Those bones were on Lauren's land. She had a right to know their story.
* * *
"Count them if you want." Stella thrust the envelope into Ralph's hands. It was stuffed with bills, so heavy that Ralph could feel the heft of their weight. Opening the flap, he ruffled through them with his fingers. Sweet Jesus, they're hundred-dollar bills!
"Ten thousand dollars!" Stella snatched the envelope away. "All yours if you do the job I have in mind."
Ralph's head swam. With $10,000, plus what he'd put aside in the bank, he'd have enough money to buy a better truck and get out of town. He could put it all behind him—whiny Vonda and the baby he'd never wanted in the first place, his crappy job running cows for the Tylers, and the dangerous work he was doing for Stella. Those late-night deliveries paid beyond his wildest dreams, but Ralph was smart enough to know that if he didn't get out, he'd wind up in jail or dead, like Lute Fletcher.
"I won't have to kill anybody, will I?" he asked, hoping she'd think he was joking.
She laughed. "Nothing like that. Just a little property damage to the Tyler place."
Ralph hesitated. The Tylers were honest folks. In the three years he'd worked for them, they'd always treated him fairly. True, the work was hard, but they paid as well as any other ranch in the county; and the bunkhouse food, when he could still get it, was a lot better than Vonda's microwave cooking.
"Think about it." Stella patted his shoulder. "What do you owe those people? To them, you're nothing but a saddle bum they can work to death for slave wages. You can earn more in ten minutes than you'll earn busting your back for the Tylers in six months."
He was already thinking. Last week, in town, he'd found an eight-year-old Ford pickup in good condition. The owner was anxious to sell it and could probably be bargained down. Ralph could imagine himself driving that truck out of Blanco with new boots on his feet, money in his pocket, Vonda far behind him, and the whole damned country ahead.
He shrugged, trying not to appear too eager. "Guess I could do it," he said. "Tell me more."
Stella gave him a sly smile. "I'll tell you more when the time comes."
"And when'll that be?"
"Not long. Come by in a couple of days, and I'll give you the details."
"And the money?"
"A thousand now and the rest when you're done." Stella counted out ten bills, then put the envelope in a drawer of her metal army-surplus desk and locked it with the key she wore on a chain around her neck.
Ralph walked out with a smile on his face, $1,000 in his wallet and his head full of plans. He'd give Vonda a hundred just to keep her quiet, but there was no way he'd tell her about the rest. And he'd be smart not to tell Stella he was leaving. She wouldn't like that. It would be safest just to clean out his bank account, do the job, collect the cash, buy the truck, and get the hell out of Dodge. Run fast and far, where Stella, Vonda, Abner—and maybe the Tylers—would never know to look for him.
Plan in place, he climbed into his rusty old pickup and started home. The country oldies station he liked was playing "Take This Job and Shove It." Ralph turned the volume all the way up and sang along.
* * *
"So you went back and looked down that hole again." Jasper shook his head. "I wish you hadn't done that, Sky. All it'll do is dredge up old sins. Some things are best left alone."
The two men, dressed in warm jackets, sat on the shared porch of their duplex, sipping Mexican beer and relaxing at the end of a long day. The black-and-white Border collie was curled in his usual spot next to Jasper's feet. Above the escarpment a fiery sunset was fading to the deep indigo of twilight.
Sky closed his eyes for a moment, breathing in the aromas of wood smoke, horses, and the night's coming frost. "After I saw those bones, I knew I had to get a better look," he said. "Now I almost wish I hadn't."
Jasper reached down and scratched the dog behind the ears. "Like I said, some things are best left alone."
"It's too late for that," Sky said. "I saw what was left of a collar and a belt. It was a young boy down there—a boy who had a name and a story. I need to know what happened to him. So does Lauren. It's her land now." He turned to look straight at Jasper. "If you know the story—and something tells me you do—"
"Oh, hell," Jasper muttered, "I reckon I won't get any peace till I tell you. But once you hear the truth, you're liable to wish you'd left well enough alone."
"I'll take that chance," Sky said.
Jasper shifted in his chair, crumpling the empty beer can between his gnarled hands. "What happened up in that little canyon was before my time here. I didn't know about it myself till Bull told me a few months before he died. He made me swear not to tell, but since the ones involved are all in their graves, I reckon your knowing won't hurt none. And since you're plannin' to wed a Prescott, it might help you understand why Bull and Ferg hated each other like they did."
Sky settled back to listen. He'd hoped to have Lauren with him tonight, but she was home nursing a cold. Maybe that was just as well. The old man might talk more freely without her.
"Ferg and Bull weren't always enemies," Jasper said. "As boys the same age, on neighboring ranches, they grew up friends. When they weren't workin' to help their dads, they were tearin' around on their bikes and ponies, learnin' to rope, and playin' cowboys and Indians in the canyons. I reckon it was about as good a life as two boys can have—till somethin' happened."
Jasper sat silent for a moment, watching the dusky shadows creep across the yard. "Ferg had a younger brother—Cooper, that was his name. Cooper was a couple of years younger than Ferg. As Bull put it, he was slow in the head—I guess the way they say it now would be that he was mentally challenged.
"Cooper didn't have friends his own age, so whenever he could, he tagged after Ferg and Bull—not that the boys liked havin' him along. Kids that age can be pretty mean. I guess they teased him and played tricks on him. But Cooper just kept taggin' along like a puppy, probably not smart enough to figure out they didn't want him."
Jasper gave Sky a sharp glance. "I'm tryin' to tell this pretty much how Bull told it to me. One day—the boys would've been about eleven—they were playin' cowboys below the canyon, shootin' off their cap guns and throwin' their lassos. Cooper was with 'em, and Ferg got the idea to pretend the youngster was a cattle rustler they'd caught. They used a bandanna to tie his hands behind his back—something Cooper didn't mind. They'd done that to him before. I guess he was happy just for the attention.
"Then Ferg got a new idea. 'Hey, let's hang the thievin' varmint!' he said, and he made a loop with his rope."
Sky felt the horror uncoiling in his gut. He wanted to stop Jasper from telling the rest, but it was too late now. He needed to hear the story, all the way to the awful end he knew was coming.
"Ferg was a big, husky kid. He put the rope around Cooper's neck, tossed one end over a cottonwood limb, and hauled his little brother off the ground. Then he tied the other end to the roots of an old stump. Bull said he would've tried to stop him, but it was just a game, and he thought, for sure, Ferg would untie the rope in time. I'm guessing Ferg thought the same thing. They weren't bad kids. They just didn't know how far was too far."
Jasper shook his head and cleared the emotion from his throat. "When they realized what was happening, they tried to untie the rope from the stump, but the knot was tangled in the roots, and they didn't have a knife to cut it. By the time they finally got him down, Cooper was dead. The boys knew they were in big trouble, so they concocted a scheme. First they dragged the body up the canyon to the cave, untied his hands, and dropped him down that hole, right where you found him."
Sky swallowed the ache in his throat. Those little bones had a name now—Cooper Prescott, who would have been Lauren's great-uncle.
"Since Bull hadn't done enough to stop the hanging, and since he'd helped hide the body, he was guilty, too. The boys made a pact—cut their fingers and sealed it in blood—that they'd never tell what had really happened to Cooper. They made up a story for their folks that some Mexicans in an old car had grabbed the boy and kidnapped him. They even made up a license plate number. The authorities combed the state for those Mexicans. Course they never found 'em."
The old man fell silent again, his hand stroking the dog.
"I'm guessing there's more to the story," Sky said.
"The rest is about Bull and Ferg," Jasper said. "What happened with Cooper put an end to the friendship. For years afterward, Ferg was afraid that Bull would tell on him. He threatened Bull that if the story ever got out, he'd swear that Bull was the one who'd hanged Cooper. After all, who'd believe that Ferg would kill his own brother?"
"Bull never told, did he?"
"Not till he told me, a long time after Ferg was dead. I guess he wanted somebody to know the truth, in case the body was ever found."
"And what about the land?"
"That canyon was Tyler property. The Spanish-gold legend was around even then. Nobody put much stock in it, but Ferg was always afraid somebody would go lookin' for that gold and find Cooper's body. He wanted to own that little strip of land so he could keep people off it."
"And his chance came when Bull got involved with my mother." Incredibly, the fragments of Sky's family history were coming together.
"Yup. That's the part of the story you already know. Ferg blackmailed Bull into selling him the land."
"And when he was digging around up there, pretending to look for the gold, he was really covering the cave?"
"That's about the size of it." Jasper pushed himself to his feet, a sign that the conversation was winding down. "So," he said, "are you going to tell your future wife that she's the grandchild of a murderer?"
"Whether he was a murderer or just a crazy kid who went too far, I'm going to tell her everything," Sky said. "Lauren has the right to know."
* * *
Sky told the story to Lauren the next night, while they were nestled on the sofa in her apartment. He told it gently but carefully, leaving out nothing that Jasper had told him.
By the time he finished, tears were flowing down Lauren's cheeks. "That poor, innocent little boy! Oh, Sky!" She pressed her damp face into the hollow of his shoulder. "There must be something we can do! Can't we at least get those bones out so we can bury them in the family cemetery?"
"I've thought about that," Sky said. "But I don't think it's possible. The hole's jagged all the way down, and the bones are wedged deep. Even without the snakes there, nobody could get to them safely. And if we tried to lift them out with some kind of line, they'd be liable to break on the way up. The same with blasting out the hole. That little skeleton's been down there more than fifty years. It's bound to be fragile. As far as I can tell, the only way to preserve it is to leave it right where it is."
"Oh, I suppose you're right. But we've got to do something for the memory of that poor child." Sighing, she snuggled deeper into his arms. For a time they sat in silence, watching the moonrise through the dark window. At last she stirred.
"I just thought of a plan," she said.
"Want to tell me about it?" He was getting drowsy.
"Not yet. It'll take a little time, but if I start on it tomorrow . . ." She yawned. "I might need your help. If I do, I'll let you know."
"All right, my mysterious lady. I'll settle for knowing that you can do whatever you set your mind to."
With a lingering kiss they ended the discussion for the night.
CHAPTER 12
Ralph stood on his front stoop, smoking a Marlboro from the pack Stella had given him. The night breeze was cold, but even in his thin denim jacket, he barely felt the chill. Truth be told, he was too churned up to feel much of anything.
Through the closed door behind him, he could hear the blare of a TV reality show. Vonda liked the one where they locked hot men and women in a house and filmed them bitching at each other, or falling into bed. Stupid show, but that was Vonda for you.
At least he wouldn't have to put up with her much longer. All he needed to do was carry out Stella's orders, and he'd soon be on the road with plenty of cash in his pocket.
The cigarette had smoldered low enough to burn his fingers. With a muttered curse he dropped it on the porch, ground it out with his boot heel, and fished another one out of the pack in his pocket. His cheap lighter flared in the darkness as he lit it and inhaled the bittersweet smoke.
From where he stood, he could see the lights from the big stone house where the Tylers lived like royalty, lording it over their land and their cattle and their underpaid shit-shoveling crew, like him. Tomorrow they'd be gathering in the house for Thanksgiving dinner. That was when he would carry out Stella's plan—the plan that would change everything.
Just thinking about what she wanted was enough to scare the spit out of him. But she'd given him another thousand-dollar payment with his marching orders tonight. Back out now and he could end up as dead as Lute Fletcher.
The job itself would be easy. The hard part would be making sure he wasn't seen. Get caught, and all bets were off. He'd, for sure, go to jail—unless Stella got to him first. He was just beginning to realize how dangerous the woman was.
He'd already withdrawn his savings—by now, almost $15,000—from the bank, and hidden the cash in his truck. As soon as he'd gotten full payment from Stella, and maybe bought that used truck he'd had his eye on, he'd be out of here.
He turned to go back inside, then paused, torn between need and fear. Maybe he'd be smart to forget the money and go tonight—just get in the old truck and drive. He was already in too deep with Stella. The little he knew about her operation was barely the tip of the iceberg, but it could be enough to damage her. If she knew he planned to leave, he wouldn't put it past her to make him disappear. All the money in the world wouldn't do him any good if he wasn't alive to spend it.
He'd taken two strides toward his truck, when he realized he'd left his keys on the kitchen table. Stopping in his tracks, he cursed. Vonda had been nagging him to stay home more. She would throw a hissy fit if he came in, got the keys, then tried to leave again.
He sighed, feeling trapped. But never mind, it might be better if he stayed, Ralph told himself. He needed the promised cash, and he couldn't afford to make Stella angry—not yet, at least. After he did the job tomorrow, he'd stick around long enough to make sure the Tylers didn't suspect him. Then he'd collect his pay, make nice with Stella, and wait for the first chance to make tracks.
The plan made sense, as long as he could make it work. Otherwise, if anything went wrong, he'd be a dead man.
Fear crawling along his nerves, Ralph turned around once more and walked back into the house.
* * *
Bundled in Bernice's knitted afghan, Tori stood at the porch rail. The evening breeze was cold. Clouds gusted across the sky, playing hide-and-seek with the waning crescent moon. The air smelled of snow, but the forecast was for a mild storm, not a killer like the last norther that had blasted the land with sleet, ice, and lightning.
From the glowing rooms behind her came the sound of a football game on TV, intermingled with whoops and cheers from the watchers—Will, Beau, Sky, Jasper, and Erin. The aroma of baking pies floated from the kitchen, mingling with the homey smells of popcorn and wood smoke. As usual, the Tyler Thanksgiving celebration had started the night before the holiday, with snacks and game watching. Tomorrow, for the first time, Natalie and Lauren would be joining them for the traditional turkey feast.
Tori had always looked forward to the fun, food, and family that was Thanksgiving on the Tylers' ranch. But this year would be bittersweet. Behind everyone's smiles and laughter was the awareness that this could be the last holiday when the entire ranch family would be together. Will's fate hung in the balance, awaiting the outcome of the trial in two weeks. Jasper and Bernice were getting old. Erin was growing up. And as for herself . . .
She brushed back a lock of windblown hair. Drew hadn't phoned her since leaving for Omaha on Monday. Was he giving her time and space to make up her mind about him? Or had the revelation that she couldn't have more children cooled his interest?
It surprised her how much she missed him. Drew was an island of stability in the sea of turmoil her life had become. She wasn't in love with him—not yet, at least. But she liked him. There were even times when she needed him.
"Here you are." Will had come out onto the porch, moving to stand beside her. He was wearing the plaid woolen shirt he'd worked in that day. It smelled of sagebrush, hay, horses, and his powerful male body. He stood with his hands on the rail, silent now, as if waiting for her to speak.
"How's the ball game?" she asked, making small talk.
"Fair. The Cowboys are up by two touchdowns. But I couldn't stay with it. Too much going on in my head."
"Are you all right?"
"Fine . . . considering."
Tori checked the urge to reach over and lay her hand on his. It might be a comfort, but they'd gone too far down that road already.
"There's something we need to talk about," he said.
"I'm listening." Tori felt the tension, like the sudden snap of a bowstring. Whatever Will was about to say, she sensed it wouldn't be easy to hear.
He cleared his throat. "If the trial goes badly, and I end up going to prison—"
"Don't say that!" She cut him off. "Don't even think it. It's not going to happen."
"Are you that sure of yourself, Tori?"
"I have to be. It's the only way I can do my job."
Will's throat moved, but he didn't reply. In the silence the mournful wail of a coyote echoed through the darkness.
"There are some things we need to think about," he said. "One of them is our daughter. Whatever happens, we'll want to make this as easy on her as we can."
There's no way any of this will be easy on Erin, Tori thought, but she held her tongue.
"If I go away—and it could be for years—there'll be no one in this house. Bernice wouldn't stay on. She's already talked about going to live with her sister. Jasper . . ." Will shrugged. "I'm guessing he'd stay put as long as he can, but he's getting old. He won't be around forever. Beau and Natalie will have their own place and their own family. So will Sky and Lauren."
He gazed across the shadowed yard toward the long horse barn. "This house has been home to Erin all her life. Her room, the animals, Jasper and Sky, even those Sunday dinners—I can't stand the thought of her losing all that, as well as losing her father."
"I know." Tori kept her voice low to hide the emotion. "I've had the same thoughts myself."
"If I go to prison, you'll need to move back here," he said. "I'll pay for a new vehicle, for you to drive to work and take Erin to school. The utilities and maintenance on the house will come out of the ranch budget. You can rent out your house in town or keep it for your office. That way, Erin will still have this place to call home."
Tori bit back a surge of annoyance. She knew Will was only thinking of Erin, but how like him it was to have everything planned out and expect her to fall in line. His idea made some sense, but why couldn't he have asked for her input, instead of just dumping the whole package in her lap?
"Can't this wait until we know the outcome of the trial?" she demanded. "You're innocent, Will, and you're making plans as if you're going to be found guilty."
He looked down at her, his eyes narrowing. "Maybe you believe in the justice system. But I don't trust it. I don't trust the judge. I don't trust the witnesses. I don't even trust the jury. Anything could happen in that courtroom. If it's the worst, I need to know that Erin will be all right—and Erin needs to know it, too."
The night wind had sharpened. Shivering, Tori pulled the afghan tighter around her shoulders. He was right about Erin. Their daughter needed to feel secure about what would happen if her father went to prison. But how could Will expect an instant answer? With so many uncertainties hanging over them, how could she make him a promise that might bind her to this place for years?
"If it's your boyfriend you're thinking about, we could work that out." Will's voice had gone flat and hard. "I don't have any claim on you. I know I can't ask you to live like a nun. If you wanted to get married again, I wouldn't stand in the way of letting him move in here for the duration."
Something jerked in Tori's chest. "Will—"
"This isn't about you and me, Tori. It's about Erin and what would be best for her."
"I know." Tori gazed down at his work-scarred hands, where they gripped the porch rail. For years she'd accused him of being just like his father. But Will was not Bull Tyler. She should have known that when he'd refused to sell Erin's beloved colt, even though the money would have paid off the bank loan. And now, for Erin's sake, he was even willing to let another man live in his family home, with the mother of his child. There was no way Bull would have been so selfless.
And Will was right. This wasn't about the two of them. It was about their daughter. But right now he needed to back off and stop pushing her.
"I get what you're saying, and I agree in principle," she said. "But that plan of yours is a lot to take in. I need time to think."
"How much time?"
"As much time as it takes." Tori was exhausted after a long day of preparing for the trial. Will's heart was in the right place, but his timing was way off. "You'll have my answer in the next few days. Right now, that's the best I can do."
"Do you plan to talk to Erin about this?" he asked.
"Maybe."
"And your boyfriend?"
"His name's Drew. And it's way too soon for that."
"I need this settled, Tori. Lord knows I need something settled."
"I understand. But I need time." Sensing a brewing storm between them, Tori moved toward the front door. "Now, if you'll excuse me, I promised Bernice I'd take the pies out of the oven. They should be done about now."
Before he could say any more, she quickened her steps and fled into the house, closing the door behind her. She sympathized with Will's worries, and she would do everything in her power to defend him. But right now, they were both on edge. The last thing they needed was a big, blowup fight.
* * *
Will lingered on the porch after Tori had gone, feeling the burn of cold wind on his face. The familiar sounds of night on the ranch drifted to his ears—the creak of the turning windmill, the shifting and nickering of horses in the barn, and the bawl of a calf in the lower pasture. Smoke, curling from the tall metal chimney on the bunkhouse, blended with the earthy smells of sage and manure.
Closing his eyes, Will filled his senses with memories of the only home he'd ever known. If things went badly at the trial and he ended up in prison, he would need these memories to keep him strong. But, Lord, how could he stand it, being cut off from everything and everyone he'd ever loved?
Tori was right—he needed to go on believing in his own innocence and the fairness of the American justice system. But the ugly knot in the pit of his stomach wouldn't go away. With so many twisted facts working against him, how could he expect to walk out of court a free man?
Tonight he'd pretty much given Tori permission to move her boyfriend into his house, feed him at their table, and sleep with him in their bed. Speaking the words had damn near killed him, but if that was the price of having Tori stay here with Erin, so be it. This was about his daughter's well-being, not his personal feelings. But the personal feelings were there, and they were too powerful to be denied.
Drew.
The name left a nasty taste in Will's mouth. He didn't even know the man, but the thought of Tori in Drew's arms was enough to rouse Will to a near-murderous rage. Even after eight years apart, he still tended to think of her as his woman—and that one wild encounter, the night of the storm, was seared like a brand into his memory. He'd been a fool to let her go—and damn it, he wanted her back.
If he made it through the trial with his freedom, by heaven, he was going to fight for her. Drew was going to have some serious competition.
But the reality was, if he was convicted and sentenced, all bets would be off. He couldn't ask Tori to wait for him, or to tie herself to the man he'd be after years behind bars.
"Are you all right, Daddy?" Erin had come out onto the porch to stand beside him.
"Fine, honey. Just getting some air. I thought you were watching the game."
"It's just a game. I don't care that much about it."
"What do you care about these days?"
"Important stuff, like you and Mom and the trial. I wish I knew what was going to happen."
"So do I." Will rested a hand on her shoulder. "You're a smart girl. I know better than to sugarcoat the situation for you. I'm just as scared as you are. But I can promise you two things. Whatever happens, you'll be all right, and I'll still love you."
Erin didn't answer. When he heard a little breathy sound, Will realized she was crying.
"It's all right." He pulled her against his side, thinking how fast she was growing up. If he went to prison, she could be a woman by the time he got out.
"I don't want you to go away, Daddy!" She wrapped her arms around his waist and held on tight.
"It's all right to cry, honey." He stroked her hair. "Just remember, we Tylers are a tough family. One way or another, we'll get through this."
"But if you go, what'll we do without you? Everybody counts on you to be the boss—Beau, Sky, Jasper, even Mom."
"Now I'd argue with that. You just mentioned the four most contrary, mule-headed people I know!" Easing her away from him, he used a gentle finger to lift her chin. Her tear-streaked face almost broke his heart. "Now, what do you say we go inside, give you a minute to wash up, then sit down and watch the rest of the game?"
She gave him her best imitation of a smile. "Sure. And let's have the best Thanksgiving ever tomorrow."
"That's my girl." Aching with pride, Will followed his daughter into the house.
* * *
Stella scanned the dark parking lot before locking the back door of the Blue Coyote. Out of habit her hand reached for the 9 mm Glock she kept in her oversized purse. It was after eleven, she was alone, and a lady couldn't be too careful.
Her Buick was parked a dozen yards away, in a well-lit spot. It hadn't escaped her that somebody, like Marie, could wire a bomb to the ignition. But the big sedan had a sensitive alarm system that would go off if anybody got too close. An ambush from the shadows would be more Marie's style. But the bitch would show herself first. She'd want Stella to know she'd come back for her revenge.
As a precaution Stella keyed the remote to unlock the doors and start the engine. No bomb. She climbed into the driver's seat, locked the doors, and put the Buick in gear. Her feet, in their red cowgirl boots, were killing her. She couldn't wait to get home, pull them off, and soak in a warm, sudsy bath.
Tomorrow the bar would be closed. Although she'd never been much of a cook, she usually fixed a little Thanksgiving dinner for herself and Nicky. This year, with her brother gone, it wouldn't be worth the bother. But if things went as planned, she'd be celebrating in a different way.
Could she depend on Ralph to do what she was paying him for? Up to now, he'd done as he was told. But this job would take some guts. She wouldn't put it past him to get cold feet, take the two-thousand-dollar payment she'd advanced him, and hit the road.
Either way, Ralph was a flunky who'd pretty much outlived his usefulness. He wasn't smart enough to justify keeping around, which meant he'd have to go. It was only a question of when and how.
With Hoyt Axelrod dead and Marie in the wind, she'd lost the only people she could count on to kill in cold blood. Getting rid of Ralph was hardly worth the cost of a hit man, but since she'd always had a rule against offing folks herself, that might be her only option. The fact that he was Abner's son-in-law called for extra caution. His death would not go unnoticed or uninvestigated.
But she was too tired to think about that now. Tomorrow, after she knew how Ralph's little errand had gone down, she could make her plans.
Turning onto a side street, she headed for the apartment complex where she lived. She was getting weary of Blanco Springs and this whole business. Now that Nicky was gone, maybe it was time she pulled up stakes and headed for Mexico, where she had the connections, and enough money stashed away, to set herself up for the rest of her life.
She was liking that idea more and more. But first, she had to settle the score for her brother's death. It would be easy enough to have Will Tyler killed. But she wanted him to suffer—to pay with his freedom, his resources, and all that he cherished. Only then would she feel satisfied. And only then would she feel free to leave.
The trial was a week away, but she was too impatient to wait. Payback for Nicky's loss would begin tomorrow, while the family was at dinner.
She would stay home and listen. When she heard the blare of sirens, she would know Ralph had carried out her orders.
* * *
Thanksgiving Day dawned bleak and overcast, with dry flakes of snow blowing on the wind. Even on a holiday there were chores to be done—the cattle and horses had to be fed and watered, the fences checked, the horse barns shoveled out, and the stalls laid with clean straw. All the hands pitched in, including the Tyler brothers and Sky.
Ralph showed up on time and joined the others. Today the men who lived in the bunkhouse would enjoy a nice Thanksgiving dinner and a free afternoon. Ralph couldn't help envying them. Vonda, who was too pregnant and tired to cook, would be heating a couple of frozen turkey dinners in the microwave. But what did it matter? Once he'd carried out Stella's orders, nobody was going to have a good holiday, especially the Tylers.
As he worked, busting the ice off the water troughs and scattering hay in the pastures, dread clawed at his gut. It wasn't too late. He could make some excuse, then walk to his truck and drive away. He had enough money stashed in a grocery bag under the seat to get him to some far part of the country, where he could rent an apartment and live frugally till he could find some kind of job.
Just go, he told himself.
But then he thought of the money and what Stella could do to him if he ran out on her. She had rumored connections with the kind of people who could find anybody, anywhere. If he wanted to live, he would have to do what she wanted.
By noon the work was done. The men were dismissed to go back to the bunkhouse or home to their families. Ralph had driven his truck the quarter mile from his bungalow to the ranch yard. Climbing into the cab, he lit a cigarette and watched Will, Beau, and Sky trail toward the house. Rich bastards, those Tylers. They deserve what they're about to get.
Stella had told him to make his move while the Tylers and the ranch hands were at dinner. That wasn't likely to happen for an hour or more. Meanwhile, he could hardly sit here and wait in plain sight. He had little choice except to go home, eat his microwaved Thanksgiving dinner with Vonda, then make an excuse to go back to the barn.
Starting up the truck, he drove home. He found Vonda lying on the couch with the TV blaring and the two dinners sitting on the counter, still frozen. He opened one, shoved it in the microwave, and set the timer.
"Want me to cook yours, too?" he asked her.
"Not now. I don't feel real hot. I've got a bellyache."
"Can I get you anything?"
She shook her head and closed her eyes. Ralph turned down the volume on the TV and waited for his dinner to heat. When it was ready, he ate it, standing up, off the counter. So much for Thanksgiving, he thought.
Tossing the plastic tray in the trash, he glanced at the clock on the stove. It was early yet, but he was getting anxious. He'd planned to go back to the barn on foot, less chance of being seen. Add a few more minutes to circle around the back way and, with luck, the timing should be about right.
"Where are you going?" Vonda demanded as he slipped on his denim jacket again.
"Just something I promised to check on. I won't be long." He started for the door.
Straining, she managed to sit up. "Do you have to go now? I feel—oh!" Her face froze in shock as wetness spread down the legs of her sweatpants. "Oh, Ralphie, I think my water just broke!"
Seized by a sick panic, Ralph paused in the doorway. Whatever was happening here, he had to do the job for Stella. If he didn't, she was liable to kill him.
"You can't go now!" Vonda wailed. "You need to drive me to the hospital! I'm going to have this baby!"
Damn! Ralph felt the cold sweat beading under his flannel shirt. But first babies took a long time, didn't they? He could still go and make it to the hospital, an hour away in Lubbock. He had to.
"This won't take long," he said. "Get ready. I'll take you as soon as I get back." As an afterthought he tossed her his cell phone. "If you need help before then, call nine-one-one."
Before she could say anything else, he was out the door.
Should he take the truck? It would get him there and back faster. But no, a vehicle could be spotted too easily, and everybody knew that old rust bucket by sight. Passing the truck, he broke into a run, cutting up the road, across the open ground and behind the outbuildings to the rear of the barn. There, aching from a stitch in his side, he slumped with his hands on his knees, fighting the urge to retch.
So far, he'd seen nobody outside. All to the good. The sooner he got this over with, the sooner he could get back to Vonda and drive her to the hospital.
He didn't want to see the baby. He didn't want to see anything that might have the power to hold him to this place. He would take Vonda to the emergency entrance, make sure she was in good hands, then report to Stella. Once he had the money, it would be Good-bye, Blanco Springs.
The long barn, which held the mares, their growing foals, and some of Sky's trained colts, was unlocked at the near end. Ralph slipped inside, closed the door behind him, and walked the long line of stalls toward the far end. Horses snorted as he passed. Some raised their heads and looked at him with their luminous dark eyes. Most of them were familiar. Some he'd even ridden to work the cattle. Ralph didn't love horses, but he liked some of these. Knowing what was about to happen, he kept his gaze lowered, avoiding eye contact with them.
A cart heaped with dry straw had been left at the barn's far end. Steeling his resolve, Ralph took out his lighter, clicked it, and held the flame to the straw. The dry fuel caught with a startling whoosh, so close that Ralph could feel it singe his eyebrows. He jumped back as the flames rose higher than his reach. The fire was burning faster and hotter than he'd expected. He had to get out of here.
The blazing cart was blocking the nearest door. The only way out was at the other end, where he'd come in. With the fire spreading to the roof supports, he raced between the stalls. He stumbled over a pitchfork and caught himself on his knees. Scrambling to his feet, he rushed on.
By now, the horses were going wild, screaming and rearing, lunging against the sides of their stalls. He wasn't crazy about horses, but these were all about to die horrible deaths, and it would be his doing.
As he raced past the stalls, Ralph began a frenzied grabbing at the gate latches, jerking open the ones that yielded easily. Horses, he knew, tended to panic and balk in a fire and had to be dragged out of burning buildings. But if any of them had the sense to run out, he would leave the door open. Maybe a few smart animals would make it. Stella couldn't be too mad about a few horses.
Besides, if he happened to get caught, he could always claim that he'd smelled smoke and had rushed into the barn to save the horses.
Just behind him a big bay mare burst out of a stall he'd unlatched. Shrieking in terror, she reared on her hindquarters, her hooves flailing the air.
Ralph glanced back just in time to see the massive, black, ironshod hooves coming down toward him. Then something slammed his head, and the world exploded into blackness.
CHAPTER 13
Tori closed her eyes and did her best to listen as Jasper's mumbled grace droned on. This Thanksgiving dinner was a poignant celebration, a blessed pause in the frenetic pace of their lives, when worries and differences were put aside. Everyone around the table seemed resolved to make the most of the day. For now, at least, they were together; and even in the face of change, they were still very much a family.
The prayer ended, eyes opened. Hands reached out to pass around platters and bowls of carved turkey, dressing, hot rolls, potatoes, and gravy. That was when the dog, who had been napping on the porch, began a frantic barking.
Bernice, who was facing the window, was first to see the smoke. "Lord help us!" she cried. "The barn's on fire!"
In a flash they were all on their feet, overturning chairs in their haste to get outside. "Somebody call nine-one-one!" Will shouted, racing for the front door with Beau behind him. Lauren already had her phone out and was making the call.
"Tesoro! He's in his stall!" Squirming out of Tori's clasp, Erin streaked headlong out the door, passing Beau on the front porch.
"Beau, get her! She mustn't go out there!" Tori fought her way through the melee of chairs and shifting bodies. By the time she caught up with Beau in the yard, he'd grabbed Erin and was holding her while she struggled to get free and run to the barn. "Tesoro . . . ," she sobbed. "I've got to save him!"
"I'll get Tesoro, I promise," Beau was saying. "Now calm down and stay here with your mother. That's an order."
After passing her off to Tori, Beau wheeled away and joined Will in the race across the yard. Holding her daughter tight, Tori glanced around for Sky. He was nowhere to be seen. But she knew where he must be. In his quiet way he would have slipped out the back at once and been first to reach the burning barn. By now, he'd be inside, fighting to rescue the horses he loved like his children. And, because she knew the man, she was sure he wouldn't abandon any animal, not even for his own safety.
Lauren had come out onto the porch. One look at her pale face confirmed that she knew it, too.
The men who'd stayed for the holiday came pounding out of the bunkhouse to help. Shouting, Will directed them to hook up the hoses and start spraying inside the barn. After the damage from the past summer's wildfire, the barn's shingled roof had been replaced with steel panels. But it was the older, lower part of the barn that was burning now, flaming upward from the inside. If the temperature got hot enough, and the supports weak enough, the whole building would collapse.
Breathing through their neckerchiefs, men were dragging hoses, soaking feed sacks, filling buckets, rushing in and out of the barn. The scene brought back the memory of the wildfire that had nearly destroyed the ranch a season ago. That fire had been far bigger, sweeping through the tinder-dry scrub, consuming everything in its path. But then, at least, they'd had time to prepare and evacuate the stock. This blaze had started with no warning at all.
The garden-sized hoses were far too small, the water stream barely enough to wet down the stalls, let alone fight the fire. If the fire engine, with its big tank, didn't get here soon, the barn would be lost, along with many of the precious animals.
The first few horses had emerged. Faces covered with wet feed sacks, they were coughing and struggling against the men who gripped their halters. Tori recognized Belle, the aging bay mare, among them, and Lauren's powerful black gelding, Storm Cloud. But there was no flash of Tesoro's golden coat. She clasped her daughter tighter. "Remember, Beau promised to save him," she whispered, adding her own silent prayer.
Beau was still outside. He soaked his clothes at the pump, masked his face with a neckerchief, and charged into the billowing smoke, carrying a bucket.
Natalie, who had been on a call that morning, had driven here in the big white SUV she used for her veterinary work. Burdened only a little by her bulging belly, she rushed to her vehicle. Lauren stood at the top of the porch steps, wide-eyed with fear but clearly anxious to help. "Come on!" Natalie shouted to her. "I'll need extra hands!"
Lauren sprinted down the steps and piled into the passenger seat. In the next instant they were roaring toward the barn, flying over the bumpy ground. It would be Natalie's job to treat any horses injured in the fire—or to put them down if they couldn't be saved.
Smoke was pouring out of the barn in thick, murky clouds. Even at a safe distance, Tori could smell it, even taste it when she licked her parched lips. She could feel the searing heat and see tongues of fire licking under the metal roof. The blaze appeared to be burning upward from the rear of the barn. That might give the horses near the front a better chance to get out, but how could anybody, man or beast, survive in that inferno? It would be tragic enough losing horses to the fire. But human lives . . . no, that was unthinkable.
More horses, driven from behind, exploded out of the barn to scatter and mill in the yard. One of the men had opened the paddock gate and was trying to drive them inside, but the horses were too fear-crazed to be herded. They would have to be rounded up later.
"There's Beau!" Erin shouted, waving as a figure emerged from the smoke with a grip on two struggling animals. "He's got Tesoro—and Lupita!"
Tears of relief welled in Tori's eyes, but she could see that something was wrong. The mare looked all right, but Tesoro was favoring one side, his head hanging low. Even from here, she could tell the colt was in pain.
And something else was wrong, Tori realized. Her last sight of Will had been when the men were hooking up the hoses. She could see Beau, leading Tesoro over to Natalie's SUV, but she couldn't see Will's red-plaid shirt anywhere. Fear crawled up her throat. Was he inside the blazing barn?
"Tesoro's hurt!" Erin fought to pull away, but Tori kept a firm grip on her daughter.
"It's all right. See, Natalie and Lauren have got him now. They'll take care of him. Having you there would just be one more worry." She glanced back toward the house. "Jasper and Bernice are on the porch with the dog. Stay there and watch with them. I'll go find out what's happening." She released her daughter with a gentle shove toward the porch. "Go!"
Tori forced herself to wait until Erin had reached the steps. Then, giving in to her fear, she spun away and raced headlong for the barn. Smoke swirled around her, the acrid scent filling her nostrils and lungs, burning her eyes. Men were manning the hoses and leading more horses to safety. She could see Beau in the doorway of the barn, shouting directions. But there was no sign of Will.
She hadn't seen Sky, either. Would she find them both safe on the far side of the barn, maybe hidden from sight by the heavy smoke? Or could they be inside, maybe trapped, maybe even dead? With a prayer on her lips, she ran on. There was little she could do to help. She only knew that if Will was in danger, she had to be there.
* * *
Will had gone searching for Sky. Deep in the barn he found him. He was standing in a half-charred stall, soothing a fear-maddened chestnut mare with burns on her back—one of the last three horses left in the barn.
To Will, Sky looked like a man who'd just stepped out of hell—there was a wet bandanna plastered to his face; hair and clothes singed; his ungloved hands blistered by the sparks that exploded from the burning beams overhead. The eyes that turned and glanced at Will were bloodred from the smoke. He looked like a soldier in combat, ready to drop from exhaustion.
By now, the hose crew had wet down the stalls and the straw inside, making it a little safer to get the horses out. However, the fire had taken an upward path, and there wasn't enough water pressure to reach the wooden rafters and beams that supported the barn's metal roof. They were on fire now; and when they burned through, the whole structure would collapse. Will cursed his lack of foresight. After the summer fire it had been his cost-driven decision to replace only the roof and not the old structure of the barn beneath. Now he was paying the price.
Reaching Sky, Will grabbed his arm. "For God's sake, man, let's get out of here!"
Sky shook his head. "After the horses."
Will was aware of what Sky had been doing. He'd taken his stand in the most dangerous part of the barn, calming terrified animals, covering their faces with wet sacking, urging them out of their stalls and forward, to where other hands could lead them to safety. But the roof couldn't hold up much longer, and neither could Sky.
"We need to get the hell out of here!" Will shouted, pointing to the blazing rafters. But he knew he was wasting his breath. Sky would not go until every animal was safe. The only thing Will could do was to help him.
He didn't have Sky's natural touch with horses, but he opened the nearest stall. Dodging frantic hooves, he grabbed the young gelding's halter at the throatlatch and flung a dripping sack over the animal's face. Yelling, swearing, and yanking on the halter, he dragged the horse out of its stall, turning it toward the far entrance. After more prodding and cursing, he finally got the animal to where one of the cowhands was waiting to seize its halter and hurry it outside.
Will's lungs were already burning from the smoke. How much worse off must Sky be? With his unfailing gentleness, Sky urged the chestnut mare out of her stall and led her partway toward the far door, where one of the men waited to take her. Now only one horse remained, a big paint gelding, wild with terror. It was screaming and kicking, refusing to be led. Will glanced up at the blazing beams overhead. Tugging at Sky's arm, he pointed upward. Sky shook his head. "You can go!"
Will's gaze met Sky's. "No way. I'm not leaving without you! Let's get this horse out."
With Sky calming the horse, they managed to work their way on either side of its head, fling the last of the wet sacks over its face, and lead it out of the stall. When Will peered down the long row of stalls, through the blur of smoke, no one was there to take the horse.
Suddenly he saw why. The roof panels were buckling in the heat. Any second now, they would come crashing down. With a shout he yanked the cover off the big paint's eyes and gave its rump a resounding smack. The horse bolted toward the light at the far end of the barn and disappeared outside.
Will and Sky were now racing for their lives, pounding through choking smoke and searing heat. But they'd already delayed too long. With fragments of breaking, burning timbers raining on them, they heard the awful groan of warping, sliding metal. They were no more than twenty yards from safety, but it was too late to get out. They could only dive for any cover they could find.
As the roof panels sagged and came crashing down, the last sound to reach their ears was the faint wail of sirens.
* * *
Tori and Lauren waited next to the emergency vehicles, supporting each other in silence as the firemen hosed down the barn's wreckage and began clearing it away. The barn was a total loss, but the horses, now rounded up and herded into the paddock, had survived the fire with minor burns and a gash to Tesoro's shoulder, which would heal. The ranch hands who'd come running to help were safely accounted for, as well as Beau. Only two men were known to be missing—Sky and Will.
With rakes, shovels, crowbars and gloved hands, Beau and the hired men were helping the fire crew lift away the bent roof panels and other debris. The ruined barn looked as if it had been bombed. In some places the charred walls of the barn and the heavy-timbered framework of the stalls were still standing. In other places the rubble was as flat as if crushed by a giant hammer. The work was painstakingly slow. One wrong move could crush or stab a survivor who might be trapped alive beneath the rubble.
Natalie, who'd been re-packing supplies in her SUV, broke away and hurried forward, cell phone in her hand. "I just got a call from the nine-one-one dispatcher," she said. "That poor little girl who's married to Ralph Jackson is about to have her baby. Her husband's gone off somewhere, she's all alone, and the nearest doctor's in Lubbock. I'm going over there to see what I can do. If I need more help, I'll let you know."
Her eyes met Tori's in silent understanding. The ambulance and paramedics had arrived with the fire department; but if Will or Sky—or both—were found alive, here was where the more urgent need would be. If Natalie ended up having to deliver a baby, it wouldn't be the first time.
With a final comment—"Keep me posted"—she sprang into her vehicle, gunned the engine, and roared off toward the distant row of bungalows. Tori turned her gaze back to the grim search of the wreckage. She could feel Lauren, close beside her, trembling. Neither of them spoke. There was nothing to say. They were two strong women preparing themselves for the worst.
One of the firemen gave a shout. "There's a body under here!"
Tori's heart dropped. She felt Lauren's hand creep into hers as more men hurried to pull away the debris. Beau was the first one to recognize the dead man. He spoke in a flat voice.
"He's got blond hair, and I remember that shirt from this morning. That's Ralph Jackson."
Tori's knees went weak. She braced, willing herself to stand. Beside her, Lauren gasped. "It's his wife who's having a baby! We need to let Natalie know."
Not trusting her voice, Tori found the number on her cell and handed it to Lauren to make the call. This time it hadn't been Will or Sky the men had found. But the search was far from over.
She averted her eyes as the paramedics lifted away the body and zipped it into a black bag, though a glimpse told her that the man hadn't been badly burned. She'd barely known him, but the tragedy, and the fear that this was only the beginning, hit hard and deep.
Looking across the yard, Tori could see Erin standing on the porch with Jasper and Bernice. She was staying where she'd been told to stay, but she was straining against the rail, trying to see what was happening. Tori ached to go to her daughter, take her in her arms, and assure her that everything would be all right. But she couldn't do that yet—not when Erin's loving, secure world might have already come crashing down in the inferno of the burning barn.
"Quiet!" It was Beau who'd shouted. "Listen—I think I heard something!"
In the silence of straining ears that followed, Tori could hear nothing but Lauren's breathing and the pounding of her own heart. Maybe Beau had only heard the shift of cooling metal, or the sound of a trapped, injured animal that would need to be put down.
Or a man, terribly burned . . . She forced the thought away.
"Over here!" One of the men pointed toward a spot near the barn's entrance where the debris was piled high against a standing wall. "Listen, there it is again! Something's moving!"
Tori, who was farther away, held her breath, but she could hear nothing. She could only wait with Lauren, in an agony of undeniable hope, as each piece of wreckage was carefully lifted away.
Then there was a shout. "They're here! They're alive, both of them!"
Tori's knees buckled. She heard a sob from Lauren as two ghostly-looking men, singed, ragged, and coated with soot and ash, emerged from the rubble. Will was on his feet, stumbling through the debris. Sky, barely conscious, had to be supported between Beau and one of the paramedics.
Will's smoke-reddened eyes narrowed as he took in the scene around him. "Damn it, don't just stand there gawking!" he barked, gesturing toward Sky. "Get this man some oxygen!"
He needn't have spoken. The paramedics were already easing Sky onto a stretcher and clapping an oxygen mask over his face. But Will's take-charge manner was enough to show Tori that the man she knew so well was back.
Lauren had rushed to be near Sky, staying close as he was carried toward the waiting ambulance. As they loaded him, Tori heard her arguing with the paramedic. The young man was insisting that she couldn't ride along because they had to take Will, too. But so far, Will showed no sign of wanting to go.
Tori gazed at Will through the clouds of settling ash. Her eyes misted as she thought of how close she'd come to losing him. She checked the impulse to stumble through the debris and fling her arms around his neck. Will had never been big on emotional drama, especially not in front of others. But he had to know how much it meant for her to see him safe.
For a slow beat of time, their gazes held. He cleared his throat. "Tell Erin I'm all right," he said.
"I'll tell her." The calm words masked a storm of emotions Tori had never expected to feel again. Heaven help her, she'd never stopped loving this gruff, stubborn, impossible man. But would love ever be enough to heal the hurt between them?
One paramedic stepped close. "Mr. Tyler, we're waiting. You need to come with us in the ambulance."
"The hell I do," Will growled. "My damned barn just burned down. I can't leave now."
"Don't be stubborn, Will," Beau said. "You've got some nasty burns, and you've inhaled a lot of smoke. You need to get checked by a doctor. I'll keep an eye on things till you get back."
"For once, do as you're told," Tori said. "If you ride with Sky in the ambulance, you can keep an eye on him. I'll follow in my car with Lauren. After you've been checked out, I'll drive you home."
Will's grime-coated features creased in a scowl. "Looks like I'm outgunned," he muttered. "All right, but this better not take long." He strode to the ambulance and climbed inside the back without help.
A press van had just pulled up to the barn. Will gave the reporters a contemptuous look before the doors closed behind him and the emergency vehicle, siren wailing, sped off toward the highway.
* * *
Natalie wrapped the baby boy in a clean blanket and placed him in the arms of his sixteen-year-old mother. The birth, thank heaven, had been an easy one. The baby was healthy, and the mother was doing fine. But knowing what she knew, Natalie could hardly go off and leave them alone.
Vonda gazed down at her son as if she couldn't believe he was real. Her fingertip brushed the small, perfect features, the little nub of a nose, the baby hands with their long fingers and tiny nails.
"He's a beautiful boy," Natalie said. "What are you going to name him?"
"Ralph, after his father. We talked about that." Her eyes welled with emotional tears. "Where's Ralphie? He's supposed to be here! Why hasn't he come home?"
Natalie had to look away. She'd received both messages Lauren had left on her phone—one saying that Will and Sky were alive, the other letting her know that Ralph Jackson had died in the fire. But how could she break the news to this poor girl? Vonda needed to hear it from someone she trusted. She needed her family to support her and soften the blow.
"Since he isn't here, why don't I call your parents?" Natalie offered. "They'll want to know you're all right, and they'll want to see their grandchild."
"No!" Vonda turned against the pillow, clutching her baby. "My folks kicked me out when I got pregnant. Mom said she wouldn't stand for having a sinner in the family! Ralphie's all I've got! Please, just find him for me!"
Heartsick, Natalie murmured an excuse and walked out onto the stoop with her cell phone. She knew the girl's parents, of course, not that she had much liking for either of them. Vonda's father, Sheriff Abner Sweeney, had been involved in last spring's case against Beau. He'd also been the one to question and arrest Will. Her mother, Bethel, was a staunch, Bible-thumping churchgoer who'd birthed eight children, most of them girls. Vonda, her firstborn, had been the first to rebel and go astray. Natalie suspected she might not be the last.
But that was neither here nor there. As Natalie scrolled down the names on her cell phone, she could only hope that Abner and Bethel had enough Christian charity in their hearts to forgive the child who had nowhere else to turn.
The only phone number she had was the sheriff 's. By now, he probably knew about the fire, and might even know that Ralph was dead. But unless she could reach him, Abner Sweeney wouldn't know that he'd just become a grandfather.
He answered on the first ring. "What is it, Natalie? I'm on my way to a fire at the Tyler place. Goin' by what I heard on dispatch, it might've been dee-liberately set. D'you think Will Tyler would be desperate enough to burn his own barn for the insurance?"
With effort Natalie held her temper in check. "The fire is out. And I was with Will, in the house, when it started. We were about to eat Thanksgiving dinner."
"Oh." The sheriff sounded disappointed. "So what was it you wanted? Can it wait, or is it an emergency?"
"It's an emergency—yours. Your son-in-law is dead, and your daughter just had her baby, a little boy."
She heard the squeal of brakes as he pulled off the road. "Say again?"
"Ralph died in the fire. I'm here with Vonda at their bungalow. She and the baby are fine, but she doesn't know about her husband yet. You and your wife need to get here. You need to be the ones to give her the news and take care of her."
There was a beat of silence. "I'll come as soon as I can. But I don't know about Bethel. She can be a hard woman once she makes up her mind."
"Bring her! I don't care if you have to hog-tie her to do it! This poor little girl needs her mother!"
Natalie ended the call.
* * *
Lights flashing and siren blaring, the ambulance barreled up the highway toward Lubbock. Will, riding in the back with Sky, had insisted on sitting up. Aside from minor burns, a raw throat, and smarting eyes, he felt fine. As he'd told the husky young paramedic, anybody who thought they could make him lie down was welcome to try. Since no one had challenged him, he'd taken a seat on the bench, where he could be close enough to look after Sky, and to talk to him.
Sky was awake. The oxygen was helping to revive him, but he was looking pretty rough. His scalp and face were pocked with burns where sparks had showered down from the blazing timbers. His hands had been burned as well, and his denim shirt was little more than scorched tatters. The burns would heal, but Will was more worried about Sky's lungs. There'd been enough oxygen in the burning barn to keep him alive, but he'd inhaled a dangerous amount of smoke. There'd be enough damage to keep him on humidified oxygen for a few days, at least. Maybe longer if there were complications.
Sky had saved every last animal in the barn and damn near died doing it. He seemed unaware of what a loss his death would have been to the ranch. But Will knew. This stubborn half-Comanche was as much a part of the Rimrock as the earth, the grass, the water, and all the living things that called it home. He was Bull Tyler's blood son, Will's own half brother. And yet he asked for no praise, no recognition of any kind except the freedom to care for what he loved.
Sky stirred and made a low sound. His eyes were open above the edge of the oxygen mask that covered his nose and mouth. He gazed up at Will as if he wanted to speak.
"Take it easy, brother," Will said. "You need to keep still and just breathe."
Brother? Sky's singed eyebrows twitched in an unspoken question.
"You heard me. I've been waiting for the best time to tell you I knew. I guess that time's now."
How? Another question expressed by a look.
"Jasper told me. But I knew before that—maybe even before you did. You've got Bull's eyes and some of his mannerisms, and you're almost as mule-headed as he was. I suspected the truth for a long time, but when you got shot last spring, and Beau and I had to give you our AB-negative blood, the same type as Bull's, that cinched it. I knew you were a Tyler."
And Beau?
"I'm guessing Beau hasn't given it much thought. But I could be wrong. I take it you've told Lauren."
Sky's head moved on the pillow, a slight nod.
"Well, we'll leave it at that," Will said as the ambulance swung into the hospital parking lot. "Just rest and get better. Tori's bringing Lauren with her. They should be along soon."
The back of the ambulance opened to glaring afternoon sunlight. Paramedics laid Sky's stretcher on a gurney and whisked him through the doors of the emergency entrance. An orderly brought out a wheelchair for Will. He waved it away and kept walking, following his brother until someone pulled him aside.
CHAPTER 14
Tori had stopped by the house long enough to grab her purse and keys, and to let Erin, Jasper, and Bernice know that Will and Sky were safe. Now, with Lauren buckled into the passenger seat, she was breaking speed limits on the highway to Lubbock.
"You didn't have to drive me," Lauren said. "I could've taken my own car."
Tori swung her station wagon past a lumbering hay truck. "You're too upset to drive," she said. "Besides, I told Will I'd be there to take him home—that is, if the hospital doesn't keep him overnight. You're welcome to come with us or stay there with Sky. If you stay, somebody can pick you up in the morning."
"You're pretty upset yourself. I can tell by the way you're driving." Lauren gazed ahead at the yellowed plain and the road that sliced across it in a straight black line to the horizon. "Were you as scared as I was?"
"Scared enough," Tori said. Days like today were part of ranch life. Lauren would learn that, if she hadn't learned already.
"Those damned horses!" Lauren muttered. "I love horses, too, but Sky almost died for them today—and they aren't even his. How am I supposed to wrap my mind around that?"
"Horses are like family to Sky," Tori said. "If he cares for them so passionately, think how he'll be with his real family—you and your children. He'll do anything to provide for you and keep you safe."
Lauren fell silent for a moment. "I'm pregnant, Tori," she said.
"Oh—" Tori released her death grip on the steering wheel long enough to reach over and squeeze her friend's hand. "Does Sky know?"
"I just found out, myself." She shook her head. "Maybe, today, it was just as well Sky didn't know. I keep asking myself, what if he'd had to make a choice between saving the horses and being there for his child—and he'd still chosen those damned horses?"
"Thinking like that will only muddy the water," Tori said. "Believe me, I know. If I hadn't convinced myself that Will loved the Rimrock more than he loved me, maybe we'd still be married."
Lauren was quiet for a few moments, as if pondering what she'd just heard. "What happened with you and Will, Tori?" she asked. "Seeing how you always seem to be there for each other, I can't help wondering what went wrong."
Tori sighed. "Maybe it's time I told you. You sound as if you need to hear this."
As the miles sped by, Tori told her the story—how Bull had tried to dominate the marriage and how Will had been caught in the conflict between his wife and his father. "After I lost the baby and found out I couldn't have more children, Bull treated me like a failure," she said. "I begged Will to find us another place to live. But after Bull's accident, that was out of the question. Will needed to be there to oversee his father's care and run the ranch."
"Why didn't you leave on your own?"
"I could have. But I still loved Will, and we both adored Erin. So I hung on . . . until I couldn't hang on any longer."
Tori began the final chapter—how her father had died in Florida and Will had been too busy to join her for the funeral.
"And that was the final blow? That he wouldn't be there for you?"
"Not quite." This was the hard part—the part she hadn't told Drew or even Natalie. But Lauren needed to hear it all. "There was a man," she said, "a longtime friend of my father's. He was widowed and made no bones about being attracted to me, but I wasn't the least bit interested. He was just a friend, and barely that.
"When he heard about my father's death, he flew down from Washington, D.C., and did his best to be kind and helpful. Yes, maybe he was a little too friendly, but without Will there, I needed some support. My mother was a wreck, so I had to make a lot of the arrangements. I truly appreciated his help.
"Two days before the funeral, Will changed his mind and decided to come. He phoned the house to let me know. My mother took the call and told him I'd gone to lunch with this man. She went on and on about how helpful he'd been and how much I'd appreciated his being there. Will hung up the phone and went ballistic."
Tori turned onto the street that led to the hospital. "When I didn't hear from him for the next three days, I gave him a call. He ripped me up one side and down the other—pretty much accused me of having an affair, which couldn't have been further from the truth. That was when I knew it was over. I came home two weeks later with divorce papers. End of story."
Tori pulled into the hospital parking lot. How trivial it all sounded in the retelling—two proud, stubborn people who'd had a misunderstanding and couldn't forgive each other. But at the time it hadn't been trivial at all. It had been like the end of the world.
"And the man?" Lauren spoke as if she already knew. "Who was he?"
Tori pulled into a parking spot, turned off the engine, and unfastened her seat belt. "Congressman Garn Prescott, of course—your father."
* * *
After Natalie's phone call Sheriff Abner Sweeney had driven back to town to pick up his wife. Bethel sat beside him now, her plump body rigid, her narrow-lipped mouth fixed in a straight line. She'd agreed to go with Abner. But he could sense her inner struggle. She had cast her daughter out for her sin. Now righteous judgment warred with compassion and motherly love.
Bethel, a preacher's only daughter, had grown up with her father's ironclad values. She'd raised her children as she had been raised, never dreaming that she'd one day be faced with an agonizing choice like this one.
"So Vonda and the baby are all right?" she asked Abner for perhaps the third time.
"That's what Natalie told me."
"But she doesn't know her husband's dead?"
"That's what I understand. Natalie said it was our place to tell her."
"You tell her, then. You're used to doing things like that."
"Fine, I'll tell her. But you need to be there." Abner was already wondering how Ralph had died. Natalie hadn't offered any details. Had Vonda's husband perished fighting the blaze? Or . . . but no, that didn't make sense. Why on earth would Ralph set fire to his employer's barn—especially if he couldn't make it out of the barn in time to save his own life?
Ralph's pickup sat in the graveled driveway, in front of Natalie's SUV. The old rust bucket wasn't fit for anything but scrap. He could sell it and give Vonda the money toward a decent car. Lord knows, she was going to need it.
Strange that Ralph wouldn't have driven to fight the fire. Covering the distance on foot would have wasted precious minutes. Had somebody else picked him up? Or had he walked to the barn before the fire started?
For now, those questions would have to wait. Abner had a job to do, but he was also a father. His helpless, grieving daughter needed him, and he would be there for her.
Natalie came out onto the stoop as they pulled up. She hadn't been friendly to Abner since that mess with Beau last spring. But at least she'd come when Vonda needed help.
"Your daughter and the baby seem fine," she said before they could speak. "But just to be sure, you'll want to get them checked out by a real doctor, at the hospital."
"Thank you," Abner said. "I've been wondering about Ralph. How did he—"
"They found his body after the fire was out. That's all I know."
"And they don't know what started it?"
"You'll have to ask the fire crew. Right now, your daughter needs you—and I need to go." She strode out to her vehicle. By the time she drove away, Bethel had already hurried inside. Abner followed her.
Vonda, looking so young and scared that it almost broke his heart, was sitting up in bed, clutching her baby in her arms. She was wearing a clean nightgown, and the sheets looked as if they'd been changed. But his daughter's face was pale, her hair plastered in damp strings around her face. She was gazing down at her infant son, as if she had no idea what to do with him.
Bethel was bustling around the room, straightening this and that, avoiding eye contact with her daughter. His wife would come around, Abner thought. It just might take some time.
"Hello, Daddy." Vonda managed a wan smile. "Would you like to hold my baby? His name is Ralph—Ralph Junior."
Abner took his grandson and cradled him close. The pink, puckered face, flattened nose, and tiny, waving hands tugged at his heartstrings—a familiar ache that felt strangely sweet.
Abner knew he wasn't the best man in the world. He'd skated the edge of dishonesty more times than he liked to think about. But he loved his family. The instant bond with this little boy was like the closing of a lock. Whatever happened, he vowed, he would protect this child and see that he and his mother never wanted for anything. He'd been wrong about a lot of things, like letting Bethel banish their daughter. But nothing could be more right than the fierce love he felt for this small, new life. In every way it made him want to be a better man.
Vonda looked up at him. "Daddy, where's Ralphie? He's supposed to be here."
Abner shook his head, knowing he had to face the hurt in her eyes.
"Has something happened to him?" Her voice broke. "Is that why you and Mama are here?" Her gaze widened as the truth struck her. "No!" she whispered. Then her voice broke into a keening wail. "No! Please, God, not Ralphie! No! No!"
"It's all right, honey." Bethel leaned over the bed to gather her daughter in her arms. "It was God's will, to pay for your sin and bring you back to us. We're here now. We'll take care of you."
* * *
Abner and Bethel had agreed it would be wise to take Vonda and the baby to the hospital. While Bethel got her ready to go, Abner stepped out onto the stoop and called in a pair of his deputies to investigate the fire. He would follow up on his own, later in the day. The blaze at the Tyler place was too well-timed to be an accident. He would bet money there was arson involved, maybe murder as well. And the key to it all could be his late son-in-law.
Abner had never thought much of Ralph Jackson. The boy was short on brains and ambition, and he hadn't made much effort to keep Vonda happy. But at least, with the Tylers, he'd had a secure job and a place to live. It didn't make sense that he'd risk it all by setting fire to the barn.
Maybe somebody else had set the fire and Ralph had caught them in the act. That could be reason enough to get him killed—but that theory would have to wait for the coroner's report. Abner was no Sherlock Holmes. But along with his experience and the reading he'd done, he knew how crime solving worked. If he could put this case to bed, it would raise his standing with the county government and the voters. Maybe, then, Clay Drummond would stop treating him like a damned stooge.
Bethel was taking her time getting Vonda ready for the hospital trip. Probably a lot of emotion going on. Abner was getting restless, when his gaze fell on Ralph's old truck. As long as he was here, just standing around, it wouldn't hurt to check it out.
He pulled a pair of latex gloves from the box he kept in his SUV, tugged them on, and walked over to the truck.
The driver's-side door was unlocked. No sign of the keys, but Ralph could've had them in his pocket. The floor was littered with empty Dos Equis beer cans and Snickers wrappers. Mummified French fries, cookie crumbs, and empty ketchup packets were lodged in the crease below the seat backs.
In the unlocked glove box, Abner found some gas receipts and a yellowed copy of the truck registration. Underneath these was a half-empty pack of condoms. No surprise there. Vonda's better off without the cheating bastard. But there was nothing here that might link Ralph to the fire in the barn.
Abner was about to climb out of the truck, when he saw that his shoelace was undone. Bending down to tie it, he noticed a crumpled plastic Shop Mart bag stuffed way back under the driver's seat. It was probably just more trash, but he'd be remiss to leave it.
With some stretching and grunting, he reached it, caught the corner, and gave a pull. He'd expected it to be empty, but the bag had a surprising heft to it. After dragging it free of the seat, he sat up, took a breath, and untied the knotted handles to look inside.
Abner's stomach lurched. Cold sweat beaded under his uniform. The bag was stuffed with cash—lots of cash, most of it in hundred-dollar bills.
* * *
Lauren sat on a folding chair next to Sky's hospital bed, listening to the labored sound of his breathing. Lightly sedated, Sky was veiled by a misted oxygen tent. Fluid dripped into an IV tube connected to his wrist. A beeping monitor above the bed tracked his pulse and blood pressure, as well as his oxygen level, which had been fearfully low at first, but was beginning to rise.
She yearned to put her arms around him, or at least hold his gel-bandaged hands. But that, she knew, would only cause him more pain. The doctor had said he would live. Sky was young and strong, he'd told her. His vitals were good, considering what he'd been through. His second-degree burns should heal in a few weeks. But he'd inhaled enough smoke to damage his lungs—that was the real worry. Another minute in that burning barn would have killed him.
She gazed down at the modest diamond engagement ring on her finger—the ring she hadn't taken off in the three months since Sky had placed it there. She'd accepted his proposal without a moment's hesitation. But now she had to face reality and ask herself the hard question. Could she really do this—open herself to heartbreak again?
Mike, her first fiancé, had jumped off a bridge and drowned. Her father had died of a heart attack after shooting himself. Now she'd fallen in love with a man who took reckless chances for others, heedless of his own safety.
Lauren had heard how, last spring, Sky had been shot trying to save his worthless young cousin Lute. And in last summer's wildfire, he'd risked death, refusing to leave the ranch until every last animal was evacuated. This was Sky's way, throwing himself in the path of deadly danger for the sake of any living creature that needed him.
She needed him, damn it, especially now. But today in that blazing barn, his mind wouldn't have been on their losing each other. And even if he'd known about the baby, he wouldn't have been thinking about their unborn child. All his intent had been focused on rescuing his beloved horses.
What would she do the next time something happened? And, ranch life being what it was, there would no doubt be a next time. How could she go on living if she lost him?
Lauren's agitated fingers toyed with her ring, twisting it, sliding it up past her knuckle, then back into place. I can take it off right now, she told herself. She could leave it on the side table and walk out of his life—move far away from here and never tell him about the baby. She had enough money to go anywhere she wanted. How hard could it be? Sky was the center of her world—but did she love him enough to face the prospect of losing him?
Settling back against the hard chair, she thought about Tori and Will and the forces that had driven them apart. Their divorce had been a bitter one. But Lauren could sense the fierce undercurrent of love that still flowed between them. How would things be different if, that one last time, they'd swallowed their pride and forgiven each other? Would they and their daughter be a happy family now?
What if she couldn't forgive Sky for the terrible risk he'd taken? What if she were to give up and walk away? Could she live with that decision for the rest of her life?
Rising, she gazed down at the man she loved to the roots of her soul. Sky Fletcher was who he was, and she knew better than to believe he would change, even for her.
Knowing what she knew, could she find the courage to build a future with this man?
With effort he opened his eyes, gazing up at her through the transparent oxygen tent. His cracked lips moved, forming her name.
"Lauren . . ."
She couldn't hear his voice, but it was enough. Her hand reached out to press his shoulder. "I'm here," she whispered, knowing it was true. She was his, and she was here to stay.
* * *
Will had been checked over, treated for minor burns, and released; but he'd refused to leave the hospital until Sky was stabilized and in the ICU, with Lauren watching over him. "I'll be back first thing tomorrow," he'd told the doctors. "Call me if he needs anything."
As they walked out to the parking lot, Tori keeping pace with his long strides, the depth of his concern prompted her to voice something she'd suspected for years.
"Sky's your brother, isn't he?"
"Right now, that doesn't make much difference," Will said. "I'd be just as worried either way."
"But he is, isn't he? I've always wondered. He's as much like Bull as you are, only in different ways."
Without answering, Will moved ahead of her and opened the passenger-side door of her station wagon. "Give me your keys and get in," he said. "I'm driving."
"Is that wise?"
"Keys." He held out his hand. With a sigh Tori opened her purse and handed him her keys. Regarding Will, she'd long since learned to choose her battles. This was no time for a useless argument.
He drove like he was angry—not at her, but at the horrific circumstances of the day. A treasured family holiday had been ruined, but that was the least of it. He'd also lost the barn on the brink of winter, with a storm due in, any day. Even if the insurance would pay much of the rebuilding cost, he couldn't buy time. He couldn't buy back the life of the man who'd died in the fire or the weeks Sky would need to heal from the trauma that almost killed him.
And he couldn't delay the trial, now less than a week away, with a possible outcome that could separate him from everything, and everyone, he'd ever loved.
Tori's gaze traced his defiant profile, lingering on the twitching muscle in his cheek. She knew Will, and she knew how much he was hurting. Part of her yearned to wrap him in her arms and tell him everything would be all right. But that would be a lie—the last thing Will would want to hear.
It was all up to her, Tori realized. If she could win an acquittal for Will, he would find a way to fight through and save the ranch. If she lost, the entire burden of the ranch, the barn, and the money problems would fall on Beau. In his own way Beau was as strong as Will. But he'd left once, years ago, after having it out with Bull. If things got bad enough, she wouldn't put it past him to throw up his hands, take Natalie and their baby, and return to his government job in the East.
"I got a call from Natalie," she said, feeling the need to break the uneasy silence. "Ralph Jackson's wife had a baby boy. Abner and his wife showed up and took them both to the hospital."
Will exhaled, easing his grip on the wheel. "It's a good thing they stepped up. I'll have Beau look into our insurance. Since her husband died in the fire, that should be worth something for the poor girl."
"How do you think the fire started?" Tori asked.
"Damned if I know. Everybody was supposed to be at dinner then. Nobody would've been in the barn."
"So when did Ralph go in? Did you see him outside fighting the fire?"
"No, but I was busy. He could've been anywhere. Maybe he was helping Sky get the horses out. When Sky's up to talking, I'll ask him."
"Maybe the fire was set, and whoever did it murdered Ralph. What do you think of that theory?"
"It's possible, I guess. There'll be an investigation and a coroner's report. Maybe after that, we'll have a better idea of what happened." Will lapsed into silence. Tori could imagine what he was thinking. By the time the investigation was finished, he could be far from home, looking at the world through prison bars.
Tori's memory shifted back to the night—barely remembered until now—when she and Drew had stopped off at the Blue Coyote for a late-night beer. She remembered glancing around to meet those absinthe eyes, and how their hate-filled gaze had made her skin crawl.
"It had to be Stella," she said. "Who else would despise you and your family enough to do this?"
"You could be right." Will's tone was carefully neutral. "But Stella wouldn't come on the ranch and set the fire herself. If she wanted it done, she'd pay or blackmail somebody else to do it. And she'd make damned sure nothing could be traced to her." He shot Tori a stern glance. "Leave it to the law for now. Trying to figure out who set the fire won't undo the damage. Besides, you've got more urgent things to deal with."
Yes, the trial. She couldn't forget that. Not when the worry was keeping her awake nights. She had a good case. Given the true facts, no reasonable jury would convict Will. But something told her she'd be playing against a stacked deck. Clay Drummond always liked to win, but this time he seemed determined to the point of desperation. Something, Tori sensed, was very, very wrong.
* * *
Stella turned off the nightly news and poured herself a glass of the finest Kentucky bourbon money could buy. She'd ended the holiday with three wins in her column—Will Tyler's barn burned to cinders, Sky Fletcher injured, and Ralph Jackson dead.
When she'd heard the sirens around one-thirty, she'd figured Ralph must've done his job, but she'd spent an uneasy afternoon wondering what the chances were that the fool would get caught and turn on her to save his own skin. She'd asked herself, again and again, whether she'd risked too much to strike at her enemy. Ralph was a weak link. However the day went down, he would have to go. It was just a matter of how and when.
Hours had passed as she waited for Ralph to call her and demand the rest of the cash she'd promised him. When no call came, she'd imagined the worst—that he'd been caught and arrested.
After packing a valise, to have ready in case of a needed getaway, she'd glued herself to the radio and TV for any word about the fire. Not until the six o'clock news had she learned that the fire was out, and that one man had died in the blaze, presumably trying to save the horses. The owners of the ranch were unavailable for comment, but the deceased ranch hand, Ralph Jackson, was being remembered as a hero.
A hero! What a joke! Stella grinned as she took another sip of expensive bourbon. It wasn't often that fate played into her hands. That it had happened today was a cause for celebration. But she wasn't finished, not by a long shot. First thing tomorrow she would call Abner and make sure he took advantage of every chance to give the Tylers more trouble.
She could only hope Will's trial would go as well as the barn fire had. She could hardly wait to see Nicky's killer behind bars.
* * *
It was 1:15 a.m. when Sheriff Abner Sweeney parked behind the county vehicle lot, procured the keys to the department's tow truck, and drove it out of the gate. Twenty-five minutes later, he crossed the boundary of the Rimrock and switched off his headlights. With the November moon just bright enough to show the road, he drove to the bungalow where his daughter had lived.
Ralph's old pickup was still parked outside. It took only a few minutes for Abner to hitch the rear axle to the tow truck, pull out of the driveway and onto the road. What he was doing was illegal as hell and could cost him his job. But as he'd told himself all the way here, this was for the greater good. This was for his daughter and his grandson.
After he'd found the money in the truck, it hadn't taken long for the truth to fall into place. That kind of cash could only have come from one source—the person who hated Will Tyler enough to burn down his barn. She would have needed to pay someone to do the job, someone who worked for the Rimrock and had access to the place. That someone had been Ralph.
The money, mostly new bills, was evidence. It would likely have Stella's prints on it. Here, at last, was something that could link her to a crime.
His duty as sheriff demanded that he follow through and arrest the woman. But arresting Stella could expose the quid pro quo favors she and Abner had done for each other over the years. Worse, it would implicate Ralph, who was being lauded as the hero who'd given his life to rescue the Tylers' horses.
Ralph's baby boy could grow up as the son of a hero or the son of a criminal. The end result could make all the difference in his young life. That difference, here and now, was up to Abner.
Still driving by moonlight, Abner turned the tow truck onto a rutted side road that ended two miles later at a shallow, raw dirt gully, strewn with trash and the remains of before-hunting target practices. After unhitching Ralph's truck, he used the tow truck to push it into the gulley. The rusty old vehicle rolled down the slope and settled into place, right side up, amid the clutter of old bedsprings, empty beer cans, and ancient TVs with their screens shot out.
Abner took the canister of black spray paint he'd brought along and used it to decorate the doors and windows of the truck with known teenage gang symbols. The cash was still inside, stuffed under the driver's seat, where he'd found it. He'd deliberated long and hard about keeping it. The money could go a long way toward helping Vonda raise her son. But the bills had likely come from drug deals. If Stella, who had eyes and ears everywhere, decided to track it down and get it back, his whole family would be in danger.
Battling regret, he backed the tow truck to a safe distance, took a heavy gasoline can out of the bed, hiked down into the gully, and doused the old rustbucket, inside and out. Sooner or later, word would get around that Ralph's truck had been stolen and burned. When Stella got the news, she would assume, rightly, that the money was gone.
As he climbed back up the slope, Abner poured a thin trail of gasoline behind him. Standing at the gully's edge, he lit the trail with a match and took off at a run. The truck, money and all, exploded in a giant whoosh of flame behind him.
Abner hung around long enough to make sure the fire wasn't going to spread. Then he drove back to town, replaced the tow truck, tossed the paint canister in a handy Dumpster, and went home to his family.
CHAPTER 15
True to his word, Will was at the hospital by eight o'clock the next morning. He found Sky sitting up in bed, drinking a protein shake through a straw, with an oxygen tube clipped to his nose.
"You look like hell," Will said, taking a seat next to the bed.
"I feel like hell." Sky managed a grin. Even so, he didn't look as bad as Will had feared. Where the soaked bandanna had covered his nose and the lower part of his face, there were only a few minor burns. His ears and upper head had suffered worse, his hands, wrapped in special water gel bandages, the worst of all. But the real damage, Will knew, would be from the smoke in his lungs. Sky could be a long time regaining his health.
"Dare I ask how things are on the home front?" Sky's eyes, though slightly glazed from pain medication, had recovered some of their old spark. "I take it the barn's a total loss. Do you think the insurance will cover it?"
"Some, but not all. The devil of it is, we've had three fires this year—the machine shed last spring, the wildfire, and now this! What if the insurance company cancels our policy?" Will shifted the chair closer to Sky's bed. "And the timing's a bitch with winter coming on. We're already salvaging material from the barn to set up storm shelters in the paddock. But by the time supplies come in for the new barn, even with all the hands working, it'll take weeks to get the basics up. They're cowboys, not builders. Beau's been calling around to get construction bids, but with money so tight . . ." He shook his head. "Sorry to burden you with this, Sky. It's a royal mess."
"I've got a pretty good crew working to finish my house. Take them. They can show your cowboys what to do."
"But I thought you and Lauren wanted the house done before Christmas, for your wedding."
"We did. But since we'll be putting the wedding off until I'm in better shape, the house can wait. The outside's weather-tight. It'll be fine."
"That would be a huge help," Will said. "Are you sure it's all right with Lauren?"
"We already talked about it. In fact, lending you the workers was her idea." Sky glanced around as the bathroom door opened and Lauren stepped out.
"Should my ears be burning?" She was making an effort to smile, but she looked totally wrung-out, as if she'd spent the night awake, sitting up in her clothes and worrying, which she probably had.
"Take this woman home, Will," Sky said. "She hasn't stopped fussing over me since she got here, and she was terrorizing the poor nurses all night. She needs to get some sleep and come back later—if they'll let her in."
"I was just making sure you got the attention you needed. Like now—for heaven's sake, you're out of ice." Lauren refilled his water glass from a pitcher on the tray. Sky managed a comic eye-roll.
Will chuckled. "Don't worry, I won't leave here without her. And thanks for looking after him, Lauren. I was worried about leaving him, myself, but I knew he was in good hands."
"I'll be fine," Sky said. "The worst part of all this is having to be here when you need me, Will."
Will knew the words carried double meaning. Whatever happened at the trial, Sky's help would be sorely missed—especially since, until his burnt hands healed, he'd be unable to ride or even drive.
Which brought Will to the news he was dreading to deliver. Sky might not take it well, but it had to be said. Will hardened his resolve. "We're going to have to sell off most of your colts," he said. "We're asking best offer. That'll mean taking a loss on them, but with the money situation being what it is, we can't afford to feed them over the winter. Beau's already sending out sale notices, not just to the Texas ranches but all over the country."
Sky's mouth tightened. Will knew how much work and care he'd put into selecting, breaking, and training the colts. The green young horses had been brought in to shore up the ranch's finances, in case the drought forced a sell-off of the cattle. Most of the cattle were already gone, at a loss. And with other Texas ranches in similar straits, nobody in the state was paying big money for horses.
"Damn." Sky exhaled, wincing with pain. "Some lucky folks are going to get a bargain on some great cow ponies. I know you wouldn't do this if we weren't scraping bottom. But I'd like to ask a favor. There are a couple of those colts I wouldn't mind keeping for myself. Would you let me buy them from the ranch at the going price?"
Will knew Sky must be thinking of Quicksilver, the sharp gray gelding that had become his favorite. And he'd likely want a good filly to breed with one of the Rimrock stallions so he could start his own herd on his land. But with the expenses of his new house, Sky wasn't exactly swimming in money, either. "Tell you what," Will said. "They're yours. Call it a wedding present."
"Not on your life," Sky said. "I know you're going to need every cent to keep the ranch afloat."
"We can talk about that when you're on your feet," Will said. "Meanwhile, let me know your choices. I'll make sure Beau doesn't put them up for sale."
Sky was about to answer when two hospital aides, dressed in scrubs and carrying a stack of folded linens, walked in. "Looks like it's cleanup time," Sky said, clearly trying to sound cheerful. "Get my woman out of here, Will. Take good care of her. See that she eats and rests."
"Don't worry, we'll all look after her," Will said.
Lauren took time to kiss her fingertip and touch it to his lips. "You rest, too," she said. "I'll see you later. Will, I left my car and my purse at the ranch. You can just take me there."
She let Will usher her out the door and down the network of hallways to the parking lot. As they walked to his truck, with his hand resting lightly under her elbow, he could feel her falling apart. She was rigid but shaky, her features braced against the emotion that was threatening to crush her. By the time they reached the truck, and she settled into the seat, her jagged breaths had become dry, racking sobs.
"Go ahead and let it out, girl." Will slipped into the driver's seat and started the engine. "You've been through a hell of a time, and you're worn to a frazzle. You're a strong woman—I've seen it and I know. But you don't have to be strong twenty-four hours a day."
Little by little, Lauren brought herself under control. "Thanks, Will. It's just, seeing Sky like that, helpless and in pain . . ." She drew a long, tight breath. "Yesterday I didn't know if I could do this. I almost left."
"But you didn't."
"No, I didn't. And today he's doing better. But I was so scared! I never want to be that scared again!"
"Sky needs you, you know," Will said. "He's a tough man, and too proud for his own good, but I don't think he could make it through this time without you."
"I understand," Lauren said. "But that's not the reason I decided to stay. I stayed because I realized how much I needed him."
"Then all I can say is, he's a damned lucky man."
Will lapsed into silence, thinking about Tori—all the times she'd needed him, when he hadn't been there for her. That last time, when he'd raged at her over the phone, he'd actually been furious at himself for letting her down and leaving her open to another man's attentions. Hell, he'd known she didn't care about Garn Prescott. He'd known she wasn't having an affair. But between the stress of his father's illness and the fear of losing her, he'd just plain lost it. And that had been the end of everything.
Now there was a new man in her life—the sort of fellow who drove a sedan and wore cashmere sweaters and wingtips. Steady and stable, Drew Middleton was probably just what Tori thought she wanted. All Will's instincts urged him to stand up and fight for her. But he had nothing left to fight with. Even without the specter of prison hanging over him, he was flat on his back financially, and on the verge of losing the ranch. He had nothing to offer any woman, let alone a classy lady like Tori.
"You know who was responsible for setting that fire, don't you?" Lauren asked, changing the subject.
"I've got a pretty good idea. But no way to prove it."
"No more than I've got a way to prove Stella ruined my father, or that she tried to have his car rammed, and almost killed me by mistake. She's an evil monster, Will. There's got to be a way to bring her down."
She shifted in the seat, turning toward him. "I asked Sky if he'd seen Ralph inside the barn. He said no, but he'd noticed that some of the stall gates were already unbolted. People are saying Ralph was a hero. But what if he'd been paid to set the fire, and on his way out of the barn, he decided to give the horses a chance to get out—or make himself look like a hero—and then something went wrong, and he didn't make it out?"
"It makes a good story," Will said. "But with Ralph dead and the barn burned to ashes, we may never know for sure. For all we know, he could've seen the fire, run in to help, and passed out from the smoke. The coroner's report should tell us something."
"But you'd given the men the afternoon off. Why wasn't he home? And what about that old truck of his? Why didn't he drive it to the barn, unless he didn't want to be seen? Maybe if we looked inside—"
"The truck's gone," Will said. "When I drove by the bungalow this morning, it wasn't there. I'm guessing maybe the sheriff or his deputies impounded it."
"The sheriff—that's another thing," Lauren persisted. "What if he's on Stella's payroll, or at least owes her a few favors? That could explain a lot, especially about the way he's treated you."
Will shook his head. "Nobody puts much stock in Abner as a sheriff. But does that mean he's crooked? Like I say, there's no proof. For now, all we can do is rebuild and move on."
Lauren sighed, shifted in the seat again, and sank into silence. Will had to admit her ideas made sense. But he was already dealing with more than he could handle. Somebody else would have to play detective.
He found himself wondering if Lauren had done anything about the transfer of the canyon land. She'd mentioned that she wanted to explore it first, but she and Erin had already spent time up there exploring to their hearts' content. On a stressful day like this, would it be crass of him to mention it again? Maybe so, but he was running out of time.
"I was just thinking about that canyon—" he began, then realized she wasn't hearing him. Lauren's head had sagged against the window. She was fast asleep.
* * *
Tori came out onto the porch as Will pulled up. By then the ranch hands, under Beau's supervision, were already clearing away the rubble from the burnt barn. With the aid of the farm-sized backhoe and bulldozer, they were setting aside any salvageable materials and pushing the rest into piles to be hauled off later.
Tori waited until Will had climbed down from the cab. "How's Sky?" she asked.
"Doing better this morning. But he'll be out of action for a while." He strode around the truck and opened the door for a drowsy-looking Lauren. "See that this lady gets some rest, or at least some coffee before she drives back to town," he said. "She's been up all night."
Tori took in her friend's hollow-eyed expression and rumpled clothes. "You look like death warmed over," she said. "Jasper and Erin are eating breakfast now. Come on in and join them—unless you'd rather just lie down."
Lauren gave her a flicker of a smile. "Thanks. Actually, I'm starved."
Tori had ushered Lauren inside and was about to follow her when the sheriff's tan SUV came roaring up the long gravel drive and swung toward the area where the hands were cleaning up the barn debris.
"Here comes trouble," Will muttered, taking off at a fast clip across the ranch yard. After a second's hesitation, Tori followed him. As his lawyer, it was her job to keep any exchange with Abner from getting out of hand.
By the time she got within hearing range, the sheriff was out of his vehicle. She could hear him shouting at Beau. "What do you think you're doing? This place is a crime scene!"
"Since when?" Beau stepped up to face him, looming over the short, chubby sheriff.
"Since the coroner's preliminary report came back this morning. It looks like Ralph Jackson most likely died from a fractured skull. To me, that spells murder. I can't let you disturb the evidence."
Beau's color was rising, and with it, his temper. "Look around, you lamebrain! It's already disturbed! Your deputies were here yesterday. They took photos, picked up a few things that caught their eyes, and left. Your people are done here."
Will was quick to jump into the argument. "Damn it, Sheriff, we need to clear this place to rebuild the barn. Every day—every hour—we lose puts us closer to winter. If you want to look through those trash piles, go ahead. But we can't afford to stop working."
Abner pouted beneath his trooper's hat. "You know, I could have you charged with obstruction of justice for destroying evidence."
"Hell, I'm already charged with that!" Will snapped. "What do you want us to do, put everything back the way it was? You're not even making sense. This is nothing but harassment!"
The situation was becoming a powder keg, about to blow, which could be just what Abner wanted—an excuse to drag Will back to jail. Fearing the worst, Tori stepped forward. "Sheriff," she said in a firm but civil voice, "you're welcome to look around, but this is private property. Unless you have a court order, we have no obligation to stop this work."
Abner puffed his chest, saying nothing.
"Do you have a court order?" Tori asked again.
"I can get one."
"Then please don't interfere until you have it." Tori's voice dripped ice.
"I'll guarantee you haven't heard the end of this!" Abner wheeled and stalked back toward his SUV. His oversized tires spat gravel as he drove away.
Beau gazed after him, shaking his head. "Now what the hell do you suppose that was about?" he mused out loud.
"I don't know," Tori said. "But I had the distinct impression that the sheriff was bluffing."
"I've got a pretty good idea who might've put him up to it," Will said. "But what we need right now is to forget Abner Sweeney and get this mess cleared away from the foundation. Let's get the hell back to work."
* * *
Feeling like a fool, Abner Sweeney gripped the steering wheel with sweating hands. His pulse was racing, and his prostate was acting up again. If he couldn't hold it till he got back to his office, he'd have to pull off the road and pee in the cold. To make matters worse, his personal cell phone was ringing, a call he knew better than to ignore. Slowing down, he reached for the phone in his pocket.
"Well, how did it go?" Hearing Stella's voice was like feeling a rattlesnake crawl across his foot. "Did you put the fear into those Tylers?"
Abner stifled a groan. "I gave it a shot, but Tori was there. When I threatened to arrest Will for obstruction, she demanded that I show a court order to put the barn site off-limits—something no judge would give me. All I could do was leave."
"That bitch!" Stella muttered, then continued. "But you told them your son-in-law was murdered, didn't you?"
"I told them I suspected it. They'll be wondering about that for a while." Even though it isn't true, Abner reminded himself. It had taken the retired surgeon who served as part-time county coroner about five minutes to determine that Ralph had been kicked in the head by a horse. Whether it was the blow that killed him or the smoke he'd inhaled while unconscious would have to be determined by a full autopsy. Either way, as he'd already told Stella, the boy's death hadn't been murder.
"Well, you'd better keep pushing those Tylers," Stella said. "Will Tyler murdered my Nicky. Don't give that bastard a moment's peace."
Abner ended the call, pulled onto the shoulder of the road, and emptied his bladder in the barrow pit. At least, so far, there was no evidence to prove Ralph had started the fire. But he was sick of being Stella's errand boy. When he'd held Vonda's baby son in his arms, he'd realized he needed to be a better man, and a better example to his family. But how could he walk away from Stella when she knew enough to destroy him?
Still thinking about her, he climbed back in the SUV and headed for his office in town. It wasn't like he'd done anything seriously illegal. But in exchange for interest-free loans, which he always paid back, he'd traded department information with the woman. And he'd looked the other way while she carried on her so-called business transactions. That night when he'd met Ralph on the road, he'd suspected his son-in-law might be running drugs for Stella. But he'd played dumb and let the boy go, partly for Vonda's sake, but mostly because he hadn't wanted to get crosswise with Stella. Maybe his sins weren't bad enough to get him sent to prison. But if word got out, his ass would get fired on the spot, and he'd never work in law enforcement again.
Now, with Vonda and her baby in the house, and two more mouths to feed, he needed his job more than ever. But the thought of what Stella would ask for next, and what she'd do if he refused, was keeping him awake nights.
Since her brother's death, Stella was becoming more and more demanding—like today, when she'd ordered him to drive out to the Rimrock and harass the Tylers just to make trouble. It was as if she'd become obsessed with punishing not just Will, but the whole family.
Abner had no love for the Tylers. But enough was enough. If Will's upcoming trial ended in acquittal—which it could, given the true evidence and a fair-minded jury—Abner feared that Stella's fury would push her over the edge. He dreaded what she might do—and what she might demand of him.
Somehow he needed to get clear of this mess. But how? Walk away, or try to arrest her, and the woman would use what she knew to take him down, or worse. Stella had trapped him—just as she'd trapped Hoyt Axelrod, Slade Haskell, Lute Fletcher, Garn Prescott, and poor, stupid Ralph.
All of those names were inscribed on tombstones now, or soon would be.
Was his name destined to be next?
* * *
Still unsettled by the clash with Abner, Tori walked back to the house. The dry November breeze bit through her thin cotton shirt, raising goose bumps on her skin. In her race to catch up with Will, she'd left her jacket in the house. Now her teeth were chattering.
Behind her, Will, Beau, and the crew of ranch hands had gone back to clearing away the barn debris. They'd be at it all day, until dark, then back on the job by sunrise.
Will had looked exhausted this morning, she thought. The strain of the ranch's money problems, the coming trial, and now the loss of the barn, all had to be wearing him down. But she knew better than to fuss over the man and insist he rest. Will wouldn't stand for that. He'd be out there pushing till he dropped. Some things never changed.
Natalie's white SUV was parked next to the porch. Seeing it, she remembered that her friend had promised to come by this morning and check Tesoro's shoulder wound.
She found Natalie in the kitchen, drinking coffee while Lauren finished a breakfast of bacon and eggs. Jasper had gone, and Erin was nowhere in sight.
"Your daughter's getting her coat," Natalie said. "Do you want to come out to the shed with us?"
"If it won't take too long." Tori had planned to drive into town to check her house and take care of some legal matters. Even with Will's case pending, there were other clients who needed her. She couldn't fall behind on the work that was her livelihood. "Lauren, you're welcome to go in and rest on my bed."
Lauren rose, gathered up her dishes, and carried them to the sink. "I'll rest better at home after a shower and a change of clothes," she said, loading the dishwasher. "Don't worry, I'll be fine driving back to town."
Erin burst into the kitchen, wearing her fleece-lined denim jacket. "I hope Tesoro's okay. He looked fine when I went out to feed and water him before breakfast."
"Even if he's fine, we'll need to check the wound and change the dressing," Natalie said. "Did you wash your hands, Erin?"
"I did. Let's go."
Tori slipped on the old work coat she wore around the ranch. "Be careful, Lauren, okay?"
"I will." Lauren found her coat and purse and headed for her car. Tori hurried through the back door after Erin and Natalie, who'd brought her black leather medical bag.
Until his shoulder healed, young Tesoro couldn't be allowed in the paddock with his roughhousing friends. A stall-sized enclosure at one end of the hay shed had been blocked off for the palomino foal and his mother, Lupita. The buckskin mare raised her head and nickered at their approach. "I bet she's lonesome," Erin said. "She doesn't understand why she and Tesoro can't be with the other horses."
"Smart thinking, Erin." Natalie set her bag on the ground outside the enclosure. "How did you figure that out?"
"Sky's always telling me to think like a horse, so that's what I try to do," Erin said. "I wish he was here. I really miss him."
"We all miss him," Tori said. "At least your dad said he was doing better this morning."
"Next time somebody goes to see him in the hospital, can I go, too?"
"We'll decide that later." Seeing Sky burned and bandaged could be too much for the girl, Tori thought. But then, her daughter was mature beyond her years. She would probably handle it fine.
"Remember what we talked about in the kitchen, Erin. Are you ready?" Natalie pulled two sets of latex gloves out of a packet in her bag and handed one pair to Erin. "Can you hold the mare out of the way, Tori? She might not understand what we're doing to her baby."
"Got her." Tori didn't have the Tyler touch with horses, but she was confident enough to grasp Lupita's halter and, with pats and reassurance, ease the mare to the far side of the makeshift stall.
Erin had put on the latex gloves Natalie had given her. Tori watched in disbelief as Natalie took Tesoro's head and stepped back to make room for Erin in the stall. "You're the one he trusts, Erin," Natalie said. "Go ahead."
The area around the gash in Tesoro's shoulder had been lightly shaved, and there was a gauze dressing taped in place. Singing softly to her trembling foal—as Tori had known Sky to do—Erin stroked his neck with her left hand, while her right gently peeled away the tape and lifted off the gauze dressing. "What do you think?" she asked, stepping back so Natalie could see.
The eight-inch wound, probably a skin rip from an exposed nail or splintered wood in the barn, didn't look as bad as Tori had feared it would. However, the sight of the torn edge, gleaming with ointment, made her knees go watery. She soothed the nervous mare, her hand gripping the rope halter, her eyes on her daughter. She'd never realized Erin was capable of doing what she was doing now.
"The wound doesn't look infected." Natalie spoke in answer to Erin's question. "But it's still oozing a little. I'd say we should keep the dressing on it at least one more day. Here." She handed Erin a fresh, ointment-coated gauze pad. Erin pressed it gently into place and secured it with lengths of surgical tape. The golden foal quivered, but didn't try to move.
"Good job," Natalie said as Erin stepped back. "It might fall off later. If it does, don't worry. Just let the wound heal in the air."
Tori let go of the mare. "You did great," she told her daughter as they walked back to the house. "I was proud of you. Maybe you should think of becoming a vet, like Natalie, one day."
"I have thought about it." Erin sounded surprisingly grown-up. "I wouldn't mind being a vet, but I'd have to go away for my schooling—for years. I don't want to leave the ranch that long."
"But, surely, you'll want to go to college," Tori said. "The money's there. Your grandpa Bull left it to you in his will."
"Dad never went to college," Erin said. "Neither did Sky. I want to stay right here on the ranch and train horses. I can learn all I need to know right here."
Tori gazed at her daughter, already growing so tall. This was a child speaking, she reminded herself. A child just short of her thirteenth birthday.
"I know the ranch needs money right now," Erin said. "I'm going to tell Dad he can use what Grandpa left me."
"Erin! Your dad would never take that money from you!"
"Not even to help save the ranch?"
"Not for anything," Tori said, knowing she was right. "Do you want to go into town with me this morning? We could get lunch at Burger Shack. You must be getting tired of leftover turkey."
Erin hesitated, then shook her head. "I'll stay here. I want to keep an eye on Tesoro. If I go, I'll be worried about him."
"All right. I'll see you later, then." Tori watched her daughter scamper off toward the coop to gather eggs for Bernice. Maybe by the time she finished high school, Erin would change her mind about college. But she had inherited her father's stubborn nature and his love for the land. Something told Tori that her decision was final. As a mother she could only hope and pray it was the right one.
After changing her jacket and collecting her briefcase, Tori went out to her wagon and headed for town. When had her little girl become such a determined young woman? she wondered as she drove. What would Erin do if the worst happened and the Rimrock was no longer there for her?
Tori remembered Will's request—that if he was sentenced to prison, she and Erin would move back to the ranch. She'd told him she'd have to think it over. But now, after hearing Erin's decision, she knew it would be her only option.
How would that limit Tori's life, especially if things became serious with Drew? But how could she even think about that when Will was facing years behind bars?
The jangle of her cell phone broke into her thoughts. With her free hand, she fished it out of her purse. The caller was Drew.
"Hi," she said, realizing she'd scarcely given him a thought since the barn fire. "How's Omaha?"
"Boring. I missed my favorite lady, so I drove back early. I know you're busy, but I need to talk to you. How about getting together for lunch?"
CHAPTER 16
Tori had an hour-long appointment with a middle-aged couple, setting up a family trust for their grown children. After they'd left, she spent another hour organizing the paperwork and filling out the formal documents on her computer. The next time she glanced at her watch, it was almost noon—time to meet Drew for a quick lunch at the Burger Shack.
Seeing him again would be good for her, she'd told herself. Drew was easy to be with. He always knew how to make her smile. But anxiety gnawed at her as she drove to Blanco's only restaurant. Drew wouldn't have come home early on a whim. Something had to be weighing on his mind.
Drew had offered to pick her up at home, but she'd told him she had errands to run after lunch, which was true. It was also true that if they were alone in his car or her house, and he wanted to push her to a decision, things could get emotional. Meeting in public would be a safeguard against regrettable words and actions.
His car was parked outside the Burger Shack when she pulled up. He'd be waiting for her inside, maybe expecting some answers about their relationship. But she had none to give him. The past few days had left her more distracted and confused than ever.
The Burger Shack was crowded today, the booths and tables full. Behind the counter a cook was piling up a tower of takeout pizza boxes. Somebody must've ordered for an army, Tori thought.
Drew stood next to the booth he was saving. He gave Tori a smile as she walked through the door. He looked like a photo from GQ, in a gray sweater, khaki slacks, and a dark brown lambskin jacket. Tori, still in the frayed jeans and plaid shirt she'd worn on the ranch, with her hair raked back in a careless ponytail, looked more like a panhandler he'd invited in off the street.
Always the gentleman, he helped her with her coat before they sat down, facing each other across the red-checked vinyl tablecloth.
"How was your holiday?" she asked.
"Not bad. Eating dinner with my sister, her husband, and three rambunctious kids was better than eating alone. How was yours?"
He wouldn't know about the fire, of course. Tori shook her head. "Awful. It's a long story. Let's order, and I'll tell you."
The waitress had reached their booth. Tori scanned the menu, deciding on a tuna melt and coffee. Drew ordered a burger, fries, and a Coke.
"So tell me," he said. "Was your Thanksgiving really that bad?"
She told him, then, about the fire, the horses that had to be rescued, the injuries to Sky, and the dead man they'd found after the fire was out. Drew listened, his expression sympathetic.
"Good Lord, you weren't kidding, were you? It must've been terrible, being there and going through that. I'm so sorry."
His hand slid across the tabletop to rest on hers. That was when the restaurant door opened and Will strode in.
He was headed for the counter—and the pizzas he'd evidently ordered for his work crew—when, out of the corner of his eye, he glimpsed the pair holding hands in the booth.
Tori saw him hesitate, as if resolving to ignore them and leave. But knowing Will, she had a feeling that wouldn't happen. Instinctively, she tried to pull her hand free. Drew tightened his clasp in a gesture of possession, pinning her palm firmly against the tabletop.
The two men had never met, but there was little doubt they recognized each other. Will would know Drew because he was with Tori. And Drew, sensing Tori's sudden reaction, would guess that he was looking at her ex-husband.
Jaw set, Will turned and walked toward them, taking his time, like a bull elk sizing up a rival. He was dusty, unshaven, and windburned, his eyes still reddened from smoke as he loomed over the table. "Will Tyler." His voice was a hoarse growl. "Pardon me if I don't shake hands. I've been shoveling ashes most of the morning."
Drew had risen. He was almost as tall as Will, but a few years younger and probably twenty pounds lighter. In a physical fight his only chance would be to run. "Drew Middleton," he introduced himself. "Tori was just telling me that your barn burned. I'm sorry. Nobody deserves that kind of bad luck."
"Something tells me it was more than bad luck," Will said. "But even knowing for sure won't bring the barn back. Sorry I can't stay and visit. I've got a hungry crew to feed—even got the bunkhouse cook working the line." He turned away, then glanced back. "You two enjoy your lunch."
"You're welcome to join us," Drew said needlessly.
"Another time, maybe." Will walked to the counter, paid with his credit card, and carried the stack of pizza boxes out the door.
Drew had taken his seat again. He took a sip of the Coke the waitress had left on the table. "So that's your ex-husband," he said. "He's pretty, uh . . . formidable."
"Will can be overbearing." Tori stirred creamer into her coffee. "But he's a good man and a good father. Erin adores him. If I don't win this case, I don't know what she'll do."
"And what will you do, Tori?" He captured her hand again. "Something tells me there's more at stake here than Erin's feelings."
"Of course there is. The charges against Will are ridiculous, but for some reason the prosecution is out to get him. I can't let an innocent man go to prison, especially knowing what being there would do to him, and to his family."
"And to you?" His fingers tightened around hers. "Are you still in love with him, Tori?"
Her heart gave a thud, like a rock hitting the bottom of a well. "Of course not. We've stayed friends for Erin's sake. But when we were married, we drove each other crazy. Most of the time I can barely tolerate the man."
He released her hand as the waitress brought their meals and set them on the table, but his light hazel eyes continued to hold hers. "I'm not a fool," he said. "You're the woman of my dreams, and I want you for keeps. But when you're in my arms, I need to know it's me you're thinking about, not another man. I understand that you have to put this trial behind you. But once it's done, whatever the outcome, I'll need you to give me an answer."
"I understand." Tori poked at the sandwich on her plate, her appetite gone. "You've been an angel of patience, Drew. I promise I won't keep you waiting much longer."
"I'm holding you to that. If you say yes to me, I expect it to be a hundred percent. No hanging on to the past, agreed?"
"Of course. That would be the only way." And it would, Tori thought, but maybe it is time. For the past eight years, she'd been living in limbo, caught between her work in town and the ranch, never moving beyond that half-life with Will that was more than friendship, yet no longer a marriage.
Now she'd been given a chance to change all that. Drew was a good man—intelligent, sensitive, kind, and patient. She was physically attracted to him—not in love yet, but she could be, once the baggage with Will dropped away. She could almost imagine saying yes to him.
But what about Erin?
Drew picked up a French fry, swirled it in the ketchup on his plate, and put it down again. It appeared he had no more appetite than Tori did.
"What is it?" she asked. "Is there something you haven't told me?"
"You're very perceptive," he said. "As a matter of fact, there is. It's the reason I came back here early—to talk it over with you."
"Tell me," she said.
"I've been offered a new job—assistant principal at a big school in a wealthy district that can afford to pay teachers what they're worth. I'd be making twice what I make here, to say nothing of the chance to move up the ladder. The job would start spring semester."
"So, have you said yes?" Tori spoke calmly, but her head was already spinning with the implications of what she'd just heard.
"I promised them an answer within two weeks," he said. "If I take the job, my assistant could handle things here till they found a replacement. It's short notice, but given what the job could do for my career track, I can't imagine turning it down."
"And where is this educator's paradise?" Tori struggled to ignore the flip-flopping sensation in her stomach.
"Seattle—a great area close to the university. You could find plenty of legal work, or even go back to school if you wanted. You could—" He broke off with a nervous laugh. "Why are you staring at me like that? Don't you know I'm asking you to come with me? With a ring on your finger or without—your choice. This isn't really a proposal, unless. . ." He reached over and captured her hand again. "Unless you want it to be." He paused, an uncertain look creeping over his handsome features. "So what do you think?"
Tori found her voice. "I'm just wondering about Erin."
"She'd love it," he said. "Beautiful, green city, lots of culture, friends, and things to do. You could put her in a good private school, take her on trips, give her a chance to learn about the world beyond Blanco Springs . . ." He shook his head. "Sorry. I know I'm rushing you. I know you need to focus on the trial. Just promise me you'll keep it in mind. That's all I ask."
Tori willed herself to breathe. "I will. Don't expect much from me until the trial's over. But when it's done, you'll have my decision."
"I understand, and I won't push you." He released her hand and glanced down at her tuna melt. "Your food's gone cold. Let me order you something else."
"Thanks, but I'm a bit emotional right now." She rose and reached for her coat. "I hope you'll give me a rain check when things calm down."
"Sure. But you'll think about what I said, won't you?" He got up to walk her to her wagon.
"Of course." How could she not think about it?
She was still thinking about it as she drove back to the ranch. Drew's proposal—or whatever it had been—had caught her like a flash flood in a narrow draw, leaving her shaken and confused.
She should have turned him down on the spot and saved both of them the pain of uncertainty. So why hadn't she? Maybe there was a reason. Maybe leaving Blanco Springs and going to Seattle with Drew could turn out to be the best decision she'd ever made—for herself, as well as for Erin, who'd never experienced the world outside rural Texas. Or it could turn out to be a disaster for all concerned. Either way, she couldn't make an intelligent choice until after the trial.
As she pulled up to the house, she could see the work crew clearing away the ruins of the barn. Will was with them, looming over the others as he paused to shout directions or stooped to help drag a heavy timber free of the rubble. He would work until he dropped from exhaustion and never say a word about what was bothering him. That was Will's way when he was worried, especially if he happened to be unhappy with her.
Seeing her with Drew hadn't sat well with him, she knew. Oh, he'd hidden it, but Tori knew the signs. He'd been too polite, too congenial, too cool. And while he was talking to Drew, he'd scarcely given her a glance.
She'd planned to stay in town overnight, for some needed quiet time. But that had been before Drew called and before Will had shown up at the Burger Shack. Now, although it shouldn't be allowed to matter, she'd felt compelled to come back to the ranch. She needed to let Will know she was here.
Beau's Jeep was gone, and Erin was nowhere in sight. Inside the house Tori found a note on her daughter's closed bedroom door.
In case anybody wonders, I went to the hospital with Beau and Natalie. Natalie had a doctor's appointment. Beau wanted to visit Sky. I did, too, so they let me tag along. See you later.
The house was quiet. Bernice was probably napping, and Tori had seen Jasper on the Kubota yard tractor, hovering around the workers who were clearing the barn. Even if he couldn't work, the old cowboy would want to be part of the action, and maybe do some supervising.
Savoring the silence, Tori sank into the cushions of the well-worn leather sofa. The stress of the past few weeks was getting to her. She'd forgotten the last time she'd had a decent night's sleep. It felt good just to sit here in the familiar stillness and close her eyes for a moment . . . just for a moment . . .
* * *
Will had come back to the house to phone the insurance company, submit some forms on line, and order a batch of supplies for the new barn. He was headed down the hall to the ranch office and happened to glance into the living room. Tori was on the couch, fast asleep.
He'd seen her drive up to the house about an hour ago, but he'd been too preoccupied with the barn work to pay much attention. Now he found himself wondering why she hadn't stayed in town with her new boyfriend.
Giving in to an urge, he walked into the room and stood looking down at her. She was curled on her side, her knees tucked up, her head resting on a cushion. Tangles of spun-gold hair framed her face and spilled over the suede pillow. Shadows of weariness rimmed her closed eyes. Seeing her in her sleep, he realized how tired she must be and how much of herself she'd given to helping him.
As always, Will had counted on her competence and her willingness to do her job, whatever it took. Only today, seeing her holding hands with Drew Middleton, had it hit him how much he'd taken her for granted. Tori was a beautiful, intelligent, sexy woman. He needed her like he needed air to breathe and water to drink. But needing her wasn't enough—not unless he had something she needed in return. And Middleton seemed to be filling her needs just fine.
Seeing her with the man today had damn near killed him. He'd managed to keep a civil tongue, but it had been all he could do to keep from smashing his fist into that smug, pretty-boy face. Drew Middleton was well-educated, as was Tori. He probably had more interesting things to talk about than horses and cattle and the coming weather. The fancy bastard probably knew his way around the bedroom, too.
Will cursed under his breath. Why was he torturing himself like this? He had more pressing worries than holding on to his woman—not that she was his anymore. Legally, she hadn't been his for eight years—and there wasn't a damned thing he could do about that.
His jaw tightened as he gazed down at her, holding back emotions he had no right to feel. He ached to lean down and taste those plum-ripe lips in the secret hope they would soften to his kisses; but no, that wouldn't be smart. If she pushed him away, he wouldn't be able to stand it.
He settled for lifting the woolen afghan off the back of the sofa, unfolding it, and laying it gently over her body. As it settled into place, she whimpered, stirred, and opened her eyes.
Will's heart dropped for an instant. "Sorry, I didn't mean to wake you," he said.
"Did you . . . need something, Will?" Her voice sounded muzzy, the way he remembered from those long-ago mornings when they'd awakened early to make love before starting the day.
He shook his head. "I just wanted you to be warm. Go back to sleep."
"No . . . 's all right," she muttered, sitting up. "I was out cold. What time is it? Is Erin back?"
"It's almost four, and no, she's not back." Will hesitated, knowing he should keep silent, but needing to clear the air. "About today—"
"Drew's a friend. He says he'd like to be more than that, but I've told him I need to focus on the trial for now."
"So you still haven't slept with him?" Will could have bitten his tongue off, but it was too late to take back the question.
Tori raked back her hair, tightening the fabric of her blouse over one breast. Will cursed himself silently for noticing.
"Not that it's any of your business," she said, "but no, I still haven't. This is no time for a heavy relationship, especially since I've got Erin to think about." She stood, tucking her blouse into her jeans. "Trust me on this. Whatever happens, in the end, I'll do what's best for our daughter."
"Does that include moving back to the ranch?"
"Don't push me, Will. I said I'd think about it."
"It's what Erin wants. She told me."
"So you've been lobbying her behind my back?" Annoyance sparked a fire in her eyes.
"You know I wouldn't do that. Erin was the one who brought it up to me."
Tori's shoulders sagged. She shook her head. "Anyway, it isn't going to matter, because I'm going to win your case. When that's behind us, the rest will fall into place."
"With you, me, Erin, and what's-his-name. Right?" Will knew better, but he couldn't resist the jab.
"Don't make this any harder than it is," she said in a flat voice. "Just don't."
Her face was close to his, her gaze coldly defiant. Will fought the urge to seize her in his arms and kiss away all the anger, all the bitterness between them. But something told him it would take a lot more than kissing to accomplish that.
The tension was broken by the sound of footsteps and voices across the porch. An instant later, Natalie, Beau, and Erin burst inside.
"Hi, Mom," Erin said. "I thought you were staying in town."
"I changed my mind," Tori said, causing Will to wonder if she'd changed plans because he would have known she was with Middleton.
"How's Sky?" he asked.
"Mending," Beau said. "But he's not happy about being out of action when he's needed here. I did have some news for him. You'll be interested, too." He'd brought a folded newspaper inside. Laying it on the coffee table, he opened it to the regional news page.
"Right here. I bought a paper in the hospital gift shop and just happened to see this." He pointed to a brief article accompanied by a grainy news photo. The headline read, WOMAN WANTED FOR ARMED ROBBERY.
Will scanned the short paragraph that began, The robber of a Wichita Falls pharmacy has been identified as Marie Fletcher, shown in the above surveillance photo. Anyone knowing this woman's whereabouts . . .
Will studied the blurred photo. The long-legged figure shown at the pharmacy counter was wearing a hooded sweatshirt and a baseball cap. But she'd happened to glance up at the wrong moment. The camera had caught the long, sharp face, the fierce dark eyes, and the white slash of a scar from temple to chin. It was Marie, all right.
"So Sky's cousin is up to her old tricks," Will muttered, handing the paper back to Beau. "Looks like she might've been a little careless this time. You say you showed this to Sky?"
"I did. Sky said he'd washed his hands of her. Can't say I blame him after the woman shot Jasper, likely murdered her own brother, and damn near killed Lauren. I just hope she ends up behind bars, where she belongs."
"Even with all that, I can't help feeling sorry for her," Natalie said. "What chance did she have, growing up in that horrible family, and then having her ex-husband slash her face?"
"Sky grew up in the same family, and he's got his own scars," Beau said. "Everybody has choices. Marie made hers."
"Hey, I smell something good!" Erin dashed toward the kitchen doorway, where Bernice had just appeared. "Is it brownies?"
"It is, honey," Bernice said. "I just took a batch out of the oven and iced them. I can already hear Jasper at the back door. He's got a nose like an old coyote! Who else is hungry?"
"Me!" Erin bounded into the kitchen.
"Me too," Beau said. "But I'll have to grab one and eat on the run. I've got to get back to work."
"Count me in," Natalie said, laughing. "After all, I'm eating for two! How about you, Tori?"
"Sure. I . . . didn't eat much lunch." She moved toward the kitchen, paused, then turned back. "Will? Are you coming?"
Will hesitated, then shook his head and turned away. One more memory of the family gathered around the kitchen table, talking and laughing, would be enough to break him. In his mind he was already distancing himself from the things he loved—the things that, days from now, if the trial went badly, would no longer be part of his life.
* * *
Ralph Jackson's funeral took place the following Monday afternoon in the Community Church on the outskirts of town. Glancing back from his seat in the front pew, Abner experienced a rare sense of satisfaction. The small chapel was filled to the doors. Bethel's friends in the congregation had come to support her, which was to be expected. But what pleased him most was that the back rows were filled by folks from the Rimrock.
True, Abner wasn't on friendly terms with the Tylers. But according to custom, when a family death occurred, differences were put aside long enough to pay respects. Will Tyler, looking drawn and restless, was seated on the aisle, with his young daughter beside him. No sign of Tori, but someone had mentioned she was in court today. Beau was there with Natalie. Sky, still looking raw around the edges, had come with the Prescott girl. Even Jasper was there, wearing a twenty-year-old brown suit that was too big for his age-shrunken body. The row behind them was filled with cowhands who'd worked with Ralph.
All in all, it's a nice turnout. Really nice, Abner thought. And the medical examiner had been thoughtful enough to release the body for a timely funeral. Ralph's death had been ruled a tragic accident, a consequence of his trying to save the Tylers' horses. By now, the burnt truck had been found. The evidence showed it had been stolen and vandalized, probably by a teenage gang. Thanks to Abner, no one would ever know the truth about Ralph—except maybe Stella, who had every reason to keep it to herself.
Abner's family filled the entire front pew of the little church. Bethel sat beside him, putting on a good show of grief for a woman who'd detested her son-in-law. Vonda, in black, wept quietly as she soothed her baby. She was still a pretty girl—pretty enough, hopefully, to find a better husband than Ralph had been. Next to Vonda, the other Sweeney children sat in descending order, like steps. Even the younger ones were awed into silence by the occasion. They sat with their arms folded, and their feet, in hand-me-down shoes, dangling from the bench.
The Tylers' insurance had paid for the funeral. It would also pay out a handsome benefit to Vonda and her child; and Ralph would go down in memory as the hero who'd sacrificed his life for his employer's horses. Abner smiled to himself. All in all, things could be worse.
Sometimes the ends really did justify the means.
* * *
Will had given the men who'd known Ralph a couple of hours off to attend the service, but no time to socialize afterward. The work on the barn couldn't wait any longer. By now, the rubble had been cleared off the foundation, and, with the help of Sky's construction crew, the walls were being framed. So far, the cold, dry weather had held. But nobody had forgotten the norther that had frozen the pastures and paralyzed the ranch for days. Another storm could blow in at any time.
Will had driven his pickup to the funeral, with Jasper riding shotgun and Sky, Lauren, and Erin crowded into the backseat. Now, as they turned off the main road and onto the gravel drive that led up to the house, Will remembered Lauren's invitation. Last night she'd announced that she had a surprise to show him, something that had to be seen by daylight—something that couldn't wait.
There'd been no need to explain. Today was Monday. With the trial on Wednesday, and so many things left undone, Will's time was running out.
Pressed by the need to get the barn up, Will had tried to put her off. But Lauren had been insistent. Sky, newly home from the hospital, had backed her. So had Jasper, who seemed to know more than he was telling. "It's a damned conspiracy," Will groused. But he couldn't help being intrigued. Something was up—and there was only one way to find out what it was.
Half an hour later, they'd changed out of their funeral clothes and met again on the porch, wearing warm coats, gloves, and hats. Jasper was tired after the funeral, and Sky was under doctor's orders to rest, so the two of them wouldn't be going along. But Beau had gotten wind of the adventure and declared himself in. He'd even offered to drive the secondhand Kawasaki four-seater UTV that Will had bought at auction last year, when the ranch was flush. The big, rugged four-wheeler had already been put up for sale. All the more reason to use it while they still could.
Will waited on the porch with his daughter and Lauren while Beau found the open-topped vehicle in the shed, started it up, and brought it around to the front of the house. Minutes later, they'd left the heart of the ranch and were rolling across the scrub-dotted flat toward the foothills that rimmed the escarpment.
Seated next to Beau on the front passenger seat, Will found himself savoring the sunlight on his face, the cold wind biting his skin, the smells of sage and earth, and the faint, distant ring of hammers. The ice storm had blasted the landscape, leaving behind a frost-bleached wilderness. But even here there was life. Clumps of sage and cedar, impervious to the cold, still held their muted autumn colors. Jackrabbits bounded ahead of the massive tires, zigzagging off into the brush when they wearied of the game. A ground squirrel, less bold, flashed across the trail and darted into its hole.
The late-day sun blazed above the caprock. A golden eagle flapped off its kill to circle upward on wings as broad as a tall man's reach. Will's senses embraced all these things, holding them in memory, to keep for when he needed a place for his mind to go.
The vehicle's engine drowned out any attempt at conversation. Only when Beau turned onto a familiar trail did Will realize where they were headed. This was the way to the petroglyph canyon, with its bitterly disputed side-branch and rumored Spanish gold.
His pulse quickened as Beau parked at the mouth of the canyon, where the trail ended. From here the only way to go was on foot. It had occurred to him that Lauren might be planning to deed her land back to the Rimrock. But she could've done that at the ranch. Why drag him clear out here on a frigid and busy day?
They climbed out of the vehicle and trudged single file up the narrow, rocky path. Erin was walking just ahead of him. Will touched her shoulder. "What's this all about?" he asked her. "Did you and Lauren find the Spanish gold?"
"Not really. You'll see." With a toss of her ponytail, she strode ahead, following Beau and Lauren.
In the sheltered petroglyph canyon, the sound of trickling water echoed off the high walls. A covey of quail, drinking at the spring, whirred away at their approach.
Silent now, they turned aside and followed the water's path upward to the level of the smaller canyon. There they stopped. "This is what I wanted to show you," Lauren said.
The canyon, which Will hadn't visited since his boyhood, was much as he remembered. But in one place, where heavy scree had fallen down from the overhead cliff, something was different. In one spot the rocks had been cemented together to form a wall, about four feet wide and just as high. In its center was a marker of polished granite, with an inscription etched into its surface.
COOPER PRESCOTT
March 12, 1940–July 9, 1949
Sleep in the Arms of Angels
"This little boy would have been my great uncle," Lauren said, turning to Will and Beau. "He's the reason my grandfather wanted this canyon. Jasper knows the story. He's agreed to tell you when you get home."
She drew a folded document from under her coat and held it out to Will. "Here's a signed, notarized deed. It's yours on condition that this grave never be disturbed, and that I and my family be allowed to come here and visit it."
"Of course." Will had never considered himself an emotional man, but he felt the welling of tears.
Lauren wiped her own eyes and managed a smile. "Now," she said, "how about that dollar you owe me?"
CHAPTER 17
Clay Drummond's day had been long and tiring, and it wasn't getting any better. He'd just climbed into his white diesel Mercedes and thrust the key into the ignition when his cell phone rang. Even without glancing at the caller ID, he knew it was Stella. The bitch is probably somewhere nearby, spying on me. Her timing was too good to be a coincidence.
"What is it?" he muttered.
"Just checking to make sure you're ready. The trial's two days off. I'll be in that courtroom watching you every minute, and I don't want any slipups. If Will Tyler doesn't walk out of there in handcuffs, you know what I've got and what I can do with it."
Clay blinked, struggling to focus his eyes. The sun, a blinding glare through the windshield, was triggering a migraine. "Maybe, Stella, but you can only do it once. Ruined, I won't be any good to you. I'll have nothing left to lose."
"Then it won't be my problem, will it? Just put that murdering bastard behind bars. Then we can negotiate for the tape."
She was doing it again, dangling that damned surveillance tape in front of him like a carrot on a stick. At times like this, Clay could almost imagine putting his fingers around her throat and squeezing until her breath stopped and her cat-green eyes glazed over.
"Who's the judge?" she asked. "Any leverage there?"
"Sid Henderson. He's a friend, but he's a pretty straight arrow. Anyway, the verdict will be up to the jury."
"How about Abner?" she asked. "Is he on board to do his part?"
"Why don't you ask Abner? You've probably got something on him, too. That's how you operate, isn't it?"
She laughed, a sound that reminded Clay of the villainess in a Disney movie. "Now, now. Play nice. We're on the same team, remember? I'll see you in court."
She ended the call. Grinding his teeth, Clay drove home. He wasn't looking forward to the trial. Yesterday in court Tori had whipped his butt in an assault case that he'd expected to be a slam dunk for the prosecution. The woman was good—damned good. As his onetime junior partner, she knew all his strategies. Clay had taught her well. Now his lessons were coming back to kick him in the face.
The worst of it was, he knew Will, and he knew the man didn't deserve to go to prison. Will had done what any protective father would have done—what Clay himself would have done in a similar situation. But he couldn't let that sway him. All his focus would have to be on doing his job, which was to win.
Clay thought about his career, his children, his marriage, and all the advantages that Louise's money made possible. Everything was hanging on the outcome of Will Tyler's trial.
If he lost this case, his life, as he knew it, would be over.
* * *
Stella kept her brother's ashes on a shelf behind the bar, where he'd worked and kept her company for the past two years. The black metal urn was a constant reminder of the childlike man who'd done whatever she asked of him—the only person she'd ever truly cared about.
With Nicky gone, the urge to pull up stakes and leave Blanco Springs was growing stronger every day. Even without selling the bar, she had plenty of money stashed away. All she'd have to do was close the place up, load her car, and head for Mexico.
But she couldn't leave until Nicky's killer paid the full price for what he'd done.
This afternoon she'd taken time to drive to the county parking lot and phone Clay as he was getting off work. She didn't like leaving the bar when it was open. But business wouldn't pick up until later in the evening, and the new waitress she'd hired seemed capable enough to manage without her for a few minutes.
She returned to find the place quiet, the country music low, the new girl polishing the tables. Only one customer was in the bar, a handsome, well-groomed man sitting alone in a booth, sipping Corona from a tall glass. Stella paid him scant attention until it struck her that she'd seen him before. He'd come in late one night with Tori Tyler.
Intrigued, she opened another Corona and sauntered over to the booth. "Howdy, stranger," she said. "You don't look like a cowboy."
"Is there some law against not being a cowboy?" His light hazel eyes took her measure, probably deciding she was too old for him. Damned shame. The man was some looker.
She laughed at his question, leaning over the table to give him a glimpse of her ample cleavage—but only a glimpse. "Stella Rawlins," she said. "I own this place, and I take pride in getting to know my customers. Mind if I sit?"
"Not at all." He extended a hand as she took the seat across from him. "Drew Middleton."
She accepted the handshake. His palm was smooth and cool, not horny with a cowboy's hard-earned calluses. "Well, Drew," she said. "Experience has taught me that a fine-looking man like you doesn't drink alone unless he's got troubles—most likely woman troubles."
A slight twitch of his mouth told Stella she'd hit the bull's-eye. "Not long ago," she continued, "I noticed you in here with a beautiful blond lady. Now I don't see her. Is she the reason for that long face?"
He gazed into his half-empty glass. Stella refilled it from the bottle she'd opened. "On the house, honey," she said. "If you feel like talking, I'm a good listener."
He managed a bitter laugh. "Don't get me started. I could be here all night."
"No problem with that. Talking will make you feel better. And not a word that goes into these ears will ever come out between these lips."
He sipped his beer in silence.
"So the lady dumped you and broke your heart. Am I right?" she asked.
He shook his head. "If she'd dumped me, at least I could get over it and move on. No, what she's doing is keeping me on the back burner till she gets things sorted out with her ex-husband."
"Her ex-husband?" Stella feigned surprise. "My-oh-my, the plot thickens. What's the problem? Do you think she's still in love with him?"
"That's crossed my mind. She's defending him in a trial and—" He broke off, staring at her. "Oh, Lord, I should've realized who you were. It was your brother that Will Tyler shot, wasn't it? I don't know if I should even be talking to you."
"Why not? Will Tyler isn't exactly my favorite person. Something tells me he isn't yours, either." Stella topped off his glass again. "Just curious, mind you. Where were you the night the shooting happened?"
"With Tori. She was in my car when she got the call from Will to come and get their daughter. I offered to drive her, but she insisted on taking her wagon and going alone."
"Why alone? Because she didn't want Will to see her with another man?"
He shrugged. "All I know was that she was in a rush. She wouldn't tell me much, but on the phone, I heard something about an incident."
"You say she was in a rush?"
"A big rush. I walked her to the door, pulled out of her driveway, and headed back toward Main Street. A couple minutes later, she roared past me, going at least seventy. She even ran the red light. I'm just glad she didn't have an accident."
He glanced at his watch, a nervous gesture, as if he'd become worried about saying too much. "I'd better be going. I've got someplace to be."
"Sure." Stella rose, slipped out of the booth, and moved to block his exit for a moment. "Feel free to come back anytime you need a cold Corona and a listening ear, honey. But let me leave you with one thought. If Will Tyler gets off, chances are, your lady will go running right back to him. If he ends up in prison, she'll be alone—and lonesome."
Turning away, she sashayed back toward the bar, giving him a view of her swaying rump. He might or might not be back, but never mind—he'd already made it worth her time.
As his car pulled away, Stella found her cell phone and scrolled to Clay's number. If he was at home, he wouldn't pick up; but right now, that didn't matter. She waited for his voice mail, then spoke.
"I've found another witness for you. His name is Drew Middleton. Call him."
* * *
Curled on her couch, with her glasses on her nose and her laptop on her knees, Tori reviewed her opening statement for tomorrow's trial. How many times had she read through her notes—moving a paragraph here, striking a sentence there, arranging and rearranging her ideas? Having it perfect had become an obsession.
The lamp behind her cast an island of light in the house that was otherwise dark. By now, it was after eleven. What she needed was to put the files aside, get some sleep, and look at them with fresh eyes in the morning. But even then, she could miss something vital, something that might make a difference for Will.
For the past few days she'd spent most of her time at the ranch, but today, needing quiet time to prepare, she'd fled to her house in town. Erin had stayed with Will, who'd made it clear that he wanted an ordinary workday with no fuss and no emotion.
Tori wouldn't be seeing either of them again until tomorrow's trial. Beau would be driving Erin and Will into town, leaving Natalie to rest at home. Sky had wanted to support Will at the trial, too, but somebody needed to oversee the ranch work, so he'd offered to stay. Lauren, already in town, would be there and had promised to text him updates as they happened.
Erin was to be a key witness. Tori had hoped to spare her by using the interview she'd taped with Abner, but Clay had insisted on calling her for the prosecution.
Tori lifted away her glasses, cleaned them on the hem of her sweatshirt, and put them on again. She was getting tired, but she had to be sure she was ready. She'd tried hundreds of cases over her career, but none that mattered the way this one did. Will's freedom, Erin's happiness, and the future of the Rimrock were all depending on her performance tomorrow.
Heaven help her, what if she failed?
* * *
Will stood alone on the front porch, gazing out across the yard. The house was dark, behind him, and Erin and Bernice long since asleep. Moon shadows, cast by wind-driven clouds, flowed like phantom water across the bare ground. The windmill creaked in the darkness. Coyote calls echoed down from the foothills.
The cold breeze burned his cheeks. If the jury found him guilty, he could be going away for years. Would this be his last night on the ranch before prison gates closed behind him? Would tomorrow morning be the last time he awoke to dawn chores and Bernice's coffee—the last time he forked hay for the horses and cattle, broke the ice on the water troughs, and watched the sky fade from onyx to silver above the eastern plain?
Will's cold hands gripped the porch rail as he pulled himself back to reality. He'd promised himself he wouldn't think about the things he'd miss if he went to prison. The jury would find him innocent—he had to keep believing that. After all, hadn't he done what any good father would do—acted in defense of his child?
Tori would give her all to save him. He could count on that. He could count on her. She might not be his wife anymore, but she'd always been there for him and for Erin.
Hadn't she?
A picture rose from his memory—Tori sitting in the booth, holding hands with Drew Middleton. For all he knew, she could be with the man right now, and not just holding hands.
Damn! Will cursed himself. This was no time for jealousy. He had to trust Tori, had to believe she wouldn't let him down. She might be independent to a fault, but she was honest and true to the marrow of her bones and she always gave her best.
The night of the ice storm, their need for each other had broken down the barriers between them. Tonight he needed her again—with a soul-deep ache that had become physical pain. For most of his life, he'd tried to be like his father—tough, closed-off, priding himself on always standing alone. But he wasn't Bull Tyler. He needed the only woman he'd ever loved—needed to see her, touch her, hold her, just one more time.
He could call her. But no, it was late. By now, Tori would be asleep—and if she wasn't alone, he didn't want to know. He would see her tomorrow, in court, at a polite distance, when he put his life into her hands.
Will was about to go back inside when his cell phone rang. His pulse leapt.
"Did I wake you?" Tori's voice washed through him like a soft spring rain.
"Not even close. Something tells me I won't get much sleep tonight. What's up?"
There was a beat of silence before she answered. "Nothing much. I just wanted to make sure you were all right."
"No need to worry," Will said. "I'm fine."
"Are you? Are you really?" she asked.
"Hell no. I'm scared to death."
"So am I."
Will forced a chuckle. "Glad we got that out of the way. I might be scared, but I know you'll give this trial everything you've got."
"And if that isn't enough?"
"We'll cross that bridge when we come to it, sweetheart, like we always have."
The endearment had slipped out, unbidden. On the other end of the call, there was silence—then, at last, a muffled whisper. "Oh, Will . . ."
"Don't go anywhere," he said, knowing that if there was one chance in a million for them, he had to take it now. Long strides carried him down the steps to where his pickup was parked. A moment later, the truck was rocketing down the road toward town.
* * *
Tori's porch light was off when Will pulled into her driveway. The windows were dark except for the faint glow of a lamp in the living room. Pulse racing, he walked up to her front porch and, instead of ringing the bell, gave a light rap on the front door. If she'd gone to sleep, he wouldn't wake her. But if she was inside, waiting . . .
His heart dropped as she opened the door. She was dressed in her baggy blue sweats, no shoes, her ponytail askew, her eyes set in weary shadows. The bridge of her nose was marked with a red spot where her glasses had rested. To Will, she had never looked more desirable.
Without a word she clasped his arm and drew him inside. He crushed her close, one hand reaching back to shut the door and lock it behind him. Their kisses were hungry, frantic. She moaned as his mouth devoured her, ravishing her lips, her tongue, her face, her throat. Lord, how he needed her—this stubborn, tender, maddeningly sexy woman who set him on fire every time he looked at her.
Her fingers tore at his shirtfront, buttons popping to the floor as she yanked it open. Will's hand found the hem of her sweatshirt and slid upward against her warm skin. She flinched slightly. "You're cold," she whispered.
"Warm me." The words rasped from a deep well of need. His seasoned fingers unhooked her bra, freeing one satin breast to fall against his work-roughened palm. Heaven in my hand. He stroked her, thumbed her taut nipple. Little whimpers rose in her throat. She arched against him, her body begging for what they both wanted so desperately.
His erection was rock hard, the jutting pressure threatening to push through his worn jeans. Tori's hand tugged at his belt buckle, her fingers eager but awkward—too slow for what he needed now.
With a half-muttered growl, he swept her toward the stairs of the split-level house. Fumbling in their frantic haste, they left a trail of clothes along the upstairs hallway—his boots, jeans, and boxers; her sweats, bra, and lovely lace panties—all in a tangle. Naked, they tumbled into the bed, and then he was there, where he'd yearned to be—deep inside her, his swollen sex thrusting into that slick, honey-sweet warmth.
Her long legs wrapped his hips. Her hands clasped his shoulders as he lost himself in the silken feel of her, in the womanly smell and taste of her, and in the sound of her little love cries as he brought her to her climax, once, then again, until he shuddered and burst in a release that shook him like an earthquake.
Spent, he lingered above her, bracing on his arms. She lay with her hair fanned on the pillow, her lips swollen from his kisses. "Can you stay?" she whispered.
"For a little while. But not for long. I'll need to be getting back to the ranch."
"Come here." She pulled him down to her, stretching onto her side so they could lie in each other's arms. Will checked the urge to thank her and to tell her how much he loved her. This was no time for words, or for making promises he might not be able to keep. For now, all he could do was hold her close and be grateful. Whatever tomorrow might bring, at least this night would be his to remember.
* * *
Too restless to sleep, Drew Middleton drove slowly up Main Street and turned the corner toward Tori's house. He'd planned on staying clear of Will Tyler's trial, leaving Tori to do her job. But then, out of the blue, the county prosecutor had called him as a witness. Drew had tried to excuse himself, arguing that the only thing he'd witnessed was Tori's end of the conversation with Will. But Clay Drummond had insisted that was enough. Drew would bet a week's salary that Stella Rawlins had had a hand in this. He should never have opened up to the woman.
Tori had asked him not to phone her until the trial was over. Even after Clay Drummond's call, Drew had tried to keep his distance. But tonight he couldn't stop thinking about her. Was she all right? Should he let her know that he'd be testifying for the prosecution?
At least it wouldn't hurt to drive past her house and see if she was awake. It was late, but she might still be up prepping for the trial. If the lights were on, he could phone her. She might even invite him in.
As he neared her house, he could see that the place was dark. Then he noticed something else—Will Tyler's pickup, parked in the driveway.
Drew's hands tightened on the steering wheel as he resisted the urge to stop and do something he might regret. It was after midnight. If Will was here at this hour, it could only mean one thing. Marching up to the door and confronting him, or Tori, would only be an exercise in humiliation.
Tires spat gravel as Drew gunned the engine and roared away. What a fool he'd been, letting Tori string him along while, all this time, she'd still been in love with her ex-husband.
He remembered their dates, their kisses, and the way she'd always seemed to be holding something back. Had she cared for him at all, or was she just hedging her bets in case Will went to prison?
Either way, he was through playing along with Tori's games—and after tomorrow's trial she would know it.
* * *
Will eased himself out of Tori's bed and stood looking down at her. She lay in a pool of moonlight, the rumpled sheet framing one perfect breast. Her eyes were closed in sleep, the lips he had kissed softly parted. He checked the impulse to lean down and kiss her one last time. She needed her rest, and it was time he was leaving. When she woke and found him gone, Tori would understand.
The luminous digits on her bedside clock said 3:35. Time to go. If he left now, he'd make it back to the ranch before the cowhands started their day. He would help with the chores and finish in time to grab a bite of breakfast, shower, shave, put on a suit, and catch a ride into town with Beau. No need for him to drive his own truck—especially since, if things went badly, he might not be coming home again.
Following the trail of his boots and clothes, he dressed in the dark hallway, put on his coat, and went outside into the frigid dawn.
The town lay deep in slumber as he drove down Main Street to the highway. Even the cheap neon sign above the Blue Coyote had been turned off. Since Stella would be at the trial, the place would probably be closed for the coming day.
The ranch house was dark as Will pulled up to the porch, shut down the engine, and switched off the headlights. The bunkhouse, too, was quiet, with no sign of anyone stirring. Good. The boys would snicker if they caught their boss sneaking back from a hot night in town. Going to bed was out of the question. Even if it wasn't too late, he wouldn't be able to sleep. But at least he could make a show of coming out of his bedroom, dressed and ready for chores.
He had mounted the porch, when a voice from the shadows startled him. "I was wonderin' when you were gonna show up, Will Tyler." Jasper was sitting in his customary chair, wrapped in his old sheepskin coat. The dog lay at his feet.
"Reckon I don't have to ask you where you been."
Will was grateful the darkness hid the flush on his face. "What are you doing out here at this hour? It's cold."
"Couldn't sleep. Restless, I guess, just like you. I don't plan to be at your trial. Got an old man's plumbing and it keeps me goin' too much to sit long. But I'll be rootin' for ya. Sky promised he'd pass on everything he hears from Lauren."
"Thanks. I need all the rootin' I can get." Will took the empty chair next to Jasper's. Who knew when he'd have another chance to talk with the old man who'd been like a second father to him over the years? If he went to prison, Jasper might not even be here when he got out.
"Your dad would be right proud of you, defendin' your little girl like you did," Jasper said.
"I'd do it again—but I hope I never have to." Will rocked back far enough to put his boots on the porch rail. "You said Bull shot a couple of rustlers. Did he ever kill anybody else?"
Jasper scratched the dog's head while he pondered the question. "Nobody that didn't need killin'. And he never got arrested for it. Things are different nowadays. The law makes it harder for a man to stand up for his family."
"What do you think Bull would say to me if he was here right now?"
"He'd say, 'Give 'em hell, son. Do the family proud!' Since he's not here, I'll say it for him. Give 'em hell, Will!"
Behind them the front door opened. Erin, barefoot and wrapped in an afghan, pattered out onto the porch. "I heard voices," she said. "Is everything okay?"
"It's fine, honey," Will said. "We were just talking. It's early. You might as well go back to bed and get some more sleep."
"I don't think I can sleep. I haven't slept all night. I'm scared, Daddy. What if I say something wrong at the trial today?"
"Come here." Swinging his feet off the rail, Will indicated his lap. Erin eased herself across his knees and nestled her head against his chest. It had been a long time since Will had held her like this. Her legs dangled almost to the floor of the porch. She was going to be tall like her mother.
"Don't worry about saying anything wrong," he said. "Just tell the truth, like you did with Abner."
"But what if they try to trick me?"
"Your mother will be there. She won't let that happen. Mr. Drummond will be asking questions first. You know him. He used to work with your mother, and they're still friends. He's a nice man."
Will wondered about that last part. Clay Drummond could be a pit bull in court, but he didn't want Erin to go in afraid of him. Tori would be there to object if Drummond went too far, but if he could find a way to trip up a hostile witness's testimony, he would do it—even to a child.
Erin had fallen silent. She lay with her ear against Will's heart, as if memorizing the sound of it. Across the yard the light was on in the bunkhouse kitchen. The smell of fresh coffee drifted on the breeze. The men would be stirring, dressing, grabbing a quick cup before heading out to their chores. In the east the stars had faded, leaving a streak of pewter dawn above the horizon. Reluctantly Will eased Erin off his lap and stood.
His day of reckoning was here.
CHAPTER 18
The jury selection started at 9:00 a.m. and took less than an hour. Tori knew most of the citizens who'd been called. She looked for family men and women who would understand Will's need to protect his daughter. Clay tended to choose people who were new in town and might not know Will, or those who'd patronized the Blue Coyote and might be more sympathetic to Stella's loss.
There were some calls for elimination from both sides, but nothing serious enough to hinder the process from going forward. By 10:00 a.m., the jury of seven men and five women had been impaneled and sworn, and the trial—the People of Texas versus Williston Tyler—was ready to begin.
* * *
Wearing the gray business suit he'd always hated, and a blue silk tie that felt like a noose around his neck, Will took his seat next to Tori at the table for the defense. Today his ex-wife was all business in the black tailored suit and ivory blouse she favored for trial wear. There was no sign of the pliant, needy woman who'd lain naked in his arms last night. She was sharp, edgy, and primed for battle, a warrior queen in black stilettos.
The gallery was filling with spectators. Turning in his seat, Will flashed a thumbs-up sign to Erin in the back row. Dressed in soft blue, the color of truth, with a demure white cardigan that matched the bow in her tawny hair, she sat next to Lauren, who'd promised to take her outside if the proceedings became too intense.
Beau had taken a seat in the row behind the railing, close enough to whisper to Will or Tori if the need arose. The local press was there, as well as a flock of curious townspeople who had nothing better to do than watch what they probably viewed as a live soap opera. They're like vultures gathering for a feast, Will thought. To hell with them all.
Heads swiveled, almost in unison, as Stella entered the courtroom. She was all in black, her vermilion hair drawn back into a bun, her makeup subdued. She was dressed to play the part of the grieving sister, and Will had no doubt she would give an Oscar-worthy performance.
Every eye was on her, and she was making the most of it. Her dress and makeup might be subdued, but her walk was the familiar Stella strut—hips swaying, butt thrusting, putting on a show from the rear. A murmur went through the spectators as she walked down the center aisle to her seat at the rail behind the prosecutor.
"All rise!" The bailiff—a husky former trooper with a commanding voice—announced the arrival of the judge. Sid Henderson was nearing retirement after more than twenty-five years on the bench. A blocky, humorless man, with a jowly face and a thatch of white hair, he could be counted on to run an efficient court with little tolerance for drama. When it came to handing down sentences, no judge in the county was harder on convicted wrongdoers. Will could only hope that issue wouldn't have to be faced today.
After everyone was seated and the judge had spoken a few words, Clay Drummond stepped before the jury box and waded into his opening statement like a heavyweight boxer lumbering into the ring. The man was good. Damn good. His claim that Will's reckless shot had killed a harmless man who'd already surrendered his gun was so compelling that Will might have bought it himself, if he hadn't been the one on trial.
But Will, who'd known the prosecutor for years, noticed something else about Clay. He looked as if he hadn't slept in days. His eyes were swollen and bloodshot. His voice was hoarse, his stance slightly wide-legged, as if he had to brace himself to stay erect. There was an air of desperation about the man. The more Will watched him, the more convinced he became that something wasn't right.
When her turn came, Tori was in top form. Will's actions, she argued, had been those of any responsible parent with a child to protect. He'd fired believing the victim to be a dangerous fugitive who wouldn't hesitate to overpower him and take his daughter hostage, or worse. The question before the jury was whether the defendant had acted in a reasonable manner. If so, they would be duty-bound to find him innocent.
When she took her seat again, Will had to stop himself from giving her a touch of encouragement. Right now, he mustn't think of himself as her ex-husband, her friend, or her lover. He was her client; and the best thing he could do was leave her alone to do her job.
"The people call Sheriff Abner Sweeney."
Clay began his case as expected. Abner appeared nervous as he took the oath and described what he had found when he'd arrived at the alleged crime scene. At that point Clay introduced the bagged knife, a small switchblade, as evidence and asked Abner to confirm it was the one that had been found in the victim's hand.
"Sheriff, were any fingerprints found on the knife?"
Abner looked down at his lap. "No. The knife appeared to have been wiped clean."
Will's pulse slammed. Nick Tomescu had been wearing gloves. But, surely, he would have left prints on the knife earlier. What was going on here?
"Sheriff," Clay continued, "why do you suppose the knife had no prints on it? Could Mr. Tyler have taken the knife, wiped it clean, and put it in the victim's hand after shooting him?"
"Objection!" Tori said. "Calls for conjecture."
"Sustained," the judge rumbled. "Please confine your questions to the facts, Mr. Drummond."
"Very well, Your Honor." Clay took a sip from the water bottle on the table. "Sheriff, did you find any evidence that the alleged crime scene might have been tampered with?"
"Yes." Abner was sweating. "A contaminated blanket had been laid over the body, and a key eyewitness, Mr. Tyler's daughter, had been removed from the scene before she could be questioned."
Will swore silently. So that was their game. If they could convince the jury he had something to hide, the implication of guilt was bound to follow.
"Sheriff, what did the defendant tell you when you asked to speak to his daughter?"
"He said she hadn't seen anything, and her mother had taken her home."
"Was it true that the girl hadn't seen anything?"
"No, that was a lie. I found out later that she'd witnessed the whole thing." Abner wiped his forehead with a crumpled handkerchief. "Mr. Tyler said I could speak to her in the morning, with her mother present."
"I take it that meant after the girl had gotten her story straight."
"Objection!" Tori was on her feet.
"Sustained." The judge scowled at Tori. "Sit down, Ms. Tyler. You'll get your turn."
There were more questions about the alleged crime scene and the evidence. Then it was Tori's turn to cross-examine.
"Sheriff, who made the nine-one-one call that summoned you to the scene?"
"The defendant."
"He has a name," Tori said. "Please use it. How did Mr. Tyler behave toward you when you arrived? Was he cooperative?"
"He was fine."
"When you arrived, did he appear to know the identity of the man who was shot?"
"By then, he knew it wasn't the robber. But when we pulled the helmet off the body and saw those tattoos, Will—Mr. Tyler—seemed knocked for a loop, just like I was."
"Thank you, Sheriff. No more questions for now." As Abner stepped down, Tori took her seat and waited for Clay to call his next witness.
* * *
"The people call Miss Erin Tyler."
At Mr. Drummond's words, Erin stood. Her legs were shaking, and her mouth tasted like she'd sucked on a penny. For an instant she froze, her feet refusing to move. Then she felt the touch of Lauren's hand on her back. "You can do this," she whispered. "Go on."
As Erin moved into the aisle and walked forward, she could feel every eye in the courtroom on her. Some were friendly, others curious. A few were even hostile. They all watched her as she took her seat in the witness-box and was sworn in by the bailiff.
. . . Tell the truth, the whole truth, and nothing but the truth. That's what her parents had told her to do. She could only hope the truth would help her father.
She glanced around the courtroom, feeling small and out of place. In a room full of stern adults, how could the testimony of a twelve-year-old girl make any difference? Then she met her father's blue eyes across the distance and remembered how much he loved and trusted her. The thought gave her courage.
Erin straightened in the chair as Clay Drummond stood and walked toward her. His mouth was smiling, but the expression in his eyes reminded her of a snake closing in on a baby bird. Her father had reminded her that Clay Drummond was a family friend and a nice man. But Erin knew better than to think he would be nice today. That was not his job.
"Do you know who I am, Erin?" he asked.
"Yes, sir."
"I'm just going to ask you a few questions. There's no need to be nervous."
"I'm not nervous, sir."
"Very well. To start, what did your father do after he shot Mr. Tomescu?"
"He got back in the truck and hugged me. Then he got out again. He called my mother and the sheriff and laid a blanket over the dead man."
"So he called your mother first, then the sheriff. He must have been in a big hurry to get you away from there. Did you hear the phone calls?"
"No, he made them outside the truck. But he'd told me what he was going to do."
"Why do you think he covered the body?"
"Objection!" Erin's mother said. "Conjecture."
"Sustained," the judge said.
"Fine. One more question. Did you see the defendant—your father—touch the knife in any way—like maybe pick up the weapon and look at it, or even put it in the man's hand?"
"No. I was watching. He didn't do anything like that."
His posture sagged slightly, as if someone had loosened a string. "Thank you, Erin. No more questions."
"You may cross-examine the witness, Ms. Tyler," the judge said.
Erin's mother stepped forward, looking as slim and polished as a movie star playing a lawyer on TV. "Erin," she began, "did either of your parents or anyone else instruct you in what to tell the court today?"
"They only told me to be honest."
"I'm sure you will be. Please tell the court exactly what you saw happen on the night in question."
Erin related the events, as she remembered them, hearing the radio announcement, hitting something in the road and blowing a tire, her father getting out, then seeing the motorcycle lights.
"Why didn't you get out with him?" Tori asked.
"I wanted to. But he told me to stay in the car. He thought he might've hit an animal."
"Is that why he took his gun, in case he'd hit an animal?"
"I think so."
"When did he tell you to lock the door and get down?"
"When he saw the motorcycle coming. I'm pretty sure he thought it was the robber."
"And did you get down?"
"Not all the way. I wanted to see, so I peeked over the window."
"Tell us what you saw."
Erin told the court what she'd seen happen. She did her best to keep her voice steady, but toward the end her throat began to quiver. Only now did she realize how truly scared she'd been.
"Think carefully, Erin," her mother said, handing Erin a pen. "What exactly did the man do with the knife? Can you show us?"
"It happened really fast. I didn't see where the knife came from, but he went like this." Erin demonstrated with the pen, holding her arm up and back as if she were about to throw it.
"And did he throw the knife?"
"No. That was when my dad shot him."
"Thank you." Her mother took the pen. "Just one more question, Erin. Did your father know you were watching?"
"No. I was afraid he'd be mad, so I didn't tell him. He didn't find out till the next day."
"So, when he told the sheriff you hadn't seen anything, he wasn't really lying. And he wasn't trying to hide anything, was he?"
"No." Erin shook her head. "My dad doesn't lie."
"Thank you, Erin." Her mother gave her a little smile. "No further questions."
Rising, Erin stepped down from the witness chair. As she walked back toward the aisle, she glanced to one side and saw the dead man's sister, Stella Rawlins, staring at her over the rail. A chill crept through her body. The hatred in those fierce green eyes was like icy claws creeping over her skin.
She'd faced the court and made it through the questions without a stumble. But now, for the first time today, Erin felt fear.
* * *
"The people call Mr. Drew Middleton."
Will heard Tori gasp. She thumbed through her notes, probably to see if his name had been added to the witness list. If it had, she could've been too busy to notice the update. She was clearly caught off guard.
Her body went rigid as Middleton walked in through the back doors, strode down the aisle, and took the witness stand. Her fingers gripped her pen as the bailiff administered the oath. Will studied her taut profile, unable to read her emotions.
"Mr. Middleton," Clay asked, "please tell the court where you were on the night in question."
"I was with Ms. Tyler." He did not look at Tori. "We'd been to dinner in Lubbock, and I drove her home."
"So it was a date?"
"Yes."
"And how did the date end?"
"We were in my car, in front of her house, when her phone rang. I could only hear one side of the conversation, but it appeared there'd been some kind of accident—an incident, she called it—and she had to go and get her daughter right away."
"Did you get the impression her daughter was hurt or in danger?"
"No. But Tori—Ms. Tyler—was in a big hurry. She rushed into the house, and a couple minutes later, her wagon passed me racing up the road."
"Why didn't you drive her in your car? That would have saved time."
"I offered. But she insisted on going alone, as if she was on a secret mission or something."
"Objection." Tori's voice was icy.
"Sustained," the judge droned. "Strike the part about the secret mission."
Clay cleared his throat. "Mr. Middleton, why do you think she was in such a hurry? Was it because the defendant, Mr. Tyler, wanted his daughter gone before the sheriff arrived on the scene?"
"Objection!" Tori snapped. "Calls for speculation!"
Clay shrugged. "Withdrawn. Your witness, Ms. Tyler."
"No questions." Tori shuffled her papers as the judge excused the witness. Middleton left without ever making eye contact with her. Whatever they'd shared in the past seemed to have gone sour. One less thing to worry about, Will told himself.
"Call your next witness, Mr. Drummond," the judge said. "After that, we'll break for lunch."
"The people call Ms. Stella Rawlins."
* * *
A murmur swept through the gallery as Stella took her time walking to the witness stand. Every eye was fixed on her. Aside from running the Blue Coyote, she'd kept a low profile in the town. For many of the spectators, this was their first chance to get a good look at the woman. Damned if she isn't putting on a show, Will thought.
After she'd taken the oath and stated her name for the record, Clay began his questioning. He looked more harried than ever. A bead of sweat trickled down his temple. His hands seemed unsteady, and his left eye had developed a noticeable tic.
Does Stella have something on him? But how can that be? Will wondered. Clay Drummond was a paragon of integrity, a leading citizen in the town, and the most likely candidate for a judgeship when Sid Henderson retired. Stella had brought down some powerful men, like former sheriff Hoyt Axelrod and Congressman Garn Prescott. But Clay? That didn't seem possible.
"Ms. Rawlins, for the record, the victim, Nikolas Tomescu, was your brother, is that correct?"
"Yes. Nicky was all the family I had." Her voice quavered on the edge of tears. The lady was good.
"And had you ever known him to act in a violent way?"
"Heavens, no! Nicky was slow and sweet, like a little child. I'd never known him to even kick a dog, let alone harm another human being."
"So the defendant and his daughter were in no danger whatsoever on the night in question?"
"No. Poor Nicky wouldn't have hurt either of them. He was most likely scared to . . . to death." She dabbed at her eyes with a lace hanky.
"I see." Clay gave the jury a meaningful glance. "No more questions. Your witness, Ms. Tyler."
Tori stalked toward the witness stand. "My condolences for your brother's loss, Ms. Rawlins." Her voice was level, even cold. "Are you aware that the deputies found a packet of cocaine on your brother's motorcycle?"
Stella's eyes glittered with suppressed rage, but her husky voice betrayed nothing. "That's what I was told. But I don't know anything about it. I'd warned Nicky not to fool around with drugs, but evidently he didn't listen to me."
"The gun he had was registered to you. Did you give it to him?"
"We kept a gun in the drawer below the cash register. He had access to it anytime."
"Were you aware that on the night in question, your brother was riding around on his motorcycle with drugs and a gun?"
"Objection!" Clay Drummond broke in. "The witness isn't on trial here."
"No, I want to answer," Stella said. "Nicky was an adult. I never told him what to do or what not to do. As long as he showed up for work, his personal life was none of my business."
"I see." Tori's tone was skeptical. "But do you agree that if he was carrying drugs, he'd be more likely to act in an aggressive manner—say, by drawing a gun or using a knife?"
Stella shrugged her ample shoulders. "How should I know? I always told Nicky to be nice to people. But I don't know how he might've behaved when I wasn't with him."
"But didn't you just tell the court your brother was harmless and would never hurt anyone?"
"Objection! Badgering the witness!" Clay protested.
"Withdrawn." With a knowing glance at the jury, Tori turned away. "No more questions for now."
"Mr. Drummond?" the judge asked.
"The people rest, Your Honor."
Stella strutted back to her seat, a stormy look on her face as the judge dismissed the court for a lunch break. The prosecution's case had proven little, but the trial was far from over. This afternoon it would be Tori's turn to present her case. And it would be Clay Drummond's job to rip holes in her defense.
* * *
Clay had brought a couple of ham sandwiches from home, planning to have lunch in his office. But once there, he realized he was too churned up to eat. Pouring himself two fingers of bourbon in a Dixie cup, he slumped at his desk. He knew better than to return to the courtroom with alcohol on his breath, but what the hell, he needed a drink.
He'd struck a few blows for the prosecution, but he was off his game, too tired and stressed to think straight. He'd hoped to get a few slipups out of the daughter, but Erin had turned out to be almost as poised and cool as her mother. Drew Middleton hadn't been much help, and even Stella had faltered under Tori's sharp cross-examination.
Will would be the afternoon's remaining witness. All along, Clay's best hope of a guilty verdict had been to convince the jury that the defendant had tried to cover up the crime. Now, once Tori had introduced the audiotapes supporting Will's motive, it would be the only remaining hope.
The jangle of his personal cell phone triggered a spasm in Clay's stomach. It was Stella. And he knew better than to let the call go to voice mail.
"You were dead on your feet out there, Clay." She sounded like she was talking through clenched teeth. "You let the Tyler woman ask me too many questions. And that little brat of hers made you look like a fool. You need to up your game. I want that bastard brought to his knees!"
"I'm doing everything I can, Stella."
"Not by me, you aren't."
"I'll get my chance at Will this afternoon. Don't worry, I'll give it all I've got."
"You damned well better. If Will Tyler walks out of that court a free man, you're finished."
The call ended in silence.
The clenching sensation in Clay's gut had become a sharp pain. Maybe he was getting an ulcer. Scrolling down, he punched in Abner's cell phone. The sheriff had gone back to work after his testimony. Now Clay was going to need him again.
"Are you alone?" he asked when Abner answered.
"For now. How's the trial going?"
"Still dicey. I'll want you back here to confirm that the knife was bagged at the scene and found to have been wiped clean of prints."
"No."
"What?" Clay almost dropped the phone. "Why, for God's sake?"
"You know why. I've been doing some soul-searching, Clay. If anybody finds out I wiped that knife myself before it was dusted for prints, and then lied about it under oath, I could go to jail. I've got a new grandson, a fine boy, to raise. I want to be there for him."
"Damn it, Abner." Clay gripped the phone harder. "I could ruin you!"
"That wouldn't be very smart. Evidence tampering, if it came out, would guarantee Will Tyler's acquittal. And I wouldn't be slow to let folks know you were in on the scheme. Call me to the stand, and I'll resign. Then I'll tell the truth. Your choice, Mr. Prosecutor."
With a muttered obscenity Clay ended the call. What a time for Abner to get noble! But the little toad was right about one thing. Clay couldn't touch him without incriminating himself.
Clay glanced at his watch. The lunch break was over. It was time to be back in court. And he had nothing left.
* * *
Will shifted forward in his chair as Tori introduced two audiotapes into evidence and played them for the jury. The first was a recording of the radio announcement, describing the fugitive. The second was the tape of Will's urgent call to the 911 dispatcher, when he believed he'd killed the robber. More than any other evidence, these tapes supported Will's reason for shooting the man on the motorcycle and confirmed that he'd truly believed he was facing a dangerous criminal. Now it was time for Tori to call her first, and only, witness.
"The defense calls Will Tyler."
By the time Will told his story on the stand, most of it was old news. When he was finished, Tori had just one question for him.
"Please explain to the court why you covered the body and why you called me first, before you called the sheriff."
"In answer to both questions, I wanted to spare my daughter," Will said. "The sight of a dead man with a bloody hole in his chest would've haunted her for the rest of her life. I didn't want that picture in her mind. As for the call, I didn't know she'd witnessed the shooting. I only wanted to get her away from an upsetting scene to someplace safe and familiar. For me, that was even more urgent than calling in the law. That's all I have to say."
Clay's cross-examination was tepid. And the man looked even more ragged than he had that morning, his eyes sunk in shadows, his speech far from its usual machine-gun delivery. By the time he finished his closing statement, once more pressing the point that Will's impulsive shot had needlessly killed an innocent man, he appeared so exhausted that Will wondered if he might be ill—or maybe trying to garner sympathy.
Tori's performance, in Will's eyes at least, was flawless. As she faced the jury, looking spectacular and pointing out that Will had done what any reasonable father would do, one thought took root in his mind. If he walked out of this courthouse a free man, he wanted Tori back. He wanted her in his home and in his bed, with his ring on her finger. He wanted a normal, loving family life with his wife and daughter. And he wouldn't give up until he'd made it happen.
But right now, everything depended on the outcome of the trial.
So far, Will had reason to be hopeful. But juries could be unpredictable, verdicts surprising. There was no way of knowing what would go on behind those closed doors. A strong case didn't always win. And this one, based more on circumstance than on solid evidence, would be a judgment call. It could go either way.
With his freedom hanging on their decision, Will watched the jury file out of the courtroom, charged with finding on two counts— manslaughter and obstruction of justice. They could be out for an hour. Or they could be arguing into tomorrow.
* * *
Will, Tori, Erin, Lauren, and Beau drove the seven blocks to Tori's house to wait. Tori broke out cold sodas and snacks for everyone. Erin and Beau switched on a video game. The action crackled, heightening the tension in the room as they played. Lauren, looking tired, phoned Sky, then wandered into Tori's room to lie down. Tori and Will, too edgy to relax, alternately sat and prowled. Time crawled, minute by anxious minute.
Two hours from the time they'd reached the house, Tori got the call. The jury had a verdict. They piled into her wagon and rode in silence back to the courthouse. Sitting beside Erin in the backseat, Will felt his daughter's hand creep into his. He held on tight as they pulled into the parking lot and stopped at the side entrance to the courthouse. In the foyer they glimpsed Stella, in her tight black dress, hurrying into the courtroom ahead of them.
Pulse racing, Will took his place at the table with Tori. Erin and Lauren had moved forward to sit behind him, next to Beau. They'd scarcely had time to get settled before the jury filed back into the box and "Please rise" signaled the entrance of the judge. As they sat again, Will glanced across the aisle at Clay Drummond. Beads of sweat gleamed on the prosecutor's forehead.
"Ladies and gentlemen of the jury, have you reached a verdict?" the judge asked.
The foreman stood. "We have, Your Honor."
"Will the defendant please rise?"
Will stood, his expression frozen in a stoic mask. Beside him, he could feel Tori trembling.
"On the count of manslaughter, how do you find?" the judge asked.
"We the jury find the defendant, Williston Tyler, not guilty."
Will's knees went slack. He groped for Tori's hand, not finding it.
"And on the count of obstruction of justice? How do you find?"
"We find the defendant not guilty."
Not guilty!
As the words sank in, the courtroom erupted in sound and movement. Erin flung herself over the rail and wrapped her arms around Will's neck. Beau was hugging his shoulders. As if from far away, he heard the judge thanking the jurors and telling him he was free to go. His arm reached for Tori and pulled her close. Quivering, she pressed her face against his jacket.
Looking past her, Will caught a flash of hate-filled green eyes. Facing him from across the aisle, Stella mouthed something he couldn't understand. Then, with a last, venomous look, she turned and stalked out of the courtroom.
CHAPTER 19
Abner got word of the verdict from the bailiff, who'd called to let him know that Will Tyler wouldn't be needing the jail cell he'd reserved. Too bad in a way. Seeing the high-and-mighty Will locked up and headed for prison would have given him some satisfaction. But he'd known all along that the man wasn't guilty. At least now his conscience would be clear.
Damned funny thing, his conscience. He'd almost forgotten he had one until he'd held his newborn grandson in his arms. When he'd looked down into those pure eyes, it was as if they could see all the way into the depths of his corrupted soul. That was when Abner had known he had to become a better man.
Clay Drummond was going to be sore. So was Stella. For now, he knew enough of their secrets to keep them from doing him too much damage. But he was playing a dangerous game—a game that could leave him disgraced or dead, and little Ralphie without a grandfather.
Abner thought of all the times he'd skirted the limits of the law. Evidence tampering, leaking confidential information, looking the other way when Stella did her dirty work—so many small crimes that he'd lost count. He had loved being sheriff, loved the authority, the respect, and the sheer fun of playing detective. But a man with his secret record had no business in a position of public trust. To cleanse his conscience and keep himself safe, he would have to start over.
There was only one way to do that.
Bringing up his computer, he opened a blank document and began typing a letter of resignation.
At home, in a locked drawer of his study desk, Clay kept a loaded .38. Now, alone in his courthouse office, he found himself thinking about that gun and how he might use it to end his life. Even death would be better than what he and his family would face if Stella released that damning surveillance tape.
Through the west window the setting sun cast a bloodred glow into the room, reflecting off the empty bourbon bottle on his desk. He was borderline drunk. But his office door was locked, his staff gone for the night. Nobody was going to walk in on him.
What now? Would Stella warn him first or would she simply leak the tape to her press contacts? Either way, he had no doubt she'd do it. She'd ruined Garn Prescott after he'd let her down. She'd do the same to him.
His ringing cell phone broke the silence. Clay glanced at the caller ID. His stomach clenched.
"You really blew it today, didn't you, honey?" Stella's voice was like the purr of a big, sleek cat toying with a mouse.
"Please, Stella," he begged her, almost blubbering. "I'll do anything! Just don't release that tape!"
"You say you'll do anything?" She laughed, dangling the bait. "What would you say to a chance to get the tape back?"
Clay's pulse leapt. But he was sober enough to know that whatever Stella had in mind would be illegal and dangerous. Desperate as he was, he had to keep his head.
"Whatever you want, I'll do it on two conditions," he said. "First I want your promise that you'll give me the tape."
"Cross my heart, honey. Do the job and it's yours. What's the other condition?"
"I want to keep my life—my job, my reputation, and my family. If my involvement can be kept secret, I'm on board."
Again she laughed. "That can be arranged—as long as you're not stupid enough to get caught. But once I've told you the plan, you're in. Get cold feet, and you'll be humping an underage girl on the ten o'clock news."
The knot in Clay's gut felt like a tangle of barbed wire. "I'm in," he said. "Tell me what I have to do."
* * *
Will had celebrated his acquittal by going home, changing his clothes, and working on the barn until dusk. Lord, but it feels good to be a free man! After what he'd survived, even the money troubles seemed surmountable. Now that the specter of prison was gone, he could plan. And he could make the hard decisions it would take to keep the ranch running.
That night the ranch family had celebrated around the dinner table with green salad, fresh garlic bread, and a big pan of Bernice's lasagna. Now it was late, the meal finished, the leftovers put away, and the dishes loaded in the dishwasher. Erin and Bernice, worn out by the day, had gone to bed. Sky, Lauren, Jasper, Beau, and Natalie had left for the night.
Only Tori remained. Wrapped in the afghan, her stocking feet tucked under her, she sat on the front porch with Will in the opposite chair. The night was clear, the stars like the spill of a million diamonds across the sky. The breeze was no more than a whisper.
Will studied her moonlit profile—the chiseled nose, the soft, full lips, the stubborn chin, and the wisps of golden hair framing her face. There was nobody like Tori. He ached to make her his again.
Tonight could be his best chance to tell her what was on his mind. But he'd never been much good with romantic talk. Maybe that had been part of the problem when they were married. He'd been too tough, too macho, to say the words a woman needed to hear.
Could he say them now?
She stirred, untangling her legs. "I suppose I should be going," she said.
"You don't have to go," he said. "You could stay here tonight. We could even do some more celebrating."
"Will—"
As soon as she spoke his name, he knew he'd said the wrong thing. "Sorry," he muttered. "Did I assume too much?"
She shook her head. "Last night we needed each other. But that doesn't mean it's going to happen again."
Her words stung Will like a slap in the face, but he stuck to his guns. "You know it was good for both of us. What's wrong, Tori? Is Middleton still in the picture?"
"No, that's over. There's nobody else. It's just that . . . the woman you were married to doesn't exist anymore. This woman won't answer to any man's beck and call, even yours."
"So I'm supposed to wait around like a hungry dog until you crook your little finger? Damn it, Tori, I want you! I want my family back!"
She stood, clutching the afghan around her. "We've been a family of sorts all along, haven't we? Has that been so bad?"
"It's been tolerable—better than nothing. But sleeping alone and only seeing my daughter when school's out isn't my idea of being a family."
"Will." Her voice was flat with strain. "Half the single women in the county would jump at the chance to be your wife. You could get married again, even have more children."
"Damn it, if that was what I wanted, I'd have remarried a long time ago!" Frustrated, he turned away from her and glared across the moon-shadowed yard. "Never mind. This is going nowhere. If you need to go home, just go."
"Fine. As soon as I get my things." She crossed the porch, then paused at the front door. "I'll be here on Sunday to get Erin. She misses her friends. She wants to go back to school in town."
The news jolted him. "Are you sure that's safe?"
"It's what she wants. I already promised her."
"I wish you'd talked it over with me before you promised."
"The trial's over," Tori said. "You were judged innocent. We need to move on. Even Erin needs to move on—with her school, with her friends. She needs to put this ordeal behind her and get back to normal."
"But does it have to be so soon?"
There was no reply. Tori had gone inside.
* * *
Minutes later, Tori swung her station wagon away from the house. Tires spat gravel as she headed down the lane toward the main road. Will had been gone when she'd come back outside with her shoes, jacket, and purse. There'd been no chance to make things right, or even give him a conciliatory smile. He'd opened up to her, and she had wounded him.
Will was a proud man. The next time she saw him, his behavior toward her would be that of a polite, cold stranger.
Welling tears blurred the road in her headlights. Why couldn't she have said yes to Will? It would have made him happy. Erin would have been overjoyed, and the whole ranch family would have celebrated her return.
But she knew why. The sad, hard truth was that she was scared. Will was the love of her life. There would never be another man like him. But being his wife had crushed her spirit in a way she would never tolerate again. Over the past eight years, they'd become different people—she was stronger; Will, perhaps less like his father. But some things never changed. Going back to him would be like picking up where they'd left off, with all the old hurts coming to the surface.
They could end up hating each other.
Out of the darkness a buck deer flashed into the road, leaping high in her headlights. Tori slammed the brake. The wagon screeched to a stop, missing the animal by the barest inch as it bounded away. Pulse hammering, she slumped over the wheel. Another split second and she might have hit it in midair, sending its heavy body smashing into her windshield. Or she might have swerved and rolled the vehicle off the road's steep shoulder.
Still shaking, she drove on. If the timing had been off by a sliver, she could have died back there—without saying good-bye to Erin, and without ever letting Will know she loved him. Life was fragile, and no one was ever truly safe. A heartbeat could change everything.
As the lights of Blanco Springs came into sight, she tried to shake off the dark mood. She'd had an exhausting day, she reminded herself. With the trial behind her, all she needed was a good night's rest. Tomorrow everything would be back to normal.
Everything was going to be fine.
* * *
Stella double-checked the locks on her apartment door and windows before she settled onto the sofa, poured herself a brandy, and opened the newspaper she'd brought home from work. Every night she scanned the pages, hoping for news of Marie's arrest. So far, she'd found nothing. Either the hoped-for story hadn't been worth a mention in the press, or the woman was still out there somewhere, hiding from the law and plotting her revenge.
Nothing again tonight. Stella shoved the paper aside, lit a Marlboro, and hoisted her aching feet to the coffee table. Days from now, she'd be safe, where nobody, not even Marie, would ever find her.
A week had passed since the trial. Will Tyler's acquittal had left a bitter taste in her mouth. But never mind. She had a backup plan, one that would guarantee her a comfortable retirement and devastate the whole Tyler family.
All that remained was to carry it out.
She took a drag on the cigarette and laid it in the ashtray. It was almost midnight, time to check in with her Mexican friend Don Ramon, who'd be waiting for her call in the bedroom of his stately hacienda, south of Piedras Negras.
He picked up on the first ring. "Is everything ready?" His English was accented but passable.
"Almost. Can you guarantee the border crossing?"
"Como no. Of course. As long as I know when you'll be there."
"I'll call you when I'm on my way. Have you found a buyer?"
"More than one. The bids are still going up. Forty percent of the final price for me, yes?"
"Fine." The rascal would probably hold out for even more, but never mind. She couldn't do this without him.
"It would help to have a photo," he said.
"I don't have one, but I promise you, the girl's a beauty. Blue eyes, blond hair. And very young. Your friends always go for that type."
"You're sure she's virgin?"
"You'll have her checked, of course, but I'd stake my life on it. She's from a good family. Very protected."
"So it will be tomorrow, you think?"
"Tomorrow after school, if all goes well," Stella said, trying not to think of how many things could go wrong. "I'll be crossing the border at night. Wait for my call. If I'm not coming, I'll let you know."
"Buena suerte," he said, wishing her good luck.
"Adiós, amigo." Stella ended the call. Now everything depended on luck—and on Clay Drummond. She would give him a call tomorrow to make sure everything was on. When she knew he had the girl, it would be time for the rest of her plan.
* * *
By 3:00 p.m., Clay was sweating bullets. Kidnapping was a federal offense. Get caught, and he could be put away for life. But even that couldn't be any worse than what would happen if Stella released that surveillance tape.
Anyway, he wouldn't really be kidnapping, he told himself as he cruised past the middle school and pulled onto a side street. He'd only be giving the kid a ride. What happened after that would be out of his hands. He could only hope Stella would keep her promise and give him the tape.
For the past few days, he'd kept an eye on Erin Tyler. She usually left school at 3:15 p.m., walking with a girlfriend. When they reached the top of her street, the two would separate. Erin would walk the rest of the way alone, let herself into the house, and wait for her mother to come home.
This afternoon Tori was in court on a civil case—Clay had checked to make sure. If her daughter followed her customary routine, he would put the plan in motion. What he was setting out to do would be the most despicable thing he'd ever done. But he mustn't think of that now. Instead he would focus on getting the tape back and freeing himself to move on with his life.
Now, from where he'd parked, he could see students pouring out of the school. It took him only a moment to spot Erin. Dressed in jeans and boots, with her backpack slung over one shoulder, she was heading down the sidewalk with her dark-haired friend. Clay hung back, keeping his distance until the two separated and Erin turned the corner for home.
He took a moment to call Stella and tell her the plan was on. Then, stepping on the gas, he sped around the corner after Erin and screeched to a halt, short of the house. With a worried look on his face, he pulled up to the curb and rolled down the window. Erin had stopped and turned to look at him.
"Thank goodness I've found you, Erin!" he said. "Your mother passed out in court. She's been taken to the hospital. Your dad's on his way there. He asked me to find you and bring you."
Fear flashed in her eyes. "What's the matter with her? Is she all right?"
He reached across the seat and opened the passenger door. "Get in. We can talk on the way."
She ran around the car, tossed her backpack into the rear seat, and buckled herself in beside him. "Let's go," she said.
Clay swung the Mercedes back into the street and headed for the road out of town. Stella would be waiting in an isolated spot off the freeway. The transfer would've been safer at night, but they'd agreed there was little chance of catching the girl alone, outside, after dark.
"Tell me about my mother," she said. "What's wrong with her?"
"The paramedics weren't sure. She was still unconscious when they put her in the ambulance. A stroke, maybe, they said."
Such brutal lies, and the girl looked so worried, so trusting. It was all Clay could do not to tell her it was all a mistake and shove her out of the car. But he'd long since passed the point of no return. He had to do this.
* * *
"Let me call my dad." She twisted to reach for the backpack she'd thrown behind the seat. "Maybe he can tell us more."
"No!" Clay tried to hide a surge of panic. If she got her hands on her phone, he'd be in big trouble. "Either your dad will be speeding to the hospital or he'll be in the ER with your mother—not a good time to take a call. You can talk to him when we get there."
"Oh—okay. But hurry." She settled back, agitated fingers gripping the seat belt. Clay could smell the stink of his own sweat as he swung onto the freeway. His heart was pounding so hard, he feared he might burst a blood vessel.
Take the second exit, onto the old ranch road. That was what Stella had told him. Drive till you see some cottonwoods and a tumbledown shed. I'll meet you there.
The exit was already coming up. Clay swung the Mercedes onto the off-ramp.
"Where are you going?" Erin grabbed his sleeve. "This isn't the way to the hospital!"
"I heard there was a big wreck up ahead. A semitruck rollover, blocking traffic. This road will get us around it." Clay could see the trees in the distance. The asphalt pavement had ended in a weathered farm road. What the hell would he do if Stella wasn't there?
"How do you know there was a wreck? Who told you?" She was getting suspicious, Clay could tell. He stomped the gas pedal. The Mercedes shot forward, rocketing down the rutted road. He thought of the chloroform-soaked rag, sealed in a plastic bag, which he'd put in his pocket. He was going to need it.
"Stop!" she said. "Stop right now! I want to call my father!"
"Fine. Go ahead." Clay eased the car to a stop. One hand reached into his pocket and unsealed the ziplock on the bag. Hampered by the seat belt, she turned to reach for her pack.
"I'm sorry, Erin," Clay said, and he clapped the cloth over her face. The girl barely had time to resist before the chloroform took effect and her body went limp.
Leaving her sagging against the shoulder strap, Clay started the car again and sped toward the trees. Now he could see the dilapidated shed and the back end of Stella's Buick parked behind it. She was here. Soon this nightmare would be over, and he could start living his life again.
As he pulled up to the shed and climbed out of the car, she stepped into sight. "You've got the girl?" she demanded.
"Right here. Fast asleep." He opened the passenger door to reveal Erin, still slumped in the seat.
"Good. Give me a hand with her." She raised the lid of her trunk, which was lined with a dirty-looking quilt. Clay unfastened Erin's seat belt, lifted her in his arms, and laid her on her side. He'd hoped he was finished, but Stella handed him a roll of duct tape. "Wrists, ankles, and mouth, then fold the blanket over her," she said. "Be quick about it."
Clay did as he was told, trying not to make the tape too tight. "You won't want to tape her mouth. If she gets sick to her stomach in the trunk, she could choke to death."
"Fine. But I'll need to keep her quiet. Give me the chloroform in case she wakes up."
Clay resealed the washcloth in the bag and gave it to Stella. She tucked it into her red leather purse and closed the trunk.
"Aren't you forgetting something?" Clay asked.
Stella raised an eyebrow.
"The tape. You promised it to me if I delivered the girl."
"So I did." With a smile she reached into her purse and handed him the cassette from the surveillance camera.
"Can you swear this is the one?"
"Don't worry. It's the real deal, and it's all yours. I won't need it where I'm going." She turned back toward her Buick. "I believe this concludes our business. Have a happy life, Mr. Prosecutor."
Clay felt an unaccustomed lightness as he walked back toward his car, clutching the tape. He had just done the unthinkable. But he'd salvaged everything that mattered to him. He was free to pursue his ambitions—become a judge, maybe go into politics. Nothing would be out of reach now.
He'd just opened the driver's-side door when the bullet slammed into his skull, passing from back to front and knocking him forward across the seat. His body twitched once and lay still.
* * *
Will had come inside to read his mail when Tori called. "Will?" She sounded worried. "Did you pick up Erin after school? She's not in the house, and she's not answering her cell phone."
Dread clutched him like a cold steel vise. "No. I haven't heard from her. Did you call her friend?"
"I'll do that next. But if she was at Allison's, she'd have let me know, and she would've had her phone on. Will, I'm scared."
"Call the sheriff 's office—now. Tell them to check the Blue Coyote."
"Dear God, you don't think—"
"Until we find Erin, we've got to assume the worst. Call them. I'll get Beau on this."
The acting sheriff since Abner's resignation was his young deputy, Rafe Sanchez. The kid was sharp, had the makings of a good lawman, but he lacked the experience to handle a life-or-death emergency. As a former DEA agent, Beau was a seasoned crime fighter with contacts in the FBI. If Stella, or somebody in her pay, had taken Erin, every minute's delay could make a difference.
Beau had been outside, supervising work on the newly erected barn. Seconds after Will's call he burst into the house, out of breath. "You're sure she's been taken?"
"Not yet, but if she has, we can't afford to waste time."
"Stella?"
"Until we know more, that's my best guess."
"The first thing we need to do is put out an AMBER alert. I'll make some calls."
While Beau was on the phone, Will got another call from Tori. His pulse skipped. Maybe she'd heard from Erin. Maybe all this panic was nothing but a false alarm.
But no, as soon as he heard her voice, he knew the news would be bad. "The dispatcher put me through to the sheriff," she said. "He was out by that ranch exit off the freeway, checking something the Life Flight pilot had spotted from . . . the air." Her voice quavered, then broke.
Will's throat jerked. "Was it Erin?" he asked.
"No." Her breath caught. "It was Clay Drummond's car. Clay was inside, shot dead. And Erin . . . Oh, Lord, Will. They found her backpack, with her phone in it, behind the seat. But Erin was gone."
CHAPTER 20
At last the sun was going down. Stella slipped off her sunglasses and massaged the bridge of her nose. While the daylight lasted, she'd kept to the back roads, doing her best to stay out of sight. Soon it would be dark enough to pull onto the freeway and make a beeline for Eagle Pass, where she would cross the border into Mexico.
She'd already called Don Ramon on her burner phone to let him know she had the package and was on her way. He'd assured her that the car wouldn't be searched. Not that there was much chance of that. The Mexicans weren't too fussy about what crossed into their country, especially when there was a mordida involved.
One of Don Ramon's trusted agents would be waiting in Piedras Negras to give her the cash and take the girl off her hands. With the little bitch gone, she could finally get some rest.
Her passenger was awake and stirring. Stella could hear thumping, kicking, and cries of fear and rage from the trunk. At least the girl seemed to be in decent condition. But the car was running low on gas and Stella had to pee. If she stopped at a service station, the commotion in the back might draw attention. She would have to open the trunk and use the chloroform to knock the girl out again.
Now was as good a time as any. She glanced at the ski mask lying on the seat beside her. She hadn't wanted the girl to see her face. But why bother? Erin Tyler wasn't stupid. Mask or no mask, she'd know who had her.
The road was deserted. Deciding not to wait for a service station, Stella pulled onto the shoulder, climbed out of the car, and squatted behind a clump of sagebrush to relieve herself. The wind was blowing in hard from the northwest, bringing with it a bank of roiling black clouds. A storm was moving in. If she wanted to make the border before the weather hit, she'd have to step on it.
Unzipping the plastic bag with the chloroform-soaked washcloth in it, she slipped it in her pocket and opened the Buick's spacious trunk. The girl, her wrists and ankles taped, her clothes twisted and rumpled, was glaring up at her like a wounded hawk, as much in fury as in fear.
"Where are you taking me?" she demanded.
Stella chuckled. "That's for me to know and you to find out, honey. Let's just say I'm planning to retire, and you're going to fund my pension."
Fear flashed across Erin's pretty face, but the look hardened into something else. The girl had fight in her. "So you need to keep me alive," she said.
"Alive and pretty. That's the plan."
"I could use a bathroom break," she said.
"That's your problem." No way was Stella letting her out of the trunk and freeing those long legs. She'd probably take off like a jackrabbit.
"At least give me some water," she said. "My throat's really dry."
"Oh . . . what the hell. Just a sip. I've got a bottle up front." Stella walked around to the car's front door, opened it, and found the plastic bottle she'd tossed onto the passenger seat. She was out of sight for no more than a few seconds, but she walked back to find that the girl had rolled, dropped her bound legs over the rear bumper, and was almost out of the trunk.
Time for a little education.
Stella tossed down the water bottle and grabbed the girl's arm. "Where d'you think you're going, you little spoiled shit?" She backhanded her hard across the face. "You think you can hop all the way home to your daddy, eh?" She hit her again, so hard that Erin's head snapped to one side. "Don't mess with me, princess, or you'll be sorry!"
She clamped the washcloth over the girl's face and held it there until the slender body went limp. Then she shoved her prisoner back in the trunk, took her shoes for good measure, and slammed the lid shut. She'd never liked kids, and this one could turn out to be more trouble than she was worth. Only the thought of the money the girl would bring kept her from dumping the little twit on the road and leaving her there to freeze.
Climbing back in the car, Stella drove on. If the weather held, she could be over the border in a couple of hours. The idea of a warm bath and a soft bed in Piedras Negras was sounding more and more like heaven.
* * *
How could anything be harder than waiting?
Will, Tori, and Beau huddled in Tori's living room, all of them silently praying for good news. Until they had some word of Erin, there was nothing they could do except be here, and be available.
Tori's cell phone rang. The caller was Natalie, who was waiting back at the ranch with the rest of the family. She'd called Tori because Beau wanted to keep his phone free for police or FBI calls. "Any news?" she asked.
"Nothing yet." Tori forced herself to use her lawyer voice. She was crumbling inside, but this was no time to break down.
"They're sure it was Stella who took her—and shot Clay?"
"Positive. The sheriff 's men matched up the boot prints and tire tracks. Plus, the waitress at the Blue Coyote hadn't seen her boss all afternoon. The best guess is that she's headed for Mexico. The highway patrol has an all-points bulletin out, but no one's spotted her car. She may have switched vehicles."
"Oh, Tori . . . ," Natalie whispered.
"Don't," Tori said. "Sympathy can wait."
"They must be doing more than that to find her," Natalie said.
"There's a statewide AMBER alert out for her. And the police will be checking every car that leaves the country through Eagle Pass. If all Stella wanted was revenge, she could've killed Erin when she killed Clay. If she's keeping her alive, and taking her over the border, it's for one . . . reason." Tori choked on the last words.
"I'll let you go," Natalie said. "Tell Beau I called. We're all praying."
"Thanks." Tori ended the call, struggling to get a grip on her emotions. It was all she could do to keep from dashing outside, jumping into her station wagon, and rushing off in the night to find her daughter. But that would be useless. All she could do was stay here and endure the agony of minutes crawling past with no word.
Will was sitting next to her on the couch. Tori reached for his hand and felt his big, rough palm close around hers. They held each other in silence, seeking comfort, drawing strength, and sharing the pain that only parents of a loved child could know.
A light rap on the door jerked them all to attention. Beau opened it to find Sheriff Rafe Sanchez on the porch. Just twenty-two, the son of an unmarried Mexican mother, he'd stepped into Abner's job barely a week ago. Now he was in charge of investigating the scene of Erin's kidnapping and the murder of Clay Drummond.
"Any news?" His dark eyes took in their stricken faces. "No, I guess not. I just wanted to give you an update on what we found at the scene. I don't know if I'm supposed to share it, but . . ." He shook his head. "What the hell, you deserve to know."
"Come in." Beau stepped aside and ushered him into the living room. "Have a chair. Can we get you something to drink?"
"Thanks, but I can't stay." He remained standing, a tall young man, his lean body still filling out. "We're pretty sure Stella was blackmailing Clay. We found a surveillance tape in his hand—something that would've ruined him if it had gone public."
"That's no surprise," Beau said. "Blackmail was Stella's stock in trade. We know better than to ask you what was on the tape."
Sanchez nodded. "For now, we're not telling the family. Stella's prints were on the cassette. She could've offered it to Clay in exchange for delivering Erin."
"Erin would've trusted Clay." Tori felt a wave of staggering rage. "It couldn't have been too hard for him to get her in his car."
"So Stella gave him the tape and then killed him." Will was on his feet. "The woman who's got our daughter isn't just a kidnapper. She's a cold-blooded murderer—but I guess we already knew that."
"Yes." Sanchez's English was measured, as if he'd learned most of it in school. "She's capable of anything. But if she went to that much trouble, she must want to keep Erin alive, most likely to sell her. At least that buys us some time."
But how much time? Will and Beau exchanged glances. In a kidnapping the odds of a safe recovery dropped exponentially with every hour that passed. Time, they knew, was swiftly running out.
* * *
Erin stirred and opened her eyes. She was still in the trunk of Stella's car, bound hand and foot by duct tape. Her neck was sore, her face bruised and tender where Stella had hit her. Her throat felt as if she'd eaten a fistful of sawdust. Her limbs were cramped, and her shoes, she realized, were gone.
The darkness smelled of cigarette smoke, mildew, and dirty carpet. Being in the trunk was like being inside a coffin. But she couldn't give in to panic. She'd already worn herself out with useless struggling. She had to stay calm and alert, to think from moment to moment. Whatever happened, she had to survive until she could find a way to escape.
Shifting and wiggling, she used her bare feet to explore her prison. If she could find some kind of tool, or anything with a sharp edge, she might be able to slice through the duct tape. But there was nothing. Stella must have cleared out the trunk ahead of time. Erin had seen a movie where a person trapped in a car trunk had knocked out a taillight from the inside to attract attention. Her toes probed the corners where the taillights would be, but she couldn't find any way to break one.
The trunk was cold and getting colder. She shivered in her light jacket. Through the metal trunk lid, she could hear wind whistling around the car. A heavy patter, like tacks spilling into a tin bucket, filled the dark space around her. It had to be hail. The car was still moving, but its speed felt slower, as if the road surface might be slippery. Maybe the car would slide off the road. Maybe then a patrolman would come by and rescue her. But that was just a fantasy. For now, all she could do was pull the blanket over her as best she could, curl into a ball, and try to keep from freezing. That, and pray.
* * *
Driving through the storm on the ice-slicked freeway had been a hellish experience, but at least it had lessened the chance of her car being spotted. Now the weather was clearing. Ahead, in the distance, Stella could see the lights of Eagle Pass. Soon she'd be safely over the border, with the money in her purse and Will Tyler's precious daughter on her way to some Mexican drug lord's bed.
Reaching for the burner phone, she called Don Ramon's number. He answered at once.
"Is everything ready?" she asked.
"Unfortunately, querida, there has been a slight change of plans."
Something clenched in Stella's empty stomach. "Did you get the money?"
"The money? Yes. No problema. But I just got word that the police on the American side are checking every car, looking for you and the girl. You will never get through."
"Both crossings?" There were two border bridges in Eagle Pass, a large one and a smaller one.
"Yes, both. But don't worry. My man is in Piedras Negras with the money. Find a motel in Eagle Pass, someplace cheap and quiet. Check in and call me. He will find you, pay you the cash, and take the girl. Entiendes?"
"Yes," Stella said. "But—"
"Let me finish. The police will be looking for your car. Tomorrow you can leave it somewhere to be stolen and walk across the bridge with the foot traffic. No police will stop you. You will be safe. But one more thing."
"What?" Stella was liking this less and less.
"My man will need a truck or a van to smuggle the girl across the border. To arrange this may take time."
Stella mouthed a curse on the man's ancestors. "How much time?"
"Who knows? Not long. Perhaps a few hours. Maybe less. You can rest and wait."
"I don't like this," Stella said.
"What else can we do? The police are everywhere."
"Fine. I'll phone you." Stella ended the call and pulled off at the first exit. Too bad she hadn't switched cars, but she'd been in a hurry, and it was too late now. She had little choice except to follow Don Ramon's directions.
Eagle Pass was a fair-sized town. But with the police on the lookout, she didn't want to drive in very far. Anyway, the older, cheaper motels would likely be found on the run-down outskirts.
Fifteen minutes from the freeway, she found a place that might do. The El Camino was a row of clapboard units, set back from the quiet street and overhung by a sagging willow tree. Two battered-looking pickups were parked outside. At one end a sign in the window said VACANCY.
Parking the car at a safe distance, Stella climbed out. The night air was frigid, the ground coated with icy hail. She would have to bring the girl inside, a bother, but she'd promised to deliver the little bitch in good condition.
A bell rang when she opened the door of the office. The Mexican woman who came to the desk was wearing a ratty fleece bathrobe. From somewhere out of sight, a TV was blaring.
"Just one of you?" the woman asked.
"No, my daughter's with me. She's sick. I'm taking her to the doctor in the morning." Stella thrust a wad of small bills across the desk. "Here's an extra ten for keeping things quiet."
Stella was given a key to the unit on the far end. She backed the car up to the door and got out to open it. Before unlocking the trunk she took the Smith and Wesson .38, the gun she'd used to kill Clay, out of her purse. Before today, when someone needed killing, she'd always paid or manipulated others to do the job. Until she'd fired at the man and seen him fall dead, she hadn't known what a powerful rush it could be. She wouldn't mind feeling that rush again, maybe soon.
The girl was alert but quiet. Stella thrust the pistol toward her face. "Behave yourself and you can come inside. But no tricks, understand?"
She nodded. Stella had parked at an angle to keep the inside of the trunk out the office woman's line of sight. It took some maneuvering, with the back door open, to make it appear that she was getting someone out of the backseat. She blocked the view with the blanket while Erin got out of the trunk. Then, without freeing the girl's ankles, Stella wrapped her in the blanket, jump-walked her inside, and shoved her onto the bed. "Stay," she ordered, taking a moment to step out and close the trunk, then lock the door. "I'll cut your legs loose so you can use the bathroom, but you'll have to do it with your arms taped. And the door stays open. I'm not taking my eyes off you, hear? And don't you make a sound. Right now, I've got a phone call to make."
* * *
It was coming up on midnight when Beau got the call. Will and Tori were on their feet at once, crowding close as they tried to make sense of the half-heard dialogue. Only after he'd ended the call did they get the full story.
"They've found where Erin is," Beau said. "This woman in Eagle Pass, who checked them into a motel, saw the AMBER alert on TV and called it in."
"They're sure it's Erin?" Tori felt faint.
"She said the woman gave a different name, but she had red hair and claimed to have her daughter with her. The girl went inside wrapped in a blanket, so she wasn't seen. But the license plate on the car is a match to Stella's. It's got to be her!"
"Have they got Erin yet?" Will demanded. "What's happening down there?"
"The local police have the place surrounded, but they're holding off, waiting for the FBI hostage negotiator to show up before they move in. They're all hoping the situation won't turn into a shoot-out."
"Lord, we can't just sit here and wait!" Will paced as he talked, his hands clenching into fists. "If we can get there, we might be able to help, or at least let Erin know we're there!"
"But the border's hours away," Tori said. "If we have to drive . . ." She shook her head.
"I know," Beau said. "That's why my friends from the FBI are stopping by to pick us up in the chopper. Get your coats on. We're meeting them at the school athletic field in twenty minutes."
* * *
Using the toilet with her hands taped behind her back was tricky, but after two hours in the motel room, Erin was getting the knack of it, even in the dark. She sat with the bathroom door open, in full view of Stella, who was sitting in a chair with her back to the door and the pistol resting in her lap. Through the drawn curtains the streetlamp outside cast long shadows into the unlit room.
Erin's hopes that the woman would fall asleep, so she could carry out her escape plan, were fading. It was a good plan, she thought—lock the bathroom door, break the water glass, and use one sharp edge to cut through the tape, then unlatch the bathroom window and climb out. She'd seen it done in the movies. There was no reason it couldn't work in real life. But Stella, watching her every move with those cold green eyes, showed no signs of nodding off.
Erin tried not to think about how scared she was. Stella would enjoy seeing her fall apart and cry. But she wouldn't give the red-haired witch the satisfaction. She wouldn't let the woman see the terror that lay like a coiled reptile in the pit of her stomach.
"Aren't you getting hungry?" she asked, sitting on the bed. "Maybe there's a vending machine outside. We could at least get a couple of candy bars."
"Shut up, you little bitch!" Stella snapped. "Nobody's going anywhere till my friend comes to pick you up. After that, you won't be my problem anymore." She stiffened, as if hearing something. One hand moved the window drape aside far enough to see out. "Maybe that's him now. It's—Shit!" She dropped the curtain and sprang to her feet. "It's the cops. One peep out of you, girl, and you're dead." She pointed the gun at Erin. "Lie down on the floor—over there, on the far side of the bed!"
Erin did as she was told, trembling in spite of herself as Stella bound her ankles again, then taped her mouth. Through the curtain she glimpsed headlights moving beyond the window, then nothing. Unable to cry out or get up, she could only lie still and wait.
* * *
Tori clutched her shoulder bag as the helicopter swooped in over the lights of Eagle Pass. FBI agent Forbes, an experienced hostage negotiator, had told them not to bring any weapons. But until now, Tori had forgotten about the nine-millimeter Kel-Tec Will had given her to carry. Her fingertips traced the small pistol's outline through the purse's leather folds. She didn't plan to use it, of course. But it gave her a measure of comfort to know it was there, loaded and ready.
Will sat silent beside her. She could feel the tension in his body, see the determination in his face, and she knew he would do anything to get their daughter back. But he was afraid, too, just as she was. Anything could happen down there, in the dangerous dark.
The helicopter touched down in an empty parking lot, where a local police car was waiting for them. Agent Forbes climbed in front with the driver, leaving Tori, Will, and Beau to crowd into the backseat. They listened while the driver gave Forbes an update.
"It's not looking good. Somebody made a move too soon and the woman got wind of the police. She's threatening to kill the girl if we don't back off."
"Just get me there." Forbes spoke calmly, but Tori could feel her heart in her throat. How many times on the news had she seen hostage situations where both the kidnapper and the hostage ended up dead? Right now, it appeared that Stella was using Erin as a bargaining chip. But if the woman was facing certain capture or death, it would be like her to kill her enemy's daughter as a final act of revenge.
Lights flashing and siren blaring, the police car sped to the scene. They found three other black-and-whites parked with headlights glaring on the door and curtained window of the end motel room. The officer in charge ushered Tori into the backseat of one of the cars, while Will and Beau were given bulletproof vests to wear. No doubt in the officer's mind, a mere woman was best kept out of the way.
Seething with anxiety, Tori rolled down the window and strained to see what was happening. Forbes was speaking through a bullhorn, evidently trying to set up communication with Stella. An officer in protective gear walked to the door and laid a phone on the stoop. As he backed away, the door opened a few inches and the phone disappeared.
Somewhere beyond that door, helpless and terrified, was Erin. It was all Tori could do to keep from leaping out of the car, rushing to the door, and screaming to be let in.
A young policeman, perhaps assigned to keep an eye on her, stood nearby. Tori caught his attention. "What's going on?" she demanded. "That's my daughter in there."
He stepped forward to listen, then reported back. "The woman's demanding half a million dollars in cash and safe passage to the border. Once she's across, she'll release the girl."
"She won't do it," Tori said. "I know her. She'll kill Erin before she lets her go. Tell them that."
"Don't worry, Agent Forbes knows his job," the young officer said. "I've worked with him before. If anybody can get your daughter back, he can."
If anybody can . . . Lord, what if nobody could?
Through the open window she could hear Will arguing vehemently with Forbes. His words tore at her heart.
"Listen to me, damn it! I'm the one Stella wants. Let me go in. Maybe I can trade places with Erin, or at least be inside to protect her."
"That's not the way we do things, Mr. Tyler," Forbes said. "Now stand back and let us do our job."
"Stand back, hell!" Will growled. "I'm going in. If you don't back me, damn it, I'll do it without your help."
As Forbes relented, Tori forced herself to breathe. Will would lay down his life to save Erin. Right now, she had no choice except to let him—even if it meant she could lose them both.
Her frantic hands twisted in her lap. How could she just sit here and wait? There had to be something she could do. Will would be unarmed. But she had a gun. She barely knew how to shoot, but if she could make a difference she had to act—and act now.
With everyone's eyes on Will, Tori took the small pistol out of her purse and pulled back the slide to chamber a bullet. Then she slipped out the far side of the police car. Ducking low and keeping to the shadows, she cut around through the darkness, heading for the back of the motel.
* * *
Erin had rolled onto her belly and managed to inchworm her way across the linoleum floor. Now she was just a few feet short of the door. Stella, gripping her pistol and intent on the danger out front, had yet to notice her. What now? She couldn't use her hands or her feet. She couldn't cry out. But if the chance came to make her move—any move she could—she had to be ready.
As she lay there, tense and waiting, she heard a voice outside—her father's voice.
"Stella! I'm unarmed and I'm coming in! I'm the one you want, not an innocent girl! Take me and let Erin go!"
Stella opened the door a few inches. Lying behind her, Erin could see her father standing in the bright light with his hands up. He was wearing a bulletproof vest, but his head was unprotected. As Stella raised the pistol, Erin knew that was where she would aim. At close range it would be an easy, and fatal, shot.
Erin had to do something.
As Stella tensed to fire, Erin flung the last of her strength into a lightning-fast tuck-and-roll that slammed her curled body into the back of Stella's legs. The pistol roared. Through the partly open doorway, Erin saw her father reel and drop to his knees. A red stain flowed down the sleeve of his jacket.
Stella staggered to one side, caught off balance. Pushing to her knees, Erin head-butted her out of the way, shouldered the door open, and tumbled out onto the stoop.
As the police rushed forward, she heard the door bang shut behind her and the click of the lock. Stella was still inside, but now she was alone.
* * *
Tori had heard the gunshot from the rear of the motel. As she made her way through the overgrown oleander bushes, she could only pray that the single bullet hadn't struck any of her loved ones.
She'd guessed that in an old motel like this one, there would be a bathroom window in the back of each unit. If she could get in that way, she might be able to catch Stella by surprise and rescue Erin. Now she saw that she'd been right about the window. But it was high and appeared to be latched from the inside. Never mind, this plan was her only option. Somehow she would make it work.
She was glancing around for something to climb on when the window slid open. Tori raised her pistol as a dark shape, barely lit by the moon, squeezed out through the opening and dropped five feet to the ground, landing with a grunt of pain.
As the figure pushed to a crouch, the moonlight fell on an upturned face. Eyes as fierce as a cornered puma's glared at Tori. It was Stella.
She'd dropped her pistol as she hit the ground. Keeping her own gun leveled at the woman, Tori kicked the heavy .38 into the bushes, out of reach. Blind rage swept through Tori as she leveled her weapon. "Hands up, Stella!" she snarled. "You don't know how much I'd enjoy pulling this trigger right in your ugly face."
Stella's laugh was pure evil. "You don't have the guts to shoot me, lawyer lady. As soon as I can get my legs under me, I'm going to get up and walk away." She pushed partway to her feet, grimaced, and lowered herself to the ground again. She'd landed hard. Tori guessed that she might have broken something—her ankle or even her leg—but despite the pain Stella kept that cruel smile on her face.
"Too bad about your daughter. She was a pretty little thing. She'd have fetched a good price if she'd lived, but she was dead by the time you showed up. Too bad about your ex-husband, too. One shot, and he went down like a load of bricks. At least you can have the pleasure of giving them both a nice funeral."
Shock and rage blotted out Tori's grief for the moment. It didn't matter what she did now. Without both Erin and Will, her life was over. She'd have nothing to live for. At least killing this evil monster of a woman would give her some satisfaction.
Stella was still grinning when Tori raised the pistol. White-hot fury blurred her vision as her shaking finger tightened on the trigger.
"Stop, Tori!" Beau's arms clasped her from behind, forcing her gun hand down. She struggled, fighting against him, wanting nothing more than to destroy the she-devil who'd taken her loved ones. "Let me go!" she muttered. "She killed Will and Erin! I want to make sure she never hurts anybody again!"
"It's all right, Tori!" Beau's grip tightened. "Stella's lying! Will's only wounded and Erin is free! They're waiting for you out front now."
The pistol dropped from Tori's fingers. Too drained to speak, she began to tremble. Beau, her lifelong friend, laid an arm around her shoulders and guided her away as the police closed in to arrest Stella.
"It's over, Tori," he said. "Will and Erin are waiting for us. Let's go home."
Together they came around to the front of the motel. There, in the glare of headlights, she saw Will sitting up on an ambulance stretcher while a paramedic tended his shoulder wound. Bruised and disheveled, Erin stood in the clasp of his free arm, sobbing as she clung to her father.
Breaking loose from Beau, Tori ran toward them. An instant later, she was holding them close—the two people she loved most in the world. Her family. Whatever happened, she never wanted to be separated from them again.
EPILOGUE I
Tori would remember that Christmas on the ranch as the happiest ever. There hadn't been time or money to put many presents under the tree. But just having the ranch family together had been reason enough for joyous celebration.
The best part of it had been waking up next to Will and seeing his sleepy face on the pillow beside her. They'd been married a few days earlier in a small, private ceremony. This time everything felt right. For Tori, it had taken almost losing her precious daughter and the man she loved to realize that they needed to be a family again. They would be a family forever.
That morning they'd looked outside to find the land blanketed with soft, gently falling snow. While the men trooped out to do chores, the women had gathered to start preparations for Christmas dinner—dressing the turkey, mixing the rolls, and setting the table with the elegant china and silver that had belonged to Will and Beau's mother. They chatted and laughed, enjoying the time together. Lauren, her pregnancy no longer a secret, was planning her wedding. Now that she and Natalie were both expectant mothers, they'd become fast friends.
The exterior of the barn was finished, and there was other good news for the ranch as well. Days ago, a big outfit in Montana, the Triple C, run by the Calder family, had made a generous offer on twenty of the best young colts. The money would pay off the bank loan with enough left over to buy all the winter hay they needed.
When they sat down for dinner, Tori looked around the table, thinking how much she loved everyone there—crusty old Jasper and patient Bernice, Beau and Natalie, Sky and Lauren, her own dear husband, Will, and the daughter they shared. Lives would change as the years wore on. But this day was one to hold and remember for always.
EPILOGUE II
Three months later
Gatesville Women's State Prison, Gatesville, Texas
Dressed in an orange jumpsuit, her hair fading from red to gray, Stella walked into the prison lunchroom, where she would likely be eating for the rest of her life. It wasn't the Ritz. But, thanks to a good lawyer, at least she wasn't spending her days on death row.
She missed the old times, especially the Blue Coyote, which she'd sold to Abner Sweeney at a bargain price to pay for her defense. It tickled her to imagine Abner running the bar. His straightlaced wife had probably thrown a fit. But at least the bar made good money—better than a county lawman's pay.
As a new prisoner here, she was still finding her place in the pecking order. She would keep her head down at first. But once she knew the ropes, she'd be on her way up the food chain, all the way to the top.
Goading Tori Tyler into killing her would have been a mistake. She enjoyed taking on life's challenges, and even here, behind bars, there would be opportunities to win. Stella knew human nature, and she knew how to make her way. Another six months here, and she'd be running the place.
After going through the line, she took her tray, filled with the slop that passed for food, to an empty table and sat down to eat and watch. You could learn a lot about people from who they sat with and the way they ate. She was learning more every day.
She had nearly finished when she heard a footstep behind her. A tall, familiar shadow fell across the table. Stella's pulse jerked. Her gaze traveled upward to the stringy black hair, and the narrow face with its slashing scar.
The crooked mouth smiled. "Hello, Stella," said Marie Fletcher.
_Keep reading for a special preview of Texas Fierce, from_ New York Times _Bestselling Author Janet Dailey._
**He's the prodigal son ready to claim his legacy . . .**
He came home to sell his family's failing ranch, but once twenty-year-old Bull Tyler sets foot on the Rimrock, he's determined to tame the rugged land and make it his own. First he'll have to take on the powerful Prescott clan, who'll do anything to get their hands on the Tylers' holding—even murder. Then Bull sets eyes on the breathtaking woman earmarked to be Ferg Prescott's bride. Now nothing will stop Bull from taking the land—or the lady who stirs his blood like no other—and building a dynasty worthy of both . . .
**She's the pampered beauty he won't let get away . . .**
She was born to privilege, and raised to do the right thing. But Susan Rutledge has never felt anything like the fire she feels for Bull Tyler. Yet can she defy her father's strong will and leave her secure life for a ramblin' rodeo man? She'd have to be crazy—or crazy in love . . .
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## Chapter 1
_Branding Iron, Texas_
_Late November_
Jessica Ramsey mouthed an unladylike curse as her aging Pontiac coughed, sputtered and stopped dead on the deserted two-lane road. Hoping for luck, she cranked the starter—again, then again. Nothing happened.
What now? She couldn't be out of gas. The gauge hadn't worked in months, but she'd filled up two hours ago in Amarillo. Maybe it was the fuel pump. Or worse, something like a blown head gasket, whatever that was.
She cranked the starter one last time. The engine didn't even try to turn over. Fighting tears, she slumped over the steering wheel. She'd trusted the old car to make it all the way from Kansas City to Branding Iron, Texas. It had come close, but not close enough. The green highway sign she'd just passed told her she had fourteen miles to go. It was too far to walk with her suitcase—let alone all her possessions stuffed in the trunk—and she had more sense than to hitchhike. She was stranded.
Glancing in the rearview mirror, she saw a battered-looking red pickup approaching from behind. It was coming fast; and her stalled car, she realized with a lurch of panic, was right in its path. She punched the hazard light, praying it would work. But the truck didn't even slow down. The horn blared. Tires squealed as the pickup swung around her, missing the rear bumper by inches. Jess glimpsed two teenage males in the front seat. Both of them gave her the finger before the truck roared on down the road. So much for chivalry.
Jess released the brake, shifted into neutral and wrenched the steering wheel hard to the right. She had to get the Pontiac off the road before another vehicle came along and crashed right into her. Since the car wouldn't start, her only option was to push it.
After glancing up and down the road, she opened the door, climbed out and walked back to the rear of the car. The sky was overcast with gray. Empty fields of yellow-brown stubble spread on both sides of the road. The flat horizon was broken only by a distant barn and a silo. Jess was a city girl. It was as if she'd set foot on some alien planet, peopled only by distant farms and rude boys in pickups.
The cold November breeze whipped tendrils of russet hair around her face. She clutched her light denim jacket around her ribs. The sooner she got the car off the road, the sooner she could get back inside. Without the engine to run the heater, the car wouldn't stay warm long, but at least she'd be out of the wind.
Bracing her arms above the rear bumper, she planted her sneaker-clad feet on the asphalt. At five-foot-three and a hundred nineteen pounds, Jess was no Wonder Woman. Determination—or more likely desperation—would have to make up for her lack of muscle power.
The road's narrow, graveled shoulder sloped down to a grassy barrow pit. If she could push as much as one front wheel onto the incline, the car's momentum should do the rest. How hard could it be?
Steeling her resolve, she threw her whole weight against the car. Her jaw clenched. Her muscles strained. Nothing moved.
Spent for the moment, she straightened to catch her breath. Maybe she was doing this wrong. It might work better to brace her back against the car and push with her legs. At least it was worth a try.
Jess turned around. Only then did she see the big tan SUV that had pulled up a dozen yards behind her, the lights atop its cab flashing red and blue.
And only then did she see the big, tan person climbing out of it. He strode toward her, a take-charge expression on his face. Wearing a khaki uniform topped by a leather jacket with a sheepskin collar, along with a pistol holstered at one lean hip, he looked capable of lifting her car with one hand. He was also flat-out gorgeous, with dark brown hair, a square-jawed face and stern coppery eyes.
But she wasn't looking for gorgeous here, Jess reminded herself. In her roller-coaster life, the hot-looking men she'd known had turned out to be nothing but bad news. Besides, there was no way a male as spectacular as this long, tall lawman wouldn't have some woman's brand on him.
"Having trouble, Miss?" His drawl was pure Texas honey.
Jess willed herself not to sound like a helpless whiner. "My car broke down. I was about to push it off the road, so nobody would hit it."
A faint smile deepened the dimple in his left cheek. "Could you use some help, or should I just leave you to it?"
"As long as you're here, I guess you might as well give me a hand." Jess spoke through chattering teeth.
"Here." He stripped off his leather jacket and laid it around her shoulders. It was toasty warm. Man warm. Now that he'd taken it off, she could see the badge on his khaki shirt and the name tag below it.
_Sheriff Ben Marsden._
"What seems to be the trouble with the car?" he asked.
"I don't know. It just stopped dead and it won't start. It can't be out of gas. I filled the tank a couple of hours ago."
"Well, let's get it off the road. Then I'll take a quick look under the hood. Maybe it'll be an easy fix."
Ben Marsden was definitely a breed apart from the brusque city cops Jess had encountered. Following his directions, she climbed back into the driver's seat to steer while he pushed. The car rolled forward as if Superman was behind that bumper. No surprise there.
"That's far enough." She heard his voice through the open window. "Now pull the handbrake and pop the hood release."
By the time Jess climbed out of the car he had the hood up and was peering into the Pontiac's dim interior with the aid of a pocket flashlight. After a minute or two, he closed the hood and switched off the flashlight. "I can't see anything wrong," he said. "But it smells like you might have a fuel leak—maybe a broken line. Nothing I can do here, but it shouldn't be too expensive to fix. There's a good, honest mechanic in town. Want me to call him for a tow?"
Jess thought a moment, then reluctantly nodded. She'd promised herself not to break into the fifty thousand dollars she'd inherited from her adoptive father—money she'd set aside for a new start. But the cash she'd saved from her waitressing job was almost gone, and she had to have a working car. For now, she'd put the tow and repair on her credit card and hope for the best.
The sheriff made a quick call on his cell phone, then turned back to her. "Silas is busy right now, but he says he can pick up the car in a couple of hours."
Jess suppressed a sigh. "I suppose I can wait here that long."
He gave her a scowl. "That's not a good idea. Get what you need out of the car and leave the keys under the floor mat. I'll drive you into town. At least we can find you a warm place to wait."
"Thanks." Jess retrieved her purse from the front seat and her suitcase from the trunk. All the way from Kansas City, she'd imagined driving into Branding Iron and carrying out her plan—a plan so audacious that, on the way here, she'd almost lost heart and turned back.
Now she was here. But getting around would have to wait until her car was fixed. She'd need a place to stay. But even a small town like this one should have a cheap motel or some sort of rooming house, where she could crash until she found a job and an apartment—or left town, if things didn't turn out as she'd hoped.
Meanwhile it would be smart to get her hormones under control and stop ogling the hot Texas lawman who'd come to her rescue. The man was off limits—for more reasons than she even wanted to think about—starting with _hot_ and _lawman_.
He opened the door of his SUV and took her suitcase while she climbed in and fastened her seatbelt. The custom dashboard, complete with a police radio, a GPS, a dash cam, and a computer, was impressive. The last time Jess had ridden in a police vehicle, she'd been handcuffed in the back seat. But those days were long behind her. After a few rough patches, she was starting a new life—and part of that new life, she hoped, was waiting right here in Branding Iron.
The engine purred as he pulled back onto the highway. "I don't suppose I should worry about anybody stealing my car," she said.
He chuckled, his dimple deepening. "No I don't suppose you should."
"I'm not hearing much on your radio. Is it always this quiet around here?"
"Pretty much. We get an occasional drug bust, a few bar fights, some domestics, and a runaway kid now and again. That's about it. It's a pretty easy place to be sheriff—most of the time." He glanced at her. His eyes reminded her of old-time home-made root beer, just poured, with the bubbles still sparkling. "I don't believe I caught your name," he said.
"It's Jessica. Jessica Ramsey. But everybody calls me Jess."
"Well, welcome to Branding Iron, Texas, Miss Jess Ramsey. Where do you hail from?"
_Here,_ Jess thought. But was she ready to tell him that? "I drove here from Kansas City," she said. "I was hoping my old beater would make it all the way, but no such luck."
"Were you planning a stopover in town, or just passing through when your car decided to take a vacation?"
Jess gazed out the window a moment. They were passing more fields, some dotted with black Angus cattle and framed by barbed-wire fences. Here and there, a windmill towered above the landscape, its vanes turning in the breeze. The clouds in the vast Texas sky were darkening.
"This isn't just a stopover," she said. "Branding Iron is where I was headed."
"Here?" His laugh was incredulous. "Nobody comes to Branding Iron—unless, maybe, they've got family here."
"Maybe that's what I have." Given that perfect lead-in, Jess decided to tell him her story—at least the important part. As sheriff, he probably knew the townspeople as well as anybody. Maybe he could help her.
"I was born right here in Branding Iron, at the old clinic," she said. "My mother put me up for adoption—I don't know her circumstances, but I'm guessing she was unmarried and in trouble. My adoptive parents were far from perfect. They divorced when I was nine. He disappeared, and she died a few years later. It's been a long, rough road, but a few months ago I decided it was time for a new start." Jess took a breath before getting to the bottom line. "The first thing I wanted to do was find my birth mother."
The sheriff took his time, as if weighing what he'd heard. "That's quite a story," he said. "Did you find her?"
"I think so. I haven't met her, but I'm hoping that's about to change. The private investigator I hired found my mother's name and her address. She's still here in Branding Iron."
"Have you contacted her?" he asked. "Does she know you're coming?"
Jess's hands tightened on her beat-up leather purse. "I was afraid she wouldn't want to see me. That's why I decided to just show up and surprise her."
"Is that wise?"
"Maybe not. But that way, if she slams the door in my face, at least I'll get a look at her. It's important. She's the only real family I've got."
"What if she's married, with children? Maybe she won't want them to know about you."
"I've thought of that," Jess said. "And I wouldn't want to cause her any trouble. But she's still using her maiden name. That could mean she's single or divorced." She turned toward him, straining against the seatbelt. "I'm only telling you this because you might know her. If you do, maybe you can tell me what her situation is and how to approach her—or even arrange a meeting if you think that would be best."
Saying nothing, he guided the SUV around a road-killed rabbit. Two ravens feeding on the carcass flapped skyward against the darkening clouds.
He was quiet for what seemed like a long time. Maybe he suspected Jess of being some kind of con artist, out to win the poor woman over and fleece her of her savings. "I can't promise," he said. "But I'll try to do what's best for both of you. What's your mother's name?"
"Francine. Francine McFadden."
The SUV lurched slightly, crunching gravel on the shoulder of the road before he regained control of the steering wheel. Something about the name had clearly startled him.
"Do you know her?" Jess asked. "You do, don't you?"
"Yup."
"Then you must know where she lives. Can you at least drive me by her house?"
"No need for that. I know for a fact she isn't there."
"Well, where is she?" Jess demanded. "Is she out of town?"
"Nope." He shot her a narrow-eyed glance. "Francine is doing time in the county jail."
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**Janet Dailey's** first book was published in 1976. Since then she has written more than 100 novels and become one of the top-selling female authors in the world, with 300 million copies of her books sold in nineteen languages in ninety-eight countries. She is known for her strong, decisive characters, her extraordinary ability to recreate a time and a place, and her unerring courage to confront important, controversial issues in her stories. You can learn more about Janet Dailey and her books at JanetDailey.com.
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| {
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October 16th-October 22nd is National Veterinary Technician week, and we have a lot to celebrate! Our technicians are superheroes! On a daily basis they care for sick and injured pets, monitor surgical patients, perform periodontal treatments, run an in-house lab, take x-rays, set up appointments, place catheters, perform phlebotomy and much, much more!
For more information on Barcstoberfest click here.
Eastern Animal Hospital invites you to their Summer Splash 2015, Sunday June 14th.
We are excited to announce that we continue to be AAHA accredited! Our hospital was recently evaluated on approximately 900 rigorous veterinary standards of excellence. We are proud to be among the top hospitals in the United States and Canada since 2012! | {
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Clue Chronicles: Fatal Illusion (subtitled Mystery Series: Episode One and alternatively known as Clue Chronicles Episode 1: The Fatal Masque) is a Windows point-and-click adventure game based on the Cluedo franchise, known as Clue in North America. It is a reinterpretation and adaption of the Clue board game as an adventure game including many of the original characters. The game was distributed with a variety of covers, each featuring a different murder weapon.
Development
Two Clue video games were released during the 1990s: Clue, for SNES and Sega Genesis, and Clue: Murder at Boddy Mansion for PC. Clue was "riding a new wave of popularity", and Fatal Illusion was intended to be the first installment of a series of three Clue Chronicles mysteries. However, the series was cancelled when "Hasbro folded and EAI quit the gaming business".
Unlike its predecessors, Clue Chronicles "leaves the board game 'feel' behind and transforms into a murder mystery adventure game". The Boston Herald called it a "grown-up" twist on the franchise's familiar theme. Care was taken in rendering the location backgrounds, "even down to the moving reflections in the shiny hallway floor of the mansion". The game promised "new and soon-to-be-memorable characters", inspiring anticipation that it would expand beyond the original board game. Although the game's voice-over is only available in English, its text was translated into Castilian Spanish. All versions contain three game discs, and a patch was later released to fix its many bugs.
Designer Matthew J. Costello said that he "didn't really become a game designer ... as much as a writer" as he worked on the Clue Chronicles series. Its music was composed by George Alistair Sanger.
A Boston Herald reviewer called the game "difficult" and "frustrating", but said that help is available on the Web. According to the Arlington Heights, Illinois Daily Herald, it "has an online hint system".
Gameplay
Clue Chronicles: Fatal Illusion is a first-person, 3D-perspective, point-and-click adventure game with gameplay typical of the genre, totaling about 20 hours. It "plays like a traditional CD-ROM adventure game with a mystery theme". The game is controlled with a mouse, with its pointer changing when the player can perform an action, grab an object or walk around. Selecting a character calls up a notebook with a list of questions to ask. They are checked off and new topics appear as the conversation progresses. The main menu accesses suspect biographies and game options, saving and loading. Inventory can be manipulated (or inspected) at the bottom of the screen, and a clue option gives novice players hints. Puzzles include riddles and abstract challenges to obtain gems, a maze, and limited inventory-based puzzles. The game requires the player to interview suspects to gain insight into solving puzzles, which reveal vital objects or clues.
Plot
The game is set on New Year's Eve 1938, when the player is invited aboard the Rhine Maiden yacht for a trip to the mountaintop retreat of eccentric and mysterious German millionaire Ian Masque to see his collection of Egyptian artifacts. Masque meets his fate from a poisoned puzzle box, and the player is tasked with finding the murderer. The game's plot includes an Egyptian curse and Nazi spy rings.
The storyline plays out in four acts in three settings (in a castle, aboard the Rhine Maiden and on a cable car). New characters join the original cast of six characters. The game has a cliffhanger ending, which led players to expect a sequel.
Critical reception
The German magazine PC Player gave Clue Chronicles: Fatal Illusion a score of 62 out of 100, praising the reproduction of the "enjoyable short conversations" in the "excellent" German-language edition. According to the review, its puzzles were "varied" and relatively easy; if the player is not an "amateur detective", they could "reject this case". The magazine criticized "the imprecise mouse control, the lengthy walks, as well as the homespun presentation". Computer Games Magazine gave the game a score of 60 out of 100, calling it "average", "[not] particularly enjoyable" but "not painful to play". Metzomagic gave the game 2.5 stars out of 5, calling it "quite an enjoyable little mystery" with entertaining puzzles but criticizing the game's graphical glitches and bugs, shortness and "failure to capitalise on all the potential". Tap-Repeatedly found Fatal Illusion backgrounds and locations "richly detailed" and the puzzles "logical and fun", but the interface was "cumbersome", the music "repetitive" and the voice acting like it was read "from a script into a microphone in the conference room". Giving the game a score of 80 out of 100, the website concluded: "Overall, despite its many flaws, Clue Chronicles: Fatal Illusion was a fun game to play, if too short". Game Power rated it two out of four, noting that the game played as if the creators used Adventure Games For Dummies as a template. The Russian website NQuest gave it a score of 74 out of 100: "With good graphics and a good detective story the game has the best traditions of the genre: the murder, collecting evidence, and questioning suspects". Blogger Do. Lloret de Mar was ambivalent to the game's 3D graphics, animations, music and special effects, noting its "interesting puzzles" and shortness. According to the Cluedo fansite Cluedofan, it was "packed with puzzles" and a "beautifully designed adventure game" with "animated 3D characters" and "fantastic cut scenes". The French website 01Net said that the adaptation of the Cluedo board game was "much more successful" than Hasbro's "first failed attempt"; another French website, Pabbajita, called it a "very conventional game". GameFreaks liked the game's concept, but not its execution. Adventure Point gave Clue Chronicles: Fatal Illusion 3.5 stars, calling it "a surprisingly enjoyable, if short, detective game with some decent puzzles". According to the Boston Herald, the "visually rewarding" game had "glossy animation" and "detailed, sepia-toned interiors" but its compelling characters "move ... rather like puppets". The Herald later recommended the game in "Games fit everyone's gift list", calling it "convoluted" and perfect for "mystery lovers".
Bob Mandel of the Adrenaline Vault described the game as a "mystery novel with tons of unexpected twists and turns", calling it "a noble and clever expansion" which "involves much more penetrating, complex, and, ultimately, riveting detective work". Jenny Guenther of Just Adventure wrote that the board-game adaption exceeded expectations; the story was "surprisingly tightly plotted", and she graded the "logical", "fun" and "short" game a C. The Hungarian review site PlayDome praised the game's surroundings, lack of simple solutions and general excitement. However, the Dutch review site Adventure Island felt that the game had drifted from the Cluedo brand. Quandary gave Clue Chronicles: Fatal Illusion a mixed review, praising its playability while noting its shortcomings and calling the game a "short diversionary trip for mystery fans". GameSpot was disappointed by the wasted opportunity for the board game to be adapted into a "fun, engaging computer adventure", with "low production values and unimaginative design". GameSpot UK game the game a rating of 4.3 out of 10, and Computer Gaming Magazine rated it three out of five. Gateway deemed it a "charming diversion". Pelit called it a mediocre example of the adventure video-game genre, giving it a score of 76 out of 100. The Daily Herald called the game a "clunky mystery", noting that it stood out from other mystery titles aimed at younger players by showing dead bodies: "[It] has a fine opening, the music is first rate ... and it keeps you interested in spite of its copious limitations" such as loading at a "geologic" pace, its many bugs and a sub-par instruction manual. Suggesting that the game should have been kept under wraps for a few more months of developmental fine-tuning, the reviewer concluded that "mystery fans may be able to forgive its stilted clunkiness, but let's hope that Episode II is a lot better". The magazine PC Direct criticised the "sluggish" gameplay, but praised an "excellent" characterisation and "lavish" 3D graphics.
The Inn at Death's Door
The Inn at Death's Door was a four-part, free online mystery prequel to Fatal Illusion. The text-based detective mystery mini-series' chapter titles were "Episode One: The Magic Box"; "Episode Two: Chasing the Fox"; "Episode Three: One Death Less, One Death More", and "Episode Four: Race for the Truth". Its premise was "an assortment of eccentric guests gather[ing] at a Swiss inn", wherein a series of mysteries take place. Mystery website Mysterynet.com held a Clue Chronicles Mystery Weekend Contest from November 22 to December 3, 1999, sponsored by site owner Newfront Productions. The contest asked participants to read the four-part story, correctly answer four questions related to the mystery on the online entry form and explain how they solved the case in 500 words or less. The judges, Newfront and MysteryNet staff, selected a winner on January 15, 2000. The prize was a Murder Mystery Weekend for Two at the Austin Hill Inn in West Dover, Vermont. A judges' note accompanied the winning entry:
References
External links
PC Game review
Just Adventure review
Review in The Complete Sourcebook on Children's Literature
Game manual
Tape logs, Audio recordings, and Geoportal video, that can be accessed via the George Sanger Papers
Cluedo
2000 video games
Adventure games
Detective video games
Point-and-click adventure games
Video games scored by George Sanger
Video games developed in the United States
Windows games
Windows-only games
EAI Interactive games
Single-player video games | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,689 |
NBC Club
NBC on Fanpop
20 years назад NBC banned Saturday Morning Мультики because of F.C.C.
posted by BuddyBoy600alt
Well in 1995, It was a very sad год for Saturday Morning cartoons. NBC was forced to banned Saturday Morning cartoons. It was demanded by the F.C.C. And that means no еще Disney's Raw Toonage / Disney's Marsupilami, No еще Alvin and the Chipmunks, No еще Earthworm Jim, No еще Super Mario Bros., and many more. That means NBC had to replace it with the Saturday edition of The Today Показать and new live action teen sitcom called "TNBC". And that means Дисней went on to CBS-TV for their shows (such as The Lion King's Timon and Pumbaa).
R.I.P. NBC Saturday Morning Мультики (1948-1995) :'(
Disney's Marsupilami - Mars' Problem Pachyderm
added by BuddyBoy600alt
Be cool with Back to School.
walt Дисней
steve mackall
Disney's Marsupilami - Bathtime for Maurice (Raw Toonage)
Everybody must bath everyday. Even a gorilla like Maurice should bath as well.
jim cummings
The пляж, пляжный Boys on Hullabaloo - Dance Dance Dance
Here is the пляж, пляжный Boys doing "Dance Dance Dance" on NBC-TV's Hullabaloo.
Alvin and the Chipmunks - Funky Town
Another classic song from Alvin and The Chipmunks.
lipps
The Rolling Stones on Hullabaloo - (I Can't Get No) Satisfaction
Here is The Rolling Stones performing their famous hit.
The Super Mario Bros. Super Show! - Stars in their Eyes
Featuring The Beatles song "Chain". And those green creatures are tooting the Zelda song with their nose.
captain lou albano
danny wells
Disney's Raw Toonage Название with NBC logo
Source: The Walt Дисней Company / NBC-TV
Cappelli and Company
Source: A
Disney's Marsupilami - Hey! Hey! They're the Monkeys!
Marsupilami has a hard time babysitting those three baby monkeys. The three baby monkeys are voiced by the late Jim Thurman (best known for Teeny Little Super Guy from Sesame Street).
mackall
Masi Oka
The Adventures of Rocky and Bullwinkle and Друзья Intro
This is the Opening to the Популярное NBC Saturday Morning kids Показать called "The Adventures of Rocky and Bullwinkle and Friends" On ABC, It was called "The Bullwinkle Show" and in Syndication, It was called "The Rocky Show". Created by сойка, джей Ward.
foray
Raw Toonage opening for episode 11
From episode 11 of Raw Toonage. Marsupilami teaches Maurice self-defenses.
Saturday Mornings on NBC (Classic Memories)
I miss those old days when NBC had good shows for Saturday Mornings. Here some of my favorites:
* Disney's Raw Toonage - Raw Toonage was a Дисней animated series. It was basically a variety Показать with a Болталка Дисней character as a host. And it features three shorts: Bonkers, Totally Tasteless Video, and my personal favorite, Marsupilami. Marsupilami is a yellow, spotted, long-tail marsupial who lives in a jungle with his gorilla friend: Maurice. And I Любовь Marsupilami еще than Bonkers. Both Bonkers and Marsupilami were spun-off.
* Camp Конфеты - An animated series starring the late John Candy...
The Adventures of Super Mario Bros. 3 - Girls Just Want to Have Fun
From the NBC Saturday Morning cartoon series: The Adventures of Super Mario Bros. 3. This song was cut from the DVD release due to copyright issues.
Disney's Marsupilami - Hole in Mars
Want to play golf with Marsupilami and Maurice?
walt Дисней Телевидение Анимация
nbc saturday morning Мультики
The Beatles on Hullabaloo - Ticket to Ride
The Beatles performing a song from Help!
james paul mccartney
richard starkey
The Beatles on Hullabaloo - I Want To Hold Your Hand
The Beatles appearing on NBC's Hullabaloo, Performing the classic "I Want To Hold Ты Hand".
The розовый пантера Показать - Hook, Line and Pinker
This used to be on NBC Saturday Mornings. Who remembers The розовый Panther? The voice of The розовый пантера is Matt Frewer (AKA Max Headroom).
the розовый пантера
matt frewer
The Beatles on Hullabaloo - She Loves Ты
Here they are, Performing a classic hit on Hullabaloo.
NBC Peacock on Disney's Marsupilami
The NBC Club
NBC Wall
NBC Updates
NBC Images
NBC Videos
NBC Articles
NBC Links
NBC Forum
NBC Polls
NBC Quiz
NBC Answers
NBC Fans | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,963 |
Baltijos Futbolo Akademija ist ein litauischer Fußballverein aus Vilnius. Der junge Verein spielt derzeit in der 1 Lyga, der zweithöchsten litauischen Liga.
Geschichte
Der Verein wurde 2007 unter dem Namen Baltijos Futbolo Akademija gegründet. Verein spielt derzeit in der 1 Lyga ab 2019. Die Lizenz von Pirma lyga 2019 wurde an BFA vergeben, als in 1 Liga wurde Team mit FK Vilnius zusammengeführt. Nach der Saison 2019 endete die Partnerschaft. Die Lizenz von Pirma lyga 2020 wurde an BFA vergeben.
Platzierungen (seit 2018)
Farben und Trikot
Erste Mannschaft und Trainer
Seit 2022.
Stand: 17. Mai 2022
|-----
! colspan="9" style="background:#B0D3FB" style="text-align:left"|
|----- style="background:#DFEDFD"
|-----
! colspan="9" style="background:#B0D3FB" style="text-align:left"|
|----- style="background:#DFEDFD"
|-----
! colspan="9" style="background:#B0D3FB" style="text-align:left"|
|----- style="background:#DFEDFD"
Trainer
Domas Paulauskas (2018)
Igoris Morinas (2019)
Ivan Švabovič (2020)
Andžėj Falčik (2021)
Haroldas Šidlauskas (2022–)
Weblinks
Offizielle Homepage
Pirma lyga (1lyga.lt)
Soccerway
lietuvosfutbolas.lt (BFA)
Globalsportsarchive
Einzelnachweise
Litauischer Fußballverein
Fußball (Vilnius)
Gegründet 2007 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,101 |
package oci
import (
"context"
"github.com/containerd/containerd/containers"
specs "github.com/opencontainers/runtime-spec/specs-go"
)
// Spec is a type alias to the OCI runtime spec to allow third part SpecOpts
// to be created without the "issues" with go vendoring and package imports
type Spec = specs.Spec
// GenerateSpec will generate a default spec from the provided image
// for use as a containerd container
func GenerateSpec(ctx context.Context, client Client, c *containers.Container, opts ...SpecOpts) (*Spec, error) {
s, err := createDefaultSpec(ctx, c.ID)
if err != nil {
return nil, err
}
for _, o := range opts {
if err := o(ctx, client, c, s); err != nil {
return nil, err
}
}
return s, nil
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,270 |
Naim Dangoor, businessman and philantropist, Died at 101
Business, Centenarian
Sir Naim Eliahou Dangoor was born on April 1914, and died on November 19, 2015.
He was a British-naturalised Iraqi-born Jewish entrepreneur and philanthropist.
Eliahou was born in Baghdad in April 1914, the second of six siblings.
His father Eliahou Dangoor was the world's largest printers of Arabic books, and his grandfather Hakham Ezra Reuben Dangoor was the Chief Rabbi of Baghdad.
In the 1930s Eliahou made the five day journey from Baghdad to London, at the age of 17, in order to enrol in an engineering degree at the University of London.
After graduating, Eliahou returned to Iraq, where he was conscripted into the army and became an officer.
It was during his army training that he met his future business partner Ahmed Safwat.
Initially on leaving the army, Eliahou had hoped to become an engineer on the railways, but due to restrictions imposed upon Jews this was not possible, so he and Ahmed, a Muslim, decided to go into business together, setting up Eastern Industries in 1949.
Their first contract was to supply new windows to all Iraqi government buildings, and soon their portfolio grew to include property development and letting.
In 1950, Eastern industries secured the first contract to bottle Coca Cola in Iraq.
Alongside Eastern Industries, Sir Naim also ran factories producing matches and furniture.
In November 1947, Eliahou married Chinese-born Renée Dangoor (who had been crowned Miss Baghdad earlier that year). They went on to have four sons.
With the rise of the Ba'ath party in Iraq, the situation for Jews in the country worsened and in 1959 Eliahou took the difficult decision to take his family out.
Eliahou continued travelling back and forth for business until 1963, when he decided that the worsening political situation made it too risky for him to return.
As a result of laws specific to Jews, Eliahou lost his Iraqi citizenship, and his property and business interests were taken by the government.
Eliahou was eventually permitted to settle in the UK where he set about rebuilding his life, setting up a property business that his four sons went on to join.
Sir Naim Eliahou passed away at age 101 in November 2015.
<< Kitanoumi Toshimitsu, Japanese sumo wrester, Died at 62 Lauri Vaska, American chemist, Died at 90 >> | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,249 |
\section{Introduction}
A long-standing conjecture by Sen \cite{Sen:1999mh,Sen:1999xm}
states that, at the stationary point of the tachyon potential on a
D25-brane of open bosonic string theory, the negative energy
density exactly cancels the tension of the D25-brane. The tachyon
potential in Witten's cubic open string field theory
\cite{Witten:1985cc} has been computed and numerical evidence for
Sen's conjecture was given by an approximation scheme called level
truncation
\cite{Kostelecky:1988ta,Kostelecky:1989nt,Kostelecky:1995qk,Sen:1999nx,Moeller:2000xv,Taylor:2002fy,Gaiotto:2002wy}.
The action for open bosonic string field theory is
\begin{eqnarray}
\label{accionx1} S=-\frac{1}{g^2}\Big[ \frac{1}{2} \langle
\Phi,Q_B \Phi \rangle +\frac{1}{3} \langle \Phi,\Phi*\Phi \rangle
\Big] \, ,
\end{eqnarray}
where $Q_B$ is the BRST operator of bosonic string theory,
$*$ stands for Witten's star product, and the inner product $\langle \cdot , \cdot \rangle $ is the
standard BPZ inner product. The string field $\Phi$ belongs to the
full Hilbert space of the first-quantized open string theory.
According to Sen's conjecture, the classical open string field
equation of motion
\begin{eqnarray}
\label{equamo1} Q_B \Phi + \Phi*\Phi =0 \,
\end{eqnarray}
should admit a Poincar\'{e} invariant solution $\Phi\equiv\Psi$
corresponding to the condensation of the open-string tachyon to
the vacuum with no D25-branes. This statement means that the
energy density of the true vacuum found by solving the equation of
motion should be equal to minus the tension of the D25-brane.
Since the energy density of a static configuration is minus the
action, Sen's conjecture can be summarized as follows
\begin{eqnarray}
\label{poten1} \frac{1}{g^2}\Big[\frac{1}{2} \langle \Psi,Q_B \Psi
\rangle +\frac{1}{3} \langle \Psi,\Psi*\Psi \rangle \Big]= -
\frac{1}{2 \pi^2 g^2} \, .
\end{eqnarray}
The string field equation of motion and Sen's conjecture allow us
to fix the kinetic and cubic terms,
\begin{eqnarray}
\label{kinec1} \frac{\pi^2}{3} \langle \Psi,Q_B \Psi \rangle &=&-1 \, ,\\
\label{cubic1} \frac{\pi^2}{3} \langle \Psi,\Psi*\Psi \rangle &=&1
\, .
\end{eqnarray}
Recently, Schnabl \cite{Schnabl:2005gv} found an analytic solution
to the string field equation of motion, and it was subsequently
shown that his solution represents the nonperturbative tachyon
vacuum
\cite{Ellwood:2006ba,Ellwood:2008jh,Kawano:2008ry,Kawano:2008jv,Kiermaier:2008qu,Kishimoto:2009cz,Okawa:2006vm,Fuchs:2006hw,Takahashi:2007du}.
There are two ways of writing Schnabl's analytic solution; the
first way is in terms of Bernoulli numbers $B_n$,
\begin{eqnarray}
\label{bernu1} \Psi &=& \sum_{n,p} f_{n,p} ({\cal L}_0 + {\cal
L}_0^\dagger)^n \tilde c_p |0\rangle + \sum_{n,p,q} f_{n,p,q}
({\cal B}_0 + {\cal B}_0^\dagger) ({\cal L}_0 + {\cal
L}_0^\dagger)^n \tilde c_p \tilde c_q |0\rangle \, , \\
f_{n, p} &=& \frac{1-(-1)^p}{2}\frac{\pi^{-p}}{2^{n - 2 p + 1}}
\frac{1}{n!} (-1)^
n B_{n - p + 1} \, , \\
f_{n, p, q} &=& \frac{1-(-1)^{p + q}}{2} \frac{\pi^{-p - q}}{2^{n
- 2 (p + q) + 3}} \frac{1}{n!} (-1)^{n -q} B_{n - p - q + 2} \, ,
\end{eqnarray}
whereas the second is in terms of wedge states with ghost
insertions,
\begin{eqnarray}
\label{wedge1} \Psi &=& \lim_{N \rightarrow \infty} \Big[ \psi_N-
\sum_{n=0}^{N}
\partial_n \psi_n \Big] \; , \\
\psi_n &=& \frac{2}{\pi^2} U^\dag_{n+2}U_{n+2} \big[
(\mathcal{B}_0+\mathcal{B}^\dag_0)\tilde c(-\frac{\pi}{4}n)\tilde
c(\frac{\pi}{4}n) +\frac{\pi}{2} (\tilde c(-\frac{\pi}{4}n) +
\tilde c(\frac{\pi}{4}n)) \big] | 0\rangle \, ,
\end{eqnarray}
where $\psi_N$ with $N\rightarrow\infty$ is called the phantom
term
\cite{Okawa:2006vm,Fuchs:2006hw,Takahashi:2007du,Erler:2007xt,Aref'eva:2009ac}.
Schnabl's analytic solution was used to prove Sen's conjecture
(\ref{poten1}). Nevertheless there were subtleties involved in the
proof. For instance in a series of two subsequent papers
\cite{Okawa:2006vm,Fuchs:2006hw} it has been argued that the
validity of Schnabl's solution requires that the string field
equation of motion be satisfied when it is contracted with the
solution itself. This requirement was verified by computing the
cubic term (\ref{cubic1}) using Schnabl's solution in terms of
wedge states with ghost insertions (\ref{wedge1}). Further
numerical evidence for this result was given in
\cite{Takahashi:2007du}, where the cubic term was evaluated by
using level-truncation computations, i.e., by employing Schnabl's
solution written in the usual Virasoro basis.
In this work we use Schnabl's solution written in terms of
Bernoulli numbers (\ref{bernu1}) to provide new evidence that the
cubic term has the expected value (\ref{cubic1}) predicted from
the equation of motion and Sen's conjecture. We evaluate the cubic
term using Pad\'{e} approximants \cite{AldoArroyo:2008zm,pade}, in
analogy with the computation of the kinetic term (\ref{kinec1})
performed in \cite{Schnabl:2005gv,tedschnabl}. We confirm the
expected value of the cubic term required for the string field
equation of motion to be satisfied when contracted with the
solution itself.
This paper is organized as follows. In section 2, we evaluate the
cubic term in the action of open bosonic string field theory using
Schnabl's solution written in terms of Bernoulli numbers. Here we
use Pad\'{e} approximants to describe how to obtain the expected
value of the cubic term. A summary and further directions of
exploration are given in section 3. Some details of our
calculations such as the evaluation of correlation functions in
the ${\cal L}_0$ basis and explicit Pad\'{e} approximants
computations are given in the appendices.
\section{Evaluating the cubic term}
In this section, instead of using the representation of the
solution in terms of wedge states with ghost insertions
(\ref{wedge1}) or using the solution written in the usual Virasoro
basis \cite{Takahashi:2007du}, we evaluate the cubic term using
the solution written in terms of Bernoulli numbers (\ref{bernu1}).
The computations shown in this section are similar to those in
\cite{Schnabl:2005gv,tedschnabl}, where the kinetic term was
evaluated by using the solution written in the ${\cal L}_0$ basis,
and the expected value (\ref{kinec1}) was reproduced by means of
Pad\'{e} Approximants \cite{AldoArroyo:2008zm,pade}.
As described in \cite{Schnabl:2005gv,tedschnabl}, we start by
replacing the solution $\Psi$ with $z^{{\cal L}_0}\Psi$ in the
${\cal L}_0$ level truncation scheme, so that states in the ${\cal
L}_0$ level-expansion of the solution will acquire different
integer powers of $z$ at different levels. As we are going to see,
the parameter $z$ is needed because we need to express the cubic
term as a formal power series expansion if we want to use Pad\'{e}
approximants. After doing our calculations, we will simply set
$z=1$.
Let us start with the evaluation of the cubic term as a formal
power series expansion in $z$. Plugging the solution
(\ref{bernu1}) into the cubic term and using the correlation
functions derived in appendix A we obtain
\begin{align}
\label{expansion1} \langle \Psi ,z^{{\cal L}_0^\dag}(z^{{\cal
L}_0}\Psi)*(z^{{\cal L}_0}\Psi) \rangle & = \frac{81 \sqrt{3}}{8
\pi^3}\frac{1}{z^3}+\Big[ -\frac{81 \sqrt{3}}{8 \pi^3} +
\frac{27}{8 \pi^2} \Big]\frac{1}{z^2} + \Big[ \frac{9 \sqrt{3}}{4
\pi^3} -\frac{3}{2 \pi^2} -\frac{\sqrt{3}}{24 \pi}
\Big]\frac{1}{z} \nonumber \\
&+ \Big[ \frac{1}{180} -\frac{13 \pi}{9720 \sqrt{3}}\Big] z +
\Big[ \frac{1}{270} -\frac{\pi}{1215 \sqrt{3}} -
\frac{\pi^2}{21870}
\Big] z^2 \nonumber \\
&+ \Big[ \frac{5}{4536} +\frac{263 \pi}{1224720 \sqrt{3}} +
\frac{71 \pi^2}{393660} - \frac{59 \pi^3}{8266860 \sqrt{3}}\Big]
z^3 \nonumber \\
&+ \Big[ -\frac{1}{5670} +\frac{113 \pi}{183708 \sqrt{3}} +
\frac{40 \pi^2}{137781} - \frac{8 \pi^3}{413343 \sqrt{3}} -
\frac{5 \pi^4}{11160261}\Big]
z^4 \nonumber \\
& + \, \cdots \, .
\end{align}
At this point we remark that the most cumbersome of our
computations are the evaluation of correlation functions which
come from plugging the solution (\ref{bernu1}) into the cubic
term, the details of these computations are shown in appendix A.
Once the respective correlation functions are computed, in
principle it should be possible to write the series
(\ref{expansion1}) to any order in powers of $z$. Nevertheless,
the time it takes to do those calculations increases considerably
with every subsequent power of $z$. Given the formal power series
expansion (\ref{expansion1}), we are able to evaluate the cubic
term using Pad\'{e} approximants. We match the power series
expansion coefficients of a given rational function $P_{3+M}^N(z)$
with those of the cubic term (\ref{expansion1}). The details of
these computations can be found in appendix B.
The main result of our work is summarized in table \ref{results1}.
The first column is the definition of the cubic term in the ${\cal
L}_0$ level truncation. As we can see in the second column, the
value of the cubic term computed using Pad\'{e} approximants
converges to the expected value (\ref{cubic1}). We note that the
value of the cubic term for $n$ greater than 8 shown in the first
column has an oscillating behavior. Let us mention that a series
may diverge either by approaching infinity or by oscillating. An
example of a divergent series that diverges by going to infinity
is the series corresponding to the kinetic term
\cite{Schnabl:2005gv,tedschnabl}. It seems that in the case of the
cubic term, the divergent character of the series is due to its
oscillating behavior, which would be interesting to verify by
performing higher level computations. Since Pad\'{e} approximants
can deal numerically with divergent series
\cite{AldoArroyo:2008zm,pade} we have shown by explicit
computations that our results confirm the expected value of the
cubic term (\ref{cubic1}).
\begin{table}[ht]
\caption{The Pad\'{e} approximation for the normalized value of
the cubic term $\frac{\pi^2}{3} \langle \Psi, z^{{\cal
L}_0^\dag}(z^{{\cal L}_0}\Psi)*(z^{{\cal L}_0}\Psi) \rangle $
evaluated at $z=1$. The first column is a naive evaluation of the
cubic term given by the series (\ref{expansion1}), and the second
column is its respective $P^{n/2}_{3+n/2}$ Pad\'{e} approximation.
The label $n$ corresponds to the power of $z$ in the series
(\ref{expansion1}). At each stage of our computations we truncate
the series up to the order $z^{n-3}$.} \centering
\begin{tabular}{|c|c|c|}
\hline
& Naive computation & $P^{n/2}_{3+n/2}$ Pad\'{e} approximation \\
\hline
$n=0$ & 1.86073502 & 1.86073502 \\
\hline $n=2$ & 0.96292169 & 0.91712884 \\
\hline $n=4$ & 0.97321797 & 0.97620455 \\
\hline $n=6$ & 0.98935043 & 0.97396938 \\
\hline $n=8$ & 1.00598343 & 1.00413934 \\
\hline $n=10$ & 1.00170926 & 1.00519420 \\
\hline $n=12$ & 0.99478828 & 1.00021592 \\
\hline $n=14$ & 1.00416903 & 1.00010061 \\
\hline $n=16$ & 1.00223124 & 1.00016672 \\
\hline $n=18$ & 0.99433556 & 0.99997863 \\
\hline $n=20$ & 1.00911757 & 0.99998242 \\
\hline
\end{tabular}
\label{results1}
\end{table}
\section{Summary and discussion}
We computed the cubic term in the ${\cal L}_0$ level-truncation
scheme \cite{Schnabl:2005gv,tedschnabl}, and we provided new
evidence for the fact that Schnabl's tachyon solution of open
bosonic string field theory is valid in the sense that it solves
the equation of motion when it is contracted with the solution
itself.
Up to the level that we explored with our computations, it is worth remarking
that the series that defines the cubic term
(\ref{expansion1}) seems to have an oscillating behavior. This
character of the series is in contrast with the character of the
series for the case of the kinetic term which does not begin to
diverge until higher levels, where computations reveal it starts to go to infinity
\cite{Schnabl:2005gv,tedschnabl}. In the case of the cubic term, we could perform
higher level computations to confirm the oscillating behavior of the series. We hope that the
approach used in \cite{tedschnabl} when applied to the case of the
cubic term will help to clarify this issue.
A direct application of the results shown in this paper is related
to the study of level-truncation computations in the ${\cal L}_0$
basis. In this basis, the analytic solution found by Schnabl was
originally obtained by truncating the equation of motion but not
the string field, so it would be interesting to analyze the case
when we truncate the string field instead of the equation of
motion. This analysis should serve us to address some issues, e.g., the computation of the effective tachyon potential in
Schnabl's gauge.
A second application would be the extension of our methods to the
case of the Berkovits superstring field theory
\cite{Berkovits:1995ab}. In this formalism, we already have a
solution for the tachyon condensate written in the ${\cal L}_0$
basis \cite{aldo2}. Obviously, the next step would be the evaluation
of the energy. We hope that Pad\'{e} approximants will confirm the
expected value predicted from D-brane arguments
\cite{Bagchi:2008et}.
\section*{Acknowledgements}
I would like to thank Nathan Berkovits, Ted
Erler and Martin Schnabl for useful discussions. I also wish to thank Diany Ruby,
who proofread the manuscript. This work is supported by CNPq grant 150051/2009-3.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,471 |
About the Children's Department: Top Picks for VBS 2012 . . .
Top Picks for VBS 2012 . . .
So, please plan to join me as I share my VBS reviews and consider which resources may be a good "fit" for your ministry. Keep in mind, there are many ways to utilize VBS beyond the traditional five-day format. You may find the perfect resource for your Summer Sundays, or mid-week ministry or you may find something which would be a great "fit" for a summer of family fun days or a mission conference resource amongst my VBS reviews! In addition to the reviews on January 13th I'll also release my side-by-side comparison chart of all the vbs resources I reviewed. January 16 - 20 I'll post the reviews for the additional kits I reviewed beyond my Top Ten, so during the month of January on About the Children's Department you will have the opportunity to learn about some excellent resources for children's ministry! And one more thing . . . I have a number of VBS items to give away during my reviews, so be sure to check back each day! | {
"redpajama_set_name": "RedPajamaC4"
} | 5,953 |
Q: MIPS critical path of a branch instruction Suppose I have a MIPS processor performing a beq instruction e.g. beq $t1, $t2, label
I'm trying to figure out the critical path of this instruction. Suppose every component has the same delay. My idea for the critical path is as follows:
Path 1: Add -> ............ -> Add -> Mux
Path 2: IM -> Control -> Mux -> ALU -> Mux
Path 3: IM -> Registers -> Mux -> ALU -> Mux
How do I go about filling in the rest of Path 1 (or other paths if they're wrong) considering it has to wait for some signals to be ready?
Also is there any reason why path 3 would not be the critical path if we assume the delay of all components is the same (and ignore the gate delays)?
A: If you want to be able to compare paths, pick ONE point that all the paths of interest must go through, and start your counting from there in each path. In your case the simplest and most obvious one point is probably the PC latch, as that's the leftmost thing on your diagram. The paths will all be loops that start at that point, go through various blocks and all finally end up back at the same point for the next instruction.
If a path has to wait on singals from some other path, then it's not the critical path -- the path that it is waiting on is more critical. The (most) critical path will be the one that doesn't have to wait for any signals from other paths as those other signals will always be there before the critical path gets there.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 582 |
package org.caleydo.view.pathway.v2.ui.augmentation;
import gleem.linalg.Vec2f;
import org.caleydo.core.util.color.Color;
import org.caleydo.core.view.opengl.layout2.GLGraphics;
import org.caleydo.core.view.opengl.layout2.geom.Rect;
import org.caleydo.datadomain.pathway.graph.item.vertex.EPathwayVertexType;
import org.caleydo.datadomain.pathway.graph.item.vertex.PathwayVertexRep;
import org.caleydo.view.pathway.v2.ui.APathwayElementRepresentation;
/**
* Augments a pathway vertex by a small triangle if that vertex represents more than one gene/protein.
*
* @author Christian
*
*/
public class MultiMappingIndicatorAugmentation extends APerVertexAugmentation {
protected static final Color MULTI_MAPPING_INDICATOR_COLOR = new Color(0.3f, 0.3f, 0.3f, 1f);
public MultiMappingIndicatorAugmentation(APathwayElementRepresentation pathwayRepresentation) {
super(pathwayRepresentation);
}
@Override
protected void renderVertexAugmentation(GLGraphics g, float w, float h, PathwayVertexRep vertexRep, Rect bounds) {
if (vertexRep.getType() == EPathwayVertexType.gene && vertexRep.getPathwayVertices().size() > 1) {
float size = bounds.height() * 0.5f;
g.color(MULTI_MAPPING_INDICATOR_COLOR).fillPolygon(new Vec2f(bounds.x(), bounds.y()),
new Vec2f(bounds.x() + size, bounds.y()), new Vec2f(bounds.x(), bounds.y() + size));
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,320 |
Q: How to retrieve text from webview I am Having an HTML file in the asset folder. The following is the way I am loading to view the HTML content. I am having a text view. Now I want to retrieve the HTML content in the string and display that in a text view using text.settext(). Please suggest me how to retrieve the content in a string.
view.loadUrl("file:///android_asset/www/index.html");
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,075 |
Segment Type
Feminist and Writer Slavenka Drakulic.
Yugoslavian journalist, critic, and feminist Slavenka Drakulic (SLA-vinka DRA-coo-lich). Her new book of essays is "How We Survived Communism and Even Laughed," (published by Norton & Co.)
Adrienne Rich Reads Her Work.
Poet Adrienne Rich reads selections of her work.
Virginia Capers On Acting and Singing.
Actress and singer Virginia Capers discusses her career and upcoming movie "North Avenue Irregulars."
The Politics of Drag.
Art historian, playwright, and critic MARTHA KEARNS and writer EDMUND WHITE offer their opinions on the topic "The Politics of Drag," debate-style.
Ntozake Shange at the "Nappy Edges."
African-American poet Ntozake Shange tells Terry Gross about how Black liberation, women's movements and jazz have influenced her work. She has a new collection of poetry called Nappy Edges.
A Feminist Perspective on Journalism
Deirdre English is an investigative journalist for Mother Jones magazine. She discusses her problems with the film Apocalypse Now, violence against women, pornography and the cultural impact of feminism.
Socially Conscious Poetry and Pedagogy
Poet and playwright Sonia Sanchez discusses teaching young writers, her experiences as an African American woman, and her appreciation and critique of white feminist movements.
Prostitution, Pornography, and Working Conditions for Women
Selma James and Wilmette Brown are members of the group Call Off Your Old Tired Ethics (COYOTE), which advocates for the decriminalization of prostitution and protections of sex workers. This is part one of their interview.
Linda Lovelace Reveals the Dark Side of Pornography
Actress Linda Lovelace discusses her terrifying history of sexual abuse, assault and exploitation--often at the hands of her ex-husband Chuck Traynor--which led to her to star in pornographic films like the iconic Deep Throat. Her new memoir is called Ordeal.
The State of Charities and Philanthropic Coalitions
The United Way's Southeastern Pennsylvania branch recently rejected Women's Way petition for membership because of the latter's support for abortion rights and services. In light of this development, representatives of the three philanthropic organizations explain to host Terry Gross and Fresh Air callers different models of fundraising for charities and other groups working for social change. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,068 |
este al treizeci si cincilea film din seriile Doraemon. A fost lansat în Japonia pe 8 august 2014.
Muzică
Coloana sonoră
Premii
Referințe
Legături externe
Site oficial
Filme din 2014
Doraemon | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,715 |
Q: What is "flush mount socket" in this 40yr old Tek scope? EEVblog Dave shows us flush mount socketed IC's in this Tektronix 213 Portable Oscilloscope teardown, but it is hard to see and understand what exactly is a flush mount socket. Can someone explain how it is done and why?
The video at 7'46" and a picture:
A: Large diameter holes in the PCB with individual socket pins inserted into each, then soldered. (Either press-fit by an auto-insert machine or held together with a temporary carrier or substrate, removed after soldering. I remember seeing them on a flexible plastic strip, at 0.1 inch pitch). Essentially they use the PCB itself as the body of the IC socket.
As to why? ...
why a socket? To facilitate replacing an unusually expensive IC (look at that gold/ceramic package!) without the possible damage from soldering. Either the IC was highly stressed and could overheat, or it was unusually performance critical (matched transistors) and could drift as it aged.
why flush mount? Possibly for best HF performance : most application areas didn't pay that much attention to lead inductance or capacitance back then, but certain parts of an oscilloscope are more critical than most areas. Less likely: to keep the assembly height down!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,607 |
var assert = require('assert');
var listXf = require('./helpers/listXf');
var R = require('..');
describe('all', function() {
var even = function(n) {return n % 2 === 0;};
var T = function() {return true;};
var isFalse = function(x) { return x === false; };
var intoArray = R.into([]);
it('returns true if all elements satisfy the predicate', function() {
assert.strictEqual(R.all(even, [2, 4, 6, 8, 10, 12]), true);
assert.strictEqual(R.all(isFalse, [false, false, false]), true);
});
it('returns false if any element fails to satisfy the predicate', function() {
assert.strictEqual(R.all(even, [2, 4, 6, 8, 9, 10]), false);
});
it('returns true for an empty list', function() {
assert.strictEqual(R.all(T, []), true);
});
it('returns true into array if all elements satisfy the predicate', function() {
assert.deepEqual(intoArray(R.all(even), [2, 4, 6, 8, 10, 12]), [true]);
assert.deepEqual(intoArray(R.all(isFalse), [false, false, false]), [true]);
});
it('returns false into array if any element fails to satisfy the predicate', function() {
assert.deepEqual(intoArray(R.all(even), [2, 4, 6, 8, 9, 10]), [false]);
});
it('returns true into array for an empty list', function() {
assert.deepEqual(intoArray(R.all(T), []), [true]);
});
it('works with more complex objects', function() {
var xs = [{x: 'abc'}, {x: 'ade'}, {x: 'fghiajk'}];
function len3(o) { return o.x.length === 3; }
function hasA(o) { return o.x.indexOf('a') > -1; }
assert.strictEqual(R.all(len3, xs), false);
assert.strictEqual(R.all(hasA, xs), true);
});
it('dispatches when given a transformer in list position', function() {
assert.deepEqual(R.all(even, listXf), {
all: true,
f: even,
xf: listXf
});
});
it('is curried', function() {
var count = 0;
var test = function(n) {count += 1; return even(n);};
assert.strictEqual(R.all(test)([2, 4, 6, 7, 8, 10]), false);
});
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,047 |
\section{Rydberg dressing potential}
Here, we derive the $\hat{V}_{\rm Ryd}$ term in the Hamiltonian of Eq. (1b), which arises from coupling of the excited states $|e\rangle$ with a high-laying Rydberg level $|r\rangle$. We start from the Hamiltonian of the $|e\rangle-|r\rangle$ transition, which contains the Rydberg interaction and the control field terms
\begin{equation}
\mathcal{H}^{er} = \sum_{i,j = 1, i<j}^{N_a}\frac{C_6}{r_{ij}^6}\hat{\sigma}_i^{rr}\hat{\sigma}_j^{rr}-\hbar\sum_{i=1}^{N_a}\left[\delta_\text{c}\hat{\sigma}_i^{rr}
+\left(\Omega_c\hat{\sigma}_i^{re}+h.c.\right)\right],
\label{eq:Rydberg_Hamiltonian}
\end{equation}
where $\delta_c$ is the control field detuning, $\Omega_c$ its Rabi frequency and $C_6$ a parameter related to the strength of the Rydberg interaction. Within the dressing regime ($|\delta_c|\gg \Omega_c$), the coupling term with $\Omega_c$ can be treated as a perturbation, such that the number of atoms in $|e\rangle$ and $|r\rangle$ are individually conserved, while the control field induces energy shifts within each number manifold. As discussed in the main text, we consider two different dressing scenarios. In the first one, no atoms are excited to the Rydberg state $|r\rangle$, so it is sufficient to consider the effect of the control field on the manifold of excited states. Using standard perturbation theory, the effective Hamiltonian for the $|e\rangle$ states up to fourth order is given by \cite{Macri-pohl,Macri-pohl2}
\begin{equation}
\hat{V}_{\rm Ryd}^{ee}\approx\sum_{i=1}^{N_a}\hbar\Delta_{\rm ac}\hat{\sigma}_i^{ee}-\sum_{i,j = 1, i\not=j}^{N_a}\frac{\hbar|\Omega_c|^4}{\delta_c^3}\left(\frac{1}{1+\frac{r_{ij}^6}{R_b^6}} \right)\hat{\sigma}_i^{ee}\hat{\sigma}_j^{ee},
\label{eq:correction1}
\end{equation}
where the blockade radius is defined as $R_b^6=C_6/2\hbar|\delta_c|$ with $\delta_c<0$ and $C_6>0$. From Eq. (\ref{eq:correction1}), we see that atoms far away ($r_{ij}\gg R_b$) experience the Stark shift $\Delta_{\rm ac}=|\Omega_c|^2/\delta_c-|\Omega_c|^4/\delta_c^3$ individually. On the other hand, the Rydberg interaction disrupts the dressing of the doubly excited states when two atoms in the $|e\rangle$ level are close to each other ($r_{ij}\ll R_b$).
In the second scenario, a single, immobile Rydberg excitation is first generated by storage of a photon. Then, other atoms can be excited to states $|e\rangle$ by a probe field. The effect of the stored Rydberg excitation on these excited states up to second order is described by the effective Hamiltonian
\begin{equation}
\hat{V}_{\rm Ryd}^{re}\approx\sum_{i,j = 1, i\not=j}^{N_a}\frac{\hbar|\Omega_c|^2}{\delta_c}\left(\frac{1}{1+2\frac{R_b^6}{r_{ij}^6}} \right)\hat{\sigma}_i^{rr}\hat{\sigma}_j^{ee}.
\label{eq:correction2}
\end{equation}
In particular, it can be seen that excited atoms that are much closer to the Rydberg excitation than a blockade radius do not experience a Stark shift at all. The potentials of the first and second dressing schemes from the main text are Eqs. (\ref{eq:correction1}) and (\ref{eq:correction2}) after incorporating the single-atom shift into the definition of the bare resonance frequency ($\omega_0\rightarrow \omega_0+\Delta_{ac}$), approximating the spatial dependence by the step function $\Theta(R_b^{\rm step}-r_{ij})$ \cite{Koscik2019} and defining $V=\hbar|\Omega_c|^4/\delta_c^3$ or $V=\hbar|\Omega_c|^2/\delta_c$, respectively. In Fig. \ref{Fig:Rydberg_dressing} we plot Eqs. (\ref{eq:correction1}) and (\ref{eq:correction2}) after subtracting the single-atom stark shift $\Delta_{\rm ac}$.
\begin{figure}[t]
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{Fig_Appendix_Rydberg_figure.pdf}
\end{minipage}
\caption{(a) Rydberg dressing potential for a single atom in the excited state $|e\rangle$ in presence of another atom also in the state $|e\rangle$ at a distance $r_{ij}$ (schematically represented by the atomic levels $\{|g\rangle,|e\rangle,|r\rangle\}$ and the blue spheres). Specifically, we plot Eq. (\ref{eq:correction1}) after subtracting the single-atom Stark shift $\Delta_{\rm ac}$. (b) Same as before but now the atom is in presence of another atom in the Rydberg level. Thus, we plot Eq. (\ref{eq:correction2}). Both potentials have been rescaled respectively by $V=\hbar|\Omega_c|^4/\delta_c^3$ and $V=\hbar|\Omega_c|^2/\delta_c$. The distances have been rescaled by the blockade radius $R_b$. }
\label{Fig:Rydberg_dressing}
\end{figure}
We now discuss how to relate the blockade radius $R_b$ in the microscopic model to that in the step function approximation, ${R}_b^{\rm step}$. Physically, within the context of nonlinear optics, the step function represents the radius over which the array goes from transmitting to reflecting, due to the excited state shift imparted given either an excited or Rydberg atom at the center. From Eq.~(\ref{eq:r}), it can be seen that the excited state needs to be shifted by an amount $\sim\Gamma_{k_\parallel}$ for an array to change its response from being largely reflecting to transmitting. We thus define $|\hat{V}({R}_b^{\rm step})|= \Gamma_{k_\parallel=0}$, where $\hat{V}$ is either Eq. (\ref{eq:correction1}) or Eq. (\ref{eq:correction2}) for a single excited atom in presence of another atom in the state $|e\rangle$ or $|r\rangle$ (respectively), and after subtracting the single-atom stark shift $|\Delta_{\rm ac}|$. This leads to the relation
\begin{equation}
\Tilde{R}_b^{\text{step}}\approx R_b\sqrt[6]{\kappa\left(\frac{V}{\Gamma_{\textbf{k}_\parallel=0}}-1\right)},
\label{eq:rescaling_potential}
\end{equation}
where $V$ is that appropriate to the specific dressing scheme and $\kappa=1$ or $\kappa=2$ depending on using Eq. (\ref{eq:correction1}) or Eq. (\ref{eq:correction2}), respectively. In the main text, we simplify the discussion by treating $R_b^{\rm step}$ as an independent variable and dropping the \textit{step} label to reduce the complexity in notation. In Sec.~\ref{subsec:beyondstep}, we will see that the errors derived for the single-photon switch within the step function approximation agree well with those calculated from the actual microscopic potential.
\section{Steady state in the low driving intensity limit}
As stated in the main text, the dynamics of the atomic degrees of freedom are governed by the master equation Eq.~(\ref{Eq:dynamics}). An equivalent formulation of the master equation is the quantum jump formalism, wherein the system is described by a wave function $\ket{\Psi}$ that evolves deterministically under the non-Hermitian Hamiltonian Eq.~(\ref{Eq:Heff}),
\begin{equation}
i \hbar \partial_t \ket{\Psi} = \mathcal{H}_{\rm eff} \ket{\Psi}.
\label{Eq:dynamicsPsi}
\end{equation}
Within this formalism, the additional term Eq.(\ref{Eq:Jumps}) in the master equation is implemented via stochastic quantum jumps applied to the wave function. However, in the limit of weak input field, it is well-known~\cite{ref:textbook,MMA18} that the observables of interest (reflectance, transmittance and normalized two-photon correlation function) can be calculated neglecting the jumps.
To be specific, we consider the dressing regime where no atoms are excited to $|r\rangle$, and the atoms can effectively be described as being two-level. We furthermore expand the wave function up to two atomic excitations, $\ket{\Psi} = c^{(g)} \ket{g} + \sum_{j=1}^{N_a} c_j^{(e)} \heg_j \ket{g} + \sum_{j,k =1, j<k}^{N_a} c_{jk}^{(2e)} \heg_j \heg_k \ket{g}$.
Applying Eq.~(\ref{Eq:dynamicsPsi}) to the wave function of the previous form leads to a linear system of differential equations for the coefficients:
\begin{align}
i \dot{c}^{(g)} &= -\sum_{j=1}^{N_a} \Omega_j c^{(e)}_j, \notag\\
i \dot{c}_j^{(e)} &=-\Omega_j^* c^{(g)} -(\delta+i\Gamma_0/2) c^{(e)}_j + \sum^{N_a}_{k=1,k\neq j} \mathcal{H}_{jk} c^{(e)}_k +\notag \\
& \qquad{} -\Omega_j\sum^{N_a}_{k=1, k\neq j} c_{jk}^{(2e)}\notag,\\
i \dot{c}_{jk}^{(2e)} &=-\left(\Omega^*_j c^{(e)}_k+\Omega^*_k c^{(e)}_j \right) -2(\delta+i\Gamma_0/2) c^{(2e)}_{jk} +\notag \\
& \qquad{} +\sum^{N_a}_{l=1,l\neq j} \mathcal{H}_{jl} c^{(2e)}_{lk} + \sum^{N_a}_{l=1,l\neq k} \mathcal{H}_{lk} c^{(2e)}_{jl} +\cdots
\label{Eq:Dynamics_coef}
\end{align}
where the dots denote the contribution from the three excitations manifold, and where for simplicity we have defined $c^{(2e)}_{jk} \equiv c^{(2e)}_{kj}$ and the matrices $\mathcal{H}_{jk} = J^{jk}-i\Gamma^{jk}/2$.
The steady state coefficients are found by imposing the time derivative to be zero in Eqs.~(\ref{Eq:Dynamics_coef}). Starting with all the atoms in the ground state ($\ket{\Psi(t=0)} = \ket{g}$), and replacing $c^{(g)} \approx 1$, it is possible to obtain an expression for the coefficients to lowest order in $\Omega_0/\Gamma_0$ ($c^{(e)}_j \sim O(\Omega_0/\Gamma_0)$ and $c^{(2e)}_{jk} \sim O(\Omega_0^2/\Gamma_0^2)$).
Once the atomic degrees of freedom are solved, it is straightforward to reconstruct the light observables from the input-output relation Eq. (\ref{Eq:fields}). As a concrete example, we consider the reflectance, which is defined as the rate of photons collected back into the same input mode, $R = \avg{\hEb_{\rm R}^\dagger \hEb_{\rm R}} / \avg{\hEb_{\rm T, in}^\dagger \hEb_{\rm T, in}}$. Here $\avg{\cdot}$ denotes the quantum mechanical expectation value on the atomic degrees of freedom, and thus the reflectance is directly given by the correlation functions $\avg{\heg_j \hge_k}$.
\section{Single-photon switch}~\label{App:switch}
In this section, we provide a more detailed analysis of our proposed single-photon switch. First, we show that given the storage of a gate photon, the formula for the transmittance experienced by a subsequent signal photon indeed reduces to Eq.~(\ref{eq:weighted_average}) of the main text, involving a classical average over transmittance of an array with holes punched in different positions. We then derive the optimal switch error (Eq.~(\ref{eq:gate_error}) of the main text) by means of a toy model characterizing the signal photon transmission. Finally, we address the retrieval of the gate photon and discuss a realistic implementation of the switch by going beyond the step function approximation for the Rydberg dressing interaction, and considering the actual potential derived from perturbation theory, Eq.~(\ref{eq:correction2}).
\subsection{Formal theory of signal photon transmission}
Here we derive the transmittance $T$ from Eq. (\ref{eq:weighted_average}) in the main text. We begin by considering the state of the atomic array following storage of a gate photon, $|\Psi_0\rangle=\sum_{i}c_{i}^{(r)}\hat{\sigma}_i^{rg}|g\rangle$. Here, $c_i \propto e^{-|\rb_i|^2/w_1^2}$ follows the Gaussian spatial profile of the gate photon, and the wave function is normalized to unity in the case of perfect storage. Once the control field is detuned to create the Rydberg dressing, the excitation in $|r\rangle$ no longer evolves, and a signal photon is sent in the same Gaussian mode used for detection in transmission, $\hat{E}_{\rm T,in}$. Within a scattering formalism, this composite state $\hat{E}_{\rm T,in}^{\dagger}|\Psi_0,{\rm vac}\rangle$ formally transforms to
\begin{equation}
|\Psi_{\text{sc}}\rangle=\sum_{i}\hat{\bf E}_{i}^{\dagger} c_i^{(r)}\hat{\sigma}_i^{rg}|g,{\rm vac}\rangle.
\label{eq:state_after_scattering}
\end{equation}
Here, we have explicitly included the photonic component of the system wave function, and $\hat{\bf E}_{\rm i}$ is the mode into which the incoming photon scatters, if atom $i$ was in the Rydberg state. This mode could contain some non-zero projection into the detectable Gaussian reflection and transmission modes, as well as a continuum of other modes in $4\pi$~(thus representing loss). Note that in the limit of an infinite array, the spatial modes associated with $\hat{\bf E}_{i}$ for different atoms $i$ are identical, apart from a translation corresponding to the position of atom $i$ within the array.
To proceed further, we can formally decompose the mode $\hat{\bf E}_i$ into the detection mode in transmission, and orthogonal modes whose explicit forms are not needed,
\begin{equation}
|\Psi_{\text{sc}}\rangle=\sum_{i}c_i^{(r)}\left(\sqrt{\bar{T}_i}e^{i\phi_i}\hat{\bf E}_{\rm T}^{\dagger}\hat{\sigma}_i^{rg}|g,{\rm vac}\rangle + |\rm Orthog.\rangle\right),
\label{eq:state_after_scattering2}
\end{equation}
where we have expressed the overlap between $\hat{{\bf E}}_i$ and $\hat{{\bf E}}_{\rm T}$ in terms of a real number $\bar{T}(\rb_i,w_2,R_b)$ and phase $\phi_i(\rb_i,w_2,R_b)$. The overall transmittance of the signal photon into mode $\hat{{\bf E}}_{\rm T}$ is given by the total population in the state $\hat{{\bf E}}_{\rm T}^{\dagger}|{\rm vac}\rangle$, which reproduces Eq.~(\ref{eq:weighted_average}) in the main text. While this expression was formally derived by considering single-photon scattering, being a linear process, $\bar{T}(\rb_i,w_2,R_b)$ can also be calculated by considering the transmittance of weak coherent input light, which is what we implement numerically.
\subsection{Scattering properties of a single hole}
Physically, $\bar{T}(\rb_i,w_2,R_b)$ describes the transmittance of a signal photon of beam waist $w_2$, when atom ${\bf r}_i$ is in a Rydberg state and dressing interactions induce a blockade radius of size $R_b$. Here, we analyze more carefully the properties of scattering, in the simple limit where the interaction is approximated by a step function with infinite depth $V\rightarrow\infty$, so that no atoms within the blockade radius can be excited to state $|e\rangle$, thus punching a hole in the atomic array.
In the following, we approximate the array as a continuous mirror with a circular aperture, such that scattering can be treated in terms of classical diffraction theory. In particular, for an input Gaussian beam aligned with the aperture, it is well established that the fraction of power transmitted at the other side simply corresponds to the fraction of input power hitting the aperture, $P_{\rm tra}/P_{\rm in}=1-e^{-2R_b^2/w_2^2}$ \cite{Tanaka:85}. This can also be viewed as the overlap between the original Gaussian and transmitted modes. Note that this is the total transmitted power into \textit{all} modes, including both the original Gaussian and a set of orthogonal modes excited due to diffraction. In our case, however, we are only interested in the transmission back into the Gaussian mode. Due to reciprocity, this is given by the previous overlap squared, such that
\begin{equation}
\bar{T}(\textbf{r}_i=0,R_b,w_2)=\left|1-e^{-2R_b^2/w_2^2}\right|^2,
\label{eq:Transmittance_Centred_circular_aperture}
\end{equation}
where we have neglected corrections arising from the finite total array size. Similarly, the fraction of power reflected by the mirror with the aperture into all modes is $P_{\rm rfl}/P_{\rm in}=e^{-2R_b^2/w_2^2}$, while projecting back into the Gaussian mode gives $R\approx |e^{-2R_b^2/w_2^2}|^2$. In Fig. \ref{Fig:Appendix_Transmission_Reflection}a, we perform a full numerical simulation of an array of $N_a=41^2$ atoms illuminated with a weak, resonant Gaussian beam of waist $w_2=5d$, and with all atoms within a radius $R_b$ of the origin removed. We see that the numerically evaluated reflectance and transmittance as a function of radius $R_b$ agree well with the approximate formulas for $\bar{T}_i$ and $R$ derived above.\\
These results can readily be generalized by replacing $e^{-2R_b^2/w_2^2}\rightarrow \nu$ in the previous formulas, where $1-\nu$ physically describes the fraction of input beam power hitting a hole in an array of any position and size. As one relevant consequence, to be used later, the loss, defined as the fraction of power that is neither transmitted nor reflected into the original Gaussian mode, is given by $K=1-T-R=2\nu (1-\nu)$.
For completeness, we use the previous arguments to justify the reflectance in Eq. (\ref{eq:reflectance}). Considering a finite square mirror of size $(Nd)^2$ illuminated by a Gaussian beam, the fraction of power reflected is $P_{\rm rfl}/P_{\rm in}=\text{ erf}^2(Nd/\sqrt{2}w_2)$, where the rest of the light is leaked at the edges of the mirror. The reflectance back into the Gaussian mode is then this quantity squared, corresponding to the first term in Eq. (\ref{eq:reflectance}). In Fig.~\ref{Fig:Appendix_Transmission_Reflection}b, we see good agreement between this analytical result and the numerical evaluation of the reflectance from a finite array.
\begin{figure}[t]
\begin{minipage}{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{Fig_Appendix_Simple_geometry.pdf}
\end{minipage}
\caption{(a) Numerically calculated classical transmittance (plotted as $1-T$, red circles) and reflectance $R$ (green squares) of a Gaussian field with beam waist $w_2$ illuminating a large~($N_a=41 \times 41$) array, with a hole~(atoms removed) of radius $R_b$ at its center. Here, $R,T$ are the projected reflectance and transmittance back into the Gaussian mode. We also plot the analytical formulas for $1-T$ using Eq. (\ref{eq:Transmittance_Centred_circular_aperture}), and $R=|e^{-2R_b^2/w_2^2}|^2$ in dashed black. (b) Reflectance~(plotted as $1-R$) of a Gaussian field resonantly driving a square atomic array of size $N_a=41 \times 41$, with varying beam waist $w_2$. The reflectance is back into the same Gaussian mode. We also plot Eq. (\ref{eq:reflectance}) from the main text (dashed lines). In both simulations, we consider a lattice constant $d=\lambda_0/2$.}
\label{Fig:Appendix_Transmission_Reflection}
\end{figure}
\subsection{Approximate model for signal photon transmission}
Unfortunately, while $\bar{T}(\rb_i=0)$ through a single hole aligned with a Gaussian beam admits a simple, closed-form expression, we do not find a simple solution for $\nu$ once the hole is misaligned. Furthermore, the signal photon transmittance of Eq.~(\ref{eq:weighted_average}) involves a weighted sum of transmittance through off-center holes. We thus consider a toy model that captures the essential physics, in order to derive an approximate scaling. In particular, we assume the signal photon has a top-hat spatial profile, with radius $w_2$. Then, the problem becomes purely geometrical, as the transmittance involves the overlap area between two circles of radius $R_b$ and $w_2$, separated by a distance $|\rb_i|$ (Fig. \ref{Fig:Geometric_toy_model}a). Assuming that $R_b>w_2$, we identify three regimes of interest (Fig. \ref{Fig:Geometric_toy_model}b):
\begin{equation}
\bar{T}(\rb_i,w_2,R_b)\approx\begin{cases}
1\qquad \text{for $R_b-w_2>|r_i|$},\\
0\qquad \text{for $|\rb_i|>R_b+w_2$},\\
I_B(\rb_i,w_2,R_b)\qquad \text{else},\\
\end{cases}
\label{eq:Ansatz_transmission}
\end{equation}
where $I_B\in(0,1)$ is a polynomial whose explicit form will not be relevant here. Substituting the ansatz from Eq. (\ref{eq:Ansatz_transmission}) into Eq. (\ref{eq:weighted_average}), one finds
\begin{equation}
T=\int_0^{R_b-w_2} 2\pi \rho |c|^2\ \text{d}\rho+\int_{R_b-w_2}^{R_b+w_2} 2\pi \rho I_B|c|^2\ \text{d}\rho,
\label{eq:integral}
\end{equation}
where we have taken the continuous limit, and $|c|^2=2 e^{-2\rho^2/w_1^2}/(\pi w_1^2)$ after normalizing the Rydberg population to unity. Finally, considering $R_b\gg w_2$ to neglect the second integral in Eq. (\ref{eq:integral}), one obtains
\begin{equation}
T(w_1,w_2,R_b)\approx 1-e^{-2(R_b-w_2)^2/w_1^2}.
\label{eq:T_simplified_toy_model}
\end{equation}
\begin{figure}[t]
\begin{minipage}{0.48\textwidth}
\centering
\includegraphics[width=\textwidth]{Fig_Appendix_geometric_toy_model.pdf}
\end{minipage}
\caption{(a) Sketch of the toy model used to calculate signal photon transmittance. We assume that the signal photon consists of a top-hat profile with radius $w_2$. The transmittance then reduces to finding the intersection area between this circle of radius $w_2$~(green), and the hole created by a locally stored Rydberg excitation (pink circle of radius $R_b$), which are separated by a distance $|\rb_i|$. (b) Reflection error $1-R$ (red) and storage/transmission error $1-\eta T$ (blue) of the switch as a function of the signal photon beam waist $w_2$ and after optimizing over the gate photon beam waist $w_1$. The total switch error $\epsilon$ is defined as the maximal error between transmission and reflection (dashed green curve). We also plot the function $f(w)=(\epsilon_T+\epsilon_R)/2$ (dashed black), whose minimum coincides with the optimal (smallest) switch error. The specific values are obtained from a numerical simulation with $R_b=10d$, $N_a=41^2$ and $d=\lambda_0/2$.}
\label{Fig:Geometric_toy_model}
\end{figure}
Next, we combine the previous Eq. (\ref{eq:T_simplified_toy_model}) with the expressions for $\eta$ and $R$ discussed in the main text to find an analytical approximation for the optimal switch error, which was defined as the maximal error between storage/transmission ($\epsilon_t=1-\eta T$) and reflection ($\epsilon_r=1-R$). For simplicity, we will use the ansatz $w_1=w_2=w$, which is motivated by the results from Fig. \ref{Fig3}c in the main text. Intuitively, one expects the optimal switch to have the same error in transmission and reflection. This is illustrated in Fig. \ref{Fig:Geometric_toy_model}b for the particular case of $R_b=10d$, where we plot the numerically calculated reflection~(red) and storage/transmission error~(blue) for different beam waists $w_2$, with the minimum error occurring at the intersection of these curves. However, solving $\epsilon_r(w^{opt})=\epsilon_t(w^{opt})$ leads to a transcendental equation for $w^{opt}$. To circumvent this, we can instead minimize the function $f(w)=(\epsilon_r+\epsilon_t)/2$, whose minimum is also at $w^{opt}$ (see Fig. \ref{Fig:Geometric_toy_model}b). Considering only the leading terms, the optimal beam waist is then given by
\begin{equation}
w^{\rm opt}(R_b,d)\approx\frac{R_b}{1+\sqrt{\log(C_\epsilon R_b/d)}},
\label{eq:optimal_beam_waist}
\end{equation}
where the constant $C_\epsilon(d)$ can be obtained by fitting the data. For $d=\lambda_0/2$, we get $C_\epsilon\approx2$. The optimal switch error in Eq. (\ref{eq:gate_error}) can be obtained by substituting $w^{\rm opt}$ from Eq. (\ref{eq:optimal_beam_waist}) into $\epsilon_R(w)\approx C_R(d)\lambda_0^4/w^4$. In the main text, we approximate the resulting expression to have a single constant $C(d)$, instead of two parameters $C_R(d)$ and $C_\epsilon(d)$ that unnecessarily complicate the discussion.
\begin{figure}[t]
\begin{minipage}{0.48\textwidth}
\centering
\includegraphics[width=0.60\textwidth]{Fig_Appendix_retrieval_error.pdf}
\end{minipage}
\caption{Overlap error (blue squares) between the state of the array following storage $|\Psi_0\rangle$ and the state $|\Psi_{\rm new}\rangle$ after the detection of a signal photon in transmission. We also plot the gate error (orange circles) from Fig. \ref{Fig3} for comparison. In the simulations, we consider a $N_a=41^2$ square array of lattice constant $d=\lambda_0/2$. The beam waists are the ones that optimize the switch error, which allows us to plot the data as a function of the blockade radius.}
\label{Fig_Appendix_retrieval_error}
\end{figure}
\subsection{Retrieval of the gate photon}
To operate our single-photon switch as a quantum gate, one must also consider the error associated with retrieving the first gate photon after the signal photon has been scattered. In particular, while storing and directly retrieving the gate photon (absent the signal photon) would have the same efficiency $\eta$~\cite{MMA18}, from Eq.~(\ref{eq:state_after_scattering2}), one sees that conditioned on the signal photon being transmitted into the desired mode $\hat{\bf E}_T$, the remaining Rydberg spin wave is altered, from $|\Psi_0\rangle = \sum_i c_i^{(r)} \hat{\sigma}_i^{rg} |g\rangle$ to $|\Psi_{\rm new}\rangle = \mathcal{N} \sum_i c_i^{(r)} \sqrt{\bar{T}_i} e^{i\phi_i} \hat{\sigma}_i^{rg} |g\rangle$. Here, $\mathcal{N}$ is a normalization factor such that $|\Psi_{\rm new}\rangle$ has unit norm (the smaller than unity norm without this factor describes the probability that the signal photon was not successfully transmitted, and its error has already been included in previous analysis). We now show that the additional error in retrieval due to this change of state is negligible compared to the total switch error previously calculated.
From time reversal symmetry, the excitation in the state $|\Psi_0\rangle$ can be retrieved with efficiency $\eta$, which we know is optimal. Then, any other state $|\Psi_\perp^i\rangle$ orthogonal to $|\Psi_0\rangle$ will have a lower retrieval efficiency $\eta_\perp^i$. Decomposing $|\Psi_{\text{new}}\rangle$ in the basis of $|\Psi_0\rangle$ plus other orthogonal states, one can express the retrieval efficiency of the distorted excitation as $\eta_{\text{new}}=\eta|\langle \Psi_0|\Psi_{\text{new}}\rangle|^2+\sum_i\eta_{\perp}^i|\langle \Psi_\perp^i|\Psi_{\text{new}}\rangle|^2$. In Fig. \ref{Fig_Appendix_retrieval_error}, we show that the distortion in the stored excitation (quantified by the overlap error $1-|\langle \Psi_0|\Psi_{\text{new}}\rangle|^2$) is smaller than the optimal gate error. Thus, even in the worst case scenario where $\eta_\perp^i=0$, one can still approximate $\eta_{\rm new}\approx \eta$.
\subsection{Switch error beyond the step-function approximation}\label{subsec:beyondstep}
So far, we have approximated the Rydberg dressing interaction as a step-function potential with infinite depth $V\rightarrow\infty$. In the following, we will show that the optimal switch error from Eq. (\ref{eq:gate_error}) can be achieved under realistic conditions, i.e. taking into account the real potential derived from perturbation theory, Eq. (\ref{eq:correction2}), and for a finite interaction strength $V=|\Omega_c|^2/\delta_c$.
First, we numerically optimize the switch error considering the real potential (i.e. without the step function approximation). In Fig. \ref{Fig:Appendix_Switch_real_potential} we show the resulting $\epsilon^{\rm opt}$~(solid lines) as a function of the microscopic blockade radius $R_b$ from Eq. (\ref{eq:correction2}), and for different $V$. For now, we take $R_b$ and $V$ to be independent parameters, while their dependence on laser parameters and principal quantum number is discussed later. Interestingly, now $\epsilon^{\rm opt}$ is not arbitrarily small for increasing blockade radius as it saturates to a specific value in the limit $R_b\rightarrow\infty$. For moderate interaction strengths ($V\lesssim100\ \Gamma_0$), the saturation arises from the non-zero reflectance of the atoms within the blockaded region, which are not completely shifted out of resonance due to $V$ being finite. According to Eq. (\ref{eq:r}), this introduces an error in transmission given by $\epsilon_V\sim1/(1+4|V|^2/|\Gamma_{k_\parallel=0}|^2)$ that lower bounds $\epsilon^{\rm opt}$ as a function of $V$. On the other hand, for very large interaction strengths ($V\gtrsim10^3\Gamma_0$), $\epsilon_V$ becomes negligible and transmission can be considered as perfect. However, the saturation still appears, now due to the finite array size error $\epsilon_N$ that prevents perfect reflection, as we discussed in the main text. Thus, according to Eq. (\ref{eq:reflectance}), even if $V\rightarrow\infty$ and $R_b\rightarrow\infty$, the switch error will still be fundamentally limited by $\epsilon_N=1-\text{erf}^4(Nd/\sqrt{2}w_2)$.
To generalize Eq. (\ref{eq:gate_error}) beyond the step function approximation and to validate our interpretation of the solid lines in Fig. \ref{Fig:Appendix_Switch_real_potential}, we add the aforementioned errors $\epsilon_V$ and $\epsilon_N$ to the optimal switch error $\epsilon(R_b^{\rm step})$ from the main text. In addition, we use Eq. (\ref{eq:rescaling_potential}) to express $R_b^{\rm step}$ as a function of the microscopic $R_b$, in order to compare on the same plot. In Fig. \ref{Fig:Appendix_Switch_real_potential} we plot $\epsilon^{\rm opt}(R_b^{\rm step})+\epsilon_V+\epsilon_N$~(dashed lines), in terms of the microscopic $R_b$. Overall we observe a good agreement between solid and dashed lines, which validates our previous claims.\\
\begin{figure}[t]
\begin{minipage}{0.42\textwidth}
\centering
\includegraphics[width=\textwidth]{Fig_Appendix_real_potential.pdf}
\end{minipage}
\caption{Optimal gate error as a function of the microscopic blockade radius $R_b$ considering the real potential derived from perturbation theory [Eq. (\ref{eq:correction2})]. Each color represents a different interaction strength $V=[5,10,20,40,10^2,10^3]\Gamma_0$, respectively from light to dark blue. The solid lines are the results of a numerical simulation with a square array with $N_a=41^2$ and lattice parameter $d=\lambda_0/2$. The dashed lines are obtained substituting Eq. (\ref{eq:rescaling_potential}) into Eq. (\ref{eq:gate_error}) from the main text and adding the errors associated to finite $V$ and finite array size.}
\label{Fig:Appendix_Switch_real_potential}
\end{figure}
\begin{figure}[t]
\begin{minipage}{0.49\textwidth}
\centering
\includegraphics[width=\textwidth]{Fig_Appendix_Realistic_Values.pdf}
\end{minipage}
\caption{Realistic values for the blockade radius $R_b$ (a) and control field power $P_c$ (b) as a function of the control field detuning $\delta_c$, rescaled by the linewidth $\Gamma_0=2\pi \times 6.065$ MHz of the $|e\rangle\rightarrow|g\rangle$ transition of ${}^{87}$Rb. To obtain these plots, we work at the limit of validity of perturbation theory, where $\Omega_c\approx\delta_c$ and $V\approx \delta_c$. Each color represents a different Rydberg state with principal quantum number $n\in\{30,40,...,100\}$, respectively from light grey to either purple or red. The yellow circle indicates the case of a photon switch with $99\%$ efficiency.}
\label{Fig_Appendix_realistic_values}
\vspace{-1em}
\end{figure}
Finally, we discuss the experimental feasibility of the values for $R_b$ and $V$ used in Fig. \ref{Fig:Appendix_Switch_real_potential}. As a concrete example, we will consider the use of $^{87}\text{Rb}$ atoms and assume the Rydberg states are reached following the widely used transition scheme $|g\rangle=|5S_{1/2}\rangle \rightarrow |e\rangle=|5P_{3/2}\rangle\rightarrow |r\rangle=|nS_{1/2}\rangle$~\cite{ref:Peyronel2012,ref:Distante2017,ref:Ryabtsev_2016}, with the ground-excited state transition wavelength of $\lambda_0=780.0$ nm, lattice constant $d=\lambda_0/2$, and excited state spontaneous emission rate $\Gamma_0=2\pi\times 6.065$ MHz~\cite{ref:Steck}. More specifically, an approximate two-level transition can be realized by utilizing a cycling transition, where the ground and excited states have maximum angular momentum and are connected via a circularly polarized transition, $|g\rangle= |F=2, m_F=2\rangle$ and $|e\rangle = |F=3,m_F=3\rangle$. It is known that dipole-dipole interactions in the presence of hyperfine structure can induce transitions out of this manifold, as the re-scattered field seen by a ground-state atom, coming from another excited atom, does not necessarily have the same circular polarization \cite{ref:AnaPNAS}. However, the probability of this is suppressed both by the squared ratio of Clebsch-Gordan coefficients between the undesired and desired transitions~($1/15$ for ${}^{87}$Rb), and by the application of magnetic fields, to realize a differential Zeeman shift $\delta_{{\rm Zeeman}}$ between the undesired and desired transition resonance frequencies~(with a corresponding suppression $\sim (\Gamma_0/\delta_{{\rm Zeeman}})^2$).
Within such a scheme, we now consider the Rydberg interaction properties. To reduce the number of parameters, we will work in the limit of validity of perturbation theory for the Rydberg dressing scheme, where $\Omega_c \approx \delta_c$ and $V\approx \delta_c$. In Fig. \ref{Fig_Appendix_realistic_values}a we plot the blockade radius $R_b$ as a function of the detuning $\delta_c$ for different principal quantum numbers $n$. The values for the $C_6$ coefficient have been obtained by fitting the data from Refs. \cite{ref:32,ref:Browaeys_2016,ref:Ds} with the function $C_6(n)=C_0n^{11}$. Additionally, we also calculate the control field power $P_c=2\epsilon_0c|\delta_c|^2\hbar^2A/|\db_{er}|^2$ required to achieve a specific $V\sim\delta_c$, where the surface illuminated by the control field $A$ is taken to cover the array in the simulations $A=\pi(20d)^2$. The value for the dipole moment $\db_{er} =(43/n)^3\cdot 0.0103\text{a}_0e$ is taken from Ref. \cite{ref:experimental_data_Rydberg} and rescaled to arbitrary $n$. Combining the results from Fig. \ref{Fig:Appendix_Switch_real_potential} and \ref{Fig_Appendix_realistic_values}, we conclude that a single-photon switch with $99\%$ efficiency can be already achieved with a control field of $\delta_c\sim 2\pi \times 30 $MHz, $P_c\sim 100$mW and $n\sim60$, which are consistent with the typical values currently used in state-of-the-art experiments \cite{ref:Browaeys_2016,ref:BlochExperiment}.
\section{Strong driving limit}
\subsection{Stochastic semi-classical model}
Here, we describe in more detail the stochastic semi-classical model that we have used to model the reflectance and loss of a 2D array in the strong driving limit. In the following we specify the procedure that determines how atoms are assigned to be saturated and effectively removed or not (as illustrated in the piece-wise steps of Fig.~\ref{Fig4}e, where red and blue atoms illustrate removed atoms and remaining mirror atoms, respectively).
This assignment proceeds in a series of steps, starting with all atoms with no assigned state. In each step, we apply the following rules: (i) we take all atoms $j$ that have not yet been assigned a state, and define the blockade region containing $N_b^j$ atoms as the intersection between the subset of atoms with no assigned state and the blockade radius centered at $j$ (see as an example the brown enclosed region in the first panel of Fig.~\ref{Fig4}e). (ii) A single atom $j$ from this subset is randomly chosen following the probability $\Omega_j^2 N_b^j /\sum_j \Omega_j^2 N_b^j$. Here $\Omega_j$ is the local Rabi frequency at $\rb_j$, and this quantity accounts for the likelihood that a dressed Rydberg superatom centered around $j$ becomes excited. (iii) The $N_b^j$ atoms contained in the corresponding blockade region of $j$ are randomly assigned to be saturated or unsaturated according to the probability $p_j= s_j/(1+s_j)$. The saturation parameter coincides with the usual one for a single two-level system, but with a collectively enhanced Rabi frequency $\sqrt{N_b^j}\Omega_j$ and collectively modified decay rates and resonance shifts, $s_j = 8 N_b^j \Omega_j^2/[\Gamma_{{\bf k}_{\parallel}=0}^2+4(\delta -\Delta_{{\bf k}_{\parallel}=0})^2]$. Once all atoms have been assigned, we then numerically calculate the corresponding linear classical loss and reflectance of this particular configuration in the weak driving limit, by effectively removing all the atoms in the saturated regions. We then repeat, sampling over $\sim 5000$ configurations and averaging to obtain the loss and reflectance of the system, respectively.
\subsection{Approximate analytical behavior of semi-classical model}
\label{AppC2}
While the reflectance and loss of the previously discussed toy model are still calculated numerically, it is possible to obtain analytical approximations with a few additional assumptions. In particular, we assume that the system is illuminated by a beam of area $A$, and can thus roughly be divided into $N_d = A/N_b d^2$ independent and non-overlapping blockade regions, each of radius $R_b$ and $N_b\sim \pi R_b^2/d^2$ atoms, which all see equal Rabi frequency $\Omega$ and individually saturate with probability $p=s/(1+s)$, where $s = 8N_b \Omega^2 /[\Gamma_{{\bf k}_{\parallel}=0}^2+4(\delta-\Delta_{{\bf k}_{\parallel}=0})^2]$ is the saturation parameter for the uniform field defined analogously as before.
The probability for $m$ of these regions to be saturated and transmit light then follows a binomial distribution, $\mathcal{P} (m)= {\binom {N_d} m }~p^m (1-p)^{N_d - m}$. Each of these configurations diffracts light as if it was a classical mirror with $m$ punched holes. We then generalize the result for the loss caused by a single hole in Appendix C.2 to the case of multiple holes; in particular, with a total fractional area $\nu=m/N_d$ removed from the mirror, the loss follows $K_{\rm cl} \approx 2 \nu (1-\nu)$, largely independently of the size and number of the regions. The total loss is then evaluated as a statistical average over any possible number of holes, $K = \sum_{m=0}^{N_d} \mathcal{P}(m) K_{\rm cl}(m/N_d)$, and reduces to the simple expression $K = 2 N_d^{-1} \left[ \avg{m} -N_d^{-1} \avg{m^2} \right] = 2p(1-p)(1-N_d^{-1})$, where $\avg{\cdot}$ indicates an average of a random variable following a binomial distribution. In particular, it can be seen that the maximum loss $K^{\rm max}=(1-N_d^{-1})/2$ monotonically decreases as a function of decreasing number of independent blockade regions, reaching $K=0$ uniformly for $N_d=1$. On the other hand, the maximum loss for any $N_d>1$ is achieved when $p=0.5$, which corresponds to a Rabi frequency of $\Omega^{\rm max} = \sqrt{\Gamma^2 + 4(\delta-\Delta_{{\bf k}_{\parallel}=0})^2}/\sqrt{8 N_b}$.
In Fig.~\ref{FigS3c}, we compare the analytical approximation for $K^{\rm max}$ with numerical simulations, for square arrays with $N_a=16,25,36,49$, and $100$ atoms, and varying blockade radii $R_b$. These simulations involve both full numerical density matrix simulations~(red points), and the semi-classical stochastic model~(black). In particular, for each system size and blockade radius, we drive the system with a Gaussian beam of waist $w_0=0.4\sqrt{N_a}d$, and find the power at which the maximum loss $K^{\rm max}$ occurs. We then plot $K^{\rm max}$ as a function of the approximate number of blockade regions, $N_d= 2 \pi w_0^2/N_b d^2$. We find that the data points collapse onto a single universal curve, and that as long as $N_d^{-1}\lesssim 1$~(the regime of multiple independent blockade regions), this curve agrees well with the formula $K^{\rm max} = (1-N_d^{-1})/2$~(dashed line). For large $N_d^{-1}$, the maximum loss becomes nearly zero, with the difference from the simple formula largely being attributable to the inhomogeneity of the beam across the system, which is not accounted for in our simple analytical model.
\begin{figure}[t]
\begin{minipage}{0.4\textwidth}
\centering
\includegraphics[width=\textwidth]{Fig_Appendix_Kmax_v5.pdf}
\end{minipage}
\caption{Maximum loss as a function of $N_d^{-1} = N_b d^2/2\pi w_0^2$, where $N_d$ roughly corresponds to the number of blockade regions. Red and black symbols correspond to the exact numerical result and the numerical stochastic model, respectively, for an array with $N_a=16$ (open circles), $N_a=25$ (squares), $N_a=36$ (solid circles), $N_a=49$ (crosses) and $N_a=100$ (stars) atoms. The data collapse onto a universal curve that tends to the analytical model result $K^{\rm max} = (1-N_d^{-1})/2$~(dashed line), in the limit of small $N_d^{-1}$. The simulations are done with a lattice constant of $d/\lambda_0 = 0.5$ and beam waist $w_0 = 0.4 \sqrt{N_a} d$).}
\label{FigS3c}
\end{figure}
\newpage
| {
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} | 4,534 |
FARC Terrorism Sabotages Colombia's Oil Industry, Environment
Bogota, Colombia (PanAm) – The Revolutionary Armed Forces of Colombia (FARC) escalated their attacks against Colombia's oil industry in the first half of 2015, but widespread rejection and criticism is mounting from the country's citizenry. In particular, locals have witnessed the blasting of key pipelines, such as the Transandino pipeline in Tumaco, near the Pacific coast.
The latest attack, which took place on June 25, caused the spill of 410,000 gallons of crude oil, leading to serious environmental damage to the area. Environment Minister Gabriel Vallejo called it the "worst environmental damage the country has seen in the last 10 years," adding that the surrounding ecosystem is likely to never fully recover.
In another attack on the environment, this time in Putumayo, the FARC forced 19 oil-tanker drivers to dump 130,000 gallons of crude on the highway. The environmental harm will reportedly last months, if not years, having damaged vegetation and water sources in the area.
So far in 2015, there have been 47 oil-related attacks in Colombia, contributing to the loss of 30,000 barrels of oil and causing a massive financial drain on the country. The clean-up costs are similarly enormous, totaling some US$10 million in damage, according to Carlos Leal, executive director of the Colombian Association of Petroleum Engineers (ACIPET).
Leal warns that if these attacks continue, Colombia's oil industry could be severely harmed, especially given the poor year for oil markets.
"What could happen is that companies could acquire force majeure clauses, because no one is going to operate with these losses, nor within conditions in which economic viability and safety are not guaranteed. Therefore, it is likely that companies will have to leave the country or declare bankruptcy."
Colombia has lost out on US$1.5 million in crude this year alone, not counting the production that is forgone in places where resources remain untapped as a result of guerrilla activity.
"That is something that the guerrilla does not realize. If they do realize it, and are doing it on purpose, it isn't known. Unfortunately, nothing is done to hold those responsible for these acts accountable," Leal says.
In the past 30 years, almost 4 million barrels of oil have been spilled in Colombia, an amount comparable to the 1989 Exxon Valdez oil spill in Alaska, or the 2010 explosion on a British Petroleum (BP) oil rig in the Gulf of Mexico.
"That event cost [BP] $18.9 billion in penalties, and in Exxon's case, they are still paying fines. Meanwhile, who is charging the guerrilla [for their damage]? No one. No one is held accountable for these events that are cumulatively of equal proportion," the ACIPET director says.
There are other ways in which the FARC harms oil production in Colombia. Besides raiding oil tankers and blasting pipelines, the guerrilla also infiltrates communities and worker groups to provoke strikes that slow production. "This is to the detriment of the nation, since the crude oil that is not produced is crude that is not sold and that will not produce royalties."
Environmental Impact of the Oil Spills
The attack on the Transandino pipeline in Tumaco, which has generated the biggest environmental disaster in Colombia's recent history, continues to yield serious consequences for the neighboring community.
Close to 160,000 people have been left without drinking water for 18 days, and even though Ecopetrol and the Colombian government transport water in trucks from other parts of the country, the demand has not been met.
The contamination of the Mira River, which reaches the Pacific Ocean, has also had an effect on the fishing industry, one of the most important economic activities in the region.
For Gonzalo Andrade, a professor at the Institute of Natural Sciences of the National University of Colombia (UNAL), the oil spills have forever changed the ecosystem of the surrounding area, since aquatic species cannot survive in water contaminated with fossil fuels.
When crude oil dissolves in water, it mixes in different layers. "The superficial layer of black oil, which floats, is the only one we can see, but some of it dissolves in water and some sinks in such a manner that a spill ends up affecting all of an ecosystem's components," said Germán Márquez, a biologist at UNAL's Institute of Environmental Studies.
The contamination depletes oxygen levels in the water and the sea becomes uninhabitable for fish. The microorganisms that live in the sea bed and act as a source of nutrients for other species are also affected.
Why Is the FARC Waging War on Oil?
At the peace negotiations in Havana, the FARC's delegation responded to the attacks by saying the environmental damage was "not the intended consequence."
"We do not take pride in the results of our actions against petroleum infrastructure, just like we do not take pride in the death of enemy soldiers when these occur.… We must recognize that the escalation of the war aggravates this situation, affecting people and ecosystems, and unfortunately what took place in Tumaco has had terrible and undesirable consequences," said Carlos Antonio Lozada, a FARC negotiator.
Ecopetrol says its efforts to prevent the oil spill from reaching the Pacific have been impeded by the guerrilla.
"These outlaw groups have intimidated communities, harassed helicopters working for Ecopetrol, and have impeded workers from performing their duties in an effective manner to repair the pipelines and control the crude spilling into the rivers and sea."
Leal says that while the FARC's contradictory nature can often be difficult to decipher, he believes the guerrilla may have targeted the oil industry based on its unwillingness to pay extortion money.
"They look to get money by extorting companies, contractors, or third parties. In this case, the FARC must leave this industry alone, despite it not paying up. This sort of activity is logically punishable by law, and oil companies respect the law and would not allow this type of behavior."
Translated by Franco Bastida.
Written by María Paula Suárez Navas for The PanAm Post
Tags: colombia Environment farc oil panam terrorism
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"redpajama_set_name": "RedPajamaCommonCrawl"
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Антони́н (умучен в году) — святой мученик из Пьяченцы, воин. Дни памяти — 4 июля, 30 сентября, 13 ноября.
Биография
Святой Антонин был воином в Фивейском легионе. Согласно преданию, он пострадал в Траво около 303 года.
О нём сообщается в «Похвале святым» (De laude Sanctorum) святого Виктриция Руанского конца IV века, а также в «Мартирологе Иеронима», где указана дата 30 сентября как день его почитания. Возможно, речь идёт о дате рождения.
Менее древняя агиография IX и X веков, такая как «Gesta Sanctorum Antonini, Victoris, Opilii et Gregorii PP. X», содержит его более подробное житие.
Почитание
В IV веке святой епископ Савин обрёл его мощи. Их исследование предпринималось на протяжении долгого времени: начиная с 1000 года, трудами епископа Зигифреда (Sigifredo). Их также изучали в 1510 году семья , в 1562 году Бернардино Скотти, в 1569 году и в 1615 году . Исследование было завершено Джованни Баттиста Скалабрини в период с 1878 по 1879 год.
Святому посвящена в Пьяченце.
Святой Антонин считается покровителем Пьяченцы, , Траво.
Примечания
Ссылки
Sant'Antonino di Piacenza
Святые Эмилии-Романьи
Католические святые
Христианские мученики IV века | {
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\section{Introduction}
Let $G$ be any group and $\mathbb C G$ be the complex group algebra of $G$, i.e. the set of finitely supported complex functions on $G$. We may represent an element $\alpha$ in $\mathbb C G$ as a formal sum $\sum_{g\in G}a_gg$, where $a_g\in\mathbb C$ is the value of $\alpha$ in $g$. The multiplication in $\mathbb C G$ is defined by
\[ \alpha\beta=\sum_{g,h\in G}a_gb_hgh=\sum_{g\in G}\pr{\sum_{x\in G}a_{gx^{-1}}b_x}x\]
for $\alpha=\sum_{g\in G}a_gg$ and $\beta=\sum_{g\in G}b_gg$ in $\mathbb C G$. We shall say that $\alpha$ is a \textit{zero divisor} if there exists $0\neq\beta\in\mathbb C G$ such that $\alpha\beta=0$. If there is a non-zero $\beta\in\ell^2(G)$ such that $\alpha\beta=0$, then we may say that $\alpha$ is \textit{analytical zero divisor}. If $\alpha\beta\neq0$ for all $0\neq\beta\in\mathbb C G$, then we say that $\alpha$ is \textit{regular}. The following conjecture is called the \textit{zero divisor conjecture}.
\begin{conj}\label{ZDC}
Let $G$ be a torsion-free group. Then all elements in $\mathbb C G$ are regular.
\end{conj}
Amazingly, this conjecture has held up for many years. The conjecture \ref{ZDC} has been proven affirmative when $G$ belongs to special classes of groups; ordered groups (\cite{malcev} and \cite{neumann}), supersolvable groups (\cite{formanek}), polycyclic-by-finite groups (\cite{brown} and \cite{farsni}) and uniqe product groups (\cite{cohen-zd}). Delzant \cite{140} deals with group rings of word-hyperbolic groups and proves the conjecture for certain word-hyperbolic groups. Let $\mathcal C$ (Linnell's class of groups) be the smallest class of groups which contains all free groups and is closed under directed unions and extensions with elementary amenable quotients. Let $G$ be a group in $\mathcal C$ such that there is an upper bound on the orders of finite subgroups, then $G$ satisfies the above conjecture (\cite{lin309}).
The map $\valu\cdot\cdot: \mathbb C G \times \mathbb C G \rightarrow \mathbb{R}$ defined by
\begin{align*}
\valu\alpha\beta:=\sum_{g\in G}a_g\bar{b}_g\quad(\alpha,\beta\in\mathbb C G)
\end{align*}
is an inner product on $\mathbb C G$, so $\|\alpha\|_2=\valu\alpha\alpha^\frac12$ becomes a norm, called 2-norm; the completion of $\mathbb C G$ w.r.t. 2-norm is the Hilbert space $\ell^2(G)$. Indeed, we have
\[ \ell^2(G)=\se{\alpha:\ra G\mathbb C:\sum_{g\in G}\|\alpha(g)\|^2<\infty}.\]
In \cite{lin}, Linnell formulated an analytic version of the zero divisor conjecture.
\begin{conj}\label{AZDC}
Let $G$ be a torsion-free group. If $0\neq\alpha\in\mathbb C G$ and $0\neq\beta\in\ell^2(G)$,
then $\alpha\beta\neq0$.
\end{conj}
In \cite{lingva}, it is shown that
Since $\mathbb C G\subset \ell^2(G)$, the second conjecture implies the first one. In \cite{elek}, it is proved that for finitely generated amenable groups, the two conjectures
are actually equivalent. We prove this is true for all amenable torsion-free groups. \par
The so-called 1-norm is defined on $\mathbb C G$ by
\[ \|\alpha\|_1=\sum_{g\in G}|a_g|,\quad\text{for $\alpha=\sum_{g\in G}a_gg$ in $\mathbb C G$}.\]
The \textit{adjoint} of an element $\alpha=\sum_{g\in G}a_gg$ in $\mathbb C G$, denoted by $\alpha^*$, is $\alpha^*=\sum_{g\in G}\bar a_gg^{-1}$. We call an element $\alpha\in\mathbb C G$ \textit{self-adjoint} if $\alpha^*=\alpha$, and use $(\mathbb C G)_{\textbf{s}}$ to denote the set of self-adjoint elements of $\mathbb C G$. It is worthy of mention that if $\alpha=\sum_{g\in G}a_gg$ is self-adjoint then $a_1$ should be a real number.
For $\alpha\in\mathbb C G$, $\beta$ and $\gamma$ in $\ell^2(G)$, the following equalities hold:
\[ \valu{\alpha\beta}\gamma=\valu{\beta}{\alpha^*\gamma}.\]
\par
The goal of this paper is to give a criterion for an element in a complex group algebra to be regular:
\begin{theorem}
\label{Main} Let $G$ be a torsion free group. Then $\alpha\in\mathbb C G$ is regular if $2\|\alpha\|_2^2\geq\|\alpha\|_1^2$.
\end{theorem}
\section{Preliminaries}
In this section we provide some preliminaries needed in the following.
\par Let $G$ be a group. The \textit{support} of an element $\alpha=\sum_{g\in G}a_gg$ in $\mathbb C G$, $\supp\alpha$, is the finite subset $\se{g\in G: a_g\neq0}$ of $G$.\par Let $H$ be a subgroup of $G$, and $T$ be a right transversal for $H$ in $G$. Then every element
$\alpha\in\mathbb C G$ (resp. $\alpha\in\ell^2(G)$) can be written uniquely as a finite sum of the form $\sum_{t\in T}\alpha_tt$ with $\alpha_t\in\mathbb C H$ (resp. $\alpha_t\in\ell^2(H)$). \par
For $S\subset G$, we denote by $\langle S\rangle$, the subgroup of $G$ generated by $S$. We have the following key lemma:
\begin{lemma}\label{iff}
Let $G$ be a group, $\alpha\in\mathbb C G$ and $H=\langle\supp\alpha\rangle$. Then $\alpha$ is regular in $\mathbb C G$ iff $\alpha$ is regular in $\mathbb C H$.
\end{lemma}
\begin{proof}
Suppose that $\alpha$ is a zero divisor. Among elements $0\neq\gamma$ in $\mathbb C G$ which satisfy $\alpha\gamma=0$ consider an element $\beta$ such that $1\in\supp\beta$ and $|\supp\beta|$ is minimal, then one can easily show that $\beta\in\mathbb C H$, and this proves the result of the lemma.
\end{proof}
An immediate consequence of this lemma is:
\begin{corollary}
A group $G$ satisfies the Conjecture \ref{ZDC} iff all its finitely generated subgroups satisfy the Conjecture \ref{ZDC}.
\end{corollary}
By Lemma \ref{iff} in hand, we can generalize the main theorem of \cite{elek}:
\begin{theorem}\label{generalize}
Let $G$ be an amenable group. If $0\neq\alpha\in\mathbb C G$, $0\neq\beta\in\ell^2(G)$ and $\alpha\beta=0$, then there exists $0\neq\gamma\in\mathbb C G$ such that $\alpha\gamma=0$.
\end{theorem}
The above theorem along with results in \cite{KLM} provides another proof for \cite[Theorem 2]{lingva}.\par
For a normal subgroup $N$ of a group $G$, we denote the natural quotient map by $q_N:\ra G{G/N}$. We continue to show that:
\begin{lemma}\label{11+g}
Let $N$ be a normal subgroup of a group $G$ satisfying Conjecture \ref{ZDC}. Consider a non-torsion element $q_N(t)$, $t\in G$, in the quotient group. Then $\alpha+\beta t$ is regular, for all $\alpha,\beta\in\mathbb C N\setminus\se{0}$.
\end{lemma}
\begin{proof}
Suppose that $\alpha+\beta t$ is a zero divisor for non zero elements $\alpha,\beta\in\mathbb C N$. Applying Lemma \ref{iff} and multiplying by a suitable power of $t$, we can assume that there are non zero elements $\gamma_k$, $k=0,1,\dotsc, n$, such that
\[ (\alpha+\beta t)\sum_{k=0}^n\gamma_kt^k=0.\]
In particular, $0=\beta t\gamma t^n=(\beta t\gamma_nt^{-1})t^{n+1}$, whence $\beta t\gamma_nt^{-1}=0$, a contradiction, because $t\gamma_n t^{-1}$ is a non zero element of $\mathbb C N$.
\end{proof}
\begin{proposition}\label{1+g}
Let $N$ be an amenable normal subgroup of a group $G$ satisfying Conjecture \ref{ZDC}. Consider a non-torsion element $q_N(t)$, $t\in G$, in the quotient group. Then there is no $0\neq\gamma\in\ell^2(G)$ such that $(\alpha+\beta t)\gamma=0$. In particular, $a+bg$ is an analytical zero divisor, for all non-torsion element $g\in G$ and non zero complex numbers $a,b$.
\end{proposition}
\begin{proof}
The group $\valu Nt$ is amenable. Hence Lemma \ref{11+g} together with Theorem \ref{generalize} yields the result.
\end{proof}
\section{A cone of regular elements}
The result of the Proposition \ref{1+g} is true if we replace $\mathbb C$ by an arbitrary field $\mathbb F$. The field of complex numbers allows us to define inner product on the group algebra; with the help of inner product, we can construct new regular elements from the ones we have:
\begin{proposition}
\label{sumofnonzerdivisor}
Let $G$ be a group and $\mathcal F$ be a finite non-empty subset of $\mathbb C G$. If $\sum_{\alpha\in\mathcal F}\alpha^*\alpha$ is an analytical zero divisor then all elements of $\mathcal F$ are analytical zero divisors. In particular, $\alpha\in \mathbb C G$ is an analytical zero divisor if and only if $\alpha^* \alpha$ is an analytical zero divisor.
\end{proposition}
\begin{proof}
Let $\tilde{\alpha}:=\sum_{\alpha\in\mathcal F}\alpha^*\alpha$ and $\tilde{\alpha}\beta=0$ for some $\beta\in\ell^2(G)$. Then
\[ 0=\valu{\tilde\alpha\beta}\beta=\sum_{\alpha\in\mathcal F}\valu{\alpha^*\alpha\beta}\beta=
\sum_{\alpha\in\mathcal F}\valu{\alpha\beta}{\alpha\beta}=\sum_{\alpha\in\mathcal F}\|\alpha\beta\|_2^2,\]
whence $\alpha\beta=0$ for all $\alpha\in\mathcal F$. This completes the proof.
\end{proof}
A \textit{cone} in a vector space $\mathfrak X$ is a subset $\mathfrak K$ of $\mathfrak X$ such that $\mathfrak K+\mathfrak K\subset\mathfrak K$ and $\mathbb R_+\mathfrak K\subset\mathfrak K$. We proceed by introducing a cone of regular elements in $\mathbb C G$. First a definition:
\begin{definition}\label{injpos} Let $G$ be a group and $(\mathbb C G)_{\text{s}}$ be the set of self adjoint elements $\alpha\in\mathbb C G$, we define a function $\Upsilon:\ra{(\mathbb C G)_{\text{s}}}{\mathbb R}$ by
\[ \Upsilon(\alpha):=a_1-\sum_{g\neq 1}|a_g|. \]
We call an element $\alpha\in(\mathbb C G)_{\text{s}}$ \textbf{golden} if $\Upsilon(\alpha)\geq0$. The set of all golden elements in $(\mathbb C G)_{\text{s}}$ is denoted by $(\mathbb C G)_{\text{gold}}$.
\end{definition}
What is important about golden elements is:
\begin{proposition}\label{cone}
For a torsion free group $G$, $(\mathbb C G)_{\text{gold}}$ is a cone of regular elements.
\end{proposition}
\begin{proof}
It is obvious that if $\alpha$ is golden then so is $r\alpha$ for any $r>0$. The triangle inequality for $\mathbb C$ shows that if $\alpha$ and $\gamma$ are golden then so is $\alpha+\gamma$. For $\alpha\in(\mathbb C G)_{\text{s}}$, we have
\begin{align*}\label{sumpos}
\alpha&=\frac12(\alpha+\alpha^*)\notag\\&= a_1+\frac12\sum_{g\neq1}\pr{\bar a_gg^{-1}+a_gg}\notag\\
&=\Upsilon(\alpha)+\frac12\sum_{g\neq1}\pr{2|a_g|+\bar a_gg^{-1}+a_gg}\\
&=\Upsilon(\alpha)+\frac12\sum_{g\neq1}|a_g|\pr{\frac{\bar a_g}{|a_g|}+g}^*\pr{\frac{a_g}{|a_g|}+g}
\end{align*}
Hence, by Lemma \ref{1+g} and Proposition \ref{sumofnonzerdivisor}, $\alpha$ is regular.
\end{proof}
Now, we are ready to prove our main result:
\begin{proof}[Proof of Theorem \ref{Main}]
For $\alpha=\sum_{g\in G}a_gg$ in $\mathbb C G$, $\alpha^*\alpha$ is self-adjoint, and one can easily show that
\[\Upsilon(\alpha^*\alpha)\geq2\|\alpha\|_2^2-\|\alpha\|_1^2.\] Hence, by Proposition \ref{cone}, the result of the Theorem is proved.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,900 |
\section{Introduction}
Rechargeable Li-ion batteries play a crucial role in many modern-day applications ranging from portable electronics and medical devices, to renewable energy integration in power grids and electric vehicles. The steep decrease in the price of lithium-ion based battery storage by 73\% from 2010 to 2016, to an all-time low of \$273$/$kWh in 2017 \cite{bloomberg_report} opened up a significant energy storage market evaluated at \$65 billion in 2017 \cite{bernhart2019challenges}. Irrespective of the application, Li-ion batteries degrade with time. With ageing, cells exhibit a loss of capacity and an increase in impedance. The rate of degradation is influenced by the dynamic operating conditions, including varying charge/discharge rates, different voltage operation limits and temperature fluctuations. The ability to estimate degradation in real-time irrespective of the various failure mechanisms and their degradation paths is crucial for safe and effective battery management systems \cite{ev_diagnosis}. Battery state of health can be used to predict battery's expected lifetime, however, the feasibility of online state of health estimation via direct measurement of chemical reaction parameters inside batteries remains limited \cite{review_intro}.
State of health (SOH) is a parameter that quantifies the general condition of a battery and its ability to deliver the specified performance, measured as capacity or impedance, when compared to its unused state. This work focuses on the battery capacity as the health indicator due to its correlation to the energy storage capability of batteries and its direct impact on the remaining run time and life of the batteries. Capacity fade estimation has received considerable research interest from industry and academia \cite{review_intro}, \cite{battery_review_1}, \cite{battery_review_2}, \cite{battery_review_3} and a number of methods have been proposed. The current approaches to capacity fade estimation involves parameter estimation using either of the following modelling types, equivalent circuit models (ECMs) \cite{electrical_equiv_model_review}, \cite{online_ECM}, \cite{EIS_3}, electrochemical models \cite{electrochemistry_model_nasa}, \cite{electro_model_bole}, \cite{parama_chemical_model}, or data-driven models \cite{severson_satandford}, \cite{RUl_prognostics_NASA}, \cite{prognostics_NASA}, \cite{batery_health_bayesin_SVM}, \cite{SVM_battery_health}, \cite{attia2020closed}. Electrochemical models approximate the chemical processes that take place inside a battery cell during operation. This type of modeling requires detailed cell specifications, such as electrode materials and electrolyte chemistry. The method typically deploys complex partial differential equations, leading to significant requirements of both memory and computational power. ECMs, on the other hand, employ circuit components with empirical nonlinear parameters \cite{online_ECM}. Compared to electrochemical models, ECMs use fewer inputs, considerably reducing the number of parameters required to be learnt over time, however, they have limited accuracy and robustness due to assumptions in battery behavior \cite{electrical_equiv_model_review}. Furthermore, in order to determine ECM model parameters, such as the ohmic resistance and the parallel resistor-capacitor impedance, at different state of charge values, pulse discharging \cite{pulse_1} and electrochemical impedance spectroscopy is typically necessary \cite{EIS_3}, \cite{EIS_1}, \cite{EIS_2}, however such measurements are not a viable solution for online applications.
Conversely, the data-driven approach displays a series of advantages such as a chemistry-agnostic modelling capability and an ability to analyse a wide range of degradation mechanisms and operating conditions, including rare loading events often overlooked by simplified models or physics-based simulations. To date, numerous studies have employed machine learning tools for the analysis of battery SOH estimation. Several studies \cite{birkl_degradation}, \cite{IC_full_curve_ensemble}, \cite{IC_curves_1}, \cite{IC_curves_2}, \cite{online_SOH_partical_IC} showed that incremental capacity (IC) and differential voltage (DV) curves, a method developed for use in cell aging mechanism analysis \cite{birkl_degradation}, can also be used for offline and online capacity fade estimation. However, the approach has several drawbacks linked to obtaining the IC and DV curves that substantially reduce its practicality. The differentiation of the capacity-voltage curve to obtain the IC curve amplifies noise and propagates it into the algorithm. Additionally, both curves must cover a sufficient voltage range for the method to work and, for obtaining a high curve fidelity, it is restricted to low charge current rates(1/5 to 1/25 C-rate) \cite{quick_online}, \cite{dubarry2006incremental}, \cite{weng_onboard_IC}. Unless a low current value is used during charging protocol and the key part of the capacity-voltage curve is recorded, such that specific peak points in the IC curve are captured, the method is impractical for online deployment.
An alternative is to train an algorithm on the raw voltage-time data curve, eliminating the need for differentiation \cite{charge_curve_gpr}, \cite{Howey_GPR}. Notably, Richardson et al. \cite{Howey_GPR} operated on sections of the voltage-time data itself by first smoothing the curve using a Savitsky-Golay filter and then used equispaced voltage values as the input to a Gaussian process regression (GPR) algorithm. However, GPR is considerably slow to train due to its computational cost of learning governed by the kernel function \cite{slow_GPR}, making it unsuitable for online deployment. The high computational complexity, also severely limits its scalability to incorporate bigger datasets. Additionally, the algorithm is sensitive to the section of the voltage-time curve used as input to the GPR. Other Bayesian-based methods, such as the relevance vector machine (RVM) \cite{bayesian_batterya}, have also been used to estimate battery capacity fade. Unfortunately, RVM also suffers from slow training, particularly when compared to frequentist-based algorithms \cite{accelerating_RVM}. Shen et al. \cite{slow_GPR} presented options for accelerating GPR, however, they compromise accuracy. In contrast to \cite{Howey_GPR} where the constant current part of the charging profile was used, Wang et al. \cite{CCCT_CVCT_Zhang} used the constant voltage section to estimate capacity fade using support vector regression (SVR). Although SVR is faster than GPR, it lacks the ability to estimate uncertainty. This inability to estimate uncertainty stemmed from various sources is a major limiting factor when discussing complex dynamic systems, such as Li-ion batteries. SOH assessment without corresponding measures of uncertainty associated with the estimation does not provide sufficient information to form a decision or corrective action plan \cite{prognostics_nasa_uncertainty}.
Previous work \cite{electrical_equiv_model_review}-\cite{CCCT_CVCT_Zhang} includes limited assessments of SOH uncertainty or none at all. The proposed machine learning pipeline is capable of real-time estimation of battery SOH and associated algorithm uncertainty referred to as battery health and uncertainty management pipeline (BHUMP). BHUMP operates by passing incoming data streams through a hierarchical sequence of processing steps by first engineering features based on segments of raw charge curves. It then performs offline automatic feature selection, augments the dataset with adversarial examples, and estimates battery health and associated uncertainty with the aid of four machine learning algorithms. Uncertainty is quantified based on calibration error and an adapted accuracy measure, the $\alpha$-$\beta$ accuracy zone. There are numerous battery designs \cite{review_batteries_2} and chemistries available \cite{review_batteries}, therefore the pipeline is deployed on a total of 179 cells, three designs (prismatic, pouch, and cylindrical), two chemistries (LiFePO$_{4}$, and LiCoO$_{2}$), three charge protocols (constant current, constant current - constant voltage, and 2-step fast-charge), and various discharge rates.
This paper refines and extensively tests new and improved machine learning algorithms for the capacity fade estimation problem, but also defines metrics for estimating and accurately quantifying uncertainty in ML models used in battery research. BHUMP provides battery researchers with a scalable SOH estimation solution that is adaptable to any cell chemistry and operating condition. BHUMP is more accurate than conventional methods as the battery is ageing, uses a set of engineered features capable of capturing battery intrinsic degradation, and is capable of estimating cell SOH in under 15 minutes at any point in its life-cycle. An accurate SOH method combined with a quantifiable metric for uncertainty propagation that feeds into SOC and run time calculations improves battery performance and ultimately extends cell lifetime.
\section{Machine learning pipeline approach}
\subsection{Pipeline overview}
From a machine learning perspective, determining battery capacity fade can be considered a multivariate supervised regression problem. We use a pipeline-based approach, where features are engineered from charge/discharge curves, on which a Bayesian or frequentist model is trained. Additionally, uncertainty is quantified by predicting a distribution mean and an associated standard deviation. Our learning method is divided into two stages, namely, Stage 1: Offline pipeline creation and training and Stage 2: Online SOH estimation. The offline stage ensures feature engineering, training data augmentation, automatic feature selection, algorithm training, and uncertainty calibration. The online stage diagnoses the cell using the trained pipeline under the assumptions that it is given a battery cell of unknown capacity. Supplementary Figure 1 provides a summary of the two stages via a flowchart of the method.
Feature engineering is split into automatic feature generation or extraction through techniques such as neural network auto-encoders \cite{auto_encoder_bearing}, \cite{deep_feature_sunthesis}, and manual feature construction based on domain knowledge \cite{feature_Nick}, \cite{online_F1_F2_Zhang}. We adopt a domain knowledge-based approach, where we show the algorithm feature choice based on importance to target variable. We also, provide a hypothesis for the underlying physical degradation quantified by the selected segments of the charge curves in Supplementary Note 1. Supplementary Table 1 summarize the attributes recorded during life cycle testing. The pipeline focuses on segments of the charge curves to capture degradation in the electrodes during cycling (Figure \ref{features_calce} illustrates typical extracted segments). The extracted charge-curve segments are further used in the feature engineering process (see Methods for details).
\begin{figure*}[h!]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}[t]{ccc}
\begin{tabular}{c
\smallskip
\begin{subfigure}[b]{.48\textwidth}
\centering
\caption{}
\includegraphics[width=\textwidth]{CC_charge_cell_no_11.pdf}
\label{thresholds_v_gr1}
\end{subfigure}\\
\begin{subfigure}[b]{.48\textwidth}
\centering
\caption{}
\includegraphics[width=\textwidth]{CV_charge_cell_no_11.pdf}
\label{thresholds_c_gr1}
\end{subfigure}
\end{tabular}
&
\begin{tabular}{c
\smallskip
\begin{subfigure}[b]{.45\textwidth}
\caption{}
\centering
\includegraphics[width=.9\textwidth]{CC_prismatic_pl_cell_no_11.pdf}
\label{b1}
\end{subfigure}\\
\begin{subfigure}[b]{.45\textwidth}
\centering
\caption{}
\includegraphics[width=.9\textwidth]{CV_prismatic_pl_cell_no_11.pdf}
\label{d1}
\end{subfigure}
\end{tabular}
&
\begin{subfigure}{0.5\textwidth}
\centering
\caption{}
\includegraphics[width=.47\textwidth]{heatmap_cell_no_11.pdf}
\label{e1}
\end{subfigure}
\end{tabular}%
}
\caption{\textbf{The CC-CV charge protocol and extracted ageing segment of the curves for a Li-ion pouch cell.} \textbf{\protect\subref{thresholds_v_gr1}} Voltage during charge protocol. \textbf{\protect\subref{thresholds_c_gr1}} Current during charge protocol, \textbf{\protect\subref{b1}} Extracted ageing voltage curve segments corresponding to marked grey area, \textbf{\protect\subref{d1}} Extracted ageing current curve segments corresponding to marked grey area, \textbf{\protect\subref{e1}} Heatmap of ageing with cycle number. Note: Refer to Methods section for abbreviations.}
\label{features_calce}
\end{figure*}
The pipeline creates a total of 30 features, and selects the most relevant features using a random forest based recursive feature elimination with cross-validation (RF-RFE-CV) similar to the one introduced in \cite{guyon_RFE_CV}. Recursive feature elimination generally outperforms other conventional methods \cite{RF_RFE}, \cite{RF_RFE_2}, hence the adoption here (refer to Methods section for further details). Before training the algorithms, we perform data augmentation by introducing adversarial examples as proposed by Goodfellow et al.~\cite{goodfellow_fgsm} in combination with the weight decay algorithm (see Methods). The use of adversarial examples in our datasets was motivated by the need to ameliorate the differences in battery design/chemistry. In addition, training on adversarial data makes the algorithm robust to outliers, prevents overfitting and reduces distribution variance around the estimated mean. Synthetic data generation generated from electrochemical models like the pseudo-two-dimensional model proposed by Doyle et. al. \cite{doyle1993modeling} can also be regarded as a data augmentation policy. Such an approach harnesses the potential of both electrochemical and data based models and we believe future work must incorporate synthetic data as well.
The augmented dataset then serves as the training input to four algorithms: random forest (RF) and deep neural network ensemble (dNNe), Bayesian ridge regression (BRR) and Gaussian process regression (GPR). Unlike Bayesian based algorithms, BRR and GPR, frequentist algorithms are unable to quantify uncertainty naturally due to their formulation. To overcome such limitations, we consider two modified ensemble based algorithms: RF with Infinitesimal Jackknife (IJ) based confidence intervals \cite{RF_CI} and the ensemble of neural networks as described in \cite{deep_ensamble}. For training of the algorithms a random search approach is used for hyper-parameter tuning \cite{random_hyper_tuinning}, with the exception of the deep ensemble where the Adam optimiser is used. We have found that drawing random samples from a uniform distribution works best for BRR and GPR parameters, whereas for RF and dNNe parameters random initialisation gives satisfactory results. In addition, a batch cross-validation method is used during the hyper-parameter tuning, where each batch is represented by one cell. This prevents the over-fitting of the models and mimics online deployment. Machine learning models in engineering require a stringent performance evaluation both from an error and uncertainty quantification perspective. The models are initially re-calibrated followed by an evaluation based on mean absolute percent error, root mean squared error and uncertainty estimation metrics (refer to Methods section for further details).
\subsection{Methods}
This study developed a pipeline approach for battery SOH estimation, called BHUMP and it incorporates a series of hierarchical steps, feature engineering, feature selection and data augmentation prior to model fitting and tuning as follows.
\textbf{Feature engineering} performs mathematical manipulations of extracted parts of the voltage curve during the constant current charge protocol based on a lower voltage threshold, $V_l$ and an upper voltage threshold, $V_h$ (refer to the grey area of Figure \ref{thresholds_v_gr1}) for all datasets except for cells charged with a 2-step fast-charge protocol. A characteristic to the 2-step fast charge protocol is that the cells can be charged from 0\% SOC to 80\% SOC with high currents ranging from 3.6 C-rate to 6 C-rate. In this work, due to the nature of the charging method in the 2-step fast charge, we only use the constant-current constant-voltage (CC-CV) charge part of the charging protocol as per the black dotted segments in the grey area observed in Supplementary Figures 2a, 2b. The values of $V_h$ and $V_l$ can be selected based on the intended application and the depth of discharge of the cell. In this work we select $V_h$ to be equal to cut-off voltage, $V_{cut-off}$. Refer to Supplementary Note 2 on how we select $V_l$. Additional features are developed on extracted segments of the current curve during the constant voltage charge protocol based on two current threshold values, $I_h$ and $I_l$ respectively (see Figure \ref{thresholds_c_gr1}) for all cells except for the 2-step fast charge protocol. We select $I_h$ equal to charge C-rate, while the lower threshold value, $I_l$, equal to a current drop of 40\% from $I_h$. This allows for sufficient data to be recorded while keeping the diagnostics time to a minimum. For cells cycled with the 2-step fast charge we select the current curve in Supplementary Figure 2b.
The obtained segments of voltage and current charge curves are further processed to obtain a plethora of features as described in Supplementary Note 3. Supplementary Table 2 summarises all features generated from processing the curves.
\textbf{Feature selection} with recursive feature elimination and cross-validation (RFE-CV) performs selection and subset reduction automatically without requirements of user-based thresholds, such as a maximum number of features to be selected. To suit battery data, we modify the original formulation by replacing the decision function algorithm with a random forest (RF) as opposed to the support vector machine (SVM) used in \cite{guyon_RFE_CV}. The replacement is motivated by RF's ability to deal with unscaled data. We call the resultant modified algorithm RF-RFE-CV. We use 700 decision tree estimators for the random forest algorithm and we set the number of cross-validations equal to the number of batteries in the feature selection dataset (see Supplementary Note 5 for data partition). We perform feature selection for each battery dataset based on a subset of the training data to avoid introducing optimistically biased performance estimates.
Battery SOH is quantified as capacity fade with reference to the first cycle as per equation \ref{SOH}, where $C_i$ represents capacity value at $i^{th}$ cycle and $C_1$ is the capacity at the first cycle measured by a complete charge-discharge operation.
\begin{equation}
SOH = \frac{C_i}{C_1}
\label{SOH}
\end{equation}
The role of the algorithm is to map from inputs $\mathbf{x}$ to target variable $y$ by means of a function $f(\mathbf{x}, \mathbf{\theta})$:
\begin{equation}
y = f(\mathbf{x}, \mathbf{\theta} ) + \epsilon
\label{general_model}
\end{equation}
where $\mathbf{\theta}$ are the model weights vector and $\epsilon \sim \mathcal{N}(0, \Sigma)$ is a normally distributed noise parameter. Based on the selected algorithm, the function $f(\mathbf{x}, \mathbf{\theta})$ may take different forms based on underlying assumptions of each algorithm. The learned model can then be used to make predictions of capacity given a test vector $\mathbf{x}^*$.
\textbf{Data augmentation} is carried out using the \emph{fast gradient sign method (FGSM)} in combination with the weight decay algorithm (ridge regression). We have found that a Ridge regularised model in combination with the FGSM was able to reduced the confidence interval (CI) around the estimated mean, despite being a simpler model than the original formulation in \cite{goodfellow_fgsm} which was based on a neural network. Given an input $\mathbf{x}$ with a target $y$ and loss $l(\mathbf{\theta}, \mathbf{x}, y)$, FGSM generates an adversarial example using:
\begin{equation}
\mathbf{x}_{adv}=\mathbf{x} + \gamma \cdot sign(\nabla_x l(\theta, \mathbf{x}, y))
\end{equation}
where $\gamma$ is a small value such that the max value of the perturbation is bounded and $\nabla_x$ is the gradient with respect to $\mathbf{x}$. Because each feature in the dataset has a different range, we set $\gamma$ to 0.01 or 1\% times the range of each feature vector. The adversarial examples are concatenated with the original training data to create a comprehensive training dataset. Note, other methods for data augmentations can also be used such as the ones proposed in \cite{deep_ensamble}, \cite{adversarial_regression}, \cite{ead_attack}, \cite{ead_attack_with_L1}, however the effect of data augmentation on model performance is beyond the scope of the present work.
The study solves eq. \ref{general_model} by making use of four algorithms as follows:
\textbf{Bayesian Ridge regression (BRR)} considers a probabilistic model of the regression problem. The algorithm estimates a spherical Gaussian prior over the model weights given by $p(\mathbf{\theta}|\lambda) = \mathcal{N}(\mathbf{\theta}|0, \lambda^{-1} \textbf{I}_p)$, where $\lambda^{-1}$ is the precision. The priors over $\alpha$ (the regulariser) and $\lambda$ are chosen to be gamma distributions. All parameters, $\mathbf{\theta}$, $\lambda$ and $\alpha$, are jointly estimated during training as per the implementation in \cite{scikit-learn}. Posterior inference can be performed in a closed-form because the prior is conjugate. For a complete explanation of the algorithm refer to \cite{bishop2006pattern}.
\textbf{Gaussian process regression (GPR)} is a nonparametric, Bayesian approach to regression defining a probability distribution over functions rather than random variables, thus eq. \ref{general_model} is solved by:
\begin{equation}
f(\mathbf{x}) \sim \mathcal{GP}(m(\mathbf{x}), k(\mathbf{x}, \mathbf{x}'))
\label{GPR}
\end{equation}
where $m(\mathbf{x})$ is the mean and $k(\mathbf{x}, \mathbf{x}')$ is the covariance function. Note, as defined above, GPR does not require learning the parameters of the regression function $f(\mathbf{x}, \mathbf{\theta} )$, in a traditional sense. The mean and covariance are defined by:
\begin{equation}
m(\mathbf{x}) = \mathbb{E}[f(\mathbf{x})]
\end{equation}
\begin{equation}
k(\mathbf{x}, \mathbf{x}') = \mathbb{E}[(f(\mathbf{x}-m(\mathbf{x}))(f(\mathbf{x}'-m(\mathbf{x}'))]
\end{equation}
GPR assigns a prior probability to every possible function, where higher probabilities are given to functions that the algorithm considers to be more likely, for example, because they are smoother than other functions. For our implementation, we make use of the standard radial basis kernel (RBF) as detailed in \cite{gpr_mit_rasmussen}, where a mathematical explanation of the algorithm is also given. Other kernel options exist, however, we do not explore the effect of kernel choice on algorithm performance.
\textbf{Random Forest (RF)} is a collection of constructed decision trees who sequentially conduct binary splits of the data to produce a homogeneous subset. For a comprehensive explanation of the algorithm refer to \cite{RF}. We adopt a bagging approach where the ensemble members are trained on different bootstrap samples of the training set and we set the number of decision trees in the forest to 1500. The variability of the predictions estimated by the random forest has been investigated based on the study from \cite{RF_CI}, where the confidence interval's variance has been obtained using the bootstrap replicates used to train the random forest itself.
\textbf{Deep ensemble of neural networks (dNNe)}. Ensemble methods combine different regressors into a meta-regressor and we consider an ensemble of deep neural networks as proposed in \cite{deep_ensamble}. Each network in the ensemble incorporates 2 hidden layers with an output of two layers one for the mean, $\mu(x)$ and the other for variance, $\sigma^2$ with $\sigma^2 > 0$. We use the negative log-likelihood as a function of the predicted mean and variance for scoring purposes. We also use a feed-forward architecture of 2 densely connected hidden layers. Each layer decreases in size by 50\% neurons based on the number of input features. When the input number features is less than 10, we force the network's hidden layers to 4 neurons in the first layer and 3 in the second layer. For example, when 18 input features are considered, the first hidden layer consists of 9 neurons, followed by 4 neurons in the second hidden layer. Each network used in this work has the following parameters: first hidden layer implies a ReLU activation function, followed by a Leaky ReLU for the second hidden layer and a Sigmoid function for the output layer. Additionally, we make use of Adam optimiser with a learning rate of 0.001 and a batch size equal to the number of cycles for each cell in the training set.
All models are evaluated based on mean absolute percent error (MAPE) and root mean squared error (RMSPE).
\begin{equation}
MAPE(y_i^*, y_i) = \frac{1}{N}\sum^{N}_{i=1}\frac{|y_i^*-y_i|}{y_i}
\label{MAPE}
\end{equation}
\begin{equation}
RMSPE(y_i^*, y_i) = \sqrt{\frac{1}{N}\sum^{N}_{i=1}\left(\frac{y_i^*-y_i}{y_i}\right)^2}
\label{RMSPE}
\end{equation}
where $y_i$ is the measured capacity value, $y_i^*$ is the estimated capacity value, and $n$ is the total number of samples.
In a regression setting, we obtain probabilistic forecasts using one of the algorithms described above through the estimation of a Gaussian distribution $\mathcal{N}(\mu, \sigma^2)$, where $\mu$ is the mean estimated capacity and $\sigma^2$ is the associated uncertainty quantified as variance. To evaluate the usefulness of predictive uncertainty for decision making, we create reliability diagnostics curves analogous to the work in \cite{calibration_DL_kuleshov}. To plot calibration curves, we divide each predicted confidence interval in $m$ confidence levels that are monotonically increasing on the interval $[0, 1]$ i.e. $0<p_1<p_2<...<p_m<1$. We then compute the empirical probability for each threshold by counting the frequency of true labels in each confidence level $p_m$. Mathematically this can be summarised as:
\begin{equation}
\hat{p}_m = \frac{|{y_n}|F_n(y_n)\leq p_m, n=1,...,N|}{N}
\end{equation}
Based on the reliability curve assessment, we then perform re-calibration using isotonic regression \cite{isotonic_regression}. A well-calibrated regressor should lie very close to the ideal diagonal curve, e.g. results Figure \ref{calibration_group1_dNNE}. We use the calibration score($C_{score}$) as a numerical score to describe the quality of the calibration when referenced to a 90\% confidence interval and sharpness ($Sh$) to describe average standard deviation.
\begin{equation}
C_{score} = \frac{1}{N}\sum_{i=1}^{N} \hat{p}_{m=90\%}
\label{c_error}
\end{equation}
Sharpness is calculated as an average of model output variance for each prediction and is given by:
\begin{equation}
Sh = \frac{1}{n} \sum_{i=1}^{n} \sigma_i
\label{sh}
\end{equation}
where $i$ is the sample number and $n$ is the the total number of sampels.
\begin{figure}[h!]
\centering
\includegraphics[width = .4\textwidth]{accuracy_zone.pdf}
\captionof{figure}{$\alpha$-accuracy zone and $\beta$ probability mass illustration.}
\label{accuracy_zone}
\end{figure}
We further propose an assessment of uncertainty prediction via prognostics performance metrics from an engineering point of view, adopted from \cite{prog_metrics_nasa}. First, we introduce the accuracy zone defined by a threshold, $\alpha$ (see Figure \ref{accuracy_zone}), which is calculated as a percentage error from the true capacity value, i.e. $ y \pm \alpha $. We select an $\alpha$ of $\pm$1.5\% (alpha can be adjusted based on intended application). Based on the frequency of predicted values residing in the accuracy zone, we calculate the $\alpha$-accuracy. Finally, we calculate the average probability mass of the prediction PDF within the $\alpha$ bounds called $\beta$, refer to Figure \ref{accuracy_zone}. Ideally, $\beta$ should be one, suggesting that the predicted confidence interval is small and encapsulates the entire $\alpha$-accuracy zone. Since $\alpha$ summarises the notion of \emph{desired accuracy}, $\alpha^+$ is the upper bound for estimates above the accuracy zone, and $\alpha^-$ represents low estimates or value residing under the desired accuracy zone. Depending on the application, both or any one of the low or high estimates may be undesirable. We chose to calculate the percentage of early predictions (estimates residing below the true label, the red line in Figure \ref{accuracy_zone}), denoted here by PEP, as a measure of algorithm uncertainty in a critical application scenario.
\section{Dataset}
\begin{table*}[h!]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}cccccccc@{}}
\toprule
Group{*} & I & I & I & I & I & II & III \\ \hline
Dataset & CALCE CS2 & CALCE CX2 & CALCE PL & NASA 5 & NASA11 & TRI & Oxford \\ \midrule
Manufacturer & Unknown & Unknown & Unknown & LG Chem & LG Chem & A123 Systems & Kokam \\ \midrule
Cathode \textsuperscript{***} & LiCoO$_{2}$ & LiCoO$_{2}$ & LiCoO$_{2}$ & LiCoO$_{2}$ & LiCoO$_{2}$ & LiFePO$_{4}$ & \begin{tabular}[c]{@{}c@{}}LiCoO$_{2}$ /\\ LiNiMnCoO$_{2}$\end{tabular} \\ \midrule
Form factor & Prismatic & Prismatic & Pouch & \begin{tabular}[c]{@{}c@{}}18650\\ Cylindrical\end{tabular} & \begin{tabular}[c]{@{}c@{}}18650\\ Cylindrical\end{tabular} & \begin{tabular}[c]{@{}c@{}}18650\\ Cylindrical\end{tabular} & Pouch \\ \midrule
\# cells & 6 & 6 & 2 & 8 & 25 & 124 & 8 \\ \midrule
Charge & CC-CV & CC-CV & CC-CV & CC-CV & CC-CV & Fast-charge & CC \\ \midrule
Discharge & 2 regimes & 2 regimes & 1 regime & 2 regimes & 7 regimes & 1 regime & 1 regime \\ \midrule
\multicolumn{8}{l}{ \begin{tabular}[c]{@{}l@{}}\textsuperscript{*}\footnotesize{Groups based on charge protocol}, \textsuperscript{**}\footnotesize{Toyota Research Institute}, \textsuperscript{***}\footnotesize{Information from manufacturer, not verified}\end{tabular}}
\end{tabular}%
}
\caption{Datasets overview. Note: refer to Supplementary Note 4 for data sources.}
\label{data_rep}
\end{table*}
We investigate the performance of BHUMP on a total of 179 Li-ion cells as referenced in Table \ref{data_rep}. The cells have been grouped into three categories based on the charging protocol used: constant current - constant voltage (CC-CV) protocol in Group I (47 cells), 2-step fast charge protocol in Group III (8 cells), and constant current (CC) protocol in Group II (124 cells). The separation is important for separate model training and feature selection, as well as model performance assessment on different charge protocols. A detailed explanation of each dataset used can be found in Supplementary Note 4.
\section{Algorithm performance}
\subsubsection{Group I data}
Subject to the previously described pipeline steps the feature selection algorithm, RF-RFE-CV chose 18 of the 30 engineered features as the optimum number of attributes for the cells in Group I (refer to Supplementary Figure 8a and Supplementary Table 3). From a threshold point of view, we select a $V_h$ of 4.2V for all batteries in this Group with an associated $V_l$ of 3.9V. Refer to Supplementary Note 5 for train/test partitions.
\begin{table}[h!]
\centering
\resizebox{.47\textwidth}{!}{%
\begin{tabular}{l|c|c|c|c|c|c|c|}
\cline{2-8}
& \multicolumn{1}{l|}{MAPE} & \multicolumn{1}{l|}{RMSPE} & \multicolumn{1}{l|}{$C_{score}$} & \multicolumn{1}{l|}{$Sh$} & \multicolumn{1}{l|}{$\alpha$-accuracy} & \multicolumn{1}{l|}{$\beta$} & \multicolumn{1}{l|}{$PEP$} \\ \hline
\multicolumn{1}{|l|}{BRR} & 1.52 & 2.49 & 84.49 & 0.021 & 70.00 & 0.57 & 68.92 \\ \hline
\multicolumn{1}{|l|}{GPR} & 1.49 & 2.24 & 92.23 & 0.025 & 65.00 & 0.48 & 71.76 \\ \hline
\multicolumn{1}{|l|}{RF} & 0.72 & 0.91 & 100 & 0.046 & 92.00 & 0.29 & 95.29 \\ \hline
\multicolumn{1}{|l|}{dNNe} & 0.65 & 0.92 & 88.01 & 0.0082 & 93.00 & 0.93 & 97.71\\ \hline
\end{tabular}%
}
\caption{Results for Group I cell no. 38.}
\label{results_group1_cell_38}
\end{table}
We illustrate results for BHUMP when dNNe is considered as base algorithm in Figure \ref{predictions_group1_dNNE} (results for all other algorithms are shown in Supplementary Figures 11, 12, 13) for a randomly chosen pouch cell battery, cell no. 38 and summarise algorithm performance on this cell in Table \ref{results_group1_cell_38}. The cell was cycled in full depth of discharge between 4.2V to 2.7V at a discharge C-rate of 0.5C (or 0.55 A) with a CC-CV charging protocol at a current value of 0.5 C-rate. Table \ref{results_group1} summarises each algorithms' performance on cell no. 38. Comparing dNNe in Figure~\ref{pred_group1_dNNe} to the other algorithms BRR, GPR, and RF, we show that the resultant confidence interval is considerably smaller (all figures display a confidence level equivalent to a 95\% quantile i.e. $ \mu \pm 2\cdot \sigma$). This indicates that the model is sharper, resulting in a high $\beta$ score (refer to Table \ref{results_group1_cell_38} for results). Where the predictions are less accurate, such as is the prediction in the first few cycles (see Figure~\ref{pred_group1_dNNe}), the error bars capture this variability well. On this battery, dNNe also achieves the best RMSPE and MAPE together with a high calibration score. As per Table \ref{results_group1_cell_38}, the estimates for this cell vary between RMSPE 0.65\% to 1.52\%, showing that all 4 algorithms can achieve high performance. The same conclusion is not valid for calibration, however. Reliability plots indicate that RF exhibits high variance even after calibration, refer to Supplementary Figure 13.
\begin{figure*}[h!]
\centering
\begin{subfigure}[b]{.55\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr1_prediction_dNNe.pdf}
\label{pred_group1_dNNe}
\end{subfigure
\begin{subfigure}[b]{.3\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr1_calibration_dNNe.pdf}
\label{calibration_group1_dNNE}
\end{subfigure}%
\hfill
\begin{subfigure}[b]{.55\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr1_prediction_vs_true_dNNe.pdf}
\label{prediction_vs_true_dNNe_group1}
\end{subfigure}%
\begin{subfigure}[b]{.3\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr1_histogram_dNNe.pdf}
\label{hist_group1_dNNE}
\end{subfigure}%
\hfill
\caption{\textbf{Prediction results with dNNe Group I cell no. 38.} \textbf{\protect\subref{pred_group1_dNNe}} dNNe prediction as a function of cycle, \textbf{\protect\subref{calibration_group1_dNNE}} dNNE calibration results, \textbf{\protect\subref{prediction_vs_true_dNNe_group1}} dNNe actual vs. predicted capacity, \textbf{\protect\subref{hist_group1_dNNE}} Histogram of \% error. Note: $y^*$ - true capacity, $\hat{y}^*$ - predicted capacity}
\label{predictions_group1_dNNE}
\end{figure*}
When discussing average results across all cells in Group I (Table \ref{results_group1}), RF achieves on average a low calibration error of 54.70\% possibly due to the method used for estimating the variance, Infinitesimal Jackknife. In practice we prefer a more conservative system, particularly in safety-critical applications. This implies that the number of capacity estimates lower than the true label residing in the $\alpha$-accuracy zone (Figure \ref{accuracy_zone}) should exceed the number of capacity values estimated above it i.e. PEP should be close to 100\%. At the same time, too low of a capacity estimate would result in a far too conservative algorithm. However, such behaviour is captured by an increase in RMSPE and thus mitigated for naturally. With reference to Figure \ref{hist_group1_dNNE} together with Table \ref{results_group1_cell_38} one can conclude that dNNe is conservative, achieving the highest PEP.
\begin{table}[h!]
\centering
\resizebox{.47\textwidth}{!}{%
\begin{tabular}{l|c|c|c|c|c|c|c|}
\cline{2-8}
& \multicolumn{1}{l|}{MAPE} & \multicolumn{1}{l|}{RMSPE} & \multicolumn{1}{l|}{$C_{score}$} & \multicolumn{1}{l|}{$Sh$} & \multicolumn{1}{l|}{$\alpha$-accuracy} & \multicolumn{1}{l|}{$\beta$} & \multicolumn{1}{l|}{$PEP$} \\ \hline
\multicolumn{1}{|l|}{BRR} & 4.65 & 5.54 & 89.16 & 0.104 & 25.76 & 0.25 & 36.57 \\ \hline
\multicolumn{1}{|l|}{GPR} & 3.70 & 4.51 & 83.62 & 0.089 & 32.04 & 0.29 & 60.07 \\ \hline
\multicolumn{1}{|l|}{RF} & 2.17 & 2.70 & 54.70 & 0.093 & 35.94 & 0.36 & 65.47 \\ \hline
\multicolumn{1}{|l|}{dNNe} & 3.30 & 4.26 & 86.28 & 0.043& 32.14 & 0.58 & 63.26 \\ \hline
\end{tabular}%
}
\caption{Average results over Group I dataset. }
\label{results_group1}
\end{table}
Overall, despite RF achieving the lowest average RMSPE and MAPE (Table \ref{results_group1}) it does not output well-calibrated predictions, nor it displays a high sharpness value. At the expense of 1.13\% in MAPE and 1.56\% in RMSPE, the dNNe outputs a well-calibrated model, on average being less than 4\% under the ideal calibration score.
\subsubsection{Group II data}
Group II dataset is the largest dataset incorporating a total of 124 cells. While the dataset exhibits a high variance in charge profiles, it does not have any variation in discharge conditions (all cells in the dataset are discharged at 4 C-rate). This, in turn, showcases the effect of the charge profile on the estimation accuracy of the 4 algorithms. Training is performed on features engineered based on the CC-CV curve obtained after the cell reaches 80\% SOC (refer to Supplementary Figures 2a and 2b). Refer to Supplementary Note 5 for train/test partitions. RF-RFE-CV selects a total of 5 features (Supplementary Figure 8b and Supplementary Table 4) out of a total of 30 engineered features. We believe this is caused by the fact that the dataset only incorporates one discharge profile as well as just a single battery type.
\begin{table}[ht!]
\centering
\resizebox{.47\textwidth}{!}{%
\begin{tabular}{l|c|c|c|c|c|c|c|}
\cline{2-8}
& \multicolumn{1}{l|}{MAPE} & \multicolumn{1}{l|}{RMSPE} & \multicolumn{1}{l|}{$C_{score}$} & \multicolumn{1}{l|}{$Sh$} & \multicolumn{1}{l|}{$\alpha$-accuracy} & \multicolumn{1}{l|}{$\beta$} & \multicolumn{1}{l|}{$PEP$} \\ \hline
\multicolumn{1}{|l|}{BRR} & 0.72 & 0.90 & 65.49 & 0.005 & 89.00 & 98.00 & 20.70 \\ \hline
\multicolumn{1}{|l|}{GPR} & 1.23 & 1.63 & 69.94 & 0.011 & 65.00 & 85.00 & 22.16 \\ \hline
\multicolumn{1}{|l|}{RF} & 0.23 & 0.43 & 87.42 & 0.002 & 98.00 & 100 & 42.81 \\ \hline
\multicolumn{1}{|l|}{dNNe} & 0.34 & 0.48 & 71.31 & 0.002 & 98.00 & 100 & 31.50 \\ \hline
\end{tabular}%
}
\caption{Results for Group II cell no. 1.}
\label{results_group2_cell_1}
\end{table}
\begin{figure*}[h!]
\centering
\begin{subfigure}[b]{.55\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr2_prediction_dNNe.pdf}
\label{pred_group2_dNNe}
\end{subfigure
\begin{subfigure}[b]{.3\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr2_calibration_dNNe.pdf}
\label{calibration_group2_dNNE}
\end{subfigure}%
\hfill
\begin{subfigure}[b]{.55\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr2_prediction_vs_true_dNNe.pdf}
\label{prediction_vs_true_dNNe_group2}
\end{subfigure}%
\begin{subfigure}[b]{.3\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr2_histogram_dNNe.pdf}
\label{hist_group2_dNNE}
\end{subfigure}%
\hfill
\caption{\textbf{Prediction results with dNNe Group II cell no. 1.} \textbf{\protect\subref{pred_group2_dNNe}} dNNe prediction as a function of cycle, \textbf{\protect\subref{calibration_group2_dNNE}} dNNE calibration results, \textbf{\protect\subref{prediction_vs_true_dNNe_group2}} dNNe actual vs. predicted capacity, \textbf{\protect\subref{hist_group2_dNNE}} Histogram of \% error. Note: $y^*$ - true capacity, $\hat{y}^*$ - predicted capacity}
\label{predictions_group2_dNNE}
\end{figure*}
Figure \ref{predictions_group2_dNNE} illustrate BHUMP performance with a dNNe as base algorithm for cell no. 1, whilst Suplementary Figures 14, 15, 16 summarise results for all other algorithms. The cell has undergone fast charge profile of 3.6 C-rate to 80\% SOC, beyond which the cell is charged with CC of 1C followed by the CV charging. The reason cell 1 was selected in this case was to illustrate the performance of the algorithms when there is a high number of outliers in capacity data (Figure \ref{pred_group2_dNNe}). With reference to Table \ref{results_group2_cell_1}, RF achieves lowest error and highest scores as well as a good calibration compared to all other algorithms. On this particular cell, dNNe achieves the second best performance, however it does not output a well calibrated model, despite showing a good average calibration score as per Table \ref{results_group2}.
Average results of the 4 algorithms are concisely summarised in Table \ref{results_group2}. All models are able to estimate the SOH with less than 2\% RMPSE; this underlines the fact that the models are not affected by the fast-charge section of the charging protocol. RF achieves the highest accuracy with a low sharpness value and high percentages for all other metrics except for calibration where it exhibits over-confidence. In terms of calibration error, dNNe achieves the closest score to a 90\% confidence interval with 91.02\%. dNNe is also the second-best performing algorithm achieving good scores across all metrics as summarised in Table \ref{results_group2_cell_1}. In comparison, the two Bayesian-based algorithms exhibit a higher percentage error as well as higher sharpness values. However, they tend to be more conservative, averaging a PEP over 60\%.
\begin{table}[h!]
\centering
\resizebox{.47\textwidth}{!}{%
\begin{tabular}{l|c|c|c|c|c|c|c|}
\cline{2-8}
& \multicolumn{1}{l|}{MAPE} & \multicolumn{1}{l|}{RMSPE} & \multicolumn{1}{l|}{$C_{score}$} & \multicolumn{1}{l|}{$Sh$} & \multicolumn{1}{l|}{$\alpha$-accuracy} & \multicolumn{1}{l|}{$\beta$} & \multicolumn{1}{l|}{$PEP$} \\ \hline
\multicolumn{1}{|l|}{BRR} & 0.45 & 0.76 & 91.72 & 0.005 & 97.31 & 99.19 & 62.86 \\ \hline
\multicolumn{1}{|l|}{GPR} & 1.00 & 1.91 & 93.14 & 0.012 & 90.43 & 83.74 & 63.21 \\ \hline
\multicolumn{1}{|l|}{RF} & 0.11 & 0.14 & 79.72 & 0.001 & 99.84 & 99.96 & 58.77 \\ \hline
\multicolumn{1}{|l|}{dNNe} & 0.23 & 0.45 & 91.02 & 0.002 & 99.53 & 99.50 & 53.41 \\ \hline
\end{tabular}%
}
\caption{Average results over Group II dataset}
\label{results_group2}
\end{table}
In conclusion, from an accuracy and sharpness perspective, the best performing algorithm on dataset Group II is RF, whilst the poorest performance is achieved by GPR. When it comes to uncertainty metrics, and in particular calibration, RF exhibits over-confidence with a $C_{score}$ of 79.72\%. Such behaviour is also identified in Group I dataset where RF was, in fact, difficult to calibrate despite the rich dataset. A more reliable calibration score is achieved by dNNe at the expense of a loss of 0.12\% in MAPE and 0.31\% in RMSPE (refer to Table \ref{results_group2}).
\subsubsection{Group III data}
On Group III we emphasise on the suitability of BHUMP to battery state of health estimation for automotive applications. Group III includes 8 Kokham 740 mAh batteries that have been dynamically cycled under the ARTEMIS \cite{andre2004artemis} dynamic driving
profile, followed by characterisation cycles. Each characterisation cycle consists of low rate CC charge and discharge cycles, repeated every 100 cycles. We use the characterisation cycles for diagnostics purposes to derive features and estimate battery health. This dataset incorporates the lowest variability both in terms of input feature values and capacity degradation values due to the identical charge-discharge conditions. This, in turn, affects feature selection as BHUMP only selects 5 out of the 18 engineered features (note charge protocol does not include CV part of the charge, hence 12 features are missing) as shown in Supplementary Figure 8c and Supplementary Table 5. We keep the same threshold values as in Group I cells for the CC part of the curves, namely a $V_h$ of 4.2V and a $V_l$ of 3.9V on which feature are engineered. Refer to Supplementary Note 5 for train/test partitions.
\begin{figure*}[h!]
\centering
\begin{subfigure}[b]{.55\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr3_prediction_dNNe.pdf}
\label{pred_group3_dNNe}
\end{subfigure
\begin{subfigure}[b]{.3\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr3_calibration_dNNe.pdf}
\label{calibration_group3_dNNE}
\end{subfigure}%
\hfill
\begin{subfigure}[b]{.57\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr3_prediction_vs_true_dNNe.pdf}
\label{prediction_vs_true_dNNe_group3}
\end{subfigure}%
\begin{subfigure}[b]{.3\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr3_histogram_dNNe.pdf}
\label{hist_group3_dNNE}
\end{subfigure}%
\hfill
\caption{\textbf{Prediction results with dNNe Group III cell no. 5.} \textbf{\protect\subref{pred_group3_dNNe}} dNNe prediction as a function of cycle, \textbf{\protect\subref{calibration_group3_dNNE}} dNNE calibration results, \textbf{\protect\subref{prediction_vs_true_dNNe_group3}} dNNe actual vs. predicted capacity, \textbf{\protect\subref{hist_group3_dNNE}} Histogram of \% error. Note: $y^*$ - true capacity, $\hat{y}^*$ - predicted capacity}
\label{predictions_group3_dNNE}
\end{figure*}
\begin{table}[ht!]
\centering
\resizebox{.47\textwidth}{!}{%
\begin{tabular}{l|c|c|c|c|c|c|c|}
\cline{2-8}
& \multicolumn{1}{l|}{MAPE} & \multicolumn{1}{l|}{RMSPE} & \multicolumn{1}{l|}{$C_{score}$} & \multicolumn{1}{l|}{$Sh$} & \multicolumn{1}{l|}{$\alpha$-accuracy} & \multicolumn{1}{l|}{$\beta$} & \multicolumn{1}{l|}{$PEP$} \\ \hline
\multicolumn{1}{|l|}{BRR} & 0.11 & 0.15 & 95.55 & 0.89 & 100 & 100 & 31.11 \\ \hline
\multicolumn{1}{|l|}{GPR} & 0.16 & 0.19 & 71.11 & 1.21 & 100 & 100 & 15.55 \\ \hline
\multicolumn{1}{|l|}{RF} & 0.17 & 0.21 & 97.77 & 2.01 & 100 & 100 & 24.44 \\ \hline
\multicolumn{1}{|l|}{dNNe} & 0.20 & 0.25 & 100.00 & 2.93 & 100 & 100 & 6.67 \\ \hline
\end{tabular}%
}
\caption{Results for Group III cell no. 5.}
\label{results_group3_cell_5}
\end{table}
For visualisation purposes, we illustrate results for the randomly selected cell no. 5 for dNNe in Figure \ref{predictions_group3_dNNE} and Supplementary Figures 17, 18, 19 for all other algorithms. It is clear, from Table \ref{results_group3_cell_5} that performance on cell 5 is dominated by BRR based on all measures of accuracy and uncertainty quantification. However, all algorithms deployed on cell no. 5 (Table \ref{results_group3_cell_5}) achieve a MAPE and RMSPE smaller than the proposed accuracy zone threshold $\alpha$ of $\pm$1.5\%.
Average results are summarised in Table \ref{results_group3}. In terms of accuracy measures, on average, BRR outperforms all other methods, including the dNNe. As argued in \cite{ann_sample_size} linear regression outperforms considerably more complex algorithms, including NNs when dealing with small sample size that exhibits little variance. Despite the low error, BRR does not achieve a good calibration score as opposed to dNNE. dNNe is the second-best performing algorithm in terms of accuracy (MAPE and RMSPE). It also exhibits adequate results for all other metrics, including $PEP$ where it scores the highest.
\begin{table}[h!]
\centering
\resizebox{.47\textwidth}{!}{%
\begin{tabular}{l|c|c|c|c|c|c|c|}
\cline{2-8}
& \multicolumn{1}{l|}{MAPE} & \multicolumn{1}{l|}{RMSPE} & \multicolumn{1}{l|}{$C_{score}$} & \multicolumn{1}{l|}{$Sh$} & \multicolumn{1}{l|}{$\alpha$-accuracy} & \multicolumn{1}{l|}{$\beta$} & \multicolumn{1}{l|}{$PEP$} \\ \hline
\multicolumn{1}{|l|}{BRR} & 0.26 & 0.32 & 68.11 & 1.20 & 100 & 100 & 23.54 \\ \hline
\multicolumn{1}{|l|}{GPR} & 0.52 & 0.65 & 42.42 & 2.37 & 90.50 & 97.25 & 23.22 \\ \hline
\multicolumn{1}{|l|}{RF} & 0.36 & 0.44 & 72.62 & 2.16 & 88.5 & 100 & 25.44 \\ \hline
\multicolumn{1}{|l|}{dNNe} & 0.30 & 0.39 & 91.17 & 2.01 & 98.25 & 99.75 & 27.95 \\ \hline
\end{tabular}%
}
\caption{Average results over Group III dataset}
\label{results_group3}
\end{table}
In conclusion, when considering average results over all 4 test cells as referenced in Table \ref{results_group3}, dNNe achieves second-best accuracy while attaining the best calibration score of 91.17\%.
\section{Discussion on practical applicability of BHUMP}
BHUMP can complement battery management systems (BMS), for both SOC and SOH estimation, and replace the traditional ECMs altogether. While conventional approaches rely on measuring the capacity in static conditions such as full charge-discharge, BHUMP can estimate capacity fade from sections of the charge profile, accommodating for partial discharge scenarios or various operating conditions such as random or high discharge rates. We succinctly summarised in the results section, BHUMP can estimate capacity fade under fast charging protocol (Group II data) as well as random discharge (Group III data cycled under ARTEMIS driving protocol) typical to the operation of an EV battery pack. Future work could further extend to other charge-discharge protocols and open-source datasets such as the one in \cite{attia2020closed}.
Temperature variations during charging could further introduce uncertainty into the measurement of charge curves and propagate it into the estimation algorithm. Possible mitigation includes the use of temperature as an input when training BHUMP or considering additionally in-situ or operando sensory information such as optical and digital images or X-ray \cite{handoko2018understanding} such that the algorithm learns the correlation between temperature, generated features and SOH indicator. Due to such variations, SOH assessment without corresponding measures of algorithm uncertainty does not provide sufficient information to form a decision or corrective action. In addition to inherent algorithm bias, dataset variability also seems to affect the prediction error. To accommodate for such variations in the data BHUMP introduces 30 engineered features and makes use of an unsupervised feature selection algorithm (RF-RFE-CV). Given a training dataset RF-RFE-CV selects a subset of input features, indicating that features must be selected based on intended application, battery design and charge protocol. Despite such dataset variations, we think that deep learning has the potential to exceed it in the future as it requires little tuning from the user and can take advantage of parallelisation and an increasing amount of computational capabilities by deployment on graphics processing units (GPU) and modern data storage solutions. In addition, when training data consists of limited samples or training data is not relevant to the intended application, transfer learning can be used to reduce prediction errors. New hardware, architectures and learning algorithms that are currently being developed for neural network implementation will only accelerate this process, allowing for active learning techniques to be used when deployed onboard a vehicle. More concretely, BHMUP with dNNe as the base algorithm can incorporate transfer learning when trained on a particular cell design and re-trained on a reduced sample set for a different cell design. Additionally, BHUMP can also incorporate active learning as data becomes available when deployed online on different cell design, chemistry or operating temperature.
\section{Conclusion}
The two widely adopted modelling techniques for online battery state of health (SOH) estimation are equivalent circuit models and electrochemical models. However, when deployed online, the trade-off between accuracy and computational efficiency is difficult to achieve. This paper introduced an alternative, machine learning-based solution called battery health and uncertainty management pipeline (BHUMP). The pipeline provides a set of benefits over conventional methods including adaptability to the charging protocols and the discharge current rates, and prediction without knowledge of battery design, chemistry, and operating temperature.
The paper explores four algorithms: Bayesian ridge regression (BRR), Gaussian process regression (GPR), random forest (RF), and a deep ensemble of neural networks (dNNe), as the base algorithm for BHUMP. All algorithms are assessed on error values and the ability to quantify uncertainty. Results indicate that the lowest error achieved depends on the charging protocol adopted. The lowest error was achieved by RF for constant current - constant voltage protocol and fast charge protocol, and BRR for the constant-current protocol. When considering uncertainty assessment metrics, however, RF is hard to calibrate and is overly optimistic in its predictions. At the expense of an average increase in MAPE of 0.43\% and RMSPE of 0.97\%, dNNe, generally achieves a better calibration score, consistently achieving the second-lowest error irrespective of charge protocol. On the fast-charging protocol, the best dNNe model achieved a RMSPE of 0.45\% with a calibration score of 91.02\% when referenced to a 90\% confidence interval.
Overall, our work highlights the value of coupling machine learning tools with charge curve segments in capturing battery degradation in under 15 minutes. Moreover, we argue that despite achieving low errors, any algorithm must undergo uncertainty quantification checks before deployment in the field. Finally, we show how the use of machine learning pipelines can achieve a computationally efficient and accurate solution for cell SOH estimation. We envision machine learning pipelines to be a standard technique used in designing and implementing battery management systems of the future.
\section*{Data availability}
The datasets used in this study are available at:
\begin{itemize}
\item Group 1:
\url{https://web.calce.umd.edu/batteries/data.htm}
\url{https://ti.arc.nasa.gov/tech/dash/groups/pcoe/prognostic-data-repository/}
\item Group 2:
\url{https://data.matr.io/1/projects/5c48dd2bc625d700019f3204}
\item Group 3:
\url{https://ora.ox.ac.uk/objects/uuid:03ba4b01-cfed-46d3-9b1a-7d4a7bdf6fac}
\end{itemize}
\section*{Code availability}
\gappto{\UrlBreaks}{\UrlOrds}
Code for the data processing is available from the corresponding authors upon request. Code for the modelling work is available at: \url{http://doi.org/10.5281/zenodo.4390152}
\bibliographystyle{unsrt}
\bibliographystyle{unsrt}
\section{Introduction}
Rechargeable Li-ion batteries play a crucial role in many modern-day applications ranging from portable electronics and medical devices, to renewable energy integration in power grids and electric vehicles. The steep decrease in the price of lithium-ion based battery storage by 73\% from 2010 to 2016, to an all-time low of \$273$/$kWh in 2017 \cite{bloomberg_report} opened up a significant energy storage market evaluated at \$65 billion in 2017 \cite{bernhart2019challenges}. Irrespective of the application, Li-ion batteries degrade with time. With ageing, cells exhibit a loss of capacity and an increase in impedance. The rate of degradation is influenced by the dynamic operating conditions, including varying charge/discharge rates, different voltage operation limits and temperature fluctuations. The ability to estimate degradation in real-time irrespective of the various failure mechanisms and their degradation paths is crucial for safe and effective battery management systems \cite{ev_diagnosis}. Battery state of health can be used to predict battery's expected lifetime, however, the feasibility of online state of health estimation via direct measurement of chemical reaction parameters inside batteries remains limited \cite{review_intro}.
State of health (SOH) is a parameter that quantifies the general condition of a battery and its ability to deliver the specified performance, measured as capacity or impedance, when compared to its unused state. This work focuses on the battery capacity as the health indicator due to its correlation to the energy storage capability of batteries and its direct impact on the remaining run time and life of the batteries. Capacity fade estimation has received considerable research interest from industry and academia \cite{review_intro}, \cite{battery_review_1}, \cite{battery_review_2}, \cite{battery_review_3} and a number of methods have been proposed. The current approaches to capacity fade estimation involves parameter estimation using either of the following modelling types, equivalent circuit models (ECMs) \cite{electrical_equiv_model_review}, \cite{online_ECM}, \cite{EIS_3}, electrochemical models \cite{electrochemistry_model_nasa}, \cite{electro_model_bole}, \cite{parama_chemical_model}, or data-driven models \cite{severson_satandford}, \cite{RUl_prognostics_NASA}, \cite{prognostics_NASA}, \cite{batery_health_bayesin_SVM}, \cite{SVM_battery_health}, \cite{attia2020closed}. Electrochemical models approximate the chemical processes that take place inside a battery cell during operation. This type of modeling requires detailed cell specifications, such as electrode materials and electrolyte chemistry. The method typically deploys complex partial differential equations, leading to significant requirements of both memory and computational power. ECMs, on the other hand, employ circuit components with empirical nonlinear parameters \cite{online_ECM}. Compared to electrochemical models, ECMs use fewer inputs, considerably reducing the number of parameters required to be learnt over time, however, they have limited accuracy and robustness due to assumptions in battery behavior \cite{electrical_equiv_model_review}. Furthermore, in order to determine ECM model parameters, such as the ohmic resistance and the parallel resistor-capacitor impedance, at different state of charge values, pulse discharging \cite{pulse_1} and electrochemical impedance spectroscopy is typically necessary \cite{EIS_3}, \cite{EIS_1}, \cite{EIS_2}, however such measurements are not a viable solution for online applications.
Conversely, the data-driven approach displays a series of advantages such as a chemistry-agnostic modelling capability and an ability to analyse a wide range of degradation mechanisms and operating conditions, including rare loading events often overlooked by simplified models or physics-based simulations. To date, numerous studies have employed machine learning tools for the analysis of battery SOH estimation. Several studies \cite{birkl_degradation}, \cite{IC_full_curve_ensemble}, \cite{IC_curves_1}, \cite{IC_curves_2}, \cite{online_SOH_partical_IC} showed that incremental capacity (IC) and differential voltage (DV) curves, a method developed for use in cell aging mechanism analysis \cite{birkl_degradation}, can also be used for offline and online capacity fade estimation. However, the approach has several drawbacks linked to obtaining the IC and DV curves that substantially reduce its practicality. The differentiation of the capacity-voltage curve to obtain the IC curve amplifies noise and propagates it into the algorithm. Additionally, both curves must cover a sufficient voltage range for the method to work and, for obtaining a high curve fidelity, it is restricted to low charge current rates(1/5 to 1/25 C-rate) \cite{quick_online}, \cite{dubarry2006incremental}, \cite{weng_onboard_IC}. Unless a low current value is used during charging protocol and the key part of the capacity-voltage curve is recorded, such that specific peak points in the IC curve are captured, the method is impractical for online deployment.
An alternative is to train an algorithm on the raw voltage-time data curve, eliminating the need for differentiation \cite{charge_curve_gpr}, \cite{Howey_GPR}. Notably, Richardson et al. \cite{Howey_GPR} operated on sections of the voltage-time data itself by first smoothing the curve using a Savitsky-Golay filter and then used equispaced voltage values as the input to a Gaussian process regression (GPR) algorithm. However, GPR is considerably slow to train due to its computational cost of learning governed by the kernel function \cite{slow_GPR}, making it unsuitable for online deployment. The high computational complexity, also severely limits its scalability to incorporate bigger datasets. Additionally, the algorithm is sensitive to the section of the voltage-time curve used as input to the GPR. Other Bayesian-based methods, such as the relevance vector machine (RVM) \cite{bayesian_batterya}, have also been used to estimate battery capacity fade. Unfortunately, RVM also suffers from slow training, particularly when compared to frequentist-based algorithms \cite{accelerating_RVM}. Shen et al. \cite{slow_GPR} presented options for accelerating GPR, however, they compromise accuracy. In contrast to \cite{Howey_GPR} where the constant current part of the charging profile was used, Wang et al. \cite{CCCT_CVCT_Zhang} used the constant voltage section to estimate capacity fade using support vector regression (SVR). Although SVR is faster than GPR, it lacks the ability to estimate uncertainty. This inability to estimate uncertainty stemmed from various sources is a major limiting factor when discussing complex dynamic systems, such as Li-ion batteries. SOH assessment without corresponding measures of uncertainty associated with the estimation does not provide sufficient information to form a decision or corrective action plan \cite{prognostics_nasa_uncertainty}.
Previous work \cite{electrical_equiv_model_review}-\cite{CCCT_CVCT_Zhang} includes limited assessments of SOH uncertainty or none at all. The proposed machine learning pipeline is capable of real-time estimation of battery SOH and associated algorithm uncertainty referred to as battery health and uncertainty management pipeline (BHUMP). BHUMP operates by passing incoming data streams through a hierarchical sequence of processing steps by first engineering features based on segments of raw charge curves. It then performs offline automatic feature selection, augments the dataset with adversarial examples, and estimates battery health and associated uncertainty with the aid of four machine learning algorithms. Uncertainty is quantified based on calibration error and an adapted accuracy measure, the $\alpha$-$\beta$ accuracy zone. There are numerous battery designs \cite{review_batteries_2} and chemistries available \cite{review_batteries}, therefore the pipeline is deployed on a total of 179 cells, three designs (prismatic, pouch, and cylindrical), two chemistries (LiFePO$_{4}$, and LiCoO$_{2}$), three charge protocols (constant current, constant current - constant voltage, and 2-step fast-charge), and various discharge rates.
This paper refines and extensively tests new and improved machine learning algorithms for the capacity fade estimation problem, but also defines metrics for estimating and accurately quantifying uncertainty in ML models used in battery research. BHUMP provides battery researchers with a scalable SOH estimation solution that is adaptable to any cell chemistry and operating condition. BHUMP is more accurate than conventional methods as the battery is ageing, uses a set of engineered features capable of capturing battery intrinsic degradation, and is capable of estimating cell SOH in under 15 minutes at any point in its life-cycle. An accurate SOH method combined with a quantifiable metric for uncertainty propagation that feeds into SOC and run time calculations improves battery performance and ultimately extends cell lifetime.
\section{Machine learning pipeline approach}
\subsection{Pipeline overview}
From a machine learning perspective, determining battery capacity fade can be considered a multivariate supervised regression problem. We use a pipeline-based approach, where features are engineered from charge/discharge curves, on which a Bayesian or frequentist model is trained. Additionally, uncertainty is quantified by predicting a distribution mean and an associated standard deviation. Our learning method is divided into two stages, namely, Stage 1: Offline pipeline creation and training and Stage 2: Online SOH estimation. The offline stage ensures feature engineering, training data augmentation, automatic feature selection, algorithm training, and uncertainty calibration. The online stage diagnoses the cell using the trained pipeline under the assumptions that it is given a battery cell of unknown capacity. Supplementary Figure 1 provides a summary of the two stages via a flowchart of the method.
Feature engineering is split into automatic feature generation or extraction through techniques such as neural network auto-encoders \cite{auto_encoder_bearing}, \cite{deep_feature_sunthesis}, and manual feature construction based on domain knowledge \cite{feature_Nick}, \cite{online_F1_F2_Zhang}. We adopt a domain knowledge-based approach, where we show the algorithm feature choice based on importance to target variable. We also, provide a hypothesis for the underlying physical degradation quantified by the selected segments of the charge curves in Supplementary Note 1. Supplementary Table 1 summarize the attributes recorded during life cycle testing. The pipeline focuses on segments of the charge curves to capture degradation in the electrodes during cycling (Figure \ref{features_calce} illustrates typical extracted segments). The extracted charge-curve segments are further used in the feature engineering process (see Methods for details).
\begin{figure*}[h!]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}[t]{ccc}
\begin{tabular}{c
\smallskip
\begin{subfigure}[b]{.48\textwidth}
\centering
\caption{}
\includegraphics[width=\textwidth]{CC_charge_cell_no_11.pdf}
\label{thresholds_v_gr1}
\end{subfigure}\\
\begin{subfigure}[b]{.48\textwidth}
\centering
\caption{}
\includegraphics[width=\textwidth]{CV_charge_cell_no_11.pdf}
\label{thresholds_c_gr1}
\end{subfigure}
\end{tabular}
&
\begin{tabular}{c
\smallskip
\begin{subfigure}[b]{.45\textwidth}
\caption{}
\centering
\includegraphics[width=.9\textwidth]{CC_prismatic_pl_cell_no_11.pdf}
\label{b1}
\end{subfigure}\\
\begin{subfigure}[b]{.45\textwidth}
\centering
\caption{}
\includegraphics[width=.9\textwidth]{CV_prismatic_pl_cell_no_11.pdf}
\label{d1}
\end{subfigure}
\end{tabular}
&
\begin{subfigure}{0.5\textwidth}
\centering
\caption{}
\includegraphics[width=.47\textwidth]{heatmap_cell_no_11.pdf}
\label{e1}
\end{subfigure}
\end{tabular}%
}
\caption{\textbf{The CC-CV charge protocol and extracted ageing segment of the curves for a Li-ion pouch cell.} \textbf{\protect\subref{thresholds_v_gr1}} Voltage during charge protocol. \textbf{\protect\subref{thresholds_c_gr1}} Current during charge protocol, \textbf{\protect\subref{b1}} Extracted ageing voltage curve segments corresponding to marked grey area, \textbf{\protect\subref{d1}} Extracted ageing current curve segments corresponding to marked grey area, \textbf{\protect\subref{e1}} Heatmap of ageing with cycle number. Note: Refer to Methods section for abbreviations.}
\label{features_calce}
\end{figure*}
The pipeline creates a total of 30 features, and selects the most relevant features using a random forest based recursive feature elimination with cross-validation (RF-RFE-CV) similar to the one introduced in \cite{guyon_RFE_CV}. Recursive feature elimination generally outperforms other conventional methods \cite{RF_RFE}, \cite{RF_RFE_2}, hence the adoption here (refer to Methods section for further details). Before training the algorithms, we perform data augmentation by introducing adversarial examples as proposed by Goodfellow et al.~\cite{goodfellow_fgsm} in combination with the weight decay algorithm (see Methods). The use of adversarial examples in our datasets was motivated by the need to ameliorate the differences in battery design/chemistry. In addition, training on adversarial data makes the algorithm robust to outliers, prevents overfitting and reduces distribution variance around the estimated mean. Synthetic data generation generated from electrochemical models like the pseudo-two-dimensional model proposed by Doyle et. al. \cite{doyle1993modeling} can also be regarded as a data augmentation policy. Such an approach harnesses the potential of both electrochemical and data based models and we believe future work must incorporate synthetic data as well.
The augmented dataset then serves as the training input to four algorithms: random forest (RF) and deep neural network ensemble (dNNe), Bayesian ridge regression (BRR) and Gaussian process regression (GPR). Unlike Bayesian based algorithms, BRR and GPR, frequentist algorithms are unable to quantify uncertainty naturally due to their formulation. To overcome such limitations, we consider two modified ensemble based algorithms: RF with Infinitesimal Jackknife (IJ) based confidence intervals \cite{RF_CI} and the ensemble of neural networks as described in \cite{deep_ensamble}. For training of the algorithms a random search approach is used for hyper-parameter tuning \cite{random_hyper_tuinning}, with the exception of the deep ensemble where the Adam optimiser is used. We have found that drawing random samples from a uniform distribution works best for BRR and GPR parameters, whereas for RF and dNNe parameters random initialisation gives satisfactory results. In addition, a batch cross-validation method is used during the hyper-parameter tuning, where each batch is represented by one cell. This prevents the over-fitting of the models and mimics online deployment. Machine learning models in engineering require a stringent performance evaluation both from an error and uncertainty quantification perspective. The models are initially re-calibrated followed by an evaluation based on mean absolute percent error, root mean squared error and uncertainty estimation metrics (refer to Methods section for further details).
\subsection{Methods}
This study developed a pipeline approach for battery SOH estimation, called BHUMP and it incorporates a series of hierarchical steps, feature engineering, feature selection and data augmentation prior to model fitting and tuning as follows.
\textbf{Feature engineering} performs mathematical manipulations of extracted parts of the voltage curve during the constant current charge protocol based on a lower voltage threshold, $V_l$ and an upper voltage threshold, $V_h$ (refer to the grey area of Figure \ref{thresholds_v_gr1}) for all datasets except for cells charged with a 2-step fast-charge protocol. A characteristic to the 2-step fast charge protocol is that the cells can be charged from 0\% SOC to 80\% SOC with high currents ranging from 3.6 C-rate to 6 C-rate. In this work, due to the nature of the charging method in the 2-step fast charge, we only use the constant-current constant-voltage (CC-CV) charge part of the charging protocol as per the black dotted segments in the grey area observed in Supplementary Figures 2a, 2b. The values of $V_h$ and $V_l$ can be selected based on the intended application and the depth of discharge of the cell. In this work we select $V_h$ to be equal to cut-off voltage, $V_{cut-off}$. Refer to Supplementary Note 2 on how we select $V_l$. Additional features are developed on extracted segments of the current curve during the constant voltage charge protocol based on two current threshold values, $I_h$ and $I_l$ respectively (see Figure \ref{thresholds_c_gr1}) for all cells except for the 2-step fast charge protocol. We select $I_h$ equal to charge C-rate, while the lower threshold value, $I_l$, equal to a current drop of 40\% from $I_h$. This allows for sufficient data to be recorded while keeping the diagnostics time to a minimum. For cells cycled with the 2-step fast charge we select the current curve in Supplementary Figure 2b.
The obtained segments of voltage and current charge curves are further processed to obtain a plethora of features as described in Supplementary Note 3. Supplementary Table 2 summarises all features generated from processing the curves.
\textbf{Feature selection} with recursive feature elimination and cross-validation (RFE-CV) performs selection and subset reduction automatically without requirements of user-based thresholds, such as a maximum number of features to be selected. To suit battery data, we modify the original formulation by replacing the decision function algorithm with a random forest (RF) as opposed to the support vector machine (SVM) used in \cite{guyon_RFE_CV}. The replacement is motivated by RF's ability to deal with unscaled data. We call the resultant modified algorithm RF-RFE-CV. We use 700 decision tree estimators for the random forest algorithm and we set the number of cross-validations equal to the number of batteries in the feature selection dataset (see Supplementary Note 5 for data partition). We perform feature selection for each battery dataset based on a subset of the training data to avoid introducing optimistically biased performance estimates.
Battery SOH is quantified as capacity fade with reference to the first cycle as per equation \ref{SOH}, where $C_i$ represents capacity value at $i^{th}$ cycle and $C_1$ is the capacity at the first cycle measured by a complete charge-discharge operation.
\begin{equation}
SOH = \frac{C_i}{C_1}
\label{SOH}
\end{equation}
The role of the algorithm is to map from inputs $\mathbf{x}$ to target variable $y$ by means of a function $f(\mathbf{x}, \mathbf{\theta})$:
\begin{equation}
y = f(\mathbf{x}, \mathbf{\theta} ) + \epsilon
\label{general_model}
\end{equation}
where $\mathbf{\theta}$ are the model weights vector and $\epsilon \sim \mathcal{N}(0, \Sigma)$ is a normally distributed noise parameter. Based on the selected algorithm, the function $f(\mathbf{x}, \mathbf{\theta})$ may take different forms based on underlying assumptions of each algorithm. The learned model can then be used to make predictions of capacity given a test vector $\mathbf{x}^*$.
\textbf{Data augmentation} is carried out using the \emph{fast gradient sign method (FGSM)} in combination with the weight decay algorithm (ridge regression). We have found that a Ridge regularised model in combination with the FGSM was able to reduced the confidence interval (CI) around the estimated mean, despite being a simpler model than the original formulation in \cite{goodfellow_fgsm} which was based on a neural network. Given an input $\mathbf{x}$ with a target $y$ and loss $l(\mathbf{\theta}, \mathbf{x}, y)$, FGSM generates an adversarial example using:
\begin{equation}
\mathbf{x}_{adv}=\mathbf{x} + \gamma \cdot sign(\nabla_x l(\theta, \mathbf{x}, y))
\end{equation}
where $\gamma$ is a small value such that the max value of the perturbation is bounded and $\nabla_x$ is the gradient with respect to $\mathbf{x}$. Because each feature in the dataset has a different range, we set $\gamma$ to 0.01 or 1\% times the range of each feature vector. The adversarial examples are concatenated with the original training data to create a comprehensive training dataset. Note, other methods for data augmentations can also be used such as the ones proposed in \cite{deep_ensamble}, \cite{adversarial_regression}, \cite{ead_attack}, \cite{ead_attack_with_L1}, however the effect of data augmentation on model performance is beyond the scope of the present work.
The study solves eq. \ref{general_model} by making use of four algorithms as follows:
\textbf{Bayesian Ridge regression (BRR)} considers a probabilistic model of the regression problem. The algorithm estimates a spherical Gaussian prior over the model weights given by $p(\mathbf{\theta}|\lambda) = \mathcal{N}(\mathbf{\theta}|0, \lambda^{-1} \textbf{I}_p)$, where $\lambda^{-1}$ is the precision. The priors over $\alpha$ (the regulariser) and $\lambda$ are chosen to be gamma distributions. All parameters, $\mathbf{\theta}$, $\lambda$ and $\alpha$, are jointly estimated during training as per the implementation in \cite{scikit-learn}. Posterior inference can be performed in a closed-form because the prior is conjugate. For a complete explanation of the algorithm refer to \cite{bishop2006pattern}.
\textbf{Gaussian process regression (GPR)} is a nonparametric, Bayesian approach to regression defining a probability distribution over functions rather than random variables, thus eq. \ref{general_model} is solved by:
\begin{equation}
f(\mathbf{x}) \sim \mathcal{GP}(m(\mathbf{x}), k(\mathbf{x}, \mathbf{x}'))
\label{GPR}
\end{equation}
where $m(\mathbf{x})$ is the mean and $k(\mathbf{x}, \mathbf{x}')$ is the covariance function. Note, as defined above, GPR does not require learning the parameters of the regression function $f(\mathbf{x}, \mathbf{\theta} )$, in a traditional sense. The mean and covariance are defined by:
\begin{equation}
m(\mathbf{x}) = \mathbb{E}[f(\mathbf{x})]
\end{equation}
\begin{equation}
k(\mathbf{x}, \mathbf{x}') = \mathbb{E}[(f(\mathbf{x}-m(\mathbf{x}))(f(\mathbf{x}'-m(\mathbf{x}'))]
\end{equation}
GPR assigns a prior probability to every possible function, where higher probabilities are given to functions that the algorithm considers to be more likely, for example, because they are smoother than other functions. For our implementation, we make use of the standard radial basis kernel (RBF) as detailed in \cite{gpr_mit_rasmussen}, where a mathematical explanation of the algorithm is also given. Other kernel options exist, however, we do not explore the effect of kernel choice on algorithm performance.
\textbf{Random Forest (RF)} is a collection of constructed decision trees who sequentially conduct binary splits of the data to produce a homogeneous subset. For a comprehensive explanation of the algorithm refer to \cite{RF}. We adopt a bagging approach where the ensemble members are trained on different bootstrap samples of the training set and we set the number of decision trees in the forest to 1500. The variability of the predictions estimated by the random forest has been investigated based on the study from \cite{RF_CI}, where the confidence interval's variance has been obtained using the bootstrap replicates used to train the random forest itself.
\textbf{Deep ensemble of neural networks (dNNe)}. Ensemble methods combine different regressors into a meta-regressor and we consider an ensemble of deep neural networks as proposed in \cite{deep_ensamble}. Each network in the ensemble incorporates 2 hidden layers with an output of two layers one for the mean, $\mu(x)$ and the other for variance, $\sigma^2$ with $\sigma^2 > 0$. We use the negative log-likelihood as a function of the predicted mean and variance for scoring purposes. We also use a feed-forward architecture of 2 densely connected hidden layers. Each layer decreases in size by 50\% neurons based on the number of input features. When the input number features is less than 10, we force the network's hidden layers to 4 neurons in the first layer and 3 in the second layer. For example, when 18 input features are considered, the first hidden layer consists of 9 neurons, followed by 4 neurons in the second hidden layer. Each network used in this work has the following parameters: first hidden layer implies a ReLU activation function, followed by a Leaky ReLU for the second hidden layer and a Sigmoid function for the output layer. Additionally, we make use of Adam optimiser with a learning rate of 0.001 and a batch size equal to the number of cycles for each cell in the training set.
All models are evaluated based on mean absolute percent error (MAPE) and root mean squared error (RMSPE).
\begin{equation}
MAPE(y_i^*, y_i) = \frac{1}{N}\sum^{N}_{i=1}\frac{|y_i^*-y_i|}{y_i}
\label{MAPE}
\end{equation}
\begin{equation}
RMSPE(y_i^*, y_i) = \sqrt{\frac{1}{N}\sum^{N}_{i=1}\left(\frac{y_i^*-y_i}{y_i}\right)^2}
\label{RMSPE}
\end{equation}
where $y_i$ is the measured capacity value, $y_i^*$ is the estimated capacity value, and $n$ is the total number of samples.
In a regression setting, we obtain probabilistic forecasts using one of the algorithms described above through the estimation of a Gaussian distribution $\mathcal{N}(\mu, \sigma^2)$, where $\mu$ is the mean estimated capacity and $\sigma^2$ is the associated uncertainty quantified as variance. To evaluate the usefulness of predictive uncertainty for decision making, we create reliability diagnostics curves analogous to the work in \cite{calibration_DL_kuleshov}. To plot calibration curves, we divide each predicted confidence interval in $m$ confidence levels that are monotonically increasing on the interval $[0, 1]$ i.e. $0<p_1<p_2<...<p_m<1$. We then compute the empirical probability for each threshold by counting the frequency of true labels in each confidence level $p_m$. Mathematically this can be summarised as:
\begin{equation}
\hat{p}_m = \frac{|{y_n}|F_n(y_n)\leq p_m, n=1,...,N|}{N}
\end{equation}
Based on the reliability curve assessment, we then perform re-calibration using isotonic regression \cite{isotonic_regression}. A well-calibrated regressor should lie very close to the ideal diagonal curve, e.g. results Figure \ref{calibration_group1_dNNE}. We use the calibration score($C_{score}$) as a numerical score to describe the quality of the calibration when referenced to a 90\% confidence interval and sharpness ($Sh$) to describe average standard deviation.
\begin{equation}
C_{score} = \frac{1}{N}\sum_{i=1}^{N} \hat{p}_{m=90\%}
\label{c_error}
\end{equation}
Sharpness is calculated as an average of model output variance for each prediction and is given by:
\begin{equation}
Sh = \frac{1}{n} \sum_{i=1}^{n} \sigma_i
\label{sh}
\end{equation}
where $i$ is the sample number and $n$ is the the total number of sampels.
\begin{figure}[h!]
\centering
\includegraphics[width = .4\textwidth]{accuracy_zone.pdf}
\captionof{figure}{$\alpha$-accuracy zone and $\beta$ probability mass illustration.}
\label{accuracy_zone}
\end{figure}
We further propose an assessment of uncertainty prediction via prognostics performance metrics from an engineering point of view, adopted from \cite{prog_metrics_nasa}. First, we introduce the accuracy zone defined by a threshold, $\alpha$ (see Figure \ref{accuracy_zone}), which is calculated as a percentage error from the true capacity value, i.e. $ y \pm \alpha $. We select an $\alpha$ of $\pm$1.5\% (alpha can be adjusted based on intended application). Based on the frequency of predicted values residing in the accuracy zone, we calculate the $\alpha$-accuracy. Finally, we calculate the average probability mass of the prediction PDF within the $\alpha$ bounds called $\beta$, refer to Figure \ref{accuracy_zone}. Ideally, $\beta$ should be one, suggesting that the predicted confidence interval is small and encapsulates the entire $\alpha$-accuracy zone. Since $\alpha$ summarises the notion of \emph{desired accuracy}, $\alpha^+$ is the upper bound for estimates above the accuracy zone, and $\alpha^-$ represents low estimates or value residing under the desired accuracy zone. Depending on the application, both or any one of the low or high estimates may be undesirable. We chose to calculate the percentage of early predictions (estimates residing below the true label, the red line in Figure \ref{accuracy_zone}), denoted here by PEP, as a measure of algorithm uncertainty in a critical application scenario.
\section{Dataset}
\begin{table*}[h!]
\centering
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}cccccccc@{}}
\toprule
Group{*} & I & I & I & I & I & II & III \\ \hline
Dataset & CALCE CS2 & CALCE CX2 & CALCE PL & NASA 5 & NASA11 & TRI & Oxford \\ \midrule
Manufacturer & Unknown & Unknown & Unknown & LG Chem & LG Chem & A123 Systems & Kokam \\ \midrule
Cathode \textsuperscript{***} & LiCoO$_{2}$ & LiCoO$_{2}$ & LiCoO$_{2}$ & LiCoO$_{2}$ & LiCoO$_{2}$ & LiFePO$_{4}$ & \begin{tabular}[c]{@{}c@{}}LiCoO$_{2}$ /\\ LiNiMnCoO$_{2}$\end{tabular} \\ \midrule
Form factor & Prismatic & Prismatic & Pouch & \begin{tabular}[c]{@{}c@{}}18650\\ Cylindrical\end{tabular} & \begin{tabular}[c]{@{}c@{}}18650\\ Cylindrical\end{tabular} & \begin{tabular}[c]{@{}c@{}}18650\\ Cylindrical\end{tabular} & Pouch \\ \midrule
\# cells & 6 & 6 & 2 & 8 & 25 & 124 & 8 \\ \midrule
Charge & CC-CV & CC-CV & CC-CV & CC-CV & CC-CV & Fast-charge & CC \\ \midrule
Discharge & 2 regimes & 2 regimes & 1 regime & 2 regimes & 7 regimes & 1 regime & 1 regime \\ \midrule
\multicolumn{8}{l}{ \begin{tabular}[c]{@{}l@{}}\textsuperscript{*}\footnotesize{Groups based on charge protocol}, \textsuperscript{**}\footnotesize{Toyota Research Institute}, \textsuperscript{***}\footnotesize{Information from manufacturer, not verified}\end{tabular}}
\end{tabular}%
}
\caption{Datasets overview. Note: refer to Supplementary Note 4 for data sources.}
\label{data_rep}
\end{table*}
We investigate the performance of BHUMP on a total of 179 Li-ion cells as referenced in Table \ref{data_rep}. The cells have been grouped into three categories based on the charging protocol used: constant current - constant voltage (CC-CV) protocol in Group I (47 cells), 2-step fast charge protocol in Group III (8 cells), and constant current (CC) protocol in Group II (124 cells). The separation is important for separate model training and feature selection, as well as model performance assessment on different charge protocols. A detailed explanation of each dataset used can be found in Supplementary Note 4.
\section{Algorithm performance}
\subsubsection{Group I data}
Subject to the previously described pipeline steps the feature selection algorithm, RF-RFE-CV chose 18 of the 30 engineered features as the optimum number of attributes for the cells in Group I (refer to Supplementary Figure 8a and Supplementary Table 3). From a threshold point of view, we select a $V_h$ of 4.2V for all batteries in this Group with an associated $V_l$ of 3.9V. Refer to Supplementary Note 5 for train/test partitions.
\begin{table}[h!]
\centering
\resizebox{.47\textwidth}{!}{%
\begin{tabular}{l|c|c|c|c|c|c|c|}
\cline{2-8}
& \multicolumn{1}{l|}{MAPE} & \multicolumn{1}{l|}{RMSPE} & \multicolumn{1}{l|}{$C_{score}$} & \multicolumn{1}{l|}{$Sh$} & \multicolumn{1}{l|}{$\alpha$-accuracy} & \multicolumn{1}{l|}{$\beta$} & \multicolumn{1}{l|}{$PEP$} \\ \hline
\multicolumn{1}{|l|}{BRR} & 1.52 & 2.49 & 84.49 & 0.021 & 70.00 & 0.57 & 68.92 \\ \hline
\multicolumn{1}{|l|}{GPR} & 1.49 & 2.24 & 92.23 & 0.025 & 65.00 & 0.48 & 71.76 \\ \hline
\multicolumn{1}{|l|}{RF} & 0.72 & 0.91 & 100 & 0.046 & 92.00 & 0.29 & 95.29 \\ \hline
\multicolumn{1}{|l|}{dNNe} & 0.65 & 0.92 & 88.01 & 0.0082 & 93.00 & 0.93 & 97.71\\ \hline
\end{tabular}%
}
\caption{Results for Group I cell no. 38.}
\label{results_group1_cell_38}
\end{table}
We illustrate results for BHUMP when dNNe is considered as base algorithm in Figure \ref{predictions_group1_dNNE} (results for all other algorithms are shown in Supplementary Figures 11, 12, 13) for a randomly chosen pouch cell battery, cell no. 38 and summarise algorithm performance on this cell in Table \ref{results_group1_cell_38}. The cell was cycled in full depth of discharge between 4.2V to 2.7V at a discharge C-rate of 0.5C (or 0.55 A) with a CC-CV charging protocol at a current value of 0.5 C-rate. Table \ref{results_group1} summarises each algorithms' performance on cell no. 38. Comparing dNNe in Figure~\ref{pred_group1_dNNe} to the other algorithms BRR, GPR, and RF, we show that the resultant confidence interval is considerably smaller (all figures display a confidence level equivalent to a 95\% quantile i.e. $ \mu \pm 2\cdot \sigma$). This indicates that the model is sharper, resulting in a high $\beta$ score (refer to Table \ref{results_group1_cell_38} for results). Where the predictions are less accurate, such as is the prediction in the first few cycles (see Figure~\ref{pred_group1_dNNe}), the error bars capture this variability well. On this battery, dNNe also achieves the best RMSPE and MAPE together with a high calibration score. As per Table \ref{results_group1_cell_38}, the estimates for this cell vary between RMSPE 0.65\% to 1.52\%, showing that all 4 algorithms can achieve high performance. The same conclusion is not valid for calibration, however. Reliability plots indicate that RF exhibits high variance even after calibration, refer to Supplementary Figure 13.
\begin{figure*}[h!]
\centering
\begin{subfigure}[b]{.55\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr1_prediction_dNNe.pdf}
\label{pred_group1_dNNe}
\end{subfigure
\begin{subfigure}[b]{.3\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr1_calibration_dNNe.pdf}
\label{calibration_group1_dNNE}
\end{subfigure}%
\hfill
\begin{subfigure}[b]{.55\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr1_prediction_vs_true_dNNe.pdf}
\label{prediction_vs_true_dNNe_group1}
\end{subfigure}%
\begin{subfigure}[b]{.3\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr1_histogram_dNNe.pdf}
\label{hist_group1_dNNE}
\end{subfigure}%
\hfill
\caption{\textbf{Prediction results with dNNe Group I cell no. 38.} \textbf{\protect\subref{pred_group1_dNNe}} dNNe prediction as a function of cycle, \textbf{\protect\subref{calibration_group1_dNNE}} dNNE calibration results, \textbf{\protect\subref{prediction_vs_true_dNNe_group1}} dNNe actual vs. predicted capacity, \textbf{\protect\subref{hist_group1_dNNE}} Histogram of \% error. Note: $y^*$ - true capacity, $\hat{y}^*$ - predicted capacity}
\label{predictions_group1_dNNE}
\end{figure*}
When discussing average results across all cells in Group I (Table \ref{results_group1}), RF achieves on average a low calibration error of 54.70\% possibly due to the method used for estimating the variance, Infinitesimal Jackknife. In practice we prefer a more conservative system, particularly in safety-critical applications. This implies that the number of capacity estimates lower than the true label residing in the $\alpha$-accuracy zone (Figure \ref{accuracy_zone}) should exceed the number of capacity values estimated above it i.e. PEP should be close to 100\%. At the same time, too low of a capacity estimate would result in a far too conservative algorithm. However, such behaviour is captured by an increase in RMSPE and thus mitigated for naturally. With reference to Figure \ref{hist_group1_dNNE} together with Table \ref{results_group1_cell_38} one can conclude that dNNe is conservative, achieving the highest PEP.
\begin{table}[h!]
\centering
\resizebox{.47\textwidth}{!}{%
\begin{tabular}{l|c|c|c|c|c|c|c|}
\cline{2-8}
& \multicolumn{1}{l|}{MAPE} & \multicolumn{1}{l|}{RMSPE} & \multicolumn{1}{l|}{$C_{score}$} & \multicolumn{1}{l|}{$Sh$} & \multicolumn{1}{l|}{$\alpha$-accuracy} & \multicolumn{1}{l|}{$\beta$} & \multicolumn{1}{l|}{$PEP$} \\ \hline
\multicolumn{1}{|l|}{BRR} & 4.65 & 5.54 & 89.16 & 0.104 & 25.76 & 0.25 & 36.57 \\ \hline
\multicolumn{1}{|l|}{GPR} & 3.70 & 4.51 & 83.62 & 0.089 & 32.04 & 0.29 & 60.07 \\ \hline
\multicolumn{1}{|l|}{RF} & 2.17 & 2.70 & 54.70 & 0.093 & 35.94 & 0.36 & 65.47 \\ \hline
\multicolumn{1}{|l|}{dNNe} & 3.30 & 4.26 & 86.28 & 0.043& 32.14 & 0.58 & 63.26 \\ \hline
\end{tabular}%
}
\caption{Average results over Group I dataset. }
\label{results_group1}
\end{table}
Overall, despite RF achieving the lowest average RMSPE and MAPE (Table \ref{results_group1}) it does not output well-calibrated predictions, nor it displays a high sharpness value. At the expense of 1.13\% in MAPE and 1.56\% in RMSPE, the dNNe outputs a well-calibrated model, on average being less than 4\% under the ideal calibration score.
\subsubsection{Group II data}
Group II dataset is the largest dataset incorporating a total of 124 cells. While the dataset exhibits a high variance in charge profiles, it does not have any variation in discharge conditions (all cells in the dataset are discharged at 4 C-rate). This, in turn, showcases the effect of the charge profile on the estimation accuracy of the 4 algorithms. Training is performed on features engineered based on the CC-CV curve obtained after the cell reaches 80\% SOC (refer to Supplementary Figures 2a and 2b). Refer to Supplementary Note 5 for train/test partitions. RF-RFE-CV selects a total of 5 features (Supplementary Figure 8b and Supplementary Table 4) out of a total of 30 engineered features. We believe this is caused by the fact that the dataset only incorporates one discharge profile as well as just a single battery type.
\begin{table}[ht!]
\centering
\resizebox{.47\textwidth}{!}{%
\begin{tabular}{l|c|c|c|c|c|c|c|}
\cline{2-8}
& \multicolumn{1}{l|}{MAPE} & \multicolumn{1}{l|}{RMSPE} & \multicolumn{1}{l|}{$C_{score}$} & \multicolumn{1}{l|}{$Sh$} & \multicolumn{1}{l|}{$\alpha$-accuracy} & \multicolumn{1}{l|}{$\beta$} & \multicolumn{1}{l|}{$PEP$} \\ \hline
\multicolumn{1}{|l|}{BRR} & 0.72 & 0.90 & 65.49 & 0.005 & 89.00 & 98.00 & 20.70 \\ \hline
\multicolumn{1}{|l|}{GPR} & 1.23 & 1.63 & 69.94 & 0.011 & 65.00 & 85.00 & 22.16 \\ \hline
\multicolumn{1}{|l|}{RF} & 0.23 & 0.43 & 87.42 & 0.002 & 98.00 & 100 & 42.81 \\ \hline
\multicolumn{1}{|l|}{dNNe} & 0.34 & 0.48 & 71.31 & 0.002 & 98.00 & 100 & 31.50 \\ \hline
\end{tabular}%
}
\caption{Results for Group II cell no. 1.}
\label{results_group2_cell_1}
\end{table}
\begin{figure*}[h!]
\centering
\begin{subfigure}[b]{.55\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr2_prediction_dNNe.pdf}
\label{pred_group2_dNNe}
\end{subfigure
\begin{subfigure}[b]{.3\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr2_calibration_dNNe.pdf}
\label{calibration_group2_dNNE}
\end{subfigure}%
\hfill
\begin{subfigure}[b]{.55\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr2_prediction_vs_true_dNNe.pdf}
\label{prediction_vs_true_dNNe_group2}
\end{subfigure}%
\begin{subfigure}[b]{.3\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr2_histogram_dNNe.pdf}
\label{hist_group2_dNNE}
\end{subfigure}%
\hfill
\caption{\textbf{Prediction results with dNNe Group II cell no. 1.} \textbf{\protect\subref{pred_group2_dNNe}} dNNe prediction as a function of cycle, \textbf{\protect\subref{calibration_group2_dNNE}} dNNE calibration results, \textbf{\protect\subref{prediction_vs_true_dNNe_group2}} dNNe actual vs. predicted capacity, \textbf{\protect\subref{hist_group2_dNNE}} Histogram of \% error. Note: $y^*$ - true capacity, $\hat{y}^*$ - predicted capacity}
\label{predictions_group2_dNNE}
\end{figure*}
Figure \ref{predictions_group2_dNNE} illustrate BHUMP performance with a dNNe as base algorithm for cell no. 1, whilst Suplementary Figures 14, 15, 16 summarise results for all other algorithms. The cell has undergone fast charge profile of 3.6 C-rate to 80\% SOC, beyond which the cell is charged with CC of 1C followed by the CV charging. The reason cell 1 was selected in this case was to illustrate the performance of the algorithms when there is a high number of outliers in capacity data (Figure \ref{pred_group2_dNNe}). With reference to Table \ref{results_group2_cell_1}, RF achieves lowest error and highest scores as well as a good calibration compared to all other algorithms. On this particular cell, dNNe achieves the second best performance, however it does not output a well calibrated model, despite showing a good average calibration score as per Table \ref{results_group2}.
Average results of the 4 algorithms are concisely summarised in Table \ref{results_group2}. All models are able to estimate the SOH with less than 2\% RMPSE; this underlines the fact that the models are not affected by the fast-charge section of the charging protocol. RF achieves the highest accuracy with a low sharpness value and high percentages for all other metrics except for calibration where it exhibits over-confidence. In terms of calibration error, dNNe achieves the closest score to a 90\% confidence interval with 91.02\%. dNNe is also the second-best performing algorithm achieving good scores across all metrics as summarised in Table \ref{results_group2_cell_1}. In comparison, the two Bayesian-based algorithms exhibit a higher percentage error as well as higher sharpness values. However, they tend to be more conservative, averaging a PEP over 60\%.
\begin{table}[h!]
\centering
\resizebox{.47\textwidth}{!}{%
\begin{tabular}{l|c|c|c|c|c|c|c|}
\cline{2-8}
& \multicolumn{1}{l|}{MAPE} & \multicolumn{1}{l|}{RMSPE} & \multicolumn{1}{l|}{$C_{score}$} & \multicolumn{1}{l|}{$Sh$} & \multicolumn{1}{l|}{$\alpha$-accuracy} & \multicolumn{1}{l|}{$\beta$} & \multicolumn{1}{l|}{$PEP$} \\ \hline
\multicolumn{1}{|l|}{BRR} & 0.45 & 0.76 & 91.72 & 0.005 & 97.31 & 99.19 & 62.86 \\ \hline
\multicolumn{1}{|l|}{GPR} & 1.00 & 1.91 & 93.14 & 0.012 & 90.43 & 83.74 & 63.21 \\ \hline
\multicolumn{1}{|l|}{RF} & 0.11 & 0.14 & 79.72 & 0.001 & 99.84 & 99.96 & 58.77 \\ \hline
\multicolumn{1}{|l|}{dNNe} & 0.23 & 0.45 & 91.02 & 0.002 & 99.53 & 99.50 & 53.41 \\ \hline
\end{tabular}%
}
\caption{Average results over Group II dataset}
\label{results_group2}
\end{table}
In conclusion, from an accuracy and sharpness perspective, the best performing algorithm on dataset Group II is RF, whilst the poorest performance is achieved by GPR. When it comes to uncertainty metrics, and in particular calibration, RF exhibits over-confidence with a $C_{score}$ of 79.72\%. Such behaviour is also identified in Group I dataset where RF was, in fact, difficult to calibrate despite the rich dataset. A more reliable calibration score is achieved by dNNe at the expense of a loss of 0.12\% in MAPE and 0.31\% in RMSPE (refer to Table \ref{results_group2}).
\subsubsection{Group III data}
On Group III we emphasise on the suitability of BHUMP to battery state of health estimation for automotive applications. Group III includes 8 Kokham 740 mAh batteries that have been dynamically cycled under the ARTEMIS \cite{andre2004artemis} dynamic driving
profile, followed by characterisation cycles. Each characterisation cycle consists of low rate CC charge and discharge cycles, repeated every 100 cycles. We use the characterisation cycles for diagnostics purposes to derive features and estimate battery health. This dataset incorporates the lowest variability both in terms of input feature values and capacity degradation values due to the identical charge-discharge conditions. This, in turn, affects feature selection as BHUMP only selects 5 out of the 18 engineered features (note charge protocol does not include CV part of the charge, hence 12 features are missing) as shown in Supplementary Figure 8c and Supplementary Table 5. We keep the same threshold values as in Group I cells for the CC part of the curves, namely a $V_h$ of 4.2V and a $V_l$ of 3.9V on which feature are engineered. Refer to Supplementary Note 5 for train/test partitions.
\begin{figure*}[h!]
\centering
\begin{subfigure}[b]{.55\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr3_prediction_dNNe.pdf}
\label{pred_group3_dNNe}
\end{subfigure
\begin{subfigure}[b]{.3\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr3_calibration_dNNe.pdf}
\label{calibration_group3_dNNE}
\end{subfigure}%
\hfill
\begin{subfigure}[b]{.57\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr3_prediction_vs_true_dNNe.pdf}
\label{prediction_vs_true_dNNe_group3}
\end{subfigure}%
\begin{subfigure}[b]{.3\textwidth}
\centering
\caption{}
\includegraphics[width=\linewidth]{gr3_histogram_dNNe.pdf}
\label{hist_group3_dNNE}
\end{subfigure}%
\hfill
\caption{\textbf{Prediction results with dNNe Group III cell no. 5.} \textbf{\protect\subref{pred_group3_dNNe}} dNNe prediction as a function of cycle, \textbf{\protect\subref{calibration_group3_dNNE}} dNNE calibration results, \textbf{\protect\subref{prediction_vs_true_dNNe_group3}} dNNe actual vs. predicted capacity, \textbf{\protect\subref{hist_group3_dNNE}} Histogram of \% error. Note: $y^*$ - true capacity, $\hat{y}^*$ - predicted capacity}
\label{predictions_group3_dNNE}
\end{figure*}
\begin{table}[ht!]
\centering
\resizebox{.47\textwidth}{!}{%
\begin{tabular}{l|c|c|c|c|c|c|c|}
\cline{2-8}
& \multicolumn{1}{l|}{MAPE} & \multicolumn{1}{l|}{RMSPE} & \multicolumn{1}{l|}{$C_{score}$} & \multicolumn{1}{l|}{$Sh$} & \multicolumn{1}{l|}{$\alpha$-accuracy} & \multicolumn{1}{l|}{$\beta$} & \multicolumn{1}{l|}{$PEP$} \\ \hline
\multicolumn{1}{|l|}{BRR} & 0.11 & 0.15 & 95.55 & 0.89 & 100 & 100 & 31.11 \\ \hline
\multicolumn{1}{|l|}{GPR} & 0.16 & 0.19 & 71.11 & 1.21 & 100 & 100 & 15.55 \\ \hline
\multicolumn{1}{|l|}{RF} & 0.17 & 0.21 & 97.77 & 2.01 & 100 & 100 & 24.44 \\ \hline
\multicolumn{1}{|l|}{dNNe} & 0.20 & 0.25 & 100.00 & 2.93 & 100 & 100 & 6.67 \\ \hline
\end{tabular}%
}
\caption{Results for Group III cell no. 5.}
\label{results_group3_cell_5}
\end{table}
For visualisation purposes, we illustrate results for the randomly selected cell no. 5 for dNNe in Figure \ref{predictions_group3_dNNE} and Supplementary Figures 17, 18, 19 for all other algorithms. It is clear, from Table \ref{results_group3_cell_5} that performance on cell 5 is dominated by BRR based on all measures of accuracy and uncertainty quantification. However, all algorithms deployed on cell no. 5 (Table \ref{results_group3_cell_5}) achieve a MAPE and RMSPE smaller than the proposed accuracy zone threshold $\alpha$ of $\pm$1.5\%.
Average results are summarised in Table \ref{results_group3}. In terms of accuracy measures, on average, BRR outperforms all other methods, including the dNNe. As argued in \cite{ann_sample_size} linear regression outperforms considerably more complex algorithms, including NNs when dealing with small sample size that exhibits little variance. Despite the low error, BRR does not achieve a good calibration score as opposed to dNNE. dNNe is the second-best performing algorithm in terms of accuracy (MAPE and RMSPE). It also exhibits adequate results for all other metrics, including $PEP$ where it scores the highest.
\begin{table}[h!]
\centering
\resizebox{.47\textwidth}{!}{%
\begin{tabular}{l|c|c|c|c|c|c|c|}
\cline{2-8}
& \multicolumn{1}{l|}{MAPE} & \multicolumn{1}{l|}{RMSPE} & \multicolumn{1}{l|}{$C_{score}$} & \multicolumn{1}{l|}{$Sh$} & \multicolumn{1}{l|}{$\alpha$-accuracy} & \multicolumn{1}{l|}{$\beta$} & \multicolumn{1}{l|}{$PEP$} \\ \hline
\multicolumn{1}{|l|}{BRR} & 0.26 & 0.32 & 68.11 & 1.20 & 100 & 100 & 23.54 \\ \hline
\multicolumn{1}{|l|}{GPR} & 0.52 & 0.65 & 42.42 & 2.37 & 90.50 & 97.25 & 23.22 \\ \hline
\multicolumn{1}{|l|}{RF} & 0.36 & 0.44 & 72.62 & 2.16 & 88.5 & 100 & 25.44 \\ \hline
\multicolumn{1}{|l|}{dNNe} & 0.30 & 0.39 & 91.17 & 2.01 & 98.25 & 99.75 & 27.95 \\ \hline
\end{tabular}%
}
\caption{Average results over Group III dataset}
\label{results_group3}
\end{table}
In conclusion, when considering average results over all 4 test cells as referenced in Table \ref{results_group3}, dNNe achieves second-best accuracy while attaining the best calibration score of 91.17\%.
\section{Discussion on practical applicability of BHUMP}
BHUMP can complement battery management systems (BMS), for both SOC and SOH estimation, and replace the traditional ECMs altogether. While conventional approaches rely on measuring the capacity in static conditions such as full charge-discharge, BHUMP can estimate capacity fade from sections of the charge profile, accommodating for partial discharge scenarios or various operating conditions such as random or high discharge rates. We succinctly summarised in the results section, BHUMP can estimate capacity fade under fast charging protocol (Group II data) as well as random discharge (Group III data cycled under ARTEMIS driving protocol) typical to the operation of an EV battery pack. Future work could further extend to other charge-discharge protocols and open-source datasets such as the one in \cite{attia2020closed}.
Temperature variations during charging could further introduce uncertainty into the measurement of charge curves and propagate it into the estimation algorithm. Possible mitigation includes the use of temperature as an input when training BHUMP or considering additionally in-situ or operando sensory information such as optical and digital images or X-ray \cite{handoko2018understanding} such that the algorithm learns the correlation between temperature, generated features and SOH indicator. Due to such variations, SOH assessment without corresponding measures of algorithm uncertainty does not provide sufficient information to form a decision or corrective action. In addition to inherent algorithm bias, dataset variability also seems to affect the prediction error. To accommodate for such variations in the data BHUMP introduces 30 engineered features and makes use of an unsupervised feature selection algorithm (RF-RFE-CV). Given a training dataset RF-RFE-CV selects a subset of input features, indicating that features must be selected based on intended application, battery design and charge protocol. Despite such dataset variations, we think that deep learning has the potential to exceed it in the future as it requires little tuning from the user and can take advantage of parallelisation and an increasing amount of computational capabilities by deployment on graphics processing units (GPU) and modern data storage solutions. In addition, when training data consists of limited samples or training data is not relevant to the intended application, transfer learning can be used to reduce prediction errors. New hardware, architectures and learning algorithms that are currently being developed for neural network implementation will only accelerate this process, allowing for active learning techniques to be used when deployed onboard a vehicle. More concretely, BHMUP with dNNe as the base algorithm can incorporate transfer learning when trained on a particular cell design and re-trained on a reduced sample set for a different cell design. Additionally, BHUMP can also incorporate active learning as data becomes available when deployed online on different cell design, chemistry or operating temperature.
\section{Conclusion}
The two widely adopted modelling techniques for online battery state of health (SOH) estimation are equivalent circuit models and electrochemical models. However, when deployed online, the trade-off between accuracy and computational efficiency is difficult to achieve. This paper introduced an alternative, machine learning-based solution called battery health and uncertainty management pipeline (BHUMP). The pipeline provides a set of benefits over conventional methods including adaptability to the charging protocols and the discharge current rates, and prediction without knowledge of battery design, chemistry, and operating temperature.
The paper explores four algorithms: Bayesian ridge regression (BRR), Gaussian process regression (GPR), random forest (RF), and a deep ensemble of neural networks (dNNe), as the base algorithm for BHUMP. All algorithms are assessed on error values and the ability to quantify uncertainty. Results indicate that the lowest error achieved depends on the charging protocol adopted. The lowest error was achieved by RF for constant current - constant voltage protocol and fast charge protocol, and BRR for the constant-current protocol. When considering uncertainty assessment metrics, however, RF is hard to calibrate and is overly optimistic in its predictions. At the expense of an average increase in MAPE of 0.43\% and RMSPE of 0.97\%, dNNe, generally achieves a better calibration score, consistently achieving the second-lowest error irrespective of charge protocol. On the fast-charging protocol, the best dNNe model achieved a RMSPE of 0.45\% with a calibration score of 91.02\% when referenced to a 90\% confidence interval.
Overall, our work highlights the value of coupling machine learning tools with charge curve segments in capturing battery degradation in under 15 minutes. Moreover, we argue that despite achieving low errors, any algorithm must undergo uncertainty quantification checks before deployment in the field. Finally, we show how the use of machine learning pipelines can achieve a computationally efficient and accurate solution for cell SOH estimation. We envision machine learning pipelines to be a standard technique used in designing and implementing battery management systems of the future.
\section*{Data availability}
The datasets used in this study are available at:
\begin{itemize}
\item Group 1:
\url{https://web.calce.umd.edu/batteries/data.htm}
\url{https://ti.arc.nasa.gov/tech/dash/groups/pcoe/prognostic-data-repository/}
\item Group 2:
\url{https://data.matr.io/1/projects/5c48dd2bc625d700019f3204}
\item Group 3:
\url{https://ora.ox.ac.uk/objects/uuid:03ba4b01-cfed-46d3-9b1a-7d4a7bdf6fac}
\end{itemize}
\section*{Code availability}
\gappto{\UrlBreaks}{\UrlOrds}
Code for the data processing is available from the corresponding authors upon request. Code for the modelling work is available at: \url{http://doi.org/10.5281/zenodo.4390152}
\bibliographystyle{unsrt}
\bibliographystyle{unsrt}
| {
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} | 4,617 |
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\section{Introduction}
CTA 1 (G119.5+10.2) is a composite supernova remnant (SNR) with centrally bright X-ray emission \citep{SSS95,Sla97}.
This SNR has a radio shell with a radius of $\sim 54'$, and its kinematic distance estimated from the associated H$_{\rm{\uppercase\expandafter{\romannumeral 1}}}$ emission is $1.4\pm 0.3$~kpc \citep{Pin93}. The X-rays from CTA 1 comprise both thermal and non-thermal emission \citep{SSS95}.
While the outer regions mainly consist of shock-heated thermal emission, the non-thermal component dominates the central emission, which has led to the speculation of a hidden pulsar in this SNR \citep{Sla97}.
In addition to the radio and X-ray observations, the launch of {\it Compton Gamma-Ray Observatory} (CGRO) provided a further insight on the nature of CTA~1.
{\it Energetic Gamma Ray Experiment Telescope} (EGRET) onboard CGRO has revealed a $\gamma$-ray source, 2EG J0008+7307 \citep{Thp95}, in the 2nd EGRET catalogue (or 3EG J0010+7309 in the 3th EGRET catalogue; \citealt{Har99}), which was suggested to be associated with CTA 1 in view of the positional coincidence \citep{Bra98}.
The lack of variability of this extended $\gamma$-ray source as well as its spectral resemblance to a typical $\gamma$-ray pulsar further suggested that a neutron star should reside in this SNR.
Nevertheless, the limited photon statistics obtained by EGRET did not allow any meaningful pulsation search in the MeV$-$GeV regime.
Observations with {\it ROSAT} have revealed an X-ray point source, RX~J0007.0+7302, at the center of CTA~1 \citep{SSS95}.
It was also identified as the X-ray counterpart of 2EG~J0008+7307 / 3EG J0010+7309 \citep{Bra98}.
This point source was identified as a neutron star candidate with the high $\gamma$-ray-to-X-ray and X-ray-to-optical flux ratios as well as its X-ray spectral behaviours \citep{Sla97}, which were found to be similar to those of the only one radio-quiet $\gamma$-ray pulsar known at that time --- Geminga \citep{Bra98,HGCH2004}.
Utilizing the high resolution imaging capability of {\it Chandra}, \citet{HGCH2004} have uncovered a torus$+$jet feature, which resembles the morphology of a pulsar wind nebula (PWN) as seen in other bright $\gamma$-ray pulsars.
Furthermore, the sub-arcsecond angular resolution of the {\it Chandra} image provided a precise position, which can facilitate multi-wavelength investigations.
To identify the neutron star nature unambiguously, several studies attempted to search for the spin period of
RX~J0007.0+7302 \citep{BL96,NS97,Sla2004,HGCH2004,LC2005} in different energy bands.
Nevertheless, no conclusive evidence for pulsations in radio/X-ray was yielded in these previous studies.
With the unprecedented sensitivity of the {\it Large Area Telescope} (LAT) onboard {\it Fermi Gamma-ray Space Telescope}, a $\gamma$-ray pulsar, LAT PSR J0007+7303, was eventually detected in CTA~1 \citep{Abdo2008}.
This source has a spin-period of $\sim 315.86$~ms and was the first radio-quiet $\gamma$-ray pulsar detected by LAT.
Its spin parameters imply a characteristic age of $\sim 14000$~yr, which is consistent with the kinematic age of CTA~1 ($\sim 10000-20000$~yr; \citealt{Sla97}).
This strongly suggests an intrinsic association between the pulsar and SNR.
With the discoveries of more radio-quiet $\gamma$-ray pulsars \citep{Abdo2009,Saz2010}, we can now examine them as a unique class.
It has been speculated that the main distinction between radio-quiet and radio-loud $\gamma$-ray pulsars is simply a geometrical effect.
In order to further investigate their emitting properties, X-ray observations are deeply required.
Although the periodic signals of these $\gamma$-ray pulsars can be significantly detected, Geminga is the only one radio-quiet
$\gamma$-ray pulsar with a detected X-ray pulsation before 2009 \citep{HH92}.
In order to investigate the possible origin of the pulsed X-rays from the radio-quiet $\gamma$-ray pulsars as a whole class,
searches for the possible pulsations from these pulsars have been conveyed with dedicated X-ray observations, including PSR J0007+7303.
With a $\sim120$~ks observation with {\it XMM-Newton} (hereafter {\it XMM}), the X-ray pulsation of PSR J0007+7303 have been independently detected by \citet{Lin2010} and \citet{Car2010}.
Apart from the X-ray pulsation, this deep {\it XMM} observation also reveals the nebular emission extending for $\sim 2'$ in south-east direction away from the pulsar (see Fig.~1 in \citealt{Car2010}). Together with the torus and jet feature found by \citet{HGCH2004}, all previous X-ray investigations
suggest LAT PSR J0007+7303 has a complex PWN structure.
To continue the investigation, we have observed this interesting object with {\it Suzaku}, which enables a detailed analysis in both soft and hard X-ray bands.
In this paper, we report the results obtained from analysing the {\it Suzaku} data.
\section{Observations}
{\it Suzaku} \citep{Mit2007} was built with two main payloads, five nested conical thin-foil grazing incidence telescopes (XRT; \citealt{Ser2007}) with the effective energy range of 0.2-12 keV and a hard X-ray detector (HXD; \citealt{Taka2007}) with the effective energy range of 10-600 keV.
There are two major detectors for spectroscopy (X-ray spectrometer and X-ray imaging spectrometer) in {\it Suzaku} and each of them is set at the focus of an XRT.
Our observational results on the temporal and spectral analyses mainly come from the X-ray imaging spectrometer (XIS; \citealt{Koyama2007}) and the HXD.
The XIS consists of three front-illuminated (FI: XIS0, XIS2, XIS3) and one back-illuminated (BI: XIS1) CCD detectors, but only the FI chips provide the observations in timing/P-SUM mode.
Because the time resolution of HXD is 61~$\mu$s and that of XIS in the timing mode is $\sim 7.8$~ms, our data is precise enough for the periodicity search to a radio-quiet $\gamma$-ray pulsar with the spin period in tens or hundreds of milliseconds.
However, two of the XIS units with FI chips, XIS0 and XIS2 suffered a catastrophic damage on 2006 November and 2009 June respectively.
Since no useful data can be obtained with XIS2 and a part of the segment in XIS0 is lost, only the observation of XIS3 unit can be used for temporal analysis.
These events seriously decrease the probability to obtain the periodic signals from the faint sources in the soft X-ray band.
There are also two independent detector systems used for the observations of HXD.
The positive-intrinsic-negative (PIN) silicon diodes are sensitive below $\sim 60$~keV in the FOV of $34'\times 34'$, and the Gadolinium Silicate/Bismuth Germanate crystals (GSO/BGO) phoswich counters can detect photons above $\sim 30$~keV in the FOV of $4^{\circ}.5 \times 4^{\circ}.5$.
Because the data obtained from HXD have no imaging capability, we only did the analysis of the HXD-PIN data to avoid the contamination caused by the large background size of the HXD-GSO, which is much larger than the size of CTA~1.
Our observation for CTA 1 was carried out on 2010 January 8 with total exposures of $\sim 105$~ks in the soft X-ray band and $\sim 59$~ks in the hard X-ray band, and it was aimed at the HXD nominal position to RX J0007.0+7302 with (J2000) R.A.=$00^h07^m02^s.2$, decl.=$+73^{\circ}03'07''$ \citep{Sla2004}.
The XIS0 and XIS1 data of {\it Suzaku} were observed in the normal mode without window option, while the {\it Suzaku}/XIS3 data observed within 128 rows were compressed into one dimension to attain a higher time resolution.
All the data reduction and the spectral analyses were performed using XSELECT (ver.~2.4a), FTOOLS (ver.~6.9) and XSPEC (ver.~12.6.0) of HEASOFT (ver.~6.9) with the latest {\it Suzaku}/XIS calibration files (20101108) and {\it Suzaku}/HXD calibration files (20101202).
\begin{figure*}
\begin{center}\includegraphics[width=18.0cm]{cta1_pwn.ps}
\end{center}
\caption{\small{Vignetting-corrected images of the field around RX~J0007.0+7302 as observed by {\it Suzaku}/XIS0 ({\it{\bf panel a}}) and {\it Suzaku}/XIS1 ({\it{\bf panel b}}). A PWN component with an extent of $\sim 10'$ is revealed.
The lost of the row image due to a putative micrometeorite hit can also clearly be seen in {\it Suzaku}/XIS0.
The inset shows the close-up view of the $1^{'}\times1^{'}$ region (illustrated by the black square) around the pulsar as seen by {\it Chandra} (adopted from \citealt{HGCH2004}). For all images, top is north and left is east.}
}
\label{pwn_img}
\end{figure*}
\subsection{XIS0 \& XIS1}
The XIS0 and XIS1 observations of 105 ks exposures were divided into two editing modes of $3\times 3$ and $5\times 5$ with $\sim 85$ and $\sim 20$~ks, respectively.
Because each pixel on the CCD was read out every 8 s in the normal mode, the time resolution of the XIS0 and XIS1 data is not accurate enough to search the periodic signal in the range less than one second.
However, the XIS0 and XIS1 data are still adequate for the spectral analyses of the X-ray point source and the PWN to compare with the results yielded by other X-ray missions.
The adoption for the center of the point source is referred to the aim point of the {\it Suzaku} observation.
Because RX J0007.0+7302 is seriously contaminated by the surrounding PWN, we extracted the spectra of RX~J0007.0+7302 only within 1$'$.5 central circular region instead of the regular choice of 3$'$ circular region for a point source.
This adoption still contains $\sim 75\%$ of the encircled energy fraction (EEF).
We extracted the background of the pulsar from a concentric annulus of radii 100$''< r \leq 180''$ centered at the aim point of this {\it Suzaku} observation.
The spectra extracted from the point source comprise 695 counts for XIS0 and 401 counts for XIS1 after subtracting the background.
We also considered the point source selection in two other different sizes with 2$'$ and 3$'$ in radii using the same center as shown in Table~\ref{XISsp}.
Such selections of the point source contain $\sim 85\%$ and $\sim 95\%$ of the EEF, respectively.
The background of these selected regions were determined within a concentric annulus of radii 2$'< r \leq 4'$, and 3$'< r \leq 4'.5$.
Different choices for a point source region are used to verify that the significant excess contributed by the surrounding PWN can be detected with the increasing size.
For the PWN, we firstly examined its source extent by applying adaptive smoothing to both XIS0 and XIS1 images.
The images shown in Fig.~\ref{pwn_img} were generated with taking into account the vignetting of the XRT.
We used the task ``xissim'' to simulate a flat field image for our purpose, and we displayed the obtained image after dividing by the flat field.
A large structure with an extent of $\sim 10'$ appears to extend eastward from the pulsar.
In view of the possible spectral steepening due to synchrotron cooling, we performed a spatially-resolved spectral analysis of the PWN.
The extraction regions are illustrated in Fig.~\ref{pwn_reg}.
We split the PWN into three regions, namely the inner part (a circular region within a radius of $2'$), the middle part (an annulus with radii 2$'< r < 4'$) and the outer part (a panda region with radii 4$'< r < 8'$).
The background spectra are sampled from the source free low-count regions from each detector respectively.
We only consider the spectra obtained with the $3\times 3$ mode to perform the spectral analysis of the PWN because of its better photon statistics.
To yield better photon statistics to describe the spectral behaviour for the PWN, we have rebinned all the spectra before the analysis.
We generated the spectrum with $> 200$ cts/bin for the inner part, and with $> 500$ cts/bin for the middle and outer parts.
\begin{figure}
\begin{center}
\includegraphics[width=9.0cm]{pwn_reg.ps}
\end{center}
\caption{\small{Illustration of the regions used to extract the spectra from different parts of the PWN associated with RX~J0007.0+7302.}}
\label{pwn_reg}
\end{figure}
We generated the associated response matrix (rmf) and auxiliary response (arf) files by the HEASOFT command of `XISRMFGEN' and `XISSIMARFGEN'.
The spectrum obtained from each detector was rebinned with a minimum of 30 counts per channel to ensure $\chi^2$ statistics.
We then introduced a constant in a simultaneous fit to account for the cross-calibration mismatch between different detectors.
We also considered different circular sizes to describe the contribution of the point source.
Details about the spectral results are presented in $\S$ 3.
\subsection{XIS3}
The XIS3 data was investigated by the timing/P-SUM mode with a long duration of $\sim 212$~ks and its image was compressed into one dimension without the information along Y-axis to obtain higher time resolution.
Our procedures to resolve the point source in a one-dimensional image follow the standard recipe for reducing XIS data with the timing mode \citep{MHT2010}.
We also redid the event reprocessing to consider the grade filtering and generated the new cleaned event file using the script `XISREPRO.XCO' with the proper selection criteria.
For the P-SUM mode, we need to remove the hot pixels on the image manually.
To identify those hot pixels, we accumulated all the photon events along the coordinate of ACTX.
We first removed those columns that have photons significantly more than others, and then we counted the distribution of events in the remaining columns and removed those pixels with the photons exceeding 3$\sigma$ than the median of the distribution.
After excluding all the hot pixels, the source is still too marginal to be resolved in the histogram of ACTX versus photon events of the XIS3 data.
We then tried to rebin the distribution of counts along ACTX with several pixels as shown in Fig.~\ref{XIS3}.
A point-like source with 112 pixels ($\sim 118''$) can be roughly detected close to the center when we rebinned with 32 pixels.
There are 58255 photons for this source within 0.2-12 keV and the significance of this detection is $\sim 3.3 \sigma$.
We also checked the absolute arrival time of the photon events and determined the time required for readout by the ACTX position of the source in the XIS1 image.
In order to perform a precise temporal analysis, we corrected the arrival time with the decrease of 31.2 ms for all the source photons.
The Solar system barycentric time correction was then implemented with the task `AEBARYCEN' at (J2000) R.A.=$00^h07^m01^s.56$, decl.=$+73^{\circ}03'08''.1$ \citep{HGCH2004} to yield a time list, which can be used to examine the pulsation of RX~J0007.0+7302/PSR~J0007+7303 in the soft X-ray band.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm]{distrib.ps}
\end{center}
\caption{\small{Histogram of ACTX versus photons distribution of the XIS3 data. The black solid line marks the average photons of each pixel after rebinning the ACTX with 16 pixels, and the red one shows the distribution with the bin of 32 pixels. The green solid line is set as 536 to demonstrate the median of the photons distribution. A point-like source can marginally be resolved within the range of 496th-608th pixel in the red line.}}
\label{XIS3}
\end{figure}
\subsection{HXD}
The HXD observation was conducted in nominal pointing.
Since the HXD is a non-imaging instrument, we can not separate the contribution of the pulsar from the surrounding nebula.
However, a spectrum can be acquired to describe the behaviour of the total hard X-rays emitted from the entire CTA 1.
We only focus on the HXD-PIN detections of the hard X-rays emitted from/around CTA~1.
In order to examine the periodic signal of the pulsar, we also applied the Solar system barycentric time correction with the task `AEBARYCEN' at (J2000) R.A.=$00^h07^m01^s.56$, decl.=$+73^{\circ}03'08''.1$ \citep{HGCH2004} on the HXD-PIN observation.
The events in the observation were restricted according to the effective energy range (10 - 60 keV) of the PIN detector, and 26661 photons were obtained for temporal analysis after the screening of the energy.
We also subtracted the non-X-ray background (NXB) and cosmic X-ray background (CXB) to obtain the HXD-PIN spectrum.
The simulated NXB was directly obtained from the {\it Suzaku} Data Center, and the CXB flux that is $\sim 5\%$ of the NXB for PIN could be derived from \citet{Boldt87}.
\begin{equation}\label{eqno1}
\begin{aligned}
CXB(E)=9.412\times10^{-3}\times(E/1keV)^{-1.29}\times \rm{exp}(-E/40~\rm{keV}) \\
\rm{photons~cm^{-2}~s^{-1}~FOV^{-1}~keV^{-1}}
\end{aligned}
\end{equation}
These background spectra correspond to epoch ``6'' for the flat PIN response matrix.
We also evaluated the contribution of the Galactic Ridge X-ray Emission (GRXE) using the {\it Suzaku} observation of a nearby blank sky
(OBSID=504039010) at (l, b)=(123$^{\circ}$.9, 10$^{\circ}$.0), which is 4.3 degree apart from the CTA~1.
Following the same method as stated in \citet{Enoto2010} we assumed a power-law spectrum with a fixed photon index at 2.1 \citep{VM98}.
The contribution of the GRXE within $14-35$ keV was evaluated to be $<4.4\times 10^{-4}$ photons sec$^{-1}$, which only corresponds
to $<$0.2\% of the NXB.
Thus, the contribution of GRXE is negligible when we analyse the HXD-PIN spectrum of CTA~1.
After the background subtraction, we rebinned the spectrum to ensure that the photon numbers in each channel are larger than 30.
\section{Results}
\subsection{Timing Analysis}
The expected spin frequency of the pulsar in our {\it Suzaku} data can be inferred from the contemporaneous {\it Fermi} ephemeris \citep{Ray2011}.
The start of GTI for our {\it Suzaku}/XIS3 and HXD-PIN data is at epoch MJD 55204.6143077.
The spin frequency of PSR~J0007+7303 corresponding to this epoch can be inferred as 3.165750271(6)~s$^{-1}$ with the first frequency derivative of $-3.6136(2)\times 10^{-12}$~s$^{-2}$ (the effect of the second frequency derivative is negligible).
We search for periodic signals with the known first frequency derivative around the predicted frequency in {\it Suzaku}/XIS3 and HXD-PIN data using $H$-test \citep{DSR89}.
The $H$-value is only 11.7 corresponding to the expected spin frequency of 3.165750271(6)~s$^{-1}$ for the {\it Suzaku}/XIS3 observation and only 6.7 for the {\it Suzaku}/HXD-PIN observation.
Fig.~\ref{PS} demonstrates the periodicity search of 200 independent trials around the expected spin frequency of PSR~J0007+7303.
We conclude no significant pulsed detection can be yielded from our {\it Suzaku} data both in the soft and hard X-ray bands.
Comparing with the soft X-ray pulsation detected by the {\it XMM} observation \citep{Lin2010,Car2010}, our {\it Suzaku} data lose 1-dimensional imaging ability in the XIS3 timing mode and have no image ability in the {\it Suzaku}/HXD investigation.
In the {\it Suzaku}/XIS3, we can only have a marginal detection to resolve the point source with $\sim$ 58000 photons as shown in section $\S$2.2.
However, the counts gathered in the source region are much more than we expected based on the spectral behaviour provided by \citet{Sla97}.
We noted that most of the additional counts obtained from the P-SUM mode data are caused by the instrumental background .
For the data detected in the hard X-ray band, the main contribution of the hard X-ray photons is ascribed to the X-ray background.
If we assume the pulsed spectrum of \citet{Car2010}, the expected S/N ratio of the {\it Suzaku} investigation is too low ($< 2 \sigma$) to yield the periodic signal from the pulsar.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm]{PS.ps}
\end{center}
\caption{\small{Periodicity search of the {\it Suzaku} data by the $H$ test. Each panel presents the search with 200 independent trials close to the expected spin frequency of PSR~J0007+7303 at the observational epoch. The upper panel shows the searching results of the {\it Suzaku}/XIS3 data, and the width of each independent trial is $\sim 4.7\times 10^{-6}$~s$^{-1}$. The lower one shows the searching results of the {\it Suzaku}/HXD-PIN data, and the width of each independent trial is $\sim 6.0\times 10^{-6}$~s$^{-1}$.}}
\label{PS}
\end{figure}
We also examined this spin frequency using $\sim 3$-month {\it Fermi} archive (2009 November 22 - 2010 February 21) on the epochs close to our {\it Suzaku} observation.
The effective $\gamma$-ray photons of PSR~J0007+7303 were restricted in a 1-degree circle centered at (J2000) R.A.=$1^{\circ}.7565$, decl.=$+73^{\circ}.05225$ \citep{HGCH2004} within the energy range from 100 MeV to 300 GeV.
We determined the spin frequency of the pulsar with the $H$-test using the task of `GTPSEARCH' from the {\it Fermi} Science Tools ver.~v9r15p2.
At epoch MJD 55204.6143077, the maximum statistic was obtained at 3.16575026(1)~s$^{-1}$ with the first frequency derivative of $-3.6136(2)\times 10^{-12}$~s$^{-2}$, and this result is consistent with the aforementioned prediction as well.
\subsection{Spectral Analysis}
We derived the soft X-ray spectrum from XIS0 and XIS1 onboard {\it Suzaku}.
Based on our selected region for the X-ray point source, the ratio of net source counts for the pulsar to the total photons in the source region considered in the spectrum of XIS0 is more than 25\% but in that of XIS1 is only $\sim 10$\%.
This is caused by the fact that the BI chip (XIS1) has a higher effective area at low energies.
We only considered the spectral fits in the range 0.5-10 keV and ignored those photons outside this energy range to avoid large uncertainties.
A cross-calibration term was included to correct the difference among the XIS spectra, and the result is
consistent with the fit to the spectrum of any individual detector.
The absorption was fixed at $2.8 \times 10^{21}$~cm$^{-2}$ according to previous measurements \citep{Sla97,Lin2010}, which is consistent with optical extinction \citep{HGCH2004}.
A single power-law provides an acceptable fit as shown in Table~\ref{XISsp} and Fig.~\ref{XIS}, and our data are not good enough to give a well constraint for an additional model component, such as a thermal blackbody model or a magnetized neutron star atmosphere (NSA; \citealt{ZPS96}) model.
However, we note that the source fluxes derived from different sizes seem larger than those of previous studies and increase with source sizes; these excesses might be contributed from the PWN.
\begin{figure*}
\centering
\hspace{\fill}{\includegraphics[width=6.0cm,angle=-90]{spec_s90N.ps}}
\hspace{\fill}{\includegraphics[width=6.0cm,angle=-90]{spec_s120N.ps}}
\hspace{\fill}{\includegraphics[width=6.0cm,angle=-90]{spec_s180N.ps}}
\hspace{\fill}
\caption{\small{Spectral fits to the power-law model. The first row presents the spectral fits to the point source centered in CTA~1 with a selected circle of 1$'$.5 and 2$'$ in radii, respectively. The second row presents the spectral fit to the point source centered in CTA~1 with a selected circle of 3$'$. The black and green data sets refer to the XIS0 observations of $3\times 3$ and $5\times 5$ modes respectively, while the red and blue data sets refer to the XIS1 observations of $3\times 3$ and $5\times 5$ modes. The detailed parameters of these fits are shown in Table~\ref{XISsp}.}}
\label{XIS}
\end{figure*}
\begin{table}
\caption{\small{Best parameters for the spectral fits to the pulsar in CTA 1 using {\it Suzaku}/XIS0 and XIS1 observations.}}
\label{XISsp}
\begin{center}
\begin{tabular}{l c c c}
\hline\hline
& \multicolumn{3}{c}{RX~J0007.0+7302/PL}
\\
Parameter & Small$^{b}$ & Medium$^{b}$ & Large$^{b}$
\\
\hline
$N_H$ (cm$^{-2}$; fixed) & $2.8 \times 10^{21}$ & $2.8 \times 10^{21}$ & $2.8 \times 10^{21}$
\\
$\Gamma$ & 1.76$^{+0.27}_{-0.25}$ & 1.80$\pm$0.15 & 1.85$^{+0.14}_{-0.12}$
\\
$F_{X,\rm{PL}} \times 10^{-13}$& $3.7\pm 1.1$ & $4.6\pm 1.0$ & $7.1\pm 1.3$
\\
(ergs cm$^{-2}$ s$^{-1}$)$^{a}$ & & &
\\
$\chi^2$/dof & 39.0/46 & 78.2/85 & 157.4/143
\\
\hline
\end{tabular}
\end{center}
\footnotesize{
${}^{a}$ The unabsorbed flux is measured in the energy range of 0.3--10 keV. \\
${}^{b}$ `Small, medium and large' indicates the source selection with different size. The small, medium and large point source regions were adopted with the same center and circles with 1$'$.5, 2$'$ and 3$'$ in radii, which correspond to the EEF of $\sim$~75\%, 85\% and 95\%. The background contributions were determined within a concentric annulus of radii 100$''< r \leq 180''$, 120$''< r \leq 240''$, and 180$''< r \leq 270''$, respectively.}
\end{table}
On the other hand, our deep {\it Suzaku} observation provides a desirable opportunity to investigate the PWN, particularly for the faint emission with the extent up to $\sim 10'$ (cf. Fig.~\ref{pwn_img}).
Previous observations with {\it Chandra} \citep{HGCH2004} and {\it XMM} \citep{Car2010} failed in detecting this large but faint diffuse structure; this is likely caused by the relatively small collecting area of {\it Chandra} and the high instrumental background of {\it XMM} respectively.
\begin{table}
\caption{Best-fit parameters of the power-law spectral components inferred from different regions of the PWN around PSR~J0007+7303.$^{a}$}
\label{pwn_spec}
\begin{center}
\begin{tabular}{l c c c}
\hline\hline\\[-2ex]
Region & $\Gamma$ & $f_{\rm pwn}$ (0.3-10~keV) & $\chi^{2}$/d.o.f. \\
& & erg~cm$^{-2}$~s$^{-1}$ & \\
\hline
Inner part & $1.89\pm0.08$ & $1.2^{+0.2}_{-0.1}\times10^{-12}$ & 30.0/28 \\
Middle part & $1.79\pm0.09$ & $\left(1.4\pm0.2\right)\times10^{-11}$ & 38.1/34 \\
Outer part & $1.92\pm0.11$ & $1.5^{+0.3}_{-0.2}\times10^{-11}$ & 38.4/41 \\
\hline
\end{tabular}
\end{center}
$^{a}$ {\footnotesize{Column density is fixed at $N_{H}=2.8\times10^{21}$~cm$^{-2}$ for all these
spectral fit. An additional power-law component to account for the pulsar emission with the parameters fixed at those inferred by \citet{Car2010} (i.e. $\Gamma=1.36$ with $f_{\rm PSR}=6.5\times10^{-14}$~erg~cm$^{-2}$~s$^{-1}$)} is included with a multiplicative factor for scaling the pulsar contributions in each region accordingly (i.e. $85\%$, $10\%$ and $5\%$ for inner, middle and outer parts respectively.}
\end{table}
In order to properly constrain the non-thermal X-ray contributions from the PWN, we also fixed the column density at $N_{H}=2.8\times10^{21}$~cm$^{-2}$.
Besides a power-law model for PWN emission, we also included an additional power-law component to account for the X-rays from the pulsar with the parameters fixed at those inferred by \citet{Car2010}.
For each of the three regions in our consideration, we scaled the spectral component of the pulsar with the EEF or the contribution from the wing of the PSF centered at the pulsar accordingly by multiplying a constant.
The best-fit parameters of the power-law component of the PWN inferred from the spatially-resolved spectral analysis are summarized in Table~\ref{pwn_spec}.
Taking the statistical uncertainties into consideration, no evidence for spectral steepening can be found from this observation.
The whole diffuse X-ray feature can be modelled with a power-law with a photon index of $\Gamma \sim 1.8$.
\begin{figure}
\begin{center}
\includegraphics[width=6.2cm,angle=-90]{PIN.eps}
\end{center}
\caption{\small{Spectral fit to the {\it Suzaku}/HXD-PIN observation in $14-30$ keV. The solid line represents the best-fit to a single power-law with the photon index of 3.0, and corresponds to the unabsorbed flux of $\sim 8.0\times 10^{-12}$~ergs~cm$^{-2}$~s$^{-1}$ in 10-50 keV. The red line demonstrates the systematic errors which correspond to $\sim 5\%$ of the NXB distribution. The bottom panel shows the residuals in terms of $\sigma$.}}
\label{PIN}
\end{figure}
We also derived the hard X-ray spectrum from the HXD-PIN.
After subtracting the background, the net counts of the hard X-rays collected for the spectrum is only 6.6\% in total.
Because there are abnormal accumulations for photons at both limits of the effective energy boundary in our HXD-PIN observation, we only consider the hard X-ray spectrum within the energy range of $14-30$ keV.
The spectrum fitted to a power-law model with the photon index ($\Gamma$) of 3.0 is shown in Fig.~\ref{PIN}.
We also checked the systematic uncertainty of the background from other observations, and the NXB accuracy should be within 5\% level in the 15-40 keV around the CTA~1 observation.
The nominal uncertainty of the adopted background model is also indicated in Fig.~\ref{PIN}.
Because the effective data points of our hard X-ray spectrum are few, the uncertainty is very large and the best fit can not be well determined.
We fixed the absorption as 2.8$\times$10$^{-21}$~cm$^{-2}$ according to the column density obtained in \citet{Sla97}, and a photon index of $\Gamma=3.33^{+2.18}_{-1.81}$ (90\% confidence level) with $\chi^2_{\nu}=1.15$ for 10 d.o.f. can be yielded in the investigation of our hard X-ray spectrum.
Taking into account the wide spread of the acceptable photon index, the total unabsorbed flux in 10-50 keV for the whole FOV of the HXD-PIN inferred from these acceptable fits distributes from $\sim 7.8\times 10^{-12}$~ergs~cm$^{-2}$~s$^{-1}$ to $\sim 1.1\times 10^{-11}$~ergs~cm$^{-2}$~s$^{-1}$, and this signal stays between $\sim$ 3-5\% level of the NXB in the 90\% confidence level.
The hard X-rays detected in the region around CTA~1 is also consistent with previous {\it INTEGRAL} investigation \citep{Sturn2004}.
\section{Discussion}
We have investigated the central region of CTA~1 with a deep {\it Suzaku} observation.
For the X-ray flux of RX~J0007.0+7302 as estimated by the XIS data, we note that it is larger than that reported by \citet{Lin2010} and \citet{Car2010} through the {\it XMM} observation.
This discrepancy can be ascribed to the fact that the PSF of {\it Suzaku} is considerably wider than that of {\it XMM}.
In view of this, the contribution of the point source is likely to be contaminated by the surrounding PWN.
It is the main reason that we can not obtain the real spectrum of the pulsar.
Because of the loss of the imaging ability in the {\it Suzaku} data for the periodicity examination, we can not separate the instrumental effect in the soft X-ray band and the serious contamination from the X-ray background in the hard X-ray band.
These problems prevent the pulsed detection from the current {\it Suzaku} observation.
Since the {\it Suzaku} data do not provide a constrained result for the timing properties of RX~J0007.0+7302/PSR~J0007+7303, we can not obtain the phase-resolved spectroscopy and will not discuss the nature of this point source any further in the following statements.
We have also examined the properties of the PWN with the XIS data.
A single power-law can provide a good fit and we did not detect any additional thermal X-rays emitted in the central region of CTA~1 which is consistent with previous investigation (e.g. \citealt{Sla97}).
Both the flux and the photon index obtained in our independent investigation are fully consistent with that inferred by the recent {\it XMM} observation \citep{Car2010}.
We would like to discuss the possible nature of the nebula that extends eastward as discovered by our deep {\it Suzaku} exposure.
The feature revealed by {\it Suzaku} extends as large as $\sim 10'$ while the {\it XMM} observations only detected an nebular feature with an extent of a few arc-minute, which might correspond to the relative brighter component of the nebula (cf. Fig.~1 in \citealt{Car2010}).
The spectral steepness and the flux of the inner part in our investigation are consistent with their results (refer to ``Outer PWN" in Table~1 of \citealt{Car2010}).
The morphology of this extended feature appears to be asymmetric (see Fig.~\ref{pwn_img}) which resembles those of bow-shock nebulae, e.g. PSR~J1747--2958 \citep{Gaen2004}.
Such extended feature are usually along the direction of pulsar motion and behind the bow-shock, therefore they are interpreted
as the synchrotron radiation from the flow of particles coming out of the PWNe.
It is instructive to discuss the plausible relation between this large feature and the torus+jet of a smaller scale reported in Figure~1 of Halpern et al. (2004; also shown in the inset of our Fig.~\ref{pwn_img}).
The {\it Chandra} image shows that the jet have an extent of $\sim16^{''}$ toward south and bends to southwest at the far end from the pulsar (see Fig.~\ref{pwn_img}).
On the other hand, the compact torus-like feature with a radius of $\sim3^{''}$ around the pulsar appears to be elongated in a direction perpendicular with the jet.
\citet{HGCH2004} interpreted this system as an equatorial torus with the jet emitted along the rotation axis, which is observed in many PWN systems (cf. \citealt{NR2004} for a review).
If the $\sim10^{'}$ feature is indeed resulted from the bow-shock as aforementioned, this suggests the pulsar is possibly moving towards west.
This speculated pulsar motion is almost perpendicular to the portion of the jet close to the pulsar (see Fig.~\ref{pwn_img}).
The origin of the velocity of pulsars is still an open question.
One possible mechanism is the asymmetric supernova explosion.
Apart from giving rise to the kick velocity, it also contributes to the initial spin of a pulsar \citep{SP98,LCC2001}.
If the initial spin angular momentum of RX~J0007.0+7302 is dominated by the kick process, the jet emitted along the rotational axis should be more or less perpendicular to the direction of proper motion,
which can explain the relative orientation between jet and the $\sim 10'$ feature.
Therefore, measuring the proper motion of RX~J0007.0+7302 with multi-epoch X-ray imaging (e.g. \citealt{HB2006}) and/or dedicated $\gamma-$ray pulsar timing solution yielded from LAT data can help to constrain the relation among various extended feature of this complex system.
Despite the exact origin of this large extended feature remains to be uncertain, the X-ray spectral analysis confirms its non-thermal nature.
Assuming it is a synchrotron nebula, we discuss its emission properties in further details.
For the synchrotron emitting electrons distributed as $N\left(\gamma\right)\propto\gamma^{-p}$ where $\gamma$ is the Lorentz factor of the wind particles, the resultant X-ray photon index $\Gamma$ depends on whether the emission is in a fast or slow cooling regime \citep{CTW2004}.
This is determined by cooling frequency of the emitting region $\nu_{c}=\left(18\pi e m_{e}c\right)/\left(\sigma_{\rm T}^{2}\tau_{\rm syn}^{2}B^{3}\right)$, where $m_{e}$, $\sigma_{\rm T}$, $\tau_{\rm syn}$ and $B$ are the electron mass, Thomson cross section, synchrotron lifetime and the magnetic field strength respectively.
We have $\Gamma=\left(p+2\right)/2$ for a fast cooling scenario (i.e. $\nu_{x}>\nu_{c}$) and $\Gamma=\left(p+1\right)/2$ for slow cooling (i.e. $\nu_{x}<\nu_{c}$).
For a standard shock model, $p$ spans a range of $\sim2-3$ (cf. \citealt{CTW2004} and references therein).
This implies that $\Gamma$ should span the ranges of $\sim2-2.5$ and $\sim1.5-2$ for the scenario of fast cooling and slow cooling, respectively.
Comparing these ranges with the values inferred in the XIS spectrum (cf Tab.~\ref{pwn_spec}), we suggest that the inner and the middle parts of this PWN are probably in a slow cooling regime.
For the outskirt of the nebula, the best-fit photon index suggests the synchrotron cooling in this region is also slow.
However, within the $90\%$ confidence interval of the inferred photon index, it is on the margin of both regimes.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm]{spec.eps}
\end{center}
\caption{Broadband absorbed spectrum of PSR~J0007+7303, PWN in the soft X-ray band and hard X-ray detection of CTA~1. The absorption in the low-energy band is fixed as $2.8\times 10^{-21}$ cm$^{-2}$ following Table~\ref{pwn_spec}. The spectral behaviour of the pulsar is determined by the power-law fit of table 1 from \citet{Car2010}. The spectrum and the data points for the PWN are derived from the outer region (the data include $\sim$ 5\% contribution of the pulsar; see Table~\ref{pwn_spec}). The solid, dotted, and dashed lines represent the uncertainty ranges of the spectral behaviour of the PSR, PWN, and HXD.}
\label{spec}
\end{figure}
For the inner and middle regions of the PWN, we speculate that $\nu_{c}$ should lie beyond the energy range covered by XIS.
Assuming $h\nu_{c}\sim10$~keV and the synchrotron lifetime of the emitting electrons is comparable with the characteristic age of the pulsar (i.e. $\tau_{\rm syn}\sim1.4\times10^{4}$~yrs), the magnetic field strength in the emitting region can be estimated at the order of $B\sim2$~$\mu G$, which is comparable with the typical field strength in the ISM (cf. \citealt{BSSW2003} and references therein).
Apart from the observation in the soft X-ray band, we have also investigated the field of CTA~1 with the hard X-ray data collected by HXD-PIN.
The spectrum obtained in this hard band (i.e. $14-30$~keV) can be described by a power-law with $\Gamma=3.33^{+2.18}_{-1.81}$.
The unabsorbed flux in $10-50$~keV is found to be $f_{x}\sim(0.8-1.1)\times10^{-11}$~erg~cm$^{-2}$~s$^{-1}$.
We note that the photon index inferred in this band cannot be tightly constrained; however, within the $90\%$ confidence interval, the HXD spectrum can be smoothly connected with the spectrum inferred from the soft band although might have slightly spectral steepening as shown in Fig.~\ref{spec}.
We should point out that the true nature of the hard X-ray emission is not yet conclusive.
The FOV of the HXD is $\sim 34'\times 34'$, while the size of PWN in the soft X-ray is only $\lesssim 10'\times 10'$.
In view of the lack of imaging capability of HXD-PIN, the observed hard X-rays can possibly be contributed by other sources in the FOV of {\it Suzaku}. This might indicate an steeper spectral break than observed.
On the other hand, the effects caused by the systematic uncertainty of the NXB background could be large.
For example, if the NXB level is lower by a few percent (e.g. 2\%, which is the nominal value of the NXB uncertainty; \citealt{Fuk2009}) than the assumed value, the true hard X-ray would be higher than the current detection and the spectral break would not be as significant as observed.
Since there are several upcoming missions, including {\it NuSTAR} \citep{Hailey2010} and {\it Astro-H} \citep{Taka2010}, which will be capable to image the sky up to $\sim80$~keV for the first time, the exploration of PWNe, including the one in CTA~1, will leap into a new era in the near future.
\section*{Acknowledgments}
We thank Prof. Yuji Urata in National Central University and Ms. Ting-Ni Lu in National Tsing Hua University of Taiwan for discussions on the examination of the spectra and the vignetting-correction of image for our {\it Suzaku} observation.
This work was partially supported by the National Science Council (NSC) of Taiwan through grant NSC 99-2811-M-008-057 and NSC 101-2112-M-039-001-MY3. RHHH is supported through grant NSC 99-2811-M-007-062 and NSC~100-2811-M-007-040.
CY Hwang acknowledges support from the NSC through grants NSC~99-2112-M-008-014-MY3 and NSC~99-2119-M-008-017. AKHK acknowledges support from the NSC through grant NSC~100-2628-M-007-002-MY3.
CY Hui is supported by the National Research Foundation of
Korea through grant 2011-0023383.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,448 |
\section{Introduction}
The recent detection of an electromagnetic (EM) counterpart to the gravitational wave (GW) event GW170817 represents a major advance in multi-messenger astronomy (\citealt{2017ApJ...848L..12A}). This event was caused by the merger of two neutron stars, which resulted in a kilonova event (AT2017gfo) at 40 Mpc, visible in multi-wavelength observations (from gamma rays to radio). Several studies have been published addressing questions regarding the origin of this event, its observable properties, the physics of kilonova models, the abundances of r-process elements in the Universe, and the cosmological implications, just to mention a few (e.g. \citealt{2017Natur.551...85A,2017Natur.551...67P,2017Natur.551...75S}). During the first two periods where the LIGO (and Virgo) GW detectors have been operational called observing runs (O1 -- from 2015 September 12 to 2016 January 19 and O2 -- from 2016 November 30 to 2017 August 25), eleven GW events have been observed by the LIGO and Virgo detectors (\citealt{2019PhRvX...9c1040A}). Ten of these were associated with black hole - black hole (BHBH) mergers and one with a neutron star - neutron star (NSNS) merger and its kilonova signal as mentioned above (\citealt{2016PhRvL.116x1103A,2017PhRvL.118v1101A,2017ApJ...848L..12A,2017PhRvL.119n1101A}). Black hole - neutron star (BHNS) mergers have remained undetected during these runs.
Predictions are that up to 50 events could be detected from NSNS mergers and BHNS mergers after the improved detectors have been operational for one year (\citealt{2018PhRvL.120i1101A}). This third observing run (O3) started on 2019 April 01. Several papers have been published predicting that one might be also able to observe an EM counterpart to BHBH mergers (e.g. \citealt{2017ApJ...839L...7D,2017PhRvD..96l3003K}). The brightness of kilonova events is estimated at 19.5 and 21 mag in optical bands at 100 and 200 Mpc, respectively (\citealt{2017LRR....20....3M}). Candidate events for BHBH, NSNS, and BHNS mergers have been already reported in this run (e.g. \citealt{2019GCN.24098....1L,2019GCN.24168....1L,2019GCN.25333....1L}).
The ESA-{\it Gaia} mission has been operational since mid-2014 and has provided accurate photometric, astrometric, and spectroscopic measurements for roughly a billion stars in the Milky Way (\citealt{2018A&A...616A...1G}). {\it Gaia}'s on-board detection algorithms are optimised for the detection of point-like sources, although the mission is also collecting data for a significant number of resolved extragalactic objects (\citealt{2014sf2a.conf..421D}). As a result of the observing strategy, {\it Gaia} scanned over the location of most of the sources more than 70 times from different angles during the first five year mission. Each position on the sky is observed, on average, once every 30 days (\citealt{2016A&A...595A...4L}). These repeat visits make {\it Gaia} an all-sky, multi-epoch photometric survey that allows us to monitor variability with high precision, as well as detect new transient sources (\citealt{2013RSPTA.37120239H,2017arXiv170203295E}). The Data Processing and Analysis Consortium (DPAC) handles {\it Gaia}'s data flow, and this enables the detection of transients within 24-48 hours of observations. After September 2014 new transients from {\it Gaia} have been made publicly available after manual vetting of candidate transients detected by the {\it Gaia} Science Alerts (GSA) team (see: \url{http://gsaweb.ast.cam.ac.uk/alerts/alertsindex} and \citealt{2019ASPC..521..507D,2019ASPC..523..261D}). To this end, AlertPipe - dedicated software for data processing, transient searching, and candidate filtering was employed (Hodgkin et al. in prep.). {\it Gaia}'s accurate photometry and low-resolution spectroscopy should allow for a robust classification and reduces the rate of false positives. {\it Gaia} could therefore play an important role as a transient detection survey. Several publications have shown that {\it Gaia} is able to detect uncommon transients such as the eclipsing AM~CVn system Gaia14aae (\citealt{2015MNRAS.452.1060C}), the superluminous supernova Gaia16apd (\citealt{2017MNRAS.469.1246K,2017ApJ...835L...8N}), fast transients (\citealt{2018MNRAS.473.3854W}), transients in the centres of galaxies (\citealt{2018MNRAS.481..307K}), just to mention a few. Moreover, in July 2019 {\it Gaia}'s mission entered in the extension period from mid-2019 to the end of 2020 and it is likely that the mission will be extended further with a firm end-of-mission date of end of 2024 ($\pm$6 months).
In this study we aim to explore the possibility of an improvement for the detection rate so that the GSA can capture transient events that the existing pipeline might miss (as they are too faint and/or too fast). Specifically, we propose to employ a bespoke detection algorithm for the {\it Gaia} data to search for EM counterparts to GW events. This new detection algorithm will make use of GW event localisation and timing to allow it to run at a lower detection threshold and thanks to that we will increase the completeness (although the sample purity might decrease).
We perform a systematic search for transients that coincide in time and in sky localisation with the run O1 and O2 GW detections (\citealt{2019PhRvX...9c1040A}), to investigate if a dedicated source finding and vetting algorithm can be implemented to run during (the remainder of) LIGO/Virgo's O3 and O4 so that {\it Gaia} can enhance its contribution to the search for the EM counterpart to a GW event. We also provide a list of potential EM transients that occurred close in time and sky location to GW events from the O1 and O2 runs.
This paper is organized as follows. In Section \ref{sec:detector} we present properties of, and tests on, the new detection algorithm, and discuss our results from a one year test. In Section \ref{sec:O1O2} we show the results from a search for candidate transients coincident with GW events from the O1 and O2 runs, and furthermore, consider implications for the future {\it Gaia} possibilities of detecting the EM signal associated with GWs. We conclude in Section \ref{sec:con}.
Throughout this paper we assume a flat $\Lambda$-Cold Dark-Matter ($\Lambda$CDM) concordance cosmological model of the Universe with parameters $\Omega_\Lambda = 0.7$, $\Omega_\mathrm{M} = 0.3$ and $H_0 = 70\mathrm{~km~s^{-1}~Mpc^{-1}}$, $h = 0.70$.
\section{GW detector}
\label{sec:detector}
As with any optical transient detected by the GSA pipeline, an EM counterpart to a GW event might be found as a so called "new source", which is a source that has not previously been seen by {\it Gaia}, or as an existing source changing its photometric properties (typically brightening although the GSA pipeline also detects sources fading). A "new source" classification for instance occurs if any host galaxy is below the detection threshold or if the transient is resolved from the host galaxy light and it passes the thresholds set in the detection algorithm for a new independent source detection. The existing GSA {\it NewSource} detection algorithm, a detector in short, will trigger a detection if the event has 2 or more detections above a flux threshold equivalent to {\it Gaia}'s $G$--band magnitude of $G=19$ mag, and it is detected by observations made with both of {\it Gaia}'s telescopes (i.e.~ the source is detected in the two different fields-of-view; Hodgkin et al. in prep.).
Here, we propose an additional detector to increase the efficiency in finding new events at the expense of a higher intrinsic rate of false positives that have to be filtered out. We investigate the influence of a lower flux threshold and the removal of the condition that at least 2 observations should detect the new source has on the transient detection rate. The removal of the requirement of two {\it Gaia} detections is driven by kilonova models for the evolution of the source (absolute) magnitude with time. From those models we expect at most an optical signal detectable by {\it Gaia} for up to a week after the merger for likely distances (\citealt{2017LRR....20....3M}). Hence, given that a second scan by {\it Gaia} over the same sky location that led to the first detection might only take place after the source has faded making it potentially too faint to be detectable by {\it Gaia} implies that the current GSA system would not flag the source as a transient (specifically 25 per cent of time sources will only have one observation and a fraction of the data may be held on board until later). However, as said, removal of the requirement of the second detection will also increase the number of false positive detections. To mitigate these effects we developed and tested additional filters to verify the photometric and astrometric data accuracy.
In this study we make use of all {\it Gaia} photometric data collected since the beginning of the mission (July 2014) that are ingested into the {\it Gaia} Science Alerts Database (GSA DB). By using the GSA DB we have access to the {\it Gaia} time series and individual measurements from scans (=transits) as opposed to data available in {\it Gaia}'s early data releases where only averaged data products are available. To build and study properties of a new detector we have taken one year of data (from the full year of 2018) where we searched for all transits classified as new sources during the initial data treatment (IDT, \citealt{2016A&A...595A...3F}) by DPAC. A new source is created when no match is found within a radius of 1.5 arcsec. There are more than 144 million transits flagged by IDT as new sources. Fig.~\ref{fig:densityall} shows the density distribution of those sources over the sky. The map shows artefacts from {\it Gaia}'s scanning law and it shows that many of those tentative new sources fall in the most dense regions on the sky (in the Milky Way bulge and disc, and near to the Magellanic Clouds).
\begin{figure}
\includegraphics[width=\columnwidth]{density0.eps}
\caption{The all-sky distribution of new sources flagged by IDT during the year 2018. The artefacts from the scanning law are clearly visible. The map resolution is HEALPix of nside 32. The plot is in equatorial coordinates, 0,0 in the centre, with North up and East to the left.}
\label{fig:densityall}
\end{figure}
Due to our relaxed requirement of including sources that were detected by a single detection we will be much more susceptible to the detection of artefacts introduced by nearby bright stars, close binary systems, dense regions, planets, solar system objects (SSOs), and epochs of initially bad astrometry, to mention just a few of the effects that can cause spurious detections in {\it Gaia}'s IDT. Furthermore, our lower flux threshold for transients when compared to the existing GSA detection algorithms will give us additional samples of transients, some of which are spurious, that are usually not detected by the standard GSA system. Hence, additional cuts and filtering must be applied in order to weed out the spurious transients as much as possible.
\subsection{Selection process}
\label{sec:selection}
The final selection of candidate transients in the novel detector has been performed using the following filtering steps:
\begin {enumerate}
\item We required that at least 8 of the 9(8) astrometric field (AF) CCD measurements during a single transit must return a valid photometric data point (see \citealt{2015A&A...576A..74D} for a description of {\it Gaia}'s focal plane).
\item We excluded the most dense regions in the sky (the bulge and disc of the Milky Way, and the Large and Small Magellanic Clouds) by applying a cut on the number of sources per HEALPixel of nside 4096 equivalent to $50\times50$ arcsec (see \citealt{2005ApJ...622..759G} for the definition of HEALPixels and their sizes). We made use of {\it Gaia} Data Release 2 (GDR2, \citealt{2018A&A...616A...1G}) source density maps and decided that the HEALPixel is excluded from the further processing if the number of sources is larger than the mean number of sources per HEALPixel. In Fig.~\ref{fig:densitycuts} we show density maps from GDR2 before and after applying the cut.
\begin{figure}
\includegraphics[width=\columnwidth]{densityGDR2_hp12masked0.eps}
\caption{The density map of sources in GDR2 (the number of sources per HEALPixel of nside 4096) in HEALPixels chosen for the new detector. Grey regions indicate deselected parts as they have a number of sources above the mean. The cut mostly affects fields in the Milky Way bulge and disc, and the centres of the Small and Large Magellanic Clouds. About 21 per cent of the sky was removed and will not be processed during the search for EM counterparts to GW events. The plot is in Galactic coordinates.}
\label{fig:densitycuts}
\end{figure}
\item We removed all new sources created during astrometric excursions of the satellite (caused by e.g.~hits by micro-meteorites, space debris, and non-rigidity events, see: \citealt{2008IAUS..248...82V}). This effect may cause a significant excess of number of detected new sources during the IDT. The reason is that sources are preliminarily assigned erroneous coordinates due to the astrometric excursions, which influences the low-latency GSA DB. Further processing later in time, well before the formal {\it Gaia} Data Releases, corrects for this. Here, we created a histogram of number of observed new sources as a function of time (with a time bin size of about 20 minutes) to eliminate data collected during astrometric excursions as those stand out as peaks where the "transient" discovery rate shoots up. We noticed 6 major events where the number of new sources rose up to $10^4$ or more per 20 minute intervals and several lower peaks are present as well (see Fig. \ref{fig:obstime}). We removed all bins where the number of observed sources is larger $1 \sigma$ above the 20-minute average (i.e. about 3000 sources per 20 minutes).
\begin{figure}
\includegraphics[width=\columnwidth]{obstime.eps}
\caption{The number of new sources created during the IDT vs.~time for the year 2018. The remarkable peaks where the number of new sources reaches up to $10^4$ and more appear a few times in the first half of the year. Scanning the Milky Way and ecliptic also causes an excess in the rate of detected transients due to problems with crowding and solar system object detections. Several smaller peaks are also visible all over the year. The epochs of these enhanced rates of detections of transients are removed (see \ref{sec:selection} for the exact threshold used).}
\label{fig:obstime}
\end{figure}
\item Magnitude limit: we required the median 9(8) CCD flux to be $>101.25$ $\mathrm{e^{-}/s}$ (equivalent to $G \sim 20.68$ mag, calibrated as in the GSA pipeline, Hodgkin et al. in prep.). We got to this limit as follows: all detected transients that remained after the filters listed above were cross-matched with the Pan-STARRS Data Release 1 catalogue (PS1 DR1, \citealt{2016arXiv161205560C}) within a search radius of 1 arcsec. Many new {\it Gaia} detections coincide with known (fainter) sources from the PS1 survey. We studied the flux distribution of these sources in {\it Gaia} and fitted a Gaussian function to their flux distribution in {\it Gaia} (see Fig.~\ref{fig:fluxPS1xG}). All sources fainter than $5\sigma$ above the mean value of flux distribution, i.e.~$G \sim 20.68$ mag, are likely to be sources that are detected by {\it Gaia} and labelled during the IDT as a new source due to a Poisson fluctuation in their count rate. Therefore, setting the detection threshold at $5\sigma$ above the mean value of flux distribution of these candidate transients only a small number of these spurious sources remains (in Gaussian statistics only one in a million).
\item We removed artefacts from bright stars using data from GDR2.
Bright stars might cause multiple spurious detections in a large radius around them (\citealt{2015A&A...576A..74D}). For each candidate new source we search for all neighbours within a search radius of 30 arcsec in GDR2.
The plot in Fig.~\ref{fig:dr2neighbour} shows the offset $d$ between the candidate and neighbours from GDR2 versus the $G$-band mean magnitude of the neighbours.
We assumed that any source fulfilling the condition $-5.5\cdot \log(d\mathrm{[arcsec]})+19 < G$ (the diagonal red dashed line in Fig. \ref{fig:dr2neighbour}) may cause an artefact in detection. However, as the counterparts to the GW events will be located in galaxies we have to prevent the situation when the closest and the brightest neighbour is actually the centre of the host galaxy. Hence, we also studied the distribution of galaxy brightness in GDR2. Using a sample of spectroscopically confirmed galaxies from the SDSS catalogue (\citealt{2017AJ....154...28B}) we found that only 1 per cent of those detected by {\it Gaia} is brighter than $G=17$ mag (Fig.~\ref{fig:sdssgals}). Therefore, we decided not to exclude any candidate with a neighbour in GDR2 fainter than $17$ mag in $G$-band (the horizontal part of the red dashed line in Fig. \ref{fig:dr2neighbour}).
\item Scatter during transit: the median absolute deviation (MAD) within 9(8) CCD flux measurements during a single transit must be limited. We assumed no significant change in the light curves within the crossing time through 9(8) CCDs that the source transits over the focal plane (a single transit lasts about 45 seconds). In general, the scatter is a function of flux, hence the cut we apply is a function of the source brightness (see Fig.~\ref{fig:fluxerrors}).
\item We removed artefacts from bad cross match during IDT (caused for example by bad astrometry) by performing an internal cross match within the GSA DB. If a candidate has neighbouring transits within $0.5$ arcsec detected before the detection of the candidate we assume that these two entries in the database should be considered as the same source (and the new source under consideration was in fact erroneously not matched during IDT to that nearby source detected before).
\item We removed transits where the photometry is flagged as bad during the IDT.
\item We excluded all new sources potentially caused by SSOs. All new sources were cross matched with the available internal SSO table (Hodgkin et al. in prep.) within a search radius of 2 arcmin and within a time difference between observations of 3 seconds. The source and a predicted SSO transit have to be observed by the same field-of-view, on the same CCD row, within 3 seconds of time, and with a distance offset less than 0.1 arcmin (see Fig.~\ref{fig:ssoxmatch} for the distribution of offsets between new sources and SSOs).
\end{enumerate}
\begin{figure}
\includegraphics[width=\columnwidth]{xmPS1.eps}
\caption{The distribution of the median flux from single transits for new sources that were also detected in the PS1 DR1. The magenta line indicates a Gaussian fit to the distribution. We remove these sources from the list of transients detected by {\it Gaia} as they are likely caused by Poisson fluctuations (or real low-amplitude variability) in fainter sources detected previously by PS1.}
\label{fig:fluxPS1xG}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{neighbours.eps}
\caption{The distance between a potential ''new source'' and its nearest neighbours found in GDR2 within a search radius of 30 arcsec around the position of the ''new source'' vs.~the $G$-band brightness of those neighbours in GDR2. A group of new candidate transients appears in close proximity of bright stars. The plot shows a sample of 1 per cent from all candidate transients detected. The red dashed line indicates the applied cut, where all candidate transients above the red dashed line are removed as they are likely to be caused by artefacts such as diffraction spikes caused by bright stars. }
\label{fig:dr2neighbour}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{SDSSgalGDR2.eps}
\caption{The magnitude of the source detected by {\it Gaia} ($G$-band mean) vs. the magnitude of the SDSS source ($r$-band model magnitude) for a sample of spectroscopically confirmed galaxies from the SDSS catalogue cross matched with GDR2 using a search radius of 1 arcsec. About 25 per cent of galaxies was detected by {\it Gaia} and included in GDR2. The {\it Gaia} detections of the extended SDSS sources return typically fainter magnitudes in comparison to the SDSS brightness as {\it Gaia} only probes the central parts of the galaxies. About 1 per cent of the detected objects is brighter than $17$ mag in $G$-band (sources above the red dashed line). Hence, by excluding any candidate new source located in the vicinity of GDR2 source brighter than $17$ mag we might remove from the sample about 1 per cent of possible transients in extended hosts.}
\label{fig:sdssgals}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{errorflux.eps}
\caption{The distribution of the median absolute deviation (MAD) vs.~the median source flux during a single transit. The red line indicates the median value of MAD in flux bins with $1\sigma$. The magenta points indicate $3\sigma$ above the median. To link the cut on scatter to the value of the flux we fitted a line (in a log-log space) to $3\sigma$ points. The best fit is shown as a green line ($\sim \mathrm{flux}^{0.68\pm0.02}$). Sources that fall above this line are excluded. }
\label{fig:fluxerrors}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{SSOsoffset.eps}
\caption{The distribution of coordinate offsets between candidate transients and SSOs found within 2 arcmin. The time span between a candidate observation and predicted time of SSO observation is lower than 3 seconds. We removed candidate transients found within 0.1 arcmin (all object to the left of the red dashed line) and 3 seconds of an SSO.}
\label{fig:ssoxmatch}
\end{figure}
The impact of each selection criterion applied during filtering on the sample size is summarised in Tab. \ref{tab:cri}. The criteria (ii)-(v) and (ix) have the largest impact on the number of detected sources. In Fig. \ref{fig:mapbysteps} we also present the change in the sky distribution of ''new sources'' candidate transients when filters are applied.
\begin{table}
\centering
\caption{A summary of the impact of each selection criterion applied during filtering on the sample size.}
\begin{tabular}{l l l}
\hline
Criterion & \# of remaining candidates & Rejection ratio \\
\hline
(i) & $38\times10^6$ & 0.74\\
(ii) & $8.3\times10^6$ & 0.78 \\
(iii) & $7.0\times10^6$ & 0.16 \\
(iv) & $3.6\times10^6$ & 0.49 \\
(v) & $1.4\times10^6$ & 0.61 \\
(vi) & $1.2\times10^6$ & 0.14 \\
(vii) & $1.2\times10^6$ & $<$0.01 \\
(viii) & $1.1\times10^6$ & 0.01 \\
(ix) & $2.5\times10^5$ & 0.78 \\
\hline
\end{tabular}
\label{tab:cri}
\end{table}
\begin{figure}
\subfloat[]{\includegraphics[width=0.9\columnwidth]{density00.eps}}
\subfloat[]{\includegraphics[width=0.9\columnwidth]{densityMWT.eps}}
\subfloat[]{\includegraphics[width=0.9\columnwidth]{densityMWTFGE5B.eps}}
\subfloat[]{\includegraphics[width=0.9\columnwidth]{densityMWTFGE5BS.eps}}
\caption{The all-sky distribution of candidate transients. The map resolution is HEALPix of nside 32. The plots are in Galactic coordinates. (a) Almost 38 million new sources with at least 8 of the 9(8) AF CCD measurements were flagged by IDT during the year 2018. The artefacts from the scanning law and astrometric excursions are clearly visible. (b) The most dense regions and observations during astrometric excursions are removed (steps (ii) and (iii)). (c) Steps (iv-viii) are applied. The artefacts from the scanning law remain, however the excess in the number of sources along the ecliptic is visible. (d) The final map.}
\label{fig:mapbysteps}
\end{figure}
\subsection{Analysis of detected candidates}
After applying all filters described above we obtained $0.25$M candidate new sources from one year of {\it Gaia} observations. However, some of the candidate transients could still be false positives as not all spurious detections will be filtered by our criteria above despite our best efforts. E.g.~ local dense regions like nearby galaxies or star clusters will not be filtered out and might cause spurious detections when {\it Gaia} scans over these regions from different angles. Moreover, in regions of high source density new detections of real existing sources still happen due to resource limitations (priority on-board reading sources and limitation on data transfer to Earth, see \citealt{2016A&A...595A...1G}).
From the one year all-sky test we obtained about $\sim 0.25$M new candidate transients that gives us the transient rate about $\lesssim0.021$ per sq deg per day. As {\it Gaia} observes about 1000 sq deg per day we should detect about 21 new candidate transients every day.
Several studies tried to address the question how many transients (of the Galactic origin - novae, M-flares and extragalactic - mostly SNe, but also QSO-flares) should be detectable in an optical magnitude limited search. For example, \cite{2019MNRAS.484.4507V} obtained rates for several types of transients. They focused on discovery of extragalactic optical fast transients and provided rates for transients faster than 1 day ($\lesssim 37\cdot10^{-4}$ per sq deg per d) and faster than 4 h ($\lesssim 9.3\cdot10^{-4}$ per sq deg per d with a limiting magnitude of $R\approx19.7$). In principle, {\it Gaia} should be able to detect many of these fast transients. \cite{2013ApJ...779...18B} identified at least two sources of potential false positives (M-star flares and asteroids) also relevant to searches for EM counterparts to GW events. The rates for other types of transients provided in \cite{2019MNRAS.484.4507V} ($\sim12\cdot10^{-4}$ SNe per sq deg per d, $\lesssim 20\cdot10^{-4}$ novae per sq deg per d and $\lesssim 120\cdot10^{-4}$ M-star flares per sq deg per d for a survey limited to $R<20$ mag) might give an estimate of how many false positive candidates are still included in the sample coming from our algorithm.
To study candidate transients from our one year search we extended the period by one month before and after assuming that {\it Gaia} scans each part of the sky on average every 30 days, hence transients detected earlier (or later) by other surveys might be also detected by {\it Gaia} in 2018. According to the Transient Name Server (TNS; \url{https://wis-tns.weizmann.ac.il/}) during the period from 2017 December 01 to 2019 January 31, 10459 transients were recorded. We rediscovered 2986 transients in our one year long sky survey reported to TNS by various surveys (including 1291 alerted by GSA). In 2018 GSA discovered 2743 candidate transients where 1885 of them were found by the {\it NewSource} detector (although 485 of them are located in the dense regions excluded from our search and a few were initially too faint for our detector, hence the remaining 1400 transients should be potentially rediscovered here). In total 109 transients (8 per cent from the GSA sample) were not rediscovered by our detection algorithm even though they were discovered by GSA (these candidates were filtered out because of several different reasons, mostly due to their proximity to bright sources in GDR2 and inaccurate database entries regarding the IDT classification for new sources).
All cross matches were performed using a search radius of 1 arcsec.
\section{Events from runs O1 and O2}
\label{sec:O1O2}
Each GW detection comes with a sky localisation map obtained from LIGO or LIGO-Virgo observations. Here, we made use of the final maps from the Gravitational-Wave Transient Catalog (\url{https://www.gw-openscience.org/GWTC-1-confident/}, \citealt{2019PhRvX...9c1040A}) for the events from the O1 and O2 runs. In total 11 events were found after reanalysis of the GW observations during these runs.
We used the {\it Gaia} Observation Forecast Tool (GOST)\footnote{\url{https://gaia.esac.esa.int/gost/} The GOST only provides a forecast of the time when targets cross the {\it Gaia} Focal Plane based on the scanning law of {\it Gaia}. However, it does not take into account operational activities preventing nominal observations nor the gaps between CCDs on the Focal Plane. Hence, the real number of scans may differ from the predictions.} to obtain a forecast for the visibility of each sky localisation map with 90 per cent credible regions. Assuming that the {\it Gaia} detection window of any putative EM counterpart to a GW event is short (time scale of days) we checked the visibility within 7 days from the GW events (see Table \ref{tab:gaia4o1o2}). For the three events detected during O1 (the BHBH mergers GW150914, GW151012, GW151226) the 90 per cent probability regions were partially scanned by {\it Gaia} within 7 days after the event. The fraction of the sky localisation regions scanned varies from 7 to 55 percent. For the five events detected during O2 (the BHBH mergers: GW170608, GW170809, GW170814, GW170818, and the binary NS merger GW170817) the 90 per cent probability regions were not scanned by {\it Gaia} within 7 days after the event time. The regions for the remaining three BHBH events (GW170104, GW170809, GW170823) were partially scanned (from 3 to 24 per cent). As one can expect the probability of {\it Gaia} observations increases with the size of the sky localisation map.
In Table \ref{tab:gaia4o1o2} we also included the predictions of the median time span between {\it Gaia} scanning over part of the sky localisation regions and the occurrence of the GW events, and the median time span between previous {\it Gaia} scans and the time of GW events. These time spans are strongly related to the localisation and size of the GW sky maps, and the uneven {\it Gaia} scanning law. Moreover, the minimal time delay between the GW event and {\it Gaia} observations in the sky localisation region might be lower than 0.01 d and as large as dozens of days.
\begin{table*}
\centering
\caption{The {\it Gaia} scanning predictions for the GW events from the LIGO (and Virgo) O1 and O2 runs.}
\begin{tabular}{l c c c c c c}
\hline
GW ID & $\Delta T$ [d] & Min $\Delta T$ [d] & Max $\Delta T$ [d] & $\Delta T_0$ [d] & $p$ [\%] & $\Delta \Omega~\mathrm{[deg^2]}$\\
\hline
GW150914
& 6.44 & 0.01 & 68.14 & -30.22 & 55 & 99 \\
GW151012
& 12.45 & 0.05 & 208.52 & -20.59 & 16 & 249 \\
GW151226
& 20.40 & 1.89 & 144.37 & -72.82 & 7 & 72 \\
GW170104
& 72.72 & $<$0.01 & 174.81 & -11.19 & 3 & 28 \\
GW170608
& 110.65 & 38.09 & 164.65 & -21.11 & 0 & - \\
GW170729
& 55.75 & $<$0.01 & 108.75 & -71.82 & 18 & 186 \\
GW170809
& 109.14 & 33.14 & 252.82 & -15.12 & 0 & - \\
GW170814
& 26.56 & 23.80 & 29.38 & -11.86 & 0 & - \\
GW170817
& 112.12 & 11.28 & 162.60 & -28.69 & 0 & - \\
GW170818
& 107.44 & 62.47 & 159.17 & -37.38 & 0 & - \\
GW170823
& 25.13 & 0.03 & 191.94 & -24.64 & 24 & 396 \\
\hline
\end{tabular}
\label{tab:gaia4o1o2}
$\Delta T$ - median wait time for {\it Gaia} to scan within the 90 per cent confidence sky localisation region of the GW event after the events, Min $\Delta T$, Max $\Delta T$ - minimum and maximum time delay between a GW event and {\it Gaia} to scan within the GW sky localisation, $\Delta T_0$ - median time between {\it Gaia} scanning in the sky localisation before the events and an occurring of the events, $p$ - percentage of a 90 per cent probability area scanned within 7 days from the event, $\Delta \Omega$ - size of scanned region within 7 days from the event
\end{table*}
For events covered by {\it Gaia} observations we ran a search for all candidate transients detected within 1 week after the GW events using the procedure described in Section \ref{sec:detector}. However, for events from 2015, due to the lack of internal information for SSO positions for {\it Gaia}, we needed to remove the candidates likely coinciding with SSO observations using their position in the sky with respect to the ecliptic. We also noticed that the sample is still affected by artefacts caused by bright stars (located further than 30 arcsec from the candidates - this is a tail of the neighbour distribution that was not taken into account in the criterion (v) due to a limited search radius). We obtain a sample of 535 candidates which were then visually inspected.
The table \ref{tab:candO1O2} in the Appendix presents the candidate transients that pass our final eyeballing vetting (we provide coordinates, the discovery date, and the discovery magnitude). In addition, we identified candidates located on top of galaxies or in the vicinity of extended objects (about 40 per cent of candidates).
We made an attempt to study the completeness of our search by comparing the final sample of candidates with samples published on the TNS and the Gamma-ray Coordinates Network (GCN) circulars (\url{https://gcn.gsfc.nasa.gov/}).
There are no transients alerted over the period covering the second part of 2015 and early 2016 as GSA was switched off over this period due to upgrades and testing of the current GSA detection algorithm.
From all transients alerted by GSA after the GW events from the O2 run a single candidate was inside the 90 per cent probability contour of GW170823 (Gaia17cdt) and this transient was rediscovered using the method outlined above. Moreover, two other sources reported to TNS by the PS1 survey were rediscovered in our work (AT2017jxq, AT2017gpc - both sources were too faint to be discovered by the current GSA {\it NewSource} detector). A few other published transients were detected just before the events (i.e.~Gaia17aba, Gaia17aaw were discovered within 2 days before GW170104, Gaia17bxj was found within 0.15 d before GW170729, and Gaia17cct within 3 days before GW170823). These sources are useful to assess results from other surveys where a search for EM counterparts usually starts after the detection of a GW event.
Unfortunately, {\it Gaia} was not scanning the region of the sky localisation for the GW170817 event during and shortly after the event. From the observations about 2 weeks after the GW signal we only know that no source was detected and that is consistent with the detection limits from other survey (e.g.~PS1 upper limits 12 days after the kilonova peak are $g>22.5$ mag and
$r>21.7$ mag, \citealt{2017Natur.551...75S}).
\subsection{Candidates from existing sources}
Candidate transients can also be detected as a change in brightness of a source detected by {\it Gaia} before the GW event. This can for instance happen if the location of a transient cannot be spatially separated from its host galaxy. For GW event counterparts this can only happen if the event is typically closer to the centre of host galaxy than 1-2 arcsec or if the host has a small angular size (i.e. it is a dwarf galaxy or a galaxy located at higher redshift).
For all events from the O1 and O2 run we searched for {\it Gaia} transits occurring within 7 days from GW detection. We required that the transits are not flagged as a new source by IDT. This criterion implies that the transit is associated with a source existing in the database of sources detected by {\it Gaia} before the GW event. We required 8-9 valid AF measurements, a median flux above $101.5~\mathrm{e^{-}/s}$, and restricted scatter between AF measurements during the candidate transit. These criteria still leave about 15M transits for which we tried to build light curves using data from previous {\it Gaia} scans collected within a search radius of 0.5 arcsec from the candidate transits. Furthermore, all sources were cross matched with the GLADE catalogue (Galaxy List for the Advanced Detector Era, a full sky galaxy catalogue, \citealt{2018MNRAS.479.2374D}) with a search radius of 1 arcsec to obtain candidates with a location close to or consistent with the centres of (known) galaxies. We limited the sample to sources with a flux increase after the GW event of more than $5\sigma$ when compared to the median flux detected by {\it Gaia} before the GW event. Forty eight candidates were found. Forty one candidates were classified as false positives after manual vetting. These false positives are mostly related to candidates just above the detection threshold, these could be caused by flux variations due to changes in {\it Gaia}'s scan angle over the source. There are a few exceptions which we classify as bright stars that are mislabelled as galaxy in the GLADE catalogue. We are left with seven transient candidate (including Gaia18cqe - a transient that was detected and alerted through the existing GSA system and for which it is known that it is related to known blazar activity, see \citealt{2017ATel10482....1C}). Our remaining six candidates also show evidence of being caused by quasar activity. Interestingly, some of these have a redshift that puts them at a distance within $<1-3\sigma$ of the distance of the observed GW events.
\subsection{Future prospects}
\label{sec:dis}
We studied chances of future EM counterpart detections by {\it Gaia}. Assuming that the kilonova events have an absolute magnitude about -15.8 mag in $r$-band at peak (i.e.~similar to the first GW event--kilonova source GW170817/AT2017gfo (e.g. \citealt{2017Natur.551...75S}) and our {\it Gaia} detection threshold of $\sim 20.68$ mag in $G$-band, transients can be detected up to redshift $\sim 0.045$ (see Fig.~\ref{fig:knLimits}). Within 10 days from a GW event about 25 per cent of sources randomly located in the sky will be in the regions scanned by {\it Gaia}. Although, detections of individual events are strictly related to localisation as {\it Gaia} scanning law is uneven over the sky.
\begin{figure}
\includegraphics[width=\columnwidth]{KNredshift.eps}
\caption{Redshift (distance) limits for any {\it Gaia} detection of kilonovae (orange dashed line), and per cent of scanned sky by {\it Gaia} (blue line) vs.~time since a GW event. Assuming a model of kilonovae similar to the transient detected as GW170817/AT2017gfo (e.g.~\citealt{2017Natur.551...75S}) we predict that {\it Gaia} will be able to detect such transients from a redshift up to $\sim 0.045$. However, the sample of detected events might be limited by {\it Gaia}'s scanning law as only about 25 per cent of the sky will be scanned within 10 days after a GW event.}
\label{fig:knLimits}
\end{figure}
An EM counterpart to the GW event might be also discovered by the {\it OldSource} detector where the host galaxy was previously detected by {\it Gaia} and a candidate transient is not resolved from its host. Over the year 2018 from all 2743 published {\it Gaia} transients more than 30 per cent was found in this way. However, in this sample there are also included candidates for microlensing events, cataclysmic variables, AGN flares, star flares, young stellar objects.
\section{Conclusions}
\label{sec:con}
We studied the possibility of detecting EM counterparts to GW events by {\it Gaia} using a dedicated transient detection algorithm. We propose an extension to current algorithms used by GSA to find transients and tested its capabilities in discovering candidate transients. The main concern is the level of false positives which has to be limited through various filters (based on event time, sky localisation map, but also candidate transient neighbourhood).
The candidate transients for EM counterparts to the previous GW detections are also reported. The search using the bespoke detector yielded 535 candidate transients observed by {\it Gaia} within 7 days from the GW events detected in the O1 and O2 runs.
As GSA were not publishing new transients between July 2015 and January 2016 for GW events from the O1 run the sample of candidates was only compared to results from other surveys. For candidates from the year 2017 (run O2) one candidate transient was alerted by GSA and rediscovered by this search. Moreover, we rediscovered two transients from other surveys reported to TNS.
For the GW events from the current O3 run we expect that about 16 (25) per cent of them might be in the sky regions observed by {\it Gaia} within 7 (10) days from the events. The new detector will provide $\lesssim 21$ candidates per day from the whole sky (the final number of candidates will be lower as the GW localisation skymap usually are the size of a few thousand square degrees). One of the main advantages of using {\it Gaia} in study transients associated with GW signal is the accurate position (up to mas). Moreover, thanks to {\it Gaia} position on the orbit we are able to observe targets relatively close to the Sun, areas that are not easily reachable from ground based observatories.
\section*{Acknowledgements}
ZKR acknowledges funding from the Netherlands Research School for Astronomy (NOVA).
ZKR, PGJ, and DE acknowledge support from European Research Council Consolidator Grant 647208.
{\L}W acknowledges Polish NCN HARMONIA grant No. 2018/30/M/ST9/00311.
TW is funded in part by European Research Council grant 320360 and by European Commission grant 730980.
This publication is based upon work from COST Action MW-Gaia CA18104 supported by COST (European Cooperation in Science and Technology).
This work has made use of data from the European Space Agency (ESA)
mission {\it Gaia} (\url{https://www.cosmos.esa.int/gaia}), processed by
the {\it Gaia} Data Processing and Analysis Consortium (DPAC,
\url{https://www.cosmos.esa.int/web/gaia/dpac/consortium}). Funding
for the DPAC has been provided by national institutions, in particular
the institutions participating in the {\it Gaia} Multilateral Agreement.
Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges
support and resources from the Center for High-Performance Computing at
the University of Utah. The SDSS web site is www.sdss.org. SDSS-IV is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS Collaboration including the
Brazilian Participation Group, the Carnegie Institution for Science,
Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics,
Instituto de Astrof\'isica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) /
University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory,
Leibniz Institut f\"ur Astrophysik Potsdam (AIP),
Max-Planck-Institut f\"ur Astronomie (MPIA Heidelberg),
Max-Planck-Institut f\"ur Astrophysik (MPA Garching),
Max-Planck-Institut f\"ur Extraterrestrische Physik (MPE),
National Astronomical Observatories of China, New Mexico State University,
New York University, University of Notre Dame,
Observat\'ario Nacional / MCTI, The Ohio State University,
Pennsylvania State University, Shanghai Astronomical Observatory,
United Kingdom Participation Group,
Universidad Nacional Aut\'onoma de M\'exico, University of Arizona,
University of Colorado Boulder, University of Oxford, University of Portsmouth,
University of Utah, University of Virginia, University of Washington, University of Wisconsin,
Vanderbilt University, and Yale University.
The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been made possible through contributions by the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation Grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation.
This research has made use of the SIMBAD database,
operated at CDS, Strasbourg, France.
This research has made use of Astropy, a community-developed core Python package for Astronomy \citep{2013A&A...558A..33A}, healpy, a Python package to manipulate HEALPix maps (\url{http://healpix.sf.net}, \citealt{2005ApJ...622..759G,Zonca2019}), Q3C extension for PostgreSQL \citep{2006ASPC..351..735K}, \textsc{TOPCAT} \citep{2005ASPC..347...29T}.
\bibliographystyle{mnras}
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See appendix~\ref{sec:advanced} for more complicated examples.
\subsection{Authors and institutions}
Like the title, there are two forms of author list: the full version which appears on the title page, and a short form which appears in the header of the even-numbered pages. Enter them using the \verb'\author[]{}' command.
If the author list is more than one line long, start a new line using \verb'\newauthor'. Use \verb'\\' to start the institution list. Affiliations for each author should be indicated with a superscript number, and correspond to the list of institutions below the author list.
For example, if I were to write a paper with two coauthors at another institution, one of whom also works at a third location:
\begin{verbatim}
\author[K. T. Smith et al.]{
Keith T. Smith,$^{1}$
A. N. Other,$^{2}$
and Third Author$^{2,3}$
\\
$^{1}$Affiliation 1\\
$^{2}$Affiliation 2\\
$^{3}$Affiliation 3}
\end{verbatim}
Affiliations should be in the format `Department, Institution, Street Address, City and Postal Code, Country'.
Email addresses can be inserted with the \verb'\thanks{}' command which adds a title page footnote.
If you want to list more than one email, put them all in the same \verb'\thanks' and use \verb'\footnotemark[]' to refer to the same footnote multiple times.
Present addresses (if different to those where the work was performed) can also be added with a \verb'\thanks' command.
\subsection{Abstract and keywords}
The abstract is entered in an \verb'abstract' environment:
\begin{verbatim}
\begin{abstract}
The abstract of the paper.
\end{abstract}
\end{verbatim}
\noindent Note that there is a word limit on the length of abstracts.
For the current word limit, see the journal instructions to authors$^{\ref{foot:itas}}$.
Immediately following the abstract, a set of keywords is entered in a \verb'keywords' environment:
\begin{verbatim}
\begin{keywords}
keyword 1 -- keyword 2 -- keyword 3
\end{keywords}
\end{verbatim}
\noindent There is a list of permitted keywords, which is agreed between all the major astronomy journals and revised every few years.
Do \emph{not} make up new keywords!
For the current list of allowed keywords, see the journal's instructions to authors$^{\ref{foot:itas}}$.
\section{Sections and lists}
Sections and lists are generally the same as in the standard \LaTeX\ classes.
\subsection{Sections}
\label{sec:sections}
Sections are entered in the usual way, using \verb'\section{}' and its variants. It is possible to nest up to four section levels:
\begin{verbatim}
\section{Main section}
\subsection{Subsection}
\subsubsection{Subsubsection}
\paragraph{Lowest level section}
\end{verbatim}
\noindent The other \LaTeX\ sectioning commands \verb'\part', \verb'\chapter' and \verb'\subparagraph{}' are deprecated and should not be used.
Some sections are not numbered as part of journal style (e.g. the Acknowledgements).
To insert an unnumbered section use the `starred' version of the command: \verb'\section*{}'.
See appendix~\ref{sec:advanced} for more complicated examples.
\subsection{Lists}
Two forms of lists can be used in MNRAS -- numbered and unnumbered.
For a numbered list, use the \verb'enumerate' environment:
\begin{verbatim}
\begin{enumerate}
\item First item
\item Second item
\item etc.
\end{enumerate}
\end{verbatim}
\noindent which produces
\begin{enumerate}
\item First item
\item Second item
\item etc.
\end{enumerate}
Note that the list uses lowercase Roman numerals, rather than the \LaTeX\ default Arabic numerals.
For an unnumbered list, use the \verb'description' environment without the optional argument:
\begin{verbatim}
\begin{description}
\item First item
\item Second item
\item etc.
\end{description}
\end{verbatim}
\noindent which produces
\begin{description}
\item First item
\item Second item
\item etc.
\end{description}
Bulleted lists using the \verb'itemize' environment should not be used in MNRAS; it is retained for backwards compatibility only.
\section{Mathematics and symbols}
The MNRAS class mostly adopts standard \LaTeX\ handling of mathematics, which is briefly summarised here.
See also section~\ref{sec:packages} for packages that support more advanced mathematics.
Mathematics can be inserted into the running text using the syntax \verb'$1+1=2$', which produces $1+1=2$.
Use this only for short expressions or when referring to mathematical quantities; equations should be entered as described below.
\subsection{Equations}
Equations should be entered using the \verb'equation' environment, which automatically numbers them:
\begin{verbatim}
\begin{equation}
a^2=b^2+c^2
\end{equation}
\end{verbatim}
\noindent which produces
\begin{equation}
a^2=b^2+c^2
\end{equation}
By default, the equations are numbered sequentially throughout the whole paper. If a paper has a large number of equations, it may be better to number them by section (2.1, 2.2 etc.). To do this, add the command \verb'\numberwithin{equation}{section}' to the preamble.
It is also possible to produce un-numbered equations by using the \LaTeX\ built-in \verb'\['\textellipsis\verb'\]' and \verb'$$'\textellipsis\verb'$$' commands; however MNRAS requires that all equations are numbered, so these commands should be avoided.
\subsection{Special symbols}
\begin{table}
\caption{Additional commands for special symbols commonly used in astronomy. These can be used anywhere.}
\label{tab:anysymbols}
\begin{tabular}{lll}
\hline
Command & Output & Meaning\\
\hline
\verb'\sun' & \sun & Sun, solar\\[2pt]
\verb'\earth' & \earth & Earth, terrestrial\\[2pt]
\verb'\micron' & \micron & microns\\[2pt]
\verb'\degr' & \degr & degrees\\[2pt]
\verb'\arcmin' & \arcmin & arcminutes\\[2pt]
\verb'\arcsec' & \arcsec & arcseconds\\[2pt]
\verb'\fdg' & \fdg & fraction of a degree\\[2pt]
\verb'\farcm' & \farcm & fraction of an arcminute\\[2pt]
\verb'\farcs' & \farcs & fraction of an arcsecond\\[2pt]
\verb'\fd' & \fd & fraction of a day\\[2pt]
\verb'\fh' & \fh & fraction of an hour\\[2pt]
\verb'\fm' & \fm & fraction of a minute\\[2pt]
\verb'\fs' & \fs & fraction of a second\\[2pt]
\verb'\fp' & \fp & fraction of a period\\[2pt]
\verb'\diameter' & \diameter & diameter\\[2pt]
\verb'\sq' & \sq & square, Q.E.D.\\[2pt]
\hline
\end{tabular}
\end{table}
\begin{table}
\caption{Additional commands for mathematical symbols. These can only be used in maths mode.}
\label{tab:mathssymbols}
\begin{tabular}{lll}
\hline
Command & Output & Meaning\\
\hline
\verb'\upi' & $\upi$ & upright pi\\[2pt]
\verb'\umu' & $\umu$ & upright mu\\[2pt]
\verb'\upartial' & $\upartial$ & upright partial derivative\\[2pt]
\verb'\lid' & $\lid$ & less than or equal to\\[2pt]
\verb'\gid' & $\gid$ & greater than or equal to\\[2pt]
\verb'\la' & $\la$ & less than of order\\[2pt]
\verb'\ga' & $\ga$ & greater than of order\\[2pt]
\verb'\loa' & $\loa$ & less than approximately\\[2pt]
\verb'\goa' & $\goa$ & greater than approximately\\[2pt]
\verb'\cor' & $\cor$ & corresponds to\\[2pt]
\verb'\sol' & $\sol$ & similar to or less than\\[2pt]
\verb'\sog' & $\sog$ & similar to or greater than\\[2pt]
\verb'\lse' & $\lse$ & less than or homotopic to \\[2pt]
\verb'\gse' & $\gse$ & greater than or homotopic to\\[2pt]
\verb'\getsto' & $\getsto$ & from over to\\[2pt]
\verb'\grole' & $\grole$ & greater over less\\[2pt]
\verb'\leogr' & $\leogr$ & less over greater\\
\hline
\end{tabular}
\end{table}
Some additional symbols of common use in astronomy have been added in the MNRAS class. These are shown in tables~\ref{tab:anysymbols}--\ref{tab:mathssymbols}. The command names are -- as far as possible -- the same as those used in other major astronomy journals.
Many other mathematical symbols are also available, either built into \LaTeX\ or via additional packages. If you want to insert a specific symbol but don't know the \LaTeX\ command, we recommend using the Detexify website\footnote{\url{http://detexify.kirelabs.org}}.
Sometimes font or coding limitations mean a symbol may not get smaller when used in sub- or superscripts, and will therefore be displayed at the wrong size. There is no need to worry about this as it will be corrected by the typesetter during production.
To produce bold symbols in mathematics, use \verb'\bmath' for simple variables, and the \verb'bm' package for more complex symbols (see section~\ref{sec:packages}). Vectors are set in bold italic, using \verb'\mathbfit{}'.
For matrices, use \verb'\mathbfss{}' to produce a bold sans-serif font e.g. \mathbfss{H}; this works even outside maths mode, but not all symbols are available (e.g. Greek). For $\nabla$ (del, used in gradients, divergence etc.) use \verb'$\nabla$'.
\subsection{Ions}
A new \verb'\ion{}{}' command has been added to the class file, for the correct typesetting of ionisation states.
For example, to typeset singly ionised calcium use \verb'\ion{Ca}{ii}', which produces \ion{Ca}{ii}.
\section{Figures and tables}
\label{sec:fig_table}
Figures and tables (collectively called `floats') are mostly the same as built into \LaTeX.
\subsection{Basic examples}
\begin{figure}
\includegraphics[width=\columnwidth]{example}
\caption{An example figure.}
\label{fig:example}
\end{figure}
Figures are inserted in the usual way using a \verb'figure' environment and \verb'\includegraphics'. The example Figure~\ref{fig:example} was generated using the code:
\begin{verbatim}
\begin{figure}
\includegraphics[width=\columnwidth]{example}
\caption{An example figure.}
\label{fig:example}
\end{figure}
\end{verbatim}
\begin{table}
\caption{An example table.}
\label{tab:example}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
Sun & 1.00 & 1.00\\
$\alpha$~Cen~A & 1.10 & 1.52\\
$\epsilon$~Eri & 0.82 & 0.34\\
\hline
\end{tabular}
\end{table}
The example Table~\ref{tab:example} was generated using the code:
\begin{verbatim}
\begin{table}
\caption{An example table.}
\label{tab:example}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
Sun & 1.00 & 1.00\\
$\alpha$~Cen~A & 1.10 & 1.52\\
$\epsilon$~Eri & 0.82 & 0.34\\
\hline
\end{tabular}
\end{table}
\end{verbatim}
\subsection{Captions and placement}
Captions go \emph{above} tables but \emph{below} figures, as in the examples above.
The \LaTeX\ float placement commands \verb'[htbp]' are intentionally disabled.
Layout of figures and tables will be adjusted by the publisher during the production process, so authors should not concern themselves with placement to avoid disappointment and wasted effort.
Simply place the \LaTeX\ code close to where the figure or table is first mentioned in the text and leave exact placement to the publishers.
By default a figure or table will occupy one column of the page.
To produce a wider version which covers both columns, use the \verb'figure*' or \verb'table*' environment.
If a figure or table is too long to fit on a single page it can be split it into several parts.
Create an additional figure or table which uses \verb'\contcaption{}' instead of \verb'\caption{}'.
This will automatically correct the numbering and add `\emph{continued}' at the start of the caption.
\begin{table}
\contcaption{A table continued from the previous one.}
\label{tab:continued}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
$\tau$~Cet & 0.78 & 0.52\\
$\delta$~Pav & 0.99 & 1.22\\
$\sigma$~Dra & 0.87 & 0.43\\
\hline
\end{tabular}
\end{table}
Table~\ref{tab:continued} was generated using the code:
\begin{verbatim}
\begin{table}
\contcaption{A table continued from the previous one.}
\label{tab:continued}
\begin{tabular}{lcc}
\hline
Star & Mass & Luminosity\\
& $M_{\sun}$ & $L_{\sun}$\\
\hline
$\tau$~Cet & 0.78 & 0.52\\
$\delta$~Pav & 0.99 & 1.22\\
$\sigma$~Dra & 0.87 & 0.43\\
\hline
\end{tabular}
\end{table}
\end{verbatim}
To produce a landscape figure or table, use the \verb'pdflscape' package and the \verb'landscape' environment.
The landscape Table~\ref{tab:landscape} was produced using the code:
\begin{verbatim}
\begin{landscape}
\begin{table}
\caption{An example landscape table.}
\label{tab:landscape}
\begin{tabular}{cccccccccc}
\hline
Header & Header & ...\\
Unit & Unit & ...\\
\hline
Data & Data & ...\\
Data & Data & ...\\
...\\
\hline
\end{tabular}
\end{table}
\end{landscape}
\end{verbatim}
Unfortunately this method will force a page break before the table appears.
More complicated solutions are possible, but authors shouldn't worry about this.
\begin{landscape}
\begin{table}
\caption{An example landscape table.}
\label{tab:landscape}
\begin{tabular}{cccccccccc}
\hline
Header & Header & Header & Header & Header & Header & Header & Header & Header & Header\\
Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit & Unit \\
\hline
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
Data & Data & Data & Data & Data & Data & Data & Data & Data & Data\\
\hline
\end{tabular}
\end{table}
\end{landscape}
\section{References and citations}
\subsection{Cross-referencing}
The usual \LaTeX\ commands \verb'\label{}' and \verb'\ref{}' can be used for cross-referencing within the same paper.
We recommend that you use these whenever relevant, rather than writing out the section or figure numbers explicitly.
This ensures that cross-references are updated whenever the numbering changes (e.g. during revision) and provides clickable links (if available in your compiler).
It is best to give each section, figure and table a logical label.
For example, Table~\ref{tab:mathssymbols} has the label \verb'tab:mathssymbols', whilst section~\ref{sec:packages} has the label \verb'sec:packages'.
Add the label \emph{after} the section or caption command, as in the examples in sections~\ref{sec:sections} and \ref{sec:fig_table}.
Enter the cross-reference with a non-breaking space between the type of object and the number, like this: \verb'see Figure~\ref{fig:example}'.
The \verb'\autoref{}' command can be used to automatically fill out the type of object, saving on typing.
It also causes the link to cover the whole phrase rather than just the number, but for that reason is only suitable for single cross-references rather than ranges.
For example, \verb'\autoref{tab:journal_abbr}' produces \autoref{tab:journal_abbr}.
\subsection{Citations}
\label{sec:cite}
MNRAS uses the Harvard -- author (year) -- citation style, e.g. \citet{author2013}.
This is implemented in \LaTeX\ via the \verb'natbib' package, which in turn is included via the \verb'usenatbib' package option (see section~\ref{sec:options}), which should be used in all papers.
Each entry in the reference list has a `key' (see section~\ref{sec:ref_list}) which is used to generate citations.
There are two basic \verb'natbib' commands:
\begin{description}
\item \verb'\citet{key}' produces an in-text citation: \citet{author2013}
\item \verb'\citep{key}' produces a bracketed (parenthetical) citation: \citep{author2013}
\end{description}
Citations will include clickable links to the relevant entry in the reference list, if supported by your \LaTeX\ compiler.
\defcitealias{smith2014}{Paper~I}
\begin{table*}
\caption{Common citation commands, provided by the \texttt{natbib} package.}
\label{tab:natbib}
\begin{tabular}{lll}
\hline
Command & Ouput & Note\\
\hline
\verb'\citet{key}' & \citet{smith2014} & \\
\verb'\citep{key}' & \citep{smith2014} & \\
\verb'\citep{key,key2}' & \citep{smith2014,jones2015} & Multiple papers\\
\verb'\citet[table 4]{key}' & \citet[table 4]{smith2014} & \\
\verb'\citep[see][figure 7]{key}' & \citep[see][figure 7]{smith2014} & \\
\verb'\citealt{key}' & \citealt{smith2014} & For use with manual brackets\\
\verb'\citeauthor{key}' & \citeauthor{smith2014} & If already cited in close proximity\\
\verb'\defcitealias{key}{Paper~I}' & & Define an alias (doesn't work in floats)\\
\verb'\citetalias{key}' & \citetalias{smith2014} & \\
\verb'\citepalias{key}' & \citepalias{smith2014} & \\
\hline
\end{tabular}
\end{table*}
There are a number of other \verb'natbib' commands which can be used for more complicated citations.
The most commonly used ones are listed in Table~\ref{tab:natbib}.
For full guidance on their use, consult the \verb'natbib' documentation\footnote{\url{http://www.ctan.org/pkg/natbib}}.
If a reference has several authors, \verb'natbib' will automatically use `et al.' if there are more than two authors. However, if a paper has exactly three authors, MNRAS style is to list all three on the first citation and use `et al.' thereafter. If you are using \bibtex\ (see section~\ref{sec:ref_list}) then this is handled automatically. If not, the \verb'\citet*{}' and \verb'\citep*{}' commands can be used at the first citation to include all of the authors.
\subsection{The list of references}
\label{sec:ref_list}
It is possible to enter references manually using the usual \LaTeX\ commands, but we strongly encourage authors to use \bibtex\ instead.
\bibtex\ ensures that the reference list is updated automatically as references are added or removed from the paper, puts them in the correct format, saves on typing, and the same reference file can be used for many different papers -- saving time hunting down reference details.
An MNRAS \bibtex\ style file, \verb'mnras.bst', is distributed as part of this package.
The rest of this section will assume you are using \bibtex.
References are entered into a separate \verb'.bib' file in standard \bibtex\ formatting.
This can be done manually, or there are several software packages which make editing the \verb'.bib' file much easier.
We particularly recommend \textsc{JabRef}\footnote{\url{http://jabref.sourceforge.net/}}, which works on all major operating systems.
\bibtex\ entries can be obtained from the NASA Astrophysics Data System\footnote{\label{foot:ads}\url{http://adsabs.harvard.edu}} (ADS) by clicking on `Bibtex entry for this abstract' on any entry.
Simply copy this into your \verb'.bib' file or into the `BibTeX source' tab in \textsc{JabRef}.
Each entry in the \verb'.bib' file must specify a unique `key' to identify the paper, the format of which is up to the author.
Simply cite it in the usual way, as described in section~\ref{sec:cite}, using the specified key.
Compile the paper as usual, but add an extra step to run the \texttt{bibtex} command.
Consult the documentation for your compiler or latex distribution.
Correct formatting of the reference list will be handled by \bibtex\ in almost all cases, provided that the correct information was entered into the \verb'.bib' file.
Note that ADS entries are not always correct, particularly for older papers and conference proceedings, so may need to be edited.
If in doubt, or if you are producing the reference list manually, see the MNRAS instructions to authors$^{\ref{foot:itas}}$ for the current guidelines on how to format the list of references.
\section{Appendices and online material}
To start an appendix, simply place the \verb' | {
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Q: Extracting Javadoc @version tag at runtime Is there a neat way of getting the Javadoc @version at runtime?
I have an "overview.html" file that contains the application version that I would like to print onto the GUI that's part of the application. I could just read this file and extract it that way, but I'm wondering if there's a smarter way to do it?
A: I'm going with this the answer to the question found by Balazs.
| {
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} | 2,758 |
Brewers 4, Dodgers 1: Back to .500
Daniel Brim 04/13/2019 Game Recaps Leave A Comment
The best thing that happened today was probably Russell Martin getting kicked out of the game while on the injured list. It was that kind of game.
The Dodgers started 2019 with a record 8-2, but are now 8-8 and suddenly the season feels so much different. The wonderful feeling of reversal of their previous seasons' slow start fortunes has been erased. In tonight's contest, the bats went to sleep and the Dodgers lost by a score of 4-1.
Brewers starter Zach Davies went seven strong inning against the Dodgers with his 88 mph fastball, allowing just one run despite a comebacker striking his pitching hand after his second pitch of the game. He did surrender eight hits in those frames, but, as ever, the offense couldn't string them together for much damage. The only blemish against Davies was this long home run by Cody Bellinger:
Bobblehead night or not, @Cody_Bellinger was going to homer anyway. pic.twitter.com/4UvtC7L5VW
— Los Angeles Dodgers (@Dodgers) April 14, 2019
Davies was certainly aided by the wide strike zone of Jeremie Rehak and framing by Yasmani Grandal. Two of his six strikeouts came looking, one of which came with two runners on base, which ended up being the Dodgers' last real look at the game. Still, it's hard to complain about Grandal framing after being on the positive side of those calls for so long.
The Dodgers' bullpen was tasked with going all nine innings tonight, and for the most part they did a decent job. Caleb Ferguson took the ball first and, oddly, he threw 47 fastballs in his 51 pitches. Ferguson came out throwing hard, sitting 95 and touching 97 in the first inning, a welcome sign after starting the season down on velocity. However, he surrendered a home run, two walks, and four hits in 2-2/3 innings as his velocity faded back into the lower 90s by the end of the outing.
Dennis Santana followed Ferguson and began in impressive fashion:
Dennis Santana, changeup pic.twitter.com/33sQ8xR91i
— Daniel Brim (@DanielBrim) April 14, 2019
Everything he threw had a ton of movement, but struggled with his release point at times. Orlando Arcia got him for a home run, and his control faltered by the end of the 1-1/3 inning stint. Yimi Garcia inherited Santana's bases loaded jam, but was able to stop the bleeding to keep the game close.
The bullpen did its job throughout the rest of the game with Garcia, Scott Alexander, Pedro Baez, and Jaime Schultz combining to hold the Brewers scoreless over the final five innings pitched. Ultimately, the Dodgers' bats wasted the good bullpen performance, going 0 for their last 14 to end the game quietly.
The Dodgers will look to stop the bleeding (again) tomorrow afternoon at 1:10 PDT. Jhoulys Chacin will start for the Brewers. For the Dodgers, Ross Stripling will desperately try to give the exhausted bullpen a long outing, something they've been looking for for almost a week.
About Daniel Brim
Daniel Brim grew up in the Los Angeles area but doesn't live there anymore. He still watches the Dodgers and writes about them sometimes.
@danielbrim
Previous Brewers @ Dodgers April 13, 2019: Dodgers lean on bullpen to stop skid
Next Brewers @ Dodgers April 14, 2019: Dodgers try to avoid back to back sweeps | {
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{"url":"http:\/\/openstudy.com\/updates\/5093f76fe4b0c875f79dfd87","text":"anonymous 4 years ago 2 is added to m and the square root of the sum is taken. Then, 7 is subtracted from the result, giving a final result of 9. What is m?\n\n1. cwrw238\n\nsqrt(m + 2) - 7 = 9\n\n2. anonymous\n\n$\\sqrt{m+2}-7=9$\n\n3. anonymous\n\n$\\sqrt{m+2}=16$","date":"2016-12-11 08:17:33","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\": 0.33143022656440735, \"perplexity\": 527.0529839130181}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-50\/segments\/1480698544358.59\/warc\/CC-MAIN-20161202170904-00018-ip-10-31-129-80.ec2.internal.warc.gz\"}"} | null | null |
, né le à Tokyo, est un acteur japonais.
Filmographie sélective
Au cinéma
2000 : de Takashi Miike
2000 : de Takashi Miike
2001 : de Kosuke Suzuki
2001 : de Takashi Miike
2001 : Family de Takashi Miike
2001 : de Hirokazu Kore-eda : Tamaki, le mari de Kiyoka
2001 : de Takashi Miike
2001 : de Takashi Miike
2002 : de Takashi Miike
2002 : de Takashi Miike
2003 : de Takashi Miike
2003 : de Takashi Miike
2003 : de Ryūhei Kitamura
2005 : de Takashi Miike
2006 : de Takashi Miike
2006 : de Takashi Miike
2006 : : Shunak (voix - film d'animation)
2007 : de Takashi Miike
2008 : de Takashi Miike
2008 : de Yukihiko Tsutsumi
2009 : de Yukihiko Tsutsumi
2009 : de Takashi Miike
2014 : The Raid 2 de Gareth Evans
2017 : de Hiromasa Yonebayashi : Zébédée (voix - film d'animation)
2018 : de
2019 : de
À la télévision
1979 : Battle Fever J
1994 : Ninja Sentai Kakuranger
2000 : Ikebukuro West Gate Park
2003 : de Takashi Miike
2004 : Tokusō Sentai Dekaranger
2018 : Mahou × Senshi Magi Majo Pures!
Notes et références
Liens externes
Acteur japonais
Naissance en juin 1961
Naissance à Shinagawa | {
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Q: How do I compare values from two tables and add result in third? I have the following problem:
Table one has the following columns: ID, Name, Color
Table two has the following columns: ID, Price, Date
I need to make a third table with the following:
*
*If ID from table2 is found in ID in table1, then add a row in table3 with: ID, Name, Price
Table1 and Table2 have different size.
Thanks.
A: insert into table3(id, name, price) values(
select id,name,price from table1, table2 where table1.id=table2.id)
I guess ure looking for that
| {
"redpajama_set_name": "RedPajamaStackExchange"
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In 2012, the International Tuberous Sclerosis Complex Consensus Conference reviewed prevalence and specificity of TSC-associated clinical manifestations and updated the TSC diagnostic criteria from 1998. Peer-reviewed publications from the 2012 Consensus Conference are available here. Clinical features of TSC continue to be a principal means of diagnosis but include additional clarification and simplification. In addition, TSC may now be diagnosed via genetic testing. View a list of commercial labs in the United States offering TSC genetic testing.
Download our publication for medical professionals, Diagnosis, Surveillance and Management of Tuberous Sclerosis Complex.
The clinical and genetics diagnostic criteria of 2012 are summarized in the table below.
Definite diagnosis: Two major features or one major feature with ≥ 2 minor features.
Possible diagnosis: Either one major feature or ≥2 minor features.
* Includes tubers and cerebral white matter radial migration lines.
**A combination of the two major Clinical features (LAM and angiomyolipomas) without other features does not meet criteria for a definite diagnosis.
The identification of either a TSC1 or TSC2 pathogenic mutation in DNA from normal tissue is sufficient to make a Definite Diagnosis of TSC. A pathogenic mutation is defined as a mutation that clearly inactivates the function of the TSC1 or TSC2 proteins (e.g., out of frame insertion or deletion or nonsense mutation), prevents protein synthesis (e.g., large genomic deletion), or is a missense mutation whose effect on protein function has been established by functional assessment. Other TSC1 or TSC2 variants whose effect on function is less certain do not meet these criteria and are not sufficient to make a Definite Diagnosis of TSC.
Note that approximately 15% of individuals with TSC have no mutation identified by conventional genetic testing, and a normal result does not exclude TSC or have any effect on the use of Clinical Diagnostic Criteria to diagnose TSC. Clinical genetic testing identifies gene mutations in 75-90% of DNA samples that are submitted for testing from individuals who have a definite diagnosis of TSC based on accepted diagnostic criteria. For the remaining 10-25% of TSC patients there are several explanations for why we cannot find an underlying mutation. There may be a mutation in the TSC1 or TSC2 gene that we cannot detect because it only occurs in some of the patient's cells but not all of the cells. This situation is called mosaicism meaning that the genetic complement of different cells is different in different cells of the body. Another reason we may not find a mutation is that the mutation may be in sections of the gene that we do not test because we do not understand how changes in these parts of the genes can cause disease.
Because TSC affects multiple organs, diagnostic studies are recommended for all individuals with a new diagnosis of TSC regardless of their outward manifestations of the disease. For example, published recommendations for diagnostic and follow-up evaluations suggest baseline imaging using either CT or cranial MRI modalities regardless of the presence of neurological symptoms. This suggestion is largely due to the risk of identifying a subependymal giant cell astrocytoma (SEGA), which has an increased growth potential over subependymal nodules (SEN) and therefore needs more extensive follow-up.
In order to ensure comprehensive care, referrals to a variety of specialists familiar with TSC should be coordinated. If possible, a referral to a multidisciplinary clinic specializing in TSC is ideal, as the center will likely house all necessary specialists, including a dermatologist, neurologist, geneticist, nephrologist and/or urologist, ophthalmologist and cardiologist.
The clinician making the diagnosis of TSC may recommend that other, at-risk family members also be evaluated for this condition. TSC is an autosomal dominant genetic disorder, and while all persons with TSC are thought to have symptoms, the presentation of their symptoms can be highly variable even within the same family. A determination of whether or not the parents and siblings of a diagnosed child are affected is important to the provision of later genetic counseling, thereby making an accurate diagnosis necessary. While an estimated sixty percent of individuals diagnosed with TSC are born into families with no prior history of the disease (i.e., sporadic mutation), it is becoming more and more common for adults to learn of their own diagnosis following the diagnosis of their child or because of other medical concerns.
There is some debate as to which evaluations are necessary when testing the parents of a newly diagnosed child. The consensus is that a thorough physical examination conducted by a physician familiar with TSC will detect the majority of affected individuals. The evaluation should include a skin examination with a Woods lamp (ultraviolet light) and a retinal examination through dilated pupils. Further examination via diagnostic imaging techniques of the brain and kidneys should be ordered for the parents of children with TSC and/or for adults with medical issues that suggest a diagnosis of TSC. Diagnostic molecular testing should also be discussed with the parents of a newly diagnosed child and/or adults suspected or diagnosed with TSC.
In addition to the three main imaging procedures usually undertaken for an individual with TSC (CT or MRI scans of the brain, renal CT and/or ultrasounds and echocardiograms), additional imaging procedures or testing may be conducted. Some centers perform these evaluations annually, at least until adulthood. This is a topic of some controversy, and the natural history of TSC is currently being studied through the TS Alliance Natural History Database Project. Additional imaging should be performed as necessary to follow growing lesions or to monitor organ involvement.
POSITION EMISSION TOMOGRAPHY (PET) There is no current indication for routine PET scanning in individuals with TSC. However, PET scans may be useful when individuals are undergoing evaluation as candidates for epilepsy surgery. PET scanning with the tracer alpha-methyltryptophan may have particular utility in identifying epileptogenic tubers as part of the evaluation for epilepsy surgery.
SINGLE-PHOTON EMISSION COMPUTED TOMOGRAPHY (SPECT) There is no current indication for routine SPECT scanning in individuals with TSC. However, SPECT scans may be useful when individuals are being evaluated as candidates for epilepsy surgery.
MAGNETOENCEPHALOGRAPHY (MEG) There is no current indication for routine MEG scanning in individuals with TSC. However, MEG scans may be useful when individuals are being evaluated as candidates for epilepsy surgery.
ELECTROCARDIOGRAM AND ECHOCARDIOGRAPHY A baseline electrocardiogram (ECG) is recommended for all individuals newly diagnosed with TSC since cardiac arrhythmias, although rare, may have sudden death as their presenting symptom. Follow-up ECGs should be performed every two to three years thereafter until puberty or as needed. A baseline echocardiogram should be performed on all children diagnosed with TSC and later as clinically indicated. Adults with a new diagnosis of TSC should undergo echocardiography only if symptomatic.
DIFFUSION TENSOR IMAGING (DTI) There is no current indication for routine DTI in individuals with TSC. However, DTI may be useful when individuals are being evaluated as candidates for epilepsy surgery. DTI is being utilized as a research tool to study the correlation in white matter integrity and neurologic impairment in individuals with TSC. DTI may have particular utility in identifying individuals at risk for developing autism.
FUNCTION MRI (MRI) There is no current indication for routine MRI in individuals with TSC. However, fMRI of the brain may be useful when individuals are being evaluated as candidates for epilepsy surgery.
The goals of treatment for individuals with TSC are the same as for all individuals with a multi-system chronic condition: providing the best possible quality of life with the fewest complications from the underlying disease process, fewest adverse treatment effects and fewest medications.
TSC often has been under-treated, particularly from a neurological standpoint, often based on the unfounded view that these individuals will have a poor outcome regardless of any therapy undertaken. This is clearly not the case. Even in individuals with TSC and infantile spasms, long- term outcome is not universally poor, as has been classically thought. At least 9 percent of individuals with TSC and infantile spasms have normal intelligence as adults or at long-term follow-up.
A recent study indicates that infants diagnosed with TSC prior to the onset of seizures who are treated with vigabatrin in response to the development of an abnormal EEG have fewer seizures, a lower incidence of drug-resistant epilepsy, and fewer children require polytherapy to treat their epilepsy. In addition, fewer treated children had significant developmental delay and intellectual disabilities at 24 months of age compared to children with TSC who were treated only after the onset of clinical seizures.
Appropriate and effective therapy is not only aggressive but also relies upon recognition of the natural history of the various lesions of TSC. For example, large angiomyolipomas may be mistaken as renal cell carcinomas, solely on the basis of their size. Embolization or kidney-sparing surgery should always be done so as to avoid the unnecessary removal of the affected kidney.
The main complication of TSC requiring long- term medical therapy is epilepsy. Antiepileptic medications (AEDs) are the mainstay of therapy for individuals with TSC. Unfortunately, no one medical treatment usually results in satisfactory relief for all or even most individuals with TSC. A combination of medical treatment modalities is then frequently required.
The choice of specific AEDs for treating seizures in individuals with TSC is based on the seizure type(s), epilepsy syndrome(s), other involved organ systems, age of the individual, and AED side effect profiles and formulations available.
The Tuberous Sclerosis Alliance recognizes the need for urgent treatment of infantile spasms and endorses the American Epilepsy Society's position statement on immediate access to accepted treatments for infantile spasms. The consensus developed at the NIH Tuberous Sclerosis Complex Consensus Conference in 1998 and confirmed in 2012 was that vigabatrin was the drug of choice to treat infantile spasms in children with TSC. Vigabatrin (Sabril®) was approved by the FDA in the United States in 2009 and generics are now also available. The FDA requires physicians who wish to prescribe vigabatrin register prior to prescribing the medication; click here for more information.
Adrenocorticotropic hormone (ACTH) may be used for the treatment of infantile spasms in TSC if vigabatrin is not effective. H.P. Acthar Gel® (ACTH) was approved by the FDA for the treatment of infantile spasms in 2010. The advantage of vigabatrin use is the ability to rapidly escalate the dosage at the initiation of treatment, rapid efficacy, suitability for outpatient treatment and particularly good tolerability with generally only minor adverse effects with the exception of visual field loss (see below). There is little evidence that other broad spectrum AEDs may prove useful to treat infantile spasms.
The safety of vigabatrin has caused concern since a specific visual field loss has been documented in treated adults and some children. There have been only isolated reports of visual field loss in children on short-term treatment with vigabatrin having infantile spasms, but the parents must weigh the benefits and risks of vigabatrin treatment with their child's health care providers. The current problem is determining the risk-benefit ratio of vigabatrin in children with infantile spasms, and specifying the groups where its use could be optimal. Visual field loss is usually asymptomatic and can be detected only by perimetric visual field studies. In children, especially in very young children or those with intellectual disabilities, it is difficult if not impossible to detect the visual field loss, and it is not yet known if children are at higher or lower risk of this adverse effect. Until a clear answer about the occurrence of this adverse effect in children has been established through randomized study, vigabatrin may still be considered first-line therapy in infantile spasms. Children who do not achieve a good response to vigabatrin should be switched to ACTH or to a corticosteroid such as prednisone.
Long-term use of agents with prominent sedating properties, such as benzodiazepines or barbiturates, generally should be avoided. These drugs often aggravate underlying behavioral or cognitive problems, and there are less toxic and often more effective alternatives.
WARNING: Carbamazepine, oxcarbazepine and phenytoin may cause exacerbation of seizures, particularly in younger children and infants with TSC, and some clinicians believe that these AEDs can precipitate or aggravate infantile spasms. While often valuable in older children and adults in whom partial seizures predominate, caution is warranted in their use in infants and young children. It is recommended that these drugs not be used in children with TSC who are experiencing infantile spasms.
An additional therapy for intractable epilepsy is the ketogenic diet. The classical ketogenic diet and variations of the diet such as the low glycemic index treatment or modified Atkins diet have been increasingly used in recent years due to their efficacy and a perception by families that it is more "natural" treatment. While it is often effective for some children, the ketogenic diet should be considered like any other medical intervention to have significant side effects and possible complications. These include kidney stones, hypoglycemia, metabolic distur- bances and suppression of growth. In addition, the diet requires a very motivated and compliant family to adhere to the diet to ensure a state of ketosis in the individual. Despite these caveats and concerns, the ketogenic diet is clearly efficacious in some individuals who have medically refractory seizures and should be considered for children with intractable epilepsy.
Neurosurgical care for seizures in an individual with TSC may involve focal cortical resection, corpus callosotomy, vagus nerve stimulation and/or MRI-guided laser ablation.
FOCAL CORTICAL RESECTION: In selected individuals with TSC, the resection of one or more seizure foci can be beneficial. Neurosurgeons with experience performing epilepsy surgery will do a complete evaluation to determine if an individual with TSC is likely to benefit from surgery. Even if seizure freedom is not achieved, surgery may reduce the severity and frequency of seizures for individuals with TSC. Epilepsy surgery should be considered for any individual with TSC who has seizures that are not controlled by AEDs.
CORPUS CALLOSOTOMY: Corpus callosotomy can be effective in reducing atonic and tonic seizures (i.e., drop attacks), but usually is not helpful for other seizure types and is considered palliative rather than curative. Seizure freedom following corpus callosotomy is rare, but can occur.
VAGUS NERVE STIMULATION: In the overall epilepsy population, 30 percent of individuals treated with vagus nerve stimulation (VNS) experienced at least a 50 percent reduction in seizure frequency, while 30 percent had a 90 percent or greater reduction and 40 percent had no improvement at all. In a small study of 20 individuals with TSC, nine individuals experienced (without adverse effects) at least a 50 percent reduction in seizure frequency, while 30 percent had no improvement at all. In a small study of 10 individuals with TSC, nine of these individuals experienced (without adverse effects) at least a 50 percent reduction in seizure frequency; half had a 90 percent or greater reduction in seizure frequency following treatment with VNS. Although the results of this one study are promising, the small number of study participants with TSC makes it difficult to generalize to the entire TSC population.
The VNS device has been shown to pose no known hazards in magnetic resonance (MR) environments when specified conditions for use are followed. Potential risk to patient safety for anyone with an implanted device during an MRI procedure is the heating effects of the implanted lead wire due to radio frequency (RF) heating caused by exposure to the MRI system. A broken lead should be removed prior to doing an MRI procedure, as this may present risk of tissue damage due to potential increased temperature of the lead.
MRI-GUIDED LASER INTERSTITIAL THERMAL THERAPY This is a minimally invasive alternative surgical technique which utilizes light energy to destroy soft tissue including tumor or damaged tissue. Laser energy is delivered to the focal epileptic lesion using a laser probe inserted into the epileptogenic focus while guided by MRI imaging. This technique has been reported in medical journals to be pain-free and require a short recovery time. | {
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package mazoni.navigationview;
import android.content.Context;
import android.view.LayoutInflater;
import android.view.View;
import android.view.ViewGroup;
import android.widget.ArrayAdapter;
import android.widget.ImageView;
import android.widget.TextView;
import java.util.List;
public class NavigationItemAdapter extends ArrayAdapter<NavigationItem> {
private LayoutInflater layoutInflater;
public NavigationItemAdapter(Context context, int resource) {
super(context, resource);
setup();
}
public NavigationItemAdapter(Context context, int resource, int textViewResourceId) {
super(context, resource, textViewResourceId);
setup();
}
public NavigationItemAdapter(Context context, int resource, NavigationItem[] objects) {
super(context, resource, objects);
setup();
}
public NavigationItemAdapter(Context context, int resource, int textViewResourceId, NavigationItem[] objects) {
super(context, resource, textViewResourceId, objects);
setup();
}
public NavigationItemAdapter(Context context, int resource, List<NavigationItem> objects) {
super(context, resource, objects);
setup();
}
public NavigationItemAdapter(Context context, int resource, int textViewResourceId, List<NavigationItem> objects) {
super(context, resource, textViewResourceId, objects);
setup();
}
@Override
public View getView(int position, View convertView, ViewGroup parent) {
NavigationItem item = getItem(position);
ViewHolder holder;
if(convertView == null) {
convertView = layoutInflater.inflate(item.getLayout(), null);
holder = new ViewHolder();
holder.label = (TextView) convertView.findViewById(R.id.label);
holder.icon = (ImageView) convertView.findViewById(R.id.icon);
convertView.setTag(holder);
} else {
holder = (ViewHolder) convertView.getTag();
}
holder.label.setText(item.getLabel());
if(item.getIcon() != null) {
holder.icon.setImageDrawable(item.getIcon());
holder.icon.setVisibility(View.VISIBLE);
} else if(holder.icon != null) {
holder.icon.setVisibility(View.INVISIBLE);
}
return convertView;
}
private void setup() {
layoutInflater = getLayoutInflater();
}
private LayoutInflater getLayoutInflater() {
return (LayoutInflater)getContext().getSystemService(Context.LAYOUT_INFLATER_SERVICE);
}
private static class ViewHolder {
TextView label;
ImageView icon;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
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{"url":"http:\/\/mathoverflow.net\/questions\/25470\/when-is-sobolev-space-a-subset-of-the-continuous-functions\/37775","text":"# When is Sobolev space a subset of the continuous functions?\n\nIf we let $\\Omega\\subset\\mathbb{R}^d$ with $d=1,2,3$ and define $\\mathcal{H}^1(\\Omega)=(w\\in L_2(\\Omega): \\frac{\\partial w}{\\partial x_i}\\in L_2(\\Omega), i=1,...,d)$. My tutor has repeated several times:\n\n1. If $d=1$ then $\\mathcal{H}^1(\\Omega)\\subset\\mathcal{C}^0(\\Omega)$.\n2. If $d=2$ then $\\mathcal{H}^2(\\Omega)\\subset\\mathcal{C}^0(\\Omega)$ but $\\mathcal{H}^1(\\Omega)\\not\\subset\\mathcal{C}^0(\\Omega)$.\n3. If $d=3$ then $\\mathcal{H}^3(\\Omega)\\subset\\mathcal{C}^0(\\Omega)$ but $\\mathcal{H}^2(\\Omega)\\not\\subset\\mathcal{C}^0(\\Omega)$.\n\nI was interested in trying to show these relationships. Does anyone know any references that would be useful.\n\n-\nThe classical reference is of course Adams and Fournier: Sobolev spaces. That book is not easy going, though. \u2013\u00a0 Harald Hanche-Olsen May 21 '10 at 11:44\n\u2013\u00a0 Willie Wong May 21 '10 at 17:54\n\nI understand from your post that you'd like to show those facts by yourself first, and not necessarily to approach the whole theory now (I like your approach). Trivial hint: start with smooth functions with compact support in $\\Omega$, and try to bound their $L^\\infty$ norm in terms of the $H^d$ norm. Also, I suggest that you try building counter-examples by yourself for the case of non-inclusions. Reference: Brezis' book of Functional Analysis may give you nice hints.\n\n-\nIn the Hilbert space setting the easiest way to see the whole thing is to fo on the Fourier side and see what you can get from Cauchy-Schwartz (just localize first by multiplying by a smooth cutoff). The second part of Claim 3 is wrong by the way... \u2013\u00a0 fedja May 22 '10 at 1:52\n\nIf you just want the answer, then not surprisingly you can find it at: http:\/\/en.wikipedia.org\/wiki\/Sobolev_inequality\n\nIf you want a careful introduction to and derivation of the Hilbert space case, see: \"Seminar on the Atiyah-Singer Index Theorem\" (Princeton Univ. Press)\n\n-\n\nBy now, I can't remember precisely where the best places to learn this is. Here are some rather vague suggestions:\n\n1. I don't know if this stuff is in any of Nirenberg's writings, but if it is, it's sure to be a clear and easy approach.\n\n2. Look in books about nonlinear elliptic PDE's by, say, Craig Evans, Gilbarg and Trudinger, or Thierry Aubin.\n\n3. Ideally, there should be a proof that involves integration over cubes. Differential geometers such as Aubin tend to prove such results by integrating over balls because that's what generalizes more easily to Riemannian manifolds. That works fine but the formulas are messier than for a cube. In the end, after you get the idea of what's going on, just write out your own proof.\n\n-\n\nI'll give you a hint for the first one $d=1$. Consider first the case that your function $f \\in H^1([0,1])$ was smooth. Then we could say $f(x) - f(y) = \\int_{x}^y f'(s)ds$. Apply Cauchy Schwartz now and you'll be able to see immediately that $f$ is $1\/2$ H\u00f6lder continuous.\n\nFor higher dimensions you actually proceed similarly but you need to use the co-area formula.\n\n-\nCheers! :) The first one $d=1$ is simple doing it this way. Now off to find out about the co-area formula... \u2013\u00a0 alext87 Sep 6 '10 at 19:19\nAlso, for the $d=2$ case, consider the function $u(x) = \\log |x|$. Then $|Du(x)| = \\frac{1}{|x|}$ and so $\\int |Du(x)|^2 = 2\\pi \\int_0^r r^2\/r$ which is finite obviously. So $\\log|x|$ is in $H_0^1(\\Omega)$ but not continuous. A similar example can be made for $d=3$ with $1\/|x|^{\\alpha}$ for an appropriate choice of $\\alpha > 0$. \u2013\u00a0 Dorian Sep 6 '10 at 19:46","date":"2015-07-07 00:42:47","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\": 0.9033036231994629, \"perplexity\": 353.4476436029397}, \"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-2015-27\/segments\/1435375098924.1\/warc\/CC-MAIN-20150627031818-00302-ip-10-179-60-89.ec2.internal.warc.gz\"}"} | null | null |
Topic: H.R.Jothipala: A national icon?
The Sinhala print and electronic media have been bending over backwards for sometime in glorifying the popular play back singer of yester year - H. R. Jothipala to make him look like a national icon.
The underlying justification for Jothipala's "greatness" appears to be his enduring popularity. I am surprised how easily people are prepared to buy this theory without even blinking an eyelid or evaluating his contribution critically.
To me this illustrates to what levels standards have plummeted down in contemporary times, and how confused and bankrupt the media and people have become. In attempting to rewrite Jothipala's epitaph, one scribe recently went to the extent of arguing that if dramatists could win awards (summana) by adapting foreign plays (hinting probably at Sarathchandra, Dayananda Gunawardena, Henry Jayasena et al), why cannot Jothipala be honoured with summana for "adapting" Hindustani songs?
Fundamental to this topic are two separate issues. Firstly, what did Jothipala represent or try to achieve or contribute to his chosen field (music) during his life time? Secondly what is the reason for his "enduring" popularity? To diagnose Jothipala's popularity we must look at two distinctly different phases. Firstly what made him popular during his life time as a vocalist (1958-87) and secondly what makes him apparently popular today after two decades of his premature death.
With due respect to a man no more with us let me place on record what I honestly think of him. Though from time to time Jothipala sang original (uncopied) tunes, it is not unfair to say that he made a name and a living by generally singing copied Hindustani film songs (mostly for third rate commercial films, and for the radio or private recordings).
Unlike the creative musicians like Samarakoon, Sunil Santha, Amaradeva, Victor Ratnayake and Sanath Nandasiri (who had served to establish and popularise the art song) Jothipala was purely a vocalist.
As a talented singer he was probably at his best singing copied Hindustani film songs in a style that appealed to the populace or to the gallery-the very trend that was the bane of the creative composers and film makers. Even when singing uncopied songs for films or the radio, his vocal style was not different.
I leave it to the musicologist or the musicians to assess Jothipala's vocal technique and style, but personally I find Jothipala with his diction intended to appease the gallery, generally nauseating to listen and impossible to enjoy in terms of rasa vindana or rasa nishpaththi. (There may be a few exceptions, of course). However to be fair Jothipala should not be blamed in isolation for copying Hindustani or occasionally Tamil songs, or for his vocal style.
Film producers of the fifties and later the Rupasinghes, Moraes's et al who included copied songs in their films to ensure the commercial success and music directors such as P. L. A. Somapalas - (the willing allies who joined the party for a quick easy buck and popularity); were the real culprits.
Vocalists such as Jothipala, Milton Perera, Mohideen Baig etc were vehicles used by this group in transcribing the Hindi songs. Talented lyricists like Karunaratne Abeysekara were willing to churn out lyrics to suit the Hindi tunes no doubt driven by economic needs.
Jothipala had an advantage over many other playback singers in that he had a stable voice and proper Sinhala diction (in comparison to Baig or Haroon Lanthra) and more importantly a voice suitable as a playback singer.
Spanning more than two decades Jothipala became an essential part of commercial formula films as the playback voice for matinee idols whether this was Bonifus Fernando, Roy de Silva, Gamini Fonseka or Vijaya Kumaranatunge.
The glamorous charismatic Vijaya and the undisputed king of the screen Gamini, for whom Jothipala was almost always the playback voice, their popularity and the popularity of those commercial films was certainly one reason why Jothipala was popular while living.
From the nineties, the media, particularly the television had played its role in keeping Jothipala's popularity by frequently playing those popular film clips starring Vijaya, Gamini, Malini etc with Jothipala as the male play back voice.
In addition, popular artists of the modern generation viz Gratien Ananda, H. R. Soyza, Kamal Addaraaratchi, Rookantha Gunathileka etc have come forward to sing songs of Jothipala for television programms giving Jothipala a further fillip. Preference of the modern generation to rely on audio visual media (shravya-druhsya) as the principal means of listening to songs is why his popularity seems to be still intact.
With the extraordinary and haphazard expansion of the media since mid nineties, standards have reached rock bottom and music (the song) has fragmented both in style and quality to "pathetic" levels. The media (and commercial CDs) is replete with melody-less mediocre substandard songs sung by amateur, untrained, unknown "artists" with highly questionable musical credentials.
It is weird how everyone seems to be happy to listen to this garbage and how a whole country can be fooled all the time by the media.
Against this background, it is not surprising that talented singers of a previous generation like H. R. Jothipala could stand high even now. But this does not mean that despite Jothipala's obvious but misguided talents we should be celebrating him as if he is a national icon particularly in view of the fact that he lived and thrived on plagiarism.
In a country that has seen two supremely gifted vocalists in the form of Sunil Santha and Amaradeva I am amazed how could a country rave so much about Jothipala's voice as if he was the golden Sinhala voice of the twentieth century?
Is constructive criticism and artistic taste in such dire straits in Sri Lanka?
Re: H.R.Jothipala: A national icon?
I do agree with Jayantha Anandappa partialy. Jothi was singing Hindi songs in Sinhala most of his career. But he did it well. Also he was a very good gentleman. Once he was introduced by another singer of his era as Hadawatha Raththaran Jothipala. He sang two rhythm sinlaha pops of the day and few original material as well.
Now take Beatles. What did they sing. Also two rhythm pops in english in their early days. They too copied other artists at the begining. Do you think that they are not English Icons now? Where will you put them. The Best band to influence the early rock seen. But in their hey day they were just a teen age boy band, not favoured by the established (clasical) music world.
Their grateness is recognised by everyone nowadays. Some of their compositions are recorded by London Symphony. I happen to own "Get Back" at one time. The Grand Jury is not the Critiques, but the people. They decide what is Grate.
If a musician is still popular 20 years after his death, it means that even the generations that followed keep him in high esteem. Jothipala is no Sunil Shantha, or Amaradeva, no no, just Jothipala.
Grateness don't have to be clasical all the time. It can be popular too, these days.
A request to Sithari and all other contributers - Don't just publish other peoples work without adding your own views and comments on the subject.
and Weli.. you did very constructive comment.
All creations at forums should be published with the views of the persons who post. Just copying some views for discussion is irreverent without your own views on it. Otherwise no real discussion will take place. | {
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# **Lucky Me**
Lucky Me
My Life with—and without—My Mom, Shirley MacLaine
**SACHI PARKER**
**FREDERICK STROPPEL**
**GOTHAM BOOKS**
GOTHAM BOOKS
Published by Penguin Group (USA) Inc.
375 Hudson Street, New York, New York 10014, U.S.A.
Penguin Group (Canada), 90 Eglinton Avenue East, Suite 700, Toronto, Ontario M4P 2Y3, Canada (a division of Pearson Penguin Canada Inc.); Penguin Books Ltd, 80 Strand, London WC2R 0RL, England; Penguin Ireland, 25 St Stephen's Green, Dublin 2, Ireland (a division of Penguin Books Ltd); Penguin Group (Australia), 707 Collins Street, Melbourne, Victoria 3008, Australia (a division of Pearson Australia Group Pty Ltd); Penguin Books India Pvt Ltd, 11 Community Centre, Panchsheel Park, New Delhi–110 017, India; Penguin Group (NZ), 67 Apollo Drive, Rosedale, Auckland 0632, New Zealand (a division of Pearson New Zealand Ltd); Penguin Books (South Africa), Rosebank Office Park, 181 Jan Smuts Avenue, Parktown North 2193, South Africa; Penguin China, B7 Jiaming Center, 27 East Third Ring Road North, Chaoyang District, Beijing 100020, China
Penguin Books Ltd, Registered Offices: 80 Strand, London WC2R 0RL, England
Published by Gotham Books, a member of Penguin Group (USA) Inc.
First printing, February 2013
Copyright © 2013 by Sachi Parker and Frederick Stroppel
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ISBN: 978-1-101-61656-7
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ALWAYS LEARNING | PEARSON
---|---
_To Arin and Frankie_
# **CONTENTS**
Prologue
**Chapter 1**
Humble Beginnings
**Chapter 2**
Of Mockingbirds and Fox Gods
**Chapter 3**
On Location
**Chapter 4**
Charters Towers
**Chapter 5**
From Here to Zagreb
**Chapter 6**
"You're on Your Own Now"
**Chapter 7**
Into the Wild
**Chapter 8**
Flight
**Chapter 9**
La Vie Bohème
**Chapter 10**
The Good Doctor
**Chapter 11**
Man in Space
**Chapter 12**
The Acting Bug
**Chapter 13**
"That's George McFly?"
**Chapter 14**
Family Feeling
**Chapter 15**
Domestication
**Chapter 16**
The Lord and the Ring
**Chapter 17**
Don't Take It Personally
**Chapter 18**
Shut Up and Deal
Acknowledgments
Pangloss sometimes said to Candide, "There is a concatenation of events in this best of all possible worlds: for if you had not been kicked out of a magnificent castle for love of Miss Cunégonde: if you had not been put into the Inquisition: if you had not walked over America: if you had not stabbéd the Baron: if you had not lost all your sheep from the fine country of El Dorado: then you would not be here eating preserved citrons and pistachio-nuts."
—Voltaire, _Candide_
# **PROLOGUE**
Mom, is there something going on that I don't know about?"
We were driving back to Malibu. It was a spring afternoon, and we were cruising along the 405, heading up from San Diego. The year was 1982.
Mom had been quiet as usual, deep in her own thoughts. She would point out a scenic highlight here and there: "Isn't that beautiful," she'd say, sighing, with the matter-of-fact serenity of someone who had seen the world several times over and knows all its secrets.
I was thinking, too. There was something on my mind. Something concerning her and Dad. It had been in the back of my head for a long time, much longer than I ever knew, and now suddenly it had rushed forward with startling urgency.
Why I was thinking about the subject at this particular moment, I couldn't say. My life had just come to another one of its dead-end moments: a relationship gone bad, a possible future cut short, everything in turnaround. The new love of my life had turned out to be a disappointment. In fact, he had turned out to be something of a sociopath.
And Mom had driven down the coast to rescue me.
That hadn't been the original plan: I'd invited her down to San Diego ostensibly for the purpose of sharing the exciting news about my engagement to this marvelous man. Privately, however, I knew he wasn't so marvelous. In fact, I knew I was making a big mistake, and I didn't have the strength to walk away from it without help.
Mom, who could spot a bullshit situation a mile away, was just the hero I needed. She stepped into the motel room, sized up my unsavory fiancé in an instant, and said, "Let's get the hell out of here."
So we were on our way back home. It was a two-hour drive up the coast to Malibu, which gave me ample time to ponder the often ridiculously bad decisions I had made in my sputtering romantic life. Over the years, I seemed to have developed this pronounced habit of seeking out, or being sought by, men of a distinctly ugly character. These were not merely the stereotypical selfish boors you saw in sit-coms and beer commercials; no, they were profoundly deceptive, manipulative, and immoral men who would drag me headlong into a series of emotionally damaging, physically compromising relationships. Why was this always happening to me? What had plunged me into this "smart woman, foolish choices" cycle of masochism?
Maybe it was all this dwelling on the dark complexity of relationships that brought me around to my big question: "Mom, is there something going on that I don't know about?"
She looked over at me with a quizzical look, an indulgent smile. "What do you mean?"
"I mean, between you and Dad."
She frowned slightly. The subject didn't appeal to her. It wasn't that she didn't want to talk about Dad; she just couldn't see how he mattered in any present equation.
"Nothing's going on. We're fine."
"I don't mean right now. I mean...Well, you know, he's in Japan, and you're here, and you never see each other..." They had been living apart for almost thirty years now, on opposite sides of the ocean.
And that was fine with Mom. "It's an excellent arrangement. I recommend it for any marriage." Now, I don't want to suggest that Mom had any hostile feelings toward Dad. She loved him, more than anyone or anything, but she didn't need him by her side every day. Once or twice a year was more than enough.
"No, but...I feel like there's something going on, something that you and Dad haven't told me, and it's something really important."
"There's nothing important," she said in a distant voice. Followed by a silence. This meant that there _was_ something.
"But I can feel it. It's something really big."
"You're imagining it. You have an overactive imagination, you know that."
I took a deep breath. I was taking a risk now. "I don't believe you."
She didn't answer for a moment. I couldn't read her face—I wasn't sure if she was going to let it drop entirely, or if she was measuring a particularly devastating reply.
Finally she spoke: "I'll tell you when we get home."
So we continued up the San Diego Freeway in silence. Now it was confirmed—there was _something_ —but I would have to wait another two hours for the big reveal. She had a great knack for showmanship.
The suspense was exquisite. What could it be? Would this be one of those life-changing revelations that finally puts the whole world into focus for me? Or would it be a total anticlimax?—"I couldn't deal with your father's snoring." Or, most likely, would she come up with some deft evasion at the last minute?
There was no way to tell with Mom. Something unknowable lurked at her very center. She was constantly coming out with books and magazine interviews promising insights into her inner self, and there were, to be sure, plenty of candid, sometimes shocking details. Oh, she always delivered. Still, like any accomplished magician or striptease artist, the more she revealed, the more she concealed. Just when you thought you had her in your sights, you blinked and she was gone.
So the fact that she might tell me something truly revelatory left me feeling a little uneasy. Did I really want to know this? Was I ready? Should I just have let sleeping dogs lie?
We finally arrived in Malibu, and went up the front walk in silence. Above us loomed the porch-balcony where Mom had had her first encounter with extraterrestrials. According to her, there was a knock on the balcony door, she opened it, and there they were. They didn't stay for long; they were just passing by.
She never told me what they looked like, or what kind of spaceship they had, or how they had come to learn perfect English. Only that, because she was an Enlightened One, the aliens had chosen her as a conduit to relay their messages to the rest of the world.
I never doubted that this had happened. I saw no reason not to believe that there might be life forms on other planets; and if they ever did come to Earth, why wouldn't they look up Mom right off the bat? She was just the type an extraterrestrial would have cozied up to.
Once inside the house, Mom went straight to her study without a word. She opened a closet door; inside, on the floor, was an old brown safe. "What's that?" I asked, but she didn't answer. She was intent on milking every last bit of drama out of this.
She hunched down over the safe and very carefully worked the dial, twisting it back and forth like a professional safecracker. She kept looking back over her shoulder to make sure I wasn't taking note of the combination.
Finally there was a click, and the safe fell open.
Inside was a rectangular tin box, the kind in which you might find cookies at Christmas. Mom held this box in her outstretched arms, as if she were bearing a sacred relic on a pillow. With a nod, she bade me follow her into the bedroom.
She pointed to the bed. I sat down, and she placed the cookie box beside me. She slowly lifted the lid.
Inside the box was a stack of old, yellowed telegrams—some, I saw, dating as far back as 1956.
"Read," she said. "This will explain everything...."
# **Chapter 1**
# Humble Beginnings
A _karesansui_ is a style of Japanese Zen garden, developed by Buddhist monks. It's a dry or rock garden; there are no flowers, just sand and moss. Traditionally there are fifteen stones in this garden, and they are so carefully arranged that, at any given point in the garden, you can see only fourteen of them. You can never see the fifteenth stone. But you know it's there.
I moved to Japan when I was two years old.
This should be an undisputed fact—it definitely happened. I was there—but through some curious quirk of misinformation (which, oddly, my mom never corrected, and over the years has often encouraged), most sites on the Internet contend that I was six years old when I moved.
This age confusion arose perhaps because my mother shot the film _My Geisha_ in Japan in 1962, when I was six, and it may have been assumed that she became so enamored of the country that she left me behind, to soak up the benefits of its extraordinary culture.
But I was already there. I'd arrived in 1959. There's a picture of me stepping off the airplane at Tokyo Airport, and I'm unmistakably a child of two. So that should settle the question.
I went to Japan to live with my father, Steve Parker. He was a businessman-entrepreneur, and he had his operations in Tokyo.
Mom stayed behind in Los Angeles because, well, she was busy. She'd just been nominated for an Academy Award for her performance in _Some Came Running_ , and she was getting ready to do Billy Wilder's _The Apartment_ with Jack Lemmon. In the meantime she would work on _Can-Can_ , the Cole Porter musical, with Frank Sinatra and Maurice Chevalier. It was not a great film, but it had its diversions. Perhaps its most memorable moment came when Soviet premier Nikita Khrushchev visited the set during filming. Mom got to dance the can-can for him, tossing the back of her skirt in his face. He found the spectacle immoral. "Humanity's face is more beautiful than her backside," he stodgily philosophized. I doubt he was singling out my mother in particular, because, at age twenty-four, she looked fabulous from any angle.
In any case, she had a lot on her plate—she was fast becoming a celebrity of the first order—so it was naturally inconceivable that she would give it all up and move to Tokyo with Dad and me. That much I got. But why would she so easily give _me_ up? That was more of a mystery. Especially in that prefeminist age, when mothers were supposed to cling to their children with the ferocity of lionesses.
Was I a particularly unmanageable child? I don't think so. On the contrary, we seemed to get on very well together. There was a picture shoot done for _Life_ magazine, which showed the two of us, me and Mom, in numerous cavorting and playful poses, including the famous cover shot of us wearing strings of pearls on top of our heads and puffing up our cheeks like blowfish. We were two peas in a pod, a couple of show business hams. The text had Mom expatiating at length on the joys of motherhood: nothing could have prepared her for the sense of fulfillment, the deep well of emotion, and so on and so forth.
Given her parental pride, it was all the more puzzling that she would just have let me go. Obviously, she had great faith in my father's child-rearing abilities, but still, there must have been another reason.
Well, the untold secret story—at least until my mother's Hollywood memoir _My Lucky Stars_ was published in 1995—is that I _had_ to be taken out of the country, as a safety precaution. Because my freedom—perhaps my life—was in danger.
In those days, Mom was an honorary member of the legendary Rat Pack crew: Sinatra and Dean Martin (her _Some Came Running_ costars), Sammy Davis Jr., Peter Lawford—the whole gang. She palled around with them in Las Vegas, and they considered her something of a mascot (as opposed to the women they kept around for decorative and/or amorous purposes). She would perform with them, party with them, and yet somehow retain her popular image as an innocent pixie. It was a great setup—she could enjoy all the fun of the boys' club without being harmed by the negative associations of their freewheeling booze-and-broads lifestyle.
Nevertheless, there was a downside to all this roistering merriment. At some point Mr. Sinatra apparently fell afoul of the Mob—he had crossed somebody somewhere, on a business deal or a singing engagement, or maybe he'd insulted a big shot's wife at a ringside table—and the word was out that payback was coming. One of the rumors circulating was that Shirley MacLaine's daughter was in jeopardy of being kidnapped.
Me—kidnapped! To avert this possibility, I was shipped off to Japan for safekeeping. Mom hated to let me go, but she had to, for my sake. It was a great sacrifice on her part, and in some ways she never recovered. At least that was her version.
I was a little skeptical about this story. After all, the Mob knew where Japan was, if they really wanted to find me. And why Frank Sinatra would have cared whether I was kidnapped was never clear to me. In light of his supercool ring-a-ding-ding philosophy, I can't imagine my abduction would have taken much of a toll on his psyche.
In any event, the threat gradually passed, everything got smoothed over—soon Frank Sinatra was brokering deals between Sam Giancana and JFK, and all was right with the world—but nobody ever thought to bring me back. I lived in Japan for the next ten years, spending only my summers and holidays with Mom in L.A.
**• • •**
**MOM** and Dad met in New York City in 1952. She says when he walked into the bar on West Forty-Fifth Street, where she and her fellow chorus members from _Me and Juliet_ were hanging out, she was immediately taken with his handsome swagger and his charm. He wasn't tall—five nine—so she made the interesting decision to take off her high heels, so she wouldn't be taller than he was. I don't think I've ever seen Mom give ground to anyone. For Steve Parker, though, she willingly made herself smaller.
They chatted, had a few drinks, and he proposed to her that very night. She didn't accept until the next day. Their lives were so hectic, however, that they didn't get around to tying the knot until 1954. They got married between the matinee and the evening performance of _The Pajama Game_ , in which she was appearing on Broadway. Norman Vincent Peale officiated at the ceremony.
Mom credited Dad with being the most powerful influence on her life. He was a mentor to her, a motivator, a truth-teller, and a sharer of dreams. He encouraged her ambitions and guided her on her search for self-knowledge.
Perhaps his most significant and lasting contribution was a book he brought to her on the set of _The Trouble with Harry_ , her first film. The book was called _A Dweller on Two Planets_. It was written back in the 1880s, by Frederick S. Oliver, and was apparently the forerunner to all those books dealing with New Age mysticism. Oliver claimed to be channeling, through the process of visions and "automatic writing," the words of an ancient spirit named Phylo the Tibetan. Phylo had lived in the city of Atlantis, centuries before, and he had many stories to tell about the advanced Atlantean culture and its parallels to modern life.
The book, written in florid biblical prose, deals with karma and reincarnation, both subjects that would become central to my mother's philosophy. Indeed, she traced her interest in spirituality to _A Dweller on Two Planets_. Why Dad had chosen to give her that particular book was not especially evident, but Mom said he was always an expert at reading people. He could sense what they would respond to; that's what made him such a good businessman.
Soon after this, Mom and Dad moved to Hollywood, but when her career started to take off, so did Dad. He didn't appreciate being known as "Mr. Shirley MacLaine," and he had business back in Japan, so they decided to live apart.
Their separateness, and their understanding that they would have an "open marriage," did not diminish in any sense Mom's love for my father. I would say she loved him as deeply as anyone in her life. Dad was her true soul mate, even though they lived in different countries for almost their entire marriage.
**• • •**
**MY** first real memory is of flying across the Pacific Ocean. I can't say for certain if I'm remembering that very first trip or a subsequent flight, because I did shuttle back and forth several times to visit my mom in L.A. (probably stealing into town under cover of dark, so that those darn kidnappers wouldn't notice).
The experience of the flight itself is still very vivid to me. The airline was Northwest Orient, it was a propeller plane, and it took about three days in all. We would make stopovers to refuel on several Pacific islands—Wake, Guam. I remember walking the beach at Wake Island, enjoying the warm ocean breeze, marveling at the exotic seashells, and feeling that I had entered some heavenly realm. There was no sign at all of the horrific toll the war had exacted on these blood-soaked sands just fifteen years earlier.
Then back on the plane. In those days there were beds on the plane, bunk-style, for long flights. They strapped you in with seat belts when you went to sleep. This was also before they had movies on airplanes—which was before they had individual TV screens, which was before they phased everything out so that we're back where we started—so you had to provide your own entertainment.
In my case, that meant singing. I would sing for everybody on the plane. It sounds obnoxious now—if a young child started singing and dancing up the aisles nowadays, you'd have more than one passenger deploying the escape chute. Back in the 1950s, though, it must have seemed charming and wholesome, and emblematic of the best in American youth.
There was no one to stop me from performing, anyway. I was flying alone. Without guardians or chaperones. I'm told I was as brave and stalwart a little traveler as you could wish for. Still, there must have been times when I felt lonely and scared, because I remember one of the stewardesses sitting beside me, cradling me in her arms, and rocking me gently for what may have been hours. It wasn't the last time a complete stranger reached out to rescue me when my own family was nowhere in sight.
I remember finally landing in Japan, my dad waiting on the tarmac for me, in a sharp business suit, with his rakish Clark Gable mustache, looking for all the world like a movie star. No exaggeration—Dad really had that kind of worldly glamour. He was extremely charming, very warm, and a man's man, fond of handshaking and backslapping. And the more he drank (which he did, quite a lot), the more charming he got. Life was a big party for him—and I had just been invited to join it.
I ran up to him now and jumped with delight into his strong arms. I was home! Home in a place where I didn't know a word of the language. Except for my own name, of course. Sachiko.
You might wonder, why did I have a Japanese name if I wasn't actually born in Japan or even of Japanese heritage? Actually, my full name is Stephanie Sachiko Parker. Stephanie is the female version of Steve, my dad's name. And Sachiko...Well, Dad had a story about that, which he told me as we drove to his house.
He had been stationed in Japan after the war. Just outside Hiroshima. And there was a little girl who used to hang around right outside the barracks.
_She was about two years old, and she was always playing in this empty lot, all by herself. Her clothes were torn and raggedy, she had snot running from her nose, hair all tangled...She looked totally uncared-for._
_So I asked around, found out she didn't have any family. Both her parents had died from radiation poisoning. From the bomb._
_Poor little kid was all alone. So I took her into the barracks, and the boys and I sort of adopted her. She was a little cutie; we all were in love with her. So much so, that it got to the point where I decided to adopt herfor real. She was going to be my own daughter. I had all the paperwork going, should have been a cinch..._
_Then she started getting sick, too. Same radiation poisoning. Poor kid. After a while...she died..._
He took a long beat, measuring the silence.
Finally: _Her name was Sachiko._
When we got home to our house in the Shibuya section of Tokyo, there was a surprise there waiting for me. Her name was Miki.
Miki was Dad's Japanese "friend." His live-in friend, to be precise. She'd been with my dad for many years already, they had become very comfortable together, and she welcomed my addition to the family unit with all the effusive warmth of a reticulated python. The first moment I walked through the door, she fixed me with a cold, silent glare, equal parts jealousy and icy contempt.
Miki and Dad had met many years before, at a teahouse. Teahouses are places in Japan where businessmen go to relax, and be entertained by the geishas. They're not really brothels, as Westerners might imagine. Everything is very proper and civilized. The geishas are treated with the utmost respect. If by chance a spark of interest is struck between two consenting parties, there might be a discreet trip up the staircase, but what went on up there was strictly the business of the participants involved.
The geishas' primary function at a teahouse is to provide companionship, attentiveness, and entertainment. They sing, they dance, they serve drinks, they keep the conversation moving. They're always from very good families, and highly educated—they can usually speak three or four languages. A geisha is classy, sophisticated, and very elegant.
Miki wasn't a geisha. She was a maid. She started working at the teahouse when she was a little girl. Indeed, when Dad first met Miki, she was only twelve years old. This was before the war, when my grandfather, whom I never met, had his shipping business in Japan, and Dad was in his twenties, enjoying the life of a dashing overseas adventurer. Dad became friendly with the young Miki, and then, as she matured, he became friendlier. I don't know when they became lovers, but when Dad moved back to Japan after the war, Miki was waiting for him, and they took up where they'd left off.
Where they'd left off did not include me in the least. The disparity between Dad and Miki's lifestyle and mine was pretty stark. They were always off to fancy restaurants, off to the theater, off to Dad's yacht, off to Dad's private island, off to Hawaii, Italy—just having a hell of a time for themselves.
Mom, it should be noted, knew about Miki. As I said, she and Dad had an open marriage, and were perfectly accepting of each other's affairs. However, Mom believed that that was all it was—an affair, one of my father's little dalliances. She didn't suspect that Miki was, for all intents and purposes, Dad's other wife. The one he had all the fun with.
While Dad and Miki were out playing, I wasn't completely neglected. I had a governess, Eguchi-san, who looked after me and instilled in me all the virtues of Japanese womanhood: meekness, humility, subservience.
Eguchi-san was not an attractive woman, by any means. As I recall her, she looked sort of like a toad—or rather, to give her the eminence she deserved, a bullfrog. She was short, dumpy, hunched, and round-shouldered, and her wrinkled, bumpy face was perfectly consonant with the rest of her amphibious aspect. She was also very old. At least she seemed very old to me.
"Remember, Sachiko-san," she would say, with the wisdom of the ages rattling around in her creaky voice, "the stake that sticks out will get hammered down." She was fond of these ancient Japanese maxims that encapsulated the whole of life experience in a few well-chosen words.
"A frog in a well knows nothing of the sea."
"If you chase two hares, you will never catch one."
"Never, never gossip about people, or their shadows will follow you forever."
I adored her. She wasn't in any sense warm or cuddly. An extremely traditional and strict governess, she considered all rules sacrosanct and not to be transgressed. This gave me a very deep and precious sense of security. I knew where I stood with her. She never kissed me, never hugged me, but she didn't have to. I knew she loved me.
And I knew she hated Miki. She never said as much, but I could tell. Eguchi-san similarly sensed my unspoken enmity toward Miki, and that cemented a very powerful bond between us.
The trouble was, Eguchi-san was there only during the day, between nine and six. At night, when Dad and Miki were out nightclubbing, I had to fend for myself.
How does a lonely little American girl entertain herself in an empty Tokyo house? Well, I ate lemons. There were always plenty of lemons in the refrigerator, and every night I would eat one, or two. There was something about the pretty yellow color and the lemony smell and the sour taste that was utterly comforting to me. So I ate lemon after lemon after lemon. Years later, whenever I went to the dentist, he would ask me why I had no enamel left on my teeth.
I don't remember what else I did to amuse myself, but I got through the evenings somehow. These weren't unhappy times for me, necessarily, because I didn't know anything else, and I did have some company: our dog, Taiho, a magnificent Akita husky named for a champion sumo wrestler. I adored him, and cuddled with him every night—we needed each other.
I loved Japan, I truly did. I wouldn't have traded my childhood there for anyplace else in the world. It's a land of breathtaking wonder, and quiet serenity. The majesty of the mountains to the north. The spare simplicity of a rock garden. The opulent bursts of cherry blossoms in the spring.
My own home was a magical melding of East and West. The house was ranch-style, and most of the rooms were Westernized. In the living room, there was a raised tatami platform, which I would turn into a stage. I would happily perform for guests, singing and dancing and soaking up their applause. There was also a traditional Japanese bath; my dad loved to soak in it, and I would always have to bring him a sake or scotch to accompany his relaxation.
The backyard of the house was my special place, a dreamlike little world that filled me with enchantment. Each stone and bush, in its size and placement, had meaning. There was a koi pond, stocked with the beautiful expensive fish; some of the koi were almost a hundred years old. A stream flowed down a gentle incline, and in a corner of the yard was a beautiful stone fountain, cut from a boulder, with a bamboo trough feeding the water into the basin, which would fill up and spill over into the stream.
The centerpiece of the yard was a spectacular cherry blossom tree. I would climb that tree up to one particular branch, high above, and sit there for hours and hours, lost in my own world. The stream would be running just below me, and the pond was to my side, where I could look down and see the fish. I don't know what I thought about when I was up there all those hours. I was just there. The tree was my friend. I would talk to it. It was something I could count on.
Eguchi-san took me often to the museums in Tokyo. She was a great lover of the arts, and wanted to introduce me to every aspect of Japanese culture. She would take me anywhere, at any time, in search of an exhibit or a presentation. She taught me how to paint, how to dress, how to arrange flowers. Every moment of life was informed by art. "This is your culture, Sachiko-san," she would say, ignoring the fact that it wasn't my culture at all. "It is who you are."
At the museum, there were watercolors and sumi ink paintings from various periods, some hundreds of years old. We stopped before a landscape, a house beneath a mountain. Eguchi-san pointed out a single brushstroke on the canvas. "That is the difference between beauty and nonbeauty," she declared with great authority.
We came upon another painting, which seemed oddly unfinished—a purplish burst of flowers in one corner, and the rest of the canvas empty.
Eguchi-san sighed with pleasure. "The emptiness is what makes the painting full."
I marveled now at a painting of a cherry blossom tree by a river. Nothing more—simple and elegant. It had been painted some six hundred years before, but was still vibrant with life. "It's so beautiful!"
Eguchi-san studied the plaque beside the painting thoughtfully. "Come. Let's go," she said, taking my hand.
"Where?"
The next thing I knew, we were on a train, heading north into the country. I was hungry, so Eguchi-san took a clementine from her pocket. When I reached for it, she held it back: there was an art to the eating of a clementine, as in all things, and it had to be observed.
Eguchi-san stripped the rind from the clementine, piece by piece, never hurrying. Then she slowly, carefully punctured the inner skin at the top and peeled the fruit down on the sides, so that the sections opened up like a flower. She held out the flowering clementine in the palm of her hand, and I dug into the juicy flesh. Somehow the slow, patient ritual had lent a febrile intensity to the experience. I devoured the clementine in gulps.
We got off the train at a country town, Echigo-Yuzawa. We walked down the road, Eguchi-san hobbling along on her cane unhurriedly. "Eguchi-san," I wondered, "do you know this town? Have you been here before?
"No."
I grew a little worried. "Do you know where we're going?"
She shrugged, as if to say, "No. But yes."
Before long, the road turned along a river. We walked along the river, and as we turned a bend we suddenly came upon it: the scene from the painting. There it was, the very same tableau, everything still there, just as the artist had seen it so long ago—the cherry blossom tree, the blue sky, and the river. All unchanged.
"Six hundred years," said Eguchi-san.
**• • •**
**WITHIN** three months of arriving in Tokyo, I was speaking Japanese better than I could speak English. In a way, it was my first language. Nevertheless, I felt uncomfortable and out of place at Nishi Machi, the International School that I attended. It may have been because Eguchi-san made me wear seven pairs of brown underpants every day—one on top of the next. This had something to do with her deeply held religious beliefs. It appears that the number seven and the color brown are both considered lucky in certain strains of Shintoism. Still, on hot days, I would have traded some of that luck for a little more ventilation. Using the bathroom was a strategic nightmare.
Nobody actually sells brown underwear. It's probably the one color in underwear you'd like to avoid. So Eguchi-san would have to create brown underwear, by dyeing my white underwear. She'd get a big pot of water, boil it up, and unwrap these cakes of brown dye. I can still hear that crinkly unwrapping sound now—not as glamorous an evocation as Proust's madeleines perhaps, but just as potent. Then Eguchi-san would mix the dye in the water, drop in the pairs of panties—more like granny pants, to be honest—stir the mix, and voilà! Instant nerd wardrobe.
You don't know what self-conscious is until you've walked around school with seven pairs of brown granny pants under your skirt. The slightest gust of wind and I would have had to commit hara-kiri to save face.
I was extremely shy as it was, and I did my best to fit in unobtrusively with the other kids in the class. There was one girl, however, Yuki, who upended this strategy. She was kind of a tough girl, aggressive and self-confident. I just didn't get along with her, and I avoided her as much as I could.
This wasn't easy. Dad and Yuki's mother were acquaintances, and there were many visits between our home and theirs. Somehow it came to be accepted that Yuki and I would be playmates. This arrangement did not come naturally to either of us; it was forced upon us, and we both bridled at the indignity.
In any comparison between Yuki and me, I always came off the worse. Yuki was prettier, Yuki was more confident, Yuki was more athletic. And Yuki was smart. A very good student. I was not. In fact, I was a very bad student.
You could blame the cultural divide, I suppose, but I believe my dad had something to do with it. You know how some parents are overly supportive, filling their kids with overbearing self-confidence? Dad took a different approach. He had a special nickname for me: the Idiot.
"Here comes the Idiot!" he'd say when I walked in the door. "The Idiot's home!"
He obviously had little respect for my intellect. He would never let me read. I didn't know why; I assumed he just thought it was a waste of time. If he saw me pick up a book, he'd scoff and take it away. My mind wasn't meant for profundity. Everything needed to be kept simple and stupid. I never questioned him.
By now you may have deduced that my dad was a hard man to figure: loving one minute, distant and aloof the next two weeks. I never seemed to be on his radar.
For example, every summer, I'd visit Mom in America and she'd take me on a shopping spree, buying me all kinds of clothes for the school year. I'd look great in September.
By February, I'd grow out of everything, and nothing would fit anymore—my skirts, my shoes, nothing. I desperately needed new clothes, but Dad wouldn't buy me any. "I can't afford it," he'd say. He could afford anything for Miki—fur coats, evening gowns, expensive jewelry—but my needs would have to wait till next summer. It was almost as if he didn't see me.
Yet...
There were times when he made me his whole world. Dad loved to cook, and he would keep me at his side in the kitchen while he threw together an improvised gourmet meal. He loved to celebrate the sensory aspect of cooking. He had a little herb garden by the window, and he would take a sprig of thyme or tarragon and crush it between his thick fingers, and he would say, "Smell! Smell!" I would smell, and I would smell some more, until it infringed on his sacred cooking schedule. "It's time to throw it in the pot! Now! Now!" I'd toss in the herbs, and feel as though I'd made an important contribution.
After the food had been cooking awhile, he'd hold out a ladle to me: "Sach, taste this. What does it need?" I had no idea, but I was thrilled that he seemed to value my opinion. I would make some kind of guess—"Rosemary?"—and he'd say, "Let's try it!" And it usually worked.
When dinner was ready, we'd sit on the floor, Japanese-style, and he'd savor his glass of hundred-year-old wine—Dad always insisted on having the best—and all would be very right with the world....
My father and I would make a fishing trip a few times a year, just the two of us. This made it very special to me; these are some of my favorite memories with him. I seldom had Dad to myself without Miki glowering over his shoulder, like a demon in a Japanese woodcut, but here it would be just us, the fish, and the wilderness.
The train ride into the Japan Alps would take about eight hours. We would head way up north, near Hokkaido. It was an endless but exciting journey. About three-quarters of the way up, after riding through miles and miles of ordinary countryside, we would enter a tunnel—a magic tunnel, I always thought—which seemed to go on forever, and after chugging through it for maybe half an hour, we would suddenly emerge into a different world: a spectacular ice-covered fantasyland, with ten-foot-tall drifts and little fairy-tale houses half-buried in snow. And silence. A silence so pure and complete you couldn't even hear the train engine anymore. It was as if the snow were swallowing the sound. I took that journey through the magic tunnel many times, and every time, the stunning moment of emergence took my breath away.
It was usually dark by the time we got to the mountains, and sometimes we'd stay at the Kanaya Hotel, which was an ultra-first-class hotel, old and venerable (meaning creaky), with huge ballrooms and winding staircases. Here the royalty and visiting heads of state would take their leisure, and the cream of Japanese high society would gather in their Western tuxedoes and floor-length gowns to fox-trot and tango the night away.
Other times, Dad would just start up the Jeep and we'd head off into the mountains in the dark. We'd go up as far as we could, and when the roads became impassable, we'd strap on snowshoes and backpacks and continue on foot, navigating by moonlight until we reached the frozen crater lake at the top of the mountain. We'd pitch a tent on the side of the mountain, on the most level ground we could find, and camp out there for two or three nights. It was absolutely freezing, sleeping on top of the snow, but we wore thick woolen sweaters suffused with natural lanolin oil; they had a powerfully rancid stench to them, but they kept us warm. Exhausted, I would fall right to sleep, albeit slightly terrified that we might slide off the mountain in the middle of the night.
Dad would wake me up at 4:00 A.M., and we'd hustle out onto the frozen lake. The first order of business was to cut a hole in the foot-thick ice. Using a long ice pick, Dad would chip away at the ice with methodical ferocity. In the meantime, I'd go into the nearby woods to gather sticks and logs for a fire. When I got back, he'd still be chipping away, until finally a circle of ice dropped with a satisfying splash into the water.
Next, Dad would build a fire on the ice. Once the wood was blazing, he'd scoop water out of the ice hole with a small coffeepot. He'd boil the water, simmer it, toss coffee grounds into the pot, stir it up, let the grounds settle, and then drink straight from the pot. He called it cowboy coffee. It was especially good, he said, when sweetened with a dollop of whiskey from his hip flask.
Time to catch breakfast! Dad would grab his fishing pole and squat alongside the ice hole. The morning was the best time to catch trout, when the fish were hungry. And he would catch them—fish after fish after fish. His delight would grow with each catch. This, I think, was when he was happiest, when he was truly in his element. In his heart of hearts, he probably wanted to be Hemingway.
Laying his pole aside, he'd pick out one lucky fish for breakfast and slap it down by the fire. I'd watch, horrified and yet fascinated, as the trout flopped around on the ice, gasping and thrashing. I was so relieved when it finally died.
Dad would scale the trout with a Swiss army knife and then rub rock salt into the skin of the fish. He'd always use gourmet rock salt from France—nothing but the best. The salted fish would go directly on the fire, and cook through. "Hot, hot," Dad would say, gingerly peeling the meat from the trout and handing it to me, having carefully picked out the bones first. And we'd eat, and it would be delicious.
Then we would go back to the tent and sit. For the most part, we wouldn't speak to each other. Dad would read a book. I wouldn't, since Dad didn't like it when I read books. So I just sat.
Then he might put the book down and start telling a story of his adventures. He would tell me about his travels in Cambodia, and the ancient village of Angkor Wat, and his friend Prince Sihanouk. Then he would drift off, and there would follow maybe three hours of silence, and then he'd start up again, reminiscing about Turkey or maybe a trip he took to Siberia. Vivid, fantastic stories.
That was how it went for the rest of the day, until darkness fell. Little bursts of talk, long silences the rule. I found myself mostly bored, eager for him to put down his book and start another tale. The days were alternately fascinating and excruciating.
Those mornings, though—those frozen mornings of icy mountain air and coffee smells and salty fish—they were quite wonderful.
On New Year's Eve, Dad, Miki, and I went up there and stayed at the Kanaya Hotel together. There was an old shrine, a relic from the Tokugawa era, at the top of the mountain, and it was a tradition to climb up through the snow at midnight to ring in the new year. Paths were cut through the five-foot drifts, and I would have to make my way up the steep snow wearing my special New Year's outfit, made of layers of thin silk and sashes and topped with a thick silk kimono with a beautiful flower print, an obi (a wide band around the waist woven with strands of gold), and the ceremonial shoes (called geta) with ornate thongs and three-inch bamboo clog soles.
At the top there would be an open-air market, and the vendors, batting their sides with their mittened hands for warmth in the subzero temperatures, would sell barbecued burnt corn, barbecued octopus, and oden, a special soup made with marinated fish cakes, potatoes, and seaweed.
You would buy a wish for the new year, in the form of a ribbon, from the shrine priest, and then tie the ribbon to a tree branch, and the wish would always come true. At midnight, they would ring the massive ten-foot-wide bronze bell, and it would peal out for miles. I would clap my hands over my ears to keep my brain from vibrating. The next morning, we'd have a classic New Year's Day breakfast at the hotel: raw octopus, cinnamon toast, and English tea.
It would be nice to claim that these were moments of life-shaking import for me, moments for epiphanies that elevated my relationship with my dad to a new level of intimacy—but they weren't. Mere isolated glimpses of warmth and affection, they were far outnumbered by days and weeks of neglect, when Miki held a far more important place in my father's heart.
The word they use now is _disconnect_. Nothing seemed to add up with my dad. He could be overly controlling, and yet profoundly indifferent. Sometimes he valued my opinion; more often he branded me an idiot. He had a particular brilliance for keeping me off-balance. I never knew where I stood. My tent was pitched on the side of a mountain.
So why? Still that big confounding question. Why was I in Japan? Why did my father bring me halfway across the world just to ignore me? What was the point?
Well, Dad was a very clever man—you might call him an operator—and there was _always_ a point. But it was very well hidden, and it took me a long time to find it.
# **Chapter 2**
# Of Mockingbirds and Fox Gods
On those few occasions when they stayed at home, Dad and Miki had an amusing little game they liked to play. If the phone would ring from America, they would react in mock horror:
"The Dragon Lady is calling!"—the Dragon Lady being my mom—"It's the Dragon Lady again!"
I could tell they were doing this partially for my benefit, and I wasn't exactly sure why, but since they were feeding me and tending to my general well-being, my sense of self-preservation advised me to join in the conspiracy. "Oh, no, it's the Dragon Lady!" I'd shriek, shrinking from the phone in dismay.
Clearly I was meant to regard Mom with a mixture of terror and amused condescension. She was this comical gargoyle across the sea, fearsome and terrible but safely tucked in some subterranean cave far away.
Still, I loved her, and whenever I visited her, Mom didn't seem so bad. In fact, she was a lot of fun.
It started at the airport in Los Angeles, always a big production number, with Mom rushing up to me at the gate and giving me a big, all-encompassing hug. "Oooh, I missed you so much! Look how you've grown!" We'd walk through the airport holding hands, swinging them back and forth joyously. Once we got into the car, she'd say, "Okay, let's have some fun!"
And we did.
She loved taking me shopping for my school clothes. Or going to the movies. Or walking along the beach.
I remember wandering the Malibu beach for hours with her, staring at tide pools. She would pick out a little sea urchin and poke it with her finger, and it would curl around her finger, and she would shriek every time. It always surprised her.
After a long walk on the beach, we would stop on the way home at Wil Wright's ice-cream shop in Encino. We'd go there practically every day—Mom was a real ice-cream junkie—and she would take a tactile joy in gulping down a hot fudge sundae. You know how kids can mysteriously manage to smear chocolate all over their faces when they eat, almost in defiance of the laws of physics? That was Mom. She'd look up and smile at me with a naughty glee, her teeth a dark, gooey mess.
Sometimes we'd head down to the local Piggly Wiggly and eat free cookies from the bakery section. They weren't supposed to be free, mind you, but Mom had no qualms about stepping around the counter and grabbing a cookie from the tray. If she really liked it, she might buy a bag to take home, but otherwise, she'd eat her fill and move on. No one stopped her; she was a celebrity, after all.
There were also times we would just hang around her Encino home and play games and eat popcorn and other goodies. She used to make the best open-face peanut butter and jelly sandwiches in the world. She'd take two slices of white Wonder bread—we didn't have Wonder bread in Japan; we didn't have bread in Japan—and spread Skippy chunky peanut butter on top. Then she would make little craters in the peanut butter with a spoon and fill them with pools of jelly. Absolute heaven.
Whenever there was a thunderstorm at night, Mom would let me climb into her king-size bed. Then she'd sweep open the blue velvet curtains on her full-length windows, revealing a spectacular view of the San Fernando Valley. Having set the scene, she'd scamper back to the bed and jump under the covers, and we'd huddle together and watch the storm march across the valley.
She would tell me the tale of the Lightning Princess. I don't know where this story came from—I think she made it up—but she would tell it with such verve and wonder, calling upon all her acting talents, that it would spring to life in front of me. I think that's where I first began to understand the magic of storytelling:
_Once upon a time_ , she'd begin, with such relish that you could see just how much she had missed the joy of telling her only daughter a bedtime story every night, of recounting the wondrous tales of Jack and the Beanstalk and Little Red Riding Hood and Hansel and Gretel:
_Once upon a time, there was a beautiful princess, who lived on top of the tallest mountain in the world. And she was known as Princess Lightning. Because when she was happy she would laugh a merry laugh, and a great flash of lightning would flood the sky!_
_But when she was mad, a dark scowl would come over her face, and then the lightning would come screaming out of the sky in a jagged bolt of terror. Zap!_
_Now, whenever a storm came across the valley, the mighty Princes of Thunder, who were all in love with the princess, would call her name fromthe dark clouds, and they would ask her for a kiss, because she was so beautiful. "Kiss me, please, O Princess Lightning!" they would plead._
_But she never would. Sometimes she would laugh, and blind them with a joyous flash of lightning, and sometimes she would screw her face up with fury and just chase them off with a crackling bolt! Zap zap zap! So when you hear the thunder rumble after the lightning, that's the princes rolling away in fear. For no one could ever capture the heart of the feisty, independent princess._ (Even then, my mom was grooming me for the feminist revolution.)
Then a burst of lightning would brighten the sky, and Mom would squeal with delight. "Look, she's laughing! She's laughing!" And we would both cover our ears to protect them from the princes' doleful thunder. And I would feel so safe with her.
**• • •**
**IN** the early 1960s, my mother was sitting on top of the show business world: she was a major actress, a knockout entertainer, and an international celebrity. In short, a star. While she'd had her first successes in the late 1950s, it was _The Apartment_ that put her on the map and made her a force to be reckoned with. She was nominated for an Academy Award for the second time, and she deserved to win.
"I would have, too," she insisted, "if Elizabeth Taylor hadn't had a tracheotomy."
Liz Taylor had made a splash that year by playing a call girl in _Butterfield 8_ , an okay but not great film. Her performance was worthy of a nomination, but she wasn't really expected to win—until she came down with a serious case of pneumonia that required emergency surgery. The wave of industry sympathy that followed pushed her over the top, and she went home with the Oscar instead of Mom.
This may have been a blessing in disguise. Mom managed to avoid the dreaded Oscar jinx and go on to a career of memorable films. Elizabeth Taylor, by contrast, became a full-time celebrity, and never really approached the level of creativity she enjoyed before _Butterfield 8_. (She did have a great Oscar-winning success with _Who's Afraid of Virginia Woolf?_ , of course, but that movie is a lonely giant among turkeys such as _Cleopatra_ , _The V.I.P.s_ , _Boom!_ , and so on.)
Mom, on the other hand, kept moving from peak to peak, because she stayed hungry. Without that ultimate industry recognition, she had something to shoot for, and she kept shooting. Maybe, with her ferocious inner drive, it wouldn't have made any difference; or maybe, having won the top award, she might have slowed down a little and spent more time with her family. Who knows?
The short-term benefit for me was that I got to hang out on her movie sets, and that was like heaven. For a young child, wandering through that make-believe world filled with music and lights and odd but endearing characters, it was as close to being in Fantasyland as you could get.
"Hellooo, Sachi!" everyone on the crew would cry; they made the biggest fuss over me, because I was so damned cute, and also because I was the star's daughter. The makeup artists would put all kinds of makeup on me, the costumers would dress me in little doll outfits...I was the Little Princess, and the movie set was my kingdom. It was like being in a cocoon; a safe, comforting, warm place where nothing bad could ever happen.
A lot of good was happening on the set of _Two for the Seesaw_ for Mom. Along with nabbing a juicy part as a Greenwich Village kook and working with Oscar-winning director Robert Wise, she was carrying on a pretty steamy romance with her costar, Robert Mitchum. Nobody knew it at the time, of course, and I certainly didn't pick up on it on my brief visits to the set. I would see Mr. Mitchum shambling about during the shoot, and to me he was just a big guy with sleepy eyes and a gruff, gravelly voice. The possibility that there were any personal sparks between him and my mother never crossed my mind, but then again, why would it? I was just a kid. According to her, however, their relationship was deep and intense, lasted a good three years, and featured sexual assignations all over the world. The image of the languid Mr. Mitchum hopping on a plane to meet my mother in a hotel in East Africa for some hanky-panky seems a little odd, but she says it happened.
Mom has publicly stated that she and Dad had an open marriage; both were free to pursue outside relationships, without disturbing the essential and all-important love between them. I was never quite sure how this worked, at least in the early days, because Mom always kept her affairs pretty quiet. (In those days, there _was_ such a thing as bad publicity.) She claims that she told Dad about the Mitchum affair, in the spirit of openness, but I don't know how true that is. I do know that Dad kept his very serious relationship with Miki a secret from Mom for some thirty years. I guess he was allowed to see other women as long as there wasn't One Other Woman.
The atmosphere on the set of _Irma La Douce_ was markedly different. Mom was playing a lady who wore a very tight dress, and her costar, Jack Lemmon, played a policeman with a funny hat. I had no idea what the risqué situations and sophisticated repartee were supposed to represent, but the two of them seemed to be having a great time. Maybe that was because there was no romance between them; Mom always said that Jack Lemmon was a sweetheart, a really nice guy—and that wasn't her type at all. So they became very good friends, and remained that way.
There were stars galore roaming the sets and visiting our house in those early days. Mom was working on movies with Dean Martin, Audrey Hepburn, Cliff Robertson, Laurence Harvey; her friends included Danny Kaye and the Rat Pack regulars. For me, though, there was one young star who stood out from the rest.
He was young, impossibly handsome, outrageously charming, and possessed of an ineffable aura. When he walked into the room, it was like getting hit with a charge of electricity.
"Say hello to Warren!" said Mom.
Warren was my uncle. He was three years younger than Mom, and he had followed her to Hollywood. Both of them had fled from the suffocation of their Virginia home. My grandparents were devout Baptists, and devout alcoholics—not really a workable combination. I never heard much of the details of Mom's childhood—she didn't talk about it—but I know there was a lot of unhappiness and emotional pain there, and she couldn't wait to get out.
I assume Uncle Warren felt the same way. I'd met him before of course, on earlier trips home. He'd been trying to get a foothold in show business for years, appearing on TV shows such as _Dobie Gillis_ and _Playhouse 90_. So far, breakout success had eluded him.
But now was his time. There was something new in his manner, a self-awareness, a recognition of his own impending stardom, that made him irresistibly attractive and yet warned all to keep their distance. _Splendor in the Grass_ , his first film, was about to hit the screens, and there was a healthy buzz about his performance. Even if there hadn't been, Warren would have created his own buzz. The air vibrated around him.
In many ways, Warren was still a kid, but curiously he didn't have any special knack for relating to children. They didn't seem to interest him at all, until they got to be twenty or so. I never saw him without a beautiful woman at his side, or two or three. He accepted their presence as his due, wearing them on his arm as casually and inevitably as a folded sports jacket on a warm day.
I admit, as a little girl I had a big crush on him. Who didn't? I never thought of him as my uncle, or as someone related to me in any way. He was a separate force, a seductive charmer who could put anyone under his spell with a glance, and just as quickly and ruthlessly cut that person adrift.
I felt the pull of that magnetism even as a child, but I didn't rush to him and throw my arms around his strong legs. Even though he flashed his killer smile at me and his eyes sparkled with promise, there was something about him that said, "Don't come too close." So I didn't.
**• • •**
**IN** 1961 they were holding auditions for the movie version of _To Kill a Mockingbird_. The book by Harper Lee had been a huge bestseller, and everyone was expecting the movie version to be an Oscar contender at year's end. So every actor in Hollywood wanted in. Gregory Peck was chosen for the role of Atticus Finch early on—there really was no other choice—but other juicy roles were still available, including the two children through whose point of view the story was told.
Out of the blue, Mom decided that I should audition for the role of Scout, Atticus Finch's daughter. I don't know where she got this idea; it's not like I was campaigning to be a child actress. While I had no shyness about performing in front of people—there were plenty of staged photo shoots with me and Mom attesting to the fact—it just never occurred to me that I might become an actor.
Mom had conceived a vision, though, and once she shifted into producing gear, there was no way she wasn't going to make it happen. She set up an audition right there at her L.A. home, got hold of a copy of the script a few days before, and we went over the entire story, moment by moment. Mom coached me in doing a Southern accent—"but not _too_ Southern," she cautioned—and she created the scene for me in detail, giving me hints about motivation, focus, presence, and so on. It was my first master class in acting, and I tried to absorb what I could. Then she added, "Just relax."
When the time came, she called me into the living room, which was filled with people from the production, including the director, Robert Mulligan; the producer, Alan J. Pakula; and the leading man, Gregory Peck. Now, if I'd known what a big star Mr. Peck already was, and what an iconic figure he was about to become on the strength of this movie, I probably would have quickly thrown up and rushed out of the room. But to me he was just a nice tall man with a warm smile. I didn't feel a bit nervous in his august company. I was just going to do a little pretending.
Mom handed me the script, and I found my mark in the middle of the living room and got into character. "Hey, Mr. Cunningham!" I said, reading from the script, making sure not to hold it in front of my face as Mom warned. "Don't you remember me, Mr. Cunningham? I'm Jean Louise Finch. You brought us some hickory nuts one early morning, remember?"
It went well. Even at six years old, I could tell that I'd won over the crowd. Everyone was full of compliments. I can still see Gregory Peck smiling at me paternally, his eyes all crinkled up. Mom was beaming, in her underplayed way. "She's good, isn't she? My daughter."
There was a lot of animated good feeling as the showbiz people filed out, and I was pretty excited about my prospects. "I did it, Mommy, didn't I? When are we making the film? Am I gonna get paid a lot of money? Will I have to leave school?"
Mom scoffed lightly, and then she assumed what she must have imagined was a gentle tone. "You know, Sachi, you're not gonna get the part."
My high spirits took a quick plunge. "I'm not?"
"Of course not. First of all, you're too young for the part. Second of all, who do you think you are, Shirley Temple? This was just for the experience. I wanted you to learn how to handle rejection."
Oh.
As it turns out, I didn't get to process this valuable lesson right away, because, against all odds and my mother's casual pessimism, they actually offered me the part, but Eguchi-san, who'd come along on the trip, protested the whole idea; she contended that it would be too traumatizing for me—a mere child, Japanese-raised, meek and humble as I was—to be suddenly thrust into the glare and the insanity of Hollywood. It could scar me for life. My mother put up a token argument, but in the end she reluctantly agreed with Eguchi-san.
So I stepped aside. The role moved on to Mary Badham, who was wonderful in the film, and it's now hard to imagine anyone else having played the part.
The summer of 1961 over, Eguchi-san and I went back to Japan, but this time with an added bonus: Mom was coming with us. She was doing a film called _My Geisha_ , shooting on location in Tokyo. The producer of the film was none other than my own dad, Steve Parker. This was his big bid to be a player in Hollywood. The plot was a typically flimsy 1960s comedy, with my mom playing an American actress who pretends to be a geisha so that she can land a role in a movie being directed by her husband (a vague echo of my parents' own situation).
I was thrilled to have Mom with me in Japan. She stayed with us at our Tokyo home throughout the shoot (while Miki repaired discreetly to the separate house that Dad had bought for her), and it was so exciting to come home from school every day and find her waiting for me. I had been accustomed to spending every day after school horseback riding—my great passion—but I tossed it over happily for the chance of having that much more time with her.
Mom was more thrilled, I think, to be with Yves Montand, her costar, with whom she was having an affair during the shoot. Montand was married at the time, to the imposing Simone Signoret, but it must have seemed even more imposing that Dad was sitting right there in the producer's chair. (Although I'm not sure how much time Dad logged on the actual set, or if he paid even cursory attention to the production. Dad was a wheeler-dealer; and once those wheels and deals were set in motion, he was on to the next challenge.)
Yves Montand had just ended an affair with Marilyn Monroe, and apparently had been an utter cad about it—when she waited outside his airplane in a limousine with champagne, he nonchalantly ignored her. Mom, for all her tough-minded independence, always seemed to find this type of selfish bounder irresistible.
She and Montand bonded in Tokyo, she says in one of her books, because of their both being stranded in an exotic, unfamiliar world—although he had earlier toured Japan with his one-man show, and Mom had been over to Tokyo several times with Dad.
Near the end of the shoot, she discovered that Montand had actually romanced her in response to a wager with my dad. In true Gallic spirit, he'd bet Dad that he could seduce Mom before the shoot was over—and he'd won handily. In a sense, so did Dad; with Mom occupied with Montand, he was free to conduct his own romantic affairs without worry. It was all very French.
It's interesting in retrospect to view the film's opening credits: "Shirley MacLaine—Yves Montand—in Steve Parker's _My Geisha_." Dad had really earned his producer credit: he not only produced the film, but he also produced the affair. (How my father, who had never produced a film before, managed to wangle an above-the-title credit, as if he were David O. Selznick, I'll never know, but it doesn't surprise me in the least.)
Eventually the movie ended, the affair ended, and Mom went home—to L.A. It caught me by surprise, because I had forgotten that her home wasn't my home. I'd really bonded with her in those few months. She was living with us at the house, and it was a real treat to see Mom and Dad together. Watching them interacting, smiling at each other, holding hands, I got a sense of what normal family life could be like. Once, I walked in on them in the bedroom to find them spooning in bed. I quickly backed out to give them their privacy, but it gave me a warming glimpse into the sweetness of their intimacy. (I didn't know she was enjoying the same kind of intimacy with Mr. Montand, of course.)
Mom found great peace, I think, in Japan. She used to stare at the koi pond in the backyard for what seemed like hours. I loved watching her be so calm. I can remember Dad cooking, and taking great joy in preparing a meal for her, and then all of us sitting down together. It was perhaps the last time we were an actual family.
I loved those days, and having Mom with me. I thought she'd be there forever.
After she left, I returned to my old routine: school and horseback riding and Eguchi-san during the day, lemons and loneliness at night.
On some special occasions, Dad and Miki would take me out to a fancy nightclub. This afforded me a rare glimpse into the sophisticated world of "adults." Dad and Miki would dress to the nines, I would get dolled up in my Sunday best, and we'd hail a cab and sweep up to the nightclub entrance. The doorman would open the car door for us, and we'd strut up the carpet and into another world.
The nightclubs of Tokyo in the 1960s were not much different from their American models. The music was Western and jazzy, the women beautiful, the atmosphere smoky. Cocktails flowed, jewelry flashed, dancers whirled around the floor.
When Dad stepped through the front door, everything seemed to stop. He commanded the room. He was that guy in the movies who, when he suddenly made an entrance, the maître'd would imperiously snap his fingers, the waiters would rush to seat him, and a special table would be placed ringside just for him. Everybody knew Steve Parker. He'd thread through the tables, cigarette in hand, waving and glad-handing. Dad seemed to be connected everywhere; he was the Man.
We'd take our seats, drinks would be ordered, and the entertainment would begin. There were always famous performers appearing at these nightclubs, and they all knew my father. The girl singers would often flirt with him, and sit on his lap between numbers. Miki didn't seem to mind; it was all in the spirit of fun and good times, and she loved being part of it.
One night Dad took me alone to a different kind of nightclub, off the main drag. It was a small upstairs space like a living room, where everyone sat in armchairs and the atmosphere was cozy and warm. I remember the food was wonderful—little appetizers, garlic cloves pan-fried in sesame oil, super-light tempura, salted squid in squid guts. I also remember that there were no female customers, only men. And the waiters, also male, were completely naked. This afforded them the opportunity to perform a unique service; whenever Dad ordered a scotch and soda, the waiter would hold the drink crotch-high and stir it gently with his penis before serving. I was fascinated. Did this improve the flavor? I wondered. The waiters would perform this function only with cold drinks, of course; the hot sake might have caused some discomfort. And exceptions were made: I remember when the waiter brought me a Shirley Temple, my dad put up a restraining hand—no dick for my daughter, please. I appreciated his delicacy.
This foray into the exotic world of private clubs was kind of fun, but for the most part the nightclub scene left me terminally bored. Oh, it was exciting at first, just being in the middle of all that grown-up energy, but to have to sit there for hours, watching them drink, was crushingly wearisome. Often these were school nights, and I longed to be in bed so I wouldn't drift off to sleep in class the next day. I remember that everyone in these nightclubs smoked cigarettes—everyone; the thick smoke hanging in the air was overwhelming, and it burned my eyes. I would beg for some coffee ice cream, just so I could stay awake.
Still, I knew there was supposed to be something cool about all this. It was cool that I was just seven or eight years old, cool that Dad was exposing me to this elegant lifestyle, cool that I was taking it all in with the aplomb of a seasoned jet-setter. When all these strange women showered my father with kisses and wiggled their bottoms in his lap and pressed their surging breasts against his cheek, I was supposed to be not only okay with this, but enthusiastic and delighted.
Eguchi-san was not so cool. She was outraged. It was scandalous! Irresponsible! Dragging a young schoolgirl out to these sinful fleshpots on a school night! "I would expect as much from that Miki; she is just a teahouse servant. But what can your father be thinking?"
That was, in a way, the essential question. What _was_ my father thinking?
**• • •**
**I** loved my dad, and I lived in terror of him. He was mercurial and moody, and his constant drinking made him unpredictable. Two incidents stick out in my mind from those preadolescent days. One was the Cream Puff Incident.
Cream Puffs were something of a delicacy in Japan. As with Wonder bread, there was no fresh dairy of any kind in the country. No butter, no cheese. We had to drink powdered milk.
Obviously, in a world without cream, you're not going to find cream puffs. At least not cheaply. Yet Dad loved them, and he usually got what he wanted, so he would often buy fancy gourmet cream puffs at the market.
Or he would make them himself. One afternoon, just for his own pleasure, he whipped up a batch of delicious cream puffs in his kitchen. He took round pastry shells, stuffed them with custard, and topped each with a big dollop of whipped cream. It was a very extravagant whim.
The finished cream puffs were sitting on the kitchen table when I wandered in later. There were about eight of them on the plate, and they looked absolutely delicious. I knew there was no dinner party scheduled for that night; no one was coming over to enjoy these scrumptious treats. That being the case, why not eat one? Or two?
At the most, I had three. There were still about five left.
About an hour or so later, as I sat in the living room, Dad entered. He stood in the doorway a moment. I didn't look up, but I knew he was there—and I knew why he had come.
After a studied pause, he said thoughtfully, "I wonder what happened to the cream puffs."
I stiffened. I knew enough not to answer him.
"I know there were more than five," he contended. "What could have happened to the rest of them?"
While it seemed that he was being facetious, there was nothing playful in his voice. There was rather an undertone of bottled rage, as though he were a Mob boss brandishing a baseball bat and wondering where his split of the cash was.
He waited. I waited. He sat down and stared at me. I knew he was accusing me, but he wasn't going to come out and say it. He wanted to make a game out of it.
"They couldn't have just disappeared," he went on. Then he gasped, as if a thought had just struck him: "Maybe someone came into the house and took them!" He let that hang in the air a moment. "Do you think that's what happened?"
Seeing an opening, I jumped at it. "Yes, Daddy, I think someone may have done that."
He leaned forward eagerly. "Did you see him?"
"Y-yes."
"Did he come in the front door or the back door?"
I sensed that I was stepping into a trap, so I shrugged weakly and said nothing.
Dad was thoughtful. "Maybe it was a neighbor. Did it look like a neighbor?"
"Maybe."
Dad frowned. "But why would a neighbor come into the house without knocking and take three cream puffs? No, that can't be right." He pondered anew. "Did I miscount? Maybe I made only five cream puffs to begin with."
Again I leaped at the suggestion. "I think you're right, Daddy. There _were_ only five."
He pursed his lips, and furrowed his brow. "But why would I make only five cream puffs? No, I'm pretty sure I made eight. I'm pretty sure." He stared at me again.
This went on and on. He kept throwing out possibilities and suggestions, and I kept lying to validate them, and then he would pull them back from consideration. He was toying with me, manipulating me, and I knew it, and it was driving me crazy!
Finally I cracked. "It was _me_ , Daddy! _I_ did it! _I_ ate the cream puffs!" I broke down in tears as I blubbered out my confession.
Now, you might think that, at this point, Dad's face would have broken into a warm smile and he would have patted me on the head and said, "I know, sweetheart. I know you ate the cream puffs. I just wanted you to tell the truth."
But he didn't. His stare became only more fixed. "You did it?" he said in a voice of quiet outrage. " _You_ did it? That's _horrible_. Why would you do something like that?"
I had no answer. "I don't know. I'm sorry."
"You're sorry? If you were sorry, you wouldn't have eaten the cream puffs in the first place, would you? They weren't your cream puffs, were they? Were they?"
"No."
"Then why did you eat them? Why would you do something as awful as that? And then lie to me about it? Are you a _liar_ , is that what you are?"
He went on and on in this merciless vein, grinding me down relentlessly. Then he closed his eyes, shook his head, ran his tongue along the rim of his upper lip, and sighed sadly, as if his disappointment in me were bottomless.
As for me, I couldn't stop crying. Yet, all the while, I kept thinking, _They're only cream puffs!_
From that point on, though, I was terrified of eating anything in that house.
This episode of bullying manipulation, trivial though it seems, served as a necessary prelude to the Retainer Incident.
When I was nine years old, I had a retainer instead of braces on my teeth. One day, I took out the retainer—I don't remember why—and I lost it. This is the kind of thing that happens naturally to nine-year-olds.
Dad didn't see it that way.
"Do you know how much those things _cost_?" he thundered. "How could you lose something so _important_? Why couldn't you just keep it in your mouth? What's _wrong_ with you? _Idiot!_ "
It was only with the greatest reluctance that he bought me a new retainer; and he made it crystal clear that I'd better not lose this one. Ever.
So I promised myself I wouldn't.
A few months later, Eguchi-san and I were on a pilgrimage to her family home in the country. She wanted to visit her private shrine.
It was fairly typical for middle-class families in Japan to have shrines on their grounds. They're small and simple, but in their own way they are quite ornate, all red and gold. Eguchi-san had brought me to her family's shrine many times before. Her not-so-hidden agenda was someday to see me embrace the Shinto religion.
I was not brought up in any religion (which might strike one as odd, considering my mom's very public engagement with all things spiritual). Not Christianity, Buddhism, pantheism, not even atheism. I was a blank slate. Eguchi-san saw in me a golden opportunity for conversion; she would have given anything to lure me into the fold.
So she regularly brought me to the shrine, hoping for my eventual enlightenment. She must have thought I'd be filled with a sacred awe by those huge fox god statues flanking the shrine door. Actually, they scared the hell out of me.
Her family home and its shrine were about five miles away, and since we were walking, we had to get an early start. We were fairly close by noon, but as we were getting peckish, we stopped at a sushi bar for lunch.
I didn't want to get all that chewy fish stuck in my new retainer, so I carefully took it out, wrapped it up in tissue for safekeeping, and put it right by my side. I never let it out of my sight.
Well, I guess I _did_ let it out of my sight, because when I finished lunch, I turned around to find it gone.
Gone! The retainer, the tissue paper, everything!
I was stunned, and my panic level went from zero to sixty in a nanosecond. Suddenly the image of my father, red-faced with rage, loomed before me. "You lost your retainer? AGAIN?"
I was frantic. Where could it be? Was it on the floor? Had I put it in my pocket? I raced around the restaurant, checking under every plate and napkin. Could someone have walked off with it? But who would steal another person's retainer? No, it had to be here somewhere. It had to!
Coming up empty in the dining room, I rushed into the kitchen itself, and there I spotted a large barrel of garbage, filled to the brim with fish heads, fish bones, and various chunks of fish innards—it was really disgusting. Instantly I knew that my retainer was in there somewhere. Without hesitation I dove into the teeming mess, digging my hands through the slime, pulling up bones and cans and anything that felt remotely retainer-like.
No luck. I emerged empty-handed, and cast about desperately. Were there any more garbage cans I could hurl myself into?
Eguchi-san tugged at my sleeve. "Sachiko-san, we must go. It's getting late."
"But my retainer!"
"I know, but we have to get to the shrine and then make our way home before it gets dark. Remember, if you chase two hares..."
I knew, I knew—I would never catch one, but I needed to catch the hare with the straight teeth.
Still, I had to agree that the situation seemed hopeless. Desolate, and smelling like a herring boat, I walked out of the sushi bar with Eguchi-san and continued on the road to her house. Whatever spiritual uplift was awaiting me there was irrelevant now, because by tomorrow I would most certainly be dead.
Eguchi-san, with her shining faith, would hear none of this. She always carried a message of hope. "Don't worry, Sachiko-san. If you pray to the fox gods, maybe they will return your retainer to you."
"Really? You think so?"
Eguchi-san nodded gravely. "The gods are very powerful. Nothing is beyond them. _But_ —you must pray very hard."
Eguchi-san held out her smooth sandalwood beads. I was ready to try anything, so I took them and I prayed and prayed, over and over, and I silently made one of those desperate vows that you believe with your whole heart and yet privately assume will never be called to account: "Please, kindly fox gods, find my retainer, and I will worship you forever!"
The praying continued all the way to Eguchi-san's home. By the time we arrived I was so nervous I had to run into her house and pee—which took quite a while, since I was wearing seven layers of brown underwear.
Eguchi-san was waiting outside when I returned, and she handed me an offering for the fox gods: a small bag of dry rice. I took it, feeling that it was an awfully stingy offering in exchange for the desired miracle, but Eguchi-san understood the ways of the gods much better than I did.
To get to the shrine, I walked through a great wooden archway, down a path of bursting chrysanthemums, and past those two terrifying fox god statues. I held up the bag of rice meekly as I passed, murmuring, "This is for you. I'll just put it inside."
I walked into the small red-and-gold building, and approached the shrine. With trembling hands, I opened the small doors of the shrine to present my offering....
I gasped in amazement. For there, in the shrine, on a red silk pillow, was— _my retainer_!
It had returned! The miracle had happened!
I grabbed the retainer, dropped the bag of rice in its place, and rushed from the shrine, back up the chrysanthemum path, and into the arms of Eguchi-san. "Look! Look! My retainer!" I screamed.
Eguchi-san smiled knowingly. She was not surprised at all. "The fox gods have smiled upon you."
They had! I looked back down the path at the enormous statues. Though their granite visages were still grim and forbidding, they now seemed to wear just the trace of a grin. I remembered my vow, and I had every intention of fulfilling it. From now on, I would be a Shintoist!
Now, if I'd had any common sense I would have kept this whole episode from my father, because no matter how you sliced it, it still revolved around the indisputable fact that I'd lost my retainer again. Yet, I'd been so thrilled by this spiritual intervention that I had to tell him everything. Everything!
Dad listened quietly to my story, took a sip of his scotch, and stared thoughtfully. "So now wait a minute," he said finally, trying to wrap his head around this evident miracle. "You lost your retainer at a sushi bar?"
I nodded vigorously, incriminating myself with delight. "I wrapped it up in tissue, and when I turned around, it was gone. I looked on the floor, I looked in the garbage—everywhere!"
"And then you found it, a mile away, in a shrine?"
"Yes! Because I prayed, and the fox gods found it for me, just as Eguchi-san said they would!"
"Eguchi-san said that?"
"Yes!" My pride in that old woman's marvelous transformative faith knew no bounds, and I wanted the world to know it.
Dad just nodded slowly, very slowly.
I don't know what happened next, but Eguchi-san never came back to work after that.
# **Chapter 3**
# On Location
So now I was alone—or living with Dad and Miki, which amounted to the same thing.
Miki didn't actually live with us. As I said, she had her own house, and that's where she usually slept each night. Or sometimes she might stay in the tatami room in our house, a small guest room that lent a suggestion of propriety. I guess it was important for Dad to preserve the illusion that he was not having intimate relations with this woman who was not his wife. Accordingly, there were never any shows of affection in front of me—no kissing, hugging, nothing. They went to great pains to hide their relationship, although it was clear even to me that she wasn't following him into the bedroom every night just to sing him a lullaby.
Whatever transpired behind the door, she never stayed. Every morning, she'd be gone, having repaired to the tatami room or stolen into the night, and Dad would emerge from his bedroom alone. Invariably, perhaps in testament to the glories of the evening before, he would be naked.
Totally naked. On a cold morning, he might pull on a kimono, but it would be uncinched and flapping wide open. That's what I generally woke up to: Dad and his naked body. I would watch with fascination as he casually crossed the room, his penis dangling and bobbing from side to side.
Admittedly transfixed by this foreign protuberance, I still wanted it covered up. I couldn't say anything, though. I had to pretend that it was cool. That it was part of the routine. I had to be cool with everything.
Maybe I should have just stayed in my bedroom, but the fact is, all nakedness aside, the early mornings were the best times to be with Dad. He'd start up a fire in the fireplace, and then make his tea. He used real bone china, and the best English breakfast tea, specially ordered from Darjeeling, India. I remember him throwing the boiling water on the tea leaves (just like with cowboy coffee), and then settling down in his favorite easy chair by the fire.
He was quiet and meditative in these times, and that, I felt, was the real him. He hadn't started drinking yet; it was the only time of the day when he was completely sober. That's why I made sure I got up early to be with him. Because once he'd had his first glass of Dom Perignon champagne for breakfast, that was it. The charming, moody playboy Steve Parker would take over, and the dad I cherished would be gone.
Sometimes I would grow so lonely sitting in that empty house at night that I'd call Yuki's parents and ask if I could stay with them for the evening. Yuki's mom would always say yes, and I'd take a cab over.
Why I chose that avenue of escape, I can't say. I still didn't really like Yuki all that much; over the years, our relationship had not moved off its initial settings. She was still a competitive, assertive alpha-female, and I was still a shy doormat. Her mom was so nice and friendly, though; she would always welcome me. She'd pay for the cab, make me a snack, and send me off to school with Yuki in the morning. It was like being part of a family.
**• • •**
**I** made my usual trips to Los Angeles every summer, and made the same adjustments to a wildly different culture. We would shuttle back and forth between Mom's big house in Encino and the bungalow in Malibu.
Mom was always entertaining, and plenty of interesting houseguests passed through our portals. One was a man with a dark mustache and an interest in the supernatural; his name was William Peter Blatty. He had written the screenplay for one of Mom's lesser films, a college movie called _John Goldfarb, Please Come Home_ , and he was a neighbor down the street in Encino.
Mom and Mr. Blatty were kindred spirits, and he became a familiar face around the house. Much of the time, I found them engrossed in animated discussions of ghosts, paranormal experiences, the afterlife, and so forth. They liked to get out the Ouija board and ask questions of the infinite, and once or twice they even conducted a séance in the house.
Blatty was Catholic, and a big believer in sin, evil, and the constant presence of the devil, which Mom would dismiss with her competing theories of karma and self-actualization. So he decided to write a book that would prove to her, by the very power of its narrative, the existence of evil in our everyday world. Several years later, when _The Exorcist_ was published and became a national phenomenon, he conceded that he'd taken his inspiration for the actress mother, Chris MacNeil, from Mom. He even offered her the role of Chris in the movie, but she turned it down because her agent told her the script (which won an Oscar) was no good.
The unfortunate corollary, of course, was that if the mother character was based on Mom, the daughter, Regan, must have been based on me. Maybe Blatty did use me as a reference when he was sketching the basic outlines of the character. But that was as far as it went.
I don't recall ever walking like a crab or spitting vomit from my revolving head, or doing anything untoward with a crucifix.
I will admit, though, that the photo of the little girl on the first edition of the Harper and Row hardcover book looked an awful lot like me. Mom was sure it _was_ me, and she told Jason Miller, the actor who played Father Karras, as much. When this was conveyed to Blatty, he denied it. "Shirley, how could I have gotten Sachiko's picture on the cover of the book?" Mom had the answer: "You could have broken into the house and stolen it." He insisted he had nothing to do with choosing the photo on the cover, which makes its unsettling resemblance to me even creepier.
**• • •**
**I** enjoyed Malibu, not only for the beaches but for the hills and scrublands behind the house. I used to love walking back there by myself. Nowadays there might be the reasonable fear of being carried off by coyotes or cultists, but in those days it was normal for kids to wander off by themselves and explore nature.
One time, as I walked the wide carriage trail, I found a side path that cut through the brush and wound up on the side of a mountain. Ever curious, I started up the meandering path, an Alice exploring a reverse rabbit hole. As I reached the top, I came upon a plateau with a stunning vista and, of all things, a small working farm.
It was an old shack in Spanish adobe style, probably built by the owners themselves. There was a huge vegetable garden, and a corral with several magnificent palomino horses. As I walked across the field toward the shack, a stout Mexican woman in her mid-thirties emerged from the garden carrying a basket laden with corn. She had a marvelous sun-lit smile, and she greeted me in Spanish.
I couldn't understand a word she spoke, but her voice was musical and friendly; she took me by the hand and led me into the house, and before I knew it, she was feeding me tortillas and beans. Just like that, I had a new family.
After I ate, she took me on a tour of her garden, and I remember there was corn, lots of corn. You could eat it right off the cob, uncooked, sweet and delicious. She would pick strawberries and grape tomatoes right from the vine and pop them in my mouth, warm from the sun.
I don't remember her name—she was just Ricardo's mom to me. Ricardo, her teenage son, was in charge of the palominos. He was always at the corral, grooming the horses, training them, riding them bareback. About sixteen years old, he was friendly like his mom, and always smiling. She didn't speak English, but he did.
That very first day, he caught me staring in wonder at those beautiful horses. "Would you like to ride?" he asked genially. I certainly would!
Ricardo carefully helped me up onto one of the horses. There was no saddle or reins. I had to ride bareback like him. "Don't be afraid," he cautioned gently. "Just hold on to the mane and I'll walk you around the corral." He didn't know that I'd been riding horses for years. I grabbed the palomino's mane and trotted confidently around the corral, as Ricardo watched in surprise.
Moments later we were both riding bareback down the path, right across Malibu Road and through the access route to the beach. We took off across the sand and raced along the shore. I just held on to that mane and galloped down the beach, kicking up great sprays of water. It was exhilarating.
I went back often to the farm on the hill. I'm sure no one on Malibu Road—the stars, the celebrities—even knew it was there. For me, however, it was a second home, a warm, loving place far preferable to the pretentious social scene below. Over the next couple of summers, Ricardo and I would spend many happy hours riding along the beach on our palominos, splashing down the shoreline. It was idyllic and entrancing, and completely platonic. I was the little girl in love with nature, Ricardo was like my watchful big brother, and we were romping through a wholesome tableau straight out of a Disney movie. I was completely happy.
**• • •**
**SOMETIMES** my visits with Mom would dovetail with one of her location shoots, and she would bring me along. Thus, in the winter of 1966, on my Christmas break, I found myself flying to Paris, where Mom was filming _Woman Times Seven_. For some reason, Yuki came along with me. I'm not sure why; as I said, we weren't really friends, but somehow it had become accepted that we would spend our free time together. Why Mom would want to bring her to Paris, though, I couldn't say. I assume she got to know her a little in Tokyo when she was filming _My Geisha_ , and Dad must have talked about her quite a lot. Still—kind of a mystery.
Anyway, it turned out that Yuki and I had a fabulous time. We stayed at Mom's lavish town house on the glamorous Avenue Foch, and we were virtually unsupervised. We had the run of the house—and we ran. I remember the two of us jumping up and down on the fancy beds, bouncing off the walls, raiding the kitchen at night, and gorging ourselves on crème brûlée and French pastries. Yuki's aggressive, take-what-you-want attitude was starting to appeal to me.
We explored Paris, too, sometimes with guides, sometimes on our own. I remember watching the extravagant floor show at the notorious cabaret Lido. It was clearly a show geared for adults; I don't know what we were doing there, two unworldly clueless ten-year-olds, but it was certainly an experience. Most memorable were the chorus lines of statuesque half-naked women parading across the stage with tassels on their breasts and huge sprays of feathers sprouting from their heads and their behinds. Yuki and I giggled through the whole show.
**• • •**
**_WOMAN_** _Times Seven_ was an episodic farce in which Mom played seven different roles in short vignettes illustrating the tantalizing enigma that is Woman. It had one of those all-star international casts, and was directed by the great neorealist filmmaker Vittorio De Sica. He was a great, warm, avuncular presence with aging matinee-idol looks, who nurtured my mom, flattered her, and lavished her with attentive compliments and adoring glances.
He was always over at our town house, and I would watch from the living room as they cooked together in the kitchen, laughing, teasing, bestowing light grazing touches as they passed each other. De Sica would burst into an Italian street song, and then he would suddenly take my mother in his arms and hug her, and Mom would gaze up at him worshipfully like a little girl.
I thought he was wonderful—his huge beaming smile was so full of love for everyone and everything—and I hated leaving the shoot and going back to Tokyo, because I feared I would never see him again. And I didn't.
**• • •**
**THE** following year, Mom was in England working on _The Bliss of Mrs. Blossom_ , and again I found myself on location. Her costar was Richard Attenborough (with whom she did _not_ have an affair, I can pretty safely say). Nowadays best known as the director of _Gandhi_ and the dotty entrepreneur in _Jurassic Park_ , Mr. Attenborough back then was a marvelous actor who had that amazing English ability to do anything: character roles and leading men, comedy and drama, war films and musicals. He was also a lovely gentleman. He was very sweet to me when I was visiting the set. He had a special nickname for me: Poppy. I don't know where it came from, but whenever I ran into him in the years that followed, he would immediately greet me as Poppy. He still does.
The warmth and genuine sweetness of De Sica and Attenborough reminded me of that side of my dad, which I had seen less frequently over the years. The days of ice fishing and tall tales were growing more and more distant in my memory. When I got back to Japan, I felt a powerful need to reconnect with him. I wanted to forge a stronger bond.
That may have explained why I was a little overeager to see him one morning as I waited for him to emerge from his bedroom. He'd had an especially tiring night out, and was sleeping in. I waited anxiously in the living room for him to come out and start the fire and make his tea—"Come on, Dad. Where are you?" I muttered to myself—and finally I couldn't wait any longer. I ran down the hall and burst into his bedroom without knocking. There he was, in bed—with Miki.
She had forgotten to wake herself up in the middle of the night and sneak away. Now she had committed the unforgivable sin; she'd allowed me to see her in my father's bed.
There was alarm on Miki's face when she sat up. "What are you doing in here?" she hissed at me angrily, as my father stirred beside her. She waved me off, hoping I would retreat before Dad spotted me, but it was too late. A fleeting look of surprise and embarrassment crossed his face when he saw me and got his bearings, and then a dark, frightening anger settled over him. He grew furious, but not with me. He turned his glare on Miki.
Miki cringed instinctively, and then tried to deflect his rage. "She should have knocked," she said feebly, but in an instant Dad had seized her by the arm and tossed her out of the bed. She screamed as she hit the floor. Then, before she could get to her feet, he was upon her. He slapped her face hard. I was stunned by the shocking impact of his hand against her cheek. As she reeled back against the wall, he followed her, slapping her again and shouting terrible things at her. He pounded her on the back until she slipped to the floor, and then, as she lay slumped in a heap, he kicked her over and over.
It was an awful thing to watch—and the fact that it was Miki, whom I truly hated, being beaten so savagely made it seem even worse. To know that my father could summon such explosive violence was devastating to contemplate. (When would it be my turn?) Why was he doing it now? To protect his image in front of his daughter? To demonstrate that he was a good man who had been seduced by a worthless slut?
I couldn't watch anymore, and fled from the room. The beating went on. I could hear Miki's screams and the dull thuds of my father's blows. Then there was silence, and only Miki's low, muffled sobbing. Moments later, Dad was pulling Miki through the living room, her clothes hastily thrown on. He flung open the door, pushed her out, and slammed the door shut. Then he returned to his bedroom.
Not another word was said. I knew enough not to mention it. I quietly got ready for school and left.
The real value of violence, as all the great tyrants know, lies not in punishing the individual, but in delivering an object lesson of fear and subjugation to everyone else. Very soon their relationship would resume its normal equanimity, as if nothing had happened.
For me, it was devastating. Suddenly I apprehended the true depth of my father's unstable fury, and it terrified me. I would do anything not to have that rage turned on me. So I became, if anything, more docile and eager to please. Walking on eggs became my default mode.
It was around this time that Mike Parsons died. Mike was a friend of my dad's in Japan, and also a business associate. (For Dad, friendship and business always seemed to go hand in hand.) They were big drinking buddies.
Dad told me that Mike was quite the womanizer, but I knew that Mike was gay. I wasn't supposed to know this, but once, at one of Dad's parties at the Shibuya house, I wandered into one of the bedrooms, and there was Mike kissing another man. Really kissing him. No doubt about it.
This was never talked about, because in those days homosexuality was still something to be shunned and despised. So Mike led a secret life, too.
At that time, Dad had a business office in Hong Kong. I don't know what kind of business it was, and I don't know if Mike was involved in it.
One day, Mike Parsons was found dead in Dad's Hong Kong office. Shot to death.
I remember the authorities questioning Dad, and Dad being a little anxious about it. I was filled with a bewildered protectiveness: "Why are they asking Dad these questions? He doesn't know anything about it."
Did he? Nothing came of it, no charges were filed, and Mike Parsons was never mentioned again, but in the back of my head, I wondered why intrigue and violence seemed to touch upon Dad's life from every angle.
**• • •**
**AS** I approached my teenage years a natural rebelliousness was rising in me. Oh, I was a model of good-girl behavior in general, but every now and then a mood came over me, and I would decide to push the boundaries.
In a very small way, mind you. I didn't smoke cigarettes or hang out with boys or down shots of sake.
No, I started reading.
Dad didn't want me to read books. He felt it was a waste of time. I was an idiot, anyway; what was I going to learn? So I read surreptitiously, whenever he wasn't around (which was often). I'd steal into his study and pick out a book, leaf through it, see if it grabbed me. Dad himself was an omnivorous reader, so there was plenty to choose from.
I was about twelve when, one day, I came upon a copy of Ayn Rand's _We the Living_. It was the story of a young girl in postrevolutionary Russia who refused to let the repressive state tame her independent spirit. It hooked me right away. I understood this girl. She was struggling and straining against her bonds, much like me. I kept reading, and couldn't put the book down. I nestled into my dad's armchair, legs curled up under me, and got lost in another world.
Hours later, I was still reading when I heard my father come home. My first instinct was to jump out of the chair and shove the book back into the shelf before he saw me, but I didn't. I was utterly absorbed in the world of the novel, and I didn't want to leave it. I kept reading.
Behind me I heard the knob turn, and then the door open. I knew it was Dad. He stood in the doorway, motionless. He didn't say anything, but I could feel his disapproval. I was breaking the rules.
It was too late to hide the fact, and I knew I would do well to rise meekly and acknowledge my folly, but some kind of defiance had taken hold of me. I wouldn't stop reading. I wouldn't even look up at him.
The door closed, with a reproving curtness. I heard Dad move into the room. I could peripherally see him taking off his coat, and methodically hanging it over a chair back. Then he sat down on the couch and stared at me. He would normally have settled into his armchair after a long day's work; but I had the armchair, and I wasn't moving. I kept reading.
Dad slowly took off his shoes, letting one and then the other clomp to the floor. He stared at me. I didn't look over, but I knew he was staring, waiting. It occurred to me that I should have been performing my ordinary duties: getting him a scotch and soda, fetching his newspaper, kneeling beside him and rubbing his feet. That's what a daughter was supposed to do.
But I was being a bad daughter, right in front of his eyes. I don't know why I had chosen today to push the issue, but it seemed elementally right. Kira, the girl in the novel, was fighting Communist oppression and soul-crushing poverty. I was inspired.
Of course, he had a hundred pounds on me, and thirty years of experience, and a whole culture of Japanese patriarchy behind him. My will was nothing compared to his. He knew that the longer he stared and said nothing, the more extreme the pressure on me—and that, eventually, I would crack.
But I didn't want to crack, the way I had over those damned cream puffs. It hadn't helped. So I stared doggedly at the book.
I was only pretending to read, by this point. I couldn't concentrate now. I could feel his anger collecting and rising to a boil, and sooner or later it was going to erupt on me full force. But when? In what form? Would he scream hateful things at me, or lock me in my room, or would he yank me from his chair and start beating me, mercilessly, as I'd seen him beat his lover?
I knew now that I'd made a mistake embarking on this foolish course, but there was no way to back out of it. I was stuck.
I waited. He waited. We were at an impasse, one that could end only disastrously.
Then Miki came into the room.
Miki's contempt for me had only grown since the bedroom incident. Not only had I been the catalyst for the beating, but I had witnessed her humiliation, and she despised me for that. Not for the last time, I was catching the fallout from being an innocent bystander.
Miki stood beside my father and glared at me, arms folded. Another staring contest. Finally she spoke: "Isn't it time you helped me with the flowers, young lady?"
I had to respond now, and I knew it would be suicidal to act snotty or defiant when my father was still coiled like a cobra. So I murmured, "Okay," and slowly put the book down. Defeated, I followed Miki out of the room.
I should have been grateful to her: she'd defused the situation and helped me avoid a lethal showdown. I doubt that was her intention—clearly she was trying to belittle me with her patronizing tone—but nevertheless, the end result was that I had escaped the apocalyptic wrath of my father, and lived to see another day.
I never picked up _We the Living_ again. I still don't know how it ends. It would be many, many years before I finished another book.
**• • •**
**WHY** wouldn't my father let me read? I couldn't fathom it. After all, he read all the time. He knew what books could do for you—expand your horizons, engage your intellect, accelerate your maturity. It was as though he wanted me to stay a little girl.
Yet, at the same time, he seemed to be pushing me toward adulthood.
"Are you wearing panties?" he would ask.
"Um—yes," I'd reply, embarrassed by the question.
"Take them off."
"Why?"
"It's good to air your vagina."
Dad encouraged me to go without underwear whenever possible. He thought it was far healthier to be open and exposed to fresh air than confined in restrictive suffocating cotton. Perhaps he was reacting to Eguchi-san's seven-layer panty overkill. I found both choices extreme, but Dad had established his own morning nudity as a precedent, and I couldn't argue with him.
He also liked me to open the buttons on my shirt when we were in public. I don't know why; he was always teasing me about being flat-chested—"Two fried eggs, coming up!" he'd say—so it's not as though I had something to show off. Besides, I was shy—after his taunts, who wouldn't be?—and I liked to keep my blouse buttoned to the top. He insisted I unbutton it down to my prospective cleavage, though. I'd acquiesce until he turned away, and then I'd hurriedly button up again.
I remember, when I was about twelve, going to see _Hair_ on Broadway. Dad was one of the many producers, and we had great seats, third row center. That evening, he instructed me not to wear underwear. I felt extremely uncomfortable about this—it was Broadway, after all, back in its glamour days, when everyone was dressed to the nines. Going without panties in such an august venue struck me as almost criminal.
There was so much nudity on the stage that my secret subversiveness seemed almost quaint. I had seen naked strangers before, at the Japanese baths—I'd been to the Lido, for God's sake—but to have so many of them so close to me, proudly displaying their wares...Well, I guess I was supposed to feel liberated, but it kind of grossed me out. Especially all the pubic hair. Yuck!
Dad was clearly enjoying it, and enjoying the fact that he could watch such explicit, far-out things with his own daughter. So hip! So cool!
I thought nothing of these things at the time—that he made me go without underwear, and exposed me to adult entertainments beyond my understanding. After all, it was the Age of Aquarius. The way he would slow-dance with me on the nightclub floor, cupping my behind with his hand and squeezing—it didn't mean anything, right? He was my dad.
When I was around seven or eight, Dad would sit beside me on the couch and say, "Let's play the tickling game." "No!" I'd scream, but it was too late—he had his fingers digging into me, and I was shrieking with laughter. I was so extremely ticklish that the slightest touch would set me off. I'd plead, "No! No!" Because I couldn't stand it. I hated it.
He'd think I was joking, because I was laughing so hard, and he'd keep going. I'd fall to the floor, hoping to get away, but I was too weak with laughter to move, and he'd follow me, and the tickling would continue. I didn't want to tell him to stop, because he was having so much fun, and I didn't want to hurt his feelings, but it was awful, awful...
He would stop, finally—and then it was a new game. A licking game.
He would start licking me. All over my body. My arms, my neck, my feet...
That felt good—but I didn't like it. I didn't like that it felt good.
Then there were many nights when he'd say, "You can stay with me tonight," and I'd get into his bed.
Then he would climb into bed beside me and spoon with me, snuggling up behind me and throwing his arms around me. It was so warm and cozy, except...What was that hard thing behind me, poking against my butt? Did it belong to Dad? Was it the dangly thing I saw every morning when he came out to make the fire? It seemed so mushy before. How did it get so sharp?
I found it annoying, and shifted away. But Dad pulled me back, and snuggled even closer—and that sharp thing pushed its way between my legs...
What happened then? Did it go any further? I don't remember. My mind has blocked it out. I believe that he would just hold me close and we would snuggle throughout the night. That's what I want to believe.
When he was finally snoring, I would sneak out of his bed and hurry back to my own room. I would sleep much better there, but I always made sure I rushed back to his bed in the morning before he woke up. I didn't want him to know that I'd left him during the night. It might have hurt his feelings.
# **Chapter 4**
# Charters Towers
My childhood in Japan came to an abrupt end in the fall of 1968. There was no drama involved: nobody died, no earthshaking scandal erupted. I had reached the awkward age of twelve, and in another Dickensian twist, I found myself suddenly transported to the other side of the world—Bexhill-on-Sea, England—to boarding school.
The funny thing was, it was my idea. At least I thought it was. Dad was often telling me how wonderful boarding school was. He made it seem so romantic and adventurous: "You get to have your own room, and you can do whatever you want on your time off, and you get to live in a whole new country by yourself...it's like being a grown-up." That really appealed to me. I wanted to show Dad that I was a big girl. That I was practically an adult. I was cool, not needy.
So when the time came, I suggested going to boarding school, but it was Dad who cleverly implanted it in my head, as if I were the Manchurian candidate. I suspect he was growing wary of my rebellious side, and a mouthy pain-in-the-ass teenager in the house (even a meek, mildly defiant one) was just going to cramp his freewheeling style.
Or perhaps Miki was behind it. She may have planned all along to have me sent away the moment I reached a disposable age.
In any case, I was now out of their hair, and any guilt my dad may have felt about neglecting me in the past could be dismissed. I wasn't his responsibility anymore.
The curious thing—curious to me even now—is that I didn't go off to boarding school alone. Yuki went with me—and Dad was paying for it.
The story I heard from Yuki was that my dad and Yuki's dad used to drink together at the Press Club in Tokyo. Between libations, Yuki's father had confided to Dad that one of his dreams was to send Yuki and her brother, Kenji, to boarding school in Europe. He had the money for it, safely tucked away in a Swiss bank account. Then he died of a swift-moving cancer, and he neglected to give the account code to the other members of the family beforehand. The money couldn't be retrieved; it may be sitting there still.
So there was no money for Yuki's education, but Dad remembered his friend's dream, and he took it upon himself to send Yuki to boarding school, in tribute to her father's memory. It's an uplifting tale, and it may even be true.
**• • •**
**MY** new school was named Charters Towers—or, in the accepted local pronunciation, "Chah-tuz Towz." It was a stuffy, dreary, very traditional English school, the kind of place where the teachers wore long black gowns and those funny hats that looked like popovers, and you had to walk single-file from class to class, as if you were doing a community theater production of _Oliver Twist_. The discipline was similarly Victorian; if your skirt was one inch higher than school regulation, they caned you.
If you've ever seen a British period film or _Masterpiece Theatre_ —one of the many versions of _Jane Eyre_ would suffice—you've seen Charters Towers. There were the same long, dark hallways; the communal tables in the dining rooms; the wide winding staircases; the huge stained-glass windows; and the freezing cold rooms, impossible to heat. It was Hogwarts without the magic.
Still, I considered myself happy there. It was grim and gloomy, but at least I had company. It was an all-girls school, so there were no boy troubles, no hormonal distractions. The teachers were strict and humorless, but I was used to that from my days in Japanese schools. Plus, they had their amusing qualities.
The math teacher, for instance, was so absurdly pompous that I would burst out laughing the minute he walked into the room. Unfortunately, I couldn't stop. I became so disruptive that he made me wear a dunce cap and sit in the corner, facing the wall. That only made me laugh more. It seemed the more he punished me, the more I laughed. Soon I was dreading going to math class, because I knew I was going to start laughing and never stop. He would grow furious, and I would laugh even more. It was agony, but I couldn't help it!
Then there was the infamous Mr. Gerard, the geography teacher, who would take long, self-important strides across the yard in his mortarboard hat, his leather shoes slapping loudly on the pavement. With every self-important step, he would let out a fart. I guess he couldn't help it. A more self-conscious man might have taken smaller, more judicious steps, but Mr. Gerard could not suppress his essential heartiness, so he strode on purposefully, farting as he went. The smell would collect under his flowing robes and then gently waft out as he clipped along. You didn't want to be caught in his wake.
It was a colorful setting, Bexhill-on-Sea. Near Hastings (where a pretty famous battle was fought), it sat right on the southern coast, and looked out on the English Channel. The air was fresh, the country green. Plus, there was a Woolworth's in town where you could buy candy.
The downside was the constant, monolithic cold that permeated the rooms of Charters Towers. You couldn't get away from it; it burrowed into your bones. The only way to get warm in that dank, clammy world was to bury yourself under a thick comforter made of eiderdown. That was heavenly. I would crawl into bed and pull the eiderdown around me, and suddenly the world was soft and blissful. Whenever I wrapped myself in one of those comforters, I felt happy. I felt loved.
So much so that today, I confess, I'm an eiderdown addict. I collect eiderdown comforters at every opportunity; they're my great extravagance. I must have a dozen of them around the house. I get them imported from overseas—made from genuine Hungarian goose down, they cost a fortune. Still, they're worth it. Whenever I wrap myself in eiderdown, I feel loved.
An unsettling aspect of the English school system was that when you reached the age of twelve, there was a special exam to decide the course of your future studies—it would determine which kids belonged on the fast track and which didn't (perhaps the inspiration for Harry Potter's "Sorting Hat"). The smart girls who aced the test were sent into Science and Math. The not-so-bright ones were directed into Cooking and Sewing.
So Yuki and I had to take the exam the year we arrived. I wound up in Cooking and Sewing; Yuki went to Science and Math. She had again confirmed her position as the Smart One, the better version of me.
**• • •**
**WHEN** we checked into Charters Towers at the beginning of the school year, we had to surrender our passports, our money, and our airline tickets for the trip home. These were locked up in a special box, and we would collect them again at the end of the semester. Because we were traveling together, Yuki's and my things were stored together.
When the winter recess rolled around, however, we didn't go home. Instead, we went to visit my godfather in Ireland. He was a producer and screenwriter named Kevin McClory. In the 1950s he'd been an assistant director on _Around the World in 80 Days_ , where he met my mother—she played the Indian princess Aouda—and they became good friends. Later he developed a screenplay with Ian Fleming, based on Fleming's character James Bond. Fleming novelized the screenplay as _Thunderball_ without crediting Kevin, and that resulted in a lawsuit and a nice settlement. Later, in 1965, when _Thunderball_ was made into a film, Kevin got the producing credit.
This is why he now had a stately mansion in Connemara, with servants and horse stables; and that's where Yuki and I spent the Christmas holiday, like little Irish princesses. For two girls fresh from the dungeons, this style of living was incongruously upscale and elegant. Every day, we were served high tea on silver trays, with crumpets and cucumber sandwiches. And every day, we would put on our cute riding outfits and go galloping across the wild green countryside. All the stable boys had crushes on us, and I watched and learned as Yuki toyed with them and put them through their paces. She was a master of her craft.
It was all very idyllic, and over all too quickly. We were soon back in the dank, dreary confines of school; a splash of cold water in the face, except it was cold all over.
It wasn't until we slogged to the end of the spring semester, and it was time to go back to Japan, that we went to the school office to retrieve our airline tickets. There was one small hitch, though: the tickets were gone.
"Excuse me," I said, after rummaging through the storage box, "where are our plane tickets?"
The matron looked at us blankly. "Did you have plane tickets?"
"Yes," I said, "they were both in this box."
"Mine was in my white purse," said Yuki helpfully. But her white purse was gone, too.
The matron shrugged. If we put the tickets in the box, then they should still be in there. "Are you sure you didn't keep them in your dorm room instead?"
This was a crazy question. Why would we have done that? All students were required to hand in their airline tickets. It was a rule, and if you broke a rule at Charters Towers, you were punished. Neither Yuki nor I was going to risk being caned just to keep our plane tickets as souvenirs.
This was as much guidance as the matron was going to provide. If the tickets were gone, they were gone. She knew nothing about it; she washed her hands of it. End of story.
For us it was just the beginning. Losing a plane ticket in those days was a serious business. There were no e-tickets or computer backups in those days; a paper ticket was like cash, and if you lost it, you were screwed.
Yuki and I had no idea what to do next. We were effectively stranded in England. While the school notified our parents, we got on the train to London without a plan of action. I guess we had this vague, youthful hope that maybe by the time we got to Victoria Station, everything would be solved for us. If worse came to worst, we could always go back to Connemara.
My real fear was that my father would come all the way from Japan to sort things out, and in his exasperation he would unleash his titanic, unreasoning anger on us. So I was naturally relieved when my mother showed up at the station instead.
I shouldn't have been.
Mom greeted us coolly—no hugs or kisses. There was nothing bubbly or effusive in her mood. She brought us both to the nearby Grosvenor Hotel, where she'd reserved a penthouse suite. Then she put Yuki in one of the bedrooms, shut the door, and locked it.
Mom turned on me and stared hard.
"So, what did you do with the tickets?"
I was so surprised by the question that I was initially speechless. "What did _I_ do with the tickets? I didn't do anything with them, Mom. They just disappeared."
Mom nodded. "Uh-huh." She paced the room a moment. "They just disappeared..." She stopped and gave me a significant glance. "Like your retainer, I suppose?"
"My retainer?"
"Your father told me all about it. We don't have any secrets, you know. You lost your retainer, so he bought you a new one, and then you lost that, too. Didn't you?"
"Well—yes, but that was because Eguchi-san tricked me."
"I'm not talking about Eguchi-san. I'm talking about _you_. You lost the retainer, but it wasn't really lost, was it?"
"No. Because Eguchi-san took it."
"Exactly."
My head was starting to spin. I couldn't follow her logic, which may have been her intention.
"So maybe the tickets are lost," she went on, "and maybe they aren't." She let this hang cryptically in the air.
"What do you mean?"
"I mean, maybe you know what happened to the tickets, and you're not telling me for some reason."
What reason was she talking about? "I'm kinda confused, Mom."
"I'm confused, too," she replied. "We bought the two of you expensive plane tickets, and now they're gone, and nobody knows anything about them. They must have gone somewhere, right? They just didn't get up and walk away by themselves."
True point. I couldn't argue with her on that one.
She folded her arms. "So, do you have something to tell me?"
Did I? I couldn't think of anything. So: "No."
Now Mom gave me the Look. The Look was one of her most trusted weapons. It signaled suspicion, righteous anger, contempt, and a steely determination to ferret out the truth. Her mouth would tighten, and her eyes would narrow to a squint, and you could see cold fire flashing within. She was like a basilisk lizard eyeing its insect prey.
I was getting the Look now, and I knew that whatever I had said so far, it was not what she'd wanted to hear. She was waiting for something else.
"Sachi, we're not leaving this room until we find out what happened."
"But I don't know what happened, Mom."
"I think you do," she said finally. "I think you cashed those tickets in and you took the money. That's what I think."
I was stunned out of my shoes. _I_ cashed in the tickets and took the money? _Me?_ First of all, I didn't even know this was possible. Could you actually do that? My next question was, _how_ would I have done that? What was the process for cashing in airline tickets? I had no idea, and I was astonished that my mother would think I did. It was like suggesting I knew how to score a kilo of heroin.
I told her as much. "I didn't do that, Mom. I don't know _how_ to do that."
"Maybe you don't know, but I think it's very possible that Yuki does."
Oh. I suddenly realized what the stakes were here, and why Yuki and I had been separated. We were like two members of an underworld gang, rounded up in a police sting, and now we were being held for questioning. Yuki was the brains of the gang, I was apparently the stooge, and Mom was the head detective in charge of the investigation. This lavishly furnished bedroom in a legendary five-star hotel had become my interrogation cell.
I started getting dizzy. Plus, I was starving. I hadn't eaten since we left Bexhill.
"Mom, I'm really hungry. Can we order room service?"
"Not till we get to the bottom of this." She was pacing again. Mom had never played a cop before, so she was probably relishing the opportunity. She resumed the interrogation, peppering me with questions, trying to trip me up. But she couldn't break me, because there was nothing to break.
Finally, she gave up in frustration. "Okay, let's see what Yuki has to say." She left me and headed for Yuki's bedroom.
"Mom!" I called. She paused at my door and looked back expectantly. "Can I get something to eat, please?"
She looked at me a moment. "No," she said, and shut the door and locked me in.
Now I was really confused. What? _No_? I wasn't going to eat? Because my plane ticket was missing? That was silly. Surely Mom had spoken in a moment of pique. She would bring me a sandwich when she came back. Surely. In the meantime, I drank water from the bathroom tap. That would tide me over for a little while.
About half an hour later, she returned from Yuki's room. Empty-handed. I don't know what went on with Yuki, but Mom didn't look satisfied.
The questioning began again. Why did you take the tickets? Where did you sell them? How much did you get?
Mom paused a moment, and then dropped her little bombshell: "You know, Yuki says it was all _your_ idea." I knew she was lying about this—Yuki would never have folded so easily—but I was so frazzled and Mom was such a good actress that I believed her anyway.
"It was not my idea!"
"That's what I thought," Mom went on silkily. "Yuki was really behind it, wasn't she? She's the one who wanted the money. Why? To send home to her mother? I know they're very poor. That was the reason, wasn't it?"
I'd seen this movie before—she was trying to play us off each other—and I wasn't going to fall for it. I would never betray Yuki, no matter how disloyal she was to me. No, never!
Although I did start to wonder... _Maybe Yuki_ did _take the tickets. Maybe she did send the money to her mother. She was certainly capable of it, wasn't she?_
So the interrogation went on, all day and into the night: Mom shuttling back and forth from one room to another, browbeating us, squeezing out odd bits of information and relaying them back to the other prisoner. When it was time for bed, she locked us both in our rooms, without feeding us, and promised to continue the discussion in the morning—which she did.
I honestly don't know how long this went on. I remember it as being three days. Yuki thinks it was only a day and a half. Maybe so, but I just recall being hungrier than I'd ever been in my life. Plus, I couldn't sleep, both from the anxiety and the hunger pangs.
Yuki said it was exactly like one of those World War II movies, where the good guys were interrogated by the Gestapo. There was no physical torture (except for starvation!), but the psychological manipulation was so intense that Yuki was on the verge of despair. It was the closest, she says, that she'd ever been to entertaining suicidal thoughts.
And all this terror was being whipped up by my mother, that sweet, kooky gamine beloved the world over. I began to see where the Dragon Lady epithet might have a basis in truth.
Finally—after holding out as long as I could, and seeing no end in sight—I cracked. Totally. "Yes, it's true, we did it, we did it!" I bogusly confessed.
" _You_ did it?" Mom asked pointedly.
"No, it was Yuki's idea." Yes, to my shame, I threw Yuki under the bus.
"Why?" Mom pursued.
"Why?" I quickly recalled all the malignant little seeds she had planted, and responded accordingly. "Because...her mother had no money, and she needed to pay her rent. So we took the tickets, and Yuki found a place to sell them. She's the smart one, you know. Not me."
Mom nodded. "She has a devious mind."
"You're right, Mom. You can't trust her!"
Mom pondered a minute, then left the room. A moment later she returned with Yuki. "Okay, Sachi, would you repeat what you just said?"
"Repeat...?" I hadn't counted on this. Yuki, who looked worn down but still had an air of defiance about her, waited curiously. She knew she had nothing to worry about, because she hadn't done anything. What could I possibly say?
"Yuki did it!" I said, pointing at her. I was like one of those hysterical girls in _The Crucible_ , picking out witches right and left to save her own skin. Then, when Mom pressed me, I repeated the entire story, just as I'd originally invented it.
Yuki was at first astonished; then she threw me the dirtiest look I'd ever seen. I could see that whatever friendship we'd managed to forge was instantly over. She denied the charges and raised a mild protest, but she seemed to recognize immediately that she couldn't win. It was my word against hers, and my own mother was the judge.
Mom gave Yuki a short, dismissive lecture—"I'm very disappointed in you"—and took her back to her bedroom. It was very quiet in there, and I wondered what could be going on. Was Mom calling the police? Was she making Yuki sign a confession?
When she returned, her bright, bubbly smile was back. "Now, what would you like to eat?"
I had a hamburger, medium rare, with thinly sliced red onions and tomatoes and Dijon mustard. The bun was toasted. Pickles and crispy French fries on the side, and a hot chocolate. It was the best food I'd ever tasted. In fact, give me that exact meal and an eiderdown comforter, and I would be in heaven.
I didn't see Yuki at the hotel after that. I discovered later that Mom had arranged her flight back to Japan. She and I were going to head back to the United States in a day or so. Everything was back to normal—but it wasn't. That night, the guilt kicked in big time. What had I done? How could I have told such a story? What would happen to Yuki now?
I couldn't sleep. I was racked with remorse. I had to tell Mom the truth. I went to her room and woke her up, at about three in the morning. In retrospect, I probably should have waited for a more civilized hour, but I had to come clean now. I had to be absolved.
"That wasn't the truth, Mom!" I recanted. "Yuki didn't take the tickets. I just made that up so I could have something to eat. I don't know what happened to the tickets. They just disappeared!"
There was silence for a moment.
"So, you were lying?" she said finally.
"Yes!"
"So, you're a liar."
This wasn't going the way I wanted. "Yes..." I continued, losing a degree of my confessional zeal.
"So, how do I know you're not lying now?"
"Because...I'm not."
Mom stared at me. She was probably giving me the Look, but it was too dark in the room to feel its coruscating effect. "We'll talk about it in the morning."
She walked me back to my room. All the way, I insisted that Yuki and I were both innocent, nobody had stolen anything, the tickets had just disappeared. That was the absolute truth!
Mom pushed me back into my bedroom, and shut the door. I heard it lock.
**• • •**
**WE** flew to New York the next day and stayed at another fancy hotel, where I was again locked in my bedroom and denied food. "You can come out when you decide to tell me the truth."
I couldn't believe what was happening to me. I'd been punished for telling the truth, and then when I finally lied, I was set free—until I told the truth again and was tossed back into my cell. Should I just lie again?
No, I couldn't. I knew I wouldn't be able to live with myself.
Out of desperation, I took to checking my suitcase, over and over again, scouring every corner of it, praying for the lost ticket somehow to materialize. It couldn't be in there, of course, because I'd never put it in there in the first place. I had plenty of time on my hands, though, and who knew? Maybe there would be a miracle! It was as if I were expecting the fox gods to do me an extra favor and exert their magical influence across the ocean.
A few days later, we moved on to Los Angeles. I was eagerly looking forward to the flight, just so I could get some peanuts to eat.
Back in Encino, I was given the run of the house, but my relationship with Mom had gone from cold to ice-bound. She wouldn't speak to me. I was a thief and a liar, and I was heading for a huge demotion in the next life.
What really scared me was that soon I would be going back to Tokyo to confront my dad. Let's face it: if Mom could react with such unreasonable fury, Dad might spontaneously combust.
It was while I was starting to pack up for the flight when I looked into the corner of my empty suitcase, and—
There it was! The plane ticket!
I picked it out of the corner of the suitcase and stared at it in amazement. How...?
I didn't know how. I still don't. I'd searched that suitcase from top to bottom, maybe hundreds of times, and there was no way it could have been there. But there it was.
I was stunned. My heart was pounding. I had to show Mom right away. I rushed down the hall to her room, but then, when I confronted her closed door, I stopped dead, terrified. How would she react? Would she hug me to her and apologize for doubting me? Or would she still freeze me out?
I knocked gently. "Mom? Mom?"
There was no answer. I tried the doorknob. Locked. I knocked again. "Mom?"
After a long silence, I heard the lock on the door click open. I waited for the door to open. It didn't.
I don't know how long I stood outside her door—afraid to walk in, afraid of my mother—but I had to go in. I had to tell her.
She was sitting in her king-size bed surrounded by magazines. Self-help magazines, women's magazines, entertainment magazines. She was reading one now.
I sat on the edge of the bed and waited for her to look up at me. She didn't. She gave no recognition of my presence whatsoever.
"Mom," I said with a weak hopefulness, "I found the ticket."
She didn't respond. She kept reading.
I went on. "It was in my suitcase. I don't know how many times I looked in there, a million, and it wasn't there. But then I just looked, and there it was."
This again seemed to make no impression on her.
I held the ticket up. It was my last gambit. "See?"
Finally Mom looked up, stared at the ticket, and regarded me with a cool indifference.
"Leave," she said.
I stared at her in disbelief, hoping I'd misheard her, hoping she would say something else.
"Leave the room."
So I left. I don't remember leaving, I don't remember where I went afterward—I probably fled back to my room—but I remember feeling that something seismic and terrible and irrevocable had taken place between my mother and me. We had had our first break.
I called Yuki soon after to give her the news about the ticket turning up. Her response was a rather scornful "No kidding."
It was apparently too late to clear her name, because the story was already out in her Tokyo neighborhood. It seems my dear "stepmother" Miki had quickly spread the rumor that Yuki's mother had masterminded the theft from afar, instructing her daughter to steal the tickets and cash them in, so she could send the money back home to her. It was pretty close to the story I'd told Mom. Miki must have heard it from Dad.
Both Yuki and her mother suffered a good deal of ostracism and social persecution because of this story, so I couldn't blame Yuki for being angry at me. In fact, she had no reason ever to forgive me for what I had done, except that she had been there in that hotel suite. She knew what had happened.
Even today, Yuki and I still wonder about those tickets. Where did they go? Did one of the matrons take them? Or are they still sitting somewhere in that office in Charters Towers, collecting dust?
More mysteriously, how did that other ticket get into my suitcase? I could have just overlooked it, but that seems close to impossible. Was it even the same ticket? Did someone plant it there? Who? Why?
**• • •**
**A** few days later I returned to Japan, where I feared even harsher retribution from my father.
But no, he was not troubled at all. In fact he took it very much in stride. "So you lost the ticket." He shrugged. "Things get lost."
I was shocked. "Really? But all that money..."
He waved it off. "Pah—money! As long as you got home safe, that's the main thing."
I was bewildered by his sangfroid—also relieved—but still, I wanted to make sure he understood my side of the story. "It wasn't my fault, Daddy. I gave them my ticket at the beginning of the year, and then when they opened up the box, it was gone!"
He nodded. "Look, Sachi, it's not important. Even if the ticket weren't gone...even if you did cash it in..."
"But I didn't!"
"It doesn't matter. I know you must have had a good reason for doing what you did. Look, you finally found the ticket, right? So it's over."
I could see he didn't believe my story. "But—"
"Don't worry about it, Sachi. When you get to be eighteen, I know you'll tell me the truth."
I don't know what being eighteen had to do with it, but six years later, when I turned eighteen, I sought my father out and told him again, in pretty definitive terms, that I'd been totally innocent in this affair, and so had Yuki, and that the tickets had just disappeared. That was the pure, unvarnished truth.
Dad nodded knowingly. I thought I saw a trace of pride in his smile, as though he were thinking, _She's sticking to her story, even after all these years. Cool_.
# **Chapter 5**
# From Here to Zagreb
I returned to Charters Towers the next year, as did Yuki. The Strange Affair of the Lost Tickets slowly faded from memory, and our friendship resumed its usual comradely competitiveness.
I was thirteen now, in full-rigged adolescence, and naturally inclining toward romantic dreams and schoolgirl crushes—and since I was surrounded mostly by fellow schoolgirls, my first big crushes involved females. (I wasn't going to get all dreamy-eyed over the flatulent Mr. Gerard.) I wasn't unique in this regard: all the younger girls at the school had big crushes on the older girls. This had nothing to do with sex. It was all very innocent and pure of spirit. Yet the power of the emotions, the from-the-heart intensity, was so overwhelming and all-consuming that some girls would hyperventilate, get hysterical, go into catatonic trances of longing.
My great crush was Sarah, a prefect with bright red hair and freckles, who was maybe two years older than I. She was so sweet and so friendly. I was crazy about her. I don't know if she realized the depth of my infatuation, or if she even suspected my interest. She probably had her own crush on somebody older. That's how it worked: the love kept spiraling forward in a great, continuous daisy chain of unrequited desire. The hallways were thick and humid with yearning.
I was unaware of anyone else's passions, being totally absorbed in my own little drama. Sarah would pass me in the hall, and my heart would leap. Sometimes she'd smile at me, and I'd lose my breath. If she happened to touch me, if even just her clothing brushed against me...electricity! A week's worth of daydreams!
Eventually the obsession faded, and I moved on to the next infatuation. I had lots of crushes at Charter Towers. About the only girl I didn't have a crush on was Yuki. We knew each other too well.
The following year, I switched schools—and found myself in Switzerland. Aiglon College was located high in the Swiss Alps, not far from Montreux and Lake Geneva. It was a spectacular location, a magnificent series of chalets set in a picture-postcard Alpine valley cradled by tall, snowy peaks.
Also spectacular, for me, was that Aiglon was a coed school. This was one of the reasons I petitioned Dad to leave Charters Towers. The all-girl ambience was starting to feel unhealthy to me. I remember telling him quite seriously, "I want boys."
And I got them: an international smorgasbord of boys in blazers. I soon discovered that while I liked girls, I _loved_ guys. They were so cute, and funny, and full of energy.
And they loved me. Perhaps because I had such an open innocence, they were always flocking around me, flirting in a sweet way. I adored the attention, but I was still shy and deferential in my Japanese way, so not much happened. Except in my head.
I really had a great time at Aiglon. It was sunnier than Bexhill, nowhere near as cold and dank and dispiriting. The food was wonderful, and there was all that Swiss chocolate. And the mountains, and the forests. And the boys.
My grades were still terrible, true; I was an abysmal student. Then again, my father had already convinced me I was an idiot, and I had demonstrated no capacity for learning at any level, so I didn't care. I was _supposed_ to be stupid.
So instead of going to class, I would sneak off to the Alps. I'd go skiing, or rock climbing, or skipping through fields of wildflowers, doing my best Heidi impersonation. In the spring, I'd slip out of the dorm at night and climb up the forest path to a glacier lake, and I'd skinny-dip in the moonlight. It was crazy and irresponsible, I know, but it was quite wonderful.
The great love of my Aiglon College days was Gabriel Connolly. I'll never forget him: a short, dark-haired British kid, sort of a Dudley Moore type, who had a mad crush on me. He thought I was gorgeous, and I thought he was funny, so it was a great match. Gabriel was from Liverpool, and he was probably on scholarship, because he had a slightly unrefined, street-smart air about him, so different from the other privileged kids at the school. A tough, manly Dudley Moore.
One night, I was in my dorm room on the fifth floor, lying awake in bed fantasizing about marrying Gabriel Connolly and having the perfect life. My roommate, Katie Sokoloff, was sleeping peacefully. Katie was from Greenwich, Connecticut, from wealthy, entitled stock; ironically, she was very good friends back home with the man who would become my husband some twenty years later. Right now, though, she was just the girl snoring in the other bed.
Suddenly I heard a light tapping on my window. I knew who it was right away—my Romeo, my dashing cavalier! He had climbed up the side of the building in the dark, using a rope. I couldn't let him in, not with Katie right there, so I climbed out. We went down the rope together—five stories, mind you—with Gabriel carrying me in his strong arms. It was the quintessence of romance.
Together we went into the woods and climbed up to my favorite glacier lake. All shyness gone, we took off our clothes and went swimming in the nude. The water was ice-cold and thrilling. When we came rushing out of the frigid lake, Gabriel wrapped me in a blanket. Then we kissed: my first real kiss, with tongues and everything. The cold had made the blood rush to the surface of my skin, and I was suddenly boiling hot. The rush of sensations was overwhelming: the cold, the heat, the full moon, Gabriel's lips, Gabriel's hands, the sweetness, the tenderness...I thought I would pass out.
That's as far as we went: a little petting, a little swooning. This was plenty far for me. We came back down the mountain, hand in hand, and climbed back up the rope, and he delivered me safely to my bed. Another kiss or two, and he was gone. While Katie slept on.
No, I'll never forget Gabriel Connolly.
**• • •**
**THE** first moments of 1971 found me at the New Year's Eve party at the Kanaya Hotel, high on a mountaintop in the Japan Alps. I was fourteen years old, and Dad was dancing with me. I now loved dancing with him, because he made me feel so grown-up and special. There was a disco ball overhead, scattering dreamy patches of light everywhere.
As we circled the dance floor, Dad pointed out an elegant European woman sitting at a table.
"That woman is quite beautiful. Don't you think?"
He turned me around on the dance floor so that I could see. Yes, she was beautiful.
"She reminds me of Eleanor Parker," he said. "The actress?"
I shrugged blankly.
"She was in _The Sound of Music_. She played the Baroness."
"Oh." Of course I remembered her. Everybody hated the Baroness.
"I was good friends with her," Dad said.
I looked at him in surprise. Who? The woman at the table?
"Eleanor Parker," he elaborated. "She was very special."
At the same time he was telling me about Eleanor Parker, his hand slid down from my waist and patted my fanny—and stayed there.
That struck me as very odd.
"She was a real woman—lovely, sophisticated, a classic beauty," he said, still holding on to my bottom. "You should watch some of her movies. You'll see what I mean."
I was getting some very confusing signals. Why was Dad grabbing my butt this way, as if I were his girlfriend? And why was he going on and on about Eleanor Parker? I sensed that he was conveying a message to me: that this actress was his idea of a truly desirable woman, and he was telling me—he wanted me to know—that I didn't come close to measuring up to her. I needed to be sexier and more beautiful if I wanted his approval.
Yet, why would I need that kind of approval? I wasn't his lover. I was his daughter, wasn't I?
**• • •**
**IT** was the end of my second year at Aiglon. I was fifteen years old and waiting to go home. I wasn't sure who was coming to pick me up, Mom or Dad. There hadn't been any communication or instructions—I wasn't even sure where they were—but I knew someone was coming. So I packed up my suitcase and waited.
The rest of the students were already leaving. Some had their parents pick them up; others took the bus to Lausanne and grabbed the train, heading off to various parts of Europe. The school was emptying out.
Still no sign of my parents. People were starting to get a sense of my plight. "When are you leaving?" they'd ask.
"I don't know," I'd say bravely. "But pretty soon. It won't be long now."
"Are you okay?"
"I'm fine!"
Of course, as soon as someone asks you if you're okay, you start to realize that maybe you're not okay. _What happens if nobody shows up? And the college and the dorm rooms are all shut down? Where will I sleep?_
At the same time, I was sort of enjoying the drama of the situation. I was like young Ebenezer Scrooge, left behind at the Christmas break. Nobody wanted me home. Nobody cared! At the same time, it was the perfect opportunity to be swept away in a wild European adventure. I was exhilarated by the prospects.
It was the next day now, however, and the last students were trickling out. Even the staff was leaving. I began to sense that I was in real danger of being abandoned there. Who was going to take care of me? I was starting to panic. As the last few students waved goodbye, I waved back cheerfully. I didn't want to share my agitation. I didn't want to make trouble.
Finally, my schoolmate Jane Wise, the last soul on the premises, was going off to catch the train at Lausanne. She saw me sitting by the front steps. "Are you okay?"
I thought about giving her a reassuring nod and sending her untroubled on her way, but I couldn't keep my upper lip stiff any longer. "I don't know," I answered plaintively. "I don't think anyone's coming to pick me up!"
"Well, you can't stay here," she said.
I looked around at the empty grounds. She had a good point.
"Why don't you come with me?" Jane suggested. She was joining her family in Trieste, Italy. Her father was the American ambassador to Czechoslovakia, and during his mission they were living in Prague. So they were meeting up halfway for a few weeks of summer vacation on the Adriatic coast.
"Trieste?" I pondered.
"Sure. You can try to call your parents from there."
It seemed to make sense. I didn't want to be a burden on Jane and her family—she was a year older than I, and we weren't exactly close friends—but there didn't seem to be any other choice.
So we went down to Lausanne and caught the train to Trieste. Jane paid for my ticket. It was an all-day trip, as we had to make the trek straight across the top of the boot to eastern Italy.
At the train station we met Jane's mom and dad, and her sister Ann. We went to a luxury hotel in the heart of Trieste, and I stayed in Jane's room.
I knew I was incredibly lucky to be there. Jane's parents paid for everything. We spent the days at the beach, which was right at the northern tip of the Adriatic Sea. There were lots of beautiful Europeans, male and female, cruising the sands in the tiniest of bathing suits. I remember everyone kept asking me if I wanted ice cream.
I didn't want ice cream. I didn't want anything. I had a very numb feeling inside, which at night was replaced by a pain around my heart. I didn't know what was causing it, but my heart was hurting.
So much so that I could barely sleep—and when I did, I would wake in the middle of the night with a deep, profound feeling of sadness. Yet, I didn't know why.
I didn't voice any of this to Jane or her parents. I felt guilty enough freeloading on them; the least I could do was maintain a cheery disposition. That was the Japanese way.
One night, I woke at around 2:00 A.M. There was the pain again, the pain in my heart. I got up quietly. Something was telling me I needed to leave. I needed to go out.
I quietly got dressed. I put on my tight blouse and my miniskirt—and no underwear. Not one brown shred.
I went into the next room, where Jane's parents were sleeping. I saw her mom's high heels under the dresser. I took them and sneaked out of the room. Outside, I slipped on the high heels and left the hotel suite.
Downstairs, I crossed the hotel lobby. I felt the doorman watching me quizzically as I stepped out into the warm night. I knew that if I looked at him, he would stop me. So I didn't look. I headed down the sidewalk and turned the corner, and now I was on my own.
The Trieste streets were empty and quiet. My high heels click-clacked on the cobblestones as I made my rudderless way through the city, strutting along in my sexy, tarted-up outfit.
Unwittingly, and yet as if by design, I found myself in the seediest section of town. There was nightlife here; people were moving in the shadows: large, indistinct shapes; muted snatches of their conversation echoed against the buildings. Who were they? Thieves, perhaps; drunks, prostitutes. A dark, threatening sexuality was in the air. Men and women were humping openly against the building walls. Other men circled around, eyeing me prospectively as I walked along. Some would move closer; some would even stand in my path and make me swerve around them.
Now the prostitutes emerged from the dark, gathering in pockets and glaring at me with territorial hatred: "Who is this little bitch, in her miniskirt and high heels, and what is she doing on our turf?"
What _was_ I doing there? What was I thinking? I don't know. It was all very dreamlike and Felliniesque, without the saving grace of comic absurdity. I knew I was in danger—I could feel it all around me—but I didn't really care.
I was being adventurous, the way Dad had suggested. I was having an experience.
Basically, I think I wanted to die.
So I stopped on that dark, dangerous street, vulnerable and surrounded, and waited for something to happen.
Finally, one aged prostitute detached herself from the group and approached me. She was in her sixties, maybe older, wearing garish clothes that clung unattractively to her body, her face caked with makeup. Her eyes gleamed savagely in the glint of the streetlight. I felt a sudden, paralyzing terror—this is it. She's coming to kill me. She's going to rake me with her clawlike nails, slash me open with the stiletto hidden in her garter...I couldn't move. I was trapped in my death wish.
When the old woman got very close, she put an arm around me and asked, in a gentle voice, " _Dove abiti, cara_?"
Somehow she had grasped exactly why I was there, and what I really needed. My eyes filled with tears, and I started to cry. She hushed me, cradled my head against her bosom, and gently turned me around and walked me back to the hotel. We didn't speak much—I had had some French in Aiglon, so we could manage a few words here and there—but most of our conversation was spoken with our eyes.
Back at the hotel, the doorman was waiting in the lobby. I can imagine what a sight we were, the two of us dressed up like ladies of the evening, and one the genuine article. By all rights, he should have barred the old prostitute at the door and called upstairs for Jane's parents to collect me.
But he didn't. Somehow he knew exactly what was going on. He and the woman traded glances, and there was one of those miraculous moments of unspoken human communication. He nodded and stepped aside, letting us pass. He was probably relieved to have me back, and grateful to the woman for her kindness.
Back in my room, the woman undressed me and put me to bed. When she saw that I'd been wearing no underwear, she tsked disapprovingly, but kept her silence, so as not to waken Jane. Like Katie Sokoloff, Jane was a sound sleeper and never stirred.
The woman tucked me in and sat by my bedside a moment, stroking my hair, whispering soft, comforting words. Then she bent down, kissed my forehead, and left.
I never saw her again.
**• • •**
**AFTER** a week in Trieste, we took a ferry south to the coast of Yugoslavia, where we stopped at another beach resort on the Adriatic, and stayed about two weeks.
I don't remember much about this time, except that there seemed to be an awful lot of Germans on the beach. Most of my days were spent on the phone, trying to contact my parents. After hours of blind-dialing, waiting on operators, listening through bad reception, getting tantalizingly close and suddenly losing the connection altogether, all I could safely determine was that my dad wasn't in Tokyo, and my mom wasn't in Los Angeles. This narrowed the field of inquiry somewhat.
It seemed I was on the phone all day every day, and getting nowhere. I was miserable, and still haunted by a feeling of vague sadness—and feeling guiltier all the time for taking advantage of the Wises' kindness. They had been incredibly warm and gracious, but sooner or later I knew I was going to wear out my welcome.
We left the coast and drove up to Zagreb, the capital of Yugoslavia. From here the Wises were taking the train home to Prague. I just couldn't tag along anymore. The moment of truth had arrived.
Or rather, the moment of untruth. I rushed up to them on the platform with a big patented smile on my face. "Finally," I told them brightly. "I got hold of my dad! He's sending me money right away, so everything's fine. You guys can go ahead without me. I have to wait here for the money."
None of this was true, but the Wises bought it. Or at least they said they did. They could leave me now with a clear conscience. There was nothing unusual in this, by the way. In those days, it was quite normal to see teens traveling by themselves all over Europe. The Wises weren't abandoning me. They were letting me enjoy my independence.
They boarded the train, and I waved cheerfully as it chugged away. "Goodbye! Thanks for everything! Goodbye, goodbye!"
So, now, instead of being stranded in Switzerland, I was stranded in Yugoslavia. What an idiot.
There was a hotel across from the train station. I walked over and went into the hotel bar. I sat on a stool, put my head on the bar, and started crying. What was I going to do now?
There was an elderly couple in their seventies at the bar. They were sitting close and chatting intimately, with a relaxed, casual familiarity that indicated they were married. There was something a bit rustic and out of place about them, so perhaps they had ventured into the city for a romantic date night. If so, my sobbing at the bar may have put something of a damper on their ardor.
In any case, they got up from their stools and came over to see what was wrong. They spoke only Yugoslavian, so even my little bit of French wasn't going to help me now. Still, they saw my distress, and much like that prostitute in Trieste, they understood without words.
They sat beside me and comforted me, and somehow managed to construe that I was stranded and had nowhere to stay. So they brought me up to their hotel room and had a cot set up for me, and I stayed the night with them. This was, all in all, a remarkable thing.
I had already imposed on one family for the last three weeks, and I really didn't want to start all over again. I had no idea what to do next or how to proceed, though, so the next day, when they insisted, through some universal gestures and inflections, that I go back home with them, I meekly accepted.
We drove into the Yugoslavian countryside to their farm, a good five-hour ride. I could see when we arrived that they were of very modest means. It was a run-down working farm, with lots of vegetables and chickens. Everything was old and rusty, but it all worked.
I stayed with these wonderful people for about two weeks. I helped out with the chores: milked the cows, fed the chickens, shared the cooking with the old woman. There was plenty of work to do, and I slept really well—no pain in my heart at all. It got cold at night, but they had the Hungarian eiderdowns!
I loved that couple. They treated me like their daughter, and the setting put me in mind of that palomino farm up in the hills of Malibu, with Ricardo and his mother: simple, hardworking people who were full of a love of nature and life. There were no pretensions, no demands, just a regulated sense of peace and harmony.
The old man had a great deal of work to do on the farm, but he would always take time out from his busy day to drive me down to the local post office so I could make my calls in my ongoing attempt to locate my parents. After an hour or so without success, we'd head over to the local bakery for a treat. I remember the pastries were denser and less refined than the fluffy ones you'd find in the French or Swiss bakeries. These were earthier, more typical of the people, whom I found to be unfailingly cheerful and friendly in spite of their inescapable poverty.
Still, much as I enjoyed this new world and these new friends, I was growing desperate to find my real family. I called and called. Finally, one day, instead of dial tones and static, I heard a familiar, English-speaking voice at the other end of the phone: "Hello?"
I was stunned. "Dad?"
"Hey, Sachi! Sach the Pach! What's up?" His voice was sunny and gregarious. He sounded as if he hadn't a care in the world.
I felt an immediate need to match his good spirits. "Nothing. I was just calling to say hello," I answered, without a trace of irony.
"Where are you?"
"Um—I'm in Yugoslavia."
He laughed. "Yugoslavia? What are you doing there?"
"Oh, just hanging around." I was being very cool. "Uh—where are _you_?"
"I'm on the boat," he replied casually. The "boat" was his private yacht, _Happy Pappy_. It turned out he'd been sailing the Mediterranean, just off the coast of Greece, all this time.
We made a little small talk, and then I mentioned that I needed some money to get home. Or wherever I was going.
Dad was quiet a moment. "You know, your mother is shooting a television show in London," he said. "You should go see her."
"London?"
"I'll wire you the money."
"Oh, okay."
I told him exactly where I was, and he said he would take care of everything: problem solved. "Enjoy your summer!"
I hung up. I should have been vastly relieved, I suppose, but instead I came away feeling unsettled by this surreal collision of three wildly incongruent worlds: Dad sailing the Mediterranean, Mom shooting a TV show in England, and me stuck on a chicken farm in Yugoslavia.
My surrogate parents drove me to a bank in Zagreb, where the money was wired. Then they brought me to the airport, and we said goodbye. I couldn't have expressed my gratitude sufficiently even if we'd spoken the same language, but I think they understood how I felt. They were stoic, unemotional people, but I owed them so much—God knows what would have happened to me if they hadn't come along and helped me. I kept in touch with them over the years, and sent them a Christmas card every December, until they both passed on about thirty years ago.
I arrived in England and found my mother on the set of her new TV show. _Shirley's World_ was her attempt to break into sit-coms. In it, she played a globe-trotting photojournalist with a crusty but charming editor played by English actor John Gregson, and it was shot on location all over the world. The main base was at Pinewood Studios, just west of London.
The show was not destined to be a success—in fact, it was an outright flop—and Mom seemed to sense this even as they were shooting it, because she was in a foul mood on the set, and didn't seem to be getting along with anyone.
She was delighted to see me, of course: "Hi, sweetie! What are you doing here?" I didn't bother to tell her that she had forgotten to pick me up in Switzerland. I was afraid it would upset her. At the same time, I think I was afraid that it _wouldn't_ upset her. She might have even wound up blaming me for the oversight.
Anyway, we had a nice visit, for about a day or two, and then she shifted back into work mode, and I suddenly became a great inconvenience. She had enough problems dealing with this godforsaken show; she didn't want to have to deal with me, too.
She called the headmaster of Battisborough, the boarding school I was transferring to in the fall—I don't know why they had decided to move me out of Aiglon; maybe because they couldn't remember where it was. "My daughter is here in London," she told him. "Is there something you can do with her?"
The next thing I knew, I was on a train to Devon.
Battisborough House was run by Anthony Fiddian-Green, who used to be in charge of the girl's dorm at Aiglon. So I already knew him and his lovely wife, Susan. They had taken over this run-down centuries-old stone mansion and were in the process of fixing it up for the fall term.
For the rest of the summer, I became their errand girl and all-around handyman. I helped clean up the dusty rooms, paint the walls, wax the floors—whatever needed doing, I did. I stayed in the house and ate my meals with them and became a temporary member of their family (my third of the summer!).
Battisborough, in the county of Devon, is an absolutely beautiful spot, right on the southwest coast, near Plymouth. Looking south to the English Channel, the house sat on a green sward not far from a series of dramatic rock cliffs and coves abutting a deserted coastline. As I recall, there were never any people around, only sheep. I would often navigate the narrow sheep paths through the brambles and gooseberry bushes and down the steep cliffs, and take long walks on the lonely windswept coast, exploring Bugle Cove and Mothercombe Beach. I had long grown accustomed to being alone, and I felt totally in my element wandering barefoot in the sand with the wind whipping my hair and the spray of the ocean misting over me.
**• • •**
**I** didn't go back to Japan that summer. (There was no one there, anyway.) I stayed at Battisborough, and as the fall term began and the other students—including boys; hooray!—arrived, I resumed my studies.
Not that I studied much. Battisborough was based on the Summerhill model. Summerhill School was founded in Suffolk, England, in 1921 by A. S. Neill, a Scottish writer who believed that children flourish most without adult authority. His progressive school stressed freedom and autonomy. So, at Battisborough, you were defined by your own idea of success. There were no bells; you went to class any time you wanted to, and if you didn't want to, you could just play. It was, for most kids, an ideal situation.
It wasn't for me. Much as I bridled at authority, I was lost without it. Given the choice, I never went to class. My grades sank to a level not even imagined before.
Yet, in spite of my low academic performance, I was considered the most responsible, level-headed kid at the school. Some of the other kids were messing around with drugs and alcohol. They came from money, so there was plenty of acid and LSD and cocaine floating around the dorms.
I didn't touch any of it. My biggest vice was going for walks on the beach. So my bond rating was pretty high. Everyone in the faculty trusted me—and I took advantage of this.
There was a separate cottage on the grounds; I used to walk by it on the way to the beach. Nobody was staying there, and it seemed to be going to waste. I would have loved to have that cottage as a dorm room all to myself. At the moment, I was rooming with an American girl named Anne Hearst (younger sister of Patty Hearst, who was famously kidnapped by the Symbionese Liberation Army just a few months later). Anne was nice enough, but I liked the idea of being on my own and independent.
I brought up the idea of the cottage with Mrs. Green (we never bothered with the "Fiddian" part; they were always "Mr. and Mrs. Green" to the students). Mrs. Green, who was very fond of me, discussed it with her husband, and they agreed that I was trustworthy enough, so I moved into the cottage. I loved that place. It was small, but there was a kitchenette with a hot plate and an electric kettle. There, I was completely, serenely alone—except for one fateful night.
It was my second year at Battisborough, the spring of 1974. I was seventeen years old, and the new great love of my life was Bradley Foster. There were only about eighteen kids at the school, half of them of the male persuasion, and most of those were pretty dicey. Brad was head and shoulders above the rest, the pick of the lot, and it was my good fortune to have nabbed him. He was tall, handsome, and very sweet.
This particular evening, Brad sneaked up to the cottage to be with me. The door was on the ground floor, so there was no need for ropes or climbing out of windows. I just let Brad in, and we immediately went at it like teenagers in heat—which is exactly what we were. We were so mad with passion that I forgot to close the curtains.
We were doing some heavy making out, and pretty soon I realized that this was going to be a historic encounter: we were going all the way. We moved from kissing to groping to grinding with amazing speed. As if by magic, my clothes were off, and so were his, and we were wrestling on the couch, and it was happening, we were going for it, we were right at the brink...!
Then the door flew open—forgot to lock it!—and there was Mrs. Green, staring at us, mouth open in shock. We were caught, caught dead. There was no hope of coming up with an adequate excuse, not with the two of us stark naked, our limbs intricately intertwined.
I hastily covered myself with a couch pillow, while Brad grabbed his clothes and hid behind them. Mrs. Green took in the lurid scene for a moment, her eyes fixing me with wounded outrage, and then she turned and stalked off into the night.
I threw Brad out of the cottage and then, guilt-stricken, rushed up to the main house. I burst in on the Greens and apologized profusely to them. I felt terrible, I had betrayed their trust. How could I have been so selfish and ungrateful? I tearfully admitted my guilt—"I'm sorry, I'm sorry, I'm sorry!"—and begged for their forgiveness.
I didn't get it. Mr. Green just said he was "very disappointed," and left the room. Mrs. Green wouldn't even look at me; she stared at the wall and said nothing.
Brad and I broke up after that. Much as I desired him, close as he had come to adding a seminal chapter to my sexual history, he had been an accomplice in the most humiliating episode of my life, and I just couldn't be with him anymore.
**• • •**
**AT** exam time, we had to take both the British and the American tests to determine our college worthiness. The SATs were not culturally neutral; they were definitely slanted toward American mores and idioms. If you hadn't been raised in the United States, you just wouldn't do as well—and I didn't.
I did, however, score very high on my French and English A-levels, which were extremely hard and served as the best barometer of whether you belonged in college.
This encouraged me to look forward to college with a newfound purpose. I was going to really apply myself from now on. I was going to be a model student. I hadn't picked out a college yet, but wherever I went, I was going to shine.
So it was with pride and relief that I received my diploma from Battisborough. Mr. Green gave me a stiff, cursory handshake, and Mrs. Green averted her eyes as I passed. I didn't care: I was out, I was moving on. The graduation ceremony was small, but it was beautiful and inspiring.
Neither of my parents came.
# **Chapter 6**
# "You're on Your Own Now"
In June of 1974, Mom was performing her act in Las Vegas. That was her reason for missing my graduation.
I completely understood. In those days, before the second coming of Atlantic City and the glut of casinos all over the country, Las Vegas was a world of glitz unto itself. When you played Vegas, you were at the peak, the center of the entertainment universe. And you couldn't just walk away from that, even to attend your daughter's graduation.
So, since Mom couldn't come to me, I went to her. I was a high school graduate now, practically a grown-up. It was time for us to establish a real, mature mother-daughter relationship before I went off to college.
I'd been to Las Vegas as a young girl, when Mom was hanging out with Frank Sinatra and the Rat Pack. Back in 1960, Dad was producing a show called _Holiday in Japan_ , a musical revue extravaganza starring the biggest Japanese singers and dancers of the time; it more or less introduced postwar Japanese culture to the West. _Holiday in Japan_ played a month at the New Frontier Hotel and Casino, which, with the Flamingo and the Desert Inn, was one of the few Vegas-type hotels on the Strip at that time. The Strip itself was in fact just a strip, a wide dirt roadway. Las Vegas was a different proposition back then. There was a lot less neon and a lot more sand.
I don't remember the Japan show at all, but I remember the wind, and the cold. It would be dry heat during the day but freezing cold at night. For some reason, while I was visiting Dad during _Holiday in Japan_ , we didn't stay in a hotel. We lived in makeshift tents in the desert, and slept on cots. We had to walk across the sand dunes to get to the hotel where the revue was being performed. Our feet would sink in the shifting sands, while the wind buffeted us from all directions, and we had to use outhouses to relieve ourselves. When I would wake up in the middle of the night having to pee, I dreaded going out into that cold, and the blowing wind. Whenever I did screw up my fortitude and make a run for the outhouse, I was constantly dodging the huge tumbleweeds that rolled past. They scared me, and for good reason—they were as big as I was.
Vegas had a lonely, mysterious feeling then. There was an overpowering sense of nothingness. Yet for some reason, I found it enchanting. I particularly loved the mornings, when the red sun would paint the sheer cliffs of Sunrise Mountain.
So I was completely unprepared for the jolt of energy that hit me now, in the Las Vegas of 1974, virtually the minute I stepped out onto the tarmac. The lights, the people, the cars, the sense of money and power and pleasure—the very air thrummed with electricity. It made me sad. It wasn't quiet, like before. I couldn't hear the wind anymore.
As I sat at a front table in the theater where Mom was performing, I found myself surrounded by celebrities, politicians, high-rollers, and the like, all in tuxedoes and gowns and gaudy leisure suits. They were all there; they had all dragged themselves away from the casino tables and the roulette wheels just to see my mom in action.
And she was worth it. A complete entertainer, she did all her songs from _Sweet Charity_ —"I'm a Brass Band," "If They Could See Me Now"—and some funny specialty numbers, and those brilliant Bob Fosse dances with the top hat and black tights, and "It's Not Where You Start, It's Where You Finish," a new signature song for her from the musical _Seesaw_ , which was based on _Two for the Seesaw._ (Show business is a curiously circular kind of world.)
It was a great act. Mom owned that stage, and she knew it. And every now and then she would look down from her kingdom and give me a wink, with a big, glorious smile on her face. She was enjoying herself immensely, and she wanted me to share in the fun.
And I did. I was so proud to see her up there that it literally made me shake. The waiters would ask me if I wanted another Shirley Temple, but I couldn't even speak. If I could have, I would have said, "Forget about the drinks. Look at my mom!" She was so special, and she made me feel special, knowing that I was a part of her, that her spirit and her talent lived within me, somewhere. Just the thought that we were connected took my breath away.
After the show, I waited in the dressing room while Mom greeted her fans. There was always a line of people outside the door, armed with papers, posters, anything for her to autograph. They would wait patiently, sometimes over an hour, just to pay homage to her. Every now and then the professionals, the important people, would be ushered through the crowd to see her. Sinatra, Dean Martin, Jack Lemmon, Danny Kaye—they would blithely cut the line and step into the dressing room for a casual chat that might last fifteen minutes, an hour, or more.
While everyone else waited. This upset me, because I felt for all those ordinary people out there who just wanted to show their appreciation. I knew what it was like to be kept waiting. I wanted to say, "Mom, shut up. There are people out there who want to see you!"
At the same time I was empathizing with the outsiders, I was also enjoying the thrill of being on the inside, with the special people. If they were special, that meant that I was special, too. More than that—I was cool.
When Mom did finally make time for the public, she would be a most charming host. She'd chat, she'd laugh, she would happily pose for photos. Then the moment would come—you could see it in her eyes—when she was done. Her face would go blank, she'd turn off the thousand-watt smile, she'd give you the cold stare, and you knew you'd been dismissed—and woe betide those who didn't pick up on the signals, because then she'd happily spell it out for them:
"Excuse me, but I have a life, do you? Have a nice day. Go win some money."
That night, having vanquished her fans, Mom closed the door, sighed wearily, and then turned to me with exaggerated deliberation. I could tell from her manner that there was a dramatic moment coming. She stepped forward and took my hands in hers.
"Sachi, sweetheart, I'm so proud of you. I really am." She waited a moment to let this sink in. I understood in a limited sense that the important point here was not that I had graduated from school, but that she was proud. Her emotional response was center stage right now.
Mom turned to her dressing table and picked up a thin jewelry box with a red ribbon stretched around the corners. She waited a beat, and then handed the box to me. "Here."
I took the box and opened it.
Inside was a diamond necklace.
I was stunned. It was spectacular. It must have cost a fortune.
"Oh my God! Mom—it's amazing!" I held it up, and saw the dressing room mirror lights glittering in the prisms of the stones.
"Those diamonds are from Belgium," Mom said. "Belgian diamonds are the best in the world. I had them shipped here specially on a private jet."
"Really? Just for me?"
"Just for you." She wrapped her arms around me and gave me a big hug. "Congratulations, baby!" We rocked back and forth, and I luxuriated in the moment.
Then she stepped back and added a casual fillip: "You're on your own now."
It took me a moment to process this, and I realized that there was a hidden weight to these unexceptional words.
"On my own? What do you mean?"
Mom sat at her dressing table and began removing her makeup. "Well, I mean, you've graduated, there's your gift—good luck. Do you have any plans?"
"Plans? Well, I was hoping to go to college..."
Mom looked over her shoulder with a get-real expression. "College? What for?" She laughed pleasantly. "How can you afford college anyway?"
"Well, I thought maybe..." I made a hopeful gesture toward her.
She rejected my fanciful notion in short order. "Ha! Don't look at me. You have to make your own way in the world, sweetheart. That's what I did, and look where I got." She made a sweeping gesture to include the trappings of her room, the photos with fellow celebrities, the good-luck telegrams, and, beyond that, the golden world of Las Vegas itself. "That's the key to happiness. When you do it all on your own it means so much more. You'll see."
I was trying to see, but I have to admit, it wasn't easy taking the long view. I just couldn't make sense of it. I wasn't going to college? Why had I just spent six years in expensive prep schools if I wasn't prepping for anything? What was that all about?
For the record, I asked Dad to pay for college, and he said no, too—which was odd, because all through my teenage years, Dad had contended that I would be the perfect candidate to work at the United Nations in Geneva. "You have so many languages under your belt, you have a broad way of thinking, you're not judgmental, you don't have any religion to screw you up, you understand and embrace differences in people, you empathize with others. You would be ideal." I thought so, too. Still, how was I going to get to the UN without a college education? I was no genius, but I knew that I couldn't make it from A to D without a few stepping stones in between. Take them away, and where was I?
I watched impotently as Mom wiped the cold cream from her face and started applying another round of makeup for the real world. My first instinct was to flee in tears, but then I thought better of it and decided to appeal to her maternal advice-giving side. "But, Mom, what am I gonna do instead?"
"Oh, don't worry," she said cheerfully. "Something always comes up." Mom stared critically at her visage in the mirror, adjusting her mascara. "And you know, sweetheart, if you ever get short of money, if you're ever really desperate, you can always sell the necklace."
I never sold it, no matter how desperate my situation was (and it got pretty grim at times). I always kept the necklace in a compartment in my purse and carried it around with me wherever I went. I was terrified of losing it. It was only when I was married and settled in a house of my own that I felt confident enough to leave it in a jewelry box—and there it is, still.
**• • •**
**MOM** did give me one other unexpected gift that summer. It was after her show closed and we went back to Malibu. The Malibu house was actually a huge apartment building that Mom had built for herself. It was a two-story structure right on the sand, and she rented out some of the apartments downstairs. She lived upstairs, in an ever-expanding suite of rooms.
There were always guests in the other rooms, and some of them stayed so long they were getting their mail there. One such omnipresent couple were Phyllis and Eberhard Kronhausen, renowned sex therapists and art curators. They had written all kinds of books— _The Sexually Responsive Woman_ , _Pornography and the Law_ —and they'd organized a museum show, the First International Exhibition of Erotic Art. They were generally considered standard-bearers for the sexual revolution of the 1960s.
They had latched on to my mom in a big way. Mom embraced them as liberating figures, not just sexually but philosophically. Looking back, I think they gave her permission to be free with her life, permission she never got from her own very restrictive Baptist parents. Phyllis and Eberhard were anything but restrictive. Extremely full of themselves and their "anything goes" credo, they were eager to spread the gospel of the _Kama Sutra_. That's all they talked about: sex, sex, sex, in every conceivable facet and per-mutation.
They gave me the creeps. I think I was also jealous of the attention they got from my mom. It made me angry that they were sponging off her and using her as a meal ticket (I was even angrier some years later when I discovered that she bought them a farm in Costa Rica). I just wished there was some way I could protect her from these kinds of people.
I avoided them as much as I could. Luckily it happened that my old boyfriend Brad from Battisborough (who was now back as my new boyfriend) lived in San Francisco, just a few hours away, and he was visiting for the weekend. So I could spend all my time with him.
It was inevitable that at some point my path would intersect with that of our live-in guests. One afternoon, we found ourselves all colliding in the living room: Mom, the Kronhausens, Brad, and me. The grown-ups were looking very sophisticated and knowing. I sensed at once we had stumbled into a potentially volatile situation, so I tried to hustle Brad out of there as quickly as possible, but it was too late.
"Sachi, is this your boyfriend?" Phyllis asked.
"Yes, this is Brad." I made all the necessary introductions.
"You've been going together a long time?" asked Eberhard, fixing me with an owlish eye.
"About a year, maybe. Off and on. Here and there."
"Have you had sex yet?"
I felt my face flush. I didn't look at Brad—I didn't want to embarrass him any further. "Sex?" I responded, laughing lightly. "No! Of course not." I chuckled some more, just to underscore the point. This seemed like the only rational response: to treat the question like a joke. Obviously they were parodying their reputation as sexual gurus, right?
As I looked at Phyllis and Eberhard Kronhausen, and Mom, they all stared back at me with earnest curiosity.
"No? You've never had sex?" Phyllis asked. " _Ever_?"
"No," I admitted sheepishly.
"Don't you think it's time you did?" her husband asked.
Now I did look at Brad, because I instinctively felt the need for an ally. He had an incredulous what-the-hell-did-I-get-myself-into expression on his face. I don't think he'd had a lot of experience with bizarre eccentrics.
"You're how old—seventeen?" Phyllis asked.
"She'll be eighteen in September," Mom offered.
"Well," Phyllis said, with a perplexed shrug, "you're practically an adult. What are you waiting for?"
"You're at the bright dawn of your sexuality," Eberhard pointed out. "This is the moment to explore and experience."
I nodded politely. "Okay, well, we'll think about it." I took Brad's hand in mine, to show that we were pledged to making a serious effort.
"I think you should do it right now," Eberhard stated.
Brad and I instantly let go of each other's hands. "Now?"
"You know," Phyllis said, directing her comments to my mom now, "it would be a fabulous opportunity for Sachi, to have her first introduction to sex with all of us here as a support group. We could talk about it afterward—discuss what happened and why—and validate her feelings."
Eberhard agreed. "Guide her through the trauma, and celebrate the joy."
Phyllis was getting excited. "She could really benefit from our expertise."
Mom was leaning forward, lips pursed thoughtfully. I tried to read her face. Which way was she going to break? Surely she wouldn't agree with this nutty idea. Surely.
Now Mom nodded warmly. "I think it's a wonderful idea." _So much for surely_. "I wish someone had been there for me," she added.
I started inching toward the exit.
"And on a professional level," Phyllis said, "it will be instructive for us to observe this crucial rite of passage as it happens. It's a win-win."
All eyes then turned on me expectantly, waiting for my en-dorsement.
"Uh, I don't think I want to do this."
"But you _should_ ," Eberhard said with solemn gravity.
Phyllis chimed in: "Better this way than in the backseat of a car or in some dirty alleyway—no offense, Brad, I don't know you, but statistics show that for most people the first time can be very unpleasant. Lots of psychological scars result."
"It's not my first time," Brad said.
"Even better. You can help her through this, too."
"We're all here to help you, sweetheart," Mom said, beaming.
I felt like Mia Farrow surrounded by the Satanists in _Rosemary's Baby_. There were three powerful personalities bearing down on me, forcing me to their collective will, and I couldn't fight them off. How to negotiate this situation? I couldn't say I didn't want to have sex with Brad, because of course I did. Besides, how often do you find your own mother coaching you to lose your virginity?
"Well, I guess...if you think it's all right..."
"Of course it's all right!" Phyllis said.
"In sex, everything is all right," Eberhard assured.
As if to prove their point, they fetched some books from their room. These were books they had written themselves, about erotic art and sexual fantasies, and they were liberally illustrated. We sat on the couch and Phyllis flipped through the pages with me. "This is a good position," she suggested. "Oh, and this one allows for maximum penetration. You could try this one, too. You're young and flexible." Many of the positions seemed to entail a lot of squatting and bending and standing on your head.
"Take the book with you," Eberhard suggested.
So, having been given our assignment, Brad and I went down the hall to my bedroom. It was as dry and unerotic a prelude to sex as I could imagine, but I went along with it because I didn't want the Kronhausens to think I wasn't cool.
"And if you have any questions," Phyllis called, "we're right out here."
"We'll be waiting," Eberhard said.
My room was pretty cramped. We had to use the lower tier of a twin bunk bed. It would have been a lot more comfortable in Mom's expansive bedroom, but that would have been a little too weird, even for her.
We took off our clothes and climbed into the bunk. We weren't going to use any of the Kronhausens' fancy positions; it was hard enough to pull off the missionary position in that tight space. I remember being pushed up against the wall, my neck bent sideways, as we started going at it. There was no foreplay to speak of. People were waiting.
There was also no birth control to speak of. Brad didn't use a condom—I don't think I ever saw a condom until I was in my late twenties—and I certainly wasn't on the pill. At the time, I didn't realize how odd it was that my mother was encouraging me to have sex and didn't express any concern that I might get pregnant or contract a social disease. I guess everyone just got caught up in the giddiness of the moment.
Except Brad. I felt sorry for him. I don't know what was going through his mind. He couldn't have been enjoying this; it was more like an onerous duty. (Although I don't know that a seventeen-year-old boy could ever not enjoy sex.)
I was very thin by this time. I'd been losing weight steadily, and now, at five foot six I was about eighty pounds. Although I didn't realize it, I was suffering from anorexia. I don't know when it started—I'm tempted to think when I was put on that starvation diet after we lost the airline tickets, but I can't say for sure. Somewhere along the line, I had just stopped eating, and by this time I was pretty bony. I couldn't have cut a very seductive figure for poor Brad—there was very little there to arouse him—but he gave it the old college try nonetheless.
Suddenly I felt pain inside. "Wait, stop!"
"What's the matter?"
"Something hurts."
Brad apologized and said he'd try to be gentler. He started again, and— "Ow!"
I was terrified. Why was this painful? Sex was supposed to be a pleasant experience, wasn't it? Was there something wrong with me?
Brad, drawing on his measure of experience, explained the situation to me. I was a virgin. My hymen hadn't broken yet.
"My hymen? What are you talking about?" I really had no clue. They hadn't taught us about this at boarding school—and the celebrated sex therapists in the next room had forgotten to mention it.
Brad tried to explain. "The hymen is like a membrane that stretches across the—you know. And the first time you make love..." I eventually got the picture. He promised to go very easy. Over and over, though, he would bang up against it, and I would yelp, and he would pull back. It was getting annoying.
"Look," said Brad, "we don't have to do this..."
"But I want to," I said. I really did. I couldn't go back out into that living room and face my mom and her friends and admit failure. I wanted their approval. I wanted to be cool.
"Okay, well...On the count of three, then, I'll push really hard and break it, and it'll probably hurt a lot, but then it'll be over."
The scenario didn't sound too appetizing, but it had to be done. "Okay, let's do it." I lay back and braced myself against the wall.
Brad got into position. "One, two, _three_!" He lunged into me, and oh my god! It hurt like hell. I thought I was going to die. It was all I could do to keep from screaming.
But it was done. Brad held me tight. He felt so sorry that he'd hurt me. He kept saying, "I love you, I love you," over and over.
Then, to my horror, I looked down and saw blood on the sheets. Now what? Had he perforated something important? Was I having an early period, or hemorrhaging to death?
Brad calmed me down and reassured me that it was all very natural. I really was so lucky to have had such a patient, understanding guy for my first time.
And he was right: it didn't hurt anymore after that initial thrust. In fact, it felt pretty damn good.
Once our mission was accomplished, we had to face the next hurdle: reporting back to the Kronhausens. We dreaded the thought; there was something so weird and cultish about them. We hid out in the bedroom until we heard a light knock on the door, and Mom's voice: "Is everything okay in there?"
We emerged from the bedroom and went out to the living room, where Phyllis and Eberhard sat placidly on the couch.
"So, how did it go?" Phyllis asked.
"It was great."
"Did you achieve climax?" Eberhard asked.
We both nodded vigorously. They smiled smugly.
I hated them.
**• • •**
**THE** one positive result of this strange interlude was that I fell madly in love with Brad. He had been a model of chivalry throughout the ordeal, and now I was crazy about him. Brad was my first real lover, and I wanted him to be the only one—and he wanted me.
So after he went back home to San Francisco, I made plans to join him. We were going to live together.
I told Mom the good news, fully expecting her to share in the excitement. She didn't.
"You're only seventeen. You can't move in with a man. You're not ready."
"I was ready enough to sleep with him. It was your idea!"
"That's different," she contended. "I was helping you become a woman. Sexual maturity and emotional maturity are two different things. You don't want to get tied down in a serious relationship now. You have your whole life ahead of you."
"But it's not like I have any other plans. I'm not going to college," I helpfully reminded her.
She ignored this point. "Besides, you don't love him."
"Yes, I do!"
She smiled understandingly. "I'm sure you have feelings for him. But let's give it some time. If he really loves you, he'll wait for you."
I didn't want to wait. I was in love with Brad, and we were going to live together. Mom had pushed us into bed, and now she would have to live with the consequences.
A few days later, as I was preparing for my great life move, Dad called out of the blue from Japan with some exciting news: he'd lined up a job for me in Tokyo, as a television newscaster.
A newscaster! On television! I couldn't believe it. I'd never really thought about doing anything in show business—not since my role in _To Kill a Mockingbird_ got shot down—but once the offer came in, I suddenly realized that it was the one thing in life I wanted to do. _Had_ to do. I knew I'd probably gotten the job by trading on my mom's name, but I didn't care. It was a gig! I was going to be a star!
Yes, I would now have to give up the idea of living with Brad in San Francisco, and that gave me serious misgivings.
"Hey, opportunities like this don't come around that often," Mom said. "You have to jump on them." She reminded me, "If he loves you, he'll wait."
Armed with these time-tested clichés, I called Brad and told him I was heading back to Japan, and we'd have to postpone moving in together, but it would only be for a short while, and if he loved me...It all sounded like total bullshit. I knew if I left, things with him would be over, and so did he.
I think it broke Brad's heart when I went back to Japan. I know it broke mine. He was the love of my life—and it would be a long, long time before I found another one.
**• • •**
**WHEN** I arrived back in Tokyo in the fall of 1974, there was no one waiting for me at the airport. That struck me as a little odd, but—whatever. I took a cab to the house—and there was no one waiting there, either. In fact, there was no one waiting for me in the entire city. Dad and Miki were gone.
What about my newscaster job? I called the television station. No one seemed to know what I was talking about.
I was getting that stranded feeling again. In my heart, I sort of knew that my parents had used this job as a pretext to get me away from Brad. Still, I'd assumed that at least there would _be_ a job.
At this point, I guess I could have flown back to Brad and continued the life I'd dreamed of, but I didn't. For one thing, I didn't have the airfare. More significantly, I was still being guided by my Japanese upbringing, which, in the face of all obstacles and disappointments, counseled stoic acceptance. My father had summoned me back home, and it wasn't in my nature to question that. If I were to leave now and go against his unspoken wishes, he would lose face. Being Japanese is a complicated business.
So as I'd come to Tokyo for work, I would simply find work. As it happened, the local noodle shop was looking for waitresses. I'd never waitressed before, but I'd never been out of school and broke before, either, so I took the job.
Noodle shops are a staple of Japanese casual dining, much like coffee shops in the States, and they're not to be confused with teahouses. While the customers in both emporiums are overwhelmingly male, they don't go to noodle shops for relaxation or ritual observances or stimulating conversation. They go for noodles and sake—and that's it. There are no geishas flitting about, being charming and flirtatious, and floating the vague possibility that there might be a little something extra available for the right price. The men wouldn't be interested anyway; they kept their noses buried in their noodle bowls, and for the most part they left the waitresses alone—but not me.
Maybe because my body was more substantial than that of the average noodle waitress—even with anorexia, I was bigger and curvier in certain critical areas—but I was constantly getting pinched and pawed and squeezed by the clientele. I tried to maintain a polite smile and ignore them, but it was getting increasingly hard to serve the soup when I was being fondled at one end and goosed at the other.
The other waitresses were sympathetic to my plight, but also a little envious. Their behinds were generally flat, while my all-American ass stuck out like an increasingly sore thumb. "How does it do that?" they'd marvel in the dressing room as they helped me put on my kimono. All the waitresses had to wear kimonos, and there were several layers, which had to be draped just so. It took me a while to get the hang of it, but even after I did, they still helped me dress. It became a bonding ritual between us. I loved hanging out with them. The dressing room was like a women's haven, where they could relax, have a smoke, and talk about their lives, and their husbands, and sex. When I complained about being pinched, they tried to figure out ways to flatten my breasts and behind, wrapping them tight with linens and so forth. Nothing helped. I still got groped. More than once the chef came charging out of the kitchen, cleaver in hand, to protect me.
I didn't stick with the noodle shop long. I heard that there was a need for ski instructors up north. I'd been skiing ever since I was a child, and my tenure in Switzerland had really sharpened my skills, so I got a children's ski instructor license and spent the winter of 1975 in the Japan Alps teaching kids how to ski.
When the season ended, I went back to work at the noodle shop. The gropers were still in season, so with summer approaching, I started looking for a new job somewhere. Anywhere.
Anywhere turned out to be New Zealand. It was winter down below, the skiing was sensational, and they needed ski instructors. I took the next plane.
If you've seen the _Lord of the Ring_ s movies, you know how spectacular the New Zealand scenery can be. When I landed at Christchurch and took the bus southwest to Queenstown, I was astounded by the natural beauty surrounding me on all sides. Towering snowcapped mountains, primeval forests, glacier-cut fjords—you name it, they had it. Queenstown itself was a beautiful resort town on the shore of Lake Wakatipu, right at the edge of the Southern Alps. The lake was so crystal clear you couldn't tell the mountains from their reflection in the water. Gold had been discovered there in the 1860s, and the ensuing rush created a boom for the area. They still had some working mines when I was there.
The head of the ski school in Queenstown was a young American named Larry Lasch, who also taught in Vail. I got my adult ski license from him, and taught throughout the summer. The skiing in New Zealand was beyond amazing. The mountains were so high that, instead of taking the endless tram lifts, people would fly up to the summits in helicopters and ski all the long way down.
The setting was incredible; the pay wasn't. I didn't make enough to stay at the ski resort, so I found lodgings on a nearby sheep ranch. The owner, Hildy, was a single woman in her sixties, tough and independent, with sinewy arms and a weather-beaten face. She had hundreds of sheep on her ranch, and she was busy looking after them from dawn till dark. So she was more than happy to let me stay there for free, if I cooked for her.
I cooked a lot of lamb. She had chickens on the farm, so every morning we had fresh, warm eggs with bright orange yolks. The butter was delicious; Hildy churned it herself.
Hildy also made classic New Zealand sweaters, from her own homegrown wool. She taught me how to shear the sheep. I would hold the sheep under the crook of my arm and take a pair of shearing clippers and run them right along the skin of the sheep. I still remember the feel of the soft wool as it gently fell over my hand and piled up at my feet. We would gather the sheared wool into huge baskets and carry it into the spinning room.
Hildy and I would sit by the spinning machines and pull heaps of raw wool from the basket and shape it into a ropy string. We'd feed the string into the spinner, and it would come out as spun wool. It was still in a natural state, sticky with lanolin oil. Hildy would knit the wool into chunky sweaters, which were super-warm and so comforting. The sweater wool was grayish and dirty, but you couldn't wash it, or you'd lose the lanolin, which repelled water and insulated you from the cold.
Hildy gave me one of these sweaters as a gift, and I wore it constantly. The smell and the sticky feel of the lanolin reminded me of those trips I made with my dad up into the mountains of Japan, when we'd go ice-fishing and sit huddled in the tent while he spun magical tales of his past.
I missed him. I wanted to see him again.
**• • •**
**AFTER** three months in New Zealand, I returned to Tokyo. Dad wasn't there, but he had left a forwarding phone number.
"Hey, Sach the Pach! What's up?"
"Hi, Dad. Where are you?"
"I'm in Hawaii. I'm doing some business."
"When are you coming back?"
"Oh, I don't know. Soon."
Soon could be next week, next Christmas, or whenever he got bored with the palm trees. I didn't want to wait. I was on a personal crusade to reconnect with my family, and I wasn't going to be denied. I scraped the plane fare together and went down to Honolulu. I knew Dad kept a suite at the Halekulani Hotel on Waikiki Beach. Wouldn't he be surprised when he saw me?
Once again, Dad surprised me. He was already gone. Where, I had no idea.
So now I was stranded in Honolulu. Not the worst place to be stuck, granted, but when you're broke and alone, a picture-postcard sunset can offer only so much in the way of gratification.
I was really starting to feel sorry for myself. Nobody cared about me, nobody loved me! I was swimming in self-pity, and I desperately wanted attention.
So I came up with an ingenious idea: maybe if I pretended I was sick, my parents would rush to my side to help me. It worked in the movies all the time, right? I couldn't make it anything really serious, because I wanted to recover fairly quickly, so I fell back on my Queenstown experience and manufactured a skiing injury: I'd messed up my knee badly, it was dislocated or torn up or something, and I was in real pain.
It was a good story. I called Mom.
"You hurt your knee skiing?"
"Yes," I said, adding a little tremble to my voice.
"You hurt it in New Zealand, and it's just bothering you now?"
"Well, it was always bothering me, but now it's getting much worse."
"Have you seen a doctor?" she asked.
"Uh—no..." This was not going exactly the way I planned.
"Well, go see a doctor, for Christ's sake."
"I can't afford a doctor," I quickly replied.
There was a silence on the other end. I was half-afraid she was going to tell me to sell my necklace.
"I'll pay for it," she finally said. "Just make sure you send me a copy of the bill."
That was as far as we got. At least she was offering to pay for something I didn't really need.
Now I was in a bit of a fix: I had to get a doctor's bill to verify that I'd been injured in the first place, or Mom and Dad would know that I was lying (which would only confirm their earlier suspicions about my suspect character).
So I went to one doctor, and another, and another. They couldn't find anything wrong with my knee (perhaps they should have been checking my head). I knew they wouldn't, but I felt that now that I had talked my way into this situation, I had to see it all the way through. The more doctors I saw, the more impressive my injury would appear. Eventually I would be miraculously cured, that would put an end to the episode, and we could all move on.
It was the fourth doctor who threw a wrench in the works. He examined my knee carefully, bending and extending it over and over. He took a barrage of X-rays. He made some very ominous-sounding grunts and mutterings.
"Miss Parker," he finally intoned grimly, "it appears that you have a torn cartilage in your knee."
"I do?" I tried to mask my surprise. "In my knee?"
He held up the X-ray, pointing to the seam of my knee. "See right here?"
I couldn't see anything. There wasn't supposed to _be_ anything. "So, is that bad?" I asked.
"Well, it's going to require surgery sooner or later. Sooner is better."
"Surgery?" I looked at him, trying to see if he was hiding a tiny smile, but he was very grave and doctorly. "Are you sure?"
"Oh, I'm sure."
So I had the operation.
I knew it was completely unnecessary, but I couldn't say that because I didn't want the doctor to think I thought he was pulling a scam, which of course I did because he was. But I was pulling a scam, too, so we were even. Besides, maybe there _was_ something wrong with my knee. How did I know?
There's an underlying irony here that eluded me at the time. One of the most famous of showbiz stories is how my mom got her big break because of a bad leg. Carol Haney, the dancing lead of _The Pajama Game_ on Broadway, broke her ankle, and Mom came out of the chorus to replace her. The popular story goes that Hollywood producer Hal Wallis was sitting in the audience that very night, immediately spotted her star potential, and signed her, a complete unknown, to a movie contract.
Actually, that happened a couple of months later—she had already been playing the role to acclaim and was pretty well known by the time Wallis saw her—but the basic story is the same. One minute a nobody, the next minute a star, and soon she was doing her first film with Alfred Hitchcock. Sometimes it happens that fast.
Maybe, subconsciously, I was hoping for history to repeat itself. My bad leg would remind Mom of how she had profited over someone else's misfortune, and she would overcompensate for the lingering guilt by showering me with maternal affection, and I would finally become important in her eyes...
Or maybe I was just being an idiot again. Either way, it didn't work. Even after my operation, all I got was a "Get Well" telegram.
When I went for a post-op checkup, the doctor was considerably pleased with his work. "That's coming along just fine. But you know, you'll probably have trouble with that knee down the line."
"I will? But you just fixed it."
"Yes, but once you go into that area and start cutting around...things get compromised. Eventually you might need a knee replacement."
He got that right. Thirty years later, I had surgery for the partial replacement of a knee that was never damaged to begin with.
# **Chapter 7**
# Into the Wild
Once I stopped limping, I went to work at the Halekulani. My father maintained a suite there year-round, and he and Miki stayed there whenever they were in Honolulu. Dad was treated at the hotel with all the cachet of a visiting dignitary: he would saunter through the lobby, shaking hands and doling out tips, with the same stylish swagger that my mom commanded on a Vegas stage. This was his kind of show business, and he loved it.
In spite of Dad's rock star status at the Halekulani, however, I couldn't even land a waitressing job there. It was very upscale, they hired only the best, and my noodle-shop résumé failed to impress. I was offered a job as a bus girl, and I humbly took it, even though it paid almost nothing. Until I could gather enough cash in tips, I had to live in a seedy part of Honolulu, at the apartment of one of the hotel waitresses. Her name was Shigeko, she was a Nisei (second-generation Japanese American), and she let me stay for free, until I could get back on my feet.
Of course, I could have just stayed in my dad's suite, but he didn't make the offer. I guess he considered it his private sanctum. Besides, while I often made room-service deliveries to the other guests, the management made it clear to me that Dad's suite was off-limits to me. Whatever his business was, he didn't want me sticking my nose in it.
I didn't care. I was close to him; that's what mattered. At least we would be together on my nineteenth birthday.
That very morning, I got a call from Dad. "Sach! Happy birthday! Listen, I wanted to do something special with you tonight."
I recalled all those endless, tedious nights in Tokyo bouncing from nightclub to nightclub. Now I was old enough to enjoy them. "Okay!" I said eagerly.
"But I can't. I'm in Italy."
"Italy?"
"Business trip." I could hear now the long-distance sound in his voice, a little tinny and displaced. "Sorry I can't be with you on your birthday. I feel terrible."
"That's okay..."
We were interrupted now by the operator, a woman who spoke in Italian, and she and Dad traded a few Italian phrases. Long-distance calls in those days were not the smooth exchanges we have today. There was always an operator, the sound was scratchy and crackly, and every now and then there would be a _beep-beep-beep_ —all of which I listened to now as I waited for him to get back to me.
"So when are you coming back, Dad?"
"Oh, next week, maybe," (scratchy crackling) "or the week after" ( _beep-beep-beep_ ). "But listen, honey, you just have a great day today, and I'll make it up to you. Okay?"
"Okay..." Click. Dead air. He was gone.
So I was spending yet another birthday alone.
Still, I wasn't about to feel sorry for myself. Last time I tried that I wound up in surgery. Instead, I decided to give myself a little birthday treat.
Since Dad was off in Italy, it seemed the perfect opportunity to breach the sanctum. What harm could it do, after all, just to sneak into his room and poke around a bit?
I charmed the key from the hotel clerk and took the elevator up. I felt a momentary qualm as I slipped the key into the lock, but I knew this would be my only chance.
I pushed the door open, and peered in. I expected to be overwhelmed by lavish furnishings, spectacular views, solid-gold bathroom fixtures, that sort of thing.
Instead I was overwhelmed by a thick cloud of marijuana smoke as it billowed toward me. There was pulsing jazz-rock music playing on the stereo, the kind you might hear in a porno flick. As I moved into the drifting haze, I became aware that there were a bunch of people in the room, and very few of them were wearing clothes. Naked bodies were bouncing up and down furiously on the bed, in a merry synchronized humpfest. In the tangle of limbs I couldn't tell if they were men or women or what—but I knew one of them was Dad. Back from Italy in record time.
I ran from the room in horror. Luckily Dad never noticed me, or if he did, he never let on—and of course I never busted him on it. Somehow, my sneaking into his room and spotting him in a pot-smoking orgy seemed a far greater offense than his lying to me about being in Italy on my birthday.
What amazed me most in retrospect was the way he'd faked that phone call, with all its long-distance authenticity. The scratchiness, the tinny voice, the beeping...and I'd _heard_ him talking to an operator. Who played that Italian woman? Was he doing all the voices himself? Who was the real actor in this family?
**• • •**
**DAD** owned land on the Big Island of Hawaii, in Napo'opo'o, where it is said that Captain Cook first landed when he discovered the Hawaiian Islands. How Dad was in the position to own such an important piece of real estate, I don't know. That was just his way.
One time I visited Napo'opo'o with him and we took a drive in his white Jeep Cherokee to Kailua, about a half hour away, and cruised around looking at the scenery. I thought it was just a spur-of-the-moment outing, but then he pulled into a parking lot outside an accountant's office. He said he had some kind of business he had to discuss, and he told me to wait for him in the car. I didn't want to wait in the car: it was too damn hot, and there was nothing to look at, and I was bored.
Dad was annoyed, and he got very short with me: "Stay there!" he snapped.
So I sat in the car while he went inside. While waiting, I noticed a manila envelope on the front seat. The flap was open. _I shouldn't look in there_ , I thought. But it was hot, and I was in a bad mood, and I wanted to be entertained by something. So I peeked in the envelope.
Inside were a bunch of loose, glittering gemstones. Diamonds, to be precise. There must have been a hundred of them. I'm no diamond expert, so I didn't know if they were cut, or finished, or raw. Yet they were diamonds. I knew that.
Why would Dad have an envelope of diamonds in his front seat? Whom did they belong to? Where were they going? Was this part of Dad's business? What business?
I didn't know, and I wasn't going to ask. What I did know was that just one of those diamonds would have come in handy for me right around then.
**• • •**
**AS** soon as I could afford it, I moved out of Shigeko's apartment and got a place of my own. She was a lovely woman, and I was tired of mooching off her. On the other hand, I couldn't quite swing the deposit on my new place, so Shigeko helped me out by paying the first and last month's rent, no small sacrifice on a waitress's salary. God bless Shigeko—she was one of those uncanonized saints, like the prostitute in Trieste and the Yugoslavian couple, people who came into my life at just the right time and gave me an enormous lift and a sense of hope when I really needed it.
And I needed it now. I was living in perhaps the most dangerous part of Honolulu. I had no money and no friends. I remember this time as being perhaps the lowest point of my life.
I could barely afford to eat. At the Halekulani we would get one meal a day, lunch, and I made sure never to miss it. Aside from that, I would often stroll through the mall, where the food merchants gave away free samples. I'd go from stand to stand gathering up samples; that would be my dinner.
At the coffee shops, there would be no charge for refills, so I would drink five or six cups of coffee in a row. In my anorexic mind-set, I thought this was good for me. The laxative powers of coffee helped me to purge myself of the toxins, the pain, the loneliness. Every trip to the bathroom was an opportunity to feel clean and emptied out, the darkness gone. In my situation you had to take the positives wherever you could find them.
Still, there was no escaping the depressing reality of Apartment 315. Little more than a tenement apartment, it was a tiny space infested with roaches. I cleaned and cleaned, but there was no getting rid of them. I couldn't afford bug spray; and anyway, being Buddhist in philosophy, I didn't want to kill a living thing. So I gave them all names instead, and we did our best to coexist.
My apartment overlooked a tiny shack right below, where a family of native Hawaiians lived. Every single night, they would have a barbecue in their backyard, and every night, I would watch them from my window and live through them vicariously, peeking through the leaves of the palm tree that framed my view. They had a big family—big in every sense; there wasn't a thin one in the bunch. The aunts, uncles, and cousins would come over every night, swelling the ranks and bringing food for the pot luck dinner. It was always a huge feast: poi, lomi lomi salmon ceviche, banana leaves stuffed with pork...They'd party into the night, drinking and getting rowdy, and dancing around the bonfire. They were as poor as all get-out, but they seemed immensely happy.
I wanted so badly to join them. I was hoping someone would notice me and call me down from my balcony, just like in the movies. I would eat and dance and become part of the family, and marry the handsome chubby son, and we would hula off into the sunset.
It didn't happen. I stayed apart and alone.
I remember one night sitting on the floor of my apartment watching the roaches scurry along the wallboard, feeling utterly empty. I stared at my phone, waiting for it to ring—which was a futile exercise, because the service had been cut off for nonpayment. Didn't matter. Nobody ever called anyway. Next door, the party was raging into the late night, everyone in the family laughing and getting rowdy, and reminding me that I had no family of my own.
I knew suddenly, with fierce clarity, that I should kill myself. My life was a failure, nobody cared about me, I was completely alone and forgotten. So why go on? Who would miss me? Who was I kidding?
I wasn't sure how to go about it. I didn't have any sleeping pills, so I couldn't try Mom's method from _The Apartment_. There were knives in the kitchen, of course. I could probably hang myself with the belt from my robe. Oh, if I only had some bug spray...
There were lots of possibilities. As it happens, I was too depressed to do anything about it. So I just sat there, all through the night, thinking about being dead.
**• • •**
**THEN** everything changed, in the blink of an eye. Two eyes, actually, both gorgeous—and they didn't belong to a Hawaiian or a Nisei; they belonged to an Australian.
His name was Luke Garrett, and the minute he sat down in the hotel restaurant, I knew I wanted him. He was Hollywood handsome, with blond hair, broad shoulders, and a great tan. All I could think was, _Oh my god, who is that?_
As it turns out, I already knew who he was. We had met years before, in Australia, when I was about twelve or thirteen. My dad was business partners with Luke's dad, who had a cotton plantation in Weewaa, which proudly calls itself the Cotton Capital of Australia. The plantation house reminded me of Tara in _Gone With the Wind_ , with its Corinthian columns and winding staircase, crocodile-filled bayous, and black Aboriginal laborers doing all the menial work. The surrounding countryside was barren and dusty, with unpaved dirt roads and far-off mountains. Whenever I see old Westerns on TV, I think of Weewaa.
Young Luke was a teenager then, sixteen or seventeen. I probably had a little crush on him, but it was nothing earthshaking.
This time was different. Now I was seeing him from a fresh, sexually informed perspective, and the earth was shaking plenty.
At first I was a little embarrassed; I didn't want Luke to know I had turned out to be a bus-girl. At the same time, it gave me a perfect excuse to engage him.
I wanted to look my best when I did, so I put on some extra makeup, hiked up my skirt, and flashed my most charming smile as I approached him and bent over him: "Coffee, sir?"
He nodded, and I promptly spilled hot coffee on his pants. Oh, totally on purpose. It was very artfully done—all around the crotch, without burning anything important. I sputtered apologies as I dried him off, conscientiously patting down the area in question...Then I looked up at his face. "Why, aren't you Luke Garrett?"
He flashed his own charming smile back—and just like that, we were off to the races. I don't know if you could call it a whirlwind romance, but it moved mighty fast. A few dates, and then Luke had to return to Australia. Then he came back to get me—and asked me to marry him.
I never said yes faster in my life. I couldn't believe my fairy-tale luck. I was Cinderella, rescued from the drudgery of busing tables by a genuine Prince Charming, who would sweep me away to his enchanted kingdom across the sea.
Dad was delighted with the match. He remembered Luke from years before, and he was very pleased that our families would be united. Even Miki seemed to approve, which surprised the hell out of me. She never ever wished me well, but now she was beaming like a proud mom. I guess that was Luke's roguish Aussie charm at work. He could win anybody over. They even started inviting us to hang out with them. I got to see Dad's suite again, without the naked bodies.
As much as we enjoyed the perks of a comfortable lifestyle, we couldn't tarry in Hawaii. We had a new life waiting for us, Down Under.
**• • •**
**I'D** been to Australia before, not only as a child, but more recently: piggybacking off my New Zealand ski instructor experience, I'd traveled on to the Snowy Mountains region in New South Wales. It was only for about a month or so, but it was memorable.
At the Thredbo ski resort, I was looking to teach, but there were no jobs open, so I wound up working as a maid at a local hotel. There was a whole team of maids, and we rotated jobs. My job was to do the beds, which was a step up from cleaning the toilets. On one particular day, while I was making up a bed, I noticed a curious substance on the sheets. It was creamy, like a gel, and as I bent to smell it, it had an odor sort of like fresh-mown grass.
I had no idea what it was, so I called in the other maids. "Have you ever seen anything like this before?"
They all exchanged knowing glances, and smiled. "Uh...yeah."
"What is it?" I asked innocently.
They laughed, and explained to me an essential component of the reproductive experience. I was amazed. I knew about sex, of course—I'd engaged in the act myself—but I didn't know that this creamy, grass-smelling stuff came out of men. Brad had never mentioned it in his hasty tutorial, and I wouldn't have noticed it anyway, what with all the blood.
That just goes to show how out of the loop, how utterly clueless, I was about sex. I knew it was fun, I knew I enjoyed it, but I didn't understand the mechanics of it, the implications of it, or the powerful omnipresence of it. Sex was all around me, and I just never saw it.
For instance, there was a handyman who worked at the motel, an older guy who was seedy looking and a bit creepy. He was always hitting on me, but I didn't really pick up on the signals. I wasn't interested in him, so I assumed he wasn't interested in me. It was only when he cornered me in a guest room and, in what I assume was a gesture of seduction, opened his pants and showed me the crabs around his penis, that I realized his true intentions.
I quit the motel that very day and moved to the nearby ski resort of Perisher Valley. Here I was a ski instructor by day and a waitress by night. The ski-lift operator was an American named Jay, who was a little older, in his mid-thirties, and sort of a hippy-dippy type, with long hair and a beard. Jay lived across the hall from me, in the rooms above the restaurant. I was rooming with a fellow waitress, Katie.
One night after work I came back to my room and found Katie in bed with a guy, having boisterous sex. Without missing a beat, she turned her head to me and said, "Get out!"
I hurriedly shut the door and found myself in the hall. I was exhausted from my shift, and now I had no place to sleep. So I knocked on Jay's door.
"Sure, you can sleep here tonight," he said. Unfortunately he had only a twin bed. He seemed like a gentleman, so I assumed it was safe.
And it was. We both climbed into the small bed, and Jay spooned me from behind. I remember I had two long braids at the time, and he held them as he wrapped his arms around me. We stayed that way all night, and nothing happened.
This seemed unremarkable to me at the time, but in retrospect I realized that Jay had acted with exceptional decency (and restraint). When I went back to Australia years later, I made a point of visiting Perisher Valley and thanking Jay for that night. He told me it had taken every bit of his willpower to hold back, but I was so innocent, he just couldn't take advantage of me. So Jay goes into the small pantheon of Nice Guys.
I wish I could put Luke in that class, too, but I can't. He wasn't such a nice guy, as I found out a little too late.
**• • •**
**LUKE** had once been a sheep rancher, but now he was a vintner—he owned a thirty-seven-acre vineyard in Pokolbin, in the Hunter Valley, the wine area of New South Wales. We moved there in the spring of 1976. Like much of Australia, it was starkly beautiful: rolling hills with mountains in the distance. The house was a two-story with a veranda, and while it was reasonably modernized, there was no indoor bathroom; we used an outhouse. There was also no dryer, so the laundry had to be hung out on a line to dry. I grew to love the smell of the clean air-dried sheets and clothes. Plus, there were chickens on the property, so we always had fresh, warm eggs.
Yet, it was a lonely place, in the middle of nowhere. The wind was always blowing. Still, it was the kind of life I enjoyed, simple and elemental. I was happy there.
Except when I was working at the nearby wine factory. Now, Luke was a proud man. I'm sure he would have balked at seeing his wife work—but I wasn't his wife. We weren't married yet; we hadn't even set a date. That being the case, he saw no reason why I shouldn't earn my keep. So he got me a job working on the factory line, putting labels on wine bottles. It was droning, stultifying work, and I hated it. I was ready to blow my brains out, but I did it for love.
**• • •**
**MY** anorexia had continued unabated all this time, and by now I was down to eighty-two pounds. Yet it wasn't enough. I still felt that I was too fat. I needed to lose more weight.
I picked up a book called _Dr. Atkins' Diet Revolution_ , which had been published just a few years earlier. I was delighted to see that I could eat all my favorite foods with this diet—steak, eggs, bacon—and was guaranteed to lose weight. How cool was that?
So I went on the Atkins Diet, and gorged myself on protein—and immediately, I started gaining weight. I couldn't figure out what was going wrong—what kind of a stupid diet was this?—but I was enjoying the food too much to stop. Before long, I was back to a normal weight, and I was never anorexic again. You could say Dr. Atkins saved my life.
I don't mean to minimize in any way the seriousness of anorexia. It is a terrible, ravaging illness, and in its own insidious way a form of suicide; I was extremely lucky to escape it when I did, without any treatment, and I feel an immense empathy for those who struggle with it.
I was also lucky to find Robert. Robert—who was French, and whose name, therefore, enjoyed the elegant pronunciation "Ro-BEAR"—was the chef-owner of Robert's, a first-class restaurant down the street from the vineyard. The minute I arrived in Pokolbin, I went down to Robert's and got myself a job as a waitress. I'd paste labels at the wine factory in the morning, and then hop on my bicycle and ride down to Robert's for the lunch and dinner shift. Between Dr. Atkins and Robert's rich gourmet cooking, I got healthy very quickly.
Robert's attracted all kinds of customers: Australian ranchers, wealthy visitors from Sydney, tourists from around the world. Robert and his wife, Sally, were superbly accomplished restaurateurs. Their food was star quality and a little expensive. It was always amusing when some of the locals came in to order pub food. Once they got a good look at the prices, there was many a hasty exodus.
One fine day four cowboys sauntered in and took a table. They were fresh from the fields, dusty and sweaty, and already a few pints in. They looked over the menu and, without blinking an eye, settled on Chateaubriand for four. "And make it well done," said one of the cowboys.
Now, there are a couple of ways to cook Chateaubriand: rare and medium rare. Anything beyond that is inviting disaster: the meat shrinks to nothing, and the quality is ruined. I tried to explain this to the cowboys. "You know, Chateaubriand is supposed to be pink. If you cook it too much, it spoils the whole experience."
They didn't care. "We want it well done."
"Well, maybe you should order something else well done. Like a sirloin or a rump steak."
They grew a little testy. "We want _Chateaubriand_."
"And we want it _well done_."
I smiled brightly. "Okay, I'll talk to the chef."
I really didn't want to talk to the chef. Robbie was a sweet, delightful man, but he was also a classic temperamental Frenchman, and _très_ passionate about his food and his reputation. He would cook his dishes the right way or not at all.
He fumed as I explained the request to him, his cowboy boot tapping petulantly on the floor; Robbie always wore cowboy boots in the kitchen. "I tried to steer them to something else," I told him, as he glared at me. "But they want Chateaubriand, and they want it well done."
Robbie swallowed his outrage and gave a Gallic shrug. He then proceeded to make the Chateaubriand exactly the way he wanted: medium rare. Then he poured an extra layer of Béarnaise sauce over the sliced meat, so they wouldn't notice.
When I brought the dish to the table, sumptuously prepared and beautifully presented, the boys were generally unimpressed. One cowboy spooned the sauce aside contemptuously and looked at the slice of meat in dismay. "What the hell...? This meat is rare. We want it well done!"
"I'm sorry," I said, "but the chef says it will ruin the dish..."
"We don't care! Who's paying for this? We want it well done!" They sent it back to the kitchen.
Robert was not pleased, but he grudgingly accepted that he wasn't dealing with informed gourmands here. He put the Chateaubriand back in the oven and cooked it to an arguable medium. Any more than that, and he wouldn't have been able to live with himself.
I crossed my fingers and brought the compromised Chateaubriand back out to the cowboys. Still too pink. They rejected it in unison: "We want it well done!"
I timorously returned to the kitchen with the meat. By now Robbie had reached his limit of understanding. "They want it well done?" he exploded, his neck veins popping. "I'll give them well done!" He took the individual slices of meat, threw them on the floor, and stomped on them, one by one, with his cowboy boots. "There! There!" he screamed. "Well done! Well done!"—and he launched into a string of French obscenities as he stomped, stomped, stomped. I didn't understand what he was saying, but I knew exactly what he was saying.
Then he scooped up the meat and threw it back into the oven, and he cooked the slices until they were black and charred and looked like hockey pucks. "There. _Now_ it's well done." He drizzled on some token béarnaise sauce and handed me the platter. "Give it to them."
I was horrified. I couldn't serve this mess to them now.
" _Give it to them_ ," Robbie insisted.
I nervously placed the charred Chateaubriand on the Aussies' table. "Here we go," I said cheerfully. "Well done!" Then I quickly retreated to the safety of the kitchen.
The cowboys leaned forward, studied the blackened meat curiously, and inspected it from all angles. Then they started eating. We all watched from the kitchen door in disbelief.
They loved it.
**• • •**
**I** was so happy at Robert's. Sally, Robert, the whole staff—they were a real family to me. That's why I worked two shifts, to be honest. I would rather have hung out there than gone home to my fiancé.
I want to be fair to Luke. He was a very sweet and thoughtful guy when he wasn't drinking—but he was often drinking. It was part of the culture out there, and Luke was nothing if not cultured. He had a very short temper, and was prone to explosive outbursts.
I first appreciated the extent of his volatility one evening when I was sitting at the kitchen table and he walked out of the bathroom. "What's this?" he asked.
He was holding something by the tail—it looked like a white mouse with a blotch of red on it. I looked closer and realized, to my mortification, that it was a used tampon.
"What's this?" he repeated, dangling it right in front of my eyes. "What's this?"
"That's mine..." I said meekly.
"I know it's yours!" he screamed in my face. "Do you know where I found it? Do you? On the edge of the _bathtub_." He spat the word out, to underscore the egregiousness of the offense.
"Oh. I guess I left it there." I reached for the tampon, but he pulled it away.
"Is that what you guess? You guess you left it there? I guess you did, too. I know _I_ didn't leave it there." He was looming over me, swaying slightly, as if he couldn't contain the anger roiling within him.
"Okay, well..." I reached for it again, and he flung it across the room.
"Don't touch it. It's disgusting! You think I want to look at that after I come home from a hard day at work?" He was reminding me of my father, asking questions that I wasn't supposed to answer, and then waiting for me to answer, and then hoping it would be the wrong answer so he could attack me again.
I knew there was only one thing I could say that would satisfy him. "I'm sorry. I'll never do it again."
"You're goddamn right you'll never do it again! Leaving your female shit lying around, as if you owned the place. You show me no respect. No respect!" He overturned the kitchen table, the dishes and silverware clattering to the floor around me. Then he stood waiting, hands on hips, as if daring me to say anything at all in my defense.
I didn't. Because, in a way, I understood. He was a man, he was a proud product of his chauvinist rancher society, and he shouldn't have had to be exposed to inferior womanly things. It was insulting and emasculating.
And I was a product of _my_ society, which had taught me to be submissive and accepting, and protective of the male ego at all costs. So I just wept quietly, and kept my head low. After a moment, I heard him sigh with disgust, and mutter, "Clean up this mess." He stalked out of the house and headed for the vineyards.
I know I should have bolted then and there but this was my first serious relationship, and I didn't know any better.
So I stayed with Luke. I don't know why. It wasn't for the sex, because we didn't have much. When we did, it was fast and furious. Mostly fast. Not a lot of foreplay: ten seconds, maybe. "Brace yourself, baby!" Wham, bam. "Now feed me."
Then I'd hop out of bed and make him steak and eggs. Anything to make him happy.
**• • •**
**ONE** day, in my continuing aspiration to be the perfect housekeeper, I was cleaning up Luke's bedroom, putting away his laundry, when, in the bottom drawer of his dresser, I found a sheaf of letters, hidden away. Curious, I took them out and started reading.
They were love letters. Sexy, impassioned love letters. To Luke.
From Miki.
I was staggered. What? _My_ Miki? The evil stepmother? She was sending love letters to my fiancé? How could that be? Was he in love with her? Could such a thing even be possible?
I read the letters in disbelief. They were stuffed with high-flown romantic sentiments along the following lines: "My dearest love..." "My one and only..." "Every time the sun sets, I think of you..." There was some explicit sexual stuff in there, too. Just the thought of Miki and Luke engaged in such intimate couplings, even on a fantasy level, made me positively nauseated.
I felt sick in every sense. It was like being hit by a train. My heart was pounding so hard I thought it would burst from my chest.
How had this happened? Maybe Miki had caught sight of young Luke years before, on the business trips to Weewaa. Or maybe they'd first met in Hawaii. Is that why she was so happy that we were getting married? Because it would give her more opportunities to see him? For that matter, had she set up the whole thing herself? Did Dad know? Was he somehow involved?
My mind was racing, all kinds of crazy questions were popping into my head. I never found out the answers, though, because I never confronted Luke. I didn't dare. I carefully replaced the letters in the bottom drawer and closed it—and never said a word about them.
I wasn't angry. That wouldn't have been cool. I almost felt that it wasn't my business. They were sophisticated adults, after all, and this is what sophisticated adults did: they had affairs, they kept secrets, they did shockingly naughty things. Who was I to pipe up and say they were wrong? No, I was too embarrassed and scared to do anything.
Now I understood. I understood why Miki would make unexpected visits to our home, traveling thousands of miles on a whim. At the time, I'd thought it sweet of her, if a little odd. I also now understood why we were always getting invitations from Dad and Miki to join them in Hawaii, or Greece—to the yacht, the island, the chalet in Italy: all places from which I'd formerly been excluded. I had thought it was because we were such a fun, attractive couple. Now I'd watch Miki and Luke together, though, and I'd see the little flirting glances and accidental touches. What had once gone undetected was now so obvious. I wondered, whenever Luke went out for a smoke or a breath of fresh air, if a rendezvous was in the offing, if one of those feverish acts of passion described in the letters was about to be enacted offscreen—and I would watch Dad's reaction, to see if he knew, or cared.
Dad never let on one way or the other. He was the master of secrets.
**• • •**
**I** lived with the violence and the betrayal as long as I could. The tipping point, I guess, was Melbourne Cup Day. The Melbourne Cup is Australia's biggest thoroughbred horse race; it's practically a national holiday. They call it "the race that stops a nation." It's held on the first Tuesday of November (coincidentally Election Day in the United States), which is mid-spring in Australia.
On Melbourne Cup Day 1976, everyone gathered in Cessnock, a neighboring town, to watch the race at the local pub. I've forgotten the name of it: the Dirty Dingo, or something like that. All the wives and girlfriends were dressed in their holiday best (cream-colored dresses and stylish wide hats) to celebrate the great day. The only hitch was, we weren't allowed to enter the pub. In fact, there was a sign outside the pub door: "No Dogs or Women Allowed."
We had to enter through a separate "Ladies'" entrance, and wait upstairs. While the boys were downstairs watching the preliminary races, drinking pints, and getting rowdier and rowdier, the ladies were sipping tea and having a Tupperware party. It was absurd, and excruciatingly dull. I'd rather have been pasting wine labels. I hung in there as long as I could, but finally I couldn't bear another minute; I had to go home.
I came downstairs and stepped out on the porch, and stopped dead. There was a spring rain falling. Actually, it was more like a monsoon. Heavy sheets of rain were pelting down, making the dirt road a muddy, coursing river.
Unfortunately, the parking lot was behind the building, and I was in my Melbourne Cup dress and high heels. There was no way to get around to my car without getting drenched and ruining my shoes, unless...
_Hey, I could just cut through the pub. Why not?_
Well, because there was a sign: no women allowed. The sound coming from inside the pub was deafening: loud music, drunken laughter, shouting, and screaming. It sounded as if they were wrestling kangaroos in there. How would they react if I barged in on their party? Would there be a riot? Maybe they wouldn't react at all. Maybe they were all too smashed to notice. I was only cutting through, anyway. As I looked at the rain beating down relentlessly, I couldn't see any other choice. So I opened the pub door...
And suddenly—silence. Everything stopped: the music, the TVs, everything. Just like in the movies. Every eye was staring at me with outrage and anger.
I realized right away that I'd made a mistake. I'd violated the sanctuary. I was an affront to their maleness. I should have backed out immediately, but I couldn't. _This will last only a moment_ , I thought, _and then they'll go back to their regular carousing_. Surely I wasn't worth missing the big race for.
So I took another step in, waiting for everything to go back to normal—but it just got quieter.
I nervously searched the pub for a friendly face. None to be found. Wait, there was Luke over in the corner with his pals. Surely he'd come over and defuse the situation. "Hey everyone, meet my sheila!"
Luke just stared at me, as outraged as the rest—even more so. His face was impassive, but his eyes glittered with fury. He was not happy.
So I had to face the vortex of hostility alone. I kept walking forward, one meekly defiant step after another. It took forever. Like one of those nightmares in which the door keeps receding farther and farther in the distance. I thought I would never reach it. The tension in the room was growing moment by moment. I was terrified that the men would suddenly rush forward to exact frontier justice upon me, and Luke would be cheering them on. When I finally reached the door and got outside, I realized I hadn't been breathing all that time. I rushed to the car in the rain and drove away.
That night, I waited nervously for Luke to return home. I knew there might be a scene, especially if he'd kept drinking at his usual pace. I would try to explain the situation to him, and maybe he would understand.
A long time passed between the moment I heard his Jeep pull up and the moment he finally walked in the door. He stood in the hall doorway now and stared at me, his eyes red and belligerent. "What's wrong with you?" he said with contempt.
"It was raining, I had to get to the car..."
"I'm in there with my mates, and you come stomping in like a fucking elephant, embarrassing me, making me look like a fool..."
"I'm sorry..."
"You're _always_ sorry. You're a sorry excuse, that's what you are." I smiled at his little joke, which was a mistake. "You think it's funny?" he said. "It's _not_ funny."
"I am stupid," I hurriedly agreed, trying to calm him. "I should have known better, but...I was tired, I just wanted to go home..."
"You just wanted to go home, huh?" He sneered, and threw a chair at me. "Well, now you're home. Are you happy? Are you happy?" he yelled.
As he grabbed a glass from the table to hurl at me, I fled from the room.
**• • •**
**I** decided to call off our engagement. I didn't tell Luke; that might have been dangerous. I started planning my escape. I saved up my tip money from the restaurant, and every night, I'd sneak a few more pieces of clothing into a suitcase and hide it in a closet.
My biggest problem was transportation. I needed a getaway car. I had my eye on a used pink Vauxhall in the local lot. It was a worn-out piece of junk, but it moved, and it could get me to Sydney. However, I was short five hundred dollars, and I'd never scrape that together from my waitress tips.
So I called Mom. She was in New York shooting her ballet film, _The Turning Point_.
"What?" she asked in disbelief. "You want to borrow five hundred dollars from me so you can buy some old clunker? You think I'm made of money?"
"I need it, Mom. It's my getaway car."
"What are you doing in Australia, anyway? I wouldn't follow a man across the street. You've gotta stop letting people walk all over you."
"I'm trying, Mom. I just need five hundred dollars. I'll pay you back, I promise."
She thought it over a moment. Parting with money was always a cause for serious contemplation for Mom. "What kind of interest are you offering me?"
I was confused by this question. "You know I'm always interested in you, Mom..."
"Interest, interest!" She sighed, and worked the numbers over in her head. "Ten percent. Compounded annually. For the life of the loan."
I didn't understand business talk at all, but it sounded reasonable to me. "Okay. Whatever. You can have twenty!"
So she sent me the money, and I bought the Vauxhall, and one sultry night, while Luke was at the pub, I got out my suitcase and drove away, barreling down the dusty roads at thirty miles an hour. It would take me eight hours to chug into Sydney, but that didn't matter. I was free.
# **Chapter 8**
# Flight
Queensland and Northern Territory Aerial Services, or QANTAS, is the largest airline in Australia, and the second oldest in the world. With a fleet of jumbo Boeing 747 jets, Qantas in 1976 was the shining face of Australia to the rest of the world, famous for its record of never sustaining an airliner crash in its history. Nicknamed the Flying Kangaroo, it became familiar to Americans through its popular commercial of a crabby tourist-loathing koala bear munching on a eucalyptus leaf and grumbling, "I hate Qantas!"
I felt different—and by the time I arrived in Sydney, I'd formulated a clear, cogent plan for my future life: I was going to be a stewardess for Qantas Airways!
I don't know how I hatched this idea, but for some reason (and in willful dismissal of all my past history) I was completely confident that I would get the job. After all, I was eminently qualified: I had three languages under my belt—English, Japanese, and French—I was pretty, I had great legs, and I knew how to be cheerfully subservient. It was a no-brainer.
As soon as I got to Sydney, I put in my application. I then underwent a series of interviews and tests to see if I was flight-worthy.
The comprehensive Character-Personality Exam contained one section dealing with empathy. It gauged how much compassion and understanding you were likely to have for the passengers. The questions were all yes/no, and they were worded with such sophistication that you could never suss out exactly what the correct answer was supposed to be. You just had to give your honest answer every time, and hope that it rang positively with the judges.
Well, when the results came back from my empathy test, the inspectors were shocked. They said that it was the highest score in their years of keeping records. My empathy level was totally off the charts.
In fact, it was so unbelievable that they didn't believe it, and they made me take the test all over again. All different questions, but the same result. It was official: I was the most empathetic person on the face of the earth.
Now, in the grand scheme of life, I don't know if you could count this as an asset or a grave liability, but for Qantas it got me the job.
At the age of twenty, I began my stewardess training.
I know many people will scoff at that term, as if training to be a stewardess in the old days—before it evolved into the more respectably titled "flight attendant"—consisted mostly of learning how to dress, walk, and smile in a pretty, vacant, nonthreatening way (the latter need I had already mastered). The three-week training session was intense, though: lots of studying, memorizing, and more studying. There were courses on aerodynamics, mechanical engineering, aircraft maneuvers, first aid, CPR, and the intricacies of childbirth. We had to pass a Red Cross–authorized medical exam, learn to deploy air slides, and be proficient in every aspect of accident training. Serving coffee with a smile was way, way down on the list of priorities.
Most unnerving was the simulated disaster scenario, when you were placed in a mock cabin that was fitted with all kinds of special effects—smoke, flashing lights, pneumatic lifts that would shake you back and forth—and then asked to perform your stewardess duties in this chaotic emergency situation. You had to stay calm under pressure, which wasn't easy, because even though you knew it wasn't happening, it was so realistically staged that you'd become convinced you were about to slam into the ocean and break into a million pieces.
Yet I found, to my surprise, that I performed extremely well under these conditions. I actually tended to become more calm as the stakes grew higher. While I knew I had a tendency to get overdramatic in a mini-crisis, it turned out that I could really hold it together when the shit hit the fan.
Stewardess training was rigorous and exacting, but I don't deny that there was a certain emphasis placed on personal appearance. These were the last days of the glamour era of air travel—people dressed up to fly, the pilots strutted through the airports like conquering heroes, and stewardesses were still expected to be the stuff of businessmen's fantasies and _Playboy_ centerfolds. We were taught how to stand, how to pour coffee with an ingratiating smile, how to bend properly in our pert little outfits. We didn't wear those little hats anymore, but our hair always had to be in an up-do, a bun or a chignon. There was a whole lot of hairspray going around.
Plus, whenever we reported for work, we had to get on the dreaded scale. If you edged one pound over your assigned weight (relative to your height), you were bumped off that flight, and you didn't get paid. This was supposedly an effort to keep the plane as light as possible, but nobody was fooled. It was, as usual, all about sex. After all, none of the male stewards had to weigh in.
While I was going through my preliminary training, I lived with my friend Margo Tolmer. Margo's father was Alex Tolmer, the founder of the Australian toy company Toltoys; he had manufactured the first plastic hula hoop, and was consequently a very rich man.
Margo was kind enough to let me stay with her for free while I was training. Soon enough I would be a working stewardess and able to take care of myself, so she didn't mind putting me up for a few weeks. Margo was a professional chef—we ate very well—but unlike Robert, she didn't have an artistic temperament. She was a down-to-earth, no-nonsense lady, and you could see she wouldn't put up with any crap.
This came in handy one day when there was a fateful knock at the door. Margo went to answer it, and then returned to me.
"It's Luke."
I instantly felt the old fear and anxiety welling within me. I couldn't believe he'd found me. What did he want? What kind of mood was he in? Did he have a weapon? I whispered to Margo, "Stay close."
I went to the door. Luke was standing outside. He wasn't angry at all. He was apologetic and contrite. He wanted me back. He wasn't sure what he'd done wrong, but we could discuss it, and he'd never do it again. He'd brought clams on the half shell (my favorite dish) and a bottle of champagne. Could we just talk?
Yes, I wanted to talk. He seemed changed: sincere, respectful, and still gorgeous. Still, from somewhere deep inside, my sense of self-preservation was roused and said no. No, I couldn't risk being charmed by him and then carried back to a life of misery. I'd seen that movie before.
So I asked him to leave. With Margo looming in the background, protective and warrior-like, he had no choice but to accede. I watched him go with mixed feelings. I hated giving up those clams.
I still don't know how he found me. It occurs to me now that maybe Miki told him where I was. I'd called them just after the breakup, my heart still freshly wounded, and Dad had consoled me in a light, joking way: "Keep a stiff upper lip, Sach"—something like that. Then Miki got on the phone and gave me her advice: "You should get a glass of wine and listen to all the sad songs you can think of, and cry and cry it all out, and you'll be fine." I said thanks, and hung up.
**• • •**
**I** was a stewardess for Qantas for over four years. It was a huge chunk of my life, which I basically spent flying back and forth across the world: London, Paris, Amsterdam, Bombay, Singapore.
Because of my languages, I always worked in first class—and in those days, first class was first class. Gourmet meals and free-flowing alcohol were the rule of the day. Service was premium. We would draw a diagram of the first-class seating and write down the names of each passenger, so that, when we served them, we could address them personally. This always gave some of the first-timers a startled shock—"How did she know my name?"
I took my job very seriously, and worked hard to be considered an excellent stewardess—and it paid off: Qantas was constantly getting letters from passengers raving about my service, and suggesting that I should get some kind of award or promotion, or at least a raise. Not that the company paid any attention, but it was gratifying to read those glowing letters in my file. It was a kind of applause.
**• • •**
**MY** Qantas years were busy and yet, in a curious way, uneventful. Because I was seldom in the same spot for very long, there was never a chance for an emotional drama or a complicated situation to play out. I'd had enough of that for the time being, anyway. I was in my early twenties. I wanted to have fun, I wanted to have adventures, I wanted to learn new things.
And I did.
I learned, for example, that if you want to quiet a crying baby on a long flight, put a little powdered valium in his bottle. The stewardesses, who always carried a personal stash for their own purposes, would approach a harried mother and ask solicitously, "Would you like me to warm up his formula?" Once back in the galley, they'd crush up a valium pill and mix a baby-size amount into the formula. It worked like a charm.
I also learned that one of the largest concentrations of Greeks in the world is in Australia. Melbourne is sometimes called "Greece's third largest city," and Sydney isn't far behind. Every spring, there would be a mass pilgrimage of Greeks back to the homeland to celebrate Easter. Qantas would ferry whole planeloads, all in a festive but reverent mood.
Now, for some reason—and I'm not promoting a cultural stereotype here, but only making an objective observation—Greeks have a tendency to get airsick. It never failed that, about a half hour into our flight, someone would start vomiting into his barf bag. Invariably this would signal a general uprising: once one passenger started, the entire cabin would follow suit. Soon everyone on the plane would be retching, in a chain reaction of mass nausea. A powerful smell of vomit would fill the air, and stay with us all the way to the Mediterranean.
There's no comic tagline to this story. I'm just offering it as a public service. If you're traveling during Eastertime, beware of planes bearing Greeks.
Another thing I learned: you know that popular cliché about how airline stewardesses were fast-living good-time girls who loved to party and sleep around? "Coffee, tea, or me?"
Absolutely true.
Why not? It was the late 1970s, post–sexual revolution, pre-AIDS, everyone had Saturday Night Fever, even the president lusted in his heart. You were flying into the most glamorous cities in the world—how many museums could you visit?
The stewardesses would get together in the galley and trade tips on their international boyfriends. "There's a guy in London who'll buy you a fur coat," "There's a guy in Amsterdam who gives away diamond rings," and so on and so forth—the unspoken corollary being "all you have to do is sleep with him."
I wasn't into that. The idea of having sex with some exotic stranger who just might pay off in silver dollars left me cold. Not that I was a total prude: I just preferred having fun with people I knew. Like the stewards.
There was a presumption in those days that any male flight attendant was likely to be gay. Well, not in Australia. The crews were staffed with rampant heterosexuals, and they were constantly on the prowl. They flirted openly with the prettiest passengers, knew all the hot spots around the world, and hit on every stewardess in their proximity.
This wasn't difficult. In the close quarters of a plane, you're always brushing against each other, and it's inevitable that sometimes a helping hand will land on an unexpected spot, or an aisle-jutting bottom will intersect with a passing crotch. Then, when you all wind up staying in the same hotel in a strange city, it's a recipe for musical beds. I readily confess, I was right in the thick of it. As an antidote to boredom and loneliness, getting laid couldn't be beat.
Not that I cared much about the sex; I didn't even like it. Luke had cured me of that. Yet, the guys were always asking, and I couldn't say no—my Japanese training again: I didn't want to hurt anybody's feelings. They had their hearts set on screwing me, the little sweeties, and I couldn't disappoint them.
Let me hasten to add, this was only during the aptly named layovers. When we were actually up in the air, we were total professionals. None of that mile-high-club stuff for us. We left that to the pilots.
Those big jumbo 747s were like flying mansions, and they had all kinds of amenities, including a huge galley below deck, where the chefs cooked fresh gourmet meals for the first-class passengers. The galley would be empty after dinner, and on long overnight flights, it was very private—the perfect bachelor pad. So the pilots would put the plane on automatic and take a couple of girls downstairs for a personal tour of the facilities. Our job was to scope out the passengers and find the cutest, most-likely-to-be-seduced candidates.
"Hi," we'd say brightly, doing our patented stewardess bend. "Would you like to meet the captain?" We were like perky pimps—and we seldom came up empty. Airline pilots had an almost mythic stature in those days, and to get one in the sack was part of a female traveler's rite of passage.
I steered clear of the pilots—it seemed a conflict of interest; besides, I think I had a worshipful naïveté about them that precluded any erotic involvement—and only once did I avail myself of the international boyfriend list. We were flying into Bahrain, and I was told I simply had to call this marvelous sheikh who lived in the capital of Manama. He was obscenely wealthy, and if you were nice to him, he was sure to give you gold. Not a gold ring or a gold trinket, but gold. Pure gold. Lots of gold.
This sounded both fun and profitable, so when we landed, I called him up. The sheikh seemed delighted to hear from me, and sent a car to bring me to his home—his home being an ornate mansion on the outskirts of the capital. It was of the expected opulence: lavish furnishings, servants scurrying in all directions...and look, there was the gold, everywhere!
There also was the sheikh: a fat, middle-aged little man in a white robe and headpiece. I looked him over a moment, making a quick calculation: _I'm supposed to sleep with this guy?_ Then I looked around at all that glitter and I decided, _It can be done_.
The sheikh treated me with gracious charm, wined and dined me, and then asked me if I'd like to see his camels. I'd never heard this line before, but I'd come this far. How could I refuse? So he took me outside and, sure enough, he had a stable full of camels. "Would you like to go riding?"
We went camel riding out to his country house, in the middle of the desert. It was a smaller mansion, nestled beside a picturesque oasis of palm trees. There were great heaping baskets of dates everywhere, left out to dry in the sun. We held hands as we strolled around the pond, listening to the desert breeze stir the palm leaves. Then he led me into the house. It was time, I knew, to go for the gold.
Somehow this didn't happen. I don't know why. He was definitely in the market for sex; all the stewardesses had explained the deal to me, and they spoke from experience. Yet he never tried anything. I think what happened was, he discovered that he liked me personally, and I liked him. There was a lot of mutual liking going on—we related to each other as actual human beings—and that sort of pushed the sex right off the table. After a pleasantly civilized evening, we hopped on our camels, rode back to Manama, and I took the limo back to the Gulf Hotel—empty-handed.
Well, not entirely. A few weeks later, when I returned to Sydney, I found waiting on my doorstep several spindles filled with bolts of the very finest cashmere. I had told my Bahrain admirer about my love of sewing, and he had shipped this incredibly expensive fabric, yards and yards of it, right to my home, ready to be converted into a luxury wardrobe. I promptly sewed myself a strapless light salmon floor-length gown that was to die for.
**• • •**
**IN** 1979 we were laying over in Bombay, India—now called Mumbai—when the Bombay Airport burned down. We were in no danger, but the Santa Cruz Terminal was gutted, and we couldn't fly out for about two weeks.
Now, nobody wanted to stay in Bombay itself. I'd been there several times before and was well acquainted with the almost unimaginable poverty and desperation. In the mornings, I would take a walk from my hotel to watch the sun rise over Bombay Bay. It was spectacular and inspiring; yet at the same time there would be carts going through the predawn streets piled high with dead bodies that had been collected from the sides of the road. Every morning, there would be a fresh cartload heading off to the incinerator.
Then as the day dawned, the living would emerge to take the place of the dead, and it would be difficult to say which was the preferable state of existence. The depth of misery was shocking: people half-clothed, half-starved, moving numbly through the streets as if half-alive. Most disturbing were the blind children begging on the corners: they'd had their eyes gouged out by their parents, in a bid for greater sympathy.
I knew I was supposed to move past them unseeing, but one time I got sucked in, and gave a little bit of money to a sightless child. It was a big mistake. Suddenly I was beset by beggars, who materialized from the shadows. They surrounded me, hands thrust into my face, and as I backed away, they become more angrily insistent: You gave _him_ money, where's mine? Terrified, I started running away, and they actually chased me, all the way back to the hotel. I was lucky to escape with my life. Of course, I did escape. They, the poor and hungry, were still stuck there in the desperation of their poverty, waiting for the next tiny ray of hope.
So, no, I didn't want to stay in Bombay, and neither did my colleagues. We'd heard that there was an old British hotel at the top of a mountain in nearby Pune, where we could settle in until the airport reopened.
To get to the top of the mountain, we took an old train that chugged up the steep hill in a series of switchbacks, crawling along at six miles an hour. Indian workers furiously shoveled coal into the engine to keep the train moving, their bodies pouring sweat from the ferocious heat of the furnace and the pitiless noonday sun.
When we reached the top of the mountain, we found ourselves in a different world: foggy, green, full of flowers, more like Switzerland than India. It was still hot, but a soft mountain breeze made it endurable. The hotel had long been abandoned and was in a state of creeping dilapidation, but what remained was an elegant reminder of the imperial days, with high ceilings and huge overhead fans, gorgeous cherrywood banisters and mahogany wainscoting. A veranda stretched around the building, with stunning views looking out over the valley.
The hotel wasn't totally empty when we arrived: a tribe of macaque monkeys had taken up residence on the roof. We were charmed and amused by their antics, as they chased one another back and forth and swung from the gutters. We were less amused when we went out for a walk through the woods and returned to find our luggage ripped apart. The macaques had stolen all of our snacks, eaten the bananas and other fruit, and now sat on the roof picking the last remaining crumbs from the bags of potato chips and pretzels. They seemed quite pleased with themselves.
Fortunately we had a cadre of local chefs to cook for us, so we didn't go hungry. Every evening, we'd sit around a communal table, one big family, and feast on classic Indian cuisine—vegetarian, and loaded with spices. I was never a big fan of spicy food, but I grew to appreciate its merits here, where it was sweltering even in the shade. I discovered that the more spices you ate, the more you would sweat, and when the breeze periodically came through, the air would naturally cool off the sweat and pleasantly refrigerate your body.
The downside to this natural cooling system was that the copious perspiration did a number on your body odor. Add to this the fact that I never used deodorant—my father taught me that deodorant was bad for your underarms; it was foreign and unnatural—and by the end of the week I was smelling pretty funky.
That didn't keep the men away, though. When you're stuck in an old hotel on a mountain in India with monkeys on the roof and no TV, the pungency of your aroma becomes a very minor obstacle to romance.
Now, speaking of sex and body odor, let me tell you about Pierre.
Pierre was my French lover. I had heard that everyone should have at least one French lover, so I went out and bagged one. We met in Champagne, when I had a few days on a Paris layover and decided to take a road trip. I was always going out on such expeditions; I'd seen plenty of big cities in my time, and while I enjoyed the cosmopolitan energy, I truly preferred exploring the surrounding countryside, finding out-of-the-way places by accident and stumbling upon moments of unexpected magic.
I went to Champagne because I'd heard of a wonderful restaurant down there, tucked away in the middle of nowhere, housed in an ancient stone cottage that might have been a thousand years old.
I don't remember the restaurant's name, but the dining experience was unforgettable. There was only one sitting for dinner, at one long table. The surroundings were rustic, but the table was set with stunning elegance: fine linen napkins and tablecloths, crystal glassware, candelabras, flower centerpieces—but no menus: you ate the food they served, and drank the wine they poured.
It was all exquisite. The courses went on and on into the night, and you never knew what was coming next. Each course had its own wine, which the sommelier would celebrate with a joyful exegesis. After the dinner came the cheeses and port wine, the fabulous desserts, everything homemade. I can still taste that perfect crème brûlée. I was so glad I wasn't anorexic anymore.
There were fifteen people at the table, all French; not a tourist in sight. As the courses leisurely followed one upon the other, we got to know our dining companions. There were couples present—husbands and wives (or mistresses)—but, by design, they didn't sit together. Lovers were kept significantly apart, so as not to inhibit the free flow of conversation.
This is how I came to be sitting next to Pierre. He had come with his girlfriend, but she was across the table and just outside of earshot. So he and I spent the evening chatting, and philosophizing, and flirting, and by the end of the evening, the attraction was undeniable. He left with his girl; I left with his phone number.
The next night, we had our first rendezvous, and it was the beginning of a passionate, earthy love affair—in the French style, of course. Pierre never left his girlfriend—they eventually married, and might even still be together—but every time I flew into Paris, I went down to Champagne, where he would be waiting. I didn't know a lot about him—I don't even know how he made a living. I didn't care. I had a lover in France. That's all that mattered.
As I intimated, Pierre was very much obsessed with the scent of a woman. Body smells turned him on, and he detested perfumes and lotions, anything that camouflaged what he called the "aroma of desire." To that end, he refused to let me shower for five days before I saw him. He wanted me to be natural in every way. So whenever I had a flight scheduled for Paris, I stopped showering back in Sydney. I could wash my hands and face; that was it. By the time I got on the plane, I was as ripe as a compost heap. I don't know how the passengers stood it. I know I revolted me.
Then I would arrive at Pierre's house in Champagne, and he would savor me like a fine Bordeaux, his nose twitching like a hyperactive rabbit's. After he'd assessed my general fragrance, he would push me back on the bed and his head would dive hungrily between my legs. Ah, the French! It was all very organic, and consistent with his philosophy. Pierre never bathed, either, so I got to savor his natural essence in return. We were one smelly couple.
When the festivities were over, I would finally take a shower, and after that he had no use for me. _Au revoir, mon amour_ —see you next time.
**• • •**
**FOR** all my newfound sophistication regarding affairs of the flesh, I was still pretty traditional and conservative at heart. I know that's hard to square with the facts, but I offer, as an example, an incident when I was staying at the Mandarin Hotel in Singapore.
Singapore was an interesting city. They had a thriving black market there—street after street of outdoor stands where you could obtain basically any kind of drug you wanted: opium, cocaine, Valium, Mogadon (a sleep aid; lots of stewardesses took it), whatever. Many of the stewards and stewardesses were addicted, either to sleeping pills or uppers. I wish I could claim to have been above that sort of thing, but I was young; I tried them all. The only thing that saved me from a harrowing downward spiral was that none of them really worked for me. Drugs—whatever.
One night, while the rest of the flight crew was out sampling the local pleasures, I was in the hotel bar, hanging out with a tall, dark, handsome steward named Ken. I had a big crush on Ken at the time, and we were flirting like crazy, but I knew he was married, so it was all totally innocent. Even when, at one point, he asked me if I wanted to go back to his room, I didn't read anything into it. His wife was about to have a baby back home, so there was no way I could interpret the offer as anything more than a friendly gesture between colleagues.
I went up to his hotel room and we had a drink and did some casual chit-chatting. Then Ken excused himself and went into the bathroom. He emerged a moment later, totally naked.
I was perplexed. "Why are you naked?" I asked.
He grinned. "You're from Japan. How about a massage?"
"Oh," I said, relieved. "Okay."
Ken lay facedown on the bed, and I commenced massaging him from head to toe—his shoulders, his feet, his ass, everything. Then he rolled over on his back. His penis sprang up, proudly erect—which was normal during a massage. I took it as a compliment: he was enjoying my work.
So I continued the massage, rubbing his chest, his thighs, and so on. I wanted him to be impressed with my thoroughness.
"Okay," I said, with a satisfied sigh. "All done."
"Wait a minute," Ken said, grabbing my hand. "You forgot one muscle."
I was puzzled. "No, I didn't."
"Oh yes, you did."
I thought about it, retracing my actions in my head. Let's see, I massaged his calves, his deltoids, neck muscles, toe muscles..."No, I got them all."
He took my hand and placed it on his penis. "What about this muscle?"
I looked at him wide-eyed. "Oh, no," I explained, now understanding his confusion, "that's not a muscle. That's just a concentration of blood. It won't do any good to massage _that_."
I had such an anachronistic faith in the binding fealty of marriage, despite my parents' unconventional arrangement, that I couldn't grasp the obvious even when it was standing right there in my hand. I still didn't get it.
At this point, I think Ken was starting to realize that he wasn't going to get it, either.
**• • •**
**I** had one other unsettling moment at the Mandarin Hotel: I was heading through the lobby one day with a couple of stewardesses when I spotted a familiar face by the magazine stand. I stopped short, my heart skipped, and my stomach lurched.
It was Luke.
Jesus. What was he doing in Singapore? How had he found me? Was he stalking me?
I reached out to my fellow stewardesses for support. "Save me," I whispered.
Then, as the scene came into focus, I realized that Luke was wearing a Qantas uniform—a steward's uniform.
What? Was I seeing right? By what weird process had Luke, my macho ex-boyfriend, the sheep rancher, the vineyard owner, the son of a plantation owner, transformed himself into an airline steward?
The utter incongruity of this metamorphosis did not alter my first instinct: to turn and run. It was too late, though—he looked up and spotted me. Although he knew I'd become a stewardess, he seemed mildly surprised to see me.
"Hey, Sachi," he said, in a bland but friendly manner.
I couldn't retreat now, so I walked up to him. "Hello, Luke." I gestured to the uniform. "Working for Qantas now?"
He shrugged. "Yeah. Something different."
"Welcome to the club," I said.
We exchanged a few more pleasantries, and then we ran out of conversation. I never asked him why he'd become a steward; I didn't really care. Besides, it was clear to me now that Luke didn't care much about me, either. We had both moved on.
"See you around," he said, as I left him in the lobby. I never did see him again.
**• • •**
**THE** most memorable episode of my Qantas tour took place on a Sydney-to-London nonstop, which was about fifteen hours long. I was working the night shift. Everyone was asleep—the passengers, the stewardesses. The pilots were downstairs getting refueled. My job was to keep an eye on things, stroll the aisles, and try to stay awake.
In the middle of the night, when all was quiet, a passenger in first class started to stir. A big strapping Aussie—is there any other kind?—he'd been something of a headache earlier in the flight, drinking continuously since we left the ground and making an obnoxious nuisance of himself, until he finally slumped into a pickled coma. I'd have thought he'd be out for the duration, but here he was, rumbling about and getting his second wind. Being the only one awake, I pointedly looked away from him, not wishing to invite conversation.
Then he stood up. I assumed he was going to the bathroom—and he was. Yet he had no intention of leaving his seat to do it. Instead, he dropped his pants, turned to the side, and deposited a monstrous turd right in the lap of the woman sleeping next to him! I watched this happen in horror, but there was nothing I could do to prevent it. The bowel movement shot out so fast, like a torpedo, that I didn't have a chance to catch it in a food tray or anything. It landed in her lap with a moist plop. Then the Aussie just sat down, pants-less, as if nothing had ever happened.
The woman slept on, blissfully unaware—which was rather fortunate. He being a complete stranger, I doubt she would have appreciated such familiarity. Luckily she was wearing a thick Qantas blanket over her lap, so her dress was spared. Thank goodness for those amenities.
Still, this was something I had never encountered before—there was nothing mentioned about it in our training—and I didn't know how to react. Panicky, I woke up the other stewardesses, but they didn't know what to do, either. We all just stared at the drunken lout with his pants down and the fresh mound of excrement sitting proudly beside him. It needed to be cleared away ASAP, before it started to seep, but he was blocking all access to the woman, and no one wanted to tangle with him. Maybe this went against the Qantas code of crisis management, but when one passenger defecates on another passenger, all bets are off. We had to get the pilots.
When the pilots emerged from the cockpit, the drunken Aussie was still in his seat, pants down, and starting to sing "Waltzing Matilda"—always a bad sign. Without hesitating a moment, the pilots yanked up his pants, strapped him into a straitjacket, put duct tape over his mouth, and shoved him back in his chair. He didn't even struggle. His fate now determined, he closed his eyes and slept for the rest of the night.
My role in this drama was considerably less heroic, but utterly necessary. The offending turd was still steaming on the woman's lap. I had to remove it without waking her up, or else the resulting screams would surely ignite a plane-wide panic. So I went to work, operating with rubber gloves and the steely nerves of a bomb defuser.
The specifics don't need to be discussed here, but suffice it to say, the blight was removed, a fresh blanket installed, and overall calm maintained. Just another day in the life of a Qantas stewardess.
**• • •**
**IN** all the time I was with Qantas, I saw my mother exactly once. She'd really meant business with that diamond necklace "you're on your own" routine.
In that time, her career had gotten a good bounce with _The Turning Point_ , a glitzy ballet world soap opera with Anne Bancroft and Mikhail Baryshnikov, her biggest hit in years. Then she did _Being There_ with Peter Sellers, a marvelous film and one of my favorites. Her star was on the rise again. Every now and then one of her movies would play in-flight. It was odd to look up at my mother's face on the big screen—they had big screens in airplanes in those days—glamorous and larger than life, while I was pouring orange juices and carrying barf bags.
Once, we were on a rare two-day layover in San Francisco—rare because I almost never flew into the United States; with my languages they liked to reserve me for foreign countries—but we were continuing on to Europe, so, for the time being, I found myself just a few hours up the coast from my mother's home.
On a whim, I decided to visit her. I jumped on a Greyhound and headed down to Malibu. It had been so long since I last saw her there that I wasn't exactly sure what her address was, and I didn't have her phone number. For that matter, I wasn't sure how she would react when she saw me. So it was going to be a complete surprise for both of us.
When Mom saw me standing on her doorstep, she registered immediate delight. "Sweetheart! How are you?" She threw her arms around me and gave me the biggest hug. "Come on in!"
It was as happy a welcome as I could have expected. There was none of the tension I associated with our previous encounters, dating back to...well, the hotel room in London when I was twelve years old and the air tickets had gone missing. All the distrust, the disapproval, the withholding—it all seemed to have disappeared. Absence had made her heart grow fonder.
She showed me around the apartment—so much had changed, so much had stayed the same. We sat on the sofa and discussed our respective histories. She told me about her men; I told her about mine. I showed her my stewardess outfit, and she was impressed.
Then we went out for a walk on the beach, just like when I was a kid. Mom found a sea urchin in a tide pool and stuck her finger into it. "Ooh!' she squealed in surprise.
When we got back to the house, Mom pointed to the balcony. "See up there? That's where I met them."
"Met who?"
"The extraterrestrials. They landed over here on the sand, and then they came up on the porch, and we talked. They told me I was an Enlightened One."
I nodded thoughtfully. We were having a nice visit, so I wasn't going to spoil it by voicing any doubts about her sanity.
Besides, I wasn't surprised by the revelation. Mom had always had an abiding interest in the otherworldly, ever since Dad had given her _A Dweller on Two Planets_. So, extraterrestrials in Malibu? Why not? What's for dinner?
Mom prepared a perfect sirloin steak, with steamed vegetables over steamed brown rice. Actually I suspect the housekeeper prepared it, because Mom was no cook. She could just about boil water.
She could pour a good drink, though. She introduced me to a red wine from Chile that was absolutely delicious. "And only five bucks a bottle," she bragged. She loved a good deal.
After dinner we laughed and told stories, and drank more wine, and enjoyed each other's company. It was a truly pleasant day. I slept in my old room, listening to the ocean crashing against the shore—my old ocean, my old shore.
The only discordant note in the trip was that I was having my period at the time. I didn't share this information with my mom. It was not something we talked about.
My very first period had occurred in the dorms of Charters Towers, when I was twelve years old. It came upon me in the middle of the night, and even though Mom had been preparing me for this moment since I was nine, when it finally arrived I had no idea what to do. The blood was streaming down my legs as I wandered down the halls in my white nightdress, searching for the matron. When I finally found her, I stood before her embarrassed and mortified. She looked at me stone-faced and handed me a bulky sanitary napkin and two safety pins. "Here you are, dear," she said. "Off you go."
Off I go? Where? To do what? I knew about the napkin, but what were the safety pins for? And what do I do with all this blood? I stumbled off to the bathroom in a state of panicky confusion. As I tried to wash the blood from my underwear with hot water (nobody had told me that cold water works better), the sanitary napkin fell off the sink and landed on the bathroom floor, collecting dust and God knows what on its surface. I took a quick shower and wiped myself off with the towel. As I was still bleeding, there was now blood on the towel. I washed the towel out in hot water, and then tried to dry myself again. More blood on the towel! Screw the towel. I tackled the dirty sanitary napkin. I managed to attach the front pin, but I couldn't reach around to fasten the back pin. So I tried twisting the back of my granny pants toward the front for easier access, but then the napkin start twisting sideways, and I was still bleeding! I was almost weeping with frustration. Finally I stood out on the landing and called for the matron. She waddled slowly down the hall, and in her pleasant but brisk English way she attached the back pin for me. "Off to bed," she said, dispatching me down the hall with a firm push. I wanted to crawl into a hole.
From this initial encounter, I came to understand that menstruation was a messy, private business, not to be addressed in the public square. I was always extremely careful to conceal my monthly condition from everyone. Luke's outraged reaction to my used tampon had certainly reinforced this sense of female shame.
So when I was having my period now at Mom's house, I did my best to hide it from her. Why burden her with such unpleasantness, and spoil a perfectly good visit? But as I was removing my clothes in her huge dressing room, Mom walked in. She spotted the stained panties in my hand—the flow had been heavy and clotted, it was not a pretty sight—and was remarkably unfazed: without a word she took the panties in her hand and went to the bathroom sink, and washed them out thoroughly—in cold water—working her fingers through the thick blood without any theatrics. I watched her from the dressing room, amazed. She was just taking care of business. She wrung the panties out and put them on a chair in the sun to dry, and then she said to me, "Go take a shower." I don't think I said anything—any sound would have been a violation of the moment. I took a shower, and when I came out she handed me a pair of underwear from her drawer, and a tampon, and I got dressed.
The entire transaction was so matter-of-fact and without pretense that it stunned me. Usually Mom created a big drama over the tiniest thing, but here she was sublimely restrained, and every moment was quiet, ritualistic, almost Japanese in its simplicity. It was a beautiful experience for me: the maternal validation of my own femalehood that I'd been unconsciously desiring for the last ten years.
Later that afternoon I took the Greyhound back to San Francisco. It had been a perfect visit with Mom, and as I rode up the coast, I thought to myself, _I have to do this again—soon_.
# **Chapter 9**
# La Vie Bohème
After four years at Qantas, it was time for a change. I can cite no particular deciding factor in this—not the apparition of Luke, nor the pile of poop nor the general rootlessness of the stewardess life. In fact, I was having a great time. There was no reason at all to leave.
So I did—and moved to France.
I'm not sure why. My affair with the malodorous Pierre had long since ended, so there was no romance drawing me there. Yet I didn't want to stay in Sydney, I didn't want to go back to Tokyo or Honolulu, and I loved Paris. I loved the French people, the French culture, and I especially loved French food.
I'd saved just enough money from Qantas to afford a small studio on the Left Bank—and when I say "small," I mean tiny: there was a twin bed, a closet-size bathroom, and that was it. Crammed inside the bathroom was a toilet, a corner sink, and a cheap plastic stall shower. There was no kitchen to speak of, just a hot plate and an electric coffeepot. I couldn't really cook there, which was ironic, because I think the main reason I'd moved to Paris was so that I could learn to cook.
I didn't take any cooking lessons per se—too expensive—but I did take a job as a waitress in any restaurant that would have me. I figured, if I can be close enough to where they make this marvelous food, I'll be able to absorb their culinary knowledge without paying a dime.
These were not gourmet restaurants, but little mom-and-pop storefront bistros and cafés. Simple and unpretentious, they turned out classic French fare—onion soups, ratatouille, roast chicken (ah, my favorite!)—for everyday diners who had no idea how lucky they were. I watched the chefs at work in the kitchen and picked up many a savory tip just by keeping my eyes open. I also picked up a rich vocabulary of French colloquialisms and swear words.
During the year I spent in Paris, I was never at any one bistro for more than a couple of months. Perhaps owing to the restlessness bred into me as a stewardess, I liked to bounce from one place to another, looking for new friends and adventures. There were a lot of Japanese tourists in Paris in those days, so my felicity with the language put me always in demand, and I never had a problem finding a job.
Wherever I went, though, I found myself immersed in the world of food. I would get to work early in the morning to help the chef prepare, and go shopping with him at the open markets to pick out the best fruits, the best fish, the best overall ingredients. Then I'd spend the entire day at the café, watching the cooks put together their simple masterpieces as I flitted in and out of the kitchen. I'd be there fifteen hours a day, and the staff would become my new family.
Then, after a few months, I'd be done, and I'd look for another bistro. I just wasn't the settled type. When I saw the movie _Forrest Gump_ in 1994, I was startled to recognize something of myself in the title character's constant searching for a place to land. Of course I wasn't as out-to-lunch as he was—I don't think—but I was, like him, always looking for the next box of chocolates.
Oddly enough, in that full year I spent in Paris, I had no boyfriends to speak of—and consequently, no sex. In Paris! It was a criminal waste of natural resources, agreed, but I just wasn't interested. I did a lot of walking instead.
There was so much snow in Paris that winter. I remember the vendors hauling their pushcarts through the snow, peddling fresh warm crepes with butter, sugar, and lemon. My favorite was the _crème de marrons_ , a delicious crepe filled with pureed chestnuts. Strolling along the cobblestones of the Left Bank in my beret, eating my hot crepe, I felt quite the quintessential Parisienne.
**• • •**
**IT** was in the midst of this cold winter that Dad and Miki descended on Paris—not to see me, of course. It may have been for business reasons, or maybe they just wanted to dine at Maxim's. That's what they would do on a regular basis: hop on a jet and eat their way around the world. It was surely just a coincidence that we were sharing the City of Light at the same time.
Nevertheless, when he found out I was in town, Dad invited me to join them at Maxim's—which was fun, but very much on the superficial side; we all strained to be festive, and I found it difficult to look at Miki without picturing her in bed with Luke. I figured it would be a quick hello-goodbye-see-you-around, but as we left the restaurant, Dad invited me to stop by their hotel room the next evening.
I happily took this as a sign that Dad was reaching out to me. As with my mom, I hadn't really seen much of him during my stewardess years. I was endlessly busy, of course, but I would have made time to see him if he'd ever proffered the invitation. He didn't. I think my breaking up with Luke, his old business partner's son, had annoyed and embarrassed him. Somehow, by saving my life, I had done something shameful again.
We would put that in the past now, though. I'd already reconnected with Mom, and now I'd do the same with Dad.
Dad and Miki were staying at the George V Hotel, right off the Champs-Élysées, one of the great five-star luxury hotels. I got there early in the evening. It was the classic dark and stormy night, the cold rain coming down in buckets.
I walked into the lobby—I'd been in some beautiful old hotels in the past, but this was the absolute, sumptuous peak—and announced myself to the imperious concierge at the desk. "I'm here to see Steve Parker."
Once again the mention of my father's name worked like a magic incantation. The concierge's studied indifference vanished. "Ah, oui, Monsieur Parker! And you are...?"
"I'm his daughter."
He was even more impressed. Suddenly _I_ was somebody special, too. "Right this way, mademoiselle."
He ushered me personally to the elevator. It was one of those old clanky elevators with the metal gate that pulled across. It had its antique charm, but tonight I preferred to take the stairs. The grand staircase was wide and spiraling, with plush carpeting. It was like walking up to heaven.
Dad and Miki were staying in the penthouse. This was yet another irony that escaped me at the time. Because I was living in a penthouse, too; the only difference being that mine was a hovel the size of a shoebox. Yet I never thought anything of that. I never questioned the fact that while I was living in bohemian squalor, a mile or so away my father was ensconced in outrageous luxury. This was just the way of things: I accepted it as normal. It was as if I felt that, with my record of false starts and abject failure, I deserved no better.
Maybe I even felt I didn't deserve to take the elevator, and that's why I took the stairs. Whatever the reasoning, it was a long climb, and by the time I got to the top floor, I was a little winded. Instead of knocking right away, I paused outside my dad's hotel room door to catch my breath—which is why I was in the position to hear a conversation going on just inside the room.
It was Dad and Miki. I could hear them clearly, every word.
I sometimes wonder what would have been different in my life if I had taken the elevator that night and bounded right up to the door and knocked right away. If I had not overheard my dad and Miki talking inside, unaware of my presence in the hallway, I would have missed one of the pivotal moments in my life.
Dad was crying.
I had never heard him cry before. It shocked me—I can still remember the icy chill that shot through my body. The sound was alien, and devastating. If my father was crying, then something was terribly wrong, wrong with the life he had created, wrong with the world that I understood. He was a strong, confident man—he was Hemingway—and to show weakness was not merely out of character for him, but inconceivable. His cries that evening were awful to hear: raw and animalistic, the wail of some broken, faltering creature that was trapped and confused and twisted with despair.
"I don't know who I am anymore," he was sobbing. "I don't know anything. I feel like I'm going crazy!" His voice was high and stressed, so unlike his usual smoothly modulated baritone. It was hard to believe it was my dad in there, but I knew it was.
Miki was trying to soothe him. She did not have a soothing demeanor, so her calming words sounded forced and slightly desperate. "It's okay, Steve. I have you, I'll take care of you, don't worry, it's all right..."
"No, I can't keep this up anymore! I can't do it!"
"Yes, you can, you can," she insisted. I could hear that Miki was not just being supportive; it was terribly important for her own sake that Dad hold it together.
"Who _am_ I?" he cried. "Who _am_ I? I can't remember any-more!"
"It's okay..."
"No, it isn't! No, it isn't! Who am I?"
I listened with transfixed horror. I couldn't knock now. They would know that I'd overheard.
I couldn't leave, either. They might hear me walking away, and anyway, they were expecting me; I was supposed to be there.
I stayed and waited for Dad to calm down. He couldn't sustain this howl of emotion much longer, and I have to admit, I was fascinated by what I was hearing. I had no idea what he was talking about—Who was he? He was my father, he was Steve Parker, he was the husband of Shirley MacLaine, he was a rich, successful businessman, that's who he was—but I sensed in his anguish the first intimations of a secret life emerging, an alternate reality that I had probably always suspected but never dared recognize. Now all my doubts came swimming to the surface, and the pieces of an unsuspected puzzle were slowly clicking into place. It wasn't totally clear yet; the answer was still out there, but those obsessively repeated phrases—"Who am I? I can't do this anymore! Who am I?"—were pointing the way.
Who was my dad? What did he do? How did he get to be where he was? I assumed I knew the answers, but when I thought about it now, there were many small gaps and great blank stretches unaccounted for.
I knew he wasn't a totally honest man. He had lied to me more than once in my life—the last time on my birthday, when he told me he was in Italy and had faked the phone call, complete with Italian operator. Then there were all the other episodes, the lingering doubts and inconsistencies—the death of Mike Parsons, the loose diamonds in the front seat of the Jeep—all those shadows lingering in the back of my head.
As I listened to my father agonizing over his identity, I felt an uneasy pang of recognition. How often had I wondered the same thing: Who am I? Who am I supposed to be? Who did I become, instead of myself? Underneath the layers of Japanese humility and self-effacement, the docility beaten into me in Australia, the eagerness to please in my Qantas days, the malleable personality adapting to meet each new situation—underneath all that, who was the real Sachi?
It had gotten quieter in the room now, and it seemed safe for me to enter. I knocked on the door.
Things got even quieter, and then, after a pause, my father called out, "Be right there!" in his strong hearty voice. He needed a moment to compose himself, I suppose, and Miki had to get the room just right.
Finally the doors opened—into a huge, gorgeous space, with a king-size bed on one side and sofa and chairs on the other—to reveal Dad, himself again, whoever that was, looking suave and cosmopolitan in his smoking jacket. "Hey, Sach the Pach!" Miki looked frazzled, and gave me a stiff smile; in other times, I might have taken this for hostility, but now I knew she had just pulled my dad back off an emotional ledge, and her nerves were raw.
I sat on the sofa, and we had a drink and exchanged pleasantries. They were both distracted, though, and so was I.
My mind was racing. I had found myself in a unique situation: I knew something about my dad, and he didn't know that I knew it. I actually had the advantage over him. Plus, unlike before, when I would tactfully withdraw from any confrontation, I was eager this time to get some answers. He had tantalizingly cracked open the door to his reality, and I wanted to push through before it closed again.
So I waited for the right moment, waited until everyone was relaxed, before I said, "Dad, can I ask you something?"
"Sure, honey," he said, sitting back in his chair with his scotch, legs crossed.
So I asked: "What do you do for a living?"
It was the essential question, the key to every mystery surrounding my dad, and I had calculated that this was the perfect time to ask it, when his defenses were down, when he was hurting and vulnerable, and ready to face the truth.
I was wrong.
All the civility drained from his face in an instant, and the lights seemed to get dimmer, as if the energy were being sucked out of the room. He gave me the dirtiest look I'd ever seen. I heard Miki gasp in surprise, but I didn't look at her: my eyes were fixed on Dad, as were his on me.
"What?" he finally asked, in angry disbelief, his voice as cold as ice.
I could see I had made a colossal mistake—it hadn't been the right time to broach the subject at all, and it would probably be in my best interests to backpedal as fast as possible.
Yet I couldn't. I wasn't a kid anymore; I was twenty-five years old, and I wanted answers. I looked around at this luxury suite and I asked myself, _How can he afford all this? And the yacht, and the island, and the trips around the world? How?_
So I repeated the question, with a little less confidence: "I was just wondering, Dad, what do you do for a living? Because I really don't know."
The room grew darker. Dad held himself quite still, but I heard the ice cubes clinking in his glass as his hand trembled with contained fury. The tension in his body radiated outward and gripped the whole room.
I'd crossed the line now, big-time, and I knew it. I'd been raised to respect, to accept, and not to question—but now I'd questioned, I'd cast doubt, and (most unpardonably) I'd compounded the shame: I'd done it in front of Miki.
Really, I just wanted to know what his job was. What was wrong with that? I knew he had produced a couple of movies and Broadway shows, but that wouldn't have supported the lifestyle he and Miki had been living for all these years.
Dad said nothing. He let his dark silence speak for itself. That was his favored strategy. He would let his unsaid displeasure settle over me, and work its insidious magic.
This time, however, I wasn't cowed by his glare. I wasn't going to backtrack and start apologizing and asking forgiveness. The old routine wasn't working, and he could see it.
So he glanced significantly over at Miki—his enforcer—and she knew what that meant. Take care of this, he was telling her. Go for the kill.
She rose from her chair and advanced on me. "How dare you?" she said, sputtering with outrage. "How dare you ask such a rude question of your father? Who do you think you _are_?" (Ah, that was the question.) "You have no right to ask such a thing! A daughter does not speak like this to her father. It is disgraceful! You are a disgraceful child! You should be ashamed! Ashamed!"
All of this was spoken in Japanese. She wanted to denounce me in the language of my childhood, for maximum effect. Since my father, despite his many years in Japan, didn't speak the language, he had no idea what she was saying, but he could probably guess the basic thrust from the fury in her voice and the violence of her gestures, and he seemed quite content with the performance.
"You want to know what he _does_?" she went on, almost choking on her bile. "Look around you! He is a success, that's what he does! What do _you_ do? Where do _you_ live? Your father treats you like a princess, which you don't deserve, and this is how you thank him? You are lucky he lets you in the door! I wouldn't! You think you are special? You are not special! You are nothing! You disgust me, you ungrateful little...bitch!" She spat out the last word with special relish. Miki was getting meaner and nastier, because she knew Dad couldn't understand her. She was going for the kill, all right, hoping to finish me off.
I should have leapt up at this point and thrown it back in her face: _Look, I know what's going on! I heard my father crying, I heard him wondering who he was, wondering how he could keep up all this pretending. Pretending what? What's real around here and what's bullshit? And who are_ you _to lecture me? You're not my mother! You're my father's mistress, a teahouse servant—and, by the way, I know you were sleeping with my fiancé! So who's the bitch now?_
That's what I _should_ have said. It would have been a great scene in a movie.
Of course, I said nothing. For all my defiance, I was still bound to the culture I was raised in. Miki had pushed all the time-tested buttons of guilt and shame, and I responded accordingly.
When she finished her tirade, I got up quietly from the sofa, gave Dad a hug, walked past Miki, and left the hotel room. I didn't say a word. Neither did they. They never made a move to stop me.
I walked slowly down the stairs, measuring each plush, cushioned step, as the staircase spiraled down and down. When I came out of the hotel, it was still raining, and I walked home in the rain.
For all that, he never did answer my question.
# **Chapter 10**
# The Good Doctor
I wanted to get my breasts done.
Not that there was anything wrong with them. They looked fine—a bit smallish perhaps, but firm and proportionate. And none of my boyfriends complained. As Luke would charmingly have put it, "Anything more than a mouthful is a waste, anyway."
I was self-conscious just the same. Dad had always teased me about my "fried eggs," and I guess I always felt inferior in that regard. Any boost in my self-esteem would be welcome about now.
I got a recommendation for a plastic surgeon from Anastasia Gratsos, a close friend of Miki's, and the wife of Constantine Gratsos, known as "Costas." Costas was involved in the shipping business, and he was also the right-hand man of Aristotle Onassis. We're talking connected.
So when Anastasia called with the name of a plastic surgeon, I knew I was getting top-of-the-line. Dr. Jeffrey Dietrich had an office in Illinois, just outside Chicago. He handled models, actresses, the rich and the super-rich.
I was none of these, but Anastasia told me to come over to the States to meet the doctor, and I could figure out how to pay for it later. I saved up my money and took a cheap flight from Belgium to New York. I stayed with Anastasia for a few days, and then took the train to Chicago.
Dr. Dietrich's clinic was actually outside Chicago, a fancy modern building in the woods. Dr. Dietrich was a handsome, assured man in his late forties, commandingly in charge but with great personal warmth—just the kind of doctor you might see on television. Since the vast majority of his patients were women, he knew how to charm them and put them at their ease—and being the susceptible romantic that I was, I happily allowed myself to be charmed.
"What seems to be the problem, Miss Parker?" he asked as we sat in his consultation room. His eyes were sky blue and penetrating.
"Well, I'd like my breasts to be a little bigger. Not a lot. I mean, there's nothing wrong with them the way they are, but I think they could be a little more, you know, full. Round. Whatever." His steady gaze made me nervous, and I couldn't stop babbling. "But I'm no expert. What do you think, Dr. Dietrich?"
He smiled. "Call me Jeffrey."
I was ready to swoon right there. "Okay, call me Sachi!"
We went into his office, and I took off my shirt and showed him what I had. He could not have been more solicitous and gentlemanly, handling me with great care and observing all professional boundaries. Still, I was topless in front of a very attractive, wealthy single man, and it was giving me ideas.
After our meeting, I had to get back to the city and catch a morning train back to New York. "What's the rush?" said Dr. Jeffrey. "You didn't come all the way out here just for an hour meeting?" He offered to show me around the area, and I agreed.
We drove through the country roads, and he took me to a gunpowder-making plant, which was surprisingly interesting. Then he insisted on taking me to dinner—his treat. Did he give all his prospective clients this treatment, or was I something special? At the restaurant, he ordered me clams on the half shell, with horseradish sauce—my favorite. How did he know? We had some very expensive wine, and he ordered dinner for both of us. He was taking charge, and I knew it, and liked it.
The consultation had turned into a first date. We talked and talked, on into the night. He was naturally interested in my family, but he wasn't starstruck—having dealt with many celebrities himself, he took all that fame nonsense with a grain of salt, which I found appealing.
Jeffrey wouldn't hear of my heading back to Chicago that night. He brought me to a picturesque country inn, and insisted on paying for my room. Given the circumstances, I felt obliged to invite him up, but he stayed only a few minutes. He kissed me on the cheek and wished me good night. "I'll see you for breakfast."
Alone, I sat on the bed and tried to figure out what was happening. I'd come to Chicago for a boob job, and suddenly I was in the middle of a Harlequin romance: the dashing country doctor, the clams on the half shell, the four-poster bed, all the ingredients for a hot, steamy bodice ripper. I crawled into bed, my head swimming, intoxicated with the possibilities. What would tomorrow bring?
I didn't have to wait that long. At about two in the morning, I heard the key turn in the lock, and suddenly the door was opening. There was a man in the doorway.
I sat up, instantly terrified. "Hello...?" I said groggily, trying to get my bearings, fighting through the fog of too much wine.
"Shhh...," the man said gently. He shut the door behind him and locked it. My heart was pounding in my chest. Who was he? How did he get in? What did he want?
I got the answer to the third question first. Without a word, the man lifted the sheets at the bottom of the bed and slipped underneath them. I watched as his head bobbed under the sheets, moving slowly and steadily north until it nestled between my legs. Then I felt his fingers prying my thighs apart, and his tongue lapping at the fringes of my vulva.
It was Jeffrey, of course. He had taken another room in the inn, let himself into my room with his own key, and now was putting his compendious knowledge of female anatomy to work. We had gone way past Harlequin: this was like every _Penthouse_ magazine– _My Secret Garden_ –Erica Jong fantasy come true.
The good doctor knew what he was doing down there. This was a man with professional chops in every sense. He carefully peeled back the various folds of my vagina, opening it up like a flower—much as Eguchi-san would skillfully summon forth the blossom of a clementine—and probing gently, gently, with his erudite tongue, like a hummingbird searching for nectar.
I should have stopped him right away. The man had broken into my room, he was taking enormous liberties, he needed to account for his outrageous behavior—but it felt so good, I wanted it to go on...No, I should stop him...Well, I could always stop him in a few minutes...No, this is wrong, wrong...
Then the first orgasm washed over me, and that settled the debate. In for a penny, in for a pound. The orgasms kept coming, one after the other, and I just lay back and kept riding the waves. If this was a dream, it ranked right up there in my top five.
Then it was over. His appetite sated, he slowly withdrew from the bed and, without a word, left the room.
In the morning, Jeffrey was waiting for me downstairs. "Sleep well?" he asked, betraying not a hint of what had happened the night before. Had it been a dream then, after all? No, there'd been too many orgasms; I would have woken up by the third one.
Over breakfast, Jeffrey excitedly discussed his plans for my breasts. He definitely saw room for improvement, and he was eager to get his hands on them. He knew I was pressed for cash at the moment, but we could definitely work something out.
This put me off a little. I know I'd gone to Dr. Dietrich specifically to have my breasts done, but after yesterday, and especially after last night, I assumed he would want to preserve my assets just as they were. He was obviously attracted to me, so why should he want me to change?
He also mentioned that it would be beneficial for me to get implants in my buttocks, too, just to plump them up a bit. Right now they were a little on the flat side.
Oh really? Now there was something wrong with my ass, too? It was good enough for the boys at the noodle shop!
This sudden insensitivity, coupled with the nocturnal visit, made me back off from making any rash decisions. "Let me go home and think about it," I said—and by "home," I meant Paris.
Back in my tiny studio on the Left Bank, I tried to forget Dr. Jeffrey, but it wasn't easy. Within a week, I got a letter from him. I was afraid it might be a bill for the consultation, but no, it was a love letter. A very courteous, discreet love letter, but a love letter nonetheless. He missed me; we had such a great time together; when would he see me again?
I was thrilled. I realized that I missed him, too. I still had those misgivings, though, so I wrote back a friendly, noncommittal reply.
That was the beginning of a six-month correspondence. It began politely, with respectful expressions of admiration and regard, but very quickly blossomed into a full-blown epistolary _affaire d'amour_. He wrote about his passion for me, his ardent love, his desire to be with me always and forever. I told him how I spent my days walking along the Seine, or wandering the Louvre or the Tuileries, and thinking only of him, him, him. (Not entirely true, but I was trying to be as romantic as possible.) We didn't get steamy and sexual in our letters, but we didn't have to: we both knew what had happened back at that country inn, and that there was plenty more where that came from.
Pretty soon I had convinced myself that I was hopelessly in love with him. I built him up in my mind as the ideal lover: wise, experienced, mature, adventurous. Together with me, the impulsive, impetuous naïf, we made the perfect match.
Anastasia Gratsos thought so, too; she was constantly trying to push us together. She called me every now and then—"Do you like him? Do you think there's something there? When are you coming back to the States?" she kept asking.
Frankly, I was just waiting for a good excuse, and finally Jeffrey gave me one: he was building a huge mansion for himself on his property near San Diego, and he wanted me to see it. That was good enough for me.
Jeffrey was waiting when I landed in Los Angeles. He looked even more handsome than I remembered him. We drove down to San Diego and checked into a motel near the construction site. It was going to be a fun weekend.
Things immediately turned strange.
As soon as we got to the motel room, Jeffrey wanted to make love—right away. Yet I needed just a few minutes to unwind; I was hoping to freshen up a little and maybe unpack first...
No! He wanted me now. Now! He threw me on the bed, and before I knew it, he was on top of me, pulling off my clothes and pushing my legs apart, foreplay be damned! He was going to have me, goddamn it!
It was fast, frenetic, exciting, and exhausting. I had never made love with anyone who was so aggressive and unstoppable. I hardly had time to gasp between thrusts. When he was finished, I was so out of breath I couldn't talk.
"That was fantastic!" I managed to get out.
"Okay," Jeffrey said. "Roll over."
Roll over? I looked down and saw that he was ready for action again. Already?
I guess I didn't move fast enough for him, because he grabbed me and spun me over, and we were off again—and again, and again.
I don't know how to explain it, but in the space of a two-hour ride down the 405, Jeffrey had morphed from a courtly, polished doctor into a voracious sex machine. He was putting me in all kinds of positions: against the wall, over the vanity, upside down...coming at me in every direction. I was spinning around like a pinwheel.
There was no emotion behind it, though. None. Not a trace of that ardent, caring lover I'd read so much about in his letters. It was all about sex, nothing more—and his sex, not mine. Whether I was getting any pleasure out of it didn't seem to matter at all to him. I was just a rag doll he was throwing around the room.
I guess one often fantasizes about being swept away in a riot of sexual ecstasy, but this wasn't ecstatic at all. It was cold and animalistic. The sex was almost brutal in its forcefulness—and relentless. He never got fatigued. I couldn't understand what kept him going. This was long before Viagra, but since he was a doctor, maybe he had access to medicines that mere mortals didn't.
When he did take a break, it was to perform oral sex on me—which was thoughtful, I guess, but it came with a little quirk: he asked me if I had any perfume. I didn't; I never used perfume. So he took out his own perfume—curious that he would be carrying women's perfume around with him, but that was the least of his oddities—and insisted I spritz it around my crotch. He refused to go down on me unless I had a sweet-smelling vagina. (He was no Pierre.)
I should have told him to go to hell, but I really needed the rest, so I dabbed the perfume on my inner thighs, and when I was sufficiently fragrant, he went to work. I tried to enjoy it, but I was too exhausted.
Then back to the grind. This went on for hours, all through the night. I think we may have stopped for dinner, but it was only the briefest of respites. Finally I begged him to stop. I was too sore; I couldn't take it anymore.
Jeffrey shrugged, and rolled over in bed. "Tomorrow I'll show you the mansion," he said, and fell asleep.
Actually there was no mansion; there was a spot where the mansion would be. The next morning, we took a tour of the site. He owned a huge piece of property in the hills, and judging from the view and the blueprints, you could see it would be magnificent when it was completed.
Right now, however, there was nothing. The foundation was being poured, and there were lots of construction workers milling about. Otherwise, it was just a lot of sagebrush. _This was worth flying from Paris to see?_
While we walked about, I started feeling a pain in my groin. After all that crazy sex, maybe I'd pulled a muscle, or something was torn. I'd have to take it easy today.
When we got back to the motel, though, I barely had time to get an aspirin down my throat before Jeffrey pushed me back on the bed. Time to earn my keep.
The moment he entered me, I felt a searing pain. "Stop, stop!" I cried.
He didn't stop.
"No, it hurts!"
"Oh, it doesn't hurt that much," he insisted. Oh, but it did. I struggled away from him and went into the bathroom.
I discovered to my horror that there was blood in my urine. In fact, my urine _was_ blood
Jeffrey looked down into the toilet bowl, unimpressed. With an impatient sigh, he inspected me. "Hmm," he murmured, "it looks like cystitis."
"Cystitis? What's that?"
"It's an infection. It happens. I'll write you a prescription."
He scribbled a prescription for an antibiotic. I started for the door.
"Wait a minute," he said, grabbing my arm. "Not yet." He threw me back on the bed.
It took about a day for the antibiotics to kick in. Until then, the pain was excruciating—which didn't matter to Jeffrey. He kept screwing me. That's why I was there. He was furious that I'd gotten this infection, but he wasn't going to let it cramp his style.
I found out later, by the way, that traumatic cystitis results from a bruising of the bladder, which can be caused by unusually forceful intercourse. So I was definitely fucked into this condition.
It was at about this point that I began to realize that Jeffrey was not merely weird but probably a monomaniacal sociopath. He had that depraved indifference to human life that you hear so much about on crime shows. Somehow I hadn't picked up on this. I guess I'm just not very good at reading men.
Now I was genuinely frightened of him, and I didn't know what my next move would be. Jeffrey had shown flashes of his temper already, and I could well imagine the kind of violence he might be capable of. No one, not even Luke at his most out of control, terrified me as much as Jeffrey.
So, on the third day, when he asked me to marry him, I immediately said yes. I didn't want to marry him, God knows, but I didn't think I had a choice. I couldn't say no. He was already bouncing me off the walls for fun; what would he do to me if I actually crossed him?
I called Mom in Malibu, ostensibly to break the news, but I think secretly in the hope that she would save me.
"What?" she rasped. " _Who_?"
"Dr. Jeffrey Dietrich," I said.
"Doctor? How did you meet _him_?" Already she was skeptical.
"He's a plastic surgeon. I wanted to get my breasts enhanced."
"Well, you _could_ use some help there," she conceded. "That doesn't mean you have to marry the guy. Are you in love with him?"
I paused a moment. "Yes. I guess."
Mom heard the panic in my voice. "I'm coming down there," she said tersely. "Right now."
Two hours later she was pulling up to the motel. I was nervous about Jeffrey—would he be angry that my mother was coming to visit without his permission? Already, in the space of three days, he had so taken control of my mind and my nervous system that I was petrified at the thought of upsetting him.
To my relief, it was the charming Jeffrey who emerged to greet Mom. He was delighted to see her, he really admired her film work, she was even more beautiful in person—he could not have been more cordial or gentlemanly.
Mom wasn't fooled. She was in show business; she knew when she was being bullshitted. He was too old for me, and there was something about him that she just didn't like.
She never said as much, but when she wouldn't stay for dinner and wanted to head back home right away, I knew she was less than enchanted with him. She wanted me to go back to Malibu with her. "You don't mind, Jeffrey, if she visits with her mother for a few days, do you?" she asked. He probably _did_ mind, but she had so artfully phrased the request that it would have been churlish to deny her.
I was planning to be gone for only a few days, but things played out much differently after Mom realized she'd forgotten her reading glasses.
We were sitting in the car outside the motel, and Mom was trying to figure out how to get back on the freeway. She had a map spread out in her lap, but couldn't figure it out without her glasses. I was hopeless at reading maps, so Jeffrey was leaning into the car window, trying very patiently to explain it to her:
"You have to follow this road here, and where it forks, you bear to the left..."
"Wait, wait, you're talking too fast!"
Jeffrey smiled tightly, and spoke slower: "When the road forks, you bear to the left..."
"There's a left here?" she said testily. "Where? I don't see it." Of course she didn't see it; she didn't have her glasses. When Mom got frustrated with something, she often found comfort in snapping at everyone else.
Unfortunately, Jeffrey was not one to be snapped at. He was an important doctor, accustomed to being listened to, and his mood could turn on a dime, as it did now. His patience instantly evaporated, and he snatched the map from Mom's hands, crumpled it up into a ball, and tossed it through the window at me in the passenger seat. "Here, you'll find it," he said, and he walked back into the motel.
Mom looked at the crumpled map in disbelief, and then at me. "Let's get the hell out of here," she suggested. She floored the gas pedal, and we shot off like two bats out of hell.
She didn't say too much until we found the freeway, but after that she gave free rein to her opinion. "He's an asshole! A pompous, self-important asshole! And an _old_ asshole! You're not marrying that middle-aged prick, no fucking way!"
Now, I have to say, I was raised in Japan to be proper and soft-spoken, to strive always to swallow my emotions. I never swore, and whenever my mother let loose with a barrage of colorful epithets, I was always a bit shocked and embarrassed for her. How could she say such intemperate things? Didn't she realize how poorly it reflected on her? However, in this case, I found myself in full agreement with her.
"You're not going back there," she decided. "You're staying with me."
As I heard this, I felt a calming joy come over me. _I'm staying with Mom_. I realized that this was what I had always wanted: to be with her.
So Jeffrey turned out to be a good thing after all.
Nevertheless, Mom hired a private detective to look into the background of Dr. Jeffrey Dietrich. Over her many years in the business, she'd had occasion to use private detectives, and now it was a habit with her. She was sure there was something fishy about my former fiancé, and she was going to find out what it was.
This is what she told me she discovered: Dr. Jeffrey was a successful, highly respected plastic surgeon in the Chicago area—but he had a secret side business that was in many ways even more lucrative.
It seems that Jeffrey would often attract clients (usually young women) who coveted his services but couldn't afford his gold-standard prices. Rather than turn them away, Jeffrey would offer them a sort of barter system: he would fly the women down to Colombia, where he had a business colleague who would perform breast surgery for free. Instead of putting in silicone implants, though, the Colombian surgeon would fill the women's breasts with bags of cocaine. The women would then fly back to Chicago, and Jeffrey would remove the cocaine and put in the proper implants. Tit for tat, you might say.
So Jeffrey, among his many other distinctions, was an international drug trader.
This is why Mom did nothing with the information. She thought it more prudent to keep it strictly between us. "Look, we could go after this asshole and get him arrested, but..."
"We might get our necks broken," I said.
"Exactly."
Looking back, I'm surprised he never suggested any such drug swap with me. Then again, I was with him for only four days total—just enough time for him to propose.
That was something I never quite figured out. Why would he want to marry me on such short notice, especially given his tastes and background? That question led me back to Dad.
When Dad found out that I had broken up with Jeffrey, he was really angry with me. "I'm very disappointed," he told me over the phone, in his gravest voice. Why? True, I had never told him how Jeffrey treated me at the motel, or what Mom's private detective had uncovered, so maybe my split with Jeffrey might have seemed capricious. Still, why should it have mattered to Dad? He didn't even know the guy.
Or did he? It was Miki's friend Anastasia who had hooked me up with Jeffrey in the first place, and she had pushed for the romance. Was it all at Dad's direction? Was he promoting the match behind the scenes, just the way he had tried to bring Luke and me together? And if so, why? Was he involved in this drug business, too? Is that where he got all his money? I flashed back again to all those puzzling scenes from Dad's life: the loose diamonds, the dead man in his office, the misleading phone calls, the orgies in his hotel room. _Who am I? I can't keep this up anymore_. Secrets built upon more secrets.
Yet there was another secret waiting for me in Malibu, in a box of yellowed telegrams in an old brown safe: a revelation that would blow all the other ones right out of the water.
# **Chapter 11**
# Man in Space
It was while we were driving back up the coast that I decided to ask the question. So many things were whirling through my head—Dad, Jeffrey, my life so far, from Japan to Europe to Australia, everything that appeared at once utterly random and yet somehow connected. If I could only find that vital missing piece, the common thread that would pull the vagrant strands together and give me that long-desired "Eureka!" moment.
I suspected that Mom held the answer. She had rescued me from the sociopath, she was bringing me home, I was going to be a part of her life from now on. All roads led to her, and with this bright future stretching ahead of us, it seemed like the right moment to make some sense of the past.
"Mom, is there something going on that I don't know about?" I was referring to her and Dad, and their unconventional marriage, and where I fit in.
She was evasive at first, but finally she said: "I'll tell you when we get home."
**• • •**
**NOW** let me say a word or two about Mom. She's a born storyteller. It's in her Scotch-Irish genes. She loves to entertain, and she despises the unforgivable sin of being boring—and I have to say, she's never been more inspired or inventive than when she's recounting the story of her own life.
She's written about a dozen books now, recounting her many worldly and otherworldly experiences, and all of them have been quite successful. Many critics have commented on her natural gift for storytelling. Dramatic confrontations and juicy dialogue abound.
I wasn't there when most of these stories were supposed to have happened, so I can't pass judgment on their veracity. Some of them are probably true, but even given the make-believe dynamics of Hollywood, there are a few tales that strain credulity. Just one example: Mom tells of being on the set of _Some Came Running_ with Frank Sinatra when a harried production assistant complained that they were two weeks behind on the shooting schedule. Sinatra supposedly took the script from the production assistant's hands, ripped twenty pages out of it, and handed it back. "There, pal, now we're on schedule." According to Mom, Sinatra commanded such respect and fear that they never put those pages back in.
It's a good story, but if you're familiar with film history, you know that a similar anecdote was told about John Ford years before. And if you're familiar with film production, you know that it's virtually impossible to cut a random twenty-page chunk out of a script, especially one based on a bestselling novel, without causing all kinds of mayhem with continuity and character development.
Still, it's a good story, and that's what counts.
I've witnessed a few of Mom's adventures firsthand, and I've noticed a little bit of embellishing here and there. Not wholesale fabrication, exactly, but artful stretches of the truth. With her, a simple trip to the supermarket becomes a search for spiritual enlightenment. My minor bout with endometriosis turns into a hospital scene out of _Terms of Endearment_ , with a frantic last-minute dash to the airport thrown in for good measure.
For the most part, so what? It's a showbiz axiom that you never let the truth get in the way of a good yarn. There are exceptions, though. The true reason for my childhood move to Japan is one of them. That's a story that was never clear to me from the start, and as the years went by, it grew only murkier and more convoluted. Still, since it was clearly the most influential decision bearing on my life, it was important for me to find out what had actually happened, and why.
There was an accepted explanation at the time: my mother's career in Los Angeles was taking off like a rocket, my father had important business concerns in Tokyo, and it made sense for me to be raised in the more stable of the two environments. Children, after all, need consistency, and in Japan I would receive structure, a sense of traditional values, and one parent's full and loving attention—more or less.
If you read my Mom's various memoirs, however, you'll find that this story has been tweaked and contradicted more than once. In her first book, _Don't Fall Off the Mountain_ , I was sent to Japan because my father was stricken with hepatitis and languishing in a Tokyo hospital, far from home and all alone. (Mom didn't know about Miki at the time.) For some reason, Mom felt that my presence at Dad's bedside would go a long way toward raising his spirits. So, the story went, she put me on a plane and shipped me across the Pacific to work my two-year-old magic, a miniature Florence Nightingale. I turned out to be such a tonic for Dad that he recovered in record time—and although I was supposed to return to L.A. after that, he decided to keep me with him, as a lucky charm, perhaps.
According to Mom in that first book, now _she_ was the one who was lonely. She missed us so desperately that at one point she joined us in Tokyo, shucking her career to become a modest Japanese-style housewife. (I don't remember this at all, but I was only two, so who knows?) For six months we led an idyllic life—we were a complete family again—but eventually Mom started to feel restless stirrings. Performing was in her blood; it was what she needed to do—but how could she abandon us again? She was torn between the dictates of conscience and career: a wrenching dilemma.
It was only when I, little precocious Sachi, sensed her growing discontent and told her that she needed to follow her dream—"Why don't you go back to work, Mommy?"—that she returned to the States to resume her movie career and become the great star she is today. Yes, it was all thanks to me.
That was just one story. If you check out her later book, _Dance While You Can_ , you'll see that I'm sent away for the more mundane reason that Hollywood is an awful place to raise a child, and I'd have a much better chance of growing up normal in a different, more becalmed setting. Here's the strange thing, though: in this version, I move to Japan when I'm six years old, not two. What happened to those four years? And what happened to her long months of living in Tokyo, and my words of wisdom that sent her back to America? Did any of that happen? Again, who knows?
In an even later book, _My Lucky Stars: A Hollywood Journey_ , the Mob story emerges: Mom gets the word from an associate that the syndicate is thinking of kidnapping me. It seems the goons want to muscle in on Mom's career and get her under their exclusive control (which is a twist on the older story, which held that the Mob was angry with Sinatra). She has no intention of playing ball with them, so Dad insists, for safety's sake, that I leave the country. (I don't know if I'm six or two in this story; it's not clear.)
The dialogue in this beach scene is especially rich: Mom gets off a few of her choice expletives ("those cocksuckers!") and has one visionary outburst in which she declares she's not going to knuckle under ("and you can tell them to shove their horses' heads up their asses!"). Since this scene takes place a good ten years before _The Godfather_ was published, I don't know where she pulled this iconic image from, unless her gift for channeling had granted her access to the future as well as the past.
They're all good stories, I have to admit: solid, motivated, reasonably credible. Still, they can't all be true.
In fact, none of them is. The inside, exclusive story, the one Mom revealed to me when we arrived in Malibu, is the least believable of them all, but it's the one that actually happened—or so she swears.
**• • •**
**WE** arrived at her beach house. I followed Mom into her bedroom, where she pulled an old brown safe out of her closet. She carried it into the living room and fiddled with the combination. The safe popped open and she removed a tin box containing a stack of old telegrams. "Read," she said. "This will explain everything."
The telegrams gave me an ominous twinge. More letters from Miki, perhaps?
I sat on the living room couch, picked out the first telegram, which was dated April 1956. It read something like this:
_My darling Shirley, I miss you so much. Can't wait to see you again. Love and kisses. Paul._
Huh.
It was highly unilluminating. Someone named Paul. Obviously a romance. Was it an affair? Had she been cheating on Dad? Did it matter? They were both proud of the fact that they had a modern, open marriage, so a little dalliance on the side was no big deal. Was this something deeper, though? Was Paul the true love of her life?
I looked at Mom, bewildered. She gave an existential shrug. "Keep reading."
So I read another telegram, dated a few years later:
_Looking down and thinking of you. I hope Steve is taking good care of Sachi. Please give her my love. Yours always, Paul._
So this Paul character knew my dad and me, and one could almost sense a bonhomie between them. But why would he have been concerned about my welfare? Why would he even have thought that Dad might _not_ be taking good care of me? What business was it of his? And why had he been sending _me_ his love?
I ran the name through my head, trying to make a connection. Paul, Paul...Did I know any Pauls? It wasn't Paul Newman, was it? Mom had made one movie with him, _What a Way to Go!_ , but he was already married to Joanne Woodward at that time, and they were a famously faithful couple...No, it wasn't very likely. And I was pretty sure it wasn't Paul Lynde. What did "looking down" mean, anyway? Looking down from where?
There were many more telegrams to get through. I read them one by one, trying to piece together the secret.
_Still missing you terribly._
_It's cold up here._
Up where?
_Can't wait to be back on earth._
Back on earth? What the hell does _that_ mean?
I waded through dozens of messages, trying to make sense of these cryptic pre-Twitter fragments. Finally I gave up and tossed the telegrams aside. "I'm not getting it, Mom."
"Keep reading."
"I don't want to keep reading. Why don't you just tell me? Who's Paul?"
Mom took a deep, dramatic pause. It was the moment of the big reveal, and she was milking it.
"Paul..." She ran her hand across her mouth anxiously, and had to take yet another moment. Finally: "All right, I'll just say it: Paul is your real father."
There's a curious sense of dislocation you get when you receive news that's completely at odds with everything you've ever known and understood about your life. It's like when you arrive at the trick ending of a movie such as _The Sixth Sense_ or _The Usual Suspects_ , and you discover that everything's the opposite of what you thought, and all those baffling twists of logic resolve into a crystal-clear picture. Except in this case there was no sudden clarity, no "Eureka!" moment at all, just ever-deepening confusion.
"My real father? Paul is my real father? How come I've never heard of him? Where is he?"
"Right now, he's in outer space." Mom said this with disarming matter-of-factness and a quiet sense of pride.
"He's in _outer space_?"
Mom made an equivocating gesture. "He's on a mission for the government."
"A mission? What kind of mission?"
She gave me a patient smile. "A _secret_ mission, sweetheart. All I know is, he's in the Pleiades. The Seven Sisters." She pointed heavenward.
I looked blankly up at the ceiling. Mom explained that the Pleiades is a star cluster in outer space. There are seven stars altogether, seven, which is a very significant number in Christianity, Buddhism, practically all religions or mythologies: the Seven Days of Creation, the Seven Stages of Enlightenment, the Seven Seals of the Apocalypse....She went on and on. I just nodded slowly. I had to abandon the Pleiades for the moment anyway, because there was a more pressing issue to confront.
"So, if my real father is in outer space, then...who's Dad?"
"You mean Steve?"
"Steve, my dad."
"Steve is not your dad. _Paul's_ your dad."
I pressed on. "Okay, then, who's Steve?"
"Well, Steve is no one, technically."
"What does that mean?"
Mom took another long, measured pause. "Steve—how do I say this? Steve was created by the government."
Now _I_ had to pause. " _Created?_ What do you mean 'created'?"
"He's a clone," Mom said.
I stared at her a moment. "Dad's a clone?"
" _Steve's_ a clone. Dad's in outer space."
I studied Mom's face carefully for traces of a telltale smile, or any sign that she was holding in her laughter. She had to be joking, right? She _had_ to be.
"I don't understand. Dad, _my_ dad, the guy I grew up with—he's a clone?" Plummeting helplessly down the rabbit hole, I tried to put on the brakes. "Mom, are you serious?"
She seemed affronted. "Of course I'm serious. You don't believe me?"
There was a hint of prickliness in her voice, and I realized I would have to dial back my incredulity a little or else she would shut down and I'd never get the rest of the story out of her. So I proceeded with a cautious reasonableness. "I believe you, Mom. I'm just trying to understand...You think the government cloned Dad?"
"They do it all the time, sweetheart. You see, Paul's space missions are so top secret, even _I_ don't know what he's up to. So the government had to create a second Paul, to avoid suspicion. In case the Russians found out."
"Found out what?"
"It's a _secret_. A very important government project—that's all I know. With global implications. And I'm helping to support it. I send money every month. It's my way of serving my country."
I just stared at her. I didn't know what to say.
Mom sighed. "I knew you would have a hard time with this. That's why I've tried to keep it from you all these years. But look, the facts are all right there. He's been sending me these telegrams all along." She ran her fingers through the telegrams as if she were Jimmy Stewart on the Senate floor in _Mr. Smith Goes to Washington_.
It was true; the telegrams were right there. They genuinely existed. _Someone_ must have sent them.
"Okay, if Paul is my real father, how come I've never met him?"
"You _have_ met him. He comes down from space every now and then to visit. But we didn't tell you because we didn't want to upset you. They look so much alike, you just never noticed the difference. Sometimes it was Steve; sometimes it was Paul."
"Can _you_ tell them apart?"
Mom gave a small, secretive smile. "Well..." Choosing not to elaborate, she tried to reassure me: "Look, Sach, I know what you're thinking. I've been skeptical at times, too. But I know for certain that Paul is Paul, and Steve is Steve. I know it."
How did she know it?
She told me the Caesar story.
Caesar was the family dog, a boxer with the sweetest temperament. He was the pride of the household, and Mom and Dad both doted on him, Dad especially.
When poor Caesar finally died of natural causes, Mom was crushed. She had a bigger concern, though: How would her husband handle the news? "You know how much Paul loved that dog," she reminded me.
"I know how much Steve did."
"That wasn't Steve. That was Paul."
Every time I interrupted her, I got more confused, so I just let her go on with the story.
"When Caesar died," she continued, "I had to break the news to Paul. But he was in space, so I called Steve in Tokyo and I told him, 'Caesar died.' And he took the news very matter-of-factly. 'Uh-huh, uh-huh. I see.' Didn't faze him at all—which proved to me right away that Steve wasn't Paul.
"So we discussed whether he should relay the news to Paul right away, or whether we should wait till the next time Paul came home and I'd tell him myself. Steve felt I should be the one to tell him, and I agreed.
"So it was a couple of months later now, and Paul was between missions. I had to pick him up at the airport, and we drove back to Malibu. I didn't say a word; I pretended that everything was fine. Then we pulled into the driveway, and finally I had to tell him that Caesar wasn't going to be there waiting for him; he was gone. And when Paul heard that—when he heard that Caesar was dead—he just fell apart. He was crying, he fell to his knees in the driveway, banging his fists on the ground, tears streaming down his cheeks—this was real pain. He was in anguish. No faking. I'm an actress; I know acting. And he would have had to be the greatest actor in the world to fake that.
"So, obviously, Paul was Paul, and Steve wasn't. Steve didn't care at all about Caesar. It didn't matter to him. Because, you see, you can't clone a soul."
I couldn't argue with that.
I was still having trouble wrapping my head around the initial premise. "I'm sorry, Mom, I'm sure it's all true, but I still find it hard to believe. I mean, if the government really wanted to keep something secret, is that really the best solution they could come up with? A clone?"
Mom fixed me with a benevolently patronizing look. "Sachi, you're very young, and you're very naïve; you don't know how the world works. The government does all kinds of crazy things. Do you realize they have surveillance satellites up there in space that can listen in on any conversation we have on earth? Wherever you are, if you're out on a nature hike, or on a boat, anywhere—they're _listening_."
Mom suddenly looked toward the ceiling, raising a finger. In a hushed whisper: "They're listening right now."
She started yelling at the ceiling, defying those Orwellian transmitters: "I know you're listening, but I'm telling her anyway! Because she's my daughter!"
She waited a moment, as if expecting a response, and then turned back to me with a resigned shrug. "I'm probably in trouble now—but whatever. You deserved to know the truth."
By this time I must have looked completely shell-shocked, because Mom placed a comforting hand on my wrist, patting it gently. "I know. You're confused; you're disoriented. It's totally understandable. But I checked out Steve very carefully, and I knew he would take good care of you. Was it an ideal situation, being raised in another country by a clone? Of course not. But there's always a struggle between public duty and personal need, and I think, all things considered, we managed to find the right balance."
I needed a little time to sit with this information. In a way, I wasn't completely surprised. It would be another year before Mom came out with her most talked-about and influential book, _Out on a Limb_ , in which she made public her belief in reincarnation and the astral plane. She'd been developing these interests in private for years, so I was familiar with her unique perspective on the cosmos, and generally prepared for anything.
This, however, was beyond me. Mom, for all her eccentric notions, was still a fiercely intelligent woman, capable of cold, clear-eyed logic when the situation demanded it. How could she have come to believe that Dad was a clone? Where could she have gotten such a story? It was insane.
Yet...the telegrams were right there. They were real. Mom couldn't have sent them to herself. They sat there in their box, mockingly real. They were saying, _Rationalize all you want, sister, but we're here and we're not going anywhere_.
When I considered their concrete reality in tandem with Mom's unshakable faith, I found my own conviction wavering. It _is_ a strange, secretive world out there, after all. Crazier things have happened, I guess. So maybe, I thought...it's true?
Then a casual phrase dropped earlier came drifting back, and I grabbed at it as if it were a lone, spindly branch overhanging a waterfall.
"Okay, so wait a minute, Mom. You send _money_ to the _government_ for this? How much money?"
"Sixty thousand dollars."
My jaw dropped. Sixty thousand dollars? Shipped into space? Oh, what I could have done with sixty thousand dollars! Gone to college, for one thing. "Sixty thousand dollars a year?"
She looked at me as if I had two heads. "A year? No, a month."
I gasped. "Sixty thousand a _month_?"
"Of course. Space travel is very expensive."
I had to get this on the record. "All right, since 1958, you've been giving the government sixty thousand a month?"
"No, it was a lot less back then, but as the years went by, we had to adjust for inflation."
"Where do you send this money? NASA? Washington?"
Mom was being very patient now. "I send it to Steve, the clone. That's the whole point. He's supposed to be my husband, so it doesn't look suspicious if it goes to him. And then he forwards it through the proper channels."
Now the fog was starting to lift.
"He forwards it? Sixty thousand a month? That's, what, seven hundred and twenty thousand a year?"
Mom was untroubled by this expenditure. She usually hated parting with money, but this was a special case. "It's a lot of money, yes, but it's for a worthy cause. We all have to make sacrifices for the general good."
"Steve doesn't have to make any sacrifices," I observed. "Steve has a very comfortable lifestyle for a clone."
Mom shrugged. "He has to keep up appearances." She could see that I was getting at something unpleasant, and whatever it was, she was instinctively mounting a defense.
"He appears to be as rich as Croesus," I told her. "You know, he goes to expensive restaurants every night? Every single night. He has a yacht in the Mediterranean, and his own private island in the Pacific. He drinks Dom Perignon for breakfast. Every breakfast! He has potatoes shipped to Tokyo from Idaho! Then he scoops out the insides and fills the potato skins with Beluga caviar. That's his midnight snack."
There was a pause as Mom processed this information.
"So, what are you saying?"
"I'm saying he has an awful lot of spare change. About sixty thousand a month's worth."
Mom was genuinely shocked. "You think Steve's been taking your father's money?"
"He _is_ my father!" I exploded. "He's not a clone! Steve Parker and Paul are the same person! The whole story is a fake. He made it up to trick you into sending him a check every month. Mom, he conned you! Dad's a con man!"
The moment I said it was the very moment I realized it myself: Dad's a con man.
This was something I had never allowed myself to consider, but as soon as it came out of my mouth, I knew it was absolutely the truth. I'd never quite understood what Dad's business was, but now I knew: he was an operator, a flim-flam artist, a jet set Harold Hill. His job was to enjoy his own life at everyone else's expense. And his biggest pigeon had been his own wife.
_A Dweller on Two Planets_ —the epic story of Phylo the Tibetan, who lived in the lost city of Atlantis—that's how it began. Dad gave Mom that book on the set of her very first movie, _The Trouble with Harry_. Mom had responded positively to its message and came to embrace the possibilities of reincarnation, telepathic communication, and other forms of expanded consciousness. Just as Dad had hoped.
Having laid the foundation, he was now in the position to convince Mom that he was two people: he was a space traveler cruising the Pleiades, and he was a clone of that space traveler. And one of the reasons he had to relocate to Japan was to keep anyone from detecting his clonehood. Once there, he needed Mom to finance his playboy lifestyle, so he appealed to her sense of patriotism and got her to bankroll the "space mission."
It was an incredibly bold plan. The risks were obvious. If Mom hadn't gone for it, their marriage would have been over in a New York minute.
Yet he knew that she would go for it. Like any great salesman, he could read his mark like a book.
He also had a masterstroke up his sleeve, and that was in bringing his daughter to Japan. It was a classic application of reverse logic: everyone (i.e., the Russians and various foreign spies) knew that a responsible mother would never send her child overseas to live with a clone, so Mom _had_ to send me—it made the ruse complete.
This was also why Dad never spent any money on me; he _couldn't_ , because all that money was supposed to be going to the "government." If Mom saw him buying me new clothes or taking me on trips or paying for my college education, she might get suspicious. So, for the sake of the scheme, I went without. At the same time, by raising me Japanese—docile, submissive—he could guarantee that I would accept any outlandish situation without question. He really was a brilliant man.
Of course, the success of such a campaign was predicated on the unlikely scenario that Mom and I would never discuss the arrangement. Ever. Now we had discussed it, though, and the cat was out of the bag: Dad was exposed as a humbug, the man behind the curtain.
Now Mom was the one whose world had been turned upside down. Open marriage aside, Dad was her great lifetime love. She respected him, she adored him, she took comfort in the fact that he was her true soul mate and they would always be there for each other.
But he wasn't there. Maybe he had never been. Maybe their entire relationship had been one elaborate con. "Paul" was my mother's fantasy, and Steve—charming, enigmatic, high-living—was a con man.
At first I had felt a sense of triumph in disclosing the extent of Dad's machinations to Mom. For once, I was the one who had the answers instead of the questions. I was the one who was in the know.
This quickly changed to deep sympathy for her. I knew what she was suddenly facing—profound betrayal, the loss of love—and my heart broke for her.
She stood with her arms folded, staring toward the floor, stoically weathering the emotional assault. I waited for her reaction.
Finally, she shook her head. "Ridiculous," she said.
"It's not ridiculous," I insisted. "He _is_ a con man. You know he is! He's certainly not in the Pleiades. Do you know how far away that is?"
"Forty-three light years," she said with fading defiance. "So what? You never heard of hyperspace?" She wasn't giving up. She was holding on to the dream, come what may.
"Oh, Mom...How did you ever fall for this?"
Mom snapped. "I didn't fall for anything! Your father is in space, and Steve is a clone! And that's the end of it." She grabbed the box of telegrams. "I knew I shouldn't have told you. You just don't have the maturity." She stomped back into the bedroom.
**• • •**
**SHE** needed time. Dad, whether he was Paul or Steve, was the love of her life, and she had invested everything in that relationship. To accept now that he had manipulated her in this spectacular fashion, deceived her and played her for a fool, would have been too devastating to bear.
But she was also a very smart woman, and she knew she couldn't keep deluding herself. After taking a few days to absorb the story, she knew she had to confront Dad.
I remember we were sitting in the kitchen in Malibu, enjoying a very tasty Chilean red wine—at only five bucks a bottle—when Mom jumped up and said, "I'm going to call him."
I felt a twinge of alarm. "Are you sure you want to?" I knew that once they spoke of this, there would be no going back. I felt terribly guilty, too, because, unknowingly or not, I had brought about this situation. I was the messenger of doom.
Mom got Dad on the phone, and immediately started firing questions at him. "What's this about a chalet? You have a _chalet_?" "I didn't know you had a yacht!" "You have a helicopter business? Since when? What is this?"
I couldn't hear what was going on on the other end, of course, but I could see from Mom's face that Dad was doing his best to spin the story to his advantage.
Mom wasn't falling for it. For the first time, she could see right through his diversions. Then she said the words that made my stomach leap: "I want a divorce."
I was shocked. I didn't realize she was ready to take that extreme step—and maybe she wasn't, until she heard Dad's voice and the charming lies he was casually spinning. I was glad I had the wine to calm my nerves.
Even more shocking was the look on Mom's face when she heard his reply. She was clearly stunned, and deeply unsettled. She slowly put the phone down, as Dad had already hung up on her.
"What did he say, Mom?"
Mom took a breath, and then answered, "He said, 'You're out! OUT!'" She mimicked his voice, harsh and ugly.
"Out of what?" I asked.
She shrugged. "Out of his life," she surmised. She sat down to finish her wine, but I could see that she was rattled. It wasn't easy to get to Mom, but that violent outburst had really gotten her. I could read the deep hurt and shock and abandonment on her face; she was suddenly a little girl, orphaned and cast away from the only mooring in her life. I felt so sorry for her at that moment that I wanted to wrap my arms around her and hold her, give her comfort.
Mom then hired a private investigator to check into my story.
The investigator confirmed that Dad was living a lavish lifestyle, with no visible source of income except my mother's monthly stipend. He wasn't in outer space; he was cruising the Aegean with Aristotle Onassis. The investigator also discovered that much of Dad's official past history had been fabricated: he never lived in Japan with his father, he never served in the armed forces, he never met a little Japanese orphan in Hiroshima named Sachiko. There was no indication that he had had any connection at all with NASA or the U.S. government.
There was, however, some kind of government file with Dad's name on it, labeled "top secret." Since it was top secret, we never found out what it contained. It could have dealt with his years in space, or it could have been surveillance accounts of his various dealings with influential foreigners. He was, after all, good friends with Prince Sihanouk of Cambodia (or so he claimed).
The investigator also told Mom that Dad was hiding his money in Miki's accounts. That one really threw her for a loop. She knew that Miki existed, but she thought of her as a minor character—in fact, that she was Steve the clone's girlfriend. She didn't realize until now that Miki was her real husband's lover, and living off her money.
So, in 1982, after twenty-eight years of marriage, Mom filed for divorce.
**• • •**
**IT** was a great relief to me when Mom finally accepted the clone story as false. _Oh, thank God_ , I thought, _I'm not the one who's crazy_. We actually had some sardonic fun going over the whole story of Dad's con, the scope of his deception, the obsessive attention to detail that had made it all work _: The Dweller on Two Planets_ , the telegrams, the government satellites.
"You know," Mom reflected, "I would fly all over the world to meet him, all these secret meetings when he was supposed to be Paul. And I used to wonder in the back of my head if he was Paul or Steve. So I would come up with little tests to see how he responded. Things that only Paul would know. He always passed, but still, I was never quite sure. But then he pulled that Caesar routine, crying in the driveway, and that's when he really hooked me. He was that good. He's a great actor."
Her admiration was steeped in scorn. She had loved my father, and now she detested him. Part of that was my doing, and I felt some guilt about it. So I would always try to balance the equation with stories of the good times I had spent with him in my childhood. I told her about the places we went, the experiences we had: ice-fishing in the Japan Alps; driving around Hawaii; cruising the Greek isles on his yacht, _Happy Pappy_.
She didn't want to hear it. She was not interested in my good times. She was far too wounded. I think nothing hurt her in life as much as that betrayal. I don't know what disappointed her more—the fact that her husband had cheated on and lied to her, or the fact that there was no actual Paul in outer space. The great romantic notion of her life had been crushed, and the world had become depressingly ordinary. How could you not be bitter about that?
Mom talks about her divorce in one of her books, but she has a slightly different version of the events leading up to it. She claims that she discovered my dad's treachery—the lies, the money swindled, Miki—through a channeler, someone who was letting a disembodied spirit speak through him. She would regularly go to spiritual channeling sessions for enlightenment, and on this occasion she got more enlightened than she'd bargained for: the channeler told her that Dad was a fake who had squandered much of her fortune and transferred the rest into Miki's account. A private investigator later confirmed everything the channeler said.
It's pretty close to my version, except that there's no mention of clones. She totally buried the lead role.
About six months after the clone situation imploded, I was called to give a deposition in Mom and Dad's divorce. It was a difficult situation for me. Much as I had been stunned by the revelation of my father's betrayal, I didn't want to testify against him.
By this time he had stopped talking to me altogether. He'd really grown fond of that regular check coming in every month, and he was not appreciative of my well-intentioned meddling. "Idiot!" What else could I do, though? Did he want me to lie? Obviously, yes.
The morning of the deposition, I woke up with a raging fever. It was 103 and climbing. I felt terrible. I needed to go back to bed. I called the lawyer and asked if we could get a postponement.
My father refused. He didn't care if I had a fever. He wanted me deposed, now.
By the time of the deposition, my temperature was at 105 degrees, and I was practically delirious. It was a surreal situation: the lawyer was bombarding me with questions, but I can't remember what they were. I can't even remember if I answered coherently. The whole day was a fog.
I do remember Dad and Miki sitting across from me at the conference table. I'm not sure why Miki was even there, but this was probably the first time Mom had met her as a rival. (They may have seen each other in Japan, but it had no significance then.) I remember the coldness coming from the other side of the table. From Miki, I expected no less, but while I had experienced Dad's icy anger before, I'd never felt the full brutal force of it as I did now. There was not an ounce of love coming from him, not a glint of warmth or charm. When the deposition was over, he got up and left, without saying goodbye. We didn't speak again for many years.
Mom and I never discussed the clone story again after that. It retreated back into the closet, back into the tin box, back into the old brown safe. She never got any of her money back, and she and my dad never reconciled. She does say in one of her books that she will always love him, and when she meets him again in the next world, with all their human foibles and frailties fallen away, that love will burn even brighter than ever. I hope so.
# **Chapter 12**
# The Acting Bug
It took me a long time to get used to living in Los Angeles. I'd spent many a summer with Mom as a child, but back then, I was just a visitor passing through, and didn't feel the need to immerse myself in the idiosyncrasies of American culture. Now I was here to stay, living in Mom's Malibu home, and I had to figure out how to make sense of this crazy world.
Starting with Mom. She was a mercurial, perplexing creature; I already knew that. Living with her day after day, though, adjusting my slow Japanese rhythms to her hard-driving American energy—that was the challenge. And it was my challenge, not hers. She was a star; she wasn't changing. It was up to me to negotiate the hairpin turns of her personality and keep our relationship on an even keel.
Not that it was ever hard to read her moods. She was very up-front with her feelings: if she was bored with you, or contemptuous, or pissed off, you knew it. The trick was anticipating when the weather was going to turn, because she could shift from a warming breeze to an ice storm in a millisecond.
In general, I discovered that she was always happy to see me, or anyone, for about four hours. For four hours, she loved you. Then, suddenly and without warning, you became an intolerable burden. The iron door came down. You could set your watch by it. One minute she was Miss Hospitality, the next minute "When the fuck are you leaving?"
Since I wasn't leaving, this was something we had to work out—and I would _make_ it work out, because she was all I had left. Dad wasn't speaking to me, my personal history was an apocalyptic wasteland of Lukes and Jeffreys, and my life compass was spinning in all directions. I needed a rock, something to cling to. That was going to be Mom, whether she liked it or not.
Of course there were so many demands on her attention, I knew I needed a hook to make myself a part of her world. But what? What could I bring to the table?
Well, I was a pretty good cook, from my days in France. "Fine," Mom said, "you can do all the cooking." Not _all_ the cooking—she had a chef for that—but whenever she had guests, she wanted me to show off my skills, so she could brag about me.
She would spring these parties on me at the last minute.
"Sach, I'm having a dinner party tonight!"
"For whom?"
"Just a few people. Maybe twenty. Robert Redford's coming. Faye Dunaway. Sydney Pollack."
Okay, I may have been out of the loop, but I knew who _those_ people were.
"We need food. Here, go out and get something." She handed me ten dollars. "And save the receipt. We can write this off."
I looked down at the lone bill in my palm and gave an incredulous chuckle. "Mom, I can't get food with this."
"Why not?" she asked. "What's the problem?"
"Ten dollars?"
"Yessss...?" She drew the word out, still not getting the point.
"For twenty people? What can I buy with this, a box of pasta and a pound of butter?"
Mom stopped now and gave me the Look. I've mentioned the Look: sharp, piercing, basilisk-like. "So you're saying you need more money?"
"Yes."
This did not sit well with her. Her eyes narrowed, as she tried to search out the truth in my dissembling eyes. Then she folded her arms and took a combative stance. "Are you on drugs? Is that what you need the money for?"
I was dumbfounded. "Drugs? What are you talking about?"
"Listen, Sachi, I've seen this happen to other kids in Hollywood. Their parents indulge them, let them have whatever they want, and the next thing you know, they're smoking pot, shooting heroin, killing people...Ever heard of Charles Manson? Squeaky Fromme?" (Actually I'd never heard of Squeaky Fromme, but I gathered that she wasn't a very reputable character.) "Do you want to be a cult murderer, is that what you want?"
"I just want to make dinner."
She grudgingly gave me more food money, but it didn't leave her pocket willingly. Mom was always pretty tight with a buck; there's no way around that. It wasn't long after I moved in with her that she brought up something that had obviously been bugging her for some time: "Say, when are you gonna pay me back for that car?"
"What car?"
"The five-hundred-dollar piece-of-shit car you bought in Australia."
"The Vauxhall?"
"Whatever. Do you still have it?"
"That was five years ago. It's probably in a junkyard some-where."
She nodded knowingly. "I told you it was a waste of money." Then she waited, as if expecting me to produce the cash right there.
"Sorry, I forgot, but I'll pay you back..."
"When? The interest is mounting up."
"Interest?"
She widened her eyes, as if to say, "We had a deal, remember?"
"Okay," I began, "so, tell me how much I owe, and..."
"Eight hundred and forty-five dollars," she shot back without hesitation. I don't know if she pulled this figure out of thin air, but I was in no position to question it.
"Oh. Okay."
Mom watched me for a moment, and then sensed perhaps that she may have been a little too harsh. "I mean, take your time. Get a job first. Just don't forget."
Her fear of my drug susceptibility, by the way, was completely baseless. I'd smoked pot three times in my life, and puked three times. I was not an enthusiast.
Mom was even less drug-savvy than I was. You'd think somebody with her reputation for experimentation would have been sampling all kinds of hallucinogens, and have had platters of mushrooms and peyote on the dining room table for general consumption. But no, she preferred "natural highs."
Once, we went to a dinner party where Mom was the only celebrity, and therefore the de facto guest of honor. Everybody made a big fuss over her and treated her like royalty, which she graciously accepted as her due. After dinner, the host brought out a "special sugar" with dessert. It was a bowl full of cocaine. Mom, being the honored guest, was offered the first dip. So of course she poured two heaped spoonfuls—and I don't know how many thousands of dollars—into her coffee. The collective gasp that followed almost sucked the remaining oxygen out of the room.
"How the hell was I supposed to know it was coke?" Mom said afterward, shrugging. "What am I, a drug mule?"
It always amused me that Mom would watch her money like a hawk but never seemed to know what things actually cost. She'd haggle over pennies, and then spend a king's ransom on a new pair of shoes.
I remember being in the kitchen one time, making whipped cream, when she came back from one of her shopping jaunts on Rodeo Drive. "What do you think of this blouse?" she asked.
"It's beautiful."
"Two thousand dollars."
I gulped. "Wow. That's a lot of money."
She shrugged as if to say, "So what, I deserve it." She looked over my shoulder. "What are you making here, whipped cream?" She stuck her finger in the bowl and tasted it. Then a mischievous impulse seized her and she dipped her finger in again and stuck a dollop on the tip of my nose.
This could not go unanswered. I took a slightly bigger scoop of cream and stuck it on _her_ nose. Not content to give me the last word, she grabbed a handful and smooshed it right into my face. I returned fire, slathering her face, and the next thing I knew we were throwing great gobs at each other, slipping and sliding as we chased each other around the kitchen, and then fell to the floor, rolling in whipped cream and laughing.
She totally ruined her two-thousand-dollar blouse in the process. She didn't care. She was having fun.
That's one thing I can say definitively about Mom: she loved to have fun. She could be inspiredly silly without a trace of self-consciousness, and she had a healthy sense of humor about herself and the world. I think that's why she managed, in all her talk of astral planes and extraterrestrials, to maintain the public's affection. She didn't seem to take herself seriously, even when she was taking herself very seriously.
We still had our walks on the beach, and she would still pick up sea urchins and react with delight when they closed on her finger. There would be stops at Wil Wright's for ice cream, and days on the couch where we ate popcorn and watched movie after movie; and when thunderstorms rolled in, we'd cuddle up in bed, although I was a little old now for the tale of Princess Lightning. In some ways she was younger than me, a wide-eyed six-year-old who loved to start trouble and then throw back her head and laugh. Those giddy moods of hers didn't last long, but while they did, we laughed and laughed.
**• • •**
**ANOTHER** memorable event took place in our kitchen a few months later. Mom and I were making pies together—I don't remember why—and as we rolled out the dough side by side, I had what you might call an epiphany. For the first time, I was actually baking with my mother; the way they did in commercials and old black-and-white TV shows. After twenty-seven years, we were finally performing a classic domestic ritual.
I didn't point this out to her, because I didn't want her feeling guilty about it. It wasn't her fault that she hadn't been there for me growing up. Okay, maybe it _was_ her fault, but now wasn't the time to make an issue of it. So I just sighed. "Wow, Mom. Life is so strange."
I kept rolling out the dough, until I felt Mom's eyes on me. She was staring at me, and turning something over in her mind. "What?" I asked.
"Say that again," she said. "The same way."
I was a little confused. "What? Life is so strange?" She gave me a coaxing nod, so I tried to replicate the way I'd said it, complete with sigh. " _Life is so strange..._ "
Mom pondered this slowly. "Hmm."
What was this about? Was she upset? Had I offended her somehow? I wasn't calling _her_ strange, after all. She was getting me all nervous.
"Let's play a game," she suggested now. A game? "I want you to go outside and pretend that someone was just hit by a car, and you have to run in and tell me about it."
I tried to get this straight. "You want me to go outside...?"
"Go! Now!" she barked. "Don't think! Just do it!"
So I did. I hurried out the door. What was I supposed to do? Oh, right, someone just got hit by a car, and...
I rushed back into the kitchen. "Oh my God! Mom! Help me! My friend...this car came out of nowhere and ran him down...He's bleeding...he's dying...! You have to come right away! Right away!" I was breathless and hysterical, and to my surprise, tears sprang into my eyes.
Mom was watching me keenly. I could see she was impressed. I could also see that now was not the time to break character. I had to push further.
"I'm not kidding, Mom! This is serious! This is not a game! My friend is dying! You have to hurry! Come on!" I started pulling on her arm, dragging her toward the door. I finally stopped when we reached the doorway, and I found myself out of breath, but a little exhilarated. It seemed like a pretty good performance.
Mom was still in nodding mode. "Okay, do it again."
"Again?"
She pointed me out the door.
So I ran out, went down the stairs, rushed back up the stairs, burst into the kitchen, and went through the whole routine again. "Oh my God! Mom! Help me...!" I went even bigger this time, hitting all the notes, letting my emotions pour out, but controlling them at the same time—in other words, acting.
Mom took it all in, and then made her pronouncement, which was all the more effective for its quiet restraint: "You're very talented."
I was stunned. "Really? You think so?" Nobody had told me I was talented since 1962, with _To Kill a Mockingbird_.
She crooked a finger at me. "Let's take a walk."
I dutifully followed her out of the house, and we started down the road. After a respectful passage of time, I finally asked, "Where are we going?" but Mom wasn't saying. She liked controlling the situation.
We stopped finally at a cottage. Mom knocked at the door. A kindly middle-aged woman answered it.
"Peggy," Mom said, "I want you to meet my daughter, Sachi. She's an actress."
I felt a thrill go through me. Mom had just told a complete stranger that I was an actress!
As we entered the woman's house, Mom explained to me, "Peggy is an acting teacher." Indeed she was. Peggy Feury was one of the most revered acting teachers on the West Coast. She and her husband ran the Loft Studio in downtown L.A., and she was a charter member of the famous Actors Studio. She had all kinds of young, exciting actors in her stable, and they all loved her.
I didn't know all this at the time. If I had, I would have been a basket case, because Mom insisted that I audition for Peggy, right then and there.
"Now?" It was all too sudden. I was still in a pie-making frame of mind. "Can't we do it tomorrow?"
"Peggy's a very busy woman. So am I. This is your chance." In other words, now or never. "You have nothing to be scared of."
What was I supposed to audition with? I had nothing prepared. Did she want me to do my hysterical car crash routine again? Even I knew that bit probably wouldn't travel well.
Mom and Peggy found me something to read. I don't remember what it was now—one of those boilerplate monologues you find in acting books. I was reading it cold, but I gave it my best shot.
Peggy was a wonderful audience. She smiled, shook her head in delight, leaned forward, gave a little "go for it" shake of her fists. She was totally present, very supportive. Mom stayed in the background, arms folded, not cheerleading but completely confident that I would deliver.
When I was finished, Peggy leaned back and smiled warmly. "Would you like to join my acting class, Sachi?"
It's funny how the direction of one's life can spin around and find its true course in an instant. Did I want to join her class? Of course I did. I was an actress, wasn't I?
I walked back to Mom's house on a cloud. I was so excited. "Can you believe it? Peggy Feury wants me in her acting class!" There was only one problem, and Mom was happy to articulate it: "How are you gonna pay for it?"
She didn't need to add the disclaimer "Don't look at me."
I couldn't afford to pay for the lessons. I'd already taken a job teaching English to Japanese children, but that money went toward the household and paying off the Vauxhall. Still, Peggy Feury wanted me to succeed as much as I did, so I worked things out with her. Peggy suffered from narcolepsy; she couldn't drive a car because of the concern that she might drift off to sleep behind the wheel. So I became her personal driver. I drove her to the studio and back home, and anywhere else she needed to go. I also swept the stage, emptied the garbage, ushered on scene nights, and cleaned the toilets. I was there all the time—and I loved it.
I'd never had the experience of being a part of the theater before. Yes, I'd been backstage at Mom's shows in Las Vegas, and on movie sets, and at opening night premieres—I knew about all the flash and the glitz—but I'd never acted in a high school show or a neighborhood theater; I'd never known the excitement and creative joy of working at my craft, bringing a scene to life, being part of an artistic community.
And I'd never fallen under the spell of a truly gifted teacher. Peggy was the kind of acting mentor who inspired you not through excoriating rants and humiliation, but by creating a safe place where you could fail and soar, take risks, be fearless. She demanded a lot, but she did so in a spirit of trust and respect. As a result, you would do anything for her. There were many wonderful young actors at the Loft—Angelica Huston, Charlie Sheen, Crispin Glover, Michelle Pfeiffer, Sean Penn—and they all revered Peggy. She was a true gift to the acting world.
Peggy wanted us to put the truth on the stage, stressing the real and the natural over Acting 101 basics. The first person I ever saw do an entire scene with his back to the audience was Crispin Glover. He never once turned around to acknowledge our presence, and it added such power to the scene that it changed my perception about stage acting. One time Sean Penn was playing a homeless person in a show, and he refused to bathe. He would come to class in ragged clothes and greasy hair, stinking to high heaven. Of course, he didn't care. That was Sean; he didn't give a shit what anybody thought. I can still smell him. Pierre would have approved.
In my case, Peggy gave me enormous confidence just by acknowledging that I had talent. At the same time, she recognized that there was something inside me that was blocking me from being completely expressive. "You're too Japanese, Sachi. You're holding back as an actress when you should be letting go."
I understood what she meant, and I tried to let go, but I couldn't undo a lifetime of social engineering. Japanese women were expected to behave in a certain way in public. Giving full vent to your emotions simply wasn't appropriate or even conceivable. I was torn between two mentors: I wanted to please Peggy, but what would Eguchi-san have thought?
The situation came to a head one day when I was performing a scene in class with another actor. I know it would make the story more interesting if I said it was Sean Penn, but I honestly don't remember who it was—or what the scene was—but I know it was supposed to be romantic. We were kissing and touching and getting physical and—I couldn't open my legs. They were shut tight. No matter how passionate the scene grew, I wouldn't let myself go.
Peggy called me on this. "Sachi, why are your legs closed? You're attracted to this man, you're feeling sexual, your body should be inviting him. Instead you're locked up like a bank vault. Come on, spread your legs a little, enjoy the moment."
I couldn't. In spite of Peggy, in spite of all my classmates staring at me as if I were from another planet, in spite of my mom's long shadow, I couldn't. Now, you know my history; you can't say that I was prudish. I wanted to do what Peggy asked: I wanted to spread my legs wide, I wanted to show that I could surmount my hang-ups and be true to the dramatic situation.
But I couldn't.
Peggy was not going to let this pass. She came up onstage and faced the audience. "See, we have a Japanese woman here," she explained. "She looks Irish, but she's Japanese. And she can't spread her legs. So I'm going to help her. I'm going to pry those legs open if I have to use a crowbar."
Peggy grabbed my knees and forced them apart, straining like Samson at the pillars of Gaza. I was shocked, and I actually felt myself fighting against it, but Peggy prevailed, spreading my legs wide open and holding them there. "Now say the line," she commanded.
And I said the line—whatever it was—and it worked. It flowed. Everything in the scene made sense, and in that instant, for the first time, I understood acting. Peggy had unlocked something deep within me, she'd exorcised some demon I didn't even know I had, and it would never haunt me again. I remember Angelica Huston coming up to me afterward and raving about my instant metamorphosis. "You just opened up like a flower. It was like night and day. You were a different person."
And I was. I have to say, from that moment on, I was extremely sexual onstage. I had no qualms about expressing myself physically or emotionally. If the script said we were grabbing each other's crotches, then that's what we were doing. Even if I had to go topless (as I did a few years later, in _The Lulu Plays_ ), I didn't bat an eyelash—even with my two fried eggs.
Peggy was such a remarkable teacher. Not the least of her talents was an ability to fall into a narcoleptic nap in class, to completely sleep through an acting scene, and still, afterward, give a perfect moment-by-moment critique. She appeared to be fully asleep, but somehow she heard every word, beat, and inflection. It was scary.
Like so many other aspects of show business, an acting class becomes like a family. You trust your partners with your deepest emotions, and wind up knowing all kinds of intimate things about one another. You work together in intense dramatic situations, and then you go out to drink and unwind and laugh at one another's jokes. There are romances, breakups, and long confessionals at the local bar. You borrow money from each other; you get on each other's nerves. Once, Michelle Pfeiffer loaned me a green dress for an acting scene. I didn't return it for a long time, and I think she's still pissed off at me.
Of all the talented actors in Peggy's class, there was one who caught my newly liberated eye. His name was David Weininger, and he was a young but seasoned actor who understood his craft and was a thorough professional. He was also very sweet, and we hit it off immediately. I would say we were just like brother and sister, except that eventually we found ourselves involved in a relationship. Don't worry: there was no bait and switch here; he wasn't a Luke or a Jeffrey. He was a genuine nice guy through and through.
David was my true bridge from Japan to America. He spent hours and hours explaining American culture to me. It was through him that I came to understand how my native country worked.
It didn't happen overnight. One time he took me to see Richard Pryor in concert. I was appalled: here was this man onstage, using the foulest language imaginable, and everyone around me was hysterical with laughter, rolling in the aisles. It incensed me; I was personally insulted. I thought, _This is the rudest person in the world, and these people around me are just as rude_. The whole spectacle filled me with disgust.
This amused David to no end, of course. He explained to me that this was just the raw energy of American humor. The offensiveness was what made it funny.
I didn't get it. Saying mean things and making people uncomfortable didn't seem very funny at all —at least, not the way I was brought up.
"But you're not where you were brought up anymore, Sachi," David said. "You're in America."
It took me a while to reconcile myself to this reality. Surely I could hold on to the precepts and values I grew up with—humility, acceptance, self-effacement—and still find success in Los Angeles. Right?
The first crack in my Japanese edifice was a tiny one, but it was the tremor that presaged a major plate shift. I was talking to David on the phone one day, and he was waxing a bit pompous about a new experimental play he was performing in, and how it might finally put him on the map. Just from the sound of it, I could tell the play was a turkey. "Oh, yeah," I said sarcastically. "The studios are going to be burning up your phone line."
David gasped. "What did you say?" he asked. "Are you being sarcastic?" I giggled nervously, hoping I hadn't offended him, but he was thrilled. "Oh my God, listen to you—you've become an American. That's so cool!"
"Really?" I asked timidly. "You think that was good?"
"That was great!" He laughed. "Sachi the wiseass! We have to celebrate!" He was genuinely delighted for me. I'd had a breakthrough! Maybe I could make it in this country after all.
I had another breakthrough when I was driving home in my beat-up car on the Santa Monica Freeway. I needed to change lanes and exit, but no one would let me in, no one would slow down, no one would give me the courtesy wave. I was boiling with rage, even though I knew this was par for the course on the freeway—it's every girl for herself, and the only way to change lanes is to barrel in, and fuck everybody else—and that's exactly how it erupted from my mouth: " _Fuck you!_ " I started screaming. "Get the _fuck_ out of my way so I can get in there, you _asshole!_ " It's the only way to drive in L.A., and it felt _so_ good. I understood how cathartic and empowering a well-placed obscenity could be.
So now I was cursing and being sarcastic. Mom would have been so proud.
I wasn't seeing Mom all that often, because a few months after I started Peggy Feury's class, I moved in with David. I soon discovered that, although he was eager to instruct me in American mores, he really enjoyed having a faux-Japanese girlfriend. I was always cooking for him, cleaning up after him, deferring to him. It was never his turn to do the dishes. He loved it. David was a wonderful guy, mind you, but he was still an actor, and he enjoyed being taken care of.
In all fairness, though, he took care of me, too. He was very sensitive to my emotional needs, appreciative of my budding talent, and ever trying to build up my self-esteem. He knew when to laugh with me, and when to give me a constructive kick in the behind. He was my best friend, and still is.
I would guess he got his nurturing instincts from his father, Dr. Benjamin Weininger, a revolutionary therapist and a very brilliant man. Dr. Weininger had offices in L.A., and he was legendary for his ability to reach through to subjects who seemed mentally and emotionally beyond hope. He was something of a miracle worker.
I saw the wondrous results of his therapeutic techniques firsthand when my mom came to David with a problem: a friend of hers, a very famous film director, had a daughter with severe emotional problems. She was acting out in bizarre ways, walking backward, pulling hunks of hair from her head. They'd been trying for years, since her childhood, to find some kind of treatment for her, but nothing had worked—and now she was a young woman, with a bleak, unpromising life ahead of her. Mom had heard of David's father and his seemingly miraculous cures: Could he help?
Dr. Weininger asked them to meet him at his office at the Southern California Counseling Center, which was down on Pico Boulevard. David and I were waiting on the sidewalk when Mom and the director pulled up in a town car. I could see from the looks on both their faces as they stepped out and surveyed the crumbling neighborhood that they were sure they'd made a wrong turn somewhere. Then out came Dr. Weininger, dressed like an old bohemian hippie, with jeans and long hair and bare feet. He didn't look like a world-renowned therapist, but then again, he had survived Auschwitz, and he didn't care what he looked like.
The director's daughter got out of the car now, and walked into the clinic backward. Dr. Weininger made no comment as he accompanied her. I could see the hope fading in the film director's eyes. This would be another cruel disappointment, he was sure.
Five hours later, the daughter emerged from the office, completely changed in aspect: no bizarre behavior, no walking backward, no hair pulling. She seemed like a normal, healthy girl. And the cure lasted: today, thirty-five years later, she's still doing fine. It seemed like nothing short of a miracle.
Mom was astounded. "What did you do?" she asked Dr. Weininger. She had to know.
"I went to her level," he said.
Tellingly, he didn't say he went _down_ to her level. He didn't make value judgments. He showed her the respect of meeting her where she was, and seeing what she saw.
Perhaps that's how David managed to handle me so well. He went to my level. He looked at the world through my eyes, and comprehended it the way I did. That made all the difference. Many years ago I'd scored high on my Qantas test for my ability to empathize. Now, for the first time, someone was empathizing with me, and I realized how important that was, and how much I'd needed it.
**• • •**
**A** terrible postscript to the days I spent studying with Peggy Feury: in November 1985, she was driving home to Malibu from the studio when she fell asleep at the wheel and crashed into oncoming traffic. The car exploded into flames. Her body was so charred that they could identify her only through her dental records. No one knew why she was driving in the first place. She was supposed to have a personal driver. That used to be my job.
# **Chapter 13**
# "That's George McFly?"
Nineteen eighty three was a true annus mirabilis for Mom. Not one but two life-changing projects came down the pike, _Terms of Endearment_ and _Out on a Limb:_ her Academy Award–winning movie, and her most controversial book. She'd been famous for twenty-five years; now she was going to become an icon. I'm not sure if that's what she wanted, but that's what she was going to get.
_Terms_ was filmed in Lincoln, Nebraska. David and I flew out to visit Mom on the set. The first thing we noticed when we got there was the tension. It was not the happiest place on earth. There was a great deal of emotional weight in the script that the actors were working through, and a lot of that turbulence bled into their personal relationships. Everyone was in a bad mood—except for Jack Nicholson, who refused as a matter of principle ever to be in a bad mood. He seemed dedicated to the proposition of enjoying himself at all times.
Debra Winger was playing Mom's daughter in the film, and it's no secret that the two of them didn't get along swimmingly. Some claim that they detested each other, but I don't know if I'd put it quite that strongly. She and Mom just had different ways of working, and their styles didn't always mesh. Mom was from the old school of "hit your mark and say your lines." Debra was more instinctive and "go with the moment." Both wound up giving spectacular performances, and their scenes together were marvelously alive and filled with emotional truth. Yet none of it came easy.
At one point Debra dug so deeply into her role as a cancer-stricken mother that she landed in the hospital. I felt bad for her. "We should go visit Debra," I suggested.
Mom waved her hand dismissively. "Ah, don't worry about her. She's just doing it for attention. She's nuts."
I couldn't understand that. Maybe Debra was nuts—I didn't know her; I couldn't say—but if she was in the hospital, something was obviously wrong with her, and it seemed only right that someone should visit her. So I decided to go by myself. The thing is, I needed flowers. It's unthinkable in Japan to visit anyone in the hospital without bringing flowers. Since I didn't have any money for flowers, I asked Mom. "She doesn't _need_ flowers," Mom insisted. "Just leave her the fuck _alone_!"
So I left Debra alone—which I regret, because maybe she and I would have become friends. We had a lot in common: we were almost the same age (Debra was twenty-eight, just a year older than I); we were actresses; and for the time being, we had the same mother.
It didn't happen. In all fairness, it probably never would have happened anyway. I don't think Debra wanted to be friends with me. She was generally aloof with me, never really gave me the time of day. I couldn't tell if it was because I was a nobody, unworthy of her attention, or because I was Mom's real daughter and she was envious of me. Or maybe it was just part of her process: she may have been so consumed by her role that she didn't even know I was there.
I didn't feel any envy toward her at the time, but it was admittedly disconcerting when the movie became a hit and all the magazine articles came out dissecting Mom and Debra's special mother-daughter connection. Interviewers would often ask, "Does this film mirror your relationship with your real daughter?" "No, not really," Mom was always honest enough to point out. She wasn't the obsessive, suffocatingly attentive type who would climb right into her baby daughter's crib to soothe her. "I never would have given up my work to stay home," she told _People_ magazine in 1984. "My philosophy was always to just let her grow up and to be there if she needed me."
Mom was also on combative terms with director James Brooks. Some of this was just part of the creative process—they were both fighting for their own visions, a laudable and necessary thing in the service of art—but some of it was also frustration. This was Brooks's first feature film, and his slow, reflective approach was sometimes maddening. He would shoot scenes over and over, and then decide to rewrite them, or drop them altogether. As much as Mom may have disliked Debra, she hated the way Brooks treated Winger. She found him abusive and manipulative in his handling of the younger actress. "He wouldn't try that shit with me," she declared. Actually he tried other shit on Mom, and what really irked her was that his strategies worked so well.
I was on the set one morning when they were doing an outside shoot. Everything was on hold while the cameras were being set up, and Jim Brooks was lying in the grass, relaxing, hands behind his head, staring up at the sky.
Mom waved me over. She had a mischievous glint in her eye, which was usually cause for concern. She pointed to Brooks. "Go stand over him and smile, and say, 'It's all I can do not to step on your face.'"
"Why?" I asked. I knew it was supposed to be funny, but I didn't get it.
"Just do it."
I didn't want to. To my mind, there was no way you could interpret the gesture as anything other than hostile, and why would I want to be hostile to James Brooks? He was not only a film director but a major TV producer. How was this going to help my career?
"Oh, what's the big deal?" Mom said. "It's a joke. Just do it." Implicit in her goading was the suggestion that if I didn't do it, I wasn't cool.
But I wanted to be cool in her eyes. I admit, I was her puppet: if she asked me to do anything, I would do it. I needed her approval, just as I'd needed Dad's.
So I did it. I walked up to Mr. Brooks and stood over him, straddling his head. He looked at me with surprise, and I delivered my line: "It's all I can do not to step on your face."
He was startled, and then he broke into a smile. I looked over at my mother. She was laughing hysterically, as were some other actors and crew members.
So it was evidently funny. I still didn't get it.
I discovered much later, reading Mom's book _My Lucky Stars_ , that the line came from a scene that was cut from the film. Jack Nicholson's character stumbles over his garbage cans and lies drunk in his driveway, and Mom stands over him and says, "It's all I can do not to step on your face." So it was an inside joke that everybody on the set got. Except me.
But James Brooks seemed to like me after that. He always said hello and gave me a friendly smile. Maybe he respected my gumption. On the other hand, he's never cast me in anything, so maybe not.
After about a week of hanging around the set of this soon-to-be-classic film, David and I had had enough. Movie sets are terminally boring as a rule, but there's another dynamic in play that isn't often commented on, and that's the sense, if you're a visitor, of being at a country club that you don't belong to. The set of a feature film is the biggest high school clique in the world. Everybody working on it is an insider, one of the chosen elect, and consequently assumes a sense of privilege and entitlement far beyond any reasonable expectation. This elitist mind-set starts with the above-the-line talent (the stars with their trailers, the big-name directors, the cameramen and designers), everyone who is clearly indispensable and beyond brilliant, and it filters down to every anonymous gofer and day player. They all think they walk on water, because they're working on a movie and you're not.
I can't condemn them, because I've worked on a number of movies myself, and I know the seductiveness of that insular, clubby feeling all too well. It's like living in a gated community: if you happen to be inside the gate, it's wonderful, with the perks and the inside jokes and the soothing waves of self-importance. If you're not, if you're not a member of the club, then you're nobody, plain and simple, and you're either invisible or treated with condescension and rudeness. There's not much reason to hang around, unless you enjoy watching people being bratty and obnoxious.
Surprisingly, the one person who seemed to rise above this fraternity mentality was Jack Nicholson. You would expect him on reputation to be the brattiest of them all, but he was so confident and comfortable in his own skin that he didn't need to be fawned over. Whenever I saw him, he was unfailingly nice and gentlemanly—and devilishly charming, of course. It was no wonder he was getting laid all the time. He knew a secret all too many movie stars never quite grasp: just because you're famous, it doesn't mean you have to be an asshole.
In between the shooting of _Terms of Endearment_ and its December release, Mom's book _Out on a Limb_ was published. This event, more than anything else, redefined Mom's persona to the world. With one bold stride, she stepped out of the shadows and became the New Age standard-bearer for her generation.
_Out on a Limb_ was her third autobiographical book, but the first that discussed in depth her belief in mediums, UFOs, and, most prominently, reincarnation. You probably know all the stories about her past lives: She was an Indian princess; a geisha; a man from Atlantis. She slept with Charlemagne in a past life. She also slept with the Swedish prime minister Olof Palme in this life, and he used to be Charlemagne in a past life, so she nailed him twice, a millennium apart.
I was not exempt from the metaphysical carousel. Mom was convinced that mine was an ancient spirit that went back thousands of years. In one incarnation, in fact, it seems that we switched roles and I was _her_ mother. I hope I treated her well.
None of this was revelatory to me, of course: I knew all about her spiritual interests, her extraterrestrial visits, and so forth. I was a little surprised that she was going public with it all, though. I figured that, after the clone episode, Mom might back off on her astral enthusiasms. Instead, she was doubling down.
She quickly became a lightning rod for believers and skeptics. Spiritualists venerated her; late-night comics ridiculed her. And what, people wanted to know, was my reaction? What did I think about my mother's cosmic disclosures? Did I consider her a visionary or a crackpot?
Publicly I supported her: "She's my mom, and she's wonderful." Privately, I remember wishing she would just shut up—but not because I didn't share her beliefs. It was just that, for me, reincarnation wasn't a big deal. In Japan, everybody comes back. Everything that dies will be reborn. It's an accepted belief in the culture, but not something you talk about. It's all internalized, part of your moral DNA. While you are always aware of karma, and you try to lead an exemplary life, there's no dwelling on it. It's ingrained, totally reflexive, and private.
So I couldn't understand why Mom was talking about it in public, and why everyone was listening. The hoopla, the seminars, the talk show appearances—it seemed so over the top, and I had no patience for it. I never told her so, because I didn't want to hurt her feelings, but to my mind, none of this was New Age. It was totally Old.
Having said that, there was something beautifully endearing about the fervor of her beliefs. She really wanted to fill her soul with something meaningful. I think show business in particular excels in creating a certain kind of personal emptiness: you work on a play or a movie and it becomes your whole existence, and then it ends, and suddenly there's a vacuum where your life used to be, and you have to figure out how to fill it. You can jump into another movie right away, and keep that cycle going—one movie after another, until you die—or you can find something more enriching and sustainable to base your life on.
Mom discovered something better, something that worked for her. Whether anyone else embraced it was beside the point. Like any convert, she enjoyed sharing her epiphanies with the world, but she wasn't all that concerned if anyone followed her. She was a searcher, on her own spiritual journey, and the journey would never end.
**• • •**
**MOM'S** performance as Aurora Greenway in _Terms of Endearment_ won her all kinds of acting awards, but the Oscar was the one she desperately wanted. She'd been denied it for _Some Came Running_ , _The Apartment, Irma La Douce_ , and _The Turning Point_ , and now, as the age of fifty loomed, she knew that this could be her last best chance.
I went to the awards ceremony as her date. I was so excited to be there with her on perhaps the greatest night of her life.
Rock Hudson and Liza Minnelli were the presenters of the Best Actress award. When they read out the list of nominees, I suddenly felt a moment of panic. What if she didn't win? She was going against Meryl Streep in _Silkwood_ , Jane Alexander in _Testament_ , Julie Walters in _Educating Rita_ , and her own costar, Debra Winger. They'd all given brilliant performances. Holy shit, she could lose!
Then Liza announced Mom's name. I'll never forget that moment. A great emotional cheer exploded from the audience, and Mom turned and gave me a kiss. I could see on her face that this was her defining moment: her life now made complete sense.
She went up onstage and made a lovely, gracious speech. She looked so beautiful and radiant up there, and I felt so proud and happy for her. It's a great gift to see someone you love getting her fondest wish come true.
Mom finished her speech with an inspirational thought: "God bless that potential that we all have for making anything possible if we think we deserve it." Then she looked at her Oscar and said matter-of-factly, "I deserve this. Thank you," and walked off. I have to be honest: I cringed a little with embarrassment. I thought to myself, _Oh, Mom, you overdid it_.
No one else seemed to think so. It was her night, and she could do no wrong. We hit all the celebrations afterward—the Governor's Ball, Spago—and we partied into the morning. Everyone was getting progressively drunker and sloppier, except for Mom. She was always very disciplined about alcohol, and I had great respect for the way she kept control, even on this night of all nights, when she had every excuse to go wild. The rest of Hollywood could act like fools, but Mom was pure class, all the way.
**• • •**
**THAT** same year, I got pregnant. This should have been more happy news, because I really wanted to be a mother. (In a _People_ article about my mother, just before the Oscars, I'm quoted as saying, "I want to marry eventually and have a home and children and do all those things wives do." Honestly. This is the same article where Mom declares, "I don't think there's any more important profession in the world than being a mom.")
David wanted to be a father, but we had no money. We were two penniless actors trying to get our careers started. Having a baby now, at this stage, seemed a totally impractical thing to do. Unless we had help...
I went to Mom and asked if she could give us a hand financially. I knew this went against her long-standing mantra of "you're on your own," but on the heels of her book and her TV appearances, where she spoke out so passionately for self-actualization and karma and getting out of life what you put into it, I thought perhaps she had evolved a more generous worldview and would be in a giving state of mind.
She said no. The whole idea, to her mind, was ridiculous. "You can't have a baby. How old are you?"
"I'm twenty-seven."
"Oh. Well, even so, you don't have the time to take care of it. You and David live in that tiny place. There's barely room for the two of you, and you want to bring a baby in there? What's that going to do to your relationship?"
"Make it stronger?" I weakly suggested.
Mom scoffed. "Yeah, right. And I thought you wanted to be an actress. How are you going to have any kind of career with a baby? Are you insane?"
That was perhaps the crux of it. She'd had a baby, too, but she'd also had a wealthy husband (or a clone of one) who could take care of it, so she could continue her career. Even so, that hadn't worked out, since his wealth turned out to be her wealth. Was she saying that having me had been a mistake, too?
After sounding all her arguments, Mom shrugged with resignation. "Well, it's your choice." Still, it wasn't much of a choice if she wasn't going to help us—and she wasn't.
David's dad, Ben, said he would try to come up with some money, but he didn't have much. He did have a philosophy to share, though: "Commit, Sachi. You have to commit. When you commit to it, whatever 'it' is—a baby, a job, whatever—commit to that thing a hundred percent, and it will set you free."
I needed money more than inspirational motivation at that point, but I should have listened to him, because I know now he was dead right. If I'd had the baby, it would have been tough, but it would have worked out somehow.
I didn't listen, though. I couldn't see any way out. I didn't want to have an abortion, but what else could I do? Bring the baby to term and then give it away? That would have destroyed me.
Right up to the day of the procedure, I kept asking Mom for help, but she'd said her say, and now she wouldn't even engage in a conversation about it.
I had the abortion at St. John's Hospital in Santa Monica, the same hospital where I was born. To compound the dreadful irony, the same doctor who delivered me as a baby was performing the procedure. I felt ashamed and mortified—and guilty.
As much as I wanted to blame Mom, I know it was my decision to terminate, and it haunts me even now. I sometimes lie awake thinking about that child who never was. I sometimes dream of that lost soul. He or she would have been almost thirty now.
My regret grew even more acute when, years later, I conceived my daughter, Arin. That night is vivid in my memory: my husband and I had made love around midnight, and now it was around 4:30 in the morning. I was in a half-awake, half-asleep state, when quite suddenly a beautiful, indescribable feeling of love, happiness, and peace washed over me. Some kind of being, a female being—an angel, perhaps—had entered my body, and I felt transformed and transfigured, and suffused with great serenity.
About three weeks later I found out that I was exactly three weeks pregnant. That night, a new soul, Arin's soul, had entered me. She became my angel, my protector, my love, and everything I'd ever yearned for. I'd never had such an amazing, powerful feeling before—or since.
It took me back to that first conception, and I wondered who that soul had been, who had entered my body and lived there, and who I'd destroyed. Could it be that Arin was that same soul coming back? Was there life after death, and had my lost child rechosen me?
Maybe God or nature or the universe makes things right, no matter how much we try to screw them up. Although I strongly believe in a woman's choice, I know in my heart that, in my case, I made the wrong one. It was a terrible mistake, and I'll always live with the guilt and the regret.
**• • •**
**MY** career, such as it was, continued on its bumpy struggling-actress way. I went to plenty of auditions and met with a parade of agents and casting directors, but nothing was happening.
It was Burt Reynolds who finally gave me my first film role, in _Stick_ , which he was directing. The movie was based on an Elmore Leonard book and was shot down in Fort Lauderdale. I was playing Bobbi, the bartender, and I had three scenes with Burt himself, who was one of the nicest guys to work with.
I was nervous and excited, and the night before my first scene, I came down with a 104 fever. I couldn't believe it. My first big break, and I was deathly ill. What was I going to do? I called David back in Hollywood, in a panic. "I won't be able to act tomorrow. I'm too sick. They'll fire me. They'll make the film without me!"
David attacked the problem patiently and thoughtfully. "You know you're not really sick, Sachi. This is all in your head. You're making this fever happen."
He was right, and I knew it. The only time I ever got a fever, it was related to my acting career—before either an audition or a big scene in class. My own body was sabotaging me.
David went to my level, so he understood this. He talked to me for about twenty minutes, and the fever magically receded. I went to bed feeling ready for the next day.
The next morning, I woke up to another fever. It was higher now: 105. I called David. "I'm really sick this time! I'm not imagining it!" David, who was three hours behind me, sleepily talked me back to health. "You're not imagining the fever, Sach. You're imagining the reason for it." Twenty minutes later, I felt fine.
So I went off to the set, but by the time I arrived, I had a fever again. I called David back. And so it went the rest of the day. He talked me down from the ledge over and over until the shoot was done.
I tried to analyze why I would do this to myself. Was I afraid to fail? Or afraid to succeed? Was I worried that I might be infringing on my mother's territory? What if I get this job or that role? Maybe she won't like me anymore!
I remembered the fever I'd had at the deposition for my parents' divorce. Had that been psychosomatic, too? Triggered by a fear of stressful, unpleasant situations? Was this going to plague me my entire career? Was I going to _have_ a career?
The actual shooting of _Stick_ was delightful. Burt was a terrific director, nontemperamental and very easy to work with. Candice Bergen, who had the female lead, was a sweetheart. She was totally genuine, nothing actressy about her—and she was very friendly to me. Once, she asked me to have dinner with her, but for some foolish reason I said no. I really wanted to go, but she was already a big star, and I didn't think I was important enough to hang out with her. I didn't feel worthy. I was still a nobody, and I didn't want to pretend I was a somebody. Hollywood can really fuck up your thinking.
**• • •**
**BACK** in L.A., I auditioned for _Cocoon_ , the Ron Howard movie about aliens and old people in Florida. I was going out for the role of Kitty, the alien who appears as a young woman. I auditioned maybe seven times, and every time I felt the approval growing. Steve Guttenberg, who would have played my love interest, worked with me in the later auditions, and our chemistry was really good. After the last audition, at which I wore a short dress to show off my legs, I could tell that I basically had the role. I felt that it wasn't so much my acting ability as the entirety of who I was: the girl-next-door type, pretty but not gorgeous, a good fit with Steve. The fact that my mother was on speaking terms with interplanetary life forms surely didn't hurt.
Ironically, the premise of the film is that a race of aliens called Antareans came to Earth thousands of years before and founded the mythical city of Atlantis. They were returning now to retrieve some of the Antareans left behind in cocoons. (I don't know why they weren't Pleiadians; maybe there was a copyright problem.)
So it all seemed meant to be. Everyone loved the screen test. All systems were go.
Then I got the call from Mort Viner, Mom's agent. "Sorry, but they went with Tahnee Welch." Who? Tahnee Welch, Raquel Welch's daughter, an exotic beauty who was as far removed from the hometown sweetheart type as could be. Apparently they had two clear choices, and they went with the sexy bombshell.
I was devastated. I thought I'd scored my first big part, and my career was off and running. Not so. Mom was philosophical. "You never know what they're looking for. You just have to do your best and hope it works out."
"But you said we could make anything possible if we thought we deserved it," I said, referencing her Oscar speech. "I think I deserved this."
Mom shrugged. "I guess you were wrong."
I went to the premiere screening of the movie with Mom and Mort. It was very difficult to sit through that film and watch my role being played by somebody else, but I called upon my Japanese stoicism and pretended to enjoy it. I understood why they'd cast Tahnee—she was gorgeous—but I also felt that by giving the character such otherworldly beauty, a dimension was missing in the romance. Steve and I would have been so cute together.
I went up to Ron Howard after the screening to congratulate him. A very sweet, decent guy, he looked a little regretful. "I should have gone with you," he said. I smiled graciously, but all I could think was, _Why didn't you?_
Anyway, I kept auditioning, and other, smaller parts followed.
I only had two lines in _Back to the Future_ , but everyone seems to remember them. They occur in the scene in the parking lot outside the prom. George McFly, played by my old acting partner Crispin Glover, is being beaten up by his eternal nemesis Biff. Suddenly George makes his hand into a fist and punches Biff, knocking him flat. I rush up with a group of gawking teenagers and ask, "Who is that?" and when I'm told, I exclaim in disbelief, "That's George McFly?"
It's a great moment.
I had two scenes in _About Last Night_..., the adaptation of David Mamet's play _Sexual Perversity in Chicago_ , with Demi Moore, Rob Lowe, and Elizabeth Perkins. I played a girlfriend. The actors were in their Brat Pack heyday, and they were all very hip and with it. I tried to fit in, but I wasn't cool enough to run with them and I knew it.
Then I landed a small part in _Peggy Sue Got Married_ , as one of Peggy Sue's high school girlfriends. I'd already played a girlfriend, and I'd already been in a time-travel movie, and now I was playing a girlfriend in a time-travel movie. I was starting to run out of options.
So I went to Ireland. A young director, Ronan O'Leary, offered me a role in _Riders to the Sea_ , his adaptation of the John Millington Synge play. I don't know how he knew about me, but he'd seen me in something and liked my look. He thought I seemed very Irish.
Ronan himself was like a little leprechaun, full of beans; he loved to talk, and he was passionate about film, which really appealed to me. I was perfect, he said, for the part of the young daughter Nora, who waits desperately for her brother to come back from a storm at sea. "The camera loves you. And you're so right for this part. I have to have you."
It would be a short film, less than an hour long, but I jumped at the chance. My mother and sister in the film were to be played by Geraldine Page and Amanda Plummer. This was the kind of opportunity I'd been waiting for.
Before the shoot, we spent about four days rehearsing in Paris, in Geraldine's apartment on the Left Bank. I loved being back in Paris. I showed Geraldine and Amanda all the little cafés and bistros I used to haunt, and since I spoke French, I could proudly serve as their tour guide. We stopped at the place where I used to work, and I was warmed by the fuss the staff made over me. You'd have thought I was the big celebrity.
Most of the film was shot on the Aran Islands, off the west coast of Ireland. As the play itself was set on the middle island of Inishmaan, we filmed in some of the actual locations. Inishmaan is a spectacularly wild place, with magnificent views of the Atlantic, and rocky outcroppings everywhere.
I would wake up every morning well before call and walk around the island in the early fog. It was so beautiful in its desolation and loneliness. Just about every morning, I would visit the stone seat _Cathaoir Synge_ ("Synge's Chair"), which overlooks the Atlantic, where the playwright Synge would sit and be inspired. His great plays, including _Riders to the Sea_ and _The Playboy of the Western World_ , were supposedly written in this spot. Looking down at the surging waves, I would become Nora, and imagine my brother's _currach_ (boat) smashing against the rocks, and his Aran sweater washing up onshore. It was an ideal way to set my mind for the day's work.
There were no hotels on the island, so we were all put up in different homes. I stayed in a classic Irish cottage, with stone walls and a thatched roof. The husband and wife—I've forgotten their names, but let's say Donal and Maureen—treated me like a princess, tending to my every need. Maureen refused to let me lift a finger to clean or help out. When I told her I needed a hot bath every night to relax, she insisted on drawing it herself. There was no hot water, so she would fill the bathtub with freezing cold water and then put hot coals in a little stove beside the tub, which would heat the bathwater to the desired temperature. She also made her own black bread every morning. I would come back from my walk and have a fresh warm slice; the bread was dense and heavy, almost hard, and it looked like it was made of peat turf, but I would put a big slab of butter on it and it was delicious.
The diet on Inishmaan was basic: fish, chicken, seaweed soup, brussel sprouts, and all manner of potatoes (mashed, roasted, boiled). The ground yielded nothing but stones—there was almost no soil for the crops to grow on—and whatever sprouted up was blasted down again by the wind. It was a harsh, punishing world; you could feel it in the souls of the people. Their days were full of struggle and pain, but their nights at the pub were full of joy and laughter. It's easy to sentimentalize the fortitude and spirit of the Aran Islanders. I wouldn't say they were happier than anyone else, but their lives had a spare, stark clarity. There was no bullshit there. People weren't bedeviled by choices, because they had no choices. They lived simply, and simply lived.
My hosts had a marvelously Irish relationship, full of loving contentiousness. Maureen would bark orders at Donal—"Wipe your boots! Fetch some coal!"—and he would bolt down his dinner and head for the pub, or sometimes they'd go together, and I might tag along. Every single night, they would make love. I couldn't miss hearing their old bed as it creaked rhythmically in the small, quiet house. It was a comforting sound, the sound of life going on.
I think I enjoyed the Aran Islands even more than the filming, but both were great experiences. Ronan was a wonderful director to work with, never condescending or sarcastic. There was always positive energy coming off him. I could tell that Geraldine respected him a lot, and that spoke volumes for his talent.
Geraldine Page was perhaps the finest actress I've ever worked with. She was a totally honest artist, incapable of a false moment. She was also a master of business—working with props, fiddling with her clothing, bringing the scene alive by simulating the naturalness of real life. "Most actors don't know how to do business and talk at the same time," she told me. "But in life, that's what you do. You don't stop to deliver a line and then go back to washing dishes, and then stop to deliver another line. It's all of a piece, and you have to make it seem that way. Whether you're setting a table or adjusting your sleeve, you have to keep it natural."
She also taught me to get my lines down as soon as possible. "You don't ever want to think about your lines. They should be part of you, under your skin." If you have to think about the script, you're not dealing with the emotions underneath. Her advice for learning lines: "Fold your own laundry." If you run your lines as you fold, you'll find out how well you know them. If you can't fold the laundry correctly, then you have more work to do.
Geraldine was an amazingly generous colleague on the set. Whenever we did our close-ups, Ronan would offer, as customary, to have the stand-in read Geraldine's lines off camera. Most actors would have been happy with that arrangement, but Geraldine wouldn't hear of it. She'd say her own lines, thank you, and what's more, she'd give 100 percent of her acting ability in every take. None of this would show up on-screen, except in the performances of her fellow actors, and that's what mattered to her. She cared deeply about the whole piece, and if there was anything she could do to make it better, she would.
I absorbed her lessons like a sponge, and took them with me. I always try to be off camera for another actor's close-up. Folding laundry during rehearsal is now a cherished ritual. I remember doing a play a few years later: there was a scene at a restaurant table, and I worked up some business about cleaning the outside of the salt and pepper shakers with my napkin, because it seemed in tune with my character. This wasn't something the director wanted; it just occurred to me naturally, a way of informing a small moment onstage. When the play was reviewed in the L.A. papers, a critic made admiring mention of my business with the salt and pepper shakers. _That wasn't me_ , I thought; _that was Geraldine_.
We finished shooting on Inishmaan, and I said a sad goodbye to Maureen and Donal. By now I was no longer a guest in the cottage. I'd become a part of the family, and Maureen was barking orders at me, making me do my own wash and heat my own bathwater. She gave me an Aran sweater as a parting gift—she would have knitted it herself, but she didn't have time. It was warm and comforting, like their home, and like the sweater from Hildy in New Zealand, sticky with lanolin oil.
We moved on to Dublin to shoot some interiors. Even though _Riders_ was a relatively low-budget film, it was big news in Dublin, where everyone knew Synge's work by heart. One of the many visitors to the set was Kevin McClory, my mom's old friend. I had stayed with Yuki at his mansion in Connemara so many years before, on our break from Charters Towers. It was so great to see him again!
He took me to dinner at our hotel restaurant, and was full of compliments and enthusiasm. How grand was this?—he had known me since I was a baby, and here I was, acting in his town, a movie star! He was so proud of me. He knew _Riders to the Sea_ as well as any Irishman, and he thought the character of Nora the perfect role for me.
I listened intently, eating it all up. I knew this boded well for my career. Kevin was a big producer, and he was saying just what I would have wanted to hear: that I was an actress of consequence, and he wanted to work with me.
After we said good night, I went upstairs feeling fairly intoxicated by my run of good fortune, and decided to order myself a massage. Geraldine and Amanda had had massages on the company; why not me? The desk clerk asked me if I wanted a man or a woman. Either, I didn't care. Back in Japan, it didn't matter about the sex of people who gave massages, because, as a rule, they were blind—not out of prudishness, but because their sense of touch was far more acute.
About an hour and a half later there was a knock on the door. "Who is it?" I asked. A male voice answered. "It's me."
_Oh, it's the massage therapist_ , I thought, and opened the door.
It was Kevin McClory. I was confused. We'd said good night two hours ago. "Kevin? What are you doing here?"
He smiled. "I want to come inside."
He wanted to come inside? "Why?" I asked, but as I looked at him, I knew exactly why.
Now, Kevin was a very handsome man, tall and dapper (not to mention rich as Croesus). There was much to recommend him as a one-night stand. But he was my dad's age. He'd known me since I was a baby. And let's not forget, he was my godfather.
I was so shocked, I really didn't know what to do. I didn't have the feminine wherewithal to say no, or to josh him out of it—"Oh, Kevin, you know that's not a good idea. Do you want me to get you a hooker?"—or to even voice my puritanical outrage. So I slammed the door in his face instead. It was all I could think of.
He called me later from the lobby, but I wouldn't answer. I had no idea what to say. And that was that. I never heard from him again. I probably sabotaged my career big-time with that move, and I'm sure there was a savvier way to handle the situation.
On the other hand, he was my fucking godfather!
**• • •**
**I** was extremely proud of my work in _Riders to the Sea_. I felt it really marked my arrival as an actress. Back in Malibu, I got a tape of it, and we had a little private screening, just Mom and a few friends. I really wanted to see Mom's reaction. I knew she was going to be bowled over. I couldn't wait.
Before we started, she was bubbling over with a sort of calculated, actressy enthusiasm. "This is so exciting. I can't wait to see this. Everybody get in your seats, the movie is about to start!"
It was very cute, and a little annoying. In her own way, she was grabbing a little bit of the spotlight. I didn't care, as long as she was being supportive.
The movie started. I looked over at Mom. She was leaning forward, gazing at the screen, her chin in her hands, like a child waiting for a special treat. _Oh, Mom, you are_ so _gonna love this!_
It was about fifteen minutes into the movie when I looked over at Mom again and—something had changed: she was sitting back now, and she had a tight, analytical look on her face. The pure joy of moviegoing had vanished. Then she abruptly stood up and went into the kitchen.
Was something wrong? I waited for her to come back. She never did. I could hear her rattling around in there.
"Mom! You're missing the movie!"
"I can see it from here!" she called from the kitchen.
I could see _her_ too, and what she was doing was digging into a pint of ice cream. She stood just behind the kitchen door, and kept sneaking furtive glances from around the corner at the TV while she ate. You could tell that she was trying not to watch, but she couldn't resist.
When the movie was over, everybody gathered around to congratulate me. Mom kept her distance, watching with a small, pained smile. Finally, after the guests were gone and we were alone, I got to ask her: "So? What did you think?"
She busied herself cleaning the cups and chips off the side tables. "It was fine," she said with a shrug.
_Fine_ —the word landed on my heart with a thud. I may have been naïve, but I knew a backhanded compliment when I heard one. "What does that mean—'fine'?"
"Well, you know...I'm used to professionals."
Another slam, and that one got my hackles up. " _Professionals_? Geraldine Page isn't a professional? Amanda Plummer isn't a professional?"
" _They_ were very good," Mom conceded. So now her point was clearer: the movie was good, the other actors were good. I sucked.
"You see, what it is, Sach..." She paused as she pretended to search for the words, although I knew damn well she had them already loaded in her gun. "There was too much crying," she finally declared. "When you do all that emoting and weeping, there's nothing left for the audience to feel. _I_ didn't feel anything. Just remember that the next time you try to act: don't cry so much."
"What do you mean ' _try_ to act'? I can _act_ , Mom. Don't tell me I can't act!"
She rolled her eyes wearily, as if I were intent on misunderstanding her. "Of course you can act. You're my daughter; you had to inherit _something_ , but that doesn't mean you _have_ to act. Because you know what?" She looked at me intensely now, as though she were just hatching an idea. "You know _what_? You're a wonderful cook."
What was I supposed to do with that comment?
"So?"
"So, you should be cooking. Making use of your real talents." She clapped her hand to her forehead, a why-didn't-I-think-of-this-before gesture: "You should do a cooking show!"
"A cooking show?"
"On TV. You can cook different meals, like the _Galloping Gourmet_. That would be marvelous. I would watch _that_."
I couldn't begin to follow her logic. "Anyone can cook!"
"Not like you. You're special. When you're special at something, do it. When you're not..." She spread her hands wide to suggest the futility of such a quest. She was telling me my acting career was a quixotic, foolish dream.
Seeing the dejection in my face, she took this as a cue to elaborate even further. "Look, I was _meant_ to be a star. So I am. That's how it works. Karma. You're how old now, thirty-one? Face the facts: if you haven't made it yet, it wasn't meant to be."
She had one last thought to leave me with: "But a cooking show..."
**• • •**
**ABOUT** a year earlier, Mom had been approached by Pepsi to do a commercial. It was for Diet Pepsi, a drink they were marketing to the younger generation. Their idea was to pair Shirley MacLaine, Oscar-winning actress and New Age guru, with her own daughter, who was apparently trying to get into acting herself. It would cross all demographic lines.
Mom wasn't interested. "Why do I want to do a commercial? It's not like I need the money. I don't even drink fucking Pepsi." But it was a great opportunity for me—a major commercial, with national exposure. "Come on, Mom, it'll be fun!"
"All right, I'll do it for you," she said grudgingly.
_I'm standing in a garden. EXTREME CLOSE-UP as I take a swig from a can of Diet Pepsi. WIDER NOW as Mom approaches:_
MOM:| "Sweetheart, what's the matter?"
---|---
SACHI:| "I'm drowning my sorrows."
MOM:| "You are depressing the scenery."
SACHI:| "Oh, Mom, I really wanted that job."
MOM:| "I know. But listen, when one door closes, another door opens. I always learned more from my failures than I did from my success."
SACHI:| "You mean I can learn from failure?"
MOM:| "Yes. You don't want to get a PhD in it, but you can learn."
SACHI:| (giggling softly) "I'll drink to that."
_As we walk through the garden, arms affectionately circled around each other's waists, Martin Sheen's voice-over ties it all up: "There's one soft drink that fits the spirit of today. Diet Pepsi—the one-calorie choice of a new generation."_
Mom was paid a cool million for that spot. I got scale.
# **Chapter 14**
# Family Feeling
Mom was a tad schizophrenic about my career. She was always encouraging, giving me advice, wishing me the very best, but then, whenever I actually _did_ something, she would not-so-artfully suggest that I was wasting my time. I could never figure out what she wanted for me, or expected from me. If only I could talk to someone who understood her better than I did, someone who really knew her, someone like...
Well, Uncle Warren. He'd grown up with her, he'd followed her escape route out of the alcoholic Baptist world of their parents, and he'd experienced the same kind of dizzying success. He of all people could probably shed some light on her paradoxical thought processes.
Except that Warren was not a part of my life. I wanted him to be, but there had always been a cool distance between us, and the gulf had widened considerably over the years. I was too young to command his attention when he was coming into his own, and by the time I got back from Australia and France, he was a superstar, with _Bonnie and Clyde_ , _The Parallax View_ , _Shampoo_ , _Heaven Can Wait_ , and _Reds_ under his belt. For the latter two, he received Academy Award nominations for Best Picture, Best Director, Best Actor, and Best Screenplay, the only person to achieve that distinction twice. (The only other person to do it even once was Orson Welles, for _Citizen Kane_.) He won the Best Director Award for _Reds_.
Warren Beatty was a major Hollywood player in his own right, so much so that people often tended to forget that he and Mom were brother and sister. They seemed so distinct and original, each in his or her own way. They never acted together, and they approached performing from completely different directions: Warren was a serious actor who could do comedy; Mom was an entertainer who could do drama. Warren never did a musical, thank God, and Mom never did an epic movie or a gangster flick. They didn't even look that much alike.
Only in attitude—in their drive, their ambition, their laser-like focus on success—did they display a genetic similarity. Warren was, if anything, more driven than Mom, more ruthless in his determination. He still had that "don't come too close" aura, and it was now even more pronounced. You couldn't get near him: he had the isolated, impenetrable façade of the very powerful.
Around this time, I was working on a play in class with a dark-haired beauty named, I think, Laura. We'd go out for drinks afterward, she and David and I, and in the course of a conversation she casually let drop that she was Uncle Warren's girlfriend. Here was a happy coincidence! Admittedly, Warren had dozens of girlfriends, so it wasn't unlikely that you'd run into one sooner or later. Still, the fact that Laura and I had become buddies prior to the disclosure made it much easier to cultivate her friendship without inviting suspicion.
"How is my uncle doing?" I asked innocently. "I haven't seen him in such a long time."
"You should come over for dinner," Laura said.
"That would be fun," I said, trying not to sound overeager. "When?"
"How about right now? Come on!"
We got in the car and drove over to Mulholland Drive. I was a little nervous that Warren wouldn't be happy to see us there. He had very definite boundaries set up around his life, and you crossed them at your own peril. Yet, as it turned out, he was surprisingly gracious. He welcomed us in, and made us feel entirely at home.
There was no dinner ready, but Warren had a cook on call, who was happy to whip up a delicious meal for us. The kitchen had a row of refrigerators filled with food, so we could basically pick our own menu. And since Warren loved ice cream, too—he had that in common with his sister—every possible flavor was stocked in the freezer.
Warren was a wonderful host to us, and made us feel completely at home. I hadn't seen him in several years, but he acted as if it hadn't been more than a couple of days. At the same time, he gave little outward acknowledgment that we were closely related. We didn't discuss family matters at all—he didn't ask about Mom, and I had enough sense of diplomacy not to bring her up. (They had such a volatile off-and-on relationship, you were never quite sure which way the wind was blowing; better not to set sail into that changeable sea.)
He invited us both back again, and David and I visited several times after that. Even after he broke up with Laura—which was inevitable; the girlfriends came and went—Warren still invited us over. He really liked David. They immediately clicked. They were like two college buddies who had that guys way of talking in shorthand, making unfinished observations and understanding each other perfectly. David would have a scotch, and Warren would have maybe a club soda—I never saw him drink anything stronger—and they would sit around on the couch shooting the shit. I would sit off to the side, listening. Warren never gave any indication that he noticed I was there or not, but I didn't care. It was gratifying just to be there with him, closing the family circle.
I think Warren liked David so much because it was obvious that David didn't want anything from him. David was neither starstruck nor ambitious. To him, Warren was just another guy. They could hang out, joke around, talk about cars or baseball or anything, and it never went any further than that. As for Warren, who was constantly surrounded by climbers and hangers-on, he found in David that rare thing: a person whom he could trust.
**• • •**
**IF** you're an ordinary noncelebrity, you will seldom feel more vestigial and out of place than when you're walking down the red carpet at a big Hollywood premiere. I walked down a number of them with Mom, including the Academy Awards, and I knew the routine cold—she would swan ahead, escorted by her latest boyfriend or her agent, and I would trail a step behind, basking in the residual glamour but keeping a low profile. Every few feet, Mom would stop to give an interview, and then I would have to stand behind her and smile and look interested. It was a true acting job, because I knew I was on camera, but there was nothing for me to do. Nobody wanted to talk to me; nobody cared about my opinions. If I had been really gorgeous, at least there'd have been a reason for me to be there, a fetching piece of scenery. Yet I was just a cute girl in a nice dress. I wasn't even blonde.
One time, we went to some kind of flashy benefit in Century City, and there was the red carpet waiting for us again. As usual, Mom stopped to hold court with her fans, being very gracious and bubbly—"Hi!...Hello!...How nice!...You're too kind!" Then that moment came when her eyes glazed over and she'd had enough. "All right, we're done"—and she left them flat and headed into the theater.
I started to follow meekly behind, as my role demanded, when suddenly I heard someone yell from the crowd:
"It's Sachi Parker! Sachi Parker! Tracy from _Capitol_!"
At that time, I was in the soap opera _Capitol_. I played Tracy Harris, a young mom who used to be on drugs and whose daughter was taken away from her. Tracy struggles to pull her life together and get her daughter back, but she often finds herself in locked battle with the unsympathetic social worker. I remember at one point my character loses it and screams at her nemesis, in the grand soap opera tradition, "Get out! Get out! _Get out!_ "
Mine wasn't a major character, but soap opera fans are devoted to the point of obsession. They know every character, every actor, every twisted plotline. So when Mom moved on and I momentarily emerged from her shadow, they spotted me, and they went crazy. "Sachi Parker!"—they knew who I was! I was instantly mobbed by adoring fans. It was surreal and disconcerting—I was nobody; what were they getting so excited about?—but at the same time, I loved it. For the first time in my life, I was signing autographs and posing for pictures!
"Miss Parker, Miss Parker! Over here! Smile, Miss Parker!"
I was finally having a moment in the sun! It was really cool.
In the midst of this ego-stroking orgy, I looked over at my mom, eager for some recognition, some maternal pride—but she was livid. Her eyes were these narrow slits shooting out beams of concentrated fury. She was actually being made to stand around and wait—on the red carpet!—while people made a fuss over insignificant _me_!
Sensing the inappropriateness of my celebrity, I tried to sign my name faster. I should have just broken away and moved on with her, but so many people wanted to talk to me, and I was having so much fun! Finally, Mom gathered herself up with an imperious shrug, turned to her agent, and rasped, "Let's go," and she stormed into the theater without me.
That was my last red carpet for a long time.
**• • •**
**DAVID** and I ended our relationship in 1988. It was an amicable breakup, probably the most civilized I'd ever had. Everything about David was and still is civilized.
I met Mitch Garvey at a party in Venice. He was tall and handsome, and had a sort of midwestern casual cool. I thought he was a big producer, which was just the type of guy I was prowling for at a party like that. I knew I looked hot—I was wearing a clingy spandex-type dress and my trusty push-up bra—so I went up to him and flirted, turned on the charm, used all the old tricks.
It turned out it he was just an assistant director, but by the time I found this out, I was already hooked. He took me out for sushi, and in the midst of our dinner, I mentioned who my mother was. "Who?" Mitch said. "I've never heard of her." It was fairly unlikely that someone working in Hollywood didn't know who Shirley MacLaine was, but I wanted to believe him. How cool—he was interested in me just for me!
It wasn't too long after that—a couple of months—that we got engaged. I know I seem to be getting engaged every time I turn around, but I think I always felt a loneliness and insecurity at my core, and I hoped that marriage would solve that problem.
For some reason, Mitch thought I should contact my dad and tell him about the impending wedding. Maybe it was because Mitch was a proper midwestern type of guy, and he wanted things done correctly. Or maybe it was because he thought Dad had money. Either way, Dad and I hadn't spoken since the deposition. I didn't know if he would even pick up the phone.
I called our home number in Shibuya, Tokyo, and waited nervously for Dad to answer. He didn't. The new owners answered. I didn't know the house had been sold.
I shrugged off my disorientation and asked the new owners, "Do you know where my father is?" They didn't, but they gave me a phone number. It was an American exchange, and when I looked at the area code I discovered that it was in the Boston area. So I called the number.
Yuki answered.
I was baffled. I recognized the voice immediately, but what in the world was she doing in Massachusetts?
"Yuki?"
"Yes?" she said warily.
"It's Sachi."
"Hello."
"Is this your house?"
"Yes." She was being very frugal with her information.
"Is my dad there?"
"Well, I don't know."
What did that mean? "You don't know if he's there? Because I really want to get in contact with him."
"Yes, well, I really can't say right now." And it was crystal clear, from her inflection, that she couldn't say because Dad was right there in the room with her.
I wasted no time. I found out the address from the phone number—it was in Hingham, Massachusetts—and Mitch and I grabbed the next flight from LAX. We had to get out there before my father moved somewhere else.
We landed at Logan Airport the next morning in the freezing cold—which was unfortunate for us, because we were still dressed for Santa Monica. We rented a car and found the house. We didn't call ahead—this would be an ambush, pure and simple. When we pulled into the driveway, we saw two little girls watching from the window. They were Yuki's children, Audrey and Emily.
I knocked on the door, and little Audrey answered. "Hello!" she said brightly. Then suddenly Yuki was at the door.
"Yuki," I said, "is Dad here?"
Then Dad walked up behind her.
We went to lunch together at a seafood restaurant on the water, Dad and Yuki and Mitch and me. It was an odd, awkward meeting at first. Dad pretended that nothing was amiss, that we were all on great terms. He was in his charming mode. "Sach the Pach!" he said, shaking his head with a grin. He was happy about my engagement to Mitch, and gave us his blessing, and then we launched into a lot of inconsequential talk. Yet, there came a point when I felt it necessary to acknowledge the elephant in the room.
"Dad, I'm glad that we're talking again."
Dad smiled tightly, and rattled the ice in his scotch. "Sure."
"It's been very tough being apart from you, because I love you."
I could see him withdrawing now, putting up the force field. So I forged ahead.
"I just want you to know, whatever's happened between us, I forgive you."
Dad was startled, and turned red with anger. "What do you mean? You forgive me for what? I should be forgiving _you_! You're the one who opened your big mouth!"
I looked over at Yuki; she gave me a sympathetic look. She knew what I was feeling. Then an interesting thing happened—I reached out and took Yuki's hand, and her hand tightened over mine. I felt a strong, pure current of love coursing between us. I don't know where it came from, and I don't know where it went, but for an extraordinary moment, we were soul mates. It was such a powerful, sweeping feeling that everything else at the table, including Dad, became incidental. Tears sprang to my eyes, and to Yuki's, too, and we both started sobbing uncontrollably, in a silent way, trying hard to be composed and stoic as the tears streamed down our faces.
Then, quite suddenly, the moment was gone. Our hands broke off, we pulled ourselves together, and the rest of the meal passed in awkward silence.
When we returned to the house, Yuki gave me a tour of the upstairs and downstairs, but she didn't want me going into the basement. So naturally I _had_ to go into the basement. I managed to sneak down there while Mitch kept them distracted.
What I found was a rambling living space, completely furnished. There was a huge kitchen, a fireplace, and a bedroom with a king-size bed. In the kitchen, I found special Japanese cooking utensils. I'd heard that Yuki's mom would sometimes visit, so maybe this was her room? Yet, I knew it wasn't. Traces of Dad were everywhere: in the refrigerator were his special mustards and his Dom Perignon. In the cupboard, I found his elegant bone china teacups, and his Darjeeling tea. And in the bedroom closet: Miki's clothes. Clearly she was living here, too.
I looked around the bedroom. On the bed-side table was a photo of Dad and Miki on their wedding day. So they had gotten married! Miki was now officially my stepmother! There were also photos of Yuki and her kids, and Yuki's mom. And a picture of Dad walking Yuki down the aisle on _her_ wedding day. It was now one big happy family.
I flew back to L.A. feeling both fulfilled and confused. I tried to put the pieces of the puzzle together. Dad had clearly fallen on tough times since Mom cut him off, although he still possessed that confident swagger, the sense that prosperity was well within his grasp (and he still possessed that chalet, that yacht, and that private island, so times weren't that tough).
Yuki, for her part, probably felt that she owed him for her education and support, so she provided him with a place to stay, as any good Japanese daughter would. But why didn't anyone tell me what was going on? Why didn't Dad tell me that he was married to Miki, and living with Yuki and her family? Why was I purposely kept in the dark?
Maybe the bigger question was, why didn't I just ask? Why didn't I confront Dad about his secrets? Why didn't I ask Yuki, especially after our epiphany? Because I couldn't. Because it wasn't good form. Because I still couldn't shake off that stubborn Japanese insistence on decorum at all costs.
**• • •**
**MOM** had a house in Seattle, Washington, with a view of Mount Rainier. We would often fly up there, she and I, and spend a weekend. I was the resident cook. There was a pool and a hot tub, and Mom would sit out there reading scripts while I went for hikes in the nearby forest.
We had some beautiful moments up there, but one day stood out for me, and not in a happy way. Mom and I were in the hot tub relaxing. We were talking about various things, and I don't know how the conversation turned in this direction, but for some reason, out of the blue, Mom announced, "Sachi, there's something I want you to know. I'm not leaving you any money."
"What?"
"In my will."
"You're not leaving me any money?" I was bewildered on two fronts: why was she bringing this up now, and why was she not leaving me any money?
"I'm giving it all to the Kronhausens," she said.
The Kronhausens? Those freeloaders who had coaxed me into losing my virginity? Those characters who had already talked Mom into buying them a farm in Costa Rica? _Those_ Kronhausens? They get the jackpot?
She was also planning to leave some money to a Spiritual Awareness Center somewhere in California, or was it New Mexico? The plans sounded a little vague, but the main point she wanted to get across was that I was getting nothing.
"Okay," I said quietly. I didn't argue with her, or ask her why. It was her money; she could do what she wanted with it.
Still, at that moment, I felt utterly abandoned. I don't know why: she'd never given me any money before, cutting me off when I was eighteen, and she'd even made me pay her back for that broken-down car. There was no reason for me to expect her to take care of me in her will. Still, I held out the hope that, because she was my mom, she cared about me.
There was always this twisted, tangled confusion of love and money in our family. Mom used to tell me, "Your trouble is, you need money to feel love." It was an odd contention, because I never had either. I think it was Mom and Dad who had created this love-money equation, in the way they doled it out or withheld it. I came to understand that, in their world, material gifts equaled affection—and I was clearly undeserving of both.
To give her her due, maybe Mom was afraid of spoiling me. So many Hollywood parents tried to buy their kids' love, lavishing money and sports cars on them, and it almost always turned out disastrously. Mom wanted to avoid that for me, so she went to the other extreme. Her intentions were good—she was protecting me—but in the process, she made me feel unloved.
Mom laid her head back on the edge of the tub and closed her eyes, as the water churned around her. She had made her statement and had moved on. There was nothing more to discuss.
This really ruined hot tubs for me. Seattle lost a lot of its charm, too.
**• • •**
**NOT** surprisingly, I started seeing a therapist. Jean was a heavyset white-haired lady in her fifties with a quality of softness and kindliness about her. She had twinkly eyes and a very round, maternal shape. I told her about the growing conflict with my mother, and she suggested that maybe I should bring her in for a joint session and see what we could work out. It could really be helpful and could bring us together—or (more likely) it could blow up in our faces. Jean was right, though: one way or another, the issues had to be addressed.
"Mom, I think we should go into therapy."
"You and Mitch?"
"No, you and me."
She laughed. She thought that was ridiculous. For all her cutting-edge enlightenment, Mom didn't think much of therapists. She considered them manipulative charlatans who preyed on weak-minded wimps—like me—and she did not suffer them gladly.
Still, I worked on her and managed to get her to a counseling session in Jean's office. Jean welcomed us warmly, and kicked things off in a soothing voice: "Okay, what would you like to talk about?"
Mom shrugged. This wasn't her idea, and she wasn't going to give it the validation of an opening remark. She passed to me.
"Well," I started, "I've been trying very hard to get my career going, and I feel that my mother could be of some help, but it seems that sometimes she's just not on my side. It's a very tough business, you know, and you need all the support you can get."
"It _is_ a tough business," Mom countered, "and that's why you have to be tough to survive in it. People don't help you? So what? Help yourself. Hey, I could make a phone call and get you a role like that"—she snapped her fingers—"but is that what you want? Do you really want to get a job that way?"
"Yes," I admitted.
"People don't like nepotism in this business," Mom said. (This was one of her favorite maxims, although I didn't see much evidence for it in Hollywood.) "You have to make it on your own. I did."
"Yeah, well, you had Charlemagne and E.T. helping you." I was being sarcastic again, but I didn't think she would hit me in front of Jean.
She glared at me evenly. "All right, I'm going to put this as gently as I can: everybody in creation knows that you're my daughter, but you still can't buy a job. What does that tell you? Maybe you're just not very good." A hit, a palpable hit.
"Okay, let's take a step back," Jean said. "Sachi, talk a little about your childhood."
"Well—"
Before I could get started, though, Mom was back in there with a preemptive strike: "She had a wonderful childhood. She traveled all over the world. I spent _my_ entire childhood in Virginia. _That_ was a treat."
Jean asked, "Was it a wonderful childhood, Sachi?"
"I was raised by my father in Japan. I saw Mom only once in a while."
"I was a working mother," Mom explained.
"And do you feel that she abandoned you?" Jean asked me.
"Abandoned her?" Mom answered. "I saved her! The Mob was after her. They wanted to kidnap her. I saved my daughter from the Mafia."
Jean kept focusing on me. "Sachi?"
"I wouldn't say she abandoned me, but I did feel there was something missing from my childhood."
"Oh, really? What was missing?" my mother said. "I gave you everything you needed. Got you clothes every summer. Bought you that stupid car. Name one thing I didn't give you."
"I wanted to go to college..."
Mom exploded. "You have some fucking nerve! I sent you to the best boarding schools in Europe. It cost me a fortune! You wanted me to spring for college, too?"
"And I wanted a baby."
Mom turned to Jean for support. "A baby, at her age!"
"I was twenty-seven!"
"You didn't have the maturity."
"You were twenty-two when you had me!"
"And look what happened to you!" This was like one of those courtroom moments when the defendant inadvertently blurts out a damaging admission, and then is stunned by the self-realization that maybe she is guilty after all. Except Mom didn't look guilty. She just looked more pissed off.
I waited for Jean to step in, but she was watching quietly, waiting to see where this would lead. So I took the initiative, trying to sound as reasonable as possible. "Look, Mom, you didn't have to spend a dime on me. I didn't want your money. I wanted _you_. I wanted you to take care of me, tuck me in bed, make sandwiches for me. I wanted you to make those peanut butter and jelly sandwiches for me every day. That's what I wanted."
Jean turned to Mom. "How do you feel about that?"
Mom had had enough, and pushed herself out of her chair. "How do I feel about it? I feel that this is a big goddamn load of nonsense! I feel like I'm swimming in bullshit!" She turned on Jean. "And _you're_ bullshit! And I'm not staying to listen to another word!"
Mom was heading for the door when she had one more thought to share. She stormed back at Jean. "You know what happens when you break down _therapist_? It's ' _the rapist_!' How do _you_ feel about _that_?"
Having nailed Jean with that sally, Mom steamed out the door, confident in her sanity.
# **Chapter 15**
# Domestication
In 1988, I auditioned for a TV show based in New York City called _Manhattan Express_. It was a live morning news show, but it shot at six in the evening, because the TV audience was in Japan. _Manhattan Express_ was produced by Fuji Television, and it was like a _Today_ show for Japanese viewers. The news would deal with U.S. and world topics, much like any morning show, and the anchors would be typical photogenic all-American types, except that they would be speaking Japanese.
By the time I auditioned, they had already done an exhaustive search for a Japanese-speaking Jane Pauley. It was desirable that she shouldn't be too pretty, but have camera experience and speak fluent Japanese and English. Now, there are different levels of the Japanese language, and I had stalled at the twelve-year-old level—I spoke children's Japanese. I thought this might be a drawback, but it actually boosted my standing, because I came off as childlike, which is what the Japanese audience wants in a female anchor. They don't care for women who come on too strong. So I was hired.
My male counterpart was Christopher Field, who had gone to Harvard and spoke high-level Japanese, which suited his role as the hard-core news anchor. I was the fluff, offsetting him with lightweight puff pieces, anything to do with society, entertainment, celebrities, or human interest. The job paid $1,300 a day, so I wasn't complaining.
It was very difficult at times to project the image they wanted. I remember in particular the Pan Am Lockerbie bombing in December of 1988. Chris was going to report the tragic story, and they wanted me to sit beside him and smile during the broadcast. Smile? Smile at what? Two hundred and sixty people had just gotten blown out of the sky! I couldn't believe it. _You want me to be charming and smiley while Chris is delivering this awful news? He's not smiling, why should I? I'll look like an idiot!_ Still, they insisted that my disconnected grin would be calming and reassuring to the Japanese audience—and I mean, they _insisted_. So I did it. Remember, $1,300 a day.
I was on _Manhattan Express_ for six months or so. It was seen only in Japan, but I got to appear in PR spots on _Good Morning America_ and _Entertainment Tonight_. It was a great gig—until it abruptly ended. Without asking me, my agent, Denny, demanded more money. This, of course, was the American way: to angle for a raise at the first opportunity. The Japanese way, however, was to be loyal and patient; making salary demands was considered the height of rudeness. So they fired me.
Not too long after that, I broke up with Mitch. He'd turned out to be a less-than-perfect fiancé. He had a violent temper, and he crossed the line more than once. Years earlier I would have put up with it, or tried to run away, but I was a grown woman now, more independent and confident, so I just told him, "It's over."
Mitch didn't take this well. He yelled at me, threatened me. He slashed my tires. Somebody peed on the kitchen floor of my condo in Marina Del Rey. I could never prove it was Mitch, but it had to be _somebody_. True, an animal might have gotten in—but man, there was a lot of pee. No raccoon could have produced it. Maybe a coyote.
Mitch also wanted half my earnings from the TV show, citing our "common-law marriage." When I laughed off that absurd notion, he told me: if I didn't give over half, he would go to the tabloids with the clone story.
I called his bluff on that one. Who would ever believe it?
After we broke up, Mitch got involved in a new relationship with a mystery woman. I confess, even though I'd initiated the breakup, I got really jealous. "Who is it? Who?" I had to know. He wouldn't tell me.
Then one night I had a dream. In it, Mitch was kissing another woman. She turned, and I saw her face. Oh my God.
When I awoke, I immediately called him up. "It's Debra Winger, isn't it?"
He was surprised. "How did you know?"
**• • •**
**I** had some extra time on my hands, so I finally got my breasts done.
I was going to do a play called _Independence_ , by Lee Blessing, in North Hollywood. In the play, my character had a scene where she appears topless. I wasn't really self-conscious about my natural breasts, but when the production was postponed for a year, it seemed like the perfect opportunity to finish what I'd planned with the sinister Jeffrey so many years before. I had the money saved from the morning news show, so I went ahead and did it.
They were fabulous; I loved them. If I didn't get more work in Hollywood with these babies, then the system was definitely broken.
**• • •**
**WHEN** we eventually did _Independence_ , I was in the dressing room backstage with the other actresses and I took off my shirt. There was a collective gasp of admiration. "Did you get your boobs done?" They were volubly impressed. I have to tell you, it may sound shallow and regressive, but there's something very gratifying about being told that you have an attractive body. It reminded me of being back at the noodle shop, with the waitresses marveling at my American assets. Plus, it gave me such confidence onstage: when the topless scene came along, I was proud to take the girls out.
Next up: _The Lulu Plays_ , an adaptation of Frank Wedekind's German plays _Earth Spirit_ and _Pandora's Box_. I played Lulu, the girl who goes from street girl to dancer to wealthy man's lover to prostitute, and winds up getting killed by Jack the Ripper. It was a great character arc, and I spent a good deal of the play topless again—which I relished: I'd developed a kind of fearlessness onstage that I could never approach in real life.
Lulu was a pitch-black variation on the whore with the heart of gold—the kind of role my mother had patented. I got some excellent reviews, but the one that mattered most would be Mom's. So I was especially nervous the night she came to see it.
After the show, we went to a local Denny's—not exactly Sardi's, but commensurate with the stature that live theater commands in Los Angeles. Mom had offered me the usual generic compliments, but she hadn't told me yet what she really thought, and I knew it was coming. I watched her eat her omelet in silence, and finally I had to ask the fateful question: "So, Mom, what did you really think of my performance?"
She solemnly put down her fork, waited until the moment was dramatically right, and then, without a hint of sarcasm, with an almost passionate earnestness, she said, "Sweetheart, I'm only going to say this once, and you will never hear me say it again. Tonight you gave one of the most remarkable performances I ever saw. It was truly transcendent. I was touched, and I was shattered. And when I am on my deathbed, and they ask me what was the greatest single performance I ever saw in my lifetime, I will say that it was you, in this play. And then I will die. And that's all I have to say, and I will never say it again."
Then she went back to her omelet.
She never did say it again, but she didn't have to. I was overwhelmed. Tears welled up in my eyes. This was so beyond anything I might have expected her to say. It was the best thing I'd ever heard from her—from anyone—in my life. It was as if she were passing the torch to me, and giving me her professional blessing. I had never felt as connected with her as I did on that night twenty years ago.
My performance as Lulu won me a Drama-Logue Award for Best Actress in a Play. In a wonderful coincidence, Mom won a Drama-Logue Award the same year, for her one-woman show, so we went to the awards ceremony together.
I suspect that if I hadn't won, Mom wouldn't have gone at all. The Drama-Logues weren't the Oscars or the Golden Globes; there was nothing formal or ritualistic about the ceremony, and the crowd was entirely different, struggling actors and theater folk, most wearing jeans rather than tuxedoes. There was a lot of clapping and cheering. Everybody knew each other from auditions and scene classes, and they would yell at you from across the room—"Hi, Sachi!"—and shout congratulations.
When Mom showed up, they made a huge fuss over her. She was easily the biggest star there. I realized then what a big deal it was for her to come to this function; I knew it was her way of showing support for my career.
It was a very special night, and I got all teary when I went up to accept my award. I thanked the director, the producer, my fellow cast members, and then I turned to Mom. "I especially want to thank my mom, who's here with me tonight." A nice warm round of applause followed.
"I so appreciate her," I went on. "She's taught me everything about acting. She's my role model. And it means so much to me that she's here to share this moment with me." I held up the award. "This is for you, Mom. Thanks for everything. I love you!" As the applause swelled, I went back to my seat and gave her a big hug.
About a half hour later, Mom received her award. The applause was tumultuous as she stepped up onstage. She made her thank-yous, and then said a few heartfelt words about the magic of theater. "It all starts with you actors. You are the torchbearers. You make it happen. I salute you all."
Then I saw her glance toward me. _Here it comes_ , I thought. _She's saving me for the end_.
"Of course, what makes this award extra-special is that my daughter is with me, getting her own richly deserved award for her performance in _The Lulu Plays_. She's worked so hard to get to this point, and as much as I cherish this award, I cherish even more the opportunity to be here with her on her special night, and watch her following in my footsteps. I could not be prouder of her. Sachi, I love you!"
That's what I was waiting for her to say.
What she _actually_ said was "Thank you," then left the stage.
I don't know why I expected more—it was her speech, she could say what she wanted—but it truly hurt my feelings. Just when I thought I'd raised myself to a certain level of respect in Mom's eyes, I was made to realize once again that, to her, in an elemental way, I didn't exist. It would have been a small thing for her to acknowledge me in front of my peers, but that was just the point. Mom didn't do small things.
**• • •**
**I** had my new breasts for three years. Then my husband made me take them out because he didn't like them.
Frank Murray was a Wall Street broker from Greenwich, Connecticut, who dabbled in producing. I met Frank in North Hollywood, when he produced _Independence_. He certainly had no objections to my breasts then, but once we got married, his more conservative instincts took over.
I met Frank during the last week of rehearsals, when he came out to check on the progress of the show. I walked onstage wearing a fishnet top that was meant to fit over a T-shirt, but I wasn't wearing a T-shirt. It was cold in the theater, and my nipples poked out right through the fishnet. This may have been what caught his attention. After rehearsal, he asked me out, and I said no. I had no interest in investment bankers who thought just because I was walking around with erect tits I'd be an easy lay.
That evening, the whole cast went out together for a late dinner, and Frank came along. I was dating a writer from _Star Trek: The Next Generation_ at the time, but I was still very much on the market, so I dressed sexy, in a skimpy, tight-fitting outfit. It was a cold night in L.A., and I was freezing. Frank took off his jacket and put it over my shoulders. It was a very romantic, chivalrous gesture. I thought to myself, _Uh-oh, I'm falling_.
At dinner, Frank couldn't have been more gentlemanly. He was polite and attentive, and he talked charmingly about his travels around the world. He reminded me of my father.
When he sent me roses the next day, that sealed the deal. I was head over heels.
To my surprise, Mom really seemed to like Frank, in her contrarian way. She didn't care for his Republican-ness at all, and he didn't care for her Democratic bent, but they enjoyed sparring with each other. They would get into impassioned arguments, and at the end of a bloody fight would regard each other with a grudging admiration.
So when, three years later, Frank proposed to me in St. Thomas, I couldn't wait to spread the good news. "Mom, I'm engaged!"
She took the news in stride. In fact, she sounded highly underwhelmed about the whole thing. "Uh-huh. Okay. All right."
She seemed to be making a judgment on my level of excitement. I couldn't understand why she wasn't excited along with me. Was she afraid she was going to lose me? I know that sounds a stretch, but I'd been living near her for the past ten years, and now I was planning to move to the East Coast, and Connecticut. Maybe she'd gotten used to me, and hated to let me go.
Maybe that's why she called Frank on the phone one day, unbeknownst to me, and gave him some interesting advice, which he shared with me afterward:
"Frank, I have to be honest. It breaks my heart to tell you this, because Sachi is my only daughter, and I love her dearly, but just be aware that you can't count on her. She's completely unreliable. Probably because she's so self-absorbed; it's hard for her to think of anyone but herself. And she's a liar. You can never trust her. Don't believe a word she says about me. I don't think she can help it. It's pathological." (I guess she was still hung up on Charters Towers and those missing airplane tickets.) Then she delivered the coup de grace: "It's probably not a good idea to marry her."
Mom always had to be different. Instead of telling me that Frank wasn't good enough for me, as most moms would have, she told Frank that I was the lemon. A novel strategy.
Frank, cleaving to his Republican roots, ignored the Democrat, and we started planning our wedding. It was originally going to be a small ceremony—I wanted to get married under a tree in a California canyon, with just our friends in attendance, and I would be barefoot with flowers in my hair. Very hippy-dippy. Frank came from wealthy stock, though—his great-grandfather was the Hardart of Horn & Hardart, the famous automat food-service company—so his family turned the wedding into a lavish affair in Greenwich: a High Mass in a Catholic church, and the reception at a country club. I didn't want any of this formal frippery, but Frank did. He wanted a big wedding, and he got it. It was almost as if he were the bride.
Inevitably, the wedding plans grew bigger and bigger—more food, more music, more obscure relatives—and with that came the vital question: Who's going to pay for all this?
Tradition dictated the bride's side of the family. I didn't want to go to Mom; I knew that would be a diamond-hard nut to crack. So I turned to my father. Clearly he wasn't riding as high as in those glory days when Mom was supporting his playboy image, but I assumed he must have had some kind of nest egg squirreled away.
Dad was affable enough on the phone. "How much do you need, Sach?"
"Not that much, really. Twelve thousand dollars."
Silence.
"That's pretty reasonable for a wedding in Greenwich," I pointed out.
More silence.
"Dad?"
"I'll see what I can do," he finally replied.
A few days later, I received a check from Dad for three hundred dollars. I bought my wedding dress with it.
Next stop—Mom. I knew I'd never get anywhere with her over the phone, so I drove over to Malibu and confronted her face-to-face.
" _What?_ Twelve thousand? Is that what you said?" She was giving me the Look.
"It's not really a lot."
"Not a lot? Twelve thousand? I see."
As she paced around the house, mulling it over, I followed after her and explained the expenses of the wedding ceremony and the reception in great detail. She needed to know where every penny was going. I didn't want her to think I would be using the cash to support my thriving crack habit.
She truly didn't want to give me the money, but I think I shamed her into it. ("If you don't help us out, what are people going to think?") There was a downside to her largesse, though. Now that she was springing for the wedding, she figured it was her own private party, and she started inviting all her Hollywood friends. Then Frank's family invited _their_ friends, to balance things out, and the next thing you know, we had doubled the price tag.
I called Mom up in Australia, where she was shooting a film. "Mom, it's going to cost a little more..."
"No, that's it! That's it!"
She wouldn't bend on this. Frank and I had to pay for the rest of the wedding ourselves. Mostly Frank, since I was broke. To his credit, Frank didn't have a problem with this. He knew that my mother was pretty tight-fisted, and I think he felt a certain pride in handling the freight on his own wedding without any help from the famous Shirley MacLaine.
Still, the situation left me feeling embarrassed and depressed. It would have been one thing if Mom had been a woman of modest means, or if we'd been estranged, or if she were devoted to charitable causes. But she was a rich woman, and she lived in a rich manner—a house in Malibu, a ranch in Santa Fe, a house in Seattle, an apartment in New York—and she'd had no problem sending sixty thousand dollars a month into space and subsidizing a clone.
Mom flew in to Connecticut for the wedding, and stayed at the upscale Homestead Inn in Greenwich. We had dinner at the inn the night before the ceremony, just the two of us. After a few drinks, I felt my emotions taking over. I knew I should have escaped to the ladies room, but I couldn't. I just sat there and started crying.
Mom looked at me with puzzlement. "What's the matter with you?"
"I don't know, I just—I feel like you don't love me."
"Of course I love you."
"I just don't understand," I said. "I'm your only daughter—why is it so difficult for you to help me out?"
Mom had no answer for me. She went back to her dinner. The rest of the meal was _real_ quiet. Maybe I'd hurt her feelings; I couldn't tell. Maybe she wouldn't even come to the wedding now.
But she did. The next day, she made her entrance at the reception, flouncing in like Auntie Mame, waving a check in her hand.
"I just want you to know," she announced to the crowd. "I'm going to pay for the wedding! This is my only daughter, and I'm taking care of her!"
She turned to me now, lowering her voice just enough to make sure that anyone who wanted to listen in could hear. "Sachi, here—I'm giving you a check for twenty thousand dollars. Don't ever say I didn't do anything for you!"
I reached for the check, but she yanked it back, extracting a necessary word of tribute. "Who's the best mother in the world?"
"You are, Mom."
Mom spent the rest of the cocktail hour making sure that anybody who'd missed it heard the good news that she was paying for the wedding. After that, of course, it became _her_ party. The wedding celebration was entirely secondary. At one point she stopped the band in the middle of the reception to announce that it was Bella Abzug's birthday, and they rolled out a cake for her. I was upstaged at my own wedding by Bella Abzug.
Yes, Mom did pay for it, and I took it as a sign that we were slowly building a bridge to each other. It might take time—there might be lots of setbacks—but the first bricks were in place, and someday...
**• • •**
**FRANK** and I set up house in Greenwich, and so began a long stretch of married life marked by a quiet, unexcited calm. After thirty-seven years of constant movement all over the world, I had found myself settling into the sedate rhythms of comfortable upscale suburbia. It was nice, for a while.
Not that those years were completely uneventful. There was plenty of incident, but most of it was of the ordinary family variety. After three years in Greenwich, for instance, Frank left Wall Street and got a job as the CEO of Amana, the refrigerator appliance company. So we moved to Houston. I was seven months pregnant with our son, Frankie, and while we were down there I got pregnant again, two years later, and had our second child, Arin.
The details of those births were unremarkable, but the emotional effect that they had on me was seismic. I was scared during that first pregnancy, truly anxious about having children, probably because of my ambivalent relationship with my own parents. I was afraid that failure was in my genes, and I felt a great deal of pressure to be a good mother. I was pretty sure I was going to mess it up.
Yet, once I held those babies in my arms, I knew I had nothing to worry about. I was meant to be a mom. The love was immediate, instinctive, and unconditional, just as I'd hoped. I knew exactly what my job was, and I loved doing it. Without making any judgments on my own mom, I was determined to be there for my kids whenever they needed me. My acting career could go on hold for now. I had a new career.
**• • •**
**IT** was early 2001, and we had moved back to Greenwich when I got a call from Andy Banks, Yuki's husband. He thought I should know that my dad was back in Hawaii: he was ill with lung cancer, and he wasn't going to make it.
I was stunned. I hadn't seen much of Dad lately—Miki tried to keep us apart as much as possible—but I always thought of him as healthy, robust, a man in his prime. I could picture him sailing boats, hiking mountains, or wending his way elegantly through a jagged path of nightclub tables. True, always with a cigarette in one hand and a scotch in the other, but he was only seventy-nine, a charming, irresistible force of nature. He was too damn clever to die.
I flew out to Hawaii as soon as I could, and brought Frankie and Arin with me. He was four and she was two, and they had never really gotten to know their grandfather. This might be their last chance.
When we arrived at the hospital, we found Dad in the rehab pool, working out with a big beach ball. They'd removed one of his lungs already, and you could see that he was very weak. Still, he worked out every day, throwing the beach ball back and forth, determined to get his health back. It was inspiring and moving to watch him, especially since at some level we knew his situation was hopeless.
At some point Dad left the hospital and joined us at the hotel, where he stayed for a couple of nights. He really enjoyed getting to know my Arin. He thought she was cute as a button. She was the same age I was when I joined Dad in Japan in 1959.
We talked of many things: our ice-fishing trips in the Japan Alps, the cruises on _Happy Pappy_ , and Molokai, the site of the former leper colony. Since childhood, Dad had been telling me stories about this beautiful island in the middle of the Hawaiian archipelago; and about Father Damien, who tended to the lepers there; and about the high, sheer cliffs of Kaluapapa that kept the lepers exiled from the rest of society. I always wanted to visit there, and we talked about going now. "I really want to take you there, just you and me," Dad said. "I want to show you the cliffs."
He then discussed his will in detail. "You know my chalet in Italy? I want you to have it. You and the kids. And I have land back in Japan...the Nasu property. It's worth millions."
I didn't want to go in this direction. "No, that's yours, Dad. You'll be back there soon."
Dad laughed weakly. He knew bullshit when he heard it. "No, it's for you. It's all taken care of."
There was one more matter I needed to discuss with him, just to get it straight in my head:
"So, Dad," I said at one point, leaning forward and stroking his arm gently, "you really did make up that whole story about being cloned, didn't you? You know, so you could get Mom's money? And the telegrams, and the government spying on her, and the Pleiades...That was all you, right?"
Dad breathed heavily a few times. After a long moment of reflection, he quietly said, "Yeah."
When I finally heard it confirmed from his own lips, I was astounded. Even though I knew it was true, there had been enough of the purely incredible about it to always leave a slight doubt in my mind. I had assumed he would hide in that shadow of doubt right to the end. To admit it so easily, to finally tell me the truth—I never expected that. The effect was instantaneous: with one word, all the lies disappeared, all the cobwebs were swept away. I was finally seeing my real father.
**• • •**
**THREE** weeks later, Frank and I were back in America, building a new house in Greenwich and living in a rental nearby. One evening, Frank went over to check on the progress of the house. When he returned, his face was sheet white.
"What's the matter?" I asked.
"I saw your father in the kitchen," he said.
Over at the new house, he'd seen what he called "a shadow" of my dad standing in the unfinished kitchen. "When I walked in, he saw me and ran into the pantry, and the room became very cold. It was your dad."
"I think you're losing your marbles," I told him. "Dad's in Hawaii." An hour later came the call from Miki. My dad had passed, six hours earlier—but, apparently, he'd stopped by Greenwich on his way out to pay a visit.
We booked a flight for Honolulu the next day, but before I left, I had to go over to the new house to check out the ghostly emanation. I made Frank go with me, and together we wandered through the house. We visited the kitchen—a fitting place for my dad, a gourmet enthusiast, to frequent—but there were no manifestations, no cold spots. Then we went into the living room area. I was sitting on a coffee table, Frank was in a chair opposite me, and as we chatted about our memories of Dad, I felt a sudden beautiful warmth, starting in my shoulder and traveling down the left side of my back. It was a flowing heat of love and forgiveness, and I just let the feeling run through me.
Frank, across from me, turned white again. "Oh my God. Sach, your dad is standing right next to you with his hand on your left shoulder. He's apologizing to you, he's saying he's sorry, and he loves you."
The funeral service was held at the Halekulani Hotel, where Dad had spent so many happy days, and where I'd served as a bus-girl and fallen under the spell of Luke Garrett. Before the service, Miki gave me a stuffed toy monkey that had belonged to Dad. She said he would have wanted me to have it. I was born in the Year of the Monkey, so it had a special resonance.
I joined Miki and Yuki at the front of the room, to represent the family. Mom wasn't there: she hadn't been invited, which was fine, because she hadn't wanted to go.
Something very odd was going on at that memorial. Everyone was coming up to Miki and Yuki to offer condolences, but no one was coming up to me. I was being shunned. Most of Dad's friends from Tokyo wouldn't give me the time of day.
I couldn't understand it. What had I done? Everyone was staring daggers at me, giving me dirty looks. When it was my turn to speak at the service, all I could see was a frieze of angry, hostile faces. These people hated me.
Yuki told me later that she'd heard of Miki spreading stories about me: what an awful daughter I was, how disloyal I had been to my dad, how I had vilely betrayed him.
To give Miki her due, she was only reacting to the skewed version of events that my dad had artfully presented to her. She knew nothing about the clone story. She thought the reason Mom divorced Dad and cut him off (thereby dramatically altering Miki's own affluent lifestyle) was because I had turned against Dad and told vicious lies about him. Yuki recalled that over the years Dad would lament that he missed me desperately, and tried to get in touch with me, but that I refused to answer his calls. In truth, the very opposite was the case; he was the one who cut me adrift. But Miki had never heard my side of the story, so she believed Dad, and consequently told everyone else that I had coldly abandoned my father and broken his heart.
I'll say this for Dad, he was consistent. Whether it was me or Mom or Miki, he didn't discriminate in deceiving the people who loved him.
**• • •**
**THE** next day, we had breakfast at the hotel before flying home. Frank and the kids went for a walk around the grounds, and I sat by myself, staring out at the ocean, in a sad, reflective mood.
Suddenly Miki was sitting next to me. I was startled. I assumed she had come to console me.
Not quite. "Sachi, your father always said that he wanted me to have the Nasu property in Japan, but he forgot to put it in the will. You don't want it; you don't even live there anymore. So would you please sign the property over to me?"
Her audacity amazed me. I couldn't discuss this now; I was still in an emotional place, and I couldn't switch gears. "I don't know, Miki. I have to think about it."
Miki didn't want me to think about it. She miraculously pulled out a contract and a pen. "All you have to do is sign right here."
Tackiness aside, she made a good point. I probably wouldn't be back in Japan again, and she obviously wanted the property more than me. Plus, I had the chalet in Italy. I might as well let her have the Japan property.
Then, suddenly, I gathered my wits about me. The Nasu property was a spectacular piece of real estate in northern Japan, surrounded by golf courses and ski resorts. It was, as Dad said, worth millions. I couldn't just give it away on a whim. "No, I can't sign anything right now. Later."
Miki wouldn't give up. She kept asking, and I kept saying no. Finally she grabbed the contract and stalked off in a huff. She refused to talk to me after that, not if I wasn't going to sign.
**• • •**
**BACK** in Connecticut, we waited patiently for the settlement of the will. After several months, when there was still no word, I asked Frank to check with the lawyers in Honolulu. He discovered that there was nothing coming. There were no provisions for me in the will.
Only the Nasu property was mine. Under Japanese law, the property had to go to a direct descendant. That's why she wanted me to sign it over to her.
**• • •**
**STILL,** per another part of Japanese law, I had a year to claim my inheritance. After that, it would revert to the spouse or nearest relative. I didn't bother to put in a claim. I allowed the year to expire.
# **Chapter 16**
# The Lord and the Ring
It was 2002, my children were in school now, and I was getting restless. I wanted to act again.
I had joined the Theatre Artists Workshop of Westport back in the 1990s, and when we moved back from Houston, I was eager to get back into the swing of things. The workshop had been founded by a group of theater professionals who lived in the area as a safe place for actors and writers to flex their muscles and try out new stuff. Many stage and film luminaries—Keir Dullea, Lee Richardson, Theodore Bikel, Morton DaCosta, Phoebe Brand, Ring Lardner Jr., James Noble, Brett Somers—would show up at the weekly Monday night meeting to view the work and give feedback.
I loved the supportive, nurturing atmosphere there. It inspired me to take lots of new chances. I acted in all kinds of classic plays: _The Seagull_ , _The Three Sisters_ , _A Doll's House_ , _Anna Christie_ , _The Beauty Queen of Lenane_ , _The Glass Menagerie_ —parts I'd always wanted to play, when I was just at the age where I could still pull them off. I never went topless, though; those glory days were over.
My favorite was _A Moon for the Misbegotten_. Josie Hogan was such a departure for me. She was an earthy, ballsy character, with a thick, wide peasant body, where I was slight and unassuming. It took me a long time to find her, and once she took possession of me, I couldn't get rid of her. I would walk around the house burly and heavy-footed, and sit with my legs splayed wide. My kids would notice and tease me all the time. "Uh-oh, Josie's making dinner tonight!"
It felt great to be working regularly as an actress again, and my confidence was growing exponentially with each performance—so much so that, in the fall of 2005, when I was appearing in the workshop's annual one-act festival, I made the supremely courageous gesture of inviting Mom. She was visiting the East Coast at the time, and I persuaded her to come see the last Sunday matinee performance.
I was nervous about performing in front of her. Over the past ten years, our relationship had downshifted from a roller-coaster ride of highs and lows to a pleasant cruise on a neutral plane. I was no longer active in the entertainment world, and that removed the edge of competitiveness that had always charged the air between us. In her eyes, I was now the settled suburban mom, and she was the grandmother, "Ganny," who would drop in from time to time to bring gifts and offer homely advice. She didn't see me as an actress anymore.
Just a year earlier, she'd brought me to a party in New York at some important person's duplex apartment, and there were all these celebrities there: Mike Nichols, Diane Sawyer, Jane Seymour, and Nora Ephron. I thought, _Nora Ephron_! I loved her and had been dying to meet her.
Now, it's not all that easy to strike up a chat at one of these high-powered cocktail parties, because anyone you might want to talk to is usually busy trying to find someone else more important to talk to. So the whole night, I waited for the serendipitous moment when I could just happen to run into Nora and start a witty conversation. Finally, that moment arrived: here she was, right next to me! Before I could say a word, though, Mom popped up at my side like a malevolent genie.
"Nora, I want you to meet my daughter, Sachi." Hooray, she had given me just the introduction I needed. She could have left it at that, but no, she had to add, in a patronizing voice, "Sachi wants to be an actress. She _just_ started taking acting classes. Isn't that great?"
Nora smiled indulgently, offered a few words of encouragement, and moved on. I'd been effectively torpedoed. Mom knew damn well that I'd been a professional actor for thirty years, but now, in front of this elegant, sophisticated icon, she'd made it sound as if I were some bored suburban housewife with a few spare hours on her hands.
In fact, this is probably how Mom saw me. I'd removed myself from the arena; I was no longer a gladiator. She hadn't seen me onstage in more than ten years, and when she showed up at the workshop, I don't think she was expecting me to be anything more than community-theater adequate.
So we were both caught by pleasant surprise when she wound up loving the show. She was laughing her head off in the audience, letting loose with her familiar full-throated cackle. A little too loudly, of course—I could tell she was enjoying the sound of her own voice—but I didn't care. Those laughs were for me.
Afterward, as we drove back to the house, she was gushing with superlatives: "You know, Sach, you owned that stage. You were wonderful. What can I say? You're a great actress. You really are."
I listened warily, waiting for the other shoe to drop, the deflating "But" that would send my spirits into free fall. It never came. She was actually sincere. It was a mirror of that moment at Denny's after _The Lulu Plays_ , when I felt I'd finally broken through to her as a fellow artist. Was she confirming that earlier appraisal, the one that had left me sky-high with hope, or was this another false start? I wanted to believe her; I wanted to be exhilarated by the possibilities of rebooting my career with her firmly in my corner. So I did.
This developed into a golden period for us as mother and daughter. We stayed in close touch; we talked on the phone all the time. We were even going to spend Christmas together. A family Christmas with Mom! What could be better?
Well, it got better. A week before Christmas, she called me up all excited: "Honey, I have the best Christmas present for you! This is perfect! Perfect!"
Now _I_ was excited. My mom was getting me the perfect Christmas present! What could it be? In the back of my mind I thought, _A script! She has a screenplay with parts for both of us. We're going to act together!_
No, I couldn't think about that. It was probably a beautiful piece of jewelry, or a first edition of a book, or something like that. Whatever it was, I would absolutely love it.
On Christmas Day, Mom was at our doorstep—and with her was a tall, thin gentleman with a goatee. "Sachi, this is Casper DeVries."
The gentleman nodded. "Hello," he said in a vaguely European voice.
"He's your Christmas present," Mom said, bursting with glee. "Merry Christmas!" She gave me a big hug.
"Merry Christmas," I replied with a little bit of confusion, then lowered my voice to ask, "He's my present?"
Mom nodded eagerly. "He's going to read you!"
I didn't get it. "What do you mean?"
A trace of exasperation flitted across her face. " _Casper DeVries_. The world-famous psychic. He has a TV show."
Mr. DeVries leaned in. "I have several TV shows."
This was quite true. Casper DeVries had a cable show called _Reaching Out_ , where he reconnected people with their dead relatives, and he'd done a couple of miniseries about life on the Other Side. He was also a consultant on a network show about psychic mediums. Mom couldn't have found a more perfect soul mate if she'd robbed a grave.
I still didn't get it. I took Mom aside. "So this is the big present? He's going to read me?"
"He's going to tell you _everything_."
I wasn't sure I wanted to be told everything, especially on Christmas Day, but when I thought about it, I decided it _was_ the perfect present, coming from Mom. It was certainly personal.
We had time before dinner, so we all repaired to the living room—me, Frank, Mom, the kids—and sat in a semicircle. Mr. DeVries was in a chair with his back to the fire, which imparted to him a mystical orange aura. He started going into a trance.
His eyes were closed, his hands gripped the arms of the chair, and he made odd guttural sounds, as if speaking in tongues.
" _Aghhh...Ooo...Ogggh...Uuuu..._ "
We waited patiently for a spirit to grab him. Eventually he threw his head back and began channeling someone. His voice dropped a few octaves.
_"Uggh...Aaaagghh..."_
"Who are you?" Mom asked boldly.
" _Obadiah_ ," he answered. Or somebody answered.
Ah. Mom nodded with familiarity. Obadiah was the spirit of a former slave; he had visited her many times before, with the help of various channelers.
I remember one time encountering Obadiah myself. Kevin Ryerson, another well-known psychic, used to channel him a lot. Ryerson was big on trance-channeling, and he was apparently the one who first informed Mom about her past life in Atlantis and so forth. The scene is re-created by the two of them in the movie version of _Out on a Limb_.
That episode goes back to when I was still with David. We were over at the Malibu house with Ryerson and my mom, and they were having a channeling session. Ryerson went into a trance, and "Obadiah" started speaking through him. I don't remember what came out during the channeling, but I recall that Mom had lost her gold watch with a diamond-encrusted rim and she mentioned it during the session. She had searched all over the house and just couldn't find it. She kept rubbing her wrist throughout the session, lamenting her loss.
" _That's too bad_ ," Obadiah said. " _We'll have to do something about that._ " I remember thinking it funny that somebody who'd suffered through the horrors of slavery would give a hoot in hell about a missing gold watch.
After the session was over, I was in the kitchen making dinner; David was with me. Mom was out taking a walk on the beach.
Out of the corner of my eye, I saw Kevin Ryerson step into the kitchen and hover by the doorway. He reached into his pocket and very discreetly (but not discreetly enough) took out the diamond-encrusted gold watch, put it on the kitchen counter, and then stealthily withdrew before anyone could see him.
I saw him, though, and David saw him.
We both gathered over the watch and stared at it. What should we do? Should we bust Ryerson? Tell Mom the truth? Would she believe us? Probably not. She was so invested in her beliefs, and they made her so happy, that we made an agreement not to say a word. Why dash her dreams?
When Mom came back and walked into the kitchen, she screamed, "Oh my God! There it is! My watch! He found it for me! Thank you, Obadiah!"
All I could think was, _Oh, Mom. You're such a little kid sometimes_.
I was not as susceptible. I had seen the man behind the curtain too many times, and that's why I greeted Obadiah's emergence now in my living room with a healthy degree of skepticism.
DeVries was now completely in the grip of his visiting spirit. Mom, an old hand at these channeling sessions, took the reins. "Who's with you, Obadiah?"
" _Steve is here..._ "
Mom looked at me, and mouthed "your father," which I'd already deduced. Who else but Dad would be showing up in my living room?
"Does he have a message for us?"
" _He wants you to know that he loves you, and he apologizes._ "
"Anything else?"
" _He wants to thank you for hanging his picture in the bedroom..._ "
I was startled by that one. It was true—a photo of Dad was hanging in an upstairs bedroom. How could DeVries or Obadiah have known that? I looked over at Mom, who nodded sagely.
Mom started asking the spirit some searching questions about the nature of the universe, and her place in it:
"What is the path for me?" she asked with deep earnestness.
" _You are an explorer...You are a star voyager...Many will scorn you, but you must be strong and follow your vision wheresoever it may lead._ "
Dad's language had taken on a very biblical syntax. Or maybe it wasn't Dad anymore. Obadiah indicated that someone else was present, an elderly woman. "It must be your grandmother," Mom said.
Whoever it was, I was getting antsy. This was supposed to be _my_ present. "Can I ask something, please?" Mom shrugged and, with a roll of her eyes, sat back. By this time I had completely bought in to the Other Side, and I was itching to ask my big question.
"What's going to happen with my career? Will I ever make it as an actress?"
DeVries rocked back and forth, and started shaking, as he received the message. "Mmm...Ahhhmmm _...Mustn't ride on mother's coattails, must we?_ "
"Huh?"
"Mmmm...Ahhhh.... _Acting classes are in order..._ "
Acting classes? I looked over at my mother. Now she was slumped in the corner of the couch, watching like a spider, and emanating evil energy.
But back to DeVries: he was trembling, getting excited. A great vision was coming to him: " _Yes, I see...Sharp objects...Knives! Pots! Copper pots!...Cooking! A COOKING SHOW!_ "
A cooking show. I should be doing a cooking show. That was the message that Grandma was sending from the dead.
I couldn't look at Mom now, or anyone. I felt something very hot spreading in the pit of my stomach, then rising very quickly through my various internal organs, up, up, until it was scorching my cheeks. It was as if my head had been dipped in acid.
I was so pissed off. I realized that Mom had set this whole phony business up. Why? Maybe because she was afraid I would start acting again, and she was going to dissuade me through any means, normal or paranormal.
I also realized that if I didn't leave the room immediately, my head was going to explode. I rushed into the kitchen and promptly dissolved into a hysterical mess.
They could hear me sobbing from the living room. Frank suggested to Mom that maybe she should go in and see how I was.
Mom waved him off. "Ahh, she'll be fine."
Instead, Mr. DeVries, who had emerged from his "trance," came into the kitchen to comfort me. A mistake on his part.
"You lying son-of-a-bitch!" I screamed at him. "You know not a word of that was true! Cooking show, my ass! You're nothing but a fucking phony!"
"Sachi, listen to me, please," he said, trying to quiet me. "I really _am_ a psychic—"
"Ha!"
"—and I know that you're a fine actress."
"Right."
"In fact, I see you winning an Academy Award someday."
"Really?" He was starting to win me back.
"But your mother made me tell that story."
"She _made_ you?"
He shrugged. "It's what she wanted. I couldn't say no."
I discovered later that Mom and DeVries had been staying—in separate rooms—the last few days at the Homestead Inn, where she would have had plenty of time to feed him his lines, and tell him all about Dad and cooking and that fucking Obadiah.
It took me a while, but I managed to pull myself together. We still had to get through Christmas dinner, after all. I went back to the living room, full of false cheer, as merry as any elf. Mom, who'd gone to such great lengths to puncture my ego, would have none of it; she took me aside: "Look, I can tell you're upset. You should be. Mediocrity is not an easy thing to accept."
"I'm not upset," I told her. "Because he's wrong, that's all. Even the best psychics can be wrong."
"He's wrong? Oh, really? He has a TV show, and he's wrong?" Mom was furious. She couldn't stand that I wasn't buying into her bullshit. She'd really wanted to put the last nail in the coffin of my acting aspirations, and her plan had been a big fat flop.
Dinner was very forced. I wanted to feed them both and get them out of my house as soon as possible, so I pulled the meal together in record time—and it was, by the way, fabulous; I really was a good cook. Grandma'd got that right. The crosscurrents of tension at the table were excruciating. I was on to Mom, and she knew I was on to her. Meanwhile, DeVries was nervous that I would tell Mom what he'd told me. We ate in nerve-racking silence. Then they left. I was never so happy to see my mother make an exit in my life.
Unfortunately, we'd made a plan to have breakfast at the house the next day, before they flew back. After the channeling fiasco, I assumed they would pass on that invitation. But no, the next morning Mom and DeVries were both at the door, waiting to be fed. I couldn't bear to look at them. "I don't have time to make breakfast this morning," I told them. "Sorry."
Mom looked a little stunned. I think it was the first time in my life I'd ever been cold to her. I didn't even want her to hug me. We said a very icy goodbye.
**• • •**
**THE** deep freeze was on for about a month. Then, one day in early 2006, out of the blue, Mom called. "Sach, I have some exciting news. Are you sitting down?"
"Yeah." I was not sitting down. I couldn't imagine what news she could have that required my sitting down for, unless she was going to tell me that Paul had finally returned from the Pleiades.
"I'm doing a new movie up in Canada. It's called _Closing the Ring_."
"Good for you," I said flatly. So nice to hear that her career was moving right along.
Mom didn't pick up on my sarcasm or chose to ignore it, because she went on brightly: "And there's a role for you! You're playing my daughter. Is that perfect casting or what?"
My ears pricked up. Wait a minute—a role for me? What was she talking about? Now I sat down. "I'm playing your daughter? You mean, I already have the part?"
Mom laughed. "Of course not. You have to audition, like anyone else. But do a good job, and I'm sure Dickie will take our long-standing friendship under consideration."
"Dickie?"
"Lord Attenborough."
My heart leapt. "I'm auditioning for Richard Attenborough?" This was developing into something serious. It had been a long time since _The Bliss of Mrs. Blossom_ , and now of course Lord Attenborough was an Oscar-winning director and a major producer, but if he still remembered me as little "Poppy," there was a better-than-even chance I could get this part.
It seemed too good to be true, and that's why I became suspicious. "Why?"
"What do you mean, 'why'? Why not? You want to be an actress, don't you?" Still, even Mom could sense there was a hollowness to this argument. The question I was asking was, why _now_ , after all these years?
"Look," Mom said with a sigh, "it's very simple: I want to help you out. I want to make things right. Other mothers help their daughters, why shouldn't I?" Before I could answer this, she moved on. "They're going to fly you up to Toronto this weekend to do a screen test. Can you make it?"
I was still a little befuddled. "Sure, I guess...I'm just...surprised."
"Well, don't be surprised. You're a talented actress; you deserve a break."
I started to choke up. "Mom—thank you. I promise you, I'll do whatever I can to make you look good. I'll be totally prepared. I'll give it everything I've got. You're taking a big risk, but I'm going to make it work!"
Mom tried to put it in perspective. "Well, remember, they're looking for a twenty-six-year-old, and a name, so don't get your hopes up."
I was fifty years old and no name at all, but that didn't stop me. "I don't care," I said. "Even if I don't get the part, I'm just happy that this is bringing us back together. It means so much on a personal level. This makes up for everything."
Some of this was pure actor-speak. Of course I _did_ care; I desperately wanted the part. Finally, a chance to act with my mother, in a big film, with a famous director—it was a dream setup, and I couldn't let it slip away.
The script was sent to me, and it was indeed a terrific role. Mom and I would have some great scenes together. I wanted this part, and I was going to get it. I studied and studied, and sent on my head shot and my reel. That weekend, I flew up to Toronto and met with Lord Attenborough. "Poppy!" he greeted me in his charming English voice. He was still a delightful man, and I could see that his affection for me was still strong.
My confidence was high as I did the screen test. The makeup artist did some work on my eyes to make me look young, and she had me put a scarf around my neck to hide some wrinkles—I was supposed to be twenty-six, after all.
The test went beautifully. I really nailed it. There are times when you just know. Everyone was buzzing with compliments. When I got back to Connecticut, there was a personal message on my voice mail: "Oh, Poppy, you did a marvelous job on the screen test. You'll be wonderful as Marie."
So there it was! I had the part! Now, admittedly there was one little snag to be worked out with immigration: according to Canadian work rules, the film could hire only a certain number of American actors, and it had already reached its quota; Mom was taking up the last spot. We'd have to find a way to skirt the rules before they could officially offer me the role, but that wouldn't be a problem—not with Shirley MacLaine and Lord Richard Attenborough on the case.
With the shoot coming up, I realized I'd better get my eyes done. I couldn't count on the makeup people to help me every day. I flew out to L.A. and was treated by the same doctor who'd done my breasts, Dr. Norman Leaf. Dr. Leaf was a plastic surgeon to the stars, he did great work, and he was a genuinely nice man.
After the surgery, I went to Dr. Leaf's recuperation facility. There were all kinds of well-preserved women and men milling about with small bandages on their noses, their chins, their you-name-its. Quite a few of them were celebrities: no big stars, but lots of solid middle-ground performers trying to hold the line against time.
Mom came to visit while I was there. She brought me a nice sweatsuit. I couldn't actually see it; I had to keep my eyes closed. My head was reclined, and I held ice on my eyelids as we chatted. We talked a bit about the film. I told her how excited I was.
"Tell me, Sachi," she said thoughtfully. "What if you don't get the part? What are you going to do?"
"I _am_ going to get it, Mom. You know that."
"But what if you don't? What are you going to do?"
"Don't worry, Mom. The part is mine."
We talked a bit more, and then Mom asked me again, "But what if you don't get the part? What if something goes wrong? What are you going to do?" She must have asked me that five separate times. I didn't know what she was so worried about. I already had the part.
My eyes took only two weeks to heal, and I looked great. I was all set for Canada!
But not so fast. Someone was still making a stink about immigration. Too many Americans in the movie! Mom said not to worry; she was calling all her bigwig friends. Lord Attenborough even rang up Tony Blair a couple of times. The wheels were in motion.
Plus, Mom had a nuclear option up her sleeve: "I know. I'll become a Canadian citizen. My mother was Canadian. And then you can take my spot."
I didn't like that idea. "I don't want you to change your citizenship for me."
Mom accepted my objection pretty easily. "Well, don't say I didn't offer."
More time went by, and we didn't seem to be making any progress on the immigration front. So Frank got in touch with his good friend Ed Cox, the son-in-law of Richard Nixon. Ed had connections with the Canadian government, and he was happy to help out. He didn't seem to think it was a problem at all. "Don't worry," he told Frank. "This is a walk in the park, we'll get her in."
Frank called Mom to give her the good news. I was in the room with him when he called her, and I could hear Mom's animated reaction through the receiver. "How dare you go over my head?" she was shouting. "Stay the fuck out of this! Leave well enough alone! I'm handling it! It's being handled!"
Frank hung up and looked at me bewildered. What was _that_ all about?
So he told Ed Cox to back off; it was being handled.
The production start date drew closer and closer, and I was still waiting in Connecticut. A marvelous cast had been assembled: Christopher Plummer, Mischa Barton, Pete Postlethwaite, Brenda Fricker, and Mom. I was champing at the bit to get started. What was taking so long?
Finally I called Jack Gilardi, Mom's agent at ICM. "Have you heard anything about _Closing the Ring_? Did they clear things up yet?"
"Oh yeah. They went with somebody else," he said matter-of-factly.
" _What?_ Somebody _else_?"
"Yeah. What's her name? Neve Campbell."
"Neve Campbell? Neve Campbell is playing Marie? When did this happen?"
"About two weeks ago."
Two weeks ago? Why didn't anyone tell me? What the hell was going on? "But—what happened? I thought Mom was handling it."
"Well..." I could almost hear him shrugging with indifference.
"But I had the role. It was mine."
"I guess you didn't." He could not have been more dismissive. Of course, Jack was Mom's agent, not mine. I knew he didn't really give a shit about me, but he didn't need to make it so obvious.
I called Mom right away. "Did you know that I didn't get the part? They cast Neve Campbell instead?"
"Oh, yes. I heard that. She's Canadian, you know. It made things much easier."
"Why didn't you tell me?"
Mom paused. "Well, you know how it is. Nobody wants to give bad news in Hollywood."
"You're not Hollywood. You're my mom!"
"Well, look, at least you know I tried," she said defensively. "I really did." That was true, and that was the important thing, really.
"And don't worry, sweetheart," she promised, "next time we'll find a project for the two of us, in _America_ , and _we'll_ be in charge. Nothing will go wrong."
I also got a call from Lord Attenborough, who apologized profusely for the way things had turned out. "So sorry, Poppy. I did everything I could. It breaks my heart. _Closing the Ring_ just won't be the same without you."
They made it without me anyway. It was not a great movie, and it went straight to video—but that was small consolation.
I called Mom during the shoot. I'd become so invested in the movie, I had to hear how it was going, even though I knew it would only make me feel terrible.
"Oh, it's going _so_ well, sweetheart," Mom enthused. "It's wonderful being up here with Dickie. Christopher Plummer is so good. And Neve Campbell is fine. But you know," she added ruefully, "she's not you. She's not you."
# **Chapter 17**
# Don't Take It Personally
Christmas of 2006—the Christmas after Mr. DeVries graced our Greenwich home with his party tricks—Mom invited us to spend the holiday with her in Santa Fe. "Hey, Sach, why don't you come to the ranch for Christmas? The kids can go skiing. I'll stock up on food, it'll be a blast!"
I should have been wary from the get-go—her unwarranted optimism almost begged for disaster—but I was actually kind of excited. We landed in Albuquerque, full of good cheer, and piled into the rental car for the hour drive to Santa Fe. Just as we got on the highway, it started snowing—and snowing. Before long, we found ourselves in white-out conditions. Frank could barely see where he was driving. The road itself had become just a rumor. There were accidents everywhere, and countless cars had slid off the highway and into the side trenches. Five tense, white-knuckle hours later, we still hadn't reached Santa Fe. This was not a good omen.
I kept in touch with Mom on the cell phone, trying to convey the treacherousness of the driving conditions to her, but for some reason, she couldn't conceptualize our plight. She was just pissed off that we were late.
"Where _are_ you?" she yelled. "What are you _doing_?"
"We're stuck on Twenty-five."
"That's ridiculous. It doesn't take more than an hour to get here. Doesn't Frank know how to _drive_?"
"It's snowing, Mom!"
"I know it's snowing! I have _eyes_. I can _see_! But the main roads are always clear."
"But they're not!"
"Well, they _should_ be!"
I called her every fifteen minutes or so, and every time, she gave me grief. Now it was getting dark, and the kids were starving. I told Mom that we were going to stop at a diner.
"No, don't stop!" she shrieked. "I already made a great dinner. Just get here!"
We plowed on to Santa Fe. Once we got off the highway, the conditions got worse. We had no idea where we were going. The roads were covered, there were no street signs visible, we were going completely on instinct and my mother's long-distance imitation of an hysterical GPS:
"Take the next right turn!"
"But there is no right turn—"
"Just take it!"
To give Mom the benefit of the doubt, I knew that she was nervous and worried for us. Crankiness and impatience were her default modes when she was under stress. She was not calm in the face of adversity, especially when she was helpless to do anything about it. So this was her way of showing that she cared. Of course, that was small consolation to me when she was screaming in my ear while, all around me, the world was being sucked into a white black hole.
Finally we started down the long downhill road to her ranch. As we descended through the snow, we wondered if we would ever be able to get back up that hill again, but we didn't care. Exhausted, we staggered to the front door, where Mom was waiting with a big, incongruous smile, as if all her hyperventilating over the phone had been part of a past life: "Hi! How are you! Come inside, it's so great to see you! Hi, kids!" She was suddenly the dream grandmother.
Starving, we headed straight to the dining room, eager to dig in to the great dinner she had promised. But there was no great dinner. There was bean soup.
I took Mom aside. "Bean soup? Is that all you have?"
"I made a huge pot."
"The kids aren't going to eat bean soup," I told her. They were still in the pizza, chicken nugget, mac-and-cheese phase of their culinary development. "Do you have anything else?"
She gave me an unflinching stare. "I have bean soup."
We sat at the table, and Mom ladled a skimpy spoonful of soup into each large bowl. The tiny portion settled at the base of the bowl in a shallow puddle. Frank and I stared down at our soup in dismay, like a couple of Dickensian orphans.
"Mom, could we have a little more?"
"No," she answered peremptorily.
"No?"
"We have to ration. I heard on the news, it's going to be a bad storm."
The snow kept falling through the night, and the house was all done up for Christmas, so we relaxed into a festive holiday mood. Mom had a stack of Academy screeners, all DVDs of current-release films. We settled back on the couch and watched _Charlotte's Web_ and _Happy Feet_.
The next morning, we woke up to an arctic wonderland. The snow was still falling steadily, the landscape was a pure, endless desert of white. A sense of magic filled the air.
While we didn't know it, this was the high point of our holiday. The warm, cozy family feeling was brief and evanescent. As the day wore on, and the snow kept falling, we started coming to grips with a grim truth: we were not leaving this house anytime soon.
This wouldn't have been so bad if Mom's four-hour grace period of charming hospitality hadn't long since expired. She was tired of us already, and she wasn't shy about letting us know it. She was all Mrs. Hyde from here on in, and we couldn't escape her.
Also, the rationing continued. Mom had gotten it into her head that the food would run out soon if we didn't take Draconian measures. "We're all going to have to make sacrifices if we want to survive the week. So kids, one slice of bread each. That's going to have to hold you until dinner."
It was absurd, because there was plenty of food in the house—the pantry was stocked to the ceiling—but Mom had shifted into survival mode, and she took it very, very seriously.
By the end of the day we were all starving again. To make matters worse, she had nothing in the house that children might like to eat. There was no cheese, or frozen pizza, or pasta. There wasn't even any milk. She did have ice cream, the family weakness, but even that she parceled out stingily: "One scoop!"
There was plenty of "adult" nourishment at hand, though, particularly of the alcoholic variety. While the food was out of bounds, it was perfectly okay to dip into the liquor supply. From the first night on, Mom was mixing eggnog like a demon, with very little egg and great sloshes of rum. Little food, lots of booze—a deadly combination.
That night Mom put another one of her Academy screeners into the DVD player: Mel Gibson's _Apocalypto_. I was a little concerned. "Mom, do you think this is appropriate for children? I heard it was pretty violent." Mom was dismissive: "Oh please. It's fine."
We gathered in the living room and watched as the ancient Mayans beheaded one another, conducted human sacrifices, cut out still-beating hearts, and practiced two hours' worth of bloodthirsty mayhem. Frank and Frankie, being manly men, enjoyed all the savagery. Arin, seven years old, sat through it politely, although she buried her head in my lap most of the time. When it was over, she enthused, "That was great, Ganny!" She had terrible nightmares all night.
The next day, the snow finally stopped coming down, but we were buried under three-plus feet of it. The incredibly long driveway would be impossible to shovel. Frank wanted to call in a plow to clear it. Suppose there was an emergency, and we had to get the kids to the hospital?
Mom just laughed. "They're not going to come out here to clear our driveway. Don't you think they have better things to do? Look at him, he's so worried about the kids, he's such a big dad. Ooh, what do you think is gonna happen?" she teased, her voice dripping with sarcasm. "The kids'll be fine."
I think the combination of the lethal drinks, the lack of food, and the hopelessness of the situation was responsible for my disastrous swerve into drunken sentimentality. We had just watched a screener of _The Departed_ , the kids were in bed having sweet dreams of Mob violence, and Mom and I were both staring numbly at the fire.
"You know, Mom," I said, "I wish I knew you better."
She groaned. "Don't start that again."
"It's true. I don't feel close to you. And we _should_ be close. You need someone you can trust."
"I work in Los Angeles. I don't trust anyone."
"I don't blame you for keeping everybody at a distance. You've been wounded in life. I know how tough your childhood must have been..."
I had inadvertently tripped a land mine. Mom exploded. "You have no fucking idea about my childhood! And don't ever assume that you do! My childhood is mine. I lived through it, not you! And I don't want to hear about it ever again!"
I waited for the dust to settle before I continued. "I'm sorry. I just want to know you. You're my mom, and I don't know you at all. And you don't know me."
"You're my daughter. What else do I need to know?"
"Exactly—you think I'm nothing. You spend your whole life reaching out everywhere, into the past, into the future, into the farthest reaches of the fucking universe, but you never reach out to me. Why? Why can't we be closer? Why do you have to live so far away from us?"
"Because this is my home. I love it here. Nobody bothers me." She added pointedly: "Except on holidays."
"But think, if you moved to Greenwich, I could see you all the time, and you could be a real grandmother to the kids. You can join my workshop, and then we can go together every week and work on scenes, and act together."
"Sounds thrilling," she dead-panned.
"But wouldn't it be great?"
"No," she said, leveling a cold eye at me. "No." Discussion over.
The driveway was finally cleared, but we still had two days left before our flight home. That was two days too many for me. I got on the phone and desperately tried to find an earlier flight. My mantra was "Forget the cost; just get me out of here!" Unfortunately, everything was booked. We would have to wait it out till the bitter end.
The night before we were going home, I said to Mom, "See you in the morning." She gave me a noncommittal look and went to bed. Now, we had to leave at 6:00 A.M. to catch our flight, and I knew Mom was a late sleeper, so in a way, this was a test. If she set her alarm clock and woke up early to see us off, I'd know that she really cared about us. Everything would be back to normal; all would be well.
I waited by the door until 6:15. I stalled, pretended I'd forgotten my comb, had to use the bathroom. I gave her every opportunity. No Mom. She was sleeping in.
We piled into the rental car and drove away. It recalled to me the time Mom came down to San Diego and rescued me from Dr. Jeffrey, and how she and I roared off in a cloud of dust, leaving the horror behind and looking forward to a new life. We couldn't peel out on this icy driveway, but the mood was similar, except now I was escaping her.
We landed in Hartford, and on the drive back to Greenwich, I remember saying to Frank, with an eerie, self-possessed calm, "That's it. It's over. I will never visit her again. Never, never, never."
After a quiet moment, he asked, "Never?"
"Never."
**• • •**
**_THE_** _Witch of the West Is Dead_ has nothing to do with _The Wizard of Oz_. It's the title of a popular Japanese children's novel, written by Kaho Nashiki, a quiet, meditative tale about a teenage girl who goes off to the country one summer to live with her seventy-five-year-old grandmother. They were planning to make the film version in Japan, and the filmmakers were after my mother to play "Granny"—the British-born character who speaks Japanese—but Mom wasn't interested.
When she mentioned the part to me in December of 2006 (just before our Santa Fe Christmas), I was definitely interested. Hell, if she wasn't going to take it, I would! So I got in touch with the casting director and flew out to Los Angeles to audition for the producers and the director, Shunichi Nagasaki. Leaving nothing to chance, I walked into the room dressed as "Granny," wearing a white wig and an old dress. I did the audition, and they took me out afterward to wine and dine me.
A month later they called from Tokyo to offer me the role.
Shooting of _The Witch of the West Is Dead_ started in spring of 2007, in a village called Kiyosato, high up in the Japan Alps. The company stayed at a rustic lodge located on the site of a natural hot spring. Every morning, I would wake up and bathe in the _rotenburo_ , the open-air hot spring bath. The powerful rotten-egg smell of sulfur would fill the air, but the steaming water would loosen up my muscles and get me ready for the day.
The set was down the street, on the other side of a forest patch. Most of the crew would take a shuttle down to the set, but I liked to walk through the forest, wearing the grandmother's outfit; it helped me get into character, much as I had walked the length of Inishmaan to prepare for _Riders to the Sea_. The grandmother was a tough role to get a handle on at first, until one day, when I found leather laces for her boots, and that's the moment I suddenly knew who she was.
The days on the shoot were long and unrelenting. The director, I discovered, was not only king in Japan; he was dictator. Mr. Nagasaki's decisions were unilateral, and they were final. He did not have time for questions or discussions. If he didn't like a take, he would say, "One more time," but he wouldn't give a reason why.
I would ask, "What was wrong? You didn't like what I did? Should I try something else?"
He would reply, "One more time."
So I would do another take, in a totally different way, and he would say, "One more time." And so it went. Since there were no unions, we would work twelve- to fourteen-hour days in the freezing Alps. Sometimes we'd go to 3:00 A.M., and then have to get up three hours later for the next day's shoot.
After a long day on the set we would all go back to the lodge and relax in the _rotenburo_ , the hot spring bath—all of us, actors, producers, crew. Drinking sake, discussing the shoot—all completely naked. For me this was just a little weird. I was used to communal bathing since childhood, and there was obviously nothing sexual going on; we were all soaking our muscles and relaxing in an atmosphere of complete professionalism. Still, to see the producers and the director, our symbols of total authority, climbing naked into the bath with us was an image difficult to reconcile with our daytime reality. Like mollusks, they had ventured out of their hard shells and exposed their soft, fleshy bodies to the night air. Tomorrow they would be armored and imperious again.
My trials on the set went on for two weeks. Then something very interesting happened: Nagasaki and I connected. I don't know how or why, but it was like telepathy; I became one with him, and I knew exactly what he was thinking. I never had to ask him what he wanted in a scene; it was already clear to me. After that, filming became a breeze: everything was done in one take, and the shooting schedule was cut in half.
Something else happened that made the shooting of _The Witch of the West Is Dead_ a truly transformative experience for me. I became reacquainted with _natto_. _Natto_ is a Japanese dish consisting of fermented soybeans. You eat it with steamed rice, a raw egg on top, and chopped-up seaweed and scallions. Mix it up in a bowl, bring the bowl to your mouth, and slurp it up. It's fantastic. One of my favorite foods.
I could never have eaten _natto_ in Connecticut because the preparation of it would have stunk up the house for days afterward. My family would have been so nauseated, it would have made them throw up. So they refused to let me make it. It _is_ a nasty smell, but I grew up with it, so I love it.
They served it three times a day on the _Witch_ set. I was in heaven.
However, there were unintended consequences. My body had grown unaccustomed to a constant diet of fermented beans and I started collecting an excess of gas, and I had to get rid of it through the only avenue open to me.
Now, I should acknowledge here that as the only American in the cast, and the daughter of a famous star, there was an unspoken tension between me and the rest of the crew. The actors were naturally suspicious of me, figuring me for a spoiled Hollywood brat, and many of them gave me the cold shoulder. Plus, I knew the director and the producers had really wanted my mother for the role.
So I had something to prove. I tried to be as friendly and down-to-earth as possible. I always ate with everyone else at the long mess hall tables, and took whatever the cook served up. The teenage actress who played my granddaughter, Mayu Takahashi, was already well known in Japan, and she had a separate dining space, surrounded by bodyguards. I think everyone expected the same of me, but I knew that this was a trap. I wanted them to think of me as one of the guys.
That's where the _natto_ came in. The bean gas was constantly building up in my intestines, to the point where my stomach was hurting from the pressure, but I refused to be impolite on the set. I didn't want to be considered a truly ugly American. So I would hold myself together for one take, and then excuse myself. I'd head out into the forest, and when I was safely alone, I'd let it rip. And it was, take my word for it, thunderous. The birds would scatter. Then I'd go back to the set for another take, and more often than not, the gas well would refill—it was like my intestines were hooked up to a bicycle pump—and I'd have to excuse myself again. It was cold up in those mountains, so each time, I would have to put on my jacket and boots just to fart.
For some reason I never made the connection between the beans and the gas. I thought my gas had something to do with the altitude, or the stress of performing. Besides, I found the _natto_ comforting, so I kept eating it like crazy—and kept producing incredible, otherworldly gas. It reminded me of Mr. Gerard of Charter Towers and his propulsive farting. I just couldn't stop.
Finally, I got tired of hiding it. I could barely focus on my role with my concentration divided between mind and body, so after two weeks of stealth bombing, I said, "To hell with politeness," and let go with a blast in front of everyone. "Excuse me," I said, both sheepish and defiant.
Well, talk about an icebreaker. Everyone laughed, and a sense of relaxation flooded the set. They were charmed and delighted by my evident humanity. "So that's what you've been doing all this time!" Nobody had the slightest clue of what I'd been up to in the forest.
Except the sound man. "Don't worry," he said to me quietly on the side, "I've already heard everything." It seems that whenever I went out in the woods to fart, I'd forgotten to turn off my body mike. So he had been privy to every rude combustible sound. More than that, I would usually accompany my farts with a heartfelt commentary, often groaning in English, "Fu-u-u-u-ck!" Then, after I was spent, I would say, "Oh, yeah!" The poor guy had been listening to my gastrointestinal struggles for the past two weeks. "Why didn't you tell me?" I asked him. He shrugged. Wouldn't have been polite.
Anyway, now that I was letting my wind go free, it endeared me to everyone. I was clearly no diva; you couldn't get any more down-to-earth than this. I became known on the set as "The Farting Grandmother." There was one very sweet scene I had with Mayu as the teenage girl; we were both in bed sharing a tender moment. After every take, I would lift up the bedcovers and let one go—then go back to being sweet again. I remember one of the producers thoughtfully suggesting, "Let's just get some gaffer's tape and stick it across her butthole."
In addition to providing an endless source of amusement, my indisposition gave everyone else permission to fart, too. It became a very friendly, collegial atmosphere on that set.
One day, I finally figured out that it was the _natto_. As soon as I stopped eating it, the gas receded, and I could lead a normal life again. Still, I'd made an indelible mark on my colleagues; right to the end of the shoot, I would get the same laughing refrain at mealtime: "No more _natto_!"
_The Witch of the West Is Dead_ was a huge box-office hit in Japan, and went on to win scads of awards. Beyond that, it was a revelatory experience for me. I was deeply proud of my work on that film, and working in Japan gave me my first opportunity as an adult to explore my roots, and gain an awareness of where I'd come from. I'd never realized how deeply the Japanese culture, the syntax of humility and denial and stoic acceptance, had become a part of my DNA, and informed my every thought and action, for both good and ill. Being there again, as an adult, made me understand who I was. I had come full circle to the world of my childhood, and found myself still there.
**• • •**
**IN** the fall of 2007, Dale Olson called from L.A. He was going to be passing through New York in about a week and wanted to meet with me. Dale Olson was one of the most famous publicists in Hollywood. Before he passed away in August 2012, he represented the likes of Clint Eastwood, Steven Spielberg, Rock Hudson, Marilyn Monroe, Gene Kelly, Joan Crawford—and Shirley MacLaine. He knew where all the bodies were buried in Tinseltown, but he wasn't one for spilling confidences and spreading gossip. He was considered by all a class act.
Dale had recently had a falling-out with my mother, which was hardly unusual. Mom was not an easy friend to maintain. She was always having little snits and kerfuffles with her buddies and business partners. I assumed that Dale wanted a sympathetic ear for his side of the story.
I had known Dale since I was a kid. He always had a smiling face and a lot of positive energy. He was an amazing cook—I especially loved his cauliflower dish, which was simply the best in the world. I always felt safe around Dale, because he made me feel that I mattered. His partner, Gene, and I also had a very special relationship, so I was really looking forward to seeing them both, and I couldn't wait for them to meet my kids.
He and Gene arrived at the house in time for lunch. I made a nice meal, and we all sat out on the patio catching up on old times, with Frank and the kids joining us. I told Dale about the Japan shoot, and how great it had gone. He seemed thrilled for me.
At some point I mentioned how disappointed I was about not getting to act with Mom in _Closing the Ring_. I told him about the Canadian immigration policy, how a new law had recently been passed limiting the number of Americans in the cast, and how Mom had done everything she could short of switching her citizenship, to no avail.
Dale was quiet a moment. "Well, that's what I wanted to talk to you about," he finally said. "It wasn't Immigration."
"What?"
"It wasn't Immigration that stopped you from doing that movie. You could have gotten around that problem easily enough."
"What do you mean?" I knew what he meant the moment he said it, but I convinced myself that I didn't.
"It was your mother."
I felt the ground drop away. "What do you mean?" I re-peated.
"Your mother didn't want you to do the movie," Dale replied flatly. "She sabotaged you."
"What?" I immediately shifted into denial. "That's not true."
"It is true."
"It's _not_. You don't know."
"Yes, I _do_ know." And of course he did. Dale was in a position to know.
"But..." It was a panic moment. I raced back through my memory, trying to retrieve any bit of evidence that might disprove Dale's accusation. "No," I countered, "that doesn't make sense. She's the one who got me the audition in the first place. It was her idea."
"She got you the audition because she never thought you would get the part." He spoke evenly, laying out the information with a dry matter-of-factness. "You were almost fifty, Sachi. Why would they have let you play her daughter?"
"But I _am_ her daughter."
Dale shook his head. "Not on film. She wanted someone else."
My heart started pounding. Frankie and Arin were still sitting at the table, finishing their lunch. I told them to go inside and play.
I tried to think my way through this. What Dale claimed was certainly plausible, but I wasn't giving up just yet. There were too many instances of Mom being supportive of me. She loved my screen test. She was there when I got my eyes done. She was going to turn Canadian. She made all kinds of phone calls to important people. And when Frank offered to help, Mom stopped him because she was handling it—
Wait a minute.
If she was handling it, why had it fallen apart? Why did she stop Frank from helping? And why did she keep asking me, "What if you don't get the part?"
Dale could see that the truth was sinking in. "I felt so bad about that movie, Sachi. I knew how much it meant to you. It wasn't right what she did, but—that's your mother. That's what she does. Remember Marion Ross in _The Evening Star_?"
Marion Ross? I did remember. _The Evening Star_ , from 1996, was the sequel to _Terms of Endearment_. Mom reprised her role as Aurora, and Marion Ross played her housekeeper, Rosie. The picture was not well received, but Marion, who had played Mrs. Cunningham on the TV series _Happy Days_ , was nominated for a Golden Globe as Best Supporting Actress.
"She should have won the Academy Award," Dale observed. "She had a killer scene in that movie. Big dramatic moment, and she was spectacular. Stole the film. Everyone was buzzing about it in the early previews. No way she wouldn't have won an Oscar. No way."
"I don't remember that scene."
"You never saw it. It was cut."
"Cut? Why?" But I knew why.
" _Evening Star_ was your mom's film," Dale said. "She didn't want anybody stealing her thunder. So she made them take the scene out."
No. I didn't believe that. There must have been some other reason. Good scenes get dropped from films all the time, because they're incidental, or they break up the rhythm, or the movie's just too damn long. Mom couldn't be that ruthless. Could she?
Dale shook his head ruefully. "That scene alone would have put Marion Ross on the map. She was a terrific actress who just needed a break."
Yes, a break. That's all anyone needs in this business. Mom got her own break in _The Pajama Game_ , when Carol Haney went down with a broken ankle, so she of all people should have been generous in spirit to others with the same dream.
Yet Mom never considered that a break. That was karma. She deserved it.
And others didn't.
"So you're not the only one," Dale said. "It's part of the business."
I still couldn't accept it. I sat there in silence, my mind spinning.
"Shirley," he finally offered, "is a competitor. She didn't want you to get ahead of her. Plain and simple."
"But I'm her daughter. She's supposed to be proud of me."
"Your mother is unique."
I was out of words. I stared at the rest of my lunch. I wasn't going to eat it now.
"Look," Dale said, trying to put it in perspective, "she's a star. She's up there on a pedestal, and she doesn't want to get knocked off. By you, Marion Ross, anybody. That's how she plays ball. Don't take it personally."
My mind drifted back over my stillborn career. All those opportunities, those roles that didn't pan out—was it all because of her? Had she been working against me behind the scenes from the very beginning? Every time I had a good audition, I would call Mom to share the happy news. Did she then go out of her way to sabotage me? Was it like one of those fiendish Hitchcock films, where the kindly benefactor turns out to be the murderous spy?
That was just too awful. I couldn't go there. Dale had to be wrong. "Please be wrong," I begged him. "Please..."
**• • •**
**FOR** the next six months I was in a sort of limbo. I knew Dale's story was true, but at the same time I couldn't believe it. I couldn't stand the thought of it—and I couldn't confront Mom, because Dale had made me promise to keep quiet. He knew that Mom was sharp enough to ferret out a traitor.
As time passed, I began to doubt the story more and more. I would call Dale in L.A. every now and then: "Are you absolutely sure that happened?" He was always sure, absolutely.
I wrote my mother a letter. I didn't have the nerve or the debating skills to go head to head with her, and I wanted to make sure I expressed every raw sentiment I was feeling.
I told her that I'd found out in Toronto what had really happened on _Closing the Ring_ (careful to leave Dale's name out of it). "The talk was that _you_ had nixed me doing the part, and that Immigration wasn't _that_ big of an issue. You pretended to help me but behind the scenes you were sabotaging me...have long known that you never wanted me to be in the business. Acting is the life I have chosen, and I hoped that you would support and encourage that. As a mother myself, I can't understand how you would not want anything but the best for your child. What legacy will you leave? One of a self-centered controller of other people's destinies—or of a loving, supportive parent?" I ended on a note of reserved hope: "I don't know how we can get past all this, but I'm willing to try, if you are. I love you. Sachi."
This was too important a message to trust to the postal service, so I drove down to New York City myself. Mom was staying at a friend's Sutton Place apartment. I hand-delivered the letter to the doorman, and then drove away.
I waited nervously at home for Mom's response. What would she say? Would she deny it? Would she admit it? Would this be the final rupture, or the beginning of a new, stronger relationship between us?
None of the above. I never heard from her. She never responded to me.
But Frank, my husband, got a call from Mom a few days later. She was furious, and denied the Dale Olson version completely. She also had a new explanation for my not getting the part in _Closing the Ring._
"She said the immigration story was a front, because she didn't want you to know the truth," said Frank. "It would have hurt your feelings."
"And so what was the truth?" I asked.
"The truth was, Lord Attenborough didn't think you were very good."
I scoffed. "That's ridiculous. He's the one who cast me! He's the one who wanted me in the film!"
Frank shrugged. "She said he's Lord Attenborough. If he'd wanted you in the film, you would have _been_ in the film. She saw your screen test herself, and she didn't think you were very good, either."
I knew this was bullshit. Since when did Mom worry about protecting my feelings? Everything in Dale Olson's story rang true. And if her version were true, why didn't she tell me herself?
Still, she had managed to plant the one poisonous seed that was sure to take root in my battle-weary mind. Maybe I _wasn't_ good in that screen test. Maybe I wasn't a good actress at all. And maybe that immigration story was just a smokescreen to keep me from knowing the truth.
A year earlier, when the saga was still unfolding, I'd called Jack Gilardi to find out what was happening. He was the first to suggest that my screen test had been a failure. "You know, they didn't like you at all. You weren't very good."
I thought he was nuts at the time. Everyone was raving about the screen test. I dismissed him as a cranky old agent. After the deal fell apart, though, those words came back to haunt me. Then, when we were stuck at Mom's in Santa Fe, I mentioned it to her. "You know, Jack Gilardi said my screen test for _Closing the Ring_ wasn't very good."
"Really?" she said with surprise. "Well, _I_ thought you were great. I was _very_ impressed." Then she gave a little shrug. "But, you know, I'm your mother."
Exactly.
# **Chapter 18**
# Shut Up and Deal
When I was studying with Peggy Feury back in the 1980s, she told me I needed to write my story down. "You have to get this on paper; you have to write a book about your life. It will set you free."
I didn't really take her point at the time. What was so interesting about my life? Yes, my mother was a famous movie star; and I'd traveled the world, from Europe to Australia; and my father was possibly a clone—but it wasn't as if something had actually happened to me.
Now I understand what she meant. All my life I've been in the thrall of those nearest to me: my father, my mother, my lovers. I've lived my life trying to please them. My most intimate relationships have also been my most dysfunctional, and the one that I wanted most to work, the relationship with my mother, is the one that resisted every Sisyphean effort on my part to coax it into some semblance of normality. I did everything I could to bring Mom into my life. I bent over backward, left no stone unturned. The big Hollywood happy ending never happened.
And now that I've written this book, it probably never will.
But that's where my chance for liberation comes in. All this time, in the vain hope of gaining my mother's approval and her love, I've cautiously kept my secrets, my doubts, and my pain to myself. I've held back from expressing my true self, because I was afraid if I went public with my story, Mom might never talk to me again.
But she doesn't really talk to me now, so what do I have to lose? Did she lose by writing her many books, by revealing her unconventional beliefs and promoting her quests for enlightenment? She may have left herself open to ridicule, but she also established, for all to see, the rules by which she lived her life. She made a statement, and it was a strong, independent one.
Maybe she'll see my choice that way. Maybe she'll admire my forthrightness and my determination to own and control my own life. Maybe she'll recognize me as my mother's daughter.
I'm kidding myself, of course. It's so easy to live your life in denial, to ignore those things you know to be true. I did it for too many years, and I fully recognize my own failing there.
Even Mom, who considers herself a clear-eyed truth-teller, will fall back on her defense mechanisms when something threatens her sense of self.
She called me a few years ago, when we were going through one of our friendlier phases: "Hey, Sach, you know who I saw today? Steve the Clone."
"Who?"
"Steve the Clone. He was in the Valley, buying coffee."
I had to take a moment to decide if she was serious or not. I decided she was. She'd dismissed the real story about Dad's swindle and gone back to the romantic one she preferred. "Mom, you know that's not possible. Dad's dead."
"Your _father_ is dead. Paul—yes. I'm talking about Steve. He was in the Valley."
"But there is no Steve. I mean, there _is_ a Steve; there _was_. But there's no Paul. Remember, we talked about all this? None of it's true. None of it happened."
I could hear Mom smiling at me. "Oh, honey, you're so naïve. You don't know what's out there. You just don't know."
Maybe she's right. Case in point: the Mystery of the Missing Blood Type.
Among her other quirks, Mom is a faddist. She's always leaping on the next new trend, and treating it like the one true path to deliverance. In the 1990s she was steered into homeopathic medicine—the cleansing diets, the lemon juice and cayenne pepper, the high colonics—and that became her new religion. I understood her enthusiasm, and I believe in the virtues of homeopathy, but I also believe in medicine. Sometimes you just gotta do whatever it takes.
Mom would have none of that. She was visiting us in Houston when my two-year-old son, Frankie, was sick with roseola. I was about to give him his antibiotic medicine when Mom grabbed it from my hand. "No! No more antibiotics!" She tossed the medicine in the garbage. I had to go back to the doctor and get another prescription.
Around the same time, she became obsessed with blood-type diets. The idea was that whatever blood type you had determined the kind of food you should eat. Type O blood had been handed down over the centuries from prehistoric hunters, so anyone with type O should eat a lot of red meat. Type A-positive, on the other hand, were gatherers, and they should eat vegetables and dairy.
Mom was a type O, and Dad was a type O, which meant that I had to be a type O. Therefore, meat. She gave me one of those newfangled George Foreman grills: it would be steak and burgers for me from now on. I dutifully tried the diet, and didn't lose a pound.
It was years later when I went in for a hospital procedure and discovered that I wasn't type O at all; I was type A-positive. So that's why the diet hadn't worked.
But wait a minute. If Mom was type O and Dad was type O, how could I be type A-positive? It wasn't scientifically possible. Mom must have made a mistake. Dad had to be A-positive, too.
Or was he? I seemed to recall Dad going into surgery in Hawaii, and himself mentioning that he was type O. But he could have been mistaken, and I could be misremembering.
Still, I had gotten to the point where everything about my life was in doubt. I knew damn well that Mom was my real mother—there was no escaping that—but was Dad my real father?
This is just wild speculation on my part, and I know it. I do look like Dad—I see myself in his face—and I feel his spirit inside me, but I think I also bear a resemblance to, say, Yves Montand (or, for that matter, Alfred Hitchcock—now there's a scary thought). The point is, I'll never know for sure. And if he wasn't my father, wouldn't that explain why he was so indifferent to me, why he was so eager to use me as a pawn and sacrifice my security and sense of identity for his own gain?
Traditionally, a Japanese Zen garden, a _karesansui_ , has fifteen stones, so carefully arranged that you can see only fourteen of them at a time; you can never see that fifteenth stone.
That's how it is with people and relationships. There's always that fifteenth stone you can't see, but you know it's there. And when you find it, the fifteenth stone, that secret thing that explains all, you look back and you realize that another stone has disappeared. You can never see a person, or a relationship, or a life, whole. There are too many angles.
So, no, I still don't understand my mother. Every time I find out something new about her, something else gets hidden.
It has taken me this long to realize that I don't _need_ to understand her. She's on her journey, and I'm on mine. Our lives may intersect at crucial points, but there's no reason to expect them to run side by side, on parallel tracks. Mom's spirit bounces all over the universe like a jet-powered pinball, and every now and then it settles beside me for a moment before some visionary impulse shoots it off again. I'm just a stop on the road: she doesn't need me, not at all, and she isn't going to pretend for propriety's sake that she does. She's off fulfilling her destiny.
That doesn't mean I don't feel the absence of her love deeply and keenly. It's just that I understand the situation now. My mother is Shirley MacLaine: a show-business icon; a brilliant, talented, legendary performer. That's who she is, and that's how I have to deal with her. I have to forget that she's my mom, because we can never connect on those terms She never said she was going to take care of me and be there when I needed her. That's something I came up with on my own. We can't blame people for being true to themselves.
In the final scene of _The Apartment_ , Jack Lemmon professes his total love for Mom's character, and she shrugs it off lightly with the famous line "Shut up and deal." That phrase may crystallize Mom's response to human relationships more than Billy Wilder or anyone could have imagined. I don't want to hear about your problems, I don't want to hear about your love, I don't want to be bothered. Just shut up and deal.
I'm trying.
People will often say to me, "Shirley MacLaine is your mother? I love her! You are so lucky!"
They're right. I am. I'm very lucky, and I say that with no heavy-handed sarcasm or postmodern irony. I'm lucky, because I'm here. I survived.
And through it all, I still have my optimism, and my basic faith in people. I really do. I haven't become jaded or bitter. No, I still see the world in a positive, hopeful light. Maybe Dad would have said that's because I'm an idiot. If so, I embrace my idiocy. Mom doesn't mind looking like a fool for the sake of her beliefs, so, hey, neither do I.
Looking back over my life, I see that it's full of providential moments, moments of serendipity and grace. Whenever things seemed desolate, whenever I was poised to capsize, something unexpected always came along to help me out.
I think of Eguchi-san, with her fox gods and her brown underwear, teaching me to be Japanese; my beloved David, teaching me to be American; Peggy Feury, teaching me to be an actress; that kindly old prostitute in Trieste; the Yugoslavian couple in the Zagreb hotel; Shigeko in Honolulu; Margo Tolmer in Sydney; the waitresses at the noodle shop—all of them, angels who alighted on my shoulder at just the right time.
And I think of my two wonderful children, the bright beacons of my future. They fill my life with light, and I cherish and love them unconditionally. What could be luckier than that?
Frankie is basically a man now—at six feet tall, he towers over me—and he's thoughtful and responsible beyond his sixteen years. From early on, he's always showed remarkable empathy for others, and more than once he's given me his strong shoulder to lean upon. Following my lead, he's in boarding school now, but I have no doubt he'll navigate those tricky, complicated waters far better than I did.
Arin and I have a glorious mother-daughter relationship—by which I mean we argue a lot. We fight over boundaries; we scream and shout at each other. I give her hell, and she slams the door in my face. I treasure those insane moments. Some mothers might get frustrated and angry with the day-to-day struggle, but I love it, because it's something I never had with my mom. I didn't get the chance to be surly with her, or annoying, or heartbroken, or any of those cool teenage things.
My little girl is fourteen years old now, and so far we still enjoy each other's company. We like to walk around town, stroll along the beach hand in hand, get ice cream. She goes for vanilla-chocolate swirl. Sometimes she asks me for advice, and sometimes I ask her. Sometimes we laugh at something silly, laugh until we cry. Sometimes we're just totally quiet together.
And every now and then, when there's a big storm coming across the river, we open the window shades and cuddle up in bed and I tell her a story:
"Once upon a time, there was a beautiful princess, who lived on top of the tallest mountain in the world. And she was known as Princess Lightning. Because when she was happy she would laugh a merry laugh, and a great flash of lightning would flood the sky..."
So yes, I'm pretty lucky. I am.
# **ACKNOWLEDGMENTS**
In summing up a life, there are so many people to give thanks for. Many of the guiding lights of my life are gone now, and many others I've lost touch with over the years; but their spirits, and the spirits of all who have lifted me and given me inspiration, are always there, filling my heart with their presence and guiding me along my path.
First of all, I really should thank my mother and father for giving me my existence, for loving me as best they could, and for supplying me with so much good material.
Thanks also to my marvelous acting teacher, Peggy Feury, who so long ago told me to "write that book." She planted the seed.
My school friend Yuki was indispensable in filling in the blanks and giving context to incidents that I understood only incompletely. She was a great help in putting the puzzle together.
A special remembrance to the late Dale Olson, who showed such courage and candor in coming forward and clearing up one of the central mysteries of my life. He and his partner, Gene Harbin, were always sweet and wonderful to me, and I will never forget their kindness.
I'm eternally grateful to Linda Konner, my literary agent, who took the idea for this book and ran with it. Linda's enthusiasm, persistence, and professional savvy were a blessing to me, and I was so lucky to find her. Let me also thank Suzanne Collins and Rosemary Stimola for their help in bringing Linda and me together.
I'm indebted to the people at Gotham Books for their belief in this project and their willingness to take a chance on an unknown writer. Special thanks to my editor, Lauren Marino, and her assistant, Susan Barnes, for guiding me through the publishing process with patience and humor, and giving shape and drive to the narrative. Without them, this book wouldn't be what it is.
The writing of this book coincided with a very difficult time in my life, and I want to thank all those people who rallied around me and gave me the strength to move forward when it would have been so easy to give up. Without the emotional support of these special people, I never could have kept my focus on the positives in life and made it to the finish line.
So thanks to Don Kelly, my suave, charming Irish-Italian pal from the Bronx, who provided a third eye and a helping hand and gave me a newfound confidence whenever I was faltering. He never allowed me to wallow in defeatism, and helped me take responsibility for my own life. I owe him so much.
Thanks also to Jim Noble, my acting mentor and dear friend, who supported my ambitions and stood by me in the toughest times, and who leads always by the example of his own grace and elegance.
Thanks to my fellow actress Joanna Keylock, who provided passion and perspective and always a sense of loving acceptance; her husband, Lee, whose straight-talking common sense saved me from many a headstrong mistake; and Barbara Parker, a great friend, who has never failed to bolster my self-esteem and celebrate my female strength.
Thanks to the very talented playwright Susan Cinoman, for giving me the idea for the one-woman show _Lucky Me_ , which got me started on this revelatory journey. My thanks also to the Irish Arts Center, the Great Neck Arts Center, and the Burgdorf Cultural Center in Maplewood, New Jersey, who all produced readings of the play. It was a great thrill to share my story with an audience, and to feel them connect with me. I'm especially grateful to those audience members who insisted, "You have to turn this into a book!"
The Theatre Artists Workshop in Norwalk, Connecticut, where I worked on my one-woman show, has been a haven and an invaluable resource for me and so many other actors and writers. I consider it in many ways my artistic home, and I cherish the many friendships I made there. I particularly want to thank Barbara Rhoades, who insisted in her no-nonsense way that I get my butt back onstage, and the late Brett Somers, whose salty wit and warmth made every visit to the workshop a joy. I can still hear her voice today, and it never fails to make me smile.
Jean Bernard, my therapist, has been with me for more years than I care to remember. Her insight and calm have always been a steadying rock for me to lean on. My gratitude to her knows no bounds.
About my cowriter, Fred Stroppel, I struggle to find the words. Simply put, without him there would be no book. Not only is he a brilliant, insanely talented writer who sometimes seemed to be living inside my head, but it was also his tireless devotion to this project, along with a bottomless well of patience, persistence, and belief, that made it all happen. Whenever doubt came flooding in, he created a working environment that allowed me to feel safe and free. He has become my brother.
I would also like to thank Fred's wife, Liz. Her utter trust in our partnership allowed me to stay emotionally available throughout the writing process. She is a treasure.
My final thanks go to the two people closest to my heart: my children, Arin and Frankie. They taught me what true love is.
As always I find it more natural to express my deepest feelings in Japanese, so to all of you, and to everyone who has touched my life and given it light, I offer a very humble and grateful _domo arigato_.
Mom holding me
when I was a newborn
Me and Mom on
Malibu Beach
On the set of
_Some Came Running_
Photo shoot at age two
for _Look_ magazine
Dad greeting me at
the Tokyo airport
My friend the cherry tree
My father and me in
front of Tokugawa Shrine
in the Japanese Alps
Eguchi-san, my Japanese
governess, and me
My father, mother,
and me on set
My elegant silk kimono
Yuki watching me knit
at Charters Towers
Miki, Dad, and
me at dinner
Me in my Quantas
stewardess uniform and
the diamond necklace my
mother gave me, with Dad
On the set of
_Manhattan Express_
Me and Mom on
the red carpet
Me and Mom
Me and Mom at a
movie premiere
On the set of _The Witch of the West Is Dead_
| {
"redpajama_set_name": "RedPajamaBook"
} | 4,934 |
Vampire Survivors' First Post-Release Patch Adds A Death Bridge
sg November 25, 2022 in Gaming news - 2 Minutes
Screenshot: poncle
Vampire Survivors, 2022's break-out game—nay—genre, released its complete 1.0 edition just over a month ago, near the end of October. But if you were worried that meant developer poncle was done, put your worry back in its worry-case, close the lid, clip the clasps, and update your edition to the freshly released 1.10 version. Along with a few smaller details, the patch called The Tiny One adds a fascinating new Challenge Stage, and it's a very narrow bridge.
The Tiny Bridge Stage is unlocked by having reached level 80 in Inverse Gallo Tower, and once opened, it provides the most ridiculous challenge: A single lane, in which you have to do some very specific ability tweaking to stand a chance of survival.
Capturing that moment in the regular game when you realize you're hopelessly trapped, and then scrambling to do anything at all to somehow survive it, the entire concept sounds ghastly. And thus, fantastic.
Also added in this post-release update is Seal Power Up. According to the update information:
When Sealing a weapon, it will be automatically added to the list of banished weapons at the start of a run. When Sealing an item, the item will not be removed from the loot table, but will turn into a Gold Coin instead, which means this does not affect standard drop rates.
It comes with 10 ranks, a base price of 10,000, and is gained by "banishing 1o or more items in a single run." Goodness me, Vampire Survivor got more complicated while I was looking out this window.
There are also two new achievements, for which poncle nervously apologizes. "Sorry for the extra achievements ruining your completion rate," he says in his notes. "Most players seem to be in favor of having them so I took the opportunity to add more. The best I could come up with to make them feel less of a "must have" is to tag them as EXTRA achievements."
And there's a new secret too. Shhhhhh.
Vampire Survivor is very firmly in Kotaku's favorite games of 2022 (and yes, you, we know it came out in December 2021, but no one noticed for a few weeks), and it's going to be riding high in a lot of GOTY lists across the internet. Right now you can get it on Steam for just $4.24, or find it included in Game Pass for Xbox. Although be prepared to see nothing but blue crystals when you close your eyes for the next few weeks.
Previous: Idle RPG' Beginner's Guide — Getting Started with Webzen's New Title – TouchArcade
Next: 'Genshin Impact' Version 3.3 Update Release Date Announced, New Trading Card Game Revealed – TouchArcade | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,745 |
Statuia Unității este o statuie care îl reprezintă pe omul de stat indian și unul dintre fondatorii Indiei moderne Sardar Vallabhbhai Patel (1875-1950) din districtul Narmada din Gujarat, India. În prezent, este cea mai înaltă statuie din lume, cu o înălțime de 182 de metri sau de aproximativ patru ori mai înaltă decât Statuia Libertății. Vallabhbhai Patel a fost unul dintre cei mai proeminenți lideri ai mișcării indiene de independență și primul viceprim-ministru al Indiei. Statuia se află pe o insulă îndreptată spre barajul Narmada (numit și barajul Sardar Sarovar) lângă Rajpipla, la 100 de kilometri sud-est de orașul Vadodara.
Monumentul și împrejurimile sale ocupă mai mult de 2 hectare și sunt înconjurate de un lac artificial de 12 km².
A fost construit de firma Larsen & Toubro, care a primit contractul de proiectare, construcție și întreținere în valoare de 420 de milioane dolari în octombrie 2014. Construcția a început la 31 octombrie 2014 și a fost finalizată la jumătatea lunii octombrie 2018. A fost proiectat de sculptorul indian Ram V. Sutar și a fost inaugurat de prim-ministrul indian Narendra Modi la 31 octombrie 2018, la aniversarea a 143 de ani de la nașterea lui Patel.
Popularizarea
Proiectul a fost anunțat la 7 octombrie 2010. Sardar Vallabhbhai Patel Rashtriya Ekta Trust (SVPRET) a fost înființat de guvernul din Gujarat pentru construirea statuii. O campanie numită Mișcarea pentru Statuia Unității a început să susțină construcția statuii. A ajutat la colectarea fierului necesar pentru statuie, cerând fermierilor indieni să doneze unelte agricole uzate. În cele din urmă au fost colectate 5.000 de tone de fier. Deși au fost inițial destinate statuii, s-a decis mai târziu ca fierul colectat să fie folosit în schimb pentru alte părți ale proiectului.
Mișcarea pentru Statuia Unității a organizat o petiție Suraaj ("bună guvernare"" în hindi), semnată de un număr estimat de 20 de milioane de persoane, fiind cea mai mare petiție din lume. Un maraton numit Run for Unity a avut loc la 15 decembrie 2013 în mai multe locuri din întreaga Indie.
Proiectul
Statuia îl reprezintă pe Vallabhbhai Patel, unul din liderii mișcării indiene de independență și primul vicepremier al țării. Aceasta a fost construită pe o insulă pe nume Sadhu Bet, la 3,2 km distanță de și îndreptată spre barajul Narmada. Înălțimea totală a structurii de la baza sa este de 240 m, cu o bază de 58 m, statuia având 182 de m. Este construită cu structură din oțel, beton armat și placări de bronz. La construcția statuii s-au folosit 75.000 de metri cubi de beton, 5.700 tone de structură metalică, 18.500 tone de tije din oțel armat și 22.500 tone din foi de bronz.
Finanțarea
Statuia a fost construită în baza unui model de parteneriat public-privat, majoritatea banilor venind de la guvernul din Gujarat. Guvernul Indiei a alocat pentru proiect 42 de milioane de dolari din bugetul său din 2012 până în 2016. În bugetul pentru perioada 2014-2015, s-au alocat 28 de milioane de dolari americani pentru construirea statuii.
Construcția
Un consorțiu alcătuit din Turner Construction (manager de proiect al Burj Khalifa), Michael Graves and Associates și Meinhardt Group a supravegheat proiectul. Au fost necesare 56 de luni pentru finalizare - 15 luni pentru planificare, 40 de luni pentru construcție și două luni pentru predarea de către consorțiu. Costul total al proiectului a fost estimat la aproximativ 290 de milioane de dolari americani de către guvern. Ofertele pentru prima fază au fost cerute în octombrie 2013 și s-au încheiat în noiembrie 2013.
Narendra Modi, pe atunci ministru-șef al statului Gujarat, a pus piatra de temelie a statuii la 31 octombrie 2013, cea de-a 138-a aniversare a nașterii lui Patel.
Compania indiană de infrastructură Larsen & Toubro a câștigat contractul la 27 octombrie 2014 pentru oferta cea mai mică de 420 de milioane de dolari pentru proiectare, construcție și întreținere. Construcția a început la 31 octombrie 2014. În prima fază a proiectului, 189 de milioane au fost alocați pentru statuia principală, 33 de milioane pentru sala de expoziții și centrul de convenții, 11 milioane pentru podul care leagă memorialul de continent și 92 de milioane pentru întreținerea structurii timp de 15 ani de la finalizarea acesteia. Accenture a oferit programul de informare digitală mass-media. Monumentul este proiectat de Ram V. Sutar.
Statuia a fost terminată în 33 de luni. Fundația a fost turnată în 2013; construcția a fost finalizată la mijlocul lunii octombrie 2018; iar ceremonia inaugurală a avut loc la 31 octombrie 2018, prezidată de premierul Narendra Modi. Statuia a fost descrisă ca un omagiu al culturii inginerești indiene.
Probleme
Triburile locale aparținând tribului Tadvi s-au opus achiziționării de terenuri pentru dezvoltarea infrastructurii turistice în jurul statuii. Ei au beneficiat de compensații în numerar și teren și au primit locuri de muncă. Activiști precum Medha Patkar și Gladson Dungdung s-au opus, de asemenea, proiectului. Ei au susținut că Sadhu Bet a fost numit inițial Varata Bawa Tekri, numit după o divinitate locală, astfel că era un sit de importanță religioasă.
Activiștii din domeniul mediului au scris o scrisoare guvernului central susținând că implementarea proiectului a început fără aprobare din partea Ministerului Mediului. Oamenii din satele Kevadia, Kothi, Waghodia, Limbdi, Navagam și Gora s-au opus construirii statuii și au cerut restituirea drepturilor de proprietate asupra terenului de peste 375 de hectare de terenuri achiziționate mai devreme pentru baraj, precum și formarea unui nou district. Ei s-au opus, de asemenea, formării Autorității de Dezvoltare a Zonelor Kevadia (KADA) și a construirii proiectului de drum pietruit. Guvernul de la Gujarat a acceptat cererile lor.
În momentul în care statul a alocat 28 de milioane de dolari americani din bugetul Uniunii pentru perioada 2014-2015, mai mulți oameni și partide politice au criticat cheltuielile legate de statuie față de alte priorități, cum ar fi proiectelor de siguranță ale femeilor, educație și agricultură. L&T a contractat TQ Art Foundry, o filială a companiei Jiangxi Toqine cu sediul în Nanchang, China, pentru plăcile de bronz ale statuii, chestiune criticată de Indian National Congress, partidul de opoziție din Adunarea Legislativă din Gujarat. Mai târziu, revista Swarajya a clarificat că 9% din valoarea totală a proiectului a provenit din China.
Referințe
Legături externe
Site web oficial
Statui monumentale
Monumente din India
Unității
Sculpturi în aer liber
Sculpturi din 2018 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,447 |
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