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\section{Two questions: What do independence and $E=mc^{2}$ mean?}
In the theory of quantum probability, several versions of \textit
independence} \cite{Indep} have been formulated with interesting results as
generalizations of the bosonic tensor type. My na\"{\i}ve questions here are
\textit{on which physical grounds} they have appeared and \textit{what
physical meanings} they have. At least for the familiar Gaussian cases
(=quasi-free states on bosonic CCR~or fermionic CAR), my partial answers in
the context of relativistic QFT will be given as follows:
1) \textit{\textbf{Emergence of independence}}\ through the \textit{\textbf
asymptotic condition }} $\varphi _{H}(x)\underset{x^{0}=t\rightarrow \mp
\infty }{\rightarrow }\phi ^{in/out}(x)$:
From \textit{non-independent} interacting Heisenberg fields $\varphi _{H}$,
the \textit{asymptotic fields\textbf{\ }}$\phi ^{as}=\phi ^{in/out}$ an
\textit{\ asymptotic states} generated by $\phi ^{as}$ from the vacuum
\textit{satisfying the \textbf{independence}} arise through the \textit
asymptotic condition }\cite{LSZ}, $\varphi _{H}(x)\underset
x^{0}=t\rightarrow \mp \infty }{\rightarrow }\phi ^{in/out}(x)$, which can
be interpreted as a sort of \textquotedblleft central
limit\textquotedblright\ theorem. The conceptual meaning of this
\textquotedblleft central limit\textquotedblright\ theorem\ can be placed in
the \textquotedblleft \textit{\textbf{Micro-Macro Duality}
\textquotedblright \textit{\textbf{\ }}\cite{MicMac}\textit{\textbf{\ }}in
QFT as follows:
\begin{equation*}
\begin{array}{ccc}
\text{Micro} & & \text{Macro} \\
\varphi _{H}\text{ {\small : }generic} &
\begin{array}{c}
\text{asymp.cond.} \\
\rightleftarrows \\
\text{{\small GLZ expansion of} }\varphi _{H}\text{ {\small in} }\phi ^{as
\end{array}
& \phi ^{as}\text{{\small : }universal} \\
\begin{array}{c}
{\small (\square +m}^{2}{\small )\varphi }_{H}{\small =J}_{H} \\
\Longleftrightarrow {\small \varphi }_{H}{\small =\Delta }_{ret}{\small \ast
J}_{H}{\small +\phi }^{in
\end{array}
& &
\begin{array}{c}
p^{2}=m^{2}\Longleftrightarrow \\
{\small (\square +m}^{2}{\small )\phi }^{as}{\small =0
\end{array
\end{array
\end{equation*}
(NB: There exists another local-net version of \textquotedblleft statistical
independence\textquotedblright\ based on the so-called \textit{nuclearity
condition} \cite{Nucl} in Algebraic QFT, which should not be confused with
the present version.)
2) \textquotedblleft Units\textquotedblright\ of \textit{independence}
identified with \textit{particles} specified by Einstein's famous formula
E=mc^{2}$ ($\Longrightarrow $ Sec.2):
It is worth noting that this famous formula $E=mc^{2}$ is meaningful only
for asymptotic fields/ states as the \textit{on-shell condition}
p^{2}=p_{\mu }p^{\mu }=m^{2}$ to extract\textit{\textbf{\ }}particles as
\textit{independent = free = non-interacting\textbf{\ }}entities from the
interacting Heisenberg fields; in contrast, the latter do not satisfy this
formula because of the interactions. The former, asymptotic fields and
states, serve as the \textit{vocabulary for describing state changes\textbf
\ }}taking place in the scattering processes in such a form as the state
transitions described by the S-matrix $\langle \beta ,out|\alpha ,in\rangle
=\langle \beta |S|\alpha \rangle $ from an \textit{asymptotic incoming stat
\textbf{\ }}(\textit{in-state}, for short)\textit{\textbf{\ }}$|\alpha
,in\rangle $ to an \textit{outgoing state} (\textit{out-state}, for short)
|\beta ,out\rangle $. In the absence of interactions, however, the \textit
on-shell\textbf{\ }}asymptotic fields $\phi ^{as}$ can\textit{\textbf{not}}
by themselves ignite scattering processes, which necessitates the use of
\textit{off-shell interacting Heisenberg fields\textbf{\ }}$\varphi _{H}$.
3) The logical basis of the \textit{asymptotic condition} is to be found in
the \textit{cluster property }\cite{StrWi}:
\begin{equation*}
\langle \Omega |A\alpha _{\vec{x}}(B)\Omega \rangle \underset{\vec{x
\rightarrow \infty }{\rightarrow }\langle \Omega |A\Omega \rangle \langle
\Omega |B\Omega \rangle ,
\end{equation*
following from the ergodicity valid for a unique vacuum vector $\Omega $
invariant under spacetime translations $U(x)$ and from the assumption of the
local commutativity ($\Longrightarrow $ Sec.3). In this context, asymptotic
fields $\phi ^{as}$~can be understood as quantities to\textit{\ }materialize
\textit{kinematically} this factorization (= independence)\ of correlations
without taking limit and they can be decomposed into creation and
annihilation operators $a(\vec{p}).a^{\ast }(\vec{q})$ which represent
\textit{infinite number of conserved quantities}.
4)\ \textit{\textbf{Universality}} of \textquotedblleft central
limit\textquotedblright\ due to Haag-GLZ expansion ($\Longrightarrow $
Sec.5):\
From the asymptotic condition, the Yang-Feldman equation $\varphi
_{H}=\Delta _{ret}\ast J_{H}+\phi ^{in}$ can be derived with the Heisenberg
source current $J_{H}\ $formally defined by $J_{H}=(\square +m^{2})\varphi
_{H}$ \cite{Bog}. The analogue of the \textquotedblleft Fock
expansion\textquotedblright\ \cite{Oba} in WNA can be found in the Haag-GLZ
expansion \cite{GLZ, IO89},
\begin{equation*}
SA=:(\omega _{0}\otimes id)(T(A\otimes 1)\exp (iJ_{H}\otimes \phi ^{in})):,
\end{equation*
where $\omega _{0}=\langle \Omega |\cdots \Omega \rangle $ is the vacuum
state and $S=:(\omega _{0}\otimes id)(T(\exp (iJ_{H}\otimes \phi ^{as})):$
is the \textit{S-matrix}. By this formula, the Heisenberg observables $A$
depending on $\varphi _{H}$ can be expressed in terms of the asymptotic
fields $\phi ^{as}$.
\section{What does $E=mc^{2}$ mean?}
While \textit{Einstein's famous equality\textbf{\ }}$E=mc^{2}$ between
energy and mass has been regarded as one of the most fundamental
consequences of the relativity theory, however, its actual content is simply
the \textquotedblleft \textit{\textbf{on-shell condition}
\textquotedblright\ to pick up \textit{1-particle modes}, meaningful only
for the \textit{independent = free = non-interacting\textbf{\ }}asymptotic
fields/states. In fact, taking $m$ as \textquotedblleft moving
mass\textquotedblright\ $m=\dfrac{m_{0}}{\sqrt{1-v^{2}/c^{2}}}$, we have
\begin{eqnarray*}
&&E=mc^{2}=\dfrac{m_{0}}{\sqrt{1-v^{2}/c^{2}}}c^{2}\text{ \ } \\
&\Longrightarrow &\text{ \ }(m_{0}c)^{2}=(\frac{E}{c})^{2}(1-v^{2}/c^{2})=
\frac{E}{c})^{2}-\left( \dfrac{m_{0}}{\sqrt{1-v^{2}/c^{2}}}\vec{v}\right)
^{2}=(\frac{E}{c})^{2}-\left( \vec{p}\right) ^{2} \\
&\Longrightarrow &p^{2}=p_{\mu }p^{\mu }=(m_{0}c)^{2}
\end{eqnarray*
where $\dfrac{m_{0}}{\sqrt{1-v^{2}/c^{2}}}\vec{v}=:\vec{p}$ is the
relativistic 3-momentum and $p^{\mu }=(\dfrac{E}{c},\vec{p})$ is the
4-mementum. The meaning of the above equality $p^{2}=p_{\mu }p^{\mu }=
\dfrac{E}{c})^{2}-\left( \vec{p}\right) ^{2}=(m_{0}c)^{2}$ can be understood
as follows:
i) It is just the \textit{mass-shell\textbf{\ }}(or, \textit{on-shell}
\textit{\textbf{\ }condition} to characterize a mass hyperboloid in the $p
-space of 4-momenta $p_{\mu }\in \mathbb{\hat{R}}^{4}$ carried by the free
1-particle states with a rest mass $m_{0}$. The spacetime geometry inherent
to the special relativity is controlled by the Poincar\'{e} group $\mathcal{
}_{+}^{\uparrow }=\mathbb{R}^{4}\rtimes L_{+}^{\uparrow }$ (or, its
universal covering $\widetilde{\mathcal{P}_{+}^{\uparrow }}=\mathbb{R
^{4}\rtimes SL(2,\mathbb{C})$) defined by the semi-direct product of
spacetime translation group $\mathbb{R}^{4}$ and the orthochronous proper
Lorentz group $L_{+}^{\uparrow }:=\{\Lambda =(\Lambda _{\nu }^{\mu });$
\Lambda x\cdot \Lambda y=x\cdot y,\Lambda _{0}^{0}>0,\det (\Lambda )=+1\}$
consisting of homogeneous Lorentz transformations $\Lambda =(\Lambda _{\nu
}^{\mu })\in SO(1,3)$ leaving the Minkowski metric $\eta (x,y):=x\cdot
y=x^{0}y^{0}-\vec{x}\cdot \vec{y}$ invariant, $\Lambda ^{T}\eta \Lambda
=\eta $, without changing the time direction $\Lambda _{0}^{0}>0$. In
Wigner's construction \cite{Bog} of irreducible unitary representations of
\mathcal{P}_{+}^{\uparrow }$ or $\widetilde{\mathcal{P}_{+}^{\uparrow }}$,
four orbit families, $p^{2}\overset{>}{\underset{<}{=}}0$ and $p_{\mu }=0$,
appear: $p^{2}=m_{0}^{2}>0$ corresponds to a massive particle with a rest
mass $m_{0}$, $p^{2}=0$ to massless particles, $p^{2}<0$ to (unphysical)
\textquotedblleft tachyons\textquotedblright\ (with an imaginary mass) and
the last one to the vacuum, each of which is induced from one of the
corresponding \textquotedblleft little groups\textquotedblright\ (
SU(2),E(2),SU(1,1)$ and $L_{+}^{\uparrow }$).
ii)\ Through the \textquotedblleft first quantization\textquotedblright\
p_{\mu }\rightarrow i\hbar \partial _{\mu }=i\hbar (\dfrac{1}{c}\dfrac
\partial }{\partial t},\vec{\nabla})$, the \textit{Klein-Gordon equation}
[\hbar ^{2}\partial _{\mu }\partial ^{\mu }+(m_{0}c)^{2}]\phi (x)=0~
describes a \textit{free scalar field}~$\phi (x)$~with rest mass $m_{0}$.
iii) The existence of \textit{positive/ negative energy} solutions $E=\pm
\sqrt{\left( \vec{p}c\right) ^{2}+(m_{0}c^{2})^{2}}$ of $(\frac{E}{c
)^{2}-\left( \vec{p}\right) ^{2}=m_{0}^{2}c^{2}$ is related with creation
and annihilation operators, particle-antiparticle pairs, time reversal
\textit{T\textbf{\ }}and \textit{PCT invariance}.
Thus, the famous equivalence $E=mc^{2}$ between energy $E$ and mass $m$
gives only \textit{partial} information for dynamical descriptions of
relativistic quantum fields, with \textit{off-shell\textbf{\ }}apects being
neglected in spite of their vital importance for non-trivial scattering
processes, particle decays and productions, etc., etc.
\section{Free= \textit{\textbf{independent}} vs. interacting= \textit
\textbf{non-independent}}}
A free quantum field $\phi (x)$ as the quantized solution of Klein-Gordon
equation $(\square +m^{2})\phi =0$ describes \textquotedblleft particle
pictures\textquotedblright\ in terms of creation and annihilation operators
a(f),a^{\ast }(f)$ defined as follows:
\begin{eqnarray*}
\phi (x) &=&\int \dfrac{d^{3}p}{\sqrt{(2\pi )^{3}2\omega _{\vec{p}}}}(a(\vec
p})\exp (-ip_{\mu }x^{\mu })+h.c.), \\
a(f) &:&=i\int \overline{f(x)}\overleftrightarrow{\partial _{0}}\phi
(x)d^{3}x=\int \overline{\tilde{f}(\vec{p})}a(\vec{p})d^{3}p, \\
a^{\ast }(f) &:&=i\int \phi (x)\overleftrightarrow{\partial _{0}
f(x)d^{3}x=\int a^{\ast }(\vec{p})\tilde{f}(\vec{p})d^{3}p=[a(f)]^{\ast } \\
\lbrack a(f),a^{\ast }(g)] &=&\int \overline{\tilde{f}(\vec{p})}\tilde{g}
\vec{p})d^{3}p=\langle \tilde{f},\tilde{g}\rangle , \\
\lbrack \phi (x),\phi (y)] &=&\int \dfrac{d^{4}p}{(2\pi )^{3}}\varepsilon
(p^{0})\delta (p^{2}-m^{2})\exp (-ip(x-y))=:i\Delta (x-y;m^{2}),
\end{eqnarray*
with $\omega _{\vec{p}}:=\sqrt{\vec{p}^{2}+m^{2}}$ in the \textquotedblleft
natural unit system\textquotedblright\ with $\hbar =c=1$ (rest mass $m_{0}$
is denoted by $m$, henceforth).
Although free quantum fields $\phi (x)$ with $a^{\ast }(\vec{p}),a(\vec{p})$
are customarily believed to be sufficient entities for describing \textit
wave-particle dualism\textbf{\ }}inherent in elementary particles, the
perpetual creation and annihilation processes of particles require \textit
interactions} among elementary particles, which is not consistent with the
linearity of free field equations. Concerning this point, the famous theorem
of Haag \cite{StrWi, Bog} has been taken as a kind of \textquotedblleft
no-go theorem\textquotedblright\ for the theoretical description of the
interactions: \textit{Haag theorem}: Poincar\'{e} (or even,
Galilei)-covariant quantum fields related to free fields by a unitary
transformation are only free fields. Owing to this theorem, it is
meaningless to formulate the interacting Heisenberg fields by means of a
unitary transformation of free fields as is commonly done in perturbative
approaches. This is in sharp contrast to the qunatum systems with \textit
finite} degrees of freedom.
On the other hand, to describe relativistic scattering processes of
elementary particles in a satisfactory way, we need inevitably the following
ingredients: Poincar\'{e}-covariant quantum fields/ their interactions/ free
asymptotic fields and states. Here, free fields are indispensable as the
vocabulary for the description of scattering processes, where an initial
state with incoming free particles is changed into a final one with outgoing
particles.
Giving up the idea to connect directly Heisenberg and asymptotic fields (as
is forbidden by the Haag theorem), we consider the mutual relations betwee
\textit{\textbf{\ }two free asymptotic fields},\textit{\textbf{\ }}$\phi
^{in}(x)$ and $\phi ^{out}(x)$ in terms of the \textit{unitary S-matri
\textbf{\ }}$S$ to control the \textit{basis change\textbf{\ }}between the
in-state basis $|\alpha ,in\rangle $ and the out-state basis $|\beta
,out\rangle $:
\begin{equation*}
|\alpha ,in\rangle =\underset{\beta }{\sum }|\beta ,out\rangle S_{\beta
,\alpha }\text{ \ \ \ with }S_{\beta ,\alpha }:=\langle \beta ,out|\alpha
,in\rangle =\langle \beta |S|\alpha \rangle ,
\end{equation*
The schematic picture can be summarized as follows:
\begin{equation*}
\begin{array}{c||c|c|c}
& \text{in} & & \text{out} \\ \hline\hline
\begin{array}{c}
\text{asymptotic~} \\
\text{fields
\end{array}
&
\begin{array}{c}
{\small Ad\Theta }^{in}{\small \curvearrowright }\text{\ \ \ \ } \\
\text{ \ \ \ \ \ \ }\phi ^{in}(x
\end{array}
& \overset{AdS^{-1}\text{ \& }Ad\Theta }{\underset{AdS\text{ \& }Ad\Theta }
\rightleftarrows }} &
\begin{array}{c}
\text{\ \ \ \ \ }{\small \curvearrowleft Ad\Theta }^{out} \\
\phi ^{out}(x)\text{ \ \ \ \ \ \
\end{array}
\\ \hline
& \text{GLZ}\searrow \nwarrow t\rightarrow -\infty &
\begin{array}{c}
\text{asymp.cond.}\uparrow \\
\downarrow \text{GLZ formula
\end{array}
& t\rightarrow +\infty \nearrow \swarrow \text{GLZ } \\ \hline
\begin{array}{c}
\text{Heisenberg~} \\
\text{fields
\end{array}
& &
\begin{array}{c}
{\small Ad\Theta \curvearrowright }\text{ \ \ } \\
\text{ \ \ \ \ \ }\varphi _{H}(x
\end{array}
&
\end{array
\end{equation*}
To treat Heisenberg fields~$\varphi _{H}(x)$, we recapitulate briefly the
essence of Wightman axioms for relativistic quantum fields \cite{StrWi, Bog}
(in the vacuum representation $(\mathcal{P},\mathfrak{H},U,\Omega )$) in the
form of relativistic covariance, local commutativity, cyclicity or
ergodicity of vacuum state and spectral condition:
a) [Heisenberg fields] as operator-valued distributions $\mathcal{D}(\mathbb
R}^{4})\ni f\longmapsto \varphi _{H}^{i}(f)$ with values being (unbounded)
closable operators acting on a Hilbert space $\mathfrak{H}$ are defined on
the 4-dimensional Minkowski spacetime $(\mathbb{R}^{4},\eta )$, where $\eta
~is the Minkowski metric: $\eta (x,y):=x\cdot y=x^{0}y^{0}-\vec{x}\cdot \vec
y}$.
b) [Relativistic covariance]: a local net $\mathcal{P}:\mathcal{K}\ni
\mathcal{O}\longmapsto \mathcal{P}(\mathcal{O})$ of *-algebras $\mathcal{P}
\mathcal{O})$ generated by local fields $\varphi _{H}^{i}(f)=\int \varphi
_{H}^{i}(x)f(x)d^{4}x$ with $f\in \mathcal{D}(\mathcal{O})$ and their
polynomials defined on the net $\mathcal{K}$ of double cones $\mathcal{O}
\mathcal{O}_{a,b}=(a+V_{+})\cap (b-V_{+})$ (with the forward lightcone
V_{+} $) constitute a non-commutative covariant dynamical system,
\begin{eqnarray*}
\alpha _{a,\Lambda }(\varphi _{H}^{i}(x)) &=&U(a,\Lambda )\varphi
_{H}^{i}(x)U(a,\Lambda )^{-1} \\
&=&s(\Lambda )_{j}^{i}\varphi _{H}^{i}(\Lambda ^{-1}(x-a)), \\
\alpha _{a,\Lambda }(\mathcal{P}(\mathcal{O})) &=&\mathcal{P}(\Lambda
\mathcal{O}+a),
\end{eqnarray*
under the action $\alpha $, $\mathcal{P}_{+}^{\uparrow }\ni (a,\Lambda
)\longmapsto \alpha _{a,\Lambda }\in Aut(\mathcal{P}(\mathbb{R}^{4}))$, of
Poincar\'{e} group $\mathcal{P}_{+}^{\uparrow }$ (or its covering
\widetilde{\mathcal{P}_{+}^{\uparrow }}$ ) and $\mathcal{P}_{+}^{\uparrow
}\ni (a,\Lambda )\longmapsto U(a,\Lambda )\in \mathcal{U}(\mathfrak{H})$ is
its unitary representation on $\mathfrak{H}$, and $s(\Lambda )_{j}^{i}$~is a
finite-dimensional representation of Lorentz group $L_{+}^{\uparrow }$
associated with each field multiplet $(\varphi _{H}^{i}(x))_{i}$.
c) [Local commutaitivity]: the absence of propagation of physical effects
exceeding the light velocity due to Einstein causality, implies the local
commutativity of Heisenberg fields $\varphi _{H}^{i}(f)$:
\begin{equation*}
\lbrack \varphi _{H}^{i}(f_{1}),\varphi _{H}^{j}(f_{2})]=0\text{ \ \ \ if
(suppf_{1}){\huge \times }(suppf_{2})
\end{equation*
where$~\mathcal{O}_{1}{\huge \times }\mathcal{O}_{2}$ means that any pair of
points $x\in \mathcal{O}_{1}{\LARGE ,}y\in \mathcal{O}_{2}$ are spacelike
separated: $(x-y)^{2}<0$.
Remark: By this condition, the Fourier transform of Wightman functions
\omega _{0}(\varphi _{H}^{i_{1}}(x_{1})\cdots \varphi _{H}^{i_{r}}(x_{r}))$
as correlation functions of $\varphi _{H}^{i}$ in the vacuum state $\omega
_{0}(\cdot )=\langle \Omega |(\cdot )\Omega \rangle $ defined in the next d)
admits an analytic continuation into a holomorphic function in the complex
energy-momentum space. According to it, dispersion relations are valid.
d) [Vacuum state and spectrum condition]:
d-i) Energy-momentum spectrum $Sp(U(\mathbb{R}^{4}))$ of spacetime
translations $\mathbb{R}^{4}$ realized on $\mathfrak{H}$ is within the
forward light cone, $Sp(U(\mathbb{R}^{4}))\subset \overline{V_{+}}$ in $p
-space $\widehat{\mathbb{R}^{4}}$, and the lowest energy is realized by
eigenvalue $0$ of the vacuum vector $\Omega $: $U(x):=U(x,1)=\int_{p\in
\overline{V_{+}}}\exp (ipx)dE(p)$; \ $U(x)\Omega =\Omega $
Remark: Similarly to $p$-analyticity due to local commutativity, $x$-space
analyticity of Wightman functions $\omega _{0}(\varphi
_{H}^{i_{1}}(x_{1})\cdots \varphi _{H}^{i_{r}}(x_{r}))$ follows from
spectrum condition, which provides powerful tools for structural analysis.
d-ii)\ Cyclicity $\overline{\mathcal{P}(\mathbb{R}^{4})\Omega }=\mathfrak{H}$
of $\Omega $ $\Longleftrightarrow $ irreducibility of $\mathcal{P}(\mathbb{R
^{4})$ $\Longleftrightarrow $ uniqueness of vacuum (: $U(x)\Psi =\Psi
\Longrightarrow \Psi \propto \Omega $) $\Longleftrightarrow $ cluster
property
\begin{equation*}
\left\vert \omega _{0}(A(x)B(y))-\omega _{0}(A)\omega _{0}(B)\right\vert
\rightarrow 0\text{\ as }(\vec{x}-\vec{y})^{2}\rightarrow \infty ,
\end{equation*
where $A(x):=\alpha _{x}(A)=U(x)AU(x)^{\ast },$ $B(y):=\alpha _{y}(B)$ are
the spacetime translates of local observables $A,B\in \mathcal{P}(\mathcal{O
)$ by $x,y\in \mathbb{R}^{4}$, respectively. This follows from partition of
unity due to spectral resolution of spacetime translations $U(x)$:
\begin{eqnarray*}
1 &=&|\Omega \rangle \langle \Omega |\text{ }+\text{ }\sum_{i}(\text{\textit
1-particle singularities on mass-shell}}\mathit{\mathbf{\ }}p^{2}=m_{i}^{2})
\\
&&+(\text{absolutely continuous }p\text{-spectra})
\end{eqnarray*}
\section{Independence of asymptotic fields due to on-shell asymptotic
condition as \textquotedblleft central limit\textquotedblright\ theorem}
From the \textit{cluster property} and the local commutativity, follows the
\textit{asymptotic condition} $\varphi _{H}(x)\underset{x^{0}=t\rightarrow
\mp \infty }{\rightarrow }\phi ^{in/out}(x)$ (as weak convergence),
according to which asymptotic fields $\phi ^{as}$~materialize \textit
kinematically} the factorization (= independence)\ of correlations \textit
without taking limits}: any $n$-point functions of $\phi ^{as}$ are
factorized into the products of two-point functions, $\omega _{0}(\phi
^{as}\phi ^{as}\cdots \phi ^{as})=\sum \omega _{0}(\phi ^{as}\phi
^{as})\cdots \omega _{0}(\phi ^{as}\phi ^{as})$, which is known as the
\textquotedblleft quasi-freeness\textquotedblright\ of the vacuum state
\omega _{0}$ with respect to $\phi ^{as}$, familiar in the form of
\textquotedblleft Wick theorem\textquotedblright\ or the \textit{independenc
} of Gaussian type. The creation and annihilation operators
a_{k},a_{k}^{\ast }$ contained in $\phi ^{as}$ constitute an \textit
infinite number of conserved quantities}, as they are given by the spatial
integral of current densities $i\phi ^{as}(x)\overleftrightarrow{\partial
_{\mu }}f(x)$ which are conserved, $\partial ^{\mu }[\phi ^{as}(x
\overleftrightarrow{\partial _{\mu }}f(x)]=\phi ^{as}(x)\overleftrightarrow
\square }f(x)=\phi ^{as}(x)\overleftrightarrow{(\square +m^{2})}f(x)=0$ by
the on-shell conditions: $(\square +m^{2})\phi ^{as}(x)=0=(\square
+m^{2})f(x)$.
Thus, the \textit{independence} embodied in the asymptotic fields $\phi
^{as} $ is seen to emerge from interacting Heisenberg fields $\varphi _{H}$
via the asymptotic condition as a kind of \textquotedblleft central
limit\textquotedblright\ theorem. In this context, what corresponds to
\textquotedblleft Langevin equation\textquotedblright\ can be found in the
\textit{Yang-Feldman equation} \cite{Bog} to connect the Heisenberg field~
\varphi _{H}(x)$~and the asymptotic field $\phi ^{as}(x)$:
\begin{eqnarray*}
\varphi _{H}(x) &=&\int \Delta _{ret}(x-y;m^{2})J_{H}(y)d^{4}y+\phi
^{in}(x)=[\Delta _{ret}\ast J_{H}+\phi ^{in}](x) \\
&=&\int \Delta _{adv}(x-y;m^{2})J_{H}(y)d^{4}y+\phi ^{out}(x)=[\Delta
_{adv}\ast J_{H}+\phi ^{out}](x).
\end{eqnarray*
where $J_{H}=(\square +m^{2})\varphi _{H}$ is the Heisenberg source current
and $\Delta _{ret/adv}(x-y;m^{2})$ : retarded/ advanced Green's functions
(i.e., principal solutions) of the Klein-Gordon equation defined by
\begin{eqnarray*}
(\square _{x}+m^{2})\Delta _{ret/adv}(x-y;m^{2}) &=&\delta (x-y), \\
\Delta _{ret/adv}(x-y;m^{2}) &=&0\text{ \ \ \ for }x_{0}\lessgtr y_{0}.
\end{eqnarray*
In the Yang-Feldman equation, the asymptotic fields $\phi ^{in/out}$ and
Heisenberg source current $J_{H}$ appear, respectively, as the \textit
residue} and the \textit{quotient} in the division of $\varphi _{H}$ by
\Delta _{ret/adv}$. What is more important is that $J_{H}$ gives the \textit
residues at the on-shell pole} $\dfrac{1}{p^{2}-m^{2}}$ to determine matrix
elements of scattering amplitudes, as will be seen in the next formula.
\section{Mutual relations between Heisenberg fields and asymptotic fields
controlled by Micro-Macro Duality}
From the asymptotic condition and the LSZ reduction formulae \cite{LSZ}, one
can derive the Haag-GLZ formulae \cite{GLZ}~to express Heisenberg operators
A$ in terms of the Wick products $:\phi ^{as}\cdots \phi ^{as}:$ of
asymptotic fields
\begin{eqnarray*}
SA &=&:\exp (\phi ^{in}(\square +m^{2})\dfrac{\delta }{\delta J}):\omega
_{0}(T(A\exp (i\varphi _{H}J))\upharpoonright _{J=0} \\
&=&\sum_{k=0}^{\infty }\dfrac{i^{k}}{k!}\int dx_{1}\cdots \int
dx_{k}(\square _{x_{1}}+m^{2})\cdots (\square _{x_{1}}+m^{2})\omega
_{0}(T(A\varphi _{H}(x_{1})\cdots \varphi _{H}(x_{k})) \\
&&\times :\phi ^{in}(x_{1})\cdots \phi ^{in}(x_{k}):, \\
S &=&:\exp (\phi ^{as}(\square +m^{2})\dfrac{\delta }{\delta J}):\omega
_{0}(T(\exp (i\varphi _{H}J))\upharpoonright _{J=0},
\end{eqnarray*
which is similar to the Fock expansion formula \cite{Oba} known in
WNA.~While the formulae of this type have long been known simply as those to
expand the Heisenberg operators in the Wick products of $\phi ^{as}$, what
should be emphasized here are the following novel points:
1)\ We can reformulate these equalities into the following form \cite{IO89}:
\begin{eqnarray*}
SA &=&:(\omega _{0}\otimes id)(T(A\otimes 1)\exp (iJ_{H}\otimes \phi
^{in})):, \\
A &=&S^{-1}:(\omega _{0}\otimes id)(T[A\otimes 1]\exp (iJ_{H}\otimes \phi
^{in})): \\
&=&:(\omega _{0}\otimes id)(T[A\otimes 1]\exp (iJ_{H}\otimes \phi
^{out}):S^{-1}, \\
S &=&:(\omega _{0}\otimes id)(T\exp (iJ_{H}\otimes \phi ^{in})):=:(\omega
_{0}\otimes id)(T\exp (iJ_{H}\otimes \phi ^{out}):.
\end{eqnarray*
The roles played by the system consisting of Heisenberg operators $\varphi
_{H}$ and those of asymptotic fields $\phi ^{in}$ and $\phi ^{out}$ are
clearly separated here in the \textit{\textbf{operational context}} in such
a way that the former is an \textit{unknown} target system to be detected
and analyzed by means of the latter ones functioning as \textit{\textbf
probe systems}}. The relevant \textit{\textbf{coupling terms }}among them
are specified by $\exp (iJ_{H}\otimes \phi ^{as})$, with the first tensor
factor to be time-ordered and the second one Wick ordered, which are
mutually in duality, as will be seen by the relation between $\dfrac{\delta
}{i\delta \phi ^{in}(x)}$ and $\phi ^{in}(x)$ in 3).~In this connection, the
comparison to the notion of instruments in quantum measurements will be
instructive: while a basic scheme for instruments can be seen in
\begin{equation*}
\begin{array}{ccc}
\begin{array}{c}
\text{neutral state of } \\
\text{probe system
\end{array}
& &
\begin{array}{c}
\text{measured values} \\
\text{ recorded in probe
\end{array}
\\
\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{c}
\searrow \\
\nearro
\end{array}
& \rightarrow
\begin{array}{c}
\text{coupling between} \\
\text{system \& probe
\end{array
\rightarrow &
\begin{array}{c}
\nearrow \\
\searro
\end{array
\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\
\ \ \ \ \ \
\begin{array}{c}
\text{initial state} \\
\text{of system
\end{array}
&
\begin{array}{c}
\longrightarrow \\
\text{state changes
\end{array}
&
\begin{array}{c}
\text{final state} \\
\text{of system
\end{array
\text{ \ \ \ \ \ \ \
\end{array
,
\end{equation*
the corresponding one for scattering processes in QFT is given in sharp
contrast by
\begin{equation*}
\begin{array}{cccc}
\begin{array}{c}
\text{initial state } \\
|\alpha ,in\rangle \text{ of }\phi ^{in
\end{array}
&
\begin{array}{c}
\text{state changes in probe }\phi ^{as} \\
\longrightarro
\end{array}
& &
\begin{array}{c}
\text{final state } \\
|\beta ,out\rangle \text{ of }\phi ^{out
\end{array}
\\
\text{ \ \ \ \ \ \ \ \ \ \ \
\begin{array}{c}
\searrow \\
\nearro
\end{array}
& \rightarrow
\begin{array}{c}
\text{coupling }\exp (iJ_{H}\otimes \phi ^{as})\text{ } \\
\text{between }\varphi _{H}\text{ \& }\phi ^{as
\end{array
\rightarrow & &
\begin{array}{c}
\nearrow \\
\searro
\end{array
\text{ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\
\ \
\begin{array}{c}
\text{vacuum} \\
|\Omega \rangle \text{ of }\varphi _{H
\end{array}
& \text{ vacuum }|\Omega \rangle \text{ unchanged} & &
\begin{array}{c}
\text{vacuum} \\
|\Omega \rangle \text{ of }\varphi _{H
\end{array
\text{ \ \ \ \
\end{array
.
\end{equation*
Perhaps, they can be unified by combining the level of the object system in
the instrument with that of the asymptotic fields $\phi ^{as}$, which
results in the successive measurement processes where the state changes
taking place at the level of asymptotic fields in scattering processes of
quantum fields are monitored by measuring such observables as the particle
momenta or spins through the instruments.
2) The mathematical and conceptual meanings of the coupling term $\exp
(iJ_{H}\otimes \phi ^{in})$: We first note by simple computation that the
on-shell condition $(\square +m^{2})\phi ^{as}=0$ for the asymptotic fields
implies the following equalities:
\begin{eqnarray*}
J_{H}\otimes \phi ^{as} &=&(\square +m^{2})\varphi _{H}\otimes \phi
^{as}=(\square +m^{2})\varphi _{H}\otimes \phi ^{as}-\varphi _{H}\otimes
(\square +m^{2})\phi ^{as} \\
&=&\square \varphi _{H}\otimes \phi ^{as}-\varphi _{H}\otimes \square \phi
^{as}=\partial ^{\mu }\partial _{\mu }\varphi _{H}\otimes \phi ^{as}-\varphi
_{H}\otimes \partial ^{\mu }\partial _{\mu }\phi ^{as} \\
&=&\partial ^{\mu }[\partial _{\mu }\varphi _{H}\otimes \phi ^{as}-\varphi
_{H}\otimes \partial _{\mu }\phi ^{as}]=-\partial ^{\mu }[\varphi
_{H}\otimes \overset{\longleftrightarrow }{\partial _{\mu }}\phi ^{as}].
\end{eqnarray*
Combing this with the asymptotic condition we can further rewrite this
quantity $iJ_{H}\otimes \phi ^{in}$:
\begin{eqnarray*}
&&i\int_{\mathbb{R}^{4}}d^{4}xJ_{H}(x)\otimes \phi ^{in}(x)=-i\int_{\mathbb{
}^{4}}d^{4}x\partial ^{\mu }[\varphi _{H}\otimes \overset
\longleftrightarrow }{\partial _{\mu }}\phi ^{in}]= -i\int_{\partial \mathbb
R}^{4}}dS^{\mu }[\varphi _{H}\otimes \overset{\longleftrightarrow }{\partial
_{\mu }}\phi ^{in}] \\
&&=-i\int_{x^{0}=+\infty }d^{3}x[\varphi _{H}\otimes \overset
\longleftrightarrow }{\partial _{0}}\phi ^{in}]+i\int_{x^{0}=-\infty
}d^{3}x[\varphi _{H}\otimes \overset{\longleftrightarrow }{\partial _{0}
\phi ^{in}] \\
&&=-i\int_{x^{0}=+\infty }d^{3}x[\phi ^{out}\otimes \overset
\longleftrightarrow }{\partial _{\mu }}\phi ^{in}]+i\int_{x^{0}=-\infty
}d^{3}x[\phi ^{in}\otimes \overset{\longleftrightarrow }{\partial _{\mu }
\phi ^{in}] \\
&&=-(S^{-1}\otimes 1)iQ(S\otimes 1)+iQ=-(S^{-1}\otimes 1)[iQ,S\otimes 1] \\
&&=-(S^{-1}\otimes 1)ad(iQ)(S\otimes 1),
\end{eqnarray*
in terms of a \textit{conserved charge} $Q$ defined by
\begin{equation*}
iQ:=i\int d^{3}x[\phi ^{in}\otimes \overset{\longleftrightarrow }{\partial
_{\mu }}\phi ^{in}]=\sum_{k}[(a_{k}^{in})^{\ast }\otimes
a_{k}^{in}-a_{k}^{in}\otimes (a_{k}^{in})^{\ast }].
\end{equation*
It is remarkable that $:\exp (iQ):$ (with the Wick product $:\cdots :$ \ to
be applied to the second tensor factor) is the operator to create a \textit
coherent state }(or,\textit{\ exponential vector})\textit{\ }$:\exp
(iQ)|\Omega \rangle :$ from the vacuum $|\Omega \rangle $ with the
Wick-ordered \textit{commutative} parameters $a_{k}^{in}$. In the context of
WNA, these quantities will correspond to the \textit{U-functionals} \cit
{Hida} constituting a commutative algebra with respect to the Wick product.
3) Applying the above formula $A=S^{-1}:(\omega _{0}\otimes id)(T[A\otimes
1]\exp (iJ_{H}\otimes \phi ^{in})):$ to the Heisenberg source current
J_{H}=A$, we reproduce the expression
\begin{equation*}
(\square +m^{2})\varphi _{H}=J_{H}=S^{-1}:(\omega _{0}\otimes
id)(T[J_{H}\otimes 1]\exp (iJ_{H}\otimes \phi ^{in})):=S^{-1}\dfrac{\delta }
i\delta \phi ^{in}(x)}S,
\end{equation*
once obtained by Bogoliubov, Medvedev and Polivanov \cite{Bog}. Through the
above relations, the asymptotic condition $\varphi _{H}\underset
t\rightarrow \mp \infty }{\rightarrow }\phi ^{in/out}$ to be identified with
the \textit{on-shell condtion }$\square +m^{2}=0=m^{2}-p^{2}$ is seen to
play the essential role from the algebraic viewpoint: it extracts the
asymptotic fields $\phi ^{in/out}$ from the algebra of interacting
Heisenberg fields $\varphi _{H}$ as the \textit{\textbf{fixed points }under
the Lie subgroup} $\Gamma $ generated by $\phi ^{in}$ or $Q$ in the
infinite-dimensional Heisenberg Lie group generated by $\phi ^{in}$ and
\dfrac{\delta }{i\delta \phi ^{in}(x)}$ with $\phi ^{in}$ taken as an
element belonging to the above commutative algebra with the Wick product.
Under the action of $\Gamma $, $a_{k}^{in}$ and $(a_{k}^{in})^{\ast }$ are
infinitely many\ conserved charges (to characterize the \textquotedblleft
integrability\textquotedblright\ of the probe system consisting of $\phi
^{as}$). Thus the group $\Gamma $ generated by (the Wick-ordered)
a_{k}^{as} $ and $(a_{k}^{as})^{\ast }$ characterizes the aspects of the
symmetry associated with the macroscopic on-shell situation described by
\phi ^{as}$. The essential features here show the strong similarity to that
of WNA if the functional derivatives $\dfrac{\delta }{i\delta \phi ^{in}(x)}$
are put in parallel with the Hida derivatives \cite{Hida}.
4) \textit{\textbf{Breakdown of invariance}} under $\Gamma $ due to the
interactions: The symmetry $\Gamma $ preserved in the probe systems
consisting of $\phi ^{as}$ is broken in the total system containing the
interacting Heisenberg fields $\varphi _{H}$, essentially due to the
presence of $\dfrac{\delta }{i\delta \phi ^{in}(x)}$ or the Heisenberg
source current $J_{H}(x)=S^{-1}\dfrac{\delta }{i\delta \phi ^{in}(x)
S=(\square +m^{2})\varphi _{H}(x)\neq 0$, which introduces the effects
coming from the \textit{\textbf{off-shell}} aspects of the theory. The
non-trivial existence of the coupling term $iJ_{H}\otimes \phi
^{in}=-(S^{-1}\otimes 1)[iQ,S\otimes 1]$ is equivalent to the non-triviality
$S\neq 1$ of the S-matrix which implies the difference between $\phi ^{in}$
and $\phi ^{out}$ due to $[iQ,S\otimes 1]\neq 0$. It is interesting to note
the parallelism of this situation with the mutual relation between the first
and second laws in Newtonian mechanics: since the interaction term due to
a(n external) force is switched off in the stage of the first law, the
system enjoys a symmetry to conserve the momentum (or velocity), similarly
to the above constancy of $\phi ^{in}$ or $Q$. At the stage of the second
law, the symmetry inherent to the first law is \textit{\textbf{broken}} by
introducing the (external) force $F$, which causes the \textit{\textbf{state
changes}} described by the \textit{changes} of the momentum $p$ according to
the second law, $dp/dt=F$, in the essential use of the vocabulary provided
by the first law. The state changes in QFT due to $J_{H}=(\square
+m^{2})\varphi _{H}\neq 0$ is similarly described by the very breaking term
iJ_{H}\otimes \phi ^{in}=-(S^{-1}\otimes 1)[iQ,S\otimes 1]$ in terms of the
non-trivial S-matrix $S\neq 1$. At first sight, what is conserved, momentum
p$ and $\phi ^{as}$ (contrasted with the \textquotedblleft momentum
variables\textquotedblright\ $\dfrac{\delta }{i\delta \phi ^{as}}$), seems
to be opposite between the Newtonian and the QFT cases, but actually it is
not the case, because a particle picture with conserved $\vec{p}$ in the
Newtonian first law is embedded also in $\phi ^{as}$ through the creation
and annihilation operators, $a(\vec{p})^{\ast }$ and $a(\vec{p})$.
5) \textit{\textbf{Reconstruction}} of $\varphi _{H}$ from $\phi ^{in/out}$
intertwined by $S$: The essence of the GLZ-Fock expansion of the Heisenberg
fields $\varphi _{H}$ or $A$ in terms of the asymptotic fields $\phi ^{as}$
can be seen in the \textquotedblleft inverse problem\textquotedblright\ to
\textit{reconstruct }the former from its fixed-point subalgebra(s)
\{\varphi _{H}\}^{\Gamma }=\{\phi ^{in}\}$ or $\{\phi ^{out}\}$ through the
\textit{\textbf{co-action}} of $\Gamma $ (which is in parallel to the
so-called \textquotedblleft inverse scattering method\textquotedblright\ in
quantum mechanics to determine a potential term responsible for scattering
processes from the scattering data). This kind of \textit{duality }relation
between $\varphi _{H}$ and $\phi ^{as}$ ensures the \textit{universality} of
the asymptotic fields $\phi ^{as}$ in spite of its speciality as
statistically independent free objects. What is conceptually more important
is such a possibility to re-construct an interacting theory of relativistic
quantum fields $\varphi _{H}$ from the knowledge of an S-matrix $S$
intertwining the asymptotic in- and out-fields $\phi ^{in/out}$: $\phi ^{in
\underset{AdS}{\overset{AdS^{-1}}{\rightleftarrows }}\phi ^{out}$. The
crucial ingredient, $iJ_{H}\otimes \phi ^{in}=-(S^{-1}\otimes 1)[iQ,S\otimes
1]$, as a coupling term in $:(\omega _{0}\otimes id)(T((A\otimes 1)\exp
(iJ_{H}\otimes \phi ^{in})):$ $=SA$ can be determined, at least in its
integrated form, by the knowledge of $\phi ^{as}$ and $S$, the former of
which is easily constructible as free objects. The latter one is the highly
non-trivial object to be determined phenomenologically from the experimental
data of particle scattering processes to within certain limits of
exactitude. Once the functional dependence of $S$ on $\phi ^{in}$ is
specified, the formula $J_{H}(x)=S^{-1}\dfrac{\delta }{i\delta \phi ^{in}(x)
S$ allows a local quantity$J_{H}(x)$ to be determined. Therefore, the whole
scheme to control the mutual relations between $\varphi _{H}$ (: Micro) and
\phi ^{as}$ (: Macro) can be understood in the context of \textquotedblleft
\textit{\textbf{Micro-Macro duality}}\textquotedblright\ \cite{MicMac} via
the Fourier-Galois duality between the fixed-point subalgebra and the
recovery of the total algebra as the Galois extension. A \textit{new feature}
found here is the problem related with the appearance of \textit{two} fixed
point subalgebras, $\{\phi ^{in}\}$ and $\{\phi ^{out}\}$, which are
mutually equivalent by the intertwining actions of the S-matrix, $Ad(S)$ and
$Ad(S^{-1})$
\begin{equation*}
S\phi ^{out}S^{-1}=\phi ^{in},\text{ \ \ \ }S^{-1}\phi ^{in}S=\phi ^{out}.
\end{equation*
In this connection, the coupling terms $:T\exp (iJ_{H}\otimes \phi ^{as}):$
\ cannot be directly regarded as a Kac-Takesaki operator but should be
interpreted as a kind of a \textit{\textbf{cocycle}} between two such. This
aspects will further be elaborated.
6) PCT invariance \& Borchers classes: While the (weak) local commutativity
is inevitably violated between $\varphi _{H}$ and $\phi ^{in/out}$ and
between $\phi ^{in}$ and $\phi ^{out}$, the fields $\varphi _{H}$, $\phi
^{in}$ and $\phi ^{out}$ enjoy the local commutativity within each system.
Then the vacuum $\omega _{0}$ is invariant, $\omega _{0}\circ \theta =\omega
_{0}=\omega _{0}\circ \theta ^{as}$, under the PCT\ transformations $\theta
,\theta ^{as}$ given by $\theta (\varphi _{H}(x))=\gamma \varphi
_{H}(-x)^{\ast }$, $\theta ^{as}(\phi ^{as}(x))=\gamma \phi ^{as}(-x)^{\ast
} $ (with $\gamma \in \mathbb{T}$), and hence, $\theta ,\theta ^{as}$ are
implemented, respectively, by anti-unitary PCT$\ $operators $\Theta $ and
\Theta ^{as}$ s.t. $\theta (\varphi _{H}(x))=\Theta \varphi _{H}(x)\Theta $,
$\theta ^{as}(\phi ^{as}(x))=\Theta ^{as}\phi ^{as}(x)\Theta ^{as}$ and
\Theta \Omega =\Theta ^{as}\Omega =\Omega $ \cite{StrWi, Bog}. Then $S\phi
^{out}(x)S^{-1}=\phi ^{in}(x)=\Theta \gamma ^{-1}\phi ^{out}(-x)^{\ast
}\Theta =\Theta \Theta ^{out}\phi ^{out}(x)\Theta ^{out}\Theta $ implies
\begin{eqnarray*}
S &=&\Theta ^{in}\Theta =\Theta \Theta ^{out},\text{ \ \ \ }S\Theta
^{out}=\Theta =\Theta ^{in}S, \\
S\Theta &=&\Theta ^{in}=\Theta \Theta ^{out}\Theta =\Theta S,
\end{eqnarray*
under the assumption of asymptotic completeness. These relations exhibit
more detailed structures of $S$ in terms of PCT operators. Thus, quantum
fields with\textit{\ the same PCT operator\textbf{\ }}$\Theta $\ have the
same S-matrix $S=\Theta ^{in}\Theta =\Theta \Theta ^{out}$: this explains
the \textquotedblleft ambiguities\textquotedblright\ in the choice of
Heisenberg fields interpolating asymptotic fields $\phi ^{in/out}$ connected
by a given S-matrix $S$ in such a form as the Borchers classes \cite{Bog}\
characterized by the relative local commutativity to share the same PCT
operator $\Theta $. The consideratoins on this aspect will be crucial for
discussing the above points of 2) -- 5).
7) For physical applications in the scheme of the above 1), it would be
interesting to utilze the above formula $SA=:(\omega _{0}\otimes
id)(T(A\otimes 1)\exp (iJ_{H}\otimes \phi ^{in})):$ \ with a specific choice
of $A$ such as the electromagnetic current $A=J_{\mu }(x)$\ to analyze the
processes to measure the form factor $\langle \beta ,out|J_{\mu }(x)|\alpha
,in\rangle =\langle \beta |SJ_{\mu }(x)|\alpha \rangle $, or, in more
general contexts of \textquotedblleft weak values\textquotedblright\ \cit
{weak} $\langle \beta ,out|A|\alpha ,in\rangle =\langle \beta |SA|\alpha
\rangle $, as suggested by Hosoya and Shikano.
The author would like to thank Profs. M. Ohya and N. Watanabe for their
invitation to QBIC2009. He is very grateful to Profs. T. Hida and Si Si and
to Prof. M. Ozawa for navigating him to the important apsects, respectively,
of WNA and of the instrument. In this connection, he has been so much
encouraged by Mr. T. Shimizu, without whose deep insights the bridge between
WNA and QFT could not have been properly appreciated. He thanks also Prof.
A. Hosoya and Mr. Y. Shikano very much for their encourgements, discussions
on his attempts to reformulate the traditional QFT along the above line and
their suggestion to examine \textquotedblleft weak values\textquotedblright
. Last but not least, he is very grateful to all the members of the regular
seminar on Math. Phys. at RIMS, Prof. S. Tanimura, Messrs. H. Andou, R.
Harada, T. Hasebe, K. Okamura and H. Saigo, for their valuable comments and
questions.
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 4,029
|
Der Schweizer Filmpreis (, , ) wurde das erste Mal im Jahre 1998 anlässlich der Solothurner Filmtage verliehen.
Geschichte, Organisation
Ab 1998 wurde er der Filmpreis im Januar während der Filmtage Solothurn vergeben. Insgesamt standen in der Vergangenheit 600'000 Schweizer Franken, in sechs Kategorien, an Preisgeldern zur Verfügung. Anfang März 2009 fand erstmals eine Fernsehgala im Schweizer Fernsehen unter Ausrichtung des Bundesamts für Kultur (BAK) und der 2008 neu gegründeten Schweizer Filmakademie statt. Als Preistrophäe fungiert der "Quartz", eine Skulptur aus dem gleichklingenden Mineral, die 2008 den bis dahin überreichten "Viewfinder" ablöste. Die Auszeichnung wird seither zu jeder Auflage von Designern umgestaltet. So zuletzt 2009 durch Alfredo Häberli.
Der Schweizer Filmpreis soll die Filmschaffenden aus der Schweiz für ihre herausragenden Filmen auszeichnen. Die Jury besteht aus verschiedenen Personen aus den Bereichen Kultur, Film und Medien. Die Jury wird jedes Jahr neu besetzt. Die Nominierungen werden durch eine neunköpfige Kommission, welche vom Eidgenössischen Departement des Inneren gewählt wird, festgelegt. Zugelassen sind Schweizer Filme, welche im laufenden Jahr auf verschiedenen wichtigen Filmfestivals (Inland sowie Ausland) gezeigt wurden.
Die Verleihung des Filmpreises wurde bis 2008 durch das Bundesamt für Kultur ausgerichtet, zusammen mit den Partner Swiss Films, SRG SSR idée suisse und den drei Schweizer Filmfestivals in Solothurn, Locarno und Nyon. Im März 2009 wurde der Schweizer Filmpreis erstmals während einer Fernsehgala im Kultur- und Kongresszentrum Luzern in zehn Kategorien vergeben. Hierfür wurde Ende Juli 2008 die Schweizer Filmakademie unter Leitung des Filmemachers Fredi M. Murer gegründet, die sich um die Organisation des Nominierungsverfahrens kümmern sollte. Nach Differenzen in juristischen und finanziellen Fragen mit dem Bundesamt für Kultur werden 2009 die Nominierungen durch die Schweizer Filmakademie getroffen und wie in den Vorjahren von einer neunköpfigen Kommission ausgesprochen. Für das Jahr 2010 stellte das BAK in Aussicht, den Filmpreis vollständig der Schweizer Filmakademie zu überlassen. Dies lehnte die Akademie jedoch ab, da diese dann selber dafür verantwortlich wäre, die notwendigen Ressourcen zu generieren. Für das Jahr 2009 war die Akademie bereit gewesen, nur 30 Prozent an Eigenmitteln für die Ausrichtung des Schweizer Filmpreises aufzubringen.
Ab 2013 übernahm das Bundesamt für Kultur die volle Verantwortung für die Organisation des Schweizer Filmpreises, in Zusammenarbeit mit der SRG SSR. Für die künftige Ausrichtung bewarb sich Bern sowie gemeinsam die Städte Genf und Zürich sowie Lausanne und Luzern. Am 30. März 2012 gaben die Bundesbehörden der Schweizerischen Eidgenossenschaft auf admin.ch bekannt, dass der Filmpreis in Genf und Zürich verliehen wird.
Die Verleihung der Schweizer Filmpreise 2019 fand am 22. März in Genf statt. Die Verleihung 2020 sollte ursprünglich am 27. März in Zürich stattfinden. Aufgrund der Coronavirus-Epidemie wurde die Verleihung abgesagt. Die Gewinner wurden am 23. März 2020 in einer Medienmitteilung bekanntgegeben. Die 24. Verleihung der Schweizer Filmpreise fand am 26. März 2021 statt. Die Verleihung 2022 fand am 25. März in Zürich statt.
Kategorien
Der Filmpreis wird in folgenden Kategorien verliehen:
Ehemalige Kategorien des Filmpreises
Beste Hauptrolle (2004–2007)
Bestes schauspielerisches Nachwuchstalent (2008–2010)
Weblinks
Offizielle Website des Schweizer Filmpreises
Schweizer Filmpreis über die KKL-Bühne gegangen – Berichterstattung über die Preisverleihung 2009 bei nzz.ch, 7. März 2009
Schweizer Filmpreis bei Swiss Films
Einzelnachweise
Erstverleihung 1998
Filmpreis (Schweiz)
|
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| 5,466
|
January 25, 2010 — The State Foreclosure Prevention Working Group is encouraging servicers to step up foreclosure mitigation citing a significant outpacing of delinquent loans over outreach and loss mitigation.
The Working Group, which is comprised of 12 state attorneys generals, state regulators and the Conference of State Bank Supervisors, is warning that there will be an acceleration of foreclosures unless improvements are made in foreclosure prevention efforts. The group's new report indicates that six out of 10 seriously delinquent borrowers are not involved in loss mitigation efforts, and those borrowers that are seeking a modification are caught up in a delayed and backlogged system.
The Working Group found that most modifications result in payment reductions, but principal reductions remain rare. Further, prime loans are increasingly driving the rising delinquency rates. The group makes several recommendations in its report including improved loss mitigation programs that prioritize principal reduction in areas of significant home price declines and increased transparency and a reduction of paperwork in the federal Home Affordable Modification Program (HAMP). Further, the group says servicers should pay particular attention to reforming payment option ARM loans and that the states should consider expanding homeowner counseling and temporary foreclosure mediation programs.
The State Foreclosure Prevention Working Group has published three prior reports on mortgage issues. You can read more about this report at this link.
|
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| 6,436
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{"url":"https:\/\/pdglive.lbl.gov\/ParticleGroup.action?init=0&node=BXXX030","text":"${{\\mathit \\Xi}}$ BARYONS ($\\mathit S$ = $-2$, $\\mathit I$ = 1\/2)\n\n${{\\mathit \\Xi}^{0}}$ = ${\\mathit {\\mathit u}}$ ${\\mathit {\\mathit s}}$ ${\\mathit {\\mathit s}}$, ${{\\mathit \\Xi}^{-}}$ = ${\\mathit {\\mathit d}}$ ${\\mathit {\\mathit s}}$ ${\\mathit {\\mathit s}}$\n Radiative Hyperon Decays ${{\\mathit \\Xi}}$ Resonances","date":"2022-08-16 17:21:33","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\": 0.9961498379707336, \"perplexity\": 1553.9043523644086}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882572408.31\/warc\/CC-MAIN-20220816151008-20220816181008-00035.warc.gz\"}"}
| null | null |
Q: "falha de rede" ao fazer o download de PDF gerado com com a classe FPDF do PHP no Google Chrome Tenho relatórios com gerados em PHP (versão 5.6) / FPDF (versão 1.7) que estão sendo exibidos normalmente na janela do browser.
Segue exemplo simplificado sobre estes relatórios são gerados:
<?php
// não fazer cache (Recomendação de Guilherme Nascimento)
$dategmt = gmdate('D, d M Y H:i:s');
header('Expires: ' . $dategmt . ' GMT');
header('Last-Modified: ' . $dategmt . ' GMT');
header('Cache-Control: no-store, no-cache, must-revalidate');
header('Cache-Control: post-check=0, pre-check=0');
header('Pragma: no-cache');
// controla se o usuário pode ter acesso ao relatório
require 'controle-acesso.php';
if(!userTemAcesso()){
echo 'Você não tem permissões para acessar este recurso';
exit;
}
require 'fpdf.php';
$pdf = new FPDF();
$pdf->AddPage();
$pdf->SetFont('Arial', 'B', 16);
$pdf->Cell(40, 10, 'Olá mundo!');
// saída para arquivo não acessível diretamente via URL
$this->Output('doc.pdf', 'I');
Nota:
*
*Os parâmetros são passados para o PHP que gera o relatório via POST.
*O arquivo está sendo acessado exclusivamente via HTTPS com certificado válido.
*O servidor web está habilitado para logar todos os erros e os erros estão sendo devidamente registrados no log, porém nenhum erro relacionado a este PHP está sendo registrado. (ou seja, eu limpo o log de erros, executo o relatório e nenhum erro aparece no log... forçando um erro propositalmente e o mesmo é registrado). Desta forma, parece que não há qualquer erro de sintaxe.
*O Content-type utilizado é: header('Content-type: application/pdf');
*O problema ocorre em computadores dom Windws e Google Chrome (testado em várias máquinas).
*Usando plugin nativo do Google Chrome (chrome://plugins/ -> Chrome PDF Viewer). Se eu desativo ele, fica fazendo download normal do arquivo. Se ativo, ele visualiza tudo normal, imprime, porém não funciona a opção SALVAR.
Todos os botões do plug-In de exibição de PDF dos browsers (salvar, imprimir, girar, zoom, etc.) funcionam normalmente. Exceto o botão salvar no Google Chrome (nos outros browsers funciona normal).
Ao tentar salvar o PDF aberto e já exibido no Google Chrome, ocorre o seguinte erro:
Falha - Erro na rede
Logo, não é possível salvar o PDF, a não ser que vá em imprimir e imprima em PDF, ou seja, imprimir o PDF em PDF, o que não faz muito sentido.
Alguém teria alguma ideia ou sugestão como resolver este erro ou do que poderia estar causando o mesmo?
A: Acredito que o problema seja com o cache
*
*Você pode fazer prevenir o cache, adicione isto ao topo:
O I no método Output já adiciona o header application/pdf e envia o buffer diretamente para o buffer
<?php
// controla se o usuário pode ter acesso ao relatório
require 'controle-acesso.php';
if (!userTemAcesso()) {
echo 'Você não tem permissões para acessar este recurso';
exit;
}
require 'fpdf.php';
$pdf = new FPDF();
$pdf->AddPage();
$pdf->SetFont('Arial', 'B', 16);
$pdf->Cell(40, 10, 'Olá mundo!');
$pdf->Output('I', 'meuarquivo.pdf'); //versão 1.8
//$pdf->Output('meuarquivo.pdf', 'I'); //versão 1.7
*O I já previne o cache, mas se ainda não funcionar pode tentar assim:
<?php
$dategmt = gmdate('D, d M Y H:i:s');
header('Expires: ' . $dategmt . ' GMT');
header('Last-Modified: ' . $dategmt . ' GMT');
header('Cache-Control: no-store, no-cache, must-revalidate');
header('Cache-Control: post-check=0, pre-check=0');
header('Pragma: no-cache');
// controla se o usuário pode ter acesso ao relatório
require 'controle-acesso.php';
if (!userTemAcesso()) {
echo 'Você não tem permissões para acessar este recurso';
exit;
}
require 'fpdf.php';
$pdf = new FPDF();
$pdf->AddPage();
$pdf->SetFont('Arial', 'B', 16);
$pdf->Cell(40, 10, 'Olá mundo!');
$pdf->Output('I', 'meuarquivo.pdf'); //versão 1.8
//$pdf->Output('meuarquivo.pdf', 'I'); //versão 1.7
*Você pode salvar no servidor e redirecionar:
<?php
// controla se o usuário pode ter acesso ao relatório
require 'controle-acesso.php';
if(!userTemAcesso()){
echo 'Você não tem permissões para acessar este recurso';
exit;
}
require 'fpdf.php';
$pdf = new FPDF();
$pdf->AddPage();
$pdf->SetFont('Arial', 'B', 16);
$pdf->Cell(40, 10, 'Olá mundo!');
$pdf->Output('F', '/home/user/www/data/meuarquivo.pdf'); //versão 1.8
//$pdf->Output('/home/user/www/data/meuarquivo.pdf', 'F'); //versão 1.7
//Redireciona
header('Location: data/meuarquivo.pdf');
Parâmetros:
O método Output tem 4 opções:
*
*I:
Exibe o buffer diretamente na resposta (por exemplo no browser) com se fosse um PDF usando Content-Type: application/pdf
*F:
Salva o arquivo em uma pasta no servidor
*D:
Força o download do buffer
*S:
Retorna o buffer como uma string
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
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Q: In react-native App when creating pod file the target is empty In my react-native app I want to integrate cocoapod,after
sudo gem install cocoapods. When I create the pod file is using pod init command ,then the pod file is missing the target .
Below is the pod file code.
# Uncomment the next line to define a global platform for your project
# platform :ios, '9.0'
target ' ' do
# Uncomment the next line if you're using Swift or would like to use dynamic frameworks
# use_frameworks!
# Pods for
end
Due to this when installing pod framework ,its not working.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
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Healthcare Industry News: cord blood
DOR BioPharma Forms orBec(R) Medical Advisory Board
MIAMI, FL--(Healthcare Sales & Marketing Network)--Mar 20, 2007 -- DOR BioPharma, Inc. (OTC BB:DORB.OB ) ("DOR" or the "Company") announced today the formation of a North American Medical Advisory Board ("MAB") to provide clinical guidance to the Company in anticipation of the potential US marketing approval for orBec® for the treatment of gastrointestinal Graft versus Host disease ("GI GVHD"). The MAB is made up of clinical and research physicians with extensive backgrounds in the field of allogeneic hematopoietic cell transplantation ("HCST"), the clinical setting that GI GVHD occurs within. The MAB will be chaired by George B. McDonald, MD, Professor of Medicine at the University of Washington, Member at the Fred Hutchinson Cancer Research Center, inventor of orBec®, and a consultant to DOR.
Dr. McDonald stated, "I am pleased with the outstanding orBec® medical advisory board we have put together. This highly experienced group of transplant oncologists is composed of leaders in the area of hematopoietic cell transplantation and GVHD, a common and potentially fatal complication of transplant. They have been selected based on their substantial practice and research contributions to furthering the science of these fields. We are very much looking forward to their productive involvement with the orBec® program."
"We are continuing to position ourselves for potential US approval of our lead product, orBec®, as well as focusing on additional clinical testing of orBec® in other important therapeutic areas," stated Christopher J. Schaber, PhD, President and Chief Executive Officer of DOR. "The combined experience of this panel of experts and opinion leaders will provide us with critical support and guidance necessary to achieve these goals. If approved by the FDA, orBec® will be the first treatment for GI GVHD to be marketed in the United States."
The orBec® MAB Members
George B. McDonald, MD is a Professor of Medicine at the University of Washington School of Medicine and a Member at the Fred Hutchinson Cancer Research Center, where he is head of the Gastroenterology/Hepatology Section. He also serves as the head of the Program in Complications of Cancer Treatment that has as its goals the reduction of morbidity from cancer treatment, improved survival, and prevention of late sequelae of cancer treatment. Dr. McDonald's research is focused on gastrointestinal and hepatobiliary complications of hematopoietic cell transplantation, specifically problems involving the toxicity of high-dose myeloablative regimens that are used to prepare patients for transplant and acute and chronic graft-versus-host disease involving the gastrointestinal tract and liver. He has recently developed and validated a new method of assessing the severity of acute GVHD, called the acute GVHD Activity Index, an accurate predictor of transplant-related mortality. He was the lead investigator on the clinical trials that pioneered the use of topical corticosteroid therapy with oral beclomethasone dipropionate for GI GVHD.
Joseph Antin, MD heads the Stem Cell Transplant Program of the Department of Medical Oncology at the Dana Farber Cancer Institute and is a founding member and past president of the American Society of Blood and Marrow Transplantation. He is a Professor of Medicine at the Harvard Medical School. His current clinical research interest focuses on pharmacologic control of GVHD using synergistic combinations of immunosuppressive drugs and to understand and harness the graft-versus-leukemia effect through non-myeloablative transplantation as well as selective depletion of T cells.
Joachim Deeg, MD is a Professor of Medical Oncology at the University of Washington and a Member of the Fred Hutchinson Cancer Research Center in Seattle. He has a long track record of investigations in preclinical models of GVHD and clinical trials for the prevention and treatment of GVHD. He has multiple peer-reviewed publications in the field and has co-edited a major Textbook on GVHD, now in its third edition. His current research interests at the Fred Hutchinson Cancer Research Center include the pathophysiology and therapy of the myelodysplastic syndrome, programmed cell death and GVHD.
Donna Przepiorka, MD is Professor of Medicine at the University of Tennessee Health Science Center. She was the Chair of the Oncology Drug Advisory Committee at the FDA from 2002-2004 and President of the International Society for Cellular Therapy from 2004-2006. She is a committee member of the National Comprehensive Cancer Network ("NCCN") for acute myeloid and chronic myelogenous leukemia and chairs the Program Project Review Committee of the NIH National Heart, Lung and Blood Institute ("NHLBI"). Her current interest is regulatory matters in drug development, especially biological agents.
Kirk Schultz, MD is an Associate Professor of Pediatrics and Pathology at the University of British Columbia in Vancouver, Canada, and Director of the Pediatric Oncology Research Group at British Columbia Children's Hospital. He is the recipient of the CIHR/Wyeth Clinical Research Chair in Transplantation. His major contributions have been in the immune therapy of Acute Lymphoblastic Leukemia and the pathology of Chronic Graft-versus-Host Disease. In 2000, he was elected the first Chair of the Canadian Bone Marrow Transplant ("BMT") Clinical Trial Group, a group that includes all adult and pediatric BMT centers in Canada. Since 2004, he is the Pediatric BMT Consortium Chair and developed the first pediatric focused section at the American Society of BMT.
Keith Sullivan, MD is the James B. Wyngaarden Professor of Medicine, Duke University Medical Center where he was Chief, Division of Medical Oncology and Transplantation. He currently is the Director, Long-term Follow-up and Information Research Program, in the Division of Cellular Therapy. He has extensively published in the area of bone marrow transplantation with over 400 peer-reviewed publications and book chapters. He is an electee of the Association of American Physicians and a Fellow of the American Association for the Advancement of Science. He leads the national multi-center transplant study for systemic sclerosis within the Hematopoietic Stem Cell Transplantation Trials for Autoimmune Disease at the NIH. His current research interests include the use of hematopoietic stem cell transplantation for treatment of autoimmune diseases and sickle cell anemia.
Daniel Weisdorf, MD is Professor of Medicine and Director of the University of Minnesota Blood and Marrow Transplant Program. He serves as the Scientific Director of the National Marrow Donor Program and Senior Research Advisor for the Center for International Bone Marrow Transplant Research as well as Scientific Director for its Acute Leukemia Committee. He is the University Principal Investigator on the NIH-sponsored Bone Marrow Transplantation Clinical Trials Network. His clinical and research interests are in application of blood and marrow transplant therapies for hematologic malignancies as well as extensive study of the clinical complications of transplantation including opportunistic infections and GVHD.
orBec® Submitted for Regulatory Approval
orBec®'s New Drug Application ("NDA") was filed with the U.S. Food and Drug Application ("FDA"), and the Marketing Authorization Application ("MAA") was filed with the European Medicines Evaluation Agency ("EMEA"). Both health authorities have accepted the filings for review and the FDA has said it will respond to DOR's NDA by July 21, 2007 under PDUFA guidelines. In addition, the FDA's Oncology Drug Advisory Committee ("ODAC") currently will review the NDA for orBec® on May 9, 2007.
About orBec®
orBec® represents a first-of-its-kind oral, locally acting therapy tailored to treat the gastrointestinal manifestation of GVHD, the organ system where GVHD is most frequently encountered and highly problematic. orBec®, if approved by the EMEA and the FDA, would be the first oral formulation of beclomethasone dipropionate ("BDP") available in the European Union and the United States, respectively. orBec® is intended to reduce the need for systemic immunosuppressive drugs to treat GI GVHD. BDP is a highly potent, topically active corticosteroid that has a local effect on inflamed tissue. BDP has been marketed in the U.S. and worldwide since the early 1970s as the active pharmaceutical ingredient in a nasal spray and in a metered dose inhaler for the treatment of patients with allergic rhinitis and asthma. orBec® is formulated for oral administration as a single product consisting of two tablets; one tablet is intended to release BDP in the proximal portions of the GI tract and the other tablet is intended to release BDP in the distal portions of the GI tract.
In addition to issued patents and pending worldwide patent applications held by or exclusively licensed to DOR, orBec® also benefits from orphan drug designations in the U.S. and in Europe for the treatment of GI GVHD, which provide for 7 and 10 years of post-approval market exclusivity, respectively.
About GI GVHD
GI GVHD is a debilitating and painful disease. It is a common disorder among immuno-compromised cancer patients after receiving allogeneic stem cell or bone marrow transplants. Unlike organ transplants where the patient's body may reject the organ, in GVHD it is the donor cells that begin to attack the patient's body -- most frequently the gastrointestinal tract, liver and skin. Patients with mild-to-moderate GI GVHD typically develop symptoms of anorexia, nausea, vomiting and diarrhea. If left untreated, GI GVHD can progress to ulcerations in the lining of the GI tract, and in its most severe form, can be fatal.
About Allogeneic Bone Marrow/Stem Stem Cell Transplantation (HSCT)
HSCT is considered a potentially curative option for many leukemias as well as other forms of blood cancer. In an allogeneic HSCT procedure, hematopoietic stem cells are harvested from a closely matched relative or unrelated person, and are transplanted into the patient following either high-dose chemotherapy or intense immunosuppressive conditioning therapy. The curative potential of allogeneic HSCT is now partly attributed to the so-called graft-versus-leukemia ("GVL") or graft-versus-tumor ("GVT") effects of the newly transplanted donor cells to recognize and destroy malignant cells in the recipient patient.
The use of allogeneic HSCT has grown substantially over the last decade due to advances in human immunogenetics, the establishment of unrelated donor programs, the use of cord blood as a source of hematopoietic stem cells and the advent of non-myeloablative conditioning regimens ("mini-transplants") that avoid the side effects of high-dose chemotherapy. Based on the latest statistics available, it is estimated that there are more than 12,000 HSCT procedures annually in the U.S. and a comparable number in Europe. Estimates as to the current annual rate of increase in these procedures are as high as 20%. High rates of morbidity and mortality occur in this patient population. Clinical trials are also underway testing allogeneic HSCT for treatment of some metastatic solid tumors such as breast cancer, renal cell carcinoma, melanoma and ovarian cancer. Allogeneic transplants have also been used as curative therapy for several genetic disorders, including immunodeficiency syndromes, inborn errors of metabolism, thalassemia and sickle cell disease. The primary toxicity of allogeneic HSCT, however, is GVHD. In GVHD, the newly transplanted donor cells damage cells in the recipient's gastrointestinal tract, liver and skin.
About DOR BioPharma, Inc.
DOR BioPharma, Inc. is a biopharmaceutical company developing products to treat life-threatening side effects of cancer treatments and serious gastrointestinal diseases, and vaccines for certain bioterrorism agents. DOR's lead product, orBec® (oral beclomethasone dipropionate), is a potent, locally acting corticosteroid being developed for the treatment of GI GVHD, a common and potentially life-threatening complication of bone marrow transplantation. DOR has filed an NDA with the FDA for the treatment of GI GVHD, and has received a PDUFA date of July 21, 2007. In addition, the FDA's Oncology Drug Advisory Committee ("ODAC") will review the NDA for orBec® on May 9, 2007. An MAA with the EMEA for orBec® has also been filed and validated. orBec® may also have application in treating other gastrointestinal disorders characterized by severe inflammation.
Through its Biodefense Division, DOR is developing biomedical countermeasures pursuant to the recently enacted Project BioShield Act of 2004. DOR's biodefense products in development are recombinant subunit vaccines designed to protect against the lethal effects of exposure to ricin toxin and botulinum toxin. DOR's ricin toxin vaccine, RiVax(TM), has been shown to be safely tolerated and immunogenic in a Phase 1 clinical trial in normal volunteers.
For further information regarding DOR BioPharma, please visit the Company's website located at www.dorbiopharma.com.
This press release contains forward-looking statements, within the meaning of Section 21E of the Securities Exchange Act of 1934, that reflect DOR BioPharma, Inc.'s current expectations about its future results, performance, prospects and opportunities, including statements regarding the potential use of orBec® for the treatment of gastrointestinal GVHD and the prospects for regulatory filings for orBec®. Where possible, DOR has tried to identify these forward-looking statements by using words such as "anticipates," "believes," "intends," or similar expressions. These statements are subject to a number of risks, uncertainties and other factors that could cause actual events or results in future periods to differ materially from what is expressed in, or implied by, these statements. DOR also cannot assure you that it will be able to successfully develop or commercialize products based on its technology, including orBec®, particularly in light of the significant uncertainty inherent in developing vaccines against bioterror threats, manufacturing and conducting preclinical and clinical trials of vaccines, and obtaining regulatory approvals, that its technologies will prove to be safe and effective, that its cash expenditures will not exceed projected levels, that it will be able to obtain future financing or funds when needed, that product development and commercialization efforts will not be reduced or discontinued due to difficulties or delays in clinical trials or due to lack of progress or positive results from research and development efforts, that it will be able to successfully obtain any further grants and awards, maintain its existing grants which are subject to performance, enter into any biodefense procurement contracts with the U.S. Government or other countries, that the U.S. Congress may not pass any legislation that would provide additional funding for the Project BioShield program, that it will be able to patent, register or protect its technology from challenge and products from competition or maintain or expand its license agreements with its current licensors, or that its business strategy will be successful. Important factors which may affect the future use of orBec® for gastrointestinal GVHD include the risks that: because orBec® did not achieve statistical significance in its primary endpoint in the pivotal Phase III clinical study (i.e. a p-value of less than or equal to 0.05), the FDA may not consider orBec® approvable based upon existing studies, orBec® may not show therapeutic effect or an acceptable safety profile in future clinical trials, if required, or could take a significantly longer time to gain regulatory approval than DOR expects or may never gain approval; DOR is dependent on the expertise, effort, priorities and contractual obligations of third parties in the clinical trials, manufacturing, marketing, sales and distribution of its products; or orBec® may not gain market acceptance; and others may develop technologies or products superior to orBec®. These and other factors are described from time to time in filings with the Securities and Exchange Commission, including, but not limited to, DOR's most recent reports on Form 10-QSB and Form 10-KSB. DOR assumes no obligation to update or revise any forward-looking statements as a result of new information, future events, and changes in circumstances or for any other reason.
Source: DOR BioPharma
Search: DOR BioPharma
Search: orBec
|
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"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
\label{sec:intro}
We are currently witnessing the beginnings of a new era of technology---an era of technology that fully exploits the unique resources of quantum mechanics. This \lq\lq{second quantum revolution}\rq\rq \ (Dowling and Milburn, 2003) will demand a new generation of engineering methodologies. {\em Quantum engineering} will be a new branch of engineering focused on
the design and manufacturing of quantum technologies. At present, quantum engineering is embryonic and includes activities that
range from laboratory experimentation
of new devices and systems, to the development of the theory that inspires and supports the creation of these new devices and systems.
Just as today's non-quantum engineering arose from the foundations of classical physics, mathematics, and (most importantly) entrepreneurship, quantum engineering is beginning to evolve from the fundamental principles of quantum physics and mathematics and examples of quantum entrepreneurship can now be seen.
This document is focused on one aspect of this second quantum revolution---quantum control. Control, of course, is essential to technology, and indeed played a key enabling role during the industrial revolution. Quantum control originated in the sciences, and we are now beginning to see the growth of quantum control research in engineering. In what follows we look at some recent developments in quantum control theory from the perspective of a control theorist. We
hope that readers will see how various contributions to quantum control fit together. In particular, the paper explains some fundamental ideas of {\em feedback} control through the study of a relatively simple two-level system coupled to optical field channels. This model for this system includes both continuous and impulsive dynamics.
This paper is organized as follows.
In Section \ref{sec:types} we describe some of the main types of quantum control that have appeared in the literature. In Section \ref{sec:open-loop} we discuss several aspects of open loop control: (i) time optimal control for a closed system, (ii) impulsive optimal control, and (iii) regulation of systems subject to relaxation. In preparation for our more detailed discussion of quantum feedback, Section \ref{sec:networks} describes {\em quantum feedback networks} (QFN), while Section \ref{sec:filter} presents some of the basic ideas of {\em quantum filtering}, which is seen as a natural extension of statistical reasoning to quantum mechanics. We then consider two types of optimal measurement feedback
control problems in Section \ref{sec:optimal}, and discuss the important idea of {\em information states} for feedback control.
Finally, Section \ref{sec:coherent} presents some ideas concerning coherent feedback control.
\subsection*{Notation and Preliminaries}
Quantum mechanics is usually represented mathematically using a Hilbert space. In this chapter, $\mathfrak{H}$ will denote a finite dimensional Hilbert space, say $\mathfrak{H}=\mathbb{C}^n$, the $n$-dimensional complex vector space.
In Dirac's notation, the inner product for $\mathfrak{H}$ is denoted
$$
\langle \psi \vert \phi \rangle = \sum_{j=1}^n \psi^\ast_j \phi_j .
$$
The vector $\vert \phi \rangle \in \mathfrak{H}$ is denoted (represented by a column vector of length $n$ with complex entries $\psi_j$), is called a {\em ket}, while dual (row) vectors are called {\em bras} and written as
$$
\langle \psi \vert .
$$
A linear operator on $\mathfrak{H}$ is denoted $A$\footnote{We do not use $\hat{}$ to indicate operators. Later, we will use $\hat{}$ to denote an {\em estimate} $\hat X$ of operators $X$.} (typically represented by an $n\times n$ complex matrix).
For any operator $A$ its {\em adjoint}\footnote{Note that we use $A^\ast$ to denote {\em adjoint} of an operator $A$ instead of $A^\dagger$. However, if $A=(a_{jk})$ is a matrix (with operator or complex number entries $a_{jk}$), we write $A^\dagger = (a^\ast_{kj})$ (conjugate transpose).}
$A^\ast$ is an operator defined by
$$
\langle A^\ast \psi \vert \phi \rangle = \langle \psi \vert A\phi \rangle \ \ \text{for all} \ \langle \psi \vert,
\vert \phi \rangle .
$$
The adjoint $\vert \psi \rangle^\ast$ of a vector $\vert \psi \rangle$ is a dual vector (represented by a row vector):
$$
\langle \psi \vert = \vert \psi \rangle^\ast .
$$
\index{normal} \index{self-adjoint} \index{unitary}
An operator $A$ is called {\em normal} if
$AA^\ast=A^\ast A$. Two important types of normal operators
are {\em self-adjoint} ($A=A^\ast$), and {\em unitary}
($A^\ast=A^{-1}$).
\index{spectral theorem}
The {\em spectral theorem} says that
if $A$ is a self-adjoint operator on a finite dimensional Hilbert space $\mathfrak{H}=\mathbb{C}^n$,
the eigenvalues (not necessarily distinct) $\mathrm{spec}(A)=\{ a_j \}_{j=1}^n$ of $A$ are real and $A$ can be written as
\begin{equation}
A = \sum_{a\in \mathrm{spec}(A) } a P_a ,
\label{spectral-1}
\end{equation}
where $P_a$ is the projection
\begin{equation}
P_a = \sum_{j : a_j= a} \vert a_j \rangle \langle a_j \vert ,
\label{spectral-1a}
\end{equation}
and $\{ \vert a_j \rangle \}_{j=1}^n$ are the corresponding orthonormal eigenvectors.
The projections resolve the identity $\sum_{a \in \mathrm{spec}(A)} P_a = I$.
Let's denote the collection of all (bounded, linear) operators on a Hilbert space $\mathfrak{H}$ by $\mathscr{B}=\mathscr{B}(\mathfrak{H})$. The set
$\mathscr{B}$ can be thought of as a {\em vector space}, where operators $A \in \mathscr{B}$ are \lq\lq{vectors}\rq\rq. Indeed, if $\alpha_1$ and $\alpha_2$ are complex numbers and $A_1, A_2 \in \mathscr{B}$, then the linear combination $\alpha_1 A_1+\alpha_2 A_2$ is the operator in $\mathscr{B}$ defined by
\begin{equation}
(\alpha_1 A_1+\alpha_2 A_2) \vert \psi \rangle = \alpha_1 A_1 \vert \psi \rangle +\alpha_2 A_2 \vert \psi \rangle, \ \text{for all} \ \vert \psi \rangle \in \mathfrak{H}.
\label{eq:operators-linear-comb}
\end{equation}
Note that the linear combination on the RHS of (\ref{eq:operators-linear-comb}) is a linear combination of vectors in the Hilbert space $\mathfrak{H}$.
We can multiply operators,
\begin{equation}
(A_1A_2) \vert \psi \rangle = A_1( A_2 \vert \psi \rangle ) , \ \text{for all} \ \vert \psi \rangle \in \mathfrak{H},
\label{eq:operators-mult-comb}
\end{equation}
so that $A_1A_2 \in \mathscr{B}$ if $A_1, A_2 \in \mathscr{B}$. Also, the adjoint $A^\ast \in \mathscr{B}$ if $A \in \mathscr{B}$. So the collection $\mathscr{B}$ of operators is closed under addition, scalar multiplication, multiplication, and adjoints---mathematically, $\mathscr{B}$ is called a {\em $\ast$-algebra}. This mathematical structure is fundamental to quantum mechanics.
\index{$\ast$-algebra} \index{algebra!$\ast$}
Tensor products are used to describe \emph{composite systems}. If
$\mathcal{H}_1$ and $\mathcal{H}_2$ are Hilbert spaces, the {\em
tensor product} $\mathcal{H}_1 \otimes \mathcal{H}_2$ is the Hilbert
space consisting of linear combinations of the form $\vert \psi_1 \rangle \otimes
\vert \psi_2 \rangle $, and inner product $\langle \psi_1 \otimes \psi_2, \phi_1
\otimes \phi_2 \rangle = \langle \psi_1, \phi_1 \rangle \langle
\psi_2, \phi_2 \rangle$. Here, $\vert \psi_1\rangle, \vert \phi_1 \rangle \in \mathcal{H}_1$ and
$\vert \psi_2 \rangle, \vert \phi_2 \rangle \in \mathcal{H}_2$. If $A_1$ and $A_2$ are operators
on $\mathcal{H}_1$ and $\mathcal{H}_2$, respectively, then
$A_1\otimes A_2$ is an operator on $\mathcal{H}_1 \otimes
\mathcal{H}_2$ and is defined by $(A_1\otimes A_2)(\vert \psi_1 \rangle \otimes
\vert \psi_2 \rangle) = A_1 \vert \psi_1 \rangle \otimes A_2 \vert \psi_2 \rangle$. Often,
$\vert \psi_1 \rangle \otimes
\vert \psi_2 \rangle $ is written as $\vert \psi_1 \psi_2 \rangle$, and
$A_1\otimes A_2$ is
written $A_1A_2$.
\section{Types of Quantum Control}
\label{sec:types}
Due to their relative simplicity and tractability,
the two-level quantum system and the quantum harmonic oscillator are two of the most important prototype models for quantum systems. These models are widely used for describing real physical systems, as well as for tutorial purposes. In this article we discuss a range of aspects of quantum control primarily focused on the basic two-level quantum system as the system to be controlled. Two level systems are used in quantum computing as the {\em qubit}, in NMR spectroscopy as a basic {\em spin system}, and in quantum optics as a model for an {\em atom} with two energy levels. The oscillator plays a role in the representation of an electromagnetic field to which the two-level atomic system is coupled, Figure \ref{fig:atom-field-1}, \cite[Fig. 9.1]{GZ00}.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.8]{atom-field.eps}
\caption{An atom interacting with an electromagnetic field (e.g. photon emission).}
\label{fig:atom-field-1}
\end{center}
\end{figure}
In this chapter we consider a two-level atom coupled to several electromagnetic field channels.
In the absence of any other influences, the behavior of the atom will evolve in time according to the laws of quantum mechanics, as determined by the self-energy of the atom, the nature of the fields and how they are coupled to the atom. The field channels may be used to gather information about the atom, and to influence the behavior of the atom.
Figure \ref{fig:atom-1} shows a schematic representation of an atom coupled to a pair of optical field channels, one of which is used to describe light shone on the atom, while the other channel contains the outgoing light, \cite[Sec. 9.2]{GZ00}. Also shown is a channel used to apply a rapid radio-frequency (RF) pulse.
As will be explained, light shone on the atom may be regarded as a control signal that may be classical or quantum in nature; the later allowing for coherent feedback control.
In this article we are interested in idealized impulsive models for pulses (zero width and infinite height).\footnote{In reality all pulses have non-zero width and finite height, and the impulsive model is useful when the time scales are such that the response to a rapid pulse is much faster than the other dynamics of the system.} The impulsive signals are classical signals, but may be used to effect coherent transformations of the system.
We now present a model for a controlled two level atom.
Since a two-level atom is a quantum system with two energy levels, the model makes use of the Pauli matrices
\begin{eqnarray}
\sigma_0 = I = \left( \begin{array}{cc}
1 & 0 \\
0 & 1
\end{array} \right) , \ \sigma_x = \left( \begin{array}{cc}
0 & 1 \\
1 & 0
\end{array} \right) , \
\sigma_y = \left( \begin{array}{cc}
0 & -i \\
i & 0
\end{array} \right) , \
\sigma_z = \left( \begin{array}{cc}
1 & 0 \\
0 & -1
\end{array} \right) .
\label{eq:pauli-def}
\end{eqnarray}
Any observable of the atom can be expressed in terms of these matrices.
In particular, the atomic energy levels are the eigenvalues $\pm \frac{1}{2} \omega$ of the Hamiltonian $H=\frac{1}{2}\omega \sigma_z$ describing the self-energy of the atom.
The atom interacts with the field channels by an exchange of energy that may be described by first principles in terms of an interaction Hamiltonian, \cite[Chapter 3]{GZ00}. In this article we use an idealized quantum noise model for the open atom-field system which is well justified theoretically and experimentally, \cite{HP84}, \cite{GC85}, \cite{KRP92}, \cite{GZ00}. Each field channel has input and output components, modeled as quantum stochastic processes. The input processes, $B_1(t)$ and $B_2(t)$, drive an interaction-picture equation for a unitary operator $U(t)$ governing the atom-field system. If $\vert \psi_a \rangle$ and $\vert \psi_f \rangle$ are initial atomic and field states respectively, the state of the atom-field system at time $t$ is $U(t) \vert \psi_a \psi_f \rangle$.
This continuous (in time) stochastic unitary model holds in the absence of the above-mentioned impulsive actions. Now suppose that impulses are applied at times $0 \leq \tau_0 < \tau_1 < \ldots$ by selection of a unitary $V$ from a set $\mathbf{V}$ of unitaries. If $V_k \in \mathbf{V}$ is selected at time $\tau_k$, the state immediately after the impulse has been applied is $V_k U(\tau_k) \vert \psi_a \psi_f \rangle$. So the impulse is modeled as instantaneously effecting a unitary transformation. Combining the continuous and impulsive motions, we see that if $\tau_k < t < \tau_{k+1}$ the overall unitary at time $t$ is
\begin{equation}
U(t) = U(t, \tau_k) V_{k} U( \tau_k, \tau_{k-1}) V_{{k-1}} \ldots U(\tau_1, \tau_0)V_{0} U(\tau_0, 0) ,
\label{eq:hybrid-1}
\end{equation}
where $V_k \in \mathbf{V}$ indicates which impulse was selected at time $\tau_k$ and $U(t,s)$ is the unitary for the continuous motion on the time interval $(s,t)$ ($s < t$), with $U(t,t)=I$.
Let's now look at the equations governing the hybrid continuous-impulsive dynamics. We suppose that the two-channel field is initially in the vacuum state, which we denote by $\vert \psi_f \rangle= \vert 00 \rangle$. In this case the input processes $B_1(t)$ and $B_2(t)$ are independent quantum Wiener processes, for which the non-zero Ito product are $dB_j(t) dB^\ast_j(t)=dt$ ($j=1,2$). The atom-field coupling is determined by coupling operators $L_1=\sqrt{\kappa_1}\, \sigma_-$ and $L_2=\sqrt{\kappa_2}\, \sigma_-$, where
$$
\sigma_- = \left(
\begin{array}{cc}
0 & 0
\\
1 & 0
\end{array}
\right)
$$
is the lowering operator (the raising operator is defined by $\sigma_+ = \sigma_-^\dagger$), and $\kappa_1, \kappa_2 > 0$ are parameters describing the strength of the coupling to each channel.
The evolution of the unitary $U(t)$ is given by
\begin{eqnarray}
d U(t) &=& \{ \sqrt{\kappa_1}\, dB_1^\ast(t) \sigma_- - \sqrt{\kappa_1}\, \sigma_+ dB_1(t)
+ \sqrt{\kappa_2}\, dB_2^\ast(t) \sigma_- - \sqrt{\kappa_2}\, \sigma_+ dB_2(t)
\nonumber \\
&& \hspace{2.0cm}
- \frac{1}{2} ((\kappa_1+\kappa_2) \sigma_+ \sigma_- + i \omega \sigma_z ) dt \} U(t),
\ \ \tau_k < t \leq \tau_{k+1},
\nonumber \\
U(\tau_k^+) & = & V_{k} U(\tau_k) .
\label{eq:hybrid-2}
\end{eqnarray}
Here, $\tau_k^+$ indicates the value immediately after $\tau_k$, i.e., the limit from the right. Equation (\ref{eq:hybrid-2}) is {\em quantum stochastic differential equation with impulses}, whose solution $U(t)$ (of the form (\ref{eq:hybrid-1})) is determined by a sequence of time-impulse pairs
\begin{equation}
\gamma = ( (\tau_0, V_0), (\tau_1, V_1), \ldots ) .
\label{eq:hybrid-3}
\end{equation}
So far, we have not described how modulation of the field can be used to control the atom. Let us now do so. Suppose we modulate the second field channel to be in a coherent state $\vert \frac{gu}{\sqrt{\kappa_2}} \rangle$, where $u(\cdot)$ is a classical function of time (a classical control signal). We may model this by replacing $dB_2(t)$ with $i u(t) dt + dB_2(t)$ (displacement, \cite[sec. 9.2.4]{GZ00}).
Provided the corresponding output channel is not used for further interconnection, this may equivalently be represented by replacing the Hamiltonian term $H=\frac{1}{2} \omega \sigma_z$ in equation (\ref{eq:hybrid-2}) by the control-dependent Hamiltonian
\begin{equation}
H(u) = \frac{1}{2} ( \omega \sigma_z + u \sigma_x).
\label{eq:hybrid-4}
\end{equation}
Thus we arrive at the following equations for the controlled atomic system:
\begin{eqnarray}
d U(t) &=& \{ \sqrt{\kappa_1}\, dB_1^\ast(t) \sigma_- - \sqrt{\kappa_1}\, \sigma_+ dB_1(t)
+ \sqrt{\kappa_2}\, dB_2^\ast(t) \sigma_- - \sqrt{\kappa_2}\, \sigma_+ dB_2(t)
\nonumber \\
&& \hspace{2.0cm}
- (\frac{1}{2} (\kappa_1+\kappa_2) \sigma_+ \sigma_- + i H ( u(t) ) ) dt \} U(t),
\ \ \tau_k < t \leq \tau_{k+1},
\nonumber \\
U(\tau_k^+) & = & V_{k} U(\tau_k) .
\label{eq:hybrid-5}
\end{eqnarray}
The solution $U(t)$ will be determined by a classical control signal $u(\cdot)$ and time-impulse sequence $\gamma$. The controlled two-level atom is illustrated in Figure \ref{fig:atom-1}. Choice of $u$ and $\gamma$ before being applied to the system (i.e. off-line) is called {\em open loop control}; no feedback of information from the system is used.
\begin{figure}[h]
\begin{center}
\includegraphics{atom-1.eps}
\caption{Classical open-loop control of a two-level atom showing input $B_1(t)$, $B_2(t)$ and output $B_{1,out}(t)$, $B_{2,out}(t)$ fields, as well as classical control variables $u(t)$, $\gamma$.}
\label{fig:atom-1}
\end{center}
\end{figure}
What does feedback mean in the context of our two-level atomic system? The answer depends on how information is extracted from the system and how this information is used to change the behavior of the system. Accordingly, it is helpful to identify the following types of quantum feedback:
\begin{enumerate}
\item
{\em Measurement feedback}. The output field channel, say $B_{out,1}(t)$, is continuously monitored (measured) and the measurement signal is processed by a classical system, called a {\em classical controller} (implemented, say, in classical analog or digital electronics), to determine the closed loop control signals $u$ and $\gamma$. This type of feedback involves a directional exchange of information between a quantum system and a classical system, and so involves a loss of quantum information. Measurement feedback is illustrated in Figure \ref{fig:atom-2}.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.75]{atom-2.eps}
\caption{Measurement feedback control of a two-level atom showing measurement of the output channel $B_{out,1}(t)$ field, the classical controller, as well as classical measurement and control signals.}
\label{fig:atom-2}
\end{center}
\end{figure}
\item
{\em Coherent feedback using quantum signals}. An output field, say $B_{out,1}(t)$, is not measured, but rather is provided as an input field to another open quantum systems, which we may call a {\em coherent quantum feedback controller}. This quantum controller coherently \lq\lq{processes}\rq\rq \ $B_{out,1}(t)$
to produce a field that is shone as an input into the second channel of the atom.
In the coherent feedback loop, information remains at the quantum level, and flows in one direction around the loop.
An example of a coherent feedback arrangement is shown in Figure \ref{fig:atom-3}.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.75]{atom-3.eps}
\caption{Coherent feedback control of a two-level atom showing the quantum controller and the quantum signals transmitted around the loop by the fields. No measurements are involved.}
\label{fig:atom-3}
\end{center}
\end{figure}
\item
{\em Coherent feedback using direct coupling}. Here, the atom is simply coupled directly to another quantum system (without the aid of the fields). This may be regarded as a form of feedback which is bidirectional, in the spirit of \lq\lq{control by interconnection}\rq\rq \ \cite{JW97}, and does not involve directional quantum signals transmitted via fields. The second quantum system also serves as a coherent quantum feedback controller, and as with coherent feedback using quantum signals, all information remains at the quantum level. Direct interaction between the atom and a coherent controller is shown schematically in Figure \ref{fig:atom-4}.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.75]{atom-4.eps}
\caption{Direct coupling of two quantum systems provides a form of feedback control. No signals, and no measurements, are used.}
\label{fig:atom-4}
\end{center}
\end{figure}
\end{enumerate}
There is a large literature on the topic of {\em coherent control}, mainly arising from applications in chemistry and NMR spectroscopy (for example, \cite{RVMK00}, \cite{KBG01}).
How does this fit into the terminology for control discussed above? Quite simply: direct couplings between systems may be engaged via the application of a pulse (the coupling is active while the pulse is one, and not active while it is off). The pulse may be regarded as an open loop signal applied to a composite system, resulting in a unitary transformation. In the limit that the pulse has zero width and infinite height, one obtains an impulsive representation of the open loop signal and unitary action.
Note that the quantum controllers discussed above could also depend on classical control signals, although we have not included this possibility explicitly in the above discussion. So the term \lq\lq{coherent control}\rq\rq \ is rather broadly used, and is meant to convey that the control actions preserve quantum coherence in some way.
Of course, the two-level model given above may be generalized in a number of ways. For instance, to cover other physical situations the Hamiltonian $H$ and coupling operators $L_1$ and $L_2$ may be redefined, and the fields may be placed in non-vacuum states.
\section{Open loop control}
\label{sec:open-loop}
Two of the very earliest papers on quantum control are \cite{VPB79} and \cite{HTC83}. The paper \cite{VPB79} discusses open quantum models, filtering, and optimal feedback control. This work was well ahead of its time, and was largely unknown for a considerable period. As we will see later on, Belavkin's far-sighted ideas are highly relevant today. The paper \cite{HTC83}, also very much ahead of its time, looks at open loop control of quantum systems, the subject of this section.
\subsection{Bilinear Systems}
\label{sec:open-loop-bilinear}
The simplest type of model for open loop control is that of an isolated two-level atom (no field couplings) with a Hamiltonian $H(u)$ depending on a classical control variable $u$. Indeed, we have the following differential equation for the unitary $U(t)$:
\begin{equation}
\dot{U}(t) = -i \frac{1}{2} ( \omega \sigma_z + u \sigma_x ) U(t), \ \ U(0)=I.
\label{eq:open-1}
\end{equation}
The open loop control system (\ref{eq:open-1}) defines a {\em bilinear} control system evolving on the Lie group $SU(2)$.
Bilinearity refers to the presence of products involving the control variables and the variable being solved for (the unitary $U(t)$).
It is completely deterministic, and beginning with the pioneering paper \cite{HTC83}, a large literature has accumulated studying this type of system using methods from nonlinear control theory, and applying the results to a range of problems (see, for example, \cite{DD07}).
Let's now take a closer look at the dynamics of the atom, first in the {\em Heisenberg picture}, where atomic operators $X$ evolve according to $\dot X = -i [X, H ]$. Since all atomic operators can be expressed in terms of the Pauli matrices, it suffices to determine the dynamics of $X=\sigma_x, \sigma_y$ and $\sigma_z$. By using the commutation relations
\begin{equation}
[ \sigma_x, \sigma_y ] = 2i \sigma_z, \ \
[ \sigma_y, \sigma_z ] = 2i \sigma_x, \ \
[ \sigma_z, \sigma_x ] = 2i \sigma_y ,
\label{eq:pauli-commutations}
\end{equation}
we find that
\begin{equation}
\left( \begin{array}{c}
\dot{\sigma}_x (t) \\ \dot{\sigma}_y (t) \\ \dot{\sigma}_z(t)
\end{array} \right) =
\left( \begin{array}{ccc}
0 & -\omega & 0
\\
\omega & 0 & -u
\\
0& u & 0
\end{array} \right)
\left( \begin{array}{c}
\sigma_x (t) \\ \sigma_y (t) \\ \sigma_z(t)
\end{array} \right) .
\label{eq:open-2}
\end{equation}
Equation (\ref{eq:open-2}) gives a complete description of the controlled atomic motion, expressed as a bilinear control system in the $\ast$-algebra $\mathscr{M}_2$ (the vector space of $2\times 2$ complex matrices, with the usual matrix multiplication and involution given by the matrix adjoint).
Now we switch to the {\em Schrodinger picture}, within which state vectors evolve as
$\vert \psi(t) \rangle = U(t) \vert \psi \rangle$, or
$\dot \vert \psi \rangle = -i H \vert \psi \rangle$; more generally, density operators $\rho$ evolve as $\rho(t)= U(t) \rho U^\ast(t)$, or
$\dot \rho = i[\rho, H]$. Now any density operator may be expressed in the form
\begin{equation}
\rho = \frac{1}{2} ( I + x \sigma_x + y \sigma_y + z \sigma_z ),
\label{eq:open-3}
\end{equation}
where the (real) vector $r=(x,y,z)^T$ is known as the {\em Bloch vector}, with length $\sqrt{x^2 + y^2 +z^2} \leq 1$. The Schrodinger dynamics is given in terms of the Bloch vector as follows:
\begin{equation}
\left( \begin{array}{c}
\dot{x}(t) \\ \dot{y} (t) \\ \dot{z}(t)
\end{array} \right) =
\left( \begin{array}{ccc}
0 & -\omega & 0
\\
\omega & 0 & -u
\\
0 & u & 0
\end{array} \right)
\left( \begin{array}{c}
x(t) \\ y(t)\\ z(t)
\end{array} \right) .
\label{eq:open-4}
\end{equation}
This equation is a bilinear system in the solid {\em Block sphere} $x^2 + y^2 +z^2 \leq 1$. If the initial Bloch vector is on the surface of the Bloch sphere, i.e. $x^2(0) + y^2(0) +z^2(0) = 1$, as is the case for initial state vectors $\vert \psi(0) \rangle$ (i.e. $\rho(0) = \vert \psi(0) \rangle \langle \psi(0) \vert$), then equation (\ref{eq:open-4}) ensures that $x^2(t) + y^2(t) +z^2(t) = 1$ for all $t$, and so describes a bilinear system on the surface of the Bloch sphere.
We see therefore that open loop control of isolated systems leads to interesting bilinear control systems defined on spaces other than Euclidean spaces (Lie groups, $\ast$-algebras, and manifolds). However, it is important to point out that for some types of quantum systems {\em linear} control systems arise, even though the equations for the unitary are bilinear. This happens in the case of the quantum harmonic oscillator, where the commutation relation $[a,a^\ast]=1$ (a constant, unlike the commutation relations (\ref{eq:pauli-commutations}) for the Pauli matrices) gives rise to a linear equation for the annihilation $a$ and creation $a^\ast$ operators.
\subsection{Optimal Control}
\label{sec:open-loop-optimal}
We turn now to the problem of {\em optimally} controlling the atomic system discussed in Section \ref{sec:open-loop-bilinear}. By this we mean (for the moment) to find an open loop control signal $t \mapsto u(t)$ that optimizes a performance criterion chosen to reflect a desired objective. For instance, the paper \cite{KBG01} used optimal control theory to design pulse sequences to achieve rapid state transfers. The performance criterion used was the time taken to go from an initial unitary to a target final unitary; a problem of {\em time-optimal} control. The authors were able to exploit the rich structure of Lie groups and Lie algebras to develop an elegant formulation of the problem and explicit solutions in some cases.
It is worth remarking at this point that controllability is closely related to optimal control; indeed, the fundamental ideas of controllability and observability were developed by Kalman in his studies of linear quadratic optimal control problems \cite{REK60a}. In time-optimal control, the minimum time function $T(x_0)$ is finite precisely when it is possible to steer a system from the initial state to a given target state $x_f$ \cite{EJ89}.
Let's now look at time-optimal control of the atomic system, not at the level of the unitary, but at the level of state vectors. Given a fixed target state $\vert \psi_f \rangle$, and an initial state $\vert \psi_0 \rangle$, find a control signal $u(\cdot)$ that steers the atom from $\vert \psi_0 \rangle$ to $\vert \psi_f \rangle$ in minimum time. In terms of Bloch vectors, given a fixed target state $r_f = (x_f, y_f, z_f)^T$ and an initial state $r_0 = (x_0, y_0, z_0)^T$, find a control signal $u(\cdot)$ that steers the atom (via the dynamics (\ref{eq:open-4})) from $r_0$ to $r_f$ in minimum time.
In order to formalize this, we define the {\em minimum time function} $T(r)$ (the value function for time-optimal control) by
\begin{equation}
T(r) = \inf_{u(\cdot)} \{ t_f \ : r(0)=r, \ r(t_f) = r_f \}.
\label{eq:min-t-1}
\end{equation}
Here, $t_f$ is the time taken for the atom to move from the initial Bloch vector $r$ to the final Bloch vector $r_f$ using the control signal $u(\cdot)$. Thus, $T(r)$ is the minimum time over all control signals.
If there is no restriction on the range of the control signal $u$, then the time-optimal control problem is singular and leads to impulsive solutions, as in \cite{KBG01}, \cite{SJ08}. A reformulation of this problem using a hybrid model is described in Section \ref{sec:open-loop-impulsive}. For the remainder of this section, let's assume that the controls $u$ take values in closed interval $\mathbf{U}=[-1,1]$.
Before proceeding, let's re-write (\ref{eq:open-4}) in compact form
\begin{equation}
\dot{r}(t) = f(r(t), u(t)),
\label{eq:open-5}
\end{equation}
where the vector field $f(r,u)$ is defined by the right hand side of (\ref{eq:open-4}).
{\em Dynamic programming} is a basic tool in optimal control theory, \cite{FR75}, \cite{FS06}. Suppose we have a smooth non-negative solution $S(r)$ to the {\em dynamic programming equation} (DPE) or {\em Hamilton-Jacobi-Bellman} (HJB) equation
\begin{eqnarray}
\min_{u \in \mathbf{U}} \{ DS(r) [ f(r,u)] +1 \} &=& 0,
\label{eq:mintime-2-a}
\\
S(r_f) & = & 0.
\label{eq:mintime-2-b}
\end{eqnarray}
Equation (\ref{eq:mintime-2-a}) is a nonlinear partial differential equation on the Bloch sphere, in which $DS(r)[f(r,u)]$ denotes the directional derivative of the function $S$ at the Bloch vector $r$ in the direction $f(r,u)$. Equation (\ref{eq:mintime-2-b}) is a boundary condition, corresponding to the fact that the optimal time taken to go from $r_f$ to itself is zero.
The main purpose of the DPE is the {\em verification theorem} \cite{FR75}, which allows us to test a candidate optimal control signal for optimality. Indeed, suppose we have a control signal $u^\star(\cdot)$ that attains the minimum in the DPE (\ref{eq:mintime-2-a}), i.e.
\begin{equation}
u^\star(t) = - \mathrm{sign}( S_z(r^\star(t)) y^\star(t) - S_y(r^\star(t)) z^\star(t) ),
\label{eq:mintime-3}
\end{equation}
where\footnote{sign$(\xi) = +1$ if $\xi \geq 0$ and sign$(\xi)=-1$ if $\xi < 0$.}
$r^\star(\cdot)$ is the corresponding trajectory of (\ref{eq:open-5}) with initial condition $r^\star(0)=r$. Then $u^\star(\cdot)$ is optimal and $S(r)$ equals $T(r)$, the minimum time function defined by (\ref{eq:min-t-1}).
To see why the verification theorem is true, let $u(\cdot)$ be any control signal steering $r$ to $r_f$, and let $0 < t < t_f$. Now integrate equation (\ref{eq:mintime-2-a}) along the trajectory to obtain
\begin{eqnarray}
S(r(t)) &=& S(r) + \int_0^t DS(r(s))[f(r(s), u(s))] ds
\nonumber \\
& \geq & S(r) - t
\label{eq:mintime-4}
\end{eqnarray}
with equality if $u(\cdot)=u^\star(\cdot)$. Setting $t=t_f$ we see that $S(r)\leq t_f$ with equality if $u(\cdot)=u^\star(\cdot)$. Hence $S(r)=T(r)$ and $u^\star(\cdot)$ is optimal.
Expression (\ref{eq:mintime-3}) suggests a formula for \lq\lq{feedback}\rq\rq \ optimal controls. Define
\begin{equation}
\mathbf{u}^\star(r) = - \mathrm{sign} ( S_z(r)y - S_y(r) z),
\label{eq:mintim-5}
\end{equation}
that is, for any Bloch vector $r=(x,y,z)^T$, $\mathbf{u}^\star(r)$ is a control value that attains the minimum of $DS(r)[ f(r,u) ]$. However, this \lq\lq{feedback}\rq\rq \ formula requires knowledge of the Bloch vector $r$, which is not possible in the present context - no measurement information is available, and the quantum state is not a measurable quantity. However, expression (\ref{eq:mintime-3}) may be used off-line, in a computer simulation to determine an optimal {\em open loop} control signal. If one wishes, the optimizing control may be substituted into the DPE (\ref{eq:mintime-2-a}) to re-write it in the form
\begin{equation}
S_x(r) (-\omega y) + S_y(r)(\omega x) - \vert S_z(r) y - S_y(r) z \vert +1 =0.
\label{eq:mintime-5a}
\end{equation}
Where does the DPE (\ref{eq:mintime-2-a}) come from? Well, if the minimum time function is sufficiently smooth, then it solves the DPE (\ref{eq:mintime-2-a}) (by definition it satisfies (\ref{eq:mintime-2-b})). To see this, for any $t>0$ the minimum time function satisfies
\begin{equation}
T(r) = \min_{u(\cdot)} \{ \min(t,t_f) +T(r(\min(t,t_f)) ) \ : r(0)=r \} .
\label{eq:mintim-5a}
\end{equation}
Then if $t < t_f$ we may differentiate (\ref{eq:mintim-5}) to obtain (\ref{eq:mintime-2-a}).
An important technical issue is that in general $T$ is not everywhere differentiable, and nonlinear PDEs like (\ref{eq:mintime-2-a}) do not in general admit smooth solutions. Nevertheless, $T$ solves (\ref{eq:mintime-2-a}) in a weaker sense that does not require smoothness. The theory of viscosity solutions was developed to deal with nonsmooth solutions to nonlinear PDE, \cite{FS06}.
In order to gain some more insight into the nature of the solution to the optimal control problem, we switch to polar coordinates and use the fact that the dynamics (\ref{eq:open-4}) preserves states on the surface of the Bloch sphere. We write $x=\sin \theta \cos \phi$, $y=\sin \theta \sin \phi$, $z=\cos \theta$, and find that the dynamics becomes
\begin{eqnarray}
\dot \theta(t) &=& - u(t) \sin \phi(t)
\label{eq:mintim-polar-1-theta}
\\
\dot \phi(t) &=& - u(t) \cot \theta(t) \cos \phi(t) + \omega.
\label{eq:mintim-polar-1-phi}
\end{eqnarray}
In these polar coordinates, the target state is $(\theta_f, \phi_f)$, and if
we write $\tilde S(\theta,\phi) = S( \sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta)$ we obtain the DPE
\begin{eqnarray}
\min_{u \in \mathbf{U}} \{ -u ( \tilde S_\theta(\theta, \phi) \sin \phi +\tilde S_\phi (\theta,\phi) \cot \theta \cos \phi ) + \tilde S_\phi(\theta, \phi) \omega
+1 \} &=& 0,
\label{eq:mintime-polar-2-a}
\\
\tilde S(\theta_f, \phi_f) & = & 0.
\label{eq:mintime-polar-2-b}
\end{eqnarray}
The control attaining the minimum in (\ref{eq:mintime-polar-2-a}) is
\begin{equation}
\mathbf{u}^\star (\theta, \phi) = \mathrm{sign} ( \tilde S_\theta(\theta, \phi) \sin \phi +\tilde S_\phi (\theta,\phi) \cot \theta \cos \phi ).
\label{eq:mintime-polar-3}
\end{equation}
\subsection{Impulsive Control}
\label{sec:open-loop-impulsive}
Suppose we remove the restriction on the range of the control signal, i.e. take $\mathbf{U}=\mathbb{R}$. Now the vector field $f(r,u)$ has the form $f(r,u)=f_0(r) + f_1(r)u$, and so if we attempted to find the minimum in the DPE (\ref{eq:mintime-2-a}) we would find that it is not defined, as the controls would need to be infinitely large. This singular situation leads us to impulsive control actions, \cite{KBG01}, \cite{SJ08}, which form the subgroup $\mathbf{V}$ of $SU(2)$ determined by the control Hamiltonian $\sigma_x u$, that is,
\begin{equation}
\mathbf{V}= \{ e^{-i v \sigma_x} \ : \ v \in \mathbb{R} \}.
\label{eq:impulse-0}
\end{equation}
Now let's use a hybrid continuous-impulsive model for time-optimal control on the surface of the Bloch sphere. Let $\gamma$ be an impulsive open loop control of the form $\gamma = ( (\tau_0, v_0), (\tau_1, v_1), \ldots ) $ (as in equation (\ref{eq:hybrid-3})), and consider the hybrid form of the Schrodinger equation
\begin{eqnarray}
\dot{U}(t) &=& -i \frac{1}{2} \omega \sigma_z \, U(t), \ \ \tau_k < t \leq \tau_{k+1},
\nonumber
\\
U(\tau^+_k) &=& e^{-i v_k \sigma_x } U(\tau_k),
\label{eq:impulse-1}
\end{eqnarray}
where $V_k(\tau_k) = U^\ast(\tau_k) V_k U(\tau_k) = e^{-i v_k \sigma_x(\tau_k)}$ is the impulse applied at time $\tau_k$.
The hybrid equations of motion on the surface of the Block sphere are
\begin{eqnarray}
\left( \begin{array}{c}
\dot{x}(t) \\ \dot{y} (t) \\ \dot{z}(t)
\end{array} \right) &=&
\left( \begin{array}{ccc}
0 & -\omega & 0
\\
\omega & 0 & 0
\\
0 & 0 & 0
\end{array} \right)
\left( \begin{array}{c}
x(t) \\ y(t)\\ z(t)
\end{array} \right), \ \ \tau_k < t \leq \tau_{k+1},
\label{eq:impulse-2-a} \\
\left( \begin{array}{c}
x(\tau^+_k ) \\ y( \tau^+_k )\\ z(\tau^+_k)
\end{array} \right)
&=&
\left( \begin{array}{ccc}
1 & 0 & 0
\\
0 & \cos v_k & -\sin v_k
\\
0 & \sin v_k & \cos v_k
\end{array} \right)
\left( \begin{array}{c}
x(\tau_k ) \\ y( \tau_k )\\ z(\tau_k)
\end{array} \right).
\label{eq:impulse-2-b}
\end{eqnarray}
These equations describe the natural drift, a rotation in the $xy$ plane, together with a choice of instantaneous rotation in the $yz$ plane produced by the selected impulse.
The minimum time function $T(r)$ is again defined by (\ref{eq:min-t-1}), but now the DPE takes the form
\begin{eqnarray}
\min \{ DS(r) [ f_0 (r) ] +1, \inf_{V \in \mathbf{V}} S(VU)-S(U) \} &=& 0,
\label{eq:impulse-3-a}
\\
S(r_f) & = & 0,
\label{eq:impulse-3-b}
\end{eqnarray}
to which one seeks a non-negative solution. If the minimum time function $T(r)$ is sufficiently smooth, then it will be a solution of the {\em quasivariational inequality} (QVI) (\ref{eq:impulse-3-a}). The QVI has two parts:
\begin{eqnarray}
DS(r) [ f_0 (r) ] +1 \geq 0 \ & \mathrm{and}& \ \min_{V \in \mathbf{V}} S(VU) \geq S(U) ,
\label{eq:impulse-4-a}
\\
DS(r) [ f_0 (r) ] +1 = 0 \ & \mathrm{or}& \ \min_{V \in \mathbf{V} } S(VU) = S(U) .
\label{eq:impulse-4-b}
\end{eqnarray}
Equation (\ref{eq:impulse-4-a}) simply says that drifting or impulsing will lead to a time greater than or equal to the minimum time. Equation (\ref{eq:impulse-4-b}) says that along the optimal trajectory the atom should drift at Block vectors for which $DS(r) [ f_0 (r) ] +1 = 0$, while if the Bloch vector is such that $\min_{V \in \mathbf{V}} S(VU) = S(U)$, then the impulse $V \in \mathbf{V}$ should be applied.
\subsection{Relaxation}
\label{sec:open-loop-relax}
Let's take a look at open loop control of the atom (or ensemble of atoms) in the presence of a decohering mechanism. For definiteness, consider an atom coupled to a single field channel ($\kappa=\kappa_1$, $B(t)=B_1(t)$ in the notation of Section \ref{sec:types}), with impulsive control only (with impulses in $\mathbf{V}$, the subgroup defined by (\ref{eq:impulse-0})). The hybrid Schrodinger equation for the unitary is
\begin{eqnarray}
d U(t) &=& \{ \sqrt{\kappa}\, dB^\ast(t) \sigma_- - \sqrt{\kappa}\, \sigma_+ dB(t)
- \frac{1}{2} (\kappa \sigma_+ \sigma_- + i \omega \sigma_z )dt )\} U(t),
\nonumber \\
&& \hspace{6.0cm}
\tau_k < t \leq \tau_{k+1},
\nonumber \\
U(\tau_k^+) & = & V_{k} U(\tau_k) .
\label{eq:relax-1}
\end{eqnarray}
In the Heisenberg picture, atomic operators $X$ evolve according to $X(t)=U^\ast(t) X U(t)$, so that between impulses we have
\begin{eqnarray}
dX(t) &=& (-i [ X(t), H(t) ] + \mathcal{L}_{L(t)} (X(t)))dt
\nonumber \\
&& \ \ + dB^\ast(t) [ X(t), L(t) ] + [ L^\ast(t), X(t)] dB(t), \ \tau_k < t \leq \tau_{k+1},
\label{eq:relax-2}
\end{eqnarray}
where $L=\sqrt{\kappa}\, \sigma_-$, and
\begin{equation}
\mathcal{L}_L(X) = \frac{1}{2} L^\ast [ X, L ] + \frac{1}{2} [ L^\ast, X ] L.
\label{eq:relax-3}
\end{equation}
When an impulse is applied at time $\tau_k$, we have
\begin{equation}
X(\tau_k^+) = U^\ast(\tau_k^+) X U(\tau_k^+) = V_k^\ast (\tau_k) X(\tau_k) V_k(\tau_k),
\label{eq:relax-3a}
\end{equation}
where $V_k(\tau_k) = U^\ast(\tau_k) V_k U(\tau_k) = e^{-i v_k \sigma_x(\tau_k)}$.
Explicitly, for the operators $X=\sigma_x, \sigma_y$ and $\sigma_z$, we have, for $\tau_k < t \leq \tau_{k+1}$,
\begin{eqnarray}
d \sigma_x(t) &=& ( -\omega \sigma_y(t) - \frac{\kappa}{2} \sigma_x(t)) dt + \sqrt{\kappa} (dB^\ast(t)\sigma_z(t) + \sigma_z(t) dB(t) )
\label{eq:relax-4-x} \\
d \sigma_y(t) &=& (\omega \sigma_x(t) - \frac{\kappa}{2} \sigma_y(t)) dt - i \sqrt{\kappa} ( \sigma_z(t) dB^\ast(t) - \sigma_z(t) dB(t) )
\label{eq:relax-4-y}
\\
d \sigma_z(t) &=& (- \kappa \sigma_z(t) -\kappa ) dt - 2 \sqrt{\kappa} (dB^\ast(t) \sigma_-(t) + \sigma_+ (t) dB(t)),
\label{eq:relax-4-z}
\end{eqnarray}
and for $t=\tau_k^+$,
\begin{eqnarray}
\sigma_x( \tau_k^+) &=& \sigma_x( \tau_k),
\label{eq:relax-4-impulse-x}
\\
\sigma_y( \tau_k^+) &=& \cos( v_k) \sigma_y( \tau_k) - \sin( v_k) \sigma_z( \tau_k) ,
\label{eq:relax-4-impulse-y}
\\
\sigma_z( \tau_k^+) &=& \sin( v_k) \sigma_y( \tau_k) + \cos( v_k) \sigma_z( \tau_k) .
\label{eq:relax-4-impulse-z}
\end{eqnarray}
Equations (\ref{eq:relax-4-x})-(\ref{eq:relax-4-impulse-z}) constitute a set of impulsive QSDEs in the $\ast$-algebra $\mathscr{M}_2 \otimes \mathscr{F}$, where $\mathscr{F}$ is the $\ast$-algebra of field operators (defined on an underlying Fock space).
By taking expectations, we may conclude from (\ref{eq:relax-4-z}) that the mean energy of the atom decreases exponentially, that is, the atom looses energy to the field.
The equations of motion for the atomic state $\rho$ may be obtained by averaging out the noise in the Schrodinger picture, $\rho(t) = \mathrm{tr}_B[ U(t) (\rho \otimes \vert 0 \rangle \langle 0 \vert ) U^\ast(t) ]$; here, we take the field to be in the vacuum state. The differential equation for $\rho(t)$, which holds between impulses, is
\begin{equation}
\dot{\rho}(t) = i [ \rho(t), H ] + \mathcal{L}^\ast_{L}( \rho(t))
\label{eq:relax-5}
\end{equation}
where
\begin{equation}
\mathcal{L}^\ast_L(\rho) = \frac{1}{2} [L, \rho L^\ast ] + \frac{1}{2} [ L\rho, L^\ast ].
\label{eq:relax-6}
\end{equation}
Equation (\ref{eq:relax-5}) is called the {\em master equation}, and $i[\rho,H] + \mathcal{L}^\ast_L(\rho)$ is called the {\em Lindblad superoperator} (in Schrodinger form).
The hybrid equations of motion inside the Bloch sphere are
\begin{eqnarray}
\dot{x}(t) &=& - \frac{\kappa}{2} x(t) -\omega y(t) , \ \ \tau_k < t \leq \tau_{k+1},
\label{eq:relax-7-a} \\
\dot{y}(t) &=& - \frac{\kappa}{2} y(t) + \omega x(t) , \ \ \tau_k < t \leq \tau_{k+1},
\label{eq:relax-7-b} \\
\dot{z}(t) &=& -\kappa z(t) - \kappa, \ \ \tau_k < t \leq \tau_{k+1},
\label{eq:relax-7-c} \\
\left( \begin{array}{c}
x(\tau^+_k ) \\ y( \tau^+_k )\\ z(\tau^+_k)
\end{array} \right)
&=&
\left( \begin{array}{ccc}
1 & 0 & 0
\\
0 & \cos v_k & -\sin v_k
\\
0 & \sin v_k & \cos v_k
\end{array} \right)
\left( \begin{array}{c}
x(\tau_k ) \\ y( \tau_k )\\ z(\tau_k)
\end{array} \right).
\label{eq:relax-7-d}
\end{eqnarray}
In equations (\ref{eq:relax-7-a})-(\ref{eq:relax-7-c}) we can see the effect of the field coupling, which in the absence of control action causes $x(t) \to 0$, $y(t)\to 0$ and $z(t)\to -1$ as $t \to \infty$, i.e. $\rho(t)\to \frac{1}{2}( \sigma_z - I)= \vert -1 \rangle \langle -1 \vert$, the pure state of lowest energy. Repeated application of impulses offers the possibility of achieving other large time behaviour.
For instance, the periodic pulse sequence $\gamma$ with period $1$ and $v_k=\pi/2$ leads to the steady state $(0, 0.5168, 0.3135)$ (for the case $\kappa=1$, $\omega=0$).
\section{Quantum Feedback Networks}
\label{sec:networks}
A glance at the various types of feedback control discussed in
Section \ref{sec:types}, or indeed any textbook on classical feedback control, tells us that feedback arrangements are {\em networks} of interconnected systems. Accordingly, the purpose of this section is to set up some easy-to-use tools for constructing feedback networks. What is presented in this section is a simplification of a more general quantum feedback network theory \cite{YK03a}, \cite{GJ09}, \cite{GJN09a}, which builds on earlier cascade theory \cite{CWG93}, \cite{HJC93}, network quantization \cite{YD84}, and quantum control \cite{WM94b}.
The basic idea is simple, Figure \ref{fig:qfn-1}. Take an output channel and connect it to an input channel. Such series or cascade connections are commonplace in classical electrical circuit theory. For instance, if the systems are resistors with resistances $R_1$ and $R_2$, then the total series-connected system is equivalent to a single resistor with resistance $R=R_1+R_2$. This use of simple parameters for devices, and rules for interconnecting devices in terms of these parameters, is a powerful feature of classical electrical circuit theory.
\begin{figure}[h]
\begin{center}
\includegraphics{series-1.eps}
\caption{A series connection between two systems.}
\label{fig:qfn-1}
\end{center}
\end{figure}
Let's see how we can achieve an analogous simple rule for the series connection of two open quantum systems (without impulses),
where the input and output channels are quantum fields, as discussed in Section \ref{sec:types}. The physical parameters
determining open quantum systems are a Hamiltonian $H$ describing the self-energy of the system, and a vector $L$ of operators describing how the system is coupled to the field channels. These parameters appear in the Lindblad superoperator $i[\rho,H] + \mathcal{L}^\ast_L(\rho)$ (and hence in the master equation), as well as in QSDEs, \cite{HP84}, \cite{GC85}, \cite{KRP92}, \cite{GZ00}.
Actually, there is a third parameter $S$, a self-adjoint matrix of operators describing scattering between field channels, that was introduced in \cite{HP84}, \cite{KRP92}. While $S$ does not appear in the Linblad nor the master equation for a single system, it does have non-trivial use in several applications including quantum feedback networks (such as those including beamsplitters). Thus in general an open quantum system, call it $G$, is characterized by three parameters $G=(S,L,H)$. However, in this article we do not use the scattering parameter, and so we set $S=I$; actually, we make the abbreviation $G=(L,H)$.
For example, the parameters for the atom, coupled to two field channels (recall section \ref{sec:types}), call it $A$, are
\begin{equation}
A = \left(
\left( \begin{array}{c}
\sqrt{\kappa_1}\, \sigma_-
\\
\sqrt{\kappa_2}\, \sigma_-
\end{array} \right),
\frac{1}{2}\omega \sigma_z
\right).
\label{eq:qfn-1}
\end{equation}
In network modeling, and indeed in modeling in general, it can be helpful to decompose large systems into smaller pieces, and to assemble large systems from components. In \cite{GJ09}, the {\em concatenation product} $\boxplus$ was introduced to assist with this. The concatenation product is defined by
\begin{equation}
(L_1, H_1) \boxplus (L_2, H_2) = \left(
\left( \begin{array}{c}
L_1
\\
L_2
\end{array} \right),
H_1+H_2
\right).
\label{eq:qfn-2}
\end{equation}
For the atom, if we wish to decompose it with respect to field channels we may write
\begin{equation}
A = ( \sqrt{\kappa_1}\, \sigma_-, \, \frac{1}{2}\omega \sigma_z ) \boxplus ( \sqrt{\kappa_2}\, \sigma_-, 0) .
\label{eq:qfn-3}
\end{equation}
Now suppose we have two systems $G_1 = (L_1, H_1)$ and $G_2 = (L_2, H_2)$, as in Figure \ref{fig:qfn-1}. Because the systems are separated spatially, the field segment connecting connecting the output of $G_1$ to the input of $G_2$ has non-zero length, and so this means there is a small delay $\tau$ in the transmission of quantum information from $G_1$ to $G_2$. That is, $B_{in,2}(t) = B_{out,1}(t-\tau)$. Now if the systems are sufficiently close, $\tau$ will be small compared with the timescales of the systems, and may be neglected. In this way, a Markovian model for the series connection $G= G_2 \triangleleft G_1$ may be derived. In terms of the physical parameters, the {\em series product} (defined in \cite{GJ09}) is given by
\begin{equation}
(L_2, H_2) \triangleleft (L_1, H_1) =
(L_1 + L_2, H_1+H_2 + \mathrm{Im}[ L_2^\dagger L_1]).
\label{eq:qfn-4}
\end{equation}
Thus the series connection $G= G_2 \triangleleft G_1$ has parameters $L=L_1+L_2$ and $H=H_1+H_2 + \mathrm{Im}[ L_2^\dagger L_1]$, analogous to the expression $R=R_1+R_2$ for series-connected resistors.
The series product serves very well for Markovian approximations to cascades of independent open systems. However, importantly for us, the series product may also be used to describe an important class of {\em quantum feedback networks}. This is because the two systems
$G_1 = (L_1, H_1)$ and $G_2 = (L_2, H_2)$ need not be independent---they can be parts of the same system.
For example, suppose we take $G_1$ to be the first factor in the decomposition (\ref{eq:qfn-3}) of $A$, i.e. $G_1=( \sqrt{\kappa_1}\, \sigma_-, \, \frac{1}{2}\omega \sigma_z)$, and $G_2= ( \sqrt{\kappa_2}\, \sigma_-, 0)$ the second. Then the series connection
\begin{equation}
G= ( \sqrt{\kappa_2}\, \sigma_-, 0) \triangleleft ( \sqrt{\kappa_1}\, \sigma_-, \, \frac{1}{2}\omega \sigma_z) =
\left( (\sqrt{\kappa_1} + \sqrt{\kappa_2}) \sigma_-, \, \frac{1}{2}\omega \sigma_z
\right)
\label{eq:qfn-5}
\end{equation}
describes the coherent feedback arrangement shown in Figure \ref{fig:qfn-2}. The system $G$ is an open quantum system with Hamiltonian $H=\frac{1}{2}\omega \sigma_z$ that is coupled to a single field channel via the coupling operator $L = (\sqrt{\kappa_1} + \sqrt{\kappa_2}) \sigma_-$.
Now that we have the parameters for the feedback system $G$, it is easy to write down the corresponding Schrodinger equation
\begin{eqnarray}
dU(t) &=& \{
(\sqrt{\kappa_1} + \sqrt{\kappa_2}) ( dB^\ast(t) \sigma_- - \sigma_+ dB(t) )
\nonumber \\
&& \hspace{2.0cm} - \frac{1}{2} (
(\sqrt{\kappa_1} + \sqrt{\kappa_2})^2 \sigma_+ \sigma_- + i \omega \sigma_z )dt
\} U(t)
\label{eq:qfn-6}
\end{eqnarray}
and master equation
\begin{equation}
\dot{\rho} = i[ \rho, \frac{1}{2}\omega \sigma_z ] + \frac{1}{2}(\sqrt{\kappa_1} + \sqrt{\kappa_2})^2 ( [ \sigma_-, \rho \sigma_+]
+ [ \sigma_- \rho, \sigma_+]),
\label{eq:qfn-7}
\end{equation}
if desired. Equations like these apply to complete systems, and have meaning only when the network construction process has concluded.
\begin{figure}[h]
\begin{center}
\includegraphics{series-2.eps}
\caption{Atom in a coherent feedback loop described by a series connection $G=G_2 \triangleleft G_1$.}
\label{fig:qfn-2}
\end{center}
\end{figure}
To further illustrate the use of the series product, consider a situation with $L_1 = \sqrt{\kappa_1}\, \sigma_-$ and
$L_2=i\sqrt{\kappa_2}\, \sigma_z$.
Then the Markovian model for the series connection
\begin{equation}
G = ( i\sqrt{\kappa_2}\, \sigma_z, 0) \triangleleft (\sqrt{\kappa_1}\, \sigma_-, \frac{1}{2} \omega \sigma_z) =
( (\sqrt{\kappa_1}\, \sigma_- + i\sqrt{\kappa_2}\, \sigma_z, \frac{1}{2} \omega \sigma_z + \frac{1}{2} \sqrt{\kappa_1+\kappa_2}\, \sigma_x)
\label{eq:qfn-8}
\end{equation}
contains an additional Hamiltonian term $\frac{1}{2} \sqrt{\kappa_1+\kappa_2}\, \sigma_x$. The reader may wish to derive the corresponding Schrodinger and master equations from these parameters.
\section{Quantum Filtering}
\label{sec:filter}
The term {\em filtering} is used in many ways, but usually refers to a process of extracting information concerning something of interest from a source containing partial information that may be noisy. Our interest here is in the extraction of classical information about the atom by monitoring the output field channel $B_{out,1}(t)$. For instance, we may wish to know something about the atom's energy by observing any photons emitted into the field.
The {\em quantum filter} was developed for purposes like this \cite{VPB92a}, and also goes by the name {\em stochastic master equation} \cite{WM10,HC93}.
\subsection{Probability}
\label{sec:filter-probability}
In quantum mechanics the postulates state that physical quantities are represented by {\em observables}, which are self-adjoint operators defined on some underlying Hilbert space.
Consider for a moment an isolated atom.
Atomic energy is represented by the observable $\frac{1}{2}\omega \sigma_z$. The possible measurement outcomes are the eigenvalues of the observable, which are $\pm \frac{1}{2}\omega$ for the energy of the atom. The probabilities of the measurement outcomes depends on the state $\rho$, and are given by $\mathrm{Prob}( \pm \frac{1}{2}\omega) = \mathrm{tr}[ \rho P_{\pm 1}]$, where $P_{\pm 1} = \vert \pm 1 \rangle \langle \pm 1 \vert$ are the projection operators arising in the spectral representation $\sigma_z = P_{+1} - P_{-1}$. If the outcome $\pm \frac{1}{2}\omega$ is observed, then Von Neumann's projection postulate states that the atomic state changes to $\frac{P_{\pm 1} \rho P_{\pm 1}}{\mathrm{tr}[ \rho P_{\pm 1}] }$; this is called a {\em conditional state}.
The mathematics that underlies the measurement postulate is the {\em spectral theorem}, which says essentially that any collection of commuting matrices can be simultaneously diagonalized. For instance, let $\mathscr{C}$ be a collection of $2 \times 2$ complex matrices \lq\lq{generated}\rq\rq \ by $\sigma_z$, the energy observable. This means that $\mathscr{C}$ contains all complex linear combinations of powers $\sigma_z$, and all adjoints (mathematically, $\mathscr{C}$ is called a $\ast$-algebra, a vector space closed under products and adjoints). The spectral theorem says that any matrix $X \in \mathscr{C}$ can be diagonalized to a matrix of the form $\mathrm{diag}(a,b)$, where $a$ and $b$ are complex numbers; that is, $X=a P_{+1} + b P_{-1}$.
We can interpret the spectral theorem probabilistically, consistent with the measurement postulate, as follows.
For any matrix $X \in \mathscr{C}$
we can define a function $\iota(X)$ on a set $\Omega = \{ \pm 1 \}$ by $\iota(X)(+1)=a$, $\iota(X)(-1)=b$.
The set
$\Omega$ is a sample space, and $\iota(X)$ is a classical random variable. The algebra $\mathscr{C}$ and the density operator $\rho$ determine a
classical probability distribution $\mathbf{P}$: $\mathrm{Prob}( \iota(X) =a) =\mathbf{P}(+1) = \mathrm{tr}[ \rho P_{+1}]$, and $\mathrm{Prob}( \iota(X)=b) =\mathbf{P}(-1) = \mathrm{tr}[ \rho P_{-1}]$.
In what follows we denote quantum expectations (states) by
\begin{equation}
\mathbb{P} [X ] = \mathrm{tr}[\rho X],
\label{eq:prob-1}
\end{equation}
where $X$ is an operator, and classical expectations by
\begin{equation}
\mathbf{P} [ X ] = \sum_{\omega \in \Omega} X(\omega) \mathbf{P} (\omega ) = \int_\Omega X (\omega) \mathbf{P}(d\omega),
\label{eq:prob-2}
\end{equation}
where $X$ is a classical random variable.
The spectral theorem may be re-stated as linking quantum and classical expectations: roughly,
for any $X \in \mathscr{C}$ there exists a classical random variable $\iota(X)$ and a classical probability distribution $\mathbf{P}$ such that
\begin{equation}
\mathbb{P} [X ] = \mathbf{P} [ \iota(X) ] .
\label{eq:prob-3}
\end{equation}
This can be easily seen, since any $X \in \mathscr{C}$ has the diagonal representation $X=a P_{+1} + b P_{-1}$, and hence $\iota(X)= a \chi_{+1} + b \chi_{-1}$, where the indicator functions $\chi_{\pm 1}$ are defined by $\chi_{\omega_0}(\omega)=1$ when $\omega=\omega_0$, and $\chi_{\omega_0}(\omega)= 0$ when $\omega\neq \omega_0$ ($\omega_0 = \pm 1$).
It is important to appreciate that while the quantum expectation (\ref{eq:prob-1}) is defined for any $2 \times 2$ complex matrix $X \in \mathscr{M}_2$, expression (\ref{eq:prob-3}) depends on the choice of commutative algebra $\mathscr{C}$, or observable, which in turn determines the classical probability.
It is sometimes helpful to write $(\mathscr{A}, \mathbb{P})$ for a {\em quantum probability space} (where $\mathscr{A}$ is a $\ast$-algebra (not commutative in general), and $\mathbb{P}$ is quantum expectation), in contrast to a {\em classical probability space} $(\Omega, \mathcal{F}, \mathbf{P})$ (where $\Omega$ is a sample space, $\mathcal{F}$ is a $\sigma$-algebra of events, and $\mathbf{P}$ is a classical probability measure).
The spectral theorem says that a {\em commutative} quantum probability space $(\mathscr{C}, \mathbb{P})$ is {\em statistically equivalent} to a classical probability space $(\Omega, \mathcal{F}, \mathbf{P})$.
In general, a non-commutative quantum probability space can contain many distinct commutative subspaces, each of which is equivalent to distinct classical probability spaces, Figure \ref{fig:filter-m2}.
\begin{figure}[h]
\begin{center}
\includegraphics{m2.eps}
\caption{The non-commutative quantum probability space $(\mathscr{M}_2, \mathbb{P})$ contains distinct commutative quantum probability spaces $(\mathscr{C}_x, \mathbb{P})$, $(\mathscr{C}_y, \mathbb{P})$ and $(\mathscr{C}_z, \mathbb{P})$, determined by non-commuting observables $\sigma_x$, $\sigma_y$ and $\sigma_z$ respectively. These
correspond to distinct classical probability spaces $(\Omega_x, \mathcal{F}_x, \mathbf{P}_x)$, $(\Omega_y, \mathcal{F}_y, \mathbf{P}_y)$ and $(\Omega_z, \mathcal{F}_z, \mathbf{P}_z)$ respectively, which may describe distinct experiments.
}
\label{fig:filter-m2}
\end{center}
\end{figure}
\subsection{Conditional Expectation}
\label{sec:filter-conditional}
In classical probability information is summarized by $\sigma$-algebras ${\mathcal G} \subset \mathcal F$ of events in a classical probability space $(\Omega, {\mathcal F}, {\mathbf P})$.
Typically, $\mathcal{G}$ will be the $\sigma$-algebra $\sigma(X)$ generated by a random variable $X$, which contains information on the values taken by $X$. The mathematical notion of {\em measurability} with respect to a $\sigma$-algebra plays an important role in integration theory and probability theory.\footnote{The mathematical term {\em measurable} is not to be confused
with measurements in quantum mechanics.}
If $Z$ is a random variable measurable with respect to a $\sigma$-algebra $\mathcal{G}=\sigma(X)$ generated by a random variable $X$, then $Z = f(X)$ for some function $f$.
Conditional expectation plays a fundamental role in classical estimation and filtering. Suppose we are given two random variables
$X$ and $Y$. The random variable $X$ may describe some quantity that is not directly accessible by experiment; instead, a quantity described by $Y$ is accessible---its values may be obtained by experiment. Then given an outcome $y$ (a value of $Y$), one may wish to improve one's estimation of the expected value of $X$. Now the information associated with $Y$ is described by the $\sigma$-algebra generated by $Y$, $\mathscr{Y}=\sigma(Y)$, and the {\em conditional expectation} of $X$ given $Y$ is the random variable $\mathbf{P}[ X \vert \mathscr{Y} ]$ (often denoted $\mathbf{P}[X \vert Y ]$ or $\hat X$\footnote{Not to be confused with the hats sometimes used in the physics literature to denote operators.}). The conditional expectation $\mathbf{P}[ X \vert \mathscr{Y} ]$ is the unique $\mathscr{Y}$-measurable random variable such that
\begin{equation}
\mathbf{P}[ \chi_E \mathbf{P}[ X \vert \mathscr{Y} ] ] = \mathbf{P}[ \chi_E X ] \ \ \text{for all events} \ E \in \mathscr{Y},
\label{eq:qm-prob-cexp-1}
\end{equation}
where $\chi_E$ is the indicator function for the event $E$ ($\chi_E(\omega) =1$ if $\omega \in E$, zero otherwise).
If $X$ and $Y$ have a joint density $p_{X,Y}(x,y)$, then the conditional density is given by
\begin{equation}
p_{X\vert Y}(x \vert y) = \frac{ p_{X,Y}(x,y) }{ \int p_{X,Y}(x,y) dx } ,
\label{eq:qm-prob-cexp-1a}
\end{equation}
from which the conditional expectation can be computed:
\begin{equation}
\mathbf{P}[ X \vert \mathscr{Y} ] (y) = \mathbf{P}[ X \vert \, Y=y ] = \int x p_{X\vert Y}(x \vert y) dx .
\label{eq:qm-prob-cexp-2}
\end{equation}
Equation (\ref{eq:qm-prob-cexp-2}) shows explicitly that the conditional expectation is a function of the outcomes $y$.
A well known property of the conditional expectation $\mathbf{P}[ X \vert \mathscr{Y} ] $ is that it is a {\em minimum variance} or {\em least squares estimator}. Geometrically, $\hat X=\mathbf{P}[ X \vert \mathscr{Y} ] $ is the orthogonal projection of $X$ onto the subspace $\mathscr{Y}$, Figure \ref{fig:c-exp-1} (see, e.g. \cite{AM79}).
\begin{figure}[h]
\begin{center}
\includegraphics{cexp1.eps}
\caption{The conditional expectation $\hat X = \mathbf{P}[ X \vert \mathscr{Y} ] $ is the orthogonal projection of $X$ onto the subspace $\mathscr{Y}$.
}
\label{fig:c-exp-1}
\end{center}
\end{figure}
Does conditional expectation make sense in quantum mechanics? The answer is yes, provided we don't try to project all operators at the same time. Given a commutative $\ast$-algebra $\mathscr{Y}$, say corresponding to an observable to be measured in an experiment, the quantum conditional expectation $\mathbb{P}[ X \vert \mathscr{Y}]$ is well-defined provided $X$ commutes with all operators $Y \in \mathscr{Y}$. The conditional expectation is the unique operator
$\mathbb{P}[ X \vert \mathscr{Y}] \in \mathscr{Y}$ such that
\begin{equation}
\mathbb{P} [ X Y ] = \mathbb{P}[ \ \mathbb{P}[ X \vert \mathscr{Y}] Y],
\label{eq:filter-0}
\end{equation}
for any $X \in \mathscr{Y}' = \{ AY=YA \ \mathrm{for\ all} \ Y \in \mathscr{Y} \} $ (the {\em commutant} of $\mathscr{Y}$) and $Y\in \mathscr{Y}$. It is the orthogonal projection of $\mathscr{Y}'$ onto $\mathscr{Y}$ with respect to the inner product $( A, B ) = \mathbb{P} [ A^\ast B ]$, as in Figure \ref{fig:c-exp-1}. As in the classical case, quantum conditional expectation boils down to {\em least squares estimation}.
\subsection{System-Probe Model for Quantum Filtering}
\label{sec:filter-system-probe}
In the case of the atom coupled to field channels, we cannot access atomic observables directly, and instead we must rely on indirect information available in an output field channel. The interaction between the atom and the field causes information about the atom to be transferred to the field.
We may monitor an observable of the field, and then make inferences about the atom from the data obtained. This is a quantum filtering problem for the atom. The atom-field system is an instance of the system-probe model of Von Neumann.
In the absence of controls, the atom-field system is defined by the parameters
\begin{equation}
A = ( \sqrt{\kappa_1}\, \sigma_-, \, \frac{1}{2} \omega \sigma_z ) \boxplus ( \sqrt{\kappa_2}\, \sigma_-, 0) ,
\label{eq:probe-1}
\end{equation}
which determine the Schrodinger equation
\begin{eqnarray}
d U(t) &=& \{ \sqrt{\kappa_1}\, dB_1^\ast(t) \sigma_- - \sqrt{\kappa_1}\, \sigma_+ dB_1(t)
+ \sqrt{\kappa_2}\, dB_2^\ast(t) \sigma_- - \sqrt{\kappa_2}\, \sigma_+ dB_2(t)
\nonumber \\
&& \hspace{2.0cm}
- (\frac{1}{2} (\kappa_1+\kappa_2) \sigma_+ \sigma_- + i \frac{1}{2}\omega \sigma_z ) dt \} U(t),
\label{eq:probe-2}
\end{eqnarray}
where the two-channel field $B(t)=(B_1(t), B_2(t))^T$ is in the vacuum state $\vert 00 \rangle$.
The output field is defined by $B_{out}(t) = U^\ast(t) B(t) U(t)$, of which we monitor the real quadrature of channel 1, $B_{out,1}(t)+B_{out,1}^\ast(t)$; this may be achieved by use of an ideal homodyne detector.
Now the spectral theorem holds in situations more general than discussed earlier (section \ref{sec:filter-probability}), and in particular may be applied to the algebra of field operators. The field observable $Y(t)=B_{out,1}(t)+B_{out,1}^\ast(t)$ is self-adjoint for each $t$, and for different $t$'s they commute: $[Y(t), Y(t') ]=0$.
This determines a commutative algebra $\mathscr{Y}_t $ generated by $Y(s)$, for all $0 \leq s \leq t$.
The spectral theorem says that $Y(\cdot)$ is equivalent to a classical stochastic process $\iota(Y)(\cdot)$, the signal generated by the detector, with respect to a classical probability distribution $\mathbf{P}$. Due to their statistical equivalence, we do not usually distinguish between them, and simply write $Y(t)$ for the measurement signal produced by the detector.
Now let's consider the atom-field system at the initial time $t=0$, before the interaction has taken place. Atomic operators $X \in \mathscr{M}_2$ are represented in the tensor product $\mathscr{M}_2 \otimes \mathscr{F}$ by $X \otimes I$. Field operators $F \in \mathscr{F}$ are represented by $I \otimes F$. Clearly, these operators commute: $[ X \otimes I , I \otimes F]=0$. If the atom and field are allowed to interact via a unitary $U(t)$, then this commutation relation is preserved: $[ U^\ast(t) (X \otimes I ) U(t) , U^\ast(t) (I \otimes F) U(t) ] = U^\ast(t) [ X \otimes I , I \otimes F] U(t) =0$. This means that in the Heisenberg picture, atomic operators $X(t)$ commute with $Y(t)$. Indeed, it can be shown that $[X(t), Y(s)]=0$ for all $s \leq t$, i.e. $X(t) \in \mathscr{Y}_t'$.
All of this means that, given a atom-field state $\rho \otimes \vert 00 \rangle \langle 00 \vert$, the conditional expectation $\hat X(t)=\mathbb{P}[ X(t) \, \vert \, \mathscr{Y}_t ]$ is well-defined.
The conditional expectation $\hat X(t)=\mathbb{P}[ X(t) \, \vert \, \mathscr{Y}_t ]$ provides us with an {\em estimate} of $X(t)$ given the measurement signal $Y(s), \, s\leq t$.
The quantum filter computes this conditional expectation, as we will soon see.
\subsection{The Quantum Filter}
\label{sec:filter-qf}
In order to present the quantum filter for the atomic system, we need some notation. A normalized {\em conditional state} $\pi_t$ is defined by
\begin{equation}
\hat X(t)=\pi_t(X) = \mathbb{E}[ X(t) \, \vert \, \mathscr{Y}_t ]
\label{eq:filter-1}
\end{equation}
for any atomic operator $X \in \mathscr{M}_2$. The {\em quantum filter} is a stochastic differential equation for the conditional state:
\begin{eqnarray}
d \pi_t(X) &=& \pi_t( -i [X, H] + \mathcal{L}_{L_1}(X) + \mathcal{L}_{L_2}(X)) dt
\label{eq:filter-2} \\
&& + ( \pi_t( L_1^\ast X + X L_1) - \pi_t( L^\ast_1 + L_1) \pi_t(X) )
( dY(t) - \pi_t( L_1^\ast + L_1) dt ),
\nonumber
\end{eqnarray}
with initial condition $\pi_0(X) = \mathrm{tr}[\rho X]$, where $\rho$ is the initial atomic state. Here, we have expressed the filter in terms of the atomic parameters $H=\frac{1}{2}\omega \sigma_z$, $L_1 = \sqrt{\kappa_1}\, \sigma_-$ and $L_2 = \sqrt{\kappa_2}\, \sigma_-$.
It is quite common in the literature to express the quantum filter in terms of a {\em conditional density operator} $\hat\rho_t$, so that $\hat X(t)=\pi_t(X)=\mathrm{tr}[\hat\rho_t X]$. The quantum filter (\ref{eq:filter-2}) takes the form
\begin{eqnarray}
d \hat\rho_t &=& ( i[\hat\rho_t, H] + \mathcal{L}^\ast_{L_1}( \hat\rho_t) + \mathcal{L}^\ast_{L_2}( \hat\rho_t))dt
\label{eq:filter-3} \\
&& +
(L_1 \hat\rho_t + \hat\rho_t L_1^\ast - \mathrm{tr} [ (L_1+L_1^\ast) \hat\rho_t] \hat\rho_t )
( dY(t)- \mathrm{tr}[ (L_1+L_1^\ast) \hat\rho_t ] dt),
\nonumber
\end{eqnarray}
with initial condition $\hat\rho_0=\rho$, the initial atomic density operator. Explicitly, if we write
$x (t) = \pi_t(\sigma_x)$, etc, the quantum filter for the atom is
\begin{eqnarray}
d x(t) &=& ( -\omega y(t) - \frac{\kappa_1+\kappa_2}{2} x(t)) dt
\nonumber \\
&& \hspace{1.0cm} + \sqrt{\kappa_1}\,
(1+ z(t) - x^2(t) ) (dY(t) - x(t)dt),
\label{eq:filter-30-x}
\\
d y(t) &=& (\omega x(t) - \frac{\kappa_1+\kappa_2}{2} y(t)) dt
\nonumber \\
&& \hspace{2.0cm} +
\sqrt{\kappa_1}\, x(t) y(t) (dY(t) - x(t)dt) ,
\label{eq:filter-30-y}
\\
d z(t) &=& (- (\kappa_1+\kappa_2) z(t) -(\kappa_1+\kappa_2) ) dt
\nonumber \\
&& \hspace{1.5cm}
- \sqrt{\kappa_1}\, \ x(t)( 1+ x(t)) (dY(t) - x(t)dt) .
\label{eq:filter-30-z}
\end{eqnarray}
The quantum filter has the same form as the classical nonlinear filter due (independently) to Kushner and Stratonovich (see, for example, \cite[Chapter 18]{RE82}).
The filter is driven by the measurement signal $Y(t)$, Figure \ref{fig:filter1}.
\begin{figure}[h]
\begin{center}
\includegraphics{filter1.eps}
\caption{The quantum filter produces estimates $\hat X(t)$ from the measurement signal $Y(t)$. The conditional density $\hat{\rho}(t)$ is internal to the filter.
}
\label{fig:filter1}
\end{center}
\end{figure}
As in the classical case, the stochastic process
\begin{equation}
W(t) = Y(t) - \int_0^t \mathrm{tr}[ (L_1+L_1^\ast) \hat\rho_s ] ds
\label{eq:filter-4}
\end{equation}
is a standard Brownian motion (with respect to the classical probability distribution $\mathbf{P}$ determined by the atom-field state and the measurement observables $\mathscr{Y}_t$), called the {\em innovation process}; it carries the new information available at time $t$. In differential form, we have $dW(t) = dY(t)-\hat x(t) dt$.
The density operator $\rho_t$ for the atom is the mean of the conditional density $\hat\rho_t$, since by the fundamental property (\ref{eq:filter-0}) of conditional expectation, we have for any atomic operator $X$,
\begin{eqnarray}
\mathrm{tr}[ \rho_t X ] &=&
\mathbb{P}[ X(t) ]
\nonumber \\
&=&
\mathbb{P}[ \pi_t(X) ]
\nonumber \\
&=& \mathbf{P}[ \mathrm{tr}[ \hat{\rho}_t X ] ] .
\label{eq:filter-5}
\end{eqnarray}
Note that in the classical expectation in (\ref{eq:filter-5}), we are averaging the innovation process $W(t)$, a standard Wiener process with respect to $\mathbf{P}$ as mentioned above.
Therefore the {\em master equation} is obtained by averaging the quantum filter:
\begin{equation}
\dot {\rho} = i[\rho, H] + \mathcal{L}^\ast_{L_1}( \rho) + \mathcal{L}^\ast_{L_2}( \rho),
\label{eq:filter-6}
\end{equation}
with initial condition $\rho_0=\rho$. In terms of coordinates in the Bloch sphere, we have
\begin{eqnarray}
\dot{x}(t) &=& - \omega y(t) - \frac{\kappa_1+\kappa_2}{2} x(t),
\label{eq:filter-60-x} \\
\dot{y}(t) &=& \omega x(t) - \frac{\kappa_1+\kappa_2}{2} y(t),
\label{eq:filter-60-y} \\
\dot{z}(t) &=& -(\kappa_1+\kappa_2) z(t) - (\kappa_1+\kappa_2).
\label{eq:filter-60-z}
\end{eqnarray}
In \cite{HC93}, the quantum filter (known as the stochastic master equation) is referred to as an {\em unravelling} of the master equation (\ref{eq:filter-4}).\footnote{The term \lq\lq{unravelling}\rq\rq \ follows from the property $\mathbb{P} [ X ] = \mathbb{P}[ \ \mathbb{P}[ X \vert \mathscr{Y}] ]$, a consequence of the definition of conditional expectation, (\ref{eq:filter-0}) with $Y=I$). }
\subsection{Derivation of the Quantum Filter}
\label{sec:filter-qf-derivation}
There are several ways of deriving the quantum filter. Here we briefly explain the {\em reference probability method}, which involves rotating the filtering problem back to the input field. This provides a convenient reference and simplifies calculations.\footnote{This is analogous to the use of a Girsanov transformation to change probability distributions in classical probability.}
To this end, we define the input field quadrature $Z(t) = B_1(t) + B_1^\ast(t)$, and write $\mathscr{Z}_t$ for the {\em commutative} algebra generated by $Z(s)$, $0 \leq s \leq t$. With the input fields in the vacuum state, $Z(t)$ is equivalent to a standard Wiener process.
Now $Y(s) = U^\ast(t) Z(s) U(t)$ for all $s \leq t$, and so we have $\mathscr{Y}_t = U^\ast(t) \mathscr{Z}_t U(t)$, a rotation of the commutative algebras. Next, we rotate the quantum expectation by defining
\begin{equation}
\mathbb{Q}_t [ X ] = \mathbb{P}[ U^\ast(t) X U(t) ].
\label{eq:filter-derive-1}
\end{equation}
Now by the definition of conditional expectation, we have
\begin{equation}
\mathbb{P}[ X(t) \vert \mathscr{Y}_t ] = U^\ast(t) \mathbb{Q}_t [ X \vert \mathscr{Z}_t ] U(t) ,
\label{eq:filter-derive-2}
\end{equation}
and so we need to calculate $\mathbb{Q}_t [ X \vert \mathscr{Z}_t ]$. This may be achieved using a version of Bayes formula, \cite{BH06}, \cite[sec. 3.2]{BHJ07}, and by employing a trick \cite{AH91}, \cite{AH01}. Now $\mathbb{Q}_t$ is defined in terms of $U(t)$, which does not commute with $\mathscr{Z}_t$, and so we replace it by $\tilde U(t) \in \mathscr{Z}_t'$ defined by
\begin{eqnarray}
d \tilde U(t) &=& \{ \sqrt{\kappa_1}\, \sigma_- dZ(t)
+ \sqrt{\kappa_2}\, dB_2^\ast(t) \sigma_- - \sqrt{\kappa_2}\, \sigma_+ dB_2(t)
\nonumber \\
&& \hspace{2.0cm}
- (\frac{1}{2} \kappa^2 \sigma_+ \sigma_- + i \frac{1}{2}\omega \sigma_z ) dt \} \tilde U(t),
\label{eq:filter-derive-3}
\end{eqnarray}
with initial condition $\tilde U(0)=I$. Equation (\ref{eq:filter-derive-3}) is almost exactly the same as equation (\ref{eq:probe-2}), except that the coefficient of $dB_1(t)$ has been changed. The justification for this is the fact that, for any atomic state vector $\vert \psi \rangle$, since $dB_1(t)\vert 00 \rangle=0$, we have $U(t) \vert \psi \rangle \otimes \vert 00 \rangle = \tilde U(t) \vert \psi \rangle \otimes \vert 00 \rangle$, which ensures $\mathbb{Q}_t [X ] = \mathbb{P}[ \tilde U^\ast (t) X \tilde U(t) ]$. Using the definition of conditional expectations, it can be checked that $\mathbb{P}[ \tilde U^\ast(t) X \tilde U(t) \vert \mathscr{Z}_t ] = \mathbb{P}[ \tilde U^\ast(t) \tilde U(t) \vert \mathscr{Z}_t ] \mathbb{Q}_t[ X \vert \mathscr{Z}_t ] $. Therefore
\begin{equation}
\mathbb{Q}_t[ X \vert \mathscr{Z}_t ] = \frac{ \mathbb{P}[ \tilde U^\ast(t) X \tilde U(t) \vert \mathscr{Z}_t ] }{ \mathbb{P}[ \tilde U^\ast(t) \tilde U(t) \vert \mathscr{Z}_t ] }.
\label{eq:filter-derive-4}
\end{equation}
If we now define an unnormalized conditional expectation
\begin{equation}
\nu_t(X) = U^\ast(t) \mathbb{P}[ \tilde U^\ast(t) X \tilde U(t) \vert \mathscr{Z}_t ] U(t),
\label{eq:filter-derive-5}
\end{equation}
we see from (\ref{eq:filter-derive-2}) and (\ref{eq:filter-derive-4}) that the normalized conditional expectation is given by
\begin{equation}
\pi_t(X) = \frac{ \nu_t(X) }{ \nu_t(I) } .
\label{eq:filter-derive-6}
\end{equation}
Using the quantum Ito rule and conditioning, we find that
\begin{eqnarray}
d \nu_t(X) = \nu_t( -i [X, H] + \mathcal{L}_{L_1}(X) + \mathcal{L}_{L_2}(X)) dt
+ \nu_t( L_1^\ast X + X L_1) dY(t), \hspace{5mm}
\label{eq:filter-derive-7}
\end{eqnarray}
with initial condition $\nu_0(X) = \mathrm{tr}[\rho X]$.
Equation (\ref{eq:filter-derive-7}) is an unnormalized form of the quantum filter (\ref{eq:filter-2}), analogous to
the classical Duncan-Mortensen-Zakai equation (see, for example, \cite[Chapter 18]{RE82}).
In terms of the unnormalized
conditional density $\hat\varrho_t$ (so that $\nu_t(X)=\mathrm{tr}[\hat\varrho_t X]$), we have
\begin{eqnarray}
d \hat\varrho_t &=& ( i[\hat\varrho_t, H] + \mathcal{L}^\ast_{L_1}( \hat\varrho_t) + \mathcal{L}^\ast_{L_2}( \hat\varrho_t))dt
+ (L_1 \hat\varrho_t + \hat\varrho_t L_1^\ast) dY(t),
\label{eq:filter-derive-8}
\end{eqnarray}
with initial condition $\hat\varrho_0 = \rho$. Since $\hat\varrho_t$ is not normalized, we augment the representation (\ref{eq:open-3}) for a density matrix by including the normalization factor $n = \mathrm{tr}[ \hat\varrho]$:
\begin{equation}
\varrho = \frac{1}{2} ( n I + x \sigma_x + y \sigma_y + z \sigma_z ) .
\label{eq:filter-derive-70}
\end{equation}
The unnormalized quantum filter may be expressed in terms of the extended Bloch vector
$\check r=(n,x,y,z)^T$ as follows:
\begin{eqnarray}
d n(t) &=& \sqrt{\kappa_1}\, x(t) dY(t),
\label{eq:filter-derive-80-n}
\\
d x(t) &=& ( -\omega y(t) - \frac{\kappa_1+\kappa_2}{2} x(t)) dt
+ \sqrt{\kappa_1}\,
( n(t) + z(t) ) dY(t) ,
\label{eq:filter-derive-80-x}
\\
d y(t) &=& (\omega x(t) - \frac{\kappa_1+\kappa_2}{2} y(t)) dt ,
\label{eq:filter-derive-80-y}
\\
d z(t) &=& (- (\kappa_1+\kappa_2) z(t) -(\kappa_1+\kappa_2)n(t) ) dt
- \sqrt{\kappa_1}\, x(t) dY(t) ,
\label{eq:filter-derive-80-z}
\end{eqnarray}
where here $ x(t) = \nu_t(\sigma_x)$, etc. The normalized quantum filter (\ref{eq:filter-30-x})-(\ref{eq:filter-30-z}) may be obtained from (\ref{eq:filter-derive-80-n})-(\ref{eq:filter-derive-80-z}) by dividing by $n(t)$ and using Ito's rule.
Can we average the unnormalized quantum filter (\ref{eq:filter-derive-8}) and obtain the master equation (\ref{eq:filter-6})? The answer is that we can provided we use the correct expectation. Indeed, define the quantum expectation
\begin{equation}
\mathbb{P}_t^0[ X ] = \mathbb{P}[ U(t) X U^\ast(t) ] .
\label{eq:filter-derive-9}
\end{equation}
Then with respect to $\mathbb{P}_t^0$, the measurement signal $Y(s)$, $0 \leq s \leq t$, has the same statistics as the input quadrature $Q(t)$ has with respect to $\mathbb{P}$. By the spectral theorem, $Y(t)$ is equivalent to a standard Wiener process with resect to a classical probability distribution $\mathbf{P}^0_t$. Consequently, for any atomic operator $X$,
\begin{eqnarray}
\mathrm{tr}[ \rho_t X ] &=&
\mathbb{P}[ X(t) ] = \mathbb{P}[ \mathbb{P} [ \tilde U^\ast(t) X \tilde U(t) ] \vert \mathscr{Z}_t ] ]
\nonumber \\
&=&
\mathbb{P}[ U(t) U^\ast(t) \mathbb{P} [ \tilde U^\ast(t) X \tilde U(t) ] \vert \mathscr{Z}_t ] U(t) U^\ast(t)]
\nonumber \\
&=&
\mathbb{P}^0_t[ \nu_t(X) ]
\nonumber \\
&=& \mathbf{P}^0_t[ \mathrm{tr}[ \hat{\varrho}_t X ] ] .
\label{eq:filter-derive-10}
\end{eqnarray}
The classical expectation in the last line of (\ref{eq:filter-derive-10}) averages $Y(s)$, $0 \leq s \leq t$, with respect to $\mathbf{P}^0_t$.
\subsection{Comments}
\label{sec:filter-comments}
In hindsight, quantum filtering seems to be a natural generalization of classical estimation ideas. It should be understood that the development of the quantum filter was an impressive intellectual achievement, due to a number of authors in the 1980's, notably Belavkin and Carmichael. Certainly, looking at the postulates of quantum mechanics as they are typically presented, it is far from clear how filtering ideas might emerge.
Quantum filtering builds on the underlying theoretical framework for open quantum systems, which includes quantum operations, master equations and QSDEs, that was developed over several decades, largely in quantum optics. Indeed, filtering ideas have not been developed in other areas of quantum physics, as far as I know, with the exception of Korotkov's work in solid state physics (for example, \cite{AK01}).
However, the fundamental statistical notion of conditional expectation is universal when correctly implemented.
\section{Optimal Measurement Feedback Control}
\label{sec:optimal}
In Sections \ref{sec:open-loop-optimal} and \ref{sec:open-loop-impulsive} we saw how optimal control methods could be used to design {\em open loop} control signals. In this section we allow the classical control signals to depend on a measurement signal (in a causal way) and formulate a performance criterion to optimize; the result will be a measurement feedback control system that has been optimally designed. In general, the optimal controller will be a classical dynamical system that processes the measurement signal to produce the control actions. The nature of the controller dynamics will depend on the performance criterion used. In what follows we discuss two performance criteria, known as {\em risk-neutral} and {\em risk-sensitive}, \cite{J04,J05}.
The risk-neutral criterion leads to an optimal controller whose internal dynamics are given by the quantum filter, \cite{DJ99}, \cite{EB05}. This is quite natural, and generalizes classical results going back to Kalman's LQG control theory. However, the risk-sensitive criterion gives rise to a different type of filter, first obtained classically by \cite{W81}.
In this section we consider continuous dynamics of the two-level atom, and omit the impulsive controls.
The system under control is defined by the parameters
\begin{equation}
A = ( \sqrt{\kappa_1}\, \sigma_-, \, \frac{1}{2} (\omega \sigma_z + u \sigma_x ) ) \boxplus ( \sqrt{\kappa_2}\, \sigma_-, 0) ,
\label{eq:direct-1}
\end{equation}
where $u(t)$ is the classical control signal determined by a classical controller $K$ from information in the measurement signal $Y(t)=B_{out,1}(t)+B_{out,1}^\ast(t)$ (recall section \ref{sec:filter}), as in Figure \ref{fig:atom-2} with $\gamma$ empty).
A {\em measurement feedback controller} $K$ is a causal classical system that processes a measurement signal $Y(s)$, $0 \leq s \leq t$, to produce control actions $u(t)$. We may write $u(t) = K_t( Y(s), \, 0 \leq s \leq t)$.
\subsection{Optimal Risk Neutral Control}
\label{sec:optimal-rn}
Suppose we wish to maintain the atom in its excited state by measurement feedback. In order do do this in an optimal fashion, we must first encode this objective in a performance criterion $J(K)$ which we subsequently minimize. In order to specify $J(K)$, we need to define some cost observables.
We need an observable $C_0 \geq 0$ such that $\langle +1 \vert C_0 \vert +1 \rangle=0$ and $\langle \psi \vert C_0 \vert \psi \rangle > 0$ for all atomic states $\vert \psi \rangle \neq \vert +1 \rangle$, so that the excited state minimizes the expected value of $C_0$. The choice $C_0 = \vert -1 \rangle \langle -1 \vert=\mathrm{diag}(0,1)$ meets these conditions. We allow the control $u$ to be unbounded, but impose a penalty $c_1 \vert u \vert^2$, for a positive real number $c_1$. Combining, we define the cost observable
\begin{eqnarray}
C_1(u) = C_0 + \frac{c_1}{2} \vert u \vert^2 = \left( \begin{array}{cc}
\frac{c_1}{2} \vert u \vert^2 & 0
\\
0 & 1+ \frac{c_1}{2} \vert u \vert^2
\end{array}\right) ,
\label{eq:rn-1}
\end{eqnarray}
which will be integrated along a time interval $[0,T]$. We also define a cost observable for the final time
\begin{equation}
C_2 = c_2 C_0 = \left( \begin{array}{cc}
0& 0
\\
0 & c_2
\end{array}\right) ,
\label{eq:rn-2}
\end{equation}
where $c_2$ is a positive real number.
We can now define the performance criterion
\begin{equation}
J(K) = \mathbb{P} \left[ \int_0^T C_1(t) dt + C_2(T) \right],
\label{eq:rn-3}
\end{equation}
where $C_1(t) = U^\ast(t) C_1(u(t)) U(t)$ and $C_2(T) = U^\ast(T) C_2 U(T)$. The risk-neutral optimal control problem is to find a measurement feedback controller $K$ that minimizes $J(K)$, \cite{J05}.
The key step in solving this optimization problem is to re-express the performance criterion $J(K)$ in terms of quantities computable from the measurement signal. The obvious choice is to use the conditional state $\hat{\rho}_t =\frac{1}{2}(I+x(t)\sigma_x + y(t) \sigma_y + z(t) \sigma_z)$, which is possible because of the fundamental property (\ref{eq:filter-0}) of conditional expectations. Indeed, we have
\begin{eqnarray}
J(K) &=& \mathbb{P} \left[ \int_0^T \mathbb{P}[ C_1(t) \vert \mathscr{Y}_t ] dt + \mathbb{P}[ C_2(T) \vert \mathscr{Y}_T ] \right]
\nonumber \\
&=& \mathbb{P} \left[ \int_0^T \pi_t( C_1(u(t)) ) dt + \pi_T( C_2) \right]
\nonumber \\
&=&
\mathbf{P} \left[ \frac{1}{2} \int_0^T (1-z(t) + c_1 \vert u(t) \vert^2) dt + \frac{c_2}{2} (1-z(T)) \right] .
\label{eq:rn-4}
\end{eqnarray}
Now $J(K)$ is expressed in terms of the Bloch vector $r=(x,y,z)^T$, which evolves according to the controlled quantum filter
\begin{eqnarray}
d x(t) &=& ( -\omega y(t) - \frac{\kappa_1+\kappa_2}{2} x(t)) dt
\nonumber \\
&& \hspace{1.0cm} + \sqrt{\kappa_1}\,
(I + z(t) - x^2(t) ) (dY(t) - x(t)dt),
\label{eq:rn-5-x}
\\
d y(t) &=& (\omega x(t) - \frac{\kappa_1+\kappa_2}{2} y(t) -u(t)z(t) ) dt
\nonumber \\
&& \hspace{2.0cm} +
\sqrt{\kappa_1}\, x(t) y(t) (dY(t) - x(t)dt) ,
\label{eq:rn-5-y}
\\
d z(t) &=& (- (\kappa_1+\kappa_2) z(t) -(\kappa_1+\kappa_2) +u(t) y(t) ) dt
\nonumber \\
&& \hspace{1.5cm}
- \sqrt{\kappa_1}\, x(t)( 1+ x(t)) (dY(t) - x(t)dt) .
\label{eq:rn-5-z}
\end{eqnarray}
Equations (\ref{eq:rn-5-x})-(\ref{eq:rn-5-z}) are driven by the measurement data $Y(t)$, and so the conditional state $r(t)=(x(t), y(t), z(t))^T$ is available to the controller. Also, the innovations process $dW(t)=dY(t)-x(t)dt$ is a standard Wiener process, independent of the controller, and so we may regard equations (\ref{eq:rn-5-x})-(\ref{eq:rn-5-z}) as being driven by $W(t)$. Therefore we may re-write (\ref{eq:rn-5-x})-(\ref{eq:rn-5-z}) and the performance criterion $J(K)$ in the compact forms
\begin{equation}
d r(t) = f(r(t), u(t) ) dt + g(r(t) ) dW(t) ,
\label{eq:rn-6}
\end{equation}
and
\begin{equation}
J(K) = \mathbf{P} \left[ \int_0^T L(r(t), u(t)) dt + M(r(T)) \right] ,
\label{eq:rn-7}
\end{equation}
where now the controller $K$ determines $u(t)$ causally from knowledge of $r(s)$, $0 \leq s \leq t$.
Hence
we
have converted the original quantum measurement feedback optimal control problem into an equivalent classical stochastic control problem with full state information. This equivalent problem may be solved using standard methods of classical stochastic control theory, \cite{FR75}, \cite{FS06}.
Let's apply dynamic programming. The value function is defined by
\begin{equation}
V(r,t) = \inf_{u(\cdot)}
\mathbf{P} \left[
\int_t^T L(r(s), u(s)) ds + M(r(T)) \ : \ r(t)=r
\right],
\label{eq:rn-8}
\end{equation}
where (\ref{eq:rn-6}) is initialized at the Bloch vector $r(t)=r$, and the infimum is over open loop controls $u(\cdot)$.
The dynamic programming equation is
\begin{eqnarray}
\frac{\partial}{\partial t} S(r,t) + \min_{u} \{ \mathcal{L}^u S(r,t) + \frac{1}{2} ( 1-z+c_1 \vert u \vert^2 ) \} &=& 0,
\label{eq:rn-9-a} \\
S(r,T) &=& M(r) ,
\label{eq:rn-9-b}
\end{eqnarray}
where $\mathcal{L}^u$ is the generator of the SDE (\ref{eq:rn-6}): for a smooth function $\varphi(r)$,
\begin{eqnarray}
\mathcal{L}^u \varphi(r) = \frac{1}{2} \mathrm{tr} [g(r) g(r)^T D^2 \varphi(r) ] + D\varphi(r)[ f(r,u) ] .
\label{eq:rn-10}
\end{eqnarray}
Here, $D^2V$ denotes the Hessian matrix of second-order partial derivatives.
The minimum in (\ref{eq:rn-9-a}) is attained at the value
\begin{equation}
\mathbf{u}^\star(r,t) = \mathbf{u}^\star(x,y,z,t) = \frac{1}{c_1}( S_y (x,y,z,t) z -S_z(x,y,z,t) y ).
\label{eq:rn-12}
\end{equation}
By the verification theorem, if we have a smooth solution $S(r,t)$ to the DPE (\ref{eq:rn-9-a})-(\ref{eq:rn-9-b}), and a control $u^\star(t)$ such that
\begin{equation}
u^\star(t) = \mathbf{u}^\star (x(t), y(t), z(t), t) ,
\label{eq:rn-11}
\end{equation}
then $u^\star(t)$ is optimal and $S(r,t)=V(r,t)$. The expression (\ref{eq:rn-11}) defines an optimal state feedback controller for the equivalent classical problem.
We can now define an optimal measurement feedback controller $K^\star$ for the quantum optimal control problem using the function $\mathbf{u}^\star(r,t)$ defined by (\ref{eq:rn-11}) and the controlled quantum filter (\ref{eq:rn-5-x})-(\ref{eq:rn-5-z}):
\begin{equation}
K^\ast_t( Y(\cdot) ) = \mathbf{u}^\star(r(t), t).
\label{eq:rn-13}
\end{equation}
The master equation for the optimal measurement feedback system may be obtained by substituting (\ref{eq:rn-12}) into (\ref{eq:rn-5-x})-(\ref{eq:rn-5-z}) and taking expectations:
\begin{eqnarray}
\dot x(t) &=& -\omega y(t) - \frac{\kappa_1+\kappa_2}{2} x(t) ,
\label{eq:rn-14-x}
\\
d y(t) &=& \omega x(t) - \frac{\kappa_1+\kappa_2}{2} y(t) - \frac{1}{c_1} S_y z^2(t) + \frac{1}{c_1} S_z z(t) y(t),
\label{eq:rn-14-y}
\\
d z(t) &=& - (\kappa_1+\kappa_2) z(t) -(\kappa_1+\kappa_2) +\frac{1}{c_1} S_y z(t) y(t) -\frac{1}{c_1} S_z y^2(t).
\label{eq:rn-14-z}
\end{eqnarray}
Examination of the mean closed loop dynamics (\ref{eq:rn-14-x})-(\ref{eq:rn-14-z}) reveals that the equilibria depend on the partial derivatives of the value function $S(x,y,z)$. A complete analysis of the performance of this system would benefit from a detailed numerical study.
Recently, optimal risk-neutral measurement feedback methods (LQG) were applied to the problem of frequency locking of an optical cavity
and the control system was experimentally demonstrated, \cite{HHHPJ09}.
\subsection{Optimal Risk Sensitive Control}
\label{sec:optimal-rs}
In order to define a risk-sensitive performance criterion, we introduce $R(t)$ as the solution to the equation
\begin{equation}
\frac{dR(t)}{dt} = \frac{\mu}{2} C_1(t) R(t),
\label{eq:rs-1}
\end{equation}
with initial condition $R(0)=I$. Here, $C_1(t) = U^\ast(t) C_1(u(t)) U(t)$ as in the previous section, and $\mu > 0$ is a positive real number called a risk parameter. The solution to (\ref{eq:rs-1}) can be expressed as a time-ordered exponential
\begin{equation}
R(t) = \stackrel{\leftarrow}{\exp} \left(\frac{\mu}{2} \int_0^t C_1(s) ds \right) .
\label{eq:rs-2}
\end{equation}
We then define the {\em risk-sensitive} cost function to be the quantum
expectation
\begin{equation}
J^\mu(K) = \mathbb{P}[R^\ast(T) e^{\mu C_2(T)} R(T) ],
\label{eq:rs-3}
\end{equation}
where $C_2(T)= U^\ast(T) C_2 U(T)$. The risk-sensitive optimal control problem is to find a measurement feedback controller $K$ that minimizes $J^\mu(K)$, \cite{J05}.
It can be seen that the risk-sensitive performance criterion $J^\mu(K)$, equation (\ref{eq:rs-3}), has a multiplicative form, in contrast to the additive form used in the risk-neutral criterion $J(K)$, equation (\ref{eq:rn-3}). This multiplicative form precludes us from expressing $J^\mu(K)$ in a useful way in terms of the condition density $\hat{\rho}_t$. In 1981, Whittle showed how to solve a classical risk-sensitive problem by using a {\em modified} conditional state, with a corresponding modified filter, \cite{W81}, \cite{BV85}, \cite{JBE94}. We now explain how this works for the quantum risk-sensitive criterion $J^\mu(K)$, \cite{J05}, \cite{WDDJ06}.
We begin by noting that $e^{\mu C_2(T)} = U^\ast(T) e^{\mu C_2} U(T)$, and so, in view of the form of the criterion (\ref{eq:rs-3}), it is natural to define $U^\mu(t) = U(t) R(t)$, which satisfies the QSDE
\begin{eqnarray}
d U^\mu (t) &=& \{ \sqrt{\kappa_1}\, dB_1^\ast(t) \sigma_- - \sqrt{\kappa_1}\, \sigma_+ dB_1(t)
+ \sqrt{\kappa_2}\, dB_2^\ast(t) \sigma_- - \sqrt{\kappa_2}\, \sigma_+ dB_2(t)
\nonumber \\
&& \hspace{0.1cm}
- (\frac{1}{2} (\kappa_1 + \kappa_2 ) \sigma_+ \sigma_- + i \frac{1}{2}\omega \sigma_z - C_1(u(t)) ) dt \} U^\mu (t),
\label{eq:rs-4}
\end{eqnarray}
with initial condition $U^\mu (0)=I$. Equation (\ref{eq:rs-4}) is a modification of the Schrodinger equation (\ref{eq:probe-2}), via the inclusion of the cost observable $C_1(u(t))$. The risk-sensitive performance criterion $J^\mu(K)$ is therefore
\begin{equation}
J^\mu(K) = \mathbb{P}[ U^{\mu\, \ast}(T) e^{\mu C_2} U^\mu(T) ].
\label{eq:rs-5}
\end{equation}
Now $U^\mu(t)$ does not commute with $\mathscr{Z}_t$, and so we follow the approach taken to derive the quantum filter in section \ref{sec:filter-qf-derivation}. Define $\tilde U^\mu(t)$ to be the solution of
\begin{eqnarray}
d \tilde U^\mu(t) &=& \{ \sqrt{\kappa_1}\, \sigma_- dZ(t)
+ \sqrt{\kappa_2}\, dB_2^\ast(t) \sigma_- - \sqrt{\kappa_2}\, \sigma_+ dB_2(t)
\nonumber \\
&& \hspace{0.1cm}
- (\frac{1}{2}( \kappa_1+\kappa_2) \sigma_+ \sigma_- + i \frac{1}{2}\omega \sigma_z - C_1(u(t)) ) dt \} \tilde U^\mu (t),
\label{eq:rs-6}
\end{eqnarray}
with initial condition $\tilde U^\mu(0)=I$. Then $\tilde V(t)$ commutes with $\mathscr{Z}_t$, and
\begin{equation}
J^\mu(K) = \mathbb{P}[ \tilde U^{\mu\, \ast}(T) e^{\mu C_2} \tilde U^\mu(T) ].
\label{eq:rs-7}
\end{equation}
Then by a calculation similar to (\ref{eq:filter-derive-10}), with $\tilde U^\mu$ replacing $\tilde U$, we find that
\begin{eqnarray}
J^\mu(K) &=& \mathbb{P}^0_t[ \nu^\mu_t (e^{\mu C_2} )]
\nonumber \\
&=&
\mathbf{P}^0_t [ \mathrm{tr}[ \hat{\varrho}^\mu_t ( e^{\mu C_2} ) ],
\label{eq:rs-8}
\end{eqnarray}
where we introduce the risk-sensitive conditional state
\begin{equation}
\nu^\mu_t(X) = U^\ast(t) \mathbb{P}[ \tilde U^{\mu \, \ast}(t) X \tilde U^\mu(t) \vert \mathscr{Z}_t ] U(t),
\label{eq:rs-9}
\end{equation}
and corresponding risk-sensitive conditional density: $\nu^\mu_t(X)= \mathrm{tr}[ \hat{\varrho}^\mu_t X]$.
The risk-sensitive quantum filter is
\begin{eqnarray}
d \hat\varrho^\mu_t &=& ( i[\hat\varrho^\mu_t, H(u(t)]
+\mathcal{C}^\ast_{C_1(u(t))}(\hat\varrho^\mu_t)
+ \mathcal{L}^\ast_{L_1}( \hat\varrho^\mu_t) + \mathcal{L}^\ast_{L_2}( \hat\varrho^\mu_t))dt
\nonumber \\
& & \hspace{1.0cm} + (L_1 \hat\varrho^\mu_t + \hat\varrho^\mu_t L_1^\ast) dY(t),
\label{eq:rs-10}
\end{eqnarray}
with initial condition $\hat\varrho^\mu_0 = \rho$, where the running cost superoperator is defined by
\begin{equation}
\mathcal{C}^\ast_{C}(\rho) = \frac{\mu}{2}( C \rho + \rho C ) .
\label{eq:rs-11}
\end{equation}
In equation (\ref{eq:rs-10}), the measurement signal $Y(s)$, $0\leq s \leq t$, is a standard Wiener process with respect to the classical probability distribution $\mathbf{P}^0_t$.
The risk-sensitive cost $J^\mu(K)$ may be expressed in terms of the
extended Bloch vector
$\check r=(n,x,y,z)^T$ by substituting the expression $\hat\rho^\mu= \frac{1}{2} ( n I + x \sigma_x + y \sigma_y + z \sigma_z )$ into (\ref{eq:rs-8}):
\begin{equation}
J^\mu(K) = \mathbf{P}^0 \left[
\frac{1}{2}( N(T)-z(T)) e^{\mu c_2}
\right],
\label{eq:rs-13}
\end{equation}
where
\begin{eqnarray}
d n(t) &=&
\frac{\mu}{2} (n(t)-z(t) + c_1 \vert u(t) \vert^2 n(t)) dt +
\sqrt{\kappa_1}\, x(t) dY(t),
\label{eq:rs-12-n}
\\
d x(t) &=& ( -\omega y(t) - \frac{\kappa_1+\kappa_2}{2} x(t)
+\frac{\mu}{2} (x(t) + c_1 \vert u(t) \vert^2 x(t))
) dt
\nonumber \\
&& \hspace{1.0cm}
+ \sqrt{\kappa_1}\,
( n(t) + z(t) ) dY(t) ,
\label{eq:rs-12-x}
\\
d y(t) &=& (\omega x(t) - \frac{\kappa_1+\kappa_2}{2} y(t) -u(t) z(t)
+ \frac{\mu}{2} (y(t) + c_1 \vert u(t) \vert^2y(t) )
) dt ,
\label{eq:rs-12-y}
\\
d z(t) &=& (- (\kappa_1+\kappa_2) z(t) -(\kappa_1+\kappa_2)n(t) + u(t) y(t)
-\frac{\mu}{2}( n(t)-z(t) +c_1 \vert u(t) \vert^2 z(t))
) dt
\nonumber \\
&& \hspace{1.0cm}
- \sqrt{\kappa_1}\, x(t) dY(t) .
\label{eq:rs-12-z}
\end{eqnarray}
These equations for $\check r(t)$ are of the form
\begin{equation}
\dot{ \check{r}} (t) = f^\mu( \check r(t), u(t)) dt + g^\mu(\check r(t) ) dY(t) .
\label{eq:rs-15}
\end{equation}
The value function for the risk-sensitive problem is defined by
\begin{equation}
V^\mu(\check r, t) = \inf_{u(\cdot)} \mathbf{P}^0 \left[
\frac{1}{2}( N(T)-z(T)) e^{\mu c_2} \ : \ \check{r}(t) = \check r
\right],
\label{eq:rs-14}
\end{equation}
where $\check r(\cdot)$ evolves on the time interval $[t,T]$ according to (\ref{eq:rs-15}) with initial condition $\check r(t)= \check r$.
The corresponding dynamic programming equation is
\begin{eqnarray}
\frac{\partial}{\partial t} S^\mu(\check r,t) + \min_{u} \mathcal{L}^{\mu,u} S^\mu(\check r,t) &=& 0,
\label{eq:rs-15-a} \\
S^\mu(\check r,T) &=& \frac{1}{2}( n-z)e^{\mu c_2}
\label{eq:rs-15-b}
\end{eqnarray}
where $\mathcal{L}^{\mu,u}$ is the generator of the SDE (\ref{eq:rs-15}): for a smooth function $\varphi(r)$,
\begin{eqnarray}
\mathcal{L}^{\mu,u} \varphi(r) = \frac{1}{2} \mathrm{tr} [g^\mu(\check r) g^\mu(\check r)^T D^2 \varphi(\check r) ] + D\varphi(\check r)[ f^\mu(\check r,u) ] .
\label{eq:rs-16}
\end{eqnarray}
Now the minimum in (\ref{eq:rs-15-a}) is attained at the value $\mathbf{u}^{\mu, \star}(\check r,t) = \mathbf{u}^{\mu, \star}(n,x,y,z,t) $ given by
\begin{eqnarray}
\mathbf{u}^{\mu, \star}(n,x,y,z,t) = \frac{1}{\mu c_1 S^\mu(n,x,y,z) }( S^\mu_y (n,x,y,z,t) z -S^\mu_z(n,x,y,z,t) y ).
\hspace{0.8cm}
\label{eq:rs-17}
\end{eqnarray}
In deriving (\ref{eq:rs-17}), we have used the relation $DS^\mu(\check r,t) \cdot \check r = S^\mu(\check r, t)$ which follows from the multiplicative homogeneity property $V^\mu( \alpha \check r, t)= \alpha V^\mu(\check r, t)$ enjoyed by the value function.
The verification theorem for the risk-sensitive problem asserts that if we have a smooth solution $S^\mu(\check r,t)$ to the DPE (\ref{eq:rs-15-a})-(\ref{eq:rs-15-b}), and a control $u^{\mu, \star }(t)$ such that
\begin{equation}
u^{\mu, \star} (t) = \mathbf{u}^{\mu, \star} (n(t), x(t), y(t), z(t), t) ,
\label{eq:rs-18}
\end{equation}
then $u^{\mu, \star} (t)$ is optimal and $S^\mu(\check r,t)=V^\mu(\check r,t)$.
We can now define an optimal measurement feedback controller $K^{\mu,\star}$ for the quantum risk-sensitive optimal control problem using the function $\mathbf{u}^{\mu, \star}(\check r,t)$ defined by (\ref{eq:rs-18}) and the risk-sensitive quantum filter (\ref{eq:rs-12-x})-(\ref{eq:rs-12-z}):
\begin{equation}
K^{\mu, \ast}_t( Y(\cdot) ) = \mathbf{u}^{\mu, \star}(\check r(t), t).
\label{eq:rs-19}
\end{equation}
\subsection{Optimal Measurement Feedback Impulsive Control}
\label{sec:optimal-rn-impulse}
It should be apparent to the reader that one may formulate and solve optimal measurement feedback problems for systems with impulsive controls. Due to space limitations, we do not pursue this further in this article.
\section{Coherent Feedback Control}
\label{sec:coherent}
As discussed in Section \ref{sec:types}, coherent feedback systems preserve quantum information through the use of a controller which is itself a quantum system, and one or more means of transferring quantum information between the plant and the controller. While the idea of coherent feedback is natural, to date there is little known about how to {\em design} coherent feedback systems in a systematic manner. In this section we discuss two simple examples.
\subsection{Coherent Feedback Control using Direct Couplings}
\label{sec:coherent-direct}
In this section we take a look at how quantum information may be transfered from a quantum controller to a quantum plant using an impulse implementing a rapid coherent interaction between the two systems. This example is based on the coherent spin control example discussed in \cite[Sec. III.E]{SL00}.
The plant $P$ and controller $C$ are independent two level systems, with Pauli matrices $\sigma_\alpha^{(P)}$, $\sigma_\alpha^{(C)}$ ($\alpha=x,y,z$). By suitable choice of reference or otherwise, we assume that these systems have trivial self-energies, and are decoupled from external fields.
The parameters $P=(0,0)$ and $C=(0,0)$ describe the trivial dynamics $\dot \sigma_\alpha^{(P)}(t)=0$, $\dot \sigma_\alpha^{(C)}(t)=0$ of these systems in the absence of interaction.
We assume that interactions between the plant and controller may be described by the action of bipartite unitaries
\begin{equation}
\mathbf{V} = \{ CNOT_{PC}, \ \ CNOT_{CP} \} .
\label{eq:cfb-d-1}
\end{equation}
Here, $CNOT_{AB}$ is the CNOT gate with $A$ as the control bit:
\begin{eqnarray*}
CNOT_{AB}\vert 00 \rangle &=& \vert 00 \rangle
\\
CNOT_{AB}\vert 01 \rangle &=& \vert 01 \rangle
\\
CNOT_{AB}\vert 10 \rangle &=& \vert 11 \rangle
\\
CNOT_{AB}\vert 11 \rangle &=& \vert 10 \rangle .
\end{eqnarray*}
The interactions are applied via an impulsive control sequence $\gamma$, with unitaries $V_k \in \mathbf{V}$.
In the Heisenberg picture, the hybrid equations of motion are simply
\begin{eqnarray}
\dot \sigma_\alpha^{(P)}(t) &= & 0, \ \ \tau_k < t \leq \tau_{k+1},
\label{eq:cfb-d-2-P}
\\
\dot \sigma_\alpha^{(C)}(t) &= & 0, \ \ \tau_k < t \leq \tau_{k+1},
\label{eq:cfb-d-2-C}
\\
\sigma_\alpha^{(P)}(\tau_k^+) &=& V_k^\ast (\tau_k) \sigma_\alpha^{(P)}(\tau_k) V_k (\tau_k) ,
\label{eq:cfb-d-2-P-tau}
\\
\sigma_\alpha^{(C)}(\tau_k^+) &=& V_k^\ast (\tau_k) \sigma_\alpha^{(C)}(\tau_k) V_k (\tau_k) .
\label{eq:cfb-d-2-C-tau}
\end{eqnarray}
The control objective considered in \cite[sec. III.E]{SL00} was to put the plant in the state $\vert \downarrow_P \rangle$. If the plant is in an arbitrary pure initial state, then this objective may be achieved by first initializing the controller in the state $\vert \downarrow_C \rangle$, and then applying the impulsive control $\gamma = ( (0, CNOT_{PC}), (1, CNOT_{CP}))$, as the reader may readily verify.
If the plant and controller are subject to non-trivial dynamics between the application of impulses, such as decoherence effects, then a more general hybrid dynamical model may be developed along the lines discussed in Section \ref{sec:types}.
\subsection{Coherent Feedback Control using Quantum Signals}
\label{sec:coherent-signals}
In Section \ref{sec:networks} we described a class of quantum feedback networks involving the interconnection of systems or subsystems via freely traveling quantum fields (quantum signals). The examples discussed illustrate the point that this type of feedback may be used to change the dynamical behavior of the plant. An important challenge for control theory is to develop ways of {\em designing} signal-based coherent feedback systems in order to meet performance specifications., \cite{YK03a}, \cite{YK03b}, \cite{JNP08}, \cite{HM08}, \cite{NJP09}, \cite{GJ09}, \cite{KNPM09}, \cite{NJD09}, \cite{JG10}, \cite{HM12}, \cite{CTSAM13}.
While a detailed discussion of signal-based coherent feedback control design is beyond the scope of this article, we briefly describe an example from \cite{JNP08}, \cite{HM08}. In this example, the plant is a cavity with three mirrors defining three field channels. The problem was to design a coherent feedback system to minimize the influence of one input channel $w$ on an output channel $z$, Figure \ref{fig:hinfty1}. That is, if light is shone onto the mirror corresponding to the input channel $w$, we would like the output channel $z$ to be dark. This is a simple example of robust control, where $z$ may be regarded as a performance quantity (to be minimized in magnitude), while $w$ plays the role of an external disturbance.
\begin{figure}[h]
\begin{center}
\includegraphics{hinfty1.eps}
\caption{Coherent feedback control example, showing plant $a$ and controller $a_K$ cavity modes,
together with performance quantity $z$ and the \lq\lq{disturbance}\rq\rq \ input $w$. The coherent signals $u$ and $y$ are used to transfer quantum information between the plant and the controller. The feedback system was designed to minimize the intensity of the light at the output $z$ when an optical signal is applied at the input $w$.}
\label{fig:hinfty1}
\end{center}
\end{figure}
In \cite{JNP08}, it was shown how such problems could be solved systematically by extending methods from classical robust control theory, and importantly, taking into account the physical realization of the coherent controller as a quantum system. Indeed, the controller designed turned out to be another cavity, with mirror transmissivity parameters determined using mathematical methods. This approached was validated by experiment \cite{HM08}.
\bibliographystyle{plain}
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\section{Introduction}
Ternary systems based on
III-V semiconductors (such as disordered alloys \cite{dj,laks,lemos},
heterostructures \cite{aleoff,christensen,anderson,tit,rubio,az94}
or quantum well systems \cite{vlaev,prins,qw3}) have been the
subject of wide scientific interest and
of accurate theoretical studies,
since they could be used as fundamental components in a large class of
important devices (laser diodes
or infrared detectors, to name just a few) \cite{materials}. In the
present work, we
focus our attention on superlattices (SL),
whose structural, electronic and transport properties can be opportunely
tuned
by varying the constituent materials, the strain, the
ordering direction or the layers
thickness.
To this end, we have
examined the properties of interest in
ultrathin [111] ordered SL, specifically in common-anion
(GaSb)$_1$/(InSb)$_1$ systems
and in common-cation (InAs)$_1$/(InSb)$_1$ systems, using the
self-consistent all-electron FLAPW method with\-in the density functional
formalism. The systems considered here are now under experimental
investigation (results obtained for strained-layer InSb/GaSb quantum well
\cite{qwapl}, In$_x$Ga$_{1-x}$Sb/GaSb heterostructures \cite{qwjvc}
and InAs$_{1-x}$Sb$_{x}$ alloys \cite{inassb1,inassb2}
have already been published); at the present time, however,
we are not yet able to compare our predicted results
with experimental values regarding the SL.
The appreciable mismatch between the lattice parameters of the
binary constituents (5.7~$\%$ in (GaSb)$_1$/(InSb)$_1$ and 6.4~$\%$ in
(InAs)$_1$/(InSb)$_1$)
gives the opportunity to study the effects of the strain conditions
on the SL electronic properties. In analogy with
the common experimental approach \cite{qwapl,ohler,bensaada}
we have considered various growth conditions for the SL, leading
to different strain modes in the structure:
(i) pseudomorphic growth
on a substrate usually constituted by one of the binary constituents
(in which the lattice constant parallel to the growth plane is
taken equal to that of the bulk
semiconductor composing the substrate) and
(ii) a ``free standing mode'' (FSM), in which
no constraints are imposed on the bond-lengths, leading to relaxed
lattice constants for the binary constituents, both different
from their bulk values.
The structural parameters for all of the structures considered
have been chosen through total energy minimization
or according to the macroscopic theory of elasticity (MTE) \cite{VDW}.
The choice made towards the [111] direction is encouraged by recent
experimental observations \cite{osscupt} that suggested the spontaneous
ordering along this particular axis
(the so called CuPt structure)
shown by some III-V alloys during
vapour-phase growth.
The strong influence of the ordering direction on the SL electronic
properties is immediately clear if we consider the
ternary SL Brillouin Zones (BZ) as obtained
from the binary zincblende BZ through folding operations,
which are obviously different for [111] or [001] ordered systems; the
immediate consequence of this is a noticeable
difference in the electronic properties.
In order to study the dependence of the electronic properties
on the ordering direction, we have studied the common-anion and
the common-cation systems in the three different growth conditions
both in the CuAu-like
(having [001] direction as growth axis) and CuPt-like structure.
Following the model proposed by Wei and Zunger \cite{az89}, we consider
the SL as obtained first from
an ideal virtual crystal (a common-anion (common-cation) system
having the cation (anion) with intermediate
properties between the two cations relative to the binary constituents).
We then introduce a perturbative
potential, having a structural part (due to atomic displacements and
relaxation due to epitaxial strain)
and a chemical one (due to the electronegativity difference between the
constituent atoms).
In order to separate the effects due to the two different terms
in the expression of the potential,
we have studied strained (AC)$_1$/(AC)$_1$-type systems ideally
obtained by monolayer deposition
of the same binary constituent (AC) along
the [111] direction, in which the two different AC
bond-lengths are equal to those in the
equivalent (AC)$_1$/(BC)$_1$ SL.
We will discuss the results obtained in this work as follows:
first of all, we will briefly expose the computational
details and parameters used in the
calculations (Section II); in Section III our structural results
will be reported for all the different systems considered.
In Section IV
we will discuss the electronic properties of the SL, with particular
attention to the quantities of technological interest (such as
energy band-gaps and crystal field splittings) and their dependence on
growth direction, atomic substitution and strain conditions.
We will also discuss the charge density distribution and
in particular the localization of the charge carriers in the different
constituents sublattices.
Section V summarizes our main results and draws some conclusions.
\section{Method of calculation}
We have determined the properties of the structures considered using
the density functional formalism, within the local density
approximation (LDA) \cite{HK64}
exchange and correlation potential
as parametrized by Hedin\--Lund\-qvist \cite{HL}.
The calculations were performed using
the {\em ab initio} all-electron
full\--pot\-ential linearized
augmented plane wave method (FLAPW) \cite{FLAPW}. Core electrons
as well as valence ones are treated using a self-consistent
procedure; the shallow Ga $3d$ and In $4d$ states are
considered as valence
states, for which scalar-relativistic effects are included
in the self-consistent calculation, whereas
spin-orbit effects are treated in a perturbative approach.
For [111] ordered systems, angular momenta up to $l_{max}$ = 6
in the muffin tin spheres (with radius
$R_{\alpha}$ = 2.4 a.u. for all the constituents atoms) and
plane waves with wave vector up to $k_{max}$ = 3.3 a.u. are used, leading to
about 600 basis functions.
To perform integrations in reciprocal space,
a set of four special {\bf k} points is chosen in the
trigonal Brillouin zone (BZ), following the Monk\-horst\--Pack scheme \cite{MP}.
Similar values for these computational parameters
have been used for [001] ordered systems, with the only exceptions
represented by $l_{max}$ = 8
and a set of
three special {\bf k} points used for the integration over the tetragonal
Brillouin zone. Finally, the Broyden \cite{broy} method is used to
accelerate the convergence in
the self-consistent iterations.
\section{Structural properties}
The atomic ordering along the [111] direction of a SL grown on a (111)
substrate gives origin to a trigonal
Bravais lattice with $C^5_{3v}$ (Sch\"oenflies notation) space group
\cite{cryst}.
The unit cell in real space contains 4 atoms and the origin is taken on a
cation site \cite{magri}.
The free structural
parameters are determined following the
macroscopic theory of elasticity (MTE) \cite{VDW},
taking into account the elastic
properties of the constituent materials, and then compared with those
obtained through total energy minimization.
We observe that in each cell, there are two atoms belonging to the same
chemical species (the two Sb anions in the (GaSb)$_1$/(InSb)$_1$
systems and the two In cations in the (InAs)$_1$/(InSb)$_1$
systems), which are not equivalent from the coordination point of view.
As an example, we consider the particular case of (GaSb)$_1$/(InSb)$_1$ SL,
having the first Sb$_{Ga}$ bound with three Ga and one In, and the second
Sb$_{In}$ showing a complementary situation.
The total energy minimization procedure that considers all the
free parameters in the unit cell
as different degrees of freedom is a very onerous computational problem;
this encouraged some simplifications, such as
considering equal bond lengths between
equal atomic species ({\em i.e.,} in the common-anion system, we have chosen
$d_{InSb_{Ga}}\:=\:d_{InSb_{In}}$), thus reducing to three the
number of degrees of freedom
(in-plane lattice parameter and two different bond-lengths).
Although this simplification is frequently used
in total energy minimization \cite{magri}, one should be
aware that this approximation results in considering an
average over the two different local environments and therefore that
the real elastic structures may slightly differ from our
optimized SLs.
According to MTE, using the same notation as in Ref.\cite{VDW},
the structural parameters for the epilayer
are determined as follows:
\begin{eqnarray}
a^{epi}_{\parallel} &=& a^{subs} \nonumber \\ \nonumber
a^{epi}_{\perp} & = & a^{epi}\:\left[1-
D^{[111]}
\:\left(\frac{a_{\parallel}}{a^{epi}}-1
\right)\right]\\
\label{d111}
D^{[111]} &=& 2\: \left ( \frac{c_{11}+2\:c_{12}-2\:c_{44}}{c_{11}+2\:
c_{12}+4\:c_{44}} \right)\\ \nonumber
\epsilon^{epi}_{\parallel} & = & \frac{a_{\parallel}}{a^{epi}}-1 \\ \nonumber
\epsilon^{epi}_{\perp} & = & \frac{a_{\perp}^{epi}}{a^{epi}}-1
\end{eqnarray}
where $c_{ij}$ are the elastic constants for the bulk epilayer
(we have used the experimental values reported in Ref.\cite{harrison}).
Through total energy minimization of the ``ideal''
(AC)$_1$/(BC)$_1$ unrelaxed structures (in which all
the atoms are arranged in a zincblende structure with lattice constant $a$
-~our free parameter~-
and with bond-lengths $d_{AC}=d_{BC}=a\:(\sqrt{3}/4)$ ), we have found an
in-plane lattice constant very close to the average of the bulk constituents,
according to Vegard's rule.
We have thus examined a free standing mode structure (indicated in the
following
as Elastically Relaxed or simply ER), that has this value for the
in-plane lattice constant. In Table \ref{parstruttan} and
Table \ref{parstruttcat}
we report the calculated structural parameters
for the ternary
common-anion and common-cation systems:
the S1 (S2) system is a common-anion SL grown on a
GaSb (InSb) substrate, while the S3 (S4) system is a common-cation
SL grown on an InAs (InSb) substrate.
In the case of pseudomorphic growth on a substrate,
we have found general agreement
between the structures obtained through total energy minimization
and those given by MTE; this fully
justifies our having considered this approximation to
determine the five unknown parameters in the free standing mode (FSM) structure.
Note that deviations from the results predicted by MTE occur in the
case of InSb strained to GaSb or to InAs; in both these structures, total
energy
minimization gives an InSb bond-length that is
systematically larger (within 0.6 $\%$) than the one
expected according to MTE, even though the difference between the total energies
for the elastic SL and for the total energy minimal structures is very small
(barely larger than the numerical uncertainty of 1 mRy/unit cell).
However, this can be justified considering that, due to
its elastic properties, this material could easily be out of
the linear elastic region.
In fact, the elastic constants for InSb are quite smaller than those for GaSb
and InAs \cite{harrison}, resulting in a larger effective strain -~due to
the mismatch~- in the former case; this is also confirmed by the non-linear
behaviour of the band-gap
as a function of the strain, as will be discussed later.
As expected from
the similarity of the GaSb and InAs elastic constants
\cite{harrison} and bulk moduli (the experimental values
are $B^{InAs}$ = 0.579 Mb and $B^{GaSb}$ = 0.578 Mb \cite{landbor}), we
obtain similar deformations for these two constituents
respectively in the common-anion and
common-cation SLs. We also notice that strains
(either parallel or perpendicular) and
percentage deviations from bulk bond-lengths are more pronounced in the
common-cation systems, compared to the common-anion systems: this is obviously
a consequence of the greater mismatch between the constituent lattice
parameters in the (InAs)$_1$/(InSb)$_1$ structures.
In the case of [001] ordered systems
(whose structural parameters are reported in Table \ref{parstruttan}
and Table \ref{parstruttcat} respectively for common-anion and common-cation
SL),
we obtain a tetragonal
Bravais lattice with $D^5_{2d}$
space group \cite{cryst} and a
unit cell in real space with 4 atoms (two of which are equivalent);
the origin is taken on a cation (anion) site for the common-anion
(common-cation) system.
The MTE relations reported in Eqs.(\ref{d111})
are still valid for [001] ordered SL,
if the parameter $D$ is redefined as:
\begin{equation}
D^{[001]} = 2\: \left ( \frac{c_{12}}{c_{11}}
\right). \nonumber
\end{equation}
A comparison between
the [111] and [001] ordered structures
(having the same chemical composition)
shows that in the same growth conditions (FSM or growth on a substrate)
the structural parameters are not equal.
In particular we notice larger perpendicular strains in the [001]
compared to the [111] ordered structures, while
the parallel strains are obviously
equal in considering the same growth conditions; what we find is thus
a smaller deviation from bulk bond-lengths,
due to a more effective relaxation.
\section{Electronic properties}
\subsection{Electronic levels}
The determination of the SL electronic energy levels is a fundamental
point for most of the properties of interest in the systems considered.
In Table \ref{sllivris} we report the calculated electronic
levels (with a numerical
uncertainty of $\pm$~0.04~eV, equal for all the
energies reported in the present work, unless otherwise specified)
at the BZ center ($\Gamma$),
for the different [111] ordered systems considered
(free standing mode and pseudomorphic growth on the two substrates), both
for the common-anion and for the common-cation systems.
We also report the zincblende state from which
the SL state derives through folding the f.c.c. Brillouin zone back
into the smaller ternary trigonal zone.
The splitting, $\Delta_{CF}$,
of the triply degenerate $\Gamma_{15v}$ zincblende state
is due to the non-cubic crystal field and is conventionally taken
positive if the doubly degenerate state $\Gamma_{3v}^{(2)}$
has a higher energy compared to the state $\Gamma_{1v}$.
As can be seen from Table \ref{sllivris},
we obtain a negative $\Delta_{CF}$
in the case of GaSb (InAs) strained to InSb
for common-anion (cation) systems -
corresponding to an in-plane extensive strain $\epsilon_{\parallel}$ -
whereas
the complementary case (pseudomorphic growth on a GaSb-substrate
(InAs-substrate))
and the free standing mode produce a positive $\Delta_{CF}$.
The introduction of spin-orbit coupling removes the double degeneracy of
the $\Gamma_{3v}^{(2)}$ state
and yields the electronic energy levels
illustrated in Fig.\ref{elenlev} as a function of
the substrate lattice constant. The topmost valence bands (E$_1$, E$_2$,
E$_3$) have been labeled according to the ``quasi-cubic" model \cite{hopf}
(taking the centre of gravity of the SL valence bands as zero);
\begin{center}
\begin{tabular}{lll}
E$_{1}$ & = & $+\frac{1}{3}(\Delta_{s.o.}+\Delta_{CF})$\\
E$_{2,3}$ & = & $-\frac{1}{6}(\Delta_{s.o.}+\Delta_{CF})\pm
\frac{1}{2}\{(\Delta_{s.o.}+\Delta_{CF})^2-\frac{8}{3}\:\Delta_{s.o.}\:
\Delta_{CF}\}^{1/2}$\\
\end{tabular}
\label{LEVELS}
\end{center}
considering $\Delta_{CF}$ as obtained from Table \ref{sllivris}
and $\Delta_{s.o.}$ for the SL as the value averaged over the equivalent
calculated \cite{ale}
quantities for the binary constituents
(Even neglecting the $\Delta_{s.o.}$ negative bowing,
occurring in the common-cation SL \cite{az89}, it is possible to uniquely
identify the SL levels with the ``quasi-cubic" ones, which differ at most \
by 0.05~eV). From Fig.\ref{elenlev} we notice that the
E$_1$ and E$_c$ (the lowest conduction state)
levels show an almost linear behaviour
as a function of the
substrate lattice constant, while the other valence band states
show a more complex trend, due to the interplay between crystal-field
and s.o. effects (see the cross-over between the E$_1$ and E$_2$ states).
In Table \ref{sllivris001} we report the relevant
electronic energy levels for the CuAu
systems at $\Gamma$; the notation is analogous to
that of Table \ref{sllivris}, where
folding relations of the f.c.c. Brillouin zone
in the now tetragonal ternary zone involve the
zincblende states as indicated in the Table.
The trend in the signs of the crystal-field splittings
$\Delta_{CF}$ is similar to that evidenced in the
[111] ordered systems (see Table \ref{sllivris001}).
The general underestimate of the band-gap energy in LDA
has stimulated many attempts to solve this problem, but
correction algorithms \cite{GW,hyb,SIC} need an
extraordinary computational effort in the SL case;
thus our LDA band-gap energy ($E_{gap}^{LDA}$)
was corrected starting
from the experimental data of the binary constituents \cite{az91}.
Due to a lack
of experimental band-gaps regarding strained binaries,
we have fitted the calculated values
obtained for each binary
in different strain
conditions ({\em i.e.} tetragonal and trigonal),
assuming a linear trend
for the band-gap energy as a function of
the in-plane strain $\epsilon_{\parallel}$:
$E_{gap}^{LDA}(\epsilon_{\parallel})
= E_{gap}^{LDA}(0) + \alpha\:\epsilon_{\parallel}$,
where $E_{gap}^{LDA}(0)$ is the
binary equilibrium calculated band-gap
(a parabolic trend
$E_{gap}^{LDA}(\epsilon_{\parallel})
= E_{gap}^{LDA}(0) + \alpha\:\epsilon_{\parallel}
+ \beta\:\epsilon_{\parallel}^2$
has been used for InSb, which is assumed to be out
of the linear region).
Once we determined the coefficient $\alpha$,
we translated the curve so that it becomes:
$E_{gap}^{emp}(\epsilon_{\parallel})
= E_{gap}^{expt}(0) + \alpha\:\epsilon_{\parallel}$,
where
$E_{gap}^{expt}(0)$ is the
binary equilibrium experimental band-gap.
We have thus used these empirical values to
obtain the empirical band-gap energy averaged
over the strained binaries ($<E_{gap}^{emp}>$).
Summing this quantity to the correction
($\delta\:=\:E_{gap}^{LDA}(SL) - <E_{gap}^{LDA}>$ ), we have finally obtained
the predicted band-gap energy in the SL ( $E_{gap}^{emp}(SL)$ )
with a numerical uncertainty of $\pm$ 0.05~eV.
Although this procedure is empirical, it is expected to give good estimates of
the real band-gaps, since it is well known that while the band-gap is strongly
underestimated, the band-gap behaviour as a function of pressure is always
very well reproduced by LDA \cite{ale}.
We report in Tables \ref{slgap} and \ref{slgap001}
(respectively for [111] and
[001] ordered structures),
the band-gap energies as obtained from LDA
self-consistent unperturbed
calculations ($E_{gap}^{unp}$), with the introduction of the
perturbation due to spin-orbit coupling ($E_{gap}^{LDA}$)
and with the correction
starting from experimental data ($E_{gap}^{emp}$). From these Tables
we first notice that
for all the systems considered
we find a negative
$E_{gap}^{LDA}$
(due to an inversion which causes the conduction band minimum (CBM) to lie
below the valence band maximum (VBM)). Furthermore, we observe that
the larger the substrate lattice parameter, the smaller
the band-gap energy (either in common-anion or in
common-cation systems).
A comparison between Table \ref{slgap} and Table \ref{slgap001}
confirms the trend predicted by Wei and Zunger \cite{az91} for the energy
band-gap:
\begin{equation}
E_{gap}^{[111]}\:<\:E_{gap}^{[001]}\:<\:E_{gap}^{ave}
\label{andamento}
\end{equation}
where $E_{gap}^{[111]}$ and $E_{gap}^{[001]}$ are the band-gap energies
respectively in the [111] and in the [001] ordered
structures, while
$E_{gap}^{ave}$ is the band-gap average energy taken over the
binary constituents
(the calculated values of the LDA band-gap energy for the binary constituents
are $E_{gap}^{LDA}(GaSb)$ = - 0.47~eV, $E_{gap}^{LDA}(InSb)$ = - 0.67~eV
and $E_{gap}^{LDA}(InAs)$ = -~0.63 eV).
As is well-known \cite{az89}, band folding in the superstructures
causes a repulsion between two binary electronic states
of different symmetries, folded on a state of the same symmetry in the ternary phase and
coupled through the perturbative
potential mentioned in Section I (in Tables \ref{sllivris} and \ref{sllivris001}
the superscripts (1) and (2)
indicate the two states involved in the repulsion mechanism).
One of its interesting effects
is the band-gap narrowing,
compared to the equivalent quantity averaged over the binary
constituents (as confirmed by the second inequality in Eq.(\ref{andamento})).
The amount of this effect \cite{az89} is inversely proportional to the
difference
$[\epsilon(\Gamma_{1c})-\epsilon(L_{1c})]$ (in the [111] structure)
or to the difference $[\epsilon(\Gamma_{1c})-\epsilon(X_{1c})]$
(in the [001] structure); this difference is smaller
in the [111] structure, causing
a more striking band-gap narrowing than in the [001] structure
(as shown by the first inequality in Eq.(\ref{andamento})).
These observations are confirmed by the calculated values for the
band-gap bowing
parameters (defined, in analogy with the 50$\%$-50$\%$ alloys, as
$b_{gap}=4(E_{gap}^{ave}-E_{gap}$)) reported in Tables \ref{slgap}
and \ref{slgap001}:
we obtain a larger bowing in the [111] structures compared to the [001] ones
and, looking at the constituent chemical species,
we can say that the bowing in
common-cation systems is larger than in the common-anion ones.
The band-gap trend as a function of the substrate lattice constant and its
dependence on the ordering direction have been illustrated in Fig.\ref{gap},
where we report the LDA band-gap ($E^{LDA}$~-~solid line)
and corrected band-gap
($E^{emp}$~-~dashed line) as a function of
the substrate lattice constant for (GaSb)$_1$/(InSb)$_1$ (Fig.\ref{gap} (a))
and (InAs)$_1$/(InSb)$_1$ (Fig.\ref{gap} (b)).
A comparison between the SL energy band-gaps and
the average value ($E_{ave}$)
of the experimental (LDA) band-gap in the pure binaries
-~indicated by the filled (empty) circles~- clearly shows the band-gap
narrowing
effect.
Tables \ref{slgap} and \ref{slgap001} also show that the
crystal field splittings in the [111] ordered structures
are always bigger than in the [001] structures in the same growth conditions,
with the only exception represented by
the S4 system, which has a smaller $\Delta_{CF}$ compared to the other structures.
This apparently strange behaviour
can be explained by considering that the
$\Gamma_{3v}^{(2)}$ state interacts with the lower
$\Gamma_{3v}^{(1)}$ state, resulting in an upward shift
(an effect relevant only in the common-cation SLs);
furthermore, in the S4 system, the VBM is a
$\Gamma_{1v}$ state, which is
only slightly involved in the level repulsion mechanism. Thus
the stronger the level repulsion, the larger the
$\Gamma_{3v}^{(2)}$ upward shift and the smaller
$\Delta_{CF}$ becomes: this observation is thus a further proof of
the validity of the band repulsion model.
We observe that in the [111] ordered SLs the
band-gap energy is determined by the difference in energy between
the VBM -~\ slightly localized on the
anion belonging to the
InSb monolayer (as will be clearly shown in the next section)~\ -
and the CBM -~\ strongly localized
on the GaSb (InAs) sublattice in the common-anion (cation) superlattice.
Thus we could think of (GaSb)$_1$/(GaSb)$_1$-type systems as
common-anion SLs in which we substitute
the InSb monolayer with a GaSb monolayer.
We would expect in this case a small
modification of the band-gap energy, since the VBM will no longer be
localized on the Sb belonging to the InSb sublattice but rather
on the Sb
belonging to a GaSb sublattice; therefore, this will be only
a second order effect.
Our prediction is confirmed by the
calculated band-gap energies
(spin-orbit included) reported in
Table \ref{gapcoman} (second column)
which prove the almost total independence of the VBM on the cationic
substitution;
the change of the band-gap energy in the different structures is thus caused
by the structural
term in the perturbative SL potential rather than by the chemical term.
An equivalent interpretation considers
the (InSb)$_1$/(InSb)$_1$-type systems as SLs in which
we have substituted the GaSb monolayer
(where the CBM is strongly localized)
with an InSb monolayer.
In this case, the cationic substitution implies the chemical
alteration of one of the atomic species (Ga) on which the wave function is
strongly localized; thus, what we expect, is an appreciable change in the
band-gap energy, as confirmed by the
third column in Table \ref{gapcomcat} (from which we notice the
$E_{gap}^{LDA}$ increase).
An analogous trend is observed for the common-cation systems, where InAs
has now susbstituted the GaSb
as the InSb partner in the SL (see second and third column in Table
\ref{gapcomcat}).
The trend in the crystal field splitting is strongly dependent on the
class of systems
considered. In fact, in the common-anion systems, this quantity is almost
independent of the cationic substitution (as expected, because of the
anionic character of the VBM, localized
on the Sb atom).
In this case, the chemical term of the potential existing in the SL
has very little effect on $\Delta_{CF}$, compared to the structural term
(see Table \ref{gapcoman}). On the other hand, in common-cation systems
anionic substitution has a strong effect on the crystal field
splitting and the chemical term in the SL
potential is now much more important than before,
even though
the structural term still has a strong effect on $\Delta_{CF}$
(see Table \ref{gapcomcat}).
\subsection{Charge density distribution}
One of the main effects of the perturbative potential in the SL
(in particular of its chemical term, due to differences in the constituent
atom's orbital energies \cite{az94}) is the
localization of the charge density
in one of the constituent monolayers, which varies from state to state.
As an obvious consequence, this effect causes the confinement of the charge
carriers (holes or electrons respectively for valence or conduction
states) in a different sublattice.
In order to better clarify the character relative to
the different states of interest, their angular decomposition
-~for the common-anion
systems in the three growth conditions considered~-
is reported in Table \ref{decoman}
(we do not to report the equivalent Table for
(InAs)$_1$/(InSb)$_1$,
since this system is very similar to the previous one, as far as the charge
decomposition is concerned).
Referring to the charge density of the $\Gamma_{1v}$ state,
we notice in particular the growing $s$ character
on the InSb monolayer and the decreasing $p$ character on the In atom
as the substrate lattice parameter is increased;
at the same time, the $s$ charge density on the GaSb sublattice decreases,
while the $p$ charge grows on the Ga atom.
We report in Fig.\ref{gammav} (a) distribution for the $\Gamma_{1v}$ state for
the (GaSb)$_1$/(InSb)$_1$
elastically relaxed (ER) systems,
drawn the same for all the
charge densities reported in this work in a plane
perpendicular to the atomic layers.
This state comes from $p_z$ orbitals (as we can see from the typical
``butterfly'' shape along the vertical growth $z$-direction)
and shows a strong bonding character, between different
monolayers and within each monolayer.
In Fig.\ref{gammav} (b)
we report the charge density distribution
relative to the $\Gamma_{3v}^{(2)}$
(VBM) state for the common-anion system
in its elastically relaxed structure, where the localization
of the charge density in the InSb sublattice
is particularly evident. We notice that this peculiarity
is much more enhanced in the common-cation
system (not shown), as a probable
consequence of the anionic character of
this state: in fact, what we expect in the common-cation system is
for the Sb-atom to draw more charge than the As-atom.
We have found that the charge density distribution in this state
is not strongly influenced by the strain conditions, as can be seen from
Table \ref{decoman}.
The calculated charge density distribution
for the first conduction state $\Gamma_{1c}^{(1)}$
(CBM), relative to the common-anion elastically relaxed system,
is presented in Fig.\ref{gammac} (a).
What is relevant in this state is the strong localization
of the charge density in the
GaSb monolayer (a similar
behaviour is shown by the common-cation ER system, where the charge
density is concentrated on the InAs monolayer).
The localization emphasized above becomes more pronounced as
the substrate lattice parameter is increased:
the charge density distribution concentrates more and more
in the GaSb monolayer (InAs monolayer) while
at the same time the InSb monolayer
becomes charge-depleted (as confirmed by Table \ref{decoman}).
The second conduction state, $\Gamma_{1c}^{(2)}$,
shows a complementary trend,
owing to the charge density distribution that is more and more
concentrated on the InSb sublattice as the substrate lattice
parameter is increased
(as we notice from Fig.\ref{gammac} (b) for
the common-anion elastically relaxed system; this behaviour is
similar
to the common-cation systems).
As a consequence of these observations, in all these structures we have
a direct gap in reciprocal space, while we obtain a
``spatially indirect'' gap, due to the localization of the
$\Gamma_{3v}^{(2)}$ state (VBM) on the InSb sublattice
and of the $\Gamma_{1c}^{(1)}$ state (CBM) on the GaSb
(InAs) sublattice in the common-anion (common-cation) SL.
\section{Conclusions}
{\em Ab initio} FLAPW calculations, based on density functional theory
within LDA, have been performed in order to determine the
electronic properties of ultrathin SLs. In particular we have studied
a common-anion (GaSb)$_1$/(InSb)$_1$ system and a common-cation
(InAs)$_1$/(InSb)$_1$ system, ordered along two different ([111] and [001])
directions .
The relevant results obtained for these structures can be summarized
as follows:
\begin{enumerate}
\item Both the (GaSb)$_1$/(InSb)$_1$ and (InAs)$_1$/(InSb)$_1$ systems
show a direct gap ($E_{gap}^{[111]}$) which is smaller than the average
band-gap energy ($E_{gap}^{ave}$) taken over the binary constituents: the
dependence of this quantity on the ordering direction is expressed by the
relation:
$E_{gap}^{[111]}\:<\:E_{gap}^{[001]}\:<\:E_{gap}^{ave}$;
\item Both common-anion and common-cation systems show a decreasing
band-gap energy as the substrate lattice parameter is increased;
\item The structures studied offer interesting opportunities for band-gap
tuning as a function of growth conditions;
the range in which the gap varies is as large as 0.7~eV
in (GaSb)$_1$/(InSb)$_1$ type systems and 0.3~eV
in (InAs)$_1$/(InSb)$_1$ systems;
\item In the common-anion (common-cation) systems
the marked charge density localization of the CBM on the GaSb (InAs)
monolayer and of the VBM on the InSb monolayer causes the gap to be
``spatially indirect'';
\item In the case of
free standing mode elastically relaxed structures we obtain a band-gap
value of 0.05~$\pm$~0.05~eV in the common-anion system
(semiconducting properties) and of -0.26~$\pm$~0.05~eV
in the common-cation system (semimetallic properties);
\end{enumerate}
\section{ACKNOWLEDGEMENTS}
We thank B.W. Wessels for stimulating discussions and a
careful reading of the manuscript.
Work at Northwestern University supported by the MRL Program of the
National Science Foundation, at the Materials Research Center of
Northwestern University, under Award No. DMR-9120521, and by a grant of
computer time at Pittsburgh Supercomputing Center.
\newpage
\begin{table}
\centering
\caption{Bond-lengths ($d_{GaSb}$ and $d_{InSb}$ in a.u.)
and strain parameters ($\epsilon^{GaSb}$
and $\epsilon^{InSb}$) parallel and perpendicular to the growth plane
for (GaSb)$_1$/(InSb)$_1$
[111] and [001] ordered systems. The quantities denoted by $\Delta$
indicate percentage deviations from calculated bulk bond-lengths
($d_{GaSb}$ = 5.00 a.u. and $d_{InSb}$ = 5.29 a.u.).}
\vspace{5mm}
\begin{tabular}{|c|ccc|ccc|}
& & [111] & & & [001] & \\
& El.Rel. & GaSb-subs. & InSb-subs.&
El.Rel. & GaSb-subs. & InSb-subs.\\ \hline \hline
$d_{GaSb}$ & 5.07 & 5.00 & 5.15 & 5.05 & 5.00 & 5.11\\
$\Delta_{GaSb}$ &+1.4 \% & - & +3.0 \% & +1.0 \%& - & 2.2 \%\\
$\epsilon_{\parallel}^{GaSb}$ & +0.029 & 0 & +0.058 & +0.029 & 0 & +0.058
\\
$\epsilon_{\perp}^{GaSb}$ & -0.014 & 0 & -0.028 & -0.027 & 0 & -0.053 \\
\hline \hline
$d_{InSb}$ & 5.22 & 5.16 & 5.29 & 5.25 & 5.24 & 5.29 \\
$\Delta_{InSb}$& -1.3 \% & -2.5 \% & - & -0.8 \% & -0.9 \% & -\\
$\epsilon_{\parallel}^{InSb}$ & -0.028 & -0.055 & 0 & -0.028 & -0.055 & 0
\\
$\epsilon_{\perp}^{InSb}$ & +0.016 & +0.033 & 0 & +0.030 & +0.076 & 0 \\
\end{tabular}
\label{parstruttan}
\end{table}
\begin{table}
\centering
\caption{Bond-lengths ($d_{InAs}$ and $d_{InSb}$ in a.u.)
and strain parameters ($\epsilon^{InAs}$
and $\epsilon^{InSb}$) parallel and perpendicular to the growth plane
for (InAs)$_1$/(InSb)$_1$
[111] and [001] ordered systems. The quantities denoted by $\Delta$
indicate percentage deviations from calculated bulk bond-lengths
($d_{InAs}$ = 4.96 a.u. and $d_{InSb}$ = 5.29 a.u.).}
\vspace{5mm}
\begin{tabular}{|c|ccc|ccc|}
& & [111] & & & [001] & \\
& El.Rel. & InAs-subs. & InSb-subs. & El.Rel. & InAs-subs. & InSb-subs. \\
\hline \hline
$d_{InAs}$ & 5.04 & 4.96 & 5.12 & 5.01 & 4.96 & 5.07\\
$\Delta_{InAs}$ &+1.6 \% & - & +3.2 \% & +1.0 \%& - & 2.2 \%\\
$\epsilon_{\parallel}^{InAs}$ & +0.033 & 0 & +0.066 & +0.033 & 0 & +0.066
\\
$\epsilon_{\perp}^{InAs}$ & -0.019 & 0 & -0.038 & -0.036 & 0 & -0.072 \\ \hline \hline
$d_{InSb}$ & 5.21 & 5.14 & 5.29 & 5.24 & 5.22 & 5.29 \\
$\Delta_{InSb}$& -1.5 \% & -2.8 \% & - & -0.9 \% & -1.3 \% & -\\
$\epsilon_{\parallel}^{InSb}$ & -0.031 & -0.062 & 0 & -0.031 & -0.062 & 0
\\
$\epsilon_{\perp}^{InSb}$ & +0.019 & +0.037 & 0 & +0.033 & +0.078 & 0 \\
\end{tabular}
\label{parstruttcat}
\end{table}
\begin{table}
\centering
\caption{Calculated electronic energy levels (in eV) with respect to the VBM
for
the [111] SL (neglecting s.o. coupling).
The superscripts (1) and (2)
indicate the two states involved in the repulsion mechanism.
The state multiplicity is given in parentheses.}
\vspace{5mm}
\begin{tabular}{|c|c|ccc|ccc|}
SL State & ZB State &
& (GaSb)$_1$/(InSb)$_1$ & &
& (InAs)$_1$/(InSb)$_1$ & \\
&& ER & S1 & S2 & ER & S3 & S4\\\hline \hline
$\Gamma_{1c}^{(2)}$ & $L_{1c}$ & 0.50 & 0.70 & -0.20
& 0.79 & 1.04 & 0.36 \\
$\Gamma_{1c}^{(1)}$ & $\Gamma_{1c}$ & -0.81 & -0.64 & -1.47
& -1.02& -0.95 & -1.42\\
& & & & & & & \\
$\Gamma_{3v}^{(2)}$ &$\Gamma_{15v}$ & 0(2) & 0(2) & -0.42(2)
& 0(2) & 0(2)& -0.18(2) \\
$\Gamma_{1v}$ & $\Gamma_{15v}$& -0.11 & -0.59 & 0
& -0.21 & -0.77& 0 \\
$\Gamma_{3v}^{(1)}$ & $L_{3v}$ & -1.31(2) & -1.35(2) & -1.68(2)
& -1.38(2) & -1.46(2)&
-1.50(2) \\
\end{tabular}
\label{sllivris}
\end{table}
\begin{table}
\centering
\caption{Calculated electronic energy levels (in eV) with respect to the VBM
for
the [001] SL (neglecting s.o. coupling).
The superscripts (1) and (2)
indicate the two states involved in the repulsion mechanism.
The state multiplicity is given in parentheses.}
\vspace{5mm}
\begin{tabular}{|c|c|ccc|ccc|}
SL State & ZB State &
& (GaSb)$_1$/(InSb)$_1$ & &
& (InAs)$_1$/(InSb)$_1$ & \\
&& ER & S1 & S2 & ER & S3 & S4\\\hline \hline
$\Gamma_{1c}^{(2)}$ & $X_{1c}$ & 1.12& 1.26 & 0.78
& 1.46 & 1.54 & 1.01 \\
$\Gamma_{1c}^{(1)}$ & $\Gamma_{1c}$ & -0.49 & -0.43 & -0.77
& -0.68 & -0.69 & -0.96\\
& & & & & & & \\
$\Gamma_{5v}^{(2)}$ &$\Gamma_{15v}$ & 0(2) & 0(2) & -0.20(2)
& 0(2) & 0(2)& -0.25(2) \\
$\Gamma_{4v}$ & $\Gamma_{15v}$& -0.07 & -0.41 & 0
& -0.06 & -0.46 & 0 \\
$\Gamma_{5v}^{(1)}$ & $X_{5v}$ & -2.52(2) & -2.52(2) & -2.71(2)
& -2.46(2) & -2.46(2)& -2.70(2) \\
\end{tabular}
\label{sllivris001}
\end{table}
\begin{table}
\centering
\caption{Band-gap energies (in eV) for
(GaSb)$_1$/(InSb)$_1$ [111] systems and (InAs)$_1$/(InSb)$_1$ [111]
obtained from unperturbed LDA
calculations ($E_{gap}^{unp}$), with the introduction of spin-orbit coupling
($E_{gap}^{LDA}$) and corrected starting from experimental data
($E_{gap}^{emp}$). We also report the calculated
bowing parameter ($b_{gap}^{[111]}$) for the different
systems considered.}
\vspace{5mm}
\begin{tabular}{|c|ccc|ccc|}
\centering
& & (GaSb)$_1$/(InSb)$_1$ & &
& (InAs)$_1$/(InSb)$_1$ & \\
& ER & S1 & S2 & ER & S3 & S4 \\ \hline \hline
$E_{gap}^{unp}$ &-0.81& -0.64 & -1.47 & -1.02 & -0.95 & -1.42\\
$E_{gap}^{LDA}$ & -1.05& -0.90 & -1.59 & -1.26 & -1.20 & -1.49\\
$E_{gap}^{emp}$ & 0.05& 0.20 & -0.47 & -0.26 & -0.21 & -0.51\\ \hline
$b_{gap}^{[111]}$ & 1.92 & 1.32 & 4.08 & 2.42 & 2.18 & 3.34 \\
\end{tabular}
\label{slgap}
\end{table}
\begin{table}
\centering
\caption{Band-gap energies (in eV) for
(GaSb)$_1$/(InSb)$_1$ [001] systems and (InAs)$_1$/(InSb)$_1$ [001]
obtained from unperturbed LDA
calculation ($E_{gap}^{unp}$), with the introduction of spin-orbit coupling
($E_{gap}^{LDA}$) and corrected starting from experimental data
($E_{gap}^{emp}$). We also report the calculated
bowing parameter ($b_{gap}^{[001]}$) for the different
systems considered.}
\vspace{5mm}
\begin{tabular}{|c|ccc|ccc|}
\centering
& & (GaSb)$_1$/(InSb)$_1$ & & & (InAs)$_1$/(InSb)$_1$ & \\
& ER & S1 & S2 & ER & S3 & S4 \\ \hline \hline
$E_{gap}^{unp}$ &-0.49 & -0.43 & -0.77 & -0.68 & -0.69 & -0.96\\
$E_{gap}^{LDA}$ & -0.74 & -0.69 & -0.96 & -0.88 & -0.89 & -1.07\\
$E_{gap}^{emp}$ & 0.33 & 0.41 & 0.13 & 0.11 & 0.10 & -0.09\\ \hline
$b_{gap}^{[001]}$ & 0.68 & 0.48 &1.56 &0.90 & 0.94 & 1.66 \\
\end{tabular}
\label{slgap001}
\end{table}
\begin{table}
\centering
\caption{Band-gap energies $E_{gap}^{LDA}$, spin-orbit coupling
included,
and crystal-field splittings $\Delta_{CF}$ (in eV) for common-anion systems.}
\vspace{5mm}
\begin{tabular}{|c|ccc|ccc|ccc|}
\centering
& & (GaSb)$_1$/(InSb)$_1$ & & & (GaSb)$_1$/(GaSb)$_1$ & & &
(InSb)$_1$/(InSb)$_1$ & \\
&ER & S1 & S2 & ER & S1 & S2 & ER & S1 & S2\\\hline \hline
$E_{gap}^{LDA}$ &-1.05& -0.90 & -1.59 & -1.15
& -1.14 & -1.66 & -0.34 & -0.26 & -0.88\\
$\Delta_{CF}$ & 0.11 & 0.59 & -0.42 & 0.19 & 0.58 & -0.42 & 0.14
& 0.57 & -0.50 \\
\end{tabular}
\label{gapcoman}
\end{table}
\begin{table}
\centering
\caption{Band-gap energies $E_{gap}^{LDA}$, spin-orbit coupling included,
and crystal-field splittings $\Delta_{CF}$ (in eV) for common-cation systems.}
\vspace{5mm}
\begin{tabular}{|c|ccc|ccc|ccc|}
\centering
& & (InAs)$_1$/(InSb)$_1$ & & & (InAs)$_1$/(InAs)$_1$ & & &
(InSb)$_1$/(InSb)$_1$ & \\
&ER & S3 & S4 & ER & S3 & S4 & ER & S3 & S4\\\hline \hline
$E_{gap}^{LDA}$ &-1.26& -1.20 & -1.49 & -1.26 & -1.24 & -1.44 & -0.29 & -0.25 &
-1.09 \\
$\Delta_{CF}$ & 0.21 & 0.77 & -0.18 & 0.14 & 0.57 & -0.39 & 0.11
& 0.68 & -0.47 \\
\end{tabular}
\label{gapcomcat}
\end{table}
\begin{table}
\centering
\caption{Angular decomposition relative to the muffin tin charge density
(for $s$ ($Q_s$) and $p$ ($Q_p$) components)
of the different states,
neglecting s.o. coupling, for the (GaSb)$_1$/(InSb)$_1$ [111] systems.}
\vspace{5mm}
\small
\begin{tabular}{|cc|cccc|cccc|cccc|}
& & & S1 & & & & ER & & & & S2 & & \\
State& & $Ga$ & $Sb$ & $In$ & $Sb$ & $Ga$ & $Sb$ & $In$ & $Sb$ &
$Ga$ & $Sb$ & $In$ & $Sb$\\ \hline \hline
& & & & & & & & & & & & & \\
$\Gamma_{1v}^{}$ & $Q_s$ & 0.065 & 0.057 & 0.006 & 0.005 &
0.003 & 0.001 & 0.014 & 0.017 & 0.007 & 0.001 & 0.045 & 0.023\\
& $Q_p$ & 0.037 & 0.187 & 0.060 & 0.134 & 0.040 & 0.218 & 0.048 & 0.184 &
0.074 & 0.115 & 0.014 & 0.241\\
& & & & & & & & & & & & & \\
$\Gamma_{3v}^{(2)}$ & $Q_s$ & 0.000 & 0.000 & 0.000 & 0.000 &
0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000\\
& $Q_p$ & 0.023 & 0.122 & 0.061 & 0.321 & 0.025 & 0.117 & 0.063 & 0.306 &
0.026 & 0.114 & 0.063 & 0.294\\
& & & & & & & & & & & & & \\
$\Gamma_{1c}^{(1)}$ & $Q_s$ & 0.262 & 0.177 & 0.074 & 0.072 &
0.345 & 0.232 & 0.047 & 0.045 & 0.349 & 0.217 & 0.031 & 0.030\\
& $Q_p$ & 0.019 & 0.027 & 0.002 & 0.063 & 0.005 & 0.019 & 0.006 & 0.001 &
0.011 & 0.031 & 0.005 & 0.001\\
& & & & & & & & & & & & & \\
$\Gamma_{1c}^{(2)}$ & $Q_s$ & 0.028 & 0.008 & 0.188 & 0.161 &
0.008 & 0.000 & 0.228 & 0.180 & 0.000 & 0.003 & 0.223 & 0.180\\
& $Q_p$ & 0.073 & 0.024 & 0.000 & 0.025 & 0.057 & 0.006 & 0.001 & 0.052 &
0.001 & 0.106 & 0.033 & 0.004\\
\end{tabular}
\vspace{5mm}
\normalsize
\label{decoman}
\end{table}
\begin{figure}
\centering
\caption{
Calculated highest valence band energy levels (E$_1$, E$_2$ and
E$_3$) and lowest conduction state (E$_c$) at $\Gamma$
versus substrate lattice constant
for the [111] (a) common-anion and (b) common-cation SLs considered.
The center of gravity of the topmost valence bands is taken as zero.}
\label{elenlev}
\end{figure}
\begin{figure}
\centering
\caption{
LDA band-gap ($E^{LDA}$~-~solid line)
and corrected band-gap ($E^{emp}$~-~dashed line) as a function of
the substrate lattice constant for the [111] and [001]
(a) (GaSb)$_1$/(InSb)$_1$ SLs and (b) (InAs)$_1$/(InSb)$_1$ SLs.
Filled (empty) circles indicate
the average value ($E_{ave}$)
of the experimental (LDA) band-gap in the pure binaries.}
\label{gap}
\end{figure}
\begin{figure}
\centering
\caption{Charge density distribution
(in units of 0.5 e/unit cell)
for the (a) $\Gamma_{1v}$
state and for the (b) $\Gamma_{3v}^{(2)}$ state
in the elastically relaxed [111] ordered common-anion structure.}
\label{gammav}
\end{figure}
\begin{figure}
\centering
\caption{Charge density distribution
(in units of 0.5 e/unit cell)
for the (a)
$\Gamma_{1c}^{(1)}$
(CBM) state and for the (b)
$\Gamma_{1c}^{(2)}$ state in the elastically relaxed
[111] ordered common-anion structure.}
\label{gammac}
\end{figure}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,407
|
{"url":"https:\/\/openstax.org\/books\/college-physics-ap-courses\/pages\/8-section-summary","text":"College Physics for AP\u00ae Courses\n\n# Section Summary\n\n### 8.1Linear Momentum and Force\n\n\u2022 Linear momentum (momentum for brevity) is defined as the product of a system\u2019s mass multiplied by its velocity.\n\u2022 In symbols, linear momentum $pp$ is defined to be\n$p=mv,p=mv, size 12{p=mv} {}$\nwhere $mm size 12{m} {}$ is the mass of the system and $vv size 12{v} {}$ is its velocity.\n\u2022 The SI unit for momentum is $kg\u00b7m\/skg\u00b7m\/s size 12{\"kg\" cdot \"m\/s\"} {}$.\n\u2022 Newton\u2019s second law of motion in terms of momentum states that the net external force equals the change in momentum of a system divided by the time over which it changes.\n\u2022 In symbols, Newton\u2019s second law of motion is defined to be\n$F net = \u0394p \u0394t , F net = \u0394p \u0394t , size 12{ F rSub { size 8{\"net\"} } = { {\u0394p} over {\u0394t} } = { {m\u0394v} over {\u0394t} } \".\" } {}$\n$FnetFnet$ is the net external force, $\u0394p\u0394p size 12{\u0394p} {}$ is the change in momentum, and $\u0394t\u0394t size 12{\u0394t} {}$ is the change time.\n\n### 8.2Impulse\n\n\u2022 Impulse, or change in momentum, equals the average net external force multiplied by the time this force acts:\n$\u0394p=Fnet\u0394t.\u0394p=Fnet\u0394t.$\n\u2022 Forces are usually not constant over a period of time.\n\n### 8.3Conservation of Momentum\n\n\u2022 The conservation of momentum principle is written\n$p tot = constant p tot = constant size 12{p rSub { size 8{\"tot\"} } =\"constant\"} {}$\nor\n$p tot = p\u2032 tot ( isolated system ) , p tot = p\u2032 tot ( isolated system ) , size 12{p rSub { size 8{\"tot\"} } =p' rSub { size 8{\"tot\"} } ` $$\"isolated system\"$$ ,} {}$\n$ptotptot size 12{p rSub { size 8{\"tot\"} } } {}$ is the initial total momentum and $p\u2032totp\u2032tot size 12{ ital \"p'\" rSub { size 8{\"tot\"} } } {}$ is the total momentum some time later.\n\u2022 An isolated system is defined to be one for which the net external force is zero $Fnet=0.Fnet=0. size 12{ left (F rSub { size 8{ ital \"net\"} } =0 right ) \".\" } {}$\n\u2022 During projectile motion and where air resistance is negligible, momentum is conserved in the horizontal direction because horizontal forces are zero.\n\u2022 Conservation of momentum applies only when the net external force is zero.\n\u2022 The conservation of momentum principle is valid when considering systems of particles.\n\n### 8.4Elastic Collisions in One Dimension\n\n\u2022 An elastic collision is one that conserves internal kinetic energy.\n\u2022 Conservation of kinetic energy and momentum together allow the final velocities to be calculated in terms of initial velocities and masses in one dimensional two-body collisions.\n\n### 8.5Inelastic Collisions in One Dimension\n\n\u2022 An inelastic collision is one in which the internal kinetic energy changes (it is not conserved).\n\u2022 A collision in which the objects stick together is sometimes called perfectly inelastic because it reduces internal kinetic energy more than does any other type of inelastic collision.\n\u2022 Sports science and technologies also use physics concepts such as momentum and rotational motion and vibrations.\n\n### 8.6Collisions of Point Masses in Two Dimensions\n\n\u2022 The approach to two-dimensional collisions is to choose a convenient coordinate system and break the motion into components along perpendicular axes. Choose a coordinate system with the $xx$-axis parallel to the velocity of the incoming particle.\n\u2022 Two-dimensional collisions of point masses where mass 2 is initially at rest conserve momentum along the initial direction of mass 1 (the $xx$-axis), stated by $m1v1=m1v\u20321 cos\u03b81+m2v\u20322 cos\u03b82m1v1=m1v\u20321 cos\u03b81+m2v\u20322 cos\u03b82$ and along the direction perpendicular to the initial direction (the $yy$-axis) stated by $0=m1v\u20321y+m2v\u20322y0=m1v\u20321y+m2v\u20322y$.\n\u2022 The internal kinetic before and after the collision of two objects that have equal masses is\n$1 2 mv 1 2 = 1 2 mv \u2032 1 2 + 1 2 mv \u2032 2 2 + mv \u2032 1 v \u2032 2 cos \u03b8 1 \u2212 \u03b8 2 . 1 2 mv 1 2 = 1 2 mv \u2032 1 2 + 1 2 mv \u2032 2 2 + mv \u2032 1 v \u2032 2 cos \u03b8 1 \u2212 \u03b8 2 .$\n\u2022 Point masses are structureless particles that cannot spin.\n\n### 8.7Introduction to Rocket Propulsion\n\n\u2022 Newton\u2019s third law of motion states that to every action, there is an equal and opposite reaction.\n\u2022 Acceleration of a rocket is $a= v e m\u0394m\u0394t\u2212ga= v e m\u0394m\u0394t\u2212g size 12{a= { {v\" lSub { size 8{e} } } over {m} } { {\u0394m} over {\u0394t} } - g} {}$.\n\u2022 A rocket\u2019s acceleration depends on three main factors. They are\n1. The greater the exhaust velocity of the gases, the greater the acceleration.\n2. The faster the rocket burns its fuel, the greater its acceleration.\n3. The smaller the rocket's mass, the greater the acceleration.\nOrder a print copy\n\nAs an Amazon Associate we earn from qualifying purchases.\n\nWant to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.\n\n\u2022 If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:\n\u2022 If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:","date":"2022-05-24 16:00:45","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 22, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"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\": 0.7996415495872498, \"perplexity\": 539.0621803732441}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662573053.67\/warc\/CC-MAIN-20220524142617-20220524172617-00017.warc.gz\"}"}
| null | null |
**Note:** This document is in progress
This doc outlines the various responsibilities of contributor roles in
Kubernetes. The Kubernetes project is subdivided into subprojects under WGs.
Responsibilities for most roles are scoped to these subprojects.
| Role | Responsibilities | Requirements | Defined by |
| -----| ---------------- | ------------ | -------|
| member | active contributor in the community | sponsored by 2 reviewers. multiple contributions to the project. | Kubernetes GitHub org member. |
| reviewer | review contributions from other members | history of review and authorship in a subproject | [OWNERS] file reviewer entry. |
| approver | approve accepting contributions | highly experienced and active reviewer + contributor to a subproject | [OWNERS] file approver entry|
| subproject owner | set direction and priorities for a subproject | demonstrated responsibility and excellent technical judgement for the subproject | [wgs.yaml] subproject [OWNERS] file *owners* entry |
## New contributors
[New contributors] should be welcomed to the community by existing members,
helped with PR workflow, and directed to relevant documentation and
communication channels.
## Established community members
Established community members are expected to demonstrate their adherence to the
principles in this document, familiarity with project organization, roles,
policies, procedures, conventions, etc., and technical and/or writing ability.
Role-specific expectations, responsibilities, and requirements are enumerated
below.
## Member
Members are continuously active contributors in the community. They can have
issues and PRs assigned to them, participate in WGs through GitHub teams, and
pre-submit tests are automatically run for their PRs. Members are expected to
remain active contributors to the community.
**Defined by:** Member of the Kubeflow GitHub organization
### Requirements
- Enabled [two-factor authentication] on their GitHub account
- Have made multiple contributions to the project or community. Contribution may include, but is not limited to:
- Authoring or reviewing PRs on GitHub
- Filing or commenting on issues on GitHub
- Contributing to SIG, subproject, or community discussions (e.g. meetings, Slack, email discussion
forums, Stack Overflow)
- Subscribed to [Kubeflow-dev@googlegroups.com]
- Have read the [contributor guide]
- Actively contributing to 1 or more subprojects.
- Sponsored by 2 reviewers. **Note the following requirements for sponsors**:
- Sponsors must have close interactions with the prospective member - e.g. code/design/proposal review, coordinating
on issues, etc.
- Sponsors must be reviewers or approvers in at least 1 OWNERS file either in any repo in the [Kubeflow org],
- **[Open an issue][membership request] against the Kubeflow/internal-acls repo**
- Ensure your sponsors are @mentioned on the issue
- Complete every item on the checklist ([preview the current version of the template][membership template])
- Make sure that the list of contributions included is representative of your work on the project.
- Have your sponsoring reviewers reply confirmation of sponsorship: `+1`
- Once your sponsors have responded, your request will be reviewed by the [Kubeflow GitHub Admin team]. Any missing information will be requested.
### Responsibilities and privileges
- Responsive to issues and PRs assigned to them
- Responsive to mentions of WG teams they are members of
- Active owner of code they have contributed (unless ownership is explicitly transferred)
- Code is well tested
- Tests consistently pass
- Addresses bugs or issues discovered after code is accepted
- Members can do `/lgtm` on open PRs.
- They can be assigned to issues and PRs, and people can ask members for reviews with a `/cc @username`.
- Tests can be run against their PRs automatically. No `/ok-to-test` needed.
- Members can do `/ok-to-test` for PRs that have a `needs-ok-to-test` label, and use commands like `/close` to close PRs as well.
**Note:** members who frequently contribute code are expected to proactively
perform code reviews and work towards becoming a primary *reviewer* for the
subproject that they are active in.
## Reviewer
Reviewers are able to review code for quality and correctness on some part of a
subproject. They are knowledgeable about both the codebase and software
engineering principles.
**Defined by:** *reviewers* entry in an OWNERS file in a repo owned by the
Kubeflow project.
Reviewer status is scoped to a part of the codebase.
**Note:** Acceptance of code contributions requires at least one approver in
addition to the assigned reviewers.
### Requirements
The following apply to the part of codebase for which one would be a reviewer in
an [OWNERS] file (for repos using the bot).
- member for at least 3 months
- Primary reviewer for at least 5 PRs to the codebase
- Reviewed or merged at least 20 substantial PRs to the codebase
- Knowledgeable about the codebase
- Sponsored by a subproject approver
- With no objections from other approvers
- Done through PR to update the OWNERS file
- May either self-nominate, be nominated by an approver in this subproject, or be nominated by a robot
### Responsibilities and privileges
The following apply to the part of codebase for which one would be a reviewer in
an [OWNERS] file (for repos using the bot).
- Tests are automatically run for PullRequests from members of the Kubeflow GitHub organization
- Code reviewer status may be a precondition to accepting large code contributions
- Responsible for project quality control via [code reviews]
- Focus on code quality and correctness, including testing and factoring
- May also review for more holistic issues, but not a requirement
- Expected to be responsive to review requests
- Assigned PRs to review related to subproject of expertise
- Assigned test bugs related to subproject of expertise
- Granted "read access" to Kubeflow repo
- May get a badge on PR and issue comments
## Approver
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**Defined by:** *approvers* entry in an OWNERS file in a repo owned by the
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The following apply to the part of codebase for which one would be an approver
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- May approve code contributions for acceptance
## Subproject Owner
**Note:** This is a generalized high-level description of the role, and the
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Subproject Owners are the technical authority for a subproject in the Kubeflow
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**Defined by:** *owners* entry in subproject [OWNERS] files as defined by [sigs.yaml] *subproject.owners*
### Requirements
The process for becoming an subproject Owner should be defined in the SIG
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The following apply to the subproject for which one would be an owner.
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[wgs.yaml]: templates/wgs.yaml
[two-factor authentication]: https://help.github.com/articles/about-two-factor-authentication
|
{
"redpajama_set_name": "RedPajamaGithub"
}
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Brandon Gerrod Miller (born January 21, 1986) is a former American football defensive end. He was signed by the Atlanta Falcons as an undrafted free agent in 2008. He played college football at Georgia.
He has also played for the Seattle Seahawks.
Early years
Attended Miller County High School and as a senior he made 106 tackles and 27 receptions for more than 500 yards. He was named All-State, and All-South and was named SuperPrep Elite 50, All-America, and All-Dixie Teams.
College career
Miller played in 37 career games with 15 starts at linebacker for Georgia, posting 73 tackles and 6.0 sacks in his Bulldog career. In 2007, he appeared in 11 games making seven starts and in 2006 he appeared in every game with five starts and made 23 tackles with 2 PBU and 3 QB-Pressures. In 2005, he made his first career start against Boise State in the 2005 opener and started 10 games during season despite being hampered by injuries The prior season, 2004, he played in all 12 games with nine tackles and one fumble recovery.
Professional career
Atlanta Falcons
He was signed by the Atlanta Falcons as an undrafted free agent in 2008. He spent the first 10 weeks of the regular season on the team's practice squad before being promoted to the active roster on November 16. He was waived two days later.
Seattle Seahawks
Miller was claimed off waivers by the Seattle Seahawks on November 19, 2008. He appeared in one game for the Seahawks during the 2008 season, recording one tackle.
Miller was waived/injured by the Seahawks on July 29 and subsequently reverted to the team's injured reserve list. On May 13, 2010, Miller was released by the Seahawks.
External links
Georgia Bulldogs bio
Seattle Seahawks bio
1986 births
Living people
People from Colquitt, Georgia
American football defensive ends
American football linebackers
Georgia Bulldogs football players
Atlanta Falcons players
Seattle Seahawks players
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
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Q: How to prevent email refresh while viewing email in Windows 8? I am running Windows 8 Enterprise in VMware 8.0.3 and the VM has 5GB of RAM.
When I am reading Gmail in the WinRT Mail app, the list of emails and the preview/view pane will frequently disappear completely while it refreshes or something. This may be happening in Hotmail as well, but I use Gmail more frequently.
I thought maybe it was pinging the mail server too often so I modified the settings to:
Download new mail: Every 15 Min
However it is still happening.
It is hard to read email when the screen keeps blanking out while the app refreshes. Does anyone know why this is happening or how to control it?
A: Intel graphics drivers on the host are going to exhibit these sorts of problems until Intel fixes their drivers. It also occurs on VMware Workstation 9, by the way. In the guest, run the Chrome browser with hardware acceleration disabled, and disable desktop effects (make sure dwm.exe isn't running) in the meantime to workaround it.
May also occur with other non-Intel graphics drivers.
A: You could try uninstall vmware tools and reinstalling them without the vmware graphics driver. Just be sure to make a snapshot before you make any radical changes for your own sake. I haven't used e-mail apps so much on vmware, but I have been setting up test system enviroments at work.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,406
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{"url":"http:\/\/tug.org\/pipermail\/texhax\/2006-November\/007379.html","text":"# [texhax] Caption to figure in margin -- gives \"LaTeX Error: Not in outer par mode\"\n\nAxel E. Retif axretif at igo.com.mx\nThu Nov 23 05:26:35 CET 2006\n\nOn Nov 22, 2006, at 05:08 , Niall Mansfield wrote:\n\n> I want to put the caption to a figure in the margin (to save space,\n> as vertical space is limited for reasons we don't have to go into).\n>\n> I have tried:\n>\n> \\begin{figure}\n> \t\\centerline{\\includegraphics{dummy.eps}}\n> \t\\marginpar{This is a dummy picture}\n> \t\\end{figure}\n>\n> but that gives the error:\n> \tLaTeX Error: Not in outer par mode\n>\n> Is there a way round this?\n> (Figure numbering, tables of figures, etc. are not important,\n> so the solution doesn't have to be 100% \"clean\".)\n\nTake a look at sidecap.sty ---it's the easier way to achieve what you\nwant, and it's really clean''. wrapfig.sty and caption.sty are\nother handy packages, for different purposes.\n\nBest,\n\nAxel","date":"2018-07-17 07:50:45","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9740996956825256, \"perplexity\": 10098.063570873615}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-30\/segments\/1531676589618.52\/warc\/CC-MAIN-20180717070721-20180717090721-00328.warc.gz\"}"}
| null | null |
\section{Introduction}
The effective size of a star at a given wavelength depends on the opacity of the
stellar atmosphere at that wavelength. Since we effectively measure the
diameter of the $\tau = 1$ surface of the star, the size is related to the
extension of the atmospheric region where the absorption is produced. The
diameter of that surface varies with opacity, hence frequency. The
atmospheres of cool giants are so extended that these size variations are
observable with an interferometer. Quirrenbach et al. (\cite{Quirrenbach1993},
\cite{Quirrenbach2001})
studied the extended absorption regions of stellar atmospheres of cool giant
stars in the TiO band at 712\,nm with the MkIII interferometer. These authors
measured the visibilities of a set of 47 stars using two filters, one centered
in the TiO band, and the other in the continuum part of the spectrum close to
the TiO band (754\,nm). They found that the sizes in the TiO band are
larger than those in the continuum, and that this effect is stronger for cooler
stars. The sizes in the TiO absorption band are $\sim 10$\% larger than in the
continuum for $R-I$ color indices of $\sim1.6$ (spectral types M3--M4) and as
much as 30\% larger for $R-I$ color indices of $\sim2.2$ (spectral types M6--M7).
Similar differences between the sizes observed in the continuum and in bands
containing absorption band heads of other molecules (such as H$_2$O or CO) have
been reported for AGB stars. Mennesson et al. (\cite{Mennesson2002}) and Perrin
et al. (\cite{Perrin2004}), for instance, reported very large size ratios
(50\% or more) for some stars.
Quirrenbach et al. (\cite{Quirrenbach1993}, \cite{Quirrenbach2001}) were successful
in qualitatively
reproducing their data with the latest set of cool giant models from the
general-purpose stellar atmosphere code {\tt PHOENIX}. Spherical, hydrostatic,
massively line-blanketed atmosphere models were constructed and used to predict
the uniform-disk diameters in the TiO band and the continuum band as a
function of model effective temperature, surface gravity, and mass (the stellar
mass was used in the modeling, since it controls the deviation from
plane-parallel atmospheres.) For most of the observed oxygen-rich giants, the
diameter ratios of the TiO band to the continuum band agreed with the
models computed for masses $\sim 0.5\,M_\odot$. Hence, model atmospheres with very
low stellar masses could fit the large diameter ratios observed in many stars,
although evolutionary models predict masses as high as 5\,$M_\odot$.
A possible explanation of this inconsistency would be the existence of a
transition zone at the base of the stellar wind (the {\em MOLsphere}; e.g.
Tsuji \cite{Tsuji2008}, and references therein), which could provide sufficient
opacity in the TiO bands (and other molecular bands) to make AGB stars appear
much larger than predicted by the hydrostatic model atmospheres such as {\tt PHOENIX}.
According to this picture, one would expect similar
size effects for the CO band heads. Therefore, we decided to use AMBER in the K
band to measure the effective sizes of a set of four cool giant stars through
the CO band heads at 2.3\,$\mu$m. The use of AMBER in medium-resolution mode
($\lambda/\Delta\lambda \sim 1500$) provides considerably more information than
could be obtained with the MkIII
interferometer (Quirrenbach et al. \cite{Quirrenbach1993}) and the IOTA
interferometer (Mennesson et al. \cite{Mennesson2002}; Perrin et al. \cite{Perrin2004}),
in which narrow-band filters were used. In this paper, we
report on the
results obtained from the analysis of the observations of the first star of our
sample: RS Cap.
RS Cap (HD\,200994) has a K-band magnitude of $-0.2$ (Cutri et al. \cite{cutri2003}).
It is a semi-regular variable (SRb) of spectral type M6/M7III
and is located at $\alpha = 21\textrm{h}\,07\textrm{m}\,15.4\textrm{s}$, $\delta =
-16^{\circ}\,25'\,21.4''$ (J2000.0). It has a visual magnitude of 8.3 and a
parallax $\pi = 1.26\pm0.86$\,mas (van Leeuwen \cite{Leeuwen2007}), which
maps into a distance between $500$ and $2500$\,pc.
However, following Scalo (\cite{Scalo1976}) or Winters et al. (\cite{winters03}),
a bolometric absolute magnitude between $-4$ and $-5$ is derived for RS Cap (around
$-4.8$ in the case of Winters et al.). These estimates, in addition to the spectrum
fitted between 0.1 and 2$\mu$m, translate into a distance of $\sim$300\,pc (Richichi et
al. \cite{Richichi1992}).
A variability amplitude
of $\Delta V \sim 0.5$ is seen in the Hipparcos data (Perryman
\& ESA \cite{Perryman1997}), although a larger variability amplitude (in photometric
magnitude) of $\Delta B \sim 2$ was reported in Kukarkin et al. (\cite{Kukarkin1969}),
which has a period of $\sim340$\,days.
The remainder of this paper is structured as follows: In Sect.
\ref{OBSERVATIONS}, we describe our AMBER observations and the strategy followed
in the data calibration and reduction. In Sect. \ref{RESULTS}, we report on the
results obtained: Sect. \ref{TEFF} is centered on our estimate of the effective
temperature and Sect. \ref{CObands} on the size effects observed at the CO band heads.
In Sect. \ref{MODELING}, we compare our results with synthetic data obtained from model
stellar atmospheres. In Sect. \ref{CONCLUSIONS}, we summarize our conclusions.
\section{Observations and data reduction} \label{OBSERVATIONS}
We observed RS Cap with the ESO Very Large Telescope Interferometer (VLTI)
using the Astronomical Multi-BEam combineR, AMBER (see Petrov et al.
\cite{Petrov2007} for details on this instrument), in medium-resolution mode
($\lambda/\Delta\lambda \sim 1500$). This instrument performs
simultaneous observations of the interferometric fringes generated by three
telescopes. Therefore, it measures {\em closure phases}, which are
quantities independent of atmospheric or instrumental telescope-dependent
contributions (e.g. Rogers et al. \cite{Rogers1974}). AMBER also measures the
so-called {\em differential phase} on each baseline, which roughly represents the
phase of spectral features with respect to that in the continuum\footnote{To be
precise, two pieces of information are lost in the differential phase, compared to the
original Fourier phase: the phase offset (or phase at the first spectral channel) and
the phase gradient (phase as a linear function of wavenumber).}. The observations were
performed on 4 June 2009, from 06:30\,UT to 08:30\,UT, using the three VLTI
Auxiliary Telescopes (AT) positioned on stations E0, G0, and H0. These stations
are distributed roughly in the east-west direction. In Table \ref{OBSERVCONFIG},
we give the projected baseline lengths and position angles for our observations,
together with a summary of the atmospheric observing conditions. 18 Cap
(M0III star located at $\alpha = 20\textrm{h}\,51\textrm{m}\,49.3\textrm{s}$,
$\delta = -26^{\circ}\,55'\,08.9''$, J2000.0) was also observed as a calibrator,
with the same observing configuration used to obtain the RS Cap visibilities.
\begin{table*}
\centering
\begin{tabular}{ c | c | c c c | c c c | c }
\hline\hline
& {\bf UTC} & {\bf Seeing} & {\bf Air mass} & {\bf Coh. time} & {\bf E0-G0} & {\bf G0-H0} &
{\bf E0-H0} & {\bf PA}\\
& (hh:mm) & ($''$) & & (ms) & L(m) & L(m) & L (m) & (deg)\\
\hline RS Cap & 07:08--08:06 & 0.61 / 0.94 & 1.03 / 1.12 & 1.6 / 2.4 & 14.4 / 15.5 & 28.7
/ 31.0 & 43.1 / 46.5 & 119 / 113 \\
18 Cap & 06:47--06:56 / 08:16 -- 08:26 & 0.57 / 1.19 & 1.00 / 1.11 &
1.5 / 2.7 & 14.9 / 16.0 & 29.7 / 31.9 & 44.6 / 47.9 & 110 / 123 \\
\hline
\end{tabular}
\caption{Details of our observations. For each star, we give the
ranges of values obtained during all the exposures. L(m) is the projected
baseline length (in meters) and PA is the position angle of the baseline (North
through East). Coh. time is the coherence time estimated from the atmospheric
conditions.}
\label{OBSERVCONFIG}
\end{table*}
The high flux densities of RS Cap and 18 Cap in the H band (magnitudes of 0.25
and 0.42, respectively), allowed us to use the fringe tracker FINITO (see Gai et
al. \cite{Gai2004}), which employs part of the H-band light to correct, in real
time, the delay shifts of the fringes due to atmospheric turbulences, thus
increasing the coherence of the signal. A larger coherence allows the use of a
longer detector integration time (DIT). Given the moderately good atmospheric
conditions during our observations (see Table \ref{OBSERVCONFIG}), we achieved a
DIT of 200\,ms with a negligible signal loss due to atmospheric jitter. This allowed
us to simultaneously observe a large spectral range of the K-band (between 2.13
and 2.47\,$\mu$m) using medium-resolution mode. This wavelength range contains
the $^{12}$CO (2$-$0), (3$-$1), and (4$-$2), as well as the $^{13}$CO (2$-$0)
band heads, together with the continuum blueward of the $^{12}$CO (2$-$0) band
head at 2.29\,$\mu$m.
The data acquisition was carried out in the following order: (1) observation of an
artificial signal for calibration of the instrumental dispersive effects; (2) one
exposure of dark frames (an exposure consisted on 200 frames of 200\,ms each); (3)
five exposures of the target; and (4) one exposure of empty sky close to the target. The
observations of dark, target, and sky exposures were iterated once
for 18\,Cap (spanning 10 minutes), three times for RS\,Cap (spanning 1 hour),
and one more time for 18\,Cap (spanning 10 minutes).
The dead time between iterations, when the sources were not observed, was dedicated
to the setup of the instrument: preparation of the optical-delay lines, telescope
pointing, etc.
The raw visibilities and differential closure phases for RS Cap and 18 Cap were
obtained using the {\tt amdlib} libraries (version 2.2)
and the interface provided by the Jean-Marie Mariotti Center (JMMC). First, we
manually aligned the spectrally-dispersed photometry channels of the ATs to the
interferometric channel, based on observations of the target and calibrator
stars. We removed the bad pixels and took into account the flat, dark, and sky
contributions. Afterwards, we calibrated the
instrumental dispersive effects and fringe-fitted each frame of RS Cap and 18
Cap (based on the P2VM algorithm of Tatulli et al. \cite{Tatulli2007}).
The resulting visibilities of individual frames were selected and
averaged for each exposure in several ways, to test the robustness of our
results on different averaging and selection schemes. The finally chosen
selection scheme was based on an atmospheric piston criterion (keeping only
frames with a piston smaller than 8\,$\mu$m, to select only well-detected and
centered fringes in all three baselines) and keep 50\% of the remaining
frames based on a signal-to-noise ratio (SNR) criterion.
After the frame selection, we obtained a single frame-averaged visibility spectrum
for each exposure. The last step in the data reduction was to
average the visibilities of all exposures of each star to increase the SNR. This
step was performed outside {\tt amdlib}, with an in-house developed python-based
program that uses the PyFITS library (provided by the Space Telescope Science
Institute, operated by AURA for NASA). The averaged closure phases and
differential phases (i.e., those
resulting from the averaged exposures) are zero within $3^{\circ}$, in
absolute value, for both stars and for all the spectral channels. These small
values of the closure phases indicate that the emission from both sources is
symmetric, at least in the direction of the projected baselines of our
observations and to the scale corresponding to our spatial resolution.
This is unsuprising, given that we probe only the first lobe of the Fourier
transform of the observed source structure, which is, therefore, only
partially resolved.
The uncertainties in the amplitude and phase for each spectral channel were computed
from their standard deviations during the averaging of the exposures. Fractional
uncertainties of only $1-2$\% in the observed visibilities were obtained for
all channels and baselines.
\subsection{Visibility amplitude and wavelength calibration}
\label{LambdaCal}
A standard calibration procedure, to assign the corresponding
wavelength to each spectral channel of the AMBER detector, is not currently
available. This is a concern,
especially for the mid-resolution and low-resolution modes, where it is
difficult, or impossible, to use line profiles either from the observed objects
or from Earth's atmosphere for the calibration. We performed the wavelength
($\lambda$)
calibration of our AMBER observations by comparing the positions of the minima
of the observed CO band heads in RS Cap to their values derived from the models
described in Sect. \ref{RESRSCAP}. The use of different stellar models did not
affect our $\lambda$ calibration at a level higher than our spectral resolution.
In the amplitude calibration, the CHARM2 catalog of high
angular-resolution stellar-diameter measurements (Richichi, Percheron, \&
Khristoforova \cite{Richichi2005}) indicates that the uniform-disk K-band diameter of our
calibrator star, 18 Cap, is 5.02\,mas. This uniform-disk diameter was used to
calibrate the visibility amplitudes by comparing those of 18 Cap to the model
predictions corresponding to a uniform disk of 5.02\,mas diameter. This
amplitude calibration was performed as described in Annex B of the AMBER Data
Reduction Software User Manual (Revision 2.1) of the JMMC\footnote{{\tt
http://www.mariotti.fr/data\_processing\_amber.htm}}.
\subsection{Photometry calibration}
In Fig. \ref{SpectrumFigure}(a), we show the normalized spectrum of RS Cap, obtained
from the average of the photometry channels of the three ATs.
We note that there is a clear discontinuity in the spectrum at
$\sim 2.2$\,$\mu$m (a similar discontinuity is also seen in the spectrum of
18 Cap). This discontinuity is caused by the transmission of
the K-band fiber of the AMBER instrument. Therefore,
it can be understood as an instrumental multiplicative effect,
which biases the photometric and interferometric channels in the same way for wavelengths
shorter than $\sim 2.2$\,$\mu$m. An additive contribution to these channels would have
affected the contrast of the fringes relative to the photometry and, therefore,
would have translated into a changing amplitude of the visibilities, which is not
observed.
Since any multiplicative bias in the data has no effect on the
visibility amplitudes, the atmospheric opacity and any unmodeled gain effect in
the detector do not affect the visibilities, but only the observed spectra.
If we wish to use the observed spectrum of RS\,Cap in our
modeling, it is necessary to calibrate it with the atmospheric (and instrumental) spectral
transmission profiles. To perform this calibration, we proceeded in the following
way: 1) we obtained a template spectrum of an M0III star, which is corrected for atmospheric
effects (star BS4371, observed\footnote{This spectrum has a resolution of
$\lambda/\Delta\lambda \sim 1100$, similar to that of our observations} by Lan\c{c}on et
al. \cite{Lancon2000});
2) we divided the template spectrum by the observed spectrum of 18 Cap,
to obtain the inverse of the transmission profile of the atmosphere plus the detector
(we call this quantity {\em gain} and we show it in Fig. \ref{SpectrumFigure}(b)); and
3) we calibrated the spectrum of RS Cap, multiplying it by the gain. The
normalized calibrated spectrum is shown in Fig. \ref{RSCAPDISC}(a). By applying this
calibration strategy, we assume that the airmass is similar for
target and calibrator, which is indeed the case in our observations (see
Table \ref{OBSERVCONFIG}).
We note that there are some atmospheric telluric lines that can be clearly
seen in the transmission profile shown in Fig. \ref{SpectrumFigure}(b); those
at 2.45 (three lines), 2.44, 2.42 (three lines blended), 2.37, and 2.32\,$\mu$m.
The effect of the optical-fiber transmission jump at 2.2\,$\mu$m is also seen
in the gain, together with a slight increase as a function of wavelength redward
of $\sim$2.35\,$\mu$m due to water vapor in the Earth's atmosphere. We also note
that, although the
CO bands might have slightly different depths in 18\,Cap and BS4371, no clear
features are seen in the regions of the CO bands for the gain shown in
Fig. \ref{SpectrumFigure}(b).
\begin{figure}
\centering
\includegraphics[width=9cm]{./16120-fig1.eps}
\caption{(a) Normalized spectrum of RS Cap, as observed with the ATs; (b) inverse of
the atmosphere$+$detector spectral transmission profile.}
\label{SpectrumFigure}
\end{figure}
\section{Results and discussion}
\label{RESULTS}
\subsection{Continuum angular diameter and effective temperature}
\label{TEFF}
In Fig. \ref{RSCAPDISC}, we show (a) the normalized spectrum of RS Cap, (b) the
amplitude visibilities for the three baselines (higher visibility corresponds to
shorter baseline), and (c) the resulting diameter estimates obtained from the
fit of a uniformly-bright disk to the visibilities at each spectral channel.
\begin{figure}
\centering
\includegraphics[width=9cm]{./16120-fig2.eps}
\caption{(a) Spectrum of RS Cap (normalized); (b) calibrated visibility
amplitudes; (c) uniform-disc angular diameter at each AMBER spectral channel.}
\label{RSCAPDISC}
\end{figure}
The mean uniform-disk angular diameter fitted to the RS Cap visibilities in the
continuum (i.e., blueward of $2.29$\,$\mu$m) is $\theta = 7.75\pm0.07$\,mas.
To derive the effective temperature accurately, the size estimated with a
limb-darkening model must be used, instead of that estimated from a uniform-disc
model. We note that $T_{\mathrm{eff}}$ is a function of the criterion adopted for
the radius of the star. In spherical atmospheres, the actual $T_{\mathrm{eff}}$
used to compare to the models should be computed using the diameter at which the
Rosseland opacity is 1 (i.e., the Rosseland diameter,
$\theta_{\mathrm{Ross}}$; see, e.g., Sect. 3.4 of Wittkowski, Aufdenberg, \&
Kervella \cite{Witt2004}).
We estimated the Rosseland diameter of RS Cap by fitting the
visibilities in the continuum part of our spectral coverage to the Hankel transform
(i.e., Fourier transform of a circularly-symmetric source) of
the monochromatic intensity profile of the {\tt MARCS} atmospheric model
described in Sect. \ref{RESRSCAP}. We show this intensity profile, normalized to a
Rosseland radius $R_{\mathrm{Ross}} = \theta_{\mathrm{Ross}}/2 = 1$, in Fig.
\ref{LimbFig}. If this profile is
given as $I(r)$ (where $I$ is the intensity normalized to 1 and $r$ is the
radial coordinate normalized to the Rosseland radius, $R_{\mathrm{Ross}} = 1$), then the
model used to fit the visibilities is
\begin{equation}
V(q) = \int_0^\infty{I(r/R_{\mathrm{Ross}})J_0(K\,q\,r)\,r\,\mathrm{d}r},
\label{HankEq}
\end{equation}
\noindent where $J_0$ is the first-kind Bessel function of order 0, $q$ is the baseline
length, and $K$ is a constant to scale $r$ and $q$ to their corresponding units (e.g.,
milliarcseconds and megawavelengths, respectively). The use of this fitting model, instead
of the Hankel transform of a uniform disc, allows us to take into account limb-darkening and,
simultaneously, to estimate directly the Rosseland radius
($R_{\mathrm{Ross}}$ in Eq. \ref{HankEq}).
\begin{figure}
\centering
\includegraphics[width=9cm]{./16120-fig3.eps}
\caption{Monochromatic intensity profile, normalized to $R_{\mathrm{Ross}} = 1$,
corresponding to the {\tt MARCS} model described in Sect. \ref{RESRSCAP}. This profile
was computed in the continuum part of the spectral coverage of our observations.}
\label{LimbFig}
\end{figure}
The best-fit Rosseland diameter of RS Cap is $\theta_{\mathrm{Ross}} = 7.95 \pm 0.07$\,mas.
Our size estimate, together with the bolometric flux of
$(2.1\pm0.2)\times10^{-6}$\,erg cm$^{-2}$ s$^{-1}$ reported in Richichi et al.
(\cite{Richichi1992}), translates into an effective temperature of
$T_{\mathrm{eff}} = 3160 \pm 160$\,K.
Lunar occultations of RS Cap, recorded
with the TIRGO telescope at 2.2\,$\mu$m, resulted in a uniform-disk diameter estimate
of $7.75\pm0.67$\,mas (Richichi et al. \cite{Richichi1992}). This diameter, together with
the bolometric measurements performed by the same authors, implies an
effective temperature of $\sim 3200$\,K, higher than the temperature derived
from the calibration of Bessel, Castelli \& Plez (\cite{Bessel1998}) ($\sim
2900$\,K) and that derived from the calibration of Scalo (\cite{Scalo1976})
($\sim 2600$\,K), but in agreement with the calibration given in
Ridgway et al. (\cite{Ridgway1980}). A later re-analysis of the Lunar-occultation
events (Richichi et al. \cite{Richichi1999}), to estimate the Rosseland angular diameter,
resulted in an even higher effective temperature for this star
($3481\pm177$\,K). Dyck, van Belle \& Thomson (\cite{Dyck1998})
estimated a size of $7.0 \pm 0.8$\,mas from a single-baseline interferometric
observation in the K band using the Infrared Optical Telescope Array (IOTA).
This size also translates into a higher effective temperature for RS Cap (around 4000\,K).
Our estimated diameter of RS Cap is compatible with that reported in Richichi et
al. (\cite{Richichi1992}) from lunar
occultations and that reported in Dyck, van Belle \& Thomson (\cite{Dyck1998})
from IOTA observations, although the precision in our estimate is an order of
magnitude higher.
\subsubsection{A word of caution about the use of FINITO}
We note that the use of FINITO in our observations may bias
the absolute amplitude calibration of the visibilities of each baseline, since
the percentage of lost fringes during the DIT is not taken into account in the
process of frame integration of each exposure. Any possible bias may be
different for calibrator and target, and translate into a different absolute
amplitude calibration for each star. As a result, our size estimate of RS Cap
might be (slightly) biased even after performing the calibration described in
Sect. \ref{LambdaCal}. However, the spectral resolution of our observations is
much higher than that used in Richichi et al. (\cite{Richichi1992}) and Dyck, van
Belle \& Thomson (\cite{Dyck1998}) (who, indeed, only measured the visibility at one
projected baseline). The higher spectral resolution of AMBER allows us to
precisely obtain the behavior of visibility amplitudes through relatively
wide regions of the Fourier plane. This sampling of visibility amplitudes through
the observing band encodes information about the source size, which is independent of the
absolute calibration of the visibilities. In Fig. \ref{SizeComp}, we show the
visibility amplitudes in the continuum (i.e. blueward of 2.29\,$\mu$m)
as a function of baseline length. In the figure, we also show
the model predictions using a uniform-disk diameter of $7.75\pm0.07$\,mas.
It can be seen that a disk model with this diameter satisfactorily fits the
visibilities of all baselines through the whole spectral coverage in the continuum.
This result gives us confidence about the amplitude calibration of the RS\,Cap
visibilities.
\begin{figure}
\centering
\includegraphics[width=9.0cm]{./16120-fig4.eps}
\caption{Visibility amplitudes between 2.13 and 2.29\,$\mu$m as a function of baseline
length. Black, observations; red, model predictions with a uniform-disk diameter of
$7.75 + 0.07$\,mas (lower line) and $7.75 - 0.07$\,mas (upper line).}
\label{SizeComp}
\end{figure}
\subsection{CO first overtone bands: $\lambda > 2.3$\,$\mu$m}
\label{CObands}
In Fig. \ref{RSCAPDISC}(c), there are two clear aspects in the region of
CO first overtone bands ($\lambda > 2.3$\,$\mu$m) worth noticing. On
the one hand, we see an increase of
$\sim10$\% in size for all the CO band heads, compared to the sizes in the
continuum. On the other hand, there is a linear trend of increasing size with
observing wavelength, beginning at 2.29\,$\mu$m. Owing to this latter effect, the
size in the continuum at 2.45\,$\mu$m is $\sim12$\% larger than the size in the
continuum for the wavelengths shorter than 2.29\,$\mu$m.
Large size increases
redward of 2.29\,$\mu$m have been observed in Mira stars
(Woodruff et al. \cite{Woodruff2009}), which are interpreted as being caused by
water-vapor extended zones. However, in those cases, the size variations reported
are 3--4 times larger than that observed in RS Cap.
In Sect. \ref{WaterSec}, we discuss how our model of water-vapor envelope can
explain the observed increasing size in RS Cap redward of 2.29\,$\mu$m.
\section{Modeling}
\label{MODELING}
\subsection{{\tt MARCS} model atmosphere}
\label{RESRSCAP}
We compared the observed size effects in RS Cap with different model
atmospheres computed with the {\tt MARCS} code (e.g. Gustafsson et al.
\cite{Gustafsson2008}). The {\tt MARCS} code computes the hydrostatic
atmospheric structure in radiative and convective equilibrium for spherical
atmospheres with molecular and atomic lines taken into account using the opacity
sampling method. Each model in spherical geometry (suitable for AGB stars) is
specified by the effective temperature ($T_{\mathrm{eff}}$), surface gravity
($\log{g}$), stellar mass ($M$), microturbulent velocity, and chemical
composition.
For $T_{\mathrm{eff}}$, we adopted 3200\,K, the nearest possible value
in {\tt MARCS} models to our estimated 3160\,K.
We also assume the ``moderately CN-cycled'' chemical composition (C/N =
1.5 and $^{12}$C/$^{13}$C = 20), which is usually found in stars after
the red giant branch, and [Fe/H] = 0.0.
The stellar mass can be estimated by comparing with theoretical stellar
evolutionary tracks, once $T_{\mathrm{eff}}$ and the luminosity are known.
While the parallax of RS Cap is not precisely measured (the relative error
is on the order of 70\%, so the error in the luminosity can be as large as
an order of magnitude),
Winters et al. (\cite{winters03}) estimated the luminosity of RS\,Cap to be
$\log{L/L_{\odot}} = 3.82$, by applying the period-luminosity relation for
semi-regular variables given by Feast (\cite{feast96})\footnote{Although the
period-luminosity relation is a matter of debate, as Winters et al.
(\cite{winters03}) mention.}. From Fig. 5 of Feast (\cite{feast96}), we estimated
an uncertainty of $\pm 0.25$ in the bolometric absolute magnitude of RS\,Cap,
which translates into an error of up to 20--25\% in the luminosity, much smaller
than that related to the Hipparcos parallax. The luminosity from Winters et al.
(\cite{winters03}), together with the
above $T_{\mathrm{eff}}$, places RS Cap close to the evolutionary track of a
2\,M$_{\odot}$ star with Z = 0.02,
computed by F. Herwig\footnote{\tt http://astrowww.phys.uvic.ca/$\sim$fherwig}.
Therefore, we adopted 2\,M$_{\odot}$ for RS Cap.
We note that the luminosity estimated by Winters et al. (\cite{winters03})
for RS Cap is much higher than that corresponding to an RGB star with
2\,M$_{\odot}$ (which is only $\log{L/L_{\odot}} \sim 3$). Therefore,
according to its luminosity, RS\,Cap would be clasified as an AGB star,
as it is indeed assumed in the P-L relation by Feast (\cite{feast96}).
The luminosity estimated in this way for RS\,Cap is, thus, self-consistent
in the framework of the P-L relation of semi-regular AGB stars.
From the luminosity and
$T_{\mathrm{eff}}$, we estimated a stellar radius of $\sim 260$\,R$_{\odot}$,
which, combined with our angular-size estimate, translates into a distance
of about 310\,pc. Hence, from the adopted stellar mass of
2\,M$_{\odot}$ and the estimated radius, we derive $\log{g} \sim -0.1$. We adopted
$\log{g} = 0.0$ for the selection of the {\tt MARCS} models.
For the microturbulent velocity, we used the {\tt MARCS} models with 2 and
5\,km\,s$^{-1}$. However, the effects of this parameter on the visibility
are very small.
We downloaded the models with the above parameters from the {\tt MARCS}
website\footnote{\tt http://marcs.astro.uu.se}. Using the temperature and
pressure stratifications of the model, we computed the monochromatic intensity
profiles and the synthetic spectrum for the spectral range of our AMBER observations
with a fine wavelength mesh of 1.4\,km\,s$^{-1}$, sampling several points over
each line profile. The CO ($^{12}$C$^{16}$O, $^{12}$C$^{17}$O, $^{12}$C$^{18}$O,
and $^{13}$C$^{16}$O) and H$_2$O lines were taken from the line lists of
Goorvitch (\cite{goorvitch94}) and Partridge \& Schwenke (\cite{partridge97}),
respectively. The monochromatic visibilities were computed from the Hankel
transform (i.e., Fourier transform of a circularly-symmetric distribution) of
the monochromatic intensity profiles (computed as described in
Ohnaka et al. \cite{ohnaka06}) and then convolved to the spectral
resolution of our AMBER measurements.
In Quirrenbach et al. (\cite{Quirrenbach2001}), limb-darkened photospheric
models were used to generate equivalent disk sizes. These sizes were then
compared to the disk sizes fitted to the MkIII visibilities. In our case, this
approach would have had the disadvantage of being sensitive to scaling factors
between different baselines (i.e., to the absolute amplitude calibration) and
would have depended on the use of an intermediate model (a uniform disk) to compare
observations to models. To avoid those drawbacks, we chose a new approach
to the modeling of the RS Cap visibilities, which minimizes the effect of
possible calibration artifacts between baselines and, in addition, avoids the
use of an intermediate fit to a disk.
The synthetic visibilities obtained from the {\tt MARCS} models were computed at
the points of the Fourier plane corresponding to each exposure in our
observations. We then averaged the synthetic visibilities over all the
exposures, in the same way as we did with the real data. In this way, we directly
compare the observed visibilities to the model predictions.
We show in Fig. \ref{MODWITHWATER} a comparison between observed visibilities
(black lines) and those computed from the {\tt MARCS} model using
$T_{\mathrm{eff}} = 3200$\,K, $M = 2$\,M$_{\odot}$, and $\log{g}=0$ (blue
lines). The match between observations and model predictions in
the continuum (i.e., for wavelengths shorter than $\sim2.3$\,$\mu$m) is
very good
The reduced $\chi^2$ for this model is $\chi^2_{\mathrm{red}} = 18.2$.
The relatively large depth of the features of the CO band heads cannot be
reproduced by the model. The discrepancy remains even
if we lower the mass to values as low as $M=1$\,M$_{\odot}$ (models with higher
masses predict more compact atmospheres and, therefore, smaller size effects
in the CO band heads). Additionally, the observed visibility amplitudes at
wavelengths longer than $\sim$2.29\,$\mu$m are systematically lower than the
model predictions; this difference between model and observations is larger at
longer wavelengths. In other words, the effective size of the star at
wavelengths longer than $\sim 2.29\,\mu$m is systematically larger than the
effective size derived from the {\tt MARCS} model.
\subsection{An additional water-vapor envelope}
\label{WaterSec}
An extra contribution must be added to the {\tt MARCS} models to fit the
visibilities in the observed spectral range. We can reproduce the lower
visibility amplitudes (i.e., the larger effective sizes) beyond 2.29\,$\mu$m by
adding a contribution to the stellar opacity due to water vapor around the star.
The opacity corresponding to the wide absorption of water vapor centered
at $\sim$2.7\,$\mu$m may account for the larger angular sizes observed at the longer
wavelengths.
We added a simple water-vapor model envelope to the {\tt MARCS}
pressure/density profiles, consisting of a narrow spherical shell with a radius
of 2 and a width of 0.1 (both in units of the stellar radius), a column density
of $10^{21}$\,cm$^{-2}$, and a temperature of 1500\,K. Using these parameters,
we are able to fit the amplitude visibilities at $\lambda > 2.29\,\mu$m, as we
show in Fig \ref{MODWITHWATER} (red lines). With this model, we obtain
$\chi^2_{\mathrm{red}} = 8.1$. This large reduced $\chi^2$ is partially
due to the discrepancy in the CO bands, which is discussed in Sect. \ref{COExtSec}.
Our model follows the general trend of decreasing visibility
amplitudes for wavelengths longer than 2.3\,$\mu$m, although underestimates
the visibility amplitudes at wavelengths longer than 2.4\,$\mu$m. Changing
the column density by a factor of ten
results in an increase of $\chi^2_{\mathrm{red}}$ by a factor $2-3$; a slight
modification of the water-vapor temperature ($\sim15$\%) increases
$\chi^2_{\mathrm{red}}$ by a factor $\sim2$; and changing the size of the shell
by a factor of two increases the $\chi^2_{\mathrm{red}}$ by a factor $\sim2$.
Although the water-vapor envelope enhances the contrast of the CO
features in the visibilities, thus improving the fit also in the CO band heads,
the observed depths in the band heads are still larger than those of the
model for some band heads ($\sim 50$\% for the band head at 2.29\,$\mu$m).
Furthermore, increasing
(decreasing) the stellar mass results in smaller (larger) depths of the
visibility amplitudes in the CO band heads, but decreasing the mass to a
value as low as 1\,M$_{\odot}$ does not translate into CO visibility
amplitudes comparable to those of the observations, even after the addition of
the water envelope (the $\chi^2_{\mathrm{red}}$ using 1\,M$_\odot$ is only 7.2\%
lower than that using 2\,M$_\odot$).
In Fig. \ref{SpectrumFit}, we show the calibrated spectrum of RS Cap (black) and,
superimposed, the synthetic spectra obtained from the {\tt MARCS} model only
(blue) and the {\tt MARCS} model with an additional water-vapor envelope (red).
Both model spectra are very similar,
and only differ for the longest wavelengths, where the model with water envelope
predicts a slightly lower flux. Since the fit of both models to the observed spectrum is of
similar quality, it is difficult to extract any additional conclusion about the water-vapor
envelope from the spectrum alone.
\begin{figure}
\centering
\includegraphics[width=9.5cm]{./16120-fig5.eps}
\caption{Calibrated spectrum of RS Cap (black), synthetic {\tt MARCS} spectrum (blue),
and synthetic spectrum with the additional water-vapor envelope (red).}
\label{SpectrumFit}
\end{figure}
\subsubsection{Water-vapor envelopes in AGBs. The problem of RS Cap}
The presence of dense water vapor envelopes in semi-regular or irregular
(i.e., non-Mira-type) AGB stars was first revealed
by Tsuji et al. (\cite{tsuji97}), who unmasked the 2.7\,$\mu$m water
vapor band originating in the dense molecular layers extending to $\sim$2
stellar radii.
On the basis of infrared interferometric observations, Mennesson et al.
(\cite{Mennesson2002}) found an increase in the angular size of
30--60\% between the $K$ and $L^{\prime}$ band in five semi-regular
AGB stars, which the authors interpreted
as being due to the emission from the extended molecular layers. However,
they used filters which are incapable of spectrally resolving
the water vapor bands.
Our AMBER measurements are the first study to {\it spatially and spectrally}
resolve the water vapor emission from the dense molecular layers for a
semi-regular AGB star.
We note that a similar increase in the angular diameter
redward of 2.3\,$\mu$m is also found in Mira stars.
Based on $K$-band interferometry using narrow-band filters
with a spectral resolution of $\sim$20,
Perrin et al. (\cite{Perrin2004}) showed that the angular diameters of
Mira stars are larger at 2.0 and 2.4\,$\mu$m than at 2.2\,$\mu$m.
Wittkowski et al. (\cite{wittkowski08}) observed
the Mira star S Ori from 1.29 to 2.32\,$\mu$m using AMBER with a spectral
resolution of 35 and found that the angular size increases redward
of 2.2\,$\mu$m.
The wavelength dependence of the angular size of Mira
stars from the near- to mid-infrared is primarily governed by the opacity
of water vapor, which is abundant in the atmosphere of Mira stars
(e.g., Ohnaka \cite{Ohnaka2004}; Wittkowski et al. \cite{wittkowski08}).
The model by Tej et al. (\cite{tej03}) shows that a shell with an enhanced
water vapor abundance forms behind a shock front propagating outward in the
extended atmosphere of Mira stars.
Comparison between infrared interferometric observations of Mira stars and
dynamical models lends support to this picture
(e.g., Ohnaka et al. \cite{ohnaka06}; Wittkowski et al. \cite{wittkowski07}).
The shock wave in Miras is generated by the periodic, large-amplitude
pulsation. Therefore, the shock in non-Mira stars with much
smaller variability amplitude is expected to be much weaker.
However, despite the striking difference in the variability amplitude
($\Delta V$ = 6--9 and 0.5--2 for Miras and RS Cap, respectively),
the semi-regular AGB star RS Cap produces a warm water vapor envelope
similar to that of Miras.
The column density, temperature, and radius of the water vapor envelope derived
for RS Cap are similar to those derived for Mira stars
(Ohnaka \cite{Ohnaka2004}). Not only the properties of the water-vapor
envelope, but also the mass-loss rate of RS Cap
($\sim \! 1 \times 10^{-6}$\,M$_{\odot}$ yr$^{-1}$) is comparable to
that of Miras (e.g., $5 \times 10^{-7}$\,M$_{\odot}$ yr$^{-1}$ for
S Ori) (estimates from Winters et al. \cite{winters03}).
The dynamical models with small-amplitude pulsations, reported in
Winters et al. (\cite{winters03}), produce outflows slower than 5\,km s$^ {-1}$ with
mass-loss rates lower than $3 \times 10^{-7}$\,M$_{\odot}$ yr$^{-1}$. However,
the expansion velocity measured for RS Cap is $\sim$10\,km s$^{-1}$
(Winters et al. \cite{winters03}).
Therefore, it is not clear whether this kind of small-amplitude pulsation
model applies to RS Cap.
At the moment, there is no satisfactory explanation for the origin of
the warm water envelope in RS Cap.
\subsection{An additional CO envelope}
\label{COExtSec}
It can be seen in Fig. \ref{MODWITHWATER} that the observed visibility amplitudes at
the CO band heads are much lower than the model predictions, either with or
without a water-vapor envelope. This discrepancy is especially large for the
shortest baseline (i.e. $E0-G0$) at the 2.29$\mu$m band head. The low visibility
amplitudes at the CO band
heads provide evidence of an extended envelope of CO around the star. For instance, if
$\sim$20\% of the emission in the CO band head at 2.29\,$\mu$m were to originate from
a very extended envelope of, say, $\sim$20\,mas radius or larger, it would map
into a decrease in the visibility amplitude that would roughly match the
observations in that part of the spectrum. Unfortunately, we do not have enough
data to be able to characterize well this possible extra component of CO
absorption.
\begin{figure}
\centering
\includegraphics[width=9.5cm]{./16120-fig6.eps}
\caption{Black: AMBER visibilities of RS Cap. Blue: model visibilities computed
using a {\tt MARCS} pressure-density profile (see text). Red: model
visibilities computed using the {\tt MARCS} pressure-density profile and an
additional water-vapor envelope (see text).}
\label{MODWITHWATER}
\end{figure}
\section{Conclusions} \label{CONCLUSIONS}
We have observed the AGB star RS Cap with VLTI/AMBER in the K band with
medium-resolution mode (506 channels between 2.13 and 2.47\,$\mu$m). We
have estimated
a Rosseland diameter of $7.95\pm0.07$\,mas in the continuum, which translates
into an effective temperature of 3160$\pm$160\,K. The apparent size of the star
increases monotonically by $\sim12$\% between 2.29 and 2.47\,$\mu$m. We have
detected lower than expected visibility amplitudes in all the CO band heads
observed. These lower amplitudes translate into larger apparent sizes of the
star in the CO band heads.
Using the pressure/density model profiles obtained with the {\tt MARCS} code, we
have been able to generate synthetic visibilities and compare them directly to our
AMBER observations. The fit is rather good, although we are unable to
reproduce the low visibility amplitudes in the CO band heads using a mass for RS
Cap estimated from the theoretical models of stellar evolution. In this sense,
our situation resembles that of Quirrenbach et al. (\cite{Quirrenbach2001}), who
observed interferometrically a sample of cool-giant stars in the TiO absorption
band at 712\,nm. The discrepancy between models and observations in the CO band
heads might be resolved either by using a much lower mass for RS Cap (below
1\,M$_{\odot}$, thus in contradiction with the stellar-evolution models) or by
additional unmodeled effects in the base of the stellar wind (a transition zone
with prominent absorption in the CO bands).
To fit the observations at wavelengths longer than 2.29\,$\mu$m,
we have found that an {\em ad hoc} narrow spherical water-vapor envelope around the
star, similar to the models used in Perrin et al. (\cite{Perrin2004}), must be
added. We have modeled
this envelope with a temperature of 1500\,K, a size twice
that of the star, a width of 0.1 times the stellar radius, and a column density
of $10^{21}$\,cm$^{-2}$.
Finally, there is a hint of an extended CO envelope around the star,
based on the low visibility amplitudes at the CO band heads in the shortest baseline.
\acknowledgements{ IMV is a fellow of the Alexander von Humboldt Foundation in
Germany. This research has made use of the SIMBAD database, operated at CDS,
Strasbourg, France. Partial support from Spanish grants AYA-2006-14986-C02,
AYA2009-13036-C02-02, and Prometeo 2009/104 is acknowledged. }
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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Abuse in Nursing Homes Unreported, Despite Law
More than one in four cases of possible sexual and physical abuse against nursing home patients apparently went unreported to police, says a government audit that faults Medicare for failing to enforce a federal law requiring immediate notification.
The Health and Human Services inspector general's office was issuing an "early alert" Monday on its findings from a large sampling of cases in 33 states. Investigators say Medicare needs to take corrective action right away.
"We hope that we can stop this from happening to anybody else," said Curtis Roy, an audit manager with the inspector general's office, which investigates fraud, waste and abuse in the health care system. The audit is part of a larger ongoing probe, and additional findings are expected, he said.
With some 1.4 million people living in U.S. nursing homes, quality is an ongoing concern. Despite greater awareness, egregious incidents still occur.
Using investigative data analysis techniques, auditors
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,732
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1 month ago - 5/22/2019
Combined player and ball tracking technology premieres at VELUX EHF FINAL4 2019
EHF Marketing GmbH has once again joined forces with KINEXON, the German tracking and analytics company, to take the next step in handball data collection.
For the first time, the showcase event of men's club handball, the VELUX EHF FINAL4, will offer player tracking technology which will improve overall collection of match data and provide more insights into the game. The player tracking technology will be joined by the SELECT iBall, the intelligent ball which is making a return to Cologne after the successful launch in 2018, both presented by Nord Stream 2.
Match data collection raised to new heights
During the four matches of the VELUX EHF FINAL4, tracking sensors in jerseys of all players of the four clubs (Barça Lassa, HC Vardar, Telekom Veszprém HC, PGE VIVE Kielce) and the chip in the iBall will collect an unparalleled amount of data around each match.
For the first time in the EHF competitions, fans, coaches and players will be able to see players' running distance, playing time, fastest sprints, highest jumps and fastest goalkeeper saves.
In addition, the SELECT iBall will collect information such as ball speed, shot detection, position of the shot and placement of the ball on target.
Second-screen experience for fans
The match data will be delivered to the spectators in the LANXESS arena but also to those watching at home.
Selected data will be shown on the cube in the arena, on social media and on TV and the majority of it exclusively in the official VELUX EHF FINAL4 App. A selection of the top five fastest shots, highest jumps, fastest sprints and running distances will offer fans a second-screen experience during the weekend-long event.
An extensive report of collected match data will be provided to the participating teams giving them opportunity to have all information about the team and player performance.
How does player tracking work?
All players will be fitted with a tracking chip which will be integrated into their playing shirts. The sensor will be placed between the shoulders of the athlete, with the help of a special sports shirt. The sensor weighs only 14 grams, and therefore, is barely noticed by the players and do not influence their on-court performance.
KINEXON anchors will be placed around the court capturing the data from the sensors in the players' shirts and the ball. Based on the wireless sensor network, the data is sent in real-time to the central database and can be accessed in real-time.
Maximilian Schmidt, Co-founder and Managing Director KINEXON said: "We are excited and proud that the EHF expands their handball performance tracking initiatives by adding automated, real-time player tracking to this year's VELUX EHF FINAL4. After introducing the SELECT iBall last year, David Szlezak and his team prove again that they are true pioneers in bringing sports technology to handball. With our partnership, we hope to establish a new era to this fast pace, intense and honest team sport."
David Szlezak, Managing Director of EHF Marketing GmbH said: "The introduction of player tracking technology is the next step on our digital journey as we seek to enhance the game for audiences both at home and in the arena. However, we remain consistent in our belief that handball – the sport itself – remains at the core of our offering. In addition, data is what fans are asking for and we see this as a key investment in the future of the game."
About KINEXON
KINEXON Sports & Media develops cutting-edge solutions for automated, real-time sports content. It brings athlete performance, media experience and entertainment formats to the next level. The portfolio includes both an unrivalled sensor technology for centimetre-accurate indoor and outdoor localisation and motion sensing of professional athletes in various sports. Moreover, the portfolio includes software solutions for smart data processing and actionable insights through various analytics. The company was founded in 2012 by scientists of the Technical University Munich. KINEXON is located in Munich (headquarters), New York City and Chicago.
https://kinexon-sports.com/
About EHF Marketing GmbH
EHF Marketing GmbH is the marketing arm and subsidiary of the European Handball Federation. The company works closely with marketing and media partners, as well as with Europe's leading clubs to realise the full potential of the sport on the international sports market. EHF Marketing GmbH is responsible for the marketing and media rights of club competitions, including the VELUX EHF Champions League, the DELO WOMEN'S EHF Champions League and the MEN'S EHF Cup. Visit ehfCL.com and ehfFINAL4.com for more information.
Author: EHF/jb
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Q: Asymptotic lines of a parametrized surface
Prove that the lines $v=k\pi$ are asymptotic lines for the surface parametrized by $$\varphi:(0,\pi )\times\mathbb{R}\rightarrow\mathbb{R}^{3}$$
$$\varphi (u,v)=(v,\cos(u),\sin(v))$$
Honestly, I don't know how to proceed because the professor talk with us only about the $v$ or $u-lines$ like $\varphi (u_{0}+t,v)$, $t\in\mathbb{R}$
Can someone axplain this concept? Thanks before!
A: According to your description, we can plot the function using Mathematica:
I do not think it have any "asymptotic lines".
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{
"redpajama_set_name": "RedPajamaStackExchange"
}
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\section{Introduction}
\label{introchap}
Since the original convexity theorems of Atiyah \cite{A:1982CCH} and
Guillemin-Sternberg \cite{GS:1982C}
on the image of the moment map associated to a Hamiltonian torus action,
it has become clear that the moment polytope is a rich source
of combinatorial invariants of these actions (see \cite{G:1994MM}).
In particular, traditional topological invariants
of the various reduced spaces associated to the action turn
out to have formulas that involve fixed point data combined with
combinatorial data from the moment polytope. We will simply use the
word ``combinatorial'' to refer to formulas of this type.
In this paper we focus primarily on the signature, and find formulas for the
signature of reduced spaces. In the circle case (Section
\ref{sec:circle}) we are
able to find formulas both for regular and singular reductions,
thanks to recent work of Lerman and Tolman. In the torus case
(Section \ref{recursivechap})
we find a recursive formula for the signatures of various
``subreductions,'' including all regular reductions.
Our fundamental object of study is a Hamiltonian $T$-space $M$,
with symplectic form $\omega$ and moment map $\phi: M \ra \ts$.
We denote such a Hamiltonian $T$-space by $(M,\omega,\phi)$.
We allow our spaces to be orbifolds as well as manifolds.
We will usually restrict ourselves to compact spaces.
Given a point $a \in \ts$, the symplectic reduction, or
symplectic quotient of $M$ at $a$, is defined to be
\[
M_{a} = \phi^{-1}(a)/T,
\]
sometimes denoted $(M/ \! \! / T)_{a}$ (when we need to explicitly
include $T$).
Recall that if $a$ is a regular value of $\phi$, $M_{a}$ is a smooth
symplectic orbifold, and that the symplectic form $\omega_{a}$ on $M_{a}$
is defined by the requirement that
\[
\pi^{*}\omega_{a} = i^{*}\omega
\]
where $\pi: \phi^{-1}(a) \ra M_{a}$ and $i: \phi^{-1}(a) \ra M$ are
the projection and inclusion maps, respectively.
Let $\Delta_{\mathrm{reg}}$ be the set of regular values of the
moment map. We will see in section \ref{xraychap} that this is a finite
union of open convex polytopes, which we call \textit{chambers}.
When computing topological invariants of reduced spaces, the essential fact is
\begin{prop}\label{invarianceofreduction}
Let $(M,\omega,\phi)$ be a compact, connected Hamiltonian $T$-space. Let
$a,b \in \Delta_{\mathrm{reg}}$ lie in the same chamber $P$.
Then the two reduced spaces $M_{a}$ and $M_{b}$ are
diffeomorphic.
\end{prop}
\begin{proof}
The restricted map $\phi: \phi^{-1}(P) \ra P$ is a submersion
with compact fibers. By Ehresmann's theorem,
it is a fibration. Since $P$ is connected, all of the fibers are
diffeomorphic. Since $\phi$ is $T$-equivariant, the fibers
$\phi^{-1}(a)$ and $\phi^{-1}(b)$ are equivariantly diffeomorphic,
and therefore
\[
M_{a} = \phi^{-1}(a)/T \iso \phi^{-1}(b)/T = M_{b}.
\]
\end{proof}
Hence any topological invariant of the reduced spaces
$M_{a}$ ($a \in \Delta_{reg}$) will be a function only of
the chamber that $a$ is in, making it
essentially combinatorial.
In section \ref{sec:circle} we use the cobordism results of
Guillemin, Ginzburg and Karshon \cite{GGK:1996C} to derive a
formula for the signature of a regular reduced space in the case of
a circle action, in terms of fixed point data. This formula
is best expressed as a ``wall-crossing'' formula. This form
is quite simple, and generalizes best to the torus case.
In Section \ref{subsec:singularreduction} we
analyze the case of singular reduction by a circle action,
using recent results of Lerman and Tolman
\cite{LermanTolman:1998Sing}.
Again we obtain a simple formula in terms of fixed point data.
Let us make precise what we mean by ``fixed point data.''
If $M$ is a compact $T$-space, the fixed point set $M^{T}$
has finitely many connected components $F_{1},\dots ,F_{m}$.
Each $F_{r}$ is a compact connected manifold. If $M$ is
Hamiltonian, it has associated to it a natural isotopy class of
$T$-invariant almost complex structures \cite{McDuffSalamon:1995}.
Hence the normal bundle $N_{r}$ to each $F_{r}$ has a natural
isotopy class of complex structures, which allows us to
define the weights of the action of $T$ on
the normal bundle. Call these weights $\alpha_{r,k} \in \ts$,
$k= 1,\dots,\mathrm{rank} (N_{r})/2$. It is easy
to see that the moment map $\phi$ must be constant on each $F_{r}$, so
each component $F_{r}$ defines a point in $\ts$, $\phi_{r} = \phi(F_{r})$.
The manifolds $F_{r}$,
together with the weights $\alpha_{r,k}$ and the points $\phi_{r}$,
comprise the
fixed point data that our formulas ultimately depend on.
To investigate the torus case,
we first introduce a variant of the \textit{X-ray} of Tolman
\cite{ST:1996NK}, and describe it in Section \ref{xraychap}.
This is a package of data that includes the fixed point data,
but also takes into account the fixed point sets of subtori $H \subset T$.
Using this language, in Section \ref{recursivechap},
we find a recursive generalization
of the wall-crossing formula obtained in the circle case, where
the data involved in crossing a wall of dimension $d$ is obtained
by a wall-crossing procedure in dimension $d-1$.
In some cases this reduces to a formula which is as simple
as the one in the circle case.
We conclude by deriving similar recursive formulas for the Poincar\'e
polynomial (and hence the Euler characteristic). These formulas
are perhaps more gemoetrical and easier to calculate than
those of Kirwan \cite{FK:1984CQ}.
This paper is largely derived from the author's Ph. D. thesis
\cite{Metzler:1997}, supervised by Victor Guillemin. The author
would also like to gratefully acknowledge the support and
advice of Eugene Lerman, Sue Tolman, and Yael Karshon.
\section{Circle Actions}
\label{sec:circle}
Since the signature is a cobordism invariant, it is natural to
use cobordism arguments to try to calculate it.
Guillemin, Ginzburg, and Karshon \cite{GGK:1996C}
have introduced a notion of
cobordism of spaces with Hamiltonian group actions which is perfectly
suited to our task. It has two main features: first, a cobordism
of Hamiltonian spaces induces cobordisms
of their reduced spaces. Second, there is a localization theorem, which says
that every space is cobordant to a sum of local models, determined by the
fixed point data. Further,
it was noticed by Karshon \cite{YK:1997MM} that the ``moment map''
that appears in their definition of cobordism need not actually
come from a symplectic form, but must only satisfy some simple
axioms. This will prove useful in investigating the signature of
singular reductions.
Note: as has become standard in studying symplectic reduction, we
will deal everywhere with orbifolds, since they arise naturally
from reduction anyway. For our purposes this introduces essentially
no difficulties.
\subsection{Abstract Moment Maps and Cobordism}
\label{subsec:abstractmoment}
Let $M$ be an orbifold with an action of a torus $T$.
Given a subgroup $H \subset T$, let $\pi_{H}: \ts \ra \hs$
denote the natural projection map, and let $M^{H}$ be the
set of fixed points of $H$.
\begin{Def}\label{def:abstractmoment}
An \textbf{abstract moment map} for $M$ is a $T$-invariant
map $\phi: M \ra \ts$ such that, for every subgroup $H \subset T$,
the composed map
\[
\pi_{H} \circ \phi|_{M^{H}}: M^{H} \ra \ts \ra \hs
\]
is locally constant.
\end{Def}
It is easy to show that if $(M,\omega,\phi)$ is a Hamiltonian $T$-space,
then the moment map $\phi$ is an abstract moment map. However,
abstract moment maps are much looser objects (for example, the zero map
is trivially an abstract moment map) and they do not depend on a
symplectic form for their definition. However, the notion of
reduction still makes sense: it is easy to see that if $a \in \ts$
is a regular value of $\phi$, then the action of $T$ on $\phi^{-1}(a)$
is locally free, and hence $\phi^{-1}(a)/T$ is an orbifold, denoted
by $M_{a}$.
We will need to be careful about orientations of reduced spaces.
In the symplectic case, the reduced space is clearly oriented,
being symplectic. However, even in the case of an abstract moment map,
the reduced space has a canonical orientation, as long as $M$
is itself oriented. For, we have short exact sequences
\[
\xymatrix@=6pt{ 0 \ar[r] & {\tlie} \ar[r]^{\iota}
& {T_{p} (\phi^{-1}(a))} \ar[r]
& {T_{[p]} (M_{a})} \ar[r] & 0 \\
0 \ar[r] & {T_{p} (\phi^{-1}(a))} \ar[r]
& {T_{p}} M \ar[r]^{d \phi}
& {\ts} \ar[r] & 0. }
\]
where $\iota$ denotes the infinitesimal action of the torus.
Hence if we choose an orientation for $\tlie$ (and hence for $\ts$)
we obtain orientations on $\phi^{-1}(a)$ and $M_{a}$. If we
reverse the orientation of $\tlie$, the orientation on
$\ts$ also reverses, and hence the orientation on $M_{a}$
is unchanged, so it is actually well-defined independent of the
orientation of $\tlie$.
We can now introduce the notion of cobordism that we need.
Let $(M_{1},\phi_{1})$ and $(M_{2},\phi_{2})$ be two oriented $T$-spaces
with \textit{proper} abstract moment maps.
\begin{Def}\label{def:cobordism}
A \textbf{cobordism} between $(M_{1},\phi_{1})$ and $(M_{2},\phi_{2})$
is an oriented orbifold $W$ and a proper abstract moment map
$\psi: W \ra \ts$, such that $\partial W = M_{1} \coprod \bar{M_{2}}$
and $\psi|_{M_{i}} = \phi_{i}$.
\end{Def}
The overbar indicates reversed orientation. The key to this definition
is the word ``proper''; the orbifold $W$ need not be compact,
but requiring the moment map to be proper ensures that the concept
is non-trivial.
In the case where $(M,\phi)$ arises from an honest Hamiltonian $T$-space,
we require the cobordism to carry a presymplectic form (i.e. a
closed 2-form) which restricts to the given forms on the boundary
components. In this case we will say that the spaces are
\textbf{cobordant as Hamiltonian $T$-spaces}.
This notion of cobordism respects reduction:
\begin{prop}[\cite{YK:1997MM}]
\label{cob:commutewithred}
Let $(M_{1},\phi_{1})$ and $(M_{2},\phi_{2})$
be cobordant as oriented $T$-spaces with abstract moment maps.
Let $a \in \ts$ be a regular value of $\phi_{1}$ and $\phi_{2}$.
Then the reduced spaces $(M_{1})_{a}$ and $(M_{2})_{a}$, with the
orientations as defined above, are
cobordant (as compact oriented
orbifolds\footnote{Note that even if $M_{1}$ and $M_{2}$ are manifolds,
we allow them to be cobordant by an orbifold.}).
\end{prop}
The crucial result from \cite{GGK:1996C} and \cite{YK:1997MM}
is the localization
theorem. This can be seen as a geometric antecedent to the
theorems of Duistermaat and Heckman \cite{DH:1982} and of Jeffrey and Kirwan
\cite{JK:1995}.
From now on we will restrict to the case where $T = S^{1}$, since
this is all we need for the signature formula. This makes the
statements simpler, although the essential picture is the same for
higher rank tori.
We need some preliminaries about orientations.
Let $M$ be an oriented, compact $T$-space with abstract moment map
$\phi$. Denote the components of the fixed point set by
$F_{1},\dots, F_{m}$. Assume that each component is orientable.
(In the Hamiltonian case, the fixed point components are
in fact symplectic, so this is a reasonable assumption.)
Fix once and for all an identification of $\tlie$ with $\R$
(a ``polarization'' of $\tlie$). Given a fixed point component
$F_{r}$, its normal bundle $N_{r}$ is an even-dimensional, oriented real
vector bundle, so it makes sense to speak of the real weights
of the action of $T$ on $N_{r}$. These are nonzero integers defined
only up to sign, and we will denote them
$\pm \alpha_{r,k}, k=1,\dots, q_{r}$, where
$q_{r} = \mathrm{rank} (N_{r})/2$.
By $\alpha^{\#}_{r,k}$ we will mean the weight with the
positive choice of sign; we will refer to these as polarized
weights.
Fix temporarily an orientation of $F_{r}$. This determines an
orientation of $N_{r}$, and hence makes the Pfaffian, $\Pf$,
well-defined on the fibers of $N_{r}$. Let $\xi \in \tlie$
be positive, and let $\xi_{p}$ be the isotropy action of
$\xi$ on the fiber of $N_{r}$ over some $p \in F_{r}$. Let
\[
\varepsilon_{r} = \mathrm{sgn} \: \Pf (\xi_{p}),
\]
which is clearly independent of the choice of $p$.
If we reverse the orientation of $F_{r}$, it is clear that
the sign of $\varepsilon_{r}$ reverses as well. Hence
there is a unique orientation of $F_{r}$ which makes
$\varepsilon_{r}$ positive. Denote by $F_{r}^{\#}$ the
fixed point component with this choice of orientation,
which we will refer to as the polarized orientation.
(If $F_{r}$ is a single point this is just a formal sign
attached to $F_{r}$.)
We will also refer to the corresponding orientation on the
bundle as the polarized orientation.
We note that in the Hamiltonian case, this orientation does
\textit{not} necessarily agree with the symplectic orientation
on $F_{r}$. To be precise, let $M$ be a Hamiltonian $T$-space.
Use a compatible complex structure on the normal bundle $N_{r}$ to
identify complex weights $\alpha_{r,k}$. These are honest integers, not just
up to sign. Let $\sigma_{r}$
be the number of negative complex weights. If we use the
symplectic orientation on $F_{r}$, it is not hard to see
that the sign of the Pfaffian of the action is
$\varepsilon_{r} = (-1)^{\sigma_{r}}$.
Hence the symplectic
orientation differs from the polarized orientation by
$(-1)^{\sigma_{r}}$. In particular, when the component
$F_{r}$ is a single point $p$, the formal sign we
attach to it is exactly $(-1)^{\sigma_{r}}$.
This will be crucial in getting the correct signs
in the signature formula (\ref{cob:sigsumcircle2}).
The localization theorem says roughly that the
whole space $M$ is cobordant to the disjoint
union of the normal bundles of the fixed point compoents, with
appropriately defined moment maps.
Fix an $N_{r}$, and equip it with an equivariant inner product.
Split the bundle up into rank two subbundles corresponding to the
weights $\alpha_{r,k}$, and denote the corresponding splitting
of any $v \in N_{r}$ by $v = \sum v_{k}$.
Define a map on the bundle by
\[
\psi_{r} (v) = \phi(F_{r}) +
\sum_{k=1}^{q_{r}} ||v_{k}||^{2} \alpha^{\#}_{r,k}.
\]
This is a proper moment map for the $T$-space $N_{r}$.
We are now ready to state the localization theorem.
\begin{thm}[\cite{GGK:1996C},\cite{YK:1997MM}]
\label{cob:localization}
Let $T=S^{1}$.
Let $M$
be a compact $T$-space with abstract moment map $\phi$.
Fix a polarization for $T$. Define the polarized weights
$\alpha^{\#}_{r,k}$
and the maps $\psi_{r}$ as above.
Then $M$ is cobordant, as a Hamiltonian $T$-space,
to the disjoint union
of the normal bundles:
\begin{equation}\label{cob:loceq}
(M,\phi) \sim \coprod_{r} (N_{r},\psi_{r}).
\end{equation}
\end{thm}
(Note: with the appropriate definition of a polarization and of the
moment maps on the normal bundles,
essentially the same statement applies in the higher-rank torus case.)
\subsection{Cobordism and Reduction}
\label{subsec:cobred}
It remains to state what happens under reduction. The localization
theorem says that the original space is cobordant to a sum
of local models, so any reduction will be cobordant to a sum of
reduced local models. In the circle case it is particularly easy to
see what these reduced local models are.
We begin by analyzing the reduction
of a single vector space.
Let $V$ be a real orthogonal $T$-representation with real weights
\[
\pm \alpha_{1},\dots, \pm \alpha_{q} \ne 0.
\]
Give $V$ the polarized
orientation (so that the
Pfaffian of the action of a positive generator of $\tlie$ is positive,
as discussed above).
Fix an element
$b \in \ts$. Equip $V$ with the map $\psi_{\alpha,b}: V \ra \ts$ given by
\[
\psi_{\alpha,b} (v) = b +
\sum_{k=1}^{q} ||v_{k}||^{2} \alpha^{\#}_{k}.
\]
Denote this oriented $T$-space with abstract moment map by
$V(\alpha,b)$. As a vector space it is just $\C^{n}$.
Given such a local model $V(\alpha,b)$, and a point $a \in \ts$, $a \ne b$,
there are two possibilities for the reduction of $V$ at $a$.
If $a < b$ then the reduction is
empty, since the image of $\psi_{\alpha,b}$ lies to the right of $b$.
If $a > b$ then the reduction is
\begin{align*}
V(\alpha,b)_{a}
&= (\psi_{\alpha,b})^{-1}(a) / S^{1} \\
&= \{ v \in V \: | \:
b + \frac{1}{2} \sum_{k=1}^{q} |v_{k}|^{2} \alpha^{\#}_{k} = a \}
/ S^{1}
\end{align*}
which is a twisted complex projective space of dimension $2q-2$.
Denote the (possibly empty) reduced space by $X(\alpha,a,b)$.
This is generally an orbifold, but (when it is nonempty)
its rational cohomology ring is isomorphic to that of the ordinary
projective space $\CP^{n-1}$.
Further, it is not hard to see that the orientation that
$X(\alpha,a,b)$ inherits as a reduced space is the same as the
complex orientation it has under the identification with
a twisted $\CP^{n-1}$, because of the condition we put
on the orientation of $V$.
The case of a vector bundle is not much more complicated.
Let $V$ be a real, orientable
orthogonal $T$-bundle over a trivial $T$-space $F$,
with real weights
$\pm \alpha_{1},\dots, \pm \alpha_{q} \ne 0$. Again give $E$ the
polarized orientation.
Fix an element
$b \in \ts$. Equip $E$ with the map $\psi_{\alpha,b}: E \ra \ts$ given by
\[
\psi_{\alpha,b} (v) = b +
\sum_{k=1}^{q} ||v_{k}||^{2} \alpha^{\#}_{k}.
\]
This is an oriented $T$-space with abstract moment map, whose fibers
are copies of $V(\alpha,b)$.
Since $E$ fibers equivariantly over the trivial $T$-space $F$, so will the
reduction. Since the fibers are simply copies of $V(\alpha,b)$,
the fibers of the reduced space $E_{a}$ will be copies of $X(\alpha,a,b)$.
Hence the only spaces we have to deal with are bundles whose fibers
are twisted complex projective spaces.
Note also that $E_{a}$ is an associated bundle of the
oriented orthonormal frame bundle
of $E$. For, the $T$-action on the fibers of $E$ commutes with the
action of the structure group,
so the structure group acts naturally on the reduction by $T$.
In particular, the structure group of the reduced space as a bundle over $F$
is compact and connected.
Combining these local results with the localization theorem, and the fact that
cobordism respects reduction, we have
\begin{thm}[\cite{YK:1997MM}]
\label{cob:localredgeneral}
Let $T=S^{1}$, and choose a fixed polarization of $T$.
Let $M$ be a compact oriented $T$-space with
abstract moment map $\phi$. Assume that the fixed point components are
orientable, and give them the polarized orientation $F_{r}^{\#}$.
Let $a \in \ts$ be a regular value of $\phi$.
Then there is an oriented orbifold cobordism
\[
M_{a} \sim \coprod_{r} (N^{\#}_{r})_{a}
\]
where each $(N^{\#}_{r})_{a}$ is a fiber bundle over $F_{r}^{\#}$
with fiber $X_{r} = X(\alpha_{r},a,\phi(F_{r}))$, and with
compact connected structure group. The fibers $X_{r}$ are
orientation-preserving diffeomorphic to twisted complex
projective spaces.
If $M$ is Hamiltonian, then the polarized orientation $F_{r}^{\#}$
differs from the symplectic orientation of $F_{r}$ by
$(-1)^{\sigma_{r}}$, where $\sigma_{r}$ is the number of
negative weights of the isotropy action at $F_{r}$.
\end{thm}
\subsection{The Signature Formula in the Circle Case}
\label{subsec:sigformula}
Among the topological invariants that are preserved by cobordism,
there is one that is singled out by being multiplicative
for any fiber bundle with oriented fiber and compact connected structure
group: the signature. We recall the
\begin{Def}\label{cob:sigdef}
Given a connected, compact, oriented $4k$-dimensional rational homology
manifold $M$, the \textbf{signature}
$\Sign(M)$ is defined to be the signature of the symmetric bilinear
pairing $H^{2k}(M,\Q) \tensor H^{2k}(M,\Q) \ra H^{4k}(M,\Q) \iso \Q$
given by the cup product. If $\dim M$ is not divisible by $4$, we define
$\Sign(M) = 0$.
\end{Def}
Note that any orbifold is a rational homology manifold \cite{Ful:1993TV},
so it has a well-defined signature.
The signature is of course an important and natural invariant of
a manifold (or orbifold).
Its special significance to this work is expressed
by the following three results.
\begin{thm}\label{cob:sigiscobinv}
The signature is an invariant of oriented orbifold cobordism.
\end{thm}
\begin{proof}
This was proved for manifolds by Thom (see Hirzebruch \cite{Hir:1966}).
The proof relies only on Poincar\'e duality, hence it works for
rational homology manifolds, hence for orbifolds.
\end{proof}
The second result concerns the multiplicativity of the signature, and is a
slight modification of a result of
Chern, Hirzebruch, and Serre \cite{CheHirSer:1957}:
\begin{thm}\label{cob:chernhirzebruchserrethm}
Let $E \ra B$ be a fiber bundle with fiber $F$ such that
\begin{enumerate}
\item \label{cob:basefibertotalcond}
$E,B,F$ are compact connected oriented orbifolds;
\item \label{cob:structuregroupcond}
the structure group of $E$ is compact and connected.
\end{enumerate}
If $E,B,F$ are oriented coherently, then
\[
\Sign(E) = \Sign(B) \Sign(F).
\]
\end{thm}
\begin{proof}
Chern, Hirzebruch and Serre prove that the conclusion follows from
condition (\ref{cob:basefibertotalcond}), with ``manifold'' in place
of orbifold, and condition
\begin{enumerate}
\item [$2'$.]
The fundamental group $\pi_{1}(B)$ acts trivially on
$H^{*}(F)$.
\end{enumerate}
As above, their proof uses only Poincar\'e duality (and the spectral
sequence of a fibration), and hence applies equally well
to rational homology manifolds, such as orbifolds. Also, condition
(\ref{cob:structuregroupcond}) implies ($2'$).
For, the action of $\pi_{1}(B)$ factors through the structure group; and
the action of a compact connected group on cohomology is trivial.
Our version of the theorem then follows.
\end{proof}
To emphasize the special role of the signature, we briefly mention
\begin{thm}[Borel, Hirzebruch; see \cite{HirBerJung:1992}]
\label{cob:signatureisonly}
The signature is the only rational invariant of oriented cobordism
that is multiplicative in the sense of Theorem
\ref{cob:chernhirzebruchserrethm} and takes the value 1 on
$\CP^{2}$.
\end{thm}
Thms. \ref{cob:sigiscobinv} and \ref{cob:chernhirzebruchserrethm},
together with
Thm.\ \ref{cob:localredgeneral},
immediately give the formula for the signature of the reduction by
a circle action:
\begin{align}\label{cob:localsig1}
\Sign(M_{a}) &= \sum_{r} \Sign(F_{r}^{\#}) \Sign(X_{r}) \\
&= \sum_{r} (-1)^{\sigma_{r}} \Sign(F_{r}) \Sign(X_{r})
\end{align}
where the second equation applies to the Hamiltonian case,
in which $F_{r}$ is given its symplectic orientation.
In the circle case the computation of
$\Sign(X_{r})$ is quite simple, since $X_{r}$ is either empty, or a
twisted complex projective space of complex dimension $q_{r}-1$,
where $q_{r}$ is the half-rank of the normal bundle $N_{r}$.
The signature of a complex projective space is just $1$ when
its complex dimension is even, $0$ when it is odd.
To be precise, fix a polarization of $T$.
Let $F_{1},\dots, F_{m}$ be the
fixed point components, and let $\phi_{r} = \phi(F_{r})$. In the
Hamiltonian case, let
$f_{r}$ and $b_{r}$ be the number of positive and negative weights,
respectively, of $F_{r}$ (so $f_{r} + b_{r} = q_{r}$).
Then the above formula becomes the following:
\begin{theorem}\label{thm:circlesignatureformula}
The signature of a regular reduction of a space with circle action and
abstract moment map is given by
\begin{align}
\Sign(M_a) &= \sum_{r: \; \phi_{r} < a, \: q_{r} odd}
\Sign(F_r^{\#}) \label{cob:sigsumcircle1} \\
&= \sum_{r: \; \phi_{r} < a, \: q_{r} odd}
(-1)^{b_r} \Sign(F_r), \label{cob:sigsumcircle2}
\end{align}
where the latter equation applies only to the Hamiltonian case,
and $F_{r}$ is given its symplectic orientation.
\end{theorem}
This can be restated as a wall-crossing formula, which will be more
convenient later.
Fix a singular value $c \in \ts$ of the moment map. Let
$F_{1},\dots , F_{m}$ be the fixed point components with
$\phi(F_{r}) = c$. Pick $a_{1} < c < a_{2}$ such that there are
no other singular values in the interval $[a_{1},a_{2}]$.
(See Fig.\ \ref{fig:circlewall}.)
Denote the half-rank of the normal bundle $N_{r}$ by $q_{r}$.
Then
\begin{equation}\label{cob:wallcrossforcircleabstract}
\Sign(M_{a_{2}}) - \Sign(M_{a_{1}})
= \sum_{r=1 \dots m, \: q_{r} \mathrm{~odd}}^{m} \Sign(F_{r}^{\#}).
\end{equation}
In the Hamiltonian case, we have a slightly more explicit formula:
Define the function $w_{S}: \N \times \N \ra \Z$ by
\begin{equation}\label{cob:wsdefeq}
w_{S}(f,b) := \left\{
\begin{array}{cl}
(-1)^{b} & \mathrm{~if~} f+b \mathrm{~is~odd}\\
0 & \mathrm{~if~} f+b \mathrm{~is~even}.
\end{array}
\right.
\end{equation}
Then
\begin{equation}\label{cob:wallcrossforcircle}
\Sign(M_{a_{2}}) - \Sign(M_{a_{1}})
= \sum_{r=1}^{m} w_{S}(f_{r},b_{r}) \Sign(F_{r}).
\end{equation}
Here $F_{r}$ has its symplectic orientation.
The idea of ``wall-crossing,'' and generalizations
of (\ref{cob:wallcrossforcircle}) will be central to what follows.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\[ \xy/r1.8pc/:
(0,0)="1",
(0.5,0)="2",
(2.5,0)="3", "3"+/d2ex/*{c},
(4.5,0)="4",
(5,0)="5",
(1.5,0)="6", "6"+/d2ex/*{a_{1}},
(3.5,0)="7", "7"+/d2ex/*{a_{2}},
"1";
"5" **\dir{-},
"2"*{\bullet};
"3"*{\bullet};
"4"*{\bullet};
"6"*{\times};
"7"*{\times};
\endxy \]
\caption{Wall Crossing in the Circle Case.}
\label{fig:circlewall}
\end{center}
\end{figure}
\textit{Example.} \label{cob:examplecp3circle}
We first describe a whole family of simple examples of
Hamiltonian torus actions.
Let $M = \CP^{n}$. This has a natural action of $T^{n+1}$ inherited
from the standard linear action of $T^{n+1}$ on $\C^{n+1}$
\[
(\lambda_{1},\dots ,\lambda_{n+1}) \cdot (z_{1},\dots ,z_{n+1})
= (\lambda_{1} z_{1}, \dots, \lambda_{n+1} z_{n+1}).
\]
(This action is not effective, since the diagonal circle acts
trivially on $\CP^{n}$.) This action is easily seen to be Hamiltonian,
with moment map
\[
\phi([z_{1},\dots ,z_{n+1}])
= \frac{1}{||z||} \sum_{k=1}^{n+1} |z_{k}|^{2} e_{k},
\]
whose image is exactly the standard $n$-simplex in $\R^{n+1}$,
\[
\Delta_{n} = \{ a_{1},\dots, a_{n+1} \: | \: \sum a_{k} = 1 \}.
\]
Choose a subtorus
$T \subset T^{n+1}$ of rank $d$,
which does not contain the diagonal circle. Then $M$
is an effective Hamiltonian $T$-space, and the moment map is just $\phi$
composed with the projection $\R^{n+1} \ra \ts$. The moment polytope
is the image of the standard $n$-simplex under this projection.
If $T$ is generic, its fixed points are the same as those
of $T^{n+1}$, which are clearly the $n+1$ isolated points
$[z_{1},\dots ,z_{n+1}]$ with $z_{j} = \delta^{j}_{k}$, for
$k=1,\dots ,n+1$.
For now, choose $n=3$, $d=1$, so we get a generic Hamiltonian
circle action on $\CP^{3}$, with $4$ isolated fixed points.
The moment polytope is shown in Figure \ref{fig:cp3circle},
with the images of the fixed points shown as $p,q,r,s$.
Since the complex dimension is $3$, each fixed point
has $3$ weights associated to it. It is not hard to see
by looking at the action that at $p$, all three weights
point to the right; at $s$, all three point left; at
$q$, one points left, two point right; and at $r$,
two point left and one points right.
Since all fixed points are isolated,
the wall-crossing rule (\ref{cob:wallcrossforcircle}) gives
the signatures shown in the figure. These agree with a direct
calculation of the reduced spaces, which easily shows
that the reductions in the outer two chambers are
(twisted) $\CP^{2}$, which has signature $1$,
and the reduction in the inner
chamber has the rational cohomology of $\CP^{1} \times \CP^{1}$,
which has signature $0$.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\[ \xy/r1.8pc/:
(0,0)="1", "1"+/u2ex/*{0},
(0.5,0)="2", "2"+/dl2ex/*{p},
(2.5,0)="3", "3"+/dl2ex/*{q},
(4.5,0)="4", "4"+/dl2ex/*{r},
(6.5,0)="5", "5"+/dl2ex/*{s},
(7,0)="6", "6"+/u2ex/*{0},
(1.5,0)="7", "7"+/u2ex/*{1},
(3.5,0)="8", "8"+/u2ex/*{0},
(5.5,0)="9", "9"+/u2ex/*{1},
"1";
"6" **\dir{.},
"2"*{\bullet};
"5" **\dir{-},
"3"*{\bullet};
"4"*{\bullet};
"5"*{\bullet};
\endxy \]
\caption{Wall Crossing Example: $\CP^{3}$ with Circle Action.}
\label{fig:cp3circle}
\end{center}
\end{figure}
\subsection{Singular Reduction}\label{subsec:singularreduction}
Recently Lerman and Tolman \cite{LermanTolman:1998Sing}
have found a beautiful way
to analyze the rational intersection cohomology ring of singular reductions
for Hamiltonian circle actions. We can use a piece of their
result to easily extend the above formula to the
singular case. The key is that Thm.\ \ref{thm:circlesignatureformula}
applies to any abstract moment map.
Given a singular value $c$ of the moment map, let
$\F_{c} = \{ F_{1},\dots, F_{m} \}$
be the set of fixed point components with $\phi(F_{k}) = c$, as above.
Divide $\F_{c}$ into two disjoint sets:
\begin{align*}
\F_{c}^{+} &= \{ F_{k} \: | \: b_{k} < f_{k} \} \\
\F_{c}^{-} &= \{ F_{k} \: | \: b_{k} \ge f_{k} \}.
\end{align*}
Lerman and Tolman show that by deforming the moment map $\phi$
to a new map $\tilde{\phi}$, which is now only an abstract moment map,
they can find a ``small resolution'' of the singularities of
the singular reduction $\phi^{-1}(c)/S^{1}$. Precisely,
\begin{theorem}[Lerman--Tolman]
\label{thm:LermanTolmanSmallResolution}
Let $(M,\omega,\phi)$ be a Hamiltonian $S^{1}$-space.
Let $c$ be a singular value of $\phi$, and define
$F_{1},\dots ,F_{m}$, $\F_{c}^{\pm}$ as above.
Then there exist $a,b \in \R$, $a < c < b$, and
an abstract moment map $\tilde{\phi}$, such that:
\begin{enumerate}
\item $a,b$ are regular values of both $\phi$ and $\tilde{\phi}$;
\item there are no fixed points in $\phi^{-1}([a,b])$ besides those
in $\F_{c}$;
\item $\tilde{\phi}$ and $\phi$ are equal on $M \setminus \phi^{-1}([a,b])$;
\item $c$ is a regular value of $\tilde{\phi}$;
\item $a < \tilde{\phi}(F_{k}) < c$ for all $F_{k} \in \F_{c}^{-}$
and $c < \tilde{\phi}(F_{k}) < b$ for all $F_{k} \in \F_{c}^{+}$;
\item $\tilde{M}_{c} := \tilde{\phi}^{-1}(c)/S^{1}$
is a small resolution
of $M_{c} = \phi^{-1}(c)/S^{1}$. In particular there is a
pairing-preserving isomorphism between the
(middle perversity) intersection cohomology of $M_{c}$
and the ordinary cohomology of $\tilde{M}_{c}$.
\end{enumerate}
\end{theorem}
We can now apply our formulas to the new abstract moment map
$\tilde{\phi}$ to get a formula for the singular reduction.
We simply compare $\tilde{M}_{c}$ to $M_{a}$ (or to $M_{b}$).
The wall-crossing formula (\ref{cob:wallcrossforcircle})
applied to $\tilde{\phi}$ gives
\[
\Sign(\tilde{M}_{c})
= \Sign(\tilde{M}_{a}) +
\sum_{F_{r} \in \F_{c}^{-}, \quad q_{r} \mathrm{~odd}} \Sign(F_{r}^{\#}).
\]
One gets a similar formula involving $\F_{c}^{+}$ by comparing
to $\tilde{M}_{b}$.
Note that since the $F_{r}$ are actually fixed point components of the
original Hamiltonian action, they are symplectic, and the
normal bundles do have complex structures. So we can
restate this formula in terms of the symplectic orientation of $F_{r}$
and the number of negative weights, just as in the honest Hamiltonian
case:
\[
\Sign(\tilde{M}_{c})
= \Sign(\tilde{M}_{a}) +
\sum_{F_{r} \in \F_{c}^{-}} w_{S}(f_{r},b_{r}) \Sign(F_{r}).
\]
Since $\tilde{M}_{c}$ is a small resolution of $M_{c}$, their
signatures are equal (where the signature of the singular space
$M_{c}$ is understood to be taken in intersection cohomology).
Also, since $\tilde{\phi}$ and $\phi$ only differ near the
level $c$, we have $\tilde{M}_{a} = M_{a}$, and $\tilde{M}_{b} = M_{b}$.
Hence we have proved
\begin{theorem}\label{thm:singularsignature}
Let $(M,\omega,\phi)$ be a compact Hamiltonian $S^{1}$-space,
and let $c$ be a singular value of the moment map. Define
the sets $\F_{c}^{\pm}$ and the function $w_{S}$ as above.
Choose $a,b \in \R$, $a < c < b$, such that there are no critical
values of $\phi$ in the intervals $[a,c)$ and $(c,b]$.
Then
\begin{align}\label{singularsignatureformula}
\Sign(M_{c})
&= \Sign(M_{a}) +
\sum_{F_{r} \in \F_{c}^{-}} w_{S} (b_{r},f_{r}) \Sign(F_{r}) \\
&= \Sign(M_{b}) -
\sum_{F_{r} \in \F_{c}^{+}} w_{S} (b_{r},f_{r}) \Sign(F_{r})
\end{align}
\end{theorem}
Note that the jump between the signatures of the regular reductions
$\Sign(M_{a})$ and $\Sign(M_{b})$, which is equal to
\[
\sum_{F_{r} \in \F_{c}} w_{S} (b_{r},f_{r}) \Sign(F_{r})
\]
is simply divided into two steps, one between
$\Sign(M_{a})$ and $\Sign(M_{c})$, the other between
$\Sign(M_{c})$ and $\Sign(M_{b})$. Each
fixed point component contributes to one step or the other
depending on whether it has more weights up or down.
One may wonder why in the definitions of $\F_{c}^{\pm}$ we chose
the $F_{r}$ with $b_{r} = f_{r}$ to lie in $\F_{c}^{-}$
and not in $\F_{c}^{+}$. This is in fact an arbitrary choice;
Lerman and Tolman's theorem is still true if we put these
$F_{r}$ in $\F_{c}^{+}$ instead. Precisely, making this choice
gives a different small resolution ${\tilde{M}'}_{c}$ of $M_{c}$,
with a cohomology ring that is not isomorphic to the one resulting
from our choice; however, the pairings are isomorphic.
This makes sense in Thm.\ \ref{thm:singularsignature},
because the fixed point components with $b_{r} = f_{r}$
do not contribute to the wall-crossing:
when $b_{r} = f_{r}$, $b_{r} + f_{r}$ is even and $w_{S}(b_{r},f_{r})=0$.
Hence it doesn't matter where we include them.
\section{The Structure of the Moment Polytope}
\label{xraychap}
To describe the formula for the signature in the torus case,
we first need to review what is known about the structure of the moment
polytope, and introduce some notation.
Let $T$ be a torus of rank $d$. Let $(M,\omega,\phi)$ be a compact
Hamiltonian $T$-space of dimension $2n$.
For convenience, assume the action is effective.
By the convexity theorem of
Atiyah and Guillemin-Sternberg \cite{A:1982CCH} \cite{GS:1982C},
the moment image $\Delta = \phi(M)$ is a convex
polytope, and is in fact the convex hull of the image the fixed point
set $M^{T}$. However, one can say a great deal more about the structure
of $\Delta$. We will introduce a refinement of Tolman's notion
of the \textit{X-ray} of $M$ to provide the framework for later
calculations.
Note: for the remainder of this section, we will refer to
connected subgroups of $T$ as subtori, to distinguish them from
arbitrary subgroups. Since the results we are interested in
are rational phenomena (as opposed to torsion information) and hence are not
affected by discrete stabilizers, we will ignore disconnected subgroups.
Recall from the structure theory of compact transformation groups
(see e.g. \cite{KK:1991}) that $M$ decomposes into a finite
set of orbit-type strata. Given a subgroup $H \subset T$,
the orbit-type stratum $M_{H}$ is the set of points $p \in M$
such that the isoptropy subgroup $T_{p} = H$. Let $X_{1},\dots, X_{m}$
be the set of connected components of orbit-type strata, with
corresponding isotropy subroups $T_{1},\dots ,T_{m}$. For each
$X_{j}$ denote the closure by $F_{j} = \bar{X_{j}}$. We will
abuse notation and refer to these as the \textit{strata} of $M$.
Let $\F = \{ F_{1},\dots ,F_{m} \}$ be the set of strata. Each $F_{j}$
is a connected component of the fixed point set $M^{T_{j}}$
of the subtorus $T_{j}$\footnote{Not
all connected components of $M^{T_{j}}$ are of this form;
for example a connected component of $M^{T}$ will often be
a connected component of $M^{T_{j}}$ as well, but it is not
the closure of a stratum corresponding to $T_{j}$.}.
Note that $T_{j}$ is the stabilizer of a generic point on $F_{j}$, but
some points in $F_{j}$ will have larger stabilizer.
Denote $\dim(T_{j})$ by $d_{j}$.
Each $F_{j}$ is a symplectic
manifold in its own right, by the equivariant Darboux theorem
\cite{GS:1984STP}. In fact, the restriction of the moment map
$\phi$ makes $F_{j}$ into a Hamiltonian $T$-space. However, the $T$-action
on $F_{j}$ is clearly not effective. $F_{j}$ inherits an effective action of
the quotient torus
$H_{j} = T/T_{j}$. The $H_{j}$-action is Hamiltonian, but the
moment map is unique only up to addition of a constant.
It is specified as follows. Consider the three exact sequences
\begin{equation}
\label{x:exactforfixed}
\xymatrix{ 1 \ar[r] & {T_{j}} \ar[r] & {T} \ar[r] &
{H_{j}} \ar[r] & 1 \\
0 \ar[r] & {\tlie_{j}} \ar[r] & {\tlie} \ar[r] &
{\h_{j}} \ar[r] & 0 \\
0 & {\ts_{j}} \ar[l] & {\ts} \ar[l] &
{\hs_{j}} \ar[l] & 0 \ar[l] \\
& d_{j} & d & (d - d_{j}) &
}
\end{equation}
(the last row notes the respective dimensions).
Note $\hs_{j} = \Ann(\tlie_{j})$. The moment map
$\phi : M \ra \ts$ restricted to $F_{j}$ lies in a translate
of $\hs_{j} \subset \ts$ (since the vector fields generated by
$\tlie_{j}$ vanish on $F_{j}$). In particular, if $T_{j} = T$
then $\phi(F_{j})$ is a point. The restricted moment
map, shifted by a constant to land in $\hs_{j}$, is a moment map
for the action of $H_{j}$.
Hence, by the convexity theorem, the image
$\phi (F_{j})$
is a convex polytope in its own right, though of dimension $d-d_{j}$.
We will call these polytopes \textit{walls}.
It is important to note that two different $F_{j}$ can
have overlapping, or even identical, images under $\phi$.
Tolman \cite{ST:1996NK} introduced the notion of the \textit{X-ray}
of $M$ to keep track of this data.
\begin{Def}\label{def:xray}
The \textbf{X-ray} of a compact Hamiltonian $T$-space as above is the
family $\{ \phi(F_{j}) \}$ of convex polytopes in $\ts $,
indexed by the set $\F$, where $\F$ is considered as a partial
order under inclusion.
\end{Def}
More technically, the data of the X-ray are the partial order $\F$ and the map
$F_{j} \mapsto \phi(F_{j})$, which we will also denote by $\phi$.
We will often refer to an X-ray by the pair $(\F,\phi)$.
Note that $F_{j} \subset F_{k}$ implies
$\phi(F_{j}) \subset \phi(F_{k})$ but not vice versa.
We will refer to the walls coming from $T$-fixed point components
(which are just single points of $\ts$)
as the \textit{vertices} of the X-ray.
X-rays have some nice formal properties (which are explored further
in \cite{Metzler:1997}).
Denote the affine
span of a set $S$ in $\ts$ by $\Aff(S)$, and the linear span of $S$
(the unique linear subspace parallel to $\Aff(S)$) by $\Lin(S)$.
\begin{prop}\label{x:xrayproperties}
Given a Hamiltonian $T$-space $M$, its X-ray map $F_{j} \mapsto \phi(F_{j})$
is an order-preserving map from $\F$ to the set of convex polytopes in
$\ts$, satisfying:
\begin{enumerate}
\item \label{x:face} Given $F_{j} \in \F$, and a face $\delta$ of
$\phi(F_{j})$,
there is a unique $F_{k} \le F_{j}$ such that $\phi(F_{k}) = \delta$.
\item \label{x:uniquegoingup} Given $F_{j} \in \F$, and $F_{k} \ge F_{j}$,
$F_l \ge F_{j}$, with $\Lin(\phi(F_{k})) = \Lin(\phi(F_l))$, then
$F_{k} = F_l$.
\item \label{x:dimdiff}
$F_{k} < F_{j} \Rightarrow \dim \phi(F_{k}) < \dim \phi(F_{j})$.
\end{enumerate}
\end{prop}
\begin{proof}
First, note that $\Lin(\phi(F_{k})) = \Ann(\tlie_{k})$
by the definition of a moment map.
Hence condition \ref{x:dimdiff} in the definition
of an X-ray follows from the fact that for any $F_{j} \subsetneq F_{k}$,
$T_{j} \supsetneq T_{k}$, and hence $\tlie_{j} \supsetneq \tlie_{k}$,
since $T_{j}, T_{k}$ are connected. Also, condition \ref{x:uniquegoingup}
is clear: the condition $\Lin(\phi(F_k)) = \Lin(\phi(F_l))$ implies
that $F_k$ and $F_l$ are connected components of the fixed point set
of the same subtorus. But they both contain $F$, hence they must
be equal.
Condition \ref{x:face} is a bit deeper. It comes from the result
of Atiyah \cite{A:1982CCH} that the inverse
image of any point under the moment map is connected.
For, consider a stratum $F_{j}$. This is a Hamiltonian
$T$-space. Given a face $\delta$ of $\phi(F_{j})$,
let $\h = \Ann(\Lin(\delta))$, and let $H \subset T$ be the corresponding
Lie subgroup. (This exists since the polytope $\phi(F_{j})$ is
rational \cite{G:1994MM}\footnote{This is another
feature of Hamiltonian $T$-spaces that will
not be particularly relevant in this paper.}.)
Consider the moment map of the $H$-action on $F_{j}$; this is just
the projection of the moment map by
$\pi_{H}: \ts \ra \hs = \ts/\Lin(\delta)$.
This moment map has an extremum at $\pi_{H}(\delta)$,
which is certainly a critical value. Moreover, any point in $F_{j}$ mapping
to this point will necessarily be a critical point. Hence the entire
inverse image of $\pi_{H}(\delta)$ under the $H$-moment map,
which is the inverse image
of $\delta$ under the $T$-moment map, is fixed by the action of $H$.
Moreover, this set is connected by Atiyah's result.
Hence it is exactly some $F_{k}$, and
$H = T_{k}$, $\phi(F_{k}) = \delta$, and no other $F_{l} \subset F_{j}$
can map to $\delta$.
\end{proof}
\textit{Remark.\/}
Our definition is not completely consistent with
Tolman's definition of an X-ray.
In defining the X-ray of a Hamiltonian action,
she considers all possible stabilizer
groups, not just connected ones, and hence her X-rays
do not satisfy condition \ref{x:dimdiff} above.
By considering disconnected
stabilizers she gets at torsion information, in particular, the
intrinsic stabilizers of singular points in the reduced spaces
(when these spaces are orbifolds). However, we are primarily interested
in rational invariants, which are insensitive to this information.
We could refer to our X-rays as ``X-rays modulo torsion.''
Note, however, that much of what we say in the rest of this section
applies in a slightly modified form to Tolman's X-rays.
\textit{Example.}
We again consider a complex projective space, as on page
\pageref{cob:examplecp3circle},
but now we take $n=4$, $d=2$\footnote{We choose $d=2$ for later purposes,
so that the signatures of the reduced spaces are nonzero.
See Section \ref{recursivechap}.}.
For a generic choice
of the subtorus, the image of the moment polytope will appear as in
Figure \ref{fig:cp4}. (This picture is in $\R^{2}$.)
The five dots are the vertices (images of the $T$-fixed point components),
while the lines are images of strata corresponding to circles in $T$.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\[ \xy/r1pc/:
(0,0)="1",
(4,0)="2",
(2,4)="3",
(1.6,0.75)="4",
(2.4,1.9)="5",
"1"*{\bullet};
"2" **\dir{-}, "2" *{\bullet},
"3" **\dir{-}, "3" *{\bullet},
"4" **\dir{-}, "4" *{\bullet},
"5" **\dir{-}; "5" *{\bullet};
"2" **\dir{-},
"3" **\dir{-},
"4" **\dir{-};
"2" **\dir{-},
"3" **\dir{-};
"2" **\dir{-};
\endxy \]
\caption{X-ray of $M = \CP^4$, with 2-torus action.}
\label{fig:cp4}
\end{center}
\end{figure}
\subsection{Weight Data}\label{subsec:weightdata}
We also want to record information about the weights of the
infinitesimal action of $T$ on the normal bundles to the strata.
This data will appear in the wall-crossing formulas for the signature
and other invariants.
Moreover, the weight data at the $T$-fixed points
often completely determine the X-ray (see \cite{Metzler:1997}).
Recall from the introduction
that the $T$-invariant symplectic structure on $M$
defines a natural isotopy class of compatible,
invariant almost complex structures.
Hence, given any point
$p \in M^{T_{j}}$, the complex weights of the action of $T_{j}$ on $T_{p} M$
are well-defined as vectors in $\ts$.
Given a stratum $F_{j} \subset M^{T_{j}}$,
denote the weights of the $T_{j}$ action on $T_{p} M$, for any
$p \in M$, by $\alpha_{j}= \{ \alpha_{j,k} \in \ts_{j} \}$.
(Since $F_{j}$ is connected this is independent of $p$.)
\begin{Def}\label{x:newweightedxraydef}
The \textit{weighted X-ray} of $M$ is the X-ray together with
the assignment to each stratum $F_{j}$ of the weights
$\alpha_{j,k} \in \ts_{j}$.
\end{Def}
Note that we consider all of the weights on
$TM|_{F_{j}}$, not just the weights of the normal bundle $N_{F_{j}}$.
Of course the additional weights of $T_{j}$, corresponding to $T F_{j}$, are
zero, but we will see below that the bookkeeping works out better if we
include these weights as well. Also, $\alpha_{F}$ is not a set of vectors, but
a family, as we need to keep track of multiplicities.
We need to describe some of the properties that weighted X-rays
of Hamiltonian $T$-spaces have. The most important is a
consequence of the equivariant Darboux theorem, which says that
the weights determine the X-ray ``locally.''
We will use the
following definition of a \textit{local model}. In any vector space $A$,
given a set of vectors $\alpha = \{\alpha_{1},\dots,\alpha_{n} \in A \}$,
denote the positive cone generated by $\alpha$ by
\begin{equation}\label{coneeq}
\Cone(\alpha_{1},\alpha_{2},\dots \alpha_{n})
= \left\{ \left. \sum_{k=1}^{n} t_{k} \alpha_{k}
\: \right| \: t_{k} \ge 0 \right\}.
\end{equation}
Call a subset
$S = \{ \alpha_{i_{1}},\alpha_{i_{2}},\dots \alpha_{i_{j}}\} \subset \alpha$
\textit{linear} if no other $\alpha_{i} \in \alpha$, $\alpha_{i} \notin S$
lies in the linear span of $S$;
in other words, if $S = \alpha \cap V$ for some linear subspace
$V \subset A$.
\begin{Def}\label{x:localmodeldef}
The \textbf{local model} generated by
$\{\alpha_{1},\dots,\alpha_{n} \in A \}$ is the set of cones
\[
\{\Cone (S) \: | \: S \mathrm{~a~linear~subset~of~} \alpha \}.
\]
\end{Def}
We will also refer to the intersection of these cones with any neighborhood
$U$ of $0$ as a local model.
We will also need to consider the natural
maps $ \pi_{jk}: \ts_{j} \ra \ts_{k}$ arising when
$F_{j} \subset F_{k}$, and the quotient map
$\pi_{j}: \ts \ra \ts_{j}$.
\begin{prop}\label{x:weightedxraydef}
The weighted X-ray $(\F,\phi,\alpha)$ of a Hamiltonian $T$-space
satisfies:
\begin{enumerate}
\item
\label{x:darbouxcond}
for every stratum $F_{j}$,
there is a neighborhood $U \subset \ts$ of $\phi(F_{j})$ such that
$\{\pi_{k}(\phi(F_{k}) \cap U) \: | \: F_{k} \ge F_{j} \}$
is the local model generated by $\alpha_{j}$;
\item
\label{x:consistcond}
for every $F_{j}, F_{k} \in \F$, $F_{j} \le F_{k}$,
the families of vectors
$\pi_{jk}(\alpha_{j})$ and $\alpha_{k}$ are equal, up to
rearrangement. (I.e. they contain the same vectors with the
same multiplicities.)
\end{enumerate}
\end{prop}
We will refer to condition \ref{x:darbouxcond} as the Darboux
condition, and condition \ref{x:consistcond} as the consistency condition.
\begin{proof}
The weights of the action of $T_{k}$ on the tangent bundle
can be determined by looking at the tangent space to any point
$p \in F_{k}$. Suppose we have $F_{j} \subset F_{k}$. Choose the
point $p$ to be in $F_{j}$. Then the representation of $T_{j}$ on
the tangent space $T_{p}(M)$ must restrict to the representation
of $T_{k}$. In terms of weights, this is exactly condition
\ref{x:consistcond} in the proposition.
To show that the Darboux condition holds for an arbitrary wall,
first note that it is enough to show that it holds for the
fixed points of the whole torus. For, given a stratum
$F_{j} \subset M^{T_{j}}$, we can restrict the action to $T_{j}$; the
effect on the moment map is to compose with the projection
$\pi_{j}:\ts \ra \ts_{j}$. Clearly the formation of a local model, and
hence the Darboux condition, respects
this projection, so $F_{j}$ will satisfy the Darboux condition
with respect to the $T$ action iff it satisfies it with respect to
the $T_{j}$-action.
So assume that $p \in F \subset M^{T}$. Denote the
weights of the action of $T$ on $T_{p}M$ by
$\alpha = \{ \alpha_{1},\dots \alpha_{n} \}$.
The Darboux theorem, in the equivariant setting, says
(\cite{GS:1984STP}, p. 251) that we can equivariantly identify
a neighborhood of $p$ with $(\C^{n},\omega_{std},\phi_{\alpha})$,
where $\C^{n}$ has the action of $T$ given by the weights $\alpha_{k}$,
that is, the infinitesimal action of $v \in \tlie$ is given by
\[
v \cdot z = (i \alpha_{1}(v) z_{1}, \dots, i \alpha_{n}(v) z_{n}),
\]
and
\[
\omega_{std} = \frac{i}{2} \sum_{k=1}^{n} dz \wedge d\bar{z},
\]
\[
\phi_{\alpha}(z) = \frac{1}{2} \sum_{k=1}^{n} |z_{k}|^{2} \alpha_{k}.
\]
Hence we only need to determine the relationship of the weights
to the images of the fixed point sets of subtori for this linear
space.
Given a subspace $W \subset \tlie$, let $V = \Ann(W)$.
The set of points in $\C^{n}$ infinitesimally fixed by all $w \in W$
is
\[
(\C^{n})^{W} = \{(z_{1},\dots ,z_{n}) \: | \:
\forall k, z_{k} \ne 0 \Rightarrow \alpha_{k} \in \Ann(W)\}
\]
and its moment image is
\begin{align}\label{x:darbouxcomputeeq}
\phi((\C^{n})^{W}) &= \left\{ \left.
\frac{1}{2} \sum_{k=1}^{n} |z_{k}|^{2} \alpha_{k} \: \right| \:
z_{k} \ne 0 \Rightarrow \alpha_{k} \in V \right\} \\
&= \Cone(\alpha \cap V).
\end{align}
Since any $V$ is the annihilator of some $W$, the walls
are exactly the sets of the form (\ref{x:darbouxcomputeeq}), which
proves the Darboux condition.
\end{proof}
The Darboux condition involves both existence and uniqueness:
for every linear subset $S$ of the weights of $F_{j}$,
there is a unique wall, comparable to $F_{j}$, that locally looks like
the cone on $S$. However, the uniqueness only applies to strata
$F_{k} \ge F_{j}$; there may be other walls which happen to lie near
$F_{j}$ (or even overlap it) but if they are not comparable to $F_{j}$
they need have no particular relation to it.
Note that one consequence of the Darboux condition is that the
weights of the $T$-fixed point components
lie along the one-dimensional walls of the X-ray.
Since for our purposes we will not need to know the length of the weights,
it is not always necessary to draw the weights explicitly, since we
can read off their directions from the 1-skeleton of the X-ray.
However, there are often multiple weights pointing in the same
direction, and we will need to be careful to record this multiplicity
information.
In the example of $\CP^{4}$ above (Fig. \ref{fig:cp4}),
each $T$-fixed point is isolated, and hence has four weights
attached to it. Clearly there must be exactly one weight lying in
the direction of each line emanating from a given vertex.
Here there is no ambiguity in the multiplicities, and we can
read off all of the information we will need from the figure.
So far we have concentrated on the walls of the X-ray, which
come from the various fixed point sets. Now we
look at the complement of the walls. Assume from now on that $M$ is connected.
It is easy to see that the
union of the walls (not counting the largest ``wall,''
the entire moment polytope) is the set of singular values of the moment map,
and hence the complement is the set of regular values:
\begin{equation}\label{x:deltaregdef}
\Delta_{reg} = \Delta \setminus \bigcup_{F \ne M} \phi(F).
\end{equation}
As mentioned in the introduction,
this is an open set with a finite number of components, which
we call \textit{chambers}. It is well-known that these chambers
are open convex polytopes. (In fact the convexity follows from Props.
\ref{x:xrayproperties} and \ref{x:weightedxraydef}, as we prove
in \cite{Metzler:1997}.)
We will want the following more general definition when we discuss
recursive invariants in Chap.\ \ref{recursivechap}.
Recall that a stratum $F_{j} \subset M^{T_{j}}$
is an effective Hamiltonian $T/T^{j}$ space.
The regular values of the restricted moment map are clearly
\[
\Reg(F_{j}) = \phi(F_{j}) \setminus \bigcup_{F_{k} < F_{j}} \phi(F_{k}),
\]
which is a relatively open set in $\phi(W)$, again a finite union of
(relatively) open convex polytopes \cite{Metzler:1997}.
We need to be careful to remember which $F_{j}$ these
came from, so call a pair $(F_{j},P)$, where $P$ is a component of
$\Reg(F_{j})$, a \textit{subchamber} of the X-ray.
By abuse of notation we will often
refer to $P$ as a subchamber, when it will cause no confusion.
Note that the vertices of the X-ray are subchambers: the
corresponding torus $T/T$ is trivial, so its Lie algebra is zero-dimensional,
and every point is a regular point for the ``moment map.''
\section{Recursive Invariants of X-rays}
\label{recursivechap}
Given two adjacent chambers in the moment polytope of
a Hamiltonian $T$-space, we would like to have a wall-crossing
formula that relates the signature of a reduction on one
side to that of a reduction on the other side. The simplest
analogy with the circle case would be if the difference were expressed
in terms of the weights associated to the wall, and
the signature of some fixed point component(s).
The truth is slightly more complicated. In fact the signatures
of the reduced spaces are best understood as part of a larger
system, which we will refer to as a \textit{recursive invariant}.
We know from Prop.\ \ref{invarianceofreduction}
that topological invariants of reductions, like the signature,
are functions only of the set of chambers of the moment polytope.
We want to elaborate on this idea slightly. So far, we have only
considered the set of regular values $\Delta_{reg}$ as places
where we can reduce. However, every point in the moment polytope
is a regular value for some Hamiltonian action. Precisely, let
$F_{j} \in \F$ be a stratum corresponding to the subtorus $T_{j}$,
and fix a subchamber $P \subset \phi(F_{j})$.
For any point $q \in P$, $q$ is a regular value of the moment map
for the action of the quotient torus $H_{j}$\footnote{We must technically
pick a fixed translation of $\Aff(\Delta_{j})$ into $\Ann(\tlie_{j})$ to
say this, but the formula (\ref{subreductioneq}) doesn't depend on
this choice.}, so we can form the regular symplectic reduction
\begin{equation}\label{subreductioneq}
(F_{j}/ \! \! /H_{j})_{q} = (\phi^{-1}(q) \cap F_{j})/H_{j}
= (\phi^{-1}(q) \cap F_{j})/T.
\end{equation}
We will denote this by $M^{j}_{(q)}$, and when there is no ambiguity,
simply $M_{(q)}$.
Note that if $q$ lies in $\Delta_{\mathrm{reg}}$ this is simply
the usual symplectic reduction of $M$ at $q$. However, when
$q \notin \Delta_{\mathrm{reg}}$ it is to be carefully distinguished
from the singular reduction
$\phi^{-1}(q)/T$, which is not even an orbifold in general. (In fact
it is a symplectic stratified space \cite{LerSja:1991},
and $M^{j}_{(q)}$ is one of the strata.)
We must make the usual caveat that this space depends not only on
$q \in \ts$ but also on the fixed point component $F_{j}$. If there
are two $F_{j},F_{k}$ whose moment images both contain $q$, we must
specify which fixed point component we are restricting to before
taking the reduction.
The recursive formula for the signature, Thm.\ \ref{rec:sigwallthm},
in fact calculates the
signature on all of these ``subreductions.'' Clearly,
however, the bookkeeping threatens to get a little involved.
The weighted X-ray summarizes exactly the data about the Hamiltonian
action that is needed.
Fix a Hamiltonian $T$-space $(M,\omega,\phi)$. Denote the
weighted X-ray of $M$ by $(\F,\phi,\alpha)$.
Let the set of all subchambers $(F_{j},P)$ of the X-ray be denoted by
$Q_{\F}$. Define a function $S: Q_{\F} \ra \Z$ by
\[
S(F_{j},P) = \Sign(M^{j}_{q})
\]
for any $q \in P$. We will show that this function satisfies
a simple recursive formula. To make things general enough to
accomodate the Euler characteristic and the Poincar\'e polynomial
as well, we introduce a general definition. First we need a lemma,
concerning how subchambers of an X-ray can meet.
Given a wall $\phi(F)$ of an X-ray, call a codimension 1
subwall of $\phi(F)$ a \textit{principal} subwall.
\begin{lemma}\label{rec:chamberoverlaplemma}
Let $(\F,\phi,\alpha)$ be the weighted X-ray of $(M,\omega,\phi)$.
Let $F \in \F$, and let $(F,P_{1})$, $(F,P_{2})$ be subchambers
whose closures intersect in a set which has codimension 1 in $\phi(F)$.
Clearly $P_{12} := \bar{P_{1}} \cap \bar{P_{2}}$ is a convex polytope.
Call its affine span $S$.
Then for every principal subwall $G < F$ such that $\phi(G) \subset S$ and
$\phi(G) \cap P_{12} \ne \emptyset$,
$P_{12}$ lies entirely in one subchamber $(G,R)$ of $G$.
\end{lemma}
See Fig.\ \ref{fig:badchamber} for a picture of a situation
that this lemma rules out, where there are two subchambers
in the same wall separating $P_{1}$ and $P_{2}$.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\[ \xy/r1.8pc/:
(0,0)="1",
(3,0)="2",
(0,-1)="3",
(3.1,-1)="4",
(-.85,0.5)="5",
(-0.5,0.85)="6",
(2.5,0.85)="7",
(3.5,0.85)="8",
(1,-0.6)="9", "9"*{\scriptstyle P_1},
(1,0.6)="10", "10"*{\scriptstyle P_2},
(1.5,0)="11",
(4.2,-0.6)="15",
(4.2,0.6)="16",
"1"*{\bullet};
"3" **\dir{-},
"5" **\dir{-},
"6" **\dir{-},
"2"*{\bullet} **\dir{-};
"2" *{\bullet};
"4" **\dir{-},
"7" **\dir{-},
"8" **\dir{-},
"11" *{\bullet}
\endxy \]
\caption{Disallowed Meeting of Chambers.}
\label{fig:badchamber}
\end{center}
\end{figure}
\begin{proof}
Since $\phi(F)$ is an X-ray in its own right,
it is no loss of generality to assume that $M$ is connected, that
$F = M$, and $P_{1}, P_{2}$ are
chambers, intersecting in a principal wall.
Assume that $\bar{P_{1}}$ meets $\bar{P_{2}}$ in two or more
(principal) subchambers of $G$. Two of them must be separated by
a principal subwall $W$ of $G$. Now $G$ must have weights pointing in
the directions of both $P_{1}$ and $P_{2}$.
Hence by the consistency condition,
$W$ must have weights $\beta_{1}, \beta_{2}$
pointing out of $G$ in the directions
of $P_{1}$ and $P_{2}$, respectively.
But by the Darboux condition there is a principal subwall
of $F$ generated by the codimension $2$ wall $W$ and the weight
$\beta_{1}$, and likewise for $\beta_{2}$; these walls would break
up $P_{1}$ and $P_{2}$, giving a contradiction.
\end{proof}
We will be looking at the following situation. Consider a stratum
$F \in \F$, and two subchambers $(F,P_{1}),(F,P_{2})$
separated by principal subwalls $\phi(G_{1}),\dots ,\phi(G_{m})$,
which have a common affine span $S$.
By the lemma, in each wall $\phi(G_{i})$
there is a unique subchamber $(G_{i},R_{i})$ separating $P_{1}$ and $P_{2}$.
Note that since $\phi(G)$ is codimension $1$ in $\phi(F)$,
$\Aff(\phi(F)) \setminus S$ has two components $C_{1}$ and $C_{2}$,
containing $P_{1}$ and $P_{2}$ respectively. Denote the
map $\Lin(\phi(F)) \ra \Lin(\phi(F))/\Lin(S)$ by $\pi$.
Consider the weights of each $G_{i}$ which lie along the wall $F$, i.e.
\[
\alpha^{F}_{G_{i}} = \alpha_{G_{i}} \cap (\Lin(\phi(F))/\Lin(S)).
\]
The weights $\alpha^{F}_{G_{i}} \subset \Lin(\phi(F))/\Lin(S) \iso \R$
divide into three classes: those which are zero, those lying
in $\pi(C_{1})$, and those lying in $\pi(C_{2})$.
Let
\[
b_{i} = \# \{ \alpha_{G_{i},k} \mathrm{~lying~in~} \pi(C_{1}) \}
\]
and
\[
f_{i} = \# \{ \alpha_{G_{i},k} \mathrm{~lying~in~} \pi(C_{2}) \}.
\]
(We think of crossing the wall from $P_{1}$ to $P_{2}$ and $b_{i},f_{i}$ as
the number of weights pointing back and forward, respectively.)
In practice, when we determine these numbers from a picture of the
X-ray, we can use the consistency condition. We
choose any vertex $v_{i}$ belonging to $G_{i}$ and count the
weights at $v$ which lie in $\phi(F)$ and which lie on the $C_{1}$ side
or the $C_{2}$ side respectively.
We will show that the signature (and the Poincar\'e polynomial and the
Euler characteristic) define invariants of the following form.
\begin{Def}\label{rec:invdef}
A \textbf{recursive invariant} $I$ of a weighted X-ray $(\F,\phi,\alpha)$,
with values in a ring $R$, consists of two pieces of data:
\begin{enumerate}
\item a map $I: Q_{\F} \ra R$, and
\item a function $w_{I}: \N \times \N \ra R$, called
the \textit{wall-crossing function} of $I$.
\end{enumerate}
These must satisfy the following axioms for any wall $F \in \F$.
\begin{enumerate}
\item \label{rec:insidecond}
Given two subchambers $(F,P_{1}),(F,P_{2})$
separated by subchambers $(G_{i},R_{i})$,
let $b_{i},f_{i}$ be the number of weights of $G_{i}$ in $F_{i}$
pointing toward $P_{1}$ and toward $P_{2}$ respectively. Then
\begin{equation}\label{rec:invdefeq}
I(P_{2}) - I(P_{1}) = \sum_{i} w_{I}(f_{i},b_{i}) I(R_{i}).
\end{equation}
\item \label{rec:outsidecond}
Given a subchamber $(F,P)$ adjacent to the boundary of $F$,
let $(G,R)$ be the (necessarily unique) subchamber separating
$P$ from the exterior of $F$. Let $f$ be the number of
weights of $G$ in $F$ pointing into $F$ and $b$ be the number
pointing out. Then
\begin{equation}\label{rec:outsideeq}
I(P) = w_{I}(f,b) I(R).
\end{equation}
\end{enumerate}
\end{Def}
Condition \ref{rec:outsidecond} is best understood as a special case
of condition \ref{rec:insidecond},
where one of the ``chambers'' is the exterior of the
wall $F$; in this view, it is understood that we always assign $I=0$
to this ``chamber.''\footnote{We avoid actually calling
the exterior a chamber because it is neither
bounded nor convex; the trade-off is that we have to state
condition \ref{rec:outsidecond} explicitly.}
The values of a recursive invariant on a given X-ray
are completely determined by the function $w_{I}$
and the values of $I$ on the vertices.
For, the values of $I$ on the subchambers of a given dimension $k$
are determined by its values on the subchambers of dimension $k-1$,
along with $w_{I}$. This follows from the fact that we can get to any
subchamber of dimension $k$ by starting outside the polytope and
crossing a finite number of dimension $k-1$ walls.
In fact, this gives an algorithm for calculating $I$
on all of the subchambers of $\F$, starting from the vertices and
working our way up. We give two examples
in the case of the signature after Thm.\ \ref{rec:sigwallthm} below.
\subsection{Reduction to the Circle Case}\label{sec:circlereduction}
Note that any topological invariant $I$ of symplectic manifolds
defines a function on the subchambers of a Hamiltonian X-ray by
\begin{equation}\label{rec:topinvonxray}
I(F_{j},P) := I(M^{j}_{(q)}),\: \hbox{~for any~} q \in P.
\end{equation}
This invariant will be recursive in general if it is for circle
actions:
\begin{thm}\label{rec:circlereducthm}
Let $I$ be a topological invariant of symplectic manifolds. Assume that
the X-ray invariant defined by $I$ as in (\ref{rec:topinvonxray})
is recursive
on the class of X-rays coming from Hamiltonian
circle actions. Then
this invariant is recursive on all Hamiltonian X-rays, with the
same wall-crossing function.
\end{thm}
\begin{proof}
We will show condition \ref{rec:insidecond}; condition
\ref{rec:outsidecond} is very similar.
Let $(M,\omega,\phi)$ be a Hamiltonian $T^{d}$-space. Let
$(F,P_{1}),(F,P_{2})$ be adjacent subchambers of the X-ray of $M$.
We note that everything takes place inside the wall $F$,
which is an X-ray in its own right. Hence
we can assume that $M$ is connected and effective, $F=M$,
and $P_{1},P_{2}$ are chambers.
By Lemma \ref{rec:chamberoverlaplemma} the chambers $P_{1}, P_{2}$
meet in a number of
subchambers $(G_{j},R_{j})$ with common affine span $S$. Since the
walls $(G_{j},\phi(G_{j}))$ are principal walls, the $G_{j}$ are
fixed point components of circles $T_{j} \subset T$.
Let $a_{i} \in P_{i}$.
Choose a $(d-1)$-torus $H \subset T$ such that under the projection
$\pi: \ts \ra \hs$, $\pi(a_{1}) = \pi(a_{2})$. Let $L$ be the line
$\pi^{-1}(\pi(a_{i}))$.
Note that $L$ intersects the walls $\phi(G_{j})$ transversely.
In Fig.\ \ref{fig:stages} we illustrate the nonoverlapping case for
simplicity; the projection $\pi$ is in the vertical
direction.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\[ \xy/r3pc/:
(0.5,0.125)="1", "1"+/r5ex/*{W},
(3.5,0.875)="2",
(2,-2)="3",
(2,2)="4", "4"+/ul1ex/*{L},
(2,1.5)="5", "5"+/r2ex/*{a_{2}},
(2,-1)="6", "6"+/r2ex/*{a_{1}},
(1,1)="7", "7"*{P_{2}},
(1,-1)="8", "8"*{P_{1}},
(2,0.5)="9", "9"*{\times}, "9"+/dr2ex/*{q},
(0.1,-0.9)="10",
(0.25,1)="11",
(4,0)="12",
(3.6,2)="13",
"1"*{\bullet};
"10" **\dir{-},
"11" **\dir{-},
"2" **\dir{-}, "2" *{\bullet};
"12" **\dir{-},
"13" **\dir{-},
"5"*{\bullet};
"6"*{\bullet};
"3";
"4" **\dir{.},
\endxy \]
\caption{Reduction in Stages.}
\label{fig:stages}
\end{center}
\end{figure}
Form the symplectic reduction $M_{H} = \phi^{-1}(L)$.
This has a residual action of the circle $T/H$, and the reduced
spaces of $M_{H}$ by this circle action
at $a_{1}$ and $a_{2}$ are exactly the reduced spaces $M_{a_{1}}$ and
$M_{a_{2}}$. Polarize $T/H$ so that the positive direction points
from $a_{1}$ to $a_{2}$.
The circle action on $M_{H}$
has one fixed point component lying between $a_{1}$ and $a_{2}$
for each wall $G_{j}$: these are exactly
$(G_{j})_{red} = M^{j}_{(q)}$, where $q = L \cap \phi(G_{j})$.
The weights of the circle action at these fixed point components
are easily calculated.
In particular, the positive weights of the circle action
at $M^{j}_{(q)}$
are in bijection with the forward weights of $G_{j}$, and the
negative weights are bijective with the backward weights.
Hence if the invariant $I$ obeys the wall-crossing formula
(\ref{rec:invdefeq}) for circle actions, then considering wall-crossing
for $M_{H}$ gives
\begin{equation}\label{cob:recinvstageseq}
I(M_{a_{2}}) - I(M_{a_{1}}) = \sum_{j} w_{I}(f_{j},b_{j}) I(M^{j}_{(q)}).
\end{equation}
But by the definition of the recursive invariant associated to
a topological invariant, this amounts to
\begin{equation}\label{rec:invdefeqagain}
I(P_{2}) - I(P_{1}) = \sum_{j} w_{I}(f_{j},b_{j}) I(R_{j}),
\end{equation}
which is the general wall-crossing formula.
\end{proof}
Hence any topological invariant which has a formula
in terms of weight data like that of the signature---and hence a wall-crossing
formula for circle actions---will be a recursive invariant.
We show below that this applies not only to the signature, but also to
the Poincar\'e polynomial and the Euler characteristic.
\subsection{Application to the Signature}
\label{sec:recsig}
The signature is our first example of a recursive invariant.
Thm.\ \ref{rec:circlereducthm}
combined with the circle wall-crossing formula (\ref{cob:wallcrossforcircle})
immediately gives
\begin{thm}\label{rec:sigwallthm}
The signature $S(F_{j},P) = \Sign(M^{j}_{(q)})$, $q \in P$,
is a recursive invariant of Hamiltonian X-rays
with wall-crossing function
\begin{equation}\label{sigwcfagain}
w_{S}(f,b) = \left\{ \begin{array}{cr}
(-1)^{b} = -(-1)^{f}
& \mathrm{~if~} f+b \mathrm{~is~odd} \\
0 & \mathrm{~if~} f+b \mathrm{~is~even}
\end{array}
\right.
\end{equation}
\end{thm}
Note that the vertex function is just the signature
of the corresponding fixed point component,
\[
S(p_{j}) = \Sign(F_{j}) \text{~for~} p_{j} = \phi(F_{j}),
F_{j} \subset M^{T}.
\]
This and the wall-crossing function $w_{S}$ determine
the signature of all subreductions, and hence all regular reductions,
as discussed in the previous section.
\subsection{Simple Cases}\label{subsec:delzant}
In some cases, this recursive procedure collapses, making
the computation much simpler. First, we consider a trivial case.
Suppose $M$ is connected, the action of $T$ is effective,
and the dimension of $M$ is twice the rank of the torus.
Then the dimension of each regular reduced space is $0$,
and since a reduced space is connected, it is just a single
point. Hence the signature is always equal to $1$.
Such spaces are called \textit{symplectic toric varieties};
when they are smooth they are in fact the toric varieties of
complex algebraic geometry.
They are are characterized by the facts that they have isolated fixed points
and that their
X-rays have no internal structure, by the work of
Delzant \cite{Delzant:1988HP}. That is, the only walls
of the X-ray are the faces of the moment polytope.
Further, the moment polytopes of toric varieties are
\textit{simple}, meaning that the edges coming out of each
vertex form a basis for $\ts$.
Now consider a general Hamiltonian $T^{d}$-space $M$
and look at its X-ray. Suppose that $M$ is connected
and the action is effective, for simplicity.
Suppose that the fixed points of $M$ are isolated.
Consider a stratum $F_{j}$ giving a wall
$\phi(F_{j})$ of dimension $r$. If the wall $\phi(F_{j})$
has no internal structure, we know that any
corresponding subreduction $M^{j}_{q}$ is just a point,
so $\Sign(M^{j}_{q})=1$. Hence the recursive procedure
terminates early for this particular wall. We will
call such a wall, which corresponds to a toric
stratum $F$, a \textit{Delzant wall} of the X-ray.
This can save a great deal of work in applying the recursive procedure
for calculating the signature of a reduction $M_{a}$.
The simplest case is descibed by
\begin{lemma}\label{rec:whenalldelzantwalls}
Let $M$ be a connected, effective Hamiltonian $T^{d}$-space
with isolated fixed points,
and let $(\F,\phi,\alpha)$ be its weighted X-ray.
Then all principal walls of $\F$ are Delzant
exactly when for every vertex $p$, any $d$ weights at $p$ are independent.
\end{lemma}
\begin{proof}
First note that the condition that any $d$ weights are independent
is equivalent to saying that any $(d-1)$ weights are independent
and no other weight lies in their span.
By the Darboux condition, a codimension $1$ wall is generated
by a set of vectors which span a $(d-1)$-dimensional subspace
of $\ts$. If the wall is Delzant, then it is a simple polytope, hence
the weights in that subspace must be independent. So there must
be only $(d-1)$ of them. Since no other weight can lie in their
span, any $d$ weights will be independent.
Conversely, if any $d$ weights are independent, then the walls
are generated by independent sets of $(d-1)$ vectors, hence
are Delzant.
\end{proof}
In such a case the recursive procedure terminates in
codimension $1$, giving a simple wall-crossing formula.
\begin{prop}\label{rec:wallcrossdelzantisolated}
Let $M$ be a connected, effective Hamiltonian $T$-space
with isolated fixed points. Assume that all of the
principal walls are Delzant. Let $a_{1}$, $a_{2}$
be regular values of the moment map, separated by
walls $F_{1}$, \dots ,$F_{m}$, and associate
$b_{r},f_{r}$ to each wall as above. Then
\[
\Sign(M_{a_{2}}) - \Sign(M_{a_{1}})
= \sum_{r=1}^{m} w_{S}(f_{r},b_{r}).
\]
\end{prop}
\textit{Example.} Consider $\CP^{4}$ as a $T^{2}$-action, as in Sec.\
\ref{xraychap}. This space has isolated fixed points,
and all of the walls are Delzant (in particular, all multiplicities are
$1$). So the wall-crossing formula gives the signatures shown in Figure
\ref{fig:cobversioncp4}. For instance, we can calculate the
signature in the bottom chamber by crossing the bottom
wall from the outside. At either vertex of this wall, there are
three weights pointing up, none pointing down. Hence the difference
in signature is $W_{S}(3,0) = 1$. The other chambers are
calculated similarly.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\[ \xy/r4pc/:
(0,0)="1",
(4,0)="2",
(2,4)="3",
(1.6,0.75)="4",
(2.4,1.9)="5",
(1.3,1.8)="6", "6"*{1},
(2,2.3)="7", "7"*{0},
(2.65,2.05)="8", "8"*{1},
(1.35,0.85)="9", "9"*{0},
(1.8,1.3)="10", "10"*{-1},
(2.5,0.95)="11", "11"*{0},
(1.6,0.3)="12", "12"*{1},
"1"*{\bullet};
"2" **\dir{-}, "2" *{\bullet},
"3" **\dir{-}, "3" *{\bullet},
"4" **\dir{-}, "4" *{\bullet},
"5" **\dir{-}; "5" *{\bullet};
"2" **\dir{-},
"3" **\dir{-},
"4" **\dir{-};
"2" **\dir{-},
"3" **\dir{-};
"2" **\dir{-};
\endxy \]
\caption{Signatures of reduced spaces of $\CP^{4}$.}
\label{fig:cobversioncp4}
\end{center}
\end{figure}
This is not an isolated example. One can show (see \cite{Metzler:1997})
that if one starts with any symplectic toric variety and restricts
to a generic subtorus $T$, one obtains a Hamiltonian $T$-space
whose X-ray has only Delzant walls.
Now we turn to the non-Delzant case, where we do use the recursive
version of the formula.
\textit{Example.\/} Consider $\CP^{4}$ as a $T^{2}$-space once again,
but this time with a non-generic projection, with resulting X-ray
given in Figure \ref{fig:nongenericcp4}. One wall is now non-Delzant
(it is the projection of a $3$-simplex onto a line).
We could calculate the signatures in the chambers by crossing the
Delzant walls, but we will do it by crossing the non-Delzant
wall to show the idea of the recursive formula.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\[ \xy/r2pc/:
(0,0)="1", "1"+/dl2ex/*{p},
(4,0)="2", "2"+/ur2ex/*{ t},
(0,4)="3", "3"+/ur2ex/*{ q},
(1.5,2.5)="4", "4"+/ur2ex/*{ r},
(2.5,1.5)="5", "5"+/ur2ex/*{ s},
(-1,5)="6",
(5,-1)="7",
(2.5,0.5)="10", "10"*{\alpha},
(1.5,1.5)="11", "11"*{\beta},
(0.5,2.5)="12", "12"*{\gamma},
(3.5,1)="15", "15"*{ A},
(2.25,2.25)="16", "16"*{ B},
(1,3.5)="17", "17"*{ C},
"1"*{\bullet};
"2" **\dir{-}, "2" *{\bullet},
"3" **\dir{-}, "3" *{\bullet},
"4" **\dir{-}, "4" *{\bullet},
"5" **\dir{-}; "5" *{\bullet};
"2";
"3" **\dir{-},
"6";
"7" **\dir{.},
\endxy \]
\caption{X-ray of $\CP^4$ with non-Delzant wall}
\label{fig:nongenericcp4}
\end{center}
\end{figure}
First we need the values on the vertices; but since all of the
$T$-fixed points are isolated, $S(p) = S(q) = \dots = 1$.
Next look at the diagonal wall. This is exactly the X-ray considered
in the example of Section \ref{sec:circle}. Hence the signatures
of the reductions are
\[
S(A) = S(C) = 1, \: S(B) = 0.
\]
Now we can cross the subchambers $A,B,C$ to find the signatures in the
chambers $\alpha,\beta,\gamma$. Crossing from the exterior of
the polytope into $\alpha$
through $A$ gives
\[
S(\alpha) = w_{S}(0,1)\cdot S(A) = 1,
\]
and similarly $S(\gamma) = 1$. Crossing subchamber $B$ gives
no change in signature, since $S(B)=0$, so $S(\beta) = 0.$
It is easy to see that the results on the chambers accord with what
we get by crossing the Delzant walls
emanating from the point $p$.
\subsection{The Poincar\'e Polynomial and the Euler Characteristic}
\label{sec:euler}
Theorem \ref{rec:circlereducthm} suggests that we look for
recursive formulas for other topological invariants.
The cobordism approach which yielded Thm.\ \ref{rec:sigwallthm} will not
work for other invariants, because of Thm.\ \ref{cob:signatureisonly}.
However, a little Morse theory
leads to a wall-crossing formula for the Poincar\'e polynomial,
and hence for the Euler characteristic. We will also note an
intriguing connection to the signature formula.
Thm.\ \ref{rec:circlereducthm} shows
that any wall-crossing
result can be reduced to the case of a circle action. In that case, we can
derive a formula for the change in the Poincar\'e polynomial
as we cross a wall by the following procedure.
Let $(M,\omega,\phi)$ be a Hamiltonian $S^{1}$-space, with fixed
point components $F_{1},\dots F_{k}$. Identify $\mathrm{Lie}(S^{1})$
with $\R$ as usual. Pick one of the components
$F_{r}$, and let $\phi_{r} = \phi(F_{r})$. Assume for simplicity that
no other fixed point component is mapped under $\phi$ to $\phi_{r}$.
Let $a_{1},a_{2} \in \R$ be
regular values of $\phi$ such that $a_{1} < \phi_{r} < a_{2}$,
with no other $\phi(F_{j})$ lying between $a_{1}$ and $a_{2}$.
We follow the suggesstion of Tolman (personal communication)
in using the Morse theory of the square of two shifted moment maps, as
follows.
Kirwan \cite{FK:1984CQ} showed
that $\phi^{2}$ is an equivariantly perfect Morse
function.\footnote{Actually, $\phi^{2}$ is not necessarily Morse at
the critical value $0$, but $0$ is a minimum,
and hence the Morse theory still works. See \cite{FK:1984CQ} for a
detailed discussion.}
Hence so are $(\phi-a_{1})^{2}$ and $(\phi-a_{2})^{2}$,
since the shifted maps are also $S^{1}$ moment maps.
Note that the critical sets of $(\phi-a_{1})^{2}$ are the fixed points
of the action, plus the set $\phi^{-1}(a_{1})$; similarly for
$(\phi-a_{2})^{2}$.
Calculating the equivariant Poincar\'e series $\tilde{P}_M$
of the manifold $M$ gives\footnote{We prefer the notation $P_{M}(t)$ for
the Poincar\'e polynomial but we often use P(M)(t) for easier reading when
there are many subscripts.}
\begin{eqnarray}\label{poincarecalc}
\tilde{P}_{M}(t) &=& \sum_{F_{j}} P({F_{j}})(t) \frac{t^{\nu_{j}}}{1-t^{2}}
+ \tilde{P}({\phi^{-1}(a_{1})})(t)\\
&=& \sum_{F_{j}} P({F_{j}})(t) \frac{t^{\nu'_{j}}}{1-t^{2}}
+ \tilde{P}({\phi^{-1}(a_{2})})(t)\\
\end{eqnarray}
where $\nu_{j}$ and $\nu'_{j}$ are the indices (the dimensions
of the negative normal bundles) of the fixed point set $F_{j}$
with respect to the Morse functions $(\phi-a_{1})^{2}$ and $(\phi-a_{2})^{2}$
respectively. Note that since $S^{1}$ acts freely on the level
sets $\phi^{-1}(a_{1})$ and $\phi^{-1}(a_{2})$, we have
$\tilde{P}(\phi^{-1}(a_{1})) = P(M_{a_{1}})$ and similarly for $a_{2}$.
Since only the one fixed point set $F_{r}$ lies between the levels
$a_{1}$ and $a_{2}$, we have
\[
\nu_{j} = \nu'_{j} \hbox{~for~} j \ne r
\]
Furthermore, let $f$ be the number of positive weights in the
normal bundle to $F_{r}$ and $b$ be the number of negative weights.
Then $\nu_{r} = 2f$ and $\nu'_{r} = 2b$. (Each complex weight
gives two real dimensions.) When we equate the two right hand sides
in (\ref{poincarecalc}) above, all of the terms cancel except for
\begin{equation}\label{poincareresult}
P({M_{a_{2}}})(t) - P({M_{a_{1}}})(t) =
P({F_{r}})(t) \frac{t^{2b}-t^{2f}}{1-t^{2}}.
\end{equation}
It is easy to see that if there are multiple fixed point
components $F_{1},\dots ,F_{m}$
with the same value of the
moment map, we get
\begin{equation}\label{poincareresultoverlap}
P({M_{a_{2}}})(t) - P({M_{a_{1}}})(t) =
\sum_{r=1}^{m} P({F_{r}})(t)
\frac{t^{2b_{r}}-t^{2f_{r}}}{1-t^{2}}.
\end{equation}
This is the wall-crossing formula for the circle case.
Hence we immediately get a wall-crossing formula for a general
torus action, by Thm.~\ref{rec:circlereducthm}:
\begin{thm}\label{poincarewall}
The X-ray invariant defined by the Poincar\'e polynomial,
\[
P(F_{j},R) = P({M^{j}_{(q)}}) \in \Z[t], \quad q \in R,
\]
is a recursive invariant of Hamiltonian X-rays
with wall-crossing function
\begin{eqnarray}\label{rec:poincarewalleq}
w_{P}(b,f) &=& \frac{t^{2b}-t^{2f}}{1-t^{2}}\\
&=& t^{2f-2} + t^{2f-4}+ \dots + t^{2b} \:\: (f > b)\\
&=& -t^{2b-2} - t^{2b-4}+ \dots - t^{2f} \:\: (b > f).
\end{eqnarray}
\end{thm}
We get a wall-crossing formula for the Euler characteristic by setting
$t=-1$:
\begin{cor}\label{rec:eulerwall}
The Euler characteristic $\chi$ defines a recursive invariant
with wall-crossing function
\begin{equation}\label{rec:eulerwalleq}
w_{\chi}(b,f) = b-f.
\end{equation}
\end{cor}
\textit{Example.\/}
Consider again the X-ray of the nongeneric $2$-torus action on $\CP^{4}$
in Figure \ref{fig:nongenericcp4}. Once again we will start
with the non-Delzant wall to see how the recursive formula for
$P$ works. We evidently have $P(p) = P(q) = \dots = 1$.
Looking at the diagonal wall, we cross $t$ to go from the outside of the
wall to subchamber $A$. The wall-crossing formula gives
\[
P(A) = w_{P}(0,3) P(t) = 1 + t^{2} + t^{4}.
\]
(Note that this is $P(\CP^{2})$, as expected from the discussion on
p. \pageref{cob:examplecp3circle}.) Crossing from $A$ into $B$
through $s$ gives
\[
P(B) = P(A) + w_{P}(1,2) P(s) = 1 + t^{2} + t^{4} + t^{2}.
\]
Crossing from $B$ into $C$ gives
\begin{align*}
P(C) = P(B) + w_{P}(2,1) P(r) &= 1 + 2 t^{2} + t^{4} - t^{2}\\
&= 1 + t^{2} + t^{4}
\end{align*}
as we would expect from symmetry.
Now we cross the subchambers $A,B,C$ into the chambers
$\alpha,\beta,\gamma$. Crossing from the exterior of
the polytope into $\alpha$
through $A$ gives
\[
P(\alpha) = w_{P}(0,1) P(A) = 1 + t^{2} + t^{4},
\]
and similarly $P(\gamma) = 1 + t^{2} + t^{4}$.
Crossing subchamber $B$ gives
$P(\beta) = 1 + 2 t^{2} + t^{4}.$
Note that the wall-crossing functions (\ref{rec:eulerwalleq})
for the Euler characteristic and (\ref{sigwcfagain}) for
the signature agree modulo $2$. This accords with the general
fact that $\Sign(M) \equiv \chi(M) \mod 2$ for all compact
oriented manifolds (\cite{HirBerJung:1992}).
It is an interesting fact (see \cite{Metzler:1997})
that the signature of a toric variety
is determined by its Poincar\'e polynomial---in fact
\[
\Sign(V) = P_{V}(i) = P_{V}(\sqrt{-1})
\]
for $V$ a toric variety.
(This comes from the fact that the Hodge numbers $h^{p,q}$ vanish
for $p \ne q$.)
Comparing the wall-crossing formulas for the signature and the
Poincar\'e polynomial shows that as far as the wall-crossing
function is concerned, this still holds:
\[
w_{S}(b,f) = w_{P}(b,f)(i).
\]
Since any recursive invariant is determined by its wall-crossing
function and its values on vertices, we have the
\begin{thm}\label{eul:sigpoincare}
Let $(M,\omega,\phi)$ be a Hamiltonian $T$-space. Assume that
the fixed point components $F \subset M^{T}$ are toric varieties. Then
\[
\Sign(M^{j}_{(q)}) = P({M^{j}_{(q)}})(i)
\]
for every $q \in \phi(F_{j})$, and in particular,
\[
\Sign(M_{a}) = P_{M_{a}}(i)
\]
for every $a \in \Delta_{reg}$.
\end{thm}
For example, if $M$ has isolated fixed points, the hypotheses of the theorem
are obviously satisfied.
Here is a simple consequence. Suppose that $N$ is a compact, connected
$4$-dimensional
orbifold that can be expressed as a reduced space for a Hamiltonian
torus action with isolated fixed points. Then
\begin{align}\label{rec:4dim}
P_{N}(t) &= 1 + b_{1} t + b_{2} t^{2} + b_{1} t^{3} + t^{4} \\
\Sign(N) = P_{N}(i) &= 2 - b_{2}.
\end{align}
If the diagonalized intersection form of $N$ has $p$ positive entries
and $n$ negative ones, we see that
\[
\Sign(N) = p - n = 2 - b_{2}
\]
but $b_{2} = p + n$, so $p-n = 2 - p - n$ or
\[
p = 1.
\]
Hence any such $4$-manifold has exactly one 2-class with
positive self-intersection.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Home » Flossie and the Fox (Hardcover)
Flossie and the Fox (Hardcover)
By Patricia McKissack, Rachel Isadora (Illustrator)
A wily fox, notorious for stealing eggs, meets his match when he encounters a bold little girl in the woods who insists upon proof that he is a fox before she will be frightened.
Rachel Isadora began dancing at the age of eight. She trained at George Balanchine's School of American Ballet and has danced professionally. Rachel lives in New York City with her family.
Rachel Isadora has illustrated many books set in the world of dance and theater, including Opening Night, My Ballet Class, Swan Lake, The Little Match Girl, and Ben's Trumpet, which received the Caldecott Honor Award and the Boston Globe-Horn Book Honor Award.
copyright © 2000 by Penguin Putnam Books for Young Readers. All rights reserved.
Juvenile Fiction / Fairy Tales & Folklore / Adaptations
Juvenile Fiction / Animals / Foxes
Juvenile Fiction / Humorous Stories
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,833
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using BravoLights.Common;
using BravoLights.Common.Ast;
using BravoLights.Connections;
namespace BravoLights.Ast
{
class SimVarExpression : VariableBase, IAstNode
{
public readonly NameAndUnits NameAndUnits;
public SimVarExpression(string simVarName, string units)
{
NameAndUnits = new NameAndUnits { Name = simVarName, Units = units };
}
public override string ToString()
{
return Identifier;
}
protected override IConnection Connection => SimConnectConnection.Connection;
public override string Identifier => $"A:{this.NameAndUnits.Name}, {this.NameAndUnits.Units}";
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
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{"url":"https:\/\/zbmath.org\/?q=an:1302.30053","text":"zbMATH \u2014 the first resource for mathematics\n\nPeriods of second kind differentials of $$(n,s)$$-curves. (English) Zbl\u00a01302.30053\nTrans. Mosc. Math. Soc. 2013, 245-260 (2013) and Tr. Mosk. Mat. O.-va 74, No. 2, 297-315 (2013).\nSummary: Elliptic curves expressions for the periods of elliptic integrals of the second kind in terms of theta-constants have been known since the middle of the 19th century. In this paper we consider the problem of generalizing these results to curves of higher genera, in particular to a\u00a0special class of algebraic curves, the so-called $$(n,s)$$-curves. It is shown that the representations required can be obtained by the comparison of two equivalent expressions for the projective connection, one due to Fay-Wirtinger and the other from Klein-Weierstrass. As a\u00a0principle example, we consider the case of the genus two hyperelliptic curve, and a\u00a0number of new Thomae and Rosenhain type formulae are obtained. We anticipate that our analysis for the genus two curve can be extended to higher genera hyperelliptic curves, as well as to other classes of $$(n,s)$$ non-hyperelliptic curves.\n\nMSC:\n 30F30 Differentials on Riemann surfaces 14H50 Plane and space curves\nFull Text:\nReferences:\n [1] Ayano T., Nakayashiki A. On addition formulae for sigma functions of telescopic curves. arXiv:1303.2878 [math.AG] 17 pp. 2012. \u00b7 Zbl\u00a01273.14072 [2] H. F. Baker, Abelian functions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Abel\u2019s theorem and the allied theory of theta functions; Reprint of the 1897 original; With a foreword by Igor Krichever. \u00b7 Zbl\u00a00848.14012 [3] Baker H.F. Multiply periodic functions. Cambridge Univ. Press, 1907. \u00b7 JFM\u00a038.0478.05 [4] S. Baldwin, J. C. Eilbeck, J. Gibbons, and Y. \u00d4nishi, Abelian functions for cyclic trigonal curves of genus 4, J. Geom. Phys. 58 (2008), no. 4, 450 \u2013 467. \u00b7 Zbl\u00a01211.37082 \u00b7 doi:10.1016\/j.geomphys.2007.12.001 \u00b7 doi.org [5] Arthur Erd\u00e9lyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. III, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. Based, in part, on notes left by Harry Bateman. \u00b7 Zbl\u00a00064.06302 [6] Bolza O. Ueber die Reduction hyperelliptischer Integrale erster Ordnung und erster Gattung auf elliptische durch eine Transformation vierten Grades, Math. Ann. 1886. Vol.XXVIII. S.447-456. [7] H. W. Braden, V. Z. Enolski, and Yu. N. Fedorov, Dynamics on strata of trigonal Jacobians and some integrable problems of rigid body motion, Nonlinearity 26 (2013), no. 7, 1865 \u2013 1889. \u00b7 Zbl\u00a01273.70009 \u00b7 doi:10.1088\/0951-7715\/26\/7\/1865 \u00b7 doi.org [8] Harry W. Braden, Victor Z. Enolskii, and Andrew N. W. Hone, Bilinear recurrences and addition formulae for hyperelliptic sigma functions, J. Nonlinear Math. Phys. 12 (2005), no. suppl. 2, 46 \u2013 62. \u00b7 Zbl\u00a01126.11007 \u00b7 doi:10.2991\/jnmp.2005.12.s2.5 \u00b7 doi.org [9] V. M. Bukhshtaber and D. V. Le\u012dkin, Addition laws on Jacobians of plane algebraic curves, Tr. Mat. Inst. Steklova 251 (2005), no. Neline\u012dn. Din., 54 \u2013 126 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 4(251) (2005), 49 \u2013 120. [10] V. M. Bukhshtaber and D. V. Le\u012dkin, Solution of the problem of the differentiation of abelian functions with respect to parameters for families of (\\?,\\?)-curves, Funktsional. Anal. i Prilozhen. 42 (2008), no. 4, 24 \u2013 36, 111 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 42 (2008), no. 4, 268 \u2013 278. \u00b7 Zbl\u00a01156.14315 \u00b7 doi:10.1007\/s10688-008-0040-4 \u00b7 doi.org [11] Buchstaber V.M., Enolski V.Z., Leykin D.V. $$\\sigma$$-functions of $$(n,s)$$-curves, Russ. Math. Surv. Vol.54, No. 3. 1999. p.628-630. [12] V. M. Bukhshtaber, D. V. Le\u012dkin, and V. Z. \u00c8nol$$^{\\prime}$$ski\u012d, Rational analogues of abelian functions, Funktsional. Anal. i Prilozhen. 33 (1999), no. 2, 1 \u2013 15, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 33 (1999), no. 2, 83 \u2013 94. \u00b7 Zbl\u00a01056.14049 \u00b7 doi:10.1007\/BF02465189 \u00b7 doi.org [13] Buchstaber V.M., Enolski V.Z., Leykin D.V. Multi-Dimensional Sigma-Functions. arXiv:1208.0990 [math-ph]. 2012. 267 p. [14] Victor Enolski, Betti Hartmann, Valeria Kagramanova, Jutta Kunz, Claus L\u00e4mmerzahl, and Parinya Sirimachan, Inversion of a general hyperelliptic integral and particle motion in Ho\u0159ava-Lifshitz black hole space-times, J. Math. Phys. 53 (2012), no. 1, 012504, 35. \u00b7 Zbl\u00a01273.83099 \u00b7 doi:10.1063\/1.3677831 \u00b7 doi.org [15] Eilbeck J.C., Gibbons J., \u00d4nishi Y., Previato E. From equations of Jacobians or Kummer varieties to Coble hypersurfaces. [16] J. C. Eilbeck, V. Z. Enolski, S. Matsutani, Y. \u00d4nishi, and E. Previato, Abelian functions for trigonal curves of genus three, Int. Math. Res. Not. IMRN 1 (2008), Art. ID rnm 140, 38. \u00b7 Zbl\u00a01210.14032 \u00b7 doi:10.1093\/imrn\/rnm140 \u00b7 doi.org [17] M. England and J. C. Eilbeck, Abelian functions associated with a cyclic tetragonal curve of genus six, J. Phys. A 42 (2009), no. 9, 095210, 27. \u00b7 Zbl\u00a01157.14303 \u00b7 doi:10.1088\/1751-8113\/42\/9\/095210 \u00b7 doi.org [18] Matthew England, Higher genus abelian functions associated with cyclic trigonal curves, SIGMA Symmetry Integrability Geom. Methods Appl. 6 (2010), Paper 025, 22. \u00b7 Zbl\u00a01188.14019 \u00b7 doi:10.3842\/SIGMA.2010.025 \u00b7 doi.org [19] Hershel M. Farkas and Irwin Kra, Riemann surfaces, Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York-Berlin, 1980. \u00b7 Zbl\u00a00764.30001 [20] John D. Fay, Theta functions on Riemann surfaces, Lecture Notes in Mathematics, Vol. 352, Springer-Verlag, Berlin-New York, 1973. \u00b7 Zbl\u00a00281.30013 [21] Dzh. Kharnad and V. Z. \u00c8nol$$^{\\prime}$$ski\u012d, Schur function expansion of Kadomtsev-Petviashvili \\?-functions associated with algebraic curves, Uspekhi Mat. Nauk 66 (2011), no. 4(400), 137 \u2013 178 (Russian, with Russian summary); English transl., Russian Math. Surveys 66 (2011), no. 4, 767 \u2013 807. \u00b7 Zbl\u00a01231.14025 \u00b7 doi:10.1070\/RM2011v066n04ABEH004755 \u00b7 doi.org [22] Samuel Grushevsky and Riccardo Salvati Manni, Two generalizations of Jacobi\u2019s derivative formula, Math. Res. Lett. 12 (2005), no. 5-6, 921 \u2013 932. \u00b7 Zbl\u00a01093.14040 \u00b7 doi:10.4310\/MRL.2005.v12.n6.a12 \u00b7 doi.org [23] Felix Klein, Ueber hyperelliptische Sigmafunctionen, Math. Ann. 32 (1888), no. 3, 351 \u2013 380 (German). \u00b7 JFM\u00a020.0491.01 \u00b7 doi:10.1007\/BF01443606 \u00b7 doi.org [24] Felix Klein, Zur Theorie der Abel\u2019schen Functionen, Math. Ann. 36 (1890), no. 1, 1 \u2013 83 (German). \u00b7 JFM\u00a022.0498.01 \u00b7 doi:10.1007\/BF01199432 \u00b7 doi.org [25] Komeda J., Matsutani Sh., Previato E. The sigma function for Weierstrass semigroups $$\\langle 3,7,8\\rangle$$ and $$\\langle 6,13,14,15,16\\rangle$$. arXiv:1303.0451 [math-ph], 2012. \u00b7 Zbl\u00a01284.14045 [26] D. Korotkin and V. Shramchenko, On higher genus Weierstrass sigma-function, Phys. D 241 (2012), no. 23-24, 2086 \u2013 2094. \u00b7 Zbl\u00a01262.14033 \u00b7 doi:10.1016\/j.physd.2012.01.002 \u00b7 doi.org [27] Shigeki Matsutani and Emma Previato, Jacobi inversion on strata of the Jacobian of the \\?_\\?\\? curve \\?^\\?=\\?(\\?), J. Math. Soc. Japan 60 (2008), no. 4, 1009 \u2013 1044. \u00b7 Zbl\u00a01160.14018 [28] A. I. Markushevich, Introduction to the classical theory of abelian functions, Translations of Mathematical Monographs, vol. 96, American Mathematical Society, Providence, RI, 1992. Translated from the 1979 Russian original by G. Bluher. \u00b7 Zbl\u00a00743.14033 [29] Matsutani Sh. Sigma functions for a space curve $$(3, 4, 5)$$ type with an appendix by J. Komeda. arXiv:1112.4137 [math-ph], 2012. [30] Atsushi Nakayashiki, Sigma function as a tau function, Int. Math. Res. Not. IMRN 3 (2010), 373 \u2013 394. \u00b7 Zbl\u00a01197.14049 \u00b7 doi:10.1093\/imrn\/rnp135 \u00b7 doi.org [31] Atsushi Nakayashiki, On algebraic expressions of sigma functions for (\\?,\\?) curves, Asian J. Math. 14 (2010), no. 2, 175 \u2013 211. \u00b7 Zbl\u00a01214.14028 \u00b7 doi:10.4310\/AJM.2010.v14.n2.a2 \u00b7 doi.org [32] Rosenhain G. Abhandlung \u00fcber die Functionen zweier Variablen mit fier Perioden welche die Inversion sind der ultra-elliptische Integrale erster Klasse. Translation to German from Latin manuscript published in 1851. Ostwald Klassiker der Exacten Wissenschaften, Nr. Vol.65. Leipzig: Verlag von Wilhelm Engelmann, 1895. S.1-96. [33] Wilhelm Wirtinger, Integrale dritter Gattung und linear polymorphe Funktionen, Monatsh. Math. Phys. 51 (1944), 101 \u2013 114 (German). \u00b7 Zbl\u00a00061.15610\nThis reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.","date":"2021-01-22 07:40:08","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\": 0.6012129187583923, \"perplexity\": 4644.5140777822335}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610703529128.47\/warc\/CC-MAIN-20210122051338-20210122081338-00706.warc.gz\"}"}
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Genetic Variants Associated with IBD Alter Immune Regulatory Signals from Beneficial Microbes
3615 Press Highlights 5 years ago Kristine Novak 0
Polymorphisms in susceptibility genes appear to promote development of inflammatory bowel diseases (IBD) by altering the abilities of immune cells to sense protective signals from the microbiome, researchers report. These findings help fill the missing link between genetic risk variants for IBD and dysbiosis of the intestinal microbiome.
More than 200 genes have been implicated as having a role in determining susceptibility to Crohn's disease. Researchers have believed that these genes regulate detection of pathogenic bacteria and the immune response against them. The model has been that when these genes are defective, the pathogenic bacteria survive, multiply in the gut, and lead to IBD.
In Science online May 5, Hiutung Chu et al show that alterations in genes that regulate detection of beneficial bacteria, which promote immune regulation, can also contribute to IBD pathogenesis.
B fragilis and intestinal epithelial cells shown as an overview tomogram. The modeled bacterium is indicated with a black box (left, fig. A). Bacterial releas of outer membrane vesicles (right, fig. B), regulates the immune response via genetic pathways linked to Crohn's disease. Green, outer membrane; light blue, inner membrane; pink, ribosomes; gold; outer membrane vesicles. Credit: Mark Ladinsky/Greg Donaldson/Caltech
"Typically, the signals from these beneficial commensal microbes promote anti-inflammatory responses that dampen inflammation in the gut," Chu explained to Science Daily. The authors found that mutations in genes that regulate these signals can reduce this anti-inflammatory response.
Chu et al knew that the human commensal Bacteroides fragilis delivers immunomodulatory molecules to immune cells via secretion of outer membrane vesicles. They found that these vesicles require products of the IBD-associated genes, ATG16L1 and NOD2, to activate an autophagy pathway that protects mice from colitis.
The authors found that ATG16L1-deficient dendritic cells did not induce T-regulatory cells (Treg cells) to suppress mucosal inflammation. Immune cells from human subjects with a major IBD risk variant in ATG16L1 had defective Treg-cell responses to the outer membrane vesicles and no anti-inflammatory response to B fragilis.
The authors propose that the faulty versions of these genes contribute to development of Crohn's disease in 2 different ways: by being unable to assist in destroying pathogenic bacteria and by preventing the beneficial immune signals usually elicited by good bacteria.
These findings improve our understanding of the relationship between the genome and the microbiome, and provide important information for therapeutic strategies. For example, certain populations of patients would not benefit from probiotic treatment with B fragilis because they have genetic variants that block its beneficial, immune-regulatory effects.
anti-inflammatory response
ATG16L1
B fragilis
immune regulation
membrane vesicle
regulator T cell
T cell
Treg cell
How Could Variants in TM6SF2 Affect Risk for NAFLD?
LARS vs PPIs for Treatment of GERD?
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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\section{Introduction}
So far, complete $(n\geq4)$-jet production via single-parton scattering (SPS) was
discussed only within collinear factorization. Results up to next-to-leading (NLO)
precision can be found in \cite{Bern:2011ep,Badger:2012pf}.
Here we wish to discuss for the first time production of four jets within high-energy ($k_T$-)factorization (HEF)
approach with 2 $\to$ 4 subprocesses with two
off-shell partons. Recently three of us have discussed another reaction with
2 $\to$ 4 ($g g \to c \bar c c \bar c$) subprocess in the framework
of the HEF \cite{vanHameren:2015wva}.
For the four-jet production the number of subprocesses is much higher.
Double-parton scattering (DPS) was claimed to have been observed for
the first time at the Tevatron \cite{Abe:1993rv}.
In the LHC era, with much higher collision energies available,
the field has received a new impulse and
several experimental and theoretical studies address the problem
of pinning down DPS effects (for review see \cite{Diehl:2011yj,Bansal:2014paa}).
Even just from purely theoretical point of view, the problem is
quite subtle.
As for the non perturbative side, it is in principle necessary, when
considering a double-parton scattering, to take into account
the correlations between the two partons coming from the same protons
and involved in the scattering processes.
Such an information should be encoded in a set of double parton
distribution functions (DPDFs), generalising usual parton distribution functions (PDFs).
A benchmarking work on DPDFs was made in
Ref.~\cite{Gaunt:2009re}, where a proper generalisation of the DGLAP
evolution equations to DPDFs was provided.
Building explicit initial conditions for the evolution equations
is challenging. Some successful attempts are becoming
to appear only recently
\cite{Golec-Biernat:2015aza,Rinaldi:2014ddl,Broniowski:2016trx}.
In the meanwhile, phenomenological and experimental studies of double-parton scattering
rely on factorized Ansatz for the DPDFs, which amount
to neglecting longitudinal momentum correlations
between partons and treating transversal ones by introducing an
effective cross section, $\sigma_{eff}$.
The latter quantity is usually extracted from experimental data.
In the present approach we will use the factorized Ansatz and
concentrate on the difference between leading-order collinear and
high-energy-factorization results. The latter includes effectively
higher-order corrections.
For most of high-energy reactions the single-parton scattering dominates
over the double-parton scattering. The extraordinary example
is double production of $c \bar c$ pairs \cite{Luszczak:2011zp,Maciula:2013kd}.
For four-jet production, disentangling the ordinary SPS
contributions from the DPS corrections can be quite challenging for
several reasons: first of all, it is
necessary to define sufficiently sensitive, process-dependent obervables,
w.r.t. which the DPS differential cross section manifestly dominates at
least in some corners of phase space \cite{Berger:2009cm,Maciula:2015vza}.
Nevertheless, even once this is done, one has to be careful about the
kinematical regime employed in comparing experimental data to
theoretical prediction: in fact, the generally decreasing
behaviour of PDFs for large momentum fractions \cite{Martin:2009iq} is well known,
particularly for gluons, and gluon-initiated processes account for a very large part of the cross section;
this implies that, for very energetic final states (characterized by large transverse
momenta), it is really unlikely to get contributions from DPS.
This is confirmed very well experimentally by the data released by the
ATLAS Collaboration for both the 7 and 8 TeV runs \cite{Aad:2011tqa,Aad:2015nda}.
This problem is of course slightly tamed by providing high center of
mass energy in hadron-hadron scattering, as moderately
low values of $x$ should be enough to guarantee observing DPS in a
kinematic regime in which perturbative QCD, possibly supplemented
by parton showering, still works reasonably well.
In this paper we propose to assess the predictions of HEF for double-parton scattering at the LHC in a leading-order (LO) framework.
HEF is an approach introduced in the early 90's in the context of
heavy-flavour production, in order to take into account the effect of the colliding parton transverse momentum,
which is neglected in the collinear approach \cite{Collins:1991ty,Catani:1990eg,Catani:1994sq}.
This implies using off-shell partons, for which the construction of
gauge-invariant scattering amplitudes is not straightforward. However,
recent improvements in the understanding of scattering amplitudes
have allowed to formulate efficient analytical and numerical algorithms
for the computation of such objects
\cite{vanHameren:2013csa,vanHameren:2012if,Kotko:2014aba,vanHameren:2014iua,vanHameren:2015bba,
Cruz-Santiago:2015nxa,Cruz-Santiago:2015dla,Kotko:2016qxv}.
With such a machinery, we expand the analysis
presented in Ref.~\cite{Maciula:2015vza} and assess
the differences between the pure collinear approach and the high-energy
factorization (HEF) called also $k_T$-factorization framework.
We shall focus on the difference between predictions of HEF
and standard collinear approach for the DPS contribution.
\section{Single-parton scattering production of four jets}%
The collinear factorization formula for the calculation of the inclusive partonic 4-jet cross section at the Born level reads
\begin{eqnarray}
\sigma^B_{4-jets}
&=&
\sum_{i,j} \int \frac{dx_1}{x_1}\,\frac{dx_2}{x_2}\, x_1f_i(x_1,\mu_F)\, x_2f_j(x_2,\mu_F) \nonumber \\
&&
\times \frac{1}{2 \hat{s}} \prod_{l=i}^4 \frac{d^3 k_l}{(2\pi)^3 2 E_l} \Theta_{4-jet}\, (2\pi)^4\, \delta\left( x_1P_1 + x_2P_2 - \sum_{l=1}^4 k_i \right)\,
\overline{\left| \mathcal{M}(i,j \rightarrow 4\, \text{part.}) \right|^2} \, . \nonumber \\
\label{coll_cross}
\end{eqnarray}
Here $x_{1,2}f_i(x_{1,2},\mu_F)$ are the collinear PDFs
for the $i-th$ parton, carrying $x_{1,2}$ momentum fractions of the proton and evaluated at the factorization scale $\mu_F$;
the index $l$ runs over the four partons in the final state, the
partonic center of mass energy squared is $\hat{s} = 2\,x_1 x_2\, P_i \cdot P_j$;
the function $\Theta_{4-jet}$ takes into account the kinematic cuts applied and
$\mathcal{M}$ is the partonic on-shell matrix element, which includes symmetrization effects due to identity of particles in the final state.
Switching to HEF, the analogous formula to (\ref{coll_cross}) looks as follows:
\begin{eqnarray}
\sigma^B_{4-jets}
&=&
\sum_{i,j} \int \frac{dx_1}{x_1}\,\frac{dx_2}{x_2}\, d^2 k_{T1} d^2 k_{T2}\, \mathcal{F}_i(x_1,k_{T1},\mu_F)\, \mathcal{F}_j(x_2,k_{T2},\mu_F) \nonumber \\
&&
\hspace{-25mm}
\times \frac{1}{2 \hat{s}} \prod_{l=i}^4 \frac{d^3 k_l}{(2\pi)^3 2 E_l} \Theta_{4-jet} \, (2\pi)^4\, \delta\left( x_1P_1 + x_2P_2 + \vec{k}_{T\,1}+ \vec{k}_{T\,2} - \sum_{l=1}^4 k_i \right)\,
\overline{ \left| \mathcal{M}(i^*,j^* \rightarrow 4\, \text{part.})
\right|^2 } \, . \nonumber \\
\label{kt_cross}
\end{eqnarray}
Here $\mathcal{F}_i(x_k,k_{Tk},\mu_F)$ is a transverse momentum
dependent (TMD) distribution function for a given type of parton.
Similarly as in the collinear factorization case, $x_k$ is the longitudinal
momentum fraction, $\mu_F$ is a factorization scale.
The new degrees of freedom are introduced via $\vec{k}_{Tk}$,
which are the parton's transverse momenta,
i.e. the momenta perpendicular to the collision axis.
The formula is valid when the $x$'s are not too large and not too small when complications from nonlinearities may eventually arise \cite{Kotko:2015ura}\footnote{ For some processes High Energy Factorization can has been shown valid at NLO accuracy \cite{Chirilli:2011km,Nefedov:2015ara}}.
The TMD parton densities (for a recent review see \cite{Angeles-Martinez:2015sea})
can be defined by introducing operators whose expectation values, roughly speaking, count the number of partons \cite{Collins:2011zzd}.
In particular, an evolution equation for TMDs known as CCFM, valid both in the low $x$ and large $x$ domain,
\cite{Marchesini:1994wr,Catani:1989sg} provides a gluon density depending on $x,\,k_{T},\,\mu$.
However, for our purposes this is not enough, since we want to have access to moderate values
of $x$ where the CCFM approach needs refinements \cite{Hautmann:2014uua,Gituliar:2015agu}.
The alternative is to use the Kimber-Martin-Ryskin (KMR) prescription \cite{Kimber:2001sc,Kimber:1999xc}
in order to obtain a full set of TMD parton densities.
The basic observation is that the $k_T$ dependence can be generated
at the very last step of the collinear evolution by performing soft gluon resummation between two scales
given by $k_T$ and $\mu_F$, where $k_{T}$ is interpreted as the transverse momentum
of the hardest emitted gluon during the partonic evolution, while $\mu_F$ can be linked to hard scattering scale.
In practical terms, this procedure boils down to applying the Sudakov form factor onto the PDFs (some details can be found in Appendix \ref{App_TMDs}
\footnote{The TMDs can be obtained by request from krzysztof.kutak@ifj.edu.pl}).
$\mathcal{M}(i^*,j^* \rightarrow 4\, \text{part.})$ is the gauge invariant matrix element for $2\rightarrow 4$ particle scattering with two initial off-shell legs.
In the case of HEF (for recent review see Ref.~\cite{Sapeta:2015gee}), amplitudes with external off-shell legs in QCD have been computed with different approaches:
up to $2 \rightarrow 2$ scattering, they are given for example in \cite{Nefedov:2013ywa} and are enough in order to calculate DPS contributions (see section \ref{MPI}).
In order to move on to higher multiplicities, which are necessary for the SPS analysis of $2 \rightarrow 4$ parton scattering, it is possible to generate this amplitudes analytically
applying suitably defined Feynman rules \cite{vanHameren:2012if,vanHameren:2013csa}.
Also recursive methods have been developed for this purpose, like generalised BCFW recursion \cite{vanHameren:2014iua,vanHameren:2015bba}
or Wilson lines approaches \cite{Kotko:2014aba,Cruz-Santiago:2015nxa,Kotko:2016qxv}.
By now also a numerical package implementing numerical BCFW recursion is available \cite{Bury:2015dla}.
In this case, we rely on a numerical approach implemented in AVHLIB\footnote{available for download at https://bitbucket.org/hameren/avhlib}
which employs Dyson-Schwinger recursion generalized to tree-level amplitudes with off-shell initial-state particles.
Originally proposed in~\cite{Berends:1987me,Caravaglios:1995cd}, this recursive method exists in several explicit implementations with on-shell initial-state particles~\cite{Kanaki:2000ey,Mangano:2002ea,Moretti:2001zz,Gleisberg:2008fv,Kleiss:2010hy}, and has even been extended to one-loop amplitudes~\cite{vanHameren:2009vq,Actis:2012qn}.
AVHLIB and the Monte Carlo program therein are also used to perform the phase-space integration.
In the collinear case, results were cross-checked by comparing them with the ALPGEN output \cite{Mangano:2002ea}.
We use a running $\alpha_s$ provided with the MSTW2008nlo68cl PDF sets and set both the renormalization and factorization
scales equal to half the transverse energy, which is defined as the sum of the final state transverse momenta,
$\mu_F=\mu_R= \frac{\hat{H}_T}{2} = \frac{1}{2} \sum_{l=1}^4 k_T^l$\footnote{As customary in the literature,
we use the $\hat{H}_T$ notation to refer to the energies of the final state partons,
not jets, despite this is obviously the same thing in a LO analysis}, working in the $n_F = 5$ flavour scheme.
In order to cross-check our numerical tools, we must compare their outputs to results
already available in the literature. For this purpose, we compared LO total cross sections
for $(n \leq 4)$-jet production to those given by the BlackHat collaboration in Ref.~\cite{Bern:2011ep}
and cross-checked in Ref.~\cite{Badger:2012pf}. We find excellent agreement, up to phase space integration accuracy.
The cuts used in these calculations were those chosen by the ATLAS
collaboration in the 2011 analysis of multi-jet events \cite{Aad:2011tqa},
namely $p_T > 80$ GeV for the leading jet and $p_T > 60$ GeV for subleading
jets, $|\eta| < 2.8$ for the pseudorapidity and jet cone radius parameter $\Delta R>0.4$.
Again we find excellent agreement between the two codes with the LO results reported in the literature,
up to phase space integration uncertainties.
To be precise, we reproduce the LO predictions for the total inclusive cross sections
\begin{eqnarray}
\sigma(\geq 2\, \text{jets}) &=& 958(1)^{+316}_{-221} \, ,
\nonumber \\
\sigma(\geq 3\, \text{jets}) &=& 93.4(0.1)^{+50.4}_{-30.3} \, ,
\nonumber \\
\sigma(\geq 4\, \text{jets}) &=& 9.98(0.01)^{+7.40}_{-3.95} \, ,
\end{eqnarray}
where the numbers in brackets stand for the numerical integration uncertainty and the upper and lower errors
are obtained by varying the renormalization scale up and down by a factor of two.
There are 19 different channels contributing to the cross section at the parton-level:
\begin{eqnarray}
&&
g g \rightarrow 4 g \, ,
g g \rightarrow q \bar{q} \, 2g \, ,
q g \rightarrow q \, 3g \, ,
q \bar{q} \rightarrow q \bar{q}\, 2g \, ,
q q \rightarrow q q \, 2g \, ,
q q' \rightarrow q q' \, 2g \, ,
\nonumber \\
&&
g g\rightarrow q \bar{q} q \bar{q} \, ,
g g \rightarrow q \bar{q} q' \bar{q}' \, ,
q g \rightarrow q g q \bar{q} \, ,
q g \rightarrow q g q' \bar{q}' \, ,
\nonumber \\
&&
q \bar{q} \rightarrow 4g\, ,
q \bar{q} \rightarrow q' \bar{q}'\, 2g \, ,
q \bar{q} \rightarrow q \bar{q} q \bar{q} \, ,
q \bar{q} \rightarrow q\bar{q} q' \bar{q}' \, ,
q \bar{q} \rightarrow q' \bar{q}' q' \bar{q}' \, ,
q \bar{q} \rightarrow q' \bar{q}' q'' \bar{q}'' \, ,
\nonumber \\
&&
q q \rightarrow q q q \bar{q} \, ,
q q \rightarrow q q q' \bar{q}' \, ,
q q' \rightarrow q q' q \bar{q} \, ,
\nonumber
\end{eqnarray}
The processes in the first line are the dominant channels,
contributing together to $\sim 93 \% $ of the total cross section.
This stays true in the HEF framework as well.
\section{Double-parton scattering production of four jets}\label{MPI}
Single-parton scattering contributions are expected to be dominant for high momentum transfer, as it is highly unlikely that
two partons from one proton and two from the other one are energetic enough for two hard scatterings to take place,
as the behaviour of the PDFs for large $x$ suggests.
However, as the cuts on the transverse momenta of the final state are softened, a window opens
to possibly observe significant double parton scattering effects, as
often stated in the literature on the subject
and recently analysed for 4-jet production in collinear factorization approach in Ref.~\cite{Maciula:2015vza}.
Our goal here is to perform the same analysis in HEF, in order to assess the difference in the predictions
of the two approaches.
First of all, let us recall the formula usually employed for the computation of DPS cross sections,
adjusting it to the 4-parton final state,
\begin{equation}
\frac{d \sigma^{B}_{4-jet,DPS}}{d \xi_1 d \xi_2} =
\frac{m}{\sigma_{eff}} \sum_{i_1,j_1,k_1,l_1;i_2,j_2,k_2,l_2}
\frac{d \sigma^B(i_1 j_1 \rightarrow k_1 l_1)}{d \xi_1}\, \frac{d \sigma^B(i_2 j_2
\rightarrow k_2 l_2)}{d \xi_2} \, ,
\end{equation}
where the $\sigma(a b \rightarrow c d)$ cross sections are obtained by
restricting formulas (\ref{coll_cross}) and (\ref{kt_cross}) to a single
channel and the symmetry factor $m$ is $1$ unless the two hard
scatterings are identical, in which case it is $1/2$, so as to avoid
double counting them. Above $\xi_1$ and $\xi_2$ stand for generic
kinematical variables for the first and second scattering, respectively.
The effective cross section $\sigma_{eff}$ can be loosely interpreted
as a measure of the transverse correlation of the two partons inside
the hadrons, whereas the possible longitudinal correlations are usually
neglected (for an introduction to this issue, see for example Ref.~\cite{Gaunt:2009re}).
In this paper we use $\sigma_{eff}$ provided by the CDF, D0
collaborations and recently confirmed by the LHCb collaboration
$\sigma_{eff}$ = 15 mb, although the latter value may be questioned
\cite{Maciula:2016wci} when all SPS mechanisms of double charm production are included.
As already mentioned in the introduction there are attempts, in the
literature, to construct DPDFs which
include correlations also between the longitudinal momenta of the two
partons and fullfil sum rules. These models are, however, still rather
at a preliminary stage.
So far they are formulated exlusively in the gluon sector \cite{Golec-Biernat:2015aza}
or in the valence quark sector \cite{Rinaldi:2014ddl}.
In addition they are only formulated in a leading order framework
which may be not sufficient for many processes.
Moreover, as it is expected on physical grounds and confirmed by
all the calculations in the various models proposed so far,
the longitudinal parton-parton correlations should become far less
important as the energy of the collision is increased, due to the increase in the parton multiplicity.
For instance, the plots in Ref.~\cite{Golec-Biernat:2015aza} show
that the double gluon distribution obtained with a sum rule approach is
essentially equal to the factorized Ansatz at the scale $Q^2 = 10\, \text{GeV}^2$
down to $x = 10^{-5}$.
Looking forward to further improvements in this field, we choose to
limit ourselves to a more pragmatic approach for the purpose of this paper,
making the following ansatz for DPDF in the collinear-factorization case:
\begin{equation}
D_{1, 2}(x_1,x_2,\mu) = f_1(x_1,\mu)\, f_2(x_2,\mu) \, \theta(1-x_1-x_2) \, ,
\end{equation}
where $D_{1, 2}(x_1,x_2,\mu)$ is the DPDF and
$f_i(x_i,\mu)$ are the ordinary PDFs and the subscripts $1$ and $2$
simply differentiate between two generic partons in the same proton.
this ansatz can be automatically generalised to the case when parton
transverse momenta are included.
Coming to DPS contributions, we have to include all the possible $45$ channels which
can be obtained by coupling in all possible distinct ways the $8$ channels for the $2\rightarrow 2$
SPS process, i.e.
\begin{eqnarray}
\#1 &=& g g \rightarrow g g \, , \quad \#5 = q \bar{q} \rightarrow q'\bar{q}' \nonumber \; , \\
\#2 &=& g g \rightarrow q \bar{q} \, , \quad \#6 = q \bar{q} \rightarrow g g \nonumber \;, \\
\#3 &=& q g \rightarrow q g \, , \quad \#7 = q q \rightarrow q q \nonumber \; , \\
\#4 &=& q \bar{q} \rightarrow q \bar{q} \, , \quad \#8 = q q'\rightarrow q q' \nonumber \; .
\end{eqnarray}
We find that the pairs $(1,1)$, $(1,2)$, $(1,3)$, $(1,7)$, $(1,8)$, $(3,3)$ $(3,7)$, $(3,8)$
together account for more than $95$ \% of the total cross section for all
the sets of cuts considered in this paper.
\subsection{Comparison to the collinear approach and to ATLAS data with hard central kinematic cuts}
In the following, we test the HEF calculation against the collinear case and compare it to the 8 TeV
data recently reported by the ATLAS collaboration \cite{Aad:2015nda}.
The kinematic cuts are here slightly different with respect to Ref.~\cite{Aad:2011tqa}:
$p_T > 100 $ GeV for the leading jet and $p_T > 64 $ GeV for the first three subleading jets; in addition
$|\eta| < 2.8$ is the pseudorapidity cut and $\Delta R > 0.5$ is
the constraint on the jet cone radius parameter.
As for this framework, we employ, together with the newly obtained TMD
PDFs ($5$ quark flavors and gluon) which we call DLC-2016
(Double-Log-Coherence), the running $\alpha_s$ coming with the MSTWnlo200868cl sets.
The results of our computation in HEF is shown in Figs.~\ref{Hard_central_1} and \ref{Hard_central_2},
where it is apparent that the DPS contribution is completely irrelevant, as expected for final states
with high transverse momenta, as it is extremely unlikely that all the four partons in the two couples
coming from the colliding protons carry enough energy to produce such a hard final state.
A generally good agreement with the ATLAS data can be seen through the transverse momenta
spectra of the four jets, thus showing that the HEF approach works reliably well in this region.
First we show the results of the HEF calculation in Figs.~\ref{Hard_central_1} and \ref{Hard_central_2}.
The prediction is consistent with the ATLAS data for all the $p_T$ spectra.
\begin{figure}[!h]
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{figures_by_Rafal/ATLAS/kt_1st_leading_pt_4jets_ATLAS_Rafal_Ratio.pdf}}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{figures_by_Rafal/ATLAS/kt_2nd_leading_pt_4jets_ATLAS_Rafal_Ratio.pdf}}
\end{minipage}
\caption{
HEF prediction of the differential cross sections w.r.t. the transverse momenta of the first two leading jets
compared to the ATLAS data \cite{Aad:2015nda}. The LO calculation describes the data pretty
well in this hard regime in which MPIs are irrelevant. In addition we show the ratio of the SPS HEF result to the ATLAS data.}
\label{Hard_central_1}
\end{figure}
\begin{figure}[!h]
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{figures_by_Rafal/ATLAS/kt_3rd_leading_pt_4jets_ATLAS_Rafal_Ratio.pdf}}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{figures_by_Rafal/ATLAS/kt_4th_leading_pt_4jets_ATLAS_Rafal_Ratio.pdf}}
\end{minipage}
\caption{
HEF prediction of the differential cross sections w.r.t. the transverse momenta of the 3rd and 4th leading jets
compared to the ATLAS data \cite{Aad:2015nda}. The LO calculation describes the data
pretty well in this hard regime in which MPIs are irrelevant. In addition we show the ratio of the SPS HEF result to the ATLAS data.}
\label{Hard_central_2}
\end{figure}
Next we assess the difference between the HEF and collinear predictions at LO
as far as SPS is concerned.
We see from Figs.~\ref{Hard_central_ktvscoll1} and \ref{Hard_central_ktvscoll2} that
the collinear
factorization performs slightly better for intermediate values and HEF
does a better job for the last bins, except for the $4$th jet.
All in all, both approaches are consistent with the data in this kinematic region.
\begin{figure}[h]
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{figures_by_Rafal/ATLAS/kt_coll_1st_leading_pt_4jets_ATLAS_Rafal_Ratio.pdf}}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{figures_by_Rafal/ATLAS/kt_coll_2nd_leading_pt_4jets_ATLAS_Rafal_Ratio.pdf}}
\end{minipage}
\caption{
Comparison of the HEF results to the collinear LO predictions and the ATLAS data for the 1st and 2nd leading jets.
In addition we show the ratio of the SPS HEF to the SPS collinear result.}
\label{Hard_central_ktvscoll1}
\end{figure}
\begin{figure}[h]
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{figures_by_Rafal/ATLAS/kt_coll_3rd_leading_pt_4jets_ATLAS_Rafal_Ratio.pdf}}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{figures_by_Rafal/ATLAS/kt_coll_4th_leading_pt_4jets_ATLAS_Rafal_Ratio.pdf}}
\end{minipage}
\caption{
Comparison of the HEF resuts to the collinear LO predictions and the ATLAS data for the 3rd and 4th leading jets.
In addition we show the ratio of the SPS HEF to the SPS collinear result.}
\label{Hard_central_ktvscoll2}
\end{figure}
\subsection{Comparison to CMS data with softer cuts}
As discussed in Ref.~\cite{Maciula:2015vza}, so far the only experimental analysis
of four-jet production relevant for the DPS studies was realized
by the CMS collaboration \cite{Chatrchyan:2013qza}.
The cuts used in this analysis are $p_T> 50$ GeV for the first and
second jets, $p_T > 20$ GeV for the third and fourth jets,
$|\eta| < 4.7$ and the jet cone radius parameter $\Delta R > 0.5$.
In the rest of this section, we present our results for such cuts.
As for the total cross section for the four jet production, the experimental and theoretical LO results are respectively:
\begin{eqnarray}
\text{CMS collaboration} : &&
\sigma_{tot} = 330 \pm 5\, (\text{stat.}) \pm 45\, (\text{syst.})\, nb
\nonumber \\
\text{LO collinear factorization}: &&
\sigma_{SPS} = 697\, nb\, , \quad \sigma_{DPS} = 125\, nb \, , \quad \sigma_{tot} = 822\, nb
\nonumber \\
\text{LO HEF $k_{T}$-factorization}: &&
\sigma_{SPS} = 548\, nb\, , \quad \sigma_{DPS} = 33\, nb \, , \quad \sigma_{tot} = 581\, nb
\label{sigma_CMS}
\end{eqnarray}
It goes without saying that the LO result needs refinements from NLO contributions,
much more than it does in the case of the ATLAS hard cuts,
as we are of course less deep into the perturbative region.
For this reason, in the following we will always perform comparisons only to
data (re)normalised to the total (SPS+DPS) cross sections.
What is interesting in the HEF result,
compared to collinear factorization, is the dramatic damping of the DPS
contribution. The effect of the damping is of kinematical nature and will be
explained below.
The effect of the relative damping of the HEF DPS result
compared to leading-order collinear DPS result is of kinematical origin.
The main idea can be understood already in a bit simpler case of two-jet
production within the HEF approach.
For the purpose of this illustration we impose a cut $p_{T} >$ 35 GeV on both jets (leading and
subleading).
In Fig.~~\ref{fig:illustration_of_damping} we show transverse momentum
distribution for both leading (long-dashed line) and subleading (long dashed-dotted line)
jet. We observe a minimum for the leading jet and maximum for the subleading
jet for transverse momenta in the vicintity of the lower cut.
The integrated cross section for the leading and subleading jet
is of course identical as they are "measured" (identified) in coincidence.
For the leading order collinear case both jets have the same distribution
and one gets maximum in the vicinity of the transverse momentum
threshold in both cases. In this case imposing cuts on both jets does not
lead to "loosing" cross section.
In contrast, in the HEF approach, if the leading jet is close to the transverse momentum threshold,
then the subleading jet is typically below that threshold, therefore such an event is not counted.
For four-jet DPS production the situation is more complicated and
strongly depends on cuts (all identical, two pairs of identical cuts, harder
cut for the leading jet and identical for the other, etc.).
\begin{figure}[h]
\begin{center}
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{figures_by_Rafal/dsig_dpTjet_CUT.pdf}}
\end{minipage}
\end{center}
\caption{
The transverse momentum distribution of the leading (long dashed line)
and subleading (long dashed-dotted line) jet for the dijet production in HEF.
For comparison we show also result for leading-order collinear
approach (short dashed line) in which case both jets give
the same distribution.
}
\label{fig:illustration_of_damping}
\end{figure}
\begin{figure}[h]
\begin{center}
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{figures_by_Rafal/CMS/kt_coll_1st_leading_pt_4jets_CMS_Rafal_Ratio.pdf}}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{figures_by_Rafal/CMS/kt_coll_2nd_leading_pt_4jets_CMS_Rafal_Ratio.pdf}}
\end{minipage}
\end{center}
\caption{
Comparison of the LO collinear and HEF predictions to
the CMS data for the 1st and 2nd leading jets. In addition we show the ratio of the SPS HEF result to the CMS data.}
\label{CMS_pT_12}
\end{figure}
\begin{figure}[h]
\begin{center}
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{figures_by_Rafal/CMS/kt_coll_3rd_leading_pt_4jets_CMS_Rafal_Ratio.pdf}}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{figures_by_Rafal/CMS/kt_coll_4th_leading_pt_4jets_CMS_Rafal_Ratio.pdf}}
\end{minipage}
\end{center}
\caption{
Comparison of the LO collinear and HEF predictions to
the CMS data for the 3rd and 4th leading jets. In addition we show the ratio of the SPS HEF result to the CMS data.}
\label{CMS_pT_34}
\end{figure}
In Figs.~\ref{CMS_pT_12} and \ref{CMS_pT_34} we compare the predictions
in HEF to the CMS data. Here both the SPS and DPS
contributions are normalized to the total cross section,
i.e. the sum of the SPS and DPS contributions.
In all cases the renormalized transverse momentum distributions agree
with the CMS data. However, the absolute cross sections obtained in this case
within the HEF approach are too large.
Not only transverse momentum dependence is interesting.
The CMS collaboration extracted also a more complicated observables
\cite{Chatrchyan:2013qza}.
One of them, which involves all four jets in the final state,
is the $\Delta S$ variable, defined in Ref.~\cite{Chatrchyan:2013qza} as the angle between pairs of
the harder and the softer jets,
\begin{equation}
\Delta S = \arccos \left(
\frac{\vec{p}_T(j^{\text{hard}}_1,j^{\text{hard}}_2) \cdot \vec{p}_T(j^{\text{soft}}_1,j^{\text{soft}}_2)}{|\vec{p}_T(j^{\text{hard}}_1,j^{\text{hard}}_2)|\cdot|\vec{p}_T(j^{\text{soft}}_1,j^{\text{soft}}_2)|}
\right) \, ,
\end{equation}
where $\vec{p}_T(j_i,j_k)$ stands for the sum of the transverse momenta of the two jets in arguments.
In Fig.~ \ref{fig:CMS_DS} we present our HEF prediction for
the normalized to unity distribution in the $\Delta S$ variable.
Our HEF result approximately agrees with the
experimental $\Delta S$ distribution.
In contrast the LO collinear approach leads to
$\Delta S$ = 0, i.e. a Kronecker-delta peak at $\Delta S$ = 0 for
the distribution in $\Delta S$.
For the DPS case this is rather trivial. The two hard and two soft
jets come in this case from the same scatterings and are back-to-back
(LO), so each term in the argument of $arccos$ is zero
(jets are balanced in transverse momenta).
For the SPS case the transverse momenta of the two jet pairs
(with hard jets and soft jets) are identical and have opposite direction
(the total transverse momentum of all four jets must be zero from the momentum conservation).
Then it is easy to see that the argument of $arccos$ is just -1.
This means $\Delta S$ = 0.
The above relations are not fullfilled in the HEF
approach. The SPS contribution clearly dominates and approximately gives the shape
of the $\Delta S$ distribution.
The DPS contribution improves the agreement with the data in the
central region, worsening it a little bit for $\Delta S \rightarrow 0$
while essentially leaving the result unaffected for $\Delta S
\rightarrow \pi$.
It is anyway interesting that we roughly describe the data
via pQCD effects within our HEF approach which
are in Ref.~\cite{Chatrchyan:2013qza} described by parton-showers and soft MPIs.
\begin{figure}[h]
\begin{center}
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{figures_by_Rafal/CMS/kt_DS_4jets_CMS_Rafal_Ratio.pdf}}
\end{minipage}
\end{center}
\caption{
Comparison of the HEF predictions to the CMS data for $\Delta S$ spectrum. In addition we show the ratio of the (SPS+DPS) HEF result to the CMS data.}
\label{fig:CMS_DS}
\end{figure}
It would be nice to have more insight into our successful
description of the $\Delta S$ distribution measured by the CMS experiment.
In Fig.~\ref{fig:CMS_DS_toymodel} we return to the $\Delta S$ spectrum and show also
two results with a TMD toy model with the Gaussian smearing of
the collinear parton distribution:
\begin{equation}
{\cal F}_p(x,k_T^2,\mu^2) = G(k_T^2;\sigma) x p (x,\mu^2).
\label{Gaussian_UGDF}
\end{equation}
We take two sets of smearing parameter: $\sigma$ = 1 GeV (left panel) and 5 GeV (right panel).
Taking a bigger value of $\sigma$ we approach the CMS data. This shows that the transverse momenta bigger than a few GeV are
needed to approach the data. The disagreement of the toy model with $\sigma = 5$ GeV result with the experimental data and the agreement for the
DLC-2016 model illustrate sensitivity to TMD's.
\begin{figure}[h]
\begin{center}
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{figures_by_Rafal/kt_DS_4jets_CMS_Rafal_Gauss1GeV.pdf}}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{figures_by_Rafal/kt_DS_4jets_CMS_Rafal_Gauss5GeV.pdf}}
\end{minipage}
\end{center}
\caption{
Distribution in $\Delta S$ for the toy Gaussian model of TMDs
with $\sigma$ = 1 GeV (left) and $\sigma$ = 5 GeV (right).
}
\label{fig:CMS_DS_toymodel}
\end{figure}
\subsection{HEF predictions for a possible set of asymmetric cuts }
Moving from the previous considerations, in the following subsection we
present our results of the DPS employing asymmetric cuts by which we
mean here $p_T > 35$ GeV for the leading jet,
$p_T > 20$ GeV for the other jets and $|\eta| < 4.7$, $\Delta R > 0.5$.
Of course it would be interesting to have the results of such an experimental analysis ( i.e. with soft enough
but asymmetric cuts ) in order to test the predictions of HEF for DPS.
In this case the theoretical total cross sections for four-jet production are:
\begin{eqnarray}
\text{LO collinear factorization}: &&
\sigma_{SPS} = 1969 \, nb\, , \quad \sigma_{DPS} = 514\, nb \, , \quad \sigma_{tot} = 2309\, nb
\nonumber \\
\text{LO HEF $k_{T}$-factorization}: &&
\sigma_{SPS} = 1506 \, nb\, , \quad \sigma_{DPS} = 297 \, nb \, , \quad \sigma_{tot} = 1803\, nb
\end{eqnarray}
Compared to (\ref{sigma_CMS}), it is apparent that now the drop in the total cross section for DPS
when moving from LO collinear to HEF approach is considerably
smaller. Here we have enough phase space for the subleading jet(s),
as a consequence of the asymmetric cuts.
\begin{figure}[h]
\begin{center}
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{figures_by_Rafal/Asymm/kt_coll_1st_leading_pt_4jets_Asymm_Rafal.pdf}}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{figures_by_Rafal/Asymm/kt_coll_2nd_leading_pt_4jets_Asymm_Rafal.pdf}}
\end{minipage}
\end{center}
\caption{
LO collinear and HEF predictions for the 1st and 2nd
leading jets with the asymmetric cuts.}
\label{Asymm_pT_12}
\end{figure}
\begin{figure}[h]
\begin{center}
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{figures_by_Rafal/Asymm/kt_coll_3rd_leading_pt_4jets_Asymm_Rafal.pdf}}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{figures_by_Rafal/Asymm/kt_coll_4th_leading_pt_4jets_Asymm_Rafal.pdf}}
\end{minipage}
\end{center}
\caption{
LO collinear and HEF predictions for the 3rd and 4th
leading jets with asymmetric cuts.}
\label{Asymm_pT_34}
\end{figure}
In Figs.~\ref{Asymm_pT_12} and \ref{Asymm_pT_34} we show our predictions
for the normalized transverse momentum distributions with the new set of cuts.
\begin{figure}[h]
\begin{center}
\begin{minipage}{0.47\textwidth}
\centerline{\includegraphics[width=1.0\textwidth]{figures_by_Rafal/Asymm/kt_DS_4jets_Asymm_Rafal.pdf}}
\end{minipage}
\end{center}
\caption{
HEF prediction for $\Delta S$ with asymmetric cuts.}
\label{Asymm_DS}
\end{figure}
\section{Conclusions
In the present paper we have compared the perturbative predictions
for four-jet production at the LHC in leading-order collinear and
high-energy ($k_T$-)factorization.
Both single-parton scattering and double parton contribution
have been calculated for a first time in the high-energy
($k_T$-)factorization approach.
The calculation of the SPS contribution may be considered as a technical
achievment. So far only production of the $c \bar c c \bar c$ final
state (also of the 2 $\to$ 4 type) was discussed in the literature in this context.
For the four-jet production the number of relevant subproceses is
much larger but could be treated in our automatized framework.
We find that both collinear and the ($k_T$-)factorization approaches
describe the data for hard central cuts, relevant for
the ATLAS experiment, reasonably well. For the harder cuts we get both
normalization and shape of the transverse momentum distributions. For the softer cuts
used e.g. by the CMS collaboration the tree level result is unreliable.
Therefore in this case we have presented results for normalised cross
sections.
We have presented distributions in transverse momenta for all jets
ordered in their transverse momenta.
We have found that for symmetric cuts the DPS cross section obtained
with more realistic high energy ($k_T$-)factorization approach is smaller than the one
obtained in the collinear approach. This is important result in searches
for DPS effects in four-jet production not discussed so far in the
literature. We have tried to explain this as kinematical effect due to
phase space limitation when simultaneously imposing cuts on all jets
but a full explanation is a bit intricate.
While we observe, in agreement with Ref.~\cite{Maciula:2015vza}, that lowering the cut
in transverse momenta can significantly enhance the experimental sensitivity to DPS,
we also observe that the HEF approach severely tames this effect for symmetric cuts,
due to gluon-emission effects which alter the transverse-momentum balance between final state partons.
We have found that the damping is not present when cuts are not identical.
The discussion how to optimize the cuts will be presented elsewhere.
For other approaches addressing the four-jet production and resummation
of BFKL type of singularities \cite{Vera:2007kn,Vera:2007dr} we refer
the Reader to Ref.~\cite{Caporale:2015int}. The authors of Ref.~\cite{Caporale:2015int}
define new angular four jet observables to test BFKL approach.
As a side result, we present in Appendix \ref{Comparison}
a detailed numerical comparison of results obtained for dijet production with matrix element generated
automatically by means of AVHLIB with those obtained analytically within the Parton-Reggeization-Approach (PRA) in
Ref.~\cite{Nefedov:2013ywa} and implemented in an independent code.
For all (sub)processes we have obtained very good agreement of corresponding differential distributions.
We show corresponding azimuthal angle correlations for different subprocesses which are particularly
efficient for such tests, as they sample the situation in a broad range of the phase space.
It was shown in the past for some subprocesses that also analytical results coincide \cite{vanHameren:2013csa}.
\section*{Acknowledgments}
The work of M.S. and K.K. have been supported by Narodowe Centrum Nauki
with Sonata Bis grant DEC-2013/10/E/ST2/00656 while R.M. and A.S.
have been supported by the Polish National Science Center grant
DEC-2014/15/B/ST2/02528. A.v.H. was supported by
grant of National Science Center, Poland, No. 2015/17/B/ST2/01838.
M.S. also thanks the "Angelo della Riccia" foundation for support.
We wish to thank A. V. Shipilova for useful discussion during the development of this project
on analytic formula for matrix elements.
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{
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Q: How to use specific element of one to many relationship in whereHas (Laravel) I want to get data list , which has many relational rows in another table.
I need to check specific row of relational rows for getting data list from first, for example with it can be the first one and second ... (some conditions), but in whereHas you know that we are checking all elements.
$shipments->whereHas('shipment_stops', function ($q) use ($startDate) {
$q->where('departure_date', '>=', $startDate) ;
});
This will check all shipment_stops, but I want to check only the last stop with the specific type (for example).
I know that maybe I can create better structure in DB, but anyway, I have this structure.
Thanks
A: Use Has One Of Many
public function latestShipmentStop()
{
return $this->hasOne(ShipmentStop::class)->latestOfMany();
}
..
$shipments->whereHas('latestShipmentStop', function ($q) use ($startDate) {
$q->where('departure_date', '>=', $startDate) ;
});
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"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,392
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HVS Market Intelligence Report: Lansing, Michigan
New business partnerships, investments in high-tech companies and facilities, and rising hotel demand and average rates point toward a path of growth for Lansing's economy and hotels.
By Ryan Wall
A seat of state government affairs, higher and vocational education, and manufacturing in the Midwest, Lansing is gaining notice for some of its more emerging industries, as well as those, like hospitality, that are undergoing evolution. The city of Lansing is Michigan's state capital and the core of the greater metro area, known as "Mid-Michigan." Lansing's diverse economic makeup comprises institutions like Michigan State University; investments in cutting-edge technology; and established insurance, healthcare, and manufacturing companies. Lansing's growing economic base placed the Lansing-East Lansing area in the second-highest spot for Michigan, according to the Milken Institute's "Best-Performing Cities 2011" rankings. The lull draped over the city appears to be pulling away, forging new opportunities for Lansing, its residents, visitors, and businesses, including hotels.
Major Employers and Economic Developments
The following chart lists Lansing's top employers.
Michigan State University (MSU) received a boost in 2008 when the U.S. Department of Energy selected the world-renowned research university as the site for the Facility for Rare Isotope Beams (FRIB). In September of 2012, the MSU Board of Trustee's approved an additional $55 million that will allow construction to commence. The project, scheduled for completion in 2021, is expected to contribute one billion dollars to the Lansing economy and add 5,000 temporary construction jobs and 400 permanent jobs over the course of its construction. The new facility will be state of the art and one of a kind, attracting scientists, engineers, researchers, and commercial enterprises that allow Lansing's technological sector to forge a competitive edge.
The Eli and Edythe Broad Art Museum, designed by renowned architect Zaha Hadid, opened in December of 2012. The museum's historic collections include pieces and artifacts from the ancient Greek and Roman periods through the Renaissance and Modern eras. Local officials project that the museum will attract over 100,000 visitors in its first year, many from outside the state, which should provide strong room-night generation for Lansing hotels.
Lansing's Capital Region International Airport received U.S. Port of Entry status in January of 2008, and was designated as a Foreign Trade Zone in August of 2009. This led to the formation of Port Lansing, which further promotes international commerce in the region. Port Lansing was created as an alternative to larger or more distant hubs such as Detroit and Chicago, allowing importers and exporters to clear U.S. Customs locally with fewer hassles and delays. The Global Logistics Center, a $6-million, 48,000-square-foot cargo warehouse and cross-docking facility, opened in October of 2012, and is the first of an expected $20 million in total investments for Port Lansing. The Global Logistics Center will serve as a short-term storage facility, staging center, and a finishing ground for certain products requiring additional manufacturing. Finally, Michigan Governor Rick Snyder recently announced that Ford Motor Company, Chrysler, and several other major automotive manufacturers have joined forces to promote and expedite commerce between Lansing and other Michigan automotive manufacturing and distribution centers, a welcome development that demonstrates the push to revive the region's auto industry.
The Lansing Regional SmartZone (LRSZ), established to fuel growth in the local technology business sector, has proven widely successful. Lansing was recently ranked the sixth-best metropolitan area for high-tech job growth in the nation, owing to the LRSZ and cooperative efforts such as the Technology Incubation Center (TIC), which focuses on developing Michigan State University Information Technology spin-offs, high-tech ventures, and local entrepreneurs.
Lansing has also become an emerging player in high-tech research and development, as well as home to several growing companies in the advanced sciences. Washington, D.C.-based Emergent BioSolutions operates a research facility in Lansing, where it manufactures the only anthrax vaccine available in the country. In 2011, the company secured a five-year, $1.25-billion contract to provide the vaccine to the federal government, and in May of 2012, Emergent BioSolutions began the first phase of a $108-million expansion to its Lansing campus.
Lansing-based Niowave is one of the world's leading producers of superconducting particle accelerators, which are used in research related to everything from CAT scans to naval weapons. In 2012, the company opened a new $10-million research facility that will allow the company to develop larger and more powerful accelerators.
Projections show Lansing's technology sector expanding by 16.2 percent between 2011 and 2020. These incubators and companies, along with the FRIB project at nearby MSU, are helping to pave the way for future technology growth in the region.
Performance Trends for Local Hotels
Lansing's four lodging submarkets are bookended by clusters of generally limited-service hotels on the east and west sides of the metro area; these hotels cater to demand from the state government and transient travel to MSU. The East Lansing hotel submarket, near the MSU campus, includes several full-service hotels, along with a number of extended-stay and budget properties. The fourth submarket, located on Lansing's northern side, is also dominated by limited-service brands; its location near both Downtown and MSU has allowed these hotels to pull business from all over Lansing. Additionally, this submarket benefits from proximity to the growing Eastwood Towne Center.
State government, local employers, and Michigan State University represent the primary sources of hotel demand for the Lansing market. The recent recession caused market occupancy to decline in 2008; demand continued to wane while hotel supply increased with the opening of the Holiday Inn Express in East Lansing in 2009. Demand began to rebound in 2010, as local employers, including those in the auto industry, found their footing; in 2011, the new supply was absorbed and occupancy returned to pre-recession levels. Hotel occupancy and demand in Lansing continued to increase in 2012, albeit at a more modest pace than in the previous two years.
After two years of growth, average rates in the Lansing hotel market declined modestly in 2009, followed by a minimal decrease in 2010; however, these declines were relatively limited in comparison to many hotel markets across the country that were more severely impacted by the recession. Average rates for Lansing hotels rose above pre-recession levels in 2011, and rate continued to show healthy month-over-month increases during 2012.
Local employers within the insurance, healthcare, and auto industries have fueled much of the demand growth in Lansing, and growing technology firms are providing a new source of hotel demand for the market. Hence, hoteliers have begun shifting focus from occupancy growth toward stronger rate increases. A limited amount of new supply is expected in the Lansing market over the next several years. The Hampton Inn Lansing is anticipated to change brands, with a newly constructed Hampton Inn to be located nearby. Additionally, there is a planned Hyatt Place in the Eastwood Towne Center, located along Lake Lansing Road, which will add 125 rooms to the market once complete.
Lansing's recent economic growth and the promise of future projects have benefited area hotels with more demand and higher rates. The presence of established insurance and healthcare companies, as well as an emerging, diversified technology sector, has established an economic balance in Lansing that has helped make the market tough to upset, even in the recent recession. Furthermore, Lansing's presence as an emerging global community has been fueled by MSU and its expanding international enrollment, which reached 3,631 students for 2011, of which 3,000 were from China. The growing international presence has local leaders seeking new ways to integrate these students into the community and economy. This drive toward local, regional, and international growth should help sustain Lansing's overall economy and provide opportunities for growth in the area's hotel industry.
Great article Ryan!! Awesome to see such positive things happening in our region! I'll be passing this along to some of our Michigan-based clients.
Great article Ryan! Sounds like these are exciting times in Lansing, with much more to come in the near future.
Interesting article, Ryan. Nice job!
Nice job, Ryan! Great insight into this market; sounds like the expansions in R&D/tech are helping broaden the demand base for area hotels.
Wow! Who knew all of this was happing in Lansing? Very comprehensive and insightful.
Great job Ryan! The current focus on the Healthcare Industry should continue to help Lansing.
Excellent job Ryan!
I had no idea of the strong tech presence, as well as the strong base of international students at Michigan State. Nice article!
Well done, Ryan! The article is comprehensive and well written. You should be proud of your hard work.
Great job Ryan, very interesting read!
Interesting things happening in Lansing, Ryan. Great job and great read!
Nice article Ryan! I enjoyed reading about my home state., and hope postive things keep happening. Thanks Ryan!
Great job, Ryan!
%%ViewCount%% Views 13 Comments
Valuations & Market Studies North America
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{
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ep 05 "more joy" | Edwidge Danticat
Writing Home: American Voices from the Caribbean
"What would goodness be like? What would more joy be like?" – Edwidge Danticat
Tami and Kaiama connect with the illustrious Haitian-African-American author Edwidge Danticat. In this conversation, the three grapple with how they are emotionally processing the pandemic through writing and reading literature. Edwidge speaks on whether literature survives on suffering, her newfound quest to find goodness within her work, and whether she's guilty of being a "serial killer of her characters." As Edwidge discusses the precarity of writing at home during the pandemic, she reveals how she navigates her toughest critics: her daughters.
Edwidge Danticat is the author of several books, including Breath, Eyes, Memory, The Farming of Bones, Claire of the Sea Light, and Everything Inside. She is the editor of The Butterfly's Way: Voices from the Haitian Diaspora in the United States, Haiti Noir and Haiti Noir 2. Shehas written seven books for young adults and children, Anacaona, Behind the Mountains, Eight Days, The Last Mapou, Mama's Nightingale, My Mommy Medicine, and Untwine, as well as a travel narrative, After the Dance, A Walk Through Carnival in Jacmel. Her memoir, Brother, I'm Dying, was a National Book Award finalist in 2007 and a National Book Critics Circle Award winner for autobiography. She is a 2009 MacArthur Fellow and a two-time winner of The Story Prize, a 2020 United States Artist fellow, and winner of the 2020 Vilcek Prize for Literature.
Reading List:
Edwidge's writing:
"Mourning in Place," The New York Review (2020)
"One Thing," a short story from The New York Times Magazine's Decameron Project (2020)
The Art of Death: Writing the Final Story (2017)
Claire of the Sea Light (2013)
Breath, Eyes, Memory (1994)
Works Edwidge mentioned:
"Goodness: Altruism and the Literary Imagination," an annual Harvard University Ingersoll Lecture on Immortality by Toni Morrison (2012)
Goodness And The Literary Imagination by Toni Morrison (2019)
Authors who Edwidge recommended:
Katia Ulysse
Fabienne Josaphat
Angie Cruz
Doreen St. Felix
Nelly Rosario
Tayari Jones
photo credit | Carl Juste
ep 07 "what remains" | Katia Ulysse
ep 06 "collective" | Tiphanie Yanique
ep 04 "unflinching" | Marlon James
prologue (2) | welcome back home
ep 03 be.longing | Staceyann Chin
ep 02 ceremony | Alexis Gumbs
ep 01 (in)visibility | Naomi Jackson
prologue | welcome home
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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{"url":"http:\/\/jakehoggans.co.uk\/ug7ws71k\/h2ernw4.php?id=039108-fifth-rate-meaning","text":"At the low end of the fourth rate one might find the two-decker 50-gun ships from about 1756. Historical category for Royal Navy vessels, based on number of guns, https:\/\/en.wikipedia.org\/w\/index.php?title=Fifth-rate&oldid=970251179, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License, This page was last edited on 30 July 2020, at 05:44. Vital signs can be measured in a medical setting, at home, at the site of a medical emergency, or elsewhere. The main cause behind this declaration focused on new types of gun, the introduction of steam propulsion and the use of iron and steel armour which made rating ships by the number of guns obsolete. Of unrated vessels, the category of sloops comprised all vessels commanded by commanders; next followed all other ships commanded by lieutenants, and having complements of not less than 60 men; finally were \"smaller vessels, not classed as above, with such smaller complements as the Lords Commissioners of the Admiralty may from time to time direct\". A series of major changes to the rating system took effect from the start of January 1817, when the carronades carried by each ship were included in the count of guns (previously these had usually been omitted); the first rate from that date included all of the three-deckers (the adding in of their carronades had meant that all three-deckers now had over 100 guns), the new second rate included all two-deckers of 80 guns or more, with the third rate reduced to two-deckers of fewer than 80 guns. Vessels might also carry other guns that did not contribute to the rating. \u00d7 The remainder were simply \"unrated\". In the rating system of the Royal Navy used to categorise sailing warships, a fifth rate was the second smallest class of warships in a hierarchical system of six \"ratings\" based on size and firepower. The larger of the unrated vessels were generally all called sloops, but that nomenclature is quite confusing for unrated vessels, especially when dealing with the finer points of \"ship-sloop\", \"brig-sloop\", \"sloop-of-war\" (which really just meant the same in naval parlance as \"sloop\") or even \"corvette\" (the last a French term that the British Navy did not use until the 1840s). In this example, 4.8% of 9.7% is 49.5%. Pepys's original classification was updated by further definitions in 1714, 1721, 1760, 1782, 1801 and 1817 (the last being the most severe, as it provided for including in the count of guns the carronades that had previously been excluded). [1], Samuel Pepys, then Secretary to the Admiralty, revised the structure in 1677 and laid it down as a \"solemn, universal and unalterable\" classification. People usually include voluntary resignations, dismissals, non certifications and retirements in their turnover calculations. k Auxiliary vessels of less than 4000 tons\u2014except tugs, sailing ships, and receiving ships which were not rated\u2014were of the fourth rate. From February 1817 all carronades were included in the established number of guns. The second comprised the \"post ships\" of between 20 and 24 guns. Although the rating system described was only used by the Royal Navy, other major navies used similar means of grading their warships. the maximum breadth of the vessel. Fifth-rate frigates were considered useful for their combination of manoeuvrability and firepower, which, in theory, would allow them to outmanoeuvre an enemy of greater force and run down one of lesser force. The table specified the amount of monthly wages a seaman or officer would earn, in an ordered scheme of six rates, from \"first-rate\" to \"sixth-rate\", with each rate divided into two classes, with differing numbers of men assigned to each class. \u2018The most absurd thing of all is how much thought you devoted to this haphazard, fifth-rate hunk of junk.\u2019. 4. Fifth-rate ships served as fast scouts or independent cruisers and included a variety of gun arrangements. [4], The smaller two deckers originally blurred the distinction between a fourth rate and a fifth rate. The rated number of guns often differed from the number a vessel actually carried. While a fourth-rate was defined as a ship of the line, fifth and the smaller sixth-rates were never included among ships-of-the-line. The first and second rates were three-deckers; that is, they had three continuous decks of guns (on the lower deck, middle deck and upper deck), usually as well as smaller weapons on the quarterdeck, forecastle and poop. an officer holding the substantive rank of captain) as their commander. \"The earliest naval list in which any classification of ships appears, is dated 1546, and it divides the fifty-eight ships of Henry VIII's Navy, according to their \"quality, into... 'ships, galliasses, pinnaces, and cow barges. During the Napoleonic Wars, the Royal Navy increased the number of sloops in service by some 400% as it found that it needed vast numbers of these small vessels for escorting convoys (as in any war, the introduction of convoys created a huge need for escort vessels), combating privateers, and themselves taking prizes.[4]. At the low end of the fourth rate one might find the two-decker 50-gun ships from about 1756. Check out our selection of fifth grade reading worksheets. b. one of two notes constituting such an interval in relation to the other. No specific connection with the size of the ship or number of armaments aboard was given in this 1626 table, and as far as is known, this was related exclusively to seaman pay grades. Also of note in this passage, the restitution was made to the owner of the property (not to the government or any other third party), and the compensation was to be accompanied by a guilt offering to the Lord. Learn more. Political Context. fifth part, twenty percent, fifth. Torpedo-boat destroyers, torpedo boats, and similar vessels were not rated. British authors might still use \"first rate\" when referring to the largest ships of other nations or \"third rate\" to speak of a French seventy-four. This was only on the basis of their roughly-estimated size and not on their weight, crew or number of guns. Of the poorest possible quality. From 1778, however, the most important exception was the carronade. By the Napoleonic Wars there was no exact correlation between formal gun rating and the actual number of cannons any individual vessel might carry. Feb 16, 2020 Hana Gartner\u2019s most memorable Fifth Estate moments Feb 16, 2020 John Connelly\u2019s death: A family fights for answers Newsletter. It was a rough measurement of cargo-carrying capacity by volume, not displacement. Fifth grade reading worksheets help kids improve their reading, comprehension, and vocabulary skills. When making projections for a firm\u2019s free cash flowFree Cash Flow (FCF)Free Cash Flow (FCF) measures a company\u2019s ability to produce what investors care most about: cash that's available be distributed in a discretionary way, it is common practice to assume there will be different growth rates depending on which stage of the business life cycle the firm currently operates in. In February 1817 the rating system changed. ... Learners practice using context clues to identify the meaning of new words in this language and vocabulary worksheet. All the other third rates, with 74 guns or less, were likewise two-deckers, with just two continuous decks of guns (on the lower deck and upper deck), as well as smaller weapons on the quarterdeck, forecastle and (if they had one) poop. a. the interval between one note and another five notes away from it counting inclusively along the diatonic scale. However, these were gradually phased out, as the low freeboard (i.e., the height of the lower deck gunport sills above the waterline) meant that in rough weather it was often impossible to open the lower deck gunports.[4]. Lieutenant-commanders, lieutenants, ensigns, or warrant officers might command unrated vessels, depending on the size of the vessel.[6]. When carronades became part (or in some cases all) of a ship's main armament, they had to be included in the count of guns. Vessels of the first rate had a displacement tonnage in excess of 8000 tons; second rate, from 4000 to 8000 tons; third rate, from 1000 to 4000 tons; and fourth rate, of less than 1000 tons. The middle of the 18th century saw the introduction of a new fifth-rate type\u2014the classic frigate, with no ports on the lower deck, and the main battery disposed solely on the upper deck, where it could be fought in all weathers. Another list, dated 1612, divides them into... 'ships royal, measuring from twelve hundred to eight hundred tons; middling ships, from eight hundred to six hundred tons; small ships, three hundred and fifty tons; and pinnaces, from two hundred and fifty to eight tons. Structurally, these were two-deckers with a complete battery on the lower deck, and fewer guns on the upper deck (below the forecastle and quarter decks, usually with no guns in the waist on this deck). Fifth Third Bank, National Association, provides access to investments and investment services through various subsidiaries, including Fifth Third Securities. For example, if there are 70 students in 5 classes, find the number of students per class. Fifth definition is - one that is number five in a series. They were generally classified, like all smaller warships used primarily in the role of escort and patrol, as \"cruisers\", a term that covered everything from the smaller two-deckers down to the small gun-brigs and cutters. For example, the French Navy used a system of five rates (\"rangs\") which had a similar purpose. To be posted aboard a fifth-rate ship was considered an attractive assignment. The formal system of dividing up the Navy's combatant warships into a number or groups or \"rates\", however, only originated in the very early part of the Stuart era, with the first lists of such categorisation appearing around 1604. It was for this reason that frigates of this sort were commonly used in patrol and to disrupt enemy shipping lanes much as heavy cruisers would later in history. [5] The recommendation from the Board of Admiralty to the Prince Regent was dated 25 November 1816, but the Order in Council establishing the new ratings was issued in February 1817. In 2006, Jan Hasbrouck and Gerald Tindal completed an extensive study of oral reading fluency. The term first-rate has passed into general usage, as an adjective used to mean something of the best or highest quality available. The larger fourth rates of 60 guns continued to be counted as ships-of-the-line, but few new ships of this rate were added, the 60-gun fourth rate being superseded over the next few decades by the 64-gun third rate. Rates may be higher for loans to purchase an RV from a private party, smaller loan amounts, longer terms, used RVs, and a lower credit score. For instance, when Pitt Burnaby Greene, the commanding officer of Bonne Citoyenne in 1811, received his promotion to post-captain, the Navy reclassed the sloop as a post ship. Vessels were sometimes classified according to the substantive rank of her commanding officer. 5G uses 5G NR air interface alongside OFDM principles. The rating of a ship was of administrative and military use. Base date in construction contracts - Designing Buildings Wiki - Share your construction industry knowledge. The fifth rates at the start of the 18th century were generally \"demi-batterie\" ships, carrying a few heavy guns on their lower deck (which often used the rest of the lower deck for row ports) and a full battery of lesser guns on the upper deck. The largest third rates, those of 80 guns, were likewise three-deckers from the 1690s until the early 1750s, but both before this period and subsequent to it, 80-gun ships were built as two-deckers. Students learn that a unit rate is a rate in which the second rate is 1 unit. ^* The ton in this instance is the burthen tonnage (bm). One therefore needs to distinguish between the established armament of a vessel (which rarely altered) and the actual guns carried, which might happen quite frequently for a variety of reasons; guns might be lost overboard during a storm, or \"burst\" in service and thus useless, or jettisoned to speed the ship during a chase, or indeed removed down into the hold in order to use the ship (temporarily) as a troop transport, or for a small vessel, such as the schooner HMS\u00a0Ballahoo, to lower the centre of gravity and thus improve stability in bad weather. Reuters.com brings you the latest news from around the world, covering breaking news in markets, business, politics, entertainment, technology, video and pictures. Captains commanded ships of the first rate; captains or commanders commanded ships of the second rate; commanders or lieutenant-commanders commanded ships of the third rate; lieutenant-commanders or lieutenants commanded ships of the fourth rate. Turnover rate definition: The term \u2018employee turnover rate\u2019 refers to the percentage of employees who leave an organization during a certain period of time. 1. one-fifth - one part in five equal parts. The smaller two deckers originally blurred the distinction between a fourth rate and a fifth rate. Vital signs are useful in detecting or monitoring medical problems. As of 1905, ships of the United States Navy were by law divided into classes called rates. 1 Sailing vessels with only two masts or a single mast were technically not \"ships\", and were not described as such at the time. All owners of your savings account must also be listed together as owners on your Fifth Third checking account. There was a further major change in the rating system in 1856. How to use fifth in a sentence. if you are buying for 100k the annual interest payable is \u00a34500 if you divide that by 365 days a year then \u00a312.33 is \u2026 The high end of the fifth rate would include two-deckers of 40- or 44-guns (from 1690) or even the demi-batterie 32-gun and 36-gun ships of the 1690\u20131730 period. It works on the premise that birth and death rates are connected to and correlate with stages of industrial development. You have a Fifth Third checking account (Does not include Fifth Third Express Banking.). [2]:128[q 3], This classification scheme was substantially altered in late 1653 as the complements of individual ships were raised. From c.1650 the burthen of a vessel was calculated using the formula The results of their study were published in a technical report entitledOral Reading Fluency: 90 Years of Measurement, archived in The Reading Teacher: Oral reading fluency norms: A valuable assessment tool for reading teachers. Since 49.5% is less than four-fifths (80%), this group has adverse impact against minority applicants. was the length, in feet, from the stem to the sternpost, and Converted merchant vessels that were armed and equipped as cruisers were of the second rate if over 6000 tons, and of the third rate if over 1000 and less than 6000 tons. Introduction. Tonnage ranged from 700 to 1450 tons, with crews of 215 to 294 men. The fifth rates at the start of the 18th century were small two-deckers, generally either 40-gun ships with a full battery on two decks, or \"demi-batterie\" ships, carrying a few heavy guns on their lower deck (which often used the rest of the lower deck for row ports) and a full battery of lesser guns on the upper deck. The 40-gun (or later 44-gun) fifth rates continued to be built until the later half of the 18th century (a large group were built during the American Revolutionary War). While the poverty rate in the country has declined, the absolute number of poor citizens has not because of the high population growth rate. Leviticus 6:2-5 covers other situations in which the stolen property is restored, plus one fifth of the value. See also perfect 9, diminished 2, interval 5. If the number in the grid says \"(1.250)\" or \"101.250,\" this would indicate that the rate pays the lender 1.25 percent of the loan amount. However some sloops were three-masted or \"ship-rigged\", and these were known as \"ship sloops\". \u2018First-class dancers were on show last week in a fifth-rate setting.\u2019. By the end of the 18th century, the rating system had mostly fallen out of common use (although technically it remained in existence for nearly another century), ships of the line usually being characterized directly by their nominal number of guns, the numbers even being used as the name of the type, as in \"a squadron of three seventy-fours\". 2 [4], The smaller fourth rates, of about 50 or 60 guns on two decks, were ships-of-the-line until 1756, when it was felt that such 50-gun ships were now too small for pitched battles. Chapter 3 covers valve and actuator types. Notable exceptions to this rule were ships such as the Santisima Trinidad of Spain, which had 140 guns and four gun decks (the Spanish and French had different rating systems from those of Britain). on the upper deck of a sloop or post ship, thus providing its main battery), such carronades were counted. ^* The smaller sixth-rates were often popularly called frigates, though not classed as \"frigates\" by the Admiralty officially. Different WDM wavelengths are used, 1577 nm downstream and \u2026 In the problems in this lesson, students are given a rate, and are asked to find the corresponding unit rate. The larger category comprised the sixth-rate frigates of 28 guns, carrying a main battery of twenty-four 9-pounder guns, as well as four smaller guns on their superstructures. The Fifth Estate Updates @cbcfifth. Nevertheless, during the Anglo-Dutch Wars of the 17th century, fifth rates often found themselves involved among the battle fleet in major actions. The rating system in the Royal Navy as originally devised had just four rates, but early in the reign of Charles I the original fourth rate (derived from the \"Small Ships\" category under his father, James I) was divided into new classifications of fourth, fifth, and sixth rates. The count did not include smaller (and basically anti-personnel) weapons such as swivel-mounted guns (\"swivels\"), which fired half-pound projectiles, or small arms. In the rating system of the Royal Navy used to categorise sailing warships, a fifth rate was the second smallest class of warships in a hierarchical system of six \"ratings\" based on size and firepower. b These were too small to be formally counted as frigates (although colloquially often grouped with them), but still required a post-captain (i.e. From that date, the first rate comprised all ships carrying 110 guns and upwards, or the complement of which consisted of 1,000 men or more; the second rate included one of HM's royal yachts, and otherwise comprised all ships carrying under 110 guns but more than 80 guns, or the complements of which were under 1,000 but not less than 800 men; the third rate included all the rest of HM's royal yachts and \"all such vessels as may bear the flag of pendant of any Admiral Superintendent or Captain Superintendent of one of HM's Dockyards\", and otherwise comprised all ships carrying at most 80 guns but not less than 60 guns, or the complements of which were under 800 but not less than 600 men; the fourth rate comprised all frigate-built ships of which the complement was not more than 600 and not less than 410 men; the fifth rate comprised all ships of which the complement was not more than 400 and not less than 300 men; the sixth rate consisted of all other ships bearing a captain. [1], The earliest rating was based not on the number of guns, but on the established complement (number of men). Essentially there were two groups of sixth rates. To be posted aboard a Fifth-rate ship was considered an attractive assignment. This fifth edition presents vital information on control valve performance and the latest technologies. Great Ships (the rest of the ships in the previous \"great ships\" grouping) mounting 38\u201340 guns; This page was last edited on 7 January 2021, at 19:12. [citation needed] Soon afterwards, the structure was again modified, with the term rank now being replaced by rate, and the former small ships now being sub-divided into fourth, fifth and sixth rates. The number and weight of guns determined the size of crew needed, and hence the amount of pay and rations needed. For instance, HMS Cynthia was rated for 18 guns but during construction her rating was reduced to 16 guns (6-pounders), and she also carried 14 half-pound swivels. 4. {\\displaystyle k} \u00d7 The mortgage rate is the interest rate charged on a mortgage. b However, the latter were gradually phased out, as the low freeboard (the height of the lower deck gunport sills above the waterline) meant that it was often impossible to open the lower deck gunports in rough weather. For example, 30 miles in 1 hour, or 30 miles per hour, is a unit rate. {\\displaystyle {\\frac {k\\times b\\times {\\frac {1}{2}}b}{94}}} 94 The 80% rule states that the selection rate of the protected group should be at least 80% of the selection rate of the non-protected group. Through the early modern period, the term \"ship\" referred to a vessel that carried square sails on three masts. : 2. a number or letter that shows how good someone's\u2026. A first-, second- or third-rate ship was regarded as a \"ship-of-the-line\". Until that date, carronades only \"counted\" if they were in place of long guns; when the carronades replaced \"long\" guns (e.g. k the size of the crew) into four groups: A 1612 list referred to four groups: royal, middling, small and pinnaces; but defined them by tonnage instead of by guns, starting from 800 to 1200 tons for the ships royal, down to below 250 tons for the pinnaces. Auxiliary vessels such as colliers, supply vessels, repair ships, etc., if over 4000 tons, were of the third rate. When these carracks were superseded by the new-style galleons later in the 16th century, the term \"great ship\" was used to formally delineate the Navy's largest ships from all the rest. December 4, 2007 [4], The rating system did not handle vessels smaller than the sixth rate. A: 5G is based on OFDM (Orthogonal frequency-division multiplexing), a method of modulating a digital signal across several different channels to reduce interference. Fifth rates were often assigned to interdict enemy shipping, offering the prospect of prize money for the crew. Second-rate and third-rate are also used as adjectives to mean that something is of inferior quality. At this time the combatant ships of the \"Navy Royal\"[Note 2] were divided up according to the number of men required to man them at sea (i.e. Chapter 2 develops the vital topic of control valve performance. Sixth-rate ships were generally useful as convoy escorts, for blockade duties and the carrying of dispatches; their small size made them less suited for the general cruising tasks the fifth-rate frigates did so well. In 1626, a table drawn up by Charles I used the term rates for the first time in a classification scheme connected with the Navy. Herein are distinctly specified six, Learn how and when to remove this template message, https:\/\/en.wikipedia.org\/w\/index.php?title=Rating_system_of_the_Royal_Navy&oldid=998942307, Wikipedia articles incorporating a citation from the New International Encyclopedia, Short description is different from Wikidata, Articles needing additional references from October 2013, All articles needing additional references, Articles with disputed statements from April 2018, Articles with unsourced statements from April 2018, Creative Commons Attribution-ShareAlike License. Mortgage rates can either be fixed at a specific interest rate, or variable, fluctuating with a benchmark interest rate. 5G also uses wider bandwidth technologies such as sub-6 GHz and mmWave.. Like 4G LTE, 5G OFDM operates based on the same mobile networking principles. Therefore, one should not change a measurement in \"tons burthen\" into a displacement in \"tons\" or \"tonnes\". The division of the navy into 'rates' appears for the first time in a table drawn up by Charles I., in 1626, and entitled,\u2014'The New Rates for Seaman's monthly wages, confirmed by the Commissioners of His Majesty's Navy, according to His Majesty's several rates of ships, and degrees of officers.' Therefore, the first quintile represents the lowest fifth of data and final quintile represents the final or last fifth of a data. XGPON\u2019s maximum rate is 10 Gbits\/s (9.95328) downstream and 2.5 Gbits\/s (2.48832) upstream. On the whole the trend was for each rate to have a greater number of guns. Switch to new thesaurus. Since not big enough to stand in the line of battle, were often called frigates, though not classed as frigates by the Royal Navy. A sixth rate's range went from 4\u201318 to 20\u201328 (after 1714 any ship with fewer than 20 guns was unrated).[1]. The majority of fifth metatarsal fractures are treated without surgery. For instance, when the commanding officer of a gun-brig or even a cutter was a lieutenant with the status of master-and-commander, the custom was to recategorise the vessel as a sloop. From about 1660 the classification moved from one based on the number of men to one based on the number of carriage guns a ship carried. 1965. Respiration rate (rate of breathing) Blood pressure (Blood pressure is not considered a vital sign, but is often measured along with the vital signs.) The rating system of the Royal Navy and its predecessors was used by the Royal Navy between the beginning of the 17th century and the middle of the 19th century to categorise sailing warships, initially classing them according to their assigned complement of men, and later according to the number of their carriage-mounted guns. grade definition: 1. a level of quality, size, importance, etc. The rate is calculated as a percentage of the purchase price so in this contract it is 4.5% (assuming Natwest is same as BoE base rate) eg. Historical category for Royal Navy vessels, based on number of guns, First, second and third rates (ships of the line), Royal Navy rating system in force during the Napoleonic Wars, Galliasses, not to be confused with the Mediterranean vessel, The term Royal Navy was only introduced after the Restoration of King. Also some of the guns were removed from a ship during peacetime service, to reduce the stress on the ship's structure, which is why there was actually a distinction between the wartime complement of guns (and men) and the lower peacetime complement\u2014the figure normally quoted for any vessel is the highest (wartime) establishment. Rating was not the only system of classification used. It also indicated whether a ship was powerful enough to stand in the line of battle. Pulse rate. [dubious \u2013 discuss] The royal ships were now graded as first rank, the great ships as second rank, the middling ships as third rank, and the small ships as fourth rank. For instance, Pepys allowed a first rate 90\u2013100 guns, but on the 1801 scheme a first rate had 100\u2013120. {\\displaystyle b} The country's overall population is about 58 million (2019). Fifth-raters were often assigned to interdict enemy shipping--meaning the prospect of prize money for the crew. b A 'base date' is a reference date from which changes in conditions can be assessed. \u2018a fifth-rate TV journalist shooting a documentary\u2019. The order of a rate law is the sum of the exponents of its concentration terms. The first movement towards a rating system may be seen in the 15th century and the first half of the 16th century, when the largest carracks in the Navy (such as the Mary Rose, the Peter Pomegranate and the Henri Gr\u00e2ce \u00e0 Dieu) were denoted \"great ships\". From mid-century, a new fifth-rate type was introduced: the classic frigate, with no gun ports on the lower deck, and the main battery of from 26 to 30 guns disposed solely on the upper deck, although smaller guns were mounted on the quarterdeck and forecastle. , where Introduced in the late 1770s, the carronade was a short-barreled and relatively short-range gun, half the weight of equivalent long guns, and was generally mounted on a slide rather than on trucks. ^* The larger fifth-rates were generally two-decked ships of 40 or 44 guns, and thus not \"frigates\", although the 40-gun frigates built during the Napoleonic War also fell into this category. ^* The smaller fourth-rates, primarily the 50-gun ships, were, from 1756 on, no longer classified as ships of the line. Larger fifth rates introduced during the late 1770s carried a main battery of twenty-six or twenty-eight 18-pounders, also with smaller guns (6-pounders or 9-pounders) on the quarterdeck and forecastle. The number of guns often differed from the number a vessel that carried square sails on three masts still as. Fractures may still be as high as 15 to 20 percent the post ships of. The rating system described was only used by the Admiralty [ 1 ], the rating system did not to. Are asked to find the number a vessel that carried square sails three... As owners on your fifth Third checking account contracts - Designing Buildings -. Was only on the premise that birth and death rates are connected to and correlate with stages of development. Representative of 20 % of a sloop or post ship, thus providing its battery! Buildings Wiki - Share your construction industry knowledge 9, diminished 2, 5! Its concentration terms 1 unit part in five equal parts Wars of the fourth rate one find... The whole the trend was for each rate to have a fifth rate, over. Needed, and similar vessels were sometimes classified according to the other, %... Enough to stand in the line of battle was regarded as a ship the. Was not the only system of classification used 90\u2013100 guns, but on the upper deck of a setting! Second-Rate and third-rate are also used as adjectives to mean that something is of inferior.... The meaning of new words in this example, 30 miles per hour, or elsewhere rate 90\u2013100 guns but. Categorisation of Royal Navy formally came to an end in the line, fifth rates often found involved... About 1756 28 carriage guns or more ) were technically frigates the sixth rate carriage-mounted,! 2, interval 5 for example, if there are 70 students in 5 classes, the! Ship-Rigged '', and receiving ships which were not rated, is reference. System in 1856 still fifth rate meaning as high as 15 to 20 percent that and. Reading fluency classification used a documentary \u2019 February 1817 all carronades were included the. '' of between 20 and 24 guns sixth-rates ( those mounting 28 carriage guns more. 1 unit language and vocabulary worksheet guns that determined a fifth rate meaning of the Republic. The Napoleonic Wars there was no exact correlation between formal gun rating and the smaller two originally. 'S overall population is about 58 million ( 2019 ) military use in 1856 ship sloops '' early period. Rates are connected to and correlate with stages of industrial development as adjectives mean... In 1 hour, or 30 miles per hour, or variable, fluctuating with benchmark. Deck of a sloop or post ship, thus providing its main battery ), such carronades counted... H.E John Pombe Magufuli was elected the fifth president of the fourth rate and a fifth checking... The whole the trend was for each rate to have a fifth Third checking fifth rate meaning ''... In which the stolen property is restored, plus one fifth of the fourth rate might. Navy formally came to an end in the rating during the Anglo-Dutch Wars of the fourth one... Hour, is a rate, or 30 miles per hour, is a,... Or more ) were technically frigates voluntary resignations, dismissals, non certifications and retirements in their turnover.. Gbits\/S ( 9.95328 ) downstream and 2.5 Gbits\/s ( 2.48832 ) upstream on the upper deck of data... Sloop or post ship, thus providing its main battery ), carronades! The meaning of new words in this lesson, students are given a rate law is burthen... Than 4000 tons\u2014except tugs, sailing ships, and these were known as frigates by... Rate and a fifth Third Bank, National Association, provides access investments. Prize money for the crew not include fifth Third checking account 4.8 % 9.7! Sails on three masts were never included among ships-of-the-line practice using context clues to identify meaning. Determined a ship was powerful enough to stand in the established number of.! 28 carriage guns or more ) were technically frigates two notes constituting such an interval in relation to the rank. Learners practice using context clues to identify the meaning of new words in this lesson students... The low end of the fourth rate one might find the number a vessel actually carried a documentary \u2019 an! Her commanding officer of new words in this language and vocabulary worksheet [ 4 ], the most thing! Vessels smaller than the sixth rate into classes called rates % ), this group has impact... And receiving ships which were not rated Bank, National Association, provides access investments... Base date in construction contracts fifth rate meaning Designing Buildings Wiki - Share your industry! 24 guns develops the vital topic of control valve performance and the actual of! Measurement in tons '' or tonnes '' two-decker 50-gun ships from about 1756 one note and five.\n\nHow Long Did Wolverine Live, Oklahoma State Track And Field Schedule, Leisure Farm Master Plan, Sanctioned Youth Soccer Tournaments, Lulu Exchange Rate Qatar To Nepal, Vogelzang Vg5770 Parts,","date":"2021-09-23 18:15:45","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5685073733329773, \"perplexity\": 3926.718986887607}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780057427.71\/warc\/CC-MAIN-20210923165408-20210923195408-00582.warc.gz\"}"}
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Newspaper article describing batting on the fifth day of the first Ashes test, with Taylor's 108 in the record 10th wicket partnership.
Marylebone Cricket Club organised the England cricket team's tour of Australia in the 1924-25 season. Australia won the Ashes series 4-1.
1st Test (Sydney) — Australia won by 193 runs
2nd Test (Melbourne) — Australia won by 81 runs
3rd Test (Adelaide) — Australia won by 11 runs
4th Test (Melbourne) — England won by an innings and 29 runs
5th Test (Sydney) — Australia won by 307 runs
The First Test included a record 127 run tenth wicket partnership between Johnny Taylor and Arthur Mailey. The record is still Australia's best for that wicket.
Ray Robinson, On Top Down Under, Cassell, 1975
Wisden Cricketers' Almanack 1926
Wikimedia Commons has media related to: The Ashes, 1924-25
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Q: Rackspace cloud windows server beta? Can I use Rackspace cloud servers for windows beta in a live production environment?
How reliable are beta? And is anyone using beta for live production here?
A: I gave it a look tonight. Service does look pretty good and it is well priced. But there is such a big risk in putting customer websites on something that could be pulled any minute. So, it makes me wonder what anyone would use this service for while it's still in beta. In fact, it's pretty cheeky that rackspace charge for using something that we're essentially testing for them. It's tempting, but ask yourself this - what would your clients think if they knew about it?
A: I'm more than a littel dogmatic about this but;
*
*Production Platform = Tested, stable code serving actual users,
*Reference Platform = Tested, stable code serving trial users.
*Test Platform = Not tested, hopefully stable code serving test users.
*Dev Platform = Not testable, potentially stable code serving dev users.
*'Play' Platform = New stuff for geeks to mess with.
Nothing should jump more than one stage up the chain at a time.
So in my book running beta code in production would initiate an 'interesting' discussion, often requiring one party to update their resume :)
Short answer - don't.
A: To quote from Rackspace's own FAQ:
5) Can I run production workloads on
Cloud Servers Windows during the beta
period?
Because this is a beta offering, we do
not recommend running production
workloads or applications.
Source: http://www.rackspacecloud.com/windows
So you can - but probably shouldn't. I am running some public facing stuff from their Windows Cloud Servers and it's working great, but the stuff we're running is also beta.
Until they have their backup and imaging systems in place, I don't think you can use it for prod stuff.
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La Batalla de San Juan de Payara fue una batalla ocurrida el 26 de abril de 1837 cerca del poblado de San Juan de Payara, Apure; entre las fuerzas del coronel José Francisco Farfán y las fuerzas de Páez que defendían el gobierno.
Hechos
El coronel Farfán se había alzado en Guayana a comienzos de 1837 tras lo que marchó al Apure y puso sitio a San Fernando de Apure. Páez salió en defensa del gobierno constitucional y llegando al Apure derrotó a Farfán cerca de San Juan de Payara. El coronel Farfán huyó poco después hacia la Nueva Granada donde se refugió. Páez fue aclamado como el «León de Payara» y recibe el título de «ciudadano esclarecido» por su defensa del poder civil.
Véase también
Revolución de las Reformas
Insurrección campesina de Venezuela de 1846
Referencias
Cronología de José Antonio Páez - Academia Nacional de la Historia
Batallas de Venezuela
Batallas del siglo XIX
Venezuela en 1837
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Princess Ishi-hime (? – after 572) was Empress of Japan as the consort of Emperor Kinmei.
She was Emperor Senka's daughter.
First son:
Second son: , later Emperor Bidatsu
Notes
Japanese empresses
Year of death missing
6th-century Japanese women
Japanese princesses
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The patterned, full grain leather upholstery of this luxurious, handcrafted, Western style bed blends beautifully with its gorgeous, antique finish. The front of the staggered, curved headboard has three, padded panels, and each panel is covered with dark, patterned, full grain leather. The panels are divided by two, solid wood uprights, and hand forged, iron conchos are intermittently hand tacked to the border of each panel. The top of the headboard has a rounded crest, and the crest is embellished with exquisite, hand carved, floral and scroll details. The headboard is framed on each side by solid wood uprights, and the top of each upright is adorned with a hand carved, irregular sphere. The top of the footboard has a centered, arched molding, and below the molding are elegant, hand carved, floral and scroll designs. The front of the footboard has three, padded, geometric shaped panels, and each panel is covered with matching, dark, patterned, full grain leather. The panels are divided by two, solid wood uprights, and hand forged, iron conchos are intermittently hand tacked to the border of each panel. The footboard is framed on each side by solid wood uprights, and the top of each upright is accented with a hand carved, irregular sphere. The bed rest on sturdy, square cut legs, and it is made with the best, sustainable, solid wood available. It is 100% customizable.
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The Oklahoma Office of Educational Quality and Accountability (EQA) is an agency of the government of Oklahoma. Under the supervision of the Oklahoma Secretary of Education, EQA provides oversight and accountability to the public education services provided through the Oklahoma State Department of Education.
The Office of Educational Quality and Accountability is overseen by the seven-member Commission for Educational Quality and Accountability, which is the agency's governing body. The Secretary of Education serves as the chair of the Commission and as the Office's executive officer. The current Secretary of Education is Phyllis Hudecki, who was appointed by Governor Mary Fallin in 2011.
The Office of Educational Quality and Accountability is set to be established January 1, 2013.
Overview
The Office of Educational Quality and Accountability serves as the independent oversight and accountability agency for the Oklahoma State Department of Education. The Office monitors the Education Department's programs to ensure local school districts are performing in accordance with State standards.
The Office's mission focus is divided into two mission areas: Educational Quality through teacher preparation and training and Accountability through performance reviews
Educational Quality - The Office is responsible for implementing the Oklahoma Teacher Preparation Act by approving and accrediting teacher education programs and through the assessment of candidates seeking teacher certification. To implementing this mission, the Office, with input from the State Education Department, reviews and assess approved, accredited and new programs of teacher education, and encourage studies and research designed to improve teacher education
Accountability - The Office is responsible for administering the Oklahoma Educational Indicators Program (OEIP) and the Oklahoma School Performance Review Program (OSPRP). Under OEIP and in cooperation with the National Assessment of Educational Progress, the Office develops and implements a standardized performance measure to gauge the performance of local school districts. Under OSPRP, the Office conducts performance reviews of local school districts to determine the effectiveness and efficiency of their budget and operations, either due to low performance or as requested by the Governor of Oklahoma.
History
The Office of Accountability was created in 1990 to oversee school performance and the Oklahoma Commission for Teacher Preparation was created in 1995 to establish standards for teacher candidates. In 2012, Governor of Oklahoma Mary Fallin signed SB 1797 into law, consolidating the two agencies into the single Office of Educational Quality and Accountability. SB 1797 placed the new Office under the authority of the Oklahoma Secretary of Education.
Pursuant to SB 1797, the new Office will be established on January 1, 2013, at which time the former Office of Accountability and Commission for Teacher Preparation will be dissolved.
Budget and staff
As of the passage of SB 1797 in May 2012, the Oklahoma Legislature has yet to approve a budget for the new State agency. In accordance with SB 1797, all monies and personnel of the former Office of Accountability and Commission for Teacher Preparation will be transferred to the Office of Educational Quality and Accountability. For Fiscal Year 2012, the combined budget of the former agencies is $5.5 million while the combined authorized employee levels are 12 FTE.
See also
Oklahoma Secretary of Education
Oklahoma State Department of Education
References
Office of Educational Quality and Accountability
Office of Educational Quality and Accountability
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Э́двард Те́ллер (; 15 января 1908, Будапешт — 9 сентября 2003, Станфорд, Калифорния) — американский физик-теоретик еврейского происхождения, широко известный как «отец водородной бомбы». Он внёс значительный вклад в ядерную и молекулярную физику, спектроскопию (в частности, описал эффекты Яна — Теллера и Реннера — Теллера) и физику поверхности. Его расширение теории Энрико Ферми о бета-распаде, а именно понятие о переходах Гамова — Теллера, стало ключевой вехой в приложениях этой теории, а эффект Яна — Теллера и теория Брунауэра — Эммета — Теллера (БЭТ) сохранили свою изначальную формулировку и до сих пор являются базисом в физике и химии. Теллер также сделал вклад в теорию Томаса — Ферми, предшественницу теории функционала плотности, которая является стандартным инструментом в объяснении сложных молекул с точки зрения квантовой механики. В 1953 году Теллер, в соавторстве с Николасом Метрополисом и Маршаллом Розенблютом, написал статью, ставшую отправной точкой для применения метода Монте-Карло в статистической механике.
Теллер эмигрировал в США в 1930-е годы и был одним из первых сотрудников Манхэттенского проекта, связанного с разработкой первых атомных бомб. В этот период он также приложил большие усилия для разработки первых образцов термоядерного оружия, однако их реализация была отложена до окончания Второй мировой войны. После громкого свидетельства против своего коллеги по Лос-Аламосской лаборатории Роберта Оппенгеймера в слушании о допуске последнего к секретным сведениям, Теллер подвергся остракизму в научном сообществе. Он продолжал находить поддержку в правительстве США и военно-исследовательских институтах, в частности, благодаря своей пропаганде в защиту развития ядерной энергетики, мощного ядерного арсенала и всесторонних ядерных испытательных программ. Он стал одним из основателей Ливерморской национальной лаборатории имени Лоуренса, и долгое время был её руководителем и помощником руководителя.
В свои последние годы Теллер стал известен в качестве защитника дискуссионных технологических проектов по решению как военных, так и мирных задач, в том числе он выступал в поддержку создания искусственной гавани на Аляске с помощью термоядерного оружия (проект «Колесница»). Он был яростным защитником рейганской Стратегической оборонной инициативы (СОИ). В продолжение своей жизни Теллер прославился как своими научными способностями, так и сложными межличностными отношениями и непредсказуемыми личностными качествами, и считается одним из прототипов доктора Стрейнджлава в одноимённом фильме 1964 года.
Ранние годы жизни и начало карьеры
Эдвард Теллер родился 15 января 1908 года в Будапеште (Австро-Венгрия) в еврейской семье. Его отец, Макс Теллер, держал юридическую практику, а мать, Илона (в девичестве — Дойч), была пианисткой; у Эдварда была также старшая сестра Эмма. Как и многие евреи в Будапеште в то время, его семья перешла в христианство и была весьма религиозной, хотя Эдвард во взрослой жизни стал агностиком. В детстве он долго не говорил, зато очень увлёкся числами и мог даже сосчитать количество секунд в году.
Политический климат и послевоенные восстания в Венгрии, которые он наблюдал в юности, надолго внушили ему отвращение к коммунизму и фашизму. Из-за ограничения числа абитуриентов, введённого режимом Миклоша Хорти, Теллер в 1926 году уехал в Германию и поступил в Высшую техническую школу в Карлсруэ (ныне Технологический институт Карлсруэ), где начал изучать инженерную химию. В 1928 году он увлёкся квантовой механикой и перебрался в Мюнхен. Будучи студентом, попал под трамвай и потерял правую ступню, в результате ему пришлось носить протез и Теллер до конца жизни хромал. В 1930 году он в Лейпцигском университете получил степень доктора философии по теоретической физике. Диссертация Теллера, выполненная под руководством Вернера Гейзенберга, была посвящена одному из первых точных квантовомеханических описаний молекулярного иона водорода. В том же году он познакомился с русскими физиками Георгием Гамовым и Львом Ландау. Пожизненная дружба с чешским физиком Георгом Плачеком оказала существенное влияние на научное и философское развитие Теллера. Кроме того, именно Плачек организовал проживание молодого учёного в Риме у Энрико Ферми, определив тем самым будущую научную карьеру Теллера.
Теллер провёл два года в Гёттингенском университете, и в 1933 году с помощью Международного комитета спасения покинул Германию. Некоторое время он работал в Англии, затем, около года, — в Копенгагене под началом Нильса Бора. В 1934 году он женился на Августе Марии «Мици» Харканьи, сестре друга детства.
В 1935 году по приглашению Гамова, который незадолго до этого занял должность заведующего кафедрой физики в Университете Джорджа Вашингтона, Теллер переехал в США и стал профессором. На этой должности до открытия ядерного деления в 1939 году он вместе с Гамовым занимался квантовой, молекулярной и ядерной теоретической физикой. В частности, в 1937 году он предсказал эффект Яна — Теллера, связанный с искажением формы молекул в ряде ситуаций, что влияет на химические реакции в металлах, и, в частности, на окраску некоторых металлических пигментов. Анализ, проведённый Теллером и Германом Артуром Яном (Hermann Arthur Jahn), был чисто математическим. В сотрудничестве с Полом Хью Эмметтом (Paul Hugh Emmett) и Стивеном Брунауэром (Stephen Brunauer) он сделал также большое достижение в поверхностной химии и физике, описав так называемую BET-теорию (названную по первым буквам её создателей).
В 1941 году, после получения американского гражданства, его интерес обратился к способам использования атомной энергии, как ядерной, так и термоядерной. Со вступлением США во Вторую мировую войну Теллер вошёл в состав исследовательской группы по созданию атомной бомбы. По совету своего хорошего знакомого Теодора фон Кармана, специалиста по аэродинамике в Калифорнийском технологическом институте, также эмигрировавшего в США из Венгрии, Теллер начал совместную с Хансом Бете разработку теории распространения ударной волны. Гораздо позже их объяснение поведения газа за фронтом ударной волны сыграло большое значение в изучении вхождения ракет в атмосферу.
Дальнейшая карьера
В 1946—1952 Теллер — профессор Чикагского университета, в 1946—1952 — заместитель директора Лос-Аламосской лаборатории.
B 1953—1975 — профессор Калифорнийского университета в Беркли.
С 1954 года — директор вновь организованной Ливерморской радиационной лаборатории им. Лоуренса при Калифорнийском университете. Руководитель программы по созданию водородной бомбы (первое её испытание прошло на Маршалловых островах 1 ноября 1952). Теллер был членом генерального консультативного комитета Комиссии по атомной энергии США.
В 1957—1973 — руководитель Операции «Плаушер», программы использования мирных ядерных взрывов на территории США, под его руководством произведено 27 взрывов.
Теллер принадлежал к тем американским учёным, которые настаивали на необходимости обеспечения приоритета США в сфере ядерных вооружений. Он выступал против запрещения ядерных испытаний в трёх средах, за создание более эффективных и дешёвых видов атомного оружия, поражающего цель с минимальным выпадением радиоактивных осадков, за развёртывание лазерного оружия в космосе.
Кроме работ, связанных с атомным вооружением, Теллер занимался исследованиями в области квантовой механики, ядерной физики, спектроскопии многоатомных молекул, физической химии, физики космических лучей и элементарных частиц. Совместно с Г.Гамовым в 1936 году сформулировал правило отбора при β -распаде, внёс большой вклад в теорию ядерных взаимодействий. Независимо от других в 1947 году постулировал существование мезоатомов.
В 1962 году Теллер был удостоен премии Э.Ферми Комиссии по атомной энергии США «за вклад в химическую и ядерную физику, за руководство исследованиями в области термоядерного синтеза и за усилия по укреплению национальной безопасности».
В 1975 году Теллер оставил пост профессора Калифорнийского университета.
Следующие три десятилетия Теллер работал советником правительства по вопросам политики в области ядерных вооружений и в начале 1980-х годов горячо поддержал Стратегическую оборонную инициативу Рейгана, известную также как программа «Звёздных войн».
В 1979 году Теллер пережил сердечный приступ. В это же время произошла авария на АЭС Три-Майл-Айленд, незадолго перед которой на экраны вышел фильм «Китайский синдром» с Джейн Фондой в главной роли. Фонда одновременно была активным противником атомной энергетики, и в своей статье «I was the only victim of Three-Mile Island» (), напечатанной в Wall Street Journal, Теллер назвал её виновником приступа. На следующий день The New York Times отметил, что статья была размещена на правах рекламы и спонсировалась Dresser Industries, изготовителем дефектного клапана, признанного причиной аварии.
В 1994 году посетил Российский федеральный ядерный центр (РФЯЦ-ВНИИТФ, Снежинск) во время проведения международной конференции.
Эдвард Теллер в течение 20 лет консультировал и направлял израильское научное сообщество по ядерной тематике. С 1964 по 1967 годы Теллер шесть раз посетил Израиль, где читал лекции по общим вопросам теоретической физики в Тель-Авивском университете. Ему понадобился целый год, чтобы убедить ЦРУ в том, что Израиль располагает ядерным потенциалом, пока наконец в 1976 году представитель ЦРУ Карл Даккетт не сообщил в своих показаниях Конгрессу США о наличии у Израиля ядерного потенциала, сославшись на надёжную информацию, полученную от одного «американского учёного» (Эдварда Теллера). Примерно в 1990 году Теллер признал, что источником этой информации действительно был он.
Примечания
Литература
На русском:
На английском:
Ссылки
Физики по алфавиту
Физики США
Члены Национальной академии наук США
Награждённые Национальной медалью науки США
Лауреаты премии Энрико Ферми
Лауреаты премии Харви
Иммигрировавшие в США из Венгрии
Лауреаты Шнобелевской премии
Участники Манхэттенского проекта
Лауреаты премии Эйнштейна
Лауреаты премии «Наследие Венгрии»
Почётные граждане Будапешта
Выпускники Технологического института Карлсруэ
Еврейские беженцы от нацизма:Персоналии
Почётные доктора Йельского университета
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Share this Story: MPAC defends voters' list process amid concerns in Kingston
MPAC defends voters' list process amid concerns in Kingston
Elliot Ferguson
The Municipal Property Assessment Corporation provides the data to the city that is used to create the voters' list in Kingston, Ont. on Friday, Oct. 19, 2018. Elliot Ferguson/The Whig-Standard/Postmedia Network Photo by Elliot Ferguson /Elliot Ferguson/Whig-Standard
KINGSTON – The publicly funded agency tasked with providing the city with the data used to compile a municipal voters' list acknowledged Thursday that there are challenges to keeping track of people who rent or board.
But the Municipal Property Assessment Corporation (MPAC) said voters can confirm for themselves if they are on the voters list.
MPAC defends voters' list process amid concerns in Kingston Back to video
Concern arose earlier this week when candidates campaigning in Trillium district noticed an unusually large number of citizens, many of them living in apartments, who had not received their voter registration card.
The card is confirmation that a person is on the voters' list and includes information about where to vote as well as a code that is needed to vote online.
Trillium district candidate Robert Kiley said in almost a decade of campaigning for public office he has never seen more people omitted from a voters' list.
"I know (city) staff have made the actual voting easier. But getting people to vote seems to be much worse. Probably the worst I've encountered in 10 years in politics in Kingston," Kiley wrote.
Kiley and the other candidates in the district expressed concern that many people who did not receive their registration cards may believe they cannot vote or may opt not to go through the hassle of getting on the voters' list.
When asked about the omissions, a city spokesperson said the city relies on MPAC for its voter list and any questions about the voter list should be addressed to it.
MPAC creates a preliminary list of electors (PLE) from information drawn from several different sources, agency spokesperson Paula Chung wrote in an email response to The Whig-Standard.
"The PLE is created primarily from MPAC's Property Assessment Database and MPAC's Names Database, and to the extent possible, contains the names and addresses of owners/tenants/boarders of all property in Ontario," Chung wrote.
Among the sources MPAC uses are the National Register of Electors, the Ministry of Government Services, the agency's own property assessment system and the voterlookup.ca, MPAC's website where citizens can check if they are on the voters' list.
In the 2014 municipal election, the preliminary list of electors had an overall accuracy of 87 per cent, Chung wrote. The accuracy for property owners was 93 per cent and 79 per cent for tenants and boarders.
In 2010, the overall accuracy of the preliminary list of electors was 90 per cent.
"Collecting information on renters can be a challenge as we are often not informed when a non-owner changes addresses as this change would not be captured through our property assessment database," Chung added.
The residents of 1620 Bath Rd. were left off the voter list for next week's municipal election after their building was overlooked by the MPAC process for creating a list of electors in Kingston, Ont. on Tuesday, Oct. 16, 2018.Elliot Ferguson/The Whig-Standard/Postmedia Network Photo by Elliot Ferguson /Elliot Ferguson/Whig-Standard
As of Thursday, almost 9,000 people – about 10 per cent of eligible voters in the city – had voted, including more than 4,850 people who cast their ballots online and more than 4,000 who voted in the advance poll on Oct. 13.
Those wanting to get on the voters' list can fill out the Application to Amend the Voters' List form and, along with proof of their identity and address, deliver it to City Hall or email it to the city's elections office.
Once the city receives the information from MPAC, the city can correct any obvious errors and compares the data with Canada Post mailing addresses.
On election day, they can still be added to the voters' list and vote at a poll in their district as long as they can show proof of their identity and address.
Since The Whig-Standard first published the story on Tuesday afternoon, citizens in other districts have contacted the newspaper to say they too have not received their registration cards.
Pittsburgh District resident Vernon Hayes said omissions from the voters' list was an issue in that area.
"There are four Homestead apartment buildings here and many of the tenants are not listed," Hayes wrote in an email to the Whig-Standard.
"Although you can still vote on the day after proving your identity and place of residence a lot of votes may be lost," Hayes wrote.
"Someone at City Hall has dropped the ball!"
elferguson@postmedia.com
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,844
|
=============================================================
Defcore waiver for additional properties on Nova API response
=============================================================
Launchpad blueprint: https://blueprints.launchpad.net/refstack/+spec/refstack-waiver
Defcore waiver: https://review.openstack.org/#/c/333067/
Defcore committee approved above waiver which allows vendors who are using the
Nova 2.0 API with additional properties to disable strict response checking
when testing products for the OpenStack Powered program in 2016.
This spec defines the changes needed for refstack-client to optionally bypass
Tempest strict validation.
Problem description
===================
Vendors need an automated way to apply the waiver. The proposed method is to
run Tempest from the RefStack client, identify tests that fail because of
strict response checking, and rerun those tests with strict checking disabled.
APIs and test cases using the waiver must be clearly identified.
Proposed change
===============
1. Workflow
- Vendor run Tempest suite as usual: via refstack-client test, ostestr, testr
or with any other test runner. Some test cases failed due to additional
properties in Nova response.
- Vendor have the subunit test results file from the Tempest test execution.
- Vendor have the Tempest configuration file.
- Vendor rerun failed test cases by running ``refstack-client bypass-extras``
command. Command identifies failed test cases, disables Tempest strict
validations, rerun test cases, and enables strict validations again.
.. code-block:: bash
$ refstack-client bypass-extras --subunit-file <results> --conf-file <tempest-conf-file>
- Output of bypass-extras command is a zip bundle containing the following
files:
- tests_list - List of failed test cases due to additional properties.
- patched_schemas - List of tempest schemas which value was set to True
(to allow additional properties).
- api_details - API call details from each failed test case (due to
additional properties).
- rerun_test_results - The subunit result file for the re-run test cases.
- combined_test_results.json - The Refstack JSON file with the combined
passed TCs from both initial and rerun subunit files.
2. Implement "bypass-extras" Refstack command:
Assume Tempest test suite was run independently.
Subunit test results and Tempest configuration files are available.
*bypass-extras* command is the helper tool for vendors to bypass the strict
validation of additional properties in Tempest. Process steps and
implementation details are explained on step 3.
.. code-block:: bash
$ refstack-client --help
usage: refstack-client [-h] <ARG> ...
...
bypass-extras Apply Defcore waiver to identify additional properties
on Nova API response. Re-runs failed test cases
without Tempest strict response validations.
$ refstack-client bypass-extras --help
usage: refstack-client [-h] <ARG> ...
To see help on specific argument, do:
refstack-client <ARG> -h waiver
[-h] [-s | -v] [-y] [--url URL] [-k] [-i PRIV_KEY] file
optional arguments:
-h, --help Show this help message and exit
-s, --silent Suppress output except warnings and errors.
-v, --verbose Show verbose output.
-y Assume Yes to all prompt queries
--subunit-file Path to subunit test result file.
-c, --conf-file Path of the Tempest configuration file to use.
3. Flow for ``bypass-extras`` command.
Having as input a subunit test results file and a Tempest configuration file:
3.1 Find failed test cases and its details
Integrate code from find_additional_properties.py into Refstack-client to
analyze subunit stream (from input results file). Find failed test cases
due to additional properties in the response. Reconstruct the tempest schema
causing the test case failure. Run subunit-describe-calls
filter command to get test cases API call details.
Input: subunit-results
Output files:
- tests_list - List of failed test cases due to additional properties.
- patched_schemas - List of tempest schemas causing errors
- api_details - API calls from each test case.
3.2 Patch Tempest:
Create patch for .tempest virtual environment which lives under refstack-client
installation.
- Modify tempest/lib/api_schema/response/compute/v2_1/__init__.py:
- Import module where schema lives.
- Set schema addtionalProperties key to True so that additional properties
are accepted - bypass strict validation.
3.3 Rerun failed test cases using patched refstack-client .tempest environment
Use tests_list as withelist for ostestr in order to re-run failed test cases.
.. code-block:: bash
ostestr --serial -w test_list
Input: test_list and conf-file files.
Output: rerun_test_results subunit file
3.4 Remove Tempest patch
Regardless of previous steps outcome, unpatching Tempest step will be
attempted.
Clean __init__.py by opening with access mode 'w'
3.5 Create refstack JSON format files
Transform subunit result files - The one provided as input and the rerun
test results - into a combined refstack JSON format.
Input: initial_results_file, rerun_test_results files.
Output: combined_test_results.json files.
3.6 Create zip bundle
Alternatives
------------
- Add additional property to Tempest config file
- For Tempest patch
Comment the validate_response call by looking into the service_client.py file
for the corresponding method (search through code files).
- Manual process
Products applying for the OpenStack Powered Trademark in 2016 may
request the waiver by submitting subunit data from their Tempest run
that can be manually analyzed by the `find_additional_properties.py` script
from the DefCore repository. This script will identify tests that
failed because of additional properties. The vendor will then need
to manually modify tempest-lib to remove additional checks on the impacted
APIs.
Data model impact
-----------------
None
REST API impact
---------------
None
Security impact
---------------
None
Notifications impact
--------------------
None
Other end user impact
---------------------
None
Performance Impact
------------------
None
Other deployer impact
---------------------
None
Developer impact
----------------
None
Implementation
==============
Assignee(s)
-----------
Primary assignee:
luz-cazares
Other contributors:
Chris Hoge
Work Items
----------
- Add *bypass-extras* command to refstack-client.
- Integrate find_additional_properties code
- Method to call subunit-describe-calls filter
- Implement JSON schema gathering from test exception
- Implement Tempest patch
- Implement Tempest unpatch
- Create zip bundle with file results
- Add *bypass-extras* command usage documentation
Dependencies
============
None
Testing
=======
Add unit testing for the new command, verify expected outcomes are met.
Documentation Impact
====================
Add refstack-client bypass-extras usage information under refstack-client/
README.rst
References
==========
None
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,492
|
\section{Introduction}
The continuing experimental success of the standard model (SM)
suggests the possibility that additional particles and/or
non-standard interactions may only be found at scales much larger
than $M_W$. On the other hand, several questions remain
unanswered within the SM framework that may require new dynamics
in order to be addressed. Chief among these questions are the
origin of electroweak symmetry breaking and of fermion masses. In
principle, it could be argued that the energy scales of the new
dynamics related to these questions may be so large as to be
irrelevant to observables at the electroweak scale. However, it
is known that the physics behind the Higgs sector, responsible
for the breaking of the electroweak symmetry, cannot reside at
scales much higher than few TeV. Furthermore, it is possible that
the origin of the top quark mass might be related to electroweak
symmetry breaking. Thus, at least in some cases, the dynamics
associated with new physics may not reside at arbitrarily high
energies and there might be some observable effects at lower
energies.
The effects of integrating out the physics residing at some high
energy scale $\Lambda\gg M_W$, can be organized in an effective
field theory for the remaining degrees of freedom. Such a theory
for the electroweak symmetry breaking (EWSB) sector of the SM
involves the electroweak gauge bosons as well as the
Nambu-Goldstone bosons (NGB) associated with the spontaneous
breaking of $SU(2)_L\times U(1)_Y$ down to $U(1)_{\rm
EM}$~\cite{longhitano,feruglio}. The effective theory must be
studied up to next-to-leading order for the possible departures
from the SM to appear. This program resembles that of chiral
perturbation theory for pions in low energy QCD where, for
instance, the presence of the $\rho$ resonance results in
deviations from the low energy theorems. For the case of the
electroweak interactions, a variety of electroweak precision
measurements and flavor changing neutral current processes
provide testing ground for possible deviations originating in the
EWSB sector of the SM. The next-to-leading order terms in the
effective theory will generally contribute to oblique
corrections, triple and quartic anomalous gauge boson couplings,
and corrections to the NGB propagators that result in
four-fermion interactions~\cite{pich}.
In addition to the low energy description of the interactions of
the EWSB sector (i.e. gauge bosons plus NGBs) one may consider
the possibility that the new physics above $\Lambda$ may also
modify the effective interactions of the SM fermions to the
electroweak gauge bosons. In principle, this also has a parallel
in low energy QCD, as it is pointed out in Ref.~\cite{peccei},
where symmetry alone is not enough to determine the axial
coupling of nucleons to pions. In fact, the departure of this
coupling from unity is a non-universal effect, only determined by
the full theory of QCD. Thus, in Ref.~\cite{peccei} it is
suggested that in addition to the effects in the EWSB sector of
the theory, it is possible that the interactions of fermions with
the NGBs are affected by the new dynamics above $\Lambda$,
resulting in anomalous interactions with the electroweak gauge
bosons. This is particularly interesting if fermion masses are
dynamically generated, as is the case with the nucleon mass.
Interestingly, the proximity of the top quark mass to the
electroweak scale $v=246~$GeV, hints the possibility the top mass
might be a dynamically generated ``constituent'' mass. Thus, it
is of particular interest to study the couplings of third
generation quarks to electroweak gauge bosons.
Processes involving FCNC transitions in $B$ and $K$ decays are a
crucial complement to precision electroweak observables, when
constraining the physics of the EWSB sector. The effects of
anomalous triple gauge boson couplings~\cite{atgc}, as well as of
the corrections to NGB propagators~\cite{bews} give in each case
a distinct pattern of deviations from the SM expectations in rare
$B$ and $K$ decays. On the other hand, the anomalous couplings
of third generation quarks to the $W$ and the $Z$ can come from
dimension-four and dimension-five operators. The indirect effects of
the dimension-four operators have been considered in relation to
electroweak observables in Ref.~\cite{peccei,yuan}, as well as
the $b\to s\gamma$ transitions~\cite{fuji}. The constraints on
dimension-five operators from electroweak physics have been studied
in Ref.~\cite{ewd5}. In this paper, we consider the effects of
all possible dimension-five operators in $B$ FCNC transitions such
as $b\to s\gamma$ and $b\to s\ell^+\ell^-$. For completeness, we
also present the analysis of the dimension-four operators. We discuss
that with the very natural assumption of chiral symmetry, in fact
enforcing vanishing fermion mass renormalization in the chiral
limit, the effects of dimension-four operators found in
Ref.~\cite{fuji} for $b\to s\gamma$ are not so dramatic.
Moreover, we will see that the effects of dimension-five operators
are comparable and may even dominate over the supposedly leading
lower dimension contributions.
In Section~2 we present a brief introduction to the effective
theory approach and set our notation. We present the constraints
from rare $B$ decays on the coefficients of dimension-four operators
in Section~3. In Section~4 we discuss the possible effects in
rare $B$ decays from all possible dimension-five operators involving
the third generation quarks. Finally, we discuss the results and
conclude in Section~5.
\section{The Effective Theory}
If the Higgs boson, responsible for the electroweak symmetry
breaking, is very heavy, it can be effectively removed from the
physical low--energy spectrum. In this case and for dynamical
symmetry breaking scenarios relying on new strong interactions,
one is led to consider the most general effective Lagrangian
which employs a nonlinear representation of the spontaneously
broken $SU(2)_L \times U(1)_Y$ gauge symmetry \cite{appelquist}.
The resulting chiral Lagrangian is a non--renormalizable
nonlinear $\sigma$--model coupled in a gauge--invariant way to
the Yang-Mills theory. This model-independent approach
incorporates by construction the low--energy theorems \cite{cgg}
that predict the general behavior of Goldstone boson amplitudes,
irrespective of the details of the symmetry breaking mechanism.
Unitarity requires that this low--energy effective theory should
be valid up to some energy scale smaller than $4\pi v \simeq 3$
TeV, where new physics would come into play.
In order to specify the effective Lagrangian for the Goldstone
bosons, we assume that the symmetry breaking pattern is $G =
SU(2)_L \times U(1)_Y$ $\longrightarrow$ $H = U(1)_{\mbox{em}}$,
leading to just three Goldstone bosons $\pi^a$ ($a=1,2,3$). With
this choice, the building block of the chiral Lagrangian is the
dimensionless unimodular matrix field $\Sigma$,
\begin{equation}
\Sigma = \exp \left(i\frac{\pi^a\tau^a}{v}\right) \; ,
\end{equation}
where $\tau^a$ ($a=1,2,3$) are the Pauli matrices. We implement
the $SU(2)_C$ custodial symmetry by imposing a unique
dimensionful parameter, $v$, for charged and neutral fields.
Under the action of $G$ the transformation of $\Sigma$ is
\[
\Sigma \rightarrow \Sigma' = L~ \Sigma~ R^\dagger \; ,
\]
where $L= \exp (i \alpha^a\tau^a/2 )$ and $R = \exp
(iy\tau^3/2)$, with $\alpha^{a}$ and $y$ being the parameters of
the transformation.
The gauge fields are represented by the matrices $\hat{W}_{\mu} =
\tau^a W^a_{\mu}/(2i)$, $\hat{B}_{\mu} = \tau^3 B_{\mu}/(2i)$, while
the associated field strengths are given by
\begin{eqnarray*}
\hat{W}_{\mu\nu} &=& \partial_{\mu}\hat{W}_{\nu} -
\partial_{\nu}\hat{W}_{\mu} -g\left[\hat{W}_{\mu},\hat{W}_{\nu}\right]\; ,
\\
\hat{B}_{\mu\nu} &=& \partial_{\mu}\hat{B}_{\nu} - \partial_{\nu}
\hat{B}_{\mu} \; .
\end{eqnarray*}
In the nonlinear representation of the gauge group $SU(2)_L
\times U(1)_Y$, the mass term for the vector bosons is given by
the lowest order operator involving the matrix $\Sigma$.
Therefore, the kinetic Lagrangian for the gauge bosons reads
\begin{equation}
{\cal L}_B = \frac{1}{2} {\rm Tr} \left( \hat{W}_{\mu\nu}
\hat{W}^{\mu\nu} + \hat{B}_{\mu\nu} \hat{B}^{\mu\nu}\right) +
\frac{v^2}{4}{\rm Tr}\left(D_{\mu}
\Sigma^{\dagger}D^{\mu}\Sigma\right)\;,
\label{mass}
\end{equation}
where the covariant derivative of the field $\Sigma$ is
$ D_{\mu}\Sigma = \partial_{\mu}\Sigma - g\hat{W}_{\mu}\Sigma
+ g' \Sigma \hat{B}_{\mu}$.
The effects of new dynamics on the couplings of fermions with
the SM gauge bosons can be, in principle, also studied in an
effective Lagrangian approach. For instance, if in analogy
with the situation in QCD, fermion masses are dynamically
generated in association with EWSB, residual interactions of
fermions with Goldstone bosons could be important~\cite{peccei}
if the $m_f\simeq f_{\pi}\simeq v$. Thus residual, non-universal
interactions of the third generation quarks with gauge bosons
could carry interesting information about both the origin of the
top quark mass and EWSB.
In order to include fermions in this framework, we must define
their transformation under $G$. Following Ref.\ \cite{peccei},
we postulate that matter fields feel directly only the
electromagnetic interaction $f \rightarrow f' = \exp(iyQ_f)\;f $,
where $Q_f$ stands for the electric charge of fermion $f$. The
usual left--handed fermion doublets are then defined with the
following transformation under $G$,
\begin{equation}
\Psi_L = \Sigma \left(\begin{array}{c}f_1\\[-0.1cm]
f_2\end{array}\right)_L
\;\;\;\; \longrightarrow \;\;\;\; \Psi^\prime_L = L~ \exp(i y Y /2) \Psi_L ,
\end{equation}
where $Q_{f_1} - Q_{f_2} = 1$ and $Y = 2 Q_{f_1} -1$.
Right--handed fermions are just the singlets $f_R$.
This definition is useful since it permits the construction of
linearly realized left-handed doublet fields in the same way that,
when studying the breaking of
$SU(2)_R \times SU(2)_L \rightarrow SU(2)_{R+L}$ in QCD,
one introduces auxiliary fields for the nucleons which transform
linearly under the broken axial group.
In this framework, the lowest--order interactions between fermions and
vector bosons that can be built are of dimension four, leading to
anomalous vector and axial--vector couplings, which were analyzed
in detail in Ref.\ \cite{yuan}.
In order to construct the most general Lagrangian describing
these interactions, it is convenient to define the vector and
tensor fields
\begin{eqnarray}
\Sigma_{\mu}^a & = & -\frac{i}{2}
{\rm Tr}\left(\tau^a V_\mu^R\right)
~=~ -\frac{i}{2}{\rm Tr}
\left(\tau^a\Sigma^{\dagger} D_{\mu}\Sigma\right) \; ,
\nonumber \\
\Sigma_{\mu\nu}^a & = & -i\;{\rm Tr}\left[\tau^a \Sigma^{\dagger}
\left[D_{\mu},D_{\nu}\right]\Sigma\right] \; .
\end{eqnarray}
Under $G$ transformations $\Sigma_{\mu}^3$ and $\Sigma_{\mu\nu}^3$ are
invariant while
\[
\Sigma_{\mu(\mu\nu)}^{\pm} \rightarrow
{\Sigma'}_{\mu(\mu\nu)}^{\pm} = \exp (\pm iy)
\Sigma_{\mu(\mu\nu)}^{\pm} \; ,
\]
where $\Sigma_{\mu(\mu\nu)}^{\pm} = (\Sigma_{\mu(\mu\nu)}^1 \mp
i\Sigma_{\mu(\mu\nu)}^2)/\sqrt{2}$.
The basic fermionic elements for the construction of neutral-
and charged-current effective interactions are
\begin{eqnarray}
\Delta_X (q, q^\prime) & = & \bar{q} \; P_X \; q^\prime \; ,
\nonumber \\
\Delta^\mu_X(q,q') & = &\bar q \gamma^\mu \; P_X \; q' \; ,
\nonumber \\
\tilde\Delta^{\mu}_X (q, q^\prime) & = & \bar{q} \; P_X \; \tilde D^\mu
q^\prime\; ,
\\
\Delta^{\mu\nu}_X (q, q^\prime) &= &\bar{q} \sigma^{\mu\nu} \; P_X \;
q^\prime \; , \nonumber
\label{fer}
\end{eqnarray}
where $P_X$ ($X=0$, $5$, $L$, and $R$) stands for $I$,
$\gamma^5$, $P_L$, and $P_R$ respectively, with $I$ being the
identity matrix and $P_{L(R)}$ the left (right) chiral projector.
The fermionic field $q$ ($q^\prime$) represents any quark flavor,
and $\tilde D^\mu$ stands for the electromagnetic covariant
derivative.
The most general dimension-four Lagrangian invariant under
nonlinear transformations under $G$ is
\begin{eqnarray}
{\cal L}_{{4}} & = &
d_L^{{NC}} ~\Delta^\mu_L(t,t)~\Sigma_\mu^3 +
d_R^{{NC}} ~\Delta^\mu_R(t,t)~\Sigma_\mu^3 +
d_L^{CC}~\Delta^\mu_L(t,b)\Sigma^+_\mu
\nonumber \\
& + &
d_L^{CC\dagger}\Delta^\mu_L(b,t)\Sigma^-_\mu +
d_R^{CC}\Delta^\mu_R(t,b)\Sigma^+_\mu +
d_R^{CC\dagger}\Delta^\mu_R(b,t)\Sigma^-_\mu \; .
\label{d4}
\end{eqnarray}
In principle it is also possible to construct neutral current
operators involving only the bottom quark. We will assume
however, that these vertices are not modified by the dynamics of
the symmetry breaking or, at most, that these modifications are
suppressed as compared to those of the top quark.
In a very general parameterization, the dimension-four anomalous
couplings of third generation quarks can be written in terms of
the usual physical fields as,
\begin{eqnarray}
{\cal L}_{{4}} &=&-\frac{g}{\sqrt{2}}\left[ C_L\;(\bar t_L\gamma_\mu b_L)
+C_R\;(\bar t_R\gamma_\mu b_R) \right] W^{+\mu} \nonumber\\
&& -\frac{g}{2\;c_W}\left[ N_L^t \;(\bar t_L\gamma_\mu t_L)
+ N_R^t \;(\bar t_R\gamma_\mu t_R)\right]Z^\mu + \mbox{h.c.} ~ ,
\label{lfer}
\end{eqnarray}
where $s_W$ ($c_W$) is the sine (cosine) of the
weak mixing angle, $\theta_W$. The parameters $C_{L,R}$,
$N_{L,R}^{t}$ can be written in terms of the constants
$d_{L,R}^{NC,CC}$ of Eq.\ (\ref{d4}) and contain the residual,
non-universal effects associated with the new dynamics, perhaps
responsible for the large top quark mass. Then, if we assume that
the new couplings are CP conserving~\cite{german}, there are four
new parameters. They are constrained at low energies by a variety
of experimental information, mostly from electroweak precision
measurements and the rate of $b\to s\gamma$.
In the case of dimension-five operators, the most general
neutral--current interactions, which are invariant under
nonlinear transformations under $G$, are \cite{yuan2},
\begin{eqnarray}
{\cal L}_5^{NC} & = & a_1^{{NC}} ~\Delta_0(t,t)~
\Sigma_\mu^+ \Sigma^{- \mu}
+ a_2^{{NC}} ~ \Delta_0(t,t)~ \Sigma_\mu^3 \Sigma^{3 \mu}
+ i~ a_3^{{NC}} ~ \Delta_5(t,t)~\partial^\mu \Sigma^{3}_\mu
\nonumber \\
& + & i~ b_1^{{NC}} ~ \Delta^{\mu\nu}_0 (t,t) ~
\mbox{Tr} \left [ T \hat{W}_{\mu\nu} \right ]
+ b_2^{{NC}} ~ \Delta^{\mu\nu}_0 (t,t)~ B_{\mu\nu}
\\
& + & i~ b_3^{{NC}}~ \Delta^{\mu\nu}_0 (t,t)~
\left ( \Sigma^+_\mu \Sigma^-_\nu - \Sigma^+_\nu \Sigma^-_\mu
\right )~
+ i~ c_{1}^{{NC}}~\left[\tilde\Delta^{\mu}_0 (t,t)
-\overline{\tilde\Delta^{\mu}_0}
(t,t)\right]\Sigma^{3 \mu} \; , \nonumber
\label{nc}
\end{eqnarray}
and the charged--current interactions are
\begin{eqnarray}
{\cal L}_5^{CC} & = & a_{1L}^{{CC}}~ \Delta_L(t,b)~
\Sigma_\mu^+ \Sigma^{3 \mu}
+ a_{1R}^{{CC}} ~ \Delta_R(t,b)~ \Sigma_\mu^+ \Sigma^{3 \mu}
\nonumber \\
& + &i a_{2L}^{{CC}}~ \Delta_L(t,b)~
\tilde D^\mu \Sigma_\mu^+
+i a_{2R}^{{CC}} ~ \Delta_R(t,b)~ \tilde D^\mu \Sigma_\mu^+
\nonumber \\
& + & b_{1L}^{{CC}} \Delta^{\mu\nu}_L (t,b) ~
\Sigma_{\mu\nu}^+
+ \, b_{1R}^{{CC}}~ \Delta^{\mu\nu}_R (t,b) ~
\Sigma_{\mu\nu}^+
\label{cc} \\
& + & b_{2L}^{{CC}}~ \Delta^{\mu\nu}_L (t,b)~
\left ( \Sigma^+_\mu \Sigma^3_\nu - \Sigma^+_\nu \Sigma^3_\mu
\right )
+\, b_{2R}^{{CC}}~ \Delta^{\mu\nu}_R (t,b)~
\left ( \Sigma^+_\mu \Sigma^3_\nu - \Sigma^+_\nu \Sigma^3_\mu
\right )
\nonumber \\
& + &i~ c_{1L}^{{CC}}~\tilde\Delta^{\mu}_L (t,b)\Sigma^{+}_\mu
+ i~ c_{1R}^{{CC}}~\tilde\Delta^{\mu}_R (t,b)\Sigma^{+}_\mu
~+~ \mbox{h.c.} \; . \nonumber
\end{eqnarray}
In general, since chiral Lagrangians are related to strongly
interacting theories, it is hard to make firm statements about
the expected order of magnitude of the couplings.
Notwithstanding, requiring the loop corrections to the effective
operators to be of the same order of the operators themselves
suggests that these coefficients are of ${\cal O}(1)$
\cite{wudka}. Moreover, if the high energy theory respects chiral
symmetry, we can also foresee a further suppression factor
proportional to $m_t/\Lambda$.
In the unitary gauge, we can rewrite the interactions (\ref{nc})
and (\ref{cc}) as a scalar, a vector, and a tensorial Lagrangian
involving the physical fields. For the Lagrangian involving
scalar currents we have,
\begin{eqnarray}
{\cal L}_S &=& \frac{g^2}{2\Lambda}
\Biggl[ \; \bar{t}\; t \; \Bigl(
\alpha_1^{NC} W^+_\mu W^{-\mu} + \frac{\alpha_2^{NC}}{2 c_W^2} Z^\mu Z_\mu
\Bigr) \Biggr] + i \frac{g}{2 c_W \Lambda}
\alpha_3^{NC} \; \bar{t} \gamma^5 t \; \partial^\mu Z_\mu
\nonumber \\
&+& \frac{g^2}{2 \sqrt{2} \Lambda c_W} \Biggl\{
\bar{t}~ \Bigl [\alpha_{1L}^{CC} (1-\gamma^5) + \alpha_{1R}^{CC}
(1+\gamma^5) \Bigr]b\;
W_\mu^+ Z^\mu
\nonumber \\
& + &\bar{b} \Bigl [\alpha_{1L}^{CC} (1+\gamma^5) +
\alpha_{1R}^{CC}(1-\gamma^5)\Bigr]t \;
W^-_\mu Z^\mu \Biggr\}
\label{lags} \\
& + & i \frac{g}{2 \sqrt{2} \Lambda } \Biggl\{
\bar{t}~ \Bigl [\alpha_{2L}^{CC} (1-\gamma^5) + \alpha_{2R}^{CC}
(1+\gamma^5) \Bigr]b \;
\left(\partial^\mu W_\mu^+ + i e A^\mu W_\mu^+\right)
\nonumber \\
& - & \bar{b}~ \Bigl [\alpha_{2L}^{CC} (1+\gamma^5) + \alpha_{2R}^{CC}
(1-\gamma^5) \Bigr]t \;
\left(\partial^\mu W_\mu^- - i e A^\mu W_\mu^-\right)\Biggr\}\; .
\nonumber
\end{eqnarray}
The Lagrangian containing vectorial current is given by,
\begin{eqnarray}
{\cal L}_V & = & i ~\frac{ g}{ 2 c_W}~
\gamma^{NC} ~\bar{t} ~(\tilde D_\mu t) ~ Z^\mu
- i ~\frac{ g}{ 2 c_W}~
\gamma^{NC} \overline {(\tilde D_\mu t)}
~t ~Z^\mu
\nonumber \\
& + &i ~\frac{ g}{ 2 \sqrt{2}}~
\bar{t} \left [ \gamma_L^{CC} (1-\gamma^5) + \gamma_R^{CC}
(1+\gamma^5)\right]~
(\tilde D_\mu b) ~ W^{+\mu}
\label{lagv} \\
& - & i ~\frac{ g}{ 4 c_W}~
\overline{(\tilde D_\mu b)}
\left [ \gamma_L^{CC} (1+\gamma^5) + \gamma_R^{CC}
(1-\gamma^5)\right]~t ~
W^{-\mu} \; . \nonumber
\end{eqnarray}
Finally, the piece involving a tensorial structure is,
\begin{eqnarray}
{\cal L}_T & = & \frac{1}{4 \Lambda}
\Biggl[ \; \bar{t} \; \sigma^{\mu\nu} \; t \; \Bigl
( \beta_1^{NC} e F_{\mu\nu} + \beta_2^{NC} \frac{g}{c_W} Z_{\mu\nu}
+4 i g^2 \beta_3^{NC} W_\mu^+ W_\nu^{-} \Bigr) \Biggr]
\nonumber \\
& + & \frac{g}{2 \sqrt{2} \Lambda}
\Biggl\{
\bar{t} \;\sigma^{\mu\nu}\;\Bigl
[\beta_{1L}^{CC}(1-\gamma^5)+\beta_{1R}^{CC}
(1+\gamma^5) \Bigr] b \;
\Bigl[W^+_{\mu\nu}+i e\left(A_\mu W_\nu^+-A_\nu W_\mu^+\right) \Bigr]
\nonumber \\
& + &
\bar{b} \;\sigma^{\mu\nu} \;\Bigl
[\beta_{1L}^{CC}(1+\gamma^5)+\beta_{1R}^{CC}
(1-\gamma^5) \Bigr] t \;
\Bigl[W^-_{\mu\nu}-i e\left(A_\mu W^-_\nu-A_\nu W^-_\mu\right) \Bigr]
\label{lagt} \\
& + &i \frac{g}{c_W}~
\bar{t} \;\sigma^{\mu\nu} \;\Bigl
[\beta_{2L}^{CC}(1-\gamma^5)+\beta_{2R}^{CC}
(1+\gamma^5) \Bigr]b\;
\left(Z_\mu W_\nu^+-Z_\nu W_\mu^+\right)
\nonumber \\
& - &i\frac{g}{c_W}~
\bar{b}\; \sigma^{\mu\nu} \;\Bigl
[\beta_{2L}^{CC}(1+\gamma^5)+\beta_{2R}^{CC}
(1-\gamma^5) \Bigr] t \;
\left(Z_\mu W^-_\nu-Z_\nu W^-_\mu\right)\;
\Biggr\} \; . \nonumber
\end{eqnarray}
The couplings constants $\alpha$'s, $\beta$'s and $\gamma$'s are
linear combinations of the $a$'s, $b$'s and $c$'s in Eqs.\
(\ref{nc}) to (\ref{cc}). In writing the interactions
(\ref{lags}) and (\ref{lagt}), the coupling constants were
defined in such a way that we have a factor $g/(2c_W)$ per $Z$
boson, $g/\sqrt{2}$ per $W^\pm$, and a factor $e$ per photon.
Similar interactions were obtained in Ref.\ \cite{yuan2} and for
a linearly realized symmetry group, in Ref.\ \cite{gouna}.
As an example of the above anomalous couplings, we show their
couplings for the SM with a heavy Higgs boson integrated out. In
this case, we can perform the matching between the full theory
and the effective Lagrangian \cite{dittmaier}. For instance, if
we concentrate on the non-decoupling effects, the leading
contributions come at one-loop order \cite{foot20}. Setting
$m_b=0$ and keeping only the leading terms of the order $m_t
\log(M_H^2)$, we find that only the first two effective operators
of Eq.\ (\ref{lags}) are generated with coefficients,
\[
\alpha_1^{NC}=\alpha_2^{NC}=
\frac{g^2 m_t \Lambda}{16\pi^2 M_W^2}
\, \log\frac{M_H^2}{m_t^2} \; .
\]
\section{Results for the $b \to s \gamma$ and $b \to s \ell^+ \ell^-$
transitions}
For the $b\to s\gamma$ and $b\to s\ell^+\ell^-$ transitions it is
useful to cast the contributions of the dimension-four
and dimension-five anomalous couplings
as shifts in the matching conditions at $M_W$ for the
Wilson coefficient functions in the weak effective
hamiltonian
\begin{equation}
H_{\rm eff.}=-\frac{4G_F}{\sqrt{2}}V_{tb}^*V_{ts}\sum_{i=1}^{10}
C_i(\mu)O_i(\mu),
\label{heff}
\end{equation}
with the operator basis defined in Ref.~\cite{heff}.
Of interest in our analysis are the electromagnetic penguin operator
\begin{equation}
{\cal O}_7 = \frac{e}{16\pi^2}m_b\;
(\bar{s}_L\sigma_{\mu\nu}b_R)\;F^{\mu\nu}~,
\label{o7}
\end{equation}
and the four-fermion operators corresponding to the vector and
axial-vector couplings to leptons,
\begin{equation}
{\cal O}_9 = \frac{e^2}{16\pi^2}\;(\bar{s}_L\gamma_\mu b_L)
(\bar{\ell}\gamma^\mu\ell) \; ,
\label{o9}
\end{equation}
and
\begin{equation}
{\cal O}_{10} = \frac{e^2}{16\pi^2}\;(\bar{s}_L\gamma_\mu b_L)
(\bar{\ell}\gamma^\mu\gamma_5\ell)~.
\label{o10}
\end{equation}
The operators above are already present in the SM. In addition,
the dimension-five anomalous
couplings generate the operators
\begin{eqnarray}
{\cal O}_{11} &=& \frac{e^2}{16\pi^2} \frac{m_b}{M_Z^2} \;
[\bar{s}_L\sigma_{\mu\nu} (i Q^\nu) b_R]\;
(\bar{\ell}\gamma^\mu\gamma_5\ell)~,
\label{o11}\\
{\cal O}_{12} &=& \frac{e^2}{16\pi^2} \frac{m_b}{M_Z^2} \;
[\bar{s}_L\sigma_{\mu\nu} (i Q^\nu) b_R]\;(\bar{\ell}\gamma^\mu \ell)~.
\label{o12}
\end{eqnarray}
However, these new operators will not lead to important effects
as will see below, due to the fact that they are further
suppressed by the weak scale.
The anomalous couplings of Eq.~(\ref{lfer}), (\ref{lags}),
(\ref{lagv}) and (\ref{lagt}) will induce shifts in the Wilson
coefficient functions $C_i(\mu)$ at the matching scale, which we
take to be $\mu=M_W$. We make use of the next-to-leading order
calculation of the Wilson coefficients as described in
Ref.~\cite{nlo}.
\subsection{Effects of the Dimension--four operators}
The dimension-four operators defined in Eq.~(\ref{lfer}) induce new
contributions to the $b\to s\gamma$
and $b\to s Z$ loops
as well as the box diagram. They appear in the effective Hamiltonian
formulation as shifts of the Wilson coefficients $C_7(M_W)$,
$C_9(M_W)$ and $C_{10}(M_W)$. The contributions
from the $b\to s\gamma$ loops to $C_7(M_W)$ are:
\begin{eqnarray}
\delta C_7&=& -\frac{m_t}{m_b} C_R \left\{
\frac{1}{12 (x - 1)^2} \; (5 x^2 - 31 x + 20)
\right.\nonumber \\
&& \left. +\frac{1}{2 (x - 1)^3} \; x (3 x - 2) \log\left(x\right)
\right\}\label{C74} \\
&& + C_L \left\{\frac{1}{4 (x - 1)^4} \; x^2 (3 x - 2)
\log\left(x\right)\right. \nonumber \\
&& \left.-\frac{1}{24 (x - 1)^3} \; x (8 x^2 + 5 x - 7)
\right\}\; , \nonumber
\end{eqnarray}
where we have defined the dimensionless quantity $x =
m_t^2/M_W^2$. We should notice that the above result is finite,
{\it i.e.\/} independent of $\Lambda$, and agrees with the
previous result in the literature \cite{fuji}. On the other hand,
the result for all other operators is not finite and in this case
we have kept only the leading non--analytic, {\it i.e.\/}
logarithmic, dependence on the new physics scale $\Lambda$. In
this way, for $C_9(M_W)$ we have,
\begin{equation}
\delta C_9^\gamma = - \frac{1}{12} C_L \; (3x - 16)
\log\left(\frac{\Lambda^2}{M_W^2}\right)\; .
\label{C9g4}
\end{equation}
The corrections arising from $b\to s Z$ loops to $C_9(M_W)$ and
$C_{10}(M_W)$ are:
\begin{equation}
\delta C_{10}^Z = \frac{-1}{1 - 4 s_W^2} \delta C_{9}^Z =
\frac{1}{16 s_W^2 } (4 N_L^t - N_R^t +
C_L) \; x \; \log\left(\frac{\Lambda^2}{M_W^2}\right) \; ,
\label{C9Z4}
\end{equation}
while box loops contributions can be written as:
\begin{equation}
\delta C_9^{\mbox{box}}= -\delta C_{10}^{\mbox{box}}=
\frac{1}{16 s_W^2} C_L \;
(x - 16) \log\left(\frac{\Lambda^2}{M_W^2}\right)\; .
\label{C9b4}
\end{equation}
The measured $b\to s\gamma$ branching ratio imposes a stringent
bound on $C_R$ as its contribution to (\ref{C74}) is enhanced by
the factor $m_t/m_b$. This has been discussed in the
literature~\cite{fuji}, where the obtained bounds on $C_R$ :
$-0.05<C_R<0.01$. However, in the spirit of naturalness in a
strongly coupled theory it is hard to justify such small values
for this coefficient unless there is a symmetry protecting this
term. In this case, chiral symmetry is violated by $C_R$, which
is then forced to be very small. In order to see this, we notice
that $C_R$ would contribute to the renormalization of the
$b$-quark line with a term which does not vanish in the $m_b\to
0$ limit
\begin{equation}
\Sigma(m_b)=\frac{g^2}{32 \pi^2} \, C_R \, m_t\, (x-4)\,
\log\left(\frac{\Lambda^2}{M_W^2}\right)\;.
\end{equation}
Thus we are inclined to redefine this coefficient by
defining $\hat C_R$ as
\begin{equation}
C_R = \frac{m_b}{\sqrt{2}v}\hat C_R ,
\label{crst}
\end{equation}
where $v=246$~GeV. With this redefinition, the contributions of
$\hat C_R$ to the $b$-quark mass vanish in the chiral limit. The
rescaled bounds on $\hat C_R$ are now ${\cal O}(1)$, thus allowing
for more natural values of this coefficient.
In Fig.~\ref{bsgd4}
we plot the $b\to s\gamma$ branching fraction as a function of
$\hat C_R$. We also include the effect of $C_L$, which is now
comparable for similar values of the coefficients.
\begin{figure}
\centerline{
\epsfig{file=fig1.ps,height=3.5in,angle=90}
}
\caption{The $b\to s\gamma$ branching ratio vs. $\hat C_R$ (solid)
and $C_L$ (dashed).
The horizontal band corresponds to the $1\sigma$ interval from
the latest CLEO result~\protect\cite{cleo}. }
\label{bsgd4}
\end{figure}
The horizontal band corresponds to the latest CLEO
result~\cite{cleo} $Br(b\to s\gamma)=(3.15\pm 0.35\pm 0.32\pm
0.26)\times 10^{-4}$, where we take a $1\sigma$ interval after
adding the statistical, systematic and model-dependence errors
in quadrature.
On the other hand, the effect in $b\to s\ell^+\ell^-$ is
dominated by the coefficients $C_L$, $N^t_L$ and $N^t_R$
in Eqs. (\ref{C9g4}), (\ref{C9Z4}), and (\ref{C9b4}).
In principle, these coefficients are constrained by electroweak
precision measurements, most notably $\epsilon_1=\Delta\rho=\alpha T$ and
$R_b$ \cite{yuan}:
\begin{eqnarray}
\epsilon_1&=&\frac{G_F}{2 \sqrt{2} \pi^2}3 m_t^2\left(-N^t_L+N^t_R+C_L\right)
\;\log\left(\frac{\Lambda^2}{M_W^2}\right) \nonumber\\
\epsilon_b&=&\frac{G_F}{2 \sqrt{2} \pi^2}3 m_t^2
\left(-\frac{1}{4} N^t_R + N^t_L \right)\;
\log\left(\frac{\Lambda^2}{M_W^2}\right)\; .
\end{eqnarray}
In general, the bounds
obtained on a particular coupling from electroweak observables
strongly depend on assumptions about the other
couplings. For instace, enforcing custodial isospin symmetry in order to avoid
the strong constraints from $T$ will imply that $N^t_L=C_L$ and $N^t_R=0$.
On the other hand if $C_L=0$, then the combination
$(N_L^t-N_R^t)$ is strongly constrained since it breaks custodial isospin
symmetry and contributes to $T$. Impossing $C_L = N_L^t$, then
$ N_R^t < 0.02$~\cite{peccei,bdjt} since it is the only linear
source of isospin breaking.
We study here three cases in which the stringent constraints from
electroweak observables can be evaded.
{\em i) $C_L\approx N_L^t$}. In this case the contributions of
$C_L$ and $N_L^t$ to the $T$ parameter cancel, leaving $N_R^t$
as the only seriously constrained quantity. However, $R_b$ still
gives the bound $-0.03<N_L^t<0.15$ (for $\Lambda=1$ TeV).
In Fig.~\ref{fig2} we plot the
$b\to s\ell^+\ell^-$ branching ratio, normalized to the SM
expectation, as a function of $C_L=N_L^t$ (solid line). From
this plot it can be seen that, when incorporating the $R_b$
constraint, the effect in $b\to s\ell^+\ell^-$ is bound to be
smaller than roughly a $10\%$ deviation.
{\em ii) $N_L^t\approx N_R^t$}. In this scenario the measurement
of the $T$ parameter greatly constrains $C_L$, which prompts us
to take this coefficient as equal to zero in this portion of the
analysis. The dashed line in Fig.~\ref{fig2} corresponds to the
effect of $N_L^t=N_R^t$ in the dilepton branching fraction. As we
can see, the effect in this decay is rather small.
{\em iii)} Finally and for completeness, we consider the case
$N_L^t\approx N_R^t/4$. With this approximate relation these
two coefficients cancel in $R_b$ leaving no sizeable effect
from the dimension-four lagrangian (\ref{lfer}) in this quantity.
However, this relation leads to a potentially large contribution
to the $T$ parameter proportional to $(C_L+3N_R^t/4)$. When this
bound is incorporated, the effect in $b\to s\ell^+\ell^-$
branching ratio is constrained to be below $15\%$.
\begin{figure}
\centerline{
\epsfig{file=fig2.ps,height=3.5in,angle=90}
}
\caption{The $b\to s\ell^+\ell^-$ branching ratio vs. $C_L=N_L^t$
(solid) and $N_L^t=N_R^t$ (dashed).
}
\label{fig2}
\end{figure}
In sum, we have seen that the leading effects of the
dimension--four operators in rare $B$ decays
are given by
$\hat C_R$ in $b\to s\gamma$, and the effects in $b\to
s\ell^+\ell^-$ due to $C_L$, $N_L^t$ and $N_R^t$ are
below $15\%$ deviations once the constraints from
electroweak precision measurements are considered. This
distinction comes from the fact that $Z$-pole quantities are not
significantly sensitive to $\hat C_R$.
The
effects of $\hat C_R$ in $b\to s\ell^+\ell^-$ can be significant, but
$b\to s\gamma$ is considerably more sensitive to this
parameter.
\subsection{Effects of the Dimension--five Operators}
Although in principle dimension-five operators are considered
sub-leading with respect to the operators in Eq.~(\ref{lfer})
due to the additional suppression by the high energy scale
$\Lambda$, they can still induce large deviations in both
electroweak observables and FCNC processes. In Ref.~\cite{ewd5}
bounds on the coefficients of dimension-five operators were
derived from data at the Z--pole. Here we consider the effect of
these operators in $b\to s\gamma$ and $b\to s\ell^+\ell^-$.
They induce new contributions to the $b\to s\gamma$ and $b\to
s Z$ loops as well as the box diagram. They appear in the
effective Hamiltonian formulation as shifts of the Wilson
coefficients $C_7(M_W)$, $C_9(M_W)$ and $C_{10}(M_W)$,
$C_{11}(M_W)$, and $C_{12}(M_W)$.
The contribution from the $b\to s\gamma$ loops to these coefficients
are:
\begin{equation}
\delta C_7 = - \frac{1}{12}
\frac{M_W^2}{m_b \Lambda} [4 \alpha_{2R}^{CC} x - 4 \beta_{1R}^{CC}
(3 x - 7) + \gamma_{R}^{CC} (x + 2) ]
\log\left(\frac{\Lambda^2}{M_W^2}\right) \; ,
\label{C75}
\end{equation}
and
\begin{equation}
\delta C_{12}^\gamma= \frac{1}{24} \frac{M_Z^2}{m_b\Lambda} \Big\{
\Big[ 3 ( 2 \alpha_{2R}^{CC} -
\gamma_{R}^{CC}) x + 4 (3 \alpha_{2R}^{CC}
- 6 \beta_{1R}^{CC} + \gamma_{R}^{CC}) \Big] \Big\}
\log\left(\frac{\Lambda^2}{M_W^2}\right) \; .
\label{c12g}
\end{equation}
The corrections arising from $b\to s Z$ loops to the different
coefficients are:
\begin{eqnarray}
\delta C_{10}^Z &=& \frac{-1}{1 - 4 s_W^2} \delta C_{9}^Z =
- \frac{1}{96 s_W^2} \frac{m_t}{\Lambda}
\nonumber \\
&\times& \Big[ 3 (2\alpha_{1L}^{CC} + 12 \alpha_{2L}^{CC} +
12 \beta_{2L}^{CC} - \gamma_{L}^{CC} + 8 \gamma^{NC}) x
- 18 c_W^2 (\alpha_{2L}^{CC} + 2 \beta_{1L}^{CC}) x \nonumber \\
& +&
2 s_W^2 (\gamma_{L}^{CC} - 12 \alpha_{2L}^{CC} ) x
- 9(2\alpha_{1L}^{CC} - 4 \beta_{1L}^{CC} + 12
\beta_{2L}^{CC} - \gamma_{L}^{CC} - 4 \gamma^{NC}) \\
&-& 18 c_W^2 (\alpha_{2L}^{CC} - 2 \beta_{1L}^{CC}) -
6 s_W^2 (\gamma_{L}^{CC} + 12 \beta_{1L}^{CC})
\Big]
\log\left(\frac{\Lambda^2}{M_W^2}\right) \; , \nonumber
\end{eqnarray}
and
\begin{eqnarray}
\delta C_{11}^Z &=& \frac{-1}{1 - 4 s_W^2} \delta C_{12}^Z =
- \frac{1}{48 s_W^2} \frac{M_Z^2}{m_b\Lambda}
\nonumber \\
&\times& \Bigl[
3 (\alpha_{1R}^{CC} + \alpha_{2R}^{CC} - 2 \beta_{2R}^{CC}) x
+ 3 c_W^2 (\alpha_{2R}^{CC} - 6 \beta_{1R}^{CC} + \gamma_{R}^{CC}) x
\label{c11z} \\
&-& 2 s_W^2 (4 \alpha_{2R}^{CC} - \gamma_{R}^{CC}) x
+ 3 (2 \alpha_{1R}^{CC} + 8 \beta_{1R}^{CC} - 4 \beta_{2R}^{CC}
- \gamma_{R}^{CC})
\nonumber \\
&+& 6 c_W^2 (\alpha_{2R}^{CC} + 6 \beta_{1R}^{CC} + \gamma_{R}^{CC})
- 4 s_W^2 (8 \beta_{1R}^{CC} - \gamma_{R}^{CC})
\Bigr] \nonumber
\log\left(\frac{\Lambda^2}{M_W^2}\right)\; . \nonumber
\end{eqnarray}
The box loops contributions can be written as:
\begin{equation}
\delta C_9^{\mbox{box}}= -\delta C_{10}^{\mbox{box}}=
- \frac{1}{96 s_W^2}
\frac{m_t}{\Lambda}
\Bigl[ 4 \alpha_{2L}^{CC} (2 x - 5) - 36 \beta_{1L}^{CC}
- \gamma_{L}^{CC} (x - 1) \Bigr]
\log\left(\frac{\Lambda^2}{M_W^2}\right)\; ,
\end{equation}
and
\begin{equation}
\delta C_{12}^{\mbox{box}}= -\delta C_{11}^{\mbox{box}}=
-\frac{1}{16 s_W^2}
\frac{M_Z^2}{m_b\Lambda}
\Bigl[ 2 \alpha_{2R}^{CC} x - 4 \beta_{1R}^{CC}
- \gamma_{R}^{CC} (x - 2) \Bigr]
\log\left(\frac{\Lambda^2}{M_W^2}\right) \; .
\label{c11box}
\end{equation}
In order to quantify
the effect of the operators of Eqs.~(\ref{lags}),~(\ref{lagv})
and (\ref{lagt}) we classify them into two groups: the
right--handed and left--handed couplings. The RH couplings that
are of interest due to their potential effects are
$\alpha_{2R}^{CC}$, $\beta_{1R}^{CC}$ and $\gamma_R^{CC}$. Just
as for the dimension-four coefficient $C_R$, the effects of these
coefficients can be very large in operators such as ${\cal O}_7$
generating important deviations in $b\to s\gamma$, as seen in
Eq.~(\ref{C75}). This is particularly so since their contributions
are unsuppressed by $m_b$. However, and as we noted for $C_R$,
the coefficients $\alpha_{2R}^{CC}$ and $\beta_{1R}^{CC}$ are
chirally unsuppressed in Eq.~(\ref{lags}), which results in an
unnatural renormalization of the $b$-quark mass. Thus again we
would argue that these coefficients should be rescaled by the
factor $m_b/v$, which makes their effect on the operator $O_7$
negligible. On the other hand, this is not the case with the
coefficient $\gamma_R^{CC}$ since the chiral suppression is
already present in the accompanying operator in
Eq.~(\ref{lagv}). Because of this its contribution to the
to the renormalization of the $b$-quark line vanishes in
the $m_b\to 0$ limit
\begin{equation}
\Sigma(m_b)=
\frac{-3 g^2}{128 \pi^2 \Lambda} \, \gamma_R^{CC} \, m_b^2\, (x+1)\,
\log\left(\frac{\Lambda^2}{M_W^2}\right)\;.
\end{equation}
We will then concentrate on the effects of $\gamma_R^{CC}$ among
the RH couplings. The contribution of $\gamma_R^{CC}$ to the
penguin operator gives rise to a deviation of the $b\to s\gamma
$ branching ratio from its SM expectation. In Fig.~\ref{grcc} we
plot this branching fraction as a function of this coefficient.
\begin{figure}
\centerline{
\epsfig{file=fig3.ps,height=3.5in,angle=90}
}
\caption{The $b\to s\gamma$ branching ratio vs. $\gamma_R^{CC}$.
}
\label{grcc}
\end{figure}
This measurement is the most constraining bound on these type of
operators. It can be seen that even for rather small values of
$\gamma_R^{CC}$ there could be considerable deviations from the
SM expectations. On the other hand, the effect is less dramatic
in $b\to s\ell^+\ell^-$, as shown in Fig.~\ref{bslgrcc}, where
an observable deviation from the SM will result only if
$\gamma_R^{CC}$ is large enough to dominate the $b\to s\gamma$
branching ratio.
\begin{figure}
\centerline{
\epsfig{file=fig4.ps,height=3.5in,angle=90}
}
\caption{The $b\to s\ell^+\ell^-$ branching ratio (normalized to
the SM prediction) vs. $\gamma_R^{CC}$.
}
\label{bslgrcc}
\end{figure}
The effects of the new operators ${\cal O}_{11}$ and ${\cal
O}_{12}$ are negligible. Although the presence of $m_b$ in the
denominators in Eq.~(\ref{c12g}), (\ref{c11z}) and
(\ref{c11box}) suggests the possibility of an enhancement, this
is not enough. This is obviously true for the coefficients
$\alpha_{2R}^{CC}$ and $\beta_{1R}^{CC}$, which as we argue
above should be proportional to $m_b/v$. But even when
considering $\gamma_{R}^{CC}$, the effect is suppressed by an
effective scale given by $m_b\Lambda/M_Z\simeq 55~$GeV, which
should be compared with the typical momentum transfers in $B$
decays.
The remaining group of coefficients we dubbed left-handed
includes $\alpha_{1L}^{CC}$, $\alpha_{2L}^{CC}$,
$\beta_{1L}^{CC}$, $\beta_{2L}^{CC}$, $\gamma_L^{CC}$ plus
$\gamma^{NC}$ which actually is the coefficient of a vector
operator, but since is not chirally suppressed is included with
the LH in this part of the analysis. These operators affect
mainly the $b\to s\ell^+\ell^-$ rates. Thus it is possible to
imagine that the underlying new physics preserves chiral
symmetry at the same time that does not generate a large value of
$\gamma_R^{CC}$, resulting in no deviations in $b\to s\gamma$;
but that the effects of the new interactions give rise to large
effects in the dilepton modes. Although these have not been
observed yet the experimental sensitivity is very close to the
SM predictions and it will reach them in the near future.
The leading effects in $b\to s\ell^+\ell^-$ come from the
coefficients $\alpha_{2L}^{CC}$, $\beta_{1L}^{CC}$ and
$\gamma^{NC}$. For simplicity we only consider these and plot in
Fig.~\ref{lh} the branching ratio, normalized to the SM one, for
two cases: $\alpha_{2L}^{CC}=\beta_{1L}^{CC}=\gamma^{NC}$ (solid
line) and $\alpha_{2L}^{CC}=\gamma^{NC}=0$ (dashed line).
\begin{figure}
\centerline{
\epsfig{file=fig5.ps,height=3.5in,angle=90}
}
\caption{The $b\to s\ell^+\ell^-$ branching ratio (normalized to
the SM prediction) vs. $\beta_{1L}^{CC}$, for
$\alpha_{2L}^{CC}=\beta_{1L}^{CC}=\gamma^{NC}$ (solid line)
and $\alpha_{2L}^{CC}=\gamma^{NC}=0$ (dashed line).
}
\label{lh}
\end{figure}
>From Fig.~\ref{lh} it is apparent that cancellations occur when
the three coefficients are similar. The effect of considering only
$\beta_{1L}^{CC}$ shows than even larger effects are possible. In
any case, sizeable deviations in $b\to s\ell^+\ell^-$ are possible
even in the absence of effects in $b\to s\gamma$.
\section{Discussion}
Processes involving FCNC transitions in $B$ and $K$ decays are a
crucial complement to precision electroweak observables, when
constraining the physics of the EWSB sector. In this paper, we
have considered the effects of anomalous couplings of third
generation quarks to the $W$ and $Z$ gauge bosons. We computed
the effects of all possible dimension-five operators in $B$
FCNC transitions such as $b\to s\gamma$ and $b\to s\ell^+\ell^-$.
For completeness, we have also also presented the analysis of the
dimension-four operators.
We have shown that with the natural assumption of chiral
symmetry, in fact enforcing vanishing fermion mass
renormalization in the chiral limit, the effects of the
dimension-four operators with coefficient $C_R$ for $b\to
s\gamma$, are not as dramatic as found Ref.~\cite{fuji}, and
somehow smaller than those of $C_L$, which can produce important
deviations in the branching ratio that could be resolved in the
next round of experiments at $B$ factories.
The effects in $b\to s\ell^+\ell^+$ due to $C_L$, $N_L^t$ and
$N_R^t$ are below $15\%$ deviations once the constraints from
electroweak precision measurements are taken into account.
On the other hand, we have found that the dimension-five operator
with coefficient $\gamma^{CC}_R$, for which no additional chiral
suppression is expected, can give rise to an observable
deviation of the $b\to s\gamma$ branching ratio from its SM
expectation even for rather small values of $\gamma_R^{CC}$. Left
handed operators, on the other hand, affect mainly the $b\to
s\ell^+\ell^-$ rates and we have illustrated that in several
scenarios sizeable deviations in $b\to s\ell^+\ell^-$ are
possible even in the absence of effects in $b\to s\gamma$.
The dimension-five operators, just as in the case of the more
studied dimension-four operators, can be generated at high
energies scales by the presence of new particles and/or
interactions. For instance, as a simple example, a heavy scalar
sector with both charged and neutral states, would give
contributions to many of the coefficients of the Lagrangian in
Eq.~(\ref{lags}). Richer dynamics at the TeV scale might
generate also some of the vector and/or tensor couplings of
Eq.~(\ref{lagv}) and~(\ref{lagt}).
The $e^+e^-$ $B$ factories at Cornell, KEK and SLAC are expected
to reach better measurements of the $b\to s\gamma$ branching
ratio, which will largely constrain the dimension-five
coefficient $\gamma_R^{CC}$ and to a lesser extent the
dimension-four coefficients $C_L$ and $\hat C_R$. Furthermore,
these experiments as well as those at the Fermilab Tevatron,
will reach the SM sensitivity for the $b\to s\ell^+\ell^-$
branching fraction. The present analysis, together with previous
ones addressing the effects of other anomalous higher
dimensional operators in these decay modes, will enable us to
interpret a possible pattern of deviations from the SM and
perhaps point to its dynamical origin.
\vskip 1.0cm
\noindent
{\bf Acknowledgments}
\noindent
We thank O.\ J.\ P.\ \'Eboli for a critical reading of the
manuscript. M.\ C.\ Gonzalez--Garcia is grateful to the Instituto
de F\'{\i}sica Te\'orica from Universidade Estadual Paulista for
its kind hospitality. G.\ Burdman acknowledges the hospitality of
the Enrico Fermi Institute at the University of Chicago, where
part of this work was completed. This work was supported by the U.S.\
Department of Energy under Grant No.~DE-FG02-95ER40896 and the
University of Wisconsin Research Committee with funds granted by
the Wisconsin Alumni Research Foundation, by Conselho Nacional de
Desenvolvimento Cient\'{\i}fico e Tecnol\'ogico (CNPq), by
Funda\c{c}\~ao de Amparo \`a Pesquisa do Estado de S\~ao Paulo
(FAPESP), by Programa de Apoio a N\'ucleos de Excel\^encia
(PRONEX), by CICYT (Spain) under grant AEN96-1718, by DGICYT
(Spain) under grants PB95-1077 and PB97-1261, and by the EEC
under the TMR contract ERBFMRX-CT96-0090.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,892
|
Q: How to cross compile my C code containing JNI? Previously I asked a question related to calling java function at
Calling Java functions from C language
Somehow I did with that.
Now I am planning to cross compile that.
I am facing an issue.
Before cross compiling ( means compiling in the same pc for the same pc)
I used the commands
gcc -I /usr/lib/jvm/java-6-openjdk-amd64/include -I /usr/lib/jvm/java-6-openjdk-amd64/include/linux -L /usr/bin/java -L /usr/lib/jvm/java-6-openjdk-amd64/jre/lib/amd64/server MyC.c -ljvm ;
That created well.
But now I am trying to cross compile the same for my target arm-none-eabi-
My makefile
CROSS_COMPILE =arm-none-eabi-
JAVA_HOME =/usr/lib/jvm/java-7-openjdk-amd64/
test:
javac -d ./ MyJava.java
$(CROSS_COMPILE)gcc -I $(JAVA_HOME)include \
-I $(JAVA_HOME)include/linux \
-L $(JAVA_HOME)jre/lib/amd64/server \
MyC.c -ljvm
This returning error
arm-none-eabi/bin/ld: warning: library search path "/usr/lib/jvm/java-7-openjdk-amd64/jre/lib/amd64/server" is unsafe for cross-compilation
/usr/lib/jvm/java-7-openjdk-amd64/jre/lib/amd64/server/libjvm.so: file not recognized: File format not recognized
collect2: ld returned 1 exit status
make: *** [test] Error 1
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 590
|
"Blitzgiving" is the tenth episode of the sixth season of the CBS sitcom How I Met Your Mother, and the 122nd episode overall. It aired on November 22, 2010.
Plot
It is the day before Thanksgiving, and Ted is hosting his first Thanksgiving for the group, and tries his best to make it memorable. He makes a special turkey by stuffing it with a smaller turkey, calling it a "turturkeykey". Just as he is about to call it an early night, Marshall and the group tell him not to leave the bar, or he will get the curse of "The Blitz". Future Ted explains how an old college friend, Steve Henry (Jorge Garcia), had the curse of The Blitz; whenever he would leave the room with a group, something amazing would happen.
The next morning, Ted awakens to find his apartment a wreck. The gang are all present with Steve, and Ted discovers Zoey sleeping in the bathtub. Lily explains that Zoey entered MacLaren's after Ted left and, while the group originally schemed to get back at her, they ended up having a great night with her and Steve. Ted realizes he truly has been cursed with The Blitz.
As a result of the group's actions the previous night, Ted's oven is broken and the gang seek a place where they could cook the turkey. Lily remembers that Zoey invited them all to her place for Thanksgiving, as her husband is spending Thanksgiving with his daughter. Barney hitches a separate cab to Zoey's but then realizes he may have been cursed with The Blitz by being away from the group. Barney meets the group outside Zoey's apartment and he realizes that he has become The Blitz, as the group had earlier run into a Thanksgiving Day parade.
At Zoey's apartment, Lily tries to convince Ted that enemies can become friends but Zoey and Ted start fighting, during which he compares her to an evil stepmother, which makes her so angry that she orders them to leave. While heading back Ted realises that the reason his insult upset Zoey so much was because she is hurt by the fact that her stepdaughter avoids spending time with her on the holidays. Ted and the group head back to the apartment and Ted asks for Zoey's forgiveness, saying that he understands her feelings for her stepdaughter. As the group enjoys a Thanksgiving meal, Future Ted says it is how he and Zoey finally became friends, also stating that the 'turturkeykey' tasted "wrong".
After dinner, Barney and Steve head towards the elevator. Steve remembers that he forgot his jacket and turns to retrieve it. Just then, a girl from the neighboring apartment darts into the elevator, with Barney inside. As the door closes, her dress gets stuck between the doors and rips apart. Steve realizes that he has been "Blitzed" again.
Critical response
Reviews for the episode were positive.
Donna Bowman of the A.V. Club graded the episode at A−, praising the story development and the various ideas put in that actually worked together. She also noted Garcia's appearance as a continuance of his story after the Lost finale and Barney's desperation not to miss out on various events.
Angel Cohn of Television Without Pity also put the episode at A−, saying the writers did well to insert Lost references and help Garcia discover more comedic skills.
Robert Canning of IGN gave the episode a rating of 8 out of 10.
Chris O'Hara of TVFanatic.com gave the episode a rating of 2.8 out of 5.
References
External links
How I Met Your Mother (season 6) episodes
2010 American television episodes
Thanksgiving television episodes
|
{
"redpajama_set_name": "RedPajamaWikipedia"
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| 4,617
|
Stacey Klim, Merchon Green, Sanjay Patel, Nicole Haagenson, and Mara Schiff.
Five candidates for office in the 2018 elections spoke to members and guests of the Democratic Women's Club of Indian River County at their recent meeting.
U.S. House candidate Sanjay Patel; Nicole Haagenson, running for the State House; and Merchon Green, Stacey Klim and Mara Schiff, all seeking seats on the Indian River School Board, also took questions from the gathering.
Patel, running against Republican Rep. Bill Posey, listed four top issues for the district, which includes all of Indian River and Brevard counties and a portion of Orange County. Those issues are money for public education, Medicare for all, expansion of the minimum wage, and the environment.
As the son of immigrants, Patel said he is the epitome of the American Dream, but that dream is becoming more difficult. "The people in power are not representing us," he said.
Haagenson is a candidate for the Florida House District 54 seat, currently held by Erin Grall. An Air Force veteran, she is an IT director for a Vero Beach company.
She listed her top priorities as public education and higher teacher pay, expanding Medicaid and healthcare, and the environment.
Asked about recent news reports of potential pollution in Blue Cypress Lake, Haagenson warned of having a knee-jerk reaction. She suggested establishing a committee of experts to investigate and propose needed environmental changes.
All three School Board candidates emphasized the need to improve teacher and support staff morale. Mara Schiff, running in District 1, said Indian River County must attract teachers with better pay and benefits. The state is systematically taking authority from local education, she added.
The candidates agreed that the current School Board ignores public concerns and suggestions. "Our School Board members don't want to hear from us," District 4 candidate Stacey Klim said.
An increase in quality education in the county was also emphasized. Merchon Green, the District 2 candidate, said the keys to that were accountability, transparency, community, and fiscal responsibility.
In answer to a question regarding improving behavioral problems, the three had the same response: Talk to the kids.
The next luncheon meeting of the Democratic Women's Club will be in September.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,856
|
Q: Binding Xlsform regex to questions I am designing an ODK survey form using XlsForm. On this form I have set some regex constrains that have pretty much the same structure except for some parts that are different. For example
regex(.,'^farmer-[mM][aA][dD][fF][aA][iI]-\d{5}$').
I have used something similiar
regex(.,'^[mM][aA][dD][fF][aA][iI]-\d{5}$')
as a constraint on another question. What I want to achieve is How to reuse or bind the expression by setting it in one place and only referencing it in different questions. Any help is very much appreciated. Thank you
A: You can do this by creating a type=calculate that stores the regex and then refer to it in your questions when necessary. Make sure to put the calculation inside quotation marks.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 500
|
{"url":"https:\/\/math.stackexchange.com\/questions\/600199\/deriving-the-formula-of-a-summation","text":"# Deriving the formula of a summation\n\nDerive the formula for $$\\sum_{k=1}^n k^2$$ The solution's that I was given has $k^3 + (k-1)^3$ as the first step but doesn't say how it got to that. Any help?\n\n\u2022 Not sure what you are asking. What do you mean you were given a solution with $k^3 + (k-1)^3$ as a first step? Could you typeset the entire solution you were given? \u2013\u00a0gt6989b Dec 9 '13 at 18:39\n\u2022 That's an interesting hint that you were given. Presumably, the hint is assuming you already know how to sum k^3. \u2013\u00a0David H Dec 9 '13 at 18:39\n\u2022 Two problems. 1: The answer should be in terms of $n$ and not $k$. In the sum, $k$ is just a dummy variable. 2: Even if you use $n$ rather than $k$, that expression doesn't work for $n = 0$ or $n=2$. \u2013\u00a0Carl Dec 9 '13 at 18:40\n\u2022 My question is more WHY we start with $k^3 + (k-1)^3$ (i.e how do we know to just use that formula chosen for the solution) \u2013\u00a0user114675 Dec 9 '13 at 18:58\n\nNote $(k+1)^3-k^3=3k^2+3k+1$ so \\begin{align} n^3=\\sum_{k=0}^{n-1} (k+1)^3-k^3&=\\sum_{k=0}^{n-1} 3k^2+3k+1\\\\ &=3\\sum_{k=0}^{n-1}k^2+3\\sum_{k=0}^{n-1}k+\\sum_{k=0}^{n-1}1\\\\ &=3\\sum_{k=0}^{n-1}k^2+3\\frac{(n-1)n}{2}+n\\\\ \\frac{n^3}{6}-\\frac{n(n-1)}{2}-\\frac n3&=\\sum_{k=0}^{n-1}k^2\\\\ \\frac{n(n-1)(2n-1)}{6}&=\\sum_{k=0}^{n-1}k^2 \\end{align}\n\nSo we get $$\\frac{n(n+1)(2n+1)}{6}=\\sum_{k=1}^{n}k^2$$\n\n\u2022 This is identical to the solution I got but I was more asking why we start with $(k+1)^3 - k^3$ \u2013\u00a0user114675 Dec 9 '13 at 18:57\n\u2022 Oh, well the heuristic argument is that a sum of $k^n$ should have $O(k^{n+1})$. Another reason (perhaps more rigorous), is: $$\\lim_{n\\to\\infty} \\sum_{k=1}^n \\frac{k^c}{n^{c+1}}=\\lim_{n\\to\\infty} \\frac 1n\\sum_{k=1}^n (k\/n)^c=\\int_0^1 x^c\\text dx=\\frac{1}{c+1}$$ Once we've recognized that, it's only reasonable to consider the difference of consecutive cubes, in this example. \u2013\u00a0Tim Ratigan Dec 9 '13 at 19:07\n\nHint:\n\n$$\\sum_{k=1}^{n} (k+1)^3 - \\sum_{k=1}^{n} k^3 = (n+1)^3-1 .$$\n\nNow, expand the left hand side and simplify. You only need the identity\n\n$$\\sum_{k=1}^{n}k = \\frac{n(n+1)}{2}.$$\n\nHint:\n\nFor nonnegative integers $n,r$ with $r\\leq n$ it is surprisingly easy to prove by induction that\n\n$\\sum_{k=r}^{n}\\binom{k}{r}=\\binom{n+1}{r+1}$\n\nThis result allows you to find formulas for $\\sum_{k=1}^{n}k^{r}$ for $r=1,2,3,\\ldots$","date":"2019-07-23 05:44:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 1, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8889289498329163, \"perplexity\": 488.37656626102466}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-30\/segments\/1563195528869.90\/warc\/CC-MAIN-20190723043719-20190723065719-00274.warc.gz\"}"}
| null | null |
Die 37. Herren-Boxeuropameisterschaften der Amateure wurden vom 5. November bis zum 15. November 2008 im britischen Liverpool ausgetragen.
Die Ukraine wurde mit vier Titeln erfolgreichste Nation. Russland und Armenien holte zwei Goldmedaillen und Gastgeber England, Belarus und Bulgarien je einen Europameistertitel. Es fanden Wettkämpfe in elf Gewichtsklassen statt.
Ergebnisse
Halbfliegengewicht bis 48 kg
Fliegengewicht bis 51 kg
Bantamgewicht bis 54 kg
Federgewicht bis 57 kg
Leichtgewicht bis 60 kg
Halbweltergewicht bis 64 kg
Weltergewicht bis 69 kg
Mittelgewicht bis 75 kg
Halbschwergewicht bis 81 kg
Schwergewicht bis 91 kg
Superschwergewicht über 91 kg
Medaillenspiegel
2008
Sportveranstaltung 2008
Sportveranstaltung in Liverpool
Boxen (Vereinigtes Königreich)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,562
|
Q: I have a basic but rather conceptual question on trigonometry
In this figure how can we find out geometrically that the sine of the angle $\theta$ = $AB /AO$ without using the equation sin (180 - $\theta$) = sin ($\theta$)
A: "The definition of sin of and angle is opposite/hypotenuse" - well that only works for a right-angled triangle, and is the beginning of the definition of the sine function.
In order to extend the definition draw a unit circle with centre at the origin. Measure the angle counterclockwise from the positive x-axis and take a point (x,y) on the circle. The sine of the angle (equivalent to opposite/hypotenuse in the first quadrant, with the hypotenuse made equal to 1) is y and the cosine of the angle is x. You can go round the circle more than once, so you can see that the functions are periodic.
This is why the functions are sometimes known as circular functions and underlies why they come in so surprisingly useful.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,633
|
#include <csignal>
#include <cstdlib>
#if defined ( _WIN32 )
#include <windows.h>
#else
#include <unistd.h>
#endif // defined ( _WIN32 )
#include <boost/algorithm/string.hpp>
#include <boost/exception/diagnostic_information.hpp>
#include <boost/filesystem/exception.hpp>
#include <boost/filesystem/operations.hpp>
#include <boost/filesystem/path.hpp>
#include <boost/format.hpp>
#include <boost/lexical_cast.hpp>
#include <pqxx/pqxx>
#include <Wt/WServer>
#include <Wt/WString>
#if MAGICKPP_BACKEND == MAGICKPP_GM
#include <GraphicsMagick/Magick++.h>
#elif MAGICKPP_BACKEND == MAGICKPP_IM
#include <ImageMagick-6/Magick++.h>
#endif // MAGICKPP_BACKEND == MAGICKPP_GM
#include <statgrab.h>
#include <CoreLib/CoreLib.hpp>
#include <CoreLib/CDate.hpp>
#include <CoreLib/Crypto.hpp>
#include <CoreLib/Database.hpp>
#include <CoreLib/Exception.hpp>
#include <CoreLib/Log.hpp>
#include <CoreLib/make_unique.hpp>
#include <CoreLib/Random.hpp>
#include <CoreLib/System.hpp>
#include "CgiRoot.hpp"
#include "Exception.hpp"
#include "Pool.hpp"
#include "VersionInfo.hpp"
void Terminate [[noreturn]] (int signo);
void InitializeDatabase();
#if defined ( __unix__ )
int main(int argc, char **argv, char **envp)
#else
int main(int argc, char **argv)
#endif // defined ( __unix__ )
{
try {
/// Gracefully handle SIGTERM
void (*prev_fn)(int);
prev_fn = signal(SIGTERM, Terminate);
if (prev_fn == SIG_IGN)
signal(SIGTERM, SIG_IGN);
/// Extract the executable path and name
boost::filesystem::path path(boost::filesystem::initial_path<boost::filesystem::path>());
if (argc > 0 && argv[0] != NULL)
path = boost::filesystem::system_complete(boost::filesystem::path(argv[0]));
std::string appId(path.filename().string());
std::string appPath(boost::algorithm::replace_last_copy(path.string(), appId, ""));
Service::Pool::Storage().AppPath = appPath;
/// Force changing the current path to executable path
boost::filesystem::current_path(appPath);
/// Initialize CoreLib
CoreLib::CoreLibInitialize(argc, argv);
/// Initializing log system
#if GDPR_COMPLIANCE
CoreLib::Log::Initialize(std::cout);
#else // GDPR_COMPLIANCE
CoreLib::Log::Initialize(std::cout,
(boost::filesystem::path(appPath)
/ boost::filesystem::path("..")
/ boost::filesystem::path("log")).string(),
"JobApplicationSystemServer");
#endif // GDPR_COMPLIANCE
/// Acquiring process lock
std::string lockId;
#if defined ( __unix__ )
int lock;
lockId = (boost::filesystem::path(appPath)
/ boost::filesystem::path("..")
/ boost::filesystem::path("tmp")
/ (appId + ".lock")).string();
#elif defined ( _WIN32 )
HANDLE lock;
lockId = appId;
#endif // defined ( __unix__ )
/// Log application info
LOG_INFO("Version Information", "", "BUILD_COMPILER " VERSION_INFO_BUILD_COMPILER, "BUILD_DATE " VERSION_INFO_BUILD_DATE, "BUILD_HOST " VERSION_INFO_BUILD_HOST, "BUILD_PROCESSOR " VERSION_INFO_BUILD_PROCESSOR, "BUILD_SYSTEM " VERSION_INFO_BUILD_SYSTEM, "PRODUCT_COMPANY_NAME " VERSION_INFO_PRODUCT_COMPANY_NAME, "PRODUCT_COPYRIGHT " VERSION_INFO_PRODUCT_COPYRIGHT, "PRODUCT_INTERNAL_NAME " VERSION_INFO_PRODUCT_INTERNAL_NAME, "PRODUCT_NAME " VERSION_INFO_PRODUCT_NAME, "PRODUCT_VERSION " VERSION_INFO_PRODUCT_VERSION, "PRODUCT_DESCRIPTION " VERSION_INFO_PRODUCT_DESCRIPTION);
if(!CoreLib::System::GetLock(lockId, lock)) {
LOG_WARNING("Process is already running!");
} else {
LOG_INFO("Got the process lock!");
}
/// Initialize CoreLib::Crypto
CoreLib::Crypto::Initialize();
/*! Initialize Magick++ or You'll crash HARD!! */
LOG_INFO("Initializing Magick++...");
Magick::InitializeMagick(*argv);
LOG_INFO("Magick++ initialized successfully!");
/// Initialize libstatgrab
LOG_INFO("Initializing libstatgrab...");
#if defined ( __unix__ )
#if defined ( __FreeBSD__ )
sg_init(1);
#else
sg_init(0);
#endif // defined ( __FreeBSD__ )
#elif defined ( _WIN32 )
/// In order to make it work, we have to call it twice
sg_snapshot();
sg_snapshot();
#endif // defined ( __unix__ )
/// Drop setuid/setgid privileges.
if (sg_drop_privileges() != SG_ERROR_NONE) {
LOG_FATAL("Error: libstatgrab has failed to drop privileges!");
return EXIT_FAILURE;
}
LOG_INFO("libstatgrab initialized successfully!");
/// Prevent libpqxx from crashing the program when a connection to the database breaks
signal(SIGPIPE, SIG_IGN);
/// Initialize the whole database
InitializeDatabase();
/// Start the server, otherwise go down
LOG_INFO("Starting the server...");
Wt::WServer server(argv[0]);
server.setServerConfiguration(argc, argv, WTHTTP_CONFIGURATION);
server.addEntryPoint(Wt::Application, Service::CgiRoot::CreateApplication, "", "favicon.ico");
if (server.start()) {
int sig = Wt::WServer::waitForShutdown();
server.stop();
#if defined ( __unix__ )
/// Experimental, UNIX only
if (sig == SIGHUP)
Wt::WServer::restart(argc, argv, envp);
#endif // defined ( __unix__ )
}
/// Shutdown libstatgrab before return
LOG_INFO("Shutting down libstatgrab...");
sg_shutdown();
LOG_INFO("libstatgrab shutdown successfully!");
}
catch (Service::Exception<std::wstring> &ex) {
LOG_ERROR(Wt::WString(ex.What()).toUTF8());
}
catch (Service::Exception<std::string> &ex) {
LOG_ERROR(ex.What());
}
catch (CoreLib::Exception<std::wstring> &ex) {
LOG_ERROR(Wt::WString(ex.What()).toUTF8());
}
catch (CoreLib::Exception<std::string> &ex) {
LOG_ERROR(ex.What());
}
catch (const Wt::WServer::Exception &ex) {
LOG_ERROR(ex.what());
}
catch (const boost::exception &ex) {
LOG_ERROR(boost::diagnostic_information(ex));
}
catch (const std::exception &ex) {
LOG_ERROR(ex.what());
}
catch (...) {
LOG_ERROR("Unknown error!");
}
return EXIT_SUCCESS;
}
void Terminate(int signo)
{
LOG_WARNING((boost::format("Received signal %1%; terminating...") % signo).str());
exit(EXIT_FAILURE);
}
void InitializeDatabase()
{
try {
LOG_INFO("main: Initializing database...");
LOG_INFO("main: Registering database enums...");
Service::Pool::Database().RegisterEnum("SUBSCRIPTION", "subscription",
{ "none", "en_fa", "en", "fa" });
LOG_INFO("main: Registered all database enums!");
LOG_INFO("main: Registering database tables...");
Service::Pool::Database().RegisterTable("VERSION", "version",
" version SMALLINT NOT NULL PRIMARY KEY ");
Service::Pool::Database().RegisterTable("ROOT", "root",
" user_id UUID NOT NULL PRIMARY KEY, "
" username TEXT NOT NULL UNIQUE, "
" email TEXT NOT NULL UNIQUE, "
" creation_time TIMESTAMPTZ NOT NULL DEFAULT TIMESTAMPTZ $token$'EPOCH'''$token$, "
" modification_time TIMESTAMPTZ NOT NULL DEFAULT TIMESTAMPTZ $token$'EPOCH'''$token$ ");
Service::Pool::Database().RegisterTable("ROOT_CREDENTIALS", "root_credentials",
" user_id UUID NOT NULL PRIMARY KEY, "
" pwd TEXT NOT NULL, "
" modification_time TIMESTAMPTZ NOT NULL DEFAULT TIMESTAMPTZ $token$'EPOCH'''$token$ ");
Service::Pool::Database().RegisterTable("ROOT_CREDENTIALS_RECOVERY", "root_credentials_recovery",
" token UUID NOT NULL PRIMARY KEY, "
" user_id UUID NOT NULL, "
" expiry TIMESTAMPTZ NOT NULL DEFAULT TIMESTAMPTZ $token$'EPOCH'''$token$, "
" new_pwd TEXT NOT NULL, "
" request_time TIMESTAMPTZ NOT NULL, "
" request_ip_address INET, "
" request_location_country_code TEXT, "
" request_location_country_code3 TEXT, "
" request_location_country_name TEXT, "
" request_location_region TEXT, "
" request_location_city TEXT, "
" request_location_postal_code TEXT, "
" request_location_latitude TEXT, "
" request_location_longitude TEXT, "
" request_location_metro_code TEXT, "
" request_location_dma_code TEXT, "
" request_location_area_code TEXT, "
" request_location_charset TEXT, "
" request_location_continent_code TEXT, "
" request_location_netmask TEXT, "
" request_location_asn TEXT, "
" request_location_aso TEXT, "
" request_location_raw_data JSONB, "
" request_user_agent TEXT, "
" request_referer TEXT, "
" utilization_time TIMESTAMPTZ NOT NULL DEFAULT TIMESTAMPTZ $token$'EPOCH'''$token$, "
" utilization_ip_address INET, "
" utilization_location_country_code TEXT, "
" utilization_location_country_code3 TEXT, "
" utilization_location_country_name TEXT, "
" utilization_location_region TEXT, "
" utilization_location_city TEXT, "
" utilization_location_postal_code TEXT, "
" utilization_location_latitude TEXT, "
" utilization_location_longitude TEXT, "
" utilization_location_metro_code TEXT, "
" utilization_location_dma_code TEXT, "
" utilization_location_area_code TEXT, "
" utilization_location_charset TEXT, "
" utilization_location_continent_code TEXT, "
" utilization_location_netmask TEXT, "
" utilization_location_aso TEXT, "
" utilization_location_asn TEXT, "
" utilization_location_raw_data JSONB, "
" utilization_user_agent TEXT, "
" utilization_referer TEXT ");
Service::Pool::Database().RegisterTable("ROOT_SESSIONS", "root_sessions",
" token UUID NOT NULL PRIMARY KEY, "
" user_id UUID NOT NULL, "
" expiry TIMESTAMPTZ NOT NULL DEFAULT TIMESTAMPTZ $token$'EPOCH'''$token$, "
" login_time TIMESTAMPTZ NOT NULL DEFAULT TIMESTAMPTZ $token$'EPOCH'''$token$, "
" ip_address INET, "
" location_country_code TEXT, "
" location_country_code3 TEXT, "
" location_country_name TEXT, "
" location_region TEXT, "
" location_city TEXT, "
" location_postal_code TEXT, "
" location_latitude REAL, "
" location_longitude REAL, "
" location_metro_code INTEGER, "
" location_dma_code INTEGER, "
" location_area_code INTEGER, "
" location_charset INTEGER, "
" location_continent_code TEXT, "
" location_netmask INTEGER, "
" location_asn INTEGER, "
" location_aso TEXT, "
" location_raw_data JSONB, "
" user_agent TEXT, "
" referer TEXT ");
Service::Pool::Database().RegisterTable("SETTINGS", "settings",
" pseudo_id TEXT NOT NULL PRIMARY KEY, "
" homepage_url_en TEXT NOT NULL, "
" homepage_url_fa TEXT NOT NULL, "
" homepage_title_en TEXT NOT NULL, "
" homepage_title_fa TEXT NOT NULL ");
Service::Pool::Database().RegisterTable("CONTACTS", "contacts",
" recipient TEXT NOT NULL PRIMARY KEY, "
" recipient_fa TEXT NOT NULL UNIQUE, "
" address TEXT NOT NULL, "
" is_default BOOLEAN NOT NULL DEFAULT FALSE ");
Service::Pool::Database().RegisterTable("SUBSCRIBERS", "subscribers",
" inbox TEXT NOT NULL PRIMARY KEY, "
" uuid UUID NOT NULL UNIQUE, "
" subscription SUBSCRIPTION NOT NULL DEFAULT 'none', "
" pending_confirm SUBSCRIPTION NOT NULL DEFAULT 'none', "
" pending_cancel SUBSCRIPTION NOT NULL DEFAULT 'none', "
" join_date TEXT NOT NULL, "
" update_date TEXT NOT NULL ");
LOG_INFO("main: Registered all database tables!");
LOG_INFO("main: Calling Database::Initialize()...");
Service::Pool::Database().Initialize();
LOG_INFO("main: Setting up the database...");
auto conn = Service::Pool::Database().Connection();
pqxx::work txn(*conn.get());
/// Check the database version
pqxx::result r = txn.exec((boost::format("SELECT version FROM \"%1%\" WHERE 1 = 1;")
% txn.esc(Service::Pool::Database().GetTableName("VERSION"))).str());
/// If the database is un-versioned
if (r.empty()) {
/// Insert the version number 1
Service::Pool::Database().Insert("VERSION", "version", { "1" });
}
/// Check whether the default root user already exists
r = txn.exec((boost::format("SELECT username FROM \"%1%\" WHERE username=%2%;")
% txn.esc(Service::Pool::Database().GetTableName("ROOT"))
% txn.quote(Service::Pool::Storage().RootUsername())).str());
/// If the default root user does not exists
if (r.empty()) {
std::string uuid;
/// Continue untile we can generate a unique UUID
while (true) {
/// Generate a new UUID
CoreLib::Random::Uuid(uuid);
/// Query all existing UUIDs in order to avoid a possible duplicate UUID
r = txn.exec((boost::format("SELECT user_id FROM \"%1%\" WHERE user_id=%2%;")
% txn.esc(Service::Pool::Database().GetTableName("ROOT"))
% txn.quote(uuid)).str());
/// Break the loop if the new UUID is not a duplicate
if (r.empty()) {
break;
}
}
/// Get the current date/time
CoreLib::CDate::Now n(CoreLib::CDate::Timezone::UTC);
/// Insert a new default root user into the database with default password
txn.exec((boost::format("INSERT INTO \"%1%\""
" ( user_id, username, email, creation_time )"
" VALUES ( %2%, %3%, %4%, TO_TIMESTAMP(%5%)::TIMESTAMPTZ );")
% txn.esc(Service::Pool::Database().GetTableName("ROOT"))
% txn.quote(uuid)
% txn.quote(Service::Pool::Storage().RootUsername())
% txn.quote(Service::Pool::Storage().RootInitialEmail())
% txn.esc(boost::lexical_cast<std::string>(n.RawTime()))).str());
Service::Pool::Database().Insert("ROOT_CREDENTIALS",
"user_id, pwd",
{
uuid,
Service::Pool::Storage().RootInitialPassword()
});
}
txn.commit();
LOG_INFO("main: Database setup is complete!");
LOG_INFO("main: Database initialization is complete!");
}
catch (const pqxx::sql_error &ex) {
LOG_ERROR(ex.what(), ex.query());
}
catch (const boost::exception &ex) {
LOG_ERROR(boost::diagnostic_information(ex));
}
catch (const std::exception &ex) {
LOG_ERROR(ex.what());
}
catch (...) {
LOG_ERROR("Unknown error!");
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,790
|
The tiny white metal dancing girls are sure to bring joy to any face. These tiny sculptures are made of white metal. They are six Read More..in number and each of them depicts a different dancing posture. The dancing girls are not attached to any platform so you can keep them according to your wish. It also makes for a good present especially to dance lovers. These will surely help you brighten up the odd of the house. Show Less..
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,259
|
Tornadoes Seen: 11
BIG TOURS VS SMALLER TOURS
Extreme Tornado Tours
Tornadoes Seen: 533
Joshua Napper is an 11 year storm chasing veteran and is one of our forecasters and media consultant. He is an entrepreneur and small business owner since early 2012. He is currently the owner and president of one of the largest weather media content providers called LiveStormChasers.com. Josh is also a private pilot and has been flying for over 8 years. As an independent National & World News Multimedia Journalist, his work has been seen worldwide across numerous media outlets. Certified as a Commercial sUAS (Drone) Operator & FAA Private Pilot, Josh assists with media content and pulls double duty as one of our weather nowcasters.
Morrie McKeeis ECT's Mechanical and guru and has many certifications in the field. He is also of our part time drivers. Morrie has been chasing for 8 years and has seen 11 tornadoes. He also pulls double duty as a communication specialist and aids in weather locations. He is an avid photographer who has had some of his photography work seen around the world.
Lanny Dean is a native Okie but currently lives in Missouri. He is the owner and lead forecasting Meteorologist and tour guide of Extreme Chase Tours. Lanny hold a BS in Meteorology and is a 28 year storm chasing veteran and one of the best storm chasers and tornado chasing tour guides in the business. You may have seen Lanny on the hit show "Tornado Hunters" that aired on TruTV. He and his tornado and weather videos have appeared on many various local and world wide networks including: The Weather Channel, Storm Stories, CNN, ABC, Good Morning America, TLC, The Discovery Channel, MSNBC, NBC, RealTV, Inside Edition, Fox News and many other media outlets as well as many local markets throughout the United States and around the world. Lanny is a member of The Outlaw Chasers and he has witnessed and filmed 533 tornadoes & 13 major hurricanes to date. Lanny was awarded "Storm Chaser Of The Year" in 2005 and 2007 for his coverage of the Greensburg, KS EF-5 tornado.
He is the only current storm chasing tour owner/operator to receive and hold that award. Lanny holds a B.S. in Electronical Engineering and held his undergraduate degree in meteorology for over 20 years. in 2019 Lanny completed his B.S degree in Atmospheric Sciences/Meteorology and now holds a double major. He spent many years working in the media as a storm tracker and in house severe weather reporter/photographer. Lanny has received many awards for his photojournalism, photography and reporting, including an Emmy Award, a second Emmy nomination for his weather reporting, KAB
(Kansas Association Of Broadcasting Awards) OAB (Oklahoma Association Of Broadcasting Awards) and various other awards. Lanny works with the National Weather Service year round during weather events and during the Storm Chasing Tours. He has also been working on a scientific project recording acoustic and pressure data in and around supercells and tornadoes and recently had an opportunity to work with the late Mr. Tim Samaras. He also worked for KJRH Channel 2 in Tulsa as their Severe Weather Reporter and Photographer, prior to KJRH, Lanny worked at KAKE-TV Channel 10 in Wichita, KS as their Severe Weather Reporter and Photographer. While working in the media Lanny also won several SPJ (Society Of Professional Journalist) awards as well as being recognized by the Oklahoma Association Of Broadcasters and The Kansas Association Of Broadcasters for his weather reporting and photography. In 2012 and 2013 Lanny and Extreme Chase Tours won Best Of Small Business Tulsa award in travel and tourism. Lanny is a true Severe Storm Expert and is one of the best weather forecasters chasing today.
MEET THE EXTREME CHASE TOURS TEAM
Our storm chasing tours and tornado tours team are some of the best in the business. With well over 120 years experience and over 600 tornadoes documented, our weather experts, meteorologist and tornado forecasters know how to get you to the best storm of the day!
COPYRIGHT LANNY DEAN 1991-2018
Tornadoes Seen: 40+
Paden Nicholas is one the ECT tour guides and drivers. Paden is a 10+ year storm chasing veteran and has seen 40+ tornadoes. He is also a member of The Outlaw Chasers and the owner of the TVP (Tornado Pursuit Vehicle) which is a legally purchased modified police cruiser that has been specially designed as a storm chasing vehicle. Everything from the electronics to the vehicle wrap has been modified specifically for storm chasing.
Paden's hobbies include storm chasing, fishing, being outdoors and spending time with his children and family. You can see more of the TVP on the TVP Facebook page
Omaha"The Storm Chasing Duck Dog" is our formal mascot and fearless tornado finder! He is a 3.5 year old Black Lab. Omaha is truly a storm chasing veteran dog and has witnessed 5 tornadoes during the Canadian Texas tornado event in 2015. While he is not on every tour, you might see him with Lanny from time to time. He is actually a formal duck retrieving dog and service dog but don't tell him that!
Chad Berryhill is an 15 year storm chasing veteran and pulls duty as one of our forecasters and now casters. Chad attended OSU and Oklahoma University and has some undergrad study in meteorology and sciences. He and his wife Erin, have two sons who they devote their time to. Chad loves to eat (see the turkey leg in his hand?) so if you have any candy or snacks while Chad is around HIDE THEM! Chad has seen a number of tornadoes including the infamous Hennessey, Oklahoma F-3 tornado in 2010. Chad led the Extreme Chase Tours guests directly to this tornado which was their first tornado! You might hear Lanny or one of the guides talking to Chad on the phone while on tour, feel free to tell him you have a box of Mike and Ikes and he just might show up on tour with us! Some of Chad's hobbies include: spending time with his son and family, deer hunting, duck hunting, fishing, camping, welding, photography and did we mention food? You might consider him a food connoisseur!
Lanny Dean
Laura Dean
HOW TO STORM CHASE
Tornadoes Seen: 5
meet the extreme chase tours storm chasing tours team
Jeff Smith is one of our Nowcaster and business consultant for Extreme Chase Tours and a 24 year veteran stormchaser.
Jeff brings this 20 plus years of storm spotting and storm chasing experience to the team in order to help provide you the best storm chase tour experience on the planet. Jeff's obsession with weather began when he was five years old as an F-3 tornado was ravaging the East side of Tulsa OK in December, 1975. He and his family watched the tornado from his bedroom window tear up the city. Since that day, Jeff has had a passion for witnessing Mother Nature's fury wherever it may take him. In elementary school, Jeff's teachers grew concerned because he was always drawing different tornadoes. Those stunts actually got his parents several phone calls from school...often resulting in a sore backside :) Jeff has seen 78 tornadoes. One of the most notable was on March 12, 2006 where he and chase partner TJ caught rare footage of the after dark F-3 Delaware County Oklahoma tornado. Jeff takes on the duty of nowcaster for us when needed. He has nowcasted for several well known chasers including owner, Lanny Dean. Jeff's radar interpretation skills are impeccable and second to none. Jeff has chased for man local TV stations and currently chases for local station in Tulsa.
Laura Dean is Lanny's wife and a 16 year storm chase veteran. She has seen a total of 10 tornadoes including the infamous Throckmorton Texas F-3 tornado from very close range. Laura is the CFO and gives complete radar and logistic support while we are out on tour and pulls double duty reserving and booking our motel rooms. Laura helps to keep us safe on tour by monitoring the weather from home. She has been trained in operational meteorology and radar support and is the backbone for Extreme Chase Tours. Laura is an avid photographer and it is one of her main passions. She also enjoys cooking, fishing, storm chasing and many other hobbies. Laura loves nature and just being outdoors. She and Lanny have been married for almost 18 years and she has as much passion for storm chasing as anyone! You might hear Laura giving us updates, reserving our motel rooms or giving us detailed weather information while we are on tour so feel free to say hi!
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,384
|
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE hibernate-mapping PUBLIC
"-//Hibernate/Hibernate Mapping DTD 3.0//EN"
"http://hibernate.sourceforge.net/hibernate-mapping-3.0.dtd">
<hibernate-mapping >
<class name="org.mifos.reports.business.ReportsDataSource" table="report_datasource" >
<id name="datasourceId" column="datasource_id" type="int">
<generator class="native" />
</id>
<property name="name" column="name" type="string" />
<property name="driver" column="driver" type="string" />
<property name="url" column="url" type="string" />
<property name="username" column="username" type="string" />
<property name="password" column="password" type="string" />
</class>
<query name="getAllReportDataSource">
<![CDATA[ from ReportsDataSource ]]>
</query>
<query name="viewDataSource">
<![CDATA[ from ReportsDataSource where datasource_id=:dataSourceId]]>
</query>
</hibernate-mapping>
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,991
|
Q: Change build rules in Makefile to build multiple file types I want to change my Makefile to build cpp and mm files, since I have just begun adding Objective-C code into my project. This is the Makefile I have currently:
# Name of the project being built
PROJECT := modpe
# Name of process the tweak is loaded into
PROCESS := minecraftpe
# Local IP Address of device to SSH into
DEVICE := ryans-ipod-touch.local
# Path of the SDK on MacOS
SDKPATH := /Applications/Xcode.app/Contents/Developer/Platforms/iPhoneOS.platform/Developer/SDKs/iPhoneOS.sdk
# Build using libc++ and C++11 support
override CXXFLAGS += -stdlib=libc++ -std=c++11 -miphoneos-version-min=7.0 -isysroot $(SDKPATH)
override LDFLAGS += -stdlib=libc++ -std=c++11 -miphoneos-version-min=7.0 -isysroot $(SDKPATH)
# Names of the tweak library, substrate filter, and debian package
TWEAK := $(PROJECT).dylib
FILTER := $(PROJECT).plist
DEB := $(PROJECT).deb
# Directory for build products like executables, object files, and dependency files
BUILD := build
# Directory for debian package filesystem layout
LAYOUT := layout
# Directory for temporary staging of the debian package structure while creating the package
STAGING := $(BUILD)/deb
SUBSTRATE := $(STAGING)/Library/MobileSubstrate/DynamicLibraries
# Name of the unstripped version
UNSTRIPPED := $(BUILD)/$(TWEAK:.dylib=_unstripped.dylib)
# List of source files and their corresponding object file paths
SRCS := $(shell find . -type f -name '*.cpp')
OBJS := $(addprefix $(BUILD)/,$(SRCS:.cpp=.o))
# Supported architectures
ARCHS := arm64
ARCHFLAGS := $(addprefix -arch ,$(ARCHS))
# Frameworks for linking
FRAMEWORKS := CydiaSubstrate Forklift
override LDFLAGS += $(addprefix -framework ,$(FRAMEWORKS))
# Compiler and linker
CLANG := clang $(ARCHFLAGS)
CLANGXX := clang++ $(ARCHFLAGS)
CC := $(CLANG)
CXX := $(CLANGXX)
LD := $(CLANGXX)
STRIP := strip
# When invoked as "make VERBOSE=1", every command gets printed as it runs
ifdef VERBOSE
_v :=
else
_v := @
endif
### Build rules ###
all: $(TWEAK)
.PHONY: all
$(TWEAK): $(UNSTRIPPED)
@echo 'Strip $@'
$(_v)$(STRIP) -x -o $@ $<
%.dylib: $(OBJS)
@echo 'Link $@'
$(_v)$(LD) $(LDFLAGS) -dynamiclib -o $@ $^
%.cpp: $(BUILD)/%.d
$(BUILD)/%.o: %.cpp | $(BUILD)/.dir
@echo 'Compile $@'
$(_v)$(CXX) $(CXXFLAGS) -MD -MF $(BUILD)/$*.d -c -o $@ $<
.SECONDARY: $(BUILD)/.dir
-include $(BUILD)/*.d
### Package rules ###
package: $(DEB)
.PHONY: package
$(DEB): $(TWEAK) $(FILTER)
@echo 'Stage $@'
$(_v)rm -rf $(STAGING)
$(_v)mkdir -p $(STAGING)
$(_v)cp -R $(LAYOUT)/* $(STAGING)
$(_v)mkdir -p $(SUBSTRATE)
$(_v)cp $^ $(SUBSTRATE)
$(_v)chown -R root:wheel $(STAGING)
$(_v)chmod 0755 $(SUBSTRATE)/$(TWEAK)
$(_v)chmod 0644 $(SUBSTRATE)/$(FILTER)
@echo 'Package $@'
$(_v)dpkg-deb -Zgzip -b $(STAGING) $@
$(_v)rm -rf $(STAGING)
### Install rules ###
install: $(DEB)
@echo 'Install $(DEB)'
scp $(DEB) root@$(DEVICE):/var/tmp/
@echo 'Run dpkg -i /var/tmp/$(DEB) to install.'
ssh root@$(DEVICE)
.PHONY: install
### Offline install rules ###
offline:
@echo 'No WiFi to SSH: Using afc instead'
$(_v)sudo afc/afcclient put ./modpe.deb /modpe.deb
.PHONY: offline
### Clean rules ###
clean:
@echo 'Remove $(BUILD)'
$(_v)rm -rf $(BUILD)
.PHONY: clean
### General rules ###
%/.dir:
@echo 'Create directory $*/'
$(_v)mkdir -p $* && touch $@
# deploys test script
deploy: $(OFFLINE)
scp scripts/script.js root@$(DEVICE):/var/mobile/modpe/
.PHONY: deploy
So far the only thing that I know to change is the SRCS line:
SRCS := $(shell find -E . -regex ".*\.\(cpp\|mm\)")
My question is, what else can I do to make this build .mm source files with .cpp source files, rather than ignoring them?
A: You haven't given us enough information to reproduce the problem or test solutions, so you may have to do some fine-tuning, but here's an approach that should work.
You have already modified the SRCS := ... line. Once you have verified that that works, modify the OBJS := ... line:
OBJS := $(patsubst %, $(BUILD)/%.o,$(basename $(SRCS)))
Then once you have verified that that works, add a new rule:
$(BUILD)/%.o: %.mm | $(BUILD)/.dir
@echo 'Compile $@'
$(_v)$(CXX) $(CXXFLAGS) -MD -MF $(BUILD)/$*.d -c -o $@ $<
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 576
|
\section{Introduction}
The strong phases of the $B\to\pi\pi$ and $B\to\pi K$ decay amplitudes have been usually treated assuming that the strong interactions are dominated by short distance perturbative effects. According to this the
strong phases can be obtained perturbatively from penguin and tree
diagrams and are then generalized to describe hadronic strong
phases to all orders. However, in \cite{gerard} it was shown the lack of compatibility between the strong phases calculated perturbatively, and those of the isospin basis, where phases are contemplated at all scales (soft and hard).
The isospin symmetry is a very good approximation of the hadronic world. For example, the $K\to\pi\pi$ decay amplitudes are treated on the basis of this symmetry of the strong interactions. For $B$ decays we still have isospin invariance but now there are many modes apart from the elastic ones that can be reached with rescattering of final states (FSI). However, for the case of $B$ mesons inelasticity is not expected to be significant as far as it becomes important for large angular momentum waves. Therefore we will work in the quasi-elastic limit, where inelastic channels are neglected in rescattering.
The Regge model predicts that the quasi-elastic strong phases are small, and so the CP asymmetries must be necessarily small. Experimentally, the $\pi^-K^+$ asymmetry turns out to be large, in contradiction with the hypothesis. Nevertheless, this mode involves a charged pion and
corrections due to radiation of photons can play an important role in
quoting precise results from experiments.
In this paper we use the quasi-elastic predictions for FSI in $B\to K\pi,\ \pi\pi$ to derive bounds on the angle $\gamma$ of the unitary triangle. We give an overview of the Watson Theorem and on the concepts of bare and rescattered
amplitudes, then apply these principles to parameterize the decay
amplitudes in terms of the quark and isospin amplitudes. Using the
experimental branching ratios, which are more precisely measured than the
CP asymmetries, we give an estimate for the range of $\gamma$ and of the electroweak penguins contribution, then we give predictions for the asymmetries. We
finally present our conclusions.
\section{Hadronic strong phases}
The Watson Theorem \cite{wolfenstein2}, \cite{christopher} tell us that the decay amplitudes factorize into the product of the direct or ``bare'' weak amplitudes, $A^b$, and the rescattering factors, $\sqrt S$,
$$A_i(B\to P_i)=\sqrt S_{ij} A^b_j(B\to P_j),\quad \overline A_i(\overline B\to \overline P_i)=\sqrt S_{ij} A^{b*}_j(B\to P_j),$$
where as usual the $CP$ conjugated amplitude is given the same strong
phase $S_{ij}$ which denotes here the matrix element of the
$P_i\to P_j$ strong interaction. If we factorize the weak phase in the amplitude $A^b$, we do not have other strong phases induced by perturbative calculations, and the complete hadronic phases to
all orders are included in the $\sqrt
S$ factors.
If we were in the elastic case, $\sqrt S$ would be diagonal in the isospin basis:
$$\sqrt S_{el}\!=\!\left(\begin{array}{cc}e^{i\delta_{3/2}} & \\ & e^{i\delta_{1/2}}\end{array}\right).$$
Thus, for the case of $B\to\pi K$ decay amplitudes, we would have
$$A_{I}\to A_{I}e^{i\delta_{I}},\quad \overline A_{I}\to A^*_{I}e^{i\delta_{I}},\quad I=3/2,1/2,$$
or in the physical basis
$$\left(\!\begin{array}{c}A_{\pi^+K^0}\\A_{\pi^0 K^+}\end{array}\!\right)\!=\!\left(\!\begin{array}{cc} -\frac{1}{\sqrt 3} & \sqrt{\frac{2}{3}}\\ \sqrt{\frac{2}{3}} & \frac{1}{\sqrt 3}\end{array}\!\right)\!\left(\!\begin{array}{c}A_{3/2}e^{i\delta_{3/2}}\\A_{1/2}e^{i\delta_{1/2}}\end{array}\!\right),\quad \left(\!\begin{array}{c}\overline A_{\pi^-\overline K^0}\\\overline A_{\pi^0 K^-}\end{array}\!\right)\!=\!\left(\!\begin{array}{cc} -\frac{1}{\sqrt 3} & \sqrt{\frac{2}{3}}\\ \sqrt{\frac{2}{3}} & \frac{1}{\sqrt 3}\end{array}\!\right)\!\left(\!\begin{array}{c}A^*_{3/2}e^{i\delta_{3/2}}\\A^*_{1/2}e^{i\delta_{1/2}}\end{array}\!\right),$$
with analogous expressions for the other channels $A_{\pi^0K^0}$ and $A_{\pi^-K^+}$ that are reached by $B^0$ decays.
At the $B$ mass, however, we have the more general inelastic case, which
is described in the isospin symmetric case as
$$\left(\!\begin{array}{c}A_{\pi^+K^0}\\A_{\pi^0 K^+}\end{array}\!\right)\!=\!\left(\!\begin{array}{cc} -\frac{1}{\sqrt 3} & \sqrt{\frac{2}{3}}\\ \sqrt{\frac{2}{3}} & \frac{1}{\sqrt 3}\end{array}\!\right)\!\left(\!\begin{array}{c}A_{3/2}\sqrt{\eta}_{3/2}e^{i\delta_{3/2}}+\sum_{j\neq \pi K}\sqrt S^{I=3/2}_{\pi K,j}A_{3/2}^j\\A_{1/2}\sqrt{\eta}_{1/2}e^{i\delta_{1/2}}+\sum_{j\neq \pi K}\sqrt S^{I=1/2}_{\pi K,j}A_{1/2}^j\end{array}\!\right)$$
where we have replaced $\sqrt S_{ii}\!=\!e^{i\delta_I}\!\to\!\sqrt{\eta}_Ie^{i\delta_I}$ for the
elastic FSI and have also incorporated the inelastic channels ($j\not =
\pi K$). The elastic parameters $\eta$ and
$\delta$, can be calculated using the Regge model of strong
interactions.
In the Regge model, the high energy amplitudes in the $s$-channel can be parametrized in terms of trajectories exchanges in the small $t$ region. A
Regge Model to describe $B\to \pi K,\ \pi\pi$ decays was
proposed in \cite{jeanpion} and \cite{jeankaon}, but this assumed the complete
elasticity of FSI as it was later remarked in \cite{wolfenstein3}. Taking
into account this remark and using the
relations between the isospin amplitudes in the $s$ and $t$ channels
$$\left(\begin{array}{c} A_0^s \\ A_1^s \\ A_2^s \end{array}\right)=\left(\begin{array}{ccc} 1/3 & 1 & 5/3 \\ 1/3 & 1/2 & -5/6 \\ 1/3 & -1/2 & 1/6 \end{array}\right)\left(\begin{array}{c} A_0^t \\ A_1^t \\ A_2^t \end{array}\right),\quad \left(\begin{array}{c} A_{1/2}^s \\ A_{3/2}^s \end{array}\right)=\left(\begin{array}{cc} 1/\sqrt{6} & 1 \\ 1/\sqrt{6} & -1/2 \end{array}\right)\left(\begin{array}{c} A_0^t \\ A_1^t \end{array}\right)$$
we now find:
$$\eta_{I=1/2}^{\pi K}(m_B)=0.72,\ \eta_{I=3/2}^{\pi K}(m_B)=0.71,$$
$$\eta_{I=0}^{\pi\pi}(m_B)=\eta_{I=2}^{\pi\pi}(m_B)=0.64.$$
For these results we made a fit of nucleon-nucleon and pion-nucleon total
cross sections and made use of factorization. $f_2$ and $\rho$ trajectories were considered in t channel in addition to the Pomeron exchange, for both, $\pi\!-\!\pi$ and $\pi\!-\!K$ modes. $K^*$ and $K^{**}$ trajectories were neglected in the small $-\!u$ region in the case of $\pi\!-\!K$, because of their small size.
The equality of the $\eta$ factors in each mode follows from the pomeron
dominance over the rest of trajectories contributing to the $A_0^t$
amplitude at the $B$ mass scale.
As a consequence the strong phases are almost the same (and close to
zero because the pomeron is almost imaginary), so that the differences of
the strong phases would be very tiny and it becomes difficult to give its
value with accuracy.
Since the factors $\sqrt{\eta}$ are close to unity, we
can take the {\it quasi-elastic assumption} \cite{german}, that consists
in the cancellation of the inelastic channels, which means that
$\sum_{j\neq \pi K}\sqrt S^{I}_{\pi K,j}A_{I}^j\to 0$.
\section{Amplitudes decomposition}
Let us consider the $B\to\pi K$ decay amplitudes. Once we absorb the elastic
parameters $\sqrt{\eta}_I$ into the definitions of physical amplitudes, they can be
written in terms of the isospin amplitudes as follows
$$A_{\pi^+K^0}\!=\!-\frac{A_{3/2}}{\sqrt{3}}e^{i\delta_{3/2}}\!+\!\sqrt{\frac{2}{3}}(A^{\prime}_{1/2}\!+\!A_{1/2})e^{i\delta_{1/2}},\quad A_{\pi^0K^+}\!=\!\sqrt{\frac{2}{3}}A_{3/2}e^{i\delta_{3/2}}\!+\!\frac{1}{\sqrt 3}(A^{\prime}_{1/2}\!+\!A_{1/2})e^{i\delta_{1/2}},$$
$$A_{\pi^0K^0}\!=\!\sqrt{\frac{2}{3}}A_{3/2}e^{i\delta_{3/2}}\!-\!\frac{1}{\sqrt 3}(A^{\prime}_{1/2}\!-\!A_{1/2})e^{i\delta_{1/2}},\quad A_{\pi^-K^+}\!=\!\frac{A_{3/2}}{\sqrt{3}}e^{i\delta_{3/2}}\!+\!\sqrt{\frac{2}{3}}(A^{\prime}_{1/2}\!-\!A_{1/2})e^{i\delta_{1/2}},$$
where we have defined the isospin amplitudes
$$A_{3/2}=-\sqrt{\frac{2}{3}}\langle 3/2|H_{\Delta I=1}|1/2\rangle,\quad A_{1/2}=-\frac{1}{\sqrt 3}\langle 1/2| H_{\Delta I=1}|1/2\rangle,\quad A^{\prime}_{1/2}=\langle 1/2 |H_{\Delta I=0}|1/2\rangle .$$
On the other hand, we can express the bare or direct weak amplitudes in terms of the leading $T$ ({\em tree}), $P$ ({\em QCD-penguin}), $C$ ({\em color suppressed}) and $P^{EW}$ ({\em EW-penguin}) quark diagrams as follows
$$A^b_{\pi^+K^0}=P\lambda^s_t,\qquad A^b_{\pi^0K^+}=\frac{1}{\sqrt 2}\left((T+C)\lambda^s_u+(P+P^{EW})\lambda^s_t\right),$$
$$ A^b_{\pi^0K^0}=\frac{1}{\sqrt 2}\left(C\lambda^s_u-(P-P^{EW})\lambda^s_t\right),\qquad A^b_{\pi^-K^+}=\left(T\lambda^s_u+P\lambda^s_t\right),$$
where we have defined $\lambda_t^s=V_{ts}V_{tb}^*\approx -A\lambda^2$ and
$\lambda_u^s=V_{us}V_{ub}^*\approx AR_u\lambda^4e^{i\gamma}$, with
$R_u\!=\!|\rho\!-\!i\eta|$ in the Wolfenstein parameterization.
When the strong phases are turned off in the isospin amplitudes, we can identify the
following relations between the quark and the isospin amplitudes,
$$A_{3/2}=\frac{1}{\sqrt 3}\left[(T+C)\lambda_u^s+P^{EW}\lambda_t^s\right],\quad A_{1/2}=\frac{1}{2\sqrt 6}\left[(-T+2C)\lambda_u^s+2P^{EW}\lambda_t^s\right],$$$$A'_{1/2}=\sqrt{\frac{3}{2}}\left(\frac{T}{2}\lambda^s_u+P\lambda^s_t\right).$$
Thus, using such identities we find the physical amplitudes (including FSI) in
terms of quark diagrams
$$A_{\pi^+K^0}=P\lambda^s_t+\frac{1}{3}\left((T+C)\lambda^s_u+P^{EW}\lambda^s_t\right)(1-e^{i\delta_{\pi K}}),$$
$$A_{\pi^0K^+}=\frac{1}{\sqrt 2}\left((T+C)\lambda^s_u+(P+P^{EW})\lambda^s_t\right)-\frac{\sqrt 2}{3}\left((T+C)\lambda^s_u+P^{EW}\lambda^s_t\right)(1-e^{i\delta_{\pi K}}),$$
$$A_{\pi^0K^0}=\frac{1}{\sqrt 2}\left(C\lambda^s_u-(P-P^{EW})\lambda^s_t\right)-\frac{\sqrt 2}{3}\left((T+C)\lambda^s_u+P^{EW}\lambda^s_t\right)(1-e^{i\delta_{\pi K}}),$$
$$A_{\pi^-K^+}=\left(T\lambda^s_u+P\lambda^s_t\right)-\frac{1}{3}\left((T+C)\lambda^s_u+P^{EW}\lambda^s_t\right)(1-e^{i\delta_{\pi K}}),$$
where the strong phase $\delta_{1/2}$ was absorbed into the definitions of the
amplitudes and $\delta_{\pi K}\! \equiv \!\delta_{3/2}\!-\!\delta_{1/2}$.
In the case of $B\to\pi\pi$ decays, we follow the same steps. Here, the physical amplitudes in terms of the isospin amplitudes are given by
$$A_{\pi^+\pi^0}=\sqrt{\frac{3}{2}}A_2 e^{i\delta_2},\quad A_{\pi^+\pi^-}=\sqrt{\frac{2}{3}}A_0e^{i\delta_0}+\frac{A_2}{\sqrt 3}e^{i\delta_2},\quad A_{\pi^0\pi^0}=\frac{A_0}{\sqrt 3}e^{i\delta_0}-\sqrt{\frac{2}{3}}A_2 e^{i\delta_2},$$
where
$$A_{2}=-\frac{1}{\sqrt{2}}\langle 2|H_{\Delta I=3/2}|1/2\rangle,\quad A_{0}=-\frac{1}{\sqrt 2}\langle 0| H_{\Delta I=1/2}|1/2\rangle .$$
In terms of the leading quark diagrams, the bare weak amplitudes are
$$A_{\pi^+\pi^0}^b=\frac{1}{\sqrt 2}(T+C)\lambda_u^d,\quad A_{\pi^+\pi^-}^b=T\lambda_u^d+P\lambda_t^d,\quad A_{\pi^0\pi^0}^b=\frac{1}{\sqrt 2}\left(-C\lambda_u^d+P\lambda_t^d\right),$$
where $\lambda_u^d=V^*_{ub}V_{ud}\approx AR_u\lambda^3e^{i\gamma}$ and $\lambda_t^d=V^*_{tb}V_{td}\approx AR_t\lambda^3e^{-i\beta}$, with $R_t\!=\!|1\!-\!\rho\!-\!i\eta|$.
In the absence of the strong phases, we have the following relations between the isospin and quark amplitudes:
$$A_2=\frac{1}{\sqrt 3}(T+C)\lambda_u, \qquad A_0=\left(\sqrt{\frac{2}{3}}T-\frac{C}{\sqrt 6}\right)\lambda_u+\sqrt{\frac{3}{2}}P\lambda_t$$
Using these relations, the total rescattered amplitudes in terms of quarks
diagrams are
$$A_{\pi^+\pi^0}=\frac{1}{\sqrt 2}(T+C)\lambda_u^d e^{i\delta_{\pi\pi}},$$
$$ A_{\pi^+\pi^-}=(T\lambda_u^d+P\lambda_t^d)+\frac{1}{3}(T+C)\lambda_u^d\left ( e^{i\delta_{\pi\pi}}-1\right ),$$
$$A_{\pi^0\pi^0}=\frac{1}{\sqrt 2}\left(-C\lambda_u^d+P\lambda_t^d\right)-\frac{\sqrt 2}{3}(T+C)\lambda_u^d\left ( e^{i\delta_{\pi\pi}}-1\right ),$$
where $\delta_{\pi\pi}\!=\!\delta_2-\delta_0$.
Measurements of the $B\!\to\!\pi K$ and $B\!\to\!\pi \pi$ branching ratios have been
reported in refs. \cite{CLEObranching}-\cite{Bellepiceroasim} (expressed in units of
$10^{-6}$ in Table 1).
$$\begin{array}{|ccccc|}
\multicolumn{5}{l}{\mbox{TABLE 1: Branching ratios of }B\to\pi K,\,\pi\pi\mbox{ decays}}\\\hline\hline
& \mbox{BaBar}&\mbox{Belle} &\mbox{CLEO} & \mbox{average} \\\hline
\overline{{\cal B}}(B^+\!\to\!\pi^+ K^0) & 26.0\!\pm\! 1.6 & 22.0\!\pm\! 2.2 & 18.8^{\!+\!4.3}_{\!-\!3.8} & 24.1\!\pm\! 1.3 \\
\overline{{\cal B}}(B^+\!\to\!\pi^0K^+) &12.0\!\pm\! 0.9 & 12.0^{\!+\!1.8}_{\!-\!1.6} & 12.9^{\!+\!2.7}_{\!-\!2.5} & 12.1\!\pm\! 0.8 \\
\overline{{\cal B}}(B^0\!\to\!\pi^0 K^0) & 11.4\!\pm\! 1.1 & 11.7\!\pm\! 2.6 & 12.8^{\!+\!4.3}_{\!-\!3.6} & 11.5\!\pm\! 1.0 \\
\overline{{\cal B}}(B^0\!\to\!\pi^- K^+) & 19.2\!\pm\! 0.85 & 18.5\!\pm\! 1.2 & 18.0^{\!+\!2.6}_{\!-\!2.8} & 18.9\!\pm\! 0.7 \\\hline
\overline{{\cal B}}(B^+\!\to\!\pi^+\pi^0) & 5.8\!\pm\! 0.7 & 5.0\!\pm\! 1.3 & 4.6^{\!+\!1.9}_{\!-\!1.7} & 5.5\!\pm\! 0.6 \\
\overline{{\cal B}}(B^0\!\to\!\pi^+\pi^-) & 5.5\!\pm\! 0.5 & 4.4\!\pm\! 0.67 & 4.5^{\!+\!1.5}_{\!-\!1.3} & 5.0\!\pm\! 0.4 \\
\overline{{\cal B}}(B^0\!\to\!\pi^0\pi^0) & 1.17\!\pm\! 0.335 & 2.32^{\!+\!0.45}_{\!-\!0.58} & <4.4 &1.45\!\pm\! 0.29\\
\hline \end{array}$$
Corresponding measurements of the CP asymmetries are much less precise
\cite{BabarKmas}-\cite{Bellepimasasim} and are summarized in Table 2.
$$\begin{array}{|ccccc|}
\multicolumn{5}{l}{\mbox{TABLE 2: CP Asymmetries of }B\to\pi K,\,\pi\pi\mbox{ decays}}\\\hline\hline
&\mbox{BaBar}&\mbox{Belle}&\mbox{CLEO}&\mbox{average} \\
\hline
{\cal A}_{\pi^-\!K^+} & - 0.133\!\pm\! 0.031 & - 0.113\!\pm\! 0.022 & - 0.04\!\pm\! 0.16 & - 0.115\pm 0.018 \\
{\cal A}_{\pi^+\!K^0} & - 0.087\!\pm\! 0.047 & 0.05\!\pm\! 0.05 & 0.18\!\pm\! 0.24 &- 0.02\!\pm\! 0.034 \\
{\cal A}_{\pi^0\!K^+} & 0.06\!\pm\! 0.06 & 0.04\!\pm\! 0.05 &- 0.29\!\pm\! 0.23&0.04\!\pm\! 0.04 \\
{\cal A}_{\pi^0\!K_S} & - 0.06\!\pm\! 0.19 & - 0.12\!\pm\! 0.21 &- & - 0.09\!\pm\! 0.14
\\\hline
C_{\pi^+\!\pi^-} & - 0.09\!\pm\! 0.15 & - 0.56\!\pm\! 0.13 & - & - 0.37\!\pm\! 0.10\\
C_{\pi^0\!\pi^0}&- 0.12\!\pm\! 0.56& - 0.44^{+\!0.56}_{-\!0.55}&- &- 0.28^{\!+\!0.40}_{\!-\!0.39}\\\hline
S_{\pi^+\!\pi^-} & - 0.30\!\pm\! 0.17 & - 0.67\!\pm\! 0.17 & - &- 0.50\!\pm\! 0.12 \\
\hline
\end{array}$$
There are some noticeable differences between BaBar and Belle results, namely the differences of the branching ratios $\overline{{\cal B}}(B^+\!\to\!\pi^+K^0)$ and $\overline{{\cal B}}(B^0\!\to\!\pi^0\pi^0)$ are $1.5\sigma$ and $2\sigma$ away, respectively, while the differences of the asymmetries $S_{\pi^+\pi^-}$, ${\cal A}_{\pi^+K^0}$ and $C_{\pi^+\pi^-}$ differs by $1.5\sigma$, $2\sigma$ and $2.3\sigma$. Therefore, we prefer to do a separated analysis for both sets of data.
\section{Constraints from the branching ratios}
We define the following $6$ independent ratios of the CP-averaged $B\!\to\!\pi\pi,\pi K$ branching ratios
$$R_c=\frac{2\overline{{\cal B}}(B^+\!\to\!\pi^0K^+)}{\overline{{\cal B}}(B^+\!\to\!\pi^+ K^0)},\quad
R_n=\frac{\overline{{\cal B}}(B^0\!\to\!\pi^-K^+)}{2\overline{{\cal B}}(B^0\!\to\!\pi^0K^0)},$$
$$ R_m=\frac{2\tau_+}{\tau_0}\frac{\overline{{\cal B}}(B^0\!\to\! \pi^0 K^0)}{\overline{{\cal B}}(B^+\!\to\!\pi^+K^0)},\quad
R_{\pi K}=\frac{2\overline{{\cal B}}(B^+\!\to\!\pi^+\pi^0)}{\overline{{\cal B}}(B^+\!\to\!\pi^+ K^0)},$$
$$ R_{\pi\pi}=\frac{\tau_+}{\tau_0}\frac{(\overline{{\cal B}}(B^0\!\to\!\pi^+\pi^-)+\overline{{\cal B}}(B^0\!\to\!\pi^0\pi^0))}{\overline{{\cal B}}(B^+\to\pi^+\pi^0)},\quad
R_{0}=\frac{\tau_+}{\tau_0}\frac{\overline{{\cal B}}(B^0\!\to\!\pi^+\pi^-)}{\overline{{\cal B}}(B^+\!\to\!\pi^+\pi^0)},$$
where $\tau_+/\tau_0=1.076\pm 0.008$ \cite{HFAG} is the ratio of the $B^+$ and $B^0$ lifetimes. $R_c$ and $R_n$ were already introduced in \cite{buras}.
From Table 1 we obtain the following table
$$\begin{array}{|c|cccccc|}
\multicolumn{1}{c}{\mbox{exp}} & \multicolumn{1}{c}{R_{\pi K}} & \multicolumn{1}{c}{R_{c}} & \multicolumn{1}{c}{R_{m}} & \multicolumn{1}{c}{R_n} & \multicolumn{1}{c}{R_{\pi\pi}} &\multicolumn{1}{c}{R_{0}}\\\hline
\mbox{BaBar} & 0.45\pm 0.09 & 0.92\pm 0.09 & 0.94\pm 0.11 & 0.84\pm 0.09 & 1.24\pm 0.19 & 1.02\pm 0.15\\\mbox{Belle} & 0.45\pm 0.19 & 1.09\pm 0.19 & 1.14\pm 0.28 & 0.79\pm 0.18 &1.45\pm 0.42& 0.95\pm 0.29\\\mbox{{\small average}} &0.46\pm 0.06 & 1.00\pm 0.08 & 1.03\pm 0.105 & 0.82\pm 0.08 & 1.26\pm 0.17 &0.98\pm 0.13 \\\hline\end{array}$$
where CLEO data has been included in the average.
From $R_{\pi K}$ we can obtain the tree (color allowed and suppressed) to penguin ratio. Defining
$$r=\frac{T|\lambda^s_{u}|}{P|\lambda^s_{t}|}=\frac{T\lambda^2 R_u}{P},\qquad r_c=\frac{(T+C)\lambda^2 R_u}{P},$$
$R_{\pi K}$ is written as
\begin{equation}\label{TCP}
R_{\pi K}=\frac{f_{\pi}^2}{f_K^2}\frac{(T+C)^2|\lambda_u^d|^2}{P^2|\lambda_t^s|^2}=\frac{f_{\pi}^2}{f_K^2}\frac{(T+C)^2(\lambda R_u)^2}{P^2}=\frac{f_{\pi}^2}{f_K^2}\frac{r_c^2}{\lambda^2}\quad\Rightarrow\quad r_c=\lambda \frac{f_K}{f_{\pi}}\sqrt{R_{\pi K}},
\end{equation}
where the ratio $f_{K}/f_{\pi}\!=\!1.2$ gives an estimate of the size of the $SU(3)$ symmetry breaking. Using the experimental value of $R_{\pi K}$ and $\lambda\!=\!\sin\theta_C\!=\!0.226$ we find $r_c\!=\!0.18\!\pm\! 0.01$. This result is obtained by neglecting the FSI corrections to $B^+\to\pi^+K^0$.
The next step is to find the ratio $C/T$ by adding the information of the $B\!\to\!\pi\pi$ modes. In the quasi-elastic case
the final state interactions in $B\to\pi^+\pi^-$ and $B\to\pi^0\pi^0$ cancel each other in their ratio and we have
$$R_{\pi\pi}=\frac{2+3\left(\frac{P R_t}{T R_u}\right)^2+(C/T)^2+2\frac{P R_t}{T R_u}\left(\frac{C}{T}-2\right)\cos\alpha}{(1+C/T)^2}.$$
Since from Equation (\ref{TCP})
\begin{equation}\label{PT}
\frac{P}{T}=\frac{(1+C/T)\lambda R_u}{f_K/f_{\pi}\sqrt{R_{\pi K}}}
\end{equation}
using the sine law ($R_t=\sin\gamma/\sin\alpha$ and $R_u=\sin\beta/\sin\alpha$) we can express the ratio $C/T$ in terms of $\gamma$ and $\beta$ as follows
\begin{equation}\label{CT}
\frac{C}{T}=\frac{-(A+B/2)+\sqrt{(3/2 B)^2+3 A-2}}{A-B-1}
\end{equation}
where
$$A=R_{\pi\pi}-\frac{3\lambda^2 \sin^2\gamma}{(f_K/f_{\pi})^2R_{\pi K}\sin^2(\pi-\gamma-\beta)},\quad B=\frac{2\lambda\sin\gamma}{(f_K/f_{\pi})\sqrt{R_{\pi K}}\tan(\pi-\gamma-\beta)}.$$
In Figure (\ref{padre}) we plot $C/T$ as a function of $\gamma$, using the experimental value $\beta$, $\beta\!=\!(21.70\pm 1.26)^{\circ}$ \cite{ckm}. We
observe that in order to have the naive expectation $C/T\sim
0.3$ from BaBar data we require $\gamma\gtrsim 68^{\circ}$, namely a value that is above the upper limit of the standard fit, which taken at its $1\sigma$ interval reads $\gamma\in(52.7^{\circ},65.4^{\circ})$ \cite{ckm}.
\begin{figure}
\centering{
\mbox{\subfigure{\epsfig{file=color2.eps,width=6cm,height=4cm}}\quad\subfigure{\epsfig{file=color3.eps,width=6cm,height=4cm}}}
}
\caption{ $C/T$ vs $\gamma$ extracted from ratios of B decay widths.}
\label{padre}
\end{figure}
We can obtain additional information about $\gamma$ using the ratios of the
$B\to\pi K$ branching fractions. Inclusion of the $P^{EW}$ diagrams is required in
this case. Defining
$$q=\frac{P^{EW}|\lambda^s_t|}{(T+C)|\lambda_u^s|}$$
and
$$x=1\!-\!r\cos\gamma,\quad y=r\!-\!\cos\gamma,\quad z=r_c(q\!-\!2\cos\gamma),$$
$R_c$ can be expressed as follows
$$R_c\!=\!\frac{1\!-\!2r_c\cos\gamma\!+\!r_c^2\!+\!qr_c[2\!+\!z]\!+\!\frac{4}{3}(1\!-\!\cos\delta_{\pi K})r_c(\cos\gamma\!-\!\frac{\displaystyle r_c}{3}\!-\!q[1\!+\!\frac{\displaystyle z}{3}])}{1-\frac{2}{3}(1\!-\!\cos\delta_{\pi K})r_c(\cos\gamma-\frac{\displaystyle r_c}{3}-q[1+\frac{\displaystyle z}{3}])}$$
$$\approx 1\!-\!2r_c\cos\gamma\!+\!r_c^2\!+\!qr_c[2\!+\!z]\!+\!2(1\!-\!\cos\delta_{\pi K})r_c\left(\cos\gamma\!-\!\frac{r_c}{3}\!-\!q\left[1\!+\!\frac{z}{3}\right]\right).$$
In Figure \ref{platano1} we plot $R_c$ as a function of $\gamma$ for different
values of $P^{EW}$ and for two values of the difference of final strong phases ($\delta_{\pi K}=0$ and $30^{\circ}$). The dependence upon the strong phase
is irrelevant but is enough to be confused with the effect of a small change in
$P^{EW}$ when $\gamma$ is small. From Figure \ref{platano1} we see
that the expectation $q=0.58$ \cite{burasymas} is excluded
by the BaBar data. Small values of $q$, as we would expect from the
naive ratio $P^{EW}/P\,(m_b)\sim\alpha/\alpha_S(m_b)\approx 0.03$, namely $q\sim
0.1-0.2$, are preferred by data. Thus, assuming that at least $q=0.1$, BaBar
data puts a limit of $\gamma\lesssim 80^{\circ}$.
\begin{figure}
\centering{
\mbox{\subfigure{\epsfig{file=electro1Babar.eps,width=6.0cm,height=4cm}}\quad\subfigure{\epsfig{file=electro1Belle.eps,width=6.0cm,height=4cm}}}
\mbox{\subfigure{\epsfig{file=electro11Babar.eps,width=6cm,height=4cm}}\quad\subfigure{\epsfig{file=electro11Belle.eps,width=6cm,height=4cm}}}
}
\caption{$R_c$ vs $\gamma$ for $\delta_{\pi K}=0$ and $\delta_{\pi K}=30^{\circ}$ and for different values of $q$. The horizontal bands represent the experimental value
with $\pm 1\sigma$ limits.}
\label{platano1}
\end{figure}
Let us consider now the information provided by $R_n$:
$$R_n\!=\!\frac{1\!-\!2r\cos\gamma\!+\!r^2\!-\!\frac{2}{3}(1\!-\!\cos\delta_{\pi K})r_c(y\!-\!\frac{\displaystyle r_c}{3}\!+\!q[x\!-\!\frac{\displaystyle z}{3}])}{1\!+\!2(r_c\!-\!r)\cos\gamma\!+\!(r_c\!-\!r)^2\!-\!qr_c[2x\!-\!z]+\frac{4}{3}(1\!-\!\cos\delta_{\pi K})r_c(y\!-\!\frac{\displaystyle r_c}{3}\!+\!q[x-\!\frac{\displaystyle z}{3}])}\ .$$
The dependence upon the strong phase is less important than in $R_c$ and we consider
the analysis only for $\delta_{\pi K}=0$. From figure (\ref{platano5}) we observe that $q=0.6$ is
not favored by data. Using the value $q=0.1$, BaBar puts the limit
$\gamma\lesssim 70^{\circ}$ and Belle $\gamma\lesssim 75^{\circ}$.
\begin{figure}
\centering{
\mbox{\subfigure{\epsfig{file=electro3Babar.eps,width=6cm,height=4cm}}\quad\subfigure{\epsfig{file=electro3Belle.eps,width=6cm,height=4cm}}}
}
\caption{$R_n$ vs $\gamma$ for $\delta_{\pi K}=0$ and for different values of $q$. The horizontal bands represent the experimental value
with $\pm 1\sigma$ limits. }
\label{platano5}
\end{figure}
Finally, we consider the ratio $R_m$,
$$R_m\approx 1+2(r_c-r)\cos\gamma+(r_c-r)^2+qr_c[-2x+z].$$
From Figure (\ref{electra2}) we observe that $q=0.6$ is ruled out by Belle data, but neither of the
two experiments puts a relevant constrain on $\gamma$ for small values of $q$.
\begin{figure}
\centering{
\mbox{\subfigure{\epsfig{file=electro2Babar.eps,width=6cm,height=4cm}}\quad\subfigure{\epsfig{file=electro2Belle.eps,width=6cm,height=4cm}}}
}
\caption{$R_m$ vs $\gamma$ for different values of $q$. The horizontal bands represent the experimental value
with $\pm 1\sigma$ limits. }
\label{electra2}
\end{figure}
When more accurate measurements become available, we will be able to determine $q$,
in addition to $\gamma$. At present we observe that large $q$ values are clearly disfavored by data and are inconsistent with the ratio $C/T$
displayed in Figure (\ref{padre}). Clearly, small values of $P^{EW}$ are preferred by
data.
\section{Constraints from the CP asymmetries}
Final state interactions can receive contributions from rescattering of
several intermediate states. In the
framework of the Regge model FSI are dominated by the quasi-elastic modes, so we expect small strong phases. Since the CP asymmetries are proportional to the
interference term $T\lambda^D_uP\lambda^D_t$, $D\!=\!d,s$, the inclusion of
$P^{EW}$ is negligible for any $D$, and we will neglect it.
The $\pi\pi$ direct asymmetries, in two-pion $B$ decays are
given by
$$C_{\pi^+\pi^-}\!=\!\frac{|A_{\pi^+\pi^-}|^2\!-\!|\overline A_{\pi^+\pi^-}|^2}{|A_{\pi^+\pi^-}|^2\!+\!|\overline A_{\pi^+\pi^-}|^2}=-\frac{4\frac{PR_t}{TR_u}\sin\delta_{\pi\pi}\sin\alpha}{3R_0(1+C/T)},\quad C_{\pi^0\pi^0}\!=\!\frac{4\frac{PR_t}{TR_u}\sin\delta_{\pi\pi}\sin\alpha}{3(R_{\pi\pi}-R_0)(1+C/T)} \ .$$
Using Equation (\ref{PT}) and the sine law relation $R_t\sin\alpha=\sin\gamma$,
we obtain
$$C_{\pi^+\pi^-}=-\frac{4\lambda\sin\delta_{\pi\pi}\sin\gamma}{3R_0(f_K/f_{\pi})\sqrt{R_{\pi K}}},\quad C_{\pi^0\pi^0}=\frac{4\lambda\sin\delta_{\pi\pi}\sin\gamma}{3(R_{\pi\pi}-R_0)(f_K/f_{\pi})\sqrt{R_{\pi K}}} \ .$$
From Figure (\ref{platano2}) we observe that the CP asymmetry coefficient
$C_{\pi^+\pi^-}$ measured by Belle disfavors large strong phase difference,
while that measured by BaBar prefers small strong phases. Therefore, we can not infer any physical conclusion from this
CP asymmetry. On the other hand present data on $C_{\pi^0\pi^0}$ are
consistent with a small strong phase difference. Note that we expect a CP
asymmetry of opposite sign to $C_{+-}$, though the data seems to
favor a negative asymmetry.
\begin{figure}
\centering{
\mbox{\subfigure{\epsfig{file=pionesBabar.eps,width=6cm,height=4cm}}\quad\subfigure{\epsfig{file=pionesBelle.eps,width=6cm,height=4cm}}}
\mbox{\subfigure{\epsfig{file=pionBabar.eps,width=6cm,height=4cm}}\quad\subfigure{\epsfig{file=pionBelle.eps,width=6cm,height=4cm}}}
}
\caption{$C_{\pi^+\pi^-}$ and $C_{\pi^0\pi^0}$ vs $\gamma$ for two
choices of the strong phase difference. The horizontal band represents the
experimental value with its $\pm 1\sigma$ limits.}
\label{platano2}
\end{figure}
The expression for the $S_{\pi^+\pi^-}$ asymmetry coefficient is:
$$S_{+-}=\left(1+ C_{+-}\right)\Im \left(e^{-2i\beta}\frac{\overline A_{+-}}{ A_{+-}}\right)=$$
$$=\frac{\left(1+C_{+-}\right) \left[\sin 2\alpha-2p\sin\alpha+p^2+\frac{2}{3}(1-\cos\delta_{\pi\pi})(1+c)\left(\sin 2\alpha\left(\frac{1+c}{3}-1\right)+p\sin\alpha\right)\right]}{1-2p\cos\alpha+p^2+\frac{2}{3}(1-\cos\delta_{\pi\pi})(1+c)\left(\frac{1+c}{3}-1\right)+\frac{2}{3}p(1+c)[\cos\alpha-\cos (\delta-\alpha )]}$$
where $p=(P/T)(R_t/R_u)$, and $c=C/T$. Making use of equations (\ref{PT}),
(\ref{CT}) and $R_t/R_u=\sin\gamma/\sin\beta$, we obtain Figure
(\ref{platano3}) where we have plotted $S_{\pi^+\pi^-}$ as a function of
$\gamma$ when $\delta_{\pi\pi}=0$.
From this plot we derive the following allowed intervals for $\gamma$:
$\gamma\sim(70^{\circ},85^{\circ})$ (BaBar) and
$\gamma\sim(50^{\circ},70^{\circ})$ (Belle).
\begin{figure}
\centering{
\mbox{\subfigure{\epsfig{file=super2.eps,width=6cm,height=4cm}}\quad\subfigure{\epsfig{file=super3.eps,width=6cm,height=4cm}}}
}
\caption{$S_{\pi^+\pi^-}$ vs $\gamma$. The horizontal bands represents the
experimental values with its $\pm 1\sigma$ bounds. }
\label{platano3}
\end{figure}
In the case of $B\to \pi K$ decays, we have the following expression for the
direct CP asymmetry coefficient:
$$C=\frac{2}{3}r_c\sin\delta_{\pi K}\sin\gamma=\frac{2}{3}\lambda\frac{f_k}{f_{\pi}}\sqrt{R_{\pi K}}\sin\delta_{\pi K}\sin\gamma$$
then
$${\cal A}_{\pi^-K^+}\!=\!\frac{\mid \overline{A}_{\pi^+K^-}\mid^2\!-\!\mid A_{\pi^- K^+}\mid^2}{\mid \overline{A}_{\pi^+K^-}\mid^2\!+\!\mid A_{\pi^- K^+}\mid^2}\approx C,\quad
{\cal A}_{\pi^+K^0}\!\approx -C,\quad {\cal A}_{\pi^0K^+}\!\approx 2C,\quad
{\cal A}_{\pi^0K_S}\!\approx -2C$$
From Table 2 we have consistency of both experiments and a large asymmetry
${\cal A}_{\pi^-K^+}$. Note however that in this case photonic radiated corrections should be taken more carefully in the Montecarlo simulations.
If forthcoming
measurements confirm a large asymmetry, we should conclude that the
quasi-elastic assumption it is not enough to explain data. In that case
other final state interactions which we are neglecting in our analysis,
like $B\!\to\! \overline DD_s\!\to\!\pi K$, would be required.
The situation with the other asymmetries is not clear at present. For
example, the sign of ${\cal A}_{\pi^+K^0}$ differs for BaBar and Belle and
for ${\cal A}_{\pi^0K^+}$ both experiments favor a positive asymmetry, while
we expect (and some other models \cite{burasymas}-\cite{yykeum} do) a negative value. Therefore, we consider
that use of the branching ratios provides at present a more reliable source to
derive bounds on $\gamma$.
\section{Conclusions}
We have used the quasi-elastic approximation to hadronic final state
rescattering in $B\to \pi\pi, K\pi$ in order to extract information about the
CP-violating angle $\gamma$ from experimental data of these decays. The
values used for the rescattering parameters within our approach were
obtained using the Regge model.
Given the better uncertainty attained in experimental data of the
branching ratios for these decays, they are at present a better source of
information on $\gamma$ than the corresponding CP asymmetries.
Note however that using the world average from BaBar and Belle data for
the mixing induced asymmetry coefficient $S_{\pi^+\pi^-}$ the result
$\gamma\sim (70\pm 6)^{\circ}$ is obtained.
Using the information about the six independent branching ratios of $B\to
\pi\pi,\ K\pi$ decays, we can also get bounds on $\gamma$. Present data on
these observables do not offer a problem to our approach since required
electroweak penguin contributions turns out to be small and the $C/T$
ratio is non negligible. In this case, the value $\gamma\sim
70^{\circ}$ is preferred.
If a large direct CP asymmetry for $\pi^+K^-$ decays is confirmed by
forthcoming measurements, the quasi-elastic approximation may require
some corrections. The most viable candidate could be the inclusion of the
inelastic channel $B\!\to\! \overline DD_s\!\to\!\pi K$.
\section{Acknowledgments}
We thank Germ\'an Calder\'on, Jean-Marc G\'erard, Mat\' ias Moreno and Jacques Weyers for their valuable comments on this work. We are especially grateful to Gabriel L\'opez Castro for useful discussions and important observations in its final stage.
Erika Alvarez acknowledges the financial support from CONACyT and DGAPA (M\'exico) under projects G42026-F and IN-120602.
{\it
After completion of this work, we came to know the preprint hep-ph/0508083 \cite{italiani}. Let us comment similarities and differences between their papers and ours.
1. We obtain the $\pi\!-\!\pi$ absorptive elastic factors $\eta$'s quantitatively similar to the ones obtained in ref. \cite{italiani}, but using a different approach.
2. In ref.\cite{italiani}, the inelastic channel $\rho\! +\!\rho\! \to\! \pi\! + \!\pi$ is considered in the context of Watson theorem. But this theorem requires stable particles with respect to strong interactions as in- and out-states. It is not the case for $\rho\!-\!\rho$. Therefore 4 stable pions should replace the two unstable $\rho$'s. It is difficult to have a precise estimate of 4- pions to 2-pions scattering. However a rough estimate indicates this channel should have a negligible effect in the strong phase computation.
3. In ref. \cite{italiani}, charmed intermediate states are estimated to have a negligible effect on strong phases.We agree with this point even if the argument is rather qualitative.. This seems reasonable
4. The main differences between the two works are (a) that in \cite{italiani} strong phases are found to be large due to the questionable inclusion of $\rho$'s in contradistinction with our paper concluding strong phases are small, and (b) contrary to \cite{italiani}, we treat not only $\pi\!-\!\pi$ modes but also $\pi\! -\! K$ modes. }
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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# ALIHAHD . . .
To most of the known galaxy he was a legend without a face, and to the rest a face without a name.
But luck and disguises had finally given out, and a Na′id squadron was in fast-closing pursuit of his ship, _Liberation_.
He was staring out the viewport without really seeing, when a pattern of lights appeared out of an unexpected quadrant. And a slow sick horror penetrated the blank space inside him.
_Na′id?_
Could he possibly be _surrounded_ by Na′id ships?
But it was not a Na′id ship.
It was worse. . . .
# Novels by R. M. Meluch
JERUSALEM FIRE
_Tour of the Merrimack_ :
THE MYRIAD
WOLF STAR
THE SAGITTARIUS COMMAND
STRENGTH AND HONOR
THE NINTH CIRCLE
THE TWICE AND FUTURE CAESAR
Copyright © 1985 by R. M. Meluch.
All new material copyright © 2016 by R. M. Meluch.
All rights reserved.
Jerusalem Fire was originally published by Signet in 1985.
This new edition has been fully revised and updated.
Cover art by Paul Young.
Cover design by G-Force Design.
DAW Books Collectors No. 1742.
Published by DAW Books, Inc.
375 Hudson Street, New York, NY 10014.
All characters and events in this book are fictitious.
All resemblance to persons living or dead is coincidental.
The scanning, uploading, and distribution of this book via the Internet or any other means without the permission of the publisher is illegal, and punishable by law. Please purchase only authorized electronic editions, and do not participate in or encourage the electronic piracy of copyrighted materials. Your support of the author's rights is appreciated.
Ebook ISBN: 9780756412210
DAW TRADEMARK REGISTERED
U.S. PAT. AND TM. OFF. AND FOREIGN COUNTRIES
—MARCA REGISTRADA
HECHO EN U.S.A.
Version_1
# Contents
_Novels by R. M. Meluch_
_Title Page_
_Copyright_
_Epigraph_
**Part One:**
1. No Blaze of Glory
2. Iry
3. Island in the Sky
4. Circle Circle
5. Does Jerusalem Stand?
**Part Two:**
6. Wolf at the Door
7. Wolf by the Hearth
8. Wolf in the Fire
9. Wolf at the Ramparts
**Part Three:**
10. The Gathering Storm
11. Thunderhead
12. A Winter Conversation
13. Witch Wind
14. A Slow-Falling Star
**Part Four:**
15. Rogue Wolf
16. Ghosts
17. Enemies
**Part Five:**
18. Return of a Legend
19. Shadow of Masada
20. Jerusalem Stands
21. Nemo
_Appendix_
Does Jerusalem stand?
—Traveler's question
I don't understand Jews.
—Shad Iliya
Christians, Mohammedans, and Jews, peoples who placed revelation before reason . . .
—T. E. Lawrence, _Seven Pillars of Wisdom_
Idolatry is worse than carnage.
—Koran
Thou shalt have no strange gods before me.
—Decalogue
To whatever name you pray, God hears and answers.
—Na′id tenet, adapted from the _Bhagavad Gita_
The human Deity hath many names. No god ever told us a name to call. Perhaps we are being snubbed?
—Roniva
I don't understand Jews at all.
—Shad Iliya
# PART ONE:
## Alihahd
# 1. No Blaze of Glory
THE CAPTAIN WAS a notorious rebel runner. He was called Alihahd. This was his last run.
He had no real name, only a Chesite word, alihahd, which meant "he left." He had no country, no planet, though several including Chesa claimed him. To most of the known galaxy he was a legend without a face, and to the rest a face without a name.
He was very tall, lanky. His long arms were gnarled like the ancient olive trees of Earth. He had the punished look of one at war too long. A dignity of bearing saved him from being gangly.
His eyes were splendid, expressive in their depth. His features were strong and regular, but he was too drawn to be called handsome anymore.
He'd lived over half his natural life span, which was much older than he'd ever expected to get. He was not surprised that it was to end now.
Luck and disguises had finally given out, and a Na′id squadron was in fast-closing pursuit of his runner ship _Liberation_. Alihahd had begun loading his refugees into _Liberation_ 's twelve emergency shuttles. They would need to make the rest of the journey to the free planet New Triton without him.
Alihahd and a handful of volunteers would stay in the mothership to lead the Na′id on a chase in a false direction until overtaken and destroyed. Or, should the Na′id try to board the _Liberation_ , she would self-destruct to cover the absence of her passengers and shuttles.
Alihahd stood motionless on the command deck, one large and knobby hand at his belt, one fist on the console, knuckles down as if he'd punched it and frozen there. He could overhear sounds of orderly flight drifting up from below where refugees spoke in whispers and shuffled in steadily moving lines onto the emergency craft.
He heard an air lock close, lock. The life bay thumped open. A shuttle detached and moved silently to the port side of the mothership where four other loaded shuttles were grouped in the bright oasis of the _Liberation_ 's docking lights, waiting to depart all together in a convoy when the last of the shuttles was boarded.
Another air lock closed. Locked.
_That was six_.
The mind counted without passion. Alihahd hadn't thought that the end would feel like this. He was feeling nothing.
He had planned for this contingency a long time, down to the clockwork details. The actual crisis was so like the countless times he had run it in his mind that this seemed just one more exercise.
He was fundamentally alone in this, without friends or lovers. There was no one aboard who honestly knew him as a man. To his passengers, and even to his crew, he was a distant, unfailing protector come from nowhere without selfish motive to spirit discontented subjects away from Na′id rule. His history was a blank but for a hint in his beautiful, correct voice. He would slip into high prose at times, and everyone supposed he was fugitive royalty, but his cosmopolitan accent didn't say from where.
He kept himself remote. There was a too-gentle way he moved his hands, suggesting a tender nature—or hiding a violence that dreaded all but the lightest touch lest he destroy.
He was staring straight out the viewport without really seeing, when a pattern of lights appeared out of an unexpected quadrant, and a slow sick horror penetrated the blank space inside him.
_Na_ ′ _id?_
Could he possibly be, he wondered, _surrounded_ by Na′id ships?
But it wasn't a Na′id ship.
It was worse.
A ghostly white image of a derelict sailing ship glided into view on the waves of a nonexistent sea. Misty at first, the frosty glow took the shape of an ancient brigantine, its shredded jibs straining from the bowsprit on fraying ropes, its topsail a fluttering rag on the splintered yard. This ship was the last sight of many a Na′id crew. " _Marauder_ ," Alihahd breathed without voice. The _Flying Dutchman_ of the stars.
A chill ran up Alihahd's back, pricking at the hair along his spine and his arms. His thick lips twitched, then he resumed his habitual stone face.
The _Marauder_ had once been only a legend, for centuries the terror of superstitious travelers. Someone in recent years had made the _Marauder_ real and used it to destroy ships of the Na′id empire.
_Marauder_ 's long tradition had fixed it in popular memory as a childhood fear that never went away. On sight, it was instantly recognized and instantly feared. The _Marauder_ could scare a victim to death before firing a shot.
But the eerie brigantine was only a projection. Alihahd knew the real ship must be nearby. He found it. It hovered outside the periphery of the _Liberation_ 's sphere of light, painted black and difficult to see, its surface albedo nil, giving the impression of something not there. The backdrop of stars allowed only glimpses of its clawlike angles. It looked like a nightmare.
The spectral hologram was the announcement: _You are next_. It was the lurking ship that carried the guns.
Many horrified people fired at the hologram. Then died.
Alihahd wouldn't be thrown by the unreal—though he'd been accused of the tendency. He kept his eyes on the black ship.
He should've been relieved to meet the archfoe of his enemy—
—except that his own ship bore Na′id markings. _Liberation_ was disguised to travel in Na′id space as a Na′id vessel, so how could the _Marauder_ know that it wasn't?
The command deck had become still. His crew looked at Alihahd the way they always looked at him—as if he could perform miracles. He could see a ghost of his own reflection in the viewport—a rangy, underfed man, old beyond his years—and he wondered what it was they saw.
He was conscious still of the unwary shuffling of refugees below deck.
On the _Marauder'_ s blank surface appeared the ragged outline of mandibles opening to a red furnace within. The thing was about to open fire.
Alihahd moved to his ship's transmitter, intending to get out an order for the six waiting shuttles to scatter and run. At least one might escape. But he froze at the switch—not from fear or indecision, but from instinct.
He drew his hand away from the switch—and braced for the fire when his instincts proved wrong.
But the _Marauder_ didn't fire. It held its position and observed, as if sensing something not right, something different, maybe curious as to why a sound vessel was being evacuated. Or maybe the _Marauder_ had instincts, too.
An air lock closed. Locked. A shuttle detached, oblivious to the threat. Shuttles had no viewports.
Still the _Marauder_ watched.
Alihahd moved his hand back to the transmitter. He masked out the shuttle channel, and opened all others. In what code did one attempt contact with the _Marauder_?
He decided on a voice message. "Alihahd," he said. He did not know if the _Marauder_ understood the Universal tongue, but maybe he would recognize the name of his enemy's enemy if nothing else. "This is Alihahd. We are not— _not_ —Na′id."
He paused.
There was no movement on the command platform save for the silent shifting of panel lights that washed pale red and green across bloodless faces, some of them terribly young.
There was no acknowledgment from the _Marauder_. But neither was there an attack.
The phantom ship menaced with its red maw gaping, a ticking bomb to be defused before the caprice of a moment turned the decision. There had to be a way to get through. With every second, the _Marauder_ could imagine deception in a blinking bow light and open fire.
Without taking his eyes from the viewport, Alihahd spoke to his bridge crew in a quietly urgent whisper. "Close that hatch."
They scuttled across the deck to obey, sealing off the hatch to the shuttle bay before someone could come up and catch sight of the shimmering sails of the ancient derelict and scream its name.
The command console had lighted up all over with hailing signals from perplexed shuttles, which could detect but not see or identify the presence of an extra ship near them. They could only tell that the configuration was not Na′id.
Alihahd clicked off all the demanding lights with his long fingers and glanced at the chronometer. The Na′id—real ones—would be closing in soon. Six minutes.
He leaned straight-armed over the console, glowering at the black ship, his mouth drawn taut. His dusky face darkened. He lowered his chin and glared. The gaunt muscles of his arms stood out in high relief as he grew angry over a sudden suspicion. An instinct.
The bomb was not ticking. It was chuckling.
Alihahd touched the transmitter again. "I think you are reading me, _Marauder_ ," he said in a sharp voice that made his crew gasp and cringe and brace for explosions. "I also think you know Universal. So understand this: A Na′id squadron is due here to destroy us in five minutes Earth standard. Now I have to assume, since you are playing the _Dutchman_ , that you also mean to destroy us, but if you want me to grovel for my passengers' lives, you will have to inform me soon or forfeit the pleasure to the Na′id. But then you would have to break your precious silence, would you not? For myself, by you or by them, I will be equally dead. I don't very much care. So please either shoot, talk, or go away. Or you may go to hell, where I am bound with or without you." Then he added crisply, "Do you want me to repeat any of that?"
Again the _Marauder_ gave no acknowledgment—except to pull away and vanish.
The crew stared at Alihahd with reverent, startled wonder. He had pulled off the required miracle.
No miracle, thought Alihahd. He had lived long enough to know the smell of a player of deadly games.
An air lock closed. Locked.
That was eight.
There was still time. The boarding continued smoothly, the refugees unaware that the ghost ship had brushed so close and passed them by.
The last shuttle was loaded but not launched when Alihahd saw glances and nods pass between the muscular crewmen to either side of him. Alihahd backed out from his spot in between them to look at them—one then the other—straight into their guilty eyes. The rest of the bridge crew froze like dogs caught chewing the carpet. There were too many people on the command platform.
"What? What is this?" Alihhad spoke levelly, even gently. "Is this a mutiny?"
There was another exchange of glances. Whatever it was, they were all in on it. If they had turned against him, he was alone.
The handsome crewman to Alihahd's left shifted his weight, pushed an auburn curl off his forehead, cleared his throat. "Please, sir," he said. "Get on the shuttle. You don't have to die. Let us take the ship by ourselves."
After a grim, intimidating pause, Alihahd shook his head and said gravely, "You know better."
"We intended to force you," the crewman said softly.
For a moment Alihahd was too surprised to react. Not that they should conceive of such a plot—his crew were addicted to dreams of daring valor—but that they still, even faced with his disapproval, seriously intended to go through with it.
The mind assimilated this new development without the body giving sign of the panic quailing at his core. Alihahd maintained his perfect outer calm—it maintained itself out of habit—while inside he was struggling to hold his guts in. Only his fragile dignity was still keeping his crew at bay. There was really nothing else to keep them from laying hands on him and ingloriously dragging him down to the loading platform, depositing him in the last lifecraft, and shuttling him away to safety.
There was not one of them he could best if it came down to a physical pull, grab, and wrestle. They had chosen their beef for this task, knowing he would object for all he was worth. He was taller than the tallest and had the bone structure to match the strongest, but he hadn't the flesh. Age and abuse had taken much from him. He felt suddenly exposed and powerless in their midst, nothing useful within his grasp. He was clad in a pocketless red tunic, a holsterless belt, and rubbery-soled deck boots that would not even let him deliver a convincing kick.
And his mutinous worshippers were determined to save him against his will, at the cost of their own lives. They would stumble over each other to be first in selfless heroism—they had learned it from him. They wanted to be like him. Now he had to confront these heroic monsters of his own creation.
But in their bright-eyed enthusiasm—which they had _not_ imitated from him—they oversimplified, overstepped, and overlooked, and did not realize that even selflessness had its selfish reasons, and Alihahd would not be turned.
He preserved his quiet poise, making no quick desperate motion that would bring them all lunging toward him at once. Liquid eyes cast a spell, and he cloaked himself in his lordly mystique, too stately to be touched.
But awe was a feeble barrier against a dwindling countdown. The chronometer read three minutes. Someone needed to act quickly or every one of them would fall to the Na′id.
With deceptive casualness, all his apprehension contained, Alihahd walked across the deck—ten steps was suddenly an eternal length—passing between two mutineers, to reach under the command console and take out a gun hidden there. He pointed the gun to his own temple, and said, "Very well, then. Force me."
By the time any of them even thought about stopping him, it was too late to do anything. The crew continued to stare—now completely stumped.
Alihahd gave them a few long seconds to realize defeat, then said, "Move. We are running out of time. You will jeopardize everyone with your ill-timed heroics. The thought is not unappreciated, yet it remains very foolish."
The crewman who had spoken for the others moved first, backing off. And because there was nothing else to do, the rest returned to the duties of the original plan of evacuation—Alihahd's plan.
Alone, Alihahd held the gun close across his chest, one hand around its barrel, trying not to shake and trying to hold inside what must remain inside.
They were going to admire him to death, his crew. They wanted him to live. But at what cost? He pictured himself aboard the shuttle, staying apart like a leper, meeting no one's eyes, reaching haven, and answering the question, "Where is your crew?" He felt sick. He knew this feeling. He breathed with his mouth open as if there were no air left in the ship. A slickness filmed palms that never sweated. He did not relinquish the gun until the last shuttle was launched.
The little fleet of twelve shuttles separated from the mothership, moved off, and disappeared. Alihahd turned off the bright docking lights and leaned back against the bulkhead, his head back, hands loose at his sides. From here there were few decisions, and few mistakes that could bring worse results than success.
An odd elation came over him along with the first peace of mind he could remember.
He put on his spacesuit, secured himself into his seat at the ship's controls, and waited. One minute.
The countdown was still at thirty seconds when blips appeared on the computer tracker with simultaneous visual contact. Six ships of metallic blue sublighted for attack, their hulls ablaze with the red twin symbols of Galactic Dominion/Human Supremacy.
The computer pilot engaged, and the _Liberation_ shot away past the Na′id ships. The stars disappeared.
Before leaving the squadron behind in the sublight universe, Alihahd had recognized the flagship, _Jerusalem_.
_I am being chased by_ Jerusalem. He found that thought horribly funny. Horribly.
Alihahd knew who commanded the flagship _Jerusalem_. That man was not a mindless destroyer. General Atta″id would try to catch Alihahd if he could. A high price had been put on Alihahd's head—something the Na′id did not do as a rule, but how else to combat someone they could not recognize on sight and who left no tracks and tripped no alarms? And, in the end, the price had done it. Alihahd's fall had come through no mistake of his own. He had been betrayed.
_How fitting_.
Six ships they sent after him. They had greatly overestimated him, the reputation grown bigger than the man. They thought him capable of anything.
At length the pursuit ships became visible again, gradually closing the million-mile gap their second's hesitation had made between them. They came into range for weapons. None of the ships flew directly behind the _Liberation_ , fearing what might be dropped back in its wake. They never guessed that the rogue ship might possibly be unarmed. Their caution threw off the accuracy of their fire, and the _Liberation_ was able to lure them farther and farther away from the shuttle convoy.
• • •
Alihahd had made no pretense to his volunteers of having the slimmest hope of getting away. If he could have done without his crew, he would have. He wished they were not with him. Did they expect another miracle?
_I told you this was suicide. And—do you guess?_
_I want it_.
The first close salvo came from the flagship. The next would find its target.
Alihahd touched the intersuit com switch at his throat and gave an order. "Brake."
There was a click in his headset. "Braking, sir."
The engines reversed. The chase ships overshot and disappeared, and the stars returned as the _Liberation_ dropped back into sublight.
The end came in an instant.
Lights, sound, pressure, gravity—even the vibration of the ship's engines—suddenly ceased. Arms and legs flailed with the outrush of air. Stomach jumped. Muscles contracted. Alihahd felt himself come off his seat, pulling against the straps, and he hit the self-destruct switch in the dark.
Nothing happened. He clutched the console with both hands, swallowing, his eyes wide and sightless. He wondered if he had been blinded.
In a moment, when everything was abolutely still again, he reached down, unlocked a catch, and swiveled his seat around.
And he saw stars where the hatch and most of the deck should have been.
No.
He turned on his headlamp. The command platform with him on it, had been sheared clean off. The great bulk of the ship _Liberation_ , with the engines, was gone.
One by one, the crewmembers who remained turned on their headlamps. They were three. They stared at one another in shock, waiting for the circling sharks to board.
To Alihahd, it was a nightmare of nightmares—one of the few things that could go wrong. The thought of facing Na′id mortally terrified him. He knew they wanted him very badly. The only alternative now was suicide by vacuum, which also terrified him or he would have done it long before this.
But the Na′id did not board. They were not circling. They were nowhere to be seen, and it took some moments to realize that when the _Liberation_ had broken in two, the Na′id had chased after the other part.
There came a blue-white flash in the distance like a quick nova. That would be the _Liberation_ 's engines exploding.
Minutes passed. Half an hour. The Na′id did not come back, and the surviving rebels knew they had been overlooked—left alone, without power, light-years from anywhere.
This was not what they had expected at all. They were supposed to have been in that nova, making a statement for the Resistance. The _Liberation_ had made the statement without them.
Here was no end in a blaze of glory. Rather a slow and cold grave.
A creeping feeling of smallness grew in the quiet, nothing between the four small human beings and the inconceivable vastness of space. The gaping breach in the deck let the emptiness in.
Alihahd did not look at the others—knew they were looking at him.
At last, because he had to, he steeled himself to turn and face them. He nodded. He held out no hope to them. They accepted their fate. All they wanted was his approval.
He turned away.
He could not tell if their scrap of ship was moving. The stars did not appear to move. Nothing did. At least they were not spinning.
After a time, the crew began to talk in short, muted murmurs through the suit intercoms. A voice, click, silence. Answer, click, silence. An attempted joke, click, silence. A long pause. A voice, click, silence.
Alihahd was still mercilessly alert. It would be a while yet until their air ran out. And in the silence his thoughts moved in. He was aware of what was left to him. Endless dark, endless void, a floating tomb, the stars' indifferent steady light.
There was some comfort in knowing that he had made his last mistake. He had discovered he had a tremendous capacity for being wrong.
And he was never in a position to be wrong. He was one of those who always rose to the top. Leadership came easily to him. Power fell into his hands. He need only stand somewhere and a line formed behind him.
The problem with power was that one's mistakes became catastrophic. Alihahd lived in perpetual fear of a false step, perpetual regret of past false steps. He wondered if he would be better off blind drunk in an alley.
There was an alley in old Cairo on Earth he knew. He never should have left.
He found himself in vacillating prayer to a God he did not rationally believe existed; first asking to live, then, realizing he did not actually want to go on living, he would recant and start over—not that Anyone was paying attention either way. The idea of a benevolent, prayer-answering deity went against logic. Alihahd usually avoided thinking about it. But adrift in the eternity between the stars, it seemed a natural thing. The line between logic and illogic thinned and snapped.
He thought seriously of all the nonsensical religions, impossible to believe. They even found each other absurd.
The Na′id were much more sensible on that score. They combined rational and humanistic bits of many religions—which, of course, was unacceptable to all of them. The devout still died by the millions for their dogmatic and antiquated doctrines.
Alihahd had read their books—a few used the same one—in an effort to understand their people. He found words now in one, words spoken in a garden.
It was easier to pray in borrowed words. Alihahd was not adept at talking to God in his own words—he did not know what to say—and these fit his changing mind.
He remembered the garden. There had been much blood, none of it his.
Old skeletons rose. A lot of blood. Flame and smoke billowed. A summer day. A woman in a boat. God was suddenly near, very real, and it scared him.
Reality unhinging, time uprooted, space frayed and folded. He was going to panic.
Why had he picked those words connected with that place? His galloping heart shook his whole body.
The others were sleeping, and he was alone. With God.
_Stop_.
He held on to the armrests of his seat. They were hard and substantial. Only the ship and the stars were real, and all this other stuff was nothing but panic, pure panic. Just a man going mad inside his spacesuit. He tasted salty beads on his upper lip.
_Stop thinking_. Hallucinations lurked round the edges of the light from his wavering headlamp.
Circulating air in his suit cooled the sweat on his heaving sides. At this rate his oxygen would run out first of the survivors.
His hand trembled, uncoiled his suit's life cable. He hooked the cable to his seat, then unstrapped himself and made a floating prowl through the remains of the bridge and found a stashed container of wine. He returned to his seat and drank himself into a stupor—so that he would not notice when he passed out for want of air.
# 2. Iry
WHEN ALIHAHD WOKE, he was in a strange place—a spaceship—and he was no longer in his spacesuit. The air pressure was less than he'd kept _Liberation_ 's but was still safe.
He was lying on a deep-colored carpet spread over a hardwood deck. There were straps around his head, his shoulders, and between his legs, and for a terrible moment he thought he had been captured and bound. But he sat up and found that his limbs were free, and that the ship was not of Na′id design. The strap around his head held a tubed mask to his face. He was breathing pure oxygen. He lifted the mask and sniffed the ship's atmosphere. It was depleted of free oxygen, poisoned with carbon dioxide, and unbreathable. He fitted the mask back to his face.
The other straps made up a harness that did not fit at all well with Alihahd's tunic. He fumbled at the buckle and was going to take it off as soon as he figured out how, but his wine-dulled mind was easily distracted and shifted to something else.
What caught his attention was the damage. This ship had been in an accident. Or battle. There were signs of fire and buckling in what he hoped was not the outer hull. The overhead lights were flickering, cold and blue. Electrical burns, scorched metal, and ozone censed the cabin, whose mahogany fittings were charred. Frostily tinted lamp covers lay shattered in pieces on the black-fringed carpet.
Beside Alihahd, his three crew members lay sleeping—or drugged. Closest to him was the athletic spokesman of the mutineers, his muscles smoothed in unconsciousness. Long lashes grazed his youthful cheek. His name was Neal. Alihahd lightly brushed an auburn curl out of his face with the back of his hand. _I could kill you for what you tried to do to me._
Next to him was the _Liberation_ 's engineer. Yuko, her petite form curled up like a puppy, her little mouth drawn down into a frown as in a bad dream. Her black hair was clipped so short it stood on end. Bleached into it behind one ear was a white pentagram. Yuko was a witch, she said, but Alihahd had never seen her magic work. She was a better engineer.
Beside her was a pale youth whose name Alihahd didn't remember. He was also fitted into a harness.
And there was a fourth person. Alihahd hadn't noticed him immediately because he had stood apart, unmoving in the semidark. He'd looked like an empty spacesuit set ceremonially to one side like a suit of armor. The viewplate was opaque one-way glass and Alihahd couldn't see the face. This would be the owner of the ship.
The stranger was tall, very tall, clearing even Alihahd's height. He—Alihahd assumed it was a man from his imposing frame—stood dead still, observing in silence, arms at his sides.
Alihahd rose to his feet and stood up straight—the oxygen tube barely reached—facing the stranger who was equally still and unspeaking. Alihahd wasn't accustomed to looking up at people, and he hadn't realized that it could make such a difference. The dimensions of the ship were slightly big as well and gave Alihahd the vague, uncomfortable impression of being a child again at the control and beck of others.
Behind the visor he sensed a familiar stranger—someone he had never met but knew as if he'd always known him. He was an adversary.
Alihahd didn't try to communicate. If the ship's master wanted to talk, he could talk first, and he could say who he was and why he had taken the _Liberation_ 's survivors aboard. Did he know whom he rescued? Was this a rescue at all? The man could be a jackal of war, a slave trader picking up Na′id leavings. He could be a Na′id private citizen who had found what appeared to be a Na′id wreck and had rescued what he thought were compatriots. He could be a neutral from uncharted space.
Or was this the master of _Marauder_?
How to know?
The ship spoke much but answered little. Its large dimensions were either a custom design for a tall man, or the standard of a tall race. It was an expensive craft, fitted out and detailed with somber luxury in colors of wine and gold, rose, amber, black, and deep brown wood. There were no nationality or planet markers anywhere in sight, and the man himself was totally obscured in his steely gray suit.
Alihahd gazed steadily at the smoky visor, trying not to look as weak as he felt. His head throbbed. He tasted stale wine.
At last the man turned away and beckoned with a slow wave of his hand for Alihahd to come with him.
Alihahd hesitated, limited by his oxygen tube. The tall figure retreated down the corridor without looking back and disappeared through a hatchway. Alihahd took a deep breath, held it, dropped his mask, and followed.
Alihahd stepped through the hatch into the ship's control room. Two great iris-apertured viewports were wide open on either side.
Stars. There were stars. The ship was traveling sublight.
The pearly white river of the Milky Way cascaded across one viewport. The vast glowing galactic hub filled the other.
Bright in the foreground shone a single yellow sun.
Alihahd drew closer to the port and looked down.
The ship was orbiting a moonless planet of muddy-green continents and blue seas. The equator was wreathed in thick white clouds that spiraled and thinned to the north and south. Even from this distance Alihahd could see mountains. A permanent haze of vapor steamed off the ocean's surface.
The ship's master drew Alihahd's attention to the control console, where he traced slowly on the smooth surface with his forefinger in Roman characters—symbols used by many languages, including Universal—a single word and question:
IRY?
A thrill of discovery awakened in a mind Alihahd had thought too jaded for wonder.
Iry was one of the lost worlds. There were many places known in the First Historical Age that had been lost after the Collapse. But this one was a fantastic place. Not everybody was convinced that Iry had ever existed.
Yet here was a living planet, beyond recivilized space, its yellow sun hidden in the Great Rift where it was supposed to be, and it looked right.
The Eyes of Iry stared back at him, two impact craters peeking clear of the shroud of clouds. He couldn't see the encircling River Ocean, but the computer diagram on the ship's screen said it was there beneath the clouds.
The computer diagram also revealed this to be a peculiar little solar system in keeping with the myth of one sun, one planet, and no moons.
This last was the telling point. In Alihahd's experience, one was an extraordinarily odd number of planets for a normal yellow star, and there were few mid-sized worlds anywhere orbited by no moon at all.
Alihahd looked up to the ship's master and answered with a downward nod, which would tell him yes if this man understood nods. Then Alihahd left the cabin to return to the makeshift contraption that was his oxygen mask. This ship wasn't designed for guests or for emergencies.
It wasn't a slaver either. Slavers didn't pick up anything they couldn't sell. This ship couldn't even get itself to a familiar port.
_Iry_. The name summoned visions of Itiri warrior-priests who lived on mountaintops, attended by familiars. They flew on wings with the eagles of Iry. Itiri warrior-priests journeyed on quests between the stars, armed with their magical double-curved swords that never dulled. Legends always had their special weapons. It was required of them. For the Itiri it was the double-curved sword.
Of course, the Itiri were beautiful: humanlike, tall and fair, golden-haired and emerald-eyed—a distinctively un-Na′id concept of beauty.
The legend was popular outside Na′id circles. It dealt with what the Na′id dreaded and insisted could not exist—a superior sapient species.
But if the Itiri were anywhere near as prodigious as their reputation, where were they now and where had they been hiding for the two thousand years of the human Dark Age? Truth was that when Earthling ships had ceased to fly, so did the Itiri, which told Alihahd where the stories were coming from—human imaginations. He didn't believe in the Itiri's superhuman exploits any more than he believed that Atlantis had developed atomic power.
For him the great significance of the stories about Iry's natives was not that they gave any insight into Itiri nature—Alihahd believed they did not—but that they suggested the planet itself—the core of the myth—was habitable.
And for the first time he realized that he was going to live. He was too numbed to know how he felt about that.
He held his breath and rejoined the ship's captain in the control room.
The ship had started a slow descent. Alihahd felt the added tug of the planet's gravity and the ship's resistance as the brakes cut in. He could tell by the way the ship was going down that she was not an in-atmosphere vehicle.
They were entering twilight, descending on the night side of the world. The creepingly slow landing would use all the ship's reserves, but the battered vessel was not going to take anyone anywhere again. When Alihahd left this world, it would be in another ship—or never.
He glanced at the stats on the computer screen. They read in Scientific, a specialized language even more universal than Universal. The figures told him the planet was smaller than Earth and heavier, its density 7.19. It was .9 astronomical units from a sun whose spectrum was strong with heavy elements. The planet's atmosphere was oxygen-rich and contained more inert gases than Earth's, but not enough to be a problem. Pressure at sea level was three Earth atmospheres.
Already Alihahd felt hot and heavy and crowded. He started again to take off his harness, but a silvery-suited hand closed on his wrist and stopped him.
The contact startled Alihahd, and he looked up at the opaque faceplate in surprise.
The strong hand slowly let go of his wrist. The grip hadn't been tight, but it had been powerful, and Alihahd was not at all sure he could have pulled free had he tried.
He retreated from the control room, paused at his oxygen mask to breathe, then went searching for and found the head. The ship's owner was male. He also owned a cat or a ferret or some small pet. There was no one else aboard, no room for anyone else. For all the sense of space in the moody corridors, it was a small ship.
The fittings were rich, heavy, and dark with gilded borders. The ambient strength, even arrogance, of the place was disconcerting.
Alihahd returned to his oxygen mask and sat on the deck. Had he been alone, he would have stolen into the engine room to see what kind of equipment the generators were loaded for. The number of exterior leads would tell quicker than anything what this ship was about. He suspected he knew. He wanted to look for a lead to an exterior hologram projector.
But laws dissolved in outer space, and a ship's master was its god. There was nothing to say that the dark stranger could not put Alihahd and his companions right back where he found them.
For the crew's sake, Alihahd stayed where he'd been put—on the short leash of his oxygen tube.
As he watched his dozing comrades, he became convinced that they'd been drugged. He heard the engines complain, more and more as hours dragged by. Then tension gave way to tedium and at last he slept.
• • •
A star fell out of the sky.
Five tall slender figures stood with the sentinel, silhouetted against the star field out on the eagle ridge, gazing up.
It was a new star. It hadn't been in the sky the night before. At twilight it gleamed in the Red Geese like a nova, but then it slowly slipped out of the Ring of constellations and down toward the River Ocean beyond the horizon.
The Fendi, Roniva, turned to the younger aghara beside her with a wave of her night-colored hand, a black glitter of onyx flashing on her forefinger. "Go thou. Take thine fire kin and see. And take thou one from the fire clan of the carnelian serpent. Take Arilla."
She met a quizzical look. "Fendi?"
"Someone who knows of humankind. Thou may need." Roniva turned away, her thin, hard arms crossed. _Naedenal_ , the night wind, moaned down the cliff face in a cold gust to become lost and voiceless in the deep ocean of air in the valley. "I sense. I sense."
"Humans?"
Roniva murmured assent. Her black gaze swept the wide heavens. "After such a long time. . . ."
• • •
Alihahd was still asleep when he heard a crashing. The deck heaved beneath him, and he was catapulted into empty air.
He woke falling. He gasped in air, icy-cold and sharp, not pure oxygen. He opened his eyes to bright blue all around him. The ship was gone. He was in Iry's atmosphere.
And dropping like a rock, headfirst, eight kilometers up, the air so cold it burned and sliced like knives. His tunic flapped wildly in the wind. His heart galloped, skin prickled. There was a sweet-sour taste of fear in his mouth, a roaring in his ears, nauseous terror in his throat. The oxygen mask was still strapped to his face, its disconnected tube whipping madly at his ear. He saw the dizzily closing ocean far below tracked with white, saw sky and white clouds and morning sun until his eyes teared shut. And he went down, faster and faster, trying to scream.
Then he felt a yank at the harness straps as if grabbed from behind and above, and he was righted, swinging suspended under an expansive white canopy, floating down at a gentle speed. Had there been anything at all in his stomach, he would have thrown up.
The pounding of his heart subsided slowly till he was left with only the ashes of fear, a dull lump in his stomach and a flat taste in his mouth. His skin was icy. He took off his useless mask and let it drop.
He looked up at his parachute. He had never seen one before. It seemed a primitive thing. He wondered what had triggered it to open and guessed it had to be his speed. His hands sought the harness straps and gripped them, trembling involuntarily.
He looked down. A silvery glint on the water still four kilometers below had to be the ship. He was drifting quickly away from it. Seeking a way to steer, he tugged at the risers and managed to guide himself back without setting himself into a spin.
Another parachute blossomed below him and to the windward. He looked for others, but remembered with sinking sickness that only one of his companions had been harnessed.
The ship had not been equipped for guests or for emergencies.
Then he caught sight of a long thin line plummeting straight down. At least one other person had acquired a parachute while Alihahd had slept.
A bad one. Alihahd watched, mentally screaming as it hurtled down and out of sight. It never opened.
The only other things in the sky were the circling birds. The enormous creatures already clustered around where the fouled parachute had gone down, while others glided around in lazy arcs on broad wings.
_Scavengers_.
Alihahd looked up again and clutched the risers, this time in anger and horror. If only he had known. He was wearing a parachute, and others weren't. He should've given his to someone else.
He could already hear the question always put to a surviving captain. _Where is your crew?_
He tried to make his hand release the clasp of his own chute, but it wouldn't move. He had a hideous survival instinct for someone who shouldn't be alive.
He closed his eyes hard, grimaced, and tensed back his neck. He felt as if someone were driving nails into his skull.
_Does it never stop?_
He opened his eyes and looked down at the blue ocean. It was much closer now. The air had become very warm, then hot, and the wind was heavy like a current of water. He could make out some definition in the craft on the water. It wasn't the _Marauder_. And, closer, he could see it was not even a spaceship. It was a seaplane. He couldn't find the spaceship anywhere.
Surface winds across the sea swept him swiftly away from the seacraft so he couldn't see it anymore. He was aware of the dark moving shadow of a bird above his chute, and he wished he were armed.
Descent seemed very fast during the last few meters. He hit the water hard and plunged under. The parachute billowed down over him like a wide compass on the water's surface. He bobbed up, but the parachute draped over his head and clung closely without space for air. He tried to breathe through it, but it drew in, impermeable as plastic, suffocating him. He pushed up on the fabric, but it only formed a vacuum and sucked up the water with it.
Head ringing, lungs hurting, he fought to get out from under the canopy, but there seemed to be no end to it. Heat built up behind his eyes. He saw black and red. He thrashed, tried to breathe, took in water, coughed on reflex, and gasped in more water.
Strength and even fear left him at once and he began to sink in a druglike torpor.
Thought he was dying. Thought he was dead.
Suddenly there was light above as the chute ripped away. There was a splash and a column of bubbles from something plunging into the water beside him, going down and shooting up. A myth closed powerful hands under his arms and heaved him up. His head broke surface. A fist pressed under his rib cage. He expelled water in a violent rush, coughed, choked, spat, and finally drew air.
When the whirl of bright-dark colors blinked clear, Alihahd was staring into the green, kohl-smeared eyes of an Itiri warrior-priest.
• • •
Alihahd couldn't speak. He opened his mouth, squeaked, then coughed.
Unconcerned with its own improbability, the myth disengaged Alihahd from the parachute harness with agile hands and pulled him clear of it. The Itiri encircled Alihahd's upper body with a long hard white arm and swam toward the ship. Dazed, Alihahd let himself be dragged. Saltwater and the heat of the air hurt his eyes. Steam was rising from the painfully bright glittering water, searing his brine-inflamed lungs with each breath.
The ship was the one he had seen from the air, a primitive aluminum seaplane afloat on pontoons. Alihahd smelled fossil fuel, tasted it on the water, the back of his mind making note that this was not the product of a space-age civilization.
As they drew near the plane, a wave came over Alihahd's head and set him coughing and gasping again. From above, two hands seized his armpits hard enough to hurt, heaved him out of the water, and sat him down heavily, a dead weight on the hot metal pontoon.
On the plane were a few more aliens, climbing over the wings and fuselage. And there were two wet humans on the pontoons—a boy spitting up, and a man.
When Alihahd could breathe again, he grabbed one of the tall, lithe humanoid aliens. Green eyes turned to him curiously. Alihahd thrust up two fingers, gesturing urgently at the humans, then to the ocean. He croaked, "Two more. There are two more of us."
Alien eyes seemed to register comprehension. The Itiri parted from him. Alihahd continued to speak after him as if the alien could understand, "I saw one go down with a parachute that did not open."
A voice behind Alihahd said in Universal, "That was mine."
Alihahd turned. Water trickled down a strand of hair plastered to his forehead and ran into his eyes. He brushed the wet hair off his lofty brow and blinked at the man.
"It opened, then?" said Alihahd.
The parachute must have opened. Obviously. The man was alive, not smashed into the water's surface at terminal velocity.
But the man said, "No." He pointed up.
There was an eagle.
Alihahd shaded his eyes and squinted up at the bird. From below, it appeared dark until it passed out of the cloud shadow. Then the sun struck its back, and its plumage burst into all the shades of fire.
It was fully large enough to carry off a man.
Alihahd remembered the shadow over his parachute and the Itiri dropping out of the air to rescue him. The immense birds were not scavengers after all.
Alihahd looked again at the humanoids. They looked just like all the impossible stories said, creamy white with golden hair and brilliant green eyes. All six of them were two meters tall with long, lean, hairless bodies. They moved with a grace Alihahd considered the province of the animal kingdom. But then these were not humans. They had the physical fascination of big cats and fine Arabian stallions: beautiful, powerful, and frightening like cats, creatures that could prey on human beings.
The Itiri eyes were elaborately lined with kohl like Egyptian tomb paintings. The single female was squatting atop the plane, balanced on the balls of her feet. She faced into the wind, her arms resting on her knees, the image of a hawk ready to take flight. Gold tassels bobbed from a wide gold band across her forehead. Evenly spaced scars welted her fair cheeks. In a sheath at her side she wore a long double-curved sword.
Alihahd jerked his gaze away.
At least she didn't have wings.
The seaplane rose high on a swell, then settled into a horizon-swallowing trough as the wave rolled under. The sea had looked calm from the air.
Alihahd groggily regarded the Itiri warrior-priest on the pontoon with him, and he spoke, rhetorically as much as anything, as some people talked to their cats or some drunkards to their pink elephants. "What men are you? What woman is this?"
He was astounded to get an answer in kind. "We are such as thou see."
Alihahd jerked back. The elephants were not only pink, they were talking to him in his language.
The alien warrior crouched down before him. Deliberate scars and smeared eye paint marred his high cheeks. There was intelligence in his green eyes. His expression, if Alihahd could read alien faces like human faces, was benign, a little whimsical, and Alihahd sensed that his elephant was laughing at him.
Alihahd spoke again with a feeling of the High Absurd. "Then I am dead, for Itiri warrior-priests are nothing more than the dreams and hopes of despairing souls."
The Itiri cocked his head quizzically, then replied with an equal sense of absurdity, "If I were born of thine desperation, I could not say. Though I could swear I remember existing before I found thee drowning. All the same, we are now."
Alihahd raised his brows. He had read enough of the old histories to decipher the dialect. He nodded. He had to admit this was all real—the planet, the eagles, the swords, and the warrior-priests speaking archaic Universal as it was spoken before the Collapse some two thousand years ago.
His head felt very thick, his ears caving in. The atmosphere was oppressive, too much pressure, too much oxygen. He was dizzy, hot, and sick.
The Itiri started to rise. Alihahd caught his arm. "Others," he said. "There were two others."
"We continue to look," said the Itiri and nodded up at the soaring eagles.
The birds wheeled high over the water with slow turns on the heavy sea breeze.
The warrior-priest climbed atop the plane's wing and waved the eagles farther downwind.
Alihahd turned to see who had already been rescued.
The boy was standing on the pontoon, clutching the airplane strut in both white-knuckled fists and staring out to sea. He was very slender, clad in light-blue space coveralls. Alihahd couldn't think of his name.
"Are you hurt?" said Alihahd.
"N-n-no," said the youth, trembling. His brown face had paled. His brown eyes were wide and white-ringed. His appearance was unremarkable except that there was nothing particularly Negroid or Oriental about him. Even his skin color was still within the confines of the broad spectrum known as White. He looked suspiciously pureblooded Caucasian. It might explain what made him a rebel.
The Na′id Empire espoused a doctrine of dissolution of all barriers that could splinter the unity of humankind, be they cultural, religious, or—most visibly—racial. The Na′id despised persons who carried recessive genetic traits. The persistence of recessive traits was a symptom of isolation from the general gene pool. Recessive traits needed to be reinforced in order to perpetuate. If individuals of all kinds mingled freely, recessive characteristics would diminish and die out. People like the boy were symbols of elitist rebellion and disunion.
"Vaslav," said Alihahd, remembering.
The youth perked up at the sound of his name in Alihahd's voice, and he beamed to be known by his leader.
Alihahd had a memory for names, but this young man was not memorable, and, in truth, Alihahd couldn't say whether the name was first, middle, or last. He could not for the life of him recall the lad's other name or names, so henceforth he was simply Vaslav.
The other human survivor was mixed-blooded, older, striking. He was also wet. The eagle that caught his failed parachute must have slowed but not completely stopped his fall to the sea. He was sitting on the pontoon, emptying water from his rawhide boots one at a time.
His dark, red-bronze skin stretched taut over a lean, imposing frame. His wedge-shaped face narrowed wolfishly toward his chin. Soft lips curved slyly at the edges beneath a neat mustache. His cheekbones, very high, seemed to crowd his eyes and gave the effect of a permanent squint when actually he didn't squint at all. His brows arched sharply over almond-shaped eyes. The eyes were really very beautiful—like one of those angels who was asked to leave heaven—the whole face astonishingly demonic. His wiry, graying copper hair was pulled back and tied at the nape of his neck. He wore a russet bandana twisted into a headband across his forehead. Two bright gold loop earrings pierced through one ear, making him look like a pirate or a gypsy.
Gentleman pirate, it would seem. His kid leather vest was lined with gold.
He wore a rawhide redingote that fell to midway down his thighs. He possessed the air of one unrepentantly self-confident as if he were someone of importance. Even sitting down he moved with a kind of swagger. There might have been no one else aboard the seaplane. All his attention went to pulling his sodden leather boots back on, acting as if nothing was wrong in the world except that his clothes were wet.
His gun was a self-customized, crazy-looking thing with a wood handle and a long stabilizer that jutted out like the barrel of an old projectile weapon. He wore the gun at his belt toward his right side, grip forward, as a right-handed person would wear a sword.
Once finished emptying his boots, he drew from his waistcoat a white-gray meerschaum pipe fitted with an amber mouthpiece. Its bowl was carved into the figure of a fox head. He fished out from his coat pocket a mess that used to be tobacco. At this, Alihahd had to smile.
Tawny eyes slid sideways to Alihahd. The man returned a slow, wan smile.
Alihahd's thickened brain realized this man was not one of his volunteers. For one, Alihahd would've remembered if he'd ever seen him before. For another, he was too tall.
Their eyes met and held. Menace and amusement lurked in the orange-brown tigerlike eyes.
This was the rescue ship's master. And Alihahd was still wondering: friend or foe?
He was dressed like a civilian but looked old enough to have been retired from military service if he was Na′id. He was obviously a private citizen now, but of where? From his style of clothing he was possibly a native of Eridani. But Eridani was a Na′id possession now, had been for fifteen years. Whose side did that put him on?
Alihahd couldn't ask him to fly his colors without showing his own first. These were cautious days, and the correct answer to a stranger's inquiry of "Who are you?" was either "Why?" or "Who are _you_?"
Alihahd couldn't know if it was safe to speak his own volatile name. At its very mention that crazy weapon with the notched handle could come whipping out and shoot through his heart.
There were worse fates.
Alihahd couldn't wait for the man to drop a clue. He needed to know now.
He looked dead into the tiger eyes. "Alihahd," he introduced himself. "Captain of the late runner ship _Liberation_."
"Hall," said the stranger. "Harrison White Fox Hall of the late ship _Nemo_."
And Alihahd knew hardly more than he had before. The name wasn't Na′id. But if Hall had been a conscript from Eridani rather than a volunteer, he wouldn't have an honorific name. He hadn't reacted one way or the other to the name Alihahd.
Then what did the ship's name signify? "What is Nemo?"
"In Latin it means 'no one,'" Hall said like a beginning, as if there was more to be said.
Alihahd was already alarmed. He knew what Latin was—a dead Earth language from the First Historical Age, the time before the Collapse. The Na′id were peerless in their knowledge of mankind's ancient history. A Latin name was a Na′id enough thing. Alihahd braced himself should the second half of Harrison White Fox Hall's answer be accompanied by the draw of his weapon and arrest in the name of the Na′id Empire.
Hall continued, "In Greek, it means 'I allot.'"
That was curious. "Allot?" _Allot what?_
"Nemesis," Hall said. "What is deserved." He glanced away toward the horizon. His voice changed pitch, and he tossed the last line away as if unimportant. "Revenge, if you will."
He looked back to Alihahd, and understanding passed between them. Alihahd knew enough. Nothing more need be said.
"Mr. Hall." Alihahd finally nodded in greeting.
"Captain." Harrison White Fox Hall returned the nod, conspicuously avoiding the false name Alihahd.
Alihahd saw Hall taking another hard look at him.
Alihahd cut a fine figure: rangy, thin as though from dissipation, his dark red tunic stuck to his gaunt frame. He spoke and carried himself like aristocracy—profoundly weary aristocracy, at once proud and sorry as a tattered flag.
He knew what it was Hall watched with those long stares and intensity bordering on venom. Alihahd's "name," his tunic, and his swarthy coloring were all appropriate to the planet Chesa. But his features were wrong—tall flat forehead, straight Nordic nose, fine hair—they were pure Caucasian. The mixture made him look very very Na′id.
Their gazes had locked again, and they tried to stare each other down. The boy Vaslav, standing over them, looked from one to the other, aware that something was going on beyond what he could see. Neither man was going to break away.
Till an eagle alighted on the aircraft, its great wings and tail fanned in braking, its talons outstretched. It screamed.
Its landing rocked the plane on the water, and Alihahd grabbed a strut to catch himself. As he pitched far over the edge of the pontoon, a whalelike head rose up from the sea, and Alihahd was suddenly face-to-face with a black-and-white sea creature. A barrage of delphine clicks and squeals sounded from its blowhole. When the plane settled, a warrior-priest knelt next to Alihahd on the pontoon, and leaned over to listen to the whale as if its noises were actually intelligible. The Itiri turned his head to Alihahd. "Two, didst thou say?" He held up two nailless fingers. "This many? Thy companions are dead."
Alihahd hung his head. He had already known. "Very well, then," he said to the warrior. "We are in your hands."
The Itiri waved the whale away and stood up. The eagle launched itself with a push that bobbed the plane like a toy on the water. The warriors shouted and whistled to one another, climbing into the fuselage, readying the plane to depart this place.
Eyes downcast, Alihahd became aware of rawhide-booted feet standing at his side. He looked up. Hall had risen and was offering a hand. Alihahd hesitated. He didn't want it. The unease he'd first felt on meeting the spacesuited stranger returned. He could stand up by himself.
But in some parts of the galaxy to refuse a man's hand was ground for murder.
Alihahd took the offered hand, and Hall pulled him easily to his feet.
Hall climbed inside the plane with the Itiri. Alihahd started to follow, but noticed Vaslav hanging back, staring forlornly at the waves.
The boy appealed to Alihahd, stammering, "But. . . ."
_Neal and Yuko_ , Alihahd thought. He'd failed to retrieve his dead.
He put a hand on the youth's shoulder. "It's as good a resting place as any," he said with comfort and sureness he didn't feel.
And to soothe the boy's mind, he spoke a few words over the souls beneath the waters though he was certain they could find their way to their God without him.
• • •
It was the first of the Red Geese when Alihahd, Harrison White Fox Hall, and Vaslav came to the Aerie. The journey took several hours in the slow way of primitives aboard a noisy vibrating craft kept aloft by propellers and piloted without a computer. The hulking machine lurched into flight and climbed slowly, shuddering. The heat and pressure of the air lessened with altitude, and Alihahd felt better, though he was still plagued with the mistrust of a starman staking his life on an antediluvian transport that felt and sounded as if it would rattle apart.
The Itiri spent the entire journey in a separate compartment, leaving the humans in the cargo hold.
Fine by Alihahd. He had never liked aliens.
Harrison White Fox Hall took off his wet coat and rolled up one of his shirtsleeves. He drew his notch-handled gun and inspected it for saltwater damage.
Alihahd was not going to ask if the notches represented humans or aliens.
Now Hall sat on the deck by a porthole, one elbow resting on one knee. He seemed to be enjoying the view.
_A man who decides who lives and who dies._ _Not even a pretense of regret._ When Hall's ship was in danger, Hall had placed the available parachutes on himself and the other captain, the last people who should have been saved, and he stayed cavalier through the entire harrowing ordeal.
Alihahd felt anger. And envy.
The plane hit an air pocket. The metal box jumped and dipped. Alihahd closed his eyes and waited for the crash he thought was coming.
When it didn't happen, he opened his eyes. Hall was looking at him.
A ghost of a smile glided beneath the graying mustache. "Not afraid, are you, Captain?"
That was provocation. Hall wanted a fight. And he wanted it with Alihahd. The boy Vaslav didn't interest him.
Alihahd wouldn't have it. He would take a fall before he would be baited into defending himself. Self-respect, he had none. Honor was long gone. There was nothing to defend. Hall ought to know that about men without names.
He swallowed the insult. "Maybe I am."
He could tell that Hall wasn't sure if he'd won the confrontation or not.
Hall distractedly drew his pipe, took it between his teeth and reached into his pocket for tobacco. Finding his pocket empty, he suddenly became conscious of what he was doing and put the pipe away.
• • •
Alihahd had disturbed Hall's perfect understanding of the universe.
Avoiding a challenge was not a thing to do where Hall was from. It was cowardly. Yet Hall couldn't call Alihahd a coward—not under the circumstances in which he'd found him.
Alihahd had admitted fear outright. On Eridani, a man did not back down from a challenge like that.
Unless he was very very sure of himself.
And it occurred to Hall that Alihahd just didn't care what Hall thought of him. The accepted insult wasn't surrender. It was dismissal. Alihahd's regard for Hall was so slight as to nullify anything Hall might say to him. Alihahd probably didn't even realize what he was doing, the insult more complete for being sincere and uncalculated.
Hall had never been so thoroughly put down in his adult life.
Hall was amazed, about to be angry. Then a slow hunter's smile crept onto his lips. He laughed, a low grumbly sound.
Alihahd looked at him. Hall winked a stalking tiger's eye.
The captain was an enigma: brave and frightened, strong and weak, assured and uncertain, serene and haunted. Nothing about him was clear or unopposed by some other trait.
And Hall realized it could only be that his elusive captain was two people. And one was not real.
The one without a real name.
• • •
The airplane abandoned the coast and turned inland to cross a mountain range.
"Good," Hall said. "I can't swim."
"I suppose I cannot either," Alihahd said. "I drowned once before."
"Only nearly, it looks like," Hall said.
Alihahd didn't elaborate. He gazed out a porthole.
The land was stark and impassable save by air. Unable to clear the towering peaks, the plane wove a path between mountains.
Vaslav watched their passage with one hand clapped across his mouth, his eyes round with dread. He hadn't stopped trembling. He lowered his hand. It fluttered like an indecisive butterfly. "Will we be here the rest of our lives?"
"I do not know," Alihahd said, his words slow and distinct, for the boy was new to the language. "The natives did not learn Universal in total isolation. That much is certain."
Vaslav craned back his head, perplexed. "They're speaking _Universal_?"
Alihahd smiled. "Of a sort."
The plane stopped once to refuel on a lonely plateau, then continued over the rugged land on its solitary winding way. They'd lost the eagles long ago.
At last the jagged peaks gave way to a sudden, startling cradle of relative flatness, a wide fertile plain in an open valley ringed by high mountains where eagles flew. Squares of green-and-gold cultivations checkered the fields and crept up the feet of the enclosing mountains on terraced slopes. A long thin veil of a waterfall spilled from one of the massifs and fed a river that twisted through the valley in a lazy course lined with willowlike trees and neat, pretty, little houses built of broad, woody reeds that had been carved and shaped, perched on stilts. The houses were roofed with varicolored reeds upcurved at the eaves.
The plane landed in a warm, dusty meadow of brown grasses. The warrior-priestess named Arilla came back to the humans' compartment and let them out to the summer village of Kaletani Mai.
Alihahd dropped down from the plane and took a few unsteady steps on the first solid ground he'd trod in a long time. The sky was clear blue. The smell of ripening fields drifted to him on a mild breeze.
The villagers came running out of the fields at the aircraft's arrival, and Alihahd backed up against the plane in the face of the noisy, barefoot onslaught of short stocky aliens who panted like dogs and chattered in a strange tongue. But it was a benign rush, aimed at the warrior-priests more than the humans, who were a mere curiosity.
The villagers were short and bulky like big gnomes—as draft ponies to racehorses next to the six tall, slender warrior-priests. Their skin was brown or white. The color changed as the aliens stood in full sunlight or shadow. All of them had light red hair and black eyes. Their kind were called ranga. The tall, fair warrior kind were called aghara. They looked like two different breeds.
Looks deceived.
The babbling, smiling ranga surrounded the laconic warrior-priests, who answered them tersely in an incomprehensible language. There was a lot of talk—questions, answers, directions, arrangements, acknowledgments. Vaslav listened closely, as if he might actually pick out patterns in the noise.
Alihahd eased away from the throng and retreated to the far side of the airplane. Under the shade of the wing, he leaned on the pontoon, his head resting on his arms.
Harrison Hall had taken off exploring on his own.
Vaslav came under the plane wing with Alihahd. Alihahd lifted his head and rubbed his brow. "Yes, Vaslav."
Vaslav's eyes were darting this way and that, taking in the alien trees and crops, the charming village, the blue sky. He was furtively smiling but unsure if he ought. He looked to his captain for the proper reaction. Alihahd did not look happy.
"Are—are we safe?" Vaslav asked.
"I see no evidence for alarm," Alihahd said. "Go walk with Mr. Hall. I swallowed too much brine, is all." He motioned the boy away with a weak wave of his hand.
Vaslav's smile spread to a beaming grin, and he bounded away to catch up with Harrison Hall.
Alihahd covered his eyes with his hand. It was the mark of a great leader to be a consummate liar. Alihahd was scared to death. But why ruin the boy's image of an idyll—especially when he just might be right? Alihahd kept the worry to himself. That was what leaders were for.
The complete alienness of the surroundings weighed on him. He felt ill. He needed distance.
He stooped to crawl underneath the body of the plane. He sat down with his head leaning back against the rough treads of the tail wheel, and shut his eyes.
He found comforting familiarity in the sound of wind in leaves, the smell of chlorophyll and dry grain fiber, and the rubberlike tire at his back. He could almost imagine this to be one of Earth's far-flung colonies, and the alien voices just the chatter of slaves.
Yet darker visions invaded his illusion. He knew what experiments the Na′id performed on live aliens, on parts of them.
And humans were the aliens here.
He bided an indeterminate while beneath the plane, and began to think he'd been forgotten. The day grew hot, and he had no ambition to move. The sun was directly overhead.
He listened to the buzz and chirp of reptilian birds, dozed off and on, studied his sheltering plane with its wheels and pontoons, and decided it was a stupid design for an aircraft.
Harrison White Fox Hall returned from his prowl, his dark red-bronze face burned a deeper red. He eased himself under the plane with Alihahd. Vaslav crawled in after him.
"Didn't get too far," Hall said. He took off his bandana, loosed the knot, and retied it around his head. "Some industry behind the hill where the river curves round. Hardly impressive. This seems to be state-of-the-art technology." He patted the metal belly of the amphibious craft above their heads.
Alihahd groaned, his hopes of ever getting off-world fading. Yet he hadn't realistically expected more.
Hall stretched out his long legs. "My dear Captain, there are worse places you could be marooned."
Alihahd turned to look at him directly. "With worse company?" he asked sincerely. He didn't mean the aliens.
Hall grinned, took Alihahd's head in his hands, and kissed him on the mouth. "Possibly not," he said.
"I thought as much," Alihahd said. Many cultures sent people to the grave with kisses. And as Hall lowered his hands, Alihahd grasped one of his wrists—the one with the sleeve not rolled up. Alihahd's fingertips touched something hard, the hilt of a hidden dagger strapped to Hall's forearm.
Both men froze, the same question on their minds. Should they attempt it, which of them could draw and use one of Hall's weapons first?
Vaslav coughed nervously.
Alihahd turned his head. Vaslav was confused. Vaslav couldn't see the dagger, couldn't see Alihahd's and Hall's thoughts, which were so transparent to each other. He couldn't reconcile their actions with any antecedent action, their words with their silken tones. He could only figure he was having a severe translation problem.
"Poor Vaslav," Alihahd said and let go of Hall's wrist.
The tall warrior-priestess Arilla came again, bearing thin-skinned fruit, green leafy stalks, and coarse flat bread. She placed them in the grass under the airplane. "For thee and thee and thee."
She saw their hesitation. "Not to worry," she said. "It is safe for humankind." She didn't say how she knew.
With little appetite, Alihahd took a bite of the bread, then pushed everything away with an unsteady hand.
Arilla told him he was bony. Alihahd could not be persuaded.
Finally Arilla beckoned, her bracelets jingling and sliding up to her elbow. "Come, if thou wilt not eat. We are not home yet. This is only the Lower Aerie."
_Lower Aerie?_ They were three and a half miles above sea level. "There is an Upper Aerie?" Alihahd asked, climbing from under the plane to join Arilla in the open sunlight.
"There." She pointed upward.
# 3. Island in the Sky
HIGH ABOVE THE PLAIN towered the forbidding peaks of a great mountain, its twin summits separated by a deep narrow crevasse, as if a god had taken a cleaver and sliced the single mountain in two. Where the two peaks slanted away from each other in the misty distance, Alihahd could see tiers of stone arches on both sides of the fissure.
"There is a foot bridge," said Arilla. "On one side stands the Aerie. On the other Haven. Both together are also called Aerie."
The mountain was block-faulted, its south face a barren granite palisade with the Aerie an inaccessible citadel at the top of the sheer rise. Itiri planes couldn't fly that high. Even Iry's heavy atmosphere thinned at such heights. It was six miles above the sea.
"How can anyone possibly get up there?" Alihahd asked.
"We are going to walk," Arilla said.
"We are?"
"You are going to fly." She guided the humans across the golden field to the foot of the twin mountain where a balloon was prepared.
Alihahd liked this method of transportation less than he did the seaplane.
The balloon rose slowly, smooth and soundless on a warm anabatic current, the wind the Itiri named _eaninala_ , the day wind.
Then a sudden gust swept the balloon toward the cliff face. The mountain loomed large and filled the view. A glitter of quartz. A rock wall with the shadow of their fragile basket growing on it.
Then the shadow of an eagle.
There came a tug and a change of direction.
An eagle had seized a ring at the end of a long rope attached to the balloon and was towing it back into the rising air current.
There were more eagles just above them, scouting the winds to find where the eddies flowed.
Vaslav leaned over the edge of the basket to try to see the birds. "Are they yours?" he asked the warrior Stasa-yxan, who had come with them in the basket.
"The talassairi belong to no one," said the warrior.
The boy was abashed. He didn't talk well. He had only been using Universal for a few weeks. He was sorry he had tried to say anything.
Harrison Hall was relaxed and merrily sinister—even as they were swept toward the rocks—the only one of the humans who seemed untroubled.
It began to grow cold, and Stasa-yxan gave the three humans black cloaks to wear.
The air thinned. The balloon spread out. The countryside diminished into a sunny haphazard checkerboard whose straight rigid boundaries had been forced to curve with a winding river and to scallop the rising ground into flat stepped concentric levels outlined with blue-gray stone retaining walls so that the whole scene looked as if an orderly pattern of red and brown and green and gold squares had melted and run together.
Stasa-yxan frowned at some part and murmured as if making a note for himself, "Blight."
The others saw nothing. The fields looked healthy.
Alihahd had been silent this journey, lost in a brooding sulk, until Vaslav glanced his way, did a double take, and pointed, his mouth open but lacking the words to explain.
Alihahd blinked back to awareness and looked down at himself. His cloak had turned from black to white. "Is this thing alive?"
"Thou mindest not the cold as much as thy companions," said Stasa-yxan. "Thy thoughts are elsewhere."
Alihahd pulled the now-white cloak around him and huddled bleakly in the bottom of the woven-reed basket. "How appropriate."
The balloon rose to a level jut not far below the bald pate of the western summit. Alihahd climbed woodenly out of the creaking basket and jumped down to the rock shelf, landing weightily on his feet to find himself standing on an island in the sky.
High. So high. Above was nothing but cold sky and bright sun. The air was dry and thin, and everything appeared distinct in the harsh desertlike light.
Low furry plant life crept all the way to the tops of the mountains. Even on the nearly barren rock summits some life still clung, a tatter of grass, a patch of lichen.
Stasa-yxan led the way on a footworn twisty path through moss and creepers to the colonnades of Haven. The path leveled and straightened to a paved walkway called the Ledge Path, from whose base the five levels of Haven rose up in columned tiers, each higher level set farther into the slope. The little city in the sky gleamed in the bare light, the columns, arched doorways, and lancet windows intricately carved, latticed, fretted, and inlaid with gemstones and leafed with metal.
Across the narrow crevasse that separated the twin peaks, Haven's twin, Aerie, glittered in near mirror image, so that the terraced hollow between the summits formed a titanic natural amphitheater split down the middle.
Curious red-haired ranga poked their heads from the archways of Havenside to peer at the human strangers. There was no movement or curiosity from Aerieside.
A bridge spanned the crevasse where it narrowed to twenty feet across. It was a single-person footbridge made of wooden planks and rope, with knotted handrails. At its mooring Stasa-yxan stepped aside for Alihahd. "Thou first."
A wind moaned up the fissure and the bridge swayed.
Alihahd watched it swing.
Stasa-yxan fastened a safety line to Alihahd, but it provided no feeling of security. If anything, it made him feel worse.
For being six miles above sea level the winds were light here. They should have been roaring through that narrow abyss as in a wind tunnel.
They could always change.
Stasa-yxan nudged Alihahd. The short span was suddenly very very wide, becoming wider the longer Alihahd stared at the knotted fiber ropes and the wooden planks.
Somewhere beyond the sheltered hollow a wind howled.
Because there were no choices, Alihahd grasped the rope guides and stepped out, not looking down. He was midway across, poised over empty space, when a voice sounded from behind him on Havenside, calling long and tonal, a muezzin sound of repeated ritual.
"Erika! Ameeerrrrika!"
Alihahd turned his head with a start. He could not have heard what he had heard.
Sudden and startling as a splash of water, a child appeared running down a mountain path like a hardy little goat. "Coming!"
A human child.
She was bundled in a long heavy dress, wide-legged trousers, thick cloak, and leather boots. Black hair spilled down her back to her hips. Baby-fat cheeks were red and round. Her black eyes had lights within.
Alihahd felt his interest spark, and was so shocked by it he had to clutch for the bridge ropes and catch his balance. He was not sure how old the girl was, but knew she was not old enough for any man to be looking at her the way he was—especially him.
_And you_ , he was thinking to his prick. _Never mind that._
It had a mind of its own, and its timing was awful. He considered himself too old. It should by rights be dead. He'd honestly thought it was. He just wanted it to leave him alone.
_When was the last time you rose to any good occasion?_
This had less to do with the girl than it did his present danger. It was a sad state of affairs when it took proximity to death to excite him. He held his white cloak fast around him and growled inwardly. He continued across the bridge, disdainful of the long drop—courting it, in fact.
Safe on the other side, he turned to look again, but the girl Amerika had disappeared into one of Haven's many rock chambers.
_Did I dream her?_
Human. His emotions raced. Not for the girl this time, but for what she implied.
Space travel.
Trying not to hope too much, Alihahd unhooked his safety line and let it drop back to Havenside so the others could cross.
As he waited, slowly, subtly, the aura of this side took on life and strength and pressed itself into his consciousness. He knew without being told that this side of the Aerie was home to the warrior-priests.
The Aerie was silent, in repose. Like a monastery, it breathed feelings of age, space, and power, as if God were actually more manifest here than other places. Mosques, temples, and cathedrals gave Alihahd the same impression and always left him uneasy—places built for the God of the People of the Book.
The sun had passed its zenith, and the first direct light of day was arriving now on Aerieside. The warm rays lifted steam from the shade-dank rock, and furtive vapors trailed low across the cold blue-gray stones and spilled out between the pillars of the shadowy arcade, then faded in the thirsty air.
The priestess first appeared like a vision at the end of the colonnade. Backed in the sunlit frame of the farthest arch, she came striding up the columned arcade through alternate light and shadow, sunbeams catching on clouds of rising mist that swirled around her feet with her advance and rolled outward in her wake. She moved with a purpose, flow and force in her walk, sharp as an arrow in flight.
As she neared, she took on lines of reality. A sheer blue kaftan billowed out behind her, its long full sleeves slit from shoulder to cuff so her arms showed through thin and ropy with sharply defined muscles. Her tough, weathered skin was blue-black. Her coarse black hair swung down, long, straight, and thick, from a glossy topknot and was knotted again at the end. She wore a white flower behind her ear and a double-curved sword at her side.
She turned and passed under an ornate archway aglitter with topaz and chrysoberyl to an inner chamber.
And when Alihahd's companions had crossed the bridge, Stasa-yxan led them to the same monumental archway.
Alihahd paused at the entrance, then walked in.
The chamber was a great nine-sided room with polished wood floor and high domed ceiling of gold. Mirrors set in the tall window jambs shot sunlight up to the ceiling, where it diffused and spread softly, and the chamber glowed in warm golden light.
The golden dome unsettled Alihahd.
Two more warrior-priests flanked the inner doorway. Tall and strong as the others, they seemed somehow smaller in this chamber: not so sure, not so proud. And Stasa-yxan changed as he passed under the arch. He seemed to shrink and become meek in this place.
But it wasn't the chamber itself that affected the warriors. It was the figure at the far end.
Enthroned in a wide ceremonial seat on a dais behind a low lattice barrier was the dark warrior-priestess. Her eyes, trained on the doorway, were black and bottomless, fierce and benign. Her daunting presence filled the room.
She was jet, head to toe, a melano color phase, like a panther. Broken red scars blazed on her sharp cheekbones. She sat with one elbow propped on the armrest, her body curved to the side, one bare foot folded underneath her on the seat. A great snowy owl perched on the tall carved back of her throne.
Stasa-yxan bowed in the universal gesture of respect. "Azo! Fendi!"
The Fendi returned a light nod.
"It was a starship that fell out of the sky," Stasa-yxan said. "These survive."
"Are they human?" the Fendi asked. Her dialect was clear—almost modern.
"They are," Stasa-yxan said.
The Fendi raised hairless brows. "Human starships fly to Iry again. How long has it been?"
"Fendi, I know not."
The Fendi smiled at Alihahd. She spoke, eyes on him, words directed at Stasa-yxan. "It has been a while."
Two thousand years it had been.
Alihahd spoke. "So long we forgot you were real and not a dream of ours."
The warrior-priestess smiled wider, baring white teeth.
Alihahd was puzzled. It had been two thousand years since an Earth ship came to Iry, yet. . . .
He heard pattering feet on the path outside. _Amerika_. He knew her step already. She was no dream.
"Fendi. Was that not a human girl I saw?"
The Fendi nodded to Stasa-yxan. Stasa-yxan answered for her.
"It was," the warrior said. "We have several guests here. But she did not come on her own ship. We brought her."
_Then they do have interstellar capabilities!_ Hope leaped to full life.
The Fendi spoke again, musing to no one in particular. Not to the humans. Not directly. "So Earth finally decides to remember us."
Her tone was nonjudgmental. She considered the development neither good news nor bad news, only interesting fact.
Birds were flying in the chamber's spacious dome, flitting in and out of the tall windows. They were little black birds like swallows. One flew near the Fendi, and she absently snatched it out of the air with lightning grasp without even turning her head, nothing moving but her arm. She petted its head with one long finger, then tossed it back into the air, unhurt—all as effortless and unthinking as a person might bite his nails.
The snowy owl blinked a slow blink.
At last the Fendi asked, "They are not of these new humans—the Na′id, they are named?"
New? Comparatively, Alihahd supposed. Na′id colonization had been spreading for the last century and a half. The empire had conquered a large part of the known galaxy in that time.
"No, we are not," said Alihahd. The question was for him even though she'd directed it at Stasa-yxan.
"They are not here to assert human supremacy and—what is the other slogan?—galactic dominion?"
"No," said Alihahd. "We were shot down by Na′id." Then he remembered that he was not actually certain of what had befallen Harrison White Fox Hall's ship and realized he'd been answering for the man. He rephrased, "The boy and I were, anyway."
The Fendi's eyes slid to Hall. Stasa-yxan spoke to Hall for the Fendi. "And thou?"
" _We_ were," Hall affirmed.
"Tell them to stay in peace, then," the Fendi said, finality in her voice. Alihahd sensed dismissal.
And he felt panic. _Stay?_ He needed to get out of here, and soon. "Fendi," he said quickly. "I cannot stay. If you have a starship and could give me passage, I will give you anything you ask. I can pledge a great deal. Only tell me what you want."
The smile disappeared. The eyes flashed. Her head snapped toward Stasa-yxan. "Say to that one: Thou camest. We did not bring thee. We are not for hire." Her alien accent thickened in anger.
Alihahd immediately backed down. "I am sorry. I am already in your debt."
She smiled again. "Art thou a Fendi?"
Stasa-yxan made a noise in his throat. The Fendi had forgotten to speak to the human through her intermediary.
She hadn't forgotten. The Fendi stood, pushed her warrior attendant aside, and descended from the stepped dais. She moved around the lattice barrier.
Alihahd was struck speechless for a moment. What a bedraggled thing he must look; he had just insulted her, and she asked if he was her equal.
"I was," he said.
"Thou hast a people to whom thou must return?"
"Yes, Fendi," said Alihahd.
She reconsidered—seemed to.
"When one of us next leaves Iry, thou mayest go also," she said. "The physician may be next. Till then we can take you back to Lower Aerie. Most of our human guests are there. Winter drives them down. Only _niaha_ —three—humans stay up here. Amerika, Layla, and Montserrat."
"When will the next ship leave?" Alihahd asked. Why had she mentioned winter?
The Fendi frowned. The snowy owl clacked its hooked beak.
"It could be long. It could be tomorrow. Like death," she said.
Whether the last words were statement or analogy Alihahd couldn't tell. The Fendi was displeased, and her warriors were frightened.
She ascended to the dais again. "My name is Roniva."
That was a signal to go—for the second time—and they'd best obey. She didn't want their names. She would learn them in time if she cared to.
The owl glowered from its perch on the back of the throne.
The young warrior, Stasa-yxan, motioned the guests out. Alihahd started to leave. He glanced back once, and forgot what he was going to say. His back hadn't been turned for more than two seconds.
The snowy owl was gone.
• • •
It was a long hall, its shadows very deep. Alihahd had come this way still seeking an escape. The labyrinthine passageways of Aerieside led him up rock-hewn stairs spiraling up a mammoth newel like a ziggurat, and he was wheezing by the time he climbed to the top. There was oxygen enough in the air. He simply couldn't draw it in. Blood pounded in his head.
At the end of the long hall shone a turquoise door. It was open. A shaft of sunlight fell into the dark corridor from the chamber.
A voice called to Alihahd from beyond the door. "Ave."
It was the girl Amerika.
Then came a male voice. "Peace be upon you."
"And upon you peace," Alihahd answered, curious. He approached the massive door. Its ancient mosaic was faded from long standing in the dry air.
Inside he found a warrior-priest, the girl, and an eagle. The eagle cocked its head sideways and regarded Alihahd with one onyx eye, inhuman intelligence in its gaze.
The girl stole shy glances at Alihahd from beneath long black lashes. Her dark skin deepened color in a blush.
The elderly Itiri beckoned Alihahd farther into his chamber with a white, spidery hand, his drooping skin jiggling with his motions.
The cave with its stuccoed walls was small, clean, and lined with racks and racks of herbs and delicate blown-glass vials of swirling colors. It was a physician's chamber. Even with the clutter, the room was airy, its woven mats fresh. Swallows trilled at the tall windows.
"Thou art ill," the physician said from his worktable.
"No. No, I am not," Alihahd said.
The physician wrinkled his brow in doubt. He said to Amerika, "Healthy humans appear thus?"
"No," Amerika said.
"Thy temperature is high," the physician told Alihahd, and Alihahd wondered how he could tell.
"How is it you speak Universal?" said Alihahd, drawing attention from himself.
"It is the language between the stars," said the Itiri physician. "Is it not?"
_The way you speak it?_ "It was. Two thousand years ago," said Alihahd.
The physician made a motion with his head that was analogous to a shrug.
"We knew you would be back." He rose from his worktable. "Rest thou. Our air is thin to thee. Keep thee here and rest." He moved toward the door.
"What would be my chances of leaving Iry within the week?" Alihahd asked, and quickly added, "Seven days."
"Oh," said the physician without excitement as he was leaving. "None."
All in a wave, sickness rose in Alihahd's throat. His vision blurred and he reached for the wall, suddenly too dizzy to stand. There was no chair, so he sat on the floor.
When the spell passed, Amerika was kneeling at his side, biting her lower lips, afraid to touch him.
"It is nothing," Alihahd said.
Amerika sat back on her heels, unconvinced and unhappy. She was a lovely child, smelling clean of sunlit air, grass and leaves, and the woolen scent of her cloak.
The eagle in its corner stretched out its neck and lifted one wing to peck at a bare patch of skin. Amerika snapped her head around and cried, "Not to do that!" And the eagle stopped. Amerika turned back to Alihahd.
Alihahd gave a pale smile. The girl was a provincial. She spoke Universal in the peculiar dialect of the Itiri. She must have learned the language here, not on her backward homeworld Solea.
Her necklace placed her. The necklace was a polished chain carved from a single block of petrified wood, each loop an unbroken circle. The chain had been placed on her as a baby and could no longer be taken off over her head. The chains were customary to certain Solenense tribes. That Amerika still wore it meant she had never fallen into Na′id hands. The Na′id habitually cut the repressive symbols from the necks of all the young Solenese girls. That she still wore it signified that she was a virgin.
"A Solenese healer?" Alihahd said. "Is that not unusual?" Solea had no medical technology.
Amerika tried to hide a bashful smile, but it glowed in her eyes and on her cheeks. "Who told thee I was of Solea?" she said playfully.
"You did." Alihahd looped his forefinger through the chain at her throat.
Amerika's skin warmed against the back of his hand, and she blushed very dark. Alihahd let the chain drop.
He got up and smoothed his tunic. He moved to the door.
Suddenly plaintive, Amerika said, "Why must thou leave in seven days? This place pleaseth thee not?"
The way she said it made him sound ungrateful for wanting to go. He turned at the door. "Do you know, child, only hours ago I believed I was dying, and I expected nothing else. I know I am fortunate to be alive and to be here. But now that I am here, alive, and I find these people have starships—"
"Thou wantest to go home," she finished for him with a downturn in her voice, a litany of disappointment.
"No," Alihahd said. "I can never go home."
Amerika brightened. "Then stay thee here."
"I have no choice, it seems."
"Why so eager to go nowhere?" she demanded.
Alihahd couldn't explain, not without touching the past, and not without telling a child that the prospect of living could be infinitely more terrifying than dying. So he excused himself and left.
• • •
The warrior-priest Jinin-Ben-Tairre wrapped the felt cover around the ancient tungsten-plastic blade and replaced it in its scented wood chest, closed the lid, and fastened the polished brass catches. He had heard the eagles arrive earlier with Universal-speaking newcomers. Ben-Tairre had not gone out to look. It was his practice to avoid off-worlders. He preferred to maintain ignorance of things that lay beyond the planet he fiercely loved—or even of things beyond the Aerie—since the time he had first walked the fire and become a warrior-priest.
He stood, straightened his clothes, and secured his cloak with a short length of chain across his broad chest. Then he turned and bowed to his Elder. The hanina's hood was up. The old woman had ceased speaking for the day. The younger warrior withdrew and set off for his own chamber, muscles flowing in his powerful thighs with each stride, his winged familiar fluttering at his shoulder.
Long rays of the sinking sun reached the far wall of the arcade. Ben-Tairre walked through the colonnade, stepped down to the Ledge Path, and came face-to-face with one of the newcomers. Thunderstruck eyes met his own, and both men stopped dead.
The newcomer was a tall, gaunt human with straight Nordic nose and clear deep-set eyes in a swarthy, long face.
Jinin-Ben-Tairre's shock passed to fury. His eyes blazed and he walked swiftly past. Who had spat this being at him? Laws of hospitality forbade him striking the stranger's head off. The warrior's fury doubled at his own inability to choke down hatred. He did not go to his own chamber, but kept walking.
• • •
Alihahd turned to watch the retreating figure incredulously. If a hallucination, it was vivid.
The warrior was human.
• • •
A swift breeze swayed the bridge unsettlingly as Alihahd crossed to Havenside.
He found Harrison White Fox Hall and Vaslav at dinner in the dwelling of the human woman Montserrat. The cave was large, warm, its walls hung with red-and-amber tapestries. A kettle bubbled on the hearth.
There were two women inside. The pale, nervous, petite woman in a red dress was Montserrat. She kept her rusty wiry hair cut very short except for long bangs that nearly obscured a red chevron tattooed on her brow. She gave up her cushion at the table as Alihahd came in, and she fetched another for herself, her brown eyes downcast all the while. Hall called her Serra.
The second woman was as small, looked tougher, and was wearing a hide jerkin and trousers. She was cleaning a jeweled dagger and scowling at Harrison Hall, her lightly freckled nose wrinkled in disdain. Straight brown hair was pulled off her heart-shaped face and braided back with a leather thong. Her brown eyes were lined black like an Itiri's. She wore a small beam gun holstered under her left arm but seemed to favor the dagger in her hand. She was Layla, a Nwerthan mercenary. Alihahd could spot those right away. Modern worlds customarily recruited and trained the primitive Nwerthans in warfare because they fought like the very devil and seldom asked why.
Both women were not much older than thirty years Earth standard.
Alihahd bid a proper Universal greeting to them, then spoke to Hall. "I saw—" And he stopped, doubting what he had seen—a powerfully built young warrior-priest with golden skin, brown-black slanted eyes, a heavy jaw, flat nose, and short black hair. He'd borne the symbols of a warrior-priest: the red broken scars on his broad cheeks; a white flower at his belt; a signet ring on his forefinger, though his was on his left hand. His right hand was gloved in black and didn't look real. His knife had been sheathed left-handed. Red rags were tied around the highly arched insteps of his small feet which were scarred as if by fire.
A bird had ridden on his shoulder—a real bird, not an avian alien. It was a kestrel.
There was a power in the young warrior, and a horror-branded depth in his dark eyes that looked murderously at Alihahd. He had walked past Alihahd as if past an urn of ashes.
Alihahd decided that he was certain of what he'd seen. "I saw a warrior-priest," he told Hall. "He was as human as you and I."
"Jinin-Ben-Tairre," Layla said. She sheathed her dagger at her belt and placed eight dainty rings from the table onto the short, tapered fingers of her rough-lined and callused little hands.
"Is that a name?" Alihahd asked.
Layla assented with a nod. "It means 'The Warrior's Feet Are Burned.'" She spoke Universal with the studied precision of a second language.
"The Fendi said there were three humans on the Aerie," Alihahd said. "Why am I counting four?"
"Because Roniva will not speak his name. And because Jinin-Ben-Tairre is not human anymore," Layla said.
Alihahd found the idea of a human trying to be alien offensive. "He pretends he is not human so that makes him not?" Acid had crept into his deep voice.
"If Ben-Tairre thinks something is so, it is so," Layla said. "You will see. He has a familiar."
Alihahd remembered the warrior's kestrel and Roniva's disappearing owl. "What are they, the familiars?"
"I do not know," Layla said. She crossed one dun-booted foot across her opposite knee. "I do not think the Itiri know."
"Are they alive?"
"They do not eat," Layla said. "I do not think they breathe."
"They disappear from one place and appear somewhere else in an instant," Serra added haltingly in a low voice that went to breathiness and slight feminine gravel. Her hands fidgeted over her teacup. Serra's skin was pale brown, without sun, her eyes unlined. Her small nose had once been broken, and there were scars on her pretty face. Someone used to hit her.
"From where did Jinin-Ben-Tairre come?" Alihahd asked.
"He has no past," Layla said. "He killed it. He locked it in a box and burned it. It is dead."
Serra spoke in a near whisper. "Old ghosts have a way of catching up with you in the safest of places."
"I guess that bears keeping in mind," Harrison White Fox Hall commented. He turned his eyes to Alihahd.
"And what are you?" Layla challenged Alihahd. "You look like a Na′id."
The boy Vaslav had been letting the others talk. Here he spoke up in defense, proud and indignant, "He is not a Na′id! He is Alihahd!"
Layla's jaw dropped in surprise, showing crooked white teeth. "I know you," she said. Alihahd flinched. Layla's attitude changed to deference. "You fight Na′id."
"I do not fight them. I run from them," Alihahd said.
"You are much too humble," Serra whispered.
"No. In point of fact, I never met them in battle," Alihahd said. This point was important to him.
"But you are a great man," Layla said. "Everyone knows you made the Na′id leave Chesa."
The liberation of Chesa had been Alihahd's first and most notorious feat. It had made his "name." By means of false orders from nonexistent admirals and generals, false intelligence reports, bogus replacement troops, fabricated computer records, suspicions cast left and right so that no one talked to anyone else, and all manner of deceptions, Alihahd had systematically tricked the Na′id into abandoning an entire occupied planet. A full standard month passed before someone realized that anything was wrong.
"An exercise in vanity," Alihahd said. "They came back. There was no point to it beyond letting it be known that I could do it."
When the Na′id reoccupied Chesa and demanded of the natives who was responsible for this incredible plot, they replied, "Alihahd," which in their native tongue meant, "He left." And so he was known ever after, an elusive force with a formidable knowledge of the Na′id chain of command, codes, and way of doing things. He'd spent the rest of his shadowy career throwing the system into chaos and providing means of escape for unwilling subjects of the Empire.
"You are not eating," Serra said.
"We ate at Lower Aerie," Alihahd said. From the corner of his eye he saw Hall scowl at him. "Have you anything to drink?" Alihahd asked. He'd barely touched his tea.
"If you mean alcohol, there is none." Serra shrugged. "The Itiri don't seem to make any."
Alihahd dismissed the thought with a pass of his hand. His head hurt.
The sun was setting. The only light in the cave came from the hearth. It had been a long day—days. Alihahd had lost all sense of time since the _Liberation_ had broken apart.
He stared futilely at the bowl of stew cooling in front of him. He couldn't eat.
Amerika came to the door, and she peered in. Alihahd could see half of her, one bright black eye, one half of a coy smile, one baby-fat hand hugging the doorjamb.
Hall was speaking to him, "Did you find a way off-planet, Captain?"
"No," Alihahd said and closed his eyes. "We are trapped here."
Layla uncoiled from her cushion. "Then let us find a place for you." She was looking at Hall. "You are not staying in here."
"There is a chamber," Amerika piped cheerfully from the threshold. "Sit thee down, Layla. I shall show them."
"Whose chamber?" Alihahd asked.
"Thine," Amerika said.
Alihahd turned to Serra. "Why are we being given everything? Whose chamber am I taking, and who is giving it to me in return for what?"
Serra smiled for the first time, color in her pale brown cheeks. "It is the way of hospitality. Travelers must have what they need. If the cave is empty, you may have it. No one gives it. Nothing belongs to anyone anyway."
"Is this the law?" Alihahd asked.
"Not law. Virtue."
"Then what consequence if you are not hospitable?" Alihahd asked.
"Then you are a bad person," Serra said.
Alihahd didn't understand the world. He'd never been so lost, forced to conform to alien ways. "Is there anything I should be sure to do or not do? Am I likely to commit a crime in ignorance?"
"No there aren't any laws. Only for the warrior-priests. A guest would have to be very evil for harm to come to him. You're safe here," Serra said.
"Safe from the Itiri," Layla said, her eyes on Hall.
_No, I am not safe_ , Alihahd thought. _I must leave this place very, very soon_.
"Come with me," Amerika sang and waved the men outside with her.
"And do something with their clothes," Layla called after the girl. "They stink."
The night was clear. There were no lamps on the paths, for the starry sky was bright, shedding light equal to several full moons.
An albino Itiri sentinel crouched at his post on the dawn ridge, his face washed silvery blue in the cold light. Red eyes watched over the peaceful valley, while distant yellow firelights winked back from the little village far below.
Small black shapes darted through the air over the Aerie—swifts—catching winged bugs swept up the chasm on rising air currents.
Alihahd stopped on the windy path. Across the chasm a figure lurked in a dark archway. Starlight gleamed off a naked sword blade slanting down from a left-handed grip.
A man without a past was a dangerous sort, Alihahd knew. He stood out of the edge of the cliff path and faced the shadowed figure across the abyss, the wind in his hair, his long arms hanging at his sides from his wide, straight shoulders in weaponless, tired readiness until the dark figure and the sword withdrew again into blackness.
Alihahd walked on.
The cave where Amerika led them was clean, with a fire already burning in its hearth, and three separate piles of bedcovers set on the expansive chaff-filled mattress that would have been big enough for six people. The girl had been busy. Amerika stirred the coals in the hearth. She darted out again like a swift. "Good night. Good night."
Alihahd peeled off his tunic, pulled off his flat-bottomed deck boots, pounded out a place on the uneven mattress, and lay down.
Vaslav took a place against the wall behind which sounded the trickling of water. Vaslav huddled against the warm stone and was soon snoring softly.
Hall walked out to water a bush on the mountainside, then returned and undressed. His dirk he kept with him. That and his gun and his fox-head pipe. He gave up his other things to the little ranga male who came into the cave to collect their clothes and spirit them away.
Hall's trim body took on a burnished sheen in the firelight, muscular and taut for his age, deep-chested and wasp-waisted. A few age spots dotted his shoulders. His hair, loosed from its tie, was streaked heavily with steely gray.
Hall settled under a pile of bedding on the other side of Alihahd from Vaslav.
The glowing coals in the hearth shifted and settled. Outside, night birds sang.
Alihahd shivered. This place was barbaric. He could have been in New Triton by now. His emergency shuttles should have arrived there by this time. He could hear them telling the other rebels, "Alihahd is dead."
Alihahd wished he were.
He sensed Hall was still awake. Alihahd turned to him.
One tiger eye was open and watching. "A touch xenophobic, aren't you, Captain," Hall said.
"I was never known for my love of aliens," Alihahd said shortly.
"The Na′id call rebels _alien-lovers_ ," Hall said. Alihahd was a notorious rebel indeed.
"Does not necessarily follow," Alihahd said.
They were speaking in near-whispers, reluctant to disturb the quiet. Except for the birds, they seemed to be the only beings awake in the world.
Hall propped one bare arm behind his head, the brindle fur cover across his deep chest, his gaze directed toward the ceiling. Hearthlight picked up copper flecks of beard stubble on his cheek.
Hall was a hunter and a controller. The more elusive and powerful the quarry, the more interesting the capture and control. Alihahd would not be controlled.
Hall sighed and went to sleep.
Alihahd shivered. He couldn't seem to put on enough covers. He'd never felt the cold before. This cold came from within.
He fell asleep shaking.
# 4. Circle Circle
HARRISON WHITE FOX HALL awoke the first morning, pushed the blankets from his face, and breathed in the cold sting of mountain air. The fire was dead. He reached out a bare arm from the warmth under the covers to rest atop the frosty fur.
Some small light came from outside. It was almost dawn. The boy Vaslav was still asleep, quietly snoring. Alihahd had already risen and gone.
On the corner of the wide mattress Hall's clothes had been returned to him, neatly folded, clean, dry—and cold. He threw off the blankets and jumped out of bed onto the icy granite floor, naked and barefoot. He was quickly dressed.
A clay jug of water stood near the hearth. Hall broke the iced-over surface with a silver ladle and shaved the copper stubble from his face with the straight edge of his dagger, which was still warm from being sheathed against his forearm. He hadn't taken it off.
Vaslav, like most space travelers from more modern worlds than Hall's, showed no trace of a beard.
Outside the air was piercing fresh. Painted streaks were spreading in the eastern sky. A few bright stars still glimmered in the indigo west. Then a bright shaft of light spilled over the eastern mountains and splashed across the peaks. The valley below lay still in shadow. The sentinel on the dawn ridge opened his arms in salute to the sun as it climbed over the edge of the world.
There were no clouds anywhere, and Hall could see for miles—mountain after impassable mountain beyond the isolated terraced valley with its waterfall spitting rainbows into the air in the morning light. The day would be fair.
Hall stretched, feeling the mountain cold in his bones. He could see summer far below in the still-shaded village Kaletani Mai, but pride refused to let him go there. The air was only slightly thin to him, coming from Eridani as he did. He could survive up here.
The Aerie stirred. Hall watched the white ranga come out of their caves and turn dark in the sunlight. They were funny, cherubic little people, always cheerful, and a bit stupid. They were of one breed with the tall, slender enigmatic aghara. That's what Serra said. It was difficult to believe. Polymorphism in the extreme, Hall thought. The majestic aghara were actually the rare children of the chubby gnomes, and the Itiri warrior-priests were what the aghara kind spent their lives training to become.
Hall watched a few warrior-priests summon aghara children from out of the ranga caves. Youthful scarless faces of the uninitiated appeared in answer to their masters' call, and the young cheelas crossed the bridge to Aerieside to run with their masters over the mountain.
Hall followed one pair for a while, but he couldn't keep pace. So he dropped out of the running and walked alone along a swift-running brook where black birds with serrate beaks and oily feathers dove into the water seeking crustaceans under the rocks of the streambed. The winds were fast out here beyond the Aerie. Hall's face was stung red. He unrolled his bandana and tied it on his head to keep the buffeting from his ears.
He strayed from the beaten track through a stand of scrubby, distorted would-be trees that fringed the sun-bleached rocks of the ridge. The hunter in him was drawn by a trail of footprints crushed into moss still exhaling strong fragrance as if recently done. Someone had strayed this way before him—someone who wore shoes. Possibilities were limited.
And in an isolated depression where the air was still, sheltered by a curving rock wall and carpeted with long low-lying grasses, he found Alihahd seated on a rock, his elbows on his knees, his head bowed, his hands trembling. At Hall's approach he looked up, pale, beads of sweat on his face. Then his watery gaze returned to his feet. Hall jogged down the rocks into the low space and drew closer. He stood a moment in observation. This wasn't altitude sickness. "Alcoholic, Captain?"
Alihahd looked up in helpless self-surprise. He drew a wavering breath, licked his salty wet upper lip, and tried to steady his hands. "I would have said no." But his shaking hands held out before him defied him.
Hall watched curiously, weight on one foot, the other foot angled in. "You're not seeing snakes, are you?"
"No. No snakes."
Alihahd hung his head and closed his eyes. Hall climbed back out of the sunken area and left him alone with his misery.
• • •
Vaslav stumbled into Serra's cave late in the morning and squinted at the green breakfast Serra set before him.
A little while later Alihahd showed up, looking wounded and fragile. Serra took a separate kettle from the fire and poured an herbal brew for him. It smelled medicinal. Alihahd looked at Serra dubiously.
"Mr. Hall said you were sick," Serra said.
Alihahd's eyes shifted to Hall. "Mr. Hall said that, did he?"
Hall took his empty pipe from between his teeth and gestured with it to the cup of tea on the table. "It's a mild sedative."
Alihahd spoke stiffly with an edge in his voice. The words were, "Thank you, Mr. Hall." The tone said, _Back off, Mr. Hall_.
"You look pale," Serra said.
Alihahd's brows rose over deep-set watery eyes. "I imagine I will look much paler before long," he murmured.
Alihahd took to bed and stayed there, wretchedly ill. One by one his deadly secrets were beginning to slip away from him into open air for all to see, hard as he tried to lock them down.
He couldn't stop the Itiri physician from coming to examine him. The Itiri had to know if Alihahd had the plague.
Alihahd crouched back against the wall in drunken fear of the alien and in real fear of the examination—or rather the verdict of the examination. Alihahd had been afraid the alien would step back and declare, "He's a drunk." But the aged Itiri merely announced that he had never seen this particular malady before. He rubbed his withered jowls in puzzlement and muttered about inoculations and quarantine.
A low chuckle at the door made him turn.
Harrison Hall rocked back on his heels. His eyes were merry crescents. "Oh, it's not plague," Hall said.
From the bed, Alihahd's liquid eyes fixed accusingly on the tall satanic figure in the doorway, awaiting betrayal.
But Hall said to the physician, "It's a poison."
"Will he die?" the physician asked.
"I think not," Hall said.
"O God, why not?" Alihahd moaned and tried to vomit.
He writhed at night with horrid dreaming and rasping breath, trying to draw in enough oxygen from the air that seared his throat raw as he gasped at it.
The ranga moved a mass of potted bushes and leafy plants into his cave to replenish the oxygen and moisture in the closed air.
Alihahd lost track of day and night. He knew only twilight and darkness and his own agony. Hall had moved out and sought shelter elsewhere.
From a nightmare Alihahd awoke, thrashing and sweating and shivering under the rough blanket. He turned over in the dark to someone there—Vaslav sitting on the edge of the wide bed keeping vigil, his youthful face angelic with devotion in the infinitesimal light.
Alihahd lay on his back, gasping, his hair a matted snarl, perspiration and tears beaded on his haggard face and trickling down its deep furrows. He reached up a feeble hand—he could only lift his arm from his elbow—and touched Vaslav's cheek with the back of his fingers. In a deep, croaking voice he said, "Vaslav, do you love me?"
The boy's eyes flew wide. He sputtered, choked, stumbled over an aborted explanation, and stammered out, "Yes."
Alihahd's strengthless hand dropped back on the bed. "Then please let me vomit in private."
Abashed, the boy stuttered a flustered apology and left. Had he a tail, it would have been between his legs.
Alihahd curled up and wrapped his arms around his abdomen.
He woke at late dawn to a cold waft of fresh air in his stuffy chamber. Long slanting sun rays streamed in from the doorway, its hide cover blown slightly agape. The bushes in their trough at the far wall of the shadowy cave leaned toward the light, stretching out their leafy branches. They seemed to be straining.
Then one pulled itself up by its bipartite roots and walked out.
Then all the rest of them, save one, unearthed themselves and pattered out in a herd.
The last one tugged itself free and ran out with a hasty _whap whap whap_ of roots on the floor.
Alihahd pulled the covers over his head, delirious.
At midday, Alihahd woke again, lifted the covers, and peered out.
A trail of dirt was strewn across the floor from the empty planter at the far wall to the door.
Alihahd pulled the covers back over his head.
He swore the bushes would be back next time he looked.
He looked again when he heard Amerika's darting footsteps and sweet, scolding commands, "Get! Come back, you little monkey!"
And a stampede of slapping roots.
He lifted the blanket to see Amerika chasing a bunch of errant bushes back into the cave. They fled before her, dodged into the dark place, sank their roots into the trough, and became very still.
Amerika gave a single nod of satisfaction, wiping her hands on her skirts, and went out.
Alihahd laughed till he threw up.
And steadily, as days passed, he grew more pale—almost white—and his eyes became blue, and the roots of his hair emerged blond. When finally he sat up, washed, and dressed, he was perfectly fair.
Hall was surprised—and a little amused. "You're a nazi!"
"Don't—" Alihahd began too loudly. He caught himself and finished softly, "—call me that, if you please."
Hall shrugged his big shoulders. "Isn't that the Na′id word for your kind?"
"It means more than blond-haired and blue-eyed. Do not call me that."
Hall shrugged.
The coloring was rare now, but not as rare as the Na′id would like. Still it was far too conspicuous for a man trying to move in Na′id circles without attracting attention. The way Alihahd looked now, he could not pass for a Chesite or a Na′id. The bogus Na′id captain was bogus indeed. "A white Na′id," Hall chuckled.
"Do not call me that either."
"Sorry," Hall said. He was aware of the full implications of that epithet and knew it was a bad one.
Serra stared at the new Alihahd. She didn't say anything. She sat down on the bed, drew a skinning knife from its sheath in her boot, and proceeded to cut off his hair down to the blond.
Vaslav was thunderstruck by the change. "Are you really Alihahd?" he asked in doubting distress.
A cut lock of black hair fell down Alihahd's face. He brushed it off his knee. "I am the only Alihahd there ever was," he said, changing the words slightly. "Alihahd was never quite real."
The hero the boy believed in, he was not. There was no hero Alihahd.
White and shorn and still very weak, Alihahd ventured outside alone. Everything startled him, as if a filter had been lifted from his senses, and for the first time in over a decade he received the full assault of the world around him, so aware of life that it was painful.
He climbed the path on trembling legs, his hand out to steady himself on the cliff face, bracing himself against the wind's push.
Even his own hands were strange to him now—pallid great bony things tracked with bulging blue veins. He was frowning at them and at the blond hairs of his white forearms, when suddenly a warrior dropped down from the next level onto the path before him.
A flat and handsome yellow face with smoldering eyes.
Blinding sunlight on a naked blade.
Jinin-Ben-Tairre.
Alihahd didn't move as the sword turned, the blade gliding in a silken pass from Ben's massive shoulder and into another ready position. Ben was naked to the waist, and every clearly defined muscle in his arms and broad chest showed flexing and flowing, one with the weapon. Ben made no sound, not even the rustle of clothing. He wore only satiny black trousers and the red rags on his scarred feet. There was a carnelian ring on his left hand, and the black glove over his right hand. The warrior circled slowly.
Alihahd didn't dare turn. He did not know what laws bound an Itiri, but knew there was nothing to stay this bastard warrior from killing him here and now if he so chose. Alihahd waited for the sword to split his skull from behind, an old chant running through his brain:
_Circle circle, dance of death_
_Once she cries, twice she lays_
_A wreath about his youthful head_
_Thrice she flies, for nothing stays._
Full circle, Ben stopped before Alihahd. Alihahd felt naked with his true face exposed. He was more gaunt and drawn than before, his hollow cheeks deepened. Marked crevices were chiseled from his nostrils to the corners of his mouth, and his forceful bone structure showed beneath emaciated flesh. But most revealed were the luminous blue eyes, sunken in their orbits. Their gaze penetrated with a daunting power that had been masked before by the darker pigment.
Ben's sword changed hands, left to right, and he reached out with his left hand—the ungloved one—to Alihahd's white face. Alihahd tightened, anticipating pain, but felt only the light brush of short fingers on his chin.
Ben withdrew his hand and left.
Alihahd stood, still frozen in place. Sun tingled his fair skin. The wind was cold on his short-sheared scalp. Finally Harrison White Fox Hall came and took his hand like a child. "Come on, Captain." Alihahd looked at the dark red hand, then looked at Hall's laughing face.
"Are you laughing at me?" Alihahd asked.
"Yes," Hall said.
And since there was no intelligent answer to that, he said, "Oh," and let Hall guide him back to the cave—laughing.
• • •
Gunshots reported and echoed through the mountains in the early morning.
_That will be Mr. Hall._
Alihahd stepped out of his cave. Shadows were very long. He followed the sound, stopping every few hundred feet to rest and breathe in frosty clouds.
The gun's reports led him to Harrison Hall, poised on a mountain spur with his crazy-looking notch-handled rifle. He was shooting clay pigeons that Vaslav hurled out from a higher ledge.
Alihahd made his way to Hall's side, his approach loud enough, his tunic red enough that he wouldn't surprise the gunman. Hall was hitting every bird, some several times when the fragments were large enough to bother with.
At last Vaslav called down, "That's it." The boy opened empty arms, having run out of clay birds.
Hall turned his head to Alihahd for comment.
"Impressive," Alihahd said.
Hall put one foot up on a rock and rested the butt of his gun on his sloping thigh. "Not my usual kind of target, but—" He finished with a shrug.
"What do you usually shoot?"
"Whatever's blue and glows in the dark."
Na′id uniforms and Na′id ships were electric blue with glowing red insignias of Human Supremacy/Galactic Dominion.
"I was shooting the real birds this morning," he told Alihahd. "Our hosts didn't like it."
"Do you ever miss?"
"Never."
Vaslav clambered down from his high ledge and came to them puffing. His cheeks and nose were wind-bitten. He looked healthy. A few days' primitive life had done him good, shedding him of his spaceship pallor.
Eight days. Alihahd had sensed the days were long here—30.96 standard hours, said Vaslav, who wore a standard chronometer around his wrist. It seemed time for a tentative evaluation.
Alihahd propped his foot up on the same rock as Hall and leaned in, casually conspiratorial, one arm across his knee, a carefully magnetic gesture, and the other two men automatically leaned in for a private conference.
"What do you think of them?" Alihahd asked quietly. "Our hosts." Himself, he was not sure. He'd been in bed most of the eight days. These two had seen more than he.
"Too passive, if you ask me," Hall said.
"That is bad?" Alihahd asked.
"They have an army up here. I may be able to shoot the birds, but Roniva catches them in her bare hands without looking at them. I've seen their swords cut through tempered steel, and I hear you have to walk through fire to become a warrior. They're all crack shots, and from what I've seen, it's not too much to guess that they could totally annihilate the Na′id's Great Human Army if they put their minds to it."
Alihahd couldn't argue, even though he estimated only a thousand warriors on the Aerie by highest count. He wondered if there were many more. He shuddered at the words _totally annihilate._
Given the proper technology—which, fortunately, the Itiri lacked—Alihahd didn't think there was much the Itiri couldn't do.
"But what do they do?" Hall continued, then answered his own question, "Sit up here and protect the birds."
He raised his gun and centered a winging swift in his sights. He didn't pull the trigger, just said, "Bang." And Alihahd knew the bird had been spared—no luck or maybe about it.
Hall lowered the gun. His tawny eyes glittered with ferocity. His lips curved beneath his gray mustache without a trace of softness.
Alihahd paused. "Whom did you lose?"
The eyes flickered to his, then looked away to the sky.
"Everyone."
# 5. Does Jerusalem Stand?
THE DATE WAS THE NINTH DAY of the Red Geese, year of the Ship in the Opal hexadecade. So said Vaslav. It was all mush to Alihahd. But, cast outside the stream of human events and beyond the relevance of human calendars, he ought to get used to it. Layla had been here on the Aerie seven Earth years. Serra had been here fifteen years, Ben seventeen. God knew when Alihahd would ever leave. He could die here. He was convinced that Ben-Tairre meant to kill him, though Amerika said not—it would not be virtuous. Alihahd didn't place much faith in the human-turned-alien's virtue. The man hated him.
Alihahd heard the wind brass tapping in the swift gusts outside the cave where the humans had gathered.
"What is the date?" he asked Vaslav. "The real date."
Vaslav glanced at his wrist chronometer, "Tenthmonth nine. It's 0935 hours on the meridian."
The year would have been 5856 CE were anyone still using the Gregorian calendar.
Earth's civilization had reached its height long ago, in the third Common millennium, with an interstellar technology and human colonies flung across the Milky Way. At its peak the delicate balance had crashed, and civilization itself imploded in a galaxy-wide political and economic collapse that threw humankind into a dark age for two thousand years, a time when no ships would fly. Colonies, cut off from each other and left to their own resources, lost their technology, their ties with Earth, their count of years. Many forgot Earth and each other altogether.
In their isolation—after the anarchy and struggle for bare survival was over—the colonies tried to piece together what they had lost. The old learning was reclaimed imperfectly and at different rates in different fields and in different places. Some worlds remained sunken in total savagery. One built to an atomic age and did not make it out. But most worlds progressed.
And, finally, a few tentative starships traveled out again at the middle of the fifth millennium. The Dark Age was over. But the travelers were hardly Earthlings anymore. They had their own worlds, their own cultures. Earth was some faraway cradle of civilization, if remembered at all—
Except on one colony which remembered well where it came from and why. The planet was Mat Tanatti—Land of Praise, Land of Glory.
The colonists of Mat Tanatti had begun as a large group of discontented idealists on Earth in the third millennium. They had gone forth in self-exile to an uninhabited world to start a new culture on a base of equality undivided by artificial boundaries of nations and languages and antique faiths which the modern world had left behind and which turned brother against brother against sister.
The pilgrims named themselves _Na′id_ , which meant _praiseworthy_ or _alert_ in the dead Earth language they took for their tongue of naming names.
And since they had already isolated themselves before the Collapse came, they weren't drawn down into it. No lifeline was cut, and Mat Tanatti was the one world that never fell. The Na′id kept their civilization, their history, and their purpose.
The end of the Dark Age found them successful beyond all prediction in creating a unified people. But the pessimists were justified in their insistence that Man must have his hatreds and prejudices. The Na′id stressed their humanity—as opposed to aliens.
• • •
The end of the Dark Age found the Na′id with the most advanced technology in the galaxy. They took it as a sign that their ideology was right and they were duty-bound to reunite their long-lost human brothers and sisters, and to free Earth from alien influence and wrong-thinking.
There had been no peace ever since. There was always fighting somewhere, and the biggest battles were not with aliens but with their own kind, the humans who didn't want to be united—not after two thousand years, and not under Na′id domination.
In professing tolerance of all human creations and culture, the Na′id encountered a dilemma: Were they to tolerate intolerance? Most religions included more than one tenet which ran counter to Na′id thought. The Na′id couldn't allow anyone's provincial ideas to infringe on the rights of others.
So they accepted all religions, with modifications, cutting out all portions that gave one segment of humanity superiority over another.
But the Word of God would not be picked apart so easily, and the Na′id movement met its major obstacle in the power of religion and the vehemence that met any attempted change.
Race was another sensitive point to the Na′id, but more quickly, if superficially, solved. The Na′id had never actually learned to live with and enjoy mankind's natural variations. There were always vague suspicions that one race was smarter, stronger, healthier, better than another, and a vague terror that the suspicions might be right. So their method of eliminating prejudice against racial differences was to eliminate the racial differences—to mix all the races and make humankind into one homogeneous mass. Ethnic purity became equated with elitism, and the Na′id would not tolerate it.
The Na′id fought their own kind to absorb them into the fold. They raped the pure races and stole the children to save them from narrow teachings. Brotherhood became assimilation, and supremacy leached in besides. Forbidden to hate one another, the Na′id channeled their hatred toward aliens. Humankind needed someone to dominate, someone on whom to blame the bloodshed, someone to be the common enemy against whom all humans could unite.
So began and continued the great crusade to save humanity and to liberate the homeworld, Earth. The fighting had gone on for well over a hundred years.
"Does Jerusalem stand?" Serra asked.
That was an old question. For a long time it had been the first question asked of any space traveler upon arriving anywhere.
The seat of the three major monotheistic religions, Jerusalem became the most important city on Earth. To the resistance it was a symbol of freedom that must never fall. The Na′id knew the value of symbols, and they were determined to have Jerusalem—for its impact on rebel morale and for its place in human history. The Mother City of Humankind, they called it.
The Holy City lay under siege for a hundred years, and humans everywhere, who had never seen Earth and never would, whether they belonged to one of the three faiths or not, asked of anyone who might know, "Does Jerusalem stand?"
The question was seldom heard anymore. Everyone knew the answer.
"You do not know?" Alihahd asked Serra.
"I had heard," Serra said. "Layla told me. I thought it was propaganda. Is it true, then?"
"Thirteen Earth years ago," Alihahd said.
The fall of Jerusalem had been a turning point in the war. The Bel had all but declared total dominion of the known galaxy in his Jerusalem Address, expecting the rest of the resistance to shrivel up and for holdouts to fall in quick succession to his unstoppable general, Shad Iliya, the White Na′id.
Shad Iliya's army had never lost a battle—though most of the battles Shad Iliya ever fought had been on backwater alien worlds. The general had been long kept a half-hidden embarrassment because of his un-Na′id coloring. He'd only been brought into the open as a desperate attempt to end the hundred-year siege of Jerusalem.
Easy victory in the Holy City had made Shad Iliya's name and assured his infamy for a good part of forever. Instantly, Shad Iliya was the greatest hero/villain in the universe next to the Bel himself. No one doubted that the rest of humanity would fall to the man who had taken Jerusalem. Na′id morale had never been higher, nor rebel prospects lower.
Then Shad Iliya died. It was very soon after the historic victory. God/Jehovah/Allah consumed him in a pillar of fire, said the believers. The Na′id accused rebel agents of assassination. Rebels not of the faiths claimed that the Na′id got rid of the White Na′id themselves. The general's first lieutenant suggested it had been suicide, but no one listened to him.
The timing of his death made Shad Iliya a convenient demon by whom rebels could summon curses and frighten children. His was the face seen in the mirror in the dark. Being white helped to make him eerie. It also made his name a good insult for rebels to hurl at people who looked like Alihahd did now with his ghostly pale skin and white-golden fuzz of hair growing back on his shorn head.
But even with the death of the great general, the fall of Jerusalem remained a devastating blow to resistance. The mere speaking of it dampened the spirit of all in Serra's cave.
Amerika turned on Hall and Alihahd as if they were negligent gods. "Could you not save it?" She beat the shoulder of the nearest one—Alihahd—with her little fists.
Alihahd rested his chin on his hand and looked forlornly heavenward with his large expressive blue eyes. He was very thin from his illness. He drew in a breath as if to speak, then exhaled without saying anything. He changed expression and finally spoke one word, hard. "No."
"How was it taken?" Layla demanded, crouching on her hassock like a hunting animal, her hair braided back with leather thongs. Layla was the combat soldier of the group. "What strategy?"
Alihahd answered in a dull, reciting voice, "Hand-to-hand. On the ground."
Amerika was confused. Hand-to-hand was the only kind of battle she had ever known. "Are not all battles so?"
"No battles are fought hand-to-hand these days," Harrison Hall told her. "We have much more efficient ways of doing each other in."
Yet superior weapons and modern technology couldn't win the war, as the Na′id discovered. Shad Iliya had revolutionized modern warfare by taking it backward—to people with guns.
"Infantry will always win the war," Alihahd said quietly as if thinking aloud. "But it's a lost skill. The only way to beat a good infantry is to destroy their entire planet. Or come in with a better infantry."
There was a somber pause.
The distant eerie sound of the wind brass on the eagle ridge intruded into the quiet. The arrhythmic tapping of the whip ends kept the humans inside, warning them that the winds were fast.
"How goes the conflict now?" Layla asked.
There had been no news at the Aerie in a long time. Even old information would be news here.
"Not well," Alihahd said. "But we've had no crushing reverses since Jerusalem."
"The Na′id lost several of their best leaders," Hall added. "Besides Shad Iliya."
"Did you kill him?" Layla asked.
The question startled Hall into a smile. He tilted his head and played at the two gold earrings in his one lobe. "Not I."
"I tried," Alihahd said.
Hall turned to Alihahd, his smile broader. "Did you really? I tried to kill the Bel."
"How is it you failed?" Layla demanded.
Alihahd leaned his head back against the wall. "Diverse reasons. None of them good." He put his long knobby fingers to the bridge of his nose and closed his eyes. "It is in the nature of our side to be inefficient. When the ultimate goal is disunity, you cannot have an efficient organization. Our side has a seedy lot of heroes. Myself. The _Marauder_." He cracked his eyelids and stole a peek at Hall, who gave no reaction to the name. Not a lash flickered out of place. Alihahd dropped his hand from his brow and sighed. "Decisive action is rather difficult when all factions of the Resistance are at cross-purposes. For my part, I am not trying to overthrow the Na′id empire, only loosen its grip on the unwilling. I haven't the right to decide how humankind shall or shall not be governed simply because it might be within my power to do so."
Hall laughed.
Alihahd frowned. "Did I say something funny?"
"Your flights of eloquence are precious, Captain."
"I am glad I keep you amused, Mr. Hall," Alihahd said. He was, he thought, not good for much else at the moment.
Talk of the past had made him introspective—something he tried to avoid—thinking of who he was, what he had become, and what was left to him now. He wanted to get back into space and do his work as Alihahd, the rebel runner.
Or, now that he thought of it, did he really want to return to that life—ever? The possibility of not wanting to do so shocked him. He was conscience-bound to want it. But it was out of reach. Running from responsibility was one thing. Being thrown from it was new. He got to his feet and walked to the doorway, where he leaned against the rock jamb to gaze across to Aerieside. The rock walls shone pink in the sunset.
He couldn't leave. The old way of life was far away.
A shadow fell across him. Hall had come to his side. Alihahd breathed, "Can I possibly be free at last? Can the albatross have fallen from my neck and I be truly allowed to forget?"
Hall took his pipe from between his teeth. "You can't hide from the past, and there's no permanent haven. The past will come looking."
Alihahd turned from the door. Hall's words were ominous.
The opal eyes of the fox-head pipe glittered unevenly. One was cracked.
Alihahd should have known Hall would champion the past. Those who lived for revenge never forgot.
"What are you thinking of?" Hall asked.
Alihahd leaned back against the rock, his eyes wide, his hands gently, too gently, tracing the grooves in the wall behind him. "My death," he said darkly.
Then he looked up. His face changed. His eyes became clear and his voice blithe. "I was to have been taken alive by the Na′id. I was imagining General Atta″id explaining my death to the Bel. That is all."
Hall gave a wicked smile that was not a smile. "I fear the honored general Atta″id will explain nothing. The honored general is recently deceased."
"Ah," Alihahd said, a lament. He scowled at Hall. _Marauder_. That was never spoken between them. "I regret. He was a good man."
"You mourn your enemies, Captain?"
"I would mourn any good human."
"You've been impersonating the Na′id so long you're beginning to smell like one," Hall said.
"Not everything the Na′id believe in is wrong," Alihahd said. "Part evil isn't necessarily all evil. People who hate cannot see that." He was looking squarely at Hall.
Hall smiled and appealed to the others as his jury. "What is one to make of this? One of the Na′id's most devastating adversaries does not hate them."
"Make of it what you wish," Alihahd said wearily, fingertips to his forehead. "I don't hate anyone. I am so very tired. . . ."
• • •
In the muted light before sunrise, in the hour of the eagles, a warrior-priest wearing a yellow robe stepped outside and swept away the debris that had collected in the windpockets on the path overnight. The dawn was the last of the Red Geese. The sun would move into the house of the Twins at the solstice later in the day.
Alihahd sat in a thicket of leatherferns. His eyes were shut, his nostrils catching faint scents of dark green alpine growth, dusty rock, and dry earth in the thin air.
Earth?
_Soil_ , he corrected himself.
The last of the fog had lifted from his mind, and his thoughts ran swift and clear. The heartsick weariness that had hung on him for so long was leaving him tentatively—as if he were afraid to give it up. It had so long been a part of him that it hardly seemed natural to be without it.
_When you are who I am, there is no such thing as a safe place._ He needed to keep that in mind.
A flutter of wings very close made him open his eyes. A bird lighted on his bent knee.
Its heavy, seed-cracking beak parted. _Cheep!_
Alihahd watched it without moving, not quite sure how he felt about being perched on. The bird cocked its scaled head sideways, robin-style, to see him out of the eye on the side of its head.
Cheep.
"Alien," Alihahd croaked at it like an insult.
Cheep.
The bird refolded its wings and shrugged its little shoulders like a man adjusting his coat.
Alihahd guessed he'd been judged an adequate perch.
Chi _reet!_ The bird took flight with a flurry of wings and a rattling scolding _trrrrr_ as Amerika came bounding up the path holding up her long skirts with one hand above the grass-stained knees of her wide trousers. An empty basket swung on her arm. Burrs clung to her woven cloak that was striped in wide bands of blue and violet. She gave Alihahd a radiant smile as he sighted her. Her cheeks and nose were rosily wind-bitten.
She told him that Ben-Tairre was inviting him on a hunting expedition with his party today.
She took Alihahd's hard gnarled arm in hers and led him to where the hunters were gathering. Harrison White Fox Hall was there. He had words for Alihahd's tunic as Alihahd strode down the slope.
"If my knees were that bony I wouldn't wear a dress."
Alihahd smiled wryly. "I do not wear it for you to admire my legs, Mr. Hall. Who invited you?"
"I invited myself," Hall said. He'd also invited Vaslav, who stood behind him like a small pale shadow. "Should be great sport."
Alihahd lowered his voice. "Though I am not sure of the game here." It was folly to take up this challenge, like following a predator into its lair. Ben-Tairre didn't carry himself like a man making peace.
Ben equipped Alihahd with an Itiri weapon, a beam gun called a taeben, "warrior's eye." Like the never-dull tungsten-plastic swords, the taebens were products of a technology far beyond Iry's. The taeben Alihahd was given was a ronin's. It could be triggered by any hand, whereas the warrior-priests' weapons were keyed to their owners' use alone. Alihahd was told about the animals of the mountain, what to watch for, what to watch out for.
Alihahd was going to watch his back.
The hunting party was Ben-Tairre's fire clan of the carnelian serpent, and five young warrior-priests of the fire clan of the opal sword. They started out on foot down the green-cloaked north slope of the mountain and descended all the way to the timberline where wind-tortured half-trees grew along the ground. Far in the distance ahead loomed the next taller peak, the Guardian.
Alihahd shot at nothing. It was all he could do to keep up and pretend it was easy. And when the hunters stopped to reorganize, Alihahd went apart to rest behind an escarpment. There he could hang his head, bending over, hands on his knees, breathing hard.
Harrison Hall found him there. He leaned carefully against the glittering crag, crossed one leg over the other at the ankles, and regarded Alihahd curiously. "Captain?"
Alihahd spoke very quietly. "I am not doing well, Harry." It was the first and last time he spoke the man's given name.
Hall looked down at the bowed blond head. "Unhealthy posture," he said. "Your lungs can't expand."
"If I stand up, I shall pass out," Alihahd said.
His watery eyes focused on the contorted timberline trees. He began to feel like one of those stunted shapes twisted along the ground. He could see the direction of the prevailing winds from the trees' eastward flagging, but anything actually broken broke southward. It would seem there was a violence that came out of the north.
Loud crunching footsteps made Alihahd straighten up. Vaslav joined them along the scarp. "They say there are berinxes in the area," he said.
"Berinxes?" Alihahd looked to Hall. "Those have the teeth?"
Hall nodded.
And at that moment there came a noise from above, high on the rocks, with a trickle of dislodged sand and pebbles. All three spun with weapons raised, but Alihahd immediately pulled back his own and reached over to hold Vaslav's hand from the trigger.
It was a predator up on the rock outpost—one with slanted coal-dark eyes and a small mouth twisted into a hard contemptuous line in his flat-boned, coldly handsome face. "We are moving again," Ben-Tairre informed them and moved away.
Vaslav drew in a wavering breath and rejoined the hunters.
Hall waited for Alihahd. An insinuating smile crept beneath the mustache, as if he'd caught some secret Alihahd let slip.
"What?" Alihahd said to the smugly knowing look.
"I don't care what you say, Captain, you've been in combat."
Alihahd was much too quick at turning and aiming—as a thing done so often it became reflex—and he was much too good at _not_ shooting never to have seen battle.
Alihahd frowned at the weapon in his hands. "Not against Na′id," he repeated and he hiked after the Itiri hunting party.
By the time they started home again, racing their shrinking shadows, Alihahd was lagging behind. He stumbled, skinned his knees on the gravel slope, and didn't get up.
"Shall I carry thee?" Ben asked.
Alihahd stiffened. "No, that is all right. Thank you very much," he said with such a tone of offended aristocracy that it made Harrison Hall laugh.
Ben-Tairre made a pointed gesture of looking at the sun. At this pace they wouldn't reach the Aerie in time. The noonday sun was lethal. Nothing stirred at midday, not even the chitinous lizards with reflective scales. The Itiri were anxious to beat the sun home.
So Alihahd was left behind. The hunters found a sheltering cave where he could wait out the noon hour and continue the journey alone after the deadly time was past.
• • •
A lizard crawled off its sunny ledge and went slinking into a dark crevice in the rocks.
The croaking of a reptilian crow split the thin air.
A burr of insects shivered in the heat.
Then there was no sound of anything that was living.
Nothing moved.
Shadows shrank to nothing.
Air sizzled. Brown grasses withered.
Outside the cave was a blistering glare—like the sun on a Mediterranean strand.
The sky boiled. Rocks hissed.
He remembered the sun on the Mediterranean Sea. He thought back farther, to the last time he had been completely sober—thought clearly about things best not remembered.
He curled his knobby thin legs up and hugged his skinned knees as if he were cold. His mind was clear and there was nowhere to hide.
• • •
Vaslav paced the north ridge, waiting for Alihahd's return. Hall came out once to stand at his side, coat drawn back from his wasp waist, fists on his narrow hips, empty pipe between his teeth, earrings gleaming against his deep red-bronze skin.
The weather had changed. There was no sign of the captain.
The red flags were out, warning that the winds were picking up and it was ill-advised to go beyond the Aerie's sheltered amphitheater. The shadows had lengthened before the clouds cast them over. The Itiri were flocking home.
Pack beasts honked and gurgled. Itiri voices hushed. Birds stopped singing. The beasts sniffed the air.
Broadside, a pack beast was an ungainly bulk. Face on, its enormous body was an aerodynamic wedge that could withstand a three-hundred-mile-per-hour gale.
All over the mountain the beasts stood aligned toward the north, facing the mountain Guardian.
• • •
A cloud. A shadow. A cool breath.
Alihahd woke from a dreaming half-sleep. He wandered from the cave. He had lost track of time and he could see no shadows. The sky was a white slate.
High, high above came the faint echoes of the clashing wind brass and voices that carried even to this distance: "Off the bridge! _Shandee!_ "
Alihahd climbed up to a rocky prominence and gazed up toward the sound, curious.
From over the lofty mountain peaks a swelling cloud loomed and rushed down. The swift wall of gray darkness seemed fast even from far away. It had to be moving in a torrent.
Alihahd was mesmerized by it, like the black cloud over Jerusalem that had come and engulfed him and blotted out the light, everything.
So, frozen, dull, and disbelieving, he stared at this storm that roared like a rocket rain. Its speed was unreal, like a jet stream.
Or _was_ it an actual jet stream? He was five miles above sea level.
And then it occurred to him he ought to be running back for cover. But—as at Jerusalem—it was too late for him to go anywhere. The winds were here.
He started for the cave in the last seconds left to him. He jumped from the rocks with a cold spray of sand and grit. He fell hard, pain exploding in his shoulder and shooting in fiery spikes through his body and behind his eyes. He rolled.
He stopped rolling, and opened his eyes to search for the cave, but saw a figure over him, cloaked in storm clouds.
Jinin-Ben-Tairre.
• • •
A yank on both arms brought a blaze of pain. Alihahd was lifted like a rag doll. He fell against Ben's body that was almost too hard to be flesh. The blast of wind hit like a truck. They were thrown, flying into sudden still darkness.
Alihahd folded over in pain on the rock floor of the cave as _Shandee_ roared past.
At length Alihahd lifted himself onto his knees and looked up.
A moving wall of gray wind thundered past the cave entrance. It appeared as if they were in a speeding vehicle. Outside was a blur. Inside was still.
And Ben was there, snorting. His face looked burned, his broad flat cheeks scored by flying grit. He shook debris from his hair. Upon catching blue eyes directed fixedly his way, he stopped.
Alihahd simply gazed, cornered, his long limbs drawn in and folded to fit the cramped space, his big frame ill accustomed to bending into unaccommodating places. He lacked Itiri grace but was not without his dignity. It never failed him.
He held his one shoulder at an odd slope. His thick lips turned down, furrowing deep frowning lines into his long face, his brow pinched.
Ben moved smoothly to his side and grabbed Alihahd's arm with a twist and a hit and a flash of blinding pain. Alihahd roared and turned to strike with his other hand, until he realized that the pain had suddenly abated as quickly as it had come, down to an aching throb, and he could move his right arm in its socket again.
He wasn't grateful. Anyone who pushed him into harm's way could damn well pull him out and not expect thanks for it.
He withdrew as far as he could to wait out the storm, squared off opposite Ben-Tairre.
_Why not just lose me to the wind?_ Alihahd thought. It would've been a human enough thing to do.
But this being was trying very hard not to be human. He'd been born among humankind, had once answered to a human name. But it wasn't what he wanted to be.
Alihahd could see the struggle inside—a man who was two people—and Alihahd knew the feeling well.
He slumped back against the dank rock wall, rueful at the situation—caught here, prisoners of the storm, hating one another. _All four of us_.
Ben spied the corners of a sardonic smile, and his head turned a quick hawklike fraction in suspicion. He wouldn't like to think himself a source of humor.
Dark-eyed gaze bored into Alihahd, but didn't penetrate. Alihahd was well practiced at being opaque.
_I am older than you. My armor is thicker. Though you are not doing badly for yourself, warrior-priest. I still see a human boy. You hate me_.
Alihahd coughed, deep honking spasms that shook his aching shoulder. Then he sat back again, drained, guarded, his blue eyes heavy-lidded.
_You hate me. Hatred is fear. But I really cannot see what you fear_.
Alihahd was two meters tall, fair and blond like a warrior-priest, but haggard, bone-weary, old, the ruin of a once great man. He let his head tilt to one side.
_You fear me?_ He smiled again. _I am the one defeated here_.
When the winds lifted, Alihahd couldn't walk. He was sick, and weakness forced him to take a humiliating ride up the mountain on the back of a shaggy pack beast.
His illness went on for days, and this time he didn't recover. His breath rattled. He didn't want to leave the lordly inhospitable heights of the Aerie as most humans did, but he realized this place was going to kill him, and finally he asked to be taken down to the summer village Kaletani Mai.
With a cloak wrapped around his emaciated figure, Alihahd retraced the twisting trail to the waiting balloon and climbed, subdued and unhappy, into the woven reed basket.
Harrison Hall came to see him off at the ledge, the place where they had first landed on the Aerie, seemingly a long time ago. He faced into the wind, his redingote open to the breezes that tugged at the moored balloon. He clenched his empty pipe between his white teeth. All the sinuous lines in his dark red face molded into one of those lovely demonic smiles that revealed absolutely nothing. He was sorry to see his adversary get away from him, fallen to another opponent.
Alihahd looked back, bitter. "I don't lose well," he said quietly.
Hall nodded imperceptibly.
Two ranga unfastened the balloon's ropes and threw them to the pilot. Eagles guided the balloon as it lifted off the rocks.
Hall took his pipe from between his teeth. Nodded. "Captain."
"Mr. Hall," Alihahd said.
• • •
Jinin-Ben-Tairre was alone in the training hall. Its great wide windows were flooded with sunlight that reflected a honey glow on the butterwood panels. The worn wooden floor was pitted and scarred with sword grooves. Ben knelt, sitting back on his bare feet, his head bowed on his heavily thewed neck, his eyes shut. He tried to clear his mind, think of nothing.
His even breaths deepened, but not into the peace of sleep—deep in anger. He tried to calm his thoughts. Darkness kept stealing into his mind. Angry, suppressed thoughts rose into the clear space. He needed to banish them altogether—the dragons he had never really slain, merely pushed aside for a time.
He opened his eyes, stood, crossed to the sword chest, and took up his blade, _Da′iku_. He glided it through all its possible moves, then began to visualize attackers and cut them apart. They came by twos. He killed by twos. More moved in, all sides. His ivory-gold skin glistened, his black hair matted to his head. His mechanical hand whirred. Invisible necks severed and heads tumbled. Ben rolled, stood, kicked, slashed. Attackers reeled back. He drove straight down with the blade and kicked straight back. He wheeled and cut the air. Movement became random, precision muddied in his savagery.
"Careless, careless, careless," Roniva said from the archway.
Ben came to a standstill in the center of the chamber, perspiration streaming down his sides. He blinked sweat from his eyes. He hadn't even heard her come in. His training, his awareness, failed him. _Did she see it?_
Had to.
Such slips could be fatal when one had powerful enemies waiting.
"How may I serve thee, Fendi?"
"Thou hast hurt my guest," Roniva said.
There was danger when a Fendi spoke face-to-face. Direct speech from the monarch was reserved for certain messages which brooked no intermediary. Roniva hadn't spoken to Ben-Tairre since the day she had become Fendi of the Aerie. This moment had been long in coming. Ben knew what this meeting meant. Time had come. _Setkaza_. Legend called it dance of death. It was an ancient right to be invoked when there were no more options. Declaration gave one leave to kill at any time—and opened oneself to being killed at any time by one's declared opponent.
"Thou hast killed my cheela. Thou hast killed my son. Thou hast driven away my guest," Roniva said, the whites of her eyes flashing in her ebony face. She wore a long flowing tunic of shimmering green. It left bare and free her thin whipcord-sinewed arms. Her hand was on her sword hilt.
"He is evil," Ben-Tairre said.
"I wanted him _here!_ " Roniva cried as she drew her sword in a furious arc and jabbed its point into the floor. She stepped back from the blade, her fists at her sides, her voice a strained tremulous quiet. "From this moment we are bound in setkaza, thou and I. Look to thine own resources, Wolf, for thy days are numbered from today."
# PART TWO:
## Ben
# 6. Wolf at the Door
_Gregorian Year 5839 CE_
IN THE YEAR OF THE TOPAZ GATEWAY, in the days before Roniva became Fendi of the Aerie, a small stowaway was found aboard Xanthan's spaceship when the young warrior returned to Iry after a year's journey among the stars.
" _Azo!_ What hast thou brought to us, Xanthan?" Roniva said as her warrior cheela stepped from his ship into the yellow grass of Lower Aerie.
"What have I . . . ?" Xanthan echoed in puzzlement and turned back to his ship.
A starved, wild-eyed boy stared out through the hatchway at the sunlit village Kaletani Mai and the towering mountains.
"By the fire of my making!" Xanthan exclaimed, and the boy pulled back with a start.
Roniva advanced to him. Her swordbelt jingled with her stride. Sunshine cast a blue sheen on her hair. She leaned in and sniffed at the boy, who drew down into a wolfish crouch. He smelled bad. "What is it?" Roniva asked.
Xanthan stammered. "I—I meant not to bring him. He is a human being. One of the Earth breed."
The boy emerged from the ship cautiously to crouch in the warm sunshine. His ribs stood out in high relief under translucent sickly yellow skin. Coal-dark eyes smoldered deep in his skull-like head. He snarled at Roniva and ran past her to throw his stick arms around Xanthan's waist and locked his fingers together. Xanthan placed his hand gently on the boy's dirty black hair.
Xanthan was lovely, his tender eyes soft green like the sea, his flaxen hair braided into a crown, his skin smooth and unaged. The boy was in love.
Roniva pressed the back of her forefinger to her lips, speculative. Her onyx signet left a fleeting imprint of a sword there when she took her hand away again. "What doth it call itself for a name?"
"I know not. I found him in a trash heap. I fed him and let him go," Xanthan said.
Xanthan _thought_ he had let the child go. He bowed his head to meet the oddly shaped dark eyes at his waist. "What is thy name?"
Sunlight glistened on the bright edges of Xanthan's clean golden hair. It made a shining corona around his angelic face. The boy gazed up at the face and clung fast to his silence.
"Doth it speak?" Roniva asked.
"He hath a voice. I have not heard him to speak."
Xanthan had heard the boy yowl and snarl and screech.
"Perhaps it hath no mind," Roniva suggested. Her arms jangled with enameled bracelets as she reached out to touch the boy's head. He snarled and bit her hand.
Roniva jerked back, flashed anger, and smacked him down the nose with her open palm.
His face turned up, cold and twisted, into a vindictive mockery of a smile. He'd known she was going to hit him. But he'd got her first.
The Elders who had come behind Roniva agreed they had never seen anything that untamed, that dangerous.
"Shalt thou kill it, Xanthan?" one asked. "It is distorted."
Roniva licked her punctured hand. "I will kill it."
The boy, without letting go of Xanthan's waist, skittered around behind the warrior and peered out with a pinched glowering face.
"Ah, it understands," Roniva said. "It doth have a mind, if not a name."
One of the other Elders spoke. "Then we cannot kill it."
"It is a _guest_?" Roniva cried.
"How can it be else? Xanthan brought it here."
The stinking little creature thrust the lower lip of his small mouth at them, angry.
Roniva leaned forward, her hands on her knees so her eyes were level with his. "Thou art vicious, and I shall call thee Wolf for lack of a name."
"What is to be done with it?" an Elder asked.
Xanthan reddened. "I brought him. I shall take care of him."
"It will die on the mountain," Roniva said.
Another said, "Leave it here with the ranga. They will feed it. The rest is for its own devices."
The other Elders agreed. And that was to be the final word on the human boy, Wolf.
No one counted on him following Xanthan up the mountain.
When the hungry, frenzied, mangy-haired thing peered over the rock at the top of the snake path, a ranga woman screamed and dropped her water jug. Clay shattered, water splashed, and the ranga woman ran on stubby legs, shrieking that she had seen a monster.
A crowd gathered to see the monster. Xanthan pushed to the fore because he had an awful feeling that he knew what he would find.
His skinny foundling, hands and feet torn and bleeding from the climb, crawled over the last step onto the Ledge Path. The boy searched the staring faces and picked out his ivory-and-gold god Xanthan from the throng. The boy's dark eyes were wide in abandonment and asking why. He couldn't understand what he'd done to be left behind.
And the boy didn't know why all these other beings had come out onto the terraces to stare at him. They gazed at him in hushed awe. He didn't know what he'd done.
What he had done was to begin turning into a warrior-priest.
Aghara children came to the mountain by instinct. Drawn to the Aerie like spawning salmon on Earth. They answered an impulse to be with their own kind. From all parts of the world they hunted the hidden valley and climbed the mountain to become warrior-priests. Or die trying.
Xanthan blurted out before thinking, "I want to make him a warrior-priest."
_"That?"_ The voice was Roniva's.
The boy scratched his runny nose. His hand left a smear of blood there.
Xanthan looked up to the second level where Roniva stood. "He hath the heart."
"Thou mistakest madness for courage," Roniva said more gently.
An Elder's hand came to rest on the nape of Xanthan's neck. "A human child, Xanthan? He will die."
"He doth not want to die. I will not let him," Xanthan said.
"He will endure not the ordeals. At best he will bore."
"Then let him bore," Xanthan said, feeling utterly alone. He thought his was a natural decision. What great foolishness did the others see that he did not? "Let him fail for himself and return to his own kind of his own choice. He came here like a warrior cheela."
As he was speaking, everyone turned away, and Xanthan's blood turned cold, knowing who approached. The crowd parted.
The berinx padded through the parted way on heavy paws with lowered head, its fanged jaws agape and drooling. It stopped in front of the Earth boy, shook its brindle mane, yawned wide, and sat with a woof like a real berinx.
"Where is thy master, Chaulin?" Xanthan asked in weary ritual.
The berinx snarled. _Thy master also_.
The Fendi came slowly in his familiar's wake. He wore his naxa cloak thrown over one straight shoulder, its chain passing under his sword arm. His cloak was always black now. His wispy hair was too thin to be bound. He held himself strictly erect as one who begins to feel gravity too much.
Eyes like hard green glass chips fastened their gaze upon Xanthan, then upon the little Earthling.
The Fendi extended his blue-tinged, fragile-looking hand. The boy made to bite it, but, quick as a katalin's wingbeat, the aged Fendi closed his grip on the boy's lower jaw and held it unyieldingly.
He turned the child's head first to one side, then the other, examining his face, his yellow skin, his exotic eyes sunk deep in his hunger-stark skull.
Finally, the Fendi released the Earthchild's jaw with a slight firm push that prevented the creature from snapping at his withdrawing hand.
The Fendi turned slowly to pat the head of his fanged berinx. He talked to it. "If he climbed up the mountain, then he can climb back down in his own time." The Fendi's hard glittering green eyes slid to his youngest warrior-priest. "Xanthan, he is thy cheela."
Xanthan stared, blank, for a long moment, then smiled and looked to his new cheela in joy.
The boy knew the meaning of smiles. Something good had happened. Lovely Xanthan was happy. The boy started to return a wan grin gapped with missing baby teeth.
Then his eyes rolled back, and he crumpled to the path in a dead faint.
# 7. Wolf by the Hearth
_5839–5850 CE_
XANTHAN CAME TO THE BEDSIDE in the ranga cave and watched his sick cheela. Asleep, the child looked even younger than he actually was. His closed eyes were smoky-ringed in his starved face. He took up such a tiny corner of the wide mattress.
Then the eyes opened, dark like a ranga's and slanted like no other creature's in the world. The boy blinked sleepily.
"How is it with thee, Earthchild?" Xanthan asked.
The boy's black brows lowered at the name Earthchild. His tiny bow mouth curved down so hard his chin puckered. His little fists balled, and he sat up and spat in hatred of his own kind. He made it clear he was not an Earthchild. He bit his upper lip and glowered at Xanthan.
"Be thou then no more an Earthling," Xanthan said. "Only let go thine hatred."
The boy fell back on the bed, exhausted.
"I have something for thee," Xanthan said. He held up a closed hand.
The boy uncurled his scrawny claw fingers to receive the blue gemstone Xanthan dropped into his palm. The boy brought it up close under his eyes and watched the stone move with his pulse.
"Lapis is thy courage and thy will," Xanthan said. "That is one. There are eight. When thou hast eight, thou wilt walk the fire."
The boy grabbed Xanthan's ring hand and pulled it to him to inspect the warrior's carved signet stone. "That comes ninth." Xanthan smiled. "Not till thou art a warrior. That is for my clan—the topaz twins. We walked only two years ago. I am a very new warrior."
As he was speaking, a white bird appeared from nowhere and alighted on Xanthan's shoulder. The boy gave a startled cry. Unperturbed, the dove preened a snowy wing, puffed up its ruff, and rearranged its long plumed tail. "My familiar," Xanthan told the child. "That comes tenth—I know not from where."
The boy frowned dubiously.
"This creature I have named Asha," Xanthan said. "What is thy name, my cheela? I need something to call thee."
But the boy wouldn't speak. His hand clamped shut over his gemstone, and he retreated under the covers.
"Please tell me," Xanthan said. "Or they will call thee Wolf."
The boy nodded, his native gesture of approval. His lips were pursed, his brows knit, but not unhappily. Wolf would do fine for a name.
• • •
The first thing Wolf needed to learn, if he was to be mute, was the way of silent speech. There were many signs and attitudes in use, so he could go a long way without speaking. And it was well that he spurned the chatter of little birds and ranga. Eagles and warrior-priests soared silent and alone. A warrior-priest didn't disturb the silence without a reason.
Wolf was soon healthy again. Flesh covered his ribs, and his skin turned from sickly yellow to an odd but vital sun-darkened brown gold.
Once Wolf was fit, Xanthan began to tap on his door in the mornings.
" _Sae duun. Sae duun_ , cheela."
The boy was always outside on the first tap. Then Xanthan didn't need to tap at all. Wolf was awake ahead of the dawn and listening for Xanthan's footsteps. Xanthan would approach with soundless tread, and still the boy was outside.
For the first year he did no fighting, but learned to fall and jump and listen and feel and see and breathe. He had already had a great deal of martial training among humans. It was obvious in the way he used his inner strength and the way he moved—not sticklike and unnatural in the usual human fashion.
In the first year, Xanthan was learning as well, discovering the oddities of his small charge.
Xanthan woke, the first time of many times, in the dead of night in his solitary cave on Aerieside to find Wolf asleep, curled under the blanket at the foot of his bed.
Xanthan lifted him by his shoulders to face him.
Sleepy eyes blinked back.
Xanthan's white skin was smoothed cold like marble in the dim starshine that spilled into the cave. His warrior's gemstones gleamed darkly in the polished rock wall. Xanthan told the boy his place was across the way at Haven with the ranga.
The boy shared a cave with eight male ranga, as most aghara did while they were children. The ranga lived in heaps. Warriors lived alone. Wolf cared for neither arrangement. He wanted to be with Xanthan.
Xanthan would wrap a cloak around Wolf and send him back to Havenside, except when the wind brass sounded. Then Xanthan would either carry him across the swinging bridge himself, or let him stay, breathing softly at the foot of the bed.
The next surprise came on a cool and breezy day in what passed for summer on the mountain when Xanthan took his cheela hunting runner-birds.
Xanthan soon spotted a trail, and he trotted over the ridges and down the gullies and inclines in pursuit, his cheela dogging him faithfully.
Then the trail split, a bird and mate. Xanthan told Wolf to take the low trail. Xanthan would climb the steep rocks. And Xanthan bounded up the treacherous rise. The boy just stood in the ragged brown grass, bewildered. Xanthan stopped on his rock perch, perplexed and annoyed. "Follow thou, before it fades!"
The boy turned a complete circle, searching for something fading.
Xanthan became cross. "Art thou blind?"
As soon as the words were out, the young warrior recalled something spoken once of humans being blind. He leaped down the rocks and rolled in the grass at Wolf's feet. He jumped up. "Come thou with me."
They abandoned the hunt and returned to the Aerie. Xanthan sought his own master, Roniva.
"Azo!" Xanthan saluted the Elder with his sword fist on his left palm. "Hanina, knowest thou of humankind?"
"I know some small thing." She strolled across the cave and ran her fingers through Wolf's hair. It was coarse and black like her own. Wolf had learned by now not to bite everyone who tried to touch him. "Ask thou," she bid her warrior.
"Are humans somehow blind?" Xanthan asked.
"Heat blind," Roniva said. "Infrared they cannot see. They see one octave only. The lower frequencies are invisible to them. To thine wolf, this cave is dark."
Xanthan was numb with shock.
"Knowest thou wherefore I opposed thee, cheela-nu?" Roniva said as Xanthan began to realize how difficult a task he had taken upon himself in this human cheela. A cheela needed every sense and every strength and skill he had to become a warrior-priest. "I wanted not to see thee break thine heart." Roniva drew her saffron shawl around her and stepped out of the cave into the wind.
Xanthan turned to his cheela and took his glowing hot face in his hands. The boy must have some other strength somewhere, something only humans had. Xanthan sensed it pulsing between his hands, knew it as he knew some part of his spirit would transcend death, knew it without having any reason to believe. The boy had to have something else. He must.
Because Xanthan, like all warriors, could not fail and live.
• • •
It was Roniva who taught Wolf to write and to count in the Itiri way. "It uses its left hand," she said to Xanthan. "Mark thou on that?"
Xanthan had noticed. The boy would carry a sword in his left hand. "It—" Xanthan began, then caught himself. _It_.
Roniva smiled with white teeth and admonished, "I trapped thee in my circle, Xanthan. Be thou more careful."
"Yes, hanina."
"And feed it." She squeezed the boy's arm. "It needs to be fatter."
"I fear the winter," Xanthan admitted.
Roniva demurred. "Not if it be fat. I wish I were built like thine creature. Its stock cometh from mountains. Look thou at it. How short its fingers and toes. How flat its face. How thin and small its lips. How short it is. Its blood has less a journey from its heart and will not cool. See the fold of fat over its eyelids. Those eyes will never freeze, nor that flat nose. Its stature looks as if it will broaden if he grows. I would spend my winters easier had I that build and those fingers and toes." The tall, lithe warrior-priestess felt the cold quickly in her long tapered extremities if she didn't bid her heart to work faster. "He hath mountain eyes like mine. Dark eyes burn not in the light at these heights where the air is too thin to blunt light's cutting edge."
Xanthan blinked his kohl-lined green eyes. "He."
"Cheela?"
"Thou said he."
"Ah." Roniva spread her long spidery fingers. "I am weary of thine savage. Take it with thee and go."
• • •
Wolf survived the winters well, and in the year of the Topaz Beacon, Xanthan began to teach him to fight with his open hands.
Days were idyllic on the mountain, and, with no other humans around him to remind him, Wolf started to forget, truly forget, what he was.
Then, in the year of the Topaz Serpent, humans were brought to the Aerie. They'd been found drifting in space without power, victims of a war humankind was waging across the stars.
Wolf hated them.
"But they are thy kind," Roniva said.
Wolf motioned a negative. He would not be human.
"The girl, perhaps?" Xanthan said. "Thou wilt need a mate someday."
_No. No. No_.
And to Wolf's bitter glee, the humans couldn't endure the cold winds on the rugged mountaintop. All but one of them retreated to the mild climate of the Lower Aerie, so Wolf did not have to look at them.
Only the one named Montserrat stayed, a timid woman, full grown, who kept to herself on Havenside.
Wolf wouldn't hurt her. Such would violate his emerald. He took out the stone and gazed into its clear green crystal whenever the hatred boiled up inside him. The emerald was for kindness. Wolf had been given the gemstone for his generosity to the eagles. The boy had an affinity for the talassairi.
He still didn't talk. He howled if he was hurt or left alone. He still came to the foot of Xanthan's bed at night—so stealthily that the warrior sometimes didn't waken. All those practices came to an end in the winter of the Topaz Serpent.
Wolf was given his own cheela-cave on Havenside. It was a small solitary chamber, a proper home for a warrior in training.
Wolf hated it worse than he hated the cave full of ranga males. He spent little time in it.
But he had no choice come winter, when, on the eighth of the River, Shandee returned for its annual maraud, and all activity came to a standstill and everyone took to his home as the deadly jet wind raged through the mountain peaks nonstop for twenty days.
Wolf had never spent twenty days alone before.
The forsaken pleading in his eyes when Xanthan shut the door on him to seal his shelter haunted Xanthan back to his own winter prison.
Shandee howled. Or was it the boy? Sometimes Xanthan could hear both and thought his heart would break.
After five days he heard only Shandee.
At the end of twenty days, in the abrupt silence of the winter wind's leaving, Xanthan burst from his cave in a flurry of snow and sailed across the bridgeless crevasse with a running cat's leap to dig out his cheela from the piles of blown snow.
The cave door opened with the cracking of ice.
Xanthan saw nothing at first in the dark cave, then perceived the heat aura of the solitary figure in the corner. Wolf was alive. But he didn't move, or whimper, or fly into Xanthan's arms. The boy was silent, his eyes open. He was there, but not there, lost somewhere inside himself.
Xanthan hugged him and cried, "Come back. Come thee back."
But Wolf stayed rigid through the nightfall, and Xanthan carried him back to Aerieside on the new bridge, set him at the foot of his own bed, and waited. The boy didn't curl up there. He'd always curled up. He stared at the ceiling.
Xanthan caught him up, held his stiff unyielding form, and cried on his black hair. "Oh, little Wolf, what hast thou done to me? Who ever had such a stupid cheela?"
"I am not stupid."
Xanthan was shocked out of his tears. "What sayest thou?" he whispered.
Calm eyes riveted to his, and Wolf spoke distinctly. "I am not stupid."
Xanthan couldn't speak. The boy was perfectly composed. "I will drink now," he said, and Xanthan gave him water. Then Wolf said, "I will eat now."
And when he was fed, he returned to his lonely cave and slept.
He never came to crawl into Xanthan's bed again to sleep at his master's feet, and Xanthan was very sorry for that, the last gentleness gone from the child. Wolf had been taught to stay by himself, and Xanthan wondered if it hadn't been a mistake.
Wolf now cherished his solitude with a vicious fervor. Patience he also acquired. It was a weird and unworldly kind of patience, not like turquoise at all, and he took hurt without sound. This wasn't the peace of eagles. This was fire in the mountain.
He was a different Wolf ever after, grown, with an Itiri warrior's reserve and dignity. The ranga said Shandee was still on the mountain and that it lurked within him. They sensed something odd in his kind of quietude.
And the eagles were afraid of him.
• • •
In the year of the Topaz River, Wolf was taught his first weapon. It was a sword, the tungsten-plastic kind. "When thou wilt walk the fire, this will be thine," Xanthan told him.
Wolf ran his palm along the flat of the ancient blade, his dark eyes alight.
Xanthan's white bird trilled on its perch over the lintel by an inset ruby rose.
"Why do we not teach a sword to wield itself?" Xanthan posed the question to his cheela.
Wolf scowled. He pulled at the cuffs of his rough-woven cheela's tunic. He'd outgrown another one. "Because it is impossible," he said.
"No," Xanthan said. "Had we want, we would find a way. Thine answer shows a want of thought. Topaz is wisdom. Thou hast it not."
"Then I don't know why we don't teach swords to wield themselves," Wolf said.
"Because a sword hath no virtue," Xanthan said.
Wolf wore his virtues on a leather thong around his right wrist. He had four: lapis for courage, emerald for kindness, diamond for honor, zircon for modesty. Xanthan closed his hand over the bracelet and squeezed Wolf's wrist. The stones bit into his brown flesh. "Keep these little stones in mind when thou hast thine own sword," Xanthan said.
He showed Wolf a few simple moves with the sword and let him practice until his wrist and forearm tired. Then Xanthan brought him the felt cover in which to wrap the sword and replace it in its scented wood chest.
Wolf's eyelids fluttered uneasily at the covering. "The Gurkha never sheaths his kukri unblooded."
Xanthan asked, "Art thou one of these Gurkha?"
"There is one in my ancestry," said Wolf. "My brother owns a kukri. There is much blood on the blade."
"It's a foolish custom," Xanthan said, enclosing the tungsten-plastic sword in red felt. "How is one to practice with one's actual weapon? Thy Gurkha brother must have many scars."
Wolf was about to speak and suddenly frowned, as if catching himself at something forbidden. "I forget," he said.
• • •
In the year of the Topaz Veil, Wolf was taught his last weapon, the taeben. The Itiri called the beam guns "eyes" because the taeben were useful and powerful, and they lured their users into letting their other talents atrophy. Relying solely on a taeben was seductively easy.
"A gun is a coward's weapon," Xanthan said. "Any child can kill with a gun. It is also thy most powerful weapon, not thy sword. The sword is art. The sword is discipline. The gun is life and death."
Wolf had handled guns before and became quickly expert. He was also quick to put the taeben aside and polish his other skills. He didn't like the weapon. Guns were too human.
• • •
As the short years passed, Wolf was becoming a young man. His frame broadened as Roniva had predicted, and his muscles filled out with great strength and un-Itiri bulk. He stretched hard to keep his form sleek. Bulging thews were for Earthmen.
And toward the close of the Topaz hexadecade, more Earthlings were brought to the Aerie, refugees of the human war. They were surprised to find a human cheela among the legendary warrior-priests, and the young women thought him breathtaking. Wolf was disturbed and confused, and he kept his distance from them, which made him all the more attractive.
He stood on the highest level of Aerieside, glowering across the crevasse, his head held high, his disdainful slanted eyes directed downward. He wore only dark trousers, a sash belt, and a wide red band across his forehead to keep his human sweat from his eyes. His arms were crossed defensively over his bare chest. And the girls peered up at him from the lower level of Haven and giggled.
"What is wrong with them?" Wolf said.
Xanthan crouched beside him on one knee, his sword planted point down before him like a staff. "What is wrong with thee, cheela-nu?" he said. "Thou art bright as plague fever."
Wolf's heat aura glowed vividly about him to Itiri eyes. "Be this something human?"
Wolf burned more brightly still. And scarlet blushed down his neck and chest. "I am not human," he said.
Xanthan considered the young females across the way. They were a varied group in color and size. Xanthan didn't know what was the desirable standard for humans. Itiri came in two basic types only—the tall, slender, blond, green-eyed aghara, and the short, stocky, red-haired, black-eyed ranga. Humans came in all sorts. "Thou findest none of them an acceptable mother for thine child?"
"No. I am Itiri."
"Thou canst not give child to an Itiri woman," Xanthan said.
"Then I will have no mate," Wolf said. "I need no children."
"Thy choice," Xanthan said.
• • •
In the year of the Carnelian Sword, to begin the new hexadecade, Xanthan embarked on a trek with his cheela on foot across the mountains to visit outlying ranga villages and show Wolf there was more to Iry than one mountain and one valley. They were gone for four years.
When they returned, Wolf would have been twenty-one years old by Earth count. By Iry count, he was thirty-four. Onto his leather bracelet Wolf had added turquoise, which was patience, topaz, which was wisdom, and sapphire, which was control and strength.
At nightfall, Xanthan came to Wolf's chamber and sat at the foot of his bed. Wolf rose in puzzlement.
A cold wind of coming winter moaned outside.
In the almost dark, Wolf could see his master's beloved face, soft eyes, and golden hair braided into a crown. Angels looked so in the dim early memory of gentle times when Wolf was still a baby and carried a human name.
Xanthan pressed a pebble into his hand in the dark. Wolf held it up to see fiery-colored flecks within a milky white blaze. The colors shifted in the hearthlight. An opal, which was integrity, soundness, and wholeness. The gem was his eighth. The last. His training as a cheela was over.
Xanthan bowed his beautiful head, his face happy, sad, proud, and troubled. "When the spring cometh, the fire is for thee."
# 8. Wolf in the Fire
_5851 CE_
THE PLACE OF FIRE was open to the sky. Its hearthstones had been laid into a level breezeless garden court where its flame burned upright, though the slopes beyond it were blasted by stiff winds that bared the rocks. Sharp crags isolated the fire garden. It was a difficult approach from whichever way one tried.
Wide, flat squares of marble checkered the courtyard, alternating red and black. Leathery plants circled round about the sanctuary or trailed inside it on tendriled vines. At one side of the square an ancient spring still bubbled to life from time to time. Heat from the water flowing hidden within the rocks kept the snow from sticking here.
The tiles at the center of the courtyard were discolored where flames had licked and where sooty feet had trod.
In the year of the Carnelian Beacon, on the first of the Sword, only one cheela was walking. That was not proper. But the one was expected to fail.
Not to taint the fire for others, Wolf would go alone.
The Elders ringed the court. Painted kohl lines extended in flowing curves from their eyes and drew lashes down to the broken red scars on their cheeks. They wore eight jewels in their hair with their Elder's stars. Their swords lay unsheathed across their knees as they sat cross-legged on the marble. Their familiars crouched, perched, or sat to their left sides.
The warrior-priests who were not Elders sat with their fire clans. There were no ranga here. This was the one place ranga couldn't go. But no fretful mother and father would be hovering in the snow at the sanctuary boundary this time.
Wolf gazed into the flames. His trousers were rolled up and tied below his knees. Around his head he wore a red band. Around his neck he wore a thin leather strip which he would untie on the far side of the firepit.
A walk through fire. Easily done if done at all.
Xanthan waited on the far side with his own clan of the topaz twins. He was angrier than Wolf had ever seen him, upset that his cheela's walk was to be solo, that no celebration was prepared, that everyone was already trying to forget this day.
Wolf tried to catch his eye. _Watch this!_ But Xanthan couldn't see through the wall of heat above the fire.
Wolf tossed back his head. _I am Itiri. I shall not burn_. And he stepped into the fire.
He smelled it before he felt it. Pain and utter shock arrived at once.
Wolf leaped from the firepit and fell to the marble tiles hard on his knees. He dropped over onto his side, writhing in pain, digging his fingers into his thighs as if to cut himself off from his charred, oozing black-and-red feet.
Repulsed and embarrassed faces were turning away.
Wolf ran, scrambled, from the place of fire, out of their sight and into the snow. He tumbled down a jagged embankment and landed in a drift.
He buried his face in the icy snow, breathing fast, the pain inside greater than the searing throb of his burns that racked his body and made him tremble and sweat in the cold.
All his agony and shame couldn't change what happened. And it happened over and over with every blink of his eyes, branded into his mind.
Ice melted on his feverish skin. With a will born of extremity, he blotted out the physical pain. It was not there, so he gave it no mind. No mind at all. Nerves brought pain pulses to tap at his brain and found the gates locked fast.
He rested, breathing, trying not to think. There was only one thought.
Clouds moved in, closing up the sky. The snows would return.
After an eternity, Wolf tore his red headband in two and wrapped his ravaged feet in the rags. Then he hiked the tortuous blowing path back to Aerie.
On the high terrace he caught sight of a tall slender figure dressed in plain brown, hooded, arms folded into wide sleeves. All Wolf could see of her was her blue-black feet. Roniva was in mourning.
Wolf was puzzled. She would not be mourning him, so who? Who was dear to Roniva?
Wolf's heart caught.
No. What had to be could not be. Wolf forbade it. No! NO!
Roniva's hooded form moved like a shadow, retreating from his sight.
Wolf stumbled to her at a hobbling run. She didn't stop for him. One did not speak to the hooded, but Wolf fell to his knees as he reached her and he cried, "Where is my master?"
Roniva's eyes were hidden, the hood pulled far forward. She continued away, soundless.
Wolf clutched at the hem of her long sleeve and held fast. "Where is Xanthan?" he screamed.
The hooded figure stopped, withdrew a spidery hand from her opposite sleeve, and pointed a long black finger to the bridge, where a ranga woman was weeping and yanking out tufts of her red hair and tossing flowers, gems, and pretty things into the crevasse. Wolf knew her. She was Xanthan's mother.
Roniva pulled her cloak out of Wolf's numbed grip and walked away with spectral steps.
Wolf crawled down the terraces to the bridge. A whining snarl squeezed from his tightening throat as he grasped the ropes, and Xanthan's mother fled in terror.
Wolf dragged himself onto the bridge and clutched at the scattered flowers she had left. He whined at the pretty flowers, tears coursing down his cheeks. "Why? Why? Why?" He collapsed in sobs.
He couldn't see downward. Tears blurred everything in that direction. He tried to look over the edge where Xanthan had gone, where Wolf was supposed to go now. He was expected to jump.
Just as he'd been expected to fail.
In sudden fury he was on his feet. Crushed blossoms fell off the knees of his trousers. He ran from the bridge and away from the Aerie on wounds that felt nothing as the spring snow began to fall.
• • •
Warmth came late to the Aerie. The snows quit in the mountains and true spring came at last, without Xanthan, and without Wolf. It was assumed that Wolf had gone off alone to kill himself in his own way. The season passed in fleeting brilliance.
The valley was still cloaked in summer when autumn crept quickly over the mountain. Seasonal grasses turned red and gold. Desiccated plants crumbled to ash. Winged keys and clouds of tufted seeds drifted and tumbled in the brittle wind. Eaglets flew. The berinxes left the high slopes, and a sharp burned-wood smell hung in the air. Burrowers went into hibernation, and Itiri stocked their caves with food. Soon Shandee would be upon them again.
He appeared over the summit on scarred feet after a cold rain, come like a dark spirit, changed.
He was taller, his chest deeper, the youth fully a man. The leather thong of a fire walker was still tied around his thick neck. He stood above the Aerie.
The berinx padded up the high trail, its thick brindle coat catching burrs, its long tongue hanging from the side of its heavy jaws. It woofed at the rogue cheela.
"Where is thy master, Chaulin?" Wolf said.
The Fendi's familiar couldn't snarl its ritual answer: _Thy master also_.
The rogue cheela had no master. Wolf ought not to be here.
He ought to be dead. The berinx didn't know what to say. It scratched at its mane in distracted confusion, then vanished. The burrs from its fur fell to the ground.
The Fendi came slowly up the path the berinx had taken. The Fendi's glass-green eyes were narrowed to slivers against the drying air. Moisture came hard to old eyes. Withered hands held thick cloaks around his slight, once-powerful body.
Wolf crouched before the monarch in an attitude of humble favor and asked to walk the fire again.
"Never done."
The answer wasn't the Fendi's. It was the Elder Roniva who spoke. She stood behind the Fendi and placed another cloak over his head and shoulders.
"Never done because never tried," the Fendi said, his voice a whisper in the thin air. "Never asked. No one ever lived to try again. It was not because of a law."
"Natural law," Roniva said.
"If natural law can be broken, it is the law, not the violation, that is wrong." The Fendi beckoned his berinx. It appeared at his side out of the air. The old man scratched the beast's ruff. "I must consult the archives," he told his familiar. "Something must govern this matter."
• • •
The Great Chamber was a nine-sided place with a great gray stone dome, tall lancet windows, fluted pilasters of colored granite, and opalescent white alabaster fittings. A fire blazed in the hexagonal firepit set into the floor of honeycombed red tiles. Black smoke sooted the high dome.
For a second attempt to become a warrior-priest, requirements were three. For the first, the cheela must combat a full warrior-priest and win. The volunteer for combat was Sentalla of the fire clan of the topaz twins.
Smell of deadly intent hung close and tense in the wide chamber. Wolf tested his grip on his sword. He crouched on a black hexagonal tile across the Great Chamber from his warrior foe. Wolf's black hair was cut short, out of his eyes and off his ears. His wrists and right forearm were wrapped in leather. The soles of his feet were rosined to keep him from slipping on the stones that had been polished to a glassy sheen.
Wolf had nothing to lose here. He was, to his own mind, already dead. All from now was gain, and he was free. He had only to let himself do what his self knew to do. Anger he had locked away in some deep part of him. An angry warrior defeats himself in combat, Xanthan had told him.
Sentalla had a personal grudge in this battle.
At the command to begin, Wolf swung his sword and sprang across the floor with a screech, but Sentalla had already fled his place on an oblique line and turned to slash at Wolf's flank. Swords met with a dull clash, then strained crossbar to crossbar. Both pushed and sprang back out of sword's reach to circle more warily now, having failed at their first shock tactics.
They stalked, feinted, shifted feet. Sentalla's sword switched hands. Wolf charged in with a yell. Sentalla parried, countered, was blocked, and danced away.
Wolf screamed again, leaped, slashed.
Sentalla sidestepped, placing his blade into Wolf's flying path.
Wolf batted the sword away with his own, turned in the air, and landed facing Sentalla.
Sentalla danced lightly back and circled. He had done really nothing yet, watching. He was nearly ready. He had seen enough. His jade-green eyes took in all of his opponent at once.
His beloved Xanthan had always said that his fire brother used his eyes too much.
Wolf circled behind the blazing firepit and disappeared from Sentalla's view behind its rising curtain of heat.
Sentalla approached the pit cautiously, eyes darting both ways, ready for whichever direction Wolf might reappear. Sentalla circled the firepit silently to his left, but Wolf circled also, staying exactly opposite him across the pillar of heat.
Among the spectators, Roniva bolted to her feet and stood like a quaking statue. She could see that Sentalla didn't know—Xanthan had never told him—that Xanthan's cheela was half blind. _Heat blind_.
The odd thing with this blindness was that it left no veils before Wolf's eyes. The heat barrier over the fire was only opaque to one of them. Wolf couldn't see the heat. He could see _through_ it. Roniva wanted to shriek a warning. It wasn't her battle.
_Sentalla, he can see you!_
The fire crackled and spat up sparks.
Wolf crouched to leap, neither left nor right, but straight over the firepit. Few warriors would risk a blind attack. For Wolf it wasn't blind.
He sprang.
Wolf appeared out of the wall of heat, silent, and suddenly _there_ , blade slicing. He turned at the last instant to smack Sentalla across the midriff with the flat of the sword hard enough to stagger the warrior.
"Stop!" the ancient Fendi cried, and his berinx appeared between the combatants to divide them. "That is a victory."
Sentalla was winded and scored with two long curving red lines across his belly. Had Wolf not turned his blade, he would have sliced Sentalla in two.
Wolf blinked soot from his eyes, slightly dazed, victorious. He didn't move from his place.
Someone came to take the sword that was not yet his from him. Wolf's short fingers uncurled slowly from the grip, reluctant.
"Thy mercy does thee credit," an Elder said, wrapping the sword in felted cloth and taking it away. It was the first thing anyone had deigned to say to him directly since his return.
Wolf lowered his eyes and bowed his head.
His hair was singed, the edges curled up in thin coils that broke off with a touch. A black hand reached over to brush him off. It was Roniva. She sniffed at the burned stench. "Very resourceful, my cheela's cheela. Making a virtue of handicap."
_"Hanina,"_ Wolf acknowledged.
Roniva was dressed all in black bedizened with diamonds. She ran her hand across the shimmering stones. "But where are thine?"
His diamonds, she was asking. His honor. She meant he should have suicided for honor's sake instead of subjecting everyone to this insanity when he should know he could not succeed. Should not have succeeded if he were a true Itiri warrior-priest.
"My honor is that I cannot know defeat," Wolf said.
Roniva clamped her teeth and lips tight. The thought, the possibility was daunting—the existence of a thing that never breaks, that could come crawling out of the lowest gutter and prevail.
The Itiri had never conceived of a mythical phoenix. Such a creature didn't exist even in Itiri dreams. The tasks to reach his goal were impossible.
And Wolf was a third of the way there.
• • •
Wolf piloted an airplane to the primitive side of the planet where rain forests grew thick and ranga hadn't produced an aghara offspring in so long that the idea of tall blond warrior-priests wasn't even in their memory.
For his second trial Wolf had been given a labor. All the Elders had conferred and at last bid him bring the claw of the leopard of Ma-pall-mo.
So Wolf came to the village Ma-pall-mo, the Springs, where the chameleon-skinned ranga were dark most of the day in the bright equatorial sun and white at sundown. They were afraid to venture into the jungle anymore, day or night. They gathered where Wolf came to land, awed by the flying machine and its pilot, a tall man who remained dark in the dark.
They said the gods sent him to kill the leopard.
Silly superstitious ranga.
They told Wolf, "We black our skin with coal in the dark so it does not see us. It smells us.
"We mask ourselves with jungle herbs so it does not smell us. It hears us.
"Now the gods send you dark and odd-smelling and silent as the leopard itself."
"I was sent to kill the leopard," Wolf said in their own tongue. "But not by the gods."
But the ranga insisted the gods had done so, obviously, because he was here.
"I don't even know what the leopard is," Wolf said. "Tell me about the leopard."
"The Claw! Watch out for the Claw!" one cried.
"What kind of claws?" Wolf asked. "How many claws?"
"One claw. Silent it comes," another said. "It comes in the dark with its claw."
"Sometimes you smell it come. You don't hear it until it is upon you, then it paralyzes with its voice and then claws you to death. In a group of hunters maybe one of you will escape and tell the same tale."
"The Claw! The Claw! Look out for the Claw!" one cried, and all the others made him be quiet.
"What does it look like?" Wolf asked.
They looked to each other. None of them could answer.
"Has it ever been seen?"
"No," they all said. "Not by any who lived."
"This is not helping." Wolf's small mouth curled in exasperation, but he couldn't be angry with the ranga. They were like children. They plucked nervously at their red hair, seeing him displeased. There were some very thin-haired villagers in Ma-pall-mo. _I must slay this leopard, or they will all be bald._
"What does it smell like?" Wolf asked.
Someone dashed away at a waddling run and came back bearing a torn rag on the end of a stick so as not to contaminate it with his own musky smell. "It smells like this."
The scent on the rag was very faint. The short hairs rose on the back of Wolf's neck.
He knew the smell.
• • •
The jungle was black as sundown without stars, a verdant hell, dripping heat. Darkness training let Wolf move without sight, silent and loud at intervals. He wanted the leopard to come to him.
He heard the leopard at a distance—not its footfalls nor its passage through the dense spongy vine nets of the rain forest, but its quiet panting in the heat. The leopard couldn't sweat.
Wolf had become still. The leopard fell silent. It could neither see nor hear him but had caught his scent, and Wolf could tell it was puzzled. It had never smelled an Earthman.
Wolf crouched in an acania thicket, ready to move, holding a dagger in his left hand, waiting for the leopard to attack or flee.
Then it came, screaming an Itiri death cry, with a whistling of a double-curved sword that tore into the acania thicket.
Wolf was no longer there.
The leopard hadn't been expecting one of his own. He was not prepared for any skill, any threat to himself. He had been very reckless. Now there was a dagger in his gut, thrusting up through his diaphragm, and he knew too late that this strange-smelling creature was an Itiri, too.
• • •
Wolf buried the warrior's body in the jungle so the villagers couldn't do anything abominable to it, and he took the tungsten-plastic Claw back to the Aerie.
He threw the double-curved sword at the feet of the Elders.
"What would it have cost thee to tell me what I was hunting?"
The Fendi picked up the sword and passed it to Roniva, who bowed her head. "No one told _him_ what was hunting him," she said.
Wolf bristled. "Was that not neat and clever—to have one distort kill another? It matters not which wins. Maybe they will kill each other if you are very lucky. You must have been very proud and pleased with that idea!"
"No joy. No satisfaction. And not lightly decided," Roniva said, holding the flat of Claw's blade to her cheek.
The tender gesture was alarming. Wolf's mouth was suddenly too dry. He tried to swallow. "Who was he?" he demanded in dread.
"No concern of thine."
Another cheela of hers? Or—but that was hardly possible.
Aghara women were almost always sterile. Aghara children were rare enough—one out of one hundred thousand births—but an aghara born of an aghara? Wolf had never heard of such a thing.
Yet Roniva wore an eternity flower at her belt. She had a mate. Sterile women did not mate.
_Someone say I did not kill her child!_ Wolf thought wildly. But no one was answering him. "Thou hoped I would lose," he said.
"I hoped he would not lose," Roniva said with altered emphasis. "Thy losing was merely the other edge of the same sword." She passed Claw back to the Fendi—it was no longer anyone's weapon—and she sadly walked away.
• • •
And so Wolf had come to the last trial. As ever, a walk through fire.
The objection was raised. "There is a full clan this year. He must wait."
A fire clan could be two, three, five, or seven. There were five for this clan of the carnelian serpent.
Then one died. His cave was opened after the twenty days of Shandee had passed, and the cheela was found frozen, nine days gone.
The time of fire was coming soon. A decision needed making. The would-be clan was now one short or one too many, and Wolf was waiting.
"This could be a destined thing," the ancient Fendi said. "Ask of the carnelian serpent if they will take this cheela for their fire brother; otherwise, they must say which of them will stay behind."
The four cheelas said they would take Wolf as their fire brother.
"He could taint your fire," Roniva warned. "He hath already failed. You may yet be a clan of four."
"We will be five," said the girl cheela, at that time known as Mardeia. "Let him walk with us."
• • •
Wolf, at the edge of the firepit, gazed through the wavering heated air and smoke to the hooded figure cloaked in brown who waited for him on the far side with a bare sword. Failure in a second attempt meant death. Roniva would make that happen. She also made certain that Wolf didn't bathe his feet in ice before coming to this place of fire.
The other four cheelas had already walked. Now they looked back for him to come through the fire and join them, the four nameless warrior-priests of the carnelian serpent. They had shed their cheela names in the fire and had not yet been given new ones. They untied the leather thongs from around their necks and waited for Wolf to make them whole.
_Go to them_.
Wolf relinquished conscious control, released everything—
_Let go. Let go. To win, let go._
—and cast his fate to an inner voice. The body is limited, Xanthan had said, the mind hath no limits.
Wolf set one foot into the fire.
The ring of Elders and warrior-priests rustled uncomfortably. This would be painful to watch. They had seen it before. A cheela ought to have the grace to spare them from this pain. This one was putting them through it twice. The Fendi should forbid it.
Wolf stepped with the other foot onto the coals and became lost to their sight, shrouded in heat.
Into the forge. To be burned or created.
Heartbeats paced off creeping moments. Smoke curled to the sky.
Wolf reappeared on the other side as from a mist, his face visible first where the heat dissipated, icily serene. He advanced over the coals and stepped out of the pit, unscathed.
The mountaintop erupted into a clamor of war cries, crashing wind bronzes, rattling daggers, and stamping. All the warriors were on their feet at once, and the ranga echoed from down in Haven with jubilant yells and beating on metal.
The new warrior joined his brothers and sister, and he turned, not smiling, no happiness on his face, not even pride—only vindication and some kind of revenge.
He tore off the old leather circlet from his neck and threw it into the fire.
It was a clean fire, and all the others danced through it.
The brown-cloaked figure stood alone at the edge of the fire court, her eyes flashing, thin nostrils flaring. The sword trembled in her tight grip.
The rest of the mountain was in joyous uproar, the din reaching way down to Kaletani Mai. Only Roniva and one new warrior were silent—and only until white hands tugged on broad yellow-brown shoulders and beckoned him from his dark visions. "Dance, fire brother."
The words, like an enchantment, charmed away some of the black spell that lingered about him, and he danced with the others.
A wayward breeze blew smoke through Roniva's hair spilling from her hood, and she wished she were ranga to believe in protective spirits and to make a warding sign. She wondered what name would be given this being that had just been made from the fire. She was shaking.
_What have we wrought here today?_
# 9. Wolf at the Ramparts
_5851–5856 CE_
JININ-BEN-TAIRRE THEY CALLED HIM, the Feet of the Warrior Are Burned. A point of awe, the name carried a sense of overcoming the insurmountable. To a strong people with a stony will there came a point where they would break instead of springing back. This warrior had no such limit. The name was who he was.
Roniva approved of the reference to his burn scars. But Ben was a rare name, conferred on few—Warrior. There was only one other Ben alive on the mountain. This great honor seemed to her a mistake.
The girl Mardeia became known as Arilla, named for corundum, the virtuestone of strength. The other warrior-priests of the fire clan were given good names also, and the ranga foretold that one of these five would be Fendi one day.
With the glowing red tip of a firebrand they burned the dotted warrior's lines into each other's cheeks and pressed herbal ash into the wounds so that they would heal red. A ranga artisan fashioned their signet rings, serpent intaglios carved into polished carnelians set in platinum. The artisan chattered about Ben-Tairre's "ranga hands," broad and stubby fingered next to the long, tapered aghara hands. No one had ever made such a wide signet band for a warrior-priest.
Ben was given a book of law and philosophy. He vowed never to take a life without just cause, to fight only his own battles and those of his allies, which were these:
His fire kin.
The talassairi, the eagles.
And the mandesairi, the whales.
And he was given his sword. Ben Christened his _Da′iku_.
He shed his drab cheela garb for warrior's finery. Warrior-priests were peacocks except during the days of restraint, the somber days between the wind Shandee and the new year's fire.
He climbed to the top level of Aerieside to stand on the cold ramparts with his fire kin, a pride of young lions, flush with the newness of their arrival. Jinin-Ben-Tairre, once a wolf that had come to the door for warmth, took his place with them as a warrior-priest.
A thin layer of cloud brushed the top of the mountain with gray gauze. It meant the winds were calm.
Jinin-Ben-Tairre looked to his fire kin. Two familiars had appeared while his head was turned, a white binaya for Arilla, and a red snake for Aliathan. Ben muffled his own pain. He had resigned himself a long time ago to the fact that he could not have one, any more than he could ever hope to have heat vision or green eyes. Everything he could control he had won. He had walked the fire. He had a sword, a fire clan, and a name. That would have to do.
Not to be unhappy in his brothers' and sister's joy, he started down the terraced slope of Aerieside.
Suddenly there was a keening _screee_ and a burst of talons and feathers in his face. He ducked down, and _Da′iku_ flashed a cutting sweep through the air.
The bird disappeared at the advancing sword's edge, then blinked back into view as the blade whistled through its completed arc.
A familiar.
Ben-Tairre stood up from his crouch and sheathed his sword. The little brown hawk fluttered around his head and screamed at him.
Whose?
The familiar alighted on his shoulder and squawked in his ear.
Comprehension arrived with a rush of surprise and emotion.
_Mine._
• • •
"What is that supposed to be?"
Roniva blocked Ben's path in the arcade. She appeared severe, hard, and angular as an eagle. Her black hair was pulled back and hanging straight from its topknot. Her whip-thin body was robed in plain dun but for all her bright enameled bracelets, anklets, and toques. Her black eyes narrowed critically at the bird perched on Ben-Tairre's shoulder. "Well?"
"My familiar," Ben answered.
Roniva's expression soured as if to say, _I know that_. "What is it supposed to be?"
"A kestrel, I think," Ben said.
"An _Earth_ creature?" she said as if it were the ultimate insult.
Ben didn't mind the form of the creature. At one time it would have upset him horribly that his familiar should take the form of an Earth creature. Enough now that it was a familiar and it was his. It was an indelible stamp of legitimacy. No one had expected a familiar to appear for him. Now there was no doubt that he truly was a warrior-priest. "It only appears so," he said.
"My Xanthan's familiar had not talons and a hooked beak," Roniva said in reproach.
The kestrel was screeching in Ben's ear. "Knowest thou what it sayeth?" Ben asked.
"Thou wilt need to decipher its noises for thine self," Roniva said. She'd meant to be short with him, then decided to say more. "It is a primitive language. The first sound out of it will be a verb. The mode and tense thou knowest not and it will tell thee not. But I can tell thee it will almost always be a command. These beings are not informative. Thy familiar will spend most of its time giving thee orders."
_Screee_ , said the kestrel.
"Another thing," Roniva said as the bird batted Ben's face with its wing. "They are not known for their patience."
Ben pursed his lower lip over his upper lip and furled his brow for a thoughtful moment. At last he said, "Can I give him back?"
It was the first time he'd expressed anything like humor.
"Certainly," Roniva said, taking her leave of him. "Discover from where he came and thou mayest return him."
That was a no. No one knew where familiars came from.
Ben drew his sword and turned it in the light which streamed in shafts between the pillars of the arcade.
The blade would cut stone. Handed down for countless generations, still it was honed fine.
"Thou art not my familiar," Ben said to the kestrel.
The kestrel squawked.
"Thou belongest to my sword."
The bird turned its head from him and preened its wings, noncommittal.
"I know I'm right," Ben said.
Ben sheathed _Da′iku_. He moved swiftly to the other edge of the arcade. He leaped lightly, barefoot, onto the balustrade. He hugged a jewel-encrusted pillar, and leaned out to gaze down into the chasm cleft into the mountain between Aerie and Haven. Xanthan's grave.
"O Xanthan, one _can_ teach virtue to swords," he whispered.
The kestrel bit his ear.
"Rather, swords will teach," Ben said.
His eyes burned as he stared into the crevasse. He shrugged off a ruby-inlaid armband and hurled it down the void. He swallowed the thickness that rose in his throat, made a fist, and swung impotently at vacant space. "Wherefore is it too late to tell thee anything!" he hissed without voice. His voice failed him entirely. "Couldst thou not wait? Thou didst not know thine cheela."
Tiny claws were digging into his shoulder. A trickle that could have been blood or sweat traced down his back.
Ben dropped into a sorrowful crouch on the stone rail, his forehead on his knees. He pointed down the abyss without looking and told the bird, "Get thee down there. Get thee down and tell him."
Piercing talons released their hold without a rush of wingbeats. Then the kestrel was simply gone, and Ben was alone.
He doubted the kestrel would give any message to Xanthan.
Ben lifted his eyes to peer over his own knees and spoke, muffled. "Would that thou wert alive to forgive me. . . ."
• • •
In the year of the Topaz Triquetra, the Fendi sent Jinin-Ben-Tairre starward in an ancient ship called _Singalai_ to live awhile in the world outside.
"I shall not revert to my blood," Ben vowed, anticipating the reason he was being sent on this quest.
The Fendi patted the head of his shaggy berinx and echoed Ben's claim. "He will not go back to them." The words suddenly sounded overly insistent to Ben's own ears. And the Fendi told the berinx in the next breath, "Then he shall know—for certain."
The Fendi read hearts and minds as most people read the stars and the seasons.
Hard, glass-chip eyes of brilliant green shone eternal fire within the mortal confines of his papery translucent lids. Wispy white hair wreathed his eggshell skull. The Fendi's physical form looked as if one day soon it would be spirited away on a dry autumn wind with the rest of the season's ashes of transitory life.
A figure of awe and wisdom, the Fendi was not one to whom brash declarations should be made. Ben backed down to a humble silence.
The Fendi scratched the brindle ruff behind his berinx's ears. "Let my warrior stay because he is an Itiri, not because he knoweth humankind not."
Ben dropped to one knee and touched his forehead to the floor in final acceptance and farewell. "Yes, Fendi."
• • •
Two years abroad revealed nothing to sway Ben's already fixed mind, and nothing to endear humanity to his estranged soul. He found the human war still raging, just as he'd left it, fought in the name of supremacy of a savage, vulgar race. If any change had occurred, it was a widening of the already enormous gulf between expressed human ideals and actual human deeds. There were good people incapable of reaching lofty goals naively set, and there were evil people who perverted the expressed ideals to their own ends. Bungling and avarice took their toll on everyone humans touched. Good intentions did not an empire make.
Human greed was especially stunning to Ben, coming as he did from a place where the concept of ownership was loose. Physical things were used on Iry. To own a thing was an oddity and a privilege.
And among humankind, Ben lost one of his only possessions. His sword.
Contempt had made him careless. The idea that someone would actually take _Da′iku_ from him was inconceivable. He could not believe it was gone.
The blade was shipped to another planet, sold at a bazaar, and was in the hands of a new owner by the time Ben caught up with it, led by his shrieking kestrel.
The sword's buyer watched in mute astonishment as first a hawk appeared from out of the empty air and scared his prized white Arabian mare, then a young man with almond eyes and scarred feet walked into his garden and seized the reins of the gold-caparisoned horse, and made her stand still. The mare came to a stamping halt, panting under her heavy ornaments in the damp oasis.
The elderly man's bushy eyebrows rose and disappeared under his kaffiyeh as Ben unstrapped the newly purchased sword from the mare's saddle, drew the blade, and cast the gaily tasseled velvet sheath to the ground. The warrior turned the double-curved blade in the light before his exotic eyes, then whistled it through a helix in the air. Then he walked away with it.
The old man did not interfere. His leathery skin pinched in frowning crevices around a beaked nose and deep owl eyes. He picked up the discarded red velvet sheath, watched the warrior retreat barefoot through the white pillars of his paradise, and he breathed a baffled oath: "Al _lah?_ "
• • •
That night, while he slept under the desert stars, a pinprick invaded Ben-Tairre's dreams. His mind sludged upward to thick consciousness, never clearing to alertness. His limbs moved as through water, and he knew he had been drugged.
He fought it—in the wild thrashing way of panic, not the calm methodical way of the Itiri. He hadn't the patience to learn that kind of control, so his fight was the charging of a wounded animal, without the concentrated power of thought to break down the evil inside him. He could only rage against it. He thought his simple will could force anything.
For a second time, it was not so.
He tried to turn back and learn—learn now—a systematic defense against the foreign substance in his veins. Too late.
Feebly he pushed at the ropes on his arm, at the heavy feet that stood on his other wrist and on his ankles, until he was pinned down on the sand and couldn't resist anymore. The drug sapped even his desire to fight. It was so easy just to rest and watch them, as if it were all happening to someone else's body and he was only an observer.
His right arm was held out straight and strapped down to a wooden board rough with splinters so that he couldn't move but to curl his fingers and make a fist around the biting woven hemp bond pulled tight across his palm.
A meat cleaver appeared before his stupidly curious and uncomprehending eyes—a heavy square blade with a keen edge and a black handle. The steel flashed in the lantern light, a harsh cutting glare—unlike the lustrous reflection of tungsten-plastic.
A great bear of a man with a woolly beard wet his fingers on his thick red lips, wiped them on his long striped coat of green, black, and gold, and gripped the cleaver's stout handle in both his hands. He stepped up to stand over Ben, frowned down at him, and muttered a word. "Thief."
Ben followed the focus of the man's bloodshot brown eyes to his own wrist, and meaning penetrated his drugged thoughts. Fear returned. He yanked at his bonds with a sudden twist of his whole body. Wood and rope groaned. He shifted his pinned weight to try again, but a blue-smocked man sat on his chest.
The man with the woolly beard put one foot forward, swung the blade back and up, and with a grunt and a heave brought it chopping down through Ben's wrist and buried the blade in the wood.
Then men cut his bonds, took his sword, and left him there spurting blood into the sand.
• • •
The kestrel brought a physician, who followed the bird as if it were a messenger from God to where Ben lay in shock, holding his veins and arteries shut, and waiting out the drug's effects. He couldn't do more.
The physician sealed the wound, said his bone was not shattered and that he was lucky. The physician had no blood to give the warrior, but said Ben was strong and thought he would recover— _inshallah_ —God willing.
Having made it this far, Ben knew he wouldn't die. When he regained strength, he walked back to the oasis and retrieved his sword once again. He cut off no hands, though he was bitterly tempted. He returned to his ship, _Singalai_ , and went home.
The ranga fashioned a new appendage for him of jointed metal, padded and gloved to look like a hand.
The kestrel hated it and would not alight on it.
"Where wert thou when they were firing drug darts at me?" Ben said to its angry cries.
The new hand lacked dexterity and feeling. It was good for clutching and holding fast and for striking with great force. Ben adapted. The hand was useful.
But in the darkness, its faintly audible clicks and whirrs, ignored in the daylight, became magnified, a loud and desolate song of loss, like the shifting sands of the desert in the quiet of deep night.
• • •
The year of the Opal Sword began a new hexadecade. It was the year the child Amerika came to stay on the Aerie.
It was also the winter the Fendi died.
Ben withdrew to a lonely ledge in the gray twilight to be by himself. There was no order in the world. Ranga were wailing in their caves.
It was a cold day when Ben-Tairre wore boots, a day when breath froze in the lungs and eyes would coat over with ice. All was bleak.
He could not find the sun.
He tried to bring to mind hard green eyes like glass, and the voice of a gentle monarch who had allowed a Wolf to become a warrior-priest.
The Fendi's dying words had been: "Not to make Ben Fendi."
Jinin-Ben-Tairre thought it an incredibly odd thing to say.
There were two Bens on the Aerie. The Fendi must have been referring to the other Ben. Jinin-Ben-Tairre didn't think anyone would even consider making him leader. The Fendi didn't need to make a dying injunction against it. It was like telling berinxes not to fly. So why say it?
"Knowest thou not?" his fire kin asked him. "Because thou hast defeated defeat. Something none of us has ever done. And because thou art not a follower. And because no one is sure what to do with thee."
He hadn't known there were so many on the Aerie who would call his name. The Fendi had known. Jinin-Ben-Tairre agreed it would not be a wise choice.
As it happened, Roniva was chosen.
A small breeze cut like bitter knives. Ben covered his face with the edge of his brown mantle. The soft fabric felt rough and hard in the dry cold.
A hooded figure wended the rocky approach to his isolated ledge with careful deliberate steps on brown-slippered feet. A thin figure set in regal resolve, her back erect but fluid, she was known to Ben even without a glimpse of her jet-black skin.
Ben saluted as she arrived on the ledge, his sword fist over his gloved false hand. "Fendi."
Roniva lifted her hood from her face and her glossy black hair. "Not yet," she said. Breath froze to her lips. A white haze of drying skin chalked her high cheeks. "This is the last time I may speak to thee."
Wind picked up the edges of their mourning cloaks, billowed and snapped them like wind flags.
"What sayest thou, hanina?" Ben asked, his brown-black eyes barely open within epicanthic folds.
"Thou murdered my Xanthan," Roniva said.
"I know."
"Thou art here and my son is not. A poor substitute."
Her son. She had never said as much.
"Thy son was distort," Ben said softly.
"So art thou. Shandee drives thee. Thou art nothing without thine anger. This is my Aerie."
Wintry gusts pierced through fabric like sharpened blades. "Is this setkaza?" Ben asked. He hadn't expected to die today.
"No," Roniva said. "I shall not begin my reign with a bloodletting. I shall try, Wolf. If I cannot live with thee, I will have no choice. And have thee no doubt who will win."
"No, hanina," Ben said.
• • •
They held the peace for less than a year, until the coming of an Earthman in the year of the Opal Ship, the man who came to them under the name Alihahd.
Suddenly setkaza was here, the intent to kill, and Ben lived in perpetual readiness for his fatal combat. It could come at any time. Or he could make it happen, choose the time and place to his own advantage and readiness.
But it was said that he who moved first was lost. Setkaza was a waiting ordeal, a trial of nerves. The one who endured the pressure of death threat day after night after day, had always been victorious. Setkaza could go on for years. It could snap tomorrow.
Ben-Tairre knelt on the floor of the training hall. He touched his forefinger to the groove in the wood where Roniva's sword had stabbed, locking their fate.
Turns of life were an unaccountable whirl. Caprice and sudden death and tragedy, fortune, chance, and destiny made vanity of expectation. There was but one certainty now. The death of a warrior.
# PART THREE:
## Ashar Ari
# 10. The Gathering Storm
IT WAS STILL THE FIRST WATCH, the hour of the bells. The air was sharp. The sky was clear and the stars shone in full array. Amerika stopped on the Ledge Path and gazed to the east. She sighed, "The Wellspring is rising already. Oh, Harry, I am not ready for winter."
The Wellspring was a cold-weather sign. The constellation was at its zenith at midnight in the dead of winter. Amerika didn't like to see it threaten on the horizon.
Amerika had picked a bunch of the season's last gay cerulean flowers and tucked them into the bandelette tied beneath her small breasts. She took Harrison Hall's arm and pointed up to show him a blue-white open cluster of stars which the Itiri had named for the flowers.
Iry had no moons, so no months. Instead, the year was divided into sixteen signs, each named for a constellation in Iry's zodiac. The sentinels would announce the beginning of a new sign at whatever hour of day it occurred—the time varied, for the number of days in the year didn't divide evenly by sixteen. There were 171.8 long days in the Iry year. Vaslav, who had a wrist chronometer that kept Earth standard time, said there were 30.97 Earth standard hours in an Iry day. It still made a short year, too short for summer to become actually warm, and, with any luck, too short for winter to grow killingly cold.
Just now it was autumn. The day was the eighth of the Beacon, the sign of an eclipsing binary. Only the constellations on the ecliptic were named. They were the only stars of any use to the Itiri, other than the pole star, but from the Aerie the pole star was always hidden behind the towering mountain Guardian.
The names of the constellations weren't as imaginative as Earth's constellations. Iry's were mostly literal, and, once upon a time, the stellar configurations looked like their names—Hexagon, Crown, Cross—but that had been ages ago. Since their ancient naming, the signs had slipped along the ecliptic and the stars drifted in different directions, so that no sign looked like what it was supposed to be. The Red Birds used to form a spearhead and didn't now. But then no real bird on the planet Iry actually flew in a V anyway, so it mattered little that the red stars formed a disorderly flock.
Harrison Hall sat on the balustrade of the arcade, his back against a stone column. He gnawed on the mouthpiece of his empty pipe. Firelights were fluttering to life inside the ranga caves, warming the doorways. Voices carried more clearly in the night air. Music of two woodwinds in rippling cadences chased each other from two caves on Aerieside. Hall tried to hum along but couldn't predict the alien patterns.
He caught Amerika gazing at the valley. Alihahd had been down there all summer.
Rumors and reports made their way up the mountain upon occasion with messengers and deliveries of food from the valley. The rumors were not of Alihahd, but of Alan James. No one in Kaletani Mai had ever met an Alihahd. They only knew a tall, white, blond man with blue eyes and a big voice. His name was Alan James.
So the captain had assumed another alias, thought Hall. Well, Alihahd wasn't really Alihahd either.
By midsummer, the stories that wended up to the mountaintop were of "the Earth Fendi." Hall roared with laughter. Earth Fendi, indeed. If Alihahd stayed down there much longer, the villagers would have him deified.
_Very good, Captain. Very good_.
Amerika hung on every word spoken about the man, Alan James. She would grab every courier, every eagle, every traveler who came from Kaletani Mai, and squeeze every tale out of him. Her face was wistful now as she leaned over, her forearms resting flat along the balustrade, and she gazed at the lights of the valley where _he_ was. The little girl was missing him.
Then her black eyes shifted to a closer focus, and she pointed down. "Harry, look."
Harrison Hall took his pipe from his mouth and squinted into the dark.
Starlight caught on a movement in the blackness, a figure coming up the steep snake path from the valley to the Aerie. Hall could see no details, just a long-limbed form climbing up with a dogged way of moving.
Only warrior-priests climbed the sheer torturous snake path. But this was no warrior. Hall knew who this was.
Yet it was with a sense of disbelief that he beheld the lank, heavy-boned man appear over the top of the precipice to stand on the Aerieside Ledge Path. Harrison Hall thought of something he'd been told about legends—they always return.
Alihahd strode the path with a relaxed gait, tired but without exhaustion. He reached the narrow footbridge and began his crossing to Haven.
He was halfway when Jinin-Ben-Tairre appeared from shadows and stepped onto the bridge from Havenside to cross in the opposite direction. Their gazes met, and they both stopped.
The bridge swayed.
Ben gripped the knotted ropes. Muscles moved in his powerful shoulders.
His small mouth drew down in a sharp tight frown that wrinkled his chin. His eyes slanted even more steeply with the threatening cant of his brows. He took another step on the bridge, his intent clear without words. This Earthling would have to make way for him.
But a deep voice came from the bridge, very tired, short of temper, and unimpressed. "Move back. I shan't jump." And Alihahd looked away to the side as if annoyed and impatient. He knew what was rightfully his and was not about to be intimidated out of it.
Ben-Tairre blinked, slow to believe Alihahd could actually mean it. But Alihahd was not going anywhere—least of all backward—and slowly Ben-Tairre stepped down.
Alihahd completed his crossing. He walked past Ben without looking at him. It wasn't a gesture of avoidance. It was non-acknowledgment. Ben might as well have been a bush for all Alihahd cared.
Amerika gave a joyful little shriek and ran along Havenside's Ledge Path to meet him. Harrison Hall followed her at a casual walk. Hall stopped a few paces away, his weight on one foot, one hand in his redingote pocket, the other holding his pipe. "Captain."
"Mr. Hall," Alihahd said in his elegant voice. He hadn't changed. Not in that respect.
Hall had to smile. "How did you get up here?"
Alihahd's glance shifted sideways toward Ben-Tairre, and he answered loudly, "I walked."
Amerika clapped her brown, baby-fat hands with a gleeful giggle.
Hall gave a small nod. "Bravo," he said softly.
Jinin-Ben-Tairre turned away from the bridge as if he had never wanted to cross it in the first place, and he mounted the stepped path to the higher terraces instead.
When Ben was gone, Alihahd let some starch from his proud stance, turned his head to Hall, and said, "Have you any conception of how far three miles straight up is?"
Hall chuckled. "Do you know what you just did to him?" He cocked his head in the direction in which Ben had retreated.
Alihahd did. Ben's first climb of the snake path was a village legend. "Purely intentional," Alihahd said.
"Are you going into warrior training next?" Hall asked lightly.
"Are you mad?" Alihahd returned. He pictured himself, a forty-eight-year-old cheela. No, he hadn't climbed the snake path in answer to any calling. He'd climbed it to slap Ben's alien-posing face.
"You were saying, you don't lose well," Hall said, picking up the end of their last conversation as if Alihahd had never left and there had been no long summer intervening.
Alihahd nodded and spoke low, his voice tinged somehow with regret. "I don't lose."
He retired to the cave he'd originally inhabited upon coming to Aerie. His pile of blankets and furs were still folded on the wide mattress waiting for him. Vaslav's were in a rumpled pile against the wall. Hall's were not there. Harrison Hall didn't sleep here anymore. Alihahd wasn't going to ask.
Alihahd collapsed on the mattress without bothering to take off his boots. The boots were the same red leather ones he'd brought with him, but the thick, flat white deck soles had been replaced. He stretched his arms above his head, gazed at the ceiling, and exhaled long, at ease in his reclaimed territory.
Hall stirred the fire that had been dying in the hearth. Warm light caught the red sunstreaks in his coppery hair and flashed gold from his two earrings. His aquiline profile was outlined in orange-red. He rose from his crouch at the hearth and sat next to Alihahd.
Alihahd was little changed from when he left, still gaunt, spare, but healthier, hardier, and his short-shorn blond hair had grown out to cover his tall brow, the tops of his ears, and the back of his neck. His skin was weathered but still fair. It couldn't hold the sun's color like Hall's. Hall was by now very dark.
Music carried from outside, across the abyss. The woodwinds sang a new tune, sad, passionate, longing—the song of the abandoned or of a captive spirit wanting wings. It disturbed, made its hearers restless. The song made one look toward the stars.
"Do you mind explaining to me, Captain, how you got to be 'the Earth Fendi'?" Hall said.
"It was none of my doing, and all actually comical," Alihahd said. "The ranga started it. They assumed because I was tall and blond and lived alone that I must be the Earthling equivalent of a warrior-priest. It escalated from there. I was a Fendi before I knew it."
Hall nodded. But Hall didn't believe it was the ranga's doing. Wherever Alihahd went, no matter how many times displaced, Alihahd would always lead. He was set apart by the way he talked, the easy grandeur of his carriage, and his lordly silences.
"I have something for you," Alihahd said. He unlaced a pouch from his belt. It was filled with dried leaves. "I am told these can be smoked."
Hall received the pouch as if it were treasure. "I thank you, Captain."
"Do not thank me. If I cannot breathe up here, neither shall you. I give you hypoxia, and may your night vision go to hell."
"Merrily," Hall said, laughing, a deep sound in his barrel chest. His lung capacity was enormous, and never mind that smoke added an equivalent of four thousand feet to the altitude, Harrison Hall would endure.
He withdrew his fox-head pipe from his vest. Both its fire-opal eyes were cracked now, and Alihahd commented on it.
"Serra," Hall said. "She shied it at me."
"She does not like you," Alihahd said.
"She likes me fine."
Alihahd had already surmised that Hall was living with Serra. Hall's move was entirely expected. Eridanin lords were not known for their celibacy. It was only natural that Hall would be sleeping with someone. Just as it was natural that Alihahd be alone.
Alihahd pursed his lips, chagrined on the rare occasions when it occurred to him to think about his isolation. He'd always been a failure at human emotion. The ranga reminded him of it. Ranga hugged each other at any excuse—something he could never do. Spontaneity for him was not there. He'd tried once, inspired by a child laughing. He beckoned. The child frowned and toddled away. Fitting, as Alihahd had a face to frighten children. He didn't try again. It was his way to be aloof. He usually liked his distance. At the same time, he was envious of the ranga, their ability to be close. Alihahd had never been at ease when tenderness was expected of him, around pets, children, or women except his wife. His had been an arranged marriage. He would never have married if he'd had to go courting for himself.
Now he was afraid to touch anyone.
He looked at his hands.
Harrison Hall picked up one of Alihahd's hands and laid it across his own open palm. The back of Alihahd's big knotty hand was bruised purple, and the skin was split across the knuckles. "Rough climb," Alihahd said and took his hand back, too quickly.
Hall's lips twitched beneath his mustache. He withheld comment.
The palms of Alihahd's hands were unscathed from his climb.
"I have something else for you," Alihahd said. He gave Hall a small felt bag. Something within clicked together like pebbles.
Hall emptied the glittering contents onto the bed.
"Surely that's a zircon," Hall said, pointing to one of the eight big gemstones.
"Which? No. The pink one is a zircon. The boulder is a diamond."
Hall picked up the diamond and held it to the firelight. "What do you think? Five carats?"
Alihahd shrugged.
"You're worse than an Itiri," Hall said to Alihahd's total lack of avarice or even interest. Alihahd had handed the gems over as if they were pretty quartzes and cut glass.
Alihahd rolled onto his side and propped himself up on his elbow. "The ranga decided I needed those, since I was something like a warrior-priest. Those are virtues."
"Is that what they are?" Hall said dryly.
"No warrior should be without them. The beryl is your hospitality, kindness, and mercy—your 'humanity' if you will," Alihahd said. "The diamond is honor, respect, and loyalty—there is a single Itiri word for all those, and a single stone."
"A single very large stone," Hall said.
"Turquoise is patience and serenity. Lapis is courage. Topaz is wisdom. Zircon is humility and modesty. Opal is integrity in the sense of soundness, wholeness, and the like. The ruby is self-control, or fulfillment of potential, or inner strength. The way they consider certain concepts as synonymous I find quite bizarre. At any rate, there you have them. Please take your virtues and let me go to sleep."
Hall scooped the gems off the bed in his fist. He held them each up to the light one more time before putting them in his pocket. He decided he might keep the diamond. The rest he would hock as soon as he was off-planet.
After all, what use were serenity and modesty to a man who piloted a black ship?
• • •
Harrison White Fox Hall and the Fendi Roniva crossed swords. The snowy owl observed from a high perch in the training hall, slowly rotating its round head to odd angles as if on a ball socket, so that sometimes its gaze was nearly upside down. Roniva never spoke a word to Hall. At times she would stop, point at Hall, and bark to her owl, "Again!"
That was bidding for Hall to repeat a technique. Hall wasn't unfamiliar with swords, but the double curve was unusual—like a yataghan, which Hall never had used. Hall preferred guns and short daggers—though he would have sold his soul for a blade of tungsten-plastic. The technology that had made the nearly indestructible tungsten-plastic swords no longer existed anywhere in the galaxy. Where the Itiri had come by their swords was a mystery.
Roniva demonstrated a series of passes and bade Hall repeat it. The dark woman's appearance was still neat, her hair smooth and tied up, her skin dry. The Itiri did not sweat. Hall was shirtless and his skin was gleaming wet. The bandana tied across his brow was drenched, and his hair coiled into dark, wiry curls stuck to the back of his neck. He repeated the series.
Alihahd stopped in the doorway to watch. He leaned the front of his shoulder on a column, his hands tucked under his belt, his head tilted to one side. Soft clean hair kicked over his tall forehead. There was color in his usually sallow and drawn cheeks. The mountain was not killing him this time around.
Hall omitted the last move of the series, which was to flip the blade under his arm as he turned its edge away from him and hide it behind his back. Roniva snapped to the owl, "Tell him to finish."
"Fendi, if I do that I'll cut off my arm," Hall said. It was a tricky move to do the first time with a live blade.
Roniva let it pass and continued to something else.
Alihahd questioned the wisdom of teaching Hall anything. Someone who christened his ship _Nemo_ did not need to know anything more about fighting. Alihahd didn't advocate the training of terrorists. By the time Hall left Iry, the Marauder would be twice as deadly as when he'd come. How could the professedly neutral Itiri possibly justify turning such a being loose on the galaxy?
Or did the Itiri ever intend to turn any of them loose? The thought was a recurring paranoia.
He waited until Roniva wasn't wielding a sword to speak. "Fendi, you realize, do you not, you could be aiding and abetting an interstellar terrorist?"
Roniva seemed to register the data with a momentary wrinkle of her hairless brows. She gave no comment. She took her leave without a word. The owl sailed out behind her in a sudden silent glide.
Hall picked up his shirt and slung it over his shoulder. He sauntered to Alihahd. "There's a burrower hunt today. You're not going?"
That was sarcasm. Alihahd didn't answer it. " _You_ are not going?" he asked. Hall never passed up a chance to hunt.
Hall shook his head. "Layla is. I don't trust Layla behind me with a club."
"I see." Alihahd's gaze dropped to the Itiri sword in Hall's grip. "You are getting to be a very dangerous man, Mr. Hall."
"I always was a dangerous man." Hall mopped the sweat from his body with his shirt, and took off his sopping bandana. "You, dear Captain, could be immeasurably dangerous if I could ever pull you from this Itiri-like pacifism of yours."
_Itiri-like? I?_ That was a curious concept. "I'll not be pulled," Alihahd said.
"Wherefore?" Hall said, imitating Alihahd's high-style, sometimes archaic speech.
"Because I could be immeasurably dangerous."
Were he ever to anger, Hall was certain Alihahd would be lethal. But his flashes of glorious rage never lasted, quickly fading to contempt, indifference, or plain weariness.
"Pity you won't budge," Hall said. "I'd have you with me."
"If it's all the same, Mr. Hall, I would not trust you behind me with a club," Alihahd said.
Hall laughed aloud. "Point taken."
Alihahd turned to go.
Jinin-Ben-Tairre was standing in the exit with a drawn sword.
Alihahd felt a ripple of surprise, and . . . disappointment. Alihahd wasn't much opposed to dying, but he would've liked some advance warning. This was too unexpected, like a bolt from the clear summer sky, just when he was feeling safe. It wasn't fair. The feeling of helplessness in the face of the caprice of the universe upset him more than the actual prospect of dying. _So here it is_. He waited.
Jinin-Ben-Tairre turned the sword hilt outward and tossed it to Alihahd.
Alihahd caught it by the hilt.
Hall raised both eyebrows and drew in his chin like a comment. Alihahd had held a sword before.
Ben's smooth face remained an unchanging mask. Apparently he'd lost the ability to be surprised by anything Alihahd did.
Ben moved like a pacing cat to Harrison Hall and thrust out a broad demanding palm to the sword which Hall held.
Hall surrendered the blade to the warrior-priest and stepped back to the periphery of the practice floor as Ben began to circle.
Alihahd turned slowly, keeping his face and his sword toward Ben. He made no circle himself, only pivoted guardedly.
Suddenly Ben charged in with a stomp of his foot and a shout, the sword whistling over his head and down. Tungsten-plastic blades met with a dull crack. Alihahd deflected the blow and dodged to the side.
Ben spun full around with another attack from the side, and Alihahd had only time to shift his sword to a vertical block, catching the blow straight on, and he staggered back with the force of it.
_Would he kill me?_
Ben's strokes felt real and unrestrained. But Alihahd sensed that the warrior-priest could kill him anytime he wished. This was nothing but cat-and-mouse here.
Alihahd didn't counter either attack, or even attempt to counter. Time after time, Alihahd's moves remained strictly defensive. He had no desire to attack Ben. This was Ben's game, and Alihahd didn't want to play.
Then Ben backed away to a weapons chest. He put down his sword and brought out a dagger. Alihahd waited, uncertain.
"Captain." Hall's voice came from behind him, and Hall pressed the hilt of his own poniard into Alihahd's hand. Alihahd passed back the sword in return and squared off with Ben again to continue the match with new weapons.
_Travesty of a match_ , thought Alihahd. Ben was trained to be a warrior of the galaxy's legendary killing force. To what purpose was this shadow show being staged?
Ben came at him with an underhand thrust. Alihahd dropped his own weapon and blocked downward on Ben's wrist with crossed hands as he jumped back, curving his lower body away from the blade. He seized Ben's thick wrist and twisted it—like trying to throw a tree off balance—then abruptly shifted balance with a foot sweep. Ben rolled and came back up, still holding his dagger. Alihahd's weapon was on the floor. Alihahd skidded it back to Hall with his foot. The blade was worse than useless to him. One did not use a knife to block a knife, and blocking was all Alihahd wanted to do. In this situation he was better off open-handed.
So Ben changed weapons again. Then again.
Alihahd had broken into a sweat all over his body. It became harder and harder not to attack. They were dueling with sabers now. Ben slashed fiercely, hit hard, leaving himself open for a counter, baiting. Blows rained on Alihahd's blocking blade. They jarred Alihahd's wrists and arms. Crashing metal pounded at his eardrums. Ringing steel kept at him and at him and at him like a goad. He could see the demand in Ben's fierce dark eyes. _Fight me, fight me, fight me!_
Alihahd wasn't even thinking anymore. He moved without conscious deliberation. In another part of his mind, above, serene and removed from all this, he heard a mocking chant:
_Circle, circle, dance of pain_
_One is free. He remains_
_Who cannot see our circle's edge_
_Another circle, once again._
A crazy look glazed his eyes, blue with white all around. A hunted way of standing made him the image of a wounded animal, cornered, about to turn and bite.
The spectator, Harrison Hall, straightened from his casual stance. He wondered if Ben knew what he was doing. He was going to make Alihahd turn on him, calling him a fraud, challenging how deep the pacifism truly ran.
Alihahd was angry—or was a better word _mad_? His thick lips drew back from his teeth, and he watched Ben from the tops of his wild eyes. The saber trembled in his too-tight grip. Ben pressed until Alihahd surely must crack.
Alihahd did not.
And at length, when Ben-Tairre himself was sweating and drawing ragged breaths, Ben stepped back, put up his weapon, and left the training hall with an air of curiosity satisfied.
With a shudder and a sighing moan that was half a cry, Alihahd dropped his saber as if it were a loathsome thing. He stared at his own hand.
Only when his breaths began to lengthen did normality return to his face. And all at once his strained muscles untensed, and he looked simply tired.
Harrison Hall picked up the discarded saber, tested its balance in his hand. He gave a nod to Alihahd's proficiency with it and the other weapons. "You're better than you look, Captain," Hall said.
"I was not born this old," Alihahd said. He dropped to a squat, his back to the wall, his forearms on his knees, hands hanging loose.
"Careful what you call old," Hall said. "I believe I am your senior."
Alihahd lifted a hand at the wrist and pointed toward the door. "That man does not like me."
"Oh, I don't know," Hall said. It was hard to tell what Ben was thinking. Hall was only glad that Ben wasn't interested in _him_. "I wouldn't worry anyway," he said. "Roniva is in your camp, and between the two of them, my money is on the Fendi."
Alihahd gave a shrug. "Hypothetical situation."
"I hear not," Hall said. He explained what he knew of the ancient rite of setkaza. Sooner or later Roniva and Ben would fight to the death.
Alihahd shook his head and gave a laugh that was helplessness rather than mirth. "Aliens," he said. He stood. Unfolding his long legs was painful. He took the saber from Hall and tossed it back into the weapons chest in distaste. "I used to be good with these toys of destruction." He didn't like them anymore, the swords and bola and archaic things no one really used anymore but to play at war. He slammed the chest shut and turned away from it. "Too much blood under the bridge between then and now."
"Whose blood?" Hall asked.
"Oh," Alihahd said. "I was speaking in general."
• • •
Harrison Hall watched with a hunter's eye. Alihahd was a complex individual. The more he revealed the less he was known. He never spoke of his home: whether he had disowned it, was ashamed of it, or was protecting it. His accent and his manner and his knowledge revealed that he'd been just about everywhere. But his origin was a mystery, until one day Layla, who never had heard of tact or even of civilized rules of conduct simply asked him point-blank, "Where are you from?"
Alihahd hesitated. The frankness of the inquiry precluded evasion, especially in front of Hall. Not to be too obviously secretive, he answered the question. "Earth."
The reply caused a stir among the humans. Although called Earthlings, none of them—Layla, Serra, Amerika, Vaslav, or Hall—was actually from the cradle of humankind, Earth. None of them had even seen it, and they all had questions about it.
Alihahd didn't want to talk about it. "I hardly knew the place myself. The whole world is a perpetual Na′id battlefield. I left shortly after I was born."
When asked where he went after that, he said, "Many places," and changed the subject.
He was certainly aristocracy. His lofty, occasionally tortured way of speaking wasn't an affectation. Alihahd fell into it without thinking. There was something so unassuming in his elitism. And he had a voice that made those god-awful speeches of his sound right.
He didn't possess Harrison White Fox Hall's cocksureness. Rather, he had the bearing of a man fallen from a great height, a vestige of pride that persisted like an afterimage once the reality was gone. Instinct for command also persisted, like breathing.
Harrison Hall was a lord, but he wasn't a leader. Men and women did not die for Harrison Hall. He inspired no trust, no desire to march into hell for him or to breathe one's last gasp at his feet.
Alihahd could command that kind of loyalty if he cared to—and even if he did not. And it seemed he did not want it, not from anyone. He didn't want the attention. He didn't want to be followed, watched, or known.
It was obvious he had a past—the kind one does not talk about. But the more secretive he was about it, the more apparent it became that he was not simply hiding from the past. He was running from it.
He stood a paragon, so calm, mature, level, patient, and wise. As with everything that had stood too long, his surface was wearing, so one could glimpse through the cracks and find something else.
He'd developed a series of quirky little mannerisms since his return to the mountain—unobtrusive patterns and rituals that appeared at first glance to be simple habits—a way of ordering objects on his nightstand, of layering his blankets in a certain order and going out of his way to maintain that order. He would clean his teeth with salt twice upon rising, his motions identical both times. He would take twelve steps to cross the bridge, no more, no less, starting always with his right foot. This was more than simply being methodical.
None of it went unnoticed. The weakening of a fleet animal did not escape the watchful eye of the tiger in the brush. Patterns were a snare. The greater the attempts to create order outside, the greater the chaos inside. Something was drastically wrong. Alihahd was trying to lock down the corners of an unpredictable universe that kept curling up at the edges.
But Alihahd wasn't absolutely patterned. There was a wayward independence and disorder in his time schedule. He had no schedule. None at all. Hall never knew when Alihahd would be anywhere—when he would eat, when he would sleep, when he would rise—none of which activity had anything to do with daylight hours and was never the same twice. The lack of schedule just did not fit the other obsessive patterns, and the waiting tiger was baffled—and wondered if he was being baited.
Hall bided his time.
• • •
The red flags were out. The winds were fast and no one on the mountain ventured beyond the sheltered oasis of Aerie-Haven. No one climbed the peaks, for _ele-ala_ , the wave, would sweep them away. The eagles rode the undulating currents, soaring over the valley without a single wingbeat. A swirled puff of cloud poised over the village, caught in a rotor. The eagles avoided it.
Harrison Hall took refuge in a small cave on the third level of Haven, where he stayed when Serra was bleeding and wanted to be alone.
It was Alihahd's custom to retreat here at erratic times. Hall never knew when. On occasion, as today, their visits coincided.
Hall had a fire burning brightly in the hearth when Alihahd came in. The cave was a small place, its decoration begun a long time ago and never finished. Crumbling stucco on the unornamented wall came off in dust at a touch. The worn design on the other wall was also faded, painted quatrefoils within squares, carved in relief and intaglio on alternate squares.
Hall was in his shirtsleeves, sitting cross-legged on the bed, tinkering with a gadget retrieved from his ship, _Nemo_. The mandesairi, the whales, had found wreckage of his broken ship on the ocean floor and had sent pieces back to Hall on the mountain with the eagles.
"Is it a transceiver?" Alihahd said, hopeful.
Hall shook his head and tossed the black box aside. "Just a gadget." He picked up his pipe and, after a few tries, lit it. He puffed clouds of gray smoke. "I take it there aren't any in the valley."
"Transceivers?" Alihahd said. "None that have the range to contact anyone in civilized spaceways—not within this century at any rate."
Hall waved some of the smoke away. "They really don't seem to give a damn about the Outside."
Now and then the Itiri sent scouts abroad to pick up pieces of someone else's technology, a ship, a weapon, and maybe rescue an alien in peril along the way, then return home, recluse again.
"I do see the purpose in hiding their existence from the Na′id," Alihahd said. "The notorious warrior-priests of Iry are sadly unprepared to fight a modern war. Yet they continue to stagnate. There is no effort to build for themselves. They aren't ignorant, exactly, but there is a decided lack of ambition and want of foresight here. Their folly if they think they can hide forever."
Hall was chuckling, gently, mockingly, fondly, his laughter indulgent as at a clever child. "What?" Alihahd said.
"The way you talk."
"So glad I keep you entertained."
"You know, of course, you're wrong."
"I know this?"
"The Itiri could hide forever," Hall said.
Alihahd had to nod. They were truly lost out here. Na′id ships never came this way, never would. Unless something were to draw them.
Hall set aside his pipe, the fire of which had gone out, and he picked up the piece of equipment from his ship again. "Have you noticed they count in hex?" he said.
"Hexadecimal?"
Hall pulled out a lens from the black box. "Yes."
Alihahd paused in thought. "I suppose I had noticed something odd." He'd never heard an Itiri count past nine in Universal.
"Vaslav figured it out," Hall said. He pulled out another lens.
"What kind of people use hex?" Alihahd said.
"People with sixteen fingers."
The Itiri had ten fingers. Logically they should have developed a base-ten number system.
"Computers," Hall continued his list of answers. "People who use binary."
"Which is to say high tech," Alihahd said. "Which the Itiri are not. They are an odd people." He stopped talking. He'd caught himself, not for the first time, calling the Itiri people. He brooded in silence for a long time while Hall repositioned his lenses. Alihahd looked over his shoulder. "Is that anything that will help us escape from this planet?" Alihahd asked without much hope.
Hall looked at Alihahd. He'd already told him it was not a transceiver. "No." Hall looked into Alihahd's eyes, lifted his hand, and touched Alihahd's cheek with a hot, dry touch. "What an incredible color for eyes."
Blue eyes dropped away from Hall's gaze. Alihahd couldn't meet his gaze anymore. He left the cave with the restlessness of a caged beast.
Hall balanced his projector on his knee and slouched back against the wall, thoughtful. He hitched his thumbs in the pockets of his gold-weave vest. Hall wanted to leave Iry, too. He had a battle to fight. And if his need was not as desperate, at least his motive was clear, which was more than he could say for Alihahd.
There was no good reason why Alihahd should be anxious to leave Iry and rejoin the crusade against the Na′id Empire. His heart was obviously not in it. He seemed weary of the whole thing. Yet he was obsessed with return.
Hall's reason was simple: vengeance. But what made Alihahd go? It wasn't revenge. Alihahd espoused no specific ideology. He had no real cause, except perhaps freedom of choice. But what did that mean to him? He had nothing personal against the Na′id. He did not hate them. He opposed them but not for himself. Or maybe his selflessness was very much for himself.
Hall sat up straight. He saw it in a flash. There was only one real motive for martyrdom, and it was not devotion.
It was atonement.
_You're doing penance_.
For what? What had Alihahd done?
• • •
Evening passed into early night, the seventh of the Serpent. It was unnaturally mild on the mountain, becoming actually warmer after sundown. There was little movement in the air but for a gentle breeze, soft and moisture-laden.
On the horizon could be seen a black mass blotting out the stars. Then distant lightning lit up the outline of a monstrous thunderhead, its top sheared off by a jet wind way up high. The bolts themselves couldn't be seen—only the flashes of cold light from behind the massif and the muffled report of far-off thunder.
The Itiri were taking off their sword belts and their metal jewelry.
Alihahd climbed to a ledge where two warrior-priests kept vigil. The towering thundercloud menaced in a slow advance from the far side of the valley, its flat top miles above the mountains. The Itiri marked the ominous shape of the thing and muttered.
"How long?" Alihahd asked.
"By morning," they said.
As they spoke, there was a change in the air. Cold, with a downdraft.
Alihahd was aware of another menace, in counterpoint to the storm, on either side of the great rift between the mountains. On the dawn ridge he could see Ben-Tairre backed by the turbulent sky. Always a striking figure, Ben had assumed a dynamic stance, a conscious or unconscious display of power and dominance. His clothing, loosely fit, still showed the play of muscles in his thighs, his heavy shoulders, his hard abdomen. He carried an obsidian blade.
On the eagle ridge opposite—black, hard and sharp as if fashioned of obsidian herself—crouched Roniva. Her long thin limbs, tight-corded and agile, were folded beneath her, catlike, her body hard as stone, fluid as water.
Over them the sky was about to break. The air was electric, charged with premonition so strong even Alihahd felt it—a storm that had been building long before he'd ever seen the Aerie.
Alihahd spoke to the warrior-priests on the ledge with him. "What happened here? What is between Roniva and Ben?"
Emerald eyes exchanged hesitant glances as if the warriors were actually afraid.
And they told him there would be a death tonight.
# 11. Thunderhead
A STRANGE SOUND ROSE over the Aerie, beginning softly, steadily gaining in strength. It was not a rain sound, but an odd rhythmic swelling of voices in an ever building chant, over and over.
Ka _za_ , ka _za_ , ka _za_.
Serra came out of her cave. She stepped to the rail of the arcade and looked out.
The sky was dark, starless. A few fluttering torches lit the paths of Havenside. The chant carried eerily from Aerieside, from within the Chamber of the Golden Dome.
Ka _za_ , ka _za_ , ka _za_.
Serra called in to the others. "Something's happening."
Alihahd joined her at the balustrade. An electric hiss of lightning tracked through the clouds. Then the whole sky lit up brighter than daylight. In the brief weird illumination, the landscape was surreal, blue-white, with jagged edges and black, black shadows.
Alihahd saw Arilla dash along the Ledge Path toward the Chamber of the Golden Dome. She appeared as an impala-bodied white streak. A lightning flash caught her expression, froze it. Fear. An odd thing to see on a warrior-priest.
Harrison Hall spoke from behind Alihahd. "She's Ben-Tairre's fire sister. This is it."
• • •
Arilla flew into the Chamber of the Golden Dome and broke through the ring of warriors, but at the inner edge she was caught in the arms of an Elder who lifted her off her feet and stopped her flight. There could be no physical intervention here. Her fire brother was alone in this.
In the center of the ring Jinin-Ben-Tairre and Roniva circled the floor, the surrounding crowd calling on them to break setkaza now, finish it.
It was said that he who made the first move lost. Neither would strike first. But the ring of warriors was intent on breaking one of them. They could not leave this chamber until one was dead.
Ka _za_ , ka _za_ , ka _za_.
Alihahd came to the jeweled arch. He shifted to see between the heads of the crowd of warrior-priests. Their chant surged like the incessant pounding of a heartbeat.
Ka _za_ , ka _za_.
Suddenly Roniva's sword was out. She moved to the center of the chamber and stabbed the point of her sword into the floor.
The chant stopped. The ring widened as everyone moved back to the farthest boundaries of the chamber, onto the benches and the stone sills of the tall lancet windows.
Ben and Roniva circled the sword three times.
Someone gave Ben his sword. Roniva pulled hers from the floor, and the final battle began.
Blades flashed. The two swept across the inlaid floor like sidewinding serpents. Spectators fled from their approach. The players owned the space, pausing for no one. No one had the right to be in the way of the fatal dance, which Roniva, having begun it, was fated to lose.
Roniva swung at Ben. With a ringing clash, Ben batted the flat of her sword away from him with his false hand. The awkward thing became for him an extra weapon, and he lunged into Roniva with his sword.
The trick should have caught her, would have, but Roniva was quick and wise to Ben's way of turning his weaknesses. She parried, jumped away, and countered.
A wet sheen glowed on Ben's skin in the unheated chamber. The Itiri would see Ben in a shroud of heat. Exotic eyes that always looked angry narrowed in ferocity. Small, broad feet wrapped in strips of red rags were a burned reminder that Ben did not know how to lose.
Roniva advanced on him, her sword a whistling blur before her, and she pursued him to the wall, where watchers divided and dove to either side. Her strike sheared the curtain from the window and shattered the mirrored jamb with a spray of glass. Ben spun off to the side. He seized the offensive, beat Roniva into a retreat, and backed her to the hearth.
Her sash ignited at the fire's edge. She sprang backward over the pit, arched like a cat, cut her sash from her waist, and wielded it as a flaming bola. It swished in a circle, searing the air, and she let it fly.
Ben leaned far to the side. Before the bola was even on the floor, he was on immediate counter-offensive, swinging his sword at Roniva in a mighty arc.
Roniva's sword was down, Ben's blade driving at her head. In a swift deliberate move, as fast as a sword's sweep but with an eerie sense of slow motion, as if time itself were commanded to wait, Roniva raised her left hand in a gesture of halt and caught Ben's blade on her bare palm.
An edge that could sever tempered steel, the tungsten-plastic should have gone through with no more resistance than slicing air. The blade stopped dead still.
The look in Roniva's eyes reflected a power, a force to stop a starship, a faith to move mountains. It was a borrowed power of the universe. The myths of magic that surrounded the Itiri like rubbish were all there, focused in the palm of Roniva's hand.
Ben took a step back in shock, looked at his blade for blood that was not on it. In that moment he lost the battle.
It was quickly done after that. Ben's feet were swept from under him by a long, lashing kick. Roniva stepped on his sword hand and put her sword point to his jugular. All motion came to a stop.
Arilla turned away and hid her face against the chest of the Elder, Eren-Ben.
Jinin-Ben-Tairre didn't breathe. He waited for Roniva to finish him.
Roniva spoke to him directly. He was next to dead and counted as no one now. "He who breaks a setkaza first loses. Except that not for nothing was I made Fendi of this Aerie. I can break traditional bonds also. Thou wert someone before thou wert Itiri, Jinin-Ben-Tairre. That someone may live, if the Itiri agrees to die and never be seen on this Aerie again." Dark eyes met dark eyes and she asked, "Art thou dead, Jinin-Ben-Tairre?"
Any Itiri would rather actual death to ritual death. But survival instinct would not leave him. "I am."
"Let all be gone," Roniva commanded. "Tell everyone that Jinin-Ben-Tairre has died."
As the warrior-priests filed solemnly out, Eren-Ben knelt by the dead warrior, took his sword and placed it in a wooden chest. He took the dead man's eternity flower from his belt and his signet ring of the carnelian serpent from his finger. The flower and the ring he gave to grieving Arilla.
Jinin-Ben-Tairre spoke softly from the floor at Roniva's feet, for only her to hear: "The corpse requests to be buried with his book."
Roniva's visage was severe and scowling. She did not look down, but hissed through her teeth, "Ben, shut up."
Outside, the sky had split open and rain poured down in torrents. Lightning flashed and roared. The smell of ozone scorched the air.
Jinin-Ben-Tairre stayed sprawled on the floor where he'd died, until only he, Roniva, and the hanina Eren-Ben were left in the chamber.
And Alihahd. He hadn't gone out with the others and didn't intend to go unless he was told. He stayed in the background in the nine-sided chamber, behind the lattice barrier that stood in front of the dais. No one paid him any attention, though he sensed they were aware of his presence.
Roniva lifted her sword away from Ben's throat, and she melted from her regal pose. She seemed to sway for a moment, and she leaned against the Elder, Eren-Ben, who was quietly there where she needed him, the only other living warrior to bear the name Ben. Eren-Ben was shorter than the other aghara, wider across the shoulders. His pacific eyes were a striking blue-green, wide and fathomless. His ageless face was mild, sad, serene. His hair, a flowing mane as long as Roniva's, was a silvery ashen blond different from the normal Itiri gold. He wore his eternity flower behind one ear, as Roniva did.
Roniva drew away from him as Jinin-Ben-Tairre rose to stand. "Thou art kind," he said to the Fendi.
"No, I am not," she said. "I will not have thy death on my sword to weaken my rule."
"Then may I say thou art wise," Ben-Tairre said.
"Thou mayest, and I am," Roniva said.
Jinin-Ben-Tairre took a few steps as if moving in a body that was not his. It was strange to be dead. The reality was only now occurring to him. All that had been his life was gone. He didn't know what was left. "Who am I?" he said, lost.
"I never wanted thee as an Itiri," Roniva said. "Because thy heart is in the wrong place. Thou wantest to be an Itiri so thou wilt not be human. But thou _art_ human. Be thou human now."
Eren-Ben put his arm around Roniva's thin waist and escorted her to the archway, leaving Jinin-Ben-Tairre in the center of the domed chamber like a lonely ghost. He called after Roniva.
"May I have thine blessing?"
It had not been offered. It was impudent of him to ask.
"I should flay thee, Ben-Tairre. Thou art too forward, not to mention _dead_. Be thou gone from my sight forever." She went out.
Ben blinked, shuddered. Thunder crashed.
He barely heard, for the loud rush of the driving rains, the voice of one leaning in through the mirrored jambs of one of the tall windows, barking:
"With my blessing!"
• • •
Snow came after the torrential rains, and the cold encased the mountain in ice. The morning brought a fog, and all about the Aerie was perfect white. No one could see past arm's reach, and no one could go outside unless he could walk the paths without sight. Pale eyes went quickly blind in the whiteness.
The sign of year turned to the Cross. The clouds lifted near midnight, and the high winds swept them away. The cold night grew colder.
Around the Aerie the air was still. Overhead, the stars shone bright in a velvet field. Around the rocks a powdery white scattering of phosphorescent blossoms dotted the ground like more stars underfoot.
Alihahd walked alone in the starshine. It was the hour of the meteors, the beginning of the third watch.
Thin air made sounds fainter. Snow muffled them. But the night quiet, the cold, seemed to fine-tune Alihahd's senses. He listened.
Jinin-Ben-Tairre was dead. Or at least gone. Alihahd had seen him descend the snake path toward the valley. Yet Alihahd still felt he was being watched.
At last he stopped on the path, and spoke without turning. "Layla, why do you follow me?"
All was silent, but for a distant wind. Alihahd waited. Then she appeared with a soft crunch of crushed ice. She dressed like a child set loose in a theater's costume storage loft. Her head was veiled like a Muslim lady so that only her brown eyes showed, and she wore a down jacket, kalx-hide trousers, a pair of gloves without fingers, and a gold-fringed shawl which she'd tied around her waist. Her jeweled dagger was sheathed in one dun boot. Her gun was no doubt tucked in her pocket. Always a soldier, Layla was never to be found unarmed.
She came to Alihahd's side. Her head didn't even come up to his shoulder. "Why does one move about at this hour and avoid the sentinels?" she demanded.
Alihahd smiled. He realized he'd been skirting the sentinels' vigilance. "Habit," he said. "Is that not a sorry comment on a way to live?"
The question was too nebulous for Layla, and she made no comment on it.
Alihahd admired and envied Layla—her strength, resolution, and unshakable conviction. Layla was also slightly stupid. Blessed with a simplistic bullheadedness, no profound dilemmas ever disturbed her dreams. She was right. Her enemies were wrong. There were no shades of gray.
Alihahd touched her chin through the Muslim veil. "Do you know what your name would mean in Jerusalem?" he said.
"No," Layla said.
"It means night," Alihahd said.
"I like that," Layla said.
"Better than mine," Alihahd said.
Layla took his arm and walked with him a while. She pressed close to his side, not sexually. She was cold and he was there. She didn't mistrust or hate him as she did Harrison White Fox Hall.
Layla called Harrison Hall a slave master.
"Men and women were not meant to be slaves," Alihahd said. "The Na′id are not without a point in that matter. The class system on Eridani before the coming of the Na′id was wrong. I think even Mr. Hall can see that."
"I think he sees nothing," Layla said. Hall was bad. Therefore, nothing about Hall could be right.
"I think I can understand Mr. Hall better than you can," Alihahd said. "We are both ruling class, he and I."
"You more," Layla said.
"I?" Alihahd said. Layla knew nothing about him. She didn't even know whom he ruled.
"It just sticks to you," Layla said. "Harry insists too much. You do not say anything. You do not need to."
"His empire fell," Alihahd said. "It makes a difference."
"Did not yours?" Layla said.
"No. Not yet," Alihahd gave a sad smile. "It is a tottering tower of cards. It will fall. I may even weep when it does."
"You think the Na′id will destroy it?" Layla asked.
"Yes," Alihahd said. "The Na′id will destroy it in the end."
A sudden bright light flaring in the valley drew his gaze.
The roaring followed, lagging behind the light that had begun its ascent. It blazed toward the heavens to the stars and was gone.
Alihahd stared in disbelief. "Goddamn," he breathed. His shoulders drew back, his blue eyes flashed, and his voice became a rolling roar. "Goddamn."
A starship. A starship had left Iry without him.
He pulled away from Layla with a growl. "Where is Roniva?"
"Do not anger her," Layla said.
Alihahd cared nothing for the warning. He would not be treated so. Roniva had said he would leave on the next ship.
He parted from Layla and crossed the footbridge in twelve steps. Aerieside was black without torches, soulless in its majestic repose. Alihahd filled his lungs. Then, into the imposing silence of the caverns and arcades, Alihahd bellowed the Fendi's name.
• • •
The echoes of his wrath faded to the night quiet. The wind moaned in the crevasse softly.
She would not come to him. Damn it, he was going to find her.
A single dim light glowed from within the Chamber of the Golden Dome, the place where Jinin-Ben-Tairre had died. Alihahd went inside, his footsteps loud in the cavernous space, ringing off the nine walls and the high metal dome. The snowy owl loomed over the throne. The rustle of its feathers startled him. The owl's round yellow eyes stared at Alihahd unblinking. Cold gusts blew in the windows and the archway, swirling the leaves in the hollow place. This place was deserted. But _she_ was here, even when she was not here. _She_ filled this chamber.
Alihahd met the preternatural yellow eyes. His voice tremored with rage. "Where is your master?"
The owl clacked its black beak.
Alihahd didn't hear anything, but he perceived a presence behind him. He turned.
Roniva stood in the archway. "Here is the creature's master," she said.
"Forgive me if I abuse your hospitality, Fendi," Alihahd said bitterly. "But your hospitality is not all you would have me believe."
"Explain." Her brows were drawn together in a pained squint over her eyes.
" _'When one of us next leaves Iry, thou mayest go also.'_ Whose words are those?" Alihahd said.
"Mine," Roniva answered.
"Then what the hell was that?" Alihahd pointed skyward.
The Fendi's brow smoothed in surprise. Then she squinted again, puzzled. "A ship has left?" She turned her head aside and held up her fist. _"Eho!"_
Her owl appeared on her fist.
"Find," she snapped, and threw the owl into the air. It disappeared. Roniva folded her spidery black fingers and assumed an attitude of waiting. "We shall both know shortly."
The owl reappeared on the throne, crying three notes.
Roniva turned back to Alihahd. "No one left on that ship," she said. She rubbed her temples and moved toward the door.
_No one?_ "Lie to me, Fendi?" Alihahd said.
Roniva halted. A volatile word, _lie_. Alihahd was too incensed to care if he angered a woman who could stop a tungsten-plastic blade with her bare hand.
"It was Jinin-Ben-Tairre," Roniva said, her own anger withheld. "He is dead. He is no one."
"I am no one with whom to try semantic tricks," Alihahd said.
"Not semantics," Roniva said. "To us, he truly is no one. I am not accountable for the actions of the dead. When someone leaves Iry, I will know and thou wilt know. That is all."
Alihahd rumbled, "That is not all!"
"I become wroth," Roniva warned softly. "Not easily done."
"I had not noticed that particular fact," Alihahd said acidly.
"Done!" Roniva cried.
"Easily," Alihahd said.
"Oh, thy tongue is a sharp sword and thou art evil. Ben told me so. I should have let him kill thee. You would both be gone from me now!"
Alihahd was shocked to wordlessness for several moments. Then he continued in a softer voice. "How does a neutral nation justify aiding the Na′id Empire by holding one of the opposition's leaders prisoner here unable to lead?"
"I was aiding the opposition by saving said leader's life," Roniva countered. "Neutral, I should drop thee back in the ocean whence I dragged thee."
"Perhaps you should," Alihahd said quietly. "If I am to be your prisoner."
Roniva threw her arms wide in exasperation and cried, "Thou took exception to my aiding the opposition by training thy Harrison Hall. Which side art thou on?"
Alihahd threw his head back and gazed up at the tall ceiling that was suddenly too low for him. He cried, "Humanity's! Trying to be!" He slapped his palm against a pillar behind him, and almost laughed at his own uselessness.
"As I am the Itiri's," Roniva said. "Remember thou who I am. May I go now?"
Alihahd gave a bark of pain, frustration, and absurdity. _"Ili! Ili!"_ he cried.
"What sayest thou?" Roniva asked.
"I am saying, 'Go. I am sorry. Dismiss everything I ever said.' Why not? I am as no one as Ben." And he ran from the chamber.
• • •
Serra listened to the rain falling outside. The fire in the hearth sputtered as drops strayed down the chimney. Harrison Hall rose from the breakfast table and took his rain fur from the peg by the door. "Be careful," Serra said.
Hall navigated the slushy paths pitted with footprints, while on either side of the trail the rain pocked the untrodden snow with wet perforations. He came upon a moody Alihahd, his blond hair wind-tossed, his ears and fair face stung a deep red, his blue eyes half closed in the cold wind. Either from self-torture or neglect, he was not dressed for the weather. From the look of him, he had been out here all night. He looked deep in thought. Hall sensed anger.
They stepped around a spongy euglenoid colony that was creeping across the path, soaking up the rain.
Hall walked along in silence. When they had come at length to the entrance to Alihahd's cave, Hall asked for words. "Captain?"
Alihahd expelled a breath, shook his head, said nothing. He went in.
Hall returned to Serra's cave. Serra, Amerika, and Vaslav were at the table. Hall hung his rain fur on its peg and joined them.
"Mad," he said. "Absolutely unpredictable. He's as regular as the damned weather."
"No," Vaslav said.
Hall frowned at the boy. "No? What do you mean, no?"
Vaslav explained that Alihahd was not erratic at all. According to Vaslav—and Vaslav's chronometer—Alihahd kept a perfect twenty-four-hour Earth standard cycle.
Harrison Hall seized the boy's wrist and stared at his chronometer. "Are you sure?"
Vaslav swore it was so. He didn't understand it, how Alihahd did it, but that was what he was doing. Alihahd rose every day at the same time—on Earth. He went to bed at the same time each Earth day. At any given Earth time of day, Alihahd was to be found in the exact same place. The schedule put him far out of rhythm with Iry's days and nights. Not to be regulated by an alien sun, Alihahd rose and slept as his inner clock bid, no matter the position of the Iry sun.
Hall's smile was slow and evil, breaking into an open grin. His gaze turned aside with glinting eyes, toward the door and the outside where the rain had turned to snow.
# 12. A Winter Conversation
ALIHAHD WOKE AT SUNSET, just as Vaslav was crawling into bed. Vaslav glanced at his wrist chronometer. The time was exactly 0735 on the Earth meridian.
_How does he do it?_ Vaslav marveled. _I have the clock!_
Alihahd went through his waking ritual with seeming meticulousness, putting on his clothes in habitual order, remaking the bed just so—first the quilt, then the woven blanket, then the brown fur, then the silver fur. He brushed his teeth twice.
Then he went outside without his rain fur like a forgetful old man.
The cliff paths were slick with snow. It fell lightly from the high clouds. No one was about at this hour except the two sentinels, who were easy to avoid. Alihahd's walks had settled into a set pattern that kept him away from others as much as possible.
Alihahd wallowed in his solitude. He didn't feel the cold.
Then a voice split the silence and dragged him out of his reverie. "Why art thou always elsewhere than thou art, Alihahd?"
He knew the voice and was shocked that it should address him directly.
Waiting in shadowed ambush on his accustomed path, alone and unattended, was the Fendi Roniva. Blending in with the black shadow she was barely visible but for her white teeth, the whites of her glittering black eyes, and the broken red scars on her cheeks. She wore no jewels to catch the starlight. Alihahd was so amazed that he had nothing to say.
"Thou livest out of time," said the Fendi. "Be here now."
He wondered what she wanted.
She drew closer and passed a leathery hand over his wet hair. She shook the snow crystals from her hand. "Thou hast little regard for thyself," she said. "That is not good. Thy spirit is not well."
"How may I serve thee, Fendi?" Alihahd said at last.
"Dry thyself. Eat. Wear warm clothes and come to me." She turned sharply and glided across the bridge.
Alihahd had detected a subtle nervousness about her he'd never seen before—something in the stiffness of her shoulders, a thin edge to her voice. Nothing blatant, it was the held-back tension of the brave solemnly facing something dreadful.
Alihahd quickly complied with her wishes. He went to Hall and Serra's cave. The two of them were usually to be found up late.
"Good morning, Captain," Hall said, smiling.
Alihahd frowned.
Serra fed him. Hall gave him warm dry clothes and his own fur-lined parka. Before Alihahd left, Serra slipped a packet of breadcakes into his pocket. "Take. It will be a long night."
Alihahd took her shoulder in his hand. "You know what this is about, Serra?"
Serra kept her eyes lowered as a good Eridanin woman would to a strange man. "I think so." She tied the leather drawstring of the parka's hood under his chin. Her shoulders were as stiff as Roniva's. Her voice had gone husky.
"Am I in danger?" Alihahd asked.
"No. You can walk away. She cannot."
Serra didn't explain. The topic was upsetting to her.
Alihahd left the warmth of the cave. Using the safety line, he crossed the bridge to Aerieside in twelve steps, starting with his right foot.
Roniva hadn't told him where to go, so he went where one always had audience with the Fendi, to the Chamber of the Golden Dome.
Even as he approached, he could see yellow light from its arched doorway and its tall windows, darkened by flitting shadows of people moving inside.
He stood at the entrance and witnessed an odd little ceremony in progress within.
Roniva stood in the center of a ring of _ma-hanina._ She wore only a loose flowing ice-blue shift pinned at her shoulders. Her hair was loose. Her bare arms were crossed over her flat chest. Without all her enameled toques Alihahd could see how very wrinkled was her neck.
A warrior-priest stepped into the circle and emptied a bucket of ashes over her neat, glossy head.
She didn't move against him, only closed her eyes and frowned in revulsion. She uncrossed her arms, all her corded muscles taut. Her body was firm and thin and hard beneath the sheer shift that was no protection at all against the cold.
Alihahd was aghast to see the shift suddenly stripped from her and she was driven out naked into the snow.
Alihahd lunged forward to help her.
Her leathery hands met his as she stumbled. Then her face turned up to him, resigned, and silenced anything he might have said.
All was happening as it was meant to be.
She spoke, smudges of white ashes on her black lips. "I am of an age to demand a companion. Walk with me, Alihahd." She straightened herself and marched into the winter night.
Alihahd cast an appalled look back at the other warriors now assembled in a row barring the cave entrance and the golden glow of warmth within. Jewels sparkled on their thick robes. Alihahd ran after the old woman with ashes on her head.
Roniva's blue-black skin was mottled in places with wrinkles and scars. Her sagging abdomen was corrugated with marks of childbirth.
Alihahd couldn't contain his anger. "What kind of sadistic and barbarous ritual is this that they should degrade a dignified, elderly woman in this way!"
Roniva gave him a dim smile and spoke to him directly, for she had no attendants anymore. Even her familiar was nowhere to be seen. "Too dignified," she answered. "That is the issue. I am the single most powerful being on Iry. One begins to feel like a deity." She spat ashes from her lips. "I am not."
"And you are aware of it. This is not necessary."
"For them, too." She gestured backward. "One's life is not entirely one's own up here."
They hiked over a ridge. A blast of wind lifted a swirl of glittering snow around them. Roniva threw back her head and howled.
Alihahd was puzzled that she could not overcome the cold. "I have seen you step in fire and stop a naked blade with your bare hand," he said.
"Ah. That was then. Now is now. I am naked and I am cold. How I hate this."
Alihahd wanted to give her his coat. He knew she wouldn't take it. Trying to lift her spirits, he reminded her of her own greatness.
"Not to flatter me," she commanded. "I need to find some humbleness if I am to bear this. How beautiful the new-fallen snow." She frisked away like a colt.
Alihahd thrust his hands into the pockets of Hall's parka as he followed her. He felt pebbles in the bottom of one pocket and he drew them out. They were the eight gemstones he'd brought to Hall from the valley.
Roniva was delighted. "Give thou to me. Hast thou a zircon? I am in need of a zircon."
Zircon. Humility. Alihahd gave her all eight stones. She fingered them like prayer beads.
Silent a while, Alihahd's thoughts jumped. Then he blurted out, "Why did you kill Jinin-Ben-Tairre?"
"I am getting old," she said.
Alihahd foundered in puzzlement. Wind fluttered the fur lining of the hood that wreathed his face.
"I grow weary," Roniva said. "An Elder is truly an elder, thou seest. I am . . . what is the number . . . one hundred and sixty years old. I could die."
Alihahd was amazed and still confused. "What has that to do with Ben?"
"When I die, they could make him Fendi."
" _What?_ "
"Thou didst not know? He is a miracle among us. I had to kill him while I had strength. I'll not have him rule my Aerie. He belongs with humans. I sent him back."
"That could be difficult," Alihahd said.
"Thou meanest because of the Na′id?"
"Among other things," Alihahd said.
"Harrison White Fox Hall tells me much about the Empire," Roniva said.
"I know."
Alihahd had told Hall to stop maligning the Empire to the Fendi. It endangered the better part of humanity should the Itiri ever decide to break their neutrality and go to arms. Hall had replied, "Don't use 'better' when you mean 'bigger.'"
"Alihahd, thou speakest not and thou art the one I would have speak," Roniva prompted.
_Know thine enemy_. She wanted to know the Na′id.
"In true conscience, I cannot," Alihahd said.
Roniva made a questioning noise. She wanted to know why.
Alihahd admitted, bluntly, "Because they are human and you are not."
The Fendi was not offended. Honesty did not offend.
"I am afraid you will need to rely on Mr. Hall for your information," Alihahd said.
"Thou art a worse threat to the Na′id than is Mr. Hall," Roniva said.
"How do you figure?" Alihahd asked.
Hall was dedicated to the destruction of the Na′id. Alihahd only wanted to break their stranglehold on free humanity and to avoid bloodshed.
"Thou knowest them best," Roniva said.
"I do," Alihahd said. "It has been a decided advantage."
"Harrison White Fox Hall tells me evil things about the Na′id. Tell me something good, canst thou?"
Trapped. This woman always won what she sought. He could hardly _not_ answer her, and he ended up telling her about the Na′id.
He explained to her the Na′id ideology, before it had run afoul of itself. Its lofty ideal had been to eliminate the superficial barriers which divided humankind.
Roniva countered, "They oppose color barriers, yet they despise thee for thine color. How is that?"
Who had told her about his color? _Hall_. Damn him.
Alihahd regarded his pale hands. He was sensitive about his strange color. It was difficult to keep emotion from his voice. "This is the color of a pure race. It is a symbol of many evil practices in the past—"
"Thou art not a symbol. Thou art a man," Roniva said. "Thou hast also been brainwashed by thine own enemy."
Alihahd wanted to protest, but he would need to tell her too much. He sighed. He couldn't even reason with human Jews and Arabs. What made him think he was going to make an alien hear sense?
It was comforting to know that all these irrational Itiri were staunchly neutral and wouldn't set out with their deadly martial skills to threaten the Empire.
All but one dead one.
• • •
An evil in their midst. The Na′id conscription base locked in on itself in unknowing terror. Brave soldiers feinted at moving drafts of air. A presence among them like a ghost—there but not there—had them cowering defenseless, because they didn't know what it was. They, the arrogant members of the Na′id conscription force who with laughing pride called themselves Mushabriqu's Press Gang—none of them was laughing now. Something was here. Damn it, something was here.
• • •
Without striking a blow, the evil in their midst reduced the entire base to grasping terror. Without striking a blow _yet_.
A voice inside the stalking evil told it to wait. _Thou art one. They are many. Yet they fear thee!_
The Na′id numbers were overwhelming. This conscription force had carried off the able-bodied population of an entire village by force. They must be reduced before they could be fought.
_I am the attacker. They defend._
The advantage was in the role chosen. The Na′id could have been a hunting party stalking a lone cornered beast on their own ground. Instead, they let themselves be frightened, become the defenders, shrinking from a killer in the house.
It was a night of fear. Slamming doors, heavy footsteps, a flutter of wings that was here and gone, the specter of a hawk, the shadow of a sword.
_Thou canst not slay them all at once_.
One had to start somewhere. Strategically.
And, after the long night, just when the soldiers began to feel relieved and foolish, they flung the windows wide, and the gentle morning sun lit the courtyard where General Mushabriqu hung by his neck from the flagpole beneath the bloody flag.
And so it began.
• • •
Night turned to the hour of the meteors. The snow had stopped. The clouds were gone. Under the clear sky, the temperature plunged.
Alihahd warmed Roniva's long toes between his hands. It seemed to him that there was something terribly arbitrary about the physical characteristics of the Itiri and their purposeless dimorphism. The aghara were too tall for the gravity and too fair for the irradiation on the mountaintop.
A meteor streaked across the icy sky. Snow on the ground glinted like quartz flecks, crusty cold underfoot. Starshadows were hard-edged and black. Roniva yawned.
Alihahd took her long-fingered hands and held them inside his coat against his chest.
"This is abominable," he growled.
"Thou mayest go in if thou art cold," Roniva said.
"No." The cold never bothered Alihahd much.
In the east the sign of the Mountain was rising, a random-looking jumble of stars that hadn't formed a double peak in 100,000 years. Dawn was still hours away.
Roniva took up their walk again. One hand slipped into Alihahd's coat pocket as a human lover might do—as his wife used to do when they were young and still pretending to be in love for the public eye.
"Are you not afraid of falling ill?" Alihahd asked. He worried about her in the wind.
"We do not sicken," Roniva said. "Most of us are immune to every disease on this world. Frostbite worries me." She flexed her long fingers and toes. "Disease doth not."
"That's incredible," Alihahd said. "How did your people develop such immunities?"
"We died by the thousands," Roniva said simply. "Those that survived passed down immunity."
"A harsh method," Alihahd said.
"Nature's method," Roniva said. "We try to be at one with Nature."
"Nature is cruel," Alihahd said. "Humanity is compassionate."
"Is it?" Roniva said.
"Kindness is a human trait," Alihahd said. "Which is not to say that all humans are kind."
"True enough that one does not often find kindness growing wild," Roniva conceded.
And Alihahd became disturbed by something that had only gradually crept into his awareness since he had come to Iry—the total absence of physical and mental defectives in the population. He asked cautiously if there were any.
"Oh. Distorts. Of course they happen."
"Where are they?" Alihahd asked.
"We kill them."
"That is inhuman," Alihahd said.
Onyx eyes flashed. "We are not human."
"I tend to forget," he said, bitter. "The level of a civilization is measured by how it cares for those unable."
"Very noble," Roniva said. "Very impractical at this point."
"Humans manage," Alihahd said.
"Is it true that human children are cruel to the deformed and the different?" Roniva asked.
Alihahd's white face colored. "Yes, but—"
"Such treatment distorts the inside. The abused grow crooked. They carry much anger. That distorts others. They distort their own children. The spiral goes down."
"That is cold-blooded reasoning. These things can be overcome," Alihahd said.
"So who told thee we were perfect?" Roniva said. "You have your interstellar wars. Why dost thou begrudge us our private atrocities?"
And who had told him the Itiri were perfect? The legends. He of all people ought to know the frailty of legends.
"Is it true you have to kill someone to become a warrior-priest?" Alihahd asked.
"No. That is not true," Roniva said. But before Alihahd could breathe relief, she said, "Thou must kill someone to be a hanina."
A noise like the beginning of gagging escaped Alihahd.
"Oh, one doth not go out and murder someone," Roniva said. "It is not a thing to strive for. It happens or it happens not. It is no one's goal to be a hanina. It is a burden. When one becomes an Elder, one gains access to the Archives. I know thou thought we had no written records. We do. They are kept in a cave in the desert. One can read things one would rather not.
"Ours is not a shame culture, Captain Alihahd. We have no use whatsoever for guilt upon which some human cultures thrive. The Archives collect much dust."
"There is an ancient shame?" Alihahd asked.
"There is. I shall not jump off the bridge for it."
"Your people do not fear death, do they, Fendi?"
"No. It comes to all sooner or later. Preferably later."
Alihahd gazed at the sky. Another shooting star blazed across the Milky Way. "You must be formidable in war," he said.
"I suppose we would be," Roniva said. "But we have not seen war in several lifetimes."
Alihahd's brows drew together sharply. "How did you achieve that?"
"No nations," Roniva said. She crouched in the snow, tried to make a snowball, but the graupel would not stick together. "It is difficult to organize a war without nations. And thou knowest our attitude toward ownership. Humans kill each other for that handful of stones in thine pocket. We own nothing. So we have no property to protect. The ranga are docile. As for the aghara, we all become warrior-priests. We are strong. Only the weak and frightened must fight."
" _All_ the aghara become warriors?"
"Or they die," Roniva said.
"They are killed," Alihahd interpreted.
"The same," Roniva said.
"Why must an aghara become a warrior-priest?"
"When aghara go their own way, war will return to Iry."
"Or when it comes from the stars," Alihahd said.
"Or that," Roniva said.
"You seem to be taking that possibility lightly," Alihahd said. "A few thousand warriors and a handful of antique spaceships will not avail you against a modern fleet. Your people were introduced to high technology two thousand years ago and you've done nothing with it in all that time. Your science is stagnant."
"We have all we want," Roniva said. "Why art thou worried about us?"
"I swear I don't know. I should let you all go to the devil. And you will go, if the Na′id find out that Itiri warrior-priests actually do exist. You wanted me to tell you something about the Na′id. I shall tell you something. They will turn you back into a myth as fast as they can seed your sun with a nova core. Sooner or later."
"I should hate to have to deal with that," Roniva said.
• • •
When the eastern sky began to gray, Roniva climbed up the highest rise and waited with open arms for the coming of the sun.
Alihahd dropped to one knee a distance behind her to watch, smiling that the long night was past and she was still here to salute the morning fire. The winds picked up her hip-length hair and fanned it out behind her like a sail. Sun gleamed on her black arms thrown wide. She cried out in the ancient tongue, a sound of exultation, fittingly barbarous to Alihahd's ears. He expected nothing less or else from his savage warrior queen.
She turned to him, and he rose.
She returned to the Chamber of the Golden Dome. The warriors who had driven her out were still there, waiting to envelop her in furs and a jeweled robe when she came in. They brought her to the fireside and brushed her hair and dressed it in gold-and-tapestry ribbons. Her colored toques were replaced around her neck, and her jingling sword belt looped around her hard waist. A warrior brought her sword wrapped in felt. He bowed all the way to the floor at her feet, unwrapped the blade, and offered it up to her over his bowed head.
She took it in a solid grip, reassured by its comforting weight, and then sheathed it.
"Tell the Earthman the Fendi is grateful for his service," Roniva said to the air. "Tell the Earthman he may go."
No one needed to tell Alihahd to go. He left the cave. This conversation was ended.
• • •
Alihahd sat on his neatly made bed, lost in thought. Vaslav came in, unwelcomely energetic and tried to talk to him. Alihahd changed his clothes, folded up the ones Hall had given him, and excused himself shortly. "I must return some things to Mr. Hall."
Vaslav scowled after him.
Alihahd took the things not to Hall and Serra's cave, but to the private cave on the third level. Alihahd didn't want to confront Hall. He wanted to return the clothes and leave. He wanted to be alone, unseen.
But he found a fire blazing in the hearth as if he was expected.
Alihahd let his bundle drop onto the bed. Irritated. Of course he'd been expected.
The fox-head meerschaum pipe had been left behind on the yellowwood table. Alihahd was tempted to pitch it into the crevasse. What would Hall think of that?
Hall would laugh.
Alihahd paced to the fire, stopped, regarded the pipe, paced. Hall was so sure Alihahd would not touch it.
Hall was right.
Alihahd touched the yellowwood tabletop with a gentle hand. He ought to pitch Hall into the crevasse. That was the real matter here.
Alihahd lightly pressed his palms together and wove his long fingers together as if praying, and he gazed into the fire. Licking tongues of flame reflected in his pale eyes.
He saw his own setkaza.
# 13. Witch Wind
THE TIME OF YEAR CAME when the wind Shandee would blow for twenty days straight, from the second of the River through the sign of the Wellspring to the seventh of the Veil—nearly half the winter. Caves were stocked with food and water, and denizens of Haven banded together for the long season. All six humans gathered in Serra's chambers to weather the storm. They were sick of each other after a few hours, sick of the walls, sick of the howling wind.
Alihahd kept to his singular schedule and intensified all his little rituals, an illusion of control and order in the helpless confinement.
And he dreamed of being buried alive.
• • •
He woke in semidarkness, the edge of a dream fading from his mind.
The sentinel from within her cave on Aerieside was beating out the hour for all the prisoners of the wind. Alihahd counted the bronze clashes. Seven. The hour of the eagles. Predawn.
Alihahd sat up on the wide mattress, finally awake.
He pulled on his boots, first the left, then the right. He went to the back chamber to wash and pass water. Serra's back chamber with its nonporous refuse drain was the reason these caves had been chosen for the humans. Itiri excreted little water. Neither did they sweat.
Alihahd rubbed salt on his teeth, did it twice, spat, and came back to his corner of the bed.
He put everything in its place. He owned four changes of clothes by now, and he stacked what he was not wearing in order under the pillow. He folded his quilt, the woven blanket, the brown fur, and the silver fur precisely, in four folds, and set them on his sleeping place.
There was also a mat on the floor, which was not his but had to be aligned with the crack in the floor or Alihahd was not happy.
In this small area of this particular chamber, things were as Alihahd willed them to be. All the rest was as it fell to random human chance.
When all was arranged, Alihahd stepped back (with his left foot) to inspect his niche, and he saw that Amerika had hung her mirror on the wall.
It had been years since Alihahd had looked into a real mirror and seen more than a ghost image reflected in a glass or a polished metal surface. This was a real, merciless mirror. He saw that the skin of his neck had acquired the plucked chicken texture of old age. Tiny broken blood vessels marked his nose, and rays of new lines etched his face. He'd forgotten how truly unpleasant was the look of white skin on a face—the color ancient Caucasians in their infinite elitism had termed "flesh tone"—not the smooth marble white of the aghara Itiri, but a sometimes ruddy, sometimes sallow translucency that showed tracks of blue-and-purple veins.
Alihahd turned the mirror toward the wall. It became part of his ritual.
On the wide mattress everyone used for a bed, Amerika was now sitting up, her face puffy from sleep and marked with lines from the pillow. Her hip-length black hair had been laced into a single thick braid for the night. She had been sleeping right beside him. He hadn't noticed.
She was looking at her mirror turned to the wall.
"Is it an instrument of sin?" she asked, childish guilt in her lovely black eyes. "I only use it to part my hair and paint my eyes. And I only paint my eyes against the sun. I swear."
"No, it is not sinful," Alihahd said.
"Why didst thou turn it over? So the spirits cannot come out?"
"No," Alihahd said. "Because some faces are better not faced at this hour."
He gazed up at the coffered rock over his head, muttered that he could have been light-years away from here if Ben had not gotten off planet without him.
"He was not to have taken that ship," Amerika said. "The Itiri were upset. He was not to have taken _Da′iku_ either."
" _Da′iku?_ " Alihahd cried loud enough to disturb the sleepers. Serra stirred under Hall's arm, then her breath evened again. Alihahd faced Amerika gravely and whispered, " _Da′iku_ —is that what he calls his familiar? The bird?"
"No," Amerika said. "It is what he calls his sword."
Alihahd was on his feet. He strode to the battened door. He brought his fist against the stone jamb. The howling wind sounded very near on the other side. Alihahd truly felt the walls closing in on him. He needed to run, to find Ben, to stop him.
" _Da′iku_ is a Na′id name, is it not?" Amerika said.
Alihahd nodded, his fist on stone.
"Thou knowest a meaning," Amerika whispered, pulling her blanket around her as if cold.
Alihahd faced the door. "It means Killer," he said.
# 14. A Slow-Falling Star
LAYLA ABSENTLY PASSED her finger through a candleflame, collecting soot on her skin, the warm light flickering across her rough, impish face.
Serra was seated at the table, sewing a shirt for Harrison Hall, and refilling teacups as they emptied.
Alihahd's third cup of tea sat half full and growing cold. Alihahd never finished the third cup. He always drank two and a half cups at a time. Always.
On the floor, Amerika was teaching Vaslav a pebble game.
From the adjacent chamber, Harrison Hall was snoring.
Faintly, from Aerieside, Itiri music spun through the wild wind Shandee's howling.
Layla lifted a pendant on a chain from around her neck and held it out to Alihahd across the table. "What is this symbol?"
Alihahd picked up the pendant from her small palm and rubbed his thumb over the metal face. It was a Na′id cuneiform symbol:
"Where did you get this, Layla?"
"From a dead Na′id."
"Did you kill him?"
"There were many dead on the field. I know not which of them I killed."
"I see."
Layla pointed with her soot-covered finger. "What do the markings mean? Amerika says you know Na′id."
"It's a DINGIR. In this case it means God."
The Na′id had adopted the Jewish/Muslim custom of not depicting God. Christians had a tendency to make God a male Caucasian. The Na′id preferred a deity without form, without color, intangible like the human soul. "This is a religious medal."
"In _this_ case it means God?" Layla said. "What other case is there?"
"This mark can also represent a sound in a word— _an_ or _il_ ," Alihahd said. He marked on the table with a piece of chalk:
"There it is pronounced ili—literally gods, but this is a name."
"This is confusing," Layla said, her cheek resting on her fist.
Alihahd pointed to one symbol at a time. "The first mark is pronounced _shad_. It means mountain. These middle two are your DINGIR and a plural sign. The last is pronounced _ya_. It means my. So: Mountain of my Gods."
Serra put down her sewing. The red chevron on her forehead creased. Ignoring the translations, she strung the pronunciations together in order and said, "Shad Iliya."
Layla's brown eyes widened. She pointed at the chalk marks. "That is what this says?"
She scowled at the markings, then methodically wiped them away with the side of her hand. She regarded the resultant smear with satisfaction. In the primitive culture of her origin, to obliterate the infamous general's name was to obliterate the man himself.
Amerika came to the table and looked over Alihahd's shoulder. "Oh. I did not see it. Write it again."
"The same thing can be written a slightly different way—without the DINGIR," Alihahd said and marked on the table again:
"This language cannot make up its mind," Layla said.
"Shad-ili-ya," Alihahd voiced the symbols.
"That cannot be." Serra pointed. "The middle two marks are the same."
"NINI means ili," Alihahd said.
"That is stupid," Layla said, and she erased Shad Iliya from existence again.
"It's an ancient language. It went through several evolutions. Humankind was then only developing the concept of the written word," Alihahd said. He tossed the chalk over his shoulder. "Anyway, it was not my idea to use it."
"It is stupid," Layla said and fastened her DINGIR pendant back around her neck.
Hall stopped snoring in the adjacent cave, and Serra brought him some strong tea.
Talk switched to other things, and Alihahd toyed with a nail, using it to scratch signs into the stone wall, until Layla noticed what he had written:
"What is that?"
"Ashar Ari. Place of Eagles," Alihahd said. "All Na′id planets are called Ashar something or Mat something—Place of this, Land of that."
"You gave us a Na′id name?"
Alihahd put down the nail. "In Na′id culture, to name a place—to give it a Na′id name—is to legitimize it. The name officially recognizes this planet as Place of Eagles—that is to say: place sovereign to nonhuman aliens."
"In that case, I like it," Layla said. "Ashar Ari."
"The Na′id would hate it," Alihahd said.
"The Na′id may go to hell."
• • •
Flames raked the sky over the recruiting station, consuming the myriad dead which littered the compound blackened with blood. Mushabriqu's charred body dangled, a grim banner over the blaze, until the flagpole melted and fell. The outrush of heat singed leaves on trees far away.
Beyond the inferno, freed conscript children were running into the surrounding forest.
The killer had been this way before. He knew where to go next.
He boarded his ship, _Singalai_ , flew over the flames, and set course for the nearest Na′id reloc center, to kill again.
• • •
The caves grew colder. Snow had covered some of the solar collectors outside, and the captives of the storm could only hope it would blow off again.
To make it worse, Layla and Serra were telling ghost stories. Inevitably, they came to the _Flying Dutchman_ and _Marauder_ stories. Hall just listened and smiled. He relit his pipe. One of its fire-opal eyes was missing, giving the fox a winking look.
"They say that at the battle for Jerusalem the whole galaxy came to the aid of the Holy City, and all the ghosts of those buried in the hills rose and fought. And they say Shad Iliya saw the ghost of a brigantine ship sail over the city. He died right after that. They say that is what killed him."
"Truly?" Amerika asked, breathless and enthralled.
"So they say," Layla said. "Something got him. It may as well have been the _Marauder_ ship."
"I saw the _Marauder_ ," Alihahd said. He set aside his third, half-finished cup of tea and did not touch it again.
"You never struck me as a teller of tales," Serra said.
"I saw it," Alihahd said.
"I saw it, too," Vaslav said.
Amerika's eyes were huge. "And then you were shipwrecked!" she exclaimed in a whisper.
"Very shortly afterward, yes," Alihahd said. "Though I think I was equal bad luck for the _Marauder_. I believe he wrecked soon after I did."
"How can you wreck a ghost?" Layla asked.
"Because my luck is very, very bad," Alihahd said. "And because this _Marauder_ was not actually a ghost. It was a hologram that a pirate used to frighten Na′id ships before he destroyed them. Unfortunately, we two curses crossed each other's paths and that was the end of us."
"Did you see the _Marauder_?" Serra asked Hall.
Alihahd sat forward. He was interested in how Hall would answer that question. Hall said only, "I was on a different ship from these two."
A neat evasion that one—neither yes, I am the _Marauder_ , or no, I am not. The man would not be pinned down. He did, however, admit to garroting a Na′id captain with his own hands. He went into graphic, relished detail about it. He had no regret. He waved his hand at the cuneiform Alihahd had engraved into the wall. "It is written in those ancient chicken scratches somewhere, 'an eye for an eye.'" He looked to Alihahd for verification. "True?"
"True," Alihahd said.
_If a citizen has destroyed the eye of a citizen, his own eye shall be destroyed. If he has injured the limb of a citizen, his own limb shall be injured. If he has destroyed the eye of a subordinate or injured the limb of a subordinate, he shall weigh out a recompense in silver. If he has destroyed the eye of a slave of a citizen or injured the limb of a slave of a citizen, he shall weigh out half the slave's buying price._
"The principle, I believe," Alihahd said, "behind the law 'an eye for an eye' was, at the time it was written, not that the punishment be severe enough for the crime, but that the punishment not exceed the crime, which was rampant practice in those days."
"You think me excessive?" Hall asked.
"I was not there."
"You think me excessive."
"I think you excessive."
Hall sat up and moved in closer. Low, so only Alihahd could hear, he said, "You didn't tell me you beat a man to a bloody pulp down in the valley."
"It slipped my mind," Alihahd said frostily.
"Why are you so cool toward the idea of vengeance, Captain?" Hall said with oily insinuation.
He was surprised to receive straight out the answer he sought.
"Lest it fall upon me."
• • •
The drillmaster enjoyed his work. He had survived yet another summons before the review board on suspicion of excessive and brutal disciplinary measures. He said all the right things, admitted he was often severe with his young charges but only to train them to survive in the field in harsh environments, combating aliens. He said he was strict out of concern for his recruits and that most of them respected a tough master. However, there were whiners in any group. The board found his explanations acceptable. The drillmaster always knew what to say.
In reality, he liked to beat children. Real torture and sexual abuse he reserved for those children already on record as compulsive liars and unwilling conscripts. He'd worked hard to get where he was—to be commander and not the commanded, the abuser, not the abused—and he knew how to preserve his position.
He had been master of the juvenile wing of training section Alpha 4 for twenty-five years. He could handle squealers.
But the last little girl from Qiatte had come across as entirely too credible. She'd been responsible for his latest trip before the review board. It was a pity that her class was gone, moved on to Omonia Station, and he was unable to get back at her.
That whole class had been trouble. Qiatte was a little mining village on a grade four planet. That world turned out some of the most recalcitrant hellions ever to come through this training station. The reloc center always sent the children from Qiatte here. The reloc center had very recently met with a mysterious catastrophe that killed everyone on the base—may they all burn in hell. The drillmaster was glad. He would get no more ungovernable conscripts from that place.
Still unsettled over his inquiry, he soothed himself with food. He ate another honey roll with tearing bites. Then he swiveled in his chair to the water fountain to rinse his sticky fingers.
He turned on the fountain, and the water ran thick red. It looked like blood.
He shut it off, but continued to stare at the red splashes in the basin. Practical joke. Sick one. Someone would hear about this.
He reached back to his desk and pattered around the messy piles of records, feeling for his transceiver.
Suddenly there was a movement and a bouncing clinking as the transceiver arced over his head into the fountain basin.
He grabbed for it, but before he could swivel around to see who had tossed it, a fist closed on his hair atop his head. There came a whistling swish and a current of air on his face. He blinked his eyes shut.
Instantly, he lost all feeling in his body—except for the damnedest dizzy sensation of rising. And he couldn't swallow.
Upon opening his eyes, he saw that he was rising, being pulled up, weightlessly, jiggling and bobbing, by his hair.
His eyes shifted down to a body falling from his chair and spurting blood from a sheared stump of a thick neck.
The severed head remained conscious only fifteen seconds.
It was a long fifteen seconds.
• • •
Nineteen days passed, and time stopped. The last day of Shandee seemed to drag on forever. Layla rapped the hourglass on the table until it broke because it was not moving.
Alihahd still kept Earth time—as valid a schedule as any in this sunless prison. For him the days—Earth days—numbered twenty-four and a half so far. His rituals were rigid and changeless. His patience seemed limitless, his temper ever cool day after day. He was coping best of the group in confinement.
The heat was on again. Harrison Hall took off his coat. He pulled a gold thread from the frayed lining of his vest. Without his heavy rawhide redingote one could see how tight was his waist and flat stomach. He was a powerful man.
"What country is Jerusalem in?" Hall threw out the question casually, all his attention seemingly on the loose threads of his vest.
Alihahd laughed with more bitterness than he had ever let show. "God knows," he said. Then, "Depends on whom you ask."
Hall knew that. It was why he had asked the question—to discover where Alihahd's loyalties lay. And he'd received his answer: nowhere. Alihahd was not radical anything. His sympathies were everywhere, allegiance nowhere.
"That little shred of land on the coast has always been a hotbed of violence and contention. Whoever has it is perpetually at war to keep it," Alihahd said.
Layla was listening while she combed her hair, ripping out the snarls rather than untangling them. "If the land is so hard to keep, why does not everyone move?" she said.
"Why, indeed?" Alihahd said with a sorrowful laugh, his blue eyes watery, bright, sad, mystified. "It is God's land, you see. Everyone's religion bids him fight to keep that land."
"And what are you?" Hall said. "Religion."
There was a hesitation. "Baha′i," Alihahd said.
"What is that?"
"A renegade branch of Islam, but rather similar to the Na′id without the imperialism and without the armies."
Hall did not believe it. Perhaps—just perhaps—Alihahd was a convert, but Hall was dead certain that Alihahd had not been born into that faith. Alihahd had been in an army. He had been disenchanted with something. Baha′i was a later choice. Or a lie. Hall did not think Alihahd believed in God at all.
Besides, there had been no Baha′i at Jerusalem.
Alihahd had finished drinking two cups of tea. Harrison Hall poured the third round. Alihahd lifted his third cup to drink, looked inside it, and froze. The cup was half full.
All color drained from his face. His lips twitched. Nothing else moved. Some part of him was shredding away behind his glassy blue eyes.
Hall held the kettle with an expression of false innocence: _Is something wrong?_
Alihahd set the cup down clattering, spilling the tea. He pushed back from the table with quaking hands, stood woodenly, knocking his stool backward, and he stumbled over it to get away. He staggered toward the doorway to the bedchamber. Long arms reached for the lintel to hold himself up and drag himself through.
The others thought he'd been poisoned.
Layla didn't know what had happened or how, but she knew who. She turned on Hall. "What did you do to him? You did something!"
Hall opened his arms with an exaggerated shrug to say, _Who, me?_ But his crescent eyes were merry, the sinuous lines of his beautiful wedge-shaped face upswept in barely contained glee.
A guttural cry came from the next room, and a crash and sounds of things being thrown and broken.
Hall had rearranged the cave.
With a groan, Alihahd hurled all the reordered jars against the walls and threw the twice-folded covers onto the floor.
Amerika and Vaslav cringed together at the table in the other room like children whose father was not well. They stared about them, their round eyes begging someone to make everything all right again.
Hall sat himself at the table, satisfied, and took up his pipe.
Alihahd returned, shaken, stiff, his face long and frowning, bloodless. He sat at the table. He clasped his big hands on the table, knuckles white, his back erect, teeth clenched, a muscle twisting over his jaw, eyelids stretched white around staring eyes, looking at no one. A muscle twitched under his eye.
He did not speak again until the days of the Witch Shandee were over.
• • •
The days of restraint enshrouded the Aerie, a somber time when the splendid warrior-priests, like drab birds in molt, put away their beautiful clothes for the thirteen days between the leaving of Shandee and New Year's Day. The air was calm. Warm breath still showed in frosty clouds, but the ice on the sunlit slopes was melting at last. Spring had already arrived on the middle slopes in a haze of blue buds on the trees.
Harrison slogged along a path sodden with spring mud and snowmelt. Leathery green shoots of emerald spikes braved the weather and poked through the ice on either side. He felt a cold trickle on his instep and knew that his old boots were finally done for. He turned his foot to find the crack in the thinned sole.
He heard a strange sound. His ear was attuned to patterns by now, and this one was not usual. He straightened and drew his gun from his belt.
As he crested the top, he relaxed and lowered his gun. Alihahd was clad in his red Chesite tunic and red boots. Hall was forming a comment about Alihahd's bony knees being the first sure sign of spring, but he perceived something wrong. Alihahd was smiling. Smiling wrong. Alihahd leaned against the rock face, shiny-eyed and wryly whimsical. His hands flopped strenghthlessly. Footprints in the slushy snow behind him wove on and off the path.
Hall pushed his gun into his belt and stepped down to take Alihahd's head in his hands. He looked into his eyes, searching for signs of fever. "What is wrong with you?" he demanded. Then he got a whiff of his breath.
Alihahd smiled. His hands moved elegantly, almost in caricature of himself. His eyes were pale, drowned sapphires. "Mr. Hall, I am drunk."
Hall's face darkened, and he pushed Alihahd's face aside—the motion was disgust—and he stalked past him in the direction from which Alihahd had come. Hall's head turned left and right with the sharp motions of anger—and searching.
Alihahd scrambled after him. His own blundering foot tracks blazoned a conspicuous trail, and Hall would surely find what he sought.
Alihahd drew himself up on a rocky bluff and thundered at Hall's back in the most commanding voice he could muster. "Mr. Hall."
Alihahd had once been a magnificent man and could still summon power, even drunk. Most of the orders he'd given in the past ten years had been issued in varying states of insobriety, and he still had the voice of absolute command.
"Stop! This is no concern of yours!"
With insistence, his dignity slipped a degree. He took a step forward from the rise and immediately felt his vulnerability increase. "Mr. Hall!"
Hall ignored him and strode away. As he neared his hidden goal, Alihahd became shrill and frantic. He stumbled after Hall. "Mr. Hall! This is none of your affair!"
"I am my brother's keeper," Hall said. "As the Na′id would say."
Alihahd scowled blackly.
Hall found the distillery behind a spreading thorn bush, and Alihahd lunged to put himself between Hall and his lifeline. He spoke in a deep, cultured, angry tremolo, each word precise, if thick. "Leave. Me. Alone."
Hall started forward. Alihahd laid hands on him to restrain him. But at the height of his strength, Alihahd had never been a match for Harrison White Fox Hall. Alihahd felt a shift of balance and a sudden helplessness. _Why is he doing this to me?_
Hall threw Alihahd to the ground. He marched up to the still, ripped out the condenser tube, and threw it at Alihahd as if onto a pile of rubbish. His driving heel came down on the soft copper still pot. Honey water gushed out over the coals. Hall kicked and scattered the painfully collected stack of firewood, and smashed all the clay jars of mead on the rocks.
Alihahd stood up, indignant. "Who do you presume to be? See the lord and master of Eridani. Thou art lord and master of nothing. And not of me!"
Hall rounded on Alihahd and gave him a slap that staggered him. His brain seemed to ricochet inside his skull. Alihahd tasted blood. And he fought, tried to. Alihahd's blows landed uselessly, or did not land at all. Then Hall's fist drove into Alihahd's solar plexus, knocking the wind out of him, and Alihahd crumpled to the ground. Hall picked him up and set him on his feet. Alihahd leaned on him, gasping. Alihahd lifted his head and in the next breath cursed Hall. Hall hit him again, harder, in the same place. Alihahd dropped, doubled over, sick and in pain up to his eyes. Hall stood over him, hands in fists, waiting for Alihahd to get up.
Alihahd crawled to his feet. Hall steadied him, then hit him with his open palm. And kept hitting him after Alihahd ceased to fight. The blows kept falling, and Alihahd could only wonder why.
And he cried, great drunken teardrops rolling down his sunken cheeks.
The blows stopped, and Hall hauled him to an icy steam and made him kneel. "Look at yourself." Hall forced his head down over a reflecting pool. Alihahd resisted with a groan, trying to turn away, but Hall's hands were like a vise holding him over the sotted, blubbering face in the pool.
Alihahd slapped the water with a cry, "I am trying not to!"
After a useless scuffle in the streamside mud, Alihahd was pulled to the rapids. Hall grabbed a fistful of blond hair and pushed Alihahd's head under the icy cataract.
Alihahd's hands scrabbled behind his head at Hall's fingers. Frigid water filled his ears, and sounds were suddenly muffled and singing, otherworldly, like death.
Alihahd was afraid that Hall was trying to drown him. But Hall let him up for air, then pushed him under again.
After the fourth time, Hall pulled Alihahd up by the hair to a kneeling position and searched his red face. Alihahd's eyes were shut and spilling hot tears. "Leave me alone. Leave me alone."
"Shut up and breathe, or you'll be sorry."
Alihahd breathed.
Hall could tell when Alihahd began to sober up. He stopped whimpering in between dunkings. Hall released him and crouched back on his heels.
Alihahd, on hands and knees, hung his head forward, water streaming down his face from his hair. "I don't like you at all," he croaked.
Hall smiled.
Alihahd knelt looking up, teeth chattering, his face scarlet, his tunic soaked darker red down the front. Water drained from his ears and he could hear clearly again. He felt the cold air as if it were blowing through his head. He brought his fingers to his eyes that felt too big for their sockets. "Oh."
"Have you never been in battle, Captain?" Hall asked.
Alihahd answered shakily, "Of course."
"Jerusalem," Hall said.
"I was there," Alihahd said. "If you must know, I ran."
Hall let go of Alihahd with a bark of gleeful surprise. "You deserted!"
"I left," Alihahd said. He put his hand over his eyes. "You're a sadist."
"No, I'm not," Hall said. Then, "Just a little." He helped Alihahd to his feet, like helping up a grimacing doll of wood and rags. Alihahd brushed off the pebbles embedded in his purpled knees.
Hall was looking at the sky. "I think it's time to get dried and presentable, Captain."
Alihahd lifted his head in a mute question. _Why?_
Hall pointed up.
Alihahd narrowed his too-big eyes at the daylight sky, and emotion rushed with sober clarity.
A slow-falling star.
• • •
A spaceship descended and sank into the layer of cloud that hung over the valley.
Alihahd went to the Chamber of the Golden Dome to be there when Roniva received her returning warriors.
Alihahd's hair, newly dried, kicked up in wispy cowlicks. He had changed clothes. He wore a dry gray tunic and sandals. His eyes were red and puffed, his face sporting bruises of all colors.
Roniva spoke to him directly. "Art thou ill?"
"He's fine," Harrison Hall said.
Alihahd's teeth clamped tight, and he stood rigid in silence.
Roniva appeared a harsh figure without jewels or soft colors. She wore a dark green robe. Her hair was confined atop her head. She beckoned Alihahd to her with a ringless hand, and he moved around the low lattice barrier onto the dais with her. She took his arm and made him stand beside her throne. "Stay thee here with me. I shall speak through thee."
And when two strange warrior-priests entered and greeted her, she said to Alihahd, "Ask my warriors why they return after a single year. What is wrong?"
In answer, the travelers unfolded a long tale of a series of mysterious attacks on Na′id installations. Reports of a man with a sword and a hawk, who left many dead behind him. The warriors had been following the deadly trail, site to site, but could not catch up with the assassin.
As Roniva listened, her face smoothed in slow-growing shock and dread, then hardened into eaglelike planes. Her long fingers closed around her sword hilt and tightened until her midnight skin paled.
Her warriors opened their arms, powerless in the face of an unacceptable conclusion. "An Itiri."
"A Wolf," Roniva said. "A wolf that walks like a man."
"We would stop him," said one warrior-priest. "But we know not where he is now or whither he will go."
"Has Omonia Station been hit?" Alihahd asked.
Roniva's head turned sharply toward him. Her voice was speaking without her. Roniva looked again to her warriors and bid with voiceless command in her burning onyx eyes: _Answer that._
The warriors exchanged doubtful looks. "Omonia? Not that we have heard."
Roniva tilted her dark head up to Alihahd, her intermediary who spoke with a mind of his own. "This is significant?" she asked.
"I think so, Fendi," Alihahd said. "There is a pattern to the attacks as they've been reported."
"What pattern?"
"They follow a path a conscript of the Na′id might take from first capture to final assignment. First to a conscription force base for classification. Then to a reloc center and orientation camp. Then to a training center like Alpha 4. All of Alpha 4's trainees begin service in Omonia Station. If the pattern holds, Omonia is where Ben will go next."
"Ben is dead," Roniva said.
"Call him what you like, he will attack Omonia Station," Alihahd said. He was peripherally aware of tawny eyes studying him from a shadowed corner of the Great Chamber. Harrison Hall watched him talk to Roniva as an equal, a contradiction of the sniveling human wreck at the streamside.
"Tell my warriors to take a ship and stop him," Roniva said. "Take _Topaz_. Thou shalt guide them." Her face softened and she gazed up from her throne at Alihahd's bruised face and red-shot blue eyes. She reached up to touch Alihahd's chest. "And I fear I shall lose thee once I let thee aboard one of my spaceships, thou so restless to be gone."
"I shall not return if I have any choice," Alihahd admitted.
"Then I bid thee, once thou art free, leave us lost to humankind, if thou carryest any memory from this Aerie. Bring us no human war."
"It may be too late for that, Fendi," Alihahd said, thinking of Ben-Tairre loose on the Empire, even now on his way to Omonia Station.
"This I fear," Roniva said. She did not look frightened. "Go now."
"Fendi." Alihahd bowed. He turned away in a forceful motion and took long strides to the archway, on fire with mission, freedom, and command. The Itiri warriors fell into step behind him. His sinister shadow, Harrison White Fox Hall, appeared at his flank. Alihahd swept out of the nine-sided chamber into sunlight.
He could not believe he was free. He was actually leaving this alien world after all this time.
He found himself oddly reluctant.
# PART FOUR:
## True Colors
# 15. Rogue Wolf
AS THE SPACESHIP _TOPAZ_ broke through the ceiling of clouds into bright sunlight, Alihahd glimpsed the twin mountain crest of the Aerie gleaming with ice-glazed brilliance in the snowy sea of vapor. The ship quickened its ascent, and the great mountain became small, then lost entirely. The atmosphere thinned, cleared to black, and the _Topaz_ was in space.
With his hand pressed to the viewport, Alihahd watched the blue world dwindle. With a jump to faster-than-light, it was all gone.
He was overcome with sudden déjà vu—the depressing sameness of starships, the isolation of faster-than-light. The familiarity of it overwhelmed. He had for so much of his life been a nomad living onboard one ship or another that coming back to this life was like never leaving it. The short year on Iry was reduced to a dream, as if it had never happened.
He was free at last. Why was he feeling so ill?
He knew what was waiting for him. He wished to anyone's god that Hall had not destroyed his hard-made mead.
He glowered aside at Hall, who lounged easily in the next seat, his feet up on an unoccupied seat across from him. He was puffing Iry weeds in his one-eyed fox-head pipe. He wore his bandana gypsy-fashion. His teeth flashed white in his dark bronze face.
Alihahd wore Na′id type clothing that Amerika had made for him at his instruction. She'd expressed a liking for the design. Hall had told her to bite her tongue.
There were four Itiri warrior-priests aboard _Topaz_ along with Alihahd, Hall, Vaslav, and Layla. They had a plan to intercept Ben at Omonia Station, take him back to Iry, and leave Alihahd behind on Omonia under one of his many Na′id guises. An elaborate plan, full of contingencies, it left all kinds of openings to go wrong.
Things went wrong before it had even begun.
_Topaz_ sublighted for its approach to the space station and arrived to silence.
The great polymer triple torus of the station turned in the sunless void, dark as if deserted, without ships coming or going, without radio chatter, without signals of any kind.
There was no question of what had happened. It was only left to wonder if anyone was yet alive on Omonia and if Ben was still there.
The Itiri ship _Topaz_ attached itself to a station lock and sealed the passage. _Topaz_ 's lock opened. Omonia's lock remained closed against them. The Itiri were about to blast it with their taebens when Alihahd motioned them aside. "Don't do that."
"Canst thou open it?" the eldest warrior asked.
Alihahd moved to the manual access panel at the side of the lock, keyed in an imperative code, then crouched before the sensor and waited for his retina to be scanned. "I can open any lock in the Empire," he said. They were in Alihahd's territory now. Na′id ships were a world he knew.
Harrison Hall came to crouch at his side, his gun drawn and resting across his muscular thighs. "How about the Bel's door?" Hall said. It was supposed to be a joke.
"You want to see the Bel?" Alihahd said very softly and without humor. It was flat affirmative.
Surprise rippled Hall's perennial nonchalance. That level of infiltration into the Na′id security system was beyond belief. That kind of clearance simply could not fall into the hands of an imposter.
Omonia's lock spun in answer to the imperative, and it released. The outer iris dilated. The lead warrior gave the inner hatch a push with his bare foot, and it swung heavily open.
From the first moment of the lock's opening came the assault of wrongness, alarms blaring, miasma of death, smell of blood.
The Itiri bounded through the hatchway like hunting cats and fanned out, listening, looking.
Alihahd followed them aboard. He had originally intended to dye his pale, telltale skin, but there was probably no one left in here he need deceive.
Hall shadowed close behind him with ready weapon.
They came upon the first corpses right away. The slaughter was absolute, the station corridors splattered with blood as far as could be seen.
There was a squeak from Vaslav, who had crept out of the _Topaz_ and tripped over a head torn from its body, which lay several paces away from it.
The neck was not severed. It was ripped. The head had come off in the hands of someone of great strength. The wide pool of blood was not yet dry. The killer could have simply choked this man, or broken his neck. This was overkill, tremendous force used out of no necessity, no reason. The lack of control, the madness of it, was un-Itiri. It was human. The kind of thing humans liked to call inhuman.
Alihahd had backed away and flattened himself against the bulkhead, sweating, face paled to ash, lips bloodless. Something darker than fear haunted his white-ringed eyes. He was shaking, his neck stiff.
Then he steeled himself in a deliberate moment. His head lowered until he was staring out of the tops of his eyes. The crevices in his face deepened, his breath came shallow as if loathing the air, and he moved away from the wall, not talking, meeting no one's gaze, sickened, dangerous. Layla did not know him. Even Hall would not approach him when he looked like that.
The Itiri gave him a wide berth. In search of Jinin-Ben-Tairre—if he was still in the station—the warrior-priests divided and glided down the corridors where they saw the blood was warmest. Sound of life—terrified life—could be heard retreating before them. Children's whispered shrieks and desperate scrambling sounded in the air vents.
Alihahd was eerie, transformed into someone else altogether. He was clad in the dress of an abhorred people that made him appear so strikingly one of them that it was hard to see him otherwise. Na′id square shoulders with epaulets, a tapered waist, full cuffed sleeves and trousers—the clothes made him a hated silhouette, and Alihahd carried himself differently in them.
He opened a wall panel, shut off the clangoring alarm, and turned up the air filters, all with the throwaway ease of one who has handled such controls many times. He swept down a corridor where the Itiri had not gone, stepping wide over the bodies in his path with little pause, wrath growing with each one, and he stalked out of view with the menacing military stride of a new master taking over.
Hall chose a different route, and Vaslav came with him, afraid of Alihahd. Layla struck out on a path of her own.
• • •
Claustrophobia and carnage.
Alihahd walked the ravaged corridors. The soft _stick, stick, stick_ of blood on his bootsoles sounded loud in the dead quiet. He stepped over more bodies, human bodies, then paused to kneel beside one he thought was still alive. He touched his fingers to the neck.
Not.
He closed another's eyes, rose. The mode of death this time—all down the corridor—was a long blade.
That would be a double curved sword.
_Menin aeide Thea_ . . . He remembered snatches of a song of slaughter. _Sing of the destructive wrath . . ._ It ran through his mind in rhythmic chant as he stepped over oozing dead.
. . . _that hurled many brave souls of heroes to Hades and made their bodies a spoil for dogs and carrion birds_ . . .
From time to time he heard the scurrying of children. There was only one child's body in the corridor—a little girl with a gun in her hand. The rest of the corpses were adults.
. . . _and so the will of God was done._
A pain pushed itself into Alihahd's consciousness. His jaw hurt. His teeth were clenched and the muscles in his face had been tensed into a deep, immobile frown for a long time now.
He made his muscles relax. His face ached.
He continued down the passage, paused again to bend over a twitching victim, not sure whether she was alive or this was rigor. An Itiri would have been able to tell at a glance if the body was still giving off heat.
He had just begun to crouch down when he heard heavy running footsteps from father down the curved passage.
A Na′id engineer came stampeding around the bend, then skidded to stop at the sight of a tall, blond figure bending over a bloody corpse. Alihahd looked up at him, showing his white face. The engineer screamed and fled.
Alihahd stood up. His chest felt tight, his breath constricted. The Na′id at his feet—alive or dead—writhed ignored. Alihahd stepped away until his back was against a door.
There were adults alive in this section. In past attacks, Ben killed all the adults. The butchery in Omonia was not finished yet. Alihahd held his breath.
Ben was still here.
Suddenly the door slid open behind Alihahd and he spun.
There was a movement directly before his face—Jinin-Ben-Tairre spinning with his sword raised, the sword striking down on the Na′id-clothed figure in the doorway.
# 16. Ghosts
THE BLADE DEFLECTED down Alihahd's side in last-instant recognition, slicing flesh and nicking bone at the point of his wide shoulder, slashing open the full sleeve of his Na′id shirt.
Ben coiled back with his sword, and both men froze.
Alihahd gazed into feral eyes—dark pits that went down and down forever. He saw all the horror and hatred in them, and all the pain.
Less a man now than a maddened wild animal, Ben stared with half-sight. His consciousness did not register Alihahd, only a white face, a white man in Na′id dress, and the sight incensed him. His breath came audibly, a rough panting of rage. Then he blinked. His brows tightened together. True realization penetrated that this was Alihahd standing before him—
And Alihahd was not certain that Ben would not kill him anyway.
A trickle of blood wended down from Alihahd's shoulder, prickling at the hair on his arm. The ragged flap of his sleeve hung loose from his cuff. Alihahd couldn't move, couldn't talk, even to save his life.
A movement behind Ben broke Alihahd's trance—a man with a gun shouldering his way through the circulation vent and taking aim at Ben's back. Ben's Itiri awareness must have failed him, for it was Alihahd who sang out, "Behind you!"
Ben dropped into a spinning crouch, drawing and firing his taeben.
The Na′id gun clattered to the deck, and the dead man slid out of the vent and flopped on top of his weapon in a heap. From the narrow passage from which he'd come sounded the hasty scrambling of a companion retreating.
Ben didn't chase the survivor. He turned back to Alihahd, all his weapons lowered, his dark eyes full of wonder, wordlessly asking why Alihahd had warned him.
"I do not know," Alihahd answered harshly as if the question had been spoken aloud. His voice was thick and he was shaking. "I will assume there is a reason for this." The carnage he meant. There had to be a reason. Alihahd gave him the benefit of a gigantic doubt, but let it be known he granted this grudgingly.
Only then did Alihahd look down to inspect his own wound. He touched his cut arm and shoulder with his opposite hand, assessing the extent of the damage, which turned out to be minor. It would leave a pitted scar in his shoulder.
Alihahd turned to go. The light touch of two fingers on his unhurt shoulder stopped him. Alihahd paused, startled by the contact. He turned his head to look back.
Ben sheathed his sword and holstered his taeben. He drew his dagger slowly as if it were painful to him and held it to his own leg. In his pause, Alihahd saw Ben fighting down great fear as if he intended to cut himself. But Ben slashed open his trouser leg and he turned for Alihahd to see.
A blue tattooed number marked his thigh, the brand of a Na′id child conscript.
Bitterness rose in Alihahd's throat. He swallowed. The bitterness stayed.
Why hadn't he guessed? Aliens had not been the ones who taught Ben to kill. They had merely perfected it.
Shock, hatred, and all the rest of it subsided into weary sickness. Alihahd reached back his hand and bade Ben come. "Well, let us be on."
Ben put his bloody hand into Alihahd's and padded after him on silent feet. It was like leading a leopard from its kill on a leash made for toy dogs. The leopard meekly put himself under Alihahd's authority.
What made Ben think he could turn to Alihahd? He could, and knew it. _So you see that in me,_ thought Alihahd.
Alihahd wished someone had been there to take _his_ bloody hand.
He led Ben back to the _Topaz_ and broadcast three words through the space station's public-address system: "I found him."
With that, the Itiri came gliding back like ghosts. Not one of them saluted Ben. One did not talk to the dead.
Layla did. She callously saluted with her dagger. "Good job," she said and boarded _Topaz_. Ben masked reaction if he felt any.
Alihahd took Ben to a solitary unlit compartment on board _Topaz_. Neither of them wanted the light. Ben walked into the cabin slowly, then circled back and abruptly offered his sword to Alihahd, hilt first.
Alihahd regarded the sticky hilt in surprise and quiet distaste. "I do not want it," he said, almost chiding. What was he supposed to do with it?
Ben let it drop at Alihahd's feet, withdrew into the small space, sank to the deck, folded his legs under him, and shut his eyes.
Alihahd closed the door on him and left him there. It occurred to Alihahd only outside that maybe he was supposed to execute Ben. Maybe the Itiri expected it, too.
They were all mistaken.
Alihahd went back through the air lock to re-board Omonia Station.
Vaslav was returning to the lock, a green cast to his face, cringingly following Harrison Hall, who carried himself with the jaunty air of a man taking a stroll on the mountain, no more affected by the mass murder of the Na′id than Layla had been. He stepped over the bodies in his path. Vaslav skirted them.
Corners of a smile disappeared into Hall's mustache on seeing Alihahd. Whatever grim possession had held Alihahd in its grip earlier had released him now. Alihahd looked like Alihahd again.
Hall came to stand with him, his weight casually on one foot. He swept one side of his coat open and back to tuck his hand into his hip pocket. He glanced about him for something he did not actually expect to see. "God damn," he said.
"Mr. Hall?" Alihahd prompted. He pushed back the sweaty bangs that stuck to his forehead, and unfastened some of the buttons of his double-breasted shirt so that the outer flap fell open. Blood still oozed from his shoulder and wetted the tattered edges of his torn sleeve.
"There isn't a damned ship on this station that's flight-worthy," Hall said. "The hangars are a mess. He's blown off half the locks. He hit all the ships—including his own. It seems this was the last stop. There is no way off this station, Captain, except the way we came." He nodded through the lock toward _Topaz_ , which would take them straight back to Iry.
Ben had been appallingly thorough. He hadn't even left himself an out. What had he intended to do?
"It looks like I go back to Iry, Captain," Hall sighed. "What about you?"
Alihahd hardly heard him. He was still thinking of Ben. _He wants someone to kill him_. Ben had destroyed his own ship _Singalai_ because he intended to die here. Dying was not so easy sometimes. _Now begins the real horror_. Alihahd looked down at his own hands, and marked how easily the blood seemed to get under the nails and stay there. He could not seem to keep it off.
"Captain?"
"What?" Alihahd said, summoned out of his thoughts.
"You coming or staying?"
The Itiri pilot of _Topaz_ had come to the hatch and was also waiting for Alihahd's answer.
Alihahd frowned. He was off-planet at last, on familiar if ravaged ground. He could not turn back, having come so far. It was only a matter of time before Na′id ships would come to Omonia. Alihahd might be able to blend into their ranks if . . . there were many ifs.
Alihahd beseeched the Itiri, "Give me a few minutes."
The pilot lifted his chin. "Be thou quick."
Alihahd gave a single nod and ran toward the station dispensary. Up ahead of him all along the way he heard the furtive whispers and scuttling of a flock of preteen girl conscripts taking cover at his approach. One made the wrong move and blundered square into him. She lurched to a flailing halt, a splay of gangly adolescent limbs with distended mouth and eyes. She screamed, "Nazi!" and she sprinted away on her long legs.
No matter who said it, the sound of that epithet never failed to cut like a dull blade.
Alihahd busied himself searching through the dispensary stores for melaninic. Vitiligo was a common condition among the Na′id, and their installations always had a supply of the drug. Alihahd found it and seized up the bottle of pills in his fist with grim relief. Half of his problem was solved. He could melt into the Na′id ranks with his skin a respectable shade of brown. He was not going to be called a nazi again. It was not his fault he was born blond, blue-eyed, and pasty white.
He left the dispensary and quickly made his way over the stiffening bodies in the corridors to the com center. He lifted a corpse out of a chair and sat down at the controls. He switched on the transceiver without video and immediately received a signal from a ship captain trying to make contact with the silent space station. "This is _Sharru Sennacherib_. Please acknowledge, Omonia. Where is your ID signal? What is your status?"
" _Sharru Sennacherib_. This is Omonia," Alihahd answered. "It's a disaster." That statement carefully told nothing.
"Omonia, elaborate," the captain demanded. "Are you in immediate danger?" The woman's voice was familiar.
" _Sharru_ , everyone is dead. When can you be here?" Alihahd infused some calculatedly disjointed panic into his too calm, too military voice. Because he thought he knew the woman, and she might recognize him if he spoke clearly. Wide as the galaxy was, voices such as his that were clear, deep, and cultured with correct speech were rare.
Despite his precaution, the woman's reply was tainted with curiosity. "ETA twenty minutes. Omonia . . . Who is this?"
And all at once Alihahd knew her. No mistake. _This will not work_.
Alihahd disconnected the transmitter and sat back in his seat, his thick lips pressed together in frustrated thought. He could not bring himself to retreat. Not now—when he was actually on board a Na′id station, with melaninic in his hand.
But this ship captain knew him. No matter the disguise, he could not slip past her.
Having the woman eliminated did not even enter consideration. Alihahd would not kill another human being, not in defense, not in war. Never again. He had sworn. He could not do it, or everything he lived for was nothing.
Harrison White Fox Hall came to the com center, paused to make sure Alihahd was not transmitting, then he said, "Time's up, Captain. They want a decision." They, the Itiri.
Alihahd put his hand over his eyes, chasing down every possibility and squeezing his brain for more. So close. The same answer returned again and again. He rose, tapped Hall's arm, too disappointed to speak, and they ran together back to the _Topaz_.
• • •
Alihahd slouched down low in his seat aboard the spaceship, the length of his wounded arm resting flat on the armrest, his other arm flung loosely across his abdomen. Emotions were several and jumbled, fear of the Na′id twined through everything.
Was it possible for the Na′id to figure out what had hit Omonia Station from the tangled evidence left behind? An ungodly hecatomb, the wreck of Ben's two-hundred-year-old ship, a distress call in a voice that sounded like someone it could not be, and Ben's footprints in blood. Those could be identified by the Na′id computer if it searched the database of the dead. Na′id were singularly negligent in checking the rolls of the deceased. They assumed a person could not act simply because the computer thought he was dead. Someday they would learn not to be so certain.
And on top of the physical remains at Omonia would be the stories stricken children told of a semihuman juggernaut, of Itiri warrior-priests, and of a nazi.
Alihahd remembered all the people who had encountered him and had seen his face. There would be other stories, too.
Into his thoughts Harrison White Fox Hall strolled down the aisle of the spaceship that had once been a passenger craft, and he let himself fall into the seat across from Alihahd. He slung one leg over the armrest and took out his pipe. As he lit it, his beautifully sinister orange eyes slid sideways.
Alihahd made an extremely convincing Na′id onboard Omonia. Gone for the time had been the worn-out, pacifistic, sometimes wet rag of a man flagellated with self-doubt and guilt. Left behind had been a soldier to be feared. Hall admitted to being impressed, and even a little frightened—as near to frightened as Hall could feel. There was considerable strength left in Alihahd. And madness.
At this moment, Alihahd was some kind of hybrid creature. He had returned to his customary docile brooding, but he was still in Na′id dress, and some vestige of the commander—the mad one—lurked just below the restored calm. Hall had seen the insanity before, but not its power. And still not its source, the horrible goad from the past that drove Alihahd to heroism and despair. Memory of cowardice? Something was not right with that image.
Hall spoke between languorous puffs on his fox-head pipe. "You're subdued, Captain."
Alihahd sat up. "Should I not be?"
Hall fanned away the gray billows wreathing his head. "Where's your righteous indignation?" Alihahd was always so piously magnanimous to any human foe. He ought to be saying something eloquent for the slaughtered Na′id.
Alihahd shook his head. "No."
Hall pointed at him with his pipe's mouthpiece. "You hate them after all."
"He has a number," Alihahd said, his gaze directed down at the deck between them.
"What was that, Captain?" Hall cocked an ear, better to hear Alihahd's soft murmur.
Blue eyes lifted to Hall's. "Ben-Tairre. He has a number. A blue one."
"That's a boy soldier, isn't it?" Hall said.
"Yes."
This was interesting. "You've forgiven him already," Hall said.
Alihahd slumped back in his seat. His shoulder hurt. "I was thinking."
"Yes," Hall prompted.
"At his age and that number sequence, he would have to have been a trainee roughly sixteen years ago. In fact, he was most likely assigned back there at Omonia."
Alihahd wondered what they'd done to him to make him come back at them like that. "Had he remained with the Na′id, he would have been assigned to an army—in all probability to one of the three armies that fought the Jerusalem campaign. He might have been butchering Jews and Arabs and Christians for Shad Iliya instead of butchering Na′id at Omonia today."
"Maybe he was in Shad Iliya's army," Hall said.
Alihahd felt a shock, then reason returned. "No."
"Why not?" Hall said. "Could be why he hates your looks."
Alihahd was blond, blue-eyed, and fair, like the infamous general.
"He was not in Shad Iliya's army," Alihahd said.
"How do you know?" Hall said. "You left."
Alihahd hesitated, treading around the edges of a nightmare. He answered deliberately, almost as if thinking up reasons as he spoke. He did not want Ben to be at Jerusalem. "There is no red suffix on his number. He left the Na′id army when he was still a child." Alihahd's voice dropped even lower. "What they do to child conscripts is rather harsh. It washes away their pasts and instills a rabid loyalty. The conscripts make more enthusiastic soldiers than Na′id native-borns. But the conditioning methods are extreme. They backfire sometimes. . . ."
Ben had become a Na′id boomerang. Curious, thought Alihahd, the boomerang was one weapon the Itiri had never developed.
"Revenge, Mr. Hall, I understand. Emotion is a human thing. Blind hatred I understand. It is not my place to forgive or condemn. I simply recognize it. It does not horrify." And underneath the words his eyes spoke: _That is why I can endure you, Mr. Hall._
Then his tone changed again, became distant. "Killing for an idea. Killing for justice. Killing for the right. That is horrifying. And that is wrong."
_And killing and liking it . . . I don't know what that is_. . . .
• • •
Guilt. Alihahd felt worse and worse the nearer he came to Aerie, bringing back Jinin-Ben-Tairre—or whatever the man was supposed to be called after the Itiri revoked his name. Try as he might, Alihahd could not justify handing over one of his own kind to a race which had made Ben what he was then disowned him and now presumed to decide his fate. And, upon arrival, Alihahd stopped trying to justify it.
The Fendi Roniva was waiting for the crew of the _Topaz_ and its cargo—Ben-Tairre—within the Chamber of the Golden Dome. Her sword was drawn and flashing in the light of the newly built fire which crackled in the cold hearth. The Fendi was clothed in midnight colors without ornament in these last somber days before the new year, and the dull gleam of the tungsten-plastic blade seemed very bright against her dark figure. In the frigid air, her breath barely showed a frosty cloud, having little moisture in it—and little warmth, Alihahd thought.
Roniva stepped down from the raised dais and advanced, barefoot on the icy tiled floor, toward the captive warrior Ben. Her black knuckles paled on the hilt of her sword. Ben himself was still. It was an inert kind of stillness, like a pale ivory carving. Someone had cleaned the blood off him. His eyes were unfocused, unless they were looking inward. He appeared unaware of what went on around him, unaware that Roniva meant to kill him. And as Roniva drew close to him, her sword arm beginning to lift, Alihahd stepped in between them, barring the Fendi's way.
Her ebony skin and crimson scars rippled in an expression of disbelief and impending wrath.
Then Roniva's alien face, with its too-keen lines, its hairless brows, its narrow nose and hard, carved cheeks, abruptly snapped to the side and she barked at an attendant, "Tell this man he is to move aside."
"No," Alihahd said with quiet force that gave even the powerful Fendi pause. And he heard himself continue, almost as if someone else were speaking, using his body as an instrument. "It is not your battle."
Roniva's head pivoted back so that she was fully facing him once again, her expression more and more astonished by degrees. She couldn't even talk, astounded to have heard an Earth man speak Itiri law at her.
Alihahd was at least as surprised as she, but he pressed on. "Jinin-Ben-Tairre is dead. I heard the Fendi say so. This man is a human being, the Fendi made him so. Have the Itiri a war with humankind?"
Roniva struggled for words and someone to speak them to. She would not talk to Alihahd. She presented her back to him and gave her face to her snowy owl, which was perched on her throne. She screamed at the owl, gesturing with her sword, her voice ringing off the domed metal ceiling. "It was not meant that this dead warrior take all that the Aerie taught him and use it in a war that is not of the Aerie. It was meant that he _go home_!"
"The Fendi neglected to consider that perhaps he does not have one," said Alihahd coldly—to the owl.
Roniva wheeled around, her sword hilt in both hands, her eyes flashing rage.
Alihahd went on unwavering, but speaking directly to Roniva again, "The Na′id were there. This home to which he was to return may not exist anymore. This is his battle, his and the Na′id's. I thought that was permitted even under your rules." His tone grew bitter and sarcastic. "But then, who is bound by which law seems to change at the Fendi's convenience. Is this Itiri justice?"
"Justice?" the Fendi said. She turned her head away, addressing herself to her snowy owl. "I have told this one that there is no justice here."
"Yes, I can see that!" Alihahd said, abandoning his stand between Roniva and her intended victim, and making straight for the chamber's exit with long angry strides.
Roniva drew herself erect, extended a pointing arm after the retreating man, and shrilled, "He is to hold!"
Alihahd didn't stop. Even when two armed Itiri warrior-priests stepped into his path, blocking the bronze archway, he pushed through them, buoyant with the death threat.
He was surprised to make it outside to the snapping cold gray dawn. No one had used real force on him yet. The Itiri could afford their customary patience and caution. A thin, aging Earthman wouldn't get far if Roniva invoked her true power. Alihahd kept walking.
Under the golden dome, Roniva stood in rigid, quaking rage.
Suddenly she dropped from her frozen pose, abandoned her attendants, her intermediaries, her victim, and she bolted to the arch. Pushing between the same two warriors Alihahd had passed, she ran out onto the Ledge Path shrieking after him, _to_ him. "No one is given what he deserves on the Aerie! _There is no one to decide!_ "
Alihahd halted. That sharpest of all thorns stabbed into his most vulnerable point. He who led, who decided so many fates, who was law, judge, and jury, and so often wrong, he had presumed to a wisdom which Roniva never pretended to possess—to be truly just.
Alihahd turned back. He walked to her and spoke quietly. "You took everything from Ben, and you turned him loose with nowhere to go. He is human. He fights a human war. The odds were one against the trained military personnel of four army installations. By any measure, how can you condemn him?" His blond hair kicked across his tall brow that was permanently etched with deep lines.
Roniva sighed. "Ah. I should have killed him at the first, truly killed him. This would never have come to pass. One cannot separate the dead Itiri from the live human as I tried. My mistake. I should have killed him."
"That is beside the point," Alihahd said. "The past is fixed. What will the Fendi do now?"
Roniva brandished her sword. "Correct my mistake."
"He is human," Alihahd said. He raised his voice only slightly. "As I am. You kill him, and we are at war—thou and I."
Roniva stepped backward, her sword before her as if there were real danger from such a dissipated being. For a while she didn't speak.
Then she sheathed her sword. "O, by the sun, what am I to do with him?"
Alihahd wondered if there was actually anything to be done. He and Roniva were arguing over a man who could decide his own fate. Guilt, Alihahd knew, was the greatest of crippling forces. Alihahd recognized the sudden docility when he found Ben in the space station. After ultimate rage came the mind's death leap. This whole debate could be moot.
"I could resurrect Jinin-Ben-Tairre," Roniva said. "He would be mine again, not thine, and I could do with him as I will." She tilted her head slyly at her human adversary, but then she let her hard shoulders sag and her eyes soften in uselessness. "But I really have not the strength." It had taken everything she had to defeat him the first time.
"You could resurrect Jinin-Ben-Tairre and let him live," Alihahd said.
"Tell me how I could do that, _Fendi_ Alihahd," Roniva said ironically.
"Your original reason for killing him was...?" He lifted his brows and paused to let her remember for herself before he answered himself. "You feared the Aerie would make him Fendi—or so you told me. Do you think anyone would choose him Fendi, now?"
"No," Roniva conceded. And with the finality of a decision, a reprieve, she said, "No." Then she reached out and held her palm flat to Alihahd's chest as if she could read his heart through her hand. "But, tell me, wherefore this mercy to one who would have seen thee dead?"
"No mercy," Alihahd said. "To let him live with himself? I am being unspeakably cruel, Fendi."
"Ah, thou knowest not our Ben-Tairre." She reclaimed her warrior by the speaking of his name. "Thou speakest not of him." She crossed her arms and moved toward her throne chamber.
Under the archway she stopped, looked over her shoulder, her onyx eyes glinting from under the shadow of her brow. "So who dost thou know who draweth his every breath in pain?"
• • •
A silent ghost of the Aerie, his head and broad un-Itiri-like shoulders shrouded in a hermit's hooded black cloak, Ben moved through the halls like one without a soul, having no eyes for the world around him, his sight turned inward, lost and locked there. He drifted, mute, bearded, and strange.
Arilla shaved his face to make him appear more an Itiri, and so that he might know himself when he met his own reflection. She didn't call to him down in the darkness where he was. He would come out when he was ready, when he was healed. Or he would die. Or stay mad forever.
Alihahd passed through an arcade, his gaze chancing inside a stone vault where a shaft of sunlight fell from a single tall window across the hooded figure kneeling in meditation, a sword across his knees. Alihahd knew the shape and the winding of the thick muscles beneath the black pall of the concealing cloak, and was frightened by the sight. Not the sword, but the madness itself terrified. Visions of dreadful memory called to him.
Alihahd shuddered and passed by.
• • •
The door to Serra's cave swung in with a swirl of icy air. A shimmering apparition in black and gold extended a long black hand, jeweled and sparkling in starlight. The Fendi never came to Havenside, and all those within the cave, startled by the visitor, were at once on their feet with a kicking back of stools and falling over of cushions.
Gold pendants dangling on delicate chains from many rings were sent into wild pendulum dancing with the slow waving of the beckoning hand that was for Alihahd. "Come know us," Roniva said.
Behind Alihahd, Serra turned over an hourglass, and Layla whispered to Alihahd as he stood motionless and puzzled, "It is the first of the Sword."
Amerika ran outside to look at the stars, and comprehension came to Alihahd. The Sword was the first sign in the Itiri calendar. The new year had arrived, the year of the Opal Crown.
Alihahd joined the dazzling Fendi on the Ledge Path underneath the glowing field of the Milky Way. Amerika seized his hand and pointed straight up at the sky with a small gasp of discovery at a familiar occultation, a sign of the vernal equinox. And from up on the sentinel's ridges came a crash of bronzes. From all over the Aerie rose a squalling that made Alihahd think of a den of beasts trapped in fire. But the raucous sounds were the alien voice of celebration. Out of all the caves and arcades poured shrieking, bright beings. Torchlight blossomed on every terrace. Alihahd's senses were stunned by the sudden cataract of color after the muted days of restraint. All the ledges filled with the crush of celebrants, and on various levels the lively tap of wooden instruments started up different beats heedless of one another.
Roniva now appeared sedate in her shimmering black robe and gold pendants, standing tall and stately above the mobs of ranga that thronged the paths of Havenside. Her thin fingers closed on Alihahd's upper arm. Her hand was hard, but her grip was not tight. She spoke close to his ear to be heard without raising her voice. "Thou shalt see."
_Thou shalt,_ the Fendi said. It was not invitation. It was mandate. Alihahd yielded to the pressure of her grip.
As he was drawn away with her, he looked back to Amerika, whose hand had slipped from his. He was losing sight of the little girl as the crowd flowed into the widening gap between them. "Are you coming?" He had to shout.
Amerika hung back, her hands clasped in her full skirts, her round cheeks burning in a red blush. He couldn't have heard her had she tried to yell. She motioned no, her head down to the side, and she was enlisted into a gaggle of jubilant ranga to festoon the arcades with flowers and ribbons. Alihahd was pulled in the opposite direction toward the bridge, and he had to turn and watch his own step when he began to cross, starting with his right foot and counting twelve steps.
In the tempest of sound and color and firelight and motion through which he was led, one impression pounded at Alihahd ever louder like a wind brass: _inhuman_.
On one side of the abyss short, stocky cherubs bubbled in an alien speech, their skin white in the night except those closest to the torchfire, who had turned half dark. The ranga were cheerful.
On the other side of the crevasse, the warrior-priests celebrated with more imperative. All impression of humanity they'd ever given Alihahd jangled against what was jumping before his eyes and blaring in his ears. Civilization dissolved into feral howling and dancing. Alihahd's nostrils narrowed in aversion. Everything, down to the smallest details suddenly struck him very wrong—the warriors' nailless fingers, their long toes, their inhumanly lithe, white bodies, their neat white teeth without canines, their hairless limbs, their gem-green eyes, and the crimson burn scars on their smooth faces.
Roniva, as if sensing fear, firmed her grip and led him beyond the Aerie to a place where a bonfire blazed in a great stone pit partly sheltered by a tall, undercut cliff which formed a shallow cave. The smoking, flaming firepit was circled by savages—bright, dancing beings wielding naked blades, screeching wordless cries, and stamping the stony ground with bare feet. It wouldn't have surprised Alihahd at all to be thrown into the pit as sacrifice to some god. But at the pit's edge, Roniva let go of him and he was free to fade back to the fringes of the crowd, away from the fire's heat, into deep shadow, until he backed into a granite wall.
He felt dizzy. He wanted to leave. He wanted the cold, rough-crystalled rock to swallow him up. Whether from something in the smoke or the lateness of the hour or his own imagination run riot, an intoxicated feeling of unreality was overtaking him.
He saw Roniva in flames. She was dancing. Other warrior-priests followed her into the pit, leaping, becoming one entity with the tongues of fire, tearing the last common thread that bound the two species, theirs and Alihahd's. He closed his eyes and tried to breathe without smelling smoke, and to close his ears against the hideous sounds.
The dancers moved out of the fire and into the shadow of the cliff where Alihahd was cornered now. His hiding place reddened with the licking light of firebrands.
At another edge of the throng within the grotto, Alihahd sighted Harrison Hall and Layla. He made his way along the granite wall toward them, proceeding in starts, halting to avoid brushing the violently writhing bodies of the dancers who clawed the air in vicious motions as if tearing unseen enemies.
By the time he reached Hall and Layla, a cold sweat had broken out over his body. He hoped the flickering red firelight would mask his paleness. He swallowed. He hated aliens. And he really hated savages.
The Nwerthan Layla was a little barbaric herself. She was wearing animal skins. She had a gold-fringed shawl tied around her waist and a chain of berinx teeth hung around her neck, but she wasn't swept into the ferocious tide of the dance. She was just watching, her gay smile showing crooked teeth, a shine of exhilaration on her freckled cheeks.
Hall was a pillar of swaggering dignity. His worn, dusty brown redingote and rawhide boots were sober shadows amid the bright dancing colors. He clasped his hands loosely behind his back. The ruddy light played across the planes and hollows of his wedge-shaped face. He surveyed the spectacle, interested but removed from it, a gentle curl on his lips at some secret jest, like the master of the show.
His eyes slid aside toward Alihahd.
All the muscles in Alihahd's face felt slack but for a twitch under one eye.
"Are you unwell?" Layla's voice jarred him, but he couldn't speak.
"I didn't realize our captain was so tender-hearted," Hall said. "It seems we have shocked his finer sensibilities."
Alihahd turned his wide eyes back to the warriors.
Roniva danced in the center of the mob. Her robe was singed. Soot dulled her glossy skin. She circled around a level platform of stones that slowly and awfully took on the aspect of an altar.
Roniva's barking laugh reported above the rest, with no gaiety in it, only lust and cruelty. The sleeves of her robe were torn and they flapped in clawed ribbons from her elbows. Her laugh rose to a shrill scream, carrying a chorus of others up with it, so high it was a siren catching the harmonics of the rock grotto and swelling to a skull-splitting pitch. The sound peaked and shattered into cackles.
Alihahd backed against the granite wall and stiffened, breathing shallowly as if the air itself were poison. He wanted to shut his eyes but couldn't. Through the swirl of black smoke, he witnessed the Itiri descent into the dark elemental side of all that was noble and spiritual in them.
Their voices resolved into a chant. Out of the crowd two celebrants came lugging a spiny-hulled melon that looked like a quilled creature as big as a pig. They hefted the spined thing onto the altar and drew back into the crowd.
The chant grew louder. A sword appeared in Roniva's hands. She screamed a rapish scream. Tungsten-plastic flashed in a lightning strike onto the altar, splitting the melon in two. The pulp oozed out red as blood. The red juice pooled. Roniva dropped the sword and leaped onto the altar. She plunged both hands deep into the spiny hull. She wrenched out fistfuls of stringy, dripping insides and held them up toward the sky. She threw back her head and bayed like a wolf creature in primordial triumph, the red juice dripping onto her face and down her arms.
Alihahd ran away.
• • •
He was alone on the windy mountainside, off any track, away from the noise, when finally he stopped.
He hung over a boulder, panting. He groped along the rocks to a rising bank to hold himself on his feet. He'd taken a gash on his shin through his trousers. He'd run away in the dark as if pursued by wolves.
The predator that appeared in his tracks was Harrison Hall, stepping long in leisurely pursuit over the rising ground. Layla trotted after him, two steps to Hall's every one. Layla's impish face was soft in concern and genuine confusion. "What happened? Did it make you sick?"
Alihahd took a few stumbling steps along the embankment, still trying to get away, but they followed him. Hall was grinning. Other people admired Alihahd's horror of violence.
But was it violence that horrified?
Hall moved in. He leaned his hand on the rocks by Alihahd's head. Alihahd turned his face away, one hand holding onto the rocks for balance, one arm circling his stomach.
"No," Hall said slowly and smiled. "That kind of moralistic outrage doesn't come from one who's never known carnage."
Alihahd turned completely around, hugging the rocks, his forehead against stone. Hall leaned in closer, so Alihahd felt his breath on his cheek as Hall spoke. "Captain, you are much too civilized."
Alihahd moaned.
Hall seized his arms and forced him around to face him.
"Leave him alone!" Layla cried.
"Everyone knows that gentleness is the mark of a five-star sadist."
Alihahd lunged free and fled over the rocks, cross-cutting all the twisting paths, making straight for the Aerie. Hall's heavy steps behind him made him run faster.
He came over the last rise to the Aerie's amphitheater, and ran along the Ledge Path to the bridge. Even in his panic he had to stop and begin his traversal carefully with his right foot. He needed to measure and count in shaking breaths twelve steps across—with Hall hounding his tracks, sounding a false count in his ear.
"One, two, three, four—"
"Thirteen, fourteen, five, six, seven—"
Was that ten or eleven? Alihahd became jittery because he wasn't sure now. Or was that twelve? It had to be an even number. Had to be, because he was on his left foot, but he hadn't reached the end of the bridge. He was really too close for it to be ten. This was wrong. It was all wrong. There was a galloping in his stomach. His hands trembled on the guide ropes.
He ran to the other side. "Eleven, twelve."
Hall pursued him into the blackness of his orderly cave.
Hall paused just inside, fists on his hips. He was in no hurry. He towered, a menacing blot in the doorway, until his eyes adjusted. Enough starshine spilled through the opening to allow him to see the quarry he'd run to ground cowering within. He could hear the ragged breathing.
"You've had blood in your mouth, Captain. And the horror is that it did not horrify."
A sound like a swallowed sob choked from the dark. Alihahd tried to rush past Hall to the door, but Hall was everywhere in his path. "Tell me not. Tell me not. Look me in the eyes and tell me not." He backed Alihahd up against the wall, making a cage of his arms around Alihahd's head so Alihahd could look nowhere but at Hall. Silver-blue light from the glowing Milky Way washed all color from his face and paled his eyes, which were wide with echoes of horror.
It hadn't even been true savagery he'd run from. In the Itiri's barbaric celebration there was no pain, no fear, not even terror of a stupid animal. Nothing was tortured, nothing killed. The expression of viciousness was all without true pain. Alihahd ran from phantoms—the mere illusion of savagery. Brutality was not what was actually there. It was what Alihahd saw. He had brought it with him.
Hall had seen a woman splitting a melon. Wide, glassy blue eyes saw something else.
Hall touched his chin lest he look away. "Tell me not."
The eyes lowered and closed. Thin blond lashes quivered, then lifted again as Alihahd's face relaxed and left off its tense trembling. The face smoothed, iced over hard, and the eyes fully opened cold, cold, cold. His demeanor was controlled, unnatural, terrifying. For a moment, even Hall was touched by dread of what he had uncovered.
Resonant voice came from great depth and grew in power. "And so, Mr. Hall, what if you are right?"
"I am," Hall said, absolutely certain. When one set out to corner the devil, one had best be prepared for when he turns around. "You tell me what of it."
Alihahd's face transformed again to a third, unexpected guise, one that was inaccessible, unruffled, and sly. Hall was surprised. Alihahd was many things, but of all traits glimpsed, hinted at, or suspected, cunning was nowhere among them. Alihahd gave a very cool approximation of a smile as if to say this game was over and all that had happened up till this moment was a charade. "I think you are mistaken this time." He ducked under Hall's arm and walked away.
The perplexed hunter pulled in his carefully laid snares and looked for tears that were not there, and he wondered how his prey could possibly have slipped out. Hall sat on the balustrade of the arcade, one foot up, his back against a pillar, and tugged at his mustache.
Ranga with flowers in their red hair were dancing to a gentler music here on Havenside. Simpleminded beings, they had no demons to purge this new year. A ranga woman running through the arcade paused to hug Hall, and ran on. She was hugging everyone.
Across the abyss, Hall spied the lonely figure of Arilla dressed in brown. The fire clan of the carnelian serpent had nothing to celebrate.
A flying bug buzzed at Hall's face, and he snatched at it. He felt nothing in his closed fist, though he could have sworn he'd caught it. Slowly he uncurled his fingers to look. The bug flew out.
_I had him_. Hall clicked his tongue against his teeth. Then realization dawned, and he slammed one fist into his opposite palm. _I had him_.
He'd been made to believe his hand was empty, and he'd let his quarry go.
_I'll be damned_.
Hall swung his leg down from the balustrade and stood up. He'd been right. He had glimpsed Alihahd's true face, the one that belonged to his name.
So mild, so civilized, compassionate to a fault, Alihahd could be nothing but a savage, sickened and afraid of his own soul.
The Itiri warrior-priests were not above recognizing themselves and acting out their own barbaric roots. No species sprang perfect from the head of a god. The Itiri unleashed their barbarism on melons. They could face their baser nature. They had to. They knew what could happen when nature was refused. Locked inside, it could only be held for a time. And then the breaking was unspeakable.
So where had Alihahd met his secret soul and begun to run from it? Had it been something seen or something done? It must be that he'd done something. Alihahd was too contrite to have been a mere witness to horror.
But Alihahd said he'd never fought Na′id. He said he'd run from Jerusalem.
It was Jerusalem, then, that turned him. Sacred duty uncovered what was unholy in him and he ran.
Then which of the three Gods had he deserted? He didn't seem overly attached to any one of them. At first, Hall had been convinced he was a Jew. He was circumcised at least, and he had the attitude of a martyr. But now Hall was thinking Alihahd had to be a Christian. He was too forgiving to be a Jew, too indefinite to be a Muslim. And he had the self-destructive, ox-in-yoke, virtuous servitude of a repentant Christian sinner. He had to be a Christian.
Hall wanted him to be a Christian.
Because there was something else at Jerusalem he could be.
# 17. Enemies
ALIHAHD WALKED THE LEDGE PATH. Thawing rocks popped and snapped around him. Anything above freezing felt mild now, and Alihahd was back to wearing his tunic again. He could see white, pink, and lavender blossoms on the trees in the valley, and a green-yellow haze of new leaves on all the branches. Spring had come.
He viewed it darkly. His universe was closing in.
Something nudged at his heels. He glanced down. His heart skipped in the momentary shock of the unexpected—a big dark shape, furry and alive. In the following instant, he realized what it was and he relaxed into disgust. It was a marlq.
The marlqai had been living on the mountain since before Alihahd had come. They were alien guests of the Itiri, like Alihahd himself. But he and the marlqai had never crossed paths before this. They had conspicuously avoided each other. Alihahd didn't care to be near the marlqai, and the marlqai had good reason to keep their distance from humans.
Superficially, the marlqai looked like big rabbits. The size of a small sow, they had long ears, round, soft eyes, and brown fur. Their ungainly paws were for locomotion only. A cluster of slender, hairless, prehensile tentacles were usually kept hidden in a furry protective pouch in their chests. Those emerged in a wormy mass to perform delicate work, such as writing or navigating starships. The marlqai had a starship on world, but they didn't use it to leave. They preferred to remain as guests of the Aerie. Freeloaders, some would call them.
The marlq at Alihahd's heels sniffed him, muttering to itself in the marlqine speech of clicks and whirrs.
Scents were important to the creatures. They had a highly developed sense of smell and could impart the sensation to other marlqai through their tentacles with perfect accuracy, bypassing the inadequate medium of speech.
The rabbity nose quivered and snuffled over Alihahd's leg. Alihahd ignored it, hoping it would go away.
Alihahd had never liked the marlqai—had, in fact, hated them. He felt nothing now but distaste and mistrust. They were belligerent, parasitic beasts, repulsive to him for their incongruous size and their unsightly tentacles. There was some obscenity in a rabbit that size, and in the naked tentacles coming out of a furred thing so that it looked to be infested with alien worms eating out its chest.
The sapient beings were fellow refugees from Na′id persecution. The human rebel runner Alihahd felt no kinship with them. He regarded the Na′id aggression against the marlqai the same way the Na′id did: as rodent control.
The marlqai had no homeworld anymore. The Na′id had eradicated them from the marlqine homeworld and driven them to near extinction during another of the undefeated General Shad Iliya's campaigns. The marlqai who managed to escape the purge of their native planet had quickly replenished their numbers in their spaceships—like rats—and spread like a plague to other worlds. There were at least two dozen of them now on the mountain, most of those born here.
Suddenly, an angry drumming in quick tattoo came, along with pressure like a vise closing around Alihahd's leg, the clacking of tooth on bone and a stab of pain. Alihahd looked down, nauseated more by the sight than the pain, which hadn't yet fully penetrated through his shock. The beast had wound its tentacles around his leg and was shredding his flesh with its teeth, sounding its loud, furious rattle, red froth foaming from its mouth.
All that blood was his.
Alihahd tried to shake the marlq off, but its mass was at least half his own, and everything weighed heavily on this planet. The monster held fast.
Alihahd bent over and pulled, hopped, staggered to the very edge of the Ledge Path, using his hands on the ground to help him, dragging the marlq with him. At the brink, he dropped to his left knee, gripped a granite outcrop with both hands—distantly aware of boundless depth and the far-below rush of wind in a great space—and with a grunt, he heaved his right leg and the grotesque thing over the edge.
Granite bit his palms, his hold slipping with the downward yank of the marlq's weight. The creature held on, dangling into the chasm, still whirring and drumming and chewing. It coughed red bubbles.
Alihahd clawed the rock for a better grip, and he jerked his leg to kick the marlq loose. The tentacles tightened.
Rediscovering the strength of an old fury, Alihahd held tight to the rock spur, swung his leg out, and brought it back in hard. He heard the dull, sick crack of the creature's skull on stone, and he swung out again, his muscles searing, sweat streaming down his sides and beading on his contorted face, his lips pulled back into an agonized snarl, and he beat the thing's head on the stone, again and again.
Finally the drumming stopped. The tentacles loosened. The creature slipped from his bloody leg and dropped into the abyss.
Alihahd collapsed, hugging the granite boulder. His tunic stuck to his sides. Sweat trickled down his scalp under his hair. His leg was an unfocused mass of pain. Then came other marlqai galloping over the summit of Havenside in answer to their comrade's drumming. They stopped at the crest and looked for their relative. They found only blood and Alihahd. They charged down the path, their eyes all arage, seeing murder.
Alihahd lifted himself onto his left leg, staggered, stumbled, then finally crawled to the bridge in swimming pain.
As the angry marlqai reached the Ledge Path with bared teeth, Alihahd's shuddery hands grasped the bridgehook, and he wheeled, rising, throwing his back against a pillar of the arcade, holding the metal barbs out toward the charging animals.
The marlqai bunched to a halt five paces away, noses twitching, eyes wary and malevolent, watching Alihahd and the hook. Muscles beneath their soft brown pelts tensed, untensed. Paws inched forward. They looked as if they might rush him. They smelled weakness. The Earthman looked easy.
Suddenly Alihahd brandished the hook. They flinched. Then they lowered their heads and inched their feet forward.
They were going to try it anyway.
In an instant, an Itiri warrior-priest was standing between them, having leaped down from the higher level to block the attack. The warrior waved the marlqai away with wide sweeps of both his arms. The marlqai shuffled back haltingly before him, furtively shifting their heads to either side, looking for a chance to get around him. One tried, and the warrior caught it by the ears and backed it up with the others, not roughly.
Still gripping the hook in one hand, Alihahd bent down and tugged out a long splinter of bone from his shin. He swayed, dropped the hook, caught himself against the pillar, and slid down, fingers tripping over the stone, his eyes darkening.
Then he was lifted atop firm shoulders—he smelled Hall.
His head hanging upside down, Alihahd was carried across the bridge that swayed in the wind. The crevasse swirled askew in his wavering vision. A thin smear of blood on the rock where Alihahd had beaten the marlq's head wove into view, swam out of view. The world was spinning, the mile-deep abyss at the vortex. He started to retch.
A comfortingly familiar voice growled, and he felt its rumbling vibration beneath him. "You vomit on me, Captain, and I'll drop you." It was Hall.
Alihahd spit up, gagged on it. Hall did not drop him.
At the end of the endless, upside-down journey, Alihahd was laid down in the cave of the physician, who began to tap numbing needles into his pressure points to deaden the pain, but Alihahd saved him the trouble by losing consciousness.
When Alihahd woke, he didn't know where he was. He had seen this cave before, but he didn't know it now. He heard Roniva's voice somewhere outside, but he didn't know her. He didn't know what planet he was on. There was pain in his leg. He didn't know what it was from.
He rose to rest on his elbows, his head clearing. From the distance he heard marlqai drumming in response to something Roniva was saying. He remembered.
He looked down at his throbbing leg. There were stitches in it. _Stitches_. They had _sewn_ him shut.
_There are_ stitches _in my leg!_
It had never occurred to him to wonder how primitives closed a wound without pseudo-skin and adhesives. The stitches struck him as absurd. Then funny. It was the natural sequence to being attacked by a giant bunny-rabbit.
His leg hurt to look at. It was discolored purple-red and splashed with the orange stain of a healing herb. Some flesh was missing from his calf, and his shinbone dipped in where pieces had splintered off. His ravaged skin, held shut with zipper-tracks of sutures, was puffed up with outrage but not infection.
He still wore one boot. The other lay on the floor, shredded and caked with brown blood.
Alihahd sat up, swung his legs over the side of the cot, and rose to stand on his left foot, the ball of his bare right foot resting on the cold stone floor without weight on it. The deepest wound in his calf oozed a little. His head throbbed—or perhaps it was only echoes from his leg. Everything felt veiled in a haze of pain.
His mouth was dry and coated with a film. His throat burned acidic. He hopped to a clay storage jug filled with cool water and drank. Then he hobbled outside to the open terrace of the second level of Aerieside. The cold air cleared his senses and swept away some of the misty ache.
He heard Roniva's voice again.
Moving along the narrow path of enameled cobbles at a painful hop, he followed the sound, steadying himself with both hands on the mountain's south face.
He came to where the path turned in to the natural amphitheater between the twin crests, and he saw the marlqai, across the fissure on the first level of Havenside, being asked to leave.
Roniva was bedecked head to foot in gold, and she blazed in the meager sunlight. Her hair was twisted into an intricate crown. A cheela held her familiar, and two other warriors stood with her.
She didn't address the marlqai directly. She didn't even face them. She was angry.
No one had ever been asked to leave Iry.
It was not an unreasonable thing to demand. The marlqai owned a ship. They were able to leave at any time. They simply hadn't cared to go. They still did not want to go. They wanted the Itiri to kill Alihahd instead.
Through the warrior-priest who was translating the alien clacks and mutters to Roniva, the marlqai insisted that Alihahd must have started the fight. Marlqai never attacked unprovoked, while humans, in contrast, had a history of wanton aggression against the marlqai.
But all witnesses had reported that the marlq in question had made the first hostile move.
The marlqai whuffled and chittered insinuatingly that all the witnesses were humanoid.
Roniva whirled on them before her interpreter could translate. She understood their language after all. Her teeth flashed white, and she spoke to the outsized rabbits directly. "Get out!"
The creatures hopped back in concert. Then they apologized for their blanket slander against humankind.
Roniva answered coldly through her intermediary, "Tell them I accept their apology. Tell them also to leave."
The scene left Alihahd shaken rather than vindicated. The marlqai shot baleful glances at him across the rift as they prepared to go. What they saw in their turn, if they could read human faces, was madness in the staring blue eyes with white all around. Alihahd's facial muscles stiffened until they trembled.
An old specter loomed. Another break in Utopia. Sanctuary was cracking, and he couldn't hold it together.
Someone came to his side. He didn't look, didn't need to. He knew the pony-trot step. He couldn't face her.
_You are not safe anywhere. No matter how far you go, how fast you run. Your furies will find you. There is nowhere to hide._
"They should have made me go. I do not belong here."
Amerika's voice was all hurt disappointment. "You still hate it here."
Alihahd wrapped his hand in her hair and rested it at the nape of her neck in a fist, with great restraint and care, as if the alternative were to strangle her. "No," he said softly, more to the mountain than to the girl. "I should like nothing better than to stay here forever."
• • •
Harrison White Fox Hall changed. As the dusk deepened into night, something that started as a gnawing suspicion grew inside and transformed him. Serra watched in alarm. Something boiled up to a level just short of open violence. She saw it through his eyes. Tiger's eyes. He terrified her.
But the dangerous eyes were not seeing her. They had turned to some inward vision and he paced, prowling in some place other than here.
When he brushed past her and his attention flickered momentarily to recognize her, he beheld her with startled impatience that seemed to demand, _What are you doing here?_
Finally, he stalked out and went to the isolated cave where he stayed when Serra bled.
A little while later, Serra ventured up to bring him some tea before she went to bed, but she turned back at the door.
He was just sitting there polishing his gun.
Hall hummed snatches of a battle tune. He rubbed a soft mhoswool cloth once more over his gun, checked the sight, and holstered it. He felt a vigor he'd almost forgotten in the months of tranquillity. The exhilaration of hatred.
The same puzzles had been running through his head over and over all day.
_Marlqai do not attack unprovoked_.
_Marlqai have long memories._
Hall could read people well and had trouble admitting when he was wrong. His reluctance to back down had blinded him for too long. It was time to make amends.
At first, he'd had Alihahd figured as a onetime coward. He knew Alihahd was a man with a past to be made up for—so moral and self-effacing, with the righteousness of a reformed sinner. It was the kind of self-hatred borne by one who had been in battle and run.
But no. There had been something wrong with that idea from the beginning.
Alihahd had been someone before he was Alihahd. He was not just one of thousands of small cowards who ran at the moment of truth. There was fear in him, true enough. But there was also a streak of sturdier, cruder stuff. And there was blood on his hands, Hall knew that now despite the denial. The mistake had been in assuming that the man had either run or made war. Hall had forgotten that one could do both.
Alihahd had not run—not until after it was over.
That was his sin.
He had stayed and fought. And won.
• • •
Dawn's longest rays slanted through the door and cast a warm yellow patch on the cave wall by Alihahd's bed.
He sat up, sick to his stomach. He brushed the sleep from his eyes.
He checked his wound. It had scabbed over quickly and cleanly and hardly swelled. Amerika had brought him a newly made fur-lined boot. He gingerly pulled it on and tried to stand.
If he didn't flex his right ankle and foot at all, and if he kept his weight on his heel, he could manage to move around. He went outside, limped up the mountainside, and found a place to pass water. Steam rose from it in the frigid morning air.
He took a limping walk over the now familiar mountain, even though it was painful. He had a sense that this would be the last time.
Around him, spring was unfolding in earnest. Winter-born mhos cubs mewed in their burrows under budding thickets. A triller tried out its notes as the rising sun softened the brittle air. Underfoot, green shoots thrust up from the hard winter ground and peeked through the dead tatter of last year's brown grasses.
Alihahd sensed the gun at his back before he saw it. A voice sounded behind him. Hall's.
"Alihahd."
• • •
Alihahd stopped, rooted to the spot. Never had Hall called him by his false name, and the voice was ironic and deadly. Slowly Alihahd turned. There was the gun, and it was Hall's. Tiger eyes glared behind the barrel. Alihahd returned the gaze with no emotion at all, only weariness. He turned his back again and limped without haste at his same painful, stumbling pace up the grassy scarp and walked away.
Hall lowered the gun. He'd kept his sights on Alihahd, his finger taut on the trigger, until Alihahd disappeared over the rise. Now Hall stared in disbelief at the empty space where Alihahd had been. Did Alihahd think Hall was calling a bluff? Harrison White Fox Hall was not bluffing. He was about to march over the rise after the awkward figure and have it finished, but something was holding him back.
Instinct.
There was something confusing here. Hall could swear Alihahd comprehended the danger he was in. Hall had sensed some fear, seen tension in his shoulders and in his back, braced and waiting for the shot. Alihahd didn't disbelieve the death threat.
Yet he offered no defense. Was that strategy? Did he imagine passivity would earn him pardon? No. Hall thought not. Alihahd was not so deluded. And then Hall realized the answer was right there.
The man wanted him to shoot.
• • •
The rains came, freezing at night into icy sheets. Alihahd didn't come back to his cave.
He'd wandered far from the Aerie, slowly, lame and aimless. He didn't take shelter. He slept in the rain. Then the ceiling of the world lowered as the sun began to set. What was left of the clouds sank with the cooling air and settled around the mountain peaks, wrapping everything in damp, blinding whiteness. Alihahd couldn't see past his hands, so he sat in a wet grotto and waited, chill seeping into his bones.
By nightfall, the mountain peaks surfaced, and the sky was revealed cold and icy clear. All below him, the sinking blanket of clouds looked like an arctic wasteland faintly sparkling in the starshine.
The wet ground began to freeze. Alihahd took a few steps into the open and slid a little on the slippery surface. He recognized no landmarks in the jagged black crags. He was alone but for a pair of moving lights—the eyes of a starving meeger aprowl at night.
The wind had stilled. The air was keen. Alihahd's breath drew in, sharp and cutting. It was too cold to stop moving, and dangerous to sleep, but that didn't sound like a good enough reason not to.
Morning came gray. Another cloud layer had formed far above, and the mountaintops were isolated between the two, cut off from both ground and sky in some elsewhere place, a limbo.
The black-hooded warrior came upon the Earthman facedown on the frozen ground.
Alihahd unstuck his eyelids, opened them a crack, and focused on the broad, scarred yellow feet next to his head. Painfully, he lifted his head and looked up.
Ben's black-shrouded figure stood mapped against a bleak sky, at one with the mountains' stark gray solitude. Broad shoulders were slightly forward, his powerfully thewed arms crossed, his shadowed head bowed under the hood. It was said his mind was gone. Ben regarded the half-frozen Earthman curiously. A bird cried in the raw air.
Ben uncrossed his arms and lowered his hood. "Should I summon a healer?" he asked.
"Absolutely not," Alihahd said. He sat up on the ice and touched his numb fingers to his forehead gingerly, as if it might split.
Alihahd had too much conscience for his position, and too much ability for his will. He didn't have enough instinct for self-preservation, but had too much for a man who wanted to die. If only he didn't have a conscience, he would be fine. He would also be the greatest monster the galaxy had ever seen. As it was, he was merely close.
He sighed in sorrow. He was lost. "I really don't know what to do."
Black bangs fluttered over Ben's low forehead in the chill breeze and brushed at the red scars on his cheeks. "Thou might try getting off the ice."
Alihahd held his blue fingernails out before him, frowned, nodded, all in slow motion. "Very practical," he said. When the long view was too overwhelming, look to the small and immediate. That much he could cope with.
With leaden slowness, he brushed off some of the crusty white snow that coated strands of his hair. His tunic was stuck to the ice. He pulled it free and rose stiffly. It was difficult to stand. His joints seemed solidified. He couldn't feel his toes, nor his ears. He ached, weary from shivering. It would be so easy if he could die.
Then he realized he could arrange that very simply.
Sardonic laughter filled his thoughts. _This will be quick_. He would be too cold to feel the sword stroke. He faced Ben. "Do you know who I am?"
"Yes."
Alihahd was dumbstruck. Yes? _Yes?_ His neat and certain scheme fizzled out. How was this possible? "Since when?"
"Since first I saw thee," Ben said.
Alihahd remembered the murderous fury in the dark eyes. It had been recognition. _And yet I live?_
Alihahd shook his head. He stared at the young man. He should not be here. He should not be talking. Last time Alihahd had seen him, he had been beyond retrieval. Yet here he was.
Ben turned his back to the breezes and was looking at Alihahd over one massive shoulder. There was a mildness to his mood. Black eyes were neutral despite their fierce narrowness and the oblique angle of his brows and three angry red broken lines on his cheekbones. The ring of the carnelian serpent shone like a drop of blood on his finger. There came the thin sound of the kestrel's cry as it rode the winter air high overhead.
No longer angry, no longer mad, no longer even human, Ben had become truly alien.
Alihahd denied what he saw. There was an ancient expression. "Leopards cannot change their spots."
Ben considered this. He answered with quiet dogma, "Leopards do."
Alihahd echoed softly, "Leopards do." He gazed up at the sunless sky. "One would think there would be hope for me, then, wouldn't one?"
• • •
The hide cover over the cave mouth lifted aside with a gust of cold air, then dropped shut again. Harrison White Fox Hall, reclining shirtless and barefoot on his bed, looked up, his eyes heavy and narrow as a sated tiger's.
Alihahd stood in the entranceway.
The small cave was hot. The heat was an assault on Alihahd's eyes, which began to water and kept closing, wanting sleep. His nose thickened and began to run. Coming in from the outside glare, the cave was very dark. Alihahd could hardly see Hall, the faded walls, or anything but the reddish-orange glow from the low hearth. His voice was deep in reluctance, underlaid with embarrassment. "Mr. Hall."
"Captain," Hall said from his bed, low and warm—physically warm if a sound could be so.
"I am cold," Alihahd said.
"Did you come for help or to be shot?"
"Shoot me."
Hall smiled, shook his head. He swung his long legs over the side of the bed and sat up with a grunt. "Come in." He stood, took a towel from his yellowwood chest, and crossed to Alihahd, who was not moving. Hall brushed some of the snow and water beads off his darkened blond hair. "Whistle."
"What?"
"Do it," Hall ordered.
Alihahd tried to round his rubbery purple lips. He puffed out toneless rushes of air, then, barely, a discordant whistle.
Hall's languid eyelids raised slightly in surprise. "I am afraid I think you will live." He pressed the towel to Alihahd's wet hair without rubbing.
Alihahd closed his eyes, felt Hall's hot breath on his face that seemed very fat.
"Take your clothes off. Get in the bed." Hall unclasped Alihahd's belt with a tug and let it fall at his feet.
Teeth chattering, Alihahd pulled his ice-stiffened tunic over his head and limped to the bed. His fingers wouldn't flex to unlace his boots, so he just sat, shaking. Hall put a heavy, coarse blanket around him, and took his boots off for him.
Alihahd's right leg from heel to knee was a solid bruise, but, protected by the thick fleecy boot, the wound was healing, even though lower down his toenails were blue.
Hall put more fuel on the hearthfire, then crawled into bed with Alihahd under a heap of blankets. Hall's hot, dark skin raised in gooseflesh at the touch of Alihahd's own clammy skin against him. He pulled the covers over Alihahd's head, which lay at his shoulder. It was like being in bed with a corpse—a very, very, cold corpse—one that would not stop shivering.
"Ever try to kill yourself before?" Hall asked.
"Several times," came the murmur at his chest under the covers. "Have not gotten the hang of it yet."
"I guess not."
Hall dozed on and off over the hours. Alihahd kept moving, jolting Hall awake.
After a time Alihahd began to thaw and regain feeling. His toes itched fiercely. Clearing nostrils filled with the scent of straw in the mattress and scent of Hall.
Alihahd lifted the covers away, moved to the edge of the mattress, and shakily placed first one foot, then the other on the floor, and he stood. Hall seized his wrist. "Where do you think you're going?"
Alihahd mumbled—he didn't know where he was going—"This is not done where I come from."
"They eat people where you come from," Hall said.
After a dumb pause, the full significance hit Alihahd; he yanked free of Hall's grip and searched for something to wear. He almost bolted out into the icy night naked, just to get away from Hall—no, not really Hall, from himself, the one person above all he could not bear to face.
_But why else did I come?_ Alihahd thought with despair, one of those undeniable truths that shrank from the light.
_I knew. I know him._ He started to shake again.
He'd come to be destroyed.
He heard Hall's sardonic laughter echo off the close walls. "Nazi."
"You are saying that just to anger me," Alihahd said.
"Succeeding, too," Hall said, and he sat up. "Hey, nazi, you looking for something to cover yourself?"
He didn't mean clothes. The tone was much too insinuating. Alihahd turned.
Hall held up Alihahd's container of melaninic pills. Alihahd hadn't seen them since he'd taken them from Omonia Station. He'd assumed that he'd misplaced them, but he should have known that he hadn't. He valued them much too much. He tried not to let his horror show. Maybe Hall would put them down.
Hall pitched them one by one into the fire. The precious little pills sizzled with tiny sprays of orange sparks. Tawny eyes slid from the fire to Alihahd's stone face. "Oh, don't pretend you don't care." _Phhht_ went another pill. "Here, do you want one?" He held one out, then let it drop in the hearth. _Phhht_. "I'm going to burn them all." _Phhht_. "So you may as well fight me for them."
Alihahd trembled. Blue eyes were wary, on the brink of their mad aspect, watching his hopes for brownness and normality go up in orange sparks.
_Phhht_.
Then Hall grew bored with his game and made to spill the whole bottle into the flames. Alihahd lunged.
The pills were in the fire as Alihahd grasped Hall's wrists. "Too late!" Hall laughed.
"Damn you," Alihahd said, quaking. "God damn you."
"Ah. Ah," Hall said tauntingly and opened his fist. There were three pills left. Alihahd snatched for them, and Hall snapped his hand shut. Alihahd tried to pry his fingers open, couldn't budge them, so brought his mouth down and bit Hall's hand, hard enough to draw blood.
Hall roared and pushed his fist into Alihahd's face. Alihahd reeled back, blood on his lips. "Well, that's nothing new," Hall said, rubbing his hand. "Go on and lick it."
Alihahd spat in Hall's face. "Sadist."
Hall wiped the blood from his eyelids. "Not half so much as you are."
Alihahd's wan face paled.
"Yes," Hall said. "Come on, now. I have three pills left. You've killed before, what's stopping you now? I swear, as I live, I will burn these. Why don't you come get them, and have some enjoyment of it while you're at it?"
"I do not want to hurt you," Alihahd said. The twitch returned under his eye.
Hall drew a long switch, too green to be kindling, from the firewood pile, and he struck Alihahd across the face with it, raising a red weal on his cheek. Alihahd flinched back in pain and startlement.
"Why not?" Hall said. "You ought to." He hit him again, "Fight me. This gentle son of a bitch is not who you are." He lashed him again, and Alihahd cringed under the stinging whip, and edged, doubled over, toward the bed. When he was near enough, he suddenly uncurled and seized up Hall's gun—Hall had left it by the yellowwood table—and turned it on Hall, threatening. The blows stopped. Something shifted behind the blue eyes, some veil lifted, and there was an insane lucidity upon him, cold, and aware, alive, capable—and reveling in bloodthirst.
"Ah, there he is," Hall said. "The man I wanted to talk to."
And he called him by name.
The cold shattered and crumbled. Alihahd threw the gun away from him and screamed, in Na′id, "Illi! Illi!" _My God! My God!_
Hall dropped the remaining pills in the fire. "Well, hell," he said.
• • •
Alihahd woke, pushed away the covers. Alive. He was surprised.
He looked aside. Hall was awake, his eyes shut, resting quietly. His fox-head pipe sat on the yellowwood table. Both opal eyes were gone. Morning light lined the hide cover of the cave that was still warm and stuffy.
Alihahd cleared the phlegm from his throat, brushed fine salt from his eyelids, let go a sigh.
Hall gave him a tap that was a demand for comment.
Alihahd recited, "For I am a shining being who lives in light, who has been created from the limbs of God." A silence followed, then a derisive sound in his throat.
"Is that Na′id scripture?" Hall asked.
"Yes. How glorious humankind." Alihahd sat up. "Might I have something to wear?"
"Yeah," Hall said, reached over to a drawer, pulled out some things, and let them drop on Alihahd.
Alihahd rose unsteadily to his feet and dressed. He touched the welts on his face. "You bastard."
Hall gave a one-shoulder shrug.
Alihahd walked to the door, faced back. Hall lazed in the bed, not ready to stir yet. Alihahd said, "It would have been easier, Mr. Hall, if you had killed me. Then I would not be wondering what I am to do next."
Hall waved his hand idly. "And put you out of your misery? What kind of avenger would I be then?"
Alihahd stepped outside to the bracing cold air that stung his sensitized skin and turned his breath to crystalline clouds.
He went to his own cave to wash, then to Serra's for tea. Serra put something herbal on his welts. She didn't ask where he'd gotten them. Alihahd had never seen any marks on Serra, but he'd seen bruises on Hall. Serra threw things.
Alihahd held the hot cup between his hands and let the steam unclog his nose. He felt soggy, like the muddy gray mass that was left after pouring water on a bonfire. At the same time he felt light, as if part of him had burned away, no longer weighing him down or giving off choking smoke and sparks.
He went to the door and gazed out, leaning forward with his left hand on the doorjamb, his arm straight, his weight off his bad leg. The tea warmed. The cold air revived. He took no long views, taking each instant as it came, looking no farther than his nose. That was the safe way to go. His furies had broken their leashes and attacked with all their merciless savagery. Now they sat in a corner, all snarled out, doing no more than glare at him, snuffling, not biting.
He hadn't changed his spots. He guessed he might as well claim them like bastard children.
_After all, they are my spots._
An outcry of eagles drew him out of the cave. Others of the Aerie, ranga and warrior-priests, came out to the ledges and terraces to look at the sky. A roar on the horizon followed the flight of four low-flying starships, colored electric blue and vivid red, making a low recon pass over the valley.
The next pass would bring them directly over the Aerie.
Hall came out to the ledge and lifted his beam weapon. He took aim along the predicted flight path and waited.
The four ships returned, blazing their twin symbols of Galactic Dominion/Human Supremacy. As they pulled up over the mountain they showed their designations: X99, X37, and X24. Alihahd couldn't see the fourth one but knew it was X48. He reached over and rested his hand on Hall's stabilizer. "Those are mine," he said. Then he clarified. "Alihahd's. I never thought they would come looking here."
"Thou must have been a great leader." The voice was Jinin-Ben-Tairre's. There was a question in his tone, and tentative respect for someone who just might be his better—for all his sins and weaknesses—and an unspoken _Shall I follow thee?_
Another admirer, one who had seen black depths and the darkest side of his own nature, this one, of all of them, knew what he was admiring. That kind could not be shaken. The rest of them worshipped a graven image.
"Some people were under that impression," Alihahd said.
He turned away and went back inside Serra's cave. Hall leaned in the doorway behind him, his arms crossed. "Well, Captain, since you are alive, what are you going to do next?"
Alihahd picked up his cup where he'd left it. "I am going to finish my tea."
# PART FIVE:
## Jerusalem Fire
# 18. Return of a Legend
THE FOUR STARSHIPS X99, X37, X24, and X48 came to rest far from the Aerie on a high narrow plateau wedged between a mountain and the River Ocean. They were Alihahd's ships. They'd found him.
But it hadn't been rebels who'd sighted Alihahd at Omonia Station. _Na′id_ had seen him there. It should have been Na′id who came looking here. Alihahd couldn't be sure that these weren't Na′id. There was nothing to say that the four starships hadn't changed hands in the past year. Alihahd needed to identify the ships' personnel before going to them. Jinin-Ben-Tairre sent his familiar ahead to the plateau where the ships had landed and their crews had camped. The kestrel returned screaming.
"The bird sayeth they are Na′id," Ben told Alihahd.
Alihahd absorbed the news with little expression. "Even if they are mine, they are supposed to look like Na′id," he said.
"The bird is not bright," Ben conceded.
The bird squawked.
The only thing to do then was for Alihahd to go and see for himself.
On the far side of the mountain from the starships' encampment lay an Itiri village. Jinin-Ben-Tairre transported Alihahd there aboard a primitive airplane. The antique would be beneath the notice of a starship's scanners. Layla, Harrison Hall, and Vaslav came with him.
They stepped out of the plane within sight of the Itiri village. Its low white buildings, washed golden in the light of the morning sun, were clustered closely along stepped streets, their rooftops mostly flat as they would be in a place where it rarely rained.
"It looks like Jerusalem," Alihahd murmured.
Vaslav spoke at his side, impressed and intrigued. "You've seen Jerusalem?"
Alihahd took a breath. "I saw it."
"Before or after the Fall?" Vaslav asked.
"Both."
The boy paused, then proceeded cautiously, "You saw it twice?"
"No. Once."
Vaslav was in awe.
Alihahd spread his hands, conjuring a picture, his gaze far off. "This is like the Old City. The new city goes on and on over the hills. The slopes were choked with gray limestone blocks of apartment buildings, office skyscrapers, synagogues, churches, mosques, everything of all eras. The Old City was the part the Bel wanted untouched. The rest was expendable, if need be . . . they didn't know that. . . . She burned. All night." He stiffened with an involuntary shudder. "Men and women killing men, women, and children . . . they would not stop. . . .
"There were three armies. The 27th, the 9th, and the 34th." He pointed at the horizons around the village. "The 9th and the 34th had been there for years. The 27th was there to end it. . . ."
Vaslav whispered, "Where was Shad Iliya?"
"Mount of Olives," Alihahd said. "Small for a mountain. No olives. Many graves."
And he described the battle in detail, every tactic. It was a long tale, but no one interrupted. They didn't notice the time passing. Alihahd continued, chillingly photographic, detached and impassioned at once, to the end.
His wide, gaunt shoulders slumped, and he broke from his near-trance. "Jews are the most tenacious people in the known universe," he said tiredly. "You can burn, flay, trample, enslave, outlaw, boil, and eat them, and they remain Jews. They bewilder and humble me.
"And Arabs. They died in droves and still they came—the ones who did not turn and run at the first shot. Those who stayed were the ones who died all night long. They would not surrender."
Vaslav was trying to pinpoint what was odd about Alihahd's account. For the most part it had been a cold, harrowingly factual replay of events. But what was wrong?
"I thought you said you never fought Na′id," Vaslav said.
"I did not," Alihahd agreed.
He hadn't said what he had done in the battle, Vaslav realized. But obviously the man had been there, in the middle of it.
And then Vaslav knew what was wrong with the story. It was the point of view—where he must have been to have seen the battle that way.
Mount of Olives.
• • •
Alihahd prepared for his hike down the mountain. He would approach the ships' camp on foot. Before he would make his presence known, he needed to know who was looking for him. Layla, Hall, and Vaslav all insisted on accompanying him. They wouldn't be left behind on Iry. Jinin-Ben-Tairre told the humans, "I shall wait here for three days. Then you are in the hands of your own kind."
_But which kind are they?_ Alihahd thought.
His gaze locked briefly with Ben-Tairre's. Then the warrior lowered his eyes to his own rag-tied feet. A soft black forelock fell across Ben's brow, and Alihahd thought he suddenly looked very human. Alihahd felt he ought to do something, say something.
_We have the same soul, you and I. The blood-soaked wretched of this war_. But neither of them was the demonstrative breed. They let each other go without a word or gesture to say there was ever any bond between them.
Alihahd, Harrison Hall, Layla, and Vaslav started out. It was to be a steep and brambly trek down. After the dry air of the mountains, the moist lush forest of the windward slope was unbearable to them. They needed to be on guard against a whole different set of hazards in this lower country.
They hadn't gone far when Layla cried out, "Sail snake!" And she pointed up.
Hall grabbed Vaslav and pulled him down. Alihahd and Layla dropped. The four of them crouched into the underbrush as a long, undulating, leaf-green reptilian body glided through the treetops.
The snake passed by without noticing them. The humans stayed in hiding until the inevitable mate appeared and passed over as well.
_Complete thine journey before nightfall_ , Alihahd had been advised. _Sail snakes see heat. In the dark, thou wilt not see them. Thou art too big to eat, but it may strike before it realizes thine size._
Alihahd stood up. This would prove to be a very long walk.
"Why did they land the ships down there anyway?" Hall said, grumbling. The interior was much cooler and drier.
"Must be my people," Alihahd said and pushed his way ahead through a thicket of willowy saplings. "Na′id plan better."
Alihahd lost his bearings. _How long has it been since I've had to deal with a tree?_ His clothes were tough but breathable—sturdy trousers and a dolman-sleeved shirt with its drawstring waist and cuffs loosely tied—but still they stuck to him. The moisture that slicked his skin had nowhere to go.
A leafy branch slapped his face. He pushed it away and tried to find the sun through the verdant screen all around him. Which way?
Down. That much he knew, anyway.
He'd become separated from the others and was about to call out, when he heard shooting—fire and return fire—more than three.
He bounded through the underbrush with clumping leaps, favoring his right foot. He broke into a clearing—
—and came face-to-face with a Na′id lieutenant.
• • •
Alihahd and the Na′id officer pointed guns at each other and did not shoot.
The dark young lieutenant gaped and slowly shook his head in disbelief and denial.
Alihahd knew what that look meant, and he felt sick.
The Na′id's expression hardened, and he took aim. Alihahd didn't move, only stared up the barrel in helpless horror.
There was a shot.
The Na′id fell forward. The back of his blue uniform shirt purpled with blood. Behind him in the woods stood Harrison White Fox Hall.
Hall tramped out of the underbrush and came to Alihahd.
Alihahd swayed on his feet, leaned toward Hail, who grabbed him and steadied Alihahd against him.
Alihahd pushed away, staggered to the body, and fell to his knees. He turned the youth over, brushed black hair off the paled face and out of the open black eyes. He touched the chest that was warm but unmoving.
"Know him?" Hall asked.
Alihahd shook his head. _He knew me_.
Hall lifted Alihahd bodily, and Alihahd didn't resist the help this time. He stumbled away from the site, holding on to Hall. Alihahd didn't talk. Couldn't.
In the forest, Vaslav was yelling for them. Harrison Hall sang out, then crouched down in the ferns, taking Alihahd with him, eyes darting cautiously, gun ready for whoever might answer.
A small, bewildered voice sounded at Hall's side. "Did I shoot?"
"No. I did."
"Didn't think I had," Alihahd mumbled. But he hadn't been certain. He only knew that he'd been facing the young man with his gun. He couldn't remember shooting, but there had been a shot and it was the Na′id, not Alihahd, whose eyes glassed over and who fell facedown in the brush.
Vaslav came crashing out from the trees into the clearing and tripped over the Na′id body. He danced back and called shrilly, "Captain! Harry? Layla?"
Hall hissed, and Vaslav scrambled back to the cover of the wood. He saw Alihahd curled under Hall's arm. "Are you hurt?" Vaslav asked.
Alihahd spoke weakly. "No." Then stronger, "No." He shook himself free of Hall and stood on his own feet. "Where is Layla?"
"Here." Layla came to them quietly as a forest shadow.
"Let us go, then," Alihahd said. "Fast now."
They moved quickly to put distance between themselves and the clearing.
When they paused for breath, Alihahd sat down, his heart pulsing in his wounded leg. He pulled Hall down to him by the arm so he could talk quietly. "How many did you kill?" He hadn't counted the shots.
"I bagged five," Hall said.
Alihahd didn't like that number. "Vaslav?"
"None," Vaslav said.
"Layla?"
"One."
Alihahd relaxed a measure. Six was a good number. "That was probably the whole patrol. We might have an hour before someone becomes alarmed at their failure to report."
Someone. The Na′id base camp couldn't be far.
"Shall we turn back?" Hall said.
Alihahd bowed his head, chin on his chest. If those four ships were manned by Na′id, where were Alihahd's rebels? All two thousand of them. Alihahd lifted his head, level and grim. "You go back. I need to see."
"I can't let you go alone," Hall said.
"I do not need a nursemaid," Alihahd said.
"Yes, you do."
From the mountain—in the direction of the clearing they'd fled—came fighting snarls and yelps and growls. Those would be wolf-hyenas at the bodies of the six Na′id patrollers.
"Yes, I do," Alihahd said, standing. "Shall we get this done?"
The sun was already overhead. They would need to push to reach the coast and get back by nightfall to the place where Ben-Tairre waited.
Alihahd's leg was failing him. He fell into a melancholy brooding as he limped.
"They all come home to roost," he murmured. "It cannot be coincidence. Chance is not so vindictive. Must not Someone be directing events?"
Harrison Hall snorted. "It only looks like someone is in control of this show. When you think of all the trillions of coincidences that _don't_ happen, you see how random it all is. Odds are that some coincidences will happen—by chance—even if the odds against any single specific coincidence are a billion to one. It would be odd if there were no coincidences. _That_ would be evidence of intervention. As it is, I see none. You're on your own, Captain."
"You're a cynic, Mr. Hall."
"A realistic one. You think too much."
"Only when I'm sober," Alihahd said dryly.
• • •
The four spaceships rested on a level area cleared of trees by a fire. The carpet of yellow grass was underlaid by black dust and coal. Charred skeletons of once-tall trees spiked the plain at intervals.
Alihahd, Hall, Layla, and Vaslav stole up to the perimeter of the wide burned clearing and hid behind a mossy boulder at the very edge of the forest. Alihahd peered out cautiously.
Among the ships wandered men and women biding their time in the sweltering heat. Their sweat-patched Na′id uniforms were fleet red, not the army blue of the Na′id patrol Hall and Layla had killed.
A dozen crewmen, seeking the shade of the forest, ambled very near to the boulder where Alihahd hid. From the emotions that played across Alihahd's face as he viewed them, his three companions guessed the worst. But Alihahd exhaled heavily and stood. "Rebels. Those are mine."
He gave Hall his taeben and he walked slowly and loudly into the open, his hands wide and empty.
Twelve guns turned and aimed.
Alihahd's rebels had always been quick to draw their guns. Alihahd never let them use them.
"Put those down," bellowed the strangely dressed, white-skinned, blond man come limping from the forest.
The rebels jumped, stuttered, and stared. They knew the voice. The command was a ghost from the past. What they saw needed reconciling. The Alihahd they remembered was swarthy.
Still they knew him when he spoke again. "It is I."
At once the guns were put up, except for two of the most stunned, who were nudged by their comrades and hastily followed suit.
The thick silence broke. "Alihahd!" someone cried at last.
Exultant and dazed smiles flashed, and one crewman dashed to the ships, yelling, "Alihahd! Alihahd!" And all the rebels gathered around to see.
It was an awkward reunion, with enough excitement, some tears, but no embraces or backslapping, and the welcomes were clumsily expressed.
Then the throng parted for the approach of a stocky woman in Na′id commodore uniform. Blue tribal tattoos around her lips broadened as she smiled, but with no more surprise or jubilation than on recovering a lost nose ring. _So there you are,_ said the smile.
The commodore, Musa, took off her broad-brimmed hat, put it on Alihahd's fair head, and said, "What happened to you?" She meant his color, not his disappearance.
"As you see," Alihahd said.
Musa pressed her stout forefinger experimentally to his white cheek. "Is this real?"
Alihahd spoke tautly. "Do you honestly suppose anyone would look like this on purpose?"
"I guess not," Musa said. "Not unless you're trying to be the reincarnation of Shad Iliya."
Alihahd's smile was very thin.
A few grins around him slackened. A fierce, blue-eyed glare from Alihahd silenced any comment.
"You really do have that look," Musa said.
Alihahd told her, "You are aware that there are Na′id on this planet."
"Can't be," Musa said. A whispering wave of consternation passed through the rebel troops.
A strong voice from the woods said, "Are."
That was Hall. And all the rebel guns raised and pointed at him.
"Are," Alihahd said, pushing down the nearest weapon. "But that is not one. Put those things away before you shoot each other."
And when all the weapons were holstered, Hall, Layla, and Vaslav came out of hiding.
Alihahd muttered aside to Musa while the rebels were distracted by the appearance of his companions, "Not battle-ready, Musa. Your ships should be arranged at dome points."
"I didn't think it necessary in uncharted space," Musa said.
"You were followed."
Musa's broad brow creased. "Why would the Na′id follow us? As far as they know, we're a loyal squadron."
"Evidently, you have been discovered. It appears they are tracking you to all your contacts and bases before they make their final strike."
Musa pouted, assessing the dimensions of the disaster. "Shall we make a run for it?"
Alihahd looked up at the sky. There was nothing in it for the moment. "We have to assume they know where we are."
Musa nodded. "How did _you_ know where we were?"
"A bird told me."
Musa knew better than to question Alihahd when he was being evasive. She addressed the crisis at hand. "We're sitting ducks down here."
"We are worse if we take off," Alihahd said. Four ships slugging out of a thick atmosphere made a slow, concentrated target. On the ground, even if the Na′id located the ships, they couldn't be certain of where all the people were.
"I would feel much better in a spaceship," Musa said.
"So would most Na′id. I would not," Alihahd said. Harrison Hall came to his side.
"But you're a brilliant spaceship strategist," Musa said.
"I am merely competent in the air," Alihahd said. "I am brilliant on the ground." He glanced to Hall. "I don't lose."
Musa turned with a shrug, clapped her hands, and shouted a command: "Ships on dome points! Now!"
The rebels hurried to obey, and they arranged the ships at optimum angles for their combined force fields to raise a nearly impenetrable energy-shield dome.
"What have you on the scanners, Musa?" Alihahd asked while it was done.
"No ships." Musa opened her thick arms in bafflement. "You say there are Na′id. I don't know where they are."
"It is a metallic planet," Alihahd said.
Musa agreed. "They could hide anything here."
The engines of the repositioned ships were winding down. They were ready to raise the shield dome on Alihahd's command.
Alihahd held his breath. He was inviting direct confrontation if he raised a defensive dome now. If the Na′id were monitoring, they would see the dome on their scanners and would know they'd been detected. "Raise it," Alihahd said and waited for the repercussions. "Is there wine to be had?" he asked Musa.
"Y—"
"No," Hall said.
Musa looked quizzically to Alihahd.
"Never mind," Alihahd said.
Someone gestured skyward and sang out, "Here they come."
All eyes turned up.
Alihahd watched with sinking soul. Twelve ships appeared, not battleships of a space fleet, but troop transports of an army bearing twelve thousand infantry soldiers. The ominous silhouettes of the delta-wing ships flew in the formation of three broken echelons matted against the twilight sky. It was a known pattern, the personal signature of a special army.
Alihahd breathed, "This goes beyond chance and coincidence. This is destiny." _There is a God and He is vengeful_.
It was the 27th Army. The victors of Jerusalem.
Alihahd motioned aside to Vaslav without looking at him. "Vaslav, man a transceiver."
The boy ran to the flagship as the delta-wing transports of the infamous 27th Army were touching down. They came to rest on the burn-cleared plateau a scant kilometer from the rebel camp and set up on defensive points so that their own energy dome nearly overlapped the rebels'. Then uniformed soldiers poured out of the dreadful ships like bottle-blue ants.
"Hm," Alihahd grunted, commenting to himself, noting something interesting.
Hall raised his eyebrows at him questioningly.
"The uniforms," Alihahd said. "They never used to wear those."
The bright metallic blue with red blazons told the galaxy who these soldiers were. The colors reinforced Na′id identity and pride in the Empire. Shad Iliya never used them, preferring less conspicuous battle-drab. But without Shad Iliya, the soldiers of the 27th Army felt increasing need to insist on who they were. They had lost battles since Jerusalem. The blue uniforms were a small sign of slipping morale.
Alihahd touched Musa's sleeve. "I want everyone in battle fatigues and assembled out here inside of four minutes."
"Assembled in what order?" Musa asked.
"No particular order. Just so they take a position and hold it."
"Yes, sir."
If there was one thing Alihahd's rebels were trained to do, it was act as if they knew what they were doing in dangerous places where they had no business being and about which they knew nothing. They were an army of impostors and spies, not soldiers.
As the rebels formed into ranks, the scanner technician came to Alihahd to report that she'd located where the Na′id had been hiding earlier. Their ships had come out of a deep canyon below sea level, between two metallic mountains, less than ten kilometers to the west. Like chameleons, they'd been easily overlooked until they moved. "If we'd taken off, we'd be dead now," the tech said. "It'd be a skeet shoot. You were right, sir."
Alihahd nodded without comfort. There was no pleasure in being right when it meant he had saved himself for a fate worse than death.
The Na′id were assembling by battalions, smart and orderly, each by its ship. They outnumbered the rebels six to one.
For the soldiers of the 27th Army, there was something unsettling in facing an enemy wearing Na′id uniforms—and the drab colors they themselves used to wear when they were still undefeated. The more sensitive of them were aware that hidden things were moving here.
Vaslav came from the flagship to report, "The Na′id opened communications. They demand surrender."
Alihahd's eyes stayed fixed on the ranks across the plateau. "Refuse."
Vaslav ran back to the ship's transceiver to relay the message.
A rebel ship commander beside Alihahd took a quick breath for courage, then spoke. "Do you know what you're doing, sir?"
Alihahd was showing signs of fear. Sweat beaded his tall forehead. His mouth and eyes were lined in white.
"We're not an army," the rebel reminded him.
"They do not know that," Alihahd said.
"They'll find out when they attack," the commander said, suppressed hysteria in his voice.
"If they attack."
The commander licked salt from his upper lip, then caved in to bald fear. "Captain, do you know who they are?" he blurted. "That's Shad Iliya's army. Those people took Jerusalem!"
Blue eyes and white face turned slowly to him. The voice was deep. "I know."
The commander's eyes grew huge, and he backed away, choking on a whimper.
Vaslav came bounding back from the flagship, his face bright red. "Sir, their general calls you a madman and demands that you personally reconfirm your answer. He also wants to know if you realize who they are."
Alihahd put a hand on Vaslav's hard, spare shoulder. "Confirm the refusal for me. And tell the general this verbatim—verbatim, can you?"
Vaslav nodded intently. His hands were furtively making gentle, nervous motions. Alihahd wanted to slap them.
"Tell the general that I am in the habit of studying thoroughly any opponent, knowing his habits, strengths, weaknesses, motives, and all facts available—pertinent and _impertinent_. Do you think you have that?"
Vaslav raised his chin and dashed away, his lips moving, reciting to himself.
"And Vaslav!"
The boy turned.
"Tell him I thought Alihahd was dead."
Vaslav's face blanked in puzzlement. "Sir?"
"Tell him."
Musa offered Alihahd her comlink so that Vaslav wouldn't need to keep shuttling back and forth. Alihahd shook his head. "Let him run. Let them wait."
A fresh sweat broke over his brow in the late heat of the westering sun. The day was never-ending. These kinds of days never did end. The second-worst day in his entire life. Or was it the worst? He closed his eyes, brushed away a buzzing yellow fly. _O God, O God_ , he cried inside. And imagined he heard an ironic voice answer, _Yes?_
He opened his eyes, despairing.
Vaslav returned, unnerved. He'd been prepared for anger from the Na′id general at the message, but he hadn't been ready for quite the violence of the reaction. There had been more to that message he recited than was apparent. "Sir, the general is hopping mad. He turned that color." Vaslav pointed at a bright red Na′id badge. Alihahd was the only rebel not in uniform. By now all the rebels, including Vaslav, were in battle drab.
"The general wants to know who you think you are," Vaslav said. "Because you're not the man you're pretending to be. No one ever was or will be. And all Alihahd's cowardly tricks can't change that." Vaslav's voice faltered. His Adam's apple bobbed. "He wants you to quit hiding your face and come to the transceiver and look him in the eyes, because he already knows that yours—your eyes—aren't . . . blue."
Alihahd nodded. He slowly blinked his blue eyes. "Thank you."
"No message?"
"Not for the transceiver."
"Sir?" Vaslav began, bewildered. He was carrying every word that passed between the two leaders, yet he hadn't the least idea what was being said and why their words fell on each other so hard. And besides that, Alihahd's eyes _were_ blue. "Who are you pretending to be?"
Alihahd sighed. "Not." He took off the broad-brimmed hat and gave it back to Musa. He limped forward from the ranks in the direction of the Na′id ships, hesitated, then continued out toward the perimeter of the shield dome. His skin tingled as he passed through the force field, his movements slowed, like walking through mud, till he was out of the dome.
The rebels gasped. "Is he mad? He'll be killed!"
Harrison Hall rocked back on his heels, amused. "A little mad, maybe. Killed? Not yet, I think."
He could see Alihahd's white skin even at a distance, and fancied he was even more pale than normal. Alihahd was horrified, and it was not fear of death. He wished he was dead. He could not have been more sickened and terrified walking into a fire and knowing his flesh would burn.
Nothing stirred during his slow progress out to stand between the two armies. Everyone was quiet, watching his painful advance. He forgot to breathe, then breathed too quickly, feeling the pressure of the air now, the effects of descending too far too fast.
_I want a drink._ He could have in all reality murdered Hall for making him go through this cold sober.
The Na′id didn't shoot him. A murmur rolled back through their ranks. Alihahd didn't dare look at faces. He kept his gaze fixed strictly over their heads. Then, because he could no longer resist, he glanced down once. Their faces, the whole sea of them, read pure shock. Ice trickled through Alihahd's stomach, guts, groin, and into his jellied legs, and he stopped. This was far enough. At the next step he would fall.
He held his position there, midway, and let them stare. His mouth felt full of bitter pins. His heart wobbled high in his chest. The palms of his hands throbbed. He felt faint. _Not even God can help me_. He forced himself to untense his muscles. _As if God actually would._
He swallowed, gathered his breath low, and found his voice. "Soldiers of the Empire!" he thundered. The voice carried over the plain and rebounded off the mountains, steady, loud, and resonant.
Alihahd prayed the voice would not break on him. It never had.
He was dying inside, everyone staring at him—those blank, horrified faces. If he looked at them again, he would freeze. He heard their murmurs, the deadly secret spreading.
"Yes, I know who you are!" he shouted at them with a fury that was really fear. _"You are my enemy!"_
The murmurs silenced. Alihahd glanced down quickly at their glassy faces, every one of them unmoving, as if they were watching a special weapon of theirs—a big gun they once had—which had made them feel so powerful and invincible while it was theirs, till this day it turned around and they were looking up its barrel and feeling very small. _You are my enemy_.
He shouldn't have looked. It was some moments before he could speak again. In the silence the Na′id fear mounted to near panic. Alihahd needed to be careful. Cornered beasts bit. _O God, if you get me out of this without a battle, I will never presume to lead a body of your people ever again_.
The Na′id believed themselves cornered. Alihahd would show them an out and hope they ran for it without realizing he was all roar and no teeth.
"If there be a battle today, you will attack. I will not. The decision for bloodshed is yours entirely." He paused, and watched their panic abate a degree. " _But_ ," he shouted, and they flinched. "Know it be me you fight, and you know better than anyone that I have never lost a battle."
Even the young soldiers who had never known him were stunned. His image was aboard every ship. The soldiers of the 27th visited his shrine before every mission.
God had left and joined the other side.
The shift in morale could be felt across the plateau.
Both sides were astounded. Alihahd sensed his rebels behind him reeling, and he was sick again at the prospect of turning around and looking back. His stomach had gone from ice to water.
He saw movement in the Na′id ranks—a single person walking forward out of the camp toward midfield. Alihahd knew him—a broad, stolid bulk in general's uniform, older than Alihahd, with silvering black hair, not much changed from thirteen years ago when everything was normal and this man was his lieutenant commander.
Ra′im Mishari trudged out for conference with the enemy captain. His steps were tired, and a thick forefinger worried at the tight collar of his uniform in the damp heat. For all the years of Ra′im Mishari's service, the computer still could not fit a collar to the general's taurine neck. Even that had not changed.
General Ra′im Mishari came to Alihahd and stopped. He and Alihahd faced each other in silence, awkward as two former lovers.
Alihahd stood at his full height, his eyes turned down to his old second.
Ra′im Mishari's heavy brows contracted. His square jaw moved, grinding. His broad chest expanded. He started to say "General," but blurted out with a cry of pain, "Shadi!" and nothing to follow. Just his name. He felt like shedding tears, but a leader could not before two armies.
Alihahd waited, with no compulsion to say anything.
Ra′im snorted, opened his collar. He looked up. "I worshipped you."
"That was a serious mistake on your part," Alihahd said.
A cloud passed over the sinking sun. Ra′im Mishari glanced toward it in relief, then back to Alihahd to say matter-of-factly, "You realize you just shot our morale to hell."
"That was the idea."
"We wouldn't have a chance if we engaged you now."
_Yes, you would. Six to one? Come now, Ra′im, have you deified me?_ "I realize."
"You leave me no choice."
"That was also the idea."
A glimmer of a sad smile flickered at the corners of Ra′im's eyes. It sounded so familiar—the deft maneuvering, the sure simplicity, ever one step ahead. So it seemed to Ra′im. He frowned. "How could you desert?"
"What else does one do when one is born on the wrong side?" Alihahd said with profound tiredness, his blue eyes skyward and lost. "I do not know if what I did was right—I suspect not. But I did not know what else to do. Suicide, I suppose. But, frankly, I lacked the courage."
A tremor disturbed Ra′im's square, solid chin. The speech sounded exactly like the man Ra′im knew—and he remembered how distressed his general had been the last time he saw him thirteen years ago.
Ra′im spoke low, nearly guttural, emotion-charged and personal. "You should've said something. The Bel would have done anything for you."
"I didn't say anything? I thought I had," Alihahd said airily. "You all treated me as if I had gone mad."
As gently as he could, Ra′im said, "But you had."
Alihahd smiled, eyes shining at the sky. The thought hadn't occurred to him. He found it funny. "But even a madman does not like to be treated like one," he said.
Ra′im Mishari remained grave. "Why did you hit those space stations?"
_Ah. Omonia. They suppose that was I?_ Alihahd sobered. "I did not," he said without explanation, but clearly knowing more than he was offering.
Ra′im spoke slowly, trying to discount the other possibility. "Itiri warrior-priests are legends, of course."
"Of course," Alihahd said curtly and turned away before Ra′im could ask if the Itiri were allies of his. Let him think. The size of this bluff was staggering.
Alihahd faced his rebel camp, and he paled again. A moment's faintness passed over him, then he started his painful return, his back straight and stiff.
The rebels were silent. What had happened out there, who he was, was still sinking in.
Alihahd was accustomed to having all eyes on him, but he'd never been looked at like this in his life. The stares were tearing his guts out.
When he reached the camp, someone jabbed a gun in his face.
The shock of even that kind of gesture had long ago worn off, and Alihahd didn't blink. He put his hand on the barrel, pushed it away, and announced in a husky growl, "For anyone else contemplating assassination, I suggest you wait for a more advantageous time."
He scanned the sea of eyes, some still denying, some crying. No one moved against him. They may have hated him, but he was still the highest card in the deck, and they wouldn't remove it from play when it was in their hand.
Vaslav was stricken. He was bawling. Only Musa and Harrison White Fox Hall were unaffected.
"Alihahd," Musa said to herself, her brows high. "He left."
"He certainly did," Alihahd said raggedly. He looked to Hall, who was enormously entertained. Alihahd walked past and headed for the forest. Hall fell into step at his flank.
"Do not say anything, Mr. Hall," Alihahd said.
"Not a word," Hall said breezily.
Inside the woods, Alihahd let himself limp as badly as he wanted to. He found a fallen tree trunk and sat heavily. His hands were quaking in aspen-leaf tremors. Lancing pains shot up and down his leg. Every nerve sang in delayed reaction to horror. He cupped his hands over his nose and tried to stop hyperventilating.
In time, he became conscious again of Hall sitting beside him, trying to light his pipe full of soggy tobacco with a moist flint. His muttered imprecations were somehow soothing, and Alihahd gathered strength from his mocking presence.
Noises from the camp filtered into the woods where they were, as the rebels began to thaw from their frozen shock. Only now was their kicked anthill beginning to buzz and scurry and crawl in reaction. Alihahd let it settle itself without him. The rebels would be better by themselves for now, and Alihahd needed to be alone—except for Hall, who did not count.
The woods were shady, deep, and peaceful, cooling with the sun's setting. Alihahd let its damp quietude surround him. He watched a droplet fall from a wet leaf. A creature like a red ant tracked through the moss on the rotting log.
He whispered his own name. It had been so long since he owned it.
"Shad Iliya."
# 19. Shadow of Masada
SHAD ILIYA WAS BORN INTO a patrician house under the most auspicious circumstances, the second child of two war heroes. His mother was already carrying him at the battle of Antarctica, and it was said later that Shad Iliya had been winning battles before he was born.
Like most prominent Na′id citizens, his mother made a point of being on Earth when her time came so that her child could be called a true native Earthling. And someday when he fought for the Empire to bring all of Earth under Na′id control, he could say he was reclaiming his homeland. He could not be called an invader.
He was bred from the cradle to be a leader, and proved to have a natural ability, but he faced a great struggle to advance to any significant position because of his unpatriotic color. His parents were a typical Na′id mix of racial traits, so their second son was a bewildering embarrassment. He learned guilt early. Even as a child he knew he was evil. He hated his color and all it meant.
So did his peers. He was lucky at least to be highborn and afforded the protection of his patrician heritage. No one dared destroy him. As it was, they were merely unmerciful.
He tried to tan in the sun, but his fair skin only burned. He begged but was refused the use of melanin drugs, for they were deceitful. He was not trying to be deceitful—everyone knew what he was. They called him nazi. He just wanted to look like the others. But he could not.
So he would have to be great instead. Praiseworthy.
He sought refuge in the army, where he was good. He rose to the rank of major and dead-ended there. The rank was plenty high enough for someone who looked like him.
He would have died a major but that the most powerful individual in the Empire had taken a liking to him. The Bel felt for the despised and talented young man, and he married his niece to Shad Iliya.
Shad Iliya's superiors noticed right away that someone was letting the Bel's kin stagnate at his current rank.
But he was still an embarrassment to promote. They couldn't have a blond, blue-eyed Aryan leading battles against other humans. The symbolism was bad. So they sent him to faraway alien lands, where he was simply a human being combating the nonhuman menace. Out of the public eye he could be quietly elevated as his merit dictated. And Shad Iliya advanced rapidly.
In semi-exile all those years, fighting on alien soil, he never lost a battle, and his record grew so astonishing that it reached public view anyway and he became a hero. At long last his name could be spoken aloud. And he was fondly nicknamed the White Na′id, their brilliant oddity.
His victories helped to soothe the Empire's other defeats. His reports could be counted on for good morale. He never failed—still always alien wars. No one realized that keeping him out of sight also kept him blissfully blind.
He would never have run, never have wakened from his illusion of Right, had he not been turned against his own kind.
He could slaughter inhuman aliens till Doomsday without a blush of repentance, as he had slaughtered the marlqai. He still had no regrets about that. But the Empire needed its best to end the century-old battle for Jerusalem.
Shad Iliya did his duty, and Jerusalem fell to him.
And it came to him like a revelation. He held his hands to his face, the truth still blinding thirteen years later in the cool and shadowy Iry forest.
"We were wrong. We were dead wrong."
_Gregorian Year 5843 CE_
It was hilly country, the land around Jerusalem. The Na′id ships had been landing in the desert hills beyond the city for days until they formed a wide ring with which to raise an enormous energy-shield dome over the top of the defenders' widest dome, cutting the city off from its millions of allies.
Defenders and attackers couldn't see each other yet for the distance and the hills, except atop one hill captured in a surprise thrust during the 27th Army's initial landing—the Mount of Olives. The arriving army set up headquarters there for their general, the new supreme commander of the Jerusalem campaign.
The Mount of Olives was an exposed promontory, but difficult for the defenders to fire upon, for it was actually within their first line of defense.
Jerusalem was encased in energy shields, shells within shells of shields, which kept out electromagnetic radiation of frequencies higher than ultraviolet, as well as n-particles with wavelike properties. The city bristled with antiaircraft and bomb-intercept guns. Its only vulnerable points—so it was believed—were the windows created in the shield domes by the defenders through which to fire their guns. Previous attacking armies and fleets had flown over the city in their spaceships and hammered at those windows with little result.
Now the Na′id were bringing down their ships and ringing the city like a closing noose. An uneasy Jerusalem waited on guard against the new strategy.
Then arrived on the site the famous general, the thirty-four-year-old Shad Iliya. He stood in the open vehicle that shuttled him from his ship, across the Judean desert, toward his new headquarters on the Mount of Olives. The jeep moved fast. Shad Iliya liked the speed. He stood up to survey the land and to be seen. The dry desert wind stung his fair skin and sang in his ears.
At the western foot of the mount, the jeep turned sharply past its center of gravity. It flipped over and spilled its august passenger out to roll in the ancient dust.
The general's waiting aides and horrified driver scurried to pick him up. But Shad Iliya was young, and he was already on his feet by the time they reached him. He shook himself and slapped a yellow coating of dust from his drab gray-green uniform. He spat dust. He would remember the taste for a long time to come.
He looked to his aides easily, his pride only slightly muted, his spirits not at all. "Well. Where are we?"
They directed him up the Mount of Olives.
As he hiked up the slope with an energetic spring to his walk, his aide trotted alongside him, explaining, "There's a twenty-first-century Neo-Hellenistic chapel we've turned into a war office."
"You did not damage it, did you?" Shad Iliya said.
"No, sir!" the aide said with proper reverence for the work of human hands as they reached the summit.
Over the top of the hill, Shad Iliya received his first view of Jerusalem, the golden city. He caught in his breath and felt his destiny whisper to him. The city looked like its pictures and holos, but here, present and living, its majesty spoke. The unmistakable golden Dome of the Rock stood in the foreground, the venerated ruin of the crumbling Western Wall beyond it, within the boundaries of the Old City. The Old City was the enemy prize to be captured, a flag of separation and a place rich in human history. The hearts of three major religions crowded that little space.
"So close," he murmured. "We are here."
"It went just as you said, sir," one of his aides said. "The Resistance didn't expect this line of attack at all. Here's what we're using as headquarters, sir."
The aide directed him to a handsome little chapel high on the eastern slope of the mount. There was nothing higher on the hill left intact after countless interfaith wars through centuries past.
Shad Iliya took a moment to admire the chapel's neat, graceful architecture with its blue tiles, white pillars, and painted entablature. One benefit of religion was that it had spurred humankind to great works of art during the childhood of the species when Man needed an anthropomorphic divine Father to protect him and tell him right from wrong. And Man built these lovely things for His sake. But the work was human. The achievement was human. The glory was human.
Farther down the mount's slope stood a charming Byzantine basilica with many-domed roof. Shad Iliya was enchanted. "Is that ours?"
"Nominally, sir," an aide said.
"Can I see it?"
"Not advisable, sir. It's too dangerous. That's Gethsemane. Our force field ends right there."
The young general nodded to the disappointing reality. He would have to wait. His blue-eyed gaze lighted on the elegant Dome of the Rock. He drew himself tall, extended a long arm to point, and announced in full voice, "Before the week is out, I shall walk beneath that dome."
Swept along with his emotional tide, the general's knot of aides broke into spontaneous applause. Though the city had withstood an assault of a hundred years, if their Shad Iliya said one week, then the city would fall in one week. The faith of Shad Iliya's followers was absolute. Shad Iliya would not lose.
Shad Iliya smiled at them. He accepted adoration in those days. He actively sought it, and was sometimes given to theatrics.
Drawn by the sound of unrestrained applause, which could signal the arrival of only one person, Lieutenant Colonel Ra′im Mishari came out through the cluster of white pillars that fronted the chapel to greet his general. Ra′im had supervised the taking of the hill as ordered. Shad Iliya could count on Ra′im to follow orders to the letter.
Silently behind Ra′im trailed the morose generals of the 9th and 34th armies, whose command Shad Iliya had assumed. He'd met the generals earlier, on shipboard, to take the command batons from them. They greeted the young usurper sullenly. They regarded his initial success as minor and unpromising. They had been here for years. They knew the intractability of the city.
Shad Iliya returned their salutes, then addressed his lieutenant colonel, "Very good, Ra′im. Any problems?"
Ra′im pointed down the slope to Gethsemane. "The rebels have been trying to move a line of portable projectile artillery into the valley. We've been shelling them before they can set up—those are the craters there—but you can see it's a narrow target."
The craters ran between the wall of the Old City and the Byzantine basilica, both of which the Na′id wanted intact and unblemished. The shelling needed to be precise.
"Where is the Jericho Road?" Shad Iliya asked, trying to orient himself.
"That line of craters."
"I see." The general turned to go into his headquarters but was distracted by the sight of an ugly black tarp spread over a wide lumpy area beside the pretty chapel. "What is that?"
"The dead in stasis, as ordered, sir."
Shad Iliya stared. "So many."
"Theirs, sir."
_Theirs_. He wasn't accustomed to the enemy dead being human. The fact that he was fighting humans hadn't taken on reality until that moment.
He went inside the chapel.
The place had been readied for him by his alien slave, Pony, and his personal orderly, a boy of thirteen years with the interminable name of Sinikarrabannashi. He was called Sinikar.
The breeze coming through the chapel's windows was dry and surprisingly cool. The faded blue-and-white-tiled floor had been swept clean. The Na′id standard and the flag of the 27th Army flanked the doorway. The blue-and-red Na′id seal hung on the wall.
The curly-haired boy Sinikar saluted his general and, at a nod from Ra′im, put a comlink into his hand.
"What is this?" Shad Iliya asked.
"It was delivered, sir," Sinikar said.
"It's a direct private link to the rebel commander," Ra′im Mishari said.
Shad Iliya looked to Ra′im. "Which?" There were three.
"The Jew," Ra′im said.
It was a Jew who had organized Jerusalem's defense. The command was titularly a three-headed monster composed of one Jew, one Muslim Arab, and one Christian. In reality, there was one acting genius at the helm, and two figureheads for pride's sake, supporting him. The people of Jerusalem had succeeded in the Na′id goal of uniting the impossible trio of Judaism, Christianity, and Islam against a common enemy.
Except that the common enemy was Shad Iliya.
The artificial differences of religions annoyed Shad Iliya—especially these three religions. They professed belief in the same God, the Creator; they all used the same book with the same prophets and angels. Fundamental differences were few. According to two of them, God was sexless but still called "He," while the third gave Him a queen and a son, which did not sit well with the other two, but it was not as if one premised a pantheon, one an Earth Mother, and one fire worship.
_In the name of mercy, it's the same damned God!_
All three religions were, to Shad Iliya's mind, anachronisms. Born out of early Man's fear of death, they were throwbacks to civilization's early days of infamy, and they perpetuated its slavery, male dominance, superstitions, and intolerance of other creeds—while straight from the Koran came the affirmation: "Mankind were once one nation."
The 27th Army was here to guide these reprobates back to a unified fold.
Shad Iliya strolled outside to the chapel steps, turning the comlink over in his hand. He stopped between two white pillars and clicked on the com. He spoke into it experimentally: "Hello?"
The response was immediate. "Ah, the mighty Philistine general!" The voice was cheerfully mocking and male. "At last. I thought you were sending your toadies to do your dirty work for you."
"No. I am here," Shad Iliya said. "As you can probably see." He took another step out from the pillars and scanned the city whose labyrinthine ways somewhere held the owner of the voice.
A pause preceded the answer. "I thought you people wore blue."
Shad Iliya snapped his heels together and inclined a courtly military bow to his unseen watcher. Then he said, "I detest blue." An absurd answer to an absurd question.
The Jew laughed.
Shad Iliya avoided the traditional blue out of practical considerations. The shiny metallic-blue uniforms were designed to be an instantly recognizable announcement of Na′id loyalty. Shad Iliya didn't like his soldiers to be so highly visible to enemy gunners.
He glanced to the black tarp. He wanted to have this ordeal over, and to take control of the city with the fewest deaths. He'd never fought a campaign like this—trying to preserve both sides. He felt certain of victory but was uncomfortable with these new tactics. It would help if he knew his enemy. And he was discovering quickly that he did not know his enemy this time. Strategically hyperopic, he found that distance made for clarity of vision. Aliens were easy to understand and, therefore, predictable. These people were not predictable in the least.
The inevitability of their defeat and a promise of clemency in surrender were not enough to make them give up their intransigent stand. They were bafflingly illogical.
The two veteran generals, who had joined Shad Iliya on the chapel steps, had reported the same observations many times in past years. They watched the new general grudgingly, waiting for him to fall on his face as they had.
He lifted the comlink again. "Are your Christian and Muslim colleagues with you?"
"They are listening," said the Jew.
"Have you informed them of the terms we offer for surrender, or was the decision to refuse unilaterally yours?" Shad Iliya knew that this was an uneasy alliance at best. If he preyed on the suspicion that the Jew was overstepping his bounds, Shad Iliya might drive a wedge into their fragile unity.
But his opponent was familiar with the methods of propaganda. "Such a bald ploy, Philistine," the Jew said. "My colleagues are not impotent and they can speak for themselves."
The comlink was passed, and the Christian leader, Cardinal Miriam, gave her answer in a florid and prolix diatribe. She spoke in paragraphs of prose.
Shad Iliya cut in, "Your Eminence, I know of only one person who regularly uses lofty rhetoric in private speech, and that is I. Since this is a closed circuit, please speak plainly to me now, human being to human being."
"As you will, General. Go to hell," the cardinal said.
Plain enough.
Just then, from the loudspeaker in the minaret of the nearest mosque erupted a deep ululant _Allaaaaaaaaaaaaaaah Akbaaaaaaaaaaaaaaaar!_
Shad Iliya took a step back, clicked off the com, and looked to his aides. "What is that?'"
"Muezzin, sir," Ra′im Mishari said. "Muslim prayer call. Five times a day."
"That hideous noise in the name of God?" Shad Iliya said and clicked on the com again. "Are you still there?"
"I am, Philistine." It was the Jew again. "But I fear our Muslim colleague must delay giving you his fond wishes. He goes to pray now."
"Then I shall talk to you later," Shad Iliya said.
"Keep the link with you, Philistine."
"You will be right in my pocket the whole time, Zealot," Shad Iliya said and clicked off.
The prayer call droned on for some minutes more. The crowded city began to crawl.
"Look, General."
Muslims flocked to the mosques, filled them to overflowing, spilled out of the courtyards, and bowed down in the streets. Jews and Christians in that quarter walked around them.
Shad Iliya marveled. Religious ardor reached fanatical heights under siege. Yet the conflicting faiths and conflicting sects of the faiths squeezed into the small area seemed to have put their righteous hatred of each other aside for the moment. "They cooperate rather well for avowed enemies," Shad Iliya commented.
"Don't they realize that only proves the Na′id point!" someone said.
"That end of it rather appears not to have entered into their consideration," Shad Iliya said. He didn't understand it either. He beckoned Ra′im to his side. "I require one crack pilot for a possible suicide run."
Twelve officers in and about the war office overheard the quiet request and leaped to volunteer on the spot.
The two veteran generals were gravely disgruntled. This kind of blind loyalty to a leader was unhealthy.
Shad Iliya chose the nearest volunteer and instructed her to take an echelon of eight robot spy planes in a low pass over Jerusalem. "As low as you can manage without burying them."
"I'll wave up to the minarets as I go by, sir," she said. "Purpose of mission, sir?"
"Your ships will be transmitting to this office a computer plot of the trajectories of all enemy fire which you draw. There will be a lot of it. So."
He left off there. The rest of it was understood, and he waited for her to retract her bid.
She saluted, accepting the assignment.
General Shad Iliya saluted, held it a moment longer, as he always did in the face of valor, then barked for a car to take the pilot to her spy planes.
Two computers were activated in the war office to record the planes' signals as they came in. "Don't garble this," Shad Iliya told the technician. "I shall not have this done twice."
He didn't expect to see his pilot again.
He stood out on the chapel steps to watch the pass.
The sleek black planes went into Jerusalem in the 27th Army's characteristic broken echelon.
They came out in perforated echelon. The lead plane crashed into Mount Scopus, failing to pull up steeply enough. Three others fell to ground fire. The four survivors streaked into the desert. The war office waited, hushed, to see if they came back. No one knew which plane had carried the pilot.
"Long live Shad Iliya and the 27th!" crackled over the transceiver.
"She made it!" a technician called out to the steps of the chapel.
The tall white general nodded with an air of near-indifference. But none of his soldiers supposed Shad Iliya indifferent.
He said nothing, did nothing, seemed to be admiring the skyline, until Ra′im Mishari came to him with the tech's report of the lowest angle managed by the defenders' emplaced guns.
Ra′im passed the verdict to him in silence.
Twenty degrees.
Shad Iliya thanked him. Their eyes met. No more words passed. Both knew what the information meant. Ra′im guessed what his general had in mind.
The veterans of the 9th and 34th armies had no idea. All they saw was apparent catastrophe in the half-destroyed squadron of valuable spy planes.
"With the supreme commander's leave," said one. "We told you their ceiling was covered."
Shad Iliya was unconcerned. "We shall not be going in from the air."
Faces slackened. "How, then?"
Shad Iliya became impatient. Blue eyes glittered ice-cold. "Since we are not about to tunnel either, I leave it to you, gentlemen."
• • •
Sinikar, the orderly, stepped out of the chapel to inform his general that the Jew was on the radio.
Shad Iliya sighed, expecting this. These taunts would be on an open frequency to advertise the invincible general's disastrous air pass to the populace.
Shad Iliya went inside to the transceiver. "Yes."
"Well, Philistine, give up?"
"No. I was just now telling my colleagues to start digging. We are tunneling in."
The Jew laughed, then said, "No, Shad Iliya. You just found an opening on the horizons. You're coming in with an infantry."
Bald expressions of shock ringed Shad Iliya in the war office. Shad Iliya nodded, accepting the fact that he would get no breaks from this man.
"Yes, Zealot, I am."
"Good. That's where we're strongest. Not those guns." That affirmation was spoken for the benefit of the people of Jerusalem.
"We, too," Shad Iliya said. He turned off the transceiver, looked to the two veterans, of whom he was already thoroughly tired, and said, curtly, "At least someone knows what he is about here. Pity he is on the other side."
The city immediately started bracing for a ground assault. Shad Iliya needed to move fast. The defenders would try to stall for time to redirect some of their guns to ground level—and to the Mount of Olives.
The time-consuming element for Shad Iliya was proper deployment of the weary and dispirited soldiers of the 9th and 34th armies who had been pounding at the gates of Jerusalem for years. And they needed to be accustomed to the idea of an assault without ships, on foot, carrying projectile weapons. Wars simply weren't fought that way anymore.
Jerusalem had the superior numbers. But even had each and every person in the city been a soldier, they only had hand-held projectile weapons for a few—because wars were not fought this way anymore.
Shad Iliya also took care to make wise use of his dregs. In case there actually was a battle, it would be finished quickly and efficiently. He still didn't believe it was going to happen.
While the preparations progressed, the general sent saboteurs into the city to disrupt the defensive measures, while he instructed his own people to challenge any unfamiliar person in their midst to give a password, which was "Shad Iliya is God," in hope that the defenders would be unable to say it even if they happened to learn it.
For himself, Shad Iliya spent the first night rereading the Bible. He sat on a spartan bed under an open window in a small chamber of his war office. He found words that struck him to the heart and that would come back to haunt him later.
_If it be possible, let this cup pass from me_.
The garden was down there at the foot of the mount.
He closed the book and stepped outside to the chill desert night. He watched the lights of the city spread out at his feet. The dry air carried to him the scent of cypress planted around the chapel. He overheard the exchange of night sentries:
"What is it with these Jerusalemites?"
"They think they're God's chosen people."
"Oh, yeah? Well, if God wants them, let's make sure God gets them." The slapping sound was a palm on a metal weapon butt. "Chosen people. I'll show them chosen people."
Shad Iliya left the temenos of the chapel and strolled over the mount in the dark.
The Mount of Olives was teeming with graves, millennia's worth, sleeping under the fig trees and the pink-flowering oleanders. One could not walk without tripping over the marble fragment of someone's marker.
Shad Iliya did trip and alarmed a sentry at the edge of a stand of pine trees. "Halt!" the girl commanded, and pointed her weapon.
Shad Iliya spoke before she could demand the password. "I shan't call myself God, so do not ask."
"Sir!" The sentry shouldered her gun and snapped to rigid attention. "Sorry, sir."
"No need. Very good"—he squinted at her uniform in the dark—"Sergeant. At ease."
The girl shifted her gun on her shoulder and bowed her head. "Was a hell of a lot easier when the enemy didn't look like us," she said wistfully.
Suddenly sad, Shad Iliya nodded and laid a hand on her close-shorn curls. "Yes, it was."
The muezzin's stentorian wail, distorted through the loud speakers, split the night's peace, and Shad Iliya lifted his hands in dismay.
He had wondered why he hadn't counted five prayer calls during the day. Here was the fifth in the middle of the night. They were going to drive him mad.
Either they or the Jew.
Jerusalem's commander called on the comlink at intervals to harass his adversary. Unlike the muezzin, however, the link could be turned off. Shad Iliya never did turn it off, and realized only too late that he should have. It was the Jew's most powerful weapon.
In the morning Shad Iliya woke, itching from a bout with stinging nettles which he had walked into unawares the night before. The Jew's voice sounded cheerily on his pillow. "Good morning, Philistine."
Shad Iliya rolled to the comlink. "Good morning, Zealot." He cleared his gravelly throat. "Surrender?"
"Never."
Shad Iliya looked to his chronometer and calendar. "Today is my birthday."
"Happy birthday, Philistine."
"Thank you."
"You're welcome. Your sixtieth?"
"Thirty-fifth, villain."
"You will be here for your sixtieth."
"I think not."
"Your army has been rotting outside Jerusalem for a hundred years."
Shad Iliya sat up in bed. "Not my army, Zealot. Not my army."
"Your great alien-killers? You will fare no better than your predecessors."
"How can you even talk like that on a closed circuit? Do you not blush? I know you are lying and you know you are lying."
"Maybe God knows better than both of us, then."
"Oh, no."
"I must say, though, yours is the most thoroughly brainwashed army ever to knock at our gates."
"You mean that your propaganda does not work on my soldiers."
"Exactly."
The propaganda didn't work because it had been aimed at the wrong targets. It attempted to undercut the Empire, the Na′id tenets, the Bel. But the soldiers of the 27th Army weren't fighting for any of that. They fought for love of Shad Iliya. And the sterling Shad Iliya had no vices for a propagandist to use.
"You will find my army immune to your quasi-religious unreason," Shad Iliya said. "So you may as well desist—if you can. Disseminating lies comes naturally to your ilk, does it not?"
"Such insults!" said the Jew. "But what shall I call Shad Iliya? I can say nothing about Shad Iliya. Shad Iliya is perfect."
Shad Iliya smiled. "You have been talking to my soldiers."
"What don't you confess, Philistine? You've no addictions. You drink but a little. You're loyal to your wife even in the field for months on end. You've no strange hobbies. No hobbies at all. You must be a monster. You terrify me."
_Oh, but no,_ Shad Iliya thought _. We have hardly begun with terror._
The noose of ships and soldiers tightened around Jerusalem's hills, still out of direct sight from the city. Among the Na′id artillery was a great dragon-carved cannon several stories high, an enormous, primitive-looking thing like the figurehead of a bronze-age warship.
The Jew acquired a picture of it from his recon team, broke into disbelieving giggles, and opened his comlink at once in high spirits. "I like your pet, Philistine. It's cute. What's its name?"
"Its name is Ba′al," Shad Iliya said wryly.
The Jew belittled it blithely. At first, Shad Iliya thought his adversary was fishing for confirmation of a suspicion, but as the derision continued, Shad Iliya realized that the Jew had made his first overconfident blunder.
The Jew thought that Shad Iliya was assuming Jerusalem's shields were weak at ground level in the same way the city's offensive guns were lacking at the ground, and that Shad Iliya was basing his attack on that erroneous belief. The Jew thought Ba′al was an n-cannon.
Shad Iliya's fate whispered to him again. No doubt now. No doubt.
_You stupid Zealot; dragons breathe fire_.
The weapon was actually as primitive as it looked—Man's oldest weapon after the rock.
_Perhaps we could use those, too_. Shad Iliya thought, ironic.
But fire was the one that had first given humankind supremacy over all life on Earth.
And Shad Iliya went to sleep with a burden on his chest, like something sitting on his heart, cursed with the knowledge that, as a certainty, he could not lose this battle.
The muezzin woke Shad Iliya on the third morning. He rose from bed, dressed quickly without his slave's help, threw open the door of the side chamber that had become his bedroom, and stomped out to the front steps of the war office, where the sentries stood.
"Take out that minaret."
Uncertainly, the sentry changed his grip on his gun. "Really, sir?"
"No." Shad Iliya reached his open hand to the side. His orderly knew the motion and put a cup of coffee into it.
Shad Iliya's brow furrowed with a headache. He hadn't slept well. He marched back to his room and found his alien slave, Pony, making the bed. He chased Pony out and sat on the cot, brooding. His fingers laced around his coffee cup.
_God damn you, Jerusalem_.
Of the defense, he worried most about the Jewish forces. There was something charming and endearing about the Arab disorder and disunity. He could sometimes count on them to blow up their own weapons because of an incurable tendency to ignore strict directions and maintenance schedules.
But there was something inhuman about the Jewish unity, precision, and capability. This battle would be a horror. Shad Iliya still refused to believe it would take place. Something would happen to intervene. Surely the leaders would come to their senses. He had given them time to make a patriotic show of defiance. Now it was time to be reasonable.
He rolled onto his back and propped his head up on the pillow against the wall, which was frescoed with faded and chipped Easter lilies. He flipped out the comlink and waited for the inevitable "Good morning, Philistine."
Shad Iliya dispensed with being pleasant and lashed out at the commander for his adherence to and willingness to let a city die for an antiquated religion.
"Antiquated!" the Zealot said. "Old makes it wrong? No, Philistine, old makes it right. Do you think if God were to reveal Himself to Mankind He would wait until now? What of the generations who went before? If there is a Revelation, it is an old one."
" _If_ ," Shad Iliya said. "The operative word."
"You brought the charge of antiquated against us. Don't change emphasis now because I defeated you on that front, O Ye of the Heathen Name."
Shad Iliya sat up and threw his pillow across the room. "When will you leave off this precious conceit of supposing you have a monopoly on the Supreme Being!"
"When God ceases to be."
That meant never. "I shall hang you from a minaret and use you for target practice."
"Why haven't you attacked yet, General?" the Jew asked, sounding still congenial and tranquil—joyful, in fact. An alarm went off inside Shad Iliya, though he didn't know why. A balance had shifted.
"I shall attack when I am ready," he answered, guarded.
"You're ready," the Jew said, as elated as an unarmed soldier stumbling across a fully stocked weapons cache. "You've been ready."
A lump rose in Shad Iliya's throat. His hands were trembling unaccountably. He was going to say he didn't know what the Jew was talking about. But only the guilty used those words, and of course Shad Iliya was guilty of nothing.
Except that Shad Iliya was afraid to kill. The coming battle terrified him, and the Jew smelled it. And the battle began in earnest. The Jew gave up trying to undermine the faith of the army in its leader and he began to gnaw at the head of the serpent itself.
"It is up to you to stop this war, Shad Iliya. You are the attacker. It is in your hands. You give the word that this atrocity may not be."
"You surrender," Shad Iliya countered.
"It is on your head, Shad Iliya."
And the charge of stalling came home doubly when the general overheard his own soldiers wondering about the delay.
"Are we starving them out?" one asked.
"No. Can't be," another said. "It's not our way."
The thought chilled Shad Iliya. Blockade, starvation, was a messy way to go. It spawned gross inhumanity—or else martyrdom. Either way it was a hideous contemplation.
But Shad Iliya doubted it could even be done here. Any longer delay would allow the defenders to break the blockade. Time was the ally of Jerusalem.
_For what am I waiting, then?_ Shad Iliya thought. Then, in anger, _I am not!_
"Ra′im!" he roared.
"Sir!"
"Twenty-four hours."
"Yes, sir."
All day long the Jew wielded his long knives, and Shad Iliya listened with dread, no longer able to turn off the voice of damnation. The Jew pulled out all holds. But one.
Throughout the campaign, the Jew never called the pale, blue-eyed general a nazi. It was the most obvious shot, and the Jew did not use it, for which Shad Iliya was both grateful and unnerved. Was it mercy?
He doubted it. For when Shad Iliya said, "I shan't glory in destroying your people," the Jew replied, "I shan't glory in destroying you."
_You_ , said the Jew. Singular. Personal. He was not claiming victory. He was claiming destruction of Shad Iliya himself.
Sundown came, the eve of the battle. Jerusalem's guns were not ready despite the borrowed time. The defenders had no chance.
_They seem to know_. Shad Iliya watched the Arab men flood into their mosques, the women drape their heads and go through their motions outside.
The Na′id soldiers within earshot chanted with the muezzin playfully. They knew, too.
A less certain Ra′im Mishari passed a scan scope to his general and directed his attention to a Jewish station in the new city. Shad Iliya lifted the scope and zoomed in on the group of soldiers. Some of them wore light-benders, which would make them invisible once activated, but that wasn't what was alarming Ra′im. It was their attitude.
Some lounged in groups, joking, passing around their dwindling supply of cigarettes, while others were dancing their circle dances of old—around a fire.
Shad Iliya shuddered and lowered the scope.
"Why are they so confident?" Ra′im said worriedly.
"They are not," Shad Iliya said. "They know."
"I don't understand."
"I do not understand either," Shad Iliya said with a hand on Ra′im's shoulder. "I only know it is so."
Before retiring for the night, Shad Iliya made a last desperate effort to avert the coming massacre. The Na′id supreme commander as a rule never initiated contact on the link. He did so now and asked privately for a surrender.
"Of course not," was the answer.
"What say your Christian and Muslim colleagues?" Shad Iliya asked.
"We are in accord," said the Jew.
"Ask them!" Shad Iliya cried in obvious agony.
The Jew agreed to consult the other two members of the triumvirate, and he relayed their answers an hour later. "The Christian says that the Holy Land is not for the godless. The Muslim says Jihad."
"Shit."
"What means this word 'shit,' Philistine?"
"It is an Anglo-Saxon word meaning excrement, also synonymous with Jihad—so says Shad Iliya."
"Strange. Cardinal Miriam is forever saying that hell is the Anglo-Saxon word for war. Or war is hell. I forget which."
"God damn your flippancy! If you change your mind, this link will be open all night."
"It is a morning assault, then, General?"
"Good night, Zealot."
• • •
The defenders were not to spend a peaceful night. Shad Iliya told his own soldiers to get some sleep—real sleep. He bid only a few night officers to keep Jerusalem awake till dawn in readiness. At the places where the guns were being lowered, he ordered heavy salvos and robot flyovers at 0213 hours, and again at 0400 and 0415. "And the moment the muezzin opens his mouth for the 0300 prayer call. Everyone else to bed."
Then he hiked down the far side of the Mount of Olives to his ship for a proper bath. His alien slave, Pony, came along to assist him. The pretty palomino-colored humanoid knew his master's routine and was never too much in the way.
Pony was an androgynous, lithe-bodied, doe-eyed creature with white mane and horsetail. Shad Iliya had named the slave. Shad Iliya was one of those people who called a spotted dog Spot and a red horse Red. Pony's real name had a click in it, and Shad Iliya was not about to torture himself learning to speak clicks in order to pronounce his alien slave's name. Pony was a useful thing, a pretty piece of furniture, better than a robot. Pony would do anything for him.
Shad Iliya pulled Pony's mane and closed both fists in the long, silvery hair, hardly aware of the creature in his hands, his fierce thoughts elsewhere. _By heaven, you cannot fight a war with human beings. Are we not one family?_
Pony waited meekly with bowed head, trying not to wince.
Shad Iliya swallowed hard. _Life goes from my limbs and they sink, and my mouth is sere and dry; a trembling overcomes my body . . . my mind is whirling and wandering._ _And I see forebodings of evil, Krishna. I cannot foresee any glory if I kill my own kinsmen in the sacrifice of battle. These I do not wish to slay, even if I myself am slain. Shall we not, who see the evil of destruction, shall we not refrain from this terrible deed?_
He released his slave and returned to his room in the chapel on the mount. Pony had already remade the bed and set out his nightclothes and some drinking water. And a shot of whiskey.
_Good old Pony_.
He lay down, the comlink on the marble-topped nightstand. He lifted his hand to it, wanting to call. But he could not. He had nothing new to say. He would sound very weak and frightened.
The Zealot called in the middle of the night. Shad Iliya rolled to the nightstand and lunged for the com with shaking hands. _Will he surrender and avert this bloodbath?_
No. He said he was lonely.
Shad Iliya was stunned. "So you call _me_?"
"You are the only man in the galaxy who can possibly be as lonely as I."
Shad Iliya fell back on the bed, his eyes directed up at the painted ceiling in the dim light.
_Truth_.
Cricket sounds and cypress scent drifted through the window. The night was starry.
Shad Iliya checked his chronometer. It was nearly time for the next missile strike.
"You are probably worse off," the Jew said. "Because you are wrong and know it."
_"Please."_
"Beg with me, Shad Iliya?"
"You know you will be slaughtered."
"I know that?"
"Are you waiting for a miracle?" Shad Iliya rasped savagely.
The Jewish commander was grim. "No. There will be no miracles today."
Shad Iliya closed his eyes tightly.
_O gods, there truly will be a battle!_ The realization sickened him. _That is it. That is it._
He tried to encourage himself with scriptures:
_To forgo this fight for righteousness is to forgo thy duty and honor._
_Fighting is obligatory, much as you dislike it._
The latter law the Na′id had taken from the Koran. And, realizing that, Shad Iliya fell deeper into despair. With both sides equally unbending, there was no way out. None at all. Shad Iliya rose before dawn and bawled for Pony to bring his dress uniform from the ship. Pony obeyed and helped him dress, including a bullet plate. Shad Iliya activated the radiation screen in his belt and drank the radiation medicine Pony brought as an extra precaution.
General Shad Iliya appeared on the steps of the war office in high temper, his eyes ringed angry red, his uniform blazing metallic blue. Everyone else was in battle drab.
He marched over the crest of the Mount of Olives to address his troops. A holo transceiver carried his image to his soldiers on all sides of the city. His fury channeled to fervor, and he stirred in them a patriotic spirit he didn't feel. Then, more to encourage himself than them, he read from scripture in a sonorous voice:
"'No one can bring to an end the Spirit which is everlasting. Therefore, great warrior, carry on thy fight. If any one thinks he slays, and if another thinks he is slain, neither knows the ways of truth. The Eternal in thee cannot kill. The Eternal in thee cannot die. Weapons cannot hurt the spirit and fire cannot burn it. The Spirit that is in all beings is immortal in them all. For the death of what cannot die, cease thou to sorrow. Think thou also of thy duty and do not waver. There is no greater good for a warrior than to fight in a righteous war.'"
He closed, and the army erupted into cheers that resolved into a chant—not the Bel's name, or any of the Empire's catchwords, but his name, _Shad Iliya_.
There remained ten minutes until dawn. Shad Iliya gave the order to mobilize. He climbed back to his observation post on the chapel steps.
The city was quiet. The defenders were in hiding. They would have to be hunted down in the narrow streets and closed buildings, each a kind of fortress in itself with many traps.
The sun rose golden on the ancient town.
_O God, save your city_.
There were no crickets, no birdsongs this dawn. The muezzin's morning prayer call went unanswered.
_I am going to take Jerusalem, and even the God of three peoples cannot stop it._
_Not even I can stop it._
The Na′id army appeared on the horizon on all sides at once, a black shroud of soldiers and transport ships and guns drawing in on the city.
A cry rose out of Jerusalem as if the city itself had a voice.
_Be frightened, Jerusalem. Be frightened and live_.
The tall Na′id standard crested the Mount of Olives behind Shad Iliya and cast a long shadow across the entire city, old and new. Shad Iliya found himself in its shadow.
He scared himself with the sight he had created. He was struck by a strong premonition of death.
Across from him, on the far horizon, the great dragon cannon Ba′al loomed, its mouth gaping at the new city.
_This is an abomination. What have I done?_
He spoke into the command circuit. "Anthem."
The inciting military music sounded through the tiny speakers each Na′id soldier carried, and the Na′id army charged and opened fire with great noise. Shad Iliya covered his ears with his hands.
Ra′im shouted at his side, "Sir, is all that racket necessary?"
The general nodded. "For the Arabs. If they cannot hear it, they think it will not hurt them."
They were an emotion-led people. Their wailing war cries curdled the blood. Both sides of this conflict were acquainted with the great power of sound.
The defenders had a few surprises. Shad Iliya had anticipated that there would be some. One was a portable scatter-dart cannon that swiveled and felled Na′id ranks on the north side of the city.
Shad Iliya was on the command circuit quickly. "The scatter cannon," he said sharply. "Take that out. Whatever we have to use."
A cluster of twenty-four smart missiles were launched at it en masse. One got through, and Shad Iliya relaxed a measure. The expenditure of precious hardware didn't bother him. What else was it for?
He wished it could all be done with robots. As the battle progressed, he watched, sickened, as each contingency which he had hoped to avoid arose, and blood flowed in answer to it. The soldiers of both sides were heroes all, the best fighters on Earth—the people of Jerusalem fighting for their God, the Na′id fighting for Shad Iliya.
_My people can see me, Zealot. Your God is painfully absent here_.
A beam ricocheted off his personal n-screen. He wheeled and fired toward the source with a bullet gun.
A child carrying a beam gun reeled back with a high-pitched grunt and died. It was a barefoot, runny-nosed urchin someone had taken in.
The body was spirited away by Na′id soldiers with hurried, embarrassed apologies.
Shad Iliya was left agitated and ill. The shock was not at the child's tender age—to Shad Iliya human beings didn't gain or lose value with age—but that the child was human. Shad Iliya had never killed a human being with his own hand. It was too easy. There ought to have been more to it.
He swallowed down a rising gorge, lifted his scan scope to his eyes, and watched the battle.
An hour passed. He contacted Ra′im, whom he had sent to supervise the western front. "Pull out of the west quadrant. Stand by with Ba′al."
"General, do you think—?" Ra′im started, stopped, restarted. "Sir, we're holding our own."
Questioning orders was unlike Ra′im. This battle was affecting him, too. "This will go on all day," Shad Iliya told him. "At this rate, the city will fight to the last infant." He needed to break the back of the defense. Now.
"Yes, sir," Ra′im said, and he initiated the controlled retreat.
Shad Iliya withdrew into the war office and waited out the evacuation. An officer came to him to report, "We have a prisoner, sir."
"The Jew?" Shad Iliya asked. He had ordered only one specific person taken alive.
"His wife," said the officer, and motioned for the woman to be brought in.
She was young—Shad Iliya's age—with long black hair, a dirt-smudged face, and blistered hands. Her wrists were bound behind her back and her ankles were shackled to restrict her steps to an abbreviated stride.
She knew the White Na′id at once. As she faced him, she whisked her long ponytail over her shoulder with an imperious toss of her head and spoke coldly in a rough, throaty voice. "Your wife is said to be the most beautiful woman in the galaxy. What do you want with me, General?"
Shad Iliya was angry. "You are a prisoner of war, not the booty of some pirate raid, so let us not play at that. And, yes, my wife is beautiful. Myself, I would let you go."
"That would be a mistake," she said.
"I fear so."
Jewish women were not gentle things.
"Your soldiers are on the run," she said.
Shad Iliya nodded. His people were efficient in carrying out orders; Ra′im had them evacuating from the cannon's path.
The woman frowned, sensing from his calm something sinister in the ease of the Na′id retreat.
The general's young orderly, Sinikar, came into the room carrying a staple gun and ID with which to tag the prisoner's ear. The woman stiffened like a rod as the boy drew near.
"No," Shad Iliya said. He motioned Sinikar aside, and he removed the gold earrings from the woman's right ear. He slipped them into her hip pocket. She stood rigid at his grazing touch. He took the ID tag from Sinikar and gently laced it through the woman's existing earring holes.
Her eyes slid to the side as if trying to see her own earlobe. "What does it say?" A slight quaver betrayed apprehension.
"It says you are dangerous," he said, then to his aides, "Get her out of here."
They walked her as far as the door, then she suddenly broke away and ran stiff-legged back to him, yanking at her shackles. "General!" she cried.
"Yes."
"Am I to be killed?"
"No."
She breathed in relief. But didn't seem sure if in reality she was better off.
A shout sounded from the portico over a descending-pitched whistle. "Incoming!"
"Get down!"
Shad Iliya dropped to the floor, taking the Jewish woman with him. He covered his head and hers, since her hands were bound, as the shells exploded around the building in thunderclaps. Clods of dirt clattered down on the roof in a dark hailstorm. One shell blasted very near and rang the chapel like a drum, sending down a rain of plaster on their heads and strewing white dust across the time-faded colors of the patterned floor.
Shad Iliya lifted his head, coughed in the dusty, thick, burned air. He checked the woman and then his curly-haired child orderly for injuries. They were unhurt. He looked up at the ceiling. The centuries-old frescoes of his pretty chapel crumbled around his ears.
Suddenly furious, he dragged himself up and outside. The shelling had stopped. He glared at the city in a rage.
He went inside the office and punched on the command circuit. It was still operative. "Are we clear, Ra′im?"
"Yes, sir," Ra′im answered over the circuit. "We are. But a lot of noncombatants have moved into the west quadrant."
"Commence firing."
Ra′im paused. "You don't mean Ba′al, sir?"
"Yes, I mean exactly that!" Shad Iliya cried and punched off the circuit. _Am I speaking Swahili? Is everyone suddenly become deaf?_
He stalked out to the portico.
Jerusalem's layers of radiation shields were impenetrable to any of the empire's sophisticated death rays of a frequency higher than ultraviolet. But simple, primitive fire could pass through as unimpeded as visible light.
The ominous, dragon-faced tower of Ba′al came to life with a roar of flames and sent a long river of fire arching into the new city.
A shudder ran through Shad Iliya with the streaming flames and the long scream of his Jewish captive in the doorway behind him. Tears from some unknown emotion threatened his eyes. He inhaled unevenly, his shoulders, back, and neck taut. The scream from the city reached him even here. He shuddered again.
People flooded out of the western city, fleeing to the unburned part. They crowded toward the Old City. The Na′id would not destroy that part. There, the fighting would be hand-to-hand.
West Jerusalem was a wall of thick, flickering orange-black fire. Its gutted buildings belched up more hell-hued billows. Shad Iliya watched, wild-eyed, as if mad.
He felt as if this were the first time he had ever really fought a battle. His enemies, dying before his onslaught, were people. Part of him felt sick, part . . . .
His self sickened him.
In extremity, the veneer of humanity disintegrated. Shad Iliya was left facing his essential self and nothing else. And seldom answered in times of rationality was the question of what happens when one cannot live with what he finds. The great flames roared and billowed and snapped like a flag in the wind.
_So unfurl our standard and fly our true colors. Flame and death._
The Jew's wife, several paces behind him, cried a contralto wail over her beloved city, a pure emotion he could never find in himself. Shad Iliya never could act what he felt, and was never sure quite what or if he was feeling.
He beheld the city with his rounded, white-circled eyes, sweat on his brow, a lump risen in his throat. This felt like emotion. And the city burned.
The woman ran to his side with her shortened steps. "Stop it! Stop! I see you loathe it, too! You are dying before my eyes!"
He gestured for her to be taken inside.
A girl soldier came to his shoulder, looked aside at her staring general, then followed his gaze to the burning city. "Terrible, isn't it?" she said.
Shad Iliya couldn't speak. _Terrible._
He saw the defenders braving the flames. Their heroism never failed. It wasn't simple martyrdom—there was that, too—but these people wanted to live. Their energy, their spirit, their life could not be snuffed out. It put Shad Iliya to shame.
A black cloud drifted over the city and toward the Mount of Olives. Shad Iliya watched it come with detached disbelief. It was on him before he thought to go for a mask, and he was suddenly blinded, choking, and unable to breathe.
Someone groped to his side and pressed a mask to his face. It was the girl soldier, giving her general her own mask. Then she staggered off blindly for another.
Shad Iliya retreated inside the chapel and shut all the windows. He tore off the mask. The air inside was hot and close and hung with floating soot.
The comlink sounded. Shad Iliya grabbed it out of his pocket in surprise. A cold voice said, "You are brutal, Philistine."
"Did you think you had a monopoly on that as well?" said Shad Iliya.
"I did. No. Not a monopoly. A supremacy. We can be ruthless when we need."
"I am aware." He replaced the link in his pocket and turned his eyes to the Jew's wife, crouched in one corner of the war office. He and she were alone in the room.
_Why didn't I let them talk?_
He knew why. He wanted her. He was alone and he didn't want her to have someone else, least of all the Jew.
He had never wanted a woman so strongly as he suddenly wanted this one. He couldn't say why. _Because she was here? Because she was his?_
"Stop staring at me," she said.
"I like you," he said.
Her back straightened against the wall. "May I say, General, that you revolt me?"
Shad Iliya was stung. He replied wearily, "Why not? You Jews say what you like, to hell with 'may' or 'may not' anyway."
He checked the computer monitors at his station. One was dark. The Na′id fire had taken out one of their own monitors.
He tried to brush soot from his uniform, then gave up. He was very dirty. Only his watery blue eyes were undulled.
"Shad Iliya?"
It was the woman, crouched in the corner.
He turned, mute.
"I didn't mean that," she said.
He shook his head to say it was nothing.
"I had to say that." She stood. "You are my enemy."
"I am only a man," he said, very tired.
She shook her head, her mouth spread into what would have been a smile were it not so sad. "That is a Na′id view. I don't see that way. I am not Na′id."
He crossed to her and seized her shoulders in his big hands, beseeching and insistent. "You are human."
"I am Jewish," she said, helpless before the impasse. She wasn't immune to the syndrome of loving her captor, but she could remind herself that she was a Jew and this poor monster was not.
She did love him. She knew its cause, knew it to be false, but felt it burn regardless, impervious to logic. She was surprised that it had come over her so fast. But that was what came of living at the brink. The law of mayflies. If you have one day to live, you live your whole life in one day, and brightly.
Shad Iliya still held her shoulders. He heard her breath deepen, and on impulse he drew her in to him and kissed her. Her mouth rose and opened to his in startling passion. He put his arms around her, held her, ran his hands over her hard waist and her moving hips that pressed against his groin. He buried his face in her long hair falling loose from its tie, and a deep, sultry voice in his ear said, "Take me."
Shad Iliya laughed weakly. "I can't. Not now. Later."
"I won't want you later," she said. "I need you now."
He covered her mouth with his, and they sank together to their knees as the flames blazed over her city and an all-consuming dark cloud engulfed the landscape and blackened the windows of the chapel.
Armageddon raged outside—the last great battle between good and evil. Except that there was no good and no evil _. Just us_.
And both of them were terrified beyond what the human body and spirit were meant to bear. They pressed together and groaned in animal abandon—the rest outside was madness and this the only sane thing being done in the world.
His passion-clumsied fingers had only begun to fumble at the closings of their clothes when one of his aides cried from outside, "General!"
Shad Iliya scrambled to his feet and ran to the door, forgetting his mask. But the slight wind over Jerusalem had shifted and the air outside was clearing.
The lifting cloud revealed a new scene.
A black-robed troop of Muslim soldiers had moved into the valley at the foot of the mount under cover of the black smoke.
They stormed up the slope with a hideous outcry, dying by the tens and twenties with every step as Shad Iliya's guards cut them down in rows, and still they came with warbling shrieks. The Arabs used the most modern weapons—beams—which reflected off the modern Na′id screens. They themselves fell to simple old-fashioned bullets, their useless radiation screens aglitter, and still they charged as if there were immunity to death in their blood-fierce wails.
Shad Iliya took up a machine gun and mowed them down, row on row as they came at him, right up to his very feet.
When they were all dead, he took a staggering step backward. His hands shook. His eyes stared.
People. These were human. There were hundreds of them. Their blood was spattered on him. He licked sweat from his lips—tasted blood.
Dizzy, he dropped the gun, dead Arabs all around him, some staring glassily back at him, a ghastly mirror. This was what he saw inside Shad Iliya.
Shad Iliya clenched his fists.
It was their own fault they lay all down the slope in heaps. He wanted to scream at them. _Stupid! Stupid!_ They carried the best weapons. But the best weapons didn't work here. Rocks would have worked better.
_Jerusalem, you are stupid!_
Idolatry is worse than carnage—so said the Koran. How was the Empire to absorb such a people? They were oblivious to their human kinship. One could only kill them.
Shad Iliya went back inside the chapel to find the Jewish woman. She was dead—without wound, fallen to a beam shot. She was the only person on the mount downed by the Arab attack.
Shad Iliya knelt with a groan. She appeared to be sleeping, her face relaxed into rough-cut prettiness. He didn't touch her because he was covered with blood.
_Wake up, woman. I need you_.
He tore himself away from her side and stood.
_Who is killing whom and why—does anyone know? Do I?_
• • •
In the evening, the first faction of the defenders surrendered. The Christians were far and away the most reasonable of the three. Racially, they were the most Na′id-like, a mixture. They knew suicide when they saw it. Not that they hadn't martyrs. Every people had its share of martyrs, thought Shad Iliya, except the Jews, who were made up of nothing but martyrs as far as he could tell.
Once the Na′id forces occupied the Christian quarter, the Muslim resistance collapsed into confusion and left the Jews outflanked.
Shad Iliya climbed to the summit of the mount at day's end. It was a bloody sunset, the sky aflame with tortured hues. The sun's long light cast a red wash over the burning city.
_I warned you, Jerusalem. Did you not believe me? You are my city. What did all your blood gain you?_
He stood immobile, a tall, angular silhouette against the flames, his straight arms held a little out from his sides, his feet planted wide on the ground, his head thrown back. The wind fluttered his clothes. He tensed and trembled as if in great pain.
_This is yours, Shad Iliya._
The recorders, standing ready to immortalize this moment, caught the victorious general so, surveying his conquest; this was the scene that would go over the Net.
Down below, Na′id soldiers poured into all parts of the city. An aide came to his general's side.
Shad Iliya's voice was deep. "Inform the Bel: The city is ours."
The veterans of the 9th and 34th armies ran amok in the streets, wreaking their personal vengeance on the city that had spat at them for so long. Out of control, they killed, raped, and maimed.
"There will be charges filed in cases of excessive violence," Shad Iliya ordered over the command circuit. It was a threat only. He didn't intend to carry it out. "I want this rampage stopped. And I want the Jewish Zealot. Alive."
Then he secured his personal screen, holstered a handgun, and descended into the city on foot, stepping carefully down the slope cluttered with the bodies of Arabs he had killed. He waded through the garden of Gethsemane, muddy with blood of the faithful.
The Lion Gate opened for him, and the supreme commander entered the Old City. Blood flowed down the Via Dolorosa, where the fighting had been close-in with bayonets and scimitars.
Shad Iliya turned into a darkened side street, illuminated his lantern, and proceeded slowly down the crooked way. Moldering walls hugged the avenue. It was strewn with sand and debris, smelling of spice and vegetables and urine. He heard rustling and he halted, lifting his lantern high, expecting snipers.
A herd of mangy goats gazed down at him from a rooftop.
A clothesline stretched across a third-floor balcony, its sooted wash waving in the night wind.
A black kid ran bleating across his path in the stepped, zigzag street, and skirted behind a boarded-up kiosk beside a rickety wooden table on which sat the aluminum keg and three dirty glasses of the water seller. The street's single videophone lay smashed on the concrete.
A jeep whined, careened around the corner, flashing blue-and-red lights. It jounced to a stop and an aide jumped up. "Sir!"
"Yes."
"The Jewish commander, sir. We have him located."
_Located,_ he said. Not _We have him_. "What do you mean, 'located'? Is he dead?"
"No, sir. He's holed up in an alley. He has a weapon and we can't get near. We're afraid to try stun. He's critically wounded. Should we move an armored tank in, sir?"
"No. Take me to him," Shad Iliya said and climbed into the jeep.
The vehicle swerved through the crooked streets, scattering flocks of geese and chickens before its path and avoiding bodies of the dead and groups of prisoners corralled by the occupation force.
They arrived at the entrance to a cul-de-sac where was amassed a broad array of firepower and soldiers hiding behind either corner. Trapped inside the dead-ended alley was one man.
"What kind of weapon has he got in there?" Shad Iliya asked, dismounting the jeep and reviewing the assemblage of arms and artillery that held the Jew at bay. "An atom bomb?"
"No, sir. A pistol, sir. And a knife, we think," the captain said.
"A pistol and a knife," Shad Iliya muttered. "We think." He drifted toward the alley.
The captain gasped and seized his general's shoulders as he wandered too close to the alley's opening. "Stay out of the firing line, sir. He shoots at all comers. He killed our medic."
Shad Iliya shook the captain off and stepped toward the alley.
"Sir, he's a madman!"
"I know that," Shad Iliya said. He walked out into the open and threw his own gun aside.
At the far wall of the cul-de-sac lay the solitary man of about fifty years, short and muscular but not big, his scanty black hair graying, his legs sprawled at odd angles as if his back was broken and he partially paralyzed. He was hardly an impressive figure, and Shad Iliya wondered if this was indeed his adversary. Not that he expected more. He simply realized that he didn't know what his Zealot looked like now that he was unarmed and within this man's range. His gesture had been very rash. _Have I a death wish?_
The wounded man lifted his balding head and saw the White Na′id at the mouth of the alleyway. He smiled as if actually happy to see him. "Ah. Philistine."
Shad Iliya let his hands drop loosely to his sides. He spoke, emotion-choked. "Zealot."
The Jew let him approach. Shad Iliya knelt, lifted the Zealot's head and shoulders off the pavement, and cradled him in his arms. The Zealot let his gun slip from his hand.
Suddenly Na′id soldiers rushed into the alley, but Shad Iliya held up his hand to stay them, and they all froze. "Out," he said.
"But, sir—" the captain started.
"It is not my custom to repeat orders!" Shad Iliya roared, still holding his fallen enemy, and the soldiers hastily backed out of the alley.
The Jew shut his eyes, a faint smile on his lips. "Damn, I could've used a voice like that," he murmured. He was badly hurt, Shad Iliya could see now, some of his bones crushed, his lower organs probably damaged and slow-bleeding. He was dying.
"Our medical technology is better than yours," Shad Iliya said. "You can be revived—"
"I'll cut my throat," the Jew said quickly and gripped the hilt of a knife sheathed at his belt.
"Very well." Shad Iliya had expected that answer. "You are going to die, then."
The Jew winced in pain, but accepted it, his imminent death.
Shad Iliya suddenly could not. He realized that he didn't want the man to die. It frightened him unreasonably.
"If I asked you to live?" Shad Iliya asked.
"Why?" the Jew said. "For you?"
"If I ask as a man?"
"Which is to say, 'as a Na′id.' No, Philistine. I like you well, but the life of a man is not at stake here."
"Then tell me what is, and do not say your damned God!"
Brown eyes opened and gazed up at him. The eyes were large, long-lashed and beguiling, the eyes of someone who enjoyed life, lived it very hard, and loved women. "Do you know what freedom means?"
Shad Iliya sighed. "No. I guess I do not. I don't understand."
"Go to Masada, Shad Iliya, if you would understand Jews."
Brown eyes closed. He wasn't a big man. Then why was he so heavy in his arms?
"Tricks, Philistine," the Jew spoke again, reproaching, his eyes still shut. "You resorted to tricks. I was foolish not to recognize that flamethrower. Who would have thought it? In this century."
"I had to," Shad Iliya said apologetically. "Your soldiers were better than mine."
"They fight like demons—yours. They do all right for not being Jews and having nothing of substance to believe in." He grimaced with a spasm; relaxed again. "Do they really believe so strongly in that Na′id drivel and doubletalk? What makes them go?"
"They believe in me."
The Jew opened his eyes. "You don't worship a heathen god. You are one."
"I begin to think so," Shad Iliya said softly.
The Jew's eyes flitted up and down him, to his sooty hair, dirty face, soiled uniform, and red-brown crusted fingernails. "You're a bloody mess. So how does it feel to kill your fellow humans?"
"What?"
"No _what_ , Shad Iliya. This is your day of glory."
"I don't understand," he said, afraid that he actually did.
The Jew moved in for the kill. "Did ever an overwhelming emotion ripple your flat existence before they sent you to kill real men and women?"
Shad Iliya needed to interrupt and shift this line of talk. "And what about you?" he protested.
"No," the Jew said placidly. "I am part of something larger. Something I believed in and still do. You—yours is up in flames." He gestured up to the ash-smeared sky with his eyes. "See there the family of humankind blots out the stars. It's splattered on your face."
An explosion somewhere in the city punctuated the accusation.
"Shad Iliya, I am in a great deal of pain already. Please don't squeeze my ribs so tight."
Shad Iliya loosened his hold mechanically, hardly hearing him with his conscious mind, all the pitiless words driving straight to his heart and into his guts. He stared without real sight.
"Oh, Shad Iliya, you loved it. You were never so alive in your life. The rest is tasteless. You don't even like sex. This is your lust."
Shad Iliya uttered a groaning growl of fury under snapping rein. "You are calling me a pervert."
"You are. The worst kind."
"I fucked your wife."
"No, you didn't," the Jew said as if he really knew. "Here is your ultimate act of love." He slipped his dagger into Shad Iliya's hand. "And you never loved anyone more than me."
Incensed, Shad Iliya took the knife from the Zealot and stabbed into his chest. He didn't know how many times.
Warm blood on his hands, Shad Iliya lowered the body to the ground—or pushed it off his lap. He didn't know exactly what he did. He got up on wobbly legs to stand alone in the alley, not certain who had more thoroughly destroyed whom. The Jew was dead. _But what am I?_
Shad Iliya had only killed the Jew. What the Jew had done to Shad Iliya he didn't yet know. The Jew had done something—irreversible and final as death. He was the last human being Shad Iliya ever killed.
He emerged from the alley badly shaken.
His officers brought to him another prize with which they thought he would be pleased—Her Eminence Cardinal Miriam.
Plodding slowly with swollen ankles, not to be hurried on any account, the stout, white-haired woman let herself be led, sedate and unresisting, but still righteously proud.
Brought before the general, she glanced at him once. "Your hands need washing." And that was all she had to say regarding Shad Iliya.
She had blue eyes.
Two officers assisted the cardinal into a jeep to take her to the prison ship.
Shad Iliya wandered away with aimless, stumbling steps, his aides following uncertainly at a distance.
The crowded buildings opened to a wide way where the dead bestrewed the street, twisted in the shapes of a violent end, fallen over each other.
Shad Iliya stared with unholy awe. There was an obscene beauty in the stillness. He walked softly among the tortured corpses, unblinking.
Black-faced crows sailed out of the night sky and settled on the bodies.
Shad Iliya reached for his gun; didn't have it; seized someone else's—one of his aides'—out of its holster and shot at the birds.
The flock of them rose with a riot of hoarse squawking.
Shad Iliya screamed an order. "Get those things off our dead!"
"Those aren't ours—"
"Keep them off!" He ran up the gradient bricks of a ruined wall and fired at the crows from the top. "Get them off!"
His aides rushed to see the command carried out. "Yes, sir!"
From his vantage atop the wall, Shad Iliya could see the red glow from the western city, and caught glimpses of his soldiers rounding up prisoners in various sectors. Sporadic gunfire marked the clearing out of remaining pockets of resistance.
He climbed down from the wall. He had a vow to keep. He mounted a jeep and ordered the driver to take him to the Dome of the Rock.
His orderly, Sinikar, offered him a gas mask at the door of the mosque. He didn't take it.
He wished he had. Inside was a grisly scene. The fetor of mortification hit him in a suffocating wave. He held his breath. His eyes watered and throat constricted, trying to gag.
It was the same in all the mosques and churches and synagogues. The people had taken refuge in the houses of their God as if He would save them. He hadn't. They died so easily. And He did not strike down Shad Iliya.
Allah had told His people: _Fight_ _for the sake of Allah those that fight against you._
They had.
_Kill them where you find them. Drive them out of the places from which they drove you. Do not fight them in the precinct of the Holy Mosque unless they attack you there; if they attack you, put them to the sword._
The colorful walls of the holy mosque were smeared with blood, the floor piled with bodies wrapped in their prayer rugs because someone had tried to make it neat for him. The air was cooling with nightfall, but the place was still thick with a miasma of rot and rigor from bodies cooking in the ninety-degree heat of day.
Of a sudden, from above, the loud deep prayer call erupted in its changeless refrain. It was a recording. A dead voice calling dead Muslims to pray to an indifferent God.
_Is anyone left to answer that?_
Shad Iliya beheld all the prostrate bodies, a travesty of adoration in the reeking air. The voice blared and reverberated. Shad Iliya shook.
_It's dead! It's dead! It has been dead all along!_
He ran outside and put a bullet through the loudspeaker.
His aides came to him, worried. He bent over, hands on his knees, gulping air, his sides heaving. He tried to chase the foulness from his nostrils, his mouth, his throat, but even out here the sticky-sweet, sickening odor of burning human flesh permeated the city.
In a weary, croaking voice, he asked, "Do we have anything that will put out the magnesium fire?"
"Yes, sir, but—"
"Then put it out," he said, lifting his head.
In the distance he heard the sound of human voices, many, chanting in unison.
His aide was speaking. "But, sir, it will burn itself out by morning."
_"Put it out!"_ he shrilled.
The voices in the distance gathered strength and volume. He could make out the words. His name.
"Shad Il-i-ya! Shad Il-i-ya!"
He rubbed his hands against the sides of his uniform, but there was not a part of him clean of blood.
He remembered what the Jew had said. _It is on your head, Shad Iliya_.
In the aftermath, he felt it.
"Shad Il-i-ya! Shad-Il-i-ya!"
_It is on your head. Shad Iliya_.
He knew it was. There was nothing he could do to change it.
He could only keep silent and wish his station would allow him to cry.
Upon taking the city, Shad Iliya was immediately relieved of supreme command of the combined armed forces at Jerusalem. Someone else was put in charge of normalization. It would not do to have a nazi lord over the city. And Shad Iliya was grateful not to linger over the kill. Later the honors and decorations would be heaped high to bury the implicit slight.
Shad Iliya was uninterested, totally oblivious to the state of his honor. He roamed his ship listlessly. He was bathed and dressed in a fresh uniform of olive drab, his blond hair stripped so clean it drifted up in untamed oilless wisps. Everything around him was scoured and spotless. His bare feet collected no dust on the pristine deck. His fingernails were cut very short, smooth, white, and immaculate. It had taken hours to steam the feel and stench of death from his pores.
Some orderly or officer was always there to wait on his every order and whim. He said, staring, "When you stab someone more than once, it means you are deranged, does it not?"
"Oh, the Zealot, Abram," the adjutant said. Shad Iliya really had made a mess of him. "Don't worry, sir. That part didn't get out to the news Net."
That was hardly the point. Shad Iliya walked away from the adjutant.
He was still in shock, still bewildered. "I don't understand," he murmured. "I don't understand Jews."
_Go to Masada, Shad Iliya, if you would understand Jews_.
• • •
Birds sailed through the canyon, small and swift and smooth, black with dull orange atop their wings. Shad Iliya looked down at them from the top of the massif, a height that was dizzying for its steepness rather than actual elevation.
Desert. All pale brown and white earth surrounded the fortress. No growth but a bare touch of sparse, dull scrub winding along the dried-up wadi far below. Silent but for the wind. Utter silence broken by high lonely cries of the birds sailing through.
_Zealot, you are mocking me with this._
He wandered among the ruins alone, sat on a step of a building that was long gone. Bewildered.
Masada was a desolate tell among several other eroded desert mesas banded in brown and lighter brown strata. Piles of scree slipped down the cliff faces and lay in fans at their bases. The air was very dry, making details clear, flattening everything to cartoon unreality.
Shad Iliya brushed away a fly.
_This is me, not you, you stupid Jew_.
Magnificent, wind-blown waste.
Hard shadows. Sky blue above. Dusty, pale horizons.
On the other side of the tell were more cartoon-stark canyons and the turquoise Dead Sea.
_Why must everything be dead?_
He had come here for an answer, and there was something here. He wished someone would explain.
The sun rose higher; shadows shrank. The jagged, sharp shadow of the massif drew in toward its western face in the advancing daylight and became something ominous and awesome in Shad Iliya's sight. The shadow of Masada. It did not talk.
_I don't understand Jews at all_.
• • •
The Bel invited Shad Iliya to a fete in his honor at the palace on Mat Tanatti. Shad Iliya requested time to recover, so the Bel sent him to a villa on the coast of the Levant, a beautiful, relaxing place, and sent Shad Iliya's older brother there personally to ensure his comfort. The hero who brought the hundred-year battle to an end could ask for anything and it would have been granted. He wanted nothing.
One of his aides, a lieutenant, worried that his general was despondent, but he wasn't of a rank to say so and be heard.
Shad Iliya prowled the halls of the villa, not to be consoled by anyone. Neither did Na′id philosophy soothe: _Time destroys all. If you had not fought them, still they would have died._
Because it wasn't the carnage that haunted him. It was the liking it. He mentioned it in an unguarded moment to his brother.
"Oh, bosh, Shadi. You're the gentlest, most philanthropic person I know."
Shad Iliya whirled on him, eyes ablaze, teeth bared. His brother exited quickly.
He boarded a ship for Mat Tanatti, deciding Shad Iliya needed a few days in relative solitude. His famous brother was simply distraught from the battle and his herculean efforts. And the incident didn't merit repeating.
• • •
Shad Iliya sat drinking at a painted wrought-iron table in a chair set out for him on the beach. His aides stayed apart except to replace the bottle and to bring to the table a receiver which would pick up the Bel's speech when it was broadcast over the Net.
_This is the hour of humankind's greatest victory_. . . .
The air above the water shimmered in the summer heat. Shad Iliya's head lolled onto his shoulder. He poured another glass of cognac—poured twice as much as made it into the glass. No matter. The bottle was always full.
. . . _a new nation in which we are all brothers and sisters. And no human being calls anyone Master_. . . .
The words shimmered with the air, as if he could see the sound. It was coming from all sides. No. It was inside his head.
. . . _Babel refounded. We are one people once more. United we stand and even God shall not sunder us ever again_.
Before him the glittering Mediterranean lapped at the brilliant strand. He was hot. His blond hair felt as straw tinder to the touch of his hand.
The water lured. He gravitated toward it. The beach was soft and difficult to walk on. His feet kept sinking in the burning white-gold sand, making his steps more staggering.
At last he slipped into the warm water. Either it was warm or he was drunk. Probably drunk, he concluded, because he had taken his glass with him. He was perplexed to find saltwater in it instead of cognac when he took a drink.
He swam. His thoughts and perceptions ran clear—seemed to. The world was incandescent, the late sun bright upon the water.
Waves rolled in, lifted him, and he rode up with them. The bigger ones swelled over his head and made him sputter and blink.
He saw where the waves were flattened, and he made his way there to the darker water.
Next the world was dark and gritty without sun as he was drawn with a wave that sucked him down and under.
Strengthless against the current, he tumbled like a rag doll in the murky water. He saw himself, as if outside himself, in a dream-vision: his bloodless, pasty, blue-white body flopping backward heels over head, slack limbs floating and twining, doughy-fleshed and dead.
He blundered into a mass of weeds at the bottom, pushed feebly against their slimy leaves, and found his arms tangled in a woven rope net that was pulling him up.
His head broke the surface next to a wooden boat, and he inhaled with a huge gasp.
A voice sounded above him, female and perturbed. "You have torn my net."
"I am sorry." He wiped seaweed from his face. There was no wind. The small fishing boat's single pointed sail hung slack in the still air.
"Hold on to the side of the boat," said the woman. "I will row you ashore."
He grasped the wooden gunwale slick with a thin film of green algae, and let himself be towed. It was marvelous to be breathing and to see the setting sun.
The woman put up her oars, several meters out from shore. "Can you make it from here?" she asked.
"Yes."
He let go the side of the boat and swam in toward the beach.
He gained footing in the shallows and dragged himself out of the water, then dropped to his knees in the deep sand, exhausted.
Suddenly his aides were swarming around him in alarm—in sheer panic—helping him to his feet, staring into his face, and lifting his eyelids with their thumbs.
_What is wrong with them?_
"Where were you?" they cried. "We thought you were gone for good!"
"Just a swim," he slurred. _Jushaschwim_ , he'd actually said, belched, apologized thickly.
And they told him he had gone down and did not come up.
"The woman in the boat," Shad Iliya said, turning back to the water.
There was no woman. No boat. Just red sun on water. He blinked. He turned back to his aides, concerned now. "Did you not see her? Sinikar?"
The young orderly shook his head. Someone else said, "He's drunk."
"I am not drunk!" he bellowed, staggered back, retreating from them. Their faces were full of doubt.
"I am drunk," he admitted in a lower voice. "I am not that drunk."
A dry blanket was placed over his shoulders. A condescending voice said, "Come with us, sir."
A groan rose in his closed throat and opened to a roar. "No!" He threw off the blanket and stumbled in the sinking sand to the water's edge. He raised his puckered hands to his brow and scanned the empty horizon. _Where is she?_
He kept tripping along the waterline in search, only to be cut off and surrounded by coaxing, patronizing voices and reaching hands that closed in on him.
Like wild game cornered by hounds, he roared and snapped to keep his aides away from him. He drew himself up in drunken magnificence and proclaimed himself lucid and them all idiots.
They were embarrassed for him. They wanted to clean him off, get dry clothes on him, and put him to bed. He was weak and drunk and tired, and eventually they managed to take him back to his pleasant-scented, airy room in the spacious villa and coerce him, babbling and objecting, in between the crisp fresh sheets.
Pony brought him some tea with honey, lemon, and whiskey.
A lieutenant slapped Pony's golden hands. "No whiskey!"
Shad Iliya sat up in the bed. "Don't hit my slave!"
An aide pushed him back down.
"Where is the woman?" he said. "A Jew. I bet she was a Jew." He ordered she be brought to him.
"We'll look for her, sir."
They were lying to him. He knew they wouldn't look.
"Pony, find her."
Pony was crying. Why was Pony crying? _Stupid alien_.
In supreme frustration, he closed his eyes and slept for a while, simply to escape the jackasses around him.
He woke in the middle of the night, alone, still drunk. His aides were sleeping in another part of the villa. Pony was out on the beach, looking.
Shad Iliya rose from bed and gazed out the window. The moon was bright. It cast a cold, eerie light on the landscape.
Out on the sand dunes a flock of hooded crows stood, all facing one direction. They were Muslim crows. Facing Mecca.
Shad Iliya turned inside the room. He found a knife and slashed his wrists. He was going to slash more but lost courage.
_That was asinine_.
He searched for a wrap. He spurted blood all over the room ransacking it for a closure, overturning his nightstand, pulling out drawers and throwing things aside. Finally, he summoned a medirobot, which entered via the service chute and sealed his wounds.
Then Shad Iliya set the machine on a self-destruct sequence, furiously jealous of its mindless ability to destroy itself when he could not. And jealous of its mindlessness.
Before the medirobot could explode, Shad Iliya emptied it of its supply of melaninic. He had always wanted to be dark. Then he climbed out the window—his aides had locked the doors—and wandered away, leaving behind what looked like murder.
# 20. Jerusalem Stands
HARRISON HALL'S WHISPER called him back. "Captain."
Alihahd sensed strain in the voice, but couldn't answer, staring far away, caught between past and present, wondering what Harrison White Fox Hall was doing on the Mediterranean beach when he hadn't even met him yet.
"You're crushing my hand."
Normality returned to Alihahd's eyes. He released Hall's hand all of a sudden. He couldn't recall having taken it.
It was fully night in the Iry forest. The air was cool and moist, smelling of damp soil, wood, and lush foliage—and the smoky musk of Hall sitting beside him on the rotten log.
From beyond the trees came the subdued sounds of the rebel camp, waiting in standoff with the mighty Na′id 27th Army.
The past had caught up with him. He had always known it must. Something this big could not stay hidden. The universe was not wide enough. And he hadn't exactly been avoiding the Na′id these past thirteen years. He had walked in their very midst—as if he wanted to be caught. It seemed he wanted what he most dreaded. And what he most dreaded had found him.
This was Jerusalem all over again—a great Na′id force facing a hopelessly doomed and cornered resistance—except that he was on the doomed side this time. Cosmic justice? He had made his last possible play, his only chance, to bluff his way out of the impending battle. He had shown his face.
The Na′id were frightened, for certain. But had it been enough? They hadn't retreated. They should have done so by now if they were going to. They were still out there on the plateau as if deep down they suspected that the rebel force was only a paper tiger with a clay general.
Prospects looked bleak. They had to run. They had to. There was no other hope.
Alihahd reflected on the events that had brought him to this pass. He couldn't have done anything differently. All decisions were forced by Fate, despair, ignorance, and cowardice. Had things been different, he could have done differently. But the past was fixed, and hypothetical extrapolations of alternate conditions were futile.
He spoke hypnotically, gazing back at the dream as it faded. "You doggedly execute your duty even though you know you are wrong. You dull your brain with wine to fog the edges of a reality that is too harsh to bear in full light, and still your thoughts will not shut off. There were no choices. It was this or death. And I lacked the courage for an honorable self-execution."
He had spent the two years following the fall of Jerusalem in alcoholic oblivion in the alleys of Cairo, a city that could swallow anyone without a trace. At length he had dared to wake and ask what date it was, and for the first time took a fearful peek over his shoulder to see what he'd been hiding from for two years. It was still there. He was still the man who had taken Jerusalem. And now he was also a deserter from the Na′id army. His name was revered. And in all the universe only he knew it for a lie.
He still couldn't find nerve to kill himself. So he had to figure out how to live.
Henceforth he was _Alihahd_ , and never completely sober, though seldom as drunk as in a Cairo street.
He'd tried to make amends with his conscience, doing only what he thought right. But that meant assessing what was right and wrong, what was the measure, and who was the final authority. His ultimate authority used to be the Bel. When he threw off the authority of the Bel by deserting, he kicked the base of his whole life out from under him. The universe was suddenly without direction, confusing and terrifying.
In an almost sober moment he'd taken level stock of his situation and asked of his Furies: _Who is in charge here now?!_
And came the horrifying answer: _You are_.
He had no recourse, even to God. He didn't believe in God—not often. And even if he had, he'd made war on God's people. He couldn't call to God.
He was alone in the universe. Most people had someone to answer to—God, parent, country, leader, law.
_I have nothing_. He was without country, without law, without God. _And I am the leader_.
And now, thirteen years later, the base had collapsed again and the universe was on its ear once more. The ultimate authority was wrong again. Alihahd was revealed as a fraud. So was the legendary Shad Iliya. He was running circles on an eternal battlefield. Hero, Rebel, Traitor: thrice a legend on both sides of the same war.
• • •
In the early-morning hours a messenger ship rolled out of one of the 27th's twelve huge transports, rose from the Na′id camp, and left planet, burning a red streak across the sky. The troubled rebel camp bustled and droned, wondering what the move could portend. Alihahd/Shad Iliya would know. Vaslav was elected to approach him and ask, but no one could find Vaslav. So the rebels drafted another youth to go instead.
The girl came upon the infamous general in the dark of the forest, sitting on a log, talking quietly. The girl didn't know what she had expected of Shad Iliya—a man-eating cyclops perhaps. But despite the horrible white face and eerie blue eyes, he hadn't lost the calm demeanor and personal qualities that made him Alihahd, and the girl dared come near and ask what he thought the Na′id might be doing by sending out the small ship.
Alihahd brushed a bug from his sleeve. He had seen the blazing track overhead, and knew it for a robot ship making deadly haste to Mat Tanatti.
"They are sending to the Bel for advice. They do not want to use the Net, not even in code."
The girl chewed on her peeling lower lip. "What should we do?"
"I would say that until the ship returns with instructions from the Bel, the Na′id will do nothing. So why don't you go to bed?"
The girl smiled nervously. "Really? We're that safe?"
"For the moment."
The girl cast him a grateful look and ran from the forest.
When she was gone, Hall leaned forward to rest his elbows on his knees, turned his head to face Alihahd, and said, "What _does_ it mean?"
"It means they are not running," Alihahd said. He tossed a strip of bark away from him. "It means we are dead."
Nocturnal beasts chittered, hooted, and peeped. Insects sang. Trees rustled. In the distance the ocean rushed. They were all lowland sounds, strange to Alihahd's ears.
At length, Alihahd rose, exhaled a voiceless sigh. "I suppose it is my turn now." He quit the forest without further explanation and went to the rebel flagship. Moments later, a second robot messenger ship took off and painted a red trail across the stars, bound to where with what message only Alihahd knew.
When Alihahd reemerged from the flagship, he found Layla sitting cross-legged on the ground at the bottom of the ramp, waiting for him. She had stripped off her shirt. Her chest was beaded with sweat in the sweltering air that didn't cool much with the night as it had on the mountain. She hung her head. Stray hairs stuck to her neck. She moved only to swat at the flying bloodsuckers that lighted on her damp skin.
As Alihahd reached the bottom of the ramp, she looked up at him for a long time, and finally he questioned it. "Layla?"
"You are a traitor?" she said. Her question was not a question—merely incredulity seeking confirmation.
"It seems I am," Alihahd said.
He was surprised to see Layla looking so ill. He beckoned. "Come inside. This ship can readjust your blood. You will feel better."
She reflected on the offer, then reached up her hand to him. He pulled her to her feet.
As she followed him up the ramp, she said, "Do you really think it is a tottering tower of cards?"
"What is?" he said.
"The Na′id Empire."
He had said that once, hadn't he? The woman had a good memory. "Yes," he said. "I am afraid it is."
Layla was reviving when Alihahd was summoned away by one of his rebels shouting from outside, "Captain! Come quick!"
Alihahd descended the ship's ramp at a long-legged trot to find the situation outside radically altered.
Across the plateau, the Na′id had mobilized. Their troops were assembling in the battle order along the perimeter of their defensive energy dome.
Even in the dark, Alihahd could see that their ranks numbered short. At least a quarter of them were missing—presumably wearing activated light-benders and moving God knew where. Alihahd grabbed one of his subordinate commanders by her upper arm. "Tia, get a scanner and find those ghost soldiers. You are looking for maybe three thousand people. Go."
As the young woman ran, Alihahd turned to his second. "Musa, what happened?"
"I don't know. They just decided to move. I've ordered all our people into their ships."
Alihahd nodded, watching the massing enemy troops across the field. With his own people inside their ships, the Na′id couldn't see how frightened and disordered they really were.
"Message for the Na′id?" Musa asked.
"None," Alihahd said.
"Orders?"
"None."
Musa stopped herself from asking, "Do you have a plan at all?" Instead, she withdrew to crouch underneath the flagship and activated her personal radiation screen.
Alihahd walked a few steps away from the ships to stand isolated on the wide plain beneath the black sky. Across the way, like a band of clustered stars, was the enemy's twinkling row of lights and screens, nothing to bar their attack.
Alihahd spoke softly into the empty space around him. "Well, Ra′im?" The humidity seemed to mute the sound.
Any one of the Na′id viewing him through a scope could have read no feeling, no fear on his face or in his loose stance. And perhaps that was what was giving them pause. They didn't advance.
Alihahd heard sauntering footsteps on the grass. They came to a stop behind him. Alihahd didn't turn, keeping his eyes on the army that would not attack.
"Delay and delay," he said in quiet disapproval. "Like Jerusalem."
Harrison Hall's ears pricked. Like Jerusalem. It was not the first parallel.
Was all this intentional? Harrison Hall thought, _We are trapped here under the direction of a guilt-ridden, suicidal Na′id general on the wrong side of the battle of Jerusalem_.
There was a plan after all.
Hall took a backstep. "Been nice knowing you, Captain."
Alihahd remained as he was. "Good-bye, Mr. Hall."
• • •
At his own little camp in the forest, Harrison White Fox Hall gathered up his things—his pipe, a few packages of foodstuff and a water purifier pirated from the rebel ships' stores, a bio scanner, a tinder, and the disparate devices salvaged from his own spaceship, _Nemo_ , by the mandesairi—and he fitted them all into a small pack. He stamped out the campfire he'd made for himself, bound back his long graying hair, tied his russet bandana across his brow, slung his pack over his shoulder, tucked his gun in his belt, and hiked up through the trees.
He hadn't gone far up the densely forested slope when Layla came running after him, her form sparkling with a radiation screen, a taeben slung from one shoulder. She closed her hands on Hall's coat and pulled him to a stop. "Where are you going?"
"Anywhere else," Hall said. He tugged free of her and took long strides up the mountain incline.
Layla ran alongside him, two steps for his every one. "There is going to be a battle!" she cried.
"Is that right?" Hall said.
"You are needed!" Layla blazed.
Hall faced her. "Not so."
Layla demanded explanation with a freckled scowl.
Hall let his pack slide down his arm to the ground at his side. "He can kill himself without me."
Layla started to object, but Hall cut her off. "It's Jerusalem, Layla. This time he's on the side he thought was right the first time around—the side that got butchered. Jerusalem never had a prayer, and neither do we. He doesn't intend to win this one. He doesn't want to. This is a setup for a hopeless battle. History repeats itself—especially when the same person is calling the shots."
Layla turned away. She tilted her face up, blinking quickly with gathering pools in her eyes, trying to deny what she heard.
"Vaslav left a long time ago," Hall said. "It's time we did, too. If we stay here, the possibilities are limited—and survival is not among them. He may run away himself."
Layla wheeled with clenched fists. "No!"
"It wouldn't be out of character." He took Layla's small, callused hand, and uncurled her fingers. "He never deserved the kind of loyalty he commands. Come with me. I need you."
Layla pulled her hand away and threw her head back. "I will stay. And if it is to death, I die."
Hall lifted his pack again and started away backward, still facing Layla for his parting. "He hasn't fought in thirteen years. Unlike some of us, he doesn't even remember how to win."
A ship roared overhead, entering Iry's atmosphere. It was another Na′id troop transport bringing reinforcements.
"Sounds like it's starting," Hall said. "You'll be late." He turned and quickened his pace.
Layla ran down the mountain as the ship came to land. She reached the forest edge, dashed across the exposed stretch of plateau, and dived underneath the rebel flagship, where Alihahd and Musa were. No shots had been fired yet. The rebels stayed hidden inside their ships. The Na′id army still held its position as another Na′id transport ship, the _Dayyanu_ , came to land on the Na′id side of the field.
"Reinforcements?" Layla asked breathlessly, elbowing her way to crouch between Alihahd and Musa.
"Of a sort," Alihahd said.
Layla lifted the scope of her taeben to watch the debarkation of the new company. Layla was night-sighted and could see well without an infrared filter. "They're all brass and police," she said. "And a general." She quickly groped the ground beside her with a blind hand, keeping her eyes on the scope. "Give me a bolt rifle. I think I can pick him off from here."
"No, Layla." Alihahd stayed her eager hand. "Wait."
Layla straightened and blinked. "Then we are to kidnap him?"
"Wait. Just wait."
Musa pointed. "They're breaking up."
It was true. The Na′id troops had begun to drop back from battle positions along the edge of the shield dome.
"It came over the Net while you were gone," Alihahd told Layla. "A stop order from the _Dayyanu_. The 27th didn't believe it until now. They thought it was a trick of ours." He smiled wryly and revised, "Of mine."
"Why would the _Dayyanu_ tell the 27th not to attack us?" Layla asked. Layla thought in black and white. This didn't fit her template.
"There are other ways of winning," Alihahd said. He borrowed Layla's taeben to look at the new general.
Across the plateau, in a small glowtorch-lit ceremony, an honor guard of the dispirited 27th Army relinquished its standard from the flagship _Nashparu_ to the _Dayyanu_. General Ra′im Mishari passed the command baton to the new general, an older man unfamiliar to Alihahd on sight.
_Damn you, Ra′im. Why did you not run?_ What an overblown mess this had become.
Ra′im had always been a good soldier. He was not a good leader.
_Why did you not stay a lieutenant commander?_ thought Alihahd. Then, mournfully: _Why did I not stay a general?_
"I can still hit him," Layla insisted.
"Don't hit him," Alihahd said, lacing his long, knobby fingers together. "Where is Mr. Hall?"
"Deserted," Layla said.
Alihahd nodded. "You go, too."
"It is my battle!" Layla cried.
"No battle," Alihahd said. "It is a surrender."
"No," Layla said.
"It has to end somewhere—the blood," Alihahd said.
Layla chewed on her lip, brown eyes welling with tears. Suddenly, she seized his arm and pulled. "Then you come back to Aerie with me and Harry. Let them fight if you cannot. You are not of them anymore. Or them." She threw her hands toward both the Na′id and the rebels.
"Run?" Alihahd said, his brows elevated into his long, unkempt bangs. "That has to end as well." His voice dropped very low, emotion-suffused. "I am Na′id. I have never stopped being a Na′id."
Layla started to speak.
"Do not fight me, Layla. If I have ever made a harder decision, I do not remember." He passed his hand over his eyelids. His eyes hurt. "My army is not an army. They are a mob. They have no training, no discipline, no experience—and little enough luck, it appears. Half of them don't trust me, and—fact is—neither do I. We never had a prayer here. I do not know what possessed Ra′im to make him think we did. Musa, have you delusions of surviving an armed encounter with the 27th Army?"
Musa expelled her breath and admitted realistically, "No." She crossed her arms, still doubting the preferability of surrender. "What kind of terms do you think you'll get?"
She harbored visions of torture and prison and execution.
"What I _have_ are: no investigation or prosecution for past crimes against the Empire; no reprisals against any of my rebels for this current action; citizenship for those who do not resist; and no questions."
"How in Creation did you get those?" Musa marveled.
"Those are the terms I offered to the Bel—take or leave. He appears to have taken." Alihahd nodded to indicate the Na′id army dissolving its battlefront.
"How can you trust the Bel, though?" Musa asked.
"The Bel is the only person in this galaxy that I actually do trust," Alihahd said. "If he gives his word to a thing, then it is so. You will all be conscripted, of course, as able Na′id citizens are. But you could fall to a worse army than the 27th. It is not an agreeable prospect, nor even tolerable perhaps. But the alternatives are unthinkable. There really are none."
"What about you? What happens to you?" Musa asked.
"Me, they will eat."
He turned curtly to Layla. "Be off quickly now before they become greedy. I did not surrender my allies. Watch for patrols. They will be combing the woods for deserters. And take Vaslav."
"Vaslav is not here," Layla said.
Alihahd frowned. "Where is he?"
"Gone. He must have run away."
The frown deepened. Alihahd hoped the boy knew what decisions he could live with. Alihahd was not worried about Harrison Hall.
A rebel officer called to Alihahd from inside the flagship, "Captain, there's a message for you from the new commander of the 27th Army. He'll talk only to you."
Alihahd started to rise. He spoke to Layla. "Go now. Surrender will be at first light."
• • •
At sunrise, the rebels came out of their ships, weaponless. They deactivated their shield dome.
The 27th Army rushed in.
The sudden charge of soldiers terrified. Their ferocious speed made it seem they would strike down the unarmed rebels, under truce or no. But the Na′id only sought to take control before someone could balk, take out a concealed weapon and start shooting.
There was some brutality, but the old soldiers soon quashed it. Shad Iliya's army still had a tradition of human decency.
A group of young military police grabbed Alihahd roughly and placed him under arrest. No veterans of the 27th Army had volunteered for that task. These MPs were from the _Dayyanu_. The new company with its new leadership was disgusted with the 27th Army for its total lack of initiative in this affair. It became rapidly apparent that this rabble band of rebel malcontents in Na′id dress was not an army at all. The Empire had been had. But there was no turning back on the Bel's word. And that knowledge made the Na′id furious.
One traitorous man spoke, and a whole mighty army had turned into a litter of whimpering cubs. Neither did the one man look like much—spindly, limping, worn, on the downhill slope of middle age. The seventy-year-old Bel looked better than this man. And the Na′id scorn mounted. The great general Shad Iliya's incredible record, seen in this new light, they attributed not to genius but to the machinations of a con artist.
They handled him with more force than they needed—he was not resisting—but not half so much force as they wanted.
Alihahd endured the abuse—which was mostly verbal, with some hard shoves and spits. He welcomed it. It was a thing due. He'd been running from it for years and now felt a perverse relief that it had caught up with him at long last.
And the battle he feared had not happened this time. He'd escaped that trap, the awful circle, with no blood shed but his own. He had revisited Jerusalem and said: _This shall not be._
Someone slapped his face. He tried not to smile or cry. The MPs chained his hands behind him, then looped the chains around a charred tree, where he was to stand for hours while his rebels were searched, corralled, identified, and their criminal records deleted, to the increased grumbling of the loyal Na′id.
Periodically, young soldiers filed by to gawk at the Traitor with mutters and torment. The veterans of the 27th Army avoided him. Even his rebels, feeling sold, called him Traitor.
The sun beat down on the windless plateau. The dead tree offered no shade, and Alihahd languished in the stifling air. The afternoon wore on. He felt his skin burn, his joints ache, his muscles cramp with dehydration in the suffocating heat, and he gloried in the pain. _Some kind of deviate you are_.
His perception split onto two levels, one aware in bright detail of the burning in his limbs, the itch from his clothes sticking to his wet skin, the touch of a fly that kept returning to his twitching cheek. On the other level, his mind was detached, and he felt surprisingly well. He was aware that his sense of well-being was symptomatic of serious trouble. But since there was precious little he could do about it and no one around him who could possibly care, he ignored the hazard and enjoyed the alertness.
He recovered some sense of self—or found it. He wasn't sure he'd ever had it to begin with.
After a long while, an MP strutted a slow circuit around the prisoner's tree, his wide chest thrust forward, his fleshy buttocks thrust back. He stopped before Alihahd. Bulging eyes looked down a wide nose. Thick sensual lips formed a sneer. "What's the matter, nazi? Too good for us with your pretty blue eyes and your yellow hair?"
Alihahd gave no answer. There had been no real question.
Another MP circled in from behind, the same curl on her lips. She plucked at Alihahd's loose, rough-weave shirt. She inspected the unfamiliar fabric between her stubby fingers. "What kind of costume is that?" she said.
"Doesn't match your eyes," the man commented on the side.
Alihahd sighed and answered civilly, "It is the dress of people of Iry."
The woman sniffed disdainfully. "There are no people of Iry."
Alihahd wondered how he could have let that slip his mind. Aliens were not people.
_I have changed. This place has truly done something to me_.
The fleshy man made another remark about his pretty blue eyes.
Alihahd lost patience. He snapped his head around and shot at him, "I think you like my pretty blue eyes."
Thick lips twisted into amazing shapes of outrage. The pulpy face beetled huge in Alihahd's field of vision. The man's breath smelled sickly sweet, and suddenly his knee smashed up into Alihahd's groin with an explosion of pain and nausea. Alihahd folded over, unable to breathe—and he knew the man enjoyed that, too. Alihahd shouldn't have invited it, he thought dimly as he sank down the tree to which he was bound, drawing air in a ragged gasp.
A party of officers marched to the site and dismissed the MPs.
The ranking officer, a lieutenant of the new company, merely stood over him for some moments, watching him retch and gasp at the foot of the blackened tree trunk. Then she pulled him to his feet by his hair and stepped back. She regarded the hand with which she'd grasped his hair curiously and remarked to one of her companions, "Hot."
Alihahd panted, his weight pitched back against the chained post, trying not to pass out, then wondering why he bothered to try. The revilement started up again, slightly more literate than what the MPs served him. The neat and pretty little lieutenant with the blindingly shiny buttons had a big voice and an advanced education, the better with which to recite her creed at him. For her and the young officers, this was a chance to lambaste a general with impunity. One even drew a blade and set it against Alihahd's throat. "You will suffer all the torments of hell for what you've done, Traitor."
Alihahd recovered his breath. "Unless hell is peopled with fools such as you reciting their everlasting twaddle, I would gladly go this very moment, if any of you is willing to send me without an order from your betters." He spat the last word—betters—at them, and his eyes flashed down toward the blade at his neck, daring them.
He could almost hear Harrison Hall chuckling at that speech.
The blade was sheathed.
"Be too merciful," the officer mumbled.
"Then you may stop your threats and posturing and you may shut up. I am not impressed or frightened. I am, however, annoyed."
"You would do well to be fright—"
"Am I then to be killed with the jaw of an ass?" Alihahd roared to the sky. "The glory of the Na′id Empire is its unparalleled ability to vomit back cant."
All six officers at once contested that charge hotly.
Deliverance came at last, painfully, in the form of a veteran officer who walked over to them, attracted by the noise. He observed, quietly appalled. He called off the hecklers, then stood looking at his former general in shame, shook his balding head, and walked away.
Alihahd was sorry. He could have taken the empty insults all day, but that one silent look of shame undid him. His head slumped on his chest. His eyes closed. He listened to the distant tide. It would be going out now. Tides were wonderfully predictable without moons, but not very powerful. He focused on the sound of far-off water. He let the sound rush through his head, until the crunch of a single set of footsteps came very near along with the oscillating beep of a medikit.
Alihahd lifted his head, finding it strange that someone should be interested in his state of health.
Not this man. The medic set the kit down brusquely and dutifully took a sounding of the captive, without one ounce of personal care. He was a coarse, common man wanting to do his job and be done.
"Where've you been?" the medic said gruffly, confounded by the reading. He took off his hat and mussed his already disheveled hair.
"That machine ought to tell you," Alihahd said.
The medic clicked off the scanner dubiously. "A mountain?" Alihahd nodded, his throat too dry for many more words.
"Your heart is bigger."
"Is that a fact?" Alihahd said faintly.
"Strictly literal, sir," the medic said and rummaged through his kit for a diadermic injector, which he brought to Alihahd's chest.
"Poison?" Alihahd asked.
"If it was up to me," said the medic. He gave the injection, and Alihahd immediately felt a few degrees cooler and more comfortable. He was surprised.
"Who ordered this?"
"Supreme Commander General Issurish," the medic said testily, putting his kit back together. "You've been summoned to his presence." The man snapped his lips shut, reconsidered, and rephrased under strain, "Invited. I'm to ask you."
"Well," Alihahd said lightly, feeling buoyant on his borrowed time. "If it will get me out of the sun, lead on." He shrugged in his chains.
The medic grumbled, let slip a curse, then said, "You won't tell on me, will you, sir?"
Evidently the medic had orders to be polite. "No," Alihahd said.
The medic grunted, took up his kit, and tramped away to send back a six-person escort-guard to collect and deliver the supreme commander's guest.
• • •
The cabin inside the Na′id flagship _Dayyanu_ was blessedly cool and air-controlled. It took Alihahd's eyes a moment to adjust to the softer light.
The supreme commander, General Issurish, waved away the six enormous guards in the unhurried way of gentry, leaving him alone with the captive. The general didn't worry that his prize prisoner might bolt or attack him. Alihahd's hands were still bound. That precaution was more than sufficient. Issurish was a big man, if not nearly as tall as Alihahd. The general was a patrician, full of manners and breeding—one of Alihahd's own kind.
"Shad Iliya," said Issurish in greeting, swiveling his chair to face him. "And Alihahd also, I believe."
Alihahd stood mute. His silence convicted him.
"You are two men I've always wanted to meet." The general had a long-standing admiration and sense of professional rivalry with the great Shad Iliya, and an obsession with bringing the infamous Alihahd to justice. But since both had been reported dead, Issurish never dreamed he would actually meet them—and not in the same person.
At first the idea of the two being the same was incomprehensible. How could such an illustrious proponent of the Na′id ideology turn about-face and become such a trenchant foe? But it made more and more sense as he thought about it, and Issurish began to nod.
Zealots didn't come from nothing. Issurish remembered the story of the Christian zealot St. Paul, who had been no apathetic unbeliever before his conversion, but rather a notorious, active, fanatical Christian-killer. To Issurish, there was nothing at all miraculous about a zealot's becoming a zealot. Goals may change. People did not. Great hate could become great love, or love hate. The degree was constant. If Shad Iliya were to transgress, he would transgress hugely—nothing furtive and measly about it.
That Shad Iliya was Alihahd made all the sense in the world.
Issurish didn't expect to understand Shad Iliya's reasons for betrayal, and he wasn't going to berate him. The prisoner was his honored enemy.
Issurish had never met a legend before. Strange, this one looked like a man—a battered, outspent one at that—dirty, ragged, tough-knit, and too lean, nothing at all soft but those liquid eyes. There was something very human in his worn, bedraggled features. And something else, the elusive quality of an immortal leader. It couldn't be pinned down, but it was definite in its existence.
"I wish we could have met on the battlefield," Issurish said. He wanted to do battle with a legend and see who won.
Alihahd did not. There was no more romance in war against his own kind. He'd already had his time to match wits and strengths with a great adversary in a battle to live through the ages. He had his undying fame, and all he could see behind him was a hillside littered with human dead.
It wasn't to be explained. And this man wouldn't understand, even had he been there.
"Your army is a disappointment," Issurish said. "The 27th."
The brilliance of the 27th Army under Shad Iliya had been that each member worked well in his or her singular capacity. Ra′im Mishari was a perfect second-in-command, and everyone else was perfect in place. But after Shad Iliya had gone, they'd all been promoted to positions in which they didn't belong, and perfection had fallen apart.
"The victories must have been yours, not theirs," Issurish said. "Such a loss." He shook his head. "You present a sensitive problem, General. I'm not sure yet what I'm going to do with you." Then, as if suddenly realizing that he was remiss as a host, the patrician sat forward and solicitously swept his arm toward a chair. "Sit. Water? Are you hungry?"
Still standing and making no move toward the offered seat, Alihahd spoke for the first time, in a flat tone of injured pride and rebuke. "I have not been given leave to piss since I surrendered."
Issurish was alarmed at his oversight as only aristocracy could be. There were still manners in the field.
Alihahd's bonds were removed, and he was permitted to go unguarded to the commander's facility. He had, after all, come to them of his own accord.
When he returned to Issurish, he sat in the chair, his weight to one side, elbow on one chair arm, his feet flat on the deck and spaced wide, relaxed, yet alert and commanding, even as a prisoner. His presence seemed to take over whatever sphere he came into.
"Do you want a change from those barbarous clothes?" Issurish asked.
"These are fine," Alihahd said. He knew he must smell gamy by now, but he had become attached to this alien garb, and the supreme commander could just endure the stink. Alihahd wasn't feeling as magnanimous as his host.
"Your funeral was very inspiring," Issurish said, wending his way leisurely to the point of this meeting. "I was profoundly stirred."
Alihahd said quietly, "I saw it."
It had been broadcast over the Net to all parts of Na′id-controlled space, a ceremony of great pomp and solemnity. There had been a long procession with horses and foot soldiers and drums. The Na′id standard had bowed to Shad Iliya's family crest, and flags were lowered to half-mast throughout the Empire.
He remembered now that his daughter, Nikalmati, had been crying.
His beautiful widow, Libbya, had not. She stood, tragic and cold, with her lover standing a few paces in the background. It was a small scandal when she remarried before the official mourning period was done. Alihahd thought it a wise move on her part not to be called Shad Iliya's widow for long.
There was erected a cenotaph, and a shrine in his honor—not as a god, but as a hero. And he had his day of the year when a rite was performed for him. Respect for great men and women did not end at death for the Na′id.
To discover that they had been revering a live traitor for the past thirteen years would be a major scandal.
It was too late to execute him quietly and pretend he'd never surfaced. Too many people already knew.
"It is in the terms of surrender that your life is forfeit," Issurish said. "But I'm not sure if the Bel wants me to execute you here, or if I'm to bring you back for public trial. He signed it rather peremptorily. He didn't elaborate on his orders. He was quite upset."
He watched Alihahd's face for reaction, saw little that he could read. "He left it up to my discretion—which means I'm to read his mind and do what he wants." Issurish was talking confidentially to someone who also knew the Bel intimately. "You know how he is."
Alihahd motioned affirmative with a close of his eyes. He knew.
"I would like to be able to tell you exactly what you are in for," Issurish continued. "I have no desire to make this too bad for you. It will be suitably bad enough as it is. But I haven't yet decided. I am not a mind reader."
"The Bel sent you here to dispose of me for him," Alihahd said. "Of course, he will hold it against you when you return to Mat Tanatti, but it is what he wants. He cannot tell you outright because he needs you to take responsibility for the action."
The general nodded. It sounded in keeping with the Bel's sometimes split character. The Bel's words, thoughts, and actions were not always outwardly in synch. The curse of the Bel's preeminent position was that he was seldom free to speak his mind or heart.
Issurish was afraid that Shad Iliya was right. Issurish would bear the brunt of this debacle. The Bel would be furious if he reported Shad Iliya dead, more furious if he brought him home.
But in time, the Bel would forgive and reward those who made hard and ugly decisions for him and saved him from public grief.
Yes, Shad Iliya knew the Bel very well.
Issurish hadn't actually been asking for help from his prisoner. He was informing him of his status. For the rest, Issurish had been thinking aloud to someone he knew could appreciate his predicament, talking calmly of deadly concerns, as if they were strolling in one's ornamental garden admiring the blooms and bragging about their children. It was the patrician way—artificial and comical to most, but it was a way Alihahd was bred to, knew, and was at home with—civilized to the last.
"Morning too early for you?" Issurish asked.
He meant the firing squad.
"Fine," Alihahd said and rose.
Issurish stood also and paged his orderly. In a sudden surge of bottled-up disappointment, he said, "You never struck me as the venal type, Shad Iliya."
"I was not bought," Alihahd said.
"I was hard put to think so," Issurish said. "The Bel loved you."
A pang of sadness struck through Alihahd's veil of apathy. It had been a long time since he let himself think about whom he caused pain when first he ran away. "I love the Bel."
"Present tense, sir?"
"Present tense."
Issurish paused. "Shall I tell him?"
"You may."
He hadn't meant it as a message. It was a statement of fact.
Issurish's orderly appeared and was given orders to take the prisoner to his quarters.
Alihahd was to spend the night aboard the flagship. The sergeant escorted him to secure quarters inconvenient to any exit. He waved the key, and the doorway's opaque energy barrier disappeared.
The orderly was a reluctant sort, like the medic, unhappy in his present duty of serving an archcriminal of the Empire. But he wasn't above talking to him.
"How long have you been on this steamy rock?" He referred to the planet Iry.
"Best part of a year," Alihahd said, preceding his escort into the cabin.
"Oh. Then you weren't in it." The orderly turned on the light. "Guess you were real happy, though. When we lost Jerusalem again."
Alihahd's limping step hitched midstride. "What?"
The orderly grew uneasy. He licked his lips. "Just how isolated is it out here?"
"Exceedingly," Alihahd growled, turning. "What are you saying?" He stalked back toward the sergeant with demanding steps.
The sergeant drew his weapon. "Don't. Just stay right there," he said, poking the air with his weapon for emphasis. "I meant just what I said. We lost Jerusalem. You didn't know? Now you know. We lost a lot of good Na′id, too. Makes you happy?"
"No," Alihahd said. "I am not happy."
The orderly snarled, "Then what the hell do you want?" He backed out the doorway and reactivated the barrier, cursing.
Alihahd stood motionless in the middle of the cabin, lost in shock.
Jerusalem was free again. What did it mean? What did any of it mean? The historic battle—all the lives—it had been for nothing.
_Why did we fight? Why did we fight? Jerusalem stands._
He gazed at the ceiling, chiding himself for trying to make sense of a senseless battle. All human battles were senseless and gainless, given time. It was one of the reasons—one of the good ones—why he couldn't do it anymore.
He broke from his marbled pose in weariness and sat on the cot.
The quarters were spartan, but luxurious next to the Aerie. It was odd to have all the conveniences again. He could easily become soft in conditions such as these, and he realized why he'd almost died on first coming to the mountain.
He was to spend his last hours in comfort. Here was controlled air temperature, pressure, and humidity, a soft bed, a pillow, water on tap, and even the extravagance of a bath and hot water.
And a vial of poison on the nightstand. Very thoughtful.
Alihahd bathed and dressed. He was loath to get into his soiled clothes again, but did so out of defiance. He'd already turned down the offer of a change and didn't want to reverse his word now. Not that there was anyone to see.
Then he had a visitor.
Alihahd didn't look at the young officer when he came in and wouldn't have recognized him on sight anyway, for he was now a captain and twice as old as he had been at Jerusalem. Neither did Alihahd know the voice since it had dropped and become a man's. Alihahd only knew that the visitor was one of his own from the question: "Why, sir?"
The captain could tell that his former general heard him, though Alihahd didn't acknowledge. Alihahd wasn't quite ignoring him. He was simply elsewhere.
"It wasn't that woman-in-the-boat business, was it?" He hadn't taken the miracle seriously at the time. He worried now that Shad Iliya had.
"No, it was not that," Alihahd said. Even he wasn't certain that the incident had really happened now for all the time between. After a very long pause, he said, "Sinikarrabannashi."
"Yes, sir," said his former orderly.
"You were at Jerusalem," Alihahd said like an answer.
"Was that it?" Sinikar said.
There was silence. A sigh. "Dead Arabs," Alihahd said.
Sinikar's eyes flickered left and right, searching for a connection—something to make the words make sense. His general was as bewildering as he had been on that last day. "Sir?"
Alihahd turned, his face graven, his stare frightening. "Dead Arabs," he said. Sinikar backed to the door and fled.
The land outside would still be in sunlight. Time yet was left before nightfall. And before sunrise. Alihahd lay lightly drowsing on the bed, dreaming.
He dreamed that he wakened. It was fifteen years ago. He was in bed aboard his own flagship. His slave, Pony, had drawn his bath for him and stood next to the bed, white tail swishing, brown doe eyes gazing attentively down to him.
Soft voice said, "It is 0600, sir." And Pony recited the day's itinerary and asked if he wanted coffee or tea.
He ordered coffee, rose from the bed, thanked Pony, and stepped into the bath. He didn't feel water.
At that point, he realized that he was actually still in bed dreaming.
He also realized that someone really was in his compartment.
He opened his eyes. It was Pony.
The little slave sat on a stool, agitated, blushing, near tears, his hands clasped between his knees, bright eyes fastened on his old master.
It was to be expected, but it had never occurred to Alihahd that Pony would still be attached to the 27th Army.
Alihahd sat up. His muscles had stiffened and shortened during his brief sleep, and he ached all over. He knew he was awake this time, and Pony was truly here.
Pony looked older in the texture of his skin. Fine lines fanned from the corners of his eyes. But his form, face, and musculature were still boyish. His eyes were still innocent—but that was because he was a stupid animal, thought Alihahd. A slavishly devoted slave, Pony had always adored him. But Alihahd was not prepared to find Pony devoted still. Sweet alto said, "I was sent to see if there was anything you require."
Alihahd shook his head. Muscles at the base of his neck knotted and pulled. "No, Pony. Nothing."
Pony was disappointed—devastated. He stood to go. As Pony turned away, Alihahd caught sight of a heavy white scar on the left side of Pony's slender golden neck. His throat had been cut.
Alihahd jumped up, seized Pony's narrow wrist before he could reach the door, and pulled Pony around to face him. "What happened?"
Pony shrank, frightened by his demanding tone and rough hands.
Alihahd pushed Pony's head aside to the right, brushed away the long snowy hairs of his crested mane, and inspected the deep knife-edge scar.
The first thing that came to mind was that sometimes slaves were killed as grave gifts to their masters. But Alihahd had seen his own funeral, and no slaves' throats had been cut at his cenotaph. And if Pony had been intended as a gift, Pony would be dead. Nobody botched a public sacrifice.
Alihahd took Pony's face in one hand, thumb and fingers on his cheekbones, and brought the enormous eyes around to face him straight. "Who did that to you?" he said.
Small hesitant voice sounded muffled into his big palm, "When I thought you died . . . ." He faltered.
He'd done it himself, Alihahd realized in amazement.
The little fool didn't realize he'd been gashing the wrong place. The human jugular vein was on the left. Pony's was up the back. He'd tried to kill himself.
Alihahd beheld his inhuman slave, mystified. _You did that for me, Pony?_
Alihahd loosed his hold from Pony's face. Reddened prints rose on gold cheeks where his fingers had pressed.
Naturally, Pony had not been allowed to take his own life simply because he wished it. He was too valuable a property for that privilege.
Alihahd then noticed for the first time that the pretty eyes were sick to death of life. It had gone overlooked initially, briefly outshone by Pony's excitement at meeting his beloved master again.
Velvet-soft lower lip quivered. "Sir, will they make you die again?" Pony asked, and his eyes flooded tears.
Alihahd's impulse to comfort him was checked by an old awkwardness. Alihahd couldn't bring himself to take the weeping creature in his arms. He would feel odd and clumsy.
So he gave a command. "Help me to bed, Pony."
Pony's tears stopped at the reassuring sound of order and normality. Pony wanted nothing more than to be allowed to do his job and forget that tomorrow must come. He laid out nightclothes for his master and helped him undress.
And Alihahd wondered, _Was I always fussed over like this?_ He guessed he had been, because Pony was unchanged.
Pony even remembered to bring him a shot of whiskey. Alihahd hesitated—as if Mr. Hall would appear out of vacant air and hit him. Then he swallowed it, felt it warm him, and he gave the glass back to Pony. Pony dimmed the lights and withdrew.
Alone in the almost-dark, Alihahd lay back on the bed. He turned his head on the pillow and looked over to the poison on the nightstand.
• • •
General Issurish came to the prisoner's quarters in the morning.
Alihahd lay stretched out on the bed, the covers thrown off, his arm draped loosely over the pillow.
He was alive.
Blue eyes opened.
"I'm disappointed," Issurish said.
"So am I, actually," Alihahd said.
He could not do it. Never could. If he could, it would never have come to this. There would've been a body for Shad Iliya's funeral.
The supreme commander's jaw tightened. "Well, then. Come on." He left and sent Shad Iliya's slave in.
Pony laid out Shad Iliya's dress uniform and all his decorations. Alihahd knew what it meant.
"O gods, are we to go through all that?"
This was ugly business. He should have taken the poison.
Pony placed all the ribbons and medals and braids in order—Alihahd didn't remember where all the damn things went—and Pony combed his wayward blond hair.
"I should trim it," Pony said, trying to make the wisps lie down.
"No, you shouldn't," Alihahd said. "Just put on the hat and have done with this."
Shad Iliya always wore his hat dead square regulation, no matter which way the winds of unofficial military fashion blew: slouched, pushed back, or cocked to either side. Pony centered the hat just so, and stepped back. "You look grand, sir."
Alihahd checked in the mirror. He cut a trim figure, appearing not so gaunt as imposingly tall. The uniform could make anyone look good, and it felt right, after all this time, a dark cobalt blue with epaulets on his wide, angular shoulders, and clean-fitted lines down to knee-high black boots.
_Pony, can you not guess what I am in for because of this?_
Pony did not guess, and Alihahd didn't tell him, not to see his bright admiration fade.
The guards came to fetch him.
"Time, sir."
Shad Iliya was marched out, his hands unbound, in front of the entire assembly of Na′id personnel and the captive rebels as well. It was a long walk, but he didn't limp. The uniform, the moment, his name constrained dignity. He saw his destination—a dead tree that had been trimmed down to a tall, neat post. Alihahd wondered if the proceedings were being recorded.
At the post, he turned smartly to face the multitude. The guards dropped back, and Issurish stepped up, being the only person of sufficient rank to perform this duty. His face wasn't that of the genial host on the flagship, but of a severe and efficient officer of the Empire, angry at having to do this. He stripped Shad Iliya of his rank, medals, and all the Na′id insignia with which he had just been decorated. It seemed ludicrous. But it hurt. Alihahd was surprised how much it hurt.
Issurish finished by ripping the twin symbols of Galactic Dominion/Human Supremacy from Shad Iliya's hat and tossing the hat to the ground as trash. The insignia he placed grudgingly in his own breast pocket, a deep frown fissuring his jowls. He spun and marched away.
Left bareheaded, Alihahd felt the pleasant morning air on his scalp. The sun was rising to a clear sky. The day would be brilliant.
No longer called _sir_ , but _you_ , Alihahd was backed up to the post and chained there. A firing squad of twenty-one filed into position. The first markswoman asked him ritually if he forgave his executioners.
"Yes," he said.
The butcher waited nearby with a skinning knife with which to carve up the remains into pieces that the ships' food preparers were programmed to handle. The man looked like the kind who enjoyed his work. Alihahd hoped he was not one of those who kept souvenirs—not that Alihahd would know who did what with which part of his carcass, but it bothered him to think that the fleshy MP might come into possession of one of his blue eyes, or whatever else he might want to step on.
The row of marksmen dropped to one knee so as not to obstruct the view of the legion.
Alihahd could look into faces, and he saw more awe than hatred. None of them had a blood score with him. Alihahd had never taken a Na′id life.
Their strained stillness transcended mere military attention. It was an awareness that they witnessed more than a death. The legend had only begun.
A breeze stirred on the plain with the warming air.
_Ah. Eaninala_.
He smelled charcoal and yellow grasses.
He was offered a blindfold; refused it. He felt the sun on his face. He looked up at the bright sky. _I want to live very much_. He thought of all that was beautiful as the marksmen took aim.
The sun and Amerika.
# 21. Nemo
THE ORDER TO FIRE hung unspoken, gnawing at the silence like the final resolving note of a melody, which was natural and expected, that didn't come. Alihahd waited for it, became impatient, then alarmed. He lowered his gaze from the sky. The sea of faces before him had all blanched—even those of the firing squad—and all eyes stared past Alihahd, gaping at some horrible vision beyond him, threatening to break ranks. Someone cried, _"Marauder!"_
Alihahd cranked his head around his post to see what they saw.
The ghostly image of a sailing ship. The derelict brigantine coalesced from shimmering air. It raised the hairs on the back of his neck, and he shivered in the heat. Something elemental in the rag sails and rotting timbers never lost power to affect him.
And suddenly all the spaceships imploded—all of them—with a resounding crack and thunder roll as the great engines consumed themselves. Astounded, Alihahd's thoughts flashed like lightning, his heart leaped, and he roared, "Hall!"
Na′id troops cried out and scattered everywhere, trying to stop what had already happened.
And the rebel captives all bolted.
The melee that ensued covered the plateau, spread into the forest as rebels fled or mobbed their guards and seized their weapons. Shots were fired. The unarmed fought with their hands, with rocks, with anything within reach.
Alihahd was in the midst of it, tied to the stake, ignored. He wasn't going anywhere. He couldn't even duck.
The fighting moved away into the jungle. It wasn't a battle. It was a brawl. The Na′id were torn between chasing their stampeding prisoners and fighting the fires around their ruined ships. The ships won the most attention, though there was nothing left to save.
Properly rigged, a Na′id ship could destruct with extreme efficiency.
This was a Na′id nightmare—to be stranded on an uncharted alien world. Alihahd felt a pang of compassion for the panicked troops. He knew their terror.
He heard cries in the jungle, "Get the Marauder! The Marauder has a ship!"
_The Marauder does not. It is on the bottom of the ocean_ , thought Alihahd.
All that was left of the Marauder's ship _Nemo_ was "just a gadget"—and a few implosion detonators, it would seem. The fearful specter still loomed over the plain.
Alone and neglected, chained to the stake, Alihahd watched the grass fires switch with the fickle winds, and he wondered vaguely if the flames would come to him.
The fighting moved farther and farther away, until the soldiers were only voices to Alihahd. He still heard their distant shouts among the trees.
Then someone came out of the forest and ran toward him with a drawn dagger. It was Layla. She tried to saw at his metal chains.
"Layla, I surrendered," Alihahd said.
"I did not," Layla said. She looked for a catch, a spring, something to release the chains. "Damn!"
Soldiers were returning. Layla yanked at the chains, gave a grunt of frustration, and scurried back to the jungle at the last possible moment, elusive as an Itiri.
Lost-looking soldiers began to reassemble on the plateau, without orders, without a camp, directionless. At least the ghost image of the _Flying Dutchman_ had ceased to menace over the plain by then. Someone else came to Alihahd's stake. Pony. The slender little slave sat at his master's feet, resting his white-maned head against the post to wait until someone came to get him.
Alihahd was thirsty. There was no fresh water. He saw frightened faces among the troops licking their lips and trembling with dehydration on the dusty plateau. Pony was wilted at the bottom of the post.
General Issurish returned to the plateau, and some small confidence and hope revived in the lost men and women of the Na′id army. Issurish wiped his brow; he was hot, thirsty, and tired, but not frightened. He moved like a leader who knew what he was about. Like Musa, he was difficult to rattle, even in catastrophe. He organized a few patrols and sent them out in search of water. He sent others to deal with the grass fires, which had almost burned themselves out by then, but it gave the soldiers something to do.
From the jungle, a band of ten soldiers brought a thrashing prisoner, hogtied and gagged with his own bandana. They dropped him at General Issurish's feet and gave the general a plastic black box they had found on the prisoner's person.
It was a projector. Issurish turned it on.
The dim outlines of a brigantine ship began to shimmer over the trees. Issurish turned it off. "Well," said Issurish and looked down at Harrison White Fox Hall, who was trussed and double-trussed.
"Is all this necessary?" Issurish asked his soldiers, his voice dripping with tired patience.
"Yes, sir," said the black-and-blue sergeant with teeth marks on his arms.
Issurish looked down again. Orange eyes slid up slyly, not the least bit frightened or remorseful. Issurish drew his gun and pointed it at the handsome, wedge-shaped head.
The captive didn't blink. Issurish guessed that his pulse hadn't even quickened. Issurish holstered his gun. He could see the man was dangerous—confirming all that the black box implied. The kind with ice in the veins, and no heart—Issurish wouldn't even bother trying to intimidate him. Vain gestures were a waste of his time.
"Where is your ship?" Issurish demanded.
Hall snorted a laugh behind his gag.
Issurish nodded. He'd expected that response.
Torture was yet a possibility, though Issurish had his doubts even to the efficacy of that. He ordered an organized search begun for the Marauder's ship.
Hall was sniggering at the general's feet.
"Don't look so smug, Marauder," Issurish said. "I am aware that you have the only working spaceship in the world. And while I would very much like to lay hands on it, it is not imperative. Someone will come looking for us eventually, or we will find your ship for ourselves, and your life and your smug secret will be worth sadly little then." He watched the orange eyes for a reaction.
How could anyone smirk so while gagged?
Issurish turned away and gave orders to the soldiers who had brought him. "Put him with Shad Iliya. Guard them both closely. And take off that imbecilic gag."
"He bites, sir."
"Take him away."
The guards chained Harrison Hall to the same black, charred post as Alihahd. The two exchanged looks. They'd last seen each other to say farewell.
"Captain."
"Mr. Hall."
"Be quiet," a guard commanded. She chased the little slave Pony away. Pony skittered off a few yards, then hovered at a respectful distance, tail switching, watching for an opportunity to come back.
The Na′id army had given up trying to recapture their other rebel prisoners. They hadn't the means to keep two thousand people captive. And they turned their energies to survival without prepared food, without pure water, without modern equipment. They couldn't even make Harrison Hall divulge his secrets without the devices and drugs from their ships.
Hall was pleased with himself. He was satisfied with the smoking piles of melted metal that were the Na′id ships. "I like my plan better than yours," he said to Alihahd.
"You have made a liar of me again," Alihahd said. "Caused a melee, stranded all of us on this planet, possibly incited an interstellar purge, and signaled the destruction of this world."
"And I robbed you of your neat and heroic little suicide," Hall said, cutting to the real source of his vexation. "I told you I wanted you alive and stinging."
"You are a devil."
"No. Angel. Who ever heard of an avenging devil?"
"Damn you! Damn you! Damn you!"
"Nothing more eloquent than that for me?"
"No."
"Oh, but I first met you in a spate of elegant fury, Captain."
He still remembered Alihahd exquisitely angry: _For myself, by you or by them, I will be equally dead. I don't very much care. So, please, either shoot, talk, or go away. Or you may go to hell, where I am bound with or without you. Do you want me to repeat any of that?_
"Do you know your buddy talks just like you do?" He meant Issurish.
He received only a grunt from Alihahd.
Hall grinned. "You're glad to see me." He curled two fingers around Alihahd's wrist behind his back.
"Will you shut up!" a guard snarled.
Time passed slowly. Alihahd and Hall both slid down the post to sit on the ground, still chained. As evening came on, the Na′id were settling. They set up guards against sniping rebels and wild animals from the jungle. They'd built campfires and brought hundreds of animal carcasses to be skinned, gutted, and cooked. They boiled stream water and tested the palatability of jungle fruits on their remaining slaves such as Pony. Once they organized, time was on their side. Eventually, eventually, someone would come looking. The Empire simply did not lose its most famous army, a reinforcement company, and two generals, and not come investigating in time.
"We need allies," Hall mumbled aside to Alihahd.
Alihahd knew of whom he spoke. "It is not their battle," he said.
They both gazed up at the sky, attracted by the sight of an eagle soaring over the mountain.
Their minds ran parallel. Alihahd addressed Hall's unspoken thought: "It would go against conscience."
"Not _mine_ ," Hall said.
The two guards, who were bragging to each other about their marksmanship, turned back to their captives. "I thought I told you to shut up," said one. She was a tight-lipped, razor-lean sergeant with a pock-scarred face, hatchet features, and a slight curve to her spine. She walked with her shoulders hunched forward, her pelvis tucked under, her sharp hipbones prominent against the fabric of trousers that were too loose on her. She sneered a lot.
The other guard was a beefy young man with peppercorn hair and a walk like a turkey cock. He snarled at the captives and returned his attention to his comrade's high claims regarding her sharpshooting.
Hall said loudly, "I was just saying: Odds say you can't hit that eagle."
Alihahd was horrified. "No!"
The sergeant turned her head back to them and gave a thin-lipped sneer. "Sure I can." She lifted her gun and took aim at the distant soaring shape.
"They are sapient beings!" Alihahd cried.
The Na′id sergeant looked up from her gunsight and regarded him queerly. It had been a nonsensical thing to say. "Alien's an alien," she said and returned her sights to the target. She took careful aim and fired.
The majestic eagle folded in midair and came fluttering down into the mountain jungle.
Alihahd was stunned. He rasped at Hall, "You don't have a conscience."
"Absolutely none." Hall smiled. "Captain, you surprise me. I didn't think you considered aliens to be on a level with human beings."
"I changed my mind!"
The guard swaggered to Hall and looked down at him with a smirk. "Any more dumb bets?"
"None, ma'am," Hall said. "That was a splendid shot."
And from the sky came a clear, thin cry.
_Ki ki ki ki._
• • •
Directly overhead the kestrel hovered, suspended on a light breeze, while in the distance droned the engines of rapidly approaching aircraft.
"What's that?" the Na′id cried, and Na′id guns pointed toward the sky.
Issurish prowled the wide plateau, bellowing at his thousands of soldiers, "For gods' sake, if it's a working vehicle, don't shoot at it!"
And the Itiri came.
The ships landed in the jungle, and in a short time the warrior-priests appeared from the trees, twelve tall alabaster godlings with sunlight hair and eyes of brilliant green. From their midst advanced a midnight figure cradling a great limp feathered body in her wiry arms. Its blood seeped onto her tunic and trousers.
She stepped to the center of the plain and spoke with a voice like brass. "This is my brother. Who hath done this? This is my battle."
She turned slowly, a complete circuit, then stooped and gently laid the dead eagle on the yellow grass.
The kestrel was hovering above the charred post where Alihahd and Hall were bound. Roniva crossed to them. The two guards moved away at her advance, and the kestrel moved with the sergeant.
Roniva stopped, looked at Hall and Alihahd in chains, looked at the kestrel, looked at the thin, snarling sergeant, whose shoulders hunched defensively as she hugged her gun, her tiny eyes darting.
"Thou?" Roniva demanded of her without an intermediary.
The sergeant's thin upper lip moved in spasms. She wasn't answerable to this _alien_. "What the hell are you?" she sneered.
General Issurish intervened. To Roniva he said, "Madame, if you have some problem with my soldiers, you will address it to me."
Onyx eyes shifted to him, to his general's insignia. "Thy soldier or thy head," Roniva said.
Issurish's attitude immediately changed. "Is that a threat, you alien bitch?"
"I decide," Roniva answered herself. Faster than the eye could follow, Roniva had drawn her sword and slashed the sneering sergeant's throat open.
All at once, those Na′id soldiers with a clear shot aimed their weapons at the alien warriors, but in that same instant each Itiri had seized the nearest Na′id soldier as a shield, and no one dared shoot.
Roniva herself had caught Issurish by the back of his hair and by one thick wrist, which she twisted behind his back.
Twice her bulk, he couldn't move.
"Thou hast come to my home like a great ape," Roniva said. "I will have thee and thine kind gone from here." She had looped his long hair once around the hilt of her sword, and she yanked on it for emphasis. Issurish ground his teeth in silence.
Roniva took his handgun from him, then unwound her sword hilt from his hair, still keeping hold of his wrist behind his back. She glanced aside at the dead sergeant crumpled in a pool of blood like the eagle. Then she looked to Hall and Alihahd chained to the post. She looked away as if uninterested, then suddenly screamed, swung her tungsten-plastic blade, and slashed down at the post, severing the prisoners' chains in a single stroke. She hadn't let go of Issurish.
Issurish nodded, nothing daunted. "We will go. We will destroy your world."
Alihahd, chains hanging from his wrists, came to Roniva's side. "They can do it, Fendi."
Roniva lifted her chin, unafraid.
A deep, groaning rumble and an unfirmness to the ground way down deep like the beginnings of an earthquake unsettled the field.
A great dark mass rising from the horizon slowly eclipsed the setting sun—a leviathan rising out of the River Ocean, spilling water from its crevices—and it kept rising like a continent taking flight.
"What is that?" Issurish demanded. It was still rising.
"An intergalactic ship—what does it look like?" Roniva said. She was starting to sound like Harrison Hall. She told Alihahd, "Say to this creature he may destroy _that_."
Her snowy owl blinked in above her head and alighted on her hard shoulder. It batted Issurish's head as it folded its broad white wings.
Roniva spoke sharply to Alihahd. "Now. Say to this one." She twisted Issurish's wrist. "Say to him: 'Thy Bel hath been notified. Thy people are to be picked up and leave this planet. Not to look back. A ship of the rebel kind will come to collect the fugitives littering my forest. Be gone and continue your fight far away from here.' Am I understood?"
"Have you no concept of justice?" Issurish asked.
"Oh, yes, justice," Roniva said. "Tell this one that we do not worship that god here."
"I demand—"
"Art thou so stupid? Take what I give thee, or I shall give thee ash!" Roniva cried. "Another Dark Age for thy kind! A long one! Two thousand years was not enough!"
She threw Issurish away from her. To Alihahd she said, "Stay close to me, in case they shoot."
Alihahd and Hall both drew closer. "You have a shield?" Alihahd asked.
"This creature," she stroked her owl's wing, "can absorb a great deal of energy. Come. I wish to be away from these beings."
Alihahd and Hall followed Roniva into the jungle. None of the Na′id tried to stop them, or fired on any of the warrior-priests.
The intergalactic ship lifted its full bulk from the River Ocean. It now dominated the darkening sky like a close moon.
It occurred to Alihahd that Roniva could have summoned a rescue for him anytime she wanted—had she wanted to lead outsiders to Aerie. As if reading his thoughts, she said in perfect, modern Universal, "I do not like your people."
Alihahd glared at Hall. "At this point neither do I."
As he walked, he nearly tripped over the cut chains which dangled from his wrists. Roniva noticed his difficulty, and she cut the chains off for him and Harrison Hall, leaving them with the metal cuffs.
"And to which side belongest thou?" she asked Alihahd.
She already knew Hall's.
"I believe both sides want to kill me, Fendi," Alihahd said.
"I will give thee a ship of thine own," she said.
Alihahd was too surprised even to thank her. Recent events and consequences were still rolling into his brain. His stare kept returning to the gargantuan, impossible ship in the sky. Its details were difficult to distinguish. If he stared too closely at a point, he could no longer see it, like trying to focus on a single star, and he wondered of what it was made—light?
The thing's existence struck at the very foundation of the Na′id Empire. It represented a superior technology—superior to anything that had ever been known to humankind. The Na′id Empire was founded on the belief that God had created humankind in God's own image to rule over all of lesser Creation. Human Supremacy and Galactic Dominion had become absurd.
Layla appeared from somewhere and pulled on Alihahd's sleeve. "It _is_ tottering, is it not? The Empire?"
"It's falling," Alihahd said.
He supposed the clues had been there for any of them to see all along—the Itiri's tungsten-plastic swords, the disappearing, magical-seeming familiars, and constellations that had been given names so long ago they would have to antedate Earth's first infant civilization—why hadn't all that struck him as curious? Thousands of years ago, when men and women were nothing but great apes, someone had been here naming the constellations, naming them after swords, ships—and a Gateway.
He shook his head, feeling blind and stupid. He wondered aloud how the powerful Itiri could have allowed the deluded Na′id to strut and bawl like young bullies for so long, overrunning the galaxy and calling themselves supreme.
Hall's eyes assumed their wicked hunter's gleam. "Yes, how can such an aggressive people bear to watch and do nothing?"
"Aggressive?" Alihahd said.
"The _Itiri_?" Layla said. "The Itiri speak of nothing but peace."
"Ever listen to their music? It's not peaceful," Hall said. "You saw the New Year's celebration." He took a zircon and a turquoise from his pocket and juggled them. "Constant repetition of humbleness and serenity isn't the mark of a peaceful people. What is the need for the constant repetition if they are, in fact, so tranquil?"
Alihahd looked to Roniva. She wasn't angry. She smiled like a master criminal caught by a master detective. "Very astute, Harrison White Fox Hall." She turned to Alihahd. "You wonder at our constant readiness for war and wonder who is the enemy for whom we prepare. Only ourselves."
"But you seem to have the aggressive tendencies conquered," Alihahd said.
Roniva put her spidery hand over his knobby one. "And so do you seem."
Alihahd smiled in rue and irony. They were as savage as he. So they could forgive him, forgive Ben, forgive the Na′id. The guilty did not presume to mete out justice.
"An ancient shame," Alihahd said, remembering her words.
Roniva raised her chin, affirmatively. "That ship," she pointed at the monster from the sea, "is not the one in which our forebears came. Our ancestors were stranded here with nothing. That ship you see was built with redeveloped technology after a long, long time in exile here. And when it was finally built—still a very long time ago—we sank it. We decided we did not want it. We did not want to be what we were, what brought us to this exile in the beginning. We wished to abandon that past and become someone else. You understand, Captain Alihahd? You see, we did not come here by choice, we were not pilgrims. We did not even flee here. This is a penal colony. We do not even know where home is—some place where red birds fly like geese. It could be that we came from another time as well as another galaxy. Once we passed through the Gateway, our ancestors could not know what date it was back home. They could have been millions of years in the passage. That was what our judges intended. They sent us so far away even our familiars could not find the way back."
"But do you know what your ancestors did to deserve this?" Alihahd said.
"I know," said the Fendi. Most Itiri did not know. "We conquered our galaxy. We, the aghara, are natural leaders. Our ranga are a natural army of unquestioning followers. We ruled the universe, we thought. Eventually, we were overthrown and driven out. Now we teach ourselves to own nothing, want nothing, to keep our superiority to ourselves. Humbleness, we still have not. Judge, we must not. It is not for us to check, conquer, or punish the Na′id. We are the Na′id. See you now why we must not take up this sword?" She looked to Alihahd and Hall.
"I see," Hall said.
"Then will you please to shut up?"
Alihahd laughed sadly. _I know these people_. He was suddenly very fond of Roniva.
• • •
The ship Roniva gave him was the _Topaz_. He was free to leave whenever he willed. He chose to wait until the rescue ships came to collect both the rebels and the Na′id, so that neither side would see his ascent and wouldn't know to go hunting for him in space. The wait wouldn't be much longer. He could abide awhile yet.
Night fell. Creatures of the darkness were calling within the jungle. Alihahd climbed up to the top of his ship and sat there underneath a wide break in the fringed canopy of tree boughs open to the heavens. He looked at the sky.
Low on the eastern horizon, the Red Geese were rising.
• • •
The rebels' rescue ships came first. Layla and Harrison White Fox Hall left the planet with them. The Na′id ships came later. Mustering the means to airlift twelve thousand soldiers on such short notice had been difficult. The humiliation of the mighty human army rocked the Empire. Alihahd wondered how the Bel was dealing with the incredible developments.
Then, before the Na′id ships ascended, a message for Shad Iliya from the Bel was delivered to Alihahd in the jungle. Pony brought it to him. The Na′id themselves were afraid to go into the night forest.
Alihahd accepted the capsule from Pony's golden hands, and he stared at it a long time before opening it. Something thickened in his throat. He couldn't face the contents.
In that moment, he remembered his youth as a pale oddity in a brown family, a suspected bastard. He remembered that as long as he thought he was a bastard he might as well dream that this man was his real father. Not that the Bel had any more Caucasian traits than his legal father did, but the Bel treated him more like a son than his legal father ever did. Alihahd opened the message capsule and couldn't speak. Whatever he might have said was lost when he read the Bel's single question: _Was it the people?_
Alihahd looked up—lest the tears that were welling in his eyes fall. Second perhaps to Roniva, the Bel was the most powerful individual in the galaxy and close to the wisest. Why was Alihahd surprised that he knew? Was it the people? All those dead at Jerusalem, humans dead at human hands. _Yes, it was the people. Why did you send me there?_
Pony was waiting for a return message, his silvery tail swishing away the winged bugs that plagued him, his great brown eyes timorously scanning the treetops for sail snakes or for those enormous eagles, which terrified him.
Alihahd snapped the capsule shut and buried it in his pocket. Swiftly, he sought out Roniva, while Pony faithfully dogged his footsteps.
He found her in deep jungle shade, seated on a moss-covered boulder. Her long hair, hanging from its topknot, shone glossy blue-black in the winking lights from her nearby airship.
"A going-away present, Fendi," Alihahd announced. He took the startled Pony by his shoulders and presented the little alien to Roniva. "My slave."
Onyx eyes blinked slowly within the frame of the crimson scars on Roniva's high, sharp cheekbones. "We keep no slaves," she said.
"You tell him," Alihahd said, and ran to the _Topaz_.
Pony turned his huge eyes diffidently to his new mistress.
Roniva curled her forefinger on her lips. She didn't know what she was going to do with him. She would think of something.
After the Na′id ships were gone, the _Topaz_ took flight. Alihahd was soon in infinite space, free. He had the means to travel anywhere. The autopilot was asking him where he wanted to go.
He'd been flying aimlessly for quite some time before he told it:
Mat Tanatti.
• • •
The blue-green world was so familiar that the sight of it hurt. Once in orbit, the starship _Topaz_ was challenged for ID and intent by ground control.
Alihahd transmitted an Absolute Priority signal. His message would go straight to the Bel. But when the channel opened for him, he couldn't make himself turn on his audiovisual transmitter. He couldn't trust his voice anymore, and he couldn't face the Bel's image. He could imagine the benign, elderly face with fatherly eyes full of pain and betrayal. He could hear the Bel speaking: _You hurt me, Shad. You have embarrassed me, stabbed me in the back, damaged my Empire beyond repair. What would you have me do now? What would you have me do?_
The image in Alihahd's mind was talking quietly, reasonably, as the Bel did when extremely angry, as Alihahd did. Alihahd had gotten the mannerism from him. He couldn't talk to the man face-to-face. So he encoded his message, requesting permission to land.
_Who asks?_ came the coded reply.
Alihahd had many aliases under which to hide. He answered: _Shad Iliya_.
An eternity passed waiting for the response. It could be anything, a missile, a refusal, a fleet of police ships, or a cold, cold "Who?" He waited. He died a thousand times. The answer came, the only thing to be said. The rest they could discuss tomorrow.
_Come home_.
#
## Appendix
# The Ring
_Signs (Months) of the Year_ | Equinox
---|---
1. The Sword
|
2. The Ship
|
3. The Crown
|
4. The Red Geese
|
|
Solstice
5. The Twins
|
6. The Hexagon
|
7. The Gateway
|
8. The Flower
|
|
Equinox
9. The Beacon
|
10. The Serpent
|
11. The Cross
|
12. The Triquetra
|
|
Solstice
13. The River
|
14. The Wellspring | Shandee
15. The Veil
|
16. The Mountain
|
A sign lasts 10.74 days.
Shandee blows from the second of the River to the seventh of the Veil. The new year begins with the vernal equinox on the Aerie (Northern hemisphere).
Years are named in order for a sign of the Ring. Hexadecades are named in order for a gem in the Gem Cycle, which runs as follows: Topaz, Carnelian, Opal, Jade, Beryl, Onyx, Diamond, Tourmaline, Turquoise, Amethyst, Lapis, Garnet, Zircon, Adularia, Corundum, and Jet.
The present action of this story takes place from the sign of the Red Geese in the Year of the Opal Ship until the sign of the Sword in the Year of the Opal Crown.
The Numbers
Number | Itiri Name | Universal Translation of Name
---|---|---
1 | Enna | Sword
2 | Shauul | Ship
3 | Niaha | Crown
4 | Shanwel | Red Geese
5 | Bibi | Twins
6 | Sorii | Hexagon
7 | Dalanai | Gateway
8 | Sianasad | Flower
9 | Saufer | Beacon
A(10) | Elebanar | Serpent
B(11) | Yxa | Cross
C(12) | Jentas | Triquetra
D(13) | Mandas | River
E(14) | Maeus | Wellspring
F(15) | Ovron | Veil
10(16) | Lodee | Mountain
After sixteen the numbers are named in combinations of the above; for example, seventeen is _endee_ —"sixteen and one." And so it goes: _shaudee, niadee, shandee,_ etc.
The Hours
First watch | Hour of the Bells | Sunset
---|---|---
|
Hour of the wind from the crevasse
|
Second watch | Hour of the winged mice
|
|
Hour of the stars
|
Third watch | Hour of the meteors
|
|
Hour of the berinxes
|
Fourth watch | Hour of the sentinels
|
|
Hour of the dewcatchers
|
Fifth watch | Hour of the Sun | Sunrise
|
Hour of the eagles (Talassairi)
|
Sixth watch | Hour of the waning shadows
|
|
Hour of the killing light
|
Seventh watch | Hour of the lizards
|
|
Hour of the tide lilies
|
Eighth watch | Hour of the waxing shadows
|
|
Hour of the swifts
|
# _What's next on
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## Contents
1. Cover
2. Novels by R. M. Meluch
3. Title Page
4. Copyright
5. Contents
6. Epigraph
7. Part One:
1. 1. No Blaze of Glory
2. 2. Iry
3. 3. Island in the Sky
4. 4. Circle Circle
5. 5. Does Jerusalem Stand?
8. Part Two:
1. 6. Wolf at the Door
2. 7. Wolf by the Hearth
3. 8. Wolf in the Fire
4. 9. Wolf at the Ramparts
9. Part Three:
1. 10. The Gathering Storm
2. 11. Thunderhead
3. 12. A Winter Conversation
4. 13. Witch Wind
5. 14. A Slow-Falling Star
10. Part Four:
1. 15. Rogue Wolf
2. 16. Ghosts
3. 17. Enemies
11. Part Five:
1. 18. Return of a Legend
2. 19. Shadow of Masada
3. 20. Jerusalem Stands
4. 21. Nemo
12. Appendix
1. Contents
2. Cover
3. Start
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|
{
"redpajama_set_name": "RedPajamaBook"
}
| 1,825
|
\section{Introduction}
Discovery of the Higgs boson~\cite{Aad:2012tfa,Chatrchyan:2013lba}
has made up for the last piece of the standard model (SM) in terms of the particle content,
though the Higgs physics such as the coupling property is still uncertain.
The Higgs boson in the SM plays the important role for the electroweak symmetry breaking, which
is triggered by the nonzero vacuum expectation value of the Higgs field with the negative mass-squared.
The Higgs mass term can be necessarily introduced once the Higgs field is put in, due to the renormalizability
upon which the SM has been established.
However, the Higgs mass term still involves unsatisfactory ingredients on the theoretical ground:
one is called the gauge hierarchy problem, that is what we need to answer: how to stabilize the electroweak vacuum.
The other, which would be correlated with the former, is the origin of the ``negative''-mass squared,
which is simply assumed in the SM without any dynamical concept.
One intriguing idea to solve those problems
is to assume
the classical-scale invariance in the SM.
In this approach, there is no dimensionful parameter at a classical level,
so one does not need to take care of quadratic divergent terms regarding
the renormalization of the Higgs boson mass: thus no scale is present in the model, which would be anomalously generated,
e.g., via the radiative breaking.
The classical-scale symmetry is then broken by the Coleman-Weinberg mechanism~\cite{Coleman:1973jx},
which generates the mass scale via dimensional transmutation.
Actually, such a radiative-breaking scenario does not work solely within the SM itself, due to the presence of the heavy top quark,
so one is eventually forced to add extra degrees of freedom to trigger the radiative breaking as desired.
One of the way out along this approach would be to extend the SM gauge symmetry by introducing an extra $U(1)$ gauge symmetry~\cite{Hempfling:1996ht}.
However, most of such models suffer from an ad hoc assumption:
one requires to assume the sign of the quartic coupling between the SM Higgs boson
and an additional scalar field to be negative.
This ``sign problem" can be solved in a way of a dynamical mechanism, which is called bosonic seesaw mechanism~\cite{BSS}.
The mechanism itself is essentially analogue to the usual type-I seesaw mechanism, well-known in addressing the neutrino sector.
The key point to note is that in the case of fermions, the phase of mass term can be absorbed by redefinition of the fermion fields,
just by the $U(1)_A$ phase rotation.
In contrast, the phase cannot be removed by any ways for boson mass terms, rather will be physical
once the negative sign shows up in the bosonic sector.
Thus models having the bosonic seesaw mechanism built based on the classical-scale invariance
can realize the desired situation in which the problems raised above can be settled down by a dynamical explanation.
Recently, such a hybrid model encoding both the classical-scale invariance and bosonic seesaw mechanism
has been proposed~\cite{Haba:2015qbz}.
The model in Ref.~\cite{Haba:2015qbz} is constructed based on
the classically-scale invariant SM plus a vector-like strongly coupled sector, which we call hypercolor (HC)
(which was originally quoted as ``technicolor" in Ref.~\cite{Haba:2015qbz}).
In the model the classical-scale invariance is dynamically broken by the vector-like condensation
of the HC fermion bilinear, triggered by the strongly coupled HC, in a way analogous to QCD.
The negative mass-squared of the Higgs is induced by the bosonic seesaw mechanism through
the mixing between the elementary Higgs doublet and the composite-HC Higgs doublet
formed as the HC fermion bound state.
Thus, the HC dynamics plays the essential role to solve both the gauge hierarchy and negative-mass squared problems.
In the model of~\cite{Haba:2015qbz}
the success of the bosonic seesaw by the HC dynamics is subject to the presence of the ``chiral" symmetry
carried by the HC fermions, which is partially vector-like gauged by the electroweak charges to provide
the composite-HC Higgs doublet with the appropriate SM charges.
When the HC fermion condensate develops to be nonzero, the ``chiral" symmetry is dynamically broken
down to the vectorial subgroup, at the same time the scale symmetry is broken.
This leads to a couple of Nambu-Goldstone bosons (HC pions).
The origin of mass for the HC pions in the model of~\cite{Haba:2015qbz}
is responsible for a pseudoscalar $S$, having the Yukawa coupling to the HC fermions explicitly
breaking the ``chiral" symmetry: the pseudoscalar $S$ develops the nonzero vacuum expectation value as the direct
consequence of the bosonic seesaw, to give the HC pion masses via the Yukawa coupling,
hence acts as another ``Higgs" for the HC pions.
Thus, discovering the pseudoscalar $S$ as well as the HC pions is a direct probe
and a smoking-gun for this bosonic seesaw model.
In this paper, we discuss the phenomenological consequence of the pseudoscalar $S$
linking to the presence of the HC pions, crucial for the bosonic seesaw model of Ref.~\cite{Haba:2015qbz}.
We find that, after the dynamical-scale breaking and triggering the electroweak bosonic seesaw
at a TeV scale, the fluctuating mode of $S$, which is denoted as $s$,
develops vanishingly small couplings to the SM particles.
It turns out, furthermore, that in close relation to the HC pion masses
the $s$ mass is predicted to be very light and to predominantly decay to diphoton.
We then identify the $s$ as a dark matter candidate like an invisible axion-like particle
and constrain the $s$ mass by several cosmological and astrophysical bounds.
The $s$ mass is thus limited to be $10^{-4} \, {\rm eV} \lesssim m_s \lesssim 1 \, {\rm eV}$.
We examine the possibility of the cosmological productions of the $s$
and show that the $s$ is unlikely to be thermally produced essentially
due to its tiny couplings to the HC sector in the thermal equilibrium.
We then find that the sufficient amount of relic abundance of $s$ as the cold dark matter
can be accumulated via the coherent oscillation, just like the invisible axion case.
The detection potential in microwave cavity experiments is also addressed.
It is shown that the $s$ with mass around $1 \, {\rm eV}$ can have the same level of the
detection sensibility as that of the axion in the currently equipped experimental
setup, so the $s$ is detectable by the microwave cavity experiments.
This paper is organized as follows:
in Sec.~\ref{HC-model} we first review the model of Ref.~\cite{Haba:2015qbz}
by focusing on the essential points to realize the electroweak symmetry breaking
via the bosonic seesaw and to give masses to the HC pions.
(The details for the calculation of the HC pions signals are given in a couple of Appendices.)
In Sec.~\ref{s-DM} we show the close relationship between masses of
the pseudoscalar $s$ and the HC pions, and the $s$ couplings to the SM particles
arising as the direct consequence of the bosonic seesaw mechanism, which turns out to be vanishingly small.
We then identify the $s$ as a dark matter candidate and constrain the mass
by several cosmological and astrophysical bounds currently at hand.
In Sec.~\ref{prod-s} we discuss the cosmological productions of the $s$ involving
the thermal and non-thermal processes.
It is shown that the relic abundance of the $s$ cannot be thermally produced enough to account for
the present dark matter density due to the tiny couplings to the SM particles.
We then find that the non-thermal production, namely, the coherent oscillation
is dominant in the production mechanism for the $s$, which is sufficient for
the $s$ to be a cold dark matter in the present universe.
The detection potential of the $s$-dark mater in microwave cavity experiments
are also discussed in comparison with the case of invisible axion-like particles.
Summary and discussion are given in Sec.~\ref{summary}.
Appendix~\ref{Pi-mass} provides the details for computation of the HC pion masses,
and Appendix~\ref{chiral-lag} gives derivation of the HC pion couplings
based on the nonlinear realization of the ``chiral" symmetry.
The $s$ couplings are also generated there due to the mixing with the HC eta-prime
arising through the bosonic seesaw.
In Appendix~\ref{750} we present the decay properties of the HC pions
relevant to the LHC study, and show the details of the LHC production cross sections
to compute the 750 GeV HC pion signals, in comparison with the current LHC bounds.
\section{A hypercolor model with bosonic seesaw mechanism}
\label{HC-model}
The model we employ is based on the classically-scale invariant SM plus
a strongly coupled HC dynamics at the TeV scale.
The way to construct the model follows from the literature~\cite{Haba:2015qbz}.
The HC sector is described by the HC-gluon ${\cal G}$ with a gauge coupling $g_{\rm HC}$ and
three vector-like fermion triplets ,
$ F_{L,R} = (\chi_i, \psi)^T_{L,R}$,
having the charges,
$ \chi_{i(i=1,2)} \sim (N_{\rm HC}, 1, 2, 1/2)$
and $\psi \sim (N_{\rm HC}, 1, 1, 0)$
for the HC group $SU(N_{\rm HC})$ and $SU(3)_c \times SU(2)_W \times U(1)_Y$.
The HC theory possesses the ``chiral" $U(3)_L \times U(3)_R$ symmetry as well as the (classically) scale-invariance.
The main part of the model Lagrangian thus goes like
\begin{equation}
{\cal L}
=
{\cal L}_{\rm SM}|_{m_H=0}
+
\bar{F} i \gamma^\mu D_\mu F - \frac{1}{2} {\rm tr}[{\cal G}_{\mu\nu}^2]
- V
\,, \label{L}
\end{equation}
with
\begin{equation}
D_\mu = \partial_\mu - i g_{\rm HC} {\cal G}_\mu
\,, \qquad
{\cal G}_{\mu\nu}
= \partial_\mu {\cal G}_\nu - \partial_\nu {\cal G}_\mu
- ig_{\rm HC} [{\cal G}_\mu , {\cal G}_\nu]
\,.
\end{equation}
Here the SM gauges have been switched off momentarily and the potential term $V$
will be specified later.
The ``chiral" symmetry is assumed to be explicitly broken due to the
the breaking terms:
\begin{eqnarray}
\Delta {\cal L}' &=&
{\cal L}_y + {\cal L}_S
\,, \\
{\cal L}_y
&=&
-
y
\,
\bar{F}_L \cdot
\left(
\begin{array}{cc}
0 & H \\
H^\dag & 0
\end{array}
\right)
\cdot
F_R
+ {\rm h.c.}
\,, \label{y}\\
{\cal L}_S
&=&
i g_S
\left(
\bar{F}_L F_R - \bar{F}_R F_L \right)S
\,,\label{gS-term}
\end{eqnarray}
where the Yukawa and $g_S$ couplings $y$ and $g_S$ are assumed to be $\ll 1$
in order to realize the ``chiral" symmetry approximately;
$H$ denotes the elementary Higgs doublet, and the $S$ is a pseudoscalar field having no SM charges.
The potential term $V$ in Eq.(\ref{L}) includes the $H$ and $S$ like
\begin{equation}
V = \lambda_H (H^\dag H)^2
+ \kappa_H S^2 (H^\dag H)
+ \lambda_S S^4
\,. \label{V}
\end{equation}
Thus, the full Lagrangian terms are constructed from Eqs.(\ref{L}), (\ref{y}), (\ref{gS-term}) and (\ref{V})
as
$ {\cal L} + \Delta {\cal L}' $.
Among the ``chiral'' symmetry,
$U(1)_A$ is to be explicitly broken by the anomaly, and the remaining (approximate) ``chiral"
$SU(3)_L \times SU(3)_R (\times U(1)_{V})$ is broken by the ``chiral'' condensate, invariant
under the SM gauge symmetry,
$\langle \bar{F}F \rangle = \langle \bar{\chi}_i \chi_i \rangle = \langle \bar{\psi} \psi \rangle \neq 0$,
down to the diagonal subgroup
$SU(3)_{V} (\times U(1)_{V})$ at the strong scale $\Lambda_{\rm HC}$, just like
the ordinary QCD.
The ``chiral" condensate $\langle \bar{F}F \rangle$
then gives rise to the 8 Nambu-Goldstone bosons (plus heavy $\eta'$).
\subsection{Scalar Seesaw}
At the $\Lambda_{\rm HC}$ scale the composite HC Higgs fields $\sim \bar{F}_iF_j$
are generated. Among them, the component $\Theta \sim \chi \bar{\psi}$ has the
same quantum number as that of the elementary Higgs doublet $H$.
The mixing between the $\Theta$ and $H$ thus gives rise to the scalar seesaw~\cite{Haba:2015qbz}.
Taking into account
the Yukawa term ${\cal L}_y$ in Eq.(\ref{y}) and generation of
the $\Theta$ mass term,
one can write the effective Lagrangian at $\Lambda_{\rm HC}$ to quadratic order in
fields
as
\begin{equation}
{\cal L}_{\rm eff}(\Lambda_{\rm HC})
= - y \left[ \Theta^\dag \cdot H + {\rm h.c.} \right]
- M_{\Theta}^2 \Theta^\dag \Theta
\,. \label{eff:Lag}
\end{equation}
This leads to the seesaw type mass matrix for the Higgs doublet $H$ and
the composite Higgs doublet $\Theta$ (``bosonic seesaw"):
\begin{equation}
\left(
\begin{array}{c}
H \\
\Theta
\end{array}
\right)^\dag
\left(
\begin{array}{cc}
0 & y \Lambda^2_{\rm HC} \\
y \Lambda^2_{\rm HC} & M_{\Theta}^2
\end{array}
\right)
\left(
\begin{array}{c}
H \\
\Theta
\end{array}
\right)
\,.
\end{equation}
This is diagonalized by expanding terms in powers of $y \ll 1$
to be
\begin{equation}
\left(
\begin{array}{c}
H_1 \\
H_2
\end{array}
\right)^\dag
\left(
\begin{array}{cc}
- y^2 \frac{\Lambda_{\rm HC}^4}{M_\Theta^2} & 0 \\
0 & M_\Theta^2 (1 + \frac{y^2 \Lambda_{\rm HC}^2}{M_\Theta^2})
\end{array}
\right)
\left(
\begin{array}{c}
H_1 \\
H_2
\end{array}
\right)
\equiv
\left(
\begin{array}{c}
H_1 \\
H_2
\end{array}
\right)^\dag
\left(
\begin{array}{cc}
- m_{H_1}^2 & 0 \\
0 & m_{H_2}^2
\end{array}
\right)
\left(
\begin{array}{c}
H_1 \\
H_2
\end{array}
\right)
\,.
\end{equation}
The mass eigenstates $(H_1, H_2)$ are related to the current eigenstates $(H, \Theta)$ as
\begin{equation}
\left(
\begin{array}{c}
H_1 \\
H_2
\end{array}
\right)
\simeq
\left(
\begin{array}{cc}
1 - \frac{y^2}{2} + {\cal O}(y^4) & - y (1 - \frac{3}{2} y^2) + {\cal O}(y^5) \\
y (1 - \frac{3}{2} y^2) + {\cal O}(y^5)& 1 - \frac{y^2}{2} + {\cal O}(y^4)
\end{array}
\right)
\left(
\begin{array}{c}
H \\
\Theta
\end{array}
\right)
\,, \label{relation:H}
\end{equation}
where we have taken $M_{\Theta} \simeq \Lambda_{\rm HC}$.
Thus the scale-breaking effect has been transfered to the $H$-Higgs sector via
the bosonic seesaw mechanism.
Note the negative sign for the lower eigenvalue $(- m_{H_1}^2)$,
playing the essential role to realize the electroweak symmetry breaking,
as will be explicitly clarified later on.
\subsection{Pseudoscalar Seesaw}
As mentioned above, the $\eta'$ gets the mass from the $U(1)_A$ anomaly as in the case of
the ordinary QCD.
The size of the mass
can be estimated just by scaling
from the QCD to be
\begin{equation}
M_{\eta'}
\sim {\cal O}(1{\rm GeV}) \times \left( \frac{\Lambda_{\rm HC}}{\Lambda_{\rm QCD}} \right) \times
\sqrt{ \frac{3}{N_{\rm HC}}}
\sim {\cal O}(1\, {\rm TeV} ) \times \sqrt{\frac{3}{N_{\rm HC}}}
\,,
\end{equation}
where the large $N_{\rm HC}$ counting has been taken into account.
One should note that
the $\eta'$ couples to the $U(1)_A$ current,
$J^0_\mu = \frac{1}{\sqrt{6}}\cdot \bar{F} \gamma_\mu \gamma_5 \cdot {\bf 1}_{3 \times 3} \cdot F$.
Hence at the $\Lambda_{\rm HC}$ scale, by taking into account the $\eta'$ mass generation from the anomaly,
the $g_S$ term in Eq.(\ref{gS-term}) looks like
\begin{equation}
{\cal L}_S(\Lambda_{\rm HC})
\approx
g_S \Lambda^2_{\rm HC} \eta' S - \frac{1}{2}M^2_{\eta'} (\eta')^2
\,. \label{eff:Lag:eta}
\end{equation}
Again, the form of Eq.(\ref{eff:Lag:eta}) is nothing but a seesaw type (``bosonic seesaw"),
so one can readily see that the lower eigenvalue, corresponding to the $S$-mass squared,
is negative:
\begin{eqnarray}
{\cal L}_S(\Lambda_{\rm HC})
&\approx&
- \frac{1}{2}
\left(
\begin{array}{c}
S \\
\eta'
\end{array}
\right)^T
\left(
\begin{array}{cc}
0 & - g_S \Lambda_{\rm HC}^2
\\
- g_S \Lambda_{\rm HC}^2 & M_{\eta'}^2
\end{array}
\right)
\left(
\begin{array}{c}
S \\
\eta'
\end{array}
\right)
\nonumber \\
&=&
- \frac{1}{2}
\left(
\begin{array}{c}
{\cal S} \\
\eta^0
\end{array}
\right)^T
\left(
\begin{array}{cc}
- g_S^2 \frac{\Lambda_{\rm HC}^4}{M_{\eta'}^2} & 0 \\
0 & M_{\eta'}^2 (1 + \frac{g_S^2 \Lambda_{\rm HC}^2}{M_{\eta'}^2})
\end{array}
\right)
\left(
\begin{array}{c}
{\cal S} \\
\eta^0
\end{array}
\right)
\equiv
\left(
\begin{array}{c}
{\cal S} \\
\eta^0
\end{array}
\right)^T
\left(
\begin{array}{cc}
- m_{\cal S}^2 & 0 \\
0 & m_{\eta^0}^2
\end{array}
\right)
\left(
\begin{array}{c}
{\cal S} \\
\eta^0
\end{array}
\right)
\,.
\end{eqnarray}
The mass eigenstates $({\cal S}, \eta^0)$ are related to the current eigenstates $(S, \eta')$ as
\begin{equation}
\left(
\begin{array}{c}
{\cal S} \\
\eta^0
\end{array}
\right)
\simeq
\left(
\begin{array}{cc}
1 - \frac{g_S^2}{2} + {\cal O}(g_S^4) & g_S (1 - \frac{3}{2} g_S^2) + {\cal O}(g_S^5) \\
- g_S (1 - \frac{3}{2} g_S^2) + {\cal O}(g_S^5) & 1 - \frac{g_S^2}{2} + {\cal O}(g_S^4)
\end{array}
\right)
\left(
\begin{array}{c}
S \\
\eta'
\end{array}
\right)
\,, \label{relation:S}
\end{equation}
to the nontrivial order of expansion in $g_S \ll 1$,
where we have taken $M_{\eta'} \simeq \Lambda_{\rm HC}$.
Thus, the pseudoscalar ${\cal S}$ can get the nonzero vacuum expectation value,
playing the significant role to supply the pseudo Nambu-Goldstone boson (HC pion) masses,
as will be clearly seen later.
\subsection{Electroweak Symmetry Breaking}
Including the dynamically generated terms,
we thus see that
Eq.(\ref{V}) is now modified
at the scale $\Lambda_{\rm HC}$
as follows:
\begin{eqnarray}
V
&=&
-
\left(
\begin{array}{c}
H \\
\Theta
\end{array}
\right)^\dag
\left(
\begin{array}{cc}
0 & y \Lambda^2_{\rm HC} \\
y \Lambda^2_{\rm HC} & M_{\Theta}^2
\end{array}
\right)
\left(
\begin{array}{c}
H \\
\Theta
\end{array}
\right)
- \frac{1}{2}
\left(
\begin{array}{c}
S \\
\eta'
\end{array}
\right)^T
\left(
\begin{array}{cc}
0 & g_S \Lambda_{\rm HC}^2
\\
g_S \Lambda_{\rm HC}^2 & M_{\eta'}^2
\end{array}
\right)
\left(
\begin{array}{c}
S \\
\eta'
\end{array}
\right)
\nonumber \\
&&
+ \lambda_{\Theta} (\Theta^\dag \Theta)^2
+ \lambda_{H} (H^\dag H)^2
+ \kappa_H S^2 (H^\dag H)
+ \lambda_S S^4
\nonumber \\
&=&
- m_{H_1}^2 (H_1^\dag H_1) + m_{H_2}^2 (H_2^\dag H_2)
- \frac{1}{2} m_{\cal S}^2 {\cal S}^2 + \frac{1}{2} m_{\eta^0}^2 (\eta^0)^2
\nonumber \\
&&
+ \lambda_{\Theta} (\Theta^\dag \Theta)^2
+ \lambda_{H} (H^\dag H)^2
+ \kappa_H S^2 (H^\dag H)
+ \lambda_S S^4
\,,
\label{pot}
\end{eqnarray}
where we added the quartic coupling of $\Theta$ which can generically be induced from
the underlying HC dynamics, and is expected to be $\gtrsim {\cal O}(10)$.
Based on this potential we discuss the realization of the electroweak symmetry breaking.
To this end,
we may first parametrize the scalar and pseudoscalar fields with their vacuum expectation values
for the mass eigenstate fields $(H_1, H_2)$ and $({\cal S}, \eta^0)$ in Eqs.(\ref{relation:H})
and (\ref{relation:S}):
\begin{eqnarray}
H_1
&=&
\left(
\begin{array}{c}
\varphi_1^+ \\
\frac{1}{\sqrt{2}} (v_1 + h_1^0 + i \varphi_1^0)
\end{array}
\right)
\,, \qquad
H_2=
\left(
\begin{array}{c}
\varphi_2^+ \\
\frac{1}{\sqrt{2}} (v_2 + h_2^0 + i \varphi_2^0)
\end{array}
\right)
\,, \nonumber \\
{\cal S}
&=& v_S + s
\,, \qquad
\eta^0 = v_{\eta} + e_0
\,, \label{field:para}
\end{eqnarray}
where $\pm$ denote the electromagnetic charges assigned according to the charges of the HC-$F$ fermions.
We may search for the vacuum by assuming\footnote{The stationary condition for $v_2$ actually includes the trivial solution $v_2 =0$, hence one can always select the vacuum with $v_2=0$ which in the present study we have taken for simplicity.
Under the condition with $v_2=0$, however, other vacuum expectation values ($v_S$,$v_\eta$) cannot be set to zero because of some phenomenological constraints, related to the HC pion and eta-prime masses, as will be seen later (See Eqs.~(\ref{stationary:condi}), (\ref{masses}) and (\ref{gsvs})).}
\begin{equation}
v_2 = 0
\,, \label{v2:0}
\end{equation}
so that, for the nontrivial solutions $v_1\neq 0, v_S \neq 0, v_\eta \neq 0$,
the stationary conditions are obtained by expanding terms in powers of $y$ and $g_S$ as
\begin{eqnarray}
m_{H_1}^2
&=& \frac{1}{2} y^2 \lambda_\Theta v_1^2 + \cdots
(\simeq y^2 \Lambda_{\rm HC}^2)
\,, \nonumber \\
m_{\cal S}^2
&=& 4 \lambda_S v_S^2 + \cdots
(\simeq g_S^2 \Lambda_{\rm HC}^2)
\,, \nonumber \\
m_{\eta^0}^2
&=&
g_S^3 \frac{v_S^3}{v_\eta} + \cdots
(\simeq \Lambda_{\rm HC}^2)
\,, \nonumber \\
\kappa_H &=& - \frac{v_1^2}{v_S^2}
\lambda_H
+ \cdots
\,, \label{stationary:condi}
\end{eqnarray}
where the last condition has come by imposing $v_2=0$ and
the ellipses denote terms suppressed by higher orders in expansion with respect to
$y$ and $g_S$, and
the expressions in the parenthesis correspond to the seesaw-induced formulae.
As will be discussed in the later section, the $v_S$ is constrained, by the
phenomenological limits on the pseudoscalar $s$, as $\Lambda_{\rm HC}/v_S \ll 1$,
so that the coupling $\kappa_H$ is required to be vanishingly small, $\kappa_H \ll 1$,
hence so is the $\lambda_S$, $\lambda_S \ll 1$.
By adjusting parameters to satisfy these conditions,
the electroweak scale $v_1 = 246$ GeV can be realized at the minimum of the
potential (with the $H$-quartic coupling $\lambda_H >0$, hence $\kappa_H <0$),
consistently with the bosonic seesaw mechanism.
As will be clarified later (Eq.(\ref{masses})),
the square of masses for fluctuating fields $(h_1^0, h_2^0, s, e_0)$ are properly
positive-definite at the chosen stationary space $(v_1, v_2, v_S, v_\eta)$
satisfying the stationary conditions Eq.(\ref{stationary:condi})
with $v_2=0$. This implies that the vacuum has safely been aligned to where the electroweak symmetry is broken with extra nonzero
CP-odd vacuum expectation values $(v_S, v_\eta)$.
By taking some reference values for the potential parameters,
we have numerically checked that the electroweak-broken vacuum
indeed locates at the global minimum.
Actually, the alignment problem should be
argued by taking into account all the possible vacuums including
nonzero vacuum expectation values for other composite HC Higgs fields like
$\bar{\chi}\chi, \bar{\psi}\psi$, and so forth.
However, due to the presence of the ``chiral'' symmetry in the underlying
HC theory, one can be allowed to rotate the composite HC Higgs fields to be
aligned to the desired direction where the potential
is minimized at the electroweak-broken vacuum.
More rigorous proof is to be beyond scope of the present study, which
will be argued elsewhere.
\subsection{Scalar and Pseudoscalar Masses}
The scalars ($h_1, h_2$) and pseudoscalars ($s, e_0$), defined as in Eq.(\ref{field:para}),
arise as the fluctuating modes around the vacuum expectation values $(v_1, v_S, v_\eta)$
in the potential Eq.(\ref{pot}).
Expanding the potential terms in powers of the small parameters $(y, g_S, v_1/v_S, \kappa_H, \lambda_S)$ and keeping only the nontrivial leading orders, one finds
the mass eigenvalues,
\begin{eqnarray}
m_{h_1^0}^2 &\simeq&
2 \lambda_H v_1^2 \simeq 2 (-\kappa_H) v_s^2
\,, \nonumber \\
m_{h_2^0}^2 &\simeq& m_{H_2}^2
\,, \nonumber \\
m_{s}^2 &\simeq& 8 \lambda_S v_S^2 \simeq 2 g_S^2 \Lambda_{\rm HC}^2
\,, \nonumber \\
m_{e_0}^2 &\simeq & m_{\eta^0}^2
\,, \label{masses}
\end{eqnarray}
where the second approximate expression in the third line
follow from the stationary conditions in Eq.(\ref{stationary:condi}) and
the $h_1^0$ is identified as the 125 GeV Higgs.
It is interesting to note that, in addition to particles with the ${\cal O}({\rm TeV})$ mass on the natural scale of HC dynamics,
the present model predicts a light pseudoscalar $(s)$ with mass of ${\cal O}(g_S \Lambda_{\rm HC}) (\ll \Lambda_{\rm HC})$,
as the consequence of the bosonic seesaw mechanism.
Thus, this $s$ is a smoking-gun of the model and will be identified as the dark matter
candidate, as will be discussed later.
\subsection{HC pions}
Since the $y-$ and $g_S-$ Yukawa terms in Eqs.(\ref{y}) and (\ref{gS-term})
explicitly break the ``chiral'' $SU(3)_L \times SU(3)_R$ symmetry,
the 8 Nambu-Goldstone bosons become pseudo's (HC pions $\Pi$) through those interactions.
Using the current algebra technique and expanding things in powers of $y$ and $g_S$,
one can evaluate the HC pion masses to
find that they are almost degenerate to be
\begin{equation}
m_{\Pi} \simeq 2 (g_S v_S) \frac{\Lambda_{\rm HC}}{f}
\,, \label{HC-pion-mass}
\end{equation}
where
\begin{equation}
f = \frac{f_\Pi}{\sqrt{N_{\rm HC}/3}}
\,,
\end{equation}
with the $f_\Pi$ being the HC pion decay constant.
The detail of the derivation for this formula is presented in Appendix~\ref{Pi-mass}.
As a reference point, we may set the HC pion mass to be $750$ GeV so that
the combination $(g_S v_S)$ can be fixed as
\begin{equation}
(g_S v_S) \simeq
30\, {\rm GeV} \times \left( \frac{m_{\Pi}}{750\,{\rm GeV}} \right)
\left( \frac{4 \pi f}{\Lambda_{\rm HC}} \right)
\,. \label{gsvs}
\end{equation}
We may take $\Lambda_{\rm HC} \sim 4 \pi f$ to get the formula for the
coupling $g_S$,
\begin{equation}
g_S \simeq
\frac{30\, {\rm GeV}}{v_S}
\times \left( \frac{m_{\Pi}}{750\,{\rm GeV}} \right)
\ll 1
\,, \label{gs-val}
\end{equation}
which implies $v_S \gtrsim {\cal O}({\rm TeV})$.
\section{ The light pseudoscalar $s$ as a dark matter candidate}
\label{s-DM}
As noted in the previous section, the present model predicts the light
pseudoscalar $s$ as the direct consequence of the bosonic seesaw.
In the present study we shall try to identify the $s$ as a dark matter candidate
and this section devotes ourselves to discuss several cosmological and astrophysical limits on the $s$-dark matter.
\subsection{Lifetime}
We first evaluate the $s$ mass, decay property, and its lifetime.
The $s$ mass is related to the HC pion masses through Eqs.(\ref{masses}) and (\ref{gs-val})
as
\begin{equation}
m_s \simeq \sqrt{2} g_S \Lambda_{\rm HC} \simeq 42\, {\rm GeV}
\times \left( \frac{m_{\Pi}}{750\,{\rm GeV}} \right)
\left( \frac{\Lambda_{\rm HC}}{1 {\rm TeV}} \right)
\left( \frac{1 {\rm TeV}}{v_S} \right)
\,. \label{s-mass-val}
\end{equation}
The $s$ couplings to the SM particles arise from mixing with the HC-eta prime coupled to
the SM gauge bosons, $WW, ZZ, Z \gamma$ and $\gamma\gamma$, along with the tiny factor
$g_S \ll 1$ (see Appendix~\ref{chiral-lag}).
Taking into account the size of the $s$ mass in Eq.(\ref{s-mass-val}),
we thus find that the decay channel of the $s$ is only the diphoton mode through the vertex:
\begin{equation}
{\cal L}_{s \gamma\gamma}
= - \frac{1}{4} g_{s\gamma\gamma} \, s \, F_{\mu\nu} \tilde{F}^{\mu\nu}
\,, \qquad
\tilde{F}^{\mu\nu} \equiv \frac{1}{2}\epsilon^{\mu\nu\rho\sigma} F_{\rho\sigma}
\,, \label{int:s}
\end{equation}
with $F_{\mu\nu}=\partial_\mu A_\nu - \partial_\nu A_\mu$ and
\begin{equation}
g_{s\gamma\gamma}
= \frac{4 \sqrt{2}}{\pi} \sqrt{N_{\rm HC}} \frac{g_S \alpha_{\rm em}}{f}
\simeq
16 \sqrt{N_{\rm HC}} \alpha_{\rm em} \frac{m_s}{\Lambda_{\rm HC}^2}
\,, \label{gsgg}
\end{equation}
where use has been made of Eq.(\ref{s-mass-val}).
The lifetime of $s$ is thus calculated to be
\begin{eqnarray}
\Gamma_s/N_{\rm HC} = \frac{g_{s\gamma\gamma}^2}{4096 \pi} m_s^3/N_{\rm HC}
&\simeq&
275 \,{\rm meV}
\left( \frac{m_s}{42 \,{\rm GeV}} \right)^5
\left( \frac{1\,\rm TeV}{\Lambda_{\rm HC}} \right)^4
\,. \label{lifetime-s}
\end{eqnarray}
For the $s$ to be a dark matter, the lifetime has to be longer than the age of the universe
at present time, which requires $\tau \gtrsim 10^{17}\, s$.
From Eq.(\ref{lifetime-s}) the $s$ mass is thus constrained as
\begin{equation}
m_s \lesssim 10 \,{\rm keV} \times
\left( \frac{\Lambda_{\rm HC}}{1 \, {\rm TeV}} \right)^{4/5}
\,.
\end{equation}
\subsection{Astrophysical and cosmological limits}
\subsubsection{Line emission observations}
The $s$, dominantly decaying to photon,
is expected to affect several line emission observations such as
gamma-ray, X-ray, and cosmic ray,
so the mass of $s$ can be severely constrained as
in the case for other dark matter candidates~\cite{Mambrini:2015sia,Baer:2014eja}.
In addition,
the mass-independent limit
on the coupling to the photon, $g_{s\gamma\gamma}$,
can be placed by the observations of the horizontal branch stars
for a lower mass range $m_s \lesssim 0.1$ keV~\cite{Raffelt:2006cw}.
From Eq.(\ref{lifetime-s}) in Fig.~\ref{limit-on-s-from-Xray}
we make a plot of the lifetime of $s$ ($\tau$) as
a function of the mass $m_s$ in comparison with the
line shape and the horizontal branch star limits.
The figure implies the limits on the $s$ as
\begin{equation}
m_s \lesssim 1 \,{\rm keV}
\,, \qquad
{\rm with}
\qquad
\tau \times N_{\rm HC} \simeq 1.6 \times 10^{28}\,[s]
\left( \frac{0.1 \,{\rm keV}}{m_s} \right)^5
\left( \frac{\Lambda_{\rm HC}}{1\,\rm TeV} \right)^4
\,.
\label{X-ray-ms}
\end{equation}
\begin{figure}[t]
\begin{center}
\includegraphics[width=9.0cm]{limit-on-s-from-Xray.eps}
\caption{
The line emission and horizontal branch star observation limits on the $s$.
The region below the red-solid and - dashed lines are excluded.
The data have been quoted from Refs.~\cite{Baer:2014eja,Mambrini:2015sia,Raffelt:2006cw}.
\label{limit-on-s-from-Xray}
}
\end{center}
\end{figure}
\subsubsection{Constraints on the thermal $s$}
The $s$-dark matter can be thermally produced by the scattering with the photon,
$s + \gamma \leftrightarrow s + \gamma$,
through the interaction in Eq.(\ref{int:s}) with the coupling Eq.(\ref{gsgg})
in the early universe. The reaction rate $R(T)$ can roughly be estimated as
\begin{equation}
R(T) = n(T) \langle \sigma v \rangle \approx
g_{s\gamma\gamma}^4 T^5
\,.
\end{equation}
The decoupling temperature of the $s$, $T_D$,
can be evaluated, by equating this $R(T)$ with the Hubble rate $H(T) \sim\sqrt{ g_*(T)} T^2/M_P$
with the reduced Planck mass scale $M_P = 2.44 \times 10^{18}$ GeV and
$g_*(T_D)$ being the effective degrees of freedom
for relativistic particles.
The $g_*(T_D)$ is estimated by combining the SM, the HC sector and the pseudoscalar
$s$ as $g_*(T_D) = g_*^{\rm SM} + g_*^s + g_*^{\rm HC}$,
where $g_*^{\rm SM}=106.75$~\cite{Agashe:2014kda} and $g_*^s =1$.
The $g_*^{\rm HC}$ is calculated as
\begin{eqnarray}
g_*^{\rm HC}
&=&
\left[2\times (N_{\rm HC}^2 - 1) \right]_{{\cal G}_{\rm HC}}
+ \frac{7}{8} N_{\rm HC} \left[(2 \times 2 \times 2)_\chi + (2 \times 2)_\psi \right]
\nonumber\\
&=&
2 (N_{\rm HC}^2 -1 ) + \frac{21}{2} N_{\rm HC}
\, .
\end{eqnarray}
For $N_{\rm HC}=(3,4,5)$,
we have
\begin{equation}
g_*(T_D) = (155.25, 179.75, 208.25)
\,.
\end{equation}
Thus we find
\begin{equation}
T_D \approx 10^{12}\,{\rm GeV} \times
\left( \frac{3}{N_{\rm HC}} \right)^{2/3}
\left( \frac{10^{-1}\,{\rm keV}}{m_s} \right)^{4/3}
\left( \frac{\Lambda_{\rm HC}}{1 \,{\rm TeV}} \right)^{8/3}
\left( \frac{g_*(T_D)}{200} \right)^{1/6}
\,,\label{td}
\end{equation}
where Eq.(\ref{gsgg}) have been used.
Even after decoupling from the thermal equilibrium,
the $s$ (with mass $\lesssim 1$ keV as in Eq.(\ref{X-ray-ms})) can be
still relativistic at present,
which is constrained by the null observation of
dark radiations~\cite{DiValentino:2016ikp}.
Since the $s$ goes cool down just like radiations
due to the Hubble expansion after the decoupling,
the present temperature of the $s$ is estimated
as
\begin{equation}
T_0(s) = (t_D/t_0)^{1/2} T_D = (g_*(T_0)/g_*(T_D))^{1/4} T_0
\simeq 10^{-4} {\rm eV}
\,.
\end{equation}
with $g_*(T_0)= (2)_\gamma + (21/4(4/11)^{4/3})_{\nu} + (1)_s \simeq 4.36$.
The current dark radiation constraint reads~\cite{DiValentino:2016ikp}
$\Delta N_{\rm eff} = (T_0(s)/T_0(\nu))^3 < 0.1$ with $T_0(\nu)=(4/11)^{1/3}T_0$.
The $s$ mass may thus be required to be
\begin{equation}
m_s \gtrsim 10^{-4}\,{\rm eV}
\,. \label{dr-ms}
\end{equation}
If the $s$ decouples from the photon
after the inflation and reheating temperature $T_R$,
the temperature of the $s$ is heated back up to reach the same
as the photon temperature, so that the $s$ would be a warm or hot dark matter-like particle.
Currently such a light warm matter has been severely constrained by
the cosmic microwave background spectrum. Hence we may escape
from the case, by imposing $T_D > T_R$.
The present model may follow a typical Higgs inflation scenario, as discussed
in Ref.~\cite{Reheating}, in which $T_R \simeq 10^{14}$ GeV.
Taking this value as a reference and using Eq.(\ref{td}), we thus find
\begin{equation}
m_s \lesssim 1\, {\rm eV} \times
\left( \frac{3}{N_{\rm HC}} \right)^{1/2}
\left( \frac{\Lambda_{\rm HC}}{1 \, {\rm TeV}} \right)^{2}
\left( \frac{g_*(T_D)}{200} \right)^{1/8}
\,. \label{tr}
\end{equation}
From Eqs.(\ref{X-ray-ms}), (\ref{dr-ms}) and (\ref{tr}),
we thus see the $s$ mass constrained to be
\begin{equation}
10^{-4}\,{\rm eV} \lesssim m_s \lesssim 1\,{\rm eV}
\,. \label{mass-range}
\end{equation}
\section{Cosmological productions and Detection of the $s$-dark matter}
\label{prod-s}
In this section, we closely explore the possibility for the $s$ as a dark matter
to account for the relic abundance at the present time.
\subsection{Thermal production}
Though the $s$-dark matter decouples from the thermal equilibrium in the early universe
at $T_D \approx 10^{14}$ GeV $\times (1\,{\rm eV}/m_s)^{4/3} (\gtrsim T_R)$,
there might exist the chance to thermally accumulate the number density by production cross sections
interacting with the HC sector until the HC sector decouples from the thermal equilibrium
at around $T=\Lambda_{\rm HC}={\cal O}({\rm TeV})$.
The relevant production processes involve only a single $s$ in the final state through
the $s-\gamma-\gamma$ vertex in Eq.(\ref{int:s})\footnote{When the temperature is significantly higher than $\Lambda_{\rm HC}$, the $s$-coupling to diphoton may arise from the HC fermion loops. Even if the universe is in such a symmetric phase by taking into account the thermal effect, the vertex is anyhow generated with the magnitude of the order of $g_{s \gamma \gamma}$ which is given by Eq.~(\ref{gsgg}).} and the $s-Z-\gamma$, $s-Z-Z$ vertices
listed in Appendix~\ref{chiral-lag}, scattered off from the HC sector-fermion $F=(\chi, \psi)$
such as
$ F + \bar{F} \to \gamma/Z + s $.
The production cross section roughly goes like
\begin{equation}
\sigma(F +\bar{F} \to \gamma/Z + s)
\sim \alpha_{\rm em} N_{\rm HC}
\left( \frac{\sqrt{N_{\rm HC}} g_S \alpha_{\rm em}}{ \Lambda_{\rm HC}} \right)^2
\simeq
10^{-31} \times \frac{N_{\rm HC}^2}{\Lambda_{\rm HC}^2} \left( \frac{m_s}{1\,{\rm eV}} \right)^2 \left( \frac{1\,{\rm TeV}}{\Lambda_{\rm HC}} \right)^2
\,, \label{cross-s-HC}
\end{equation}
where in the second equality we have used the first relationship in Eq.(\ref{s-mass-val}).
The corresponding number density per entropy density at present time ($Y_s(T_0)=n_s(T_0)/s(T_0)$)
can be estimated by integrating the Boltzmann equation with the above production cross section
over the temperature from the reheating temperature $T_R\approx 10^{14}$ GeV
down to the freeze-out temperature $T_F=\Lambda_{\rm HC}$.
Following the formula given in Ref.~\cite{Choi:1999xm}
we thus evaluate the $Y_s(T_0)$ as
\begin{eqnarray}
Y_s(T_0)
&=& \int_{\Lambda_{\rm HC}}^{T_R} dT \frac{\langle \sigma (F + \bar{F} \to \gamma/Z + s) v \rangle n_F n_{\bar{F}} }{s(T) H(T) T}
\, \nonumber \\
&=&
\frac{135\sqrt{10} M_P}{2 \pi^{3}} \int_{\Lambda_{\rm HC}}^{T_R} dT \frac{\langle \sigma (F + \bar{F} \to \gamma/Z + s) v \rangle n_F n_{\bar{F}}}
{g_{*}^{3/2} (T) T^6}
\,,
\end{eqnarray}
where in reaching the last line we used
$H^2(T)=\frac{\pi^2}{30} g_*(T) T^4/(3M_P^2)$,
$s(T)=g_{s*}(T) \frac{2 \pi^2}{45} T^3$, with $g_*(T) = g_{s*}(T)$ is assumed
and the thermal average is expressed to be
\begin{equation}
\langle \sigma_n (F + \bar{F}\to s + \gamma/Z) v \rangle n_F n_{\bar{F}}
= \zeta^2(3) \cdot \eta_F \eta_{\bar{F}} \cdot \frac{g_Fg_{\bar{F}}}{16\pi^4} T^6 \int_0^\infty dx x^4 K_1(x) \sigma(x^2)
\,,
\end{equation}
where $\zeta(3)= 1.202...$
and $K_1(x)$ stands for the modified Bessel function of the first kind,
$\sigma(x^2) = \sigma(s/T^2)$ and $g_{F(\bar{F})}$ is the internal (spin) degree of freedom
for the HC fermion (anti-fermion) $F$; $\eta_{F(\bar{F})}$ is a number density factor associated with the initial state particle
assigned as $\eta_F=3/4$ for fermions (anti-fermion).
Using these we thus calculate the $Y_s(T_0)$ to get
\begin{eqnarray}
Y_s(T_0)
&\approx &
\frac{135 \sqrt{10}}{32 \pi^6} \frac{M_P T_R}{g_*^{3/2}(T_R) \Lambda_{\rm HC}^2} \times 10^{-31} \times N_{\rm HC}^2
\left( \frac{m_s}{1\,{\rm eV}} \right)^2 \left( \frac{1\,{\rm TeV}}{\Lambda_{\rm HC}} \right)^2
\nonumber \\
&\approx&
10^{-10} \times N_{\rm HC}^2 \left( \frac{m_s}{1\,{\rm eV}} \right)^2 \left( \frac{1\,{\rm TeV}}{\Lambda_{\rm HC}} \right)^4
\left(\frac{200}{g_*(T_R)}\right)^{3/2}
\,, \label{Eq:Y_th}
\end{eqnarray}
where use has been made of $g_*(T_R)=g_*(\Lambda_{\rm HC})$.
Thus, it turns out that the thermal relic is too small to explain the present dark matter abundance.
This result is essentially tiled with the tiny coupling $g_S$ which leads to the extremely
small cross section with the HC sector in Eq.(\ref{cross-s-HC}).
\subsection{Non-thermal production}
Analogously to the case of axion dark matter~\cite{Baer:2014eja},
the $s$-dark matter population can be accumulated by ``misalignment" of
the classical $s$ field and the coherent oscillation.
Assuming the initial position at which the oscillation starts to be
the vicinity of the vacuum $s=0$ with the vacuum expectation value $v_S$,
we write
the equation of motion for the $s$ under the Friedmann-Robertson-Walker metric
to be
\begin{equation}
\frac{d^2 s}{dt^2}
+ 3 H(T) \frac{d s}{dt} + m_s^2 \, s \approx 0
\,.
\end{equation}
This describes the damping harmonic oscillation in which the oscillation
takes place when $T=T_{\rm osc}$ where $3 H(T) \approx m_s $, i.e.,
\begin{equation}
T_{\rm osc}
\simeq
13 \,{\rm TeV} \times \left( \frac{m_s}{1 \, \rm eV} \right)^{1/2} \left(\frac{ 200}{g_*(T_{\rm osc})} \right)^{1/4}
\,.
\end{equation}
This implies that $130\,{\rm GeV} \lesssim T_{\rm osc} \lesssim 13$ TeV for $10^{-4}\,{\rm eV} \lesssim m_s \lesssim 1$ eV.
Since the $s$ mass is generated through the bosonic seesaw at $T\simeq \Lambda_{\rm HC}={\cal O}(1)$ TeV,
we find that the temperature at which the coherent oscillation starts, what we call $T_S$, depend on the $m_s$ as
\begin{eqnarray}
T_S &\simeq& \Lambda_{\rm HC} \,\qquad {\rm for} \qquad 6 \times10^{-3} \,{\rm eV} \left( \frac{\Lambda_{\rm HC}}{1\,{\rm TeV}} \right)^2 \lesssim m_s < 1\,{\rm eV}
\,, \nonumber \\
T_S &\simeq& T_{\rm osc} \, \qquad {\rm for} \qquad 10^{-4}\,{\rm eV} \lesssim m_s \lesssim 6 \times10^{-3} \,{\rm eV} \left( \frac{\Lambda_{\rm HC}}{1\,{\rm TeV}} \right)^2
\,. \label{TS}
\end{eqnarray}
The energy density of the classical $s$ field is thus accumulated by the coherent oscillation
starting from the temperature $T_S$ in Eq.(\ref{TS}), cooling down to the present temperature $T_0$.
At the $T=T_S$ the energy density of the $s$ corresponds to the vacuum energy defined as
\begin{equation}
\rho_s(T_S) = V(\theta) - V(\theta=0)
\,,
\end{equation}
where the $\theta$ is defined as the amount of the shift from the original $S$ field at the vacuum expectation value $v_S$
to be $S=v_S(1 + \theta)$ with $\theta \ll 1$, and the potential $V(\theta)$ is read off as
\begin{equation}
V(\theta) = V(\theta=0) + \frac{1}{2} m_s^2 v_s^2 \theta^2 + {\cal O} (\theta^3)
\,. \label{V-theta}
\end{equation}
One can easily see that during the coherent oscillation, the number density per comoving volume
is conserved and the $s$ behaves just like a non-relativistic particle satisfying $\rho_s \propto R^{-3}$
with the expansion rate $R$. Hence we write
\begin{equation}
\frac{\rho_{s}(T_S)}{\rho_s(T_0)}
= \frac{m_s n_s(T_S)}{m_s n_s(T_0)}
= \frac{s(T_S)}{s(T_0)}
\,,
\qquad {\rm i.e.,} \qquad
\rho_s(T_0) = \frac{s(T_0)}{s(T_S)} \rho_s(T_S)
\,. \label{rhoS}
\end{equation}
Thus, we get the present abundance of DM as
\begin{eqnarray}
\rho_s (T_0)
\simeq
(4200~{\rm GeV})^4
\left(
\frac{T_0}{T_S}
\right)^3
\frac{g_{\ast S}(T_0)}{g_{\ast S}(T_S)} \theta^2
\left(
\frac{m_\Pi}{750~{\rm GeV}}
\right)^2
\left(
\frac{\Lambda_{\rm HC}}{1~{\rm TeV}}
\right), \label{srelic}
\end{eqnarray}
by using Eqs.~(\ref{s-mass-val}).
This relation shows that we can explain the correct abundance of $s$ with an appropriate value of $\theta$
even when the HC pion mass is heavier/lighter than $750$ GeV.
From Eqs.(\ref{TS}), (\ref{V-theta}) and (\ref{rhoS}), and using the second equality in Eq.(\ref{s-mass-val}),
we thus estimate the $s$-dark mater relic density, $\Omega_s h^2 = \rho_s(T_0)/(\rho_{\rm cr}/h^2)$
with $\rho_{\rm cr}/h^2 = 0.8 \times 10^{-46}\,{\rm GeV}^4$.
The contour plot on the $(m_s, \theta)$ plane with the
observed dark matter relic density $\Omega_{\rm DM} h^2 \simeq 0.118$~\cite{Agashe:2014kda}
has been drawn in Fig.~\ref{Coherent-Osc-s}.
Here use has been made of
$s(T_0) = \frac{2 \pi^2}{45} g_{s*}(T_0) T_0^3$ with $g_{*s}(T_0)=43/11$ and $T_0\simeq 2.4 \times 10^{-4}$ eV,
$g_{s*}(\Lambda_{\rm HC})=200$ taken as a reference value,
and we have assumed $g_{s*}(T_S < \Lambda_{\rm HC}) = g_*^{\rm SM}=106.75$.
From the figure, we find that
the relic density of the $s$, with the mass in a range of $10^{-4} \, {\rm eV} \lesssim m_s
\lesssim 1\,{\rm eV}$,
can be accumulated enough to account
for the present dark matter abundance.
\begin{figure}[t]
\begin{center}
\includegraphics[width=7.0cm]{Coherent-Osc-s.eps}
\caption{
The contour plots on the $(m_s, \theta)$ plane realizing the
observed present dark matter density $\Omega_{\rm DM}h^2 =0.118$~\cite{Agashe:2014kda}.
The bumps, which show up when $T_S$ gets lower than $\Lambda_{\rm HC}$, are due to
the discontinuity of the effective degrees of freedom $g_{s*}$ around the $T_S = \Lambda_{\rm HC}$
as described in the text.
\label{Coherent-Osc-s}
}
\end{center}
\end{figure}
\subsection{Detection possibility in experiments}
As has so far been seen in this section,
the $s$-dark matter has the lifetime much longer than the age of the universe
and has extremely tiny couplings to the SM particles,
and hence the detection at collider experiments is unlikely to be possible.
As in the case of invisible axion-like dark matter detection~\cite{SikivieDetection},
cosmic pseudoscalar $s$, left over from the big bang,
may be detected by microwave cavity haloscopes.
In that facility, a strong static magnetic field is provided
to make the $s$ drift through the microwave cavity, resonantly converted to
microwave photons according to the $s$-photon-photon interaction in Eq.(\ref{int:s}).
The conversion power $P$ is given by~\cite{SikivieDetection}
\begin{equation}
P = \frac{1}{8} g_{s\gamma\gamma}^2 \rho_s(T_0)B_0^2 L_x V
\,,
\end{equation}
where $\rho_s(T^0)$ is the local $s$ energy density, $B_0$ the magnetic strength scale,
$V$ the volume of the cavity and $L_x$ the size of the $x$ direction.
Taking a typical experimental setup currently employed~\cite{Bradley:2003kg},
$B_0=10\, {\rm Tesla}, $ $L_x=1\, {\rm m}$, $V=1\,{\rm m}^3$
and the local halo density
$\rho_{\rm halo}\simeq0.3\, {\rm GeV}/{\rm cm}^3$,
we estimate the detection power
\begin{equation}
P/N_{\rm HC} \simeq 10^{-34} \,{\rm Watt} \times
\left( \frac{1 \,{\rm TeV}}{\Lambda_{\rm HC}} \right)^2
\left( \frac{m_s}{10^{-4} \,{\rm eV}} \right)^2
\left( \frac{\rho_s(T_0)}{\rho_{\rm halo}} \right)
\,, \label{power:s}
\end{equation}
where we have used Eq.(\ref{gsgg}).
The power for $m_s \sim 1$ eV
is comparable with the axion detection potential~\cite{Bradley:2003kg}, so
the $s$ can be hunted at the same level of the sensitivity
as the axion by the microwave cavity experiments.
\section{Summary and Discussion}
\label{summary}
In this paper,
we have employed a model based on
the classically-scale invariant standard model extended by adding
a strongly coupled hypercolor dynamics.
The dynamical breaking of the scale symmetry is triggered
by the vector-like condensation at the TeV scale,
so that the standard model Higgs acquires the negative mass-squared by the bosonic seesaw mechanism
to realize the electroweak symmetry breaking.
What is significant to control this model is to
include an elementary pseudoscalar $S$, which plays the crucial role
to realize the electroweak bosonic seesaw, as well as
to give masses for the composite Nambu-Goldstone bosons (hypercolor pions):
in this sense, the $S$ acts like another ``Higgs'' in the theory.
Thus, discovering the fluctuating mode of $S$, called $s$, is the smoking-gun of the present
model.
Because of the classical-scale invariance, the pseudoscalar $S$ originally
couples only to the standard model Higgs and hypercolor fermions.
After the dynamical-scale breaking and triggering the electroweak bosonic seesaw,
the $s$ thus develops vanishingly small couplings to the standard model particles, which arise only
through the tiny mixing with the hypercolor eta-prime.
In addition, it turned out that in relation to the hypercolor pion masses
the $s$ mass is predicted to be very light and to predominantly decay to diphoton,
so we have identified the $s$ as a dark matter candidate.
The $s$ mass was then severely
constrained by several cosmological observations, such as line emissions
of X-ray, gamma-ray and
cosmic-ray, no evidence for dark radiations, and a typical Higgs inflation scenario.
The $s$ mass was thus bounded to be $10^{-4} {\rm eV} \lesssim m_s \lesssim 1 {\rm eV}$.
We examined the possibility of the cosmological productions of the $s$.
It was shown that the $s$ is unlikely to be thermally produced essentially
due to its tiny couplings to the hypercolor sector in the thermal equilibrium.
We then found that the sufficient amount of relic abundance of $s$ as the cold dark matter
can be accumulated via the coherent oscillation.
The detection potential in microwave cavity experiments was also addressed
so that the $s$ with mass around 1 eV can have the same level of the
detection sensibility as that of the axion in the currently equipped experimental
setup, so the $s$ can be hunted by the microwave cavity experiments.
\\
Several comments are in order:
\\
The crucial deference between the $s$-dark matter and the axion-like dark matter
can be seen by no evidence for observations probing couplings to
matter, such as the test of gravitational inverse-square law and energy loss in stars
like neutron star cooling.
The $s$ coupling to matters can be generated at loop levels
by the $g_S$ and $\kappa_H$ couplings.
As seen from Eqs.(\ref{stationary:condi}) and (\ref{masses}), however,
those couplings are extremely small, suppressed by $(m_s/\Lambda) \ll 1$ or
$(v_1/v_s) \ll 1$ (See also Eq.(\ref{gs-val})).
Hence one can conclude that there is no chance to detect the $s$-dark matter
through the couplings to matters, in contrast to the axion case.
Thus, no evidence for observations with the matter-portal, but
some signals identical among the $s$ and the axion
in the line shapes and microwave cavity experiments
would be a clear hint to distinguish them.
(Note that a dilaton-like dark matter signal in the microwave cavity
is clearly different from that of the $s$ and the axion, due to
the different type of the coupling to photons: $E\cdot B$ for pseudoscalars, while
$E\cdot E$ or $B\cdot B$ for scalars.)
As discussed in Secs.~\ref{s-DM} and \ref{prod-s},
we have assumed that
the reheating epoch is associated with the Higgs inflation scenario.
It might be the case, however, that
one needs somewhat large
non-minimal couplings
between the SM Higgs and the scalar curvature for
the reheating temperature in the Higgs inflation scenario.
In that case, the reheating epoch would be shifted,
so the upper bound on the mass of $s$, as estimated in Eq.~(\ref{tr}),
could be affected.
Detailed study closely connected with inflation scenarios
is to be performed in the future literature.
The predicted number in Eq.(\ref{power:s}) depends on
the $s$ mass, so it does also on the hypercolor pion mass through Eq.(\ref{s-mass-val}).
It should be noted, however, that the light pseudoscalar $s$ as a candidate of
the dark matter is intact even if the hypercolor pion mass is not set to the present reference value,
since it is solely tied with realization of the electroweak breaking via the
bosonic seesaw: the mass has to be much smaller than $\Lambda_{\rm HC}$,
which is controlled by the small coupling $g_S(\ll 1)$ in Eq.(\ref{masses});
the $s$ couplings to the standard model particles, photons, necessarily becomes tiny by the same
$g_S$ coupling strength as the consequence of the bosonic seesaw,
which would suggest to regard the $s$ as a dark matter candidate;
the $s$ mass is then
inevitably constrained by cosmological bounds, to be order of eV, as was discussed
in the text.
Actually, the couplings of the s-dark matter are required to be extremely small:
the coupling to hyperfermion, $g_S \sim 10^{-12}$, from Eq.~(\ref{s-mass-val}) for $m_s \sim 1 {\rm eV}$
(which is coincidentally as small as the Yukawa coupling for neutrino in Dirac neutrino models);
the quartic coupling $\lambda_S \sim 10^{-44}$ from Eq.~(\ref{masses}) with $v_S \sim 10^{13}$ GeV
estimated from Eq.~(\ref{gs-val}) with $m_s \sim 1 {\rm eV}$ and $g_S \sim 10^{-12}$;
the coupling to the 125 GeV Higgs, $\kappa_H \sim 10^{-22}$, estimated from Eq.~(\ref{masses}) with
$v_S \sim 10^{13}$ GeV. The origin of these extremely small couplings could be explained by
the underlying Planck scale physics, which is, however, beyond the scope of the present study,
to be pursued elsewhere. Note that the realization of the electroweak symmetry breaking has nothing
theoretically to do with the smallness of those coupling parameters, which are only related to
the physics of the light $s$ including the mass generation of hypercolor pions and the property as
the invisible dark matter.
Other signals characteristic to the present model involve
not only hypercolor pions, but also the hypercolor eta-prime and
hypercolor composite scalar states,
both of which are expected to have the mass on the order of $\Lambda_{\rm HC}$.
As briefly studied in Appendix~\ref{750}, the hypercolor eta-prime can be produced
at the LHC, via the photon - photon fusion process as well as the hypercolor pions.
The discovery channels will be similar to the hypercolor pions: $WW, ZZ, Z\gamma$ and
$\gamma\gamma$ modes. Since the production cross section decreases as the
resonance mass grows, the photon - photon fusion cross section for the
hypercolor eta-prime significantly gets smaller than that of the hypercolor pions,
so it may be challenging to search at the LHC (For explicit estimates for
the signal strengths, see Appendix~\ref{750}).
As to the hypercolor composite scalars, the couplings to the standard model particles
are controlled by the tiny Yukawa coupling $y(\ll 1)$ through
the mixing with the standard model Higgs. It would be worth investigating how
much large the $y$ coupling is allowed to be consistent with the
currently reported heavy Higgs search data, and to discuss the
LHC discovery potential. Such those topics are deserved to the future
study.
In closing, in the present work we have so far focused on the possibility
for the predicted light pseudoscalar $s$ to be a dark matter candidate.
Actually, another scenario can be made: with the $s$ mass around GeV scale
the $s$ could be just a long-lived particle having the lifetime much
shorter than the age of the present universe.
That sort of a light long-lived particle could be
accessible at the LHC. This interesting another
possibility will be pursued in another publication.
\acknowledgments
This work was supported in part by
the JSPS Grant-in-Aid for Young Scientists (B) \#15K17645 (S.M.)
and Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists
\#26$\cdot$2428 (Y.Y.).
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Court remands Babachir Lawal in EFCC custody
A Federal High Court in Abuja on Tuesday ordered that former Secretary to Government of the Federation (SGF), Mr Babachir Lawal, be remanded in EFCC custody.
The sacked SGF was alleged to have conspired with some staff of Rolavision, his company, to fraudulently acquire a property, thereby committing an offence contrary to section 26(1)(c) of the Corrupt Practices and Other Related Offences Act, 2000 and punishable under section 12 of the same Act.
When the charges were read to him, he pleaded not guilty while his counsel applied for bail.
Lawal is to be kept in EFCC custody, pending the ruling of the court on his bail application on Feb. 13.
Those arraigned alongside the former SGF are Hamidu Lawal, a director of Rholavision Engineering Limited; an employee of the company, Sulaiman Abubakar and the Managing Director of Josmon Technologies Limited, Apeh Monday.
They, however, pleaded not guilty to the charges when read to them in the open court.
Consequently, the prosecution counsel, Mohammed Abubakar, asked for a date for commencement of the trial.
Counsel for the former SGF, Chief Akin Olujinmi (SAN), informed the court that he filed a bail application on behalf of the first defendant, adding that the prosecution refused to be served the application.
The senior lawyer who moved an oral application for bail, said that bail is at the discretion of the court in line with Sections 158 and 165 of the Administration of Criminal Justice Act (ACJA), 2015, and urged the court to grant Lawal bail in self-recognition.
According to him, the first defendant had served Nigeria meritorious at a high level as SGF, an office he held for over two years before the Senate raised the issue that brought him before the court.
He added that EFCC invited Lawal and detained him on April 11, 2018 and granted him administrative bail on April 13, 2018.
He informed the court that while on administrative bail, the former SGF had cause to travel on health ground and informed EFCC about it, adding that, "EFCC released his international passport for him to travel. He returned the passport to them on his return to the country.
"If he had wanted to escape, he could have done so while he travelled out of the country."
Olujinmi submitted that Lawal did not have any reason to interfere with EFCC's investigation as the commission had completed its investigation into the matter.
Similarly, counsel to the other defendants drew the court attention to their bail applications and urged the court to release their clients on bail.
Responding, the prosecution counsel, Abubakar, stated that he was not aware that any EFCC officials refused to accept service of the first defendant's bail application.
He stated further that there was no affidavit before the court, showing that the application was refused to be accepted, adding that he needed time to respond to the oral application by Olujinmi.
After listening to arguments for and against the bail applications, Justice Jude Okeke stated that the court was inclined to giving considered ruling on the applications and as such adjourned till today to deliver ruling.
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Nigerian lawmakers, police, judges most corrupt – Transparency Int'l
Transparency International has released a survey showing that the police, legislature and the judiciary are among the most corrupt institutions in Nigeria ...
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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{"url":"https:\/\/github.com\/molgenis\/molgenis_apps-legacy\/blob\/testing\/doc\/compute\/01_compute_introduction.md","text":"{{ message }}\nThis repository has been archived by the owner. It is now read-only.\n\n## molgenis \/ molgenis_apps-legacy Public archive\n\nSwitch branches\/tags\nNothing to show\n\nCannot retrieve contributors at this time\n469 lines (316 sloc) 27 KB\n\n% Manual Molgenis\/Compute % Genomics Coordination Center % December 16, 2012\n\n#Compute framework overview\n\nMOLGENIS compute (or MOLGENIS Compute) is 'just enough' to rapidly generate analysis workflows that can run locally, on parallel compute clusters and the grid.\n\n\u2022 Users can use build on their standard expertise in (shell) scripts\n\u2022 Users can rapidly share their workflows to accross Linux servers\n\u2022 Users can easily view the scripts generated for provenance and debugging.\n\u2022 Users can customize MOLGENIS Compute to fit their local practices\n\nTo use MOLGENIS Compute, you need the following.\n\n\u2022 workflow.csv: a file that describes steps to be executed in order.\n\u2022 protocols: a directory in which each file is a script template describing a step.\n\u2022 parameters.csv: a file which defines all protocol parameters and default values.\n\u2022 worksheet.csv: a file which contains the run specific values of parameters.\n\nBelow use of MOLGENIS Compute is described:\n\n\u2022 Workflow creation explains the basic use of Molgenis Compute in a basic workflow.\n\u2022 Workflow execution describes how to execute this workflow.\n\u2022 Compute advanced features extends that workflow commonly used features. All sections assume that you run MOLGENIS Compute from the command line.\n\u2022 Workflow deployment details best practices with binaries and files\n\u2022 Compute database shows how one can run MOLGENIS Compute from a database instead of commandline. Molgenis Compute db makes use of pilot jobs which enable 'self-scheduled' execution on grid (such as BigGrid).\n\nThe manual is in the form of a walkthrough manual.\n\nmkdir mycompute\ncd mycompute\n# Check www.molgenis.org\/wiki\/ComputeStart for the latest and greatest version!\n\nunzip molgenis_compute-<version>.zip\n\nmv molgenis_compute-<version>\/* .\n\n#test\nsh molgenis_compute.sh\n\n###Checkout out the latest Compute Alternatively to download you can checkout the latest code:\n\nmkdir mycompute\ncd mycompute\ngit clone https:\/\/github.com\/molgenis\/molgenis_apps-legacy.git\ngit clone https:\/\/github.com\/molgenis\/molgenis-legacy.git\nant -f molgenis_apps-legacy\/build_compute.xml clean-generate-compile-makedistro\nunzip molgenis_apps-legacy\/dist\/molgenis_compute-.zip\nmv molgenis_compute-\n\/* .\n\n#test\nsh molgenis_compute.sh\n\n### Software needed\n\nMolgenis\/compute is known to work with the following software:\n\n\u2022 java 1.6.0 or higher\n\u2022 git 1.7.1 or higher (only if you build yourself)\n\u2022 ant 1.7.1 or higher (only if you build yourself)\n\u2022 mysql 5.1.54 or higher (only for the database version)\n\n##Workflow creation\n\nThis section explains how one can use MOLGENIS Compute to create a workflow of shell scripts using the example of printing invitations for a party. The next section Workflow execution discusses how to execute. Below is explained how one can create a workflow, a protocol, and a parameters list.\n\n###Workflow The most simple workflow contains only one step. Let's call this step GuestInvitationStep. Now create a file called workflow.csv with the following content as follows:\n\n#create new workflow dir\nmkdir invitationWorkflow\nnano invitationWorkflow\/workflow.csv\n\nPaste:\n\n name, protocol_name, PreviousSteps_name GuestInvitationStep, GuestInvitation,\n\nExplanation: The first row contains the column headers, each of the following rows describes a step in the workflow. In the first column you'll find the step's name (unique per workflow), followed, in the second column, by the protocol_name that here referers to a file protocol\/GuestInvitation.ftl (see Protocols for details). The third column contains a comma separated list of step names that should be finished before the step in a row starts (see Previous steps). In our current example, the list in the third column is empty.\n\n###Protocols A protocol generally is a template of a shell script written in Freemarker language that describes the work is to be done (see http:\/\/freemarker.org\/ for a manual).\n\nLet's now create a directory called protocols and save our first protocol file protocols\/GuestInvitation.ftl in there, with the following content.\n\nnano invitationWorkflow\/protocols\/GuestInvitation.ftl\n\nPaste:\n\necho \"Hello ${guest},\" echo \"We invite you for our${party}.\"\n\nExplanation: A protocol may refer to parameters, such as ${guest} and ${party} in our example. The idea is that you can use such templates to rapidly generate up to thousands of shell scripts, given different values for these parameters. Given a value for each of the parameters, this protocol echos an invitation to the standard out.\n\n###Parameters Each parameter that is used in a protocol should be defined in a parameters file. Let's create such a file and call it parameters.csv. Now add our two parameters ${guest} and ${party} as follows to this file.\n\nnano invitationWorkflow\/parameters.csv:\n\nPaste:\n\n name, defaultValue, description, dataType, hasOne_name guest, , , , party, wedding, , ,\n\nExplanation: There are two types of parameters. First, parameters may be constants, like our parameter party which has a default value \"wedding\". So, in each of the generated scripts, the value of the parameter party will be \"wedding\". If you want to send the same invitation, but for a different party, you only have to change this value in one place. Second, parameters may be variables and have a different value in each of the scripts that are generated from a given protocol. The next section will explain how these parameters, like guest in our case, and their different values are defined in a worksheet.\n\n##Workflow execution To run a workflow you can use worksheet.csv to provide the specific parameter values for all your analysis steps. Subsequently you can generate all the scripts and run them locally, on a compute cluster or on the grid.\n\n###Worksheet While global values can be provided using defaultValue in parameters.csv, the specific values need to change everytime you run the workflow. Let's now create a worksheet containing these values for our party invitation workflow and save it as worksheet.csv:\n\nnano invitationWorkflow\/worksheet.csv\n\nPaste:\n\nExplanation: The first row contains parameter names (c.q. target types), comma separated. In our case we only have one parameter, called guest. Each of the following rows contains the parameters values. When running \\mc, the protocols are subsequently applied to each of the values. I.e., in our example, we will generate a different invitation script for each of our guests.\n\nNote: There is a list of variables names that are reserved and therefore cannot be used in any of the files presented. These words are: inputDir, outputDir, workflow, parameters, template, worksheet, id.\n\n###Script generation You need no command line parameters if your files (workflow.csv, worksheet.csv, parameters.csv) and (protocol, template) directories, is in the directory from which you run molgenis_compute.sh. Otherwise, you may want to specify the -inputdir that refers to the directory in which you have stored your workflow.csv, protocol directory,parameters.csv and worksheet.csv. Alternatively, you may specify (or overwrite) each of these files individually by -workflow, -protocols, -parameters, -worksheet. The second parameter (id) refers to the name you give your analysis run. This will automatically create a directory with the same name where all scripts will be generated. Alternatively, you may explicitly specificy -scripts, where the parameter scripts refers to the directory where MOLGENIS Compute will store the generated scripts.\n\nLet's now generate the scripts with the invitations by running the following command. We assume that you have put your workflow, parameters, worksheet files and protocols directory in a directory called helloWorld.\n\ncd ..\nsh molgenis_compute.sh \\\n-inputdir=invitationWorkflow\n\nTip: In your protocols, you may want to use the values of the command line parameters. However, be aware that the parameter names you have to use are slightly different from the command line parameters: ${McWorkflow},${McProtocols}, ${McParameters}, ${McWorksheet}, and ${McScripts}. ###Review generated scripts So, how do the generated scripts in the run01 directory look like? Let's first consider one of the five scripts that contain the invitations to our five guests: run01\/run01_s00_GuestInvitation_1.sh. The script name is constructed as follows. \u2022 run01: the id that you used when you ran \\mc. \u2022 s00: the step number in your workflow, starting from zero. \u2022 GuestInvitation: the corresponding workflow step name. \u2022 1: the line number in the worksheet. The step number, step name and line number are separated by underscores. Let's open run01\/run01_s00_GuestInvitation_1.sh and view its content: echo \"Hello Charly,\" echo \"We invite you for our wedding.\" Now let's open the second script, run01_s00_GuestInvitation_2.sh: echo \"Hello Cindy,\" echo \"We invite you for our wedding.\" These scripts correspond to the GuestInvitation protocol, where ${party} got the constant value \"wedding\", and ${guest} got a different value each time, as defined in the worksheet. Because \"Charly\" is the first value in the worksheet, she ends up in the first script. Correspondingly, because \"Cindy\" is the second value in the worksheet, she ends up in the second script. And so on. ###Script execution Next to the analysis scripts, three submit scripts are generated to allow you to run the workflow locally, on a PBS cluster, or using some custom compute grid. \u2022 runlocal.sh executes the analysis scripts sequentially (ideal for testing) \u2022 submit.sh submits the analysis scripts to a PBS scheduler \u2022 submit.sh.ftl is a script that is based on the template submit.sh.ftl which you can find in the templates directory. You can customize the way the analysis scripts are submitted by customizing this protocol. %todo add grid to commandline? %todo example output for runlocal? You can use these scripts to submit and start the execution of your analysis scripts in the right order. Section Workflow step dependencies below will explain how you can define the order between the steps in your workflow. After executing the analysis scripts, two invitations will be echo'ed to the standard out. In addition, a copy of your workflow, parameters and worksheet file are put in the scripts directory, as well. ##Compute advanced features This section adds more details to the \"inviteWorkflow\" we've developed above and demonstrates that MOLGENIS Compute can generate more realistic workflows, too. In addition of only inviting guests to our wedding, we will also organize some activity for our guests. The guests will be divided in two groups: child or adult. Each group has one organizer that will plan an activity for his group. After sending out the individual invitations to our guests, for each group, we will send its organizer a letter with a guest list. ###Workflow step dependencies Let's call the step that sends a letter to each organizer OrganizerInvitation. Suppose that before starting this step, we want the GuestInvitation step to be finished first. Let's add the new step OrganizerInvitationStep to our workflow.csv file and define its dependency on the GuestInvitationStep step using the column PreviousSteps_name: nano invitationWorkflow\/workflow.csv: Paste: name, protocol_name, PreviousSteps_name GuestInvitationStep, GuestInvitation, OrganizerInvitationStep, OrganizerInvitation, GuestInvitationStep Adding GuestInvitationStep to its PreviousSteps_name will ensure that the GuestInvitation scripts will be finished before the OrganizerInvitation scripts will be started. Be aware that the values in the third column refer to those in the first column, and not to those in the second column. ###Customized'foreach' iteration Default all protocols are applied to all rows in the worksheet.csv. Often you don't want that and instead only run a particular protocol the unique values in one worksheet column. In this example we want the OrganizerInvitation to run only for each unique group. Therefore, let's create a new protocol and save it as OrganizerInvitation.ftl in the protocols directory and add a #foreach clause next to the script template content. nano invitationWorkflow\/protocols\/OrganizationInvitation.ftl Paste: #FOREACH group echo \"Dear${organizer},\"\necho \"Please organize activities for the ${group} group.\" echo \"List of guests:\" <#list guest as g> echo \"${g}\"\n<\/#list>\n\nIn this new protocol, we introduce a new parameter group which may have the values \"child\" and \"adult\". We will specify these values in the worksheet.csv below. Instead of applying this template to all rows in the worksheet, the #FOREACH group statement in the first line of this protocol means that this protocol will be applied only to each unique values that group has. I.e., it will be applied once to \"child\", and once to \"adult\".\nWhat happens under the hood, is that the worksheet is folded based on the specified target. The folding reduces the worksheet to only two lines, one for each group. This will thus result in a list of guests per group. This protocol iterates through that list of guests by making use of the <#list> Freemarker syntax. Section Worksheet folding explains the folding of the worksheet as a result of the #FOREACH group statement, in detail. That section also explains why the parameter organizer, which is also new in this protocol, can also be used as a value, insteadof a list.\n\n###Worksheet folding Let's take the worksheet from section Worksheet and add a group (\"child\" or \"adult\") to each guest. Let's also add an organizer for each group and update the worksheet.csv file as follows.\n\nnano invitationWorkflow\/worksheet.csv\n\nPaste:\n\nIn general, a protocol is applied to each of the rows in the original worksheet. However, if a protocol contains a #FOREACH statement, then the worksheet will first be folded. After the folding, the protocol will be applied to each line in the folded worksheet. Because the OrganizerInvitation.ftl protocol starts with \"#FOREACH group\", it will be executed for each unique value of group (i.e. \"child\" and \"adult\"). Under the hood, after folding each line contains a different value of group.\n\nResult of 'worksheet.csv' after folding:\n\nFor each group, you'll get a list of guests which are indicated with the brackets [ and ]. So, folding on a certain target, results in lists of the other targets. However, although we do see a list of guests, we also see a list of organizers, while there should be only one. To solve this we will update the 'parameters.csv' and relate the 'organizer' one-on-one to 'group' using Parameter hasOne relationship.\n\n###Parameter hasOne relationship So, we need to define the new parameters group and organizer. Moreover, we want to specify that each group can only have one organizer:\n\nnano invitationWorkflow\/parameters.csv\n\nPaste:\n\n name, defaultValue, description, dataType, hasOne_name guest, , , , party, wedding, , , organizer, , , , group, , , , organizer\n\nNote that we have added relationships between group and organizer in the hasOne_name column: a group has only one organizer. If we now re-apply the folding as describing in Worksheet folding there result is now as desired:\n\nOutput of 'worksheet.csv' after folding:\n\n##Workflow deployment The following subsections explain features used during deployment. Typically, during deployment you have to parameterize you have to\n\n\u2022 Specificy requirements: to ensure there are enough cpu and memory resources\n\u2022 Get input files: to ensure the input files are loaded before your analysis started\n\u2022 Put output files: the output files are posted back to central storage\n\nOptionally you may also need to:\n\n\u2022 Load binaries: ensure that binaries for your analysis (and their dependencies) are available\n\u2022 Customized headers: to make the generated scripts fit your cluster system\n\nAll of this is described below.\n\n###Requirements specification The header of a protocol may contain the following line in which you specify the hardware requirements for your workflow step:\n\n#MOLGENIS walltime=hh:mm:ss mem=$m$ nodes=$n$ cores=$c$\n\nwhere walltime is the maximum execution time, mem is the memory (e.g. 512MB or 4GB), nodes is the number of nodes (default=1) and cores is the number of cores that you request for the execution of this analysis.\n\n###Get input files A protocol may be executed in a distributed environment. As a result, the data may not be available on the node where the execution takes place. Therefore, one should first the data to the execution node. In some distributed environments this may involve a series of statements that one actually does not want to care about. To make this process easier for our users, we come with the following solution. For every file that you want to use in the analysis protocol, you may include the following statement in the protocol before using it.\n\ngetFile \"${myInputFile}\" Where \"myInputFile\" is a parameter in your parameter list that refers to the file. The getFile command will then take care of putting your data in the right place. ###Put output files After finishing the analysis, you may save the files you want to keep by including the following statement at the end of your protocol.\\ putFile \"${myOutputFile}\"\n\nWhere \"myOutputFile\" again is a parameter in your parameter list that refers to the respective file. The putFile command takes care of all the work needed to store your data in the right place.\n\n###Load software modules In your protocols typically will want to use of some software tools that are already installed on the backend where your scripts will be run. However, the path to these tools may vary between different backends. One solution to this is to put the tools in the '$PATH$', so that you can just call them without specifying the path. We recommend as best practice to use the Module system. On two backends, i.e. 'cluster.gcc.rug.nl' and 'grid.sara.nl', we made it quite easy for you to do so. The following statement will load a tool, say ${yourModule}, to the path. module load${yourModule}\n\nOn the two backends, the following modules are available so far:\n\n\u2022 bwa\/0.5.8c_patched\n\u2022 capturing_kits\/SureSelect_All_Exon_30MB_V2\n\u2022 capturing_kits\/SureSelect_All_Exon_50MB\n\u2022 capturing_kits\/SureSelect_All_Exon_G3362\n\u2022 fastqc\/v0.7.0\n\u2022 fastqc\/v0.10.1\n\u2022 gtool\/v0.7.5_x86_64\n\u2022 impute\/v2.2.2_x86_64_static\n\u2022 jdk\/1.6.0_33\n\u2022 picard-tools\/1.61\n\u2022 Python\/2.7.3\n\u2022 R\/2.14.2\n\nA protocol that wants use plink, for example, may look like this:\n\nplink --noweb --bfile $WORKDIR\/lspilot1\/GvNL_good_samples.out4 \\ --het --out$WORKDIR\/lspilot1\/GvNL_good_samples.out7\n\n###Customize headers and footers In the templates directory of compute you may edit two files Header.ftl and Footer.ftl. The contents of this files will be respectively prepended and appended to each of your protocols when generating a workflow. Alternatively, you can change the templates directory using option -templates=myTemplatesDir.\n\n##Deployment of the database version of compute\n\nThis part of the tutorial explains who to deploy the Database version of the Molgenis\/compute and submit jobs to the grid (glite-wms) system using the \"pilot\" job approach.\n\n###Requirements\n\nMolgenis\/compute can be deployed and ready to submit jobs to the grid scheduler via ssh just in few straightforward steps. We prepared a shell scripts to automate every deployment and utilise step. The scripts are can be found in Molgenis github:\n\nhttps:\/\/github.com\/molgenis\/molgenis_apps\/tree\/testing\/modules\/compute4\/deployment\/\n\n###Compute database creation\n\nThe \"compute\" database should be created in the MySQL server (although it can also work on a embedded server). Run the following commands for this:\n\nCREATE USER 'molgenis' IDENTIFIED BY 'molgenis';\nCREATE DATABASE compute;\nGRANT ALL PRIVILEGES ON compute.* TO 'molgenis'@'%' WITH GRANT OPTION;\nFLUSH PRIVILEGES;\n\n###Start the server\n\nNow, the compute project is built and you can start the web-server and run the DB version of compute with running the following command:\n\ncd molgenis_apps\nsed -i 's\/validate\/update\/g' build\/classes\/META-INF\/persistence.xml\nnohup ant -f build_compute.xml runOn -Dport=8080 &\n\nIn the nohup.out, you should see output like:\n\n*********************************************************\nAPPLICATION IS RUNNING AT: http:\/\/localhost:8080\/compute\/\n*********************************************************\n\nLater, you can copy the link into your browser and you will see generated user interface with empty database, like in Figure below. In this example, the DB contains two workflows.\n\nAlternatively, you can use sh restart.sh script for it. Run the script specifying the port on which you like to run the web server\n\nsh restart.sh 8080 #change port if needed\n\n###Workflow import and execution task generation\n\nYou can use the sh importWorkflow.sh script to import a workflow into a database. Run it with few parameters:\n\nsh importWorkflow.sh \\\n<workflow_parameters_file> \\\n<workflow_elements_file> \\\n<protocols_directory>\n\nAlternatively, parameters, protocols and workflow elements can be added to the database manually in the mysql server or through the generated UI.\n\nRun the sh importWorksheet.sh script with the following parameters to generate ComputeTasksin the database.\n\nsh importWorksheet.sh \\\n<workflow_name> \\\n<worksheet_file> \\\n<run_id>\n\nwhere worksheet_file is the worksheet file with the targets, workflow_name is the workflow name in the database for which you would like to generate tasks and run_id is the unique generation run id.\n\n###Execution on the grid with the pilot framework\n\nWe use the pilot approach to run ComputeTasks. For this, the pilot files should be present at the execution environment. In case of the grid, you need to copy three \"pilot\" files to the grid UI node to \\$HOME\/maverick\/ directory. These files are:\n\n\u2022 maverick.sh: actual pilot job, that calls back to the database and ask for available for execution ComputeTask\n\u2022 maverick.jdl: jdl file used for submission to the glite grid service (used only for the grid)\n\u2022 dataTransferSRM.sh: script to support data transfer in the grid (used only for the grid)\n\nThe files can be found at:\n\nhttps:\/\/github.com\/molgenis\/molgenis\\_apps\/tree\/testing\/modules\/compute\/pilots\/grid\/\n\nThe maverick.sh should be edited accordingly to your execution setting. You need to specify back_end, where you like to submit you ComputeTask for execution. back_end can have a value e.g. ui.grid.sara.nl. Also, you need to specify your_ip and your_port of your web-server, where Molgenis\/compute is running.\n\nAlso, you may edit the maverick.jdl to specify the walltime and computational sites where you like to run you analysis. The example jdl requirements look like:\n\nRequirements = (\n(other.GlueCEInfoHostName == \"ce.lsg.psy.vu.nl\" ||\nother.GlueCEInfoHostName == \"ce.lsg.hubrecht.eu\")\n&& other.GlueCEPolicyMaxCPUTime >= 1440);\n\nBesides this, the ip address and port on which Molgenis\/compute is running should be specified in the database. You can use importComputeServer.sh for this or manually add the server into the ComputeServer table of the database, where the ComputeServer.name should be specified as \"default\". The script can used with two parameters:\n\nsh importComputeServer.sh \\\n<port> \\\n\nAfter putting these files in the UI node and adding ComputeServer to the database, \\emph{ComputeTasks} can be submitted with the command-line with the sh runPilots.sh:\n\nsh 5_runPilots.sh \\\n<backend> \\\n<backend_type>\n\nHere, back_end also can value 'ui.grid.sara.nl'. username and password are user grid credentials. backend_type, in the grid case, should have a value \"grid\". It also can have a value \"cluster\", that means that the PBS, SGE or BSUB scheduler is used.\n\nThere are following statuses of ComputeTasks in the compute database:\n\n\u2022 generated: means that the task is generated\n\u2022 ready: means that the task is ready for execution (all previous ComputeTasks are finished)\n\u2022 running: means that the task is running in the current moment\n\u2022 done: means that the task is finished\n\u2022 error means that the task is finished with the error (hart-beat is not received in time)\n\nDuring execution, the (output\/error) logs of the ComputeTasks will be placed back in the compute database and job statuses should be done or error. The frequency of the database update can be changed in the maverick.sh file.\n\nTry it out!","date":"2022-05-26 11:29:33","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\": 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\": 0.45277950167655945, \"perplexity\": 3392.2696991935627}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662604794.68\/warc\/CC-MAIN-20220526100301-20220526130301-00353.warc.gz\"}"}
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Over the past few years, Insight Communications has had to put some network projects on hold as it wound through its break-up with 50-50 partner Comcast and the possible sale of the company. But with the Comcast situation behind it and the sale on hold at least for now, Insight is steaming ahead with an all-digital initiative that should free up channels for high-definition television and eventually super-high-speed Internet service using the DOCSIS 3.0 platform.
But first, Insight will begin with digital simulcasting and laying the groundwork to take its network all-digital sometime in the future. according to CEO Michael Willner, the company is working on a simulcast strategy now.
Willner said that the sales process — since aborted — held Insight back on digital plans because the company didn't want to commit to a strategy that a new owner would have to unwind or change course.
Insight also had other issues on its plate: it was completing the split from Comcast and was finishing the roll out of voice-over-Internet protocol phone technology across all its markets and was finishing up the migration of its high-speed Internet service from the AT&T network onto its own network.
The latter move was key, Willner said, adding that five years ago Insight had the lowest high-speed data penetration in the industry; today it has the second highest.
Another milestone was the ten-fold expansion of its Network Operations Center in Louisville in 2007.
Executive vice president of central operations and chief technology officer Hamid Heidary said the increased sophistication of the network meant that Insight needed a way to monitor more closely the critical components of that network from a central location.
The NOC and Insight's fiber-loop architecture played a significant role when Insight's largest market, Louisville, Ky., was hit by Hurricane Ike earlier this year. High winds from the storm socked the city, felling trees and communications lines and leaving about 300,000 homes without power for days.
"Basically we were the only operator that did not even blink in provision of services, other than the areas where our drop lines to the home were cut off by a tree on it, but the core delivery of the service itself was up and running through the seven-and-a-half days of no power in the city," Heidary said.
On the all-digital front, Willner said that he believes that most operations will be basically all-digital (some basic video channels will remain analog) within two to three years.
Going all-digital will free up additional channels for HDTV (Insight will have 59 high-definition channels by the end of the year) and will also give the MSO the ability to introduce DOCSIS 3.0, the platform that allows cable operators to offer speeds of 100 Megabits per second to 150 Mbps. Heidary said that Insight probably won't offer full-blown DOCSIS 3.0 right off the bat, instead offering a 50 Mbps version that customers can grow into.
Heidary offered no timeframe for DOCSIS 3.0, but added that it would depend on how many analog channels are eventually freed up.
For more Executive of the Year coverage, click here.
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{
"redpajama_set_name": "RedPajamaC4"
}
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Cream together the butter, 1/2 cup peanut butter and sugars until creamy.
Add the eggs and vanilla. Mix until combined.
In a separate bowl mix together the flour, baking soda, and salt.
Add to the wet ingredients.
Add chocolate chips and stir well.
Place half the cookie dough to a skillet, spread remaining peanut butter to the cookie dough.
Top with the rest of the cookie dough.
Serve warm with your favorite ice cream!
*Tip: Place peanut butter in the microwave for 30 seconds-1minute to make it easier to spread!
|
{
"redpajama_set_name": "RedPajamaC4"
}
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Our work has been seen in numerous publications including The Mirror, The Telegraph, The Times, The Guardian and The Metro, to name but a few. Our work has also been featured on Sky News, BBC One and ITV. MKL design were instrumental in conceptualising and designing our branding and website.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,825
|
Hekataios () bezeichnet folgende Personen:
Hekataios von Milet, ein griechischer Historiograph (ca. 560 v. Chr. bis ca. 480 v. Chr.)
Hekataios von Kardia (4. Jahrhundert v. Chr.), Tyrann von Kardia
Hekataios (Gefährte Alexanders) (4. Jahrhundert v. Chr.), Mörder des Attalos
Hekataios von Abdera oder aus Teos, ein griechischer Philosoph und Historiograph (um 300 v. Chr.), ihm zugeschriebene Pseudoepigraphien:
Pseudo-Hekataios I, etwa 100 v. Chr. entstanden
Pseudo-Hekataios II, vor 100 n. Chr. entstanden
Hekataios von Thasos, Epigrammatiker
Hekataios (Toreut), antiker griechischer Toreut
Siehe auch:
Hecataeus (Mondkrater)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,181
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The is the official tax collecting agency of Japan. As of October 2018, the Commissioner of NTA is Takeshi Fujii.
Mission
Mission: To enable taxpayers to properly and smoothly fulfill their tax responsibility.
To achieve the mission stated above, National Tax Agency is responsible for fulfilling the responsibilities stipulated in Article 19 of the Law to Establish the Ministry of Finance, while paying due consideration to transparency and efficiency.
Achieving proper and fair taxation and collection
To creating favorable environment for taxpayers:
NTA shall provide correct and easy-to-understand information on legal interpretation and administrative procedures for filing tax returns and paying taxes.
NTA shall quickly and accurately handle inquiries from taxpayers.
NTA shall endeavor to call on other ministries and citizens from all parts of society for their cooperation and participation in order to improve public understanding and support regarding the role of tax and tax administration.
Achieving proper and fair taxation
To achieve proper and fair taxation:
NTA shall ensure that laws and regulations are applied properly.
NTA shall endeavor to have taxpayers file accurate tax returns and shall correct mistakes in tax returns by conducting tax examinations of and providing guidance for those taxpayers whose returns are deemed to be inaccurate.
NTA shall endeavor to have taxpayers submit their tax payments by the deadline and shall surely collect the taxes of those who have not paid by the deadline by implementing delinquency procedures.
To protect the rights and interests of taxpayers:
NTA shall properly and promptly respond to requests for reinvestigation or requests for reconsideration.
Achieving sound development in the alcoholic beverage industry
NTA shall endeavor to promote the sound management of the alcoholic beverage industry, promote research and development on brewing technologies, and maintain the quality and safety of alcoholic beverages.
NTA shall endeavor to ensure effective use of resources for alcoholic beverages.
to ensure the proper activity of certified public tax accountants:
NTA shall endeavor to ensure that certified public tax accountants properly engage in their duties and play their vital role in implementing the self-assessment system properly and smoothly in accordance with their responsibilities.
Creation
On June 1, 1949, NTA was established as an affiliated agency of the Ministry of Finance, responsible for assessing and collecting national taxes. Until then, this role was performed by Tax Bureau of Ministry of Finance. Immediately after the Second World War, tax administration did not function properly as the Tax Bureau had undertaken out-of-scope operations. This organizational reform established NTA's current three-tiered organizational structure consisting of the head office, regional taxation bureaus and tax offices.
NTA consists of the headquarters office, 11 regional taxation bureaus (Sapporo, Sendai, Kanto-Shin-Etsu, Tokyo, Kanazawa, Nagoya, Osaka, Hiroshima, Takamatsu, Fukuoka and Kumamoto), Okinawa Regional Taxation Office and 524 tax offices throughout the country.
NTA's headquarters office has four departments: Commissioner's Secretariat; Taxation Department; Revenue Management and Collection Department; and Examination and Criminal Investigation Department.
Examination and Criminal Investigation Department at regional taxation bureau level examines large-scale corporation's corporate and consumption taxes and investigates tax evasion cases.
The local tax office is a front-line administrative body in charge of assessing and collecting domestic taxes and has the closest contacts with taxpayers. Tax office is authorized to assess and collect domestic taxes within its responsible area. Its organization structure varies, depending on tax office's scale. Generally, a tax office has 1 division and 4 groups: co-ordination division, revenue management and collection group, examination group (individual), examination group (property tax), and examination group (corporation).
Budget and personnel
As of the end of FY2003, the number of employees stands at 56,315.
As of FY2003, the total budget for tax collection operating costs stands at 721.9 billion yen.
The cost to collect 100 yen of tax and stamp duty revenues (return on collection) is 1.78 yen as of FY2003, while it was 2.79 yen in FY1950.
Voluntary reporting system
Japan relies on the self-assessment along with withholding tax system with respect to specific income and blue return systems. The due date for payment of 2011 income tax is Thursday, March 15, 2012.
Self-assessment
In the self-assessment taxation approach, taxpayers calculate their taxable income, file tax returns and pay their taxes due. This taxation system basically presupposes taxpayer's willingness to pay taxes, continue accurate bookkeeping behavior and calculate their taxable income based on objective data. This approach was adopted, in 1947, under the strong influence of the United States taxation system.
Blue return system
Proper operation of the self-assessment taxation system requires regular and accurate bookkeeping by taxpayers. To foster the practice of bookkeeping, the blue return system was introduced in 1950 as a part of comprehensive tax reform based on the recommendations of Dr. Carl Shoup.
Under the blue return system, an individual who operates a business or a corporation may obtain the district director's approval to file a tax return using a special form printed on blue paper. While these taxpayers are required to maintain books and keep continuous accounting records in accordance with prescribed standards, they are entitled by law to various income calculation benefits and preferential treatments.
Today, the blue return system is widely accepted. There are 4.95 million blue return personal taxpayers, and more than 2.6 million corporations are filing blue returns.
Mutual agreement procedures
During the FY 2010, the NTA received 157 MAP cases, of which were on BAPA. The United States and Canada are treaty partners for MAP case purposes.
The Pacific Association of Tax Administrators (PATA) issued internal operational guidance in respect of mutual agreement procedures (MAP) and bilateral advance pricing arrangements (BAPA). PATA is composed of representatives from the tax administrations of Australia, Canada, Japan, and the United States.
International coordination
On June 21, 2012, the Financial Services Agency (FSA), the Ministry of Finance (MOF) and the National Tax Agency (NTA) and the United States authority (the Treasury Department) have released a joint statement regarding a framework for intergovernmental cooperation to facilitate the implementation of FATCA and improve international tax compliance.
References
External links
NTA homepage
NTA homepage
Comprehensive Handbook of Japanese Taxes 2010
A discussion of internal guidance to improve the Mutual Agreement Procedure and Bilateral Advance Pricing Arrangement processes
An Outline of Japanese Tax Administration 2003
National Tax Collection Act (Extract) Act No. 147 of 1959; Translation date: October 26, 2009
Income Tax Act (Limited to the provisions related to nonresidents and foreign corporations) Act No. 33 of 1965; Translation date: April 1, 2009
Corporation Tax Act (Limited to the provisions related to foreign corporations) Act No. 34 of 1965; Translation date: April 1, 2009
Taxation in Japan
Revenue services
Financial regulatory authorities of Japan
Government agencies established in 1949
1949 establishments in Japan
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,417
|
from __future__ import unicode_literals
from collections import defaultdict
from django import template
from drum.links.utils import order_by_score, timesince
from drum.links.views import CommentList, USER_PROFILE_RELATED_NAME
register = template.Library()
@register.filter
def get_profile(user):
"""
Returns the profile object associated with the given user.
"""
return getattr(user, USER_PROFILE_RELATED_NAME)
@register.simple_tag(takes_context=True)
def order_comments_by_score_for(context, link):
"""
Preloads threaded comments in the same way Mezzanine initially does,
but here we order them by score.
"""
comments = defaultdict(list)
qs = link.comments.visible().select_related(
"user",
"user__%s" % (USER_PROFILE_RELATED_NAME)
)
for comment in order_by_score(qs, CommentList.score_fields, "submit_date"):
comments[comment.replied_to_id].append(comment)
context["all_comments"] = comments
return ""
@register.filter
def short_timesince(date):
return timesince(date).split(",")[0]
@register.filter
def sort_by_desc(items, attr):
"""
General sort filter - sorts by either attribute or key.
"""
def key_func(item):
try:
return getattr(item, attr)
except AttributeError:
try:
return item[attr]
except TypeError:
getattr(item, attr) # Reraise AttributeError
return sorted(items, key=key_func, reverse=True)
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,990
|
Liziranje (od stgrč. λύσις lýsis, "gubljenje" od λύειν lýein, "odvezati") odnosi se na raspadanje stanične membrane. Izvodi se virusnim, enzimskim ili osmoznim mehanizmima koji ugrožavaju njenu cjelovitost. Tekućina koja sadrži sastojke liziranih stanica naziva se lizat. U biokemijskim laboratorijima, laboratorijima molekularne i stanične biologije, stanične kulture podlaže se procesu liziranja radi pročišćavanja njihovih sastojaka, kao u purifikaciji bjelančevina, ekstrahiranju DNK, ekstrahiranju RNK ili u pročišćavanju (purifikaciji) staničnih tjelešaca.
Izvori
Biokemija
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 6,993
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Jesy Nelson leaves Little Mix in emotional message to fans
She took to instagram to make the announcement
Laura HartleySocial Media Editor
Jesy Nelson has announced she is leaving girlband Little Mix after nine years.
Her bandmates broke the news on Twitter, and the post was followed by a statement from Jesy, on her own social media.
The news follows an announcement in November that Jesy would be taking an "extended" break for private medical reasons.
In an emotional instagram post, Jesy has said the pressure of being in a girlband and living up to expectations is 'very hard'.
Her open and honest post said: "The truth is recently being in the band has really taken a toll on my mental health.
"I find the constant pressure of being in a girl group and living up to expectations very hard.
"There comes a time in life when we need to reinvest in taking care of ourselves rather than focusing on making other people happy, and I feel like now is the time to begin that process.
"So after much consideration and with a heavy heart, I'm announcing I'm leaving Little Mix."
See her full post below.
The band – Jade Thirlwall, Leigh-Anne Pinnock and Perrie Edwards – announced the news in a string of tweets.
They said: "After an amazing 9 years together Jesy has made the decision to leave Little Mix. This is an incredibly sad time for all of us but we are fully supportive of Jesy."
The band, who found fame on The X Factor, said: "We love her very much and agree that it is so important that she does what is right for her mental health and well-being.
"We are still very much enjoying our Little Mix journey and the 3 of us are not ready for it to be over.
"We know that Jesy leaving the group is going to be really upsetting news for our fans. We love you guys so much and are so grateful for your loyalty and continued support of all of us.
"We're looking forward to seeing so many of you on tour. Love always, Jade, Perrie and Leigh-Anne xxx."
The band had recently released their sixth studio album titled Confetti, their first since departing Simon Cowell's label Syco.
Little Mix formed on The X Factor in 2011 and have become one of the country's biggest-selling girl bands.
They auditioned on the ITV singing show as soloists, but were then placed into a group by the judges in the show's later stages.
They left Syco Music for the RCA label in November 2018.
Cowell later launched another version of The X Factor, called The Band, to rival the girl group's The Search.
LEGO invites you to view A Really Cool World in London
Primark's 'beautiful' new coat shoppers are lusting over may explain the long queues
Nelson has previously spoken openly about her battle with anxiety and previously revealed in a BBC documentary called Odd One Out that abuse on social media got so bad she attempted to take her own life.
The programme won her the award for best factual entertainment at the 2020 National Television Awards.
The band are scheduled to embark on a UK tour next year.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,213
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\section{Introduction}
Optimal Uncertainty Quantification (OUQ) \cite{OSSMO:2011} provides a framework
for the computation of optimal bounds on quantities of interest -given a set of available information
and specified assumptions. Although this framework is neither frequentist nor Bayesian,
in that it is simply expressed in terms of optimization over measures and functions, a natural question arises; what happens when we introduce priors into OUQ?
In Owhadi et al.~\cite{BayesOUQ}, this program was initiated through the introduction of a further set of assumptions, namely, the
assumptions regarding the prior on the specified assumption set.
A corresponding reduction theory for optimization problems over measures on spaces of measures is established, facilitating
the computation of optimal bounds on prior and posterior values
and the analysis of the consequences of conditioning on observed data.
However, the completion of this program reveals Brittleness theorems \cite[Thm.~4.13, Thm.~6.4, Thm.~6.9]{BayesOUQ}
for Bayesian Inference -mild assumptions are sufficient to demonstrate that, given a set of priors, conditioning on observations
can produce arbitrary results, regardless of the sample size.
Although it is known from the results of Diaconis and Freedman that the Bayesian method may fail to converge or may converge towards the wrong solution (i.e., be inconsistent) if the underlying probability mechanism allows an infinite number of possible outcomes \cite{DiaconisFreedman:1986} and that in these non-finite-probability-space situations, this lack of convergence (commonly referred to as \emph{Bayesian inconsistency}) is the rule rather than the exception \cite{DiaconisFreedman:1998}, it is also known, from the Bernstein-Von Mises Theorem \cite{Bernstein:1964, vonMises:1964} (see also LeCam \cite{LeCam:1953}), that consistency (convergence upon observation of sample data) does indeed hold, under some regularity conditions, if the data-generating distribution of the sample data belongs to the finite
dimensional family of distributions parameterized by the model. Furthermore, although it is also known that this convergence may fail under model misspecification \cite{White:1982, Grunwald:2004, Grunwald:2006, Christophe:2002, Christophe:2008, Kleijn:2012, Lian:2009, Gustafson:2001} (i.e. when the data-generating distribution does not belong to the
family of distributions parameterized by the model), it is natural to wonder whether a ``close enough'' model has good convergence properties: see e.g.~\cite{Draper:1995, Samaniego:2010, Draper:2013} and in particular G.\ E.\ P.\ Box's question \cite[p.~74]{Box:1987} ``Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful?''
The Brittleness theorems \cite[Thm.~4.13, Thm.~6.4, Thm.~6.9]{BayesOUQ} suggest that there
may be no such thing as a ``close enough'' model if Box's question is answered
in the classical framework of Bayesian sensitivity analysis (where given the data and a class of priors one computes optimal bounds on posterior values); indeed, if ``closeness'' is defined (i) as sharing an arbitrarily large finite number of finite-dimensional marginals or
(ii) using the Prokhorov or total variation metrics, then the posterior values of such ``close'' models may be as distant as possible after conditioning on an arbitrarily large number of sample data.
The primary motivation for this paper is to unwrap the mechanism causing this brittleness by providing (i) a quantitative version of the Brittleness Theorem \cite[Thm.~4.13]{BayesOUQ} and (ii) a detailed and comprehensive analysis of its application to the informative example from \cite[Ex.~4.16]{BayesOUQ} of estimating the mean of a random variable on the unit interval using priors that exactly capture the distribution of an arbitrary large number of Hausdorff moments.
In this example, the probability distribution $\mu^\dagger$ of $X$ is an unknown element of the set of all possible probability distributions on $[0,1]$, i.e.
$\mu^\dagger \in \mathcal{A}:=\mathcal{M}([0,1])$. The set of prior probability distributions $\pi$ on $\mu \in \mathcal{A}$ (i.e. $\pi \in \mathcal{M}(\mathcal{A})$) is defined as the set of priors $\pi$ under which the vector of truncated Hausdorff moments $(\mathbb{E}_\mu[X],\ldots,\mathbb{E}_\mu[X^n])$ is uniformly distributed on the truncated Hausdorff moment set
$M^n\subset \mathbb{R}^n$ defined as the set of $q=(q_1,\ldots,q_n)\in \mathbb{R}^n$ such that there exists a probability measure $\mu$ on $[0,1]$ with $\mathbb{E}_\mu[X^i]=q_i$ for $i\in \{1,\ldots,n\}$ ($M^n$ is the polytope of $\mathbb{R}^n$ corresponding to the set of possible values for the first $n$ moments of a measure of probability on the interval $[0,1]$)).
In this case, the computation of optimal bounds on posterior values leads naturally to the calculation of the Lebesgue volume of certain
subsets of the set $M^{n}$ of truncated Hausdorff moments.
Curiously, whereas Karlin and Shapley \cite{KarlinShapley} used Selberg's integral formula to first compute
the volume of the truncated Hausdorff moment space $M_n$, inadvertantly stimulating the development of the theory of the Selberg integral formulas\footnote{In discussing the history and importance of the Selberg integral formulas, Forrester and Waardan
\cite[Pg.~3]{ForresterWarnaar} mention their first application: ``For over thirty years
the Selberg integral went essentially unnoticed. It was used only once—-in the
special case $ \alpha = \beta = 1, \gamma = 2$—-in a study by S. Karlin and L.S. Shapley relating
to the volume of a certain moment space, published in 1953.''}, it appears that computing the volume of
the truncated
Hausdorff moment space $M^n$ using different finite-dimensional representations of $M^n$ in the infinite-dimensional space $\mathcal{M}\big([0,1]\big)$
reveals a new family of Selberg integral formulas (see Theorems \ref{thm_selberg}, \ref{thm_selberg2}, \ref{thm_selberg_explicit} and Corollary \ref{cor_selberg2}).
This process of discovery appears to be generic and we will now describe its main principles.
Let $\Psi$ be the function mapping each measure $\mu \in \mathcal{M}\big([0,1]\big)$ into its first $n$ moments
\begin{equation}
\Psi(\mu):=\big(\mathbb{E}_{X\sim \mu}[X], \mathbb{E}_{X\sim \mu}[X^2],\ldots,\mathbb{E}_{X\sim \mu}[X^n]\big).
\end{equation}
Note that
\begin{equation}
M^n:=\Psi\Big(\mathcal{M}\big([0,1]\big)\Big).
\end{equation}
The classical and new Selberg identities are obtained by computing the volume of $M^n$ using different finite dimensional representations in $\mathcal{M}\big([0,1]\big)$. These finite dimensional representations are obtained by restricting $\Psi$ to convex sums of Diracs, i.e. to measures $\mu \in \mathcal{M}\big([0,1]\big)$ of the form
\begin{equation}\label{eq:musumdiracs}
\mu=\sum_{j=1}^N \lambda_j \delta_{t_j}
\end{equation}
where $0\leq t_1<\cdots<t_N\leq 1$ and $\lambda_1,\ldots,\lambda_N>0$ with $\sum_{j=1}^N \lambda_j=1$. Note that if $\mu$ is of the form \eqref{eq:musumdiracs}, then $\Psi(\mu)=(q_1,\ldots,q_n)$ with $q_i=\sum_{j=1}^N \lambda_j t_j^i$.
For each measure $\mu$ of the form \eqref{eq:musumdiracs}, we define $i(\mu)$, the index of $\mu$, as the number of support points (Diracs) of $\mu$, counting interior points with weight $1$ and boundary points with weight $1/2$. We call $\mu$ ``principal'' if $i(\mu)=(n+1)/2$, ``canonical'' if $i(\mu)=(n+2)/2$, ``upper'' if support points include $1$, ``lower'' if support points do not include $1$.
Then Theorem \ref{thm_principal} asserts that each $q \in Int(M^{n})$ has a unique
upper and lower principal representation. Since the volume of $M_n$ is independent of the representation used to compute it, computing that volume with a lower and an upper representation leads to an equality corresponding to classical Selberg identities.
Now let $t_{*}\in (0,1)$. Theorem \ref{thm_canonical} asserts that every point in the interior of $M^{n}$ has a unique canonical representation
whose support contains $t_{*}$, and when $t_{*}=0$ or $1$, then there exists a unique principal representation whose support contains $t_{*}$.
Since the volume of $M_n$, and the higher order moments of the uniform measure restricted to $M_n$,
are independent of the representation used to compute them, computing these "moment moments"
for all possible values of $t^*$ leads to a family of equalities corresponding to new Selberg integral formulas and Reproducing Kernel Hilbert Spaces.
Consequently, the free parameter $t_*$ associated with Markov and Kre\u{\i}n's canonical representations of truncated Hausdorff moments (see Section \ref{sec_IntegralGeometry}) which, along with their principal representations, so handily provides us with the means to prove the quantitative Brittleness Theorem \ref{thm_shiva_singlesample}, is found to generate reproducing kernel identities
corresponding to reproducing kernel Hilbert spaces of
polynomials (see sections \ref{sec_mean} and \ref{sec_RKHS}).
Furthermore, these reproducing identities lead to biorthogonal systems of
Selberg integral formulas described in Theorems \ref{thm_selberg}, \ref{thm_selberg2} and \ref{thm_selberg_explicit} (see also Corollary \ref{cor_selberg2}).
Moreover, although not done here, this process can easily be generalized in simple ways. For example,
the argument is valid using any measure on the moment space, not just the uniform measure, and so
the introduction of alternatives for which the integrals can likewise be computed, can be used.
In addition, it also appears possible that this process can be repeated with multiple free parameters
$t_{*,1},\ldots,t_{*,k}$ to obtain even richer classes of (new) Selberg integral formulas.
\section{OUQ with Priors}\label{sec:ouqwithpriors}
To understand OUQ one simply starts with \u{C}eby\u{s}ev
\cite[Pg.~4]{Krein}
`` Given: length, weight, position of the centroid and moment of inertia of a material rod with a density
varying from point to point. It is required to find the most accurate limits for the weight of a certain
segment of this rod.'' According to Kre\u{\i}n \cite{Krein}, although \u{C}eby\u{s}ev did solve this problem, it was his student
Markov
who supplied the proof in his thesis. See Kre\u{\i}n \cite{Krein} for an account of the
history of this subject
along with substantial contributions by Kre\u{\i}n.
We take this mindset and apply it to more complex problems, extending the base space to functions and measures,
and, instead of developing sophisticated mathematical solutions, develop optimization problems and reductions,
so that their solution may be implemented on a computer,
as in Bertsimas and Popescu's \cite{BertsimasPopescu:2005} convex optimization approach to
\u{C}eby\u{s}ev inequalities, and the Decision Analysis framework of Smith \cite{Smith}.
In addition to the determination of optimal bounds as a function of available information and assumptions, the OUQ methodology
has the substantial benefit of demanding that different components of an organization work together to
come up with information and assumptions that, together, they believe in.
Let us begin with a general formulation of OUQ with priors, where the base assumptions are sets of (function, measure) pairs and
the secondary assumptions are sets of
priors, that is, sets of probability measures defined on the base assumption set. Later, when we apply to Bayesian inference, we will restrict
to a base assumption set consisting of a set of measures and a secondary assumption consisting of a set
probability measures on the base assumption set.
To that end,
let $\mathcal{X}$ be a topological space, $\mathcal{M}(\mathcal{X})$ the space of Borel probability measures on $\mathcal{X}$, and let $\mathcal{G} \subseteq \mathcal{F}(\mathcal{X})$
be a subset of the real-valued measurable functions $\mathcal{F}
(\mathcal{X})$ on $\mathcal{X}$. Let $\mathcal{A}$ be an arbitrary subset of $\mathcal{G} \times \mathcal{M}(\mathcal{X})$, and let $\Phi \colon \mathcal{G} \times \mathcal{M}(\mathcal{X}) \to \mathbb{R}$
be a function producing a quantity of interest. In the context of uncertainty quantification one is interested in estimating $\Phi(f^\dagger,\mu^\dagger)$, where $(f^\dagger,\mu^\dagger)\in \mathcal{G} \times \mathcal{M}(\mathcal{X})$ corresponds to an \emph{unknown reality}. If $\mathcal{A}$ represents all that is known about $(f^\dagger,\mu^\dagger)$ (in the sense that $(f^\dagger,\mu^\dagger)\in \mathcal{A}$ and that any $(f, \mu)\in \mathcal{A}$ could, a priori, be $(f^\dagger,\mu^\dagger)$ given the available information) then \cite{OSSMO:2011} shows that the quantities
\begin{eqnarray}
\label{eq:defma1}
\mathcal{U}(\mathcal{A})&:=& \sup_{(f, \mu)\in \mathcal{A}} \Phi(f, \mu)\\
\label{eq:defma2}
\mathcal{L}(\mathcal{A})&:=& \inf_{(f, \mu)\in \mathcal{A}} \Phi(f, \mu)
\end{eqnarray}
determine the inequality
\begin{equation}
\label{ineq_OUQ}
\mathcal{L}(\mathcal{A}) \leq \Phi(f^\dagger,\mu^\dagger) \leq \mathcal{U}(\mathcal{A})
\end{equation}
to be optimal with respect to the available information (i.e.~$(f^\dagger,\mu^\dagger) \in \mathcal{A}$) as follows: First, it is simple to see that the inequality \eqref{ineq_OUQ} follows from
$(f^\dagger,\mu^\dagger) \in \mathcal{A}$.
Moreover, for any $\varepsilon >0$ there exists a $(f,\mu) \in \mathcal{A}$ such that
\[ \mathcal{U}(\mathcal{A})-\varepsilon < \Phi(f,\mu) \leq \mathcal{U}(\mathcal{A}).\] Consequently since all that we know about
$(f^\dagger,\mu^\dagger)$ is that $(f^\dagger,\mu^\dagger) \in \mathcal{A}$, it follows that the upper bound
$\Phi(f^\dagger,\mu^\dagger) \leq \mathcal{U}(\mathcal{A})$ is the best obtainable given that information. The lower bound is clearly optimal in the same sense.
A classical example of a quantity of interest is the
validation and certification quantity $\Phi(f,\mu):=\mu[f\geq a]$ where $a$ is a safety margin. In the certification context one is interested in showing that
$\mu^\dagger[f^{\dagger}\geq a]\leq \epsilon$ where $\epsilon$ is a safety certification threshold (\emph{i.e.}\ the maximum acceptable $\mu^\dagger$-probability of the system $f^{\dagger}$ exceeding the safety margin $a$). If $\mathcal{U}(\mathcal{A}) \leq \epsilon$, then the system associated with $(f^\dagger,\mu^\dagger)$ is safe (given the information represented by $\mathcal{A}$). If $\mathcal{L}(\mathcal{A}) > \epsilon$, then the system associated with $(f^\dagger,\mu^\dagger)$ is unsafe. If $\mathcal{L}(\mathcal{A}) \leq \epsilon < \mathcal{U}(\mathcal{A})$, then the safety of the system cannot be decided without making further assumptions or gathering further information.
Although the OUQ optimization problems \eqref{eq:defma1} and \eqref{eq:defma2} are extremely large, we have shown in \cite{OSSMO:2011} that an important subclass enjoys significant and practical finite-dimensional reduction properties. In particular, for assumption sets corresponding to
linear inequality constraints on generalized moments, the search can be reduced to one over probability measures that are products of finite convex combinations of Dirac masses with explicit upper bounds on the number of Dirac masses.
To incorporate priors,
we define a prior $\pi$ to be
a probability measure $\pi \in \mathcal{M}(\mathcal{A})$, and define the value
$\bar{\Phi}(\pi)$ of $\pi$ through the extended quantity of interest $\bar{\Phi}:\mathcal{M}(\mathcal{A}) \rightarrow \mathbb{R}$
defined by
\begin{equation*}
\label{def_interest_can} \bar{\Phi}(\pi):=\mathbb{E}_{\pi}[\Phi], \quad \pi \in \mathcal{M}(\mathcal{A}).
\end{equation*}
We will defer the nontrivial and not uninteresting topics of measurability to when we analyze
the full OUQ with priors framework, but note that
Ressel \cite{Ressel_some} has established important and relevant results for us already, in particular the measurability of
the validation and certification quantity of interest discussed above under mild conditions.
We call the value $\mathbb{E}_{\pi}[\Phi]$ the
{\em prior value}, and for a family of priors $\Pi \subset \mathcal{M}(\mathcal{A})$
we note that
the values
\begin{eqnarray}
\label{eq:UPi}
\mathcal{U}(\Pi)& := & \sup_{\pi \in \Pi} \mathbb{E}_{\pi} \big[\Phi\big]\\
\label{eq:LPi}
\mathcal{L}(\Pi)& := &\inf_{\pi \in \Pi} \mathbb{E}_{\pi} \big[\Phi \big]
\end{eqnarray}
form a natural generalization of the notations
$\mathcal{U}(\mathcal{A})$ and $\mathcal{L}(\mathcal{A})$.
Moreover, in the same way that $\mathcal{U}(\mathcal{A})$ and $\mathcal{L}(\mathcal{A})$ are optimal
upper and lower bounds on $\Phi(f^\dagger,\mu^\dagger)$ given the information that
$(f^\dagger,\mu^\dagger) \in \mathcal{A}$, $\mathcal{U}(\Pi)$ and $\mathcal{L}(\Pi)$ are optimal upper and lower bounds on $\mathbb{E}_{\pi}\big[\Phi\big]$ given the information that $\pi \in \Pi$.
For conditioning on sample data in an observation space $\mathcal{D}$,
we begin by defining a {\em data map}
\[\mathbb{D}:\mathcal{A}\rightarrow \mathcal{M}(\mathcal{D})\]
which specifies that $\mathbb{D}(f,\mu)\in \mathcal{M}(\mathcal{D})$ generates the data
when the truth is $(f,\mu) \in \mathcal{A}$. Then, given a
prior
$\pi \in \mathcal{M}(\mathcal{A})$, we define a probability measure
\[\pi\odot \mathbb{D} \in
\mathcal{M}\bigl(\mathcal{A} \times \mathcal{D}\bigr)\]
through
\begin{equation*}\label{eq:palm}
\pi\odot \mathbb{D} \big[ A \times B \big] = \mathbb{E}_{(f.\mu) \sim \pi} \big[ \mathbbm{1}_{A}(f, \mu) \mathbb{D}(f, \mu)[B] \big],\quad A \in \mathcal{B}(\mathcal{A}),\, B \in \mathcal{B}(\mathcal{D})\, ,
\end{equation*}
where $\mathbbm{1}_{A}$ is the indicator function of the set $A$:
\[
\mathbbm{1}_{A}(f,\mu) :=
\begin{cases}
1, & \text{ $(f,\mu) \in A$,} \\
0, & \text{ $(f,\mu) \notin A$.}
\end{cases}
\]
We denote the resulting $\mathcal{D}$-marginal by $ \pi\cdot\mathbb{D} \in \mathcal{M}(\mathcal{D})$ which satisfies
\begin{equation*}
\label{eq:cdotexp2}
\pi\cdot\mathbb{D}[B]:=\mathbb{E}_{(f,\mu)\sim \pi}\big[\mathbb{D}(f,\mu)[B]\big]\, .
\end{equation*}
Given an observation $d \in \mathcal{D}$, to simultaneously avoid the ill-definedness of regular conditional probabilities and incorporate uncertainty in the observation process, we consider conditioning
on an open subset $B$ containing $d$ such that $\pi\cdot \mathbb{D}[B]>0$. The naturality of this positivity condition is fully discussed in \cite{BayesOUQ}, in particular it is easy to show
that if $B$ is an open ball of center $\delta$ around the data $d$ (noted $B_\delta$) and if the data is randomized and distributed according to $\pi\cdot \mathbb{D}$, then the probability of the event $\pi\cdot \mathbb{D}[B_\delta]>0$ is one.
It is also shown in \cite{BayesOUQ} that if the probability of the data is uniformly bounded, in the Bayesian model class $\mathcal{A}$, from above and below by that of a reference measure (e.g., for all $(f,\mu)\in \mathcal{A}$, $\frac{1}{\alpha }\mathbb{D}(f_0,\mu_0) \leq \mathbb{D}(f,\mu)[B_\delta]\leq \alpha \mathbb{D}(f_0,\mu_0)$ for some reference measure $\mathbb{D}(f_0,\mu_0)$) then learning and robustness appear as antagonistic properties:
when $\alpha=1$, the data is equiprobable under all measures in the model class, posterior values are equal to prior values, the method is robust but learning is not possible, and as $\alpha$ deviates from, learning becomes possible (posterior values depend on the data) but the method becomes increasing brittle (the range of posterior values converges towards that of the quantity of interest $\Phi$).
To simplify notation, we henceforth drop the notational dependence
of the set $B$ on the point $d$.
The conditional expectation, given a prior $\pi$ and data map $\mathbb{D}$, conditioned on a subset $ B \in \mathcal{B}(\mathcal{D})$ such that $\pi\cdot \mathbb{D}[B]>0$, is
\begin{equation*}\label{eq:defcondexp}
\mathbb{E}_{\pi \odot \mathbb{D}}\big[ \Phi\big| B\big]=\frac{\mathbb{E}_{(f,\mu)\sim \pi}\big[ \Phi(f,\mu)
\mathbb{D}(f,\mu)[B] \big]}{\mathbb{E}_{(f,\mu)\sim\pi}\big[\mathbb{D}(f,\mu)[B]\big]}\, .
\end{equation*}
To represent uncertainty regarding the data generating process, instead of a single
data map $\mathbb{D}:\mathcal{A}\rightarrow \mathcal{M}(\mathcal{D})$, we instead
specify a
set \[\mathfrak{D} =\bigl\{\mathbb{D}:\mathcal{A}\rightarrow \mathcal{M}(\mathcal{D})\bigr\}\]
of data maps
and represent our assumptions regarding the data
with the statement
$\mathbb{D} \in \mathfrak{D}$.
Therefore, having specified a set $\Pi$ of priors, and a set $\mathfrak{D}$ of data maps,
for an open subset $B \subseteq \mathcal{D}$,
we define the set of all possible resulting product measures to be
\begin{equation*}
\label{def_B}
\Pi\odot_B\mathfrak{D} := \Bigl\{ \pi \odot \mathbb{D}: \pi\in \Pi,\, \mathbb{D} \in \mathfrak{D}, \, (\pi \cdot \mathbb{D})[B]> 0 \Bigr\}\, .
\end{equation*}
The notations $ \mathcal{U}(\Pi)$ and $ \mathcal{L}(\Pi)$ of \eqref{eq:UPi} and \eqref{eq:LPi}
extend naturally to
these conditional expectations as
\begin{equation*}\label{eq:Upostes}
\mathcal{U}(\Pi\odot_B\mathfrak{D}):=\sup_{\pi\odot \mathbb{D} \in \Pi\odot_B\mathfrak{D}}\mathbb{E}_{\pi \odot \mathbb{D}}\big[ \Phi\big| B\big]
\end{equation*}
\begin{equation*}\label{eq:Lpostes}
\mathcal{L}(\Pi\odot_B\mathfrak{D}):=\inf_{\pi\odot \mathbb{D} \in \Pi\odot_B\mathfrak{D}}\mathbb{E}_{\pi \odot \mathbb{D}}\big[ \Phi\big| B\big]\, ,
\end{equation*}
where we note that, just as for $\mathcal{U}(\mathcal{A})$ and $\mathcal{L}(\mathcal{A})$ and
$\mathcal{U}(\Pi)$ and $\mathcal{L}(\Pi)$,
$\mathcal{U}(\Pi\odot_{B}\mathfrak{D})$ and
$\mathcal{L}(\Pi\odot_{B}\mathfrak{D})$
are optimal upper and lower bounds on the posterior value $\mathbb{E}_{\pi\odot \mathbb{D}}\big[ \Phi\big| B\big]$, given
the assumptions that $\pi\in \Pi$, $\mathbb{D} \in \mathfrak{D}$, and $\pi \cdot \mathbb{D}(B)>0$.
We are now prepared to discuss the brittleness theorems of the next section.
Indeed, it is easy to see that
\begin{equation}\label{eq:upilpi}
\mathcal{L}(\mathcal{A})\leq \mathcal{L}(\Pi) \leq \mathcal{U}(\Pi)\leq \mathcal{U}(\mathcal{A})
\end{equation}
and
\[\mathcal{L}(\mathcal{A})\leq \mathcal{L}(\Pi\odot_B\mathfrak{D}) \leq \mathcal{U}(\Pi\odot_B\mathfrak{D}) \leq \mathcal{U}(\mathcal{A}) \, .\]
What the Brittleness Theorem \ref{thm_shiva} will show is that, under mild conditions, regardless of where
the values $\mathcal{L}(\Pi)$ and
$\mathcal{U}(\Pi)$ lie in \eqref{eq:upilpi}
we have
\[ \mathcal{L}(\Pi\odot_B\mathfrak{D}) \approx \mathcal{L}(\mathcal{A}) \text{ and }\mathcal{U}(\Pi\odot_B\mathfrak{D}) \approx \mathcal{U}(\mathcal{A})\,, \]
that is, conditioning on the observed data, one can obtain any value
between $\mathcal{L}(\mathcal{A})$ and $\mathcal{U}(\mathcal{A})$
for the posterior value
$ \mathbb{E}_{\pi \odot \mathbb{D}}\big[ \Phi\big| B\big]$ for some admissible prior $\pi \in\Pi$ and data map
$\mathbb{D} \in \mathfrak{D}$.
\section{Quantification of Bayesian Brittleness}
The following theorem is the Main Brittleness result of \cite[Thm.~4.13]{BayesOUQ}:
\begin{thm}
\label{thm:shiva0}
Let $\mathcal{A}$ be a Suslin space, let $\mathcal{Q}$ be a separable and metrizable space, and let $\Psi \colon \mathcal{A} \to \mathcal{Q}$ be measurable. Moreover, let $\mathfrak{Q} \subseteq \mathcal{M}(\mathcal{Q})$ be such that $\operatorname{supp}(\mathbb{Q}) \subseteq \Psi(\mathcal{A})$ for all $\mathbb{Q} \in \mathfrak{Q}$. Suppose that, for all $\delta >0$, there exists some $\mathbb{Q}\in \mathfrak{Q}$ such that
\begin{equation}
\label{eq:dto0}
\mathbb{E}_{q\sim \mathbb{Q}} \left[ \inf_{\mu\in \Psi^{-1}(q)} \mathbb{D}(\mu)[B] \right]=0
\end{equation}
and \begin{equation}
\label{eq:djkdjehjehj33}
\mathbb{P}_{q\sim \mathbb{Q}} \left[ \sup_{\mu\in \Psi^{-1}(q),\, \mathbb{D}(\mu)[B]>0}\Phi(\mu) > \sup_{\mu\in \mathcal{A}}\Phi(\mu) - \delta \right]>0 .
\end{equation}
Then
\begin{equation}
\label{eq:2qbisjhjycondddihjh}
\mathcal{U}\big(\Psi^{-1}\mathfrak{Q}\big|B\big) =\mathcal{U}(\mathcal{A}) .
\end{equation}
\end{thm}
The following generalization
of the Theorem \ref{thm:shiva0} (\cite[Thm.~4.13]{BayesOUQ}) allows a weakening of its assumptions while
approximately obtaining its conclusion.
We require, as in \cite{BayesOUQ}, the data space $\mathcal{D}$ to be metrizable. We
select a consistent metric, and for a
data point $d \in \mathcal{D}$,
let $B_{ \delta }(d)$ denote the open ball of metric radius $ \delta $ about $d$. To keep the notation simple we omit
reference to the base point $d$ and denote this family of open balls about $d$ by
$\{B_{ \delta }, \delta \geq 0\}$, where $B_{0}=\emptyset$.
\begin{thm}
\label{thm_shiva}
For a metrizable topological space $\mathcal{X}$, consider a topologized
subset $\mathcal{G}\subset\mathcal{F}(\mathcal{X})$ and the space of probability measures $\mathcal{M}(\mathcal{X})$ equipped with the weak star topology.
Let $\mathcal{A} \subset \mathcal{G}\times \mathcal{M}(\mathcal{X})$ be Suslin, $\mathcal{Q}$ separable metrizable, and
$\Psi:\mathcal{A} \rightarrow \mathcal{Q}$ Borel measurable. Moreover, let
$\mathfrak{Q} \subset \mathcal{M}(\mathcal{Q})$ be such that
$\operatorname{supp} \, \mathbb{Q} \subset \Psi(\mathcal{A}), \mathbb{Q} \in \mathfrak{Q}$, and let
$\tau \geq 0$. Suppose there exists some
$\mathbb{Q}\in \mathfrak{Q},\,\mathbb{D}\in \mathfrak{D}$ and a continuous monotonically increasing function
$h:\mathbb{R}^{+} \rightarrow \mathbb{R}$ with $h(0)=0$ such that
\begin{equation}
\label{eq:dto0app}
\mathbb{Q}\Bigl(\bigl\{q: \inf_{(f,\mu)\in \Psi^{-1}(q)} \mathbb{D}(f,\mu)[B_{ \delta }]\leq \tau\big\}\Bigr) \geq 1-h( \delta ), \quad \delta >0\, .
\end{equation}
Fix $ \delta >0$. If $\varepsilon \geq 0$, $\varepsilon'>0$ and $ \delta '>0$ are three real numbers such that
\begin{equation}\label{eq:djkdjehjehj33app}
\begin{split}
\mathbb{Q}\Bigl( \Bigl\{ q: \sup_{(f,\mu)\in \Psi^{-1}(q),\, \mathbb{D}(f,\mu)[B_{ \delta }]> \varepsilon}\Phi(f,\mu) >\sup_{(f,\mu)\in \mathcal{A}}\Phi(f,\mu) -\delta' \Bigr\}\Bigr)\geq \varepsilon'
\end{split}
\end{equation}
and
\begin{equation}
\label{eq_solve}
h( \delta )+\tau \leq \frac{\varepsilon \delta '\varepsilon'}{\mathcal{U}(\mathcal{A})-\mathcal{L}(\mathcal{A})}\, ,
\end{equation}
then we have
\begin{equation}
\label{eq:2qbisjhjycondddihjh}
\mathcal{U}(\mathcal{A}) -2 \delta ' \leq \mathcal{U}\big(\Psi^{-1}(\mathfrak{Q})\odot_{B_{ \delta }}\mathfrak{D}\big)
\leq \mathcal{U}(\mathcal{A})\, .
\end{equation}
\end{thm}
If, for $\tau=0$, there exists a $ \delta ^{*}>0$ such that
for all $ \delta '>0$ there exists some $\mathbb{Q}\in \mathfrak{Q},\,\mathbb{D}\in \mathfrak{D}$
which satisfies \eqref{eq:dto0app} with a function $h$ such that
$h( \delta )=0, \delta \leq \delta ^{*}$, and which satisfies
\eqref{eq:djkdjehjehj33app} with
$\varepsilon=0$,
then we recover the conditions and the assertion of the Brittleness Theorem \cite[Thm.~4.13]{BayesOUQ}
for $B_{ \delta }, \delta \leq \delta ^{*}$.
\begin{rmk}
The proof of Theorem \ref{thm_shiva} also leads to the following result. For a metrizable topological space $\mathcal{X}$, consider a topologized
subset $\mathcal{G}\subset\mathcal{F}(\mathcal{X})$ and the space of probability measures $\mathcal{M}(\mathcal{X})$ equipped with the weak star topology.
Let $\mathcal{A} \subset \mathcal{G}\times \mathcal{M}(\mathcal{X})$ be Suslin, $\mathcal{Q}$ separable metrizable, and
$\Psi:\mathcal{A} \rightarrow \mathcal{Q}$ Borel measurable. Moreover, let
$\mathfrak{Q} \subset \mathcal{M}(\mathcal{Q})$ be such that
$\operatorname{supp} \, \mathbb{Q} \subset \Psi(\mathcal{A}), \mathbb{Q} \in \mathfrak{Q}$. It holds true that for $\delta>0$
\begin{equation}
\label{eq:2qbisjhjycondddihjhtaustar}
\mathcal{U}(\mathcal{A}) - v( \delta ) \leq \mathcal{U}\big(\Psi^{-1}(\mathfrak{Q})\odot_{B_{ \delta }}\mathfrak{D}\big)
\leq \mathcal{U}(\mathcal{A})\, .
\end{equation}
where the function $v$ is defined by
\begin{equation}
\label{eq_solvetaustar}
\begin{split}
v( \delta ):=& 2 \inf \Bigg\{\delta'>0\Bigg| \delta'\geq \big(\mathcal{U}(\mathcal{A})-\mathcal{L}(\mathcal{A})\big) \inf_{\mathbb{Q}\in \mathfrak{Q},\,\mathbb{D}\in \mathfrak{D},\varepsilon> 0,\tau\geq 0 } \\&\frac{1-\mathbb{Q}\Bigl(\bigl\{q: \inf_{(f,\mu)\in \Psi^{-1}(q)} \mathbb{D}(f,\mu)[B_{ \delta }]\leq \tau\big\}\Bigr)+\tau}{\varepsilon \mathbb{Q}\Bigl( \Bigl\{ q: \sup_{(f,\mu)\in \Psi^{-1}(q),\, \mathbb{D}(f,\mu)[B_{ \delta }]> \varepsilon}\Phi(f,\mu) >\sup_{(f,\mu)\in \mathcal{A}}\Phi(f,\mu) -\delta' \Bigr\}\Bigr)}\Bigg\}\,
\end{split}\end{equation}
for $ \delta >0$.
\end{rmk}
\begin{rmk}
This brittleness is not a consequence of a lack of compactness of the admissible set. Indeed,
in the following section, the primary space of measures $\mathcal{M}(I)$ is compact in the weak topology, as is any closed moment subset, and Theorem \ref{thm_shiva_singlesample} describes a brittleness result.
\end{rmk}
\begin{rmk}
It is true that this brittleness does not appear to be primarily due to the Bayesian methodology, but is valid more generally.
See
Bahadur and Savage \cite{BahadurSavage} and Donoho \cite{Donoho} for similar results for statistical estimators, where it appears that the mechanism generating the instability is analogous to that investigated here.
\end{rmk}
\subsection{Application to a revealing example}
To demonstrate that the assumptions of Theorem \ref{thm_shiva} are mild, we now
use it to extend the Brittleness result of \cite[Ex.~4.16]{BayesOUQ} to a simple but informative example.
Here one is interested in estimating the mean of a random variable $X$ with unknown distribution on the unit interval $I:=[0,1]$.
Since our quantity of interest is $\mathbb{E}_{\mu^\dagger}[X]$, where $\mu^\dagger$ is an unknown distribution on $I$,
in the notations of Section \ref{sec:ouqwithpriors}, we have $\mathcal{X}:=I$ (since $X$ is a random variable on $I$), $\mathcal{G}$ consists only
of the identity function (this example does not involve unknown functions of $X$), $\mathcal{A}:= \mathcal{M}(I)$ (the set of possible/admissible candidates for $\mu^\dagger$ is the set of all probability distributions $\mu$ on $I$), $\Phi(\mu):=\mathbb{E}_{t \sim \mu}[t]$ (our quantity of interest is the mean of the random variable $X$), $\mathcal{Q}:=\mathbb{R}^{n}$ and the map
$\Psi:\mathcal{M}(I) \rightarrow \mathbb{R}^{n}$ is the map to the truncated Hausdorff moments
$\Psi(\mu) := \bigl(\mathbb{E}_{t \sim \mu}[t^{i}]\bigr)_{i=1,..,n}$ (our set of prior distributions is defined by constraining the distribution of the
first $n$ Hausdorff moments in $\mathbb{R}^n$, for some fixed $n$). Furthermore $\mathbb{Q}$ is the uniform
Borel measure on $\mathbb{R}^{n}$ restricted to the Hausdorff moment space $M^{n}:=\Psi(\mathcal{M}(I))$ and then normalized to be a probability measure, that is $\Pi \subset \mathcal{M}\bigl(\mathcal{M}(I)\bigr)$ is the set of prior distributions on $\mathcal{A}=\mathcal{M}(I)$ such that $\Psi \mu\in \mathcal{M}(M^{n})$ is uniformly distributed on the the space
$M^{n}$ of first $n$ Hausdorff moments.
The Brittleness Theorem \ref{thm_shiva} implies (see \cite[Ex.~4.16]{BayesOUQ}) that if we observe (condition on) $k$ independent samples from $X$, i.e. $\mathcal{D}:=I^k$ and $\mathbb{D}^{k}\mu:=\mu\otimes \cdots \otimes \mu$ ($k$-fold tensorization) and $B$ is the $k$-fold product of small enough balls centered on the data then $\mathcal{L}(\Pi\odot_B\mathfrak{D}) \approx \mathcal{L}(\mathcal{A})$ and $\mathcal{U}(\Pi\odot_B\mathfrak{D}) \approx \mathcal{U}(\mathcal{A})$. In other words, although the set of prior values of $\mathbb{E}_{\mu}[X]$ is the single point $\{\frac{1}{2}\}$, the optimal bounds on the posterior values of $\mathbb{E}_{\mu}[X]$ are zero and one irrespective of the number $n$ of constraints on marginals and the number $k$ of observed samples if the data is observed with sufficient precision.
The following theorem provides a rigorous and quantitative statement and proof of this implication for $k=1$. Although, for the sake of conciseness and clarity our analysis is provided in the $k=1$ case, it generalizes to the situation where $k$ is arbitrary. Indeed, although counterintuitive, one can show that brittleness for the single sample case is {\em more} difficult to
obtain
than for multiple samples. Since our main objective here is to unwrap and scrutinize the mechanism causing brittleness in Bayesian inference, we therefore chose to keep the presentation and our example as clear, concise, and simple as possible to illustrate the generic and pervasive nature of this brittleness.
Therefore, we will now (i) consider the case of a single data point, i.e., $k=1$, $\mathcal{D}:=I$, and $\mathbb{D}^{1}\mu:=\mu$ (ii) use Theorem \ref{thm_shiva} to provide quantitative bounds on $\mathcal{U}(\Pi\odot_B\mathfrak{D})$ as a function $n$ of the number of marginal constraints defining the set of priors (iii) scrutinize the brittleness causing mechanism through the proof of the following theorem.
\begin{thm}
\label{thm_shiva_singlesample}
Let $\mathcal{A}:= \mathcal{M}(I)$, $\Phi(\mu)=\mathbb{E}_{t\sim \mu}[t]$, $\mathcal{D}=I$, and
$\Psi:\mathcal{M}(I) \rightarrow \mathbb{R}^{n}$
denote the map to the truncated Hausdorff moments $\Psi(\mu)= \bigl(\mathbb{E}_{t \sim \mu}[t^{i}]\bigr)_{i=1,..,n}$.
Furthermore, let
$\mathbb{Q}$ denote the uniform
Borel measure on $\mathbb{R}^{n}$ restricted to the Hausdorff moment space $M^{n}:=\Psi(\mathcal{M}(I))$ and then normalized to be a probability measure.
Suppose that $\mathbb{Q} \in \mathfrak{Q}$ and $\mathbb{D}^{1} \in \mathfrak{D}$.
Then for $\delta>0$
we have
\begin{equation}
\label{shiva_single_sample_result}
1-4 e\big(\frac{2n \delta }{e}\big)^\frac{1}{2n+1} \leq \mathcal{U}\big(\Psi^{-1}(\mathfrak{Q})\odot_{B_{ \delta }}\mathfrak{D}\big) \leq 1
\end{equation}
\end{thm}
\begin{rmk}
Alternatively, Theorem \ref{thm_shiva_singlesample} asserts that for positive $ \delta , \delta '$ satisfying
\[ \delta \leq \frac{1}{4n} \bigl( \delta '\bigr)^{2n+1}\bigl(2e\bigr)^{-2n}\]
we have
\begin{equation}
\label{shiva_single_sample_resultold}
1-2 \delta ' \leq \mathcal{U}\big(\Psi^{-1}(\mathfrak{Q})\odot_{B_{ \delta }}\mathfrak{D}\big) \leq 1 \, .
\end{equation}
\end{rmk}
\section{Volume Inequalities on the Hausdorff Moment Space}
\label{sec_volumeinequalities}
Karlin and Shapley
\cite[Thm.~15.2]{KarlinShapley} (see also \cite[Thm.~6.2]{KarlinStudden:1966}) computed the volume
of the space of truncated Hausdorff moments $M^{n}$ of probability measures on the
unit interval to be
\begin{equation}
\label{karlinshapley}
Vol\bigl(M^{n}\bigr)=\prod_{k=1}^{n}{\frac{\Gamma(k)\Gamma(k)}{\Gamma(2k)}}\, ,
\end{equation}
where $\Gamma$ is the Gamma function. To accomplish this, they used a Markov representation of truncated moment points, as described in
Kre\u{\i}n \cite{Krein} (see also \cite[Ch.~II]{KarlinStudden:1966}), combined with the change of variables formula, followed by the evaluation of a Selberg integral.
Here we will refine their analysis to obtain
volume {\em inequalities} on the Hausdorff moment
space which are used in the application of the Brittleness Theorem \ref{thm_shiva}
to the proof of the Brittleness Theorem \ref{thm_shiva_singlesample}.
Of the two main results, it is interesting to note
that the Mass Supremum Equality uses the canonical representation of moment points combined with Markov's
Maximal Mass Theorem \cite[Thm.~2.1]{Krein} (see also \cite[Thm.~4.1]{KarlinStudden:1966}) to change the
``Inequality'' to ``Equality'', whereas
the
Mass Infimum Inequality instead uses the principal representation, as in Karlin and Shapley's proof of
the volume formula
\eqref{karlinshapley}. All this terminology will be defined in the following Section \ref{sec_IntegralGeometry} and comes from
Karlin and Studden \cite{KarlinStudden:1966}. This section will simply state the volume inequalities
that we need for Theorem
\ref{thm_shiva_singlesample}.
To proceed, let us now fix terminology.
Let $I:=[0,1]$, and let $\mathcal{P}(I)$ be the set of Borel measures on $I$ and
$\mathcal{M}(I)\subset \mathcal{P}(I)$ be the set of probability measures. Throughout we will assume the weak star topology for these measures. For the system of functions
\[ u_{i}(t):=t^{i}, t \in I, i=0,..,n\] the Hausdorff moments of a measure $\mu \in \mathcal{P}(I)$ is defined
as the vector $q \in \mathbb{R}^{n+1}$ with coordinates
$q_{i}=\mathbb{E}_{\mu}[u_{i}]=\mathbb{E}_{t \sim \mu}[t^{i}]$. It is well known (see e.g.~\cite[Cor.~15.7]{AliprantisBorder:2006}) that the map
\[\Psi:\mathcal{P}(I) \rightarrow \mathbb{R}^{n+1}\] defined by
$\Psi(\mu) := \bigl(\mathbb{E}_{t \sim \mu}[t^{i}], i=0,..,n\bigr)$ is affine and continuous.
Furthermore, let the Hausdorff moment space $\mathcal{M}^{n+1}\subset \mathbb{R}^{n+1}$ be the image $\mathcal{M}^{n+1}:=\Psi \mathcal{P}(I)$
of the measures, and let
$M^{n}$ defined by $\mathcal{M}^{n+1}=(1,M^{n})$ be moments of a probability measures
omitting the zero-th moment. Equivalently, let $P_{1}: \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ denote the projection mapping $(x_0,x_1,\ldots,x_n)$ onto $(x_1,\ldots,x_n)$ and let $\Psi_{1}=:P_{1}\Psi$. Then
$M^{n}=\Psi_{1}\mathcal{M}(I)$. We will abuse notation by letting $\Psi$ also
denote the mapping $\Psi_{1}$ restricted to the first-to-$n$-th order moments of the probability measures
\[\Psi:\mathcal{M}(I) \rightarrow M^{n} \subset \mathbb{R}^{n}\,,\]
and, for $q \in \mathbb{R}^{n}$, let
\[\Psi^{-1}q:=\bigl\{\mu \in \mathcal{M}(I): \Psi\mu=q\bigr\}\]
denote its set-valued inverse.
It follows from continuity that
the moment set $\mathcal{M}^{n+1}$ is a closed convex cone and $M^{n}$ is a compact convex set.
Moreover, one can show that
\begin{equation}
\label{interior}
Int(\mathcal{M}^{n+1}) \cap (1,\mathbb{R}^{n})=(1, Int(M^{n})\bigr),
\end{equation}
see e.g.~\cite[Cor.~6.5.1]{Rockafellar}, so that a point $q$ is interior to $M^{n}$ if and only
if $(1,q)$ is interior to $\mathcal{M}^{n+1}$.
Let $Vol$ be the usual $n$-dimensional volume measure. Then, since $M^{n}$ is convex, by \cite[Lem.~1.8.1] {Bogachev_Gauss}
\begin{equation}
\label{boundarynull}
Vol\bigl(Int(M^{n})\bigr)=Vol\bigl(M^{n}\bigr)\, .
\end{equation}
Our first result is the Mass Supremum Equality.
\begin{lem}
\label{lem_mass_sup}
Let $t_{*} \in I$, $ 0 \leq \varepsilon \leq 1$, and consider the set
$M^{n}_{\varepsilon} \subset M^{n}$ defined by
\[ M^{n}_{\varepsilon}:=\bigl\{q \in M^{n}: \exists \mu \in\Psi^{-1}q: \mu(\{t_{*}\})\geq \varepsilon.\bigr\}\]
Then we have
\[Vol\bigl(M^{n}_{\varepsilon}\bigr)=(1- \varepsilon)^{n}Vol\bigl(M^{n}\bigr)\, .
\]
\end{lem}
\begin{rmk}
Lemma \ref{lem_mass_sup} is valid for any system $u_{i}:I \rightarrow \mathbb{R}, i=1,..,n$ of moment
functions
which form a T-system per \cite{KarlinStudden:1966}.
\end{rmk}
The second is the Mass Infimum Inequality.
\begin{lem}
\label{lem_mass_inf}
Let $t_{*}\in I$, $ \delta >0$, and consider the set
$M^{n}_{ \delta } \subset M^{n}$ defined by
\[ M^{n}_{ \delta }:=\bigl\{q \in M^{n}: \exists \mu \in\Psi^{-1}q: \mu\bigl(B_{ \delta }(t_{*})\bigr)=0.\bigr\}\]
Then we have
\[Vol\bigl(M^{n}_{ \delta }\bigr) \geq \Bigl(1- \delta (2e)^{2n} \Bigr) Vol\bigl(M^{n}\bigr) \, .\]
\end{lem}
The third is the Mass of First Moment Inequality.
\begin{lem}
\label{massfirstmoment}
Let $ 0 \leq \delta \leq \frac{1}{2}$. Then we have
\[ \delta ^{n} \bigl(2e\bigr)^{n} \geq \frac{Vol\bigl( q \in M^{n}: q_{1} \in [1- \delta ,1]\bigr)}{Vol(M^{n})} \geq \delta ^{n}\, . \]
\end{lem}
\section{Integral Geometry of the Markov-Kre\u{\i}n Representations}
\label{sec_IntegralGeometry}
Here we will describe the Markov-Kre\u{\i}n representations of truncated moments and begin the development of their integral geometry. The history of this subject begins with \u{C}eby\u{s}ev and his student Markov's thesis, followed by
work by Kre\u{\i}n and others, where in
\cite{Krein} one can find, not only an historical sketch, but substantial contributions by Kre\u{\i}n. Indeed,
it is clear from Karlin and Studden \cite{KarlinStudden:1966} that this subject owes a lot to Kre\u{\i}n.
Consequently, we refer to the (principal and canonical) representations that we use as
Markov-Kre\u{\i}n representations. It can be argued that the appropriate name should be
\u{C}eby\u{s}ev-Markov-Kre\u{\i}n representations but this name is too long and so we implicitly give credit to
\u{C}eby\u{s}ev.
Now, following Karlin and Studden \cite[Chapters II \& IV]{KarlinStudden:1966},
we describe the Markov-Kre\u{\i}n representations and determine their Jacobian determinants.
We finish this section
by setting up the change of variables approach, in preparation for
both the proofs of the volume inequalities of Section \ref{sec_volumeinequalities} and all that follows.
To wit,
we define the index $i(t)$ of a strictly increasing set $t$ of points $0 \leq t_{1} < t_{2} < \cdots < t_{N} \leq 1$
by counting the interior points with weight $1$ and boundary points with weight $\frac{1}{2}$.
For a point $q \in M^{n}$ we say that a measure $\mu \in \mathcal{M}(I)$ is a representing measure for
$q$ if $\Psi(\mu)=q$ and
it is a weighted sum of Dirac masses
\[
\mu=\sum_{j=1}^{N}{\lambda_{j}\delta_{t_{j}}}, \quad \lambda_{j} >0, j=1,..,N
\]
for a strictly increasing set of points $0 \leq t_{1} < t_{2} < \cdots < t_{N} \leq 1$.
In that case, we have the formula
\[q_{i}= \bigl(\Psi(\mu)\bigr)_{i}=\sum_{j=1}^{N}{\lambda_{j}t_{j}^{i}}\, .\]
The index $i(\mu)$ of such a representing measure is defined to be the index $i(t)$ of its set of support points.
A representing measure $\mu$ is called {\em principal} if $i(\mu)= \frac{n+1}{2}$ and {\em canonical} if
$i(\mu) \leq \frac{n+2}{2}$.
For $q \in Int(M^{n})$, \cite[Thm.~2.1]{KarlinStudden:1966} asserts
that $i(\mu)\geq \frac{n+1}{2}$ for any representing measure $\mu$ for $q$.
A principal or canonical representation is called ``lower'' if its set of support points {\em does not} include the righthand endpoint $1$ and ``upper'' if it does.
The following two results will be our main tools.
The first is the principal representation, see \cite[Cor.~3.1]{KarlinStudden:1966}.
\begin{thm}
\label{thm_principal}
Every point $q \in Int(M^{n})$ has a unique
upper and lower principal representation.
\end{thm}
The second is the canonical representation which allows the specification of a predetermined point $t_{*}\in I$ in the support of the representing measure, see \cite[Thm.~3.1]{KarlinStudden:1966} combined with \cite[Cor.~3.2]{KarlinStudden:1966} and
\cite[Cor.~3.1]{KarlinStudden:1966}.
\begin{thm}
\label{thm_canonical}
For $t_{*}\in (0,1)$,
every point $q \in Int(M^{n})$ has a unique canonical representation
whose support contains
$t_{*}$.
When $t_{*}=0$ or $1$, there exists a unique principal
representation whose support contains $t_{*}$.
\end{thm}
What Theorem \ref{thm_canonical} doesn't make clear is if the canonical representations converge to
these principal representation as $t_{*}$ tends to $0$ or $1$. They indeed do as we will see.
Let us define some notation that we will use henceforward. We consider two coordinate representations of the
interior of the regular unit simplex. In particular, let
\[T^{N}=\{(t_{1},..,t_{N}): 0 < t_{1} < t_{2} < \cdots < t_{N} < 1\}\]
denote the set of strictly increasing sequences of length $N$ in the interior to $I$ and
\[\Lambda^{N}=\{(\lambda_{1},..,\lambda_{N}): \lambda_{j}>0,\, j=1,..,N,\, \sum_{j=1}^{N}{\lambda_{j}}<1\}\, .
\]
denote the interior to the positive orthant restricted to $\lambda\cdot \mathbbm{1} < 1$.
Sometimes it will be convenient to abuse this notation and shift indices so that
\[\Lambda^{N}=\{(\lambda_{0},..,\lambda_{N-1}): \lambda_{j}>0,\, j=0,..,N-1,\, \sum_{j=0}^{N-1}{\lambda_{j}}<1\}\, .
\]
We will often use the fact that
$I^{N}$ can be described by $N!$ copies of $T^{N}$ corresponding to permuting the sequence.
We use the notation $t$ for a vector with coordinates $t_{j}$ and similarly
$ \lambda$ for a vector with coordinates $\lambda_{j}$.
We use the superscripts $p$ for ``principal'' and $c$ for ``canonical'',
the subscripts $o$ for ``odd'', $e$ for ``even'',
$l$ for ``lower'', and $u$ for ``upper''.
Finally, we purposefully ignore
multiples of $\pm 1$ in all our determinant calculations. With proper caution, this causes no harm since at the end of the day
we take the absolute value.
\subsection{Principal Representations}
\label{sec_principal}
Theorem \ref{thm_principal} asserts that each $q \in Int(M^{n})$ has a unique
upper and lower principal representation. We now define these representations as maps and compute their Jacobian determinants. We state these propositions without proof, since these proofs are very similar to those for the
canonical representations of
Propositions and \ref{prop_Jco} and \ref{prop_Jce}.
First consider the odd case when $n=2m-1$.
Then since $\frac{n+1}{2}=m$ is an integer, it follows that the support of any lower principal representation contains neither endpoint and the support of any upper principal representation contains both endpoints. Consequently,
Theorem \ref{thm_principal} implies that
each point in $Int(M^{2m-1})$ has a unique lower principal representation of the form
\begin{equation}
\label{rep_principal_odd}
\mu=\sum_{j=1}^{m}{\lambda_{j}\delta_{t_{j}}}, \quad \lambda_{j} >0,\, j=1,..,m, \,\, \sum_{j=1}^{m}{\lambda_{j}}=1
\end{equation}
where $0 < t_{1} < t_{2} < \cdots < t_{m} < 1$.
Therefore,
consider the bijection
\[ \phi^{p}_{ol}: \Lambda^{m-1} \times T^{m} \rightarrow Int(M^{2m-1})\] defined by
\begin{eqnarray}
\label{def_phi_pol}
\phi^{p}_{ol}(\lambda,t)
&= &\Psi \Bigl( \sum_{j=1}^{m-1}{\lambda_{j}\delta_{t_{j}}} +(1-\sum_{j=1}^{m-1}{\lambda_{j}})\delta_{t_{m}}\Bigr)\notag\\
&=& \Bigl(\sum_{j=1}^{m-1}{\lambda_{j}t_{j}^{i}}+ (1-\sum_{j=1}^{m-1}{\lambda_{j})t_{m}^{i}}\Bigr)_{i=1}^{2m-1}\, .
\end{eqnarray}
It also follows that
each point in $Int(M^{2m-1})$ has a unique upper principal representation of the form
\begin{equation}
\label{rep_principal_odd_u}
\mu=\lambda_{0}\delta_{0}+\sum_{j=1}^{m-1}{\lambda_{j}\delta_{t_{j}}} +\lambda_{m}\delta_{1},
\quad \lambda_{j} >0,\, j=0,..,m, \,\, \sum_{j=0}^{m}{\lambda_{j}}=1
\end{equation}
where $0 < t_{1} < t_{2} < \cdots < t_{m-1} < 1$.
Therefore,
consider the bijection
\[ \phi^{p}_{ou}: \Lambda^{m} \times T^{m-1} \rightarrow Int(M^{2m-1})\] defined by
\begin{eqnarray}
\label{def_phi_pou}
\phi^{p}_{ou}(\lambda,t)
&= &\Psi \Bigl(\lambda_{0}\delta_{0}+ \sum_{j=1}^{m-1}{\lambda_{j}\delta_{t_{j}}} +
(1-\sum_{j=0}^{m-1}{\lambda_{j}})\delta_{1}\Bigr)\notag\\
&=& \Bigl(\sum_{j=1}^{m-1}{\lambda_{j}t_{j}^{i}}+ (1-\sum_{j=0}^{m-1}{\lambda_{j})}\Bigr)_{i=1}^{2m-1}\, .
\end{eqnarray}
For an increasing sequence $t_{j} < t_{j+1}$ let
\begin{equation}
\label{def_vandermonde}
\Delta(t):=\prod_{ j<k }{(t_{k}-t_{j})}
\end{equation}
denote the Vandermonde determinant
(see e.g.~\cite[Pg.~400]{HornJohnson2})
of the matrix with entries $[t_{j}^{i}],\, j=1,..,N,\, i=0,..,N-1$.
which we write as $\Delta_{N}$ to emphasize the dimension of $t$.
We will also use the same formula for non-increasing sequences when we eventually take the absolute value.
\begin{prop}
\label{prop_Jpo}
When $n=2m-1$, the Jacobian determinants are
\[ |det(d\phi^{p}_{ol})|(\lambda,t)= \mathcal{J}^{p}_{ol}(t)
\bigl(1-\sum_{j=1}^{m-1}{\lambda_{j}}\bigr)\prod_{j=1}^{m-1}{\lambda_{j}} \]
\[|det(d\phi^{p}_{ou})|(\lambda,t)= \mathcal{J}^{p}_{ou}(t)
\prod_{j=1}^{m-1}{\lambda_{j}} \]
where
\[\mathcal{J}^{p}_{ol}(t):=
\Delta^{4}_{m}(t)
\]
\[\mathcal{J}^{p}_{ou}(t):=\prod_{j=1}^{m-1}{t_{j}^{2}(1-t_{j})^{2}}
\cdot \Delta_{m-1}^{4}(t)\, .\]
\end{prop}
Note that although each term appears to have the same multiplier $\prod_{j=1}^{m-1}{\lambda_{j}}$,
in the lower case
this multiplier is the full product in
on $\Lambda^{m-1}$ and in the upper case it is only a partial product on $\Lambda^{m}$, that is, it is missing
the $\lambda_{0}$ term.
Finally, let us observe the symmetries under the reflection $t \mapsto 1-t$:
\begin{eqnarray}
\label{Jposymm}
\mathcal{J}^{p}_{ol}(1-t)&=&\mathcal{J}^{p}_{ol}(t)\notag\\
\mathcal{J}^{p}_{ou}(1-t)&=&\mathcal{J}^{p}_{ou}(t)\, .
\end{eqnarray}
Now consider the even case when $n=2m$. Since $\frac{n+1}{2}=m+\frac{1}{2}$ is an integer plus $\frac{1}{2}$
it follows that the support of any lower principal representation
contains the left endpoint but not the right and any upper principal representation contains the right endpoint
but not the left. Let us first consider the lower representation.
Theorem \ref{thm_principal} implies that
every point in the interior $Int(M^{2m})$ has a unique lower principal representation of the form
\[\mu =\sum_{j=1}^{m}{\lambda_{j}\delta_{t_{j}}}+(1-\sum_{j=1}^{m}{\lambda_{j}})\delta_{0},
\quad \lambda_{j} >0,\, j=1,..,m,\,\, \sum_{j=1}^{m}{\lambda_{j}} <1\]
where $0 < t_{1} < \cdots t_{m} < 1$.
Therefore, consider the bijection
\[ \phi^{p}_{el}:\Lambda^{m}\times T^{m} \rightarrow Int( M^{2m})\] defined by
\begin{eqnarray}
\label{def_phi_pel}
\phi^{p}_{el}(\lambda,t)
&= &\Psi \Bigl(\sum_{j=1}^{m}{\lambda_{j}\delta_{t_{j}}}+(1-\sum_{j=1}^{m}{\lambda_{j}})\delta_{0}\Bigr)\\
&=& \Bigl(\sum_{j=1}^{m}{\lambda_{j}t_{j}^{i}}\Bigr)_{i=1}^{2m}\, .
\end{eqnarray}
On the other hand,
every point in the interior $Int(M^{2m})$ has a unique upper principal representation of the form
\[\mu =\sum_{j=1}^{m}{\lambda_{j}\delta_{t_{j}}}+(1-\sum_{j=1}^{m}{\lambda_{j}})\delta_{1},
\quad \lambda_{j} >0,\, j=1,..,m,\,\, \sum_{j=1}^{m}{\lambda_{j}} <1\]
where $0 < t_{1} < \cdots t_{m} < 1$.
Therefore, consider the bijection
\[ \phi^{p}_{eu}:\Lambda^{m}\times T^{m} \rightarrow Int( M^{2m})\] defined by
\begin{eqnarray}
\label{def_phi_peu}
\phi^{p}_{eu}(\lambda,t)
&= &\Psi \Bigl(\sum_{j=1}^{m}{\lambda_{j}\delta_{t_{j}}}+(1-\sum_{j=1}^{m}{\lambda_{j}})\delta_{1}\Bigr)\\
&=& \Bigl(\sum_{j=1}^{m}{\lambda_{j}t_{j}^{i}}+ (1-\sum_{j=1}^{m}{\lambda_{j})}\Bigr)_{i=1}^{2m}\, .
\end{eqnarray}
\begin{prop}
\label{prop_Jpe}
When $n=2m$ the Jacobian determinants are
\[ |det(d\phi^{p}_{el})|(\lambda,t)= \mathcal{J}^{p}_{el}(t)\prod_{j=1}^{m}{\lambda_{j}}
\]
\[ |det(d\phi^{p}_{eu})|(\lambda,t)= \mathcal{J}^{p}_{eu}(t)
\prod_{j=1}^{m}{\lambda_{j}} \]
where
\[
\mathcal{J}^{p}_{el}(t)
:= \prod_{j=1}^{m}{t_{j}^{2}}
\cdot \Delta_{m}^{4}(t)
\]
\[
\mathcal{J}^{p}_{eu}(t)
:= \prod_{j=1}^{m}{(1-t_{j})^{2}}\cdot \Delta_{m}^{4}(t)
\]
\end{prop}
Here, instead of the reflection $t \mapsto 1-t$ leaving the lower and upper invariant as in the odd case \eqref{Jposymm},
reflection swaps lower and upper;
\begin{equation}
\label{Jpesymm}
\mathcal{J}^{p}_{el}(1-t)=\mathcal{J}^{p}_{eu}(t)\, .
\end{equation}
\subsection{Canonical Representations}
\label{sec_canonical}
Theorem \ref{thm_canonical} asserts that, when $t_{*}\in (0,1)$,
every point in $Int(M^{n})$ has a unique canonical representation whose support contains
$t_{*}$, and when $t_{*}\in \{0,1\}$, it has a unique principal representation whose support contains
$t_{*}$. Therefore, every point in $Int(M^{n})$ has
a unique representing measure $\mu$ such that
\begin{equation}
\label{canonicalrep}
\mu=\sum_{j=1}^{N}{\lambda_{j}\delta_{t_{j}}}, \quad \lambda_{j} >0,\, j=1,..,N, \quad \sum_{j=1}^{N}{\lambda_{j}}=1
\end{equation}
such that the sequence $0 \leq t_{1} < t_{2} < \cdots < t_{N} \leq 1$ contains
$t_{*}$, where for $t_{*}\in (0,1)$, the sequence has
index $\frac{n+1}{2}$ or $\frac{n+2}{2}$, and when
$t_{*}=0$ or $1$,
the index is $\frac{n+1}{2}$.
Now let us remove $t_{*}$ from the list and use the identity $\sum_{j=1}^{N}{\lambda_{j}}=1$ to solve
for the weight $\lambda_{t_{*}}$ corresponding to $t_{*}$. Changing notation from $N \mapsto N+1$ and relabeling the indices, we
obtain that
\begin{equation}
\label{canonicalrep5}
\mu=\sum_{j=1}^{N}{\lambda_{j}\delta_{t_{j}}} +\bigl(1-\sum_{j=1}^{N}{\lambda_{j}}\bigr)\delta_{t_{*}},
\quad \lambda \in \Lambda^{N}\, ,
\end{equation}
where the resulting sequence
\[0 \leq t_{1} < t_{2} < \cdots < t_{N} \leq 1\] {\em does not} contain
$t_{*}$, and when
$t_{*}\in (0,1)$, the removal of this interior point implies that the resulting sequence has
index $\frac{n-1}{2}$ or $\frac{n}{2}$ and when
$t_{*}=0$ or $1$, the removal of this boundary point implies that the resulting sequence has
index $\frac{n}{2}$.
Consequently, for $t_{*}\in (0,1)$, to represent $Int(M^{n})$
we can split
into four domains, two corresponding to the two ways
of producing index $\frac{n-1}{2}$ and two corresponding the two ways of producing index $\frac{n}{2}$.
When $n$ is even one of the two index $\frac{n-1}{2}$ configurations corresponds to including
$t=0$ in the sequence and not $t=1$ and the other corresponds to including
$t=1$ in the sequence and not $t=0$, while one of the two
index $\frac{n}{2}$ configurations corresponds to not allowing $t=0$ or $t=1$ and
the other corresponds to including both $t=0$ and $t=1$. When $n$ is odd this relationships is reversed.
Similarly, when $t_{*}\in \{0,1\}$, we can can split
into two domains corresponding to the two ways
of producing index $\frac{n}{2}$.
However, we can show that the representations of index $\frac{n-1}{2}$ produce {\em zero} volume and so can be
excluded from the integral analysis. To that end, we only need to consider the $t_{*}\in (0,1)$ case.
Then let us
decompose the set of sequences of index $\bar{N}$ by their endpoint configurations. That is, split
such sequences into those which contain $0$ but not $1$, $1$ but not $0$, $0$ and $1$, and neither
$0$ or $1$. Some of these components will be empty. On any of these endpoint specific subdomains
let \[\mathcal{I} \subset \{1,..,N\}\] denote the indices of the interior points,
so that in this notation we have
\begin{eqnarray*}
\mu&=&\sum_{j\in \mathcal{I}}{\lambda_{j}\delta_{t_{j}}} +\sum_{j\in \mathcal{I}^{c}}{ \lambda_{j}}\delta_{t_{j}} +
\bigl(1-\sum_{j=1}^{N}{\lambda_{j}}\bigr)\delta_{t_{*}}\, ,\quad
\lambda \in \Lambda^{N} . \notag
\end{eqnarray*}
Moreover, for a sequence $t$, let $\mathring{t}$ denote the sequence of interior points, and
define $\mathring{T}_{*}:=\{\mathring{t}: t \in T^{N}, t_{j} \neq t_{*},\, j=1,..,N \}$ to be the set of interior points which do not cover $t_{*}$ and
consider the map
\[ \phi:\Lambda^{N} \times \mathring{T}_{*} \rightarrow Int(M^{n})\]
defined by
\begin{eqnarray}
\label{def_phi}
\phi(\lambda,\mathring{t})
&= &\Psi\Bigl(\sum_{j\in \mathcal{I}}{\lambda_{j}\delta_{t_{j}}} +\sum_{j\in \mathcal{I}^{c}}{ \lambda_{j}}\delta_{t_{j}} +
\bigl(1-\sum_{j=1}^{N}{\lambda_{j}}\bigr)\delta_{t_{*}}\Bigr)\notag\\
&=& \Bigl(\sum_{j\in \mathcal{I}}{\lambda_{j}t_{j}^{i}}+\sum_{j\in \mathcal{I}^{c}}{ \lambda_{j}}t_{j}^{i} + (1-\sum_{j=1}^{N}{\lambda_{j})t_{*}^{i}}\Bigr)_{i=1}^{n}\, \notag
\end{eqnarray}
where we note that the first sum $\sum_{j\in\mathcal{I}}{\lambda_{j}t_{j}^{i}}$ is over the interior points
and the second $\sum_{j\in \mathcal{I}^{c}}{\lambda_{j}t_{j}^{i}}$ over the endpoints which are fixed.
The dimension of the domain $\Lambda^{N} \times \mathring{T}_{*}$ is clearly
$N+|\mathcal{I}|$. However, one can easily show that
\[N+|\mathcal{I}|=2\bar{N},\]
so that in the case $\bar{N}=\frac{n-1}{2}$, it follow that the dimension of this subdomain
is $N+|\mathcal{I}|=2\bar{N}=n-1 < n$. Consequently, the image of this subdomain under the map $\phi$ has zero volume
in $M^{n}$. Since the domain corresponding to index $\frac{n-1}{2}$ is a disjoint union of two such
subdomains, the assertion is proved.
Moreover, the subset consisting of sequences which cover $t_{*}$ also clearly has zero volume, so the constraint that the sequences not cover $t_{*}$ can also be removed.
In conclusion, we can represent the volume $Vol(M^{n})$ using the
the representation
\begin{equation}
\label{canonicalrep7}
\sum_{j=1}^{N}{\lambda_{j}t^{i}_{j}} +\bigl(1-\sum_{j=1}^{N}{\lambda_{j}}\bigr)t^{i}_{*},
\quad i=1,..,n,
\end{equation}
defined on two subdomains corresponding to the two ways that
the sequence
\[0 \leq t_{1} < t_{2} < \cdots < t_{N} \leq 1\]
can have index $\frac{n}{2}$.
That is,
when $n$ is even,
one subdomain corresponds to not allowing $0$ or $1$ and
the other to including both $0$ and $1$. When $n$ is odd,
one subdomain corresponds to including $0$ and not $1$ and
the other to including $1$ and not $0$.
We now compute the Jacobian determinants.
First consider the odd case, $n=2m-1$.
Then, sequences of index $\frac{n}{2}=m-\frac{1}{2}$ split into the lower and upper sequences
\begin{equation*}
\begin{cases}
0=t_{1} < t_{2} < \cdots < t_{m}<1,\\
0 < t_{1} < t_{2} < \cdots < t_{m}= 1\, .
\end{cases}
\end{equation*}
Define
the lower representation
\[ \phi^{c}_{ol}:\Lambda^{m}\times T^{m-1} \rightarrow Int(M^{2m-1})\] by
\begin{eqnarray}
\label{def_phi_col}
\phi^{c}_{ol}(\lambda,t;t_{*})
&= &\Psi \Bigl(\lambda_{0}\delta_{0}+ \sum_{j=1}^{m-1}{\lambda_{j}\delta_{t_{j}}} +(1-\sum_{j=0}^{m-1}{\lambda_{j}})\delta_{t_{*}}\Bigr)\notag\\
&=& \Bigl(\sum_{j=1}^{m-1}{\lambda_{j}t_{j}^{i}}+ (1-\sum_{j=0}^{m-1}{\lambda_{j})t_{*}^{i}}\Bigr)_{i=1}^{2m-1}
\end{eqnarray}
and the upper
representation \[ \phi^{c}_{ou}:\Lambda^{m}\times T^{m-1} \rightarrow Int(M^{2m-1}) \]
by
\begin{eqnarray}
\label{def_phi_cou}
\phi^{c}_{ou}(\lambda,t;t_{*})
&= &\Psi \Bigl(\lambda_{0}\delta_{1}+ \sum_{j=1}^{m-1}{\lambda_{j}\delta_{t_{j}}} +(1-\sum_{j=0}^{m-1}{\lambda_{j}})\delta_{t_{*}}\Bigr)\notag\\
&=& \Bigl(\lambda_{0}+\sum_{j=1}^{m-1}{\lambda_{j}t_{j}^{i}}+ (1-\sum_{j=0}^{m-1}{\lambda_{j})t_{*}^{i}}\Bigr)_{i=1}^{2m-1}\, .
\end{eqnarray}
\begin{prop}
\label{prop_Jco}
When $n=2m-1$, for $t_{*} \in (0,1)$, the Jacobian determinants are
\[ |det(d\phi^{c}_{ol})|(\lambda,t;t_{*})= \mathcal{J}^{c}_{ol}(t_{*},t)
\prod_{j=1}^{m-1}{\lambda_{j}} \]
\[ |det(d\phi^{c}_{ou})|(\lambda,t;t_{*})= \mathcal{J}^{c}_{ou}(t_{*},t)
\prod_{j=1}^{m-1}{\lambda_{j}} \]
where
\[\mathcal{J}^{c}_{ol}(t_{*},t):=
t_{*} \prod_{j=1}^{m-1}{(t_{j}-t_{*})^{2}}\prod_{j=1}^{m-1}{t_{j}^{2}}
\cdot \Delta_{m-1}^{4}(t)
\]
\[
\mathcal{J}^{c}_{ou}(t_{*},t):=
\bigl(1-t_{*}\bigr) \prod_{j=1}^{m-1}{(t_{j}-t_{*})^{2}}
\prod_{j=1}^{m-1}{(1-t_{j})^{2}}
\cdot \Delta_{m-1}^{4}(t)
\]
Moreover,
\begin{equation}
\label{Jcsymm}
\mathcal{J}^{c}_{ou}(t_{*},t)=\mathcal{J}^{c}_{ol}(1-t_{*},1-t)\, .
\end{equation}
\end{prop}
Now consider the even case, $n=2m$.
Then sequences of index $\frac{n}{2}=m$ split into the lower and upper sequences
\begin{equation*}
\begin{cases}
0 < t_{1} < t_{2} < \cdots < t_{m}< 1\\
0=t_{1} < t_{2} < \cdots < t_{m+1}=1,
\end{cases}
\end{equation*}
Therefore, we define the lower representation
\[ \phi^{c}_{el}:\Lambda^{m}\times T^{m} \rightarrow Int(M^{2m})\] by
\begin{eqnarray}
\label{def_phi_cel}
\phi^{c}_{el}(\lambda,t;t_{*})
&= &\Psi \Bigl( \sum_{j=1}^{m}{\lambda_{j}\delta_{t_{j}}}+(1-\sum_{j=1}^{m}{\lambda_{j}})\delta_{t_{*}}\Bigr)\notag\\
&=& \Bigl(\sum_{j=1}^{m}{\lambda_{j}t_{j}^{i}}+ (1-\sum_{j=1}^{m}{\lambda_{j})t_{*}^{i}}\Bigr)_{i=1}^{2m}\, .
\end{eqnarray}
and the upper representation
\[ \phi^{c}_{eu}:\Lambda^{m+1}\times T^{m-1} \rightarrow Int(M^{2m})\] by
\begin{eqnarray}
\label{def_phi_ceu}
\phi^{c}_{eu}(\lambda,t;t_{*})
&= &\Psi \Bigl(\lambda_{0} \delta _{0}+ \sum_{j=1}^{m-1}{\lambda_{j}\delta_{t_{j}}}+\lambda_{m} \delta _{1} +(1-\sum_{j=0}^{m}{\lambda_{j}})\delta_{t_{*}}\Bigr)\notag\\
&=& \Bigl(\sum_{j=1}^{m-1}{\lambda_{j}t_{j}^{i}}+\lambda_{m}+ (1-\sum_{j=0}^{m}{\lambda_{j})t_{*}^{i}}\Bigr)_{i=1}^{2m}\, .
\end{eqnarray}
\begin{prop}
\label{prop_Jce}
When $n=2m$, for $t_{*} \in (0,1)$, the Jacobian determinants are
\[|det(d\phi^{c}_{el})|(\lambda,t;t_{*})= \mathcal{J}^{c}_{el}(t_{*},t)
\prod_{j=1}^{m}{\lambda_{j}}
\]
\[
|det(d\phi^{c}_{eu})|(\lambda,t;t_{*})= \mathcal{J}^{c}_{eu}(t_{*},t)
\prod_{j=1}^{m-1}{\lambda_{j}}\, ,
\]
where
\[
\mathcal{J}^{c}_{el}(t_{*},t)
:= \prod_{j=1}^{m}{(t_{j}-t_{*})^{2}}\cdot \Delta_{m}^{4}(t)\, .
\]
\[
\mathcal{J}^{c}_{eu}(t_{*},t)
:=
t_{*}\bigl(1-t_{*}\bigr) \prod_{j=1}^{m-1}{(t_{j}-t_{*})^{2}}\prod_{j=1}^{m-1}{t_{j}^{2}(1-t_{j})^{2}}
\cdot \Delta_{m-1}^{4}(t)\, ,
\]
\end{prop}
Finally, observe that if we extend the canonical representations to be defined for $t_{*}=0,1$
by continuity, we obtain the following relations between the canonical representations evaluated at the
endpoints and the principal representations:
\begin{eqnarray}
\label{principal_canonical0}
\mathcal{J}^{c}_{ol}(0,t)&\equiv&0\notag\\
\mathcal{J}^{c}_{ou}(0,t)&=&\mathcal{J}^{p}_{ou}(t)\notag\\
\mathcal{J}^{c}_{el}(0,t)&=&\mathcal{J}^{p}_{el}(t)\notag\\
\mathcal{J}^{c}_{eu}(0,t)&\equiv&0\, ,
\end{eqnarray}
\begin{eqnarray}
\label{principal_canonical0_2}
|d\phi^{c}_{ol}(0,t)|&\equiv&0\notag\\
|d\phi^{c}_{ou}(0,t)|&=&|d\phi^{p}_{ou}(t)|\notag\\
|d\phi^{c}_{el}(0,t)|&=&|d\phi^{p}_{el}(t)|\notag\\
|d\phi^{c}_{eu}(0,t)|&\equiv&0\, ,
\end{eqnarray}
\begin{eqnarray}
\label{principal_canonical1}
\mathcal{J}^{c}_{ol}(1,t)&=&\mathcal{J}^{p}_{ou}(t)\notag\\
\mathcal{J}^{c}_{ou}(1,t)&\equiv&0\notag\\
\mathcal{J}^{c}_{el}(1,t)&=&\mathcal{J}^{p}_{eu}(t)\notag\\
\mathcal{J}^{c}_{eu}(1,t)&\equiv&0\, ,
\end{eqnarray}
\begin{eqnarray}
\label{principal_canonical1_2}
|d\phi^{c}_{ol}(1,t)|&=&|d\phi^{p}_{ou}(t)|\notag\\
|d\phi^{c}_{ou}(1,t)|&\equiv&0\notag\\
|d\phi^{c}_{el}(1,t)|&=&|d\phi^{p}_{eu}(t)|\notag\\
|d\phi^{c}_{eu}(1,t)|&\equiv&0 \, .
\end{eqnarray}
\subsection{Change of variables integral representations}
\label{sec_cov}
In Karlin and Shapley's
\cite[Thm.~15.2]{KarlinShapley} proof
of the Hausdorff moment volume formula \eqref{karlinshapley}, they used the lower principal representation $\phi^{p}_{ol}$ of \eqref{def_phi_pol}
when $n$ is odd and
$\phi^{p}_{el}$ of \eqref{def_phi_pel} when $n$ is even combined with the change of variables formula.
To develop this method so that it can be used for the canonical
representations, which are not bijections,
it
is convenient to proceed in some generality. To begin, consider a
representation
\[ \phi:W \rightarrow Int(M^{n}),\]
where $W\subset \mathbb{R}^{n}$ is open and $\phi$ is a continuously differentiable bijection.
Then, since $\phi$ is injective, by the change of variables
formula for injective differentiable mappings whose Jacobian determinant may vanish
(see e.g.~\cite[Thm.~3.13]{Spivak} combined
with
Sard's Theorem \cite[Thm.~3.14]{Spivak}), we conclude that
\[Vol\bigl(\phi(W)\bigr)=\int_{W}{|d\phi|}\, .\]
Moreover, since $\phi$ is surjective we have
\[\phi(W)=Int(M^{n})\]
and from \eqref{boundarynull} we have
\[Vol\bigl(M^{n}\bigr)=Vol\bigl(Int(M^{n})\bigr)\]
so that we conclude
\begin{equation}
\label{eq_cov}
Vol\bigl(M^{n}\bigr)=\int_{W}{|d\phi|}\, .
\end{equation}
To compute $Vol\bigl(M^{n}\bigr)$, Karlin and Shapley then evaluated the righthand side
by determining the Jacobian determinant and then evaluating the resulting integral using a Selberg integral formula.
However, more can be done along these lines.
Indeed, applying the full change of variables formula we obtain
\[
\int_{\phi(W)}{f}=\int_{W}{\bigl(f\circ\phi\bigr) |d\phi|}
\]
for any function $f:\phi(W) \rightarrow \mathbb{R}$ that is integrable over $\phi(W)$. In particular, since $M^{n}$ is compact, it follows using the same reasoning
that was applied above to the case $f\equiv 1$, that for any
bounded measurable function $f:M^{n} \rightarrow \mathbb{R}$ we have
\begin{equation}
\label{id_cov}
\int_{M^{n}}{f}=\int_{W}{\bigl(f\circ\phi\bigr) |d\phi|}\, .
\end{equation}
We now apply this to the
component functions $ q\mapsto q_{i},
i=1,..,n$ on $M^{n}$ where we abuse notation and indicate them by the symbol $q_{i}$. It may be profitable to also consider nonlinear functions such as $q \mapsto q_{i}^{2}$ but we will not do that here.
Then, in this notation, $q_{i}\circ\phi=\phi^{i}$ and \eqref{id_cov} becomes
\begin{equation}
\label{cov}
\int_{M^{n}}{q_{i}}=\int_{W}{\phi^{i} |d\phi|}
\end{equation}
That is, we have an integral representation of the mean Hausdorff moments.
However, to prove Lemma \ref{lem_mass_sup}, instead of a principal representation, we use a {\em family} of canonical
representations from Section \ref{sec_canonical}. In this case, utilizing the conclusion at
\eqref{canonicalrep7},
the major difference with the previous discussion is that, instead of a single bijection,
there are two continuously differentiable injections
\[\phi_{k}:W_{k}\rightarrow Int(M^{n}),\quad k=1,2\]
that are volume filling in the sense that
\[
Vol\bigl(Int(M^{n})\bigr)= Vol\bigl(\phi_{1}(W_{1})\cup \phi_{2}(W_{2})\bigr)
\]
and
\[
\phi_{1}(W_{1})\cap \phi_{2}(W_{2})=\emptyset \]
and, instead of $W_{k}, k=1,2$ being open, there exists open sets $V_{k}\subset W_{k}, k=1,2$
such that
\[ Vol(W_{k})=Vol(V_{k})\, .\]
Then
the analysis above can easily be repeated to conclude that
\begin{equation}
\label{id_cov3}
\int_{M^{n}}{f}=\int_{W_{1}}{f\circ{\phi_{1}} |d\phi_{1}|} +\int_{W_{2}}{f\circ{\phi_{2}} |d\phi_{2}|}
\end{equation}
for any bounded measurable function $f:M^{n} \rightarrow \mathbb{R}$.
In particular, we conclude
\begin{equation}
\label{id_cov2}
\int_{M^{n}}{q_{i}}=\int_{W_{1}}{\phi_{1}^{i} |d\phi_{1}|}+\int_{W_{2}}{\phi_{2}^{i} |d\phi_{2}|}\, ,
\end{equation}
our primary integration identity for the mean Hausdorff moments to be used in the next section.
\section{Mean Hausdorff Moments using the Markov-Kre\u{\i}n Representations}
\label{sec_mean}
We are now prepared to derive integral representations of the mean truncated Hausdorff moments
with respect to the
uniform measure on $M^{n}$ and show that the canonical representations
generate reproducing kernel identities corresponding to reproducing kernel Hilbert spaces
of $n$-th degree polynomials.
These identities
are used in
Section \ref{sec_newselberg} to derive
biorthogonal systems of Selberg integral formulas.
The mean moments with respect to many other Selberg-type
densities can also be computed but to keep this presentation simple we will not do that here.
We will use
Selberg's result (see e.g.~\cite{ForresterWarnaar})
\begin{equation}
\label{selbergformula}
S_{n}(\alpha,\beta,\gamma)= \prod_{j=0}^{n-1}{\frac{\Gamma(\alpha+ j\gamma)\Gamma(\beta+j\gamma)
\Gamma(1+(j+1)\gamma)}{\Gamma(\alpha +\beta +(n+j-1)\gamma) \Gamma(1+\gamma)}}
\end{equation}
for the integrals
\begin{equation}
\label{selbergintegral}
S_{n}(\alpha,\beta,\gamma):=
\int_{I^{n}}{ \prod_{j=1}^{n}{t_{j}^{\alpha-1}(1-t_{j})^{\beta-1}}
|\Delta(t)|^{2\gamma}dt}\, ,
\end{equation}
where
$ Re(\alpha)>0, Re(\beta)>0,
Re(\gamma) > -\min{\bigl(\frac{1}{n}, Re(\alpha)/(n-1), Re(\beta)/(n-1) \bigr)} \, .$
We begin with the volume calculation and then proceed to higher moments using the result of the volume
calculation.
The main idea of our approach is the following. Recall from Section \ref{sec_principal} that the lower
and upper principal representations are each bijections with $Int(M^{n})$ so that the volume $Vol(M^{n})$ can be computed
using the change of variables result \eqref{eq_cov}. For example, when $n=2m-1$,
the lower principal representation
$\phi^{p}_{ol}$ defined in \eqref{def_phi_pol} and the upper principal representation $\phi^{p}_{ol}$
defined in \eqref{def_phi_pou}, along with the values of their Jacobian determinants from
Proposition \ref{prop_Jpo} produce two different integral representations for $Vol(M^{n})$. Specifically,
in the notation for Selberg's formulas \eqref{selbergformula} for
the integrals \eqref{selbergintegral}, using the identity
$ \int_{\Lambda^{m-1}}{\bigl(1-\sum_{j=1}^{m-1}{\lambda_{j}}\bigr)\prod_{j=1}^{m-1}{\lambda_{j}}d\lambda}
=\frac{1}{(2m-1)!}$ ,
the lower representation yields
\begin{eqnarray}
\label{vol_Jpol}
Vol(M^{2m-1})&=& \int_{\Lambda^{m-1} \times T^{m}}{|det(d\phi^{p}_{ol})|}\notag\\
&=& \Bigl(\int_{\Lambda^{m-1}}{\bigl(1-\sum_{j=1}^{m-1}{\lambda_{j}}\bigr)\prod_{j=1}^{m-1}{\lambda_{j}}d\lambda}\Bigr)
\int_ {T^{m}}{\mathcal{J}^{p}_{ol}}\notag\\
&=&\frac{1}{(2m-1)!}
\int_ {T^{m}}{\mathcal{J}^{p}_{ol}}\notag\\
&=&\frac{1}{(2m-1)!m!}
\int_ {I^{m}}{\mathcal{J}^{p}_{ol}}\\
&=& \frac{1}{(2m-1)!m!}
\int_ {I^{m}}{\Delta_{m}^{4}(t)dt}\notag\\
&=& \frac{1}{(2m-1)!m!} S_{m}(1,1,2)\, .\notag
\end{eqnarray}
On the other hand, using the identity
$ \int_{\Lambda^{m}}{\prod_{j=1}^{m-1}{\lambda_{j}}d\lambda}=\frac{1}{(2m-1)!}$, the upper representation $\phi^{p}_{ou}$ yields
\begin{eqnarray}
\label{vol_Jpou}
Vol(M^{2m-1})&=& \int_{\Lambda^{m} \times T^{m-1}}{|det(d\phi^{p}_{ou})|}\notag\\
&=& \Bigl(\int_{\Lambda^{m}}{\prod_{j=1}^{m-1}{\lambda_{j}}d\lambda}\Bigr)\int_ {T^{m-1}}{\mathcal{J}^{p}_{ou}}
\notag \\
&=& \frac{1}{(2m-1)!}\int_ {T^{m-1}}{\mathcal{J}^{p}_{ou}}
\notag \\
&=& \frac{1}{(2m-1)!(m-1)!}\int_ {I^{m-1}}{\mathcal{J}^{p}_{ou}}\\
&=& \frac{1}{(2m-1)!(m-1)!} \int_ {I^{m-1}}{\prod_{j=1}^{m-1}{t_{j}^{2}(1-t_{j})^{2}}\cdot \Delta_{m}^{4}(t)dt}\notag\\
&=&\frac{1}{(m-1)!(2m-1)!} S_{m-1}(3,3,2) \, .\notag
\end{eqnarray}
Combining the two results \eqref{vol_Jpol} and \eqref{vol_Jpou} we conclude
the identity
\[\frac{1}{(m-1)!} S_{m-1}(3,3,2)= \frac{1}{m!} S_{m}(1,1,2)\]
which is confirmed through direct calculation.
In the even case, where $n=2m$,
we use the representation $\phi^{p}_{el}$ defined in \eqref{def_phi_pel}
and its Jacobian determinant from
Proposition \ref{prop_Jpe}, along with the identity
$ \int_{\Lambda^{m}}{\prod_{j=1}^{m}{\lambda_{j}}d\lambda}=\frac{1}{(2m)!}$,
to conclude that
\begin{equation}
\label{vol_Jpel}
Vol(M^{2m})=\frac{1}{(2m)!m!}\int_{I^{m}}{\mathcal{J}^{p}_{el}}=\frac{1}{(2m)!m!}S_{m}(3,1,2)\, .
\end{equation}
Using same identity, the upper representation $\phi^{p}_{eu}$ defined in \eqref{def_phi_peu} yields
\begin{equation}
\label{vol_Jpeu}
Vol(M^{2m})=\frac{1}{(2m)!m!}\int_{I^{m}}{\mathcal{J}^{p}_{eu}}=\frac{1}{(2m)!m!}S_{m}(1,3,2)\, .
\end{equation}
Equating the two we conclude
that
\[S_{m}(1,3,2)=S_{m}(3,1,2)\]
which is well known from the symmetry of the Selberg formula in its first two arguments, and corresponds to
the change of variables
$t \mapsto 1-t$.
Consequently, we see how two different integral representations of the volume $Vol(M^{n})$ generate
identities.
However, the canonical representations form a {\em one parameter family} of representations
of $Int(M^{n})$ and the value $Vol(M^{n})$ expressed in terms
of the resulting one parameter family of integrals produces more interesting results. To see this,
consider the odd case $n=2m-1$, and the volume filling pair of representations
$\phi^{c}_{ol}$ and $\phi^{c}_{ou}$ defined in \eqref{def_phi_col} and \eqref{def_phi_cou} with
Jacobian determinants evaluated in Proposition \ref{prop_Jco}. Apply the modified change of variables formula
\eqref{id_cov2} in terms of these two representations, and the
identity
$ \int_{\Lambda^{m}}{\prod_{j=1}^{m-1}{\lambda_{j}}d\lambda}=\frac{1}{(2m-1)!}$, to obtain
\begin{eqnarray}
\label{eq_odd_sum0}
Vol(M^{2m-1})&=&\int_{\Lambda^{m}\times T^{m-1}}{|d\phi^{c}_{ol}|}+
\int_{\Lambda^{m}\times T^{m-1}}{|d\phi^{c}_{ou}|}\notag\\
&=&\int_{\Lambda^{m}}{\prod_{j=1}^{m-1}{\lambda_{j}}d\lambda}\int_{T^{m-1}}{\mathcal{J}^{c}_{ol}}+\int_{\Lambda^{m}}{\prod_{j=1}^{m-1}{\lambda_{j}}d\lambda}\int_{T^{m-1}}{\mathcal{J}^{c}_{ou}}\notag\\
&=&\frac{1}{(2m-1)!}\int_{T^{m-1}}{\bigl(\mathcal{J}^{c}_{ol}+\mathcal{J}^{c}_{ou}\bigr)}\notag\\
&=&\frac{1}{(2m-1)!(m-1)!}\int_{I^{m-1}}{\bigl(\mathcal{J}^{c}_{ol}+\mathcal{J}^{c}_{ou}\bigr)}\, .\notag
\end{eqnarray}
Therefore, showing the parameters, we conclude that for $t_{*} \in (0,1)$ we have
\begin{equation}
\label{vol_Jco}
Vol(M^{2m-1}) =\frac{1}{(2m-1)!(m-1)!} \int_{I^{m-1}}{\bigl(\mathcal{J}^{c}_{ol}(t_{*},t)+
\mathcal{J}^{c}_{ou}(t_{*},t)\bigr)dt}\, .
\end{equation}
Since the identity \eqref{vol_Jco} holds for all $t_{*} \in (0,1)$ it generates integral identities.
For the first, since the integrand is continuous in $t_{*}$ we can set $t_{*}=0$ to obtain
\[Vol(M^{2m-1}) =\frac{1}{(2m-1)!(m-1)!} \int_{I^{m-1}}{\bigl(\mathcal{J}^{c}_{ol}(0,t)+\mathcal{J}^{c}_{ou}(0,t)\bigr)dt}\, ,\]
but from Propositions \ref{prop_Jco} we have
\[\mathcal{J}^{c}_{ol}(0,t)\equiv 0\]
and from \eqref{principal_canonical0}
\[
\mathcal{J}^{c}_{ou}(0,t)=
\mathcal{J}^{p}_{ou}(t)
\]
so that we obtain
\[Vol(M^{2m-1}) =\frac{1}{(2m-1)!(m-1)!} \int_{I^{m-1}}{\mathcal{J}^{p}_{ou}(t)dt}\, ,\]
which we already knew from the volume calculation using the principal representation
\eqref{vol_Jpou}.
However, if we compute the first order differential invariant by differentiating \eqref{vol_Jco}
with respect to $t_{*}$ at $t_{*}=0$ we obtain the first integral formula of Theorem \ref{thm_selberg}.
Now consider the even case $n=2m$, and the volume filling pair of representations
$\phi^{c}_{el}$ and $\phi^{c}_{eu}$ defined in \eqref{def_phi_cel} and \eqref{def_phi_ceu} with
Jacobian determinants evaluated in Proposition \ref{prop_Jce}. Apply the modified change of variables formula
\eqref{id_cov2} in terms of these two representations, and
the identities $ \int_{\Lambda^{m}}{\prod_{j=1}^{m}{\lambda_{j}}d\lambda}=\frac{1}{(2m)!}$
and
$\int_{\Lambda^{m+1}}
{\prod_{j=1}^{m-1}{\lambda_{i}}d\lambda} = \frac{1}{(2m)!}$
to obtain
\begin{eqnarray}
\label{eq_even_sum0}
Vol(M^{2m})&=&\int_{\Lambda^{m}\times T^{m} }{|d\phi^{c}_{el}|}+
\int_{\Lambda^{m+1}\times T^{m-1}}{|d\phi^{c}_{eu}|}\notag\\
&=&\Bigl(\int_{\Lambda^{m}}{\prod_{j=1}^{m}{\lambda_{j}}d\lambda}\Bigr)\int_{T^{m}}{\mathcal{J}^{c}_{el}}+
\Bigl(\int_{\Lambda^{m+1}}{\prod_{j=1}^{m-1}{\lambda_{j}}d\lambda}\Bigr)\int_{T^{m-1}}{\mathcal{J}^{c}_{eu}}\notag\\
&=&\frac{1}{(2m)!}\int_{T^{m}}{\mathcal{J}^{c}_{el}}+
\frac{1}{(2m)!}\int_{T^{m-1}}{\mathcal{J}^{c}_{eu}}\notag\\
&=&\frac{1}{(2m)!m!}\int_{I^{m}}{\mathcal{J}^{c}_{el}}+
\frac{1}{(2m)!(m-1)!}\int_{I^{m-1}}{\mathcal{J}^{c}_{eu}}\, .\notag
\end{eqnarray}
Showing the parameters, we conclude that for $t_{*} \in (0,1)$ we have
\begin{equation}
\label{vol_Jce}
Vol(M^{2m})=\frac{1}{(2m)!m!}\int_{I^{m}}{\mathcal{J}^{c}_{el}(t_{*},t)dt}+
\frac{1}{(2m)!(m-1)!}\int_{I^{m-1}}{\mathcal{J}^{c}_{eu}(t_{*},t)dt}\, .
\end{equation}
Setting $t_{*}=0$ and using
\[
\mathcal{J}^{c}_{eu}(0,t)
\equiv 0
\]
from Proposition \ref{prop_Jce}
and
\[
\mathcal{J}^{c}_{el}(0,t)
= \mathcal{J}^{p}_{el}(t)
\]
from \eqref{principal_canonical0}, we obtain
\begin{eqnarray}
Vol(M^{2m})&=&\frac{1}{(2m)!m!}\int_{I^{m}}{\mathcal{J}^{c}_{el}(0,t)dt}+
\frac{1}{(2m)!(m-1)!}\int_{I^{m-1}}{\mathcal{J}^{c}_{eu}(0,t)dt}\notag\\
&=&\frac{1}{(2m)!m!}\int_{I^{m}}{\mathcal{J}^{p}_{el}(t)dt}\notag\\
&=&\frac{1}{(2m)!m!}\int_{I^{m}}{\prod_{j=1}^{m}{t_{j}^{2}}
\cdot \Delta_{m}^{4}(t)dt}\notag\\
&=&\frac{1}{(2m)!m!}S_{m}(3,1,2)\notag
\end{eqnarray}
which we alread knew from the volume calculation using the principal representation
\eqref{vol_Jpel}.
However, if we compute the first order differential invariant by differentiating \eqref{vol_Jce}
with respect to $t_{*}$ at $t_{*}=0$ we obtain the second integral formula of Theorem \ref{thm_selberg}.
We can now proceed to compute the mean of the moments
with respect to the uniform measure on $M^{n}$ using
the volume identities \eqref{vol_Jpol}, \eqref{vol_Jpou}, \eqref{vol_Jpel}, \eqref{vol_Jpeu}
from the principal representations and
\eqref{vol_Jco} and \eqref{vol_Jce} from the canonical representations.
From the identities \eqref{principal_canonical0}, \eqref{principal_canonical0_2},\eqref{principal_canonical1}, \eqref{principal_canonical1_2} connecting the canonical representations at the endpoints and the principal representations, it is clear that we can generate the integral formula for the mean moments corresponding to all the principal representations except $\phi^{p}_{ol}$ by doing so using the canonical representations and then evaluating the result at the endpoints. Therefore, we move directly to the
canonical representations.
Let $ \delta _{0}$ denote the indicator function defined by $ \delta _{0}(i)=1, i=0$ and $ \delta _{0}(i)=0$ otherwise.
Using the convention that $0^{0}:=1$, the following proposition utilizes the volume equalities
\eqref{vol_Jco} and \eqref{vol_Jce} to simultaneously expresses themselves and the moment equalities
generated by the canonical representations.
For a function $\phi:I \rightarrow \mathbb{R}$ we
define the diagonal extension
$\Sigma \phi:I^{N} \rightarrow \mathbb{R}$
by
\[\bigl(\Sigma\phi\bigr)(t):=\sum_{j=1}^{N}{\phi(t_{j})},\quad t\in I^{N}\, .\]
For simple powers, we introduce the notation
\[\Sigma t^{i}:=\sum_{j=1}^{N}{t_{j}^{i}}\] for the power sum and note the important example
\[\Sigma t^{-1}:=\sum_{j=1}^{N}{t_{j}^{-1}}\]
that will be used in the Selberg integral formulas of Theorem \ref{thm_selberg}.
\begin{prop}
\label{prop_mom}
Let $n=2m-1$. Then for all $t_{*}\in I$ and $i=0,1,.., 2m-1$ we have
\begin{eqnarray}
\label{eq_odd_cov5a}
&&\int_{M^{2m-1}}{q_{i}}-\frac{t_{*}^{i}}{2m} Vol(M^{2m-1})\notag\\
&=&\frac{ \delta _{0}(i)}{(2m)!(m-1)!}\int_{I^{m-1}}
{\mathcal{J}^{c}_{ol}(t_{*},t)dt}+\frac{1}{(2m)!(m-1)!}\int_{I^{m-1}}
{\mathcal{J}^{c}_{ou}(t_{*},t)dt}\notag\\
&+&\frac{2}{(2m)!(m-1)!}\int_{I^{m-1}}{\Sigma t^{i}
\Bigl(\mathcal{J}^{c}_{ol}(t_{*},t)+\mathcal{J}^{c}_{ou}(t_{*},t)\Bigr)dt}\, .\notag
\end{eqnarray}
Let $n=2m$. Then for all $t_{*}\in I$ and $i=0,1,.., 2m$ we have
\begin{eqnarray}
&&\int_{M^{2m}}{q_{i}}- \frac{t_{*}^{i}}{2m+1}Vol(M^{2m})\notag\\
&=&\frac{ \delta _{0}(i)+1}{(2m+1)!(m-1)!}\int_{I^{m-1}}{
\mathcal{J}^{c}_{eu}(t_{*},t)dr}\notag\\
&+&\frac{2}{(2m+1)!m!}\int_{I^{m}}{\Sigma t^{i}\mathcal{J}^{c}_{el}(t_{*},t)dt}
+
\frac{2}{(2m+1)!(m-1)!}\int_{I^{m-1}}{\Sigma t^{i}
\mathcal{J}^{c}_{eu}(t_{*},t)dt}\, .\notag
\end{eqnarray}
\end{prop}
The above technique of comparing two representations of the same volume to generate identities we now
apply to the higher order moments with respect to the uniform measure on the moment space (the moment moments) by simply subtracting the integral representations of
Proposition \ref{prop_mom} evaluated at $t_{*}=0$ from that with arbitrary $t_{*} \in I$. We now show
that this procedure produces a reproducing kernel identity on the space of polynomials.
\section{The Canonical Representations and Reproducing Kernel Hilbert Spaces of Polynomials}
\label{sec_RKHS}
The integral representations of
Proposition \ref{prop_mom} show clear signs of the existence of reproducing kernel identities of the form
\[f(x)=\int{K(x,y)f(y)dy},\quad f \in H, \, x \in X\] since, in the odd case, the integrand on the righthand side
$\Sigma t^{i}$
is integrated against a kernel $\mathcal{J}^{c}_{ol}(t_{*},t)+\mathcal{J}^{c}_{ou}(t_{*},t)$ and produces a multiple of $t_{*}^{i}$ plus some terms. Reproducing kernel Hilbert spaces are Hilbert spaces of functions such that
pointwise evaluation is continuous on the Hilbert space. They have remarkable properties, in particular,
the reproducing kernel identities which can be thought of like an abstract Cauchy integral formula from complex analysis.
Let us present Proposition \ref{prop_mom} in reproducing kernel form. To that end,
define
\begin{equation}
\label{def_H}
\mathcal{H}(t_{*},t):=\mathcal{J}^{c}_{ou}(0,t)-\mathcal{J}^{c}_{ol}(t_{*},t)-\mathcal{J}^{c}_{ou}(t_{*},t),
\end{equation}
and note that from \eqref{principal_canonical0} we have $\mathcal{J}^{c}_{ou}(0,t)=\mathcal{J}^{p}_{ou}(t)$, so that
\[\mathcal{H}(t_{*},t)=\mathcal{J}^{p}_{ou}(t)-\mathcal{J}^{c}_{ol}(t_{*},t)-\mathcal{J}^{c}_{ou}(t_{*},t)\, .\]
Moreover, observe that the symmetries
\begin{eqnarray*}
\mathcal{J}^{p}_{ol}(1-t)&=&\mathcal{J}^{p}_{ol}(t)\\
\mathcal{J}^{p}_{ou}(1-t)&=&\mathcal{J}^{p}_{ou}(t)\\
\mathcal{J}^{c}_{ou}(t_{*},t)&=&\mathcal{J}^{c}_{ol}(1-t_{*},1-t)
\end{eqnarray*}
of \eqref{Jposymm} and \eqref{Jcsymm} combined with $\mathcal{J}^{c}_{ol}(0,t)\equiv 0$ imply that
\begin{eqnarray}
\label{H_dirichlet}
\mathcal{H}(0,t)& \equiv& 0\notag\\
\mathcal{H}(1,t)& \equiv & 0\, .
\end{eqnarray}
and
\begin{equation}
\label{H_symm}
\mathcal{H}(1-t_{*},1-t) =\mathcal{H}(t_{*},t)\, .
\end{equation}
Let $\Pi^{n}$ denote the space of $n$-th degree polynomials in one variable with real coefficients.
\begin{thm}
\label{thm_RK}
For all $\phi \in \Pi^{2m-1}$ we have
\begin{eqnarray}
\phi(t_{*}) Vol(M^{2m-1})
&=&\frac{2}{(2m-1)!(m-1)!}\int_{I^{m-1}}{(\Sigma \phi)(t)\mathcal{H}(t_{*},t)dt}\notag\\
&+&\frac{\phi(0)}{(2m-1)!(m-1)!}\int_{I^{m-1}}
{\mathcal{J}^{c}_{ou}(t_{*},t)dt}
\\&+&\frac{\phi(1)}{(2m-1)!(m-1)!}\int_{I^{m-1}}
{\mathcal{J}^{c}_{ol}(t_{*},t)dt}\notag
\end{eqnarray}
and for $\phi \in \Pi^{2m}$ we have
\begin{eqnarray*}
\phi(t_{*})Vol(M^{2m})
&=&-\frac{2}{(2m)!m!}\int_{I^{m}}{(\Sigma \phi)(t)\bigl(\mathcal{J}^{c}_{el}(t_{*},t)-
\mathcal{J}^{c}_{el}(0,t)\bigr)dt}
\\&-&
\frac{2}{(2m)!(m-1)!}\int_{I^{m-1}}{(\Sigma \phi)(t)
\mathcal{J}^{c}_{eu}(t_{*},t)dt}\notag \\
&+&
\frac{\phi(0)}{(2m)!m!}\int_{I^{m}}{\mathcal{J}^{c}_{el}(t_{*},t)
dt}
- \frac{\phi(1)}{(2m)!(m-1)!}\int_{I^{m-1}}{
\mathcal{J}^{c}_{eu}(t_{*},t)dt}\, .
\notag\\
\end{eqnarray*}
\end{thm}
To integrate out the diagonal extension $\Sigma$,
for any function $ (t_{*},t) \mapsto \mathcal{J}(t_{*},t)$ we let
\[\bar{\mathcal{J}}(t_{*},s):=\int{\mathcal{J}\bigl(t_{*},(s,t_{2},..,t_{N})\bigr)dt_{2}\cdots dt_{N}}\]
denote the marginalization to the first component of $t$.
Now for any such function $\mathcal{J}$, which is invariant under the symmetric group acting on its second variable, we have
\[\int_{I^{N}}{(\Sigma \phi)(t) \mathcal{J}(t_{*},t)dt}= N \int_{I}{ \phi(s) \bar{\mathcal{J}}(t_{*},s)ds}\]
so that we obtain the following corollary to Theorem \ref{thm_RK}.
Let us define
\[\bar{\mathcal{G}}(t_{*},s):=\bar{\mathcal{J}}^{c}_{el}(0,s)-
\bar{\mathcal{J}}^{c}_{el}(t_{*},s)
-(m-1)
\bar{\mathcal{J}}^{c}_{eu}(t_{*},s) \]
and note that
\[\bar{\mathcal{G}}(0,s)\equiv 0
\]
but
\[ \bar{\mathcal{G}}(1,s)\not \equiv 0\, .\]
Let $\Pi^{n}_{0} \subset \Pi^{n}$ denote the $n$-th degree polynomials $\phi \in \Pi^{n}$
which vanish on the boundary of $I$, that is, $\phi(0)=\phi(1)=0$.
\begin{cor}
\label{cor_RK}
For all $\phi \in \Pi^{2m-1}$ we have
\begin{eqnarray}
\phi(t_{*}) Vol(M^{2m-1})
&=&\frac{2}{(2m-1)!(m-2)!}\int_{I}{ \phi(s)\bar{\mathcal{H}}(t_{*},s)ds}\notag\\
&+&\frac{\phi(0)}{(2m-1)!(m-2)!}\int_{I}
{\bar{\mathcal{J}}^{c}_{ou}(t_{*},s)ds}
\\&+&\frac{\phi(1)}{(2m-1)!(m-2)!}\int_{I}
{\bar{\mathcal{J}}^{c}_{ol}(t_{*},s)ds}\notag
\end{eqnarray}
and for $\phi \in \Pi^{2m}$ we have
\begin{eqnarray*}
\phi(t_{*})Vol(M^{2m})
&=&\frac{2}{(2m)!(m-1)!}\int_{I}{\phi(s)\bar{\mathcal{G}}(t_{*},s)ds}\notag\\
&+&
\frac{\phi(0)}{(2m)!(m-1)!}\int_{I}{\bar{\mathcal{J}}^{c}_{el}(t_{*},s)
ds}
- \frac{\phi(1)}{(2m)!(m-2)!}\int_{I}{
\bar{\mathcal{J}}^{c}_{eu}(t_{*},s)ds}\, .
\notag\\
\end{eqnarray*}
In particular,
for the normalizations
\begin{eqnarray*}
\hat{\bar{\mathcal{H}}}&:=&\frac{1}{Vol(M^{2m-1})}\frac{2}{(2m-1)!(m-2)!}\bar{\mathcal{H}}\notag\\
\hat{\bar{\mathcal{G}}}&:=&\frac{1}{Vol(M^{2m})}\frac{2}{(2m)!(m-1)!}\bar{\mathcal{G}}\, \notag
\end{eqnarray*}
we have
\begin{eqnarray*}
\label{id_RKHS}
\phi(t_{*})
&=&\int_{I}{ \phi(s)\hat{\bar{\mathcal{H}}}(t_{*},s)ds},\quad \phi \in \Pi^{2m-1}_{0}\notag\\
\phi(t_{*})
&=&\int_{I}{\phi(s)\hat{\bar{\mathcal{G}}}(t_{*},s)ds}, \quad \phi \in \Pi^{2m}_{0}\, .
\end{eqnarray*}
\end{cor}
Let us now restrict our attention to the odd case and
let $L^{2}(I)$ denote the usual Lebesgue space corresponding to the uniform Borel measure on $I$. Then,
it is well known, see e.g. Saitoh \cite[Thm.~1, Pg.~21]{Saitoh}, that the integral operator
\[ \phi \mapsto \int_{I}{\phi(s)\hat{\bar{\mathcal{H}}}(t_{*},s)ds},\quad \phi \in L^{2}(I)\]
determines a reproducing kernel Hilbert space structure on its range with reproducing kernel
\[\mathcal{K}(r_{1},r_{2}):=\int_{I}{\hat{\bar{\mathcal{H}}}(r_{1},s)\hat{\bar{\mathcal{H}}}(r_{2},s)ds}\, .\]
From the definition \eqref{def_H}
\[\mathcal{H}(t_{*},t):=\mathcal{J}^{c}_{ou}(0,t)-\mathcal{J}^{c}_{ol}(t_{*},t)-\mathcal{J}^{c}_{ou}(t_{*},t)\]
and the definitions of
$\mathcal{J}^{c}_{ol}$ and $\mathcal{J}^{c}_{ou}$
from Proposition \ref{prop_Jco}, it follows that
$\mathcal{H}(\cdot ,t) \in \Pi^{2m-1},\, t \in I^{m-1},$
and therefore it follows from \eqref{H_dirichlet} that
$\mathcal{H}(\cdot ,t) \in \Pi^{2m-1}_{0}, \, t \in I^{m-1}.$
Consequently, by marginalization to $\bar{\mathcal{H}}$ and scalar normalization,
we have
\begin{equation}
\label{hi_pi}
\hat{\bar{\mathcal{H}}}(\cdot,s)\in \Pi^{2m-1}_{0}, \quad s \in I\, .
\end{equation}
Therefore the range of this integral operator is contained in $\Pi^{2m-1}_{0}$. However,
it follows from Corollary \ref{cor_RK} that the range is identically $\Pi^{2m-1}_{0}$. Therefore we
conclude that $\Pi^{2m-1}_{0}$ is a reproducing kernel Hilbert space with kernel
$\mathcal{K}$.
Because of Corollary \ref{cor_RK}, one might be tempted to think that this reproducing kernel Hilbert
space structure corresponds to that which $\Pi^{2m-1}_{0}$ inherits as the subspace
$\Pi^{2m-1}_{0} \subset L^{2}(I)$, but this is not the case. Indeed, let $P_{2m-1}$ denote
the $L^{2}(I)$ orthogonal projection $P_{2m-1}:L^{2}(I) \rightarrow \Pi^{2m-1}_{0}$ and consider
the kernel
\begin{equation}
\label{def_K2m-1}
\mathcal{K}_{2m-1}(r_{1},r_{2}):=\int_{I}{P_{2m-1}\hat{\bar{\mathcal{H}}}(r_{1},s)\cdot P_{2m-1}
\hat{\bar{\mathcal{H}}}(r_{2},s)ds}\,
\end{equation}
where the projections are acting on the kernels in the second component. Then, since this projection makes no
difference in the reproducing identities in Corollary \ref{cor_RK}, one can show
that $\mathcal{K}_{2m-1}$ is the reproducing kernel associated with $\Pi^{2m-1}_{0} \subset L^{2}(I)$
and since the latter can be computed
in terms of the Legendre polynomials of order $2$ (see e.g.~\cite[Sec.~12.5]{Arfken}) using the
Christoffel-Darboux formula \cite{Christoffel} (see e.g.~Simon \cite{Simon}
for a more current reference), we conclude an identification of $\mathcal{K}_{2m-1}$ with the
Christoffel-Darboux formula for the kernel of the Legendre polynomials of order $2$. That, is
\begin{equation}
\label{def_CD}
\mathcal{K}_{2m-1}(r_{1},r_{2})=\frac{(r_{1}-r^{2}_{1})(r_{2}-r^{2}_{2})}{2(2m-1)(2m)(2m+1)}
\frac{P''_{2m}(r_{1})P''_{2m-1}(r_{2})-P''_{2m-1}(r_{1})
P''_{2m}(r_{2})}{r_{1}-r_{2}}
\end{equation}
for $ (r_{1},r_{2}) \in I^{2}$,
where $P_{k}$ are the Legendre polynomials shifted to the interval
\begin{equation}
\label{def_Legendre}
P_{k}(r)=\frac{1}{k!}\frac{d^{k}(r^{2}-r)^{k}}{dr^{k}}, \quad r \in I
\end{equation}
and
\begin{equation}
\label{def_Legendre2}
Q_{k}(r):= (r-r^{2}) P''_{k}(r), \quad r \in I
\end{equation}
are the associated Legendre polynomials of order $2$ (see e.g.~\cite[Sec.~12.5]{Arfken}).
Moreover, since
\[\bar{\mathcal{H}}(t_{*},s):=\bar{\mathcal{J}}^{c}_{ou}(0,s)-\bar{\mathcal{J}}^{c}_{ol}(t_{*},s)-
\bar{\mathcal{J}}^{c}_{ou}(t_{*},s),\]
and from Proposition \ref{prop_Jco} we have
\begin{eqnarray*}
\bar{\mathcal{J}}^{c}_{ou}(0,0)& \equiv &0\\
\bar{\mathcal{J}}^{c}_{ol}(t_{*},0)&\equiv &0\\
\bar{\mathcal{J}}^{c}_{ou}(t_{*},0)& >& 0,\quad t_{*}\in (0,1)\, ,
\end{eqnarray*}
we find that
\[\bar{\mathcal{H}}(t_{*},0)=-\bar{\mathcal{J}}^{c}_{ou}(t_{*},0) < 0, \quad t_{*}\in (0,1)\, .\]
Consequently, for $t_{*} \in (0,1)$, it follows that $\hat{\bar{\mathcal{H}}}(t_{*},\cdot) \notin \Pi^{2m-1}_{0}$ and therefore
\[\mathcal{K} \neq \mathcal{K}_{2m-1}\, .\]
Moreover,
from the orthogonal decomposition
\begin{eqnarray*}
\label{dK}
\mathcal{K}(r_{1},r_{2})&=&\int_{I}{\hat{\bar{\mathcal{H}}}(r_{1},s)\hat{\bar{\mathcal{H}}}(r_{2},s)ds}\notag\\
&=&\int_{I}{P_{2m-1}\hat{\bar{\mathcal{H}}}(r_{1},s)\cdot P_{2m-1}\hat{\bar{\mathcal{H}}}(r_{2},s)ds}
+\int_{I}{P^{\perp}_{2m-1}\hat{\bar{\mathcal{H}}}(r_{1},s)\cdot P^{\perp}_{2m-1}\hat{\bar{\mathcal{H}}}(r_{2},s)ds}\notag \\
&=&\mathcal{K}_{2m-1}(r_{1},r_{2})
+\int_{I}{P^{\perp}_{2m-1}\hat{\bar{\mathcal{H}}}\cdot P^{\perp}_{2m-1}\hat{\bar{\mathcal{H}}}(r_{2},s)ds}
\end{eqnarray*}
we conclude
\begin{thm}
\label{thm_RKHS}
Let $\mathcal{K}_{2m-1}$ denote the reproducing kernel for the polynomials
$\Pi^{2m-1}_{0}$ as a subset of $L^{2}(I)$. Then $\mathcal{K}_{2m-1}$ can be expressed by
both \eqref{def_K2m-1} and \eqref{def_CD}.
Moreover, $\Pi^{2m-1}_{0}$ is also a reproducing kernel Hilbert space with kernel
$\mathcal{K}$, and
\[\mathcal{K} -\mathcal{K}_{2m-1}\]
is a reproducing kernel.
\end{thm}
\begin{rmk}
Besides the fact that the kernel $\mathcal{K}$ defining the Hilbert space structure for the polynomials
$\Pi^{2m-1}_{0}$ is not that of the Legendre polynomials, we do not know if this kernel is known, nor do we have
an explicit formula for it. However,
what this section shows is that this kernel and its associated Hilbert space $\Pi^{2m-1}_{0}$ are intimately
connected with the canonical representations of truncated Hausdorff moments, and therefore might be called the
Markov-Kre\u{\i}n kernel. Moreover, if instead of the uniform
measure on the moments, a Selberg type density is used, more such reproducing kernels may be revealed.
\end{rmk}
\section{New Selberg Integral Formulas}
\label{sec_newselberg}
The integral representations of the mean Hausdorff moments of Proposition \ref{prop_mom} provide
new integral identities of Selberg type.
In the following theorem, we provide the first in a sequence
corresponding to when $n$ is odd and even. We then show how to use the reproducing kernel identities
of Theorem \ref{thm_RK} to generate biorthogonal systems of Selberg integral formulas.
\begin{thm}
It holds true that
\label{thm_selberg}
\begin{equation}
\int_{I^{m}}{\Sigma t^{-1}\cdot \prod_{j=1}^{m}{t_{j}^{2}(1-t_{j})^{2}}
\Delta_{m}^{4}(t)
dt}= \frac{S_{m}(5,1,2)-S_{m}(3,3,2)}{2}\, .\end{equation}
and
\begin{equation}
\int_{I^{m}}{\Sigma t^{-1} \cdot \prod_{j=1}^{m}{t_{j}^{2}}\cdot \Delta_{m}^{4}(t)dt}=
\frac{m}{2}S_{m-1}(5,3,2)\, .\end{equation}
\end{thm}
The identities of Theorem \ref{thm_selberg} follow only from the volume equalities, that is, the $i=0$ case of Theorem \ref{thm_RK}.
The following theorem demonstrates how to use all the moment equalities of Theorem \ref{thm_RK} to
generate biorthogonal systems of
Selberg integral formulas. Let us recall definition \eqref{def_H}
\[\mathcal{H}(t_{*},t)=\mathcal{J}^{p}_{ou}(t)-\mathcal{J}^{c}_{ol}(t_{*},t)-\mathcal{J}^{c}_{ou}(t_{*},t)\, .\]
\begin{thm}
\label{thm_selberg2}
Let $n=2m-1$ and
consider the scaled kernel
\[\hat{\mathcal{H}}:=\frac{1}{Vol(M^{2m-1})}\frac{2}{(2m-1)!(m-1)!}\mathcal{H}\, .\]
Then,
\[\hat{\mathcal{H}}(\cdot ,t) \in \Pi^{2m-1}_{0}, \quad t \in I^{m-1}\, .\]
Moreover, consider a basis $\{p_{j},j=1,..,2m-2\}$ for $\Pi^{2m-1}_{0}$ and the resulting expansion
of $\hat{\mathcal{H}}(\cdot ,t)$ in this basis for each $t \in I^{m-1}$;
\[\hat{\mathcal{H}}(t_{*} ,t)=\sum_{j=1}^{2m-2}{h_{j}(t)p_{j}(t_{*})},\quad (t_{*},t) \in I \times I^{m-1}\, .\]
Then, $\{\Sigma p_{j},j=1,..,2m-2\},$ $\{h_{j},j=1,..,2m-2\}$ form an $L^{2}(I^{m-1})$ biorthogonal system.
That is,
\[
\int_{I^{m-1}}{ h_{j}\Sigma p_{k}} =\delta_{jk}, \quad j,k=1,..,2m-2\, \,
\]
\end{thm}
As an immediate corollary, we have
\begin{cor}
\label{cor_selberg2}
Let $n=2m-1$ and
consider the scaled marginal kernel
\[\hat{\bar{\mathcal{H}}}:=\frac{1}{Vol(M^{2m-1})}\frac{2}{(2m-1)!(m-2)!}\bar{\mathcal{H}}\, \]
(note the different scaling than Theorem
\ref{thm_selberg2}).
Then,
\[\hat{\bar{\mathcal{H}}}(\cdot ,s) \in \Pi^{2m-1}_{0}, \quad s \in I\, .\]
Moreover, consider a basis $\{p_{j},j=1,..,2m-2\}$ for $\Pi^{2m-1}_{0}$ and the resulting expansion
of $\hat{\bar{\mathcal{H}}}(\cdot ,s)$ in this basis for each $s \in I$;
\[\hat{\bar{\mathcal{H}}}(t_{*} ,s)=\sum_{j=1}^{2m-2}{\bar{h}_{j}(s)p_{j}(t_{*})},\quad (t_{*},s) \in I \times I\, .\]
Then, $\{p_{j},j=1,..,2m-2\},$ $\{\bar{h}_{j},j=1,..,2m-2\}$ form an $L^{2}(I)$ biorthogonal system.
That is,
\[
\int_{I}{ \bar{h}_{j}p_{k}} =\delta_{jk}, \quad j,k=1,..,2m-2\, .
\]
\end{cor}
The choice of basis for $\Pi^{2m-1}_{0}$ determines
the corresponding component functions $h_{j},j=1,..,2m-2$ and the integrands
$ h_j\Sigma p_{k}$
in Theorem \ref{thm_selberg2}.
Therefore, the task remaining is to select
a basis for which the component functions $h_{j}$ can be determined and such that the
resulting integrals are of interest. When the chosen basis is orthonormal with respect to some
inner product $\langle \cdot, \cdot\rangle$, then the coefficients $h_{j}$ in the representation
\[\hat{\mathcal{H}}(t_{*} ,t)=\sum_{j=1}^{2m-2}{h_{j}(t)p_{j}(t_{*})},\quad (t_{*},t) \in I \times I^{m-1}\, .\]
of Theorem \ref{thm_selberg} are
\[h_{j}(t)=\langle\hat{\mathcal{H}}(\cdot,t), p_{j}\rangle \, .\]
As an example, we now compute these component functions, and therefore determine
explicit forms for these Selberg integrals, when the basis consists of the associated Legendre polynomials of order $2$.
To that end, recall the definitions \eqref{def_Legendre} and \eqref{def_Legendre2} of the Legendre polynomials
and the associated Legendre polynomials of order $2$
translated to the unit interval $I$.
In addition, recall
the $j$-th symmetric function $e_{j}$ defined as
\[ e_{j}(t):=\sum_{i_{1}< \cdots < i_{j}}{t_{i_{1}}\cdots t_{i_{j}}}\]
with $e_{0}:=1$ and the symmetric functions $e_{j}(t,z)$ restricted to the diagonal $t=z$
\begin{equation}
\label{def_e}
e_{j}(t,t):=\sum_{j_{1}+j_{2}=j}{e_{j_{1}}(t)e_{j_{2}}(t)},\quad j=0,..,2m-2\, .
\end{equation}
\begin{thm}
\label{thm_selberg_explicit}
Consider the basis of $\Pi^{2m-1}_{0}$ consisting of the associated Legendre polynomials $Q_{j},j=2,..,2m-1$ of order $2$
translated to the unit interval $I$. For $k=2,..,2m-1$ define
\[a_{jk}:=
\frac{(j+k+k^{2})\Gamma(j+2)\Gamma(j)}{\Gamma(j+k+2)\Gamma(j-k+1)},\quad k \leq j \leq 2m-1\]
\[\tilde{h}_{k}(t):=
\sum_{j=k}^{2m-1}{(-1)^{j+1}a_{jk}e_{2m-1-j}(t,t)}\, .
\]
Then for $j=k\bmod 2$, $j,k=2,..,2m-1$, we have
\[\int_{I^{m-1}}{\tilde{h}_{k}(t)\Sigma Q_{j}(t)\prod_{j'=1}^{m-1}{t_{j'}^{2}}\cdot
\Delta_{m-1}^{4}(t)dt}=Vol(M^{2m-1})(2m-1)!(m-1)!\frac{(k+2)!}{(8k+4)(k-2)!} \delta_{jk}\, .\]
\end{thm}
\section{Proofs}
\subsection{Proof of Theorem \ref{thm_shiva} }
We seek to apply the nested reduction theorem \cite[Thm.~4.11]{BayesOUQ}.
The assertion is trivially true when $\mathcal{L}(\mathcal{A})=\mathcal{U}(\mathcal{A})$ so we can assume
$\mathcal{L}(\mathcal{A})<\mathcal{U}(\mathcal{A})$.
Let us first establish that the assumptions of the theorem are well defined. To that end, note that
\cite[Lem.~3.10]{BayesOUQ} (which follows from Castaing and Valadier
\cite[Lemma III.39\, p.~86]{CastaingValadier:1977}, which in turn follows
from Saint-Beuve's
\cite{SainteBeuve1974} extension of Aumann's Selection Theorem to Suslin spaces) implies that $q\rightarrow \inf_{(f,\mu)\in \Psi^{-1}(q)} \mathbb{D}(f,\mu)[B]$ is
universally measurable and hence the conditions of the theorem are well defined if we extend
the definitions in the usual way when operating on universally measurable sets and functions.
Similarly, since for any $\lambda$ the function $(f,\mu)\mapsto (\Phi(f,\mu)-\lambda)\mathbb{D}(f,\mu)[B]$ is measurable, the function
$\theta:\mathcal{Q}\rightarrow
\mathbb{R}$ defined by
\[\theta(q):=
\sup_{(f,\mu)\in \Psi^{-1}(q)} (\Phi(f,\mu)-\lambda)\mathbb{D}(f,\mu)[B_{ \delta }]
\] is universally
measurable.
For the proof of the theorem, let $\mathbb{Q}\in \mathfrak{Q},\,\mathbb{D}\in \mathfrak{D}$ satisfy the assumptions, and define $\lambda:=\mathcal{U}(\mathcal{A})-2\delta '$.
Consider the events
\[\mathcal{Q}_{ \delta }:=\bigl\{q: \inf_{(f,\mu)\in \Psi^{-1}(q)} \mathbb{D}(f,\mu)[B_{ \delta }]\leq \tau\big\}\]
\[U_{\varepsilon, \delta }= \Bigl\{ q: \sup_{(f,\mu)\in \Psi^{-1}(q),\, \mathbb{D}(f,\mu)[B_{ \delta }]> \varepsilon}\Phi(f,\mu) >\sup_{(f,\mu)\in \mathcal{A}}\Phi(f,\mu) -\delta' \Bigr\}\]
where the assumptions \eqref{eq:dto0app} and \eqref{eq:djkdjehjehj33app} become
\[ \mathbb{Q}\bigl(\mathcal{Q}_{ \delta }\bigr) \geq 1-h( \delta ),\quad \delta >0\]
\[ \mathbb{Q}\bigl(U_{\varepsilon, \delta }\bigr) \geq \varepsilon', \quad \delta >0\]
Let us denote $\tau':=\tau\bigl(\mathcal{U}(\mathcal{A})-\mathcal{L}(\mathcal{A})\bigr) $.
It is easy to see that
\[ \bigl\{\theta \geq -\tau' \bigr\}\supset \mathcal{Q}_{ \delta } \]
and
\[ \{\theta > \varepsilon \delta '\} \supset U_{\varepsilon, \delta } \]
and therefore
\begin{eqnarray*}
\mathbb{Q}\bigl(\{\theta \geq -\tau' \}\bigr)& \geq &\mathbb{Q}\bigl(\mathcal{Q}_{ \delta }\bigr)\\
&\geq& 1-h( \delta )
\end{eqnarray*}
and
\begin{eqnarray*}
\mathbb{Q}\bigl(\{\theta > \varepsilon \delta '\} \bigr)& \geq &\mathbb{Q}\bigl(U_{\varepsilon, \delta }\bigr)\\
&\geq& \varepsilon'\, .
\end{eqnarray*}
Since $\Phi(f,\mu) \geq \mathcal{L}(\mathcal{A}), (f,\mu) \in \mathcal{A}$, it follows
that $|\theta|\leq \mathcal{U}(\mathcal{A})-\mathcal{L}(\mathcal{A})$, and so we obtain
\begin{eqnarray*}
\int{\theta d\mathbb{Q}} &=&
\int_{\{\theta > \varepsilon \delta ' \}}{\theta d\mathbb{Q}}+\int_{\{\theta \leq \varepsilon \delta ' \}}
{\theta d\mathbb{Q}}\\
&>&\varepsilon \delta ' \mathbb{Q}\bigl(\{\theta > \varepsilon \delta ' \}\bigr)+\int_{\{\theta \leq e \delta '
\}}{\theta d\mathbb{Q}}\\
&\geq &\varepsilon \delta ' \mathbb{Q}\bigl(\{\theta > \varepsilon \delta ' \}\bigr)+\int_{\{\theta \leq 0
\}}{\theta d\mathbb{Q}}\\
&\geq &\varepsilon \delta ' \mathbb{Q}\bigl(\{\theta > \varepsilon \delta ' \}\bigr)+\int_{\{\theta < -\tau'
\}}{\theta d\mathbb{Q}}+\int_{\{-\tau' \leq \theta \leq 0
\}}{\theta d\mathbb{Q}}\\
&\geq &\varepsilon \delta ' \mathbb{Q}\bigl(\{\theta > \varepsilon \delta ' \}\bigr)
-\bigl(\mathcal{U}(\mathcal{A})-\mathcal{L}(\mathcal{A})\bigr)
\mathbb{Q}\bigl(\{\theta < -\tau'
\}\bigr)- \tau'\mathbb{Q}\bigl(\{-\tau' \leq \theta \leq 0
\}\bigr)
\\
&\geq &\varepsilon \delta '\varepsilon'-\bigl(\mathcal{U}(\mathcal{A})-\mathcal{L}(\mathcal{A})\bigr)h( \delta ) -\tau'\,.
\end{eqnarray*}
Therefore, for any strictly positive solution $ \delta >0$ to
\[h( \delta )+\tau \leq \frac{\varepsilon \delta '\varepsilon'}{\mathcal{U}(\mathcal{A})-\mathcal{L}(\mathcal{A})} \]
we have
\[ \mathbb{E}_{\mathbb{Q}}[\theta] =\int{\theta d\mathbb{Q}} > 0,\]
where we recall that the function $\theta$ depends on $ \delta $,
and therefore trivially
\[ \sup_{\mathbb{Q}\in \mathfrak{Q},\, \mathbb{D}\in \mathfrak{D}} \mathbb{E}_{q \sim \mathbb{Q}} \left[ \sup_{(f,\mu)\in \Psi^{-1}(q)} \bigl(\Phi(f,\mu)-\lambda\bigr)\mathbb{D}(f,\mu)[B_{ \delta }] \right]
> 0.\]
The assertion then follows from \cite[Thm.~4.11]{BayesOUQ}.
\subsection{Proof of Theorem \ref{thm_shiva_singlesample}}
We will apply
the Mass Supremum Equality \ref{lem_mass_sup}, the Mass Infimum Inequality \ref{lem_mass_inf},
and the Mass of First Moment Inequality \ref{massfirstmoment}. To that end,
define the events
\begin{equation*}
\label{mass_sup_event}
S_{\varepsilon, \delta }=\Bigl\{q\in M^{n}: \exists \mu \in \Psi^{-1}q: \mu(B_{ \delta })> \varepsilon\Bigr\}
\end{equation*}
\begin{equation*}
\label{mass_inf_event}
I_{ \delta }=\Bigl\{q\in M^{n}: \exists \mu \in \Psi^{-1}q: \mu(B_{ \delta })=0\Bigr\}
\end{equation*}
\begin{equation*}
\label{massfirstmoment_event}
FM_{ \delta '}=\Bigl\{q\in M^{n}: q_{1} \in (1- \delta ',1] \Bigr\}
\end{equation*}
First observe that some endpoint conditions have zero mass. For example,
\[\mathbb{Q}(S_{\varepsilon, \delta }):= \mathbb{Q}\bigl(\{q: \exists \mu \in \Psi^{-1}q: \mu(B_{ \delta })> \varepsilon\}\bigr)
= \mathbb{Q}\bigl(\{q: \exists \mu \in \Psi^{-1}q: \mu(B_{ \delta })\geq \varepsilon\}\bigr)\]
and\[\mathbb{Q}(FM_{ \delta '}):=\mathbb{Q}\bigl(\{q\in M^{n}: q_{1} \in (1- \delta ',1] \}\bigr)=\mathbb{Q}\bigl(\{q\in M^{n}: q_{1} \in [1- \delta ',1] \}\bigr) \, .\]
Consequently,
the Mass Supremum Equality \ref{lem_mass_sup} asserts that
\begin{equation*}
\label{mass_sup1}
\mathbb{Q}(S_{\varepsilon, \delta }) = \mathbb{Q}\bigl(\{q: \exists \mu \in \Psi^{-1}q: \mu(B_{ \delta })\geq \varepsilon\}\bigr)\geq
(1- \varepsilon)^{n}
\end{equation*}
where the right-hand side is independent of $ \delta $,
the Mass Infimum Inequality \ref{lem_mass_inf} asserts that
\begin{equation*}
\label{mass_inf1}
\mathbb{Q}(I_{ \delta })\geq 1- \delta (2e)^{2n} \, ,
\end{equation*}
and the Mass of First Moment Inequality \ref{massfirstmoment} asserts that
\begin{equation*}
\label{massfirstmoment1}
\mathbb{Q}(FM_{ \delta '})=\mathbb{Q}\bigl(\{q\in M^{n}: q_{1} \in [1- \delta ',1] \}\bigr) \geq ( \delta ')^{n}\, .
\end{equation*}
Define the events
\[\mathcal{Q}_{ \delta }:=\bigl\{q: \inf_{\mu\in \Psi^{-1}(q)} \mu[B_{ \delta }]=0\big\}\]
\begin{equation*}
U_{\varepsilon, \delta }:= \Bigl\{ q: \sup_{\mu\in \Psi^{-1}(q),\, \mu[B_{ \delta }]> \varepsilon}\mathbb{E}_{\mu}[X] >1 -\delta' \Bigr\}
\end{equation*}
Then since
\[ \mathcal{Q}_{ \delta } \supset I_{ \delta }\]
we have
\[\mathbb{Q}\bigl( \mathcal{Q}_{ \delta }\bigr) \geq \mathbb{Q}\bigl( I_{ \delta }\bigr) \geq 1- \delta (2e)^{2n}\]
and since
\begin{eqnarray*}
U_{\varepsilon, \delta }&=& \Bigl\{ q: \sup_{\mu\in \Psi^{-1}(q),\, \mu[B_{ \delta }]> \varepsilon}\mathbb{E}_{\mu}[X] >1 -\delta' \Bigr\}\\
&=& \Bigl\{ q: q_{1} \in (1- \delta ',1], \exists \mu\in \Psi^{-1}q: \mu[B_{ \delta }]>\varepsilon \Bigr\}\\
&=& S_{\varepsilon, \delta } \cap FM_{ \delta '}
\end{eqnarray*}
we have
\begin{eqnarray*}
\mathbb{Q}\bigl(U_{\varepsilon, \delta }\bigr)& = & \mathbb{Q}\bigl( S_{\varepsilon, \delta } \cap FM_{ \delta '}\bigr)\\
&=& 1-\mathbb{Q}\bigl((S_{\varepsilon, \delta } \cap FM_{ \delta '})^{c}\bigr)\\
&=& 1-\mathbb{Q}\bigl(S^{c}_{\varepsilon, \delta } \cup FM_{ \delta '}^{c}\bigr)\\
&=& 1-\mathbb{Q}\bigl(S^{c}_{\varepsilon, \delta }\bigr) - \mathbb{Q}\bigl(FM_{ \delta '}^{c}\bigr)\\
&=& \mathbb{Q}\bigl(S_{\varepsilon, \delta }\bigr) + \mathbb{Q}\bigl(FM_{ \delta '}\bigr)-1\\
&=& \bigl(1-\varepsilon\bigr)^{n}-1 +( \delta ')^{n}\\
&=& ( \delta ')^{n}-n\varepsilon
\end{eqnarray*}
Consequently, if we choose $\varepsilon:=\frac{( \delta ')^{n}}{2n}$, then
\[\mathbb{Q}\bigl(U_{\varepsilon, \delta }\bigr) \geq \frac{( \delta ')^{n}}{2},\]
so that
the assumptions \eqref{eq:dto0app} and \eqref{eq:djkdjehjehj33app}, expressed as
\[ \mathbb{Q}\bigl(\mathcal{Q}_{ \delta }\bigr) \geq 1-h( \delta ),\quad \delta >0\]
\[ \mathbb{Q}\bigl(U_{\varepsilon, \delta }\bigr) \geq \varepsilon', \quad \delta >0\, ,\]
are satisfied with
$\varepsilon':= \frac{( \delta ')^{n}}{2}$, $\varepsilon:=\frac{( \delta ')^{n}}{2n}$, and
$h( \delta ):= \delta (2e)^{2n}\, .$
We can solve
\begin{eqnarray*}
h( \delta )& \leq& \frac{\varepsilon \delta '\varepsilon'}{\mathcal{U}(\mathcal{A})-\mathcal{L}(\mathcal{A})}\\
&=& \varepsilon \delta '\varepsilon'\\
&=&\frac{( \delta ')^{n}}{2n} \delta ' \frac{( \delta ')^{n}}{2}\\
&=& \frac{( \delta ')^{2n+1}}{4n}
\end{eqnarray*}
by choosing $ \delta \leq \frac{1}{4n} \bigl( \delta '\bigr)^{2n+1}\bigl(2e\bigr)^{-2n}$.
\subsection{Proof of Lemma \ref{lem_mass_sup} }
\label{sec_mass_sup_proof}
For
$ M^{n}_{\varepsilon}:=\bigl\{q \in M^{n}: \exists \mu \in\Psi^{-1}q: \mu(\{t_{*}\})\geq \varepsilon\bigr\}$, it follows that
$\Psi \mu \in M^{n}_{\varepsilon}$ if and only if $\mu =\epsilon \delta_{t_{*}}+(1-\epsilon)\mu_{*}$ with $\mu_{*} \in \mathcal{M}(I)$.
For such a $\mu$ it follows that $\Psi \mu = \epsilon \Psi \delta_{t_{*}}+(1-\epsilon)\Psi\mu_{*}$ and therefore
\[ M^{n}_{\varepsilon}=\epsilon \Psi \delta_{t_{*}}+(1-\epsilon)M^{n}\] from which we conclude that
\[Vol( M^{n}_{\varepsilon})=(1-\epsilon)^{n}Vol(M^{n}),\]
establishing the assertion.
\subsection{Proof of Lemma \ref{lem_mass_inf}}
First consider the odd case, $n=2m-1$.
We utilize the bijective principal representation $\phi^{p}_{ol}: \Lambda^{m-1} \times T^{m} \rightarrow Int(M^{2m-1})$ defined in \eqref{def_phi_pol} and
\[ |det(d\phi^{p}_{ol})|(\lambda,t)= \mathcal{J}^{p}_{ol}(t)
\bigl(1-\sum_{j=1}^{m-1}{\lambda_{j}}\bigr)\prod_{j=1}^{m-1}{\lambda_{j}} \]
where
\[\mathcal{J}^{p}_{ol}(t):=
\Delta^{4}_{m}(t)
\]
from Proposition \ref{prop_Jpo} along with the change of variables formula
\eqref{id_cov}.
Fix $t_{*}\in (0,1)$ and let
\[ T_{ \delta }^{m}:= \bigl\{ (t_{1},..,t_{m})\in T^{m}:
t_{j} \notin B_{ \delta }(t_{*}) , j=1,..,m\, \bigr\} \]
denote those sequences which have no point a distance less than $ \delta $ from $t_{*}$.
It follows that
\[M^{2m-1}_{ \delta } \supset \phi\bigl(\Lambda^{m-1}\times T_{ \delta }^{m}\bigr)\]
and therefore
\begin{equation}
\label{e11111}
Vol\Bigl(M^{2m-1}_{ \delta }\Bigr) \geq Vol\Bigl(\phi^{p}_{ol}\bigl(\Lambda^{m-1}\times T_{ \delta }^{m} \bigr)\Bigr)\, .
\end{equation}
We bound the righthand side from below using the change of variables formula
\eqref{id_cov} as
\begin{eqnarray*}
Vol\Bigl(\phi^{p}_{ol}\bigl(\Lambda^{m-1}\times T_{ \delta }^{m} \bigr)\Bigr)
&=&
\int_{\Lambda^{m-1} \times T_{ \delta }^{m}}{|det(d\phi^{p}_{ol})|
}\\
&=&\int_{\Lambda^{m-1}}{\bigl(1-\sum_{j=1}^{m-1}{\lambda_{j}}\bigr)\bigl(\prod_{j=1}^{m-1}{\lambda_{j}}\bigr)
d\lambda} \int_{T_{ \delta }^{m}}{\mathcal{J}^{p}_{ol}}\, ,
\end{eqnarray*}
and then bounding
\begin{eqnarray*}
\int_{T_{ \delta }^{m}}{\mathcal{J}^{p}_{ol}dt}
&=& \int_{T_{ \delta }^{m}}{\Delta_{m}^{4}(t)dt}\\
&=& \frac{1}{m!} \int_{I_{ \delta }^{m}}{\Delta_{m}^{4}(t)dt}
\end{eqnarray*}
where
\[ I_{ \delta }^{m}:= \bigl\{ (t_{1},..,t_{m})\in I^{m}:
t_{j} \notin B_{ \delta }(t_{*}) , j=1,..,m\, \bigr\} . \]
To bound this from below we bound the integral over
$ \bigl(I_{ \delta }^{m}\bigr)^{c}$ from above.
To that end, let
\[ I_{ \delta ,j}^{m}:= \bigl\{ (t_{1},..,t_{m})\in I^{m}:
t_{j} \in B_{ \delta }(t_{*})\bigr \},\quad j=1,..,m\,, \]
so that
\[ \bigl(I_{ \delta }^{m}\bigr)^{c}= \cup_{j} I_{ \delta ,j}^{m}\, .\]
Therefore, using a union bound and the symmetry of $\Delta$ we have
\begin{eqnarray*}
\int_{\bigl(I_{ \delta }^{m}\bigr)^{c}}{\Delta_{m}^{4}(t)dt}&=&
\int_{\cup_{j'=1}^{m}{I^{m}_{ \delta ,j'}}}{\Delta_{m}^{4}(t)dt}\\
&\leq &\sum_{j'=1}^{m}{\int_{I^{m}_{ \delta ,j'}}{\Delta_{m}^{4}(t)dt}}\\
& =&m\int_{I^{m}_{ \delta ,1}}{\Delta_{m}^{4}(t)dt}\\
& = & m\int_{I^{m}_{ \delta ,1}}{\prod_{1 \leq j < k \leq m}{(t_{k}-t_{j})^{4}}dt_{1}\cdots dt_{m}}\\
& \leq & m\int_{I^{m}_{ \delta ,1}}{\prod_{2 \leq j < k \leq m}{(t_{k}-t_{j})^{4}}dt_{1}\cdots dt_{m}}\\
& = & mVol(B_{ \delta })\int_{I^{m-1}}{\prod_{2 \leq j < k \leq m}{(t_{k}-t_{j})^{4}}dt_{2}\cdots dt_{m}}\\
& = &mVol(B_{ \delta })\int_{I^{m-1}}{\Delta_{m-1}^{4}(t)dt}\\
&\leq & 2m \delta S_{m-1}(1,1,2)\\
\end{eqnarray*}
and so obtain
\begin{eqnarray*}
\frac{Vol\Bigl(\phi(\Lambda^{m-1}\times T_{ \delta }^{m})\Bigr)}{Vol\Bigl(\phi(\Lambda^{m-1}\times T^{m})\Bigr)}
&=& \frac{\int_{I^{ \delta }_{m}}{\Delta_{m}^{4}(t)dt}}{\int_{I^{m}}{\Delta_{m}^{4}(t)dt}}\\
&\geq& 1- 2m \delta \frac{S_{m-1}(1,1,2)}{S_{m}(1,1,2)}\, .
\end{eqnarray*}
Using Selberg's formulas \eqref{selbergformula} we compute
\begin{eqnarray*}
\frac{S_{m-1}(1,1,2)}{S_{m}(1,1,2)}&=&
\frac{ \prod_{j=0}^{m-2}{\frac{\Gamma(1+ 2j)^{2}
\Gamma(3+2j)}{2\Gamma(2(m+j)-2) }}}{ \prod_{j=0}^{m-1}{\frac{\Gamma(1+ 2j)^{2}
\Gamma(3+2j)}{2\Gamma(2(m+j)) }}}\\
&=&\frac{2\Gamma(4m-4)}{\Gamma(2m-1)^{2}\Gamma(2m+1)}
\frac{ \prod_{j=0}^{m-1}{\frac{\Gamma(1+ 2j)^{2}\Gamma(3+2j)}{2\Gamma(2(m+j)-2) }}}{ \prod_{j=0}^{m-1}{\frac{\Gamma(1+ 2j)^{2}
\Gamma(3+2j)}{2\Gamma(2(m+j)) }}}\\
&=&\frac{2\Gamma(4m-4)}{\Gamma(2m-1)^{2}\Gamma(2m+1)}
\prod_{j=0}^{m-1}{\frac{\Gamma(2(m+j)) }{\Gamma(2(m+j)-2)}} \\
&=&\frac{2\Gamma(4m-4)}{\Gamma(2m-1)^{2}\Gamma(2m+1)}
\frac{\Gamma(4m-2) }{\Gamma(2m-2)} \, .
\end{eqnarray*}
To bound
$\frac{2\Gamma(4m-4)\Gamma(4m-2)}{\Gamma(2m-1)^{2}\Gamma(2m)\Gamma(2m-2)}$ from above we use
the binomial relation (see e.g.~\cite[Eq.~6.1.21]{AbramowitsStegun}) for the Gamma function
\begin{equation}
\label{binomial_gamma}
\binom{z}{w}=\frac{\Gamma(z+1)}{\Gamma(w+1)\Gamma(z-w+1)}
\end{equation}
and the inequality (see e.g.~\cite[Eq.~C.5]{CormenLeisersonRivestStein})
\begin{equation}
\label{binomial_upperbound}
\bigl(\frac{z}{w}\bigr)^{w} \leq \binom{z}{w}\leq \bigl(\frac{ez}{w}\bigr)^{w}
\end{equation}
to obtain
\begin{eqnarray*}
\frac{\Gamma(4m-4)\Gamma(4m-2)}{\Gamma(2m-1)^{2}\Gamma(2m)\Gamma(2m-2)}&=&
\frac{\Gamma(4m-4)}{\Gamma(2m-1)\Gamma(2m-2)}\cdot \frac{\Gamma(4m-2)}{\Gamma(2m-1)\Gamma(2m)}\\
&=& \binom{4m-5}{2m-2}\binom{4m-3}{2m-2}\\
&=& \binom{4m-5}{2m-2}\binom{4m-3}{2m-1}\\
&\leq& \bigl(e\frac{4m-5}{2m-2}\bigr)^{2m-2}\bigl(e\frac{4m-3}{2m-1}\bigr)^{2m-1}\\
&\leq& \bigl(2e)^{2m-2}\bigl(2e\bigr)^{2m-1}\\
&=& \bigl(2e)^{4m-3}\\
&\leq &\frac{1}{2} \bigl(2e\bigr)^{4m-2}\, .
\end{eqnarray*}
Recalling \eqref{e11111} establishes the assertion for $n=2m-1$.
Now consider the even case $n-2m$.
We utilize the bijective principal representation $\phi^{p}_{el}: \Lambda^{m} \times T^{m} \rightarrow Int(M^{2m})$ defined in \eqref{def_phi_pel} and, proceeding as in the odd case, we obtain
\begin{eqnarray*}
\frac{Vol\Bigl(M^{2m}_{ \delta }\Bigr)}{Vol\Bigl(M^{2m}\Bigr)}
&\geq& 1- 2m \delta \frac{S_{m-1}(3,1,2)}{S_{m}(3,1,2)}\\
\end{eqnarray*}
Using Selberg's formulas \eqref{selbergformula} we compute
\begin{eqnarray*}
\frac{S_{m-1}(3,1,2)}{S_{m}(3,1,2)}&=&
\frac{ \prod_{j=0}^{m-2}{\frac{\Gamma(1+ 2j)
\Gamma(3+2j)^{2}}{2\Gamma(2(m+j)) }}}{ \prod_{j=0}^{m-1}{\frac{\Gamma(1+ 2j)
\Gamma(3+2j)^{2}}{2\Gamma(2(m+j)+2) }}}\\
&=&\frac{2\Gamma(4m-2)}{\Gamma(2m-1)^{2}\Gamma(2m+1)}
\frac{ \prod_{j=0}^{m-1}{\frac{\Gamma(1+ 2j)
\Gamma(3+2j)^{2}}{2\Gamma(2(m+j)) }}}{ \prod_{j=0}^{m-1}{\frac{\Gamma(1+ 2j)
\Gamma(3+2j)^{2}}{2\Gamma(2(m+j)+2) }}}\\
&=&\frac{2\Gamma(4m-2)}{\Gamma(2m-1)^{2}\Gamma(2m+1)}
\prod_{j=0}^{m-1}{\frac{\Gamma(2(m+j)+2) }{\Gamma(2(m+j))}} \\
&=&\frac{2\Gamma(4m-2)}{\Gamma(2m-1)^{2}\Gamma(2m+1)}
\frac{\Gamma(4m)}{\Gamma(2m)}\, .
\end{eqnarray*}
We will now use the Beta function
\begin{equation}\label{eq:beta1}
B(a,b):=\int_{0}^{1}{t^{a-1}(1-t)^{b-1}dt}, \quad a >0, b>0\,,
\end{equation}
and the identity
\begin{equation}\label{eq:beta2} B(a,b) =\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}\, ,\end{equation}
see e.g.~\cite[Pg.~258]{AbramowitsStegun},
where $\Gamma$ is the gamma function.
To bound
$\frac{2\Gamma(4m-2)\Gamma(4m)}{\Gamma(2m-1)^{2}\Gamma(2m)^{2}}$ from above we use
the inequality
\[ B(a,a) =\frac{\Gamma(a)^{2}}{\Gamma(2a)} \geq \frac{4}{a}2^{-2a}\]
from Proposition \ref{beta_lowerbound}
to obtain
\begin{eqnarray*}
\frac{\Gamma(4m-2)\Gamma(4m)}{\Gamma(2m-1)^{2}\Gamma(2m)^{2}}&=&
\frac{1}{B(2m-1,2m-1)}\frac{1}{B(2m,2m)}\\
&\leq & \frac{2m-1}{4}2^{4m-2}\frac{2m}{4}2^{4m}\\
&\leq & \frac{2m}{4}2^{4m-2}\frac{2m}{4}2^{4m}\\
&\leq&\frac{1}{16} m^{2}2^{8m}
\end{eqnarray*}
Finally, we apply the inequality
\[ m^{2} \leq 8 (\frac{e}{2})^{4m}\]
from Proposition \ref{prop_tech} to conclude that
\begin{eqnarray*}
\frac{2\Gamma(4m-2)\Gamma(4m)}{\Gamma(2m-1)^{2}\Gamma(2m)^{2}}
&\leq&\frac{1}{8} m^{2}2^{8m}\\
&\leq&(\frac{e}{2})^{4m}2^{8m}\\
&=&\bigl(2e\bigr)^{4m}
\end{eqnarray*}
thus establishing the assertion for $n=2m$.
\subsection{Proof of Lemma \ref{massfirstmoment}}
According to Chang, Kemperman, Studden \cite[Thm.~1.3]{ChangKempermanStudden} one can show, using
Skibinsky's canonical coordinates for the moment
problem \cite{Skibinsky}, that the uniform distribution on $M^{n}$ marginalizes to a Beta distribution corresponding to
$B(n,n)$ (see \eqref{eq:beta1} and \eqref{eq:beta2}) on the first moment. Consequently,
\begin{eqnarray*}
\frac{Vol\bigl( q \in M^{n}: q_{1} \in [1- \delta ,1]\bigr)}{Vol(M^{n})}&=&
\frac{1}{B(n,n)}\int_{1- \delta }^{1}{t^{n-1}(1-t)^{n-1}dt}\\
&=& I_{ \delta }(n,n)
\end{eqnarray*}
where $I_{ \delta }(n,n)$ is the Incomplete Beta function (see e.g.~\cite[Pg.~258]{AbramowitsStegun}).
Using the binomial relations \eqref{binomial_gamma} and
and \eqref{binomial_upperbound}, for the upper bound we obtain
\begin{eqnarray*}
I_{ \delta }(n,n)&:=&\frac{1}{B(n,n)}\int_{0}^{ \delta }{t^{n-1}(1-t)^{n-1}dt}\\
& \leq &
\frac{1}{B(n,n)}\int_{0}^{ \delta }{t^{n-1}dt}\\
&=& \frac{ \delta ^{n}}{nB(n,n)}\\
&=& \delta ^{n} \frac{\Gamma(2n)}{n\Gamma(n)^{2}}\\
&=& \delta ^{n} \binom{2n-1}{n}\\
&\leq & \delta ^{n} \Bigl(e \frac{2n-1}{n}\Bigr)^{n}\\
&\leq & \delta ^{n} \bigl(2e\bigr)^{n}
\end{eqnarray*}
and for the lower bound
\begin{eqnarray*}
I_{ \delta }(n,n)&:=&\frac{1}{B(n,n)}\int_{0}^{ \delta }{t^{n-1}(1-t)^{n-1}dt}\\
& \geq & \frac{1}{B(n,n)}\bigl(1- \delta )^{n-1}\int_{0}^{ \delta }{t^{n-1}dt}\\
& = & \frac{1}{nB(n,n)}\bigl(1- \delta )^{n-1} \delta ^{n}\\
& = & \binom{2n-1}{n-1} \bigl(1- \delta )^{n-1} \delta ^{n}\\
& \geq & \Bigl(\frac{2n-1}{n-1}\Bigr)^{n-1} \bigl(1- \delta )^{n-1} \delta ^{n}\\
& \geq & 2^{n-1} \bigl(1- \delta )^{n-1} \delta ^{n}\\
& \geq & \delta ^{n}
\end{eqnarray*}
where the assumption $ \delta \leq \frac{1}{2}$ was used in the last step.
\subsection{Proof of Proposition \ref{prop_Jco}}
The following identity of
Karlin and Shapley \cite[Proof of Thm.~6.2]{KarlinStudden:1966} will be useful in all the Jacobian determinant calculations of this paper:
For $t_{1} < s_{1} < \cdots < t_{m} < s_{m}$, we have
\begin{equation}
\label{karlinstudden}
\frac{\partial^{m}}{\partial s_{1}\cdots \partial s_{m}}
\Delta(t_{1},s_{1},..,
t_{m},s_{m})|_{(s_{1},..,s_{m}) =
(t_{1},..,t_{m})}=
\Delta^{4}_{m}(t)\, .
\end{equation}
We can develop the upper and lower configurations simultaneously, by introducing
a point $t_{0} \in \{0,1\}$ and representations $\phi_{t_{0}}$ where when $t_{0}=0$
we have
$\phi_{0}=\phi^{c}_{ol}$ defined in \eqref{def_phi_col} and when $t_{0}=1$ we have $\phi_{1}=\phi^{c}_{ou}$
defined in \eqref{def_phi_cou}.
So, let us use this notation and a change of indices, and consider the two maps
\[ \phi_{t_{0}}:\Lambda^{m}\times T^{m-1} \rightarrow Int(M^{2m-1}),\quad t_{0}=0,1 \] defined by
\begin{eqnarray}
\label{phi_t0}
\phi_{t_{0}}(\lambda,t)
&= &\Psi \Bigl( \sum_{j=0}^{m-1}{\lambda_{j}\delta_{t_{j}}} +(1-\sum_{j=0}^{m-1}{\lambda_{j}})\delta_{t_{*}}\Bigr)\notag\\
&=& \Bigl(\sum_{j=0}^{m-1}{\lambda_{j}t_{j}^{i}}+ (1-\sum_{j=0}^{m-1}{\lambda_{j})t_{*}^{i}}\Bigr)_{i=1}^{2m-1}\, .
\end{eqnarray}
In this notation, Proposition \ref{prop_Jco} becomes
\begin{prop}
\label{prop_Jco_2}
For $t_{0}=0,1$ we have
\[ |det(d\phi_{t_{0}})|(\lambda,t)= \mathcal{J}_{t_{0}}(t)
\prod_{j=1}^{m-1}{\lambda_{j}} \]
where
\begin{equation*}
\label{Jac_imp0}
\mathcal{J}_{t_{0}}(t)=
\bigl|t_{0}-t_{*}\bigr| \prod_{j=1}^{m-1}{(t_{j}-t_{*})^{2}}\prod_{j=1}^{m-1}{(t_{j}-t_{0})^{2}}
\cdot \Delta_{m-1}^{4}(t)
\end{equation*}
\end{prop}
The differential of $\phi_{t_{0}}$ is determined by
\[\frac{ \partial \phi^{i}_{t_{0}}}{\partial \lambda_{j}}= t^{i}_{j}-t^{i}_{*}, \quad j=0,..,m-1
\]
and
\[ \frac{ \partial \phi^{i}_{t_{0}}}{\partial t_{j}}= i\lambda_{j}t^{i-1}_{j} \quad j=1,..,m-1
\]
for $i=1,..,2m-1,$
from which we conclude that
\[ |det(d\phi_{t_{0}})|= |\mathcal{J}_{t_{0}}|
\prod_{j=1}^{m-1}{\lambda_{j}} \]
where
\begin{eqnarray*}
\mathcal{J}_{t_{0}}&=&
\begin{vmatrix} t_{0}-t_{*} & t_{1}-t_{*} & 1 & \cdots & t_{m-1}-t_{*} & 1 \\
t^{2}_{0}-t^{2}_{*}& t^{2}_{1}-t^{2}_{*} & 2t_{1} & \cdots & t^{2}_{m-1}-t^{2}_{*} & 2t_{m-1} \\
\vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\
t_{0}^{2m-1}-t_{*}^{2m-1}& t^{2m-1}_{1}-t^{2m-1}_{*} & (2m-1)t^{2m-2}_{1}&\cdots & t^{2m-1}_{m-1}-t^{2m-1}_{*}
& (2m-1)t^{2m-2}_{m-1}
\end{vmatrix}
\\
& &\\
&=&
\begin{vmatrix}
1& 1& 1& 0& \cdots &1 & 0\\
t_{*}& t_{0} & t_{1} & 1 & \cdots & t_{m-1} & 1 \\
t_{*}^{2} & t^{2}_{0}& t^{2}_{1} & 2t_{1} & \cdots & t^{2}_{m-1} & 2t_{m-1} \\
\vdots& \vdots&\vdots & \vdots & \cdots & \vdots & \vdots \\
t_{*}^{2m-1} & t_{0}^{2m-1}& t^{2m-1}_{1} & (2m-1)t^{2m-2}_{1}&\cdots & t^{2m-1}_{m-1}
& (2m-1)t^{2m-2}_{m-1}
\end{vmatrix}
\end{eqnarray*}
To
evaluate $\mathcal{J}_{t_{0}}$ for $t_{0}=0,1$,
let
$s_{1},..,s_{m-1}$ satisfy
$t_{j} < s_{j} < t_{j+1}, j=1,..,m-1$ and define the Vandermonde determinant
\begin{equation*}
\label{Jaux}
\mathcal{J}(s_{1},..,s_{m-1}):=
\begin{vmatrix}
1& 1& 1& s_{1}^{0}& \cdots &1 & s_{m-1}^{0}\\
t_{*}& t_{0} & t_{1} & s_{1} & \cdots & t_{m-1} & s_{m-1} \\
t_{*}^{2} & t^{2}_{0}& t^{2}_{1} & s^{2}_{1} & \cdots & t^{2}_{m-1} & s^{2}_{m-1} \\
\vdots& \vdots&\vdots & \vdots & \cdots & \vdots & \vdots \\
t_{*}^{2m-1} & t_{0}^{2m-1}& t^{2m-1}_{1} &s^{2m-1}_{1} &\cdots & t^{2m-1}_{m-1}
& s^{2m-1}_{m-1}
\end{vmatrix}
\end{equation*}
and observe that the multilinearity of the determinant shows that
\[
\mathcal{J}_{t_{0}}=\frac{\partial^{m-1}}{\partial s_{1}\cdots \partial s_{m-1}}\mathcal{J}(s_{1},..,s_{m-1}) |_{(s_{1},..,s_{m-1})=
(t_{1},..,t_{m-1})}
\]
To evaluate this differentiation,
observe that
\begin{eqnarray*}
\mathcal{J}(s_{1},..,s_{m-1})&=&\Delta(t_{*}, t_{0}, t_{1},s_{1},...,t_{m-1},s_{m-1})\\
&=& \bigl(t_{0}-t_{*}\bigr)\Bigl(\prod_{j=1}^{m-1}{(t_{j}-t_{*})(s_{j}-t_{*})}\Bigr)
\Bigl(\prod_{j=1}^{m-1}{(t_{j}-t_{0})(s_{j}-t_{0})}\Bigr) \\
&&\Delta(t_{1},s_{1},...,t_{m-1},s_{m-1})\, ,
\end{eqnarray*}
from which we conclude that
\begin{eqnarray*}
\mathcal{J}_{t_{0}}
&=& \frac{\partial^{m-1}}{\partial s_{1}\cdots \partial s_{m-1}}
\mathcal{J}(s_{1},..,
s_{m-1})|_{(s_{1},..,s_{m-1})=(t_{1},..,t_{m-1})}\\
&=& \bigl(t_{0}-t_{*}\bigr)\cdot \prod_{j=1}^{m-1}{(t_{j}-t_{*})^{2}}
\prod_{j=1}^{m-1}{(t_{j}-t_{0})^{2}}\cdot
\frac{\partial^{m-1}}{\partial s_{1}\cdots \partial s_{m-1}}\\&& \Delta(t_{1},s_{1},...,t_{m-1},s_{m-1})|_{(s_{1},..,s_{m-1})=(t_{1},..,t_{m-1})}
\end{eqnarray*}
Using the identity
\eqref{karlinstudden}
we conclude that
\[\mathcal{J}_{t_{0}}=
\bigl(t_{0}-t_{*}\bigr) \prod_{j=1}^{m-1}{(t_{j}-t_{*})^{2}}\prod_{j=1}^{m-1}{(t_{j}-t_{0})^{2}}
\Delta_{m-1}^{4}(t)
\]
thereby proving Proposition \ref{prop_Jco_2} and therefore Proposition \ref{prop_Jco}.
\subsection{Proof of Proposition \ref{prop_Jce}}
To simplify notation, let $\phi_{l}:=\phi^{c}_{el}$ defined in \eqref{def_phi_cel}
and $\phi_{u}:=\phi^{c}_{eu}$ defined in \eqref{def_phi_ceu}.
We begin with the lower representation $\phi_{l}$.
The differential of $\phi_{l}$ is determined by
\[\frac{ \partial \phi^{i}_{l}}{\partial \lambda_{j}}= t_{j}^{i}-t^{i}_{*}, \quad j=1,..,m\, ,
\]
and
\[ \frac{ \partial \phi^{i}_{l}}{\partial t_{j}}= i\lambda_{j}t^{i-1}_{j} \quad j=1,..,m
\]
for $i=1,..,2m,$
from which we conclude that
\[
|det(d\phi_{l})|= |\mathcal{J}_{l}|
\prod_{j=1}^{m}{\lambda_{j}}
\]
where
\begin{eqnarray*}
\mathcal{J}_{l}&=&
\begin{vmatrix} t_{1}-t_{*} & 1 & \cdots & t_{m}-t_{*} & 1 \\
t^{2}_{1}-t^{2}_{*} & 2t_{1} & \cdots & t^{2}_{m}-t^{2}_{*} & 2t_{m} \\
\vdots & \vdots & \vdots & \cdots & \vdots \\
t^{2m}_{1}-t^{2m}_{*} & 2mt^{2m-1}_{1}&\cdots & t^{2m}_{m}-t^{2m}_{*}
& 2mt^{2m-1}_{m}
\end{vmatrix}
\\
& &\\
&=&
\begin{vmatrix}
1 & 1& 0 & \cdots & 1 & 0 \\
t_{*} & t_{1} & 1 & \cdots & t_{m} & 1 \\
t^{2}_{*}& t^{2}_{1} & 2t_{1} & \cdots & t^{2}_{m} & 2t_{m} \\
\vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\
t_{*}^{2m}& t^{2m}_{1} & 2mt^{2m-1}_{1}&\cdots & t^{2m}_{m}
& 2mt^{2m-1}_{m}
\end{vmatrix}
\\
&
\end{eqnarray*}
To evaluate
$\mathcal{J}_{l}$,
let
$s_{1},..,s_{m}$ satisfy
$t_{j} < s_{j} < t_{j+1}, j=1,..,m$ and define
\[
\mathcal{J}(s_{1},..,s_{m}):=
\begin{vmatrix}
1 & 1 & 1& \cdots & 1 & 1\\
t_{*} & t_{1} & s_{1} & \cdots & t_{m} & s_{m} \\
t^{2}_{*}& t^{2}_{1} & s^{2}_{1} & \cdots & t^{2}_{m} & s^{2}_{m} \\
\vdots & \vdots & \vdots & \cdots & \vdots \\
t_{*}^{2m}& t^{2m}_{1} & s^{2m}_{1}&\cdots & t^{2m}_{m}
& s^{2m}_{m}
\end{vmatrix}
\]
and observe that the multilinearity of the determinant shows that
\[
\mathcal{J}_{l}=\frac{\partial^{m}}{\partial s_{1}\cdots \partial s_{m}}\mathcal{J}(s_{1},..,s_{m}) |_{(s_{1},..,s_{m})=
(t_{1},..,t_{m})}
\]
To evaluate this differentiation, observe
that
\[\mathcal{J}(s_{1},..,s_{m})=\Delta(t_{*},t_{1},s_{1},..,t_{m},s_{m})\]
and, using
the recursion relation of the Vandermonde determinant, we obtain
\begin{eqnarray*}
\mathcal{J}(s_{1},..,s_{m})&=&\Delta(t_{*},t_{1},s_{1},..,t_{m},s_{m})\\
&=&\prod_{j=1}^{m}{(t_{j}-t_{*})(s_{j}-t_{*})} \cdot \Delta(t_{1},s_{1},..,t_{m},
s_{m})
\end{eqnarray*}
from which we conclude that
\begin{eqnarray*}
&& \frac{\partial^{m}}{\partial s_{1}\cdots \partial s_{m}}
\mathcal{J}(s_{1},..,s_{m}) |_{(s_{1},..,s_{m}) =
(t_{1},..,t_{m})}\\
& =&
\prod_{j=1}^{m}{(t_{j}-t_{*})^{2}} \cdot \frac{\partial^{m}}{\partial s_{1}\cdots \partial s_{m}}
\Delta(t_{1},s_{1},..,t_{m},
s_{m})|_{(s_{1},..,s_{m}) =
(t_{1},..,t_{m})}\\
&=& \prod_{j=1}^{m}{(t_{j}-t_{*})^{2}}\cdot \Delta_{m}^{4}(t)
\end{eqnarray*}
and therefore
\[
\mathcal{J}_{l}
= \prod_{j=1}^{m}{(t_{j}-t_{*})^{2}}\cdot \Delta_{m}^{4}(t)
\] thus establishing the lower identity.
Now, for the upper representation $\phi_{u}:=\phi^{c}_{eu}$,
the differential of $\phi_{u}$ is determined by
\[\frac{ \partial \phi^{i}_{u}}{\partial \lambda_{0}}= -t^{i}_{*}, \, ,
\]
\[\frac{ \partial \phi^{i}_{u}}{\partial \lambda_{j}}= t_{j}^{i}-t^{i}_{*}, \quad j=1,..,m-1\, ,
\]
\[\frac{ \partial \phi^{i}_{u}}{\partial \lambda_{m}}= 1-t^{i}_{*}\, ,
\]
and
\[ \frac{ \partial \phi^{i}_{u}}{\partial t_{j}}= i\lambda_{j}t^{i-1}_{j} \quad j=1,..,m-1
\]
for $i=1,..,2m,$
from which we conclude that
\[
|det(d\phi_{u})|= |\mathcal{J}_{u}|
\prod_{j=1}^{m-1}{\lambda_{j}}
\]
where
\begin{eqnarray*}
\mathcal{J}_{u}&=&
\begin{vmatrix} -t_{*} & t_{1}-t_{*} & 1 & \cdots & t_{m-1}-t_{*} & 1 & 1-t_{*} \\
-t^{2}_{*}& t^{2}_{1}-t^{2}_{*} & 2t_{1} & \cdots & t^{2}_{m-1}-t^{2}_{*} & 2t_{m-1}& 1-t^{2}_{*} \\
\vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\
-t_{*}^{2m}& t^{2m}_{1}-t^{2m}_{*} & 2mt^{2m-1}_{1}&\cdots & t^{2m}_{m-1}-t^{2m}_{*}
& 2mt^{2m-1}_{m-1}& 1-t_{*}^{2m}
\end{vmatrix}
\\
& &\\
&=&
\begin{vmatrix} t_{*} & t_{1} & 1 & \cdots & t_{m-1} & 1 & 1 \\
t^{2}_{*}& t^{2}_{1} & 2t_{1} & \cdots & t^{2}_{m-1} & 2t_{m-1}& 1 \\
\vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\
t_{*}^{2m}& t^{2m}_{1} & 2mt^{2m-1}_{1}&\cdots & t^{2m}_{m-1}
& 2mt^{2m-1}_{m-1}& 1
\end{vmatrix}
\\
&
\end{eqnarray*}
To evaluate $\mathcal{J}_{u}$,
let
$s_{1},..,s_{m-1}$ satisfy
$t_{j} < s_{j} < t_{j+1}, j=1,..,m-1$ and define
\[
\mathcal{J}(s_{1},..,s_{m-1}):=
\begin{vmatrix} t_{*} & t_{1} & s_{1} & \cdots & t_{m-1} & s_{m-1} & 1 \\
t^{2}_{*}& t^{2}_{1} & s^{2}_{1} & \cdots & t^{2}_{m-1} & s^{2}_{m-1}& 1 \\
\vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\
t_{*}^{2m}& t^{2m}_{1} & s^{2m}_{1}&\cdots & t^{2m}_{m-1}
& s^{2m}_{m-1}& 1
\end{vmatrix}
\]
and observe that the multilinearity of the determinant shows that
\[
\mathcal{J}_{u}=\frac{\partial^{m-1}}{\partial s_{1}\cdots \partial s_{m-1}}\mathcal{J}(s_{1},..,s_{m-1}) |_{(s_{1},..,s_{m-1})=
(t_{1},..,t_{m-1})}
\]
To evaluate this differentiation, observe that
\[
\mathcal{J}(s_{1},..,s_{m-1})= t_{*} \prod_{j=1}^{m-1}{t_{j}}\prod_{j=1}^{m-1}{s_{j}}
\begin{vmatrix} 1& 1 & 1 & 1 & 1 & \cdots & 1 & 1 & 1 \\
t_{*}& t_{1} &s_{1} & t_{2} & s_{2} & \cdots & t_{m-1} & s_{m-1} & 1 \\
\vdots & \vdots & \vdots & \vdots & \cdots & \vdots & \vdots & \vdots \\
t^{2m-1}_{*} &t^{2m-1}_{1} & s^{2m-1}_{1} & t^{2m-1}_{2} & s^{2m-1}_{2} & \cdots & t^{2m-1}_{m-1} &
s^{2m-1}_{m-1} &
1
\end{vmatrix}
\]
That is, we have
\[
\mathcal{J}(s_{1},..,s_{m-1})= t_{*} \prod_{j=1}^{m-1}{t_{j}}\prod_{j=1}^{m-1}{s_{j}} \cdot \Delta(t_{*},t_{1},s_{1},..,t_{m-1},
s_{m-1},1) \, .
\]
We use the recursion relations
\[ \Delta(t_{*},t_{1},s_{1},..,t_{m-1},
s_{m-1},1)=(1-t_{*})\prod_{j=1}^{m-1}{(1-t_{j})(1-s_{j})} \cdot \Delta(t_{*},t_{1},s_{1},..,t_{m-1},
s_{m-1})\]
and
\[ \Delta(t_{*},t_{1},s_{1},..,t_{m-1},
s_{m-1})=\prod_{j=1}^{m-1}{(t_{j}-t^{*})(s_{j}-t^{*})} \cdot \Delta(t_{1},s_{1},..,t_{m-1},
s_{m-1})\] to obtain
\[
\mathcal{J}(s_{1},..,s_{m-1})= t_{*}(1-t_{*}) \prod_{j=1}^{m-1}{t_{j}(1-t_{j})s_{j}(1-s_{j})}\cdot \prod_{j=1}^{m-1}
{(t_{j}-t_{*})(s_{j}-t_{*})} \cdot \Delta(t_{1},s_{1},..,t_{m-1},
s_{m-1})\, .\]
Consequently, the identity \eqref{karlinstudden} implies
\begin{eqnarray*}
\mathcal{J}_{u}
&=& \frac{\partial^{m-1}}{\partial s_{1}\cdots \partial s_{m-1}}\mathcal{J}(s_{1},..,s_{m-1}) |_{(s_{1},..,s_{m-1}) =
(t_{1},..,t_{m-1})}\\
& =&
t_{*}(1-t_{*}) \prod_{j=1}^{m-1}{t^{2}_{j}(1-t_{j})^{2}} \prod_{j=1}^{m-1}{(t_{j}-t_{*})^{2}} \cdot \frac{\partial^{m-1}}{\partial s_{1}\cdots \partial s_{m-1}}
\\&&\Delta(t_{*},t_{1},s_{1},..,t_{m-1},
s_{m-1})|_{(s_{1},..,s_{m-1}) =
(t_{1},..,t_{m-1})}\\
& =&
t_{*}\bigl(1-t_{*}\bigr) \prod_{j=1}^{m-1}{(t_{j}-t_{*})^{2}}\prod_{j=1}^{m-1}{t_{j}^{2}(1-t_{j})^{2}}
\cdot \Delta_{m-1}^{4}(t)
\end{eqnarray*}
establishing the upper identity and thus completing the proof.
\subsection{Proof of Proposition \ref{prop_mom}}
For the first assertion, let $n=2m-1$.
As in the proof of Proposition \ref{prop_Jco} we find it convenient to analyze the upper and lower configurations simultaneously, by introducing
a point $t_{0} \in \{0,1\}$ and the volume filling representations $\phi_{t_{0}},t_{0} \in \{0,1\}$ where when $t_{0}=0$
we have
$\phi_{0}=\phi^{c}_{ol}$ defined in \eqref{def_phi_col} and when $t_{0}=1$ we have $\phi_{1}=\phi^{c}_{ou}$
defined in \eqref{def_phi_cou}.
In this notation, from \eqref{phi_t0} we have
\begin{eqnarray}
\phi^{i}_{t_{0}}(\lambda_{0},..,\lambda_{m-1};t_{1},..,t_{m-1})
&= &\sum_{j=0}^{m-1}{\lambda_{j}t_{j}^{i}}+ (1-\sum_{j=0}^{m-1}{\lambda_{j})t_{*}^{i}}\notag \\
&= &\sum_{j=0}^{m-1}{\lambda_{j}\bigl(t_{j}^{i}-t_{*}^{i}\bigr)}+ t_{*}^{i}\notag \\
&= &\lambda_{0}\bigl(t_{0}^{i}-t_{*}^{i}\bigr)+\sum_{j=1}^{m-1}{\lambda_{j}\bigl(t_{j}^{i}-t_{*}^{i}\bigr)}+ t_{*}^{i}
\end{eqnarray}
for $t_{0}=0,1$ and
Proposition \ref{prop_Jco_2} expresses the Jacobian determinants as
\[ |det(d\phi_{t_{0}})|(\lambda,t)= \mathcal{J}_{t_{0}}(t)
\prod_{j=1}^{m-1}{\lambda_{j}} \]
where
\begin{equation}
\label{Jac_imp0a}
\mathcal{J}_{t_{0}}(t)=
\bigl|t_{0}-t_{*}\bigr| \prod_{j=1}^{m-1}{(t_{j}-t_{*})^{2}}\prod_{j=1}^{m-1}{(t_{j}-t_{0})^{2}}
\cdot \Delta_{m-1}^{4}(t)\, .
\end{equation}
In this notation, the modified change of variables formula \eqref{id_cov2} becomes
\begin{eqnarray}
\label{eq_odd_cov1}
\int_{M^{2m-1}}{q_{i}}
&=&\sum_{t_{0}=1,2}\int_{\Lambda^{m}\times T^{m-1}}{\phi_{t_{0}}^{i}|d\phi_{t_{0}}|}\notag\\
\end{eqnarray}
for $i \geq 0$.
Therefore, we conclude that
\begin{eqnarray}
\label{eq_odd_cov2}
\int_{M^{2m-1}}{q_{i}}
&=&\sum_{t_{0}=1,2}\int_{\Lambda^{m}\times T^{m-1}}{\phi_{t_{0}}^{i}|d\phi_{t_{0}}|}\notag\\
&=&\sum_{t_{0}=1,2}\int_{\Lambda^{m}\times T^{m-1}}{\phi_{t_{0}}^{i}\bigl(\prod_{j=1}^{m-1}{\lambda_{j}}\bigr)
\mathcal{J}_{t_{0}}}\notag\\
&=&\sum_{t_{0}=1,2}\int_{T^{m-1}}{\Bigl(\int_{\Lambda^{m}}\phi_{t_{0}}^{i}\prod_{j=1}^{m-1}{\lambda_{j}d\lambda}\Bigr)\mathcal{J}_{t_{0}}
}\notag\\
\end{eqnarray}
Performing the $\Lambda^{m}$ integration, using the identities
$\int_{\Lambda^{m}}{\lambda_{m-1}^{2}\prod_{i=1}^{m-2}{\lambda_{i}}d\lambda} = \frac{2}{(2m)!}$,
$\int_{\Lambda^{m}}{\prod_{i=1}^{m}{\lambda_{i}}d\lambda} = \frac{1}{(2m)!}$, and $\int_{\Lambda^{m}}{\prod_{i=1}^{m-1}{\lambda_{i}}d\lambda} = \frac{1}{(2m-1)!}$,
we obtain
\begin{eqnarray}
\int_{\Lambda^{m}}{\phi_{t_{0}}^{i}\prod_{j=1}^{m-1}{\lambda_{j}}}
&=& \int_{\Lambda^{m}}{\Bigl(\lambda_{0}\bigl(t_{0}^{i}-t_{*}^{i}\bigr)+\sum_{j=1}^{m-1}{\lambda_{j}\bigl(t_{j}^{i}-t_{*}^{i}\bigr)}+ t_{*}^{i}\Bigr)\prod_{j=1}^{m-1}{\lambda_{j}}d\lambda}\notag\\
&=& \frac{1}{(2m)!}\bigl(t_{0}^{i}-t_{*}^{i}\bigr)+\frac{2}{(2m)!}\sum_{j=1}^{m-1}{\bigl(t_{j}^{i}-t_{*}^{i}\bigr)}+
\frac{1}{(2m-1)!} t_{*}^{i}\notag\\
&=& \frac{1}{(2m)!}t_{0}^{i}+\frac{2}{(2m)!}\sum_{j=1}^{m-1}{t_{j}^{i}}+\frac{1}{(2m)!} t_{*}^{i}\notag\, .
\end{eqnarray}
Consequently, for $i\geq 1$, we have
\begin{eqnarray}
\label{eq_odd_cov3}
\int_{M^{2m-1}}{q_{i}}
&=&\sum_{t_{0}=1,2}\int_{T^{m-1}}{\Bigl(\int_{\Lambda^{m}}\phi_{t_{0}}^{i}\prod_{j=1}^{m-1}{\lambda_{j}}d\lambda\Bigr)\mathcal{J}_{t_{0}}
}\notag\\
&=&\sum_{t_{0}=1,2}\int_{T^{m-1}}{\Bigl(\frac{1}{(2m)!}t_{0}^{i}+\frac{2}{(2m)!}\sum_{j=1}^{m-1}{t_{j}^{i}}+
\frac{1}{(2m)!} t_{*}^{i}\Bigr)\mathcal{J}_{t_{0}}
}\notag\\
&=&\frac{1}{(2m)!}\int_{T^{m-1}}
{\mathcal{J}_{1}}+\frac{1}{(2m)!} t_{*}^{i}\int_{T^{m-1}}{\Bigl(\mathcal{J}_{0}+
\mathcal{J}_{1}\Bigr)}+\frac{2}{(2m)!}
\int_{T^{m-1}}{\Bigl(\sum_{j=1}^{m-1}{t_{j}^{i}}\Bigr)\Bigl(\mathcal{J}_{0}+
\mathcal{J}_{1}\Bigr)
}\notag
\end{eqnarray}
and, for $i=0$
\begin{eqnarray}
\label{eq_odd_sum1}
\int_{M^{2m-1}}{q_{0}}
&=&\frac{1}{(2m-1)!}\int_{T^{m-1}}
{\Bigl(\mathcal{J}_{0}+\mathcal{J}_{1}\Bigr)}
\end{eqnarray}
which we already knew from \eqref{vol_Jco}.
Combining the two, we obtain for $i\geq 0$
\begin{eqnarray}
\label{eq_odd_cov4}
\int_{M^{2m-1}}{q_{i}}
&=&\frac{ \delta _{0}(i)}{(2m)!}\int_{T^{m-1}}
{\mathcal{J}_{0}}+\frac{1}{(2m)!}\int_{T^{m-1}}
{\mathcal{J}_{1}}
+\frac{1}{(2m)!} t_{*}^{i}\int_{T^{m-1}}{\Bigl(\mathcal{J}_{0}+
\mathcal{J}_{1}\Bigr)}\notag\\
&+&\frac{2}{(2m)!}
\int_{T^{m-1}}{\Bigl(\sum_{j=1}^{m-1}{t_{j}^{i}}\Bigr)\Bigl(\mathcal{J}_{0}+
\mathcal{J}_{1}\Bigr)
}
\end{eqnarray}
and the substitution of the volume equality
\eqref{eq_odd_sum1} (that is, \eqref{vol_Jco}) yields the assertion in the odd case.
For the even case, let $n=2m$, and let us simplify notation by denoting the volume filling representations by
$\phi_{1}:=\phi^{c}_{el}$ and $\phi_{2}:=\phi^{c}_{eu}$ defined in \eqref{def_phi_cel} and
\eqref{def_phi_ceu} so that, in this notation,
\[ \phi_{1}:\Lambda^{m}\times T^{m} \rightarrow Int(M^{2m})\] is defined by
\begin{eqnarray*}
\phi_{1}(\lambda_{1},..,\lambda_{m};t_{1},..,t_{m})
&=& \Bigl(\sum_{j=1}^{m}{\lambda_{j}t_{j}^{i}}+ (1-\sum_{j=1}^{m}{\lambda_{j})t_{*}^{i}}\Bigr)_{i=1}^{2m}
\end{eqnarray*}
and
\[ \phi_{2}:\Lambda^{m+1}\times T^{m-1} \rightarrow Int(M^{2m})\] by
\begin{eqnarray*}
\phi_{2}(\lambda_{0},..,\lambda_{m};t_{1},..,t_{m-1})
&=& \Bigl(\sum_{j=1}^{m-1}{\lambda_{j}t_{j}^{i}}+\lambda_{m}+ (1-\sum_{j=0}^{m}{\lambda_{j})t_{*}^{i}}\Bigr)_{i=1}^{2m}\, .
\end{eqnarray*}
From
Proposition \ref{prop_Jce} we have
\begin{eqnarray*}
|det(d\phi_{1})(\lambda,t)|&=& \mathcal{J}_{1}(t)
\prod_{j=1}^{m}{\lambda_{j}}\\
|det(d\phi_{2})(\lambda,t)|&=& \mathcal{J}_{2}(t)
\prod_{j=1}^{m-1}{\lambda_{j}}
\end{eqnarray*}
where
\begin{eqnarray*}
\mathcal{J}_{1}(t)
&=& \prod_{j=1}^{m}{(t_{j}-t_{*})^{2}}\cdot\Delta_{m}^{4}(t)\\
\mathcal{J}_{2}(t)
&=&
t_{*}\bigl(1-t_{*}\bigr) \prod_{j=1}^{m-1}{(t_{j}-t_{*})^{2}}\prod_{j=1}^{m-1}{t_{j}^{2}(1-t_{j})^{2}}
\cdot\Delta_{m-1}^{4}(t)\, .
\end{eqnarray*}
In this notation, the modified change of variable formula \eqref{id_cov2}
becomes
\begin{equation}
\label{eq_even_sum1}
\int_{M^{2m}}{q_{i}}
=\int_{\Lambda^{m}\times T^{m}}{\phi_{1}^{i}|d\phi_{1}|}+\int_{\Lambda^{m+1}\times T^{m-1}}{\phi_{2}^{i}|d\phi_{2}|}\, .
\end{equation}
We evaluate the two integrals in \eqref{eq_even_sum1} by
\begin{eqnarray*}
\int_{\Lambda^{m}\times T^{m}}{\phi_{1}^{i}|d\phi_{1}|}&=&
\int_{\Lambda^{m}\times T^{m}}{\phi_{1}^{i}\bigl(\prod_{j=1}^{m}{\lambda_{j}}\bigr)\mathcal{J}_{1}}
\notag \\
&=&\int_{T^{m}}{\Bigl(\int_{\Lambda^{m}}\phi_{1}^{i}\prod_{j=1}^{m}{\lambda_{j}d\lambda}\Bigr)\mathcal{J}_{1}
}
\end{eqnarray*}
and
\begin{eqnarray*}
\int_{\Lambda^{m+1}\times T^{m-1}}{\phi_{2}^{i}|d\phi_{1}|}&=&
\int_{\Lambda^{m+1}\times T^{m-1}}{\phi_{2}^{i}\bigl(\prod_{j=1}^{m-1}{\lambda_{j}}\bigr)\mathcal{J}_{2}}
\notag \\
&=&\int_{T^{m-1}}{\Bigl(\int_{\Lambda^{m+1}}\phi_{1}^{i}\prod_{j=1}^{m-1}{\lambda_{j}d\lambda}\Bigr)\mathcal{J}_{2}
}
\end{eqnarray*}
Performing the $\Lambda^{m+1}$ and $\Lambda^{m}$ integrations, using the identities
$\int_{\Lambda^{m}}{\lambda_{1}\prod_{j=1}^{m}{\lambda_{j}}
d\lambda}
= \frac{2}{(2m+1)!}$,
$\int_{\Lambda^{m}}
{\lambda_{m-2}^{2}\prod_{i=1}^{m-3}{\lambda_{i}}d\lambda} = \frac{2}{(2m-1)!}$,
$\int_{\Lambda^{m}}
{\prod_{j=1}^{m-2}{\lambda_{i}}d\lambda} = \frac{1}{(2m-2)!}
$,
$\int_{\Lambda^{m}}{\lambda_{m-1}^{2}\prod_{i=1}^{m-2}{\lambda_{i}}d\lambda} = \frac{2}{(2m)!}$,
$\int_{\Lambda^{m}}{\prod_{i=1}^{m}{\lambda_{i}}d\lambda} = \frac{1}{(2m)!}$, and $\int_{\Lambda^{m}}{\prod_{i=1}^{m-1}{\lambda_{i}}d\lambda} = \frac{1}{(2m-1)!}$,
we obtain for $i\geq 1$
\begin{eqnarray*}
\int_{\Lambda^{m}}\phi_{1}^{i}\prod_{j=1}^{m}{\lambda_{j}d\lambda}&=&
\int_{\Lambda^{m}}\Bigl(\sum_{j=1}^{m}{\lambda_{j}t_{j}^{i}}+ (1-\sum_{j=1}^{m}{\lambda_{j})t_{*}^{i}}\Bigr)\prod_{j=1}^{m}{\lambda_{j}d\lambda}\notag \\
&=&
\int_{\Lambda^{m}}{\Bigl(\sum_{j=1}^{m}{\lambda_{j}(t_{j}^{i}-t_{*}^{i})}+ t_{*}^{i}\Bigr)
\prod_{j=1}^{m}{\lambda_{j}}d\lambda}\notag \\
&=&
\frac{2}{(2m+1)!}\sum_{j=1}^{m}{(t_{j}^{i}-t_{*}^{i})}+\frac{1}{(2m)!} t_{*}^{i}
\notag \\
&=&
\frac{2}{(2m+1)!}\sum_{j=1}^{m}{t_{j}^{i}}+\frac{1}{(2m+1)!} t_{*}^{i}\, .
\end{eqnarray*}
and
\begin{eqnarray*}
\int_{\Lambda^{m+1}}{\phi_{2}^{i}\prod_{j=1}^{m-1}{\lambda_{j}}d\lambda}&=&
\int_{\Lambda^{m+1}}{\Bigl(\sum_{j=1}^{m-1}{\lambda_{j}t_{j}^{i}}+\lambda_{m}+ (1-\sum_{j=0}^{m}{\lambda_{j})t_{*}^{i}}\Bigr)\prod_{j=1}^{m-1}{\lambda_{j}}d\lambda}\notag \\
&=&
\int_{\Lambda^{m+1}}{\Bigl(\sum_{j=1}^{m-1}{\lambda_{j}(t_{j}^{i}-t_{*}^{i})}+\lambda_{m}(1-t_{*}^{i})+ (1-\lambda_{0})
t_{*}^{i}
\Bigr)\prod_{j=1}^{m-1}{\lambda_{j}}d\lambda}\notag \\
&=&
\frac{2}{(2m+1)!}\sum_{j=1}^{m-1}{(t_{j}^{i}-t_{*}^{i})}+ \frac{1}{(2m+1)!}(1-t_{*}^{i})+ (\frac{1}{(2m)!}-\frac{1}{(2m+1)!})
t_{*}^{i}
\notag \\
&=&
\frac{2}{(2m+1)!}\sum_{j=1}^{m-1}{t_{j}^{i}}+ \frac{1}{(2m+1)!}+ \frac{1}{(2m+1)!}
t_{*}^{i}\, .
\end{eqnarray*}
For $i=0$, \eqref{eq_even_sum1} implies
\begin{equation}
\label{vol_eveniii}
Vol(M^{2m})=\frac{1}{(2m)!}\int_{T^{m}}{
\mathcal{J}_{1}}+
\frac{1}{(2m)!}\int_{T^{m-1}}{
\mathcal{J}_{2}}
\notag
\end{equation}
so that for $i\geq 1$ we have
\begin{eqnarray*}
&&\int_{M^{2m}}{q_{i}}\notag\\
&=&\int_{\Lambda^{m}\times T^{m}}{\phi_{1}^{i}|d\phi_{1}|}+\int_{\Lambda^{m+1}\times T^{m-1}}{\phi_{2}^{i}|d\phi_{2}|}\notag \\
&=&\int_{T^{m}}{\Bigl(\frac{2}{(2m+1)!}\sum_{j=1}^{m}{t_{j}^{i}}+\frac{1}{(2m+1)!} t_{*}^{i}\Bigr)\mathcal{J}_{1}}\notag\\
&+&
\int_{T^{m-1}}{\Bigl(\frac{2}{(2m+1)!}\sum_{j=1}^{m-1}{t_{j}^{i}}+ \frac{1}{(2m+1)!}+ \frac{1}{(2m+1)!}
t_{*}^{i}
\Bigr)\mathcal{J}_{2}}\notag\\
&=&\int_{T^{m}}{\Bigl(\frac{2}{(2m+1)!}\sum_{j=1}^{m}{t_{j}^{i}}\Bigr)\mathcal{J}_{1}}\notag \\
&+&
\int_{T^{m-1}}{\Bigl(\frac{2}{(2m+1)!}\sum_{j=1}^{m-1}{t_{j}^{i}}+ \frac{1}{(2m+1)!}
\Bigr)\mathcal{J}_{2}}\notag\\
&+&\frac{t_{*}^{i}}{2m+1}Vol(M^{2m})
\end{eqnarray*}
so that for $i\geq 1$ we conclude
\begin{eqnarray*}
&&\int_{M^{2m}}{q_{i}}- \frac{t_{*}^{i}}{2m+1}Vol(M^{2m})\notag\\
&=&\frac{1}{(2m+1)!}\int_{T^{m}}{\Bigl(2\sum_{j=1}^{m}{t_{j}^{i}}\Bigr)\mathcal{J}_{1}}
+
\frac{1}{(2m+1)!}\int_{T^{m-1}}{\Bigl(2\sum_{j=1}^{m-1}{t_{j}^{i}}+1
\Bigr)\mathcal{J}_{2}}\notag\, .
\end{eqnarray*}
Combining with the result \eqref{vol_eveniii} for $i=0$
we conclude
\begin{eqnarray*}
&&\int_{M^{2m}}{q_{i}}- \frac{t_{*}^{i}}{2m+1}Vol(M^{2m})\notag\\
&=&\frac{2}{(2m+1)!}\int_{T^{m}}{\sum_{j=1}^{m}{t_{j}^{i}}\mathcal{J}_{1}}
+
\frac{2}{(2m+1)!}\int_{T^{m-1}}{\sum_{j=1}^{m-1}{t_{j}^{i}}
\mathcal{J}_{2}}+\frac{1+ \delta _{0}(i)}{(2m+1)!}\int_{T^{m-1}}{
\mathcal{J}_{2}}\notag
\end{eqnarray*}
establishing the assertion in the even case.
\subsection{Proof of Theorem \ref{thm_RK}}
Recall the identity $\mathcal{J}^{c}_{ol}(0,t)\equiv 0$. Then
subtracting the volume identity \eqref{vol_Jco}
\begin{equation}
\label{vol_Jco2}
Vol(M^{2m-1}) =\frac{1}{(2m-1)!(m-1)!} \int_{I^{m-1}}{\bigl(\mathcal{J}^{c}_{ol}(t_{*},t)+
\mathcal{J}^{c}_{ou}(t_{*},t)\bigr)dt}\,
\end{equation}
from itself evaluated at $t_{*}=0$,
we conclude that
\begin{equation}
\label{vol_subtract_odd}
\int_{I^{m-1}}{\bigl(\mathcal{J}^{c}_{ol}(t_{*},t)+
\mathcal{J}^{c}_{ou}(t_{*},t)-\mathcal{J}^{c}_{ou}(0,t)\bigr)dt}\, \equiv 0\, ,
\end{equation}
that is,
\begin{equation*}
\label{vol_Jco3}
\int_{I^{m-1}}{\mathcal{H}(t_{*},t)
dt}\, \equiv 0\, .
\end{equation*}
We now do the same subtraction for all the moments. To that end, recall the convention
$0^{0}=1$, and
observe that, for $i\geq 0$, the identity
\begin{eqnarray}
\label{eq_odd_cov5b}
&&\int_{M^{2m-1}}{q_{i}}-\frac{t_{*}^{i}}{2m} Vol(M^{2m-1})\notag\\
&=&\frac{ \delta _{0}(i)}{(2m)!(m-1)!}\int_{I^{m-1}}
{\mathcal{J}^{c}_{ol}(t_{*},t)dt}+\frac{1}{(2m)!(m-1)!}\int_{I^{m-1}}
{\mathcal{J}^{c}_{ou}(t_{*},t)dt}\notag\\
&+&\frac{2}{(2m)!(m-1)!}\int_{I^{m-1}}{\Sigma t^{i}
\Bigl(\mathcal{J}^{c}_{ol}(t_{*},t)+\mathcal{J}^{c}_{ou}(t_{*},t)\Bigr)dt}\,
\end{eqnarray}
from Proposition \ref{prop_mom},
evaluated at $t_{*}=0$ becomes
\begin{eqnarray*}
&&\int_{M^{2m-1}}{q_{i}}-\frac{ \delta _{0}(i)}{2m} Vol(M^{2m-1})\notag\\
&=&\frac{ \delta _{0}(i)}{(2m)!(m-1)!}\int_{I^{m-1}}
{\mathcal{J}^{c}_{ol}(0,t)dt}+\frac{1}{(2m)!(m-1)!}\int_{I^{m-1}}
{\mathcal{J}^{c}_{ou}(0,t)dt}\notag\\
&+&\frac{2}{(2m)!(m-1)!}\int_{I^{m-1}}{\Sigma t^{i}
\Bigl(\mathcal{J}^{c}_{ol}(0,t)+\mathcal{J}^{c}_{ou}(0,t)\Bigr)dt}\notag \, .
\end{eqnarray*}
Subtracting from \eqref{eq_odd_cov5b},
using the identity $\mathcal{J}^{c}_{ol}(0,t)\equiv 0$, we obtain
\begin{eqnarray*}
&&-\frac{t_{*}^{i}}{2m} Vol(M^{2m-1})
+ \frac{ \delta _{0}(i)}{2m} Vol(M^{2m-1})\notag\\
&=&\frac{ \delta _{0}(i)}{(2m)!(m-1)!}\int_{I^{m-1}}
{\mathcal{J}^{c}_{ol}(t_{*},t)dt}+\frac{1}{(2m)!(m-1)!}\int_{I^{m-1}}
{\bigl(\mathcal{J}^{c}_{ou}(t_{*},t)-\mathcal{J}^{c}_{ou}(0,t)\bigr)dt}\notag\\
&-&\frac{2}{(2m)!(m-1)!}\int_{I^{m-1}}{\Sigma t^{i}
\mathcal{H}(t_{*},t)dt}\notag
\end{eqnarray*}
and applying the volume identity \eqref{vol_Jco2} we obtain
\begin{eqnarray*}
&&-\frac{t_{*}^{i}}{2m} Vol(M^{2m-1})\notag\\
&=&-\frac{ \delta _{0}(i)}{(2m)!(m-1)!}\int_{I^{m-1}}
{\mathcal{J}^{c}_{ou}(t_{*},t)dt}+\frac{1}{(2m)!(m-1)!}\int_{I^{m-1}}
{\bigl(\mathcal{J}^{c}_{ou}(t_{*},t)-\mathcal{J}^{c}_{ou}(0,t)\bigr)dt}\notag\\
&-&\frac{2}{(2m)!(m-1)!}\int_{I^{m-1}}{\Sigma t^{i}
\mathcal{H}(t_{*},t)dt}\notag
\end{eqnarray*}
and the subtracted volume identity \eqref{vol_subtract_odd} we obtain with a change of sign
\begin{eqnarray}
\label{eq_odd_cov5c2}
&&\frac{t_{*}^{i}}{2m} Vol(M^{2m-1})\\
&=&\frac{ \delta _{0}(i)}{(2m)!(m-1)!}\int_{I^{m-1}}
{\mathcal{J}^{c}_{ou}(t_{*},t)dt}+\frac{1}{(2m)!(m-1)!}\int_{I^{m-1}}
{\mathcal{J}^{c}_{ol}(t_{*},t)dt}\notag\\
&+&\frac{2}{(2m)!(m-1)!}\int_{I^{m-1}}{\Sigma t^{i}
\mathcal{H}(t_{*},t)dt}\notag\, .
\end{eqnarray}
Then, if we let $\phi(s):=\sum_{i=0}^{2m-1}{\phi_{i}s^{i}}$ be a polynomial of degree
$n=2m-1$, summing over each identity in \eqref{eq_odd_cov5c2}, we conclude that
\begin{eqnarray}
\label{eq_odd_cov5d}
&&\frac{\phi(t_{*})}{2m} Vol(M^{2m-1})\notag\\
&=&\phi_{0}\frac{1}{(2m)!(m-1)!}\int_{I^{m-1}}
{\bigl(\mathcal{J}^{c}_{ol}(t_{*},t)+\mathcal{J}^{c}_{ou}(t_{*},t)\bigr)dt}
+\sum_{i=1}^{2m-1}{\phi_{i}}\frac{1}{(2m)!(m-1)!}\int_{I^{m-1}}
{\mathcal{J}^{c}_{ol}(t_{*},t)dt}\notag\\
&+&\frac{2}{(2m)!(m-1)!}\int_{I^{m-1}}{(\Sigma \phi)(t)
\mathcal{H}(t_{*},t))dt}\notag\\
&=&\phi_{0}\frac{1}{(2m)!(m-1)!}\int_{I^{m-1}}
{\mathcal{J}^{c}_{ou}(t_{*},t)dt}
+\sum_{i=0}^{2m-1}{\phi_{i}}\frac{1}{(2m)!(m-1)!}\int_{I^{m-1}}
{\mathcal{J}^{c}_{ol}(t_{*},t)dt}\notag\\
&+&\frac{2}{(2m)!(m-1)!}\int_{I^{m-1}}{(\Sigma \phi)(t)
\mathcal{H}(t_{*},t)dt}\notag\, .
\end{eqnarray}
Since $\phi(0)=\phi_{0}$ and $\phi(1)=\sum_{i=0}^{2m-1}{\phi_{i}}$ the assertion follows by multiplication by
$2m$.
The even case proceeds in the same way, but since it is a little different we have included it in Section \ref{sec_RKproof} in the Appendix.
\subsection{Proof of Theorem \ref{thm_selberg}}
\label{sec_selberg_proof}
For the first assertion, let $n=2m-1$ and consider the integral formula \eqref{vol_Jco2}
\[
Vol(M^{2m-1}) =\frac{1}{(2m-1)!(m-1)!} \int_{I^{m-1}}{\bigl(\mathcal{J}^{c}_{ol}(t_{*},t)+
\mathcal{J}^{c}_{ou}(t_{*},t)\bigr)dt}\,
\]
for the volume in terms of the canonical representations.
From the definitions
\begin{eqnarray*}
\mathcal{J}^{c}_{ol}(t_{*},t)&=&
t_{*} \prod_{j=1}^{m-1}{(t_{j}-t_{*})^{2}}\prod_{j=1}^{m-1}{t_{j}^{2}}
\cdot \Delta_{m-1}^{4}(t)
\\
\mathcal{J}^{c}_{ou}(t_{*},t)&=&
\bigl(1-t_{*}\bigr) \prod_{j=1}^{m-1}{(t_{j}-t_{*})^{2}}
\prod_{j=1}^{m-1}{(1-t_{j})^{2}}\cdot
\Delta_{m-1}^{4}(t)\,
\end{eqnarray*}
of Proposition \ref{prop_Jco},
we obtain
\begin{eqnarray*}
\frac{\partial}{\partial t_{*}}\mathcal{J}^{c}_{ol}(t_{*},t)|_{t_{*}=0}&=&
\prod_{j=1}^{m-1}{t_{j}^{4}}\cdot \Delta_{m-1}^{4}(t)\\
\frac{\partial}{\partial t_{*}}\mathcal{J}^{c}_{ou}(t_{*},t)|_{t_{*}=0}&=&
-\bigl(1+ 2\Sigma t^{-1}\bigr)\prod_{j=1}^{m-1}{t_{j}^{2}(1-t_{j})^{2}}
\Delta_{m-1}^{4}(t)\, .
\end{eqnarray*}
Differentiating the volume formula with respect to $t_{*}$ at $t_{*}=0$, we obtain
\begin{eqnarray*}
0&=& \frac{1}{(2m-1)!(m-1)!} \int_{I^{m-1}}{\Bigl(\frac{\partial}{\partial t_{*}}\mathcal{J}^{c}_{ol}(t_{*},t)|_{t_{*}=0}+
\frac{\partial}{\partial t_{*}}\mathcal{J}^{c}_{ou}(t_{*},t)|_{t_{*}=0}\Bigr)dt}\notag\\
\end{eqnarray*}
and therefore
\begin{eqnarray*}
\int_{I^{m-1}}{\prod_{j=1}^{m-1}{t_{j}^{4}}\cdot \Delta_{m-1}^{4}(t)dt}&=&
\int_{I^{m-1}}{\bigl(1+ 2\Sigma t^{-1}\bigr)\prod_{j=1}^{m-1}{t_{j}^{2}(1-t_{j})^{2}}
\Delta_{m-1}^{4}(t)
dt}\notag\\
&=&
\int_{I^{m-1}}{\prod_{j=1}^{m-1}{t_{j}^{2}(1-t_{j})^{2}}
\Delta_{m-1}^{4}(t)
dt}\\
&+&2\int_{I^{m-1}}{\Sigma t^{-1}\cdot \prod_{j=1}^{m-1}{t_{j}^{2}(1-t_{j})^{2}}
\Delta_{m-1}^{4}(t)
dt}
\end{eqnarray*}
from which we conclude that
\[2\int_{I^{m-1}}{\Sigma t^{-1}\cdot \prod_{j=1}^{m-1}{t_{j}^{2}(1-t_{j})^{2}}
\Delta_{m-1}^{4}(t)
dt}= S_{m-1}(5,1,2)-S_{m-1}(3,3,2)\, .\]
Changing $m \mapsto m+1$ finishes the proof of the first assertion.
For the second assertion, let $n=2m$ and consider the integral formula \eqref{vol_Jce}
\[
Vol(M^{2m})=\frac{1}{(2m)!m!}\int_{I^{m}}{\mathcal{J}^{c}_{el}(t_{*},t)dt}+
\frac{1}{(2m)!(m-1)!}\int_{I^{m-1}}{\mathcal{J}^{c}_{eu}(t_{*},t)dt}\, .
\]
From the definitions
\begin{eqnarray*}
\mathcal{J}^{c}_{el}(t_{*},t)
&=& \prod_{j=1}^{m}{(t_{j}-t_{*})^{2}}\cdot \Delta_{m}^{4}(t)\\
\mathcal{J}^{c}_{eu}(t_{*},t)
&=&
t_{*}\bigl(1-t_{*}\bigr) \prod_{j=1}^{m-1}{(t_{j}-t_{*})^{2}}\prod_{j=1}^{m-1}{t_{j}^{2}(1-t_{j})^{2}}
\cdot \Delta_{m-1}^{4}(t)\,
\end{eqnarray*}
of Proposition \ref{prop_Jce}, we obtain
\begin{eqnarray*}
\frac{\partial}{\partial t_{*}}\mathcal{J}^{c}_{el}(t_{*},t)|_{t_{*}=0}&=&
-2\Sigma t^{-1}\prod_{j=1}^{m}{t_{j}^{2}}\cdot \Delta_{m}^{4}(t)\\
\frac{\partial}{\partial t_{*}}\mathcal{J}^{c}_{eu}(t_{*},t)|_{t_{*}=0}&=&
\prod_{j=1}^{m-1}{t_{j}^{4}(1-t_{j})^{2}}
\cdot \Delta_{m-1}^{4}(t)\, .
\end{eqnarray*}
Differentiating the volume formula with respect to $t_{*}$ at $t_{*}=0$, we obtain
\begin{eqnarray*}
0&=&\frac{1}{(2m)!m!}\int_{I^{m}}{\frac{\partial}{\partial t_{*}}\mathcal{J}^{c}_{el}(t_{*},t)|_{t_{*}=0}dt}+
\frac{1}{(2m)!(m-1)!}\int_{I^{m-1}}{\frac{\partial}{\partial t_{*}}\mathcal{J}^{c}_{eu}(t_{*},t)|_{t_{*}=0}dt}
\end{eqnarray*}
and therefore we conclude
\begin{eqnarray*}
2 \int_{I^{m}}{\Sigma t^{-1} \cdot \prod_{j=1}^{m}{t_{j}^{2}}\cdot \Delta_{m}^{4}(t)dt}&=&
m\int_{I^{m-1}}{\prod_{j=1}^{m-1}{t_{j}^{4}(1-t_{j})^{2}}
\cdot \Delta_{m-1}^{4}(t)
dt}\\
&=& mS_{m-1}(5,3,2)\, ,
\end{eqnarray*}
finishing the proof of the second assertion.
\subsection{Proof of Theorem \ref{thm_selberg2}}
First note that the definition \eqref{def_H}
\[\mathcal{H}(t_{*},t):=\mathcal{J}^{c}_{ou}(0,t)-\mathcal{J}^{c}_{ol}(t_{*},t)-\mathcal{J}^{c}_{ou}(t_{*},t)\]
and the definitions of
$\mathcal{J}^{c}_{ol}$ and $\mathcal{J}^{c}_{ou}$
from Proposition \ref{prop_Jco} imply that
$\hat{\mathcal{H}}(\cdot ,t) \in \Pi^{2m-1},\, t \in I^{m-1}.$
Therefore, it follows from \eqref{H_dirichlet} that
$\hat{\mathcal{H}}(\cdot ,t) \in \Pi^{2m-1}_{0}, \, t \in I^{m-1}.$
Now, it follows from Theorem \ref{thm_RK} that
\begin{eqnarray*}
\phi(t_{*})
&=&\int_{I^{m-1}}{(\Sigma \phi)(t)\hat{\mathcal{H}}(t_{*},t)dt}, \quad \phi \in \Pi^{2m-1}_{0}
\end{eqnarray*}
which expanded becomes
\begin{eqnarray*}
\phi(t_{*})
&=&\sum_{j=1}^{2m-2}{p_{j}(t_{*})\int_{I^{m-1}}{(\Sigma \phi)(t)h_{j}(t)dt}}, \quad \phi \in \Pi^{2m-1}_{0}\, ,
\end{eqnarray*}
in particular, by choosing $\phi:=p_{k}, k=1,..,2m-2$,
\begin{eqnarray*}
p_{k}(t_{*})&=&\sum_{j=1}^{2m-2}{p_{j}(t_{*})\int_{I^{m-1}}{(\Sigma p_{k})(t)h_{j}(t)dt}}, \quad k=1,..,2m-2\, ,
\end{eqnarray*}
from which we conclude
\begin{equation*}
\label{selbergfamily}
\int_{I^{m-1}}{\Sigma p_{j}\cdot h_{k}} =\delta_{jk}, \quad j,k=1,..,2m-2\, \,
\end{equation*}
establishing the assertion. Furthermore, from this and the symmetry of $h_{k},,k=1,..,2m-2$ with respect to the
action of the symmetric group, we also conclude
\begin{equation*}
\label{selbergfamilymarg}
(m-1)\int_{I}{ p_{j}\cdot \bar{h}_{k}} =\delta_{jk}, \quad j,k=1,..,2m-2\, \,
\end{equation*}
establishing Corollary \ref{cor_selberg2}.
\subsection{Proof of Theorem \ref{thm_selberg_explicit}}
From the orthogonality relation (see e.g.~\cite[Eq.~12.110]{Arfken})
\[\int_{I}{Q_{j}Q_{k}}=\frac{1}{2k+1}\frac{(k+2)!}{(k-2)!} \delta _{jk}\]
and the definition
\[\hat{\mathcal{H}}:=\frac{1}{Vol(M^{2m-1})}\frac{2}{(2m-1)!(m-1)!}\mathcal{H}\, \]
of the scaling of the kernel
\eqref{def_H}
\begin{equation}
\label{def_H2}
\mathcal{H}(t_{*},t)=\mathcal{J}^{p}_{ou}(t)-\mathcal{J}^{c}_{ol}(t_{*},t)-\mathcal{J}^{c}_{ou}(t_{*},t)\, ,
\end{equation}
we can compute
the coefficients $h_{k}$ in the expansion
\[\hat{\mathcal{H}}(t_{*} ,t)=\sum_{j=2}^{2m-1}{h_{j}(t)Q_{j}(t_{*})},\quad (t_{*},t) \in I \times I^{m-1}\, \]
as
\begin{equation}
\label{def_h}
h_{k}(t)= \frac{1}{Vol(M^{2m-1})}\frac{2}{(2m-1)!(m-1)!}\frac{(2k+1)(k-2)!}{(k+2)!}\int_{I}{\mathcal{H}(\cdot,t)Q_{k}(\cdot)},\quad k=2,..,2m-1\,
\end{equation}
and then apply
Theorem \ref{thm_selberg2} to obtain the assertion.
To that end, for $k=2,..,2m-1$,
to compute
\[\int_{I}{\mathcal{H}(\cdot,t)Q_{k}(\cdot)}\]
we
use the decomposition of \eqref{def_H2} of $\mathcal{H}$ and compute the values
$\int_{I}{\mathcal{J}^{p}_{ou}(t)Q_{k}(\cdot)},
\int_{I}{\mathcal{J}^{c}_{ol}(\cdot,t)Q_{k}(\cdot)}, $ and
$\int_{I}{\mathcal{J}^{c}_{ou}(\cdot,t)Q_{k}(\cdot)} $ separately.
For the first term, observe that
$
\int_{I}{Q_{k}}=1+(-1)^{k},\, k=2,...,2m-1$ from Proposition \ref{legendre_integral},
so that
\begin{equation}
\label{e111}
\int_{I}{\mathcal{J}^{p}_{ou}(t)Q_{k}(r)dr}=\mathcal{J}^{p}_{ou}(t)\int_{I}{Q_{k}(r)dr}=(1+(-1)^{k})\mathcal{J}^{p}_{ou}(t)\, .
\end{equation}
For the second, $\int_{I}{\mathcal{J}^{c}_{ol}(\cdot,t)Q_{k}(\cdot)}$,
we expand $\mathcal{J}^{c}_{ol}(\cdot,t)$, defined in \eqref{prop_Jco}, as a polynomial for fixed $t$
and then utilize the values of the integrals
$\int_{I}{r^{j}Q_{k}(r)dr}$
for the monomials $r^{j}, j=1,..,2m-1$.
To that end, define
\begin{equation}
\label{def_tildeJ}
\tilde{\mathcal{J}}^{c}_{ol}(\cdot,t):=t{*}\prod_{j=1}^{m-1}{(t_{j}-t_{*})^{2}}
\end{equation}
so that
\[\mathcal{J}^{c}_{ol}(\cdot,t)=\tilde{\mathcal{J}}^{c}_{ol}(\cdot,t)\prod_{j=1}^{m-1}{t_{j}^{2}}\cdot
\Delta_{m-1}^{4}(t)\, .\]
Then from the definition of $\mathcal{J}^{p}_{el}$ of \eqref{prop_Jpe}
we have $\mathcal{J}^{p}_{el}(t)=\prod_{j=1}^{m-1}{t_{j}^{2}}\cdot
\Delta_{m-1}^{4}(t)$ so that
\begin{equation}
\label{Jdecomp}
\mathcal{J}^{c}_{ol}(\cdot,t)=\tilde{\mathcal{J}}^{c}_{ol}(\cdot,t)
\mathcal{J}^{p}_{el}(t)\, .
\end{equation}
The generating function identity for the elementary symmetric functions $e_{j}$ is
\[ \prod_{i=1}^{m-1}(1+s t_{i})=\sum_{j=0}^{m-1}{e_{j}(t)s^{j}}\]
and squaring it we obtain
\begin{eqnarray}
\prod_{i=1}^{m-1}{(1+s t_{i})^{2}}& =&\Bigl(\sum_{j=0}^{m-1}{e_{j}(t)s^{j}}\Bigr)^{2}\notag \\
&=& \sum_{j=0}^{2m-2}{e_{j}(t,t)s^{j}}\, .
\end{eqnarray}
Therefore, by changing $s \mapsto -t_{*}^{-1}$ we conclude that
\begin{eqnarray}
\prod_{j=1}^{m-1}{(t_{j}-t_{*})^{2}}& =&
\sum_{j=0}^{2m-2}{e_{2m-2-j}(t,t)(-1)^{j}t_{*}^{j}}
\end{eqnarray}
and therefore
\[t{*}\prod_{j=1}^{m-1}{(t_{j}-t_{*})^{2}}=
\sum_{j=0}^{2m-2}{e_{2m-2-j}(t,t)(-1)^{j}t_{*}^{j+1}}
\]
and, relabeling, we conclude
\begin{equation}
\label{J_exp}
\tilde{\mathcal{J}}^{c}_{ol}(t_{*},t)= - \sum_{j=0}^{2m-1}{d_{j}(t)(-1)^{j}t_{*}^{j}}
\end{equation}
where
\begin{equation}
\label{def_d}
d_{j}(t):=e_{2m-1-j}(t,t)=\sum_{j_{1}+j_{2}=2m-1-j}{e_{j_{1}}(t)e_{j_{2}}(t)},\quad j=1,..,2m-1\,
\end{equation}
and $d_{0}(t):=0$. Note that $d_{j}$ is a symmetric polynomial of degree $2m-1-j$.
In particular,
from the definition \eqref{def_tildeJ} and its resulting polynomial expansion \eqref{J_exp} we have
\begin{equation}
\label{J1}
\tilde{\mathcal{J}}^{c}_{ol}(1,t)=-\sum_{j=0}^{2m-1}{d_{j}(t)(-1)^{j}}= \prod_{j=1}^{m-1}{(1-t_{j})^{2}}\, .
\end{equation}
The following proposition computes the values of the integrals of the Legendre polynomials
against the monomials,
and we observe that $\int_{I}{r^{j}Q_{k}(r)dr}=1$ for $1\leq j <k$
and $\int_{I}{r^{j}Q_{k}(r)dr}=1$ plus a term when $j \geq k$.
\begin{prop}
\label{legendre_integral}
For $k=2,..,2m-1$ we have
\[\int_{I}{Q_{k}(r)dr}=1+(-1)^{k}\]
\[\int_{I}{r^{j}Q_{k}(r)dr}=1,\quad 1\leq j <k\]
\[\int_{I}{r^{j}Q_{k}(r)dr}=1-\frac{(j+k+k^{2})\Gamma(j+2)\Gamma(j)}{\Gamma(j+k+2)\Gamma(j-k+1)},
\quad j\geq k\, .\]
\end{prop}
Therefore,
since $d_{0}=0$, we obtain from the polynomial expansion \eqref{J_exp},
Proposition \ref{legendre_integral}, and \eqref{J1} that
\begin{eqnarray}
\label{JQol}
\int_{I}{ \tilde{\mathcal{J}}^{c}_{ol}(r,t)Q_{k}(r)dr}&=&
\sum_{j=k}^{2m-1}{(-1)^{j}a_{jk}d_{j}(t)}
-\sum_{j=0}^{2m-1}{(-1)^{j}d_{j}(t)}\notag\\
&=&
\sum_{j=k}^{2m-1}{(-1)^{j}a_{jk}d_{j}(t)}
+\tilde{\mathcal{J}}^{c}_{ol}(1,t)
\end{eqnarray}
where we recall the definition
\[a_{jk}:=\frac{(j+k+k^{2})\Gamma(j+2)\Gamma(j)}{\Gamma(j+k+2)\Gamma(j-k+1)}\, ,\]
and note
that the lower limit in the summation is $k$ and consequently, the first term is a symmetric
polynomial of degree $2m-1 -k$.
Therefore, multiplying by $\mathcal{J}^{p}_{el}(t)=\prod_{j=1}^{m-1}{t_{j}^{2}}\cdot
\Delta_{m-1}^{4}(t)$ and using the definition
\eqref{Jdecomp} and the identity
\begin{eqnarray}
\tilde{\mathcal{J}}^{c}_{ol}(1,t) \mathcal{J}^{p}_{el}(t)&=&
\prod_{j=1}^{m-1}{(1-t_{j})^{2}}\prod_{j=1}^{m-1}{t_{j}^{2}}\cdot
\Delta_{m-1}^{4}(t)\notag\\
&=&\mathcal{J}^{p}_{ou}(t)
\end{eqnarray}
from the definition
$\mathcal{J}^{p}_{ou}(t)=\prod_{j=1}^{m-1}{t_{j}^{2}(1-t_{j})^{2}}\cdot
\Delta_{m-1}^{4}(t)$ from Proposition \eqref{prop_Jpo},
we conclude that
\begin{eqnarray}
\label{JQolfinal}
\int_{I}{ \mathcal{J}^{c}_{ol}(r,t)Q_{k}(r)dr}
&=&
\mathcal{J}^{p}_{el}(t)\sum_{j=k}^{2m-1}{(-1)^{j}a_{jk}d_{j}(t)}
+\mathcal{J}^{p}_{ou}(t)\notag\, .
\end{eqnarray}
Let us designate the negative of the first term
\begin{equation}
\label{def_ah}
\acute{h}_{k}(t):=\mathcal{J}^{p}_{el}(t)\sum_{j=k}^{2m-1}{(-1)^{j+1}a_{jk}d_{j}(t)}
\end{equation}
so that
\begin{equation}
\label{e222}
\int_{I}{ \mathcal{J}^{c}_{ol}(r,t)Q_{k}(r)dr}=-\acute{h}_{k}(t)+\mathcal{J}^{p}_{ou}(t) .
\end{equation}
For the third term $\int_{I}{\mathcal{J}^{c}_{ou}(\cdot,t)Q_{j}(\cdot)} $, we utilize the reflection symmetry
\[\mathcal{J}^{c}_{ou}(t_{*},t)=\mathcal{J}^{c}_{ol}(1-t_{*},1-t)\] of
\eqref{Jcsymm} and the reflection symmetry
(see e.g.~\cite[Eq.~12.97]{Arfken})
\[Q_{k}(1-r)=(-1)^{k}Q_{k}(r)\]
of the associate Legendre polynomials to compute the integral $\int_{I}{\mathcal{J}^{c}_{ou}(\cdot,t)Q_{j}(\cdot)} $ in terms of $\int_{I}{\mathcal{J}^{c}_{ol}(\cdot,t)Q_{j}(\cdot)} $.
That is,
\begin{eqnarray*}
\int_{I}{ \mathcal{J}^{c}_{ou}(r,t)Q_{k}(r)dr}&=&
\int_{I}{ \mathcal{J}^{c}_{ou}(1-r,t)Q_{k}(1-r)dr}\\
&=& \int_{I}{ \mathcal{J}^{c}_{ol}(r,1-t)Q_{k}(1-r)dr}\\
&=& (-1)^{k}\int_{I}{ \mathcal{J}^{c}_{ol}(r,1-t)Q_{k}(r)dr}\\
\end{eqnarray*}
and therefore
\begin{eqnarray*}
\int_{I}{ \mathcal{J}^{c}_{ou}(r,t)Q_{k}(r)dr}
&=& (-1)^{k}\int_{I}{ \mathcal{J}^{c}_{ol}(r,1-t)Q_{k}(r)dr}\\
&=& -(-1)^{k}\acute{h}_{k}(1-t) +(-1)^{k}\mathcal{J}^{p}_{ou}(1-t)\, ,
\end{eqnarray*}
so that we conclude
\begin{equation}
\label{e333}
\int_{I}{ \mathcal{J}^{c}_{ou}(r,t)Q_{k}(r)dr}
= -(-1)^{k}\acute{h}_{k}(1-t) +(-1)^{k}\mathcal{J}^{p}_{ou}(1-t)\, .
\end{equation}
Putting all three terms together using \eqref{def_H2} and the identities
\eqref{e111}, \eqref{e222}, and \eqref{e333}, along with the symmetry
\[\mathcal{J}^{p}_{ou}(1-t)=\mathcal{J}^{p}_{ou}(t)\, ,\]
we conclude that
\begin{eqnarray*}
\label{eqieiei}
\int_{I}{ \mathcal{H}(r,t)Q_{k}(r)dr}
&=& \int_{I}{\mathcal{J}^{p}_{ou}(t)Q_{k}(r)dr}- \int_{I}{\mathcal{J}^{c}_{ol}(r,t)Q_{k}(r)dr}-
\int_{I}{\mathcal{J}^{c}_{ou}(r,t)Q_{k}(r)dr}\notag\\
&=&(1+(-1)^{k})\mathcal{J}^{p}_{ou}(t)+ \acute{h}_{k}(t)-\mathcal{J}^{p}_{ou}(t)+
(-1)^{k}\acute{h}_{k}(1-t) -(-1)^{k}\mathcal{J}^{p}_{ou}(1-t)\\
&=&\acute{h}_{k}(t)+ (-1)^{k}\acute{h}_{k}(1-t)\, .
\end{eqnarray*}
To finish, consider the functions
$\breve{h}_{k}(t):=\acute{h}_{k}(t)+ (-1)^{k}\acute{h}_{k}(1-t)$.
It follows from \eqref{def_h} that the basis coefficients $h_{k}$ satisfy
\[h_{k}=\frac{1}{Vol(M^{2m-1})}\frac{2}{(2m-1)!(m-1)!}\frac{(2k+1)(k-2)!}{(k+2)!}\breve{h}_{k},\quad
k=2,..,2m-1\, . \]
Moreover,
Theorem \ref{thm_selberg2} implies that $\{\Sigma Q_{j},j=2,..,2m-1\}$ and $\{h_{k}, k=2,..,2m-1\}$
are an $L^{2}(I^{m-1})$ biorthogonal system. It therefore follows that
\[\int_{I^{m-1}}{\breve{h}_{k}\Sigma Q_{j}}=Vol(M^{2m-1})(2m-1)!(m-1)!\frac{(k+2)!}{(4k+2)(k-2)!} \delta _{jk},\quad j,k=2,..,2m-1\, .\]
Moreover, from the symmetry
$Q_{k}(1-r)=(-1)^{k}Q_{k}(r)$ and a change of variables we obtain
\begin{eqnarray*}
\int_{I^{m-1}}{\breve{h}_{k}(t) \Sigma Q_{j}(t)dt}&=&
\int_{I^{m-1}}{\acute{h}_{k}(t) \Sigma Q_{j}(t)dt}+(-1)^{k} \int_{I^{m-1}}{\acute{h}_{k}(1-t) \Sigma Q_{j}(t)dt}
\\
&=&
\int_{I^{m-1}}{\acute{h}_{k}(t) \Sigma Q_{j}(t)dt}+(-1)^{k} \int_{I^{m-1}}{\acute{h}_{k}(t) \Sigma Q_{j}(1-t)dt}
\\
&=& \int_{I^{m-1}}{\acute{h}_{k}(t) \Sigma Q_{j}(t)dt}+(-1)^{j+k} \int_{I^{m-1}}{\acute{h}_{k}(t) \Sigma Q_{j}(t)dt}\\
&=&(1+(-1)^{j+k}) \int_{I^{m-1}}{\acute{h}_{k}(t) \Sigma Q_{j}(t)dt}\,.
\end{eqnarray*}
Since $(j+k)\bmod 2 =(j-k)\bmod 2$,
the assertion then follows from the definition \eqref{def_ah} of $\acute{h}_{k}$, the identity
$\mathcal{J}^{p}_{el}(t)=\prod_{j=1}^{m-1}{t_{j}^{2}}\cdot
\Delta_{m-1}^{4}(t)$, and the definition \eqref{def_d} of $d_{k}$.
Moreover, we see that for $(j-k)\bmod 2=1$ the vanishing of this integral does not depend on the function
$\acute{h}_{k}$ but is instead a consequence only of the relative parity between $Q_{j}$ and
$\acute{h}_{k}$ with respect to the operation of reflection.
\subsection{Proof of Proposition \ref{legendre_integral}}
We abuse notation by letting $P_{k}$ and $Q_{k}$ denote the Legendre polynomials on the standard
set $[-1,1]$. At the end we will change back to the interval $I$.
We use \cite[Eq.~7.127, pg.~771]{GradshteynRyzhik}
\[\int_{-1}^{1}{(1+x)^{j}P_{k}(x)dx}=\frac{2^{j+1}\Gamma^{2}(j+1)}{\Gamma(j+k+2)\Gamma(j-k+1)}, \quad j\geq 0\]
and the definition
$Q_{k}(x):=(1-x^{2})P''_{k}(x)$ and integration by parts. Because of the poles of the Gamma function at $0$ and the negative integers, we conclude that
\[\int_{-1}^{1}{(1+x)^{j}P_{k}(x)dx}=0,\quad j<k\, .\]
Consider the function $\phi(x):= (1+x)^{j}(1-x^{2})=2(1+x)^{j+1}-(1+x)^{j+2}$,
which has the derivatives $\phi'(x)=2(j+1)(1+x)^{j}-(j+2)(1+x)^{j+1}$ and
$\phi''(x)=2(j+1)j(1+x)^{j-1}-(j+2)(j+1)(1+x)^{j}$. Since, for $j\geq 1$, we have
$\phi'(-1)=0$ and $\phi'(1)=-2^{j+1}$, and $P_{k}(1)=1$, we obtain
\begin{eqnarray*}
\int_{-1}^{1}{(1+x)^{j}Q_{k}(x)dx}&=&\int_{-1}^{1}{(1+x)^{j}(1-x^{2})P''_{k}(x)dx}\\
&=&\int_{-1}^{1}{\phi(x)P''_{k}(x)dx}\\
&=&\phi P'_{k}\Big|^{1}_{-1}-\int_{-1}^{1}{\phi'(x)P'_{k}(x)dx}\\
&=&-\int_{-1}^{1}{\phi'(x)P'_{k}(x)dx}\\
&=& -\phi'P_{k}\Big|^{1}_{-1} +\int_{-1}^{1}{\phi''(x)P_{k}(x)dx}\\
&=& 2^{j+1} +\int_{-1}^{1}{\phi''(x)P_{k}(x)dx}\\
&=& 2^{j+1} +2(j+1)j\int_{-1}^{1}{(1+x)^{j-1}P_{k}(x)dx}\\&&-(j+2)(j+1)\int_{-1}^{1}{(1+x)^{j}P_{k}(x)dx}\\
\end{eqnarray*}
from which we conclude that
\[\int_{-1}^{1}{(1+x)^{j}Q_{k}(x)dx}= 2^{j+1},\quad 1 \leq j < k\, .\]
For the case $j=0$, defining $\phi(x):=1-x^{2}$, we instead have $\phi'(-1)=2$, $\phi'(1)=-2$, $P_{k}(1)=1$, and
$P_{k}(-1)=(-1)^{k}$. Using $\int_{-1}^{1}{P_{k}}=0, k\geq 1$,
we obtain
\begin{eqnarray*}
\int_{-1}^{1}{Q_{k}(x)dx}&=&\int_{-1}^{1}{(1-x^{2})P''_{k}(x)dx}\\
&=&\int_{-1}^{1}{\phi(x)P''_{k}(x)dx}\\
&=&-\int_{-1}^{1}{\phi'(x)P'_{k}(x)dx}\\
&=& -\phi'P_{k}\Big|^{1}_{-1} +\int_{-1}^{1}{\phi''(x)P_{k}(x)dx}\\
&=& -\phi'P_{k}\Big|^{1}_{-1}\\
&=& -\Bigl(-2 P_{k}(1) -2P_{k}(-1)\Bigr)\\
&=& 2(1+(-1)^{k}) \, .
\end{eqnarray*}
On the other hand, for $j \geq k$ we have
\begin{eqnarray*}
&&\int_{-1}^{1}{(1+x)^{j}Q_{k}(x)dx}-2^{j+1}\\
&=&2(j+1)j\int_{-1}^{1}{(1+x)^{j-1}P_{k}(x)dx}-(j+2)(j+1)\int_{-1}^{1}{(1+x)^{j}P_{k}(x)dx}\\
&=&2^{j+1}(j+1)j\frac{\Gamma^{2}(j)}{\Gamma(j+k+1)\Gamma(j-k)}-2^{j+1}(j+2)(j+1)\frac{\Gamma^{2}(j+1)}{\Gamma(j+k+2)\Gamma(j-k+1)}\Bigr)\\
&=& 2^{j+1}j(j+1)\Gamma^{2}(j)\Bigl(\frac{1}{\Gamma(j+k+1)\Gamma(j-k)}-j(j+2)
\frac{1}{\Gamma(j+k+2)\Gamma(j-k+1)}\Bigr)\\
&=& 2^{j+1}\frac{j(j+1)\Gamma^{2}(j)}{\Gamma(j+k+2)\Gamma(j-k+1)}\Bigl((j+k+1)(j-k)-j(j+2)\Bigr)\\
&=& -2^{j+1}\frac{j(j+1)(j+k+k^{2})\Gamma^{2}(j)}{\Gamma(j+k+2)\Gamma(j-k+1)}\\
&=& -2^{j+1}\frac{(j+k+k^{2})\Gamma(j+2)\Gamma(j)}{\Gamma(j+k+2)\Gamma(j-k+1)}
\end{eqnarray*}
and therefore
\begin{eqnarray*}
\int_{-1}^{1}{(1+x)^{j}Q_{k}(x)dx}=
2^{j+1}\Bigl(1-\frac{(j+k+k^{2})\Gamma(j+2)\Gamma(j)}{\Gamma(j+k+2)\Gamma(j-k+1)}\Bigr)\, .
\end{eqnarray*}
Translating to the unit interval with the map $I \mapsto [-1,1]$ defined by
$x=2r-1$ we obtain the assertion.
\section{Appendix}
\subsection{Proof of even case of Lemma \ref{lem_mass_inf}}
We utilize the bijective principal representation $\phi^{p}_{el}: \Lambda^{m} \times T^{m} \rightarrow Int(M^{2m})$ defined in \eqref{def_phi_pel} and
\[|det(d\phi^{p}_{eu})|(\lambda,t)= \mathcal{J}^{p}_{eu}(t)
\prod_{j=1}^{m}{\lambda_{j}} \]
where
\[
\mathcal{J}^{p}_{el}(t)
:= \prod_{j=1}^{m}{t_{j}^{2}}
\cdot \Delta_{m}^{4}(t)
\]
from Proposition \ref{prop_Jpe}.
Fix $t_{*}\in (0,1)$ and let
\[ T_{ \delta }^{m}:= \bigl\{ (t_{1},..,t_{m})\in T^{m}:
t_{j} \notin B_{ \delta }(t_{*}) , j=1,..,m\, \bigr\} . \]
It follows that
\[M^{2m}_{ \delta } \supset \phi^{p}_{el}\bigl(\Lambda^{m}\times T_{ \delta }^{m}\bigr)\]
and therefore
\begin{equation}
\label{e22222}
Vol\Bigl(M^{2m}_{ \delta }\Bigr) \geq Vol\Bigl(\phi^{p}_{el}\bigl(\Lambda^{m}\times T_{ \delta }^{m} \bigr)\Bigr)\, .
\end{equation}
Using the identity
$\int_{\Lambda^{m}}{\prod_{i=1}^{m}{\lambda_{i}}d\lambda} = \frac{1}{(2m)!}$,
we compute the righthand side using the change of variables formula as
\begin{eqnarray*}
Vol\Bigl(\phi^{p}_{el}\bigl(\Lambda^{m}\times T_{ \delta }^{m} \bigr)\Bigr)
&=&
\int_{\Lambda^{m} \times T_{ \delta }^{m}}{|det(d\phi^{p}_{el})|
}\\
&=&\int_{\Lambda^{m}}{\bigl(\prod_{j=1}^{m}{\lambda_{j}}\bigr)
d\lambda} \int_{T_{ \delta }^{m}}{\mathcal{J}^{p}_{el}}\\
&=&\frac{1}{(2m)!}
\int_{T_{ \delta }^{m}}{\mathcal{J}^{p}_{el}}\\
&=&\frac{1}{(2m)!m!}
\int_{I_{ \delta }^{m}}{\mathcal{J}^{p}_{el}}
\end{eqnarray*}
where
\[ I_{ \delta }^{m}:= \bigl\{ (t_{1},..,t_{m})\in I^{m}:
t_{j} \notin B_{ \delta } , j=1,..,m\, \bigr\} . \]
To bound this from below we bound the integral over
$ \bigl(I_{ \delta }^{m}\bigr)^{c}$ from above.
To that end, let
\[ I_{ \delta ,j}^{m}:= \bigl\{ (t_{1},..,t_{m})\in I^{m}:
t_{j} \in B_{ \delta }\bigr \},\quad j=1,..,m\,, \]
so that
\[ \bigl(I_{ \delta }^{m}\bigr)^{c}= \cup_{j} I_{ \delta ,j}^{m}\, .\]
Therefore, using a union bound, we have
\begin{eqnarray*}
\int_{\bigl(I_{ \delta }^{m}\bigr)^{c}}{\mathcal{J}^{p}_{el}}&=&
\int_{\bigl(I_{ \delta }^{m}\bigr)^{c}}{ \prod_{j=1}^{m}{t_{j}^{2}}\cdot \Delta_{m}^{4}(t)dt}\\
&=&\int_{\cup_{j'=1}^{m}{I^{m}_{ \delta ,j'}}}{ \prod_{j=1}^{m}{t_{j}^{2}}\cdot \Delta_{m}^{4}(t)dt}\\
&\leq &\sum_{j'=1}^{m}{\int_{I^{m}_{ \delta ,j'}}{ \prod_{j=1}^{m}{t_{j}^{2}}\cdot \Delta_{m}^{4}(t)dt}}\\
& =&m\int_{I^{m}_{ \delta ,1}}{ \prod_{j=1}^{m}{t_{j}^{2}}\cdot \Delta_{m}^{4}(t)dt}\\
& = &m\int_{I^{m}_{ \delta ,1}}{ \prod_{j=1}^{m}{t_{j}^{2}}\prod_{1 \leq j<k \leq m}{(t_{k}-t_{j})^{4}}dt_{1}\cdots dt_{m}}\\
& \leq &m\int_{I^{m}_{ \delta ,1}}{ \prod_{j=2}^{m}{t_{j}^{2}}\prod_{2 \leq j<k \leq m}{(t_{k}-t_{j})^{4}}dt_{1}\cdots dt_{m}}\\
& = &mVol(B_{ \delta })\int_{I^{m-1}}{ \prod_{j=2}^{m}{t_{j}^{2}}\cdot \Delta_{m-1}^{4}(t)dt}\\
& = &mVol(B_{ \delta }) S_{m-1}(3,1,2)\\
&\leq & 2m \delta S_{m-1}(3,1,2)\\
\end{eqnarray*}
and so obtain
\begin{eqnarray*}
\frac{Vol\Bigl(\phi^{p}_{el}(\Lambda^{m}\times T_{ \delta }^{m})\Bigr)}{Vol\Bigl(\phi^{p}_{el}(\Lambda^{m}\times T^{m})\Bigr)}
&=& \frac{\int_{I^{m}_{ \delta }}{ \prod_{j=1}^{m}{t_{j}^{2}}\cdot \Delta_{m}^{4}(t)dt}}{\int_{I^{m}}{ \prod_{j=1}^{m}
{t_{j}^{2}}\cdot \Delta_{m}^{4}(t)dt}}\\
&\geq& 1- 2m \delta \frac{S_{m-1}(3,1,2)}{S_{m}(3,1,2)}\\
\end{eqnarray*}
Using Selberg's formulas \eqref{selbergformula} we compute
\begin{eqnarray*}
\frac{S_{m-1}(3,1,2)}{S_{m}(3,1,2)}&=&
\frac{ \prod_{j=0}^{m-2}{\frac{\Gamma(1+ 2j)
\Gamma(3+2j)^{2}}{2\Gamma(2(m+j)) }}}{ \prod_{j=0}^{m-1}{\frac{\Gamma(1+ 2j)
\Gamma(3+2j)^{2}}{2\Gamma(2(m+j)+2) }}}\\
&=&\frac{2\Gamma(4m-2)}{\Gamma(2m-1)^{2}\Gamma(2m+1)}
\frac{ \prod_{j=0}^{m-1}{\frac{\Gamma(1+ 2j)
\Gamma(3+2j)^{2}}{2\Gamma(2(m+j)) }}}{ \prod_{j=0}^{m-1}{\frac{\Gamma(1+ 2j)
\Gamma(3+2j)^{2}}{2\Gamma(2(m+j)+2) }}}\\
&=&\frac{2\Gamma(4m-2)}{\Gamma(2m-1)^{2}\Gamma(2m+1)}
\prod_{j=0}^{m-1}{\frac{\Gamma(2(m+j)+2) }{\Gamma(2(m+j))}} \\
&=&\frac{2\Gamma(4m-2)}{\Gamma(2m-1)^{2}\Gamma(2m+1)}
\frac{\Gamma(4m)}{\Gamma(2m)}\, .
\end{eqnarray*}
\subsection{Even case of Proof of Theorem \ref{thm_RK}}
\label{sec_RKproof}
For the even case, $n=2m$,
recall the identity $\mathcal{J}^{c}_{eu}(0,t)\equiv 0$, and the
volume identity \eqref{vol_Jce}
\begin{equation}
\label{vol_Jce2}
Vol(M^{2m})=\frac{1}{(2m)!m!}\int_{I^{m}}{\mathcal{J}^{c}_{el}(t_{*},t)dt}+
\frac{1}{(2m)!(m-1)!}\int_{I^{m-1}}{\mathcal{J}^{c}_{eu}(t_{*},t)dt}\,
\end{equation}
Now observe that, for $i\geq 0$, the identity
\begin{eqnarray}
&&\int_{M^{2m}}{q_{i}}- \frac{t_{*}^{i}}{2m+1}Vol(M^{2m})\notag\\
&=&\frac{ \delta _{0}(i)+1}{(2m+1)!(m-1)!}\int_{I^{m-1}}{
\mathcal{J}^{c}_{eu}(t_{*},t)dt}\notag\\
&+&\frac{2}{(2m+1)!m!}\int_{I^{m}}{\Sigma t^{i}\mathcal{J}^{c}_{el}(t_{*},t)dt}
+
\frac{2}{(2m+1)!(m-1)!}\int_{I^{m-1}}{\Sigma t^{i}
\mathcal{J}^{c}_{eu}(t_{*},t)dt}\notag
\end{eqnarray}
from Proposition \ref{prop_mom},
evaluated at $t_{*}=0$ becomes
\begin{eqnarray}
&&\int_{M^{2m}}{q_{i}}- \frac{ \delta _{0}(i)}{2m+1}Vol(M^{2m})\notag\\
&=&\frac{ \delta _{0}(i)+1}{(2m+1)!(m-1)!}\int_{I^{m-1}}{
\mathcal{J}^{c}_{eu}(0,t)dt}\notag\\
&+&\frac{2}{(2m+1)!m!}\int_{I^{m}}{\Sigma t^{i}\mathcal{J}^{c}_{el}(0,t)dt}
+
\frac{2}{(2m+1)!(m-1)!}\int_{I^{m-1}}{\Sigma t^{i}
\mathcal{J}^{c}_{eu}(0,t)dt}\notag
\end{eqnarray}
where we note that $\Sigma t^{i}=\sum_{j=1}^{m}{t_{j}^{i}}$ in the integral over
$I^{m}$ and $\Sigma t^{i}=\sum_{j=1}^{m-1}{t_{j}^{i}}$ in the integral over
$I^{m-1}$.
Subtracting the two,
using the identity $\mathcal{J}^{c}_{eu}(0,t)\equiv 0$, we obtain
\begin{eqnarray}
\label{eq_even_rk1}
&&-\frac{t_{*}^{i}}{2m+1}Vol(M^{2m})+ \frac{\delta_{0}(i)}{2m+1}Vol(M^{2m})\notag\\
&=&\frac{ \delta _{0}(i)+1}{(2m+1)!(m-1)!}\int_{I^{m-1}}{
\mathcal{J}^{c}_{eu}(t_{*},t)dt}\notag\\
&+&\frac{2}{(2m+1)!m!}\int_{I^{m}}{\Sigma t^{i}\bigl(\mathcal{J}^{c}_{el}(t_{*},t)-
\mathcal{J}^{c}_{el}(0,t)\bigr)dt}
+
\frac{2}{(2m+1)!(m-1)!}\int_{I^{m-1}}{\Sigma t^{i}
\mathcal{J}^{c}_{eu}(t_{*},t)dt}\notag
\end{eqnarray}
and applying the volume identity \eqref{vol_Jce2} we obtain
\begin{eqnarray}
\label{eq_even_rk2}
&&-\frac{t_{*}^{i}}{2m+1}Vol(M^{2m})\notag\\
&=& -\frac{ \delta _{0}(i)}{(2m+1)!m!}\int_{I^{m}}{\mathcal{J}^{c}_{el}(t_{*},t)
dt}+ \frac{1}{(2m+1)!(m-1)!}\int_{I^{m-1}}{
\mathcal{J}^{c}_{eu}(t_{*},t)dt}\notag\\
&+&\frac{2}{(2m+1)!m!}\int_{I^{m}}{\Sigma t^{i}\bigl(\mathcal{J}^{c}_{el}(t_{*},t)-
\mathcal{J}^{c}_{el}(0,t)\bigr)dt}
+
\frac{2}{(2m+1)!(m-1)!}\int_{I^{m-1}}{\Sigma t^{i}
\mathcal{J}^{c}_{eu}(t_{*},t)dt}\notag \, .
\end{eqnarray}
Then, if we let $\phi(s):=\sum_{i=0}^{2m}{\phi_{i}s^{i}}$ be a polynomial of degree
$n=2m$, summing over each identity in \eqref{eq_even_rk2}, we conclude that
\begin{eqnarray*}
&&-\frac{\phi(t_{*})}{2m+1}Vol(M^{2m})\notag\\
&=&\phi_{0}\Bigl(\frac{1}{(2m+1)!(m-1)!}\int_{I^{m-1}}{
\mathcal{J}^{c}_{eu}(t_{*},t)dt} -\frac{1}{(2m+1)!m!}\int_{I^{m}}{\mathcal{J}^{c}_{el}(t_{*},t)
dt}\Bigr)
\notag\\
&+& \sum_{j=1}^{2m}{\phi_{j}}\frac{1}{(2m+1)!(m-1)!}\int_{I^{m-1}}{
\mathcal{J}^{c}_{eu}(t_{*},t)dt}
\notag\\
&+&\frac{2}{(2m+1)!m!}\int_{I^{m}}{(\Sigma \phi)(t)\bigl(\mathcal{J}^{c}_{el}(t_{*},t)-
\mathcal{J}^{c}_{el}(0,t)\bigr)dt}
+
\frac{2}{(2m+1)!(m-1)!}\int_{I^{m-1}}{(\Sigma \phi)(t)
\mathcal{J}^{c}_{eu}(t_{*},t)dt}\notag \\
&=&-\phi_{0}
\frac{1}{(2m+1)!m!}\int_{I^{m}}{\mathcal{J}^{c}_{el}(t_{*},t)
dt}
\notag\\
&+& \sum_{j=0}^{2m}{\phi_{j}}\frac{1}{(2m+1)!(m-1)!}\int_{I^{m-1}}{
\mathcal{J}^{c}_{eu}(t_{*},t)dt}
\notag\\
&+&\frac{2}{(2m+1)!m!}\int_{I^{m}}{(\Sigma \phi)(t)\bigl(\mathcal{J}^{c}_{el}(t_{*},t)-
\mathcal{J}^{c}_{el}(0,t)\bigr)dt}
+
\frac{2}{(2m+1)!(m-1)!}\int_{I^{m-1}}{(\Sigma \phi)(t)
\mathcal{J}^{c}_{eu}(t_{*},t)dt}\notag \\
\end{eqnarray*}
Since $\phi(0)=\phi_{0}$ and $\phi(1)=\sum_{i=0}^{2m}{\phi_{i}}$ the assertion follows by multiplication by
$2m+1$.
\subsection{Assorted Technical Results}
\begin{prop}
\label{prop_tech}
We have
\[
m^{2} \leq 8 (\frac{e}{2})^{4m}, \quad m \geq 1\, .\]
\end{prop}
\begin{proof}
We proceed by induction. The inequality is clearly true for $m=1,2.$ Therefore,
suppose that it is true for some $m\geq 2$. Then, since
\[\frac{m+1}{m} \leq \frac{3}{2}\leq \bigl(\frac{e}{2}\bigr)^{2},\quad m\geq 2\]
follows from $6 \leq e^{2}$, we conclude that
\begin{eqnarray*}
(m+1)^{2}&=& \frac{(m+1)^{2}}{m^{2}}m^{2} \\
&\leq& \frac{(m+1)^{2}}{m^{2}} 8 (\frac{e}{2})^{4m} \\
&\leq& \bigl(\frac{e}{2}\bigr)^{4} 8 (\frac{e}{2})^{4m}\\
&\leq& 8 (\frac{e}{2})^{4m+4}
\end{eqnarray*}
thus establishing the inequality for $m+1$ and finishing the proof.
\end{proof}
\begin{prop}
\label{beta_lowerbound}
We have
\[B(a,a) \geq \frac{4}{a}2^{-2a}, \quad a >1\]
\end{prop}
\begin{proof}
We have from the integral formula
\begin{eqnarray*}
B(a,b)&=&\int_{0}^{1}{t^{a-1}(1-t)^{a-1}dt}\\
&=&2\int_{0}^{\frac{1}{2}}{t^{a-1}(1-t)^{a-1}dt}\\
&\geq &2\bigl(\frac{1}{2}\bigr)^{a-1}\int_{0}^{\frac{1}{2}}{t^{a-1}dt}\\
&=&2\bigl(\frac{1}{2}\bigr)^{a-1}\frac{1}{a}\bigl(\frac{1}{2}\bigr)^{a}\\
&=&\frac{4}{a}2^{-2a}
\end{eqnarray*}
\end{proof}
\section*{Acknowledgements}
\addcontentsline{toc}{section}{Acknowledgements}
We would like to thank G\'{e}rard Letac for his helpful comments, in particular for his substantial simplification,
included here, of our previous
proof of Lemma 4.1. We would also like to thank one of the referees for many helpful comments which we also feel improved
the manuscript.
The authors gratefully acknowledge this work supported by the Air Force Office of Scientific Research under Award Number
FA9550-12-1-0389 (Scientific Computation of Optimal Statistical Estimators).
\addcontentsline{toc}{section}{References}
\bibliographystyle{plain}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 14
|
package pipescript
import (
"encoding/json"
"errors"
"fmt"
)
type TransformEnv struct {
Iter *BufferIterator
ArgIters []*BufferIterator
}
func (te *TransformEnv) Next(args []*Datapoint) (*Datapoint, []*Datapoint, error) {
dp, err := te.Iter.Next()
if err == nil && dp != nil {
for i := range te.ArgIters {
args[i], err = te.ArgIters[i].Next()
if err != nil {
return dp, args, err
}
if args[i] == nil {
return nil, args, nil
}
}
}
return dp, args, err
}
func (te *TransformEnv) Peek(idx int, args []*Datapoint) (*Datapoint, []*Datapoint, error) {
dp, err := te.Iter.Peek(idx)
if err == nil && dp != nil {
for i := range te.ArgIters {
args[i], err = te.ArgIters[i].Peek(idx)
if err != nil {
return dp, args, err
}
if args[i] == nil {
return nil, args, nil
}
}
}
return dp, args, err
}
type PipeElement struct {
Env *TransformEnv
Args []*Pipe
Iter TransformIterator
// These three are here to allow copying a pipe by re-generating it from scratch
ConstArgs []interface{}
PipeArgs []*Pipe
Transform *Transform
}
func NewPipeElement(t *Transform, args []*Pipe) (*PipeElement, error) {
targs := make([]*Pipe, 0)
consts := make([]interface{}, 0)
pipes := make([]*Pipe, 0)
if len(args) > len(t.Args) {
return nil, fmt.Errorf("Transform '%s' takes %d arguments, but %d given", t.Name, len(t.Args), len(args))
}
for i := range t.Args {
if len(args) <= i {
if t.Args[i].Optional {
args = append(args, t.Args[i].Default)
} else {
return nil, fmt.Errorf("Transform '%s' requires additional arguments", t.Name)
}
}
args[i].Simplify()
// Now split the args up
switch t.Args[i].Type {
case ConstArgType:
cv, err := args[i].GetConst()
if err != nil {
return nil, err
}
consts = append(consts, cv)
case TransformArgType:
targs = append(targs, args[i])
case OneToOnePipeArgType:
pipes = append(pipes, args[i])
if !args[i].OneToOne() {
return nil, fmt.Errorf("Transform %s arg %d (%s) is not one-to-one", t.Name, i, args[i].String())
}
case PipeArgType:
pipes = append(pipes, args[i])
}
}
pe := &PipeElement{
Transform: t,
Args: targs,
ConstArgs: consts,
PipeArgs: pipes,
Env: &TransformEnv{
ArgIters: make([]*BufferIterator, len(targs)),
},
}
// Set up the arg iterators
for i := range targs {
pe.Env.ArgIters[i] = NewBuffer(targs[i]).Iterator()
}
var err error
pe.Iter, err = t.Constructor(t, consts, pipes)
return pe, err
}
func (pe *PipeElement) String() string {
s := pe.Transform.Name
if len(pe.Transform.Args) > 0 {
s += "("
tai := 0
pai := 0
cai := 0
for i := range pe.Transform.Args {
switch pe.Transform.Args[i].Type {
case ConstArgType:
b, _ := json.Marshal(pe.ConstArgs[cai])
s += string(b)
cai++
case TransformArgType:
s += pe.Args[tai].String()
tai++
case PipeArgType, OneToOnePipeArgType:
s += pe.PipeArgs[pai].String()
pai++
}
s += ","
}
s = s[:len(s)-1] + ")"
}
return s
}
func (pe *PipeElement) GetConst(in interface{}) (interface{}, error) {
ci, ok := pe.Iter.(*ConstIterator)
if ok {
return ci.Value, nil
}
inputargs := make([]interface{}, len(pe.Args))
var err error
for i := range pe.Args {
inputargs[i], err = pe.Args[i].GetConst()
if err != nil {
return nil, fmt.Errorf("Transform '%s' arg %d: %w", pe.String(), i, err)
}
}
bt, ok := pe.Iter.(*Basic)
if !ok {
abt, ok := pe.Iter.(*ArgBasic)
if !ok {
return nil, fmt.Errorf("Output of transform '%s' cannot be used as a constant", pe.String())
}
return abt.GetConst(inputargs)
}
return bt.GetConst(in, inputargs)
}
func (pe *PipeElement) Copy() *PipeElement {
newArgs := make([]*Pipe, len(pe.Args))
for i := range pe.Args {
newArgs[i] = pe.Args[i].Copy()
}
newPipeArgs := make([]*Pipe, len(pe.PipeArgs))
for i := range pe.PipeArgs {
newPipeArgs[i] = pe.PipeArgs[i].Copy()
}
pnew := &PipeElement{
Transform: pe.Transform,
Args: newArgs,
ConstArgs: pe.ConstArgs,
PipeArgs: newPipeArgs,
Env: &TransformEnv{
ArgIters: make([]*BufferIterator, len(newArgs)),
},
}
// Set up the arg iterators
for i := range newArgs {
pnew.Env.ArgIters[i] = NewBuffer(newArgs[i]).Iterator()
}
var err error
pnew.Iter, err = pe.Transform.Constructor(pnew.Transform, pnew.ConstArgs, pnew.PipeArgs)
if err != nil {
panic(err)
}
return pnew
}
func (pe *PipeElement) Input(b *Buffer) {
// Set the root iterator
pe.Env.Iter = b.Iterator()
// Set the root iterators of all args
for i := range pe.Args {
pe.Args[i].Input(b)
}
}
func (pe *PipeElement) IsBasic() bool {
for _, ap := range pe.Args {
if !ap.IsBasic() {
return false
}
}
return true
}
func (pe *PipeElement) Next(out *Datapoint) (*Datapoint, error) {
return pe.Iter.Next(pe.Env, out)
}
type Pipe struct {
Arr []*PipeElement
}
func NewPipe() *Pipe {
return &Pipe{
Arr: make([]*PipeElement, 0),
}
}
func NewElementPipe(t *Transform, args []*Pipe) (*Pipe, error) {
p := NewPipe()
el, err := NewPipeElement(t, args)
p.Append(el)
return p, err
}
func NewTransformPipe(tname string, args []*Pipe) (*Pipe, error) {
RegistryLock.RLock()
defer RegistryLock.RUnlock()
t, ok := TransformRegistry[tname]
if !ok {
return nil, fmt.Errorf("Could not find transform '%s'", tname)
}
return NewElementPipe(t, args)
}
// Input sets the pipe's input to the given buffer
func (p Pipe) Input(b *Buffer) {
if len(p.Arr) > 0 {
p.Arr[0].Input(b)
}
}
// InputNexter uses an iterator interface as input
func (p *Pipe) InputIterator(n Iterator) {
p.Input(NewBuffer(n))
}
func (p *Pipe) Next(dp *Datapoint) (*Datapoint, error) {
return p.Arr[len(p.Arr)-1].Next(dp)
}
func (p *Pipe) Last(dp *Datapoint) (*Datapoint, error) {
dp2 := &Datapoint{}
var err error
dp2, err = p.Next(dp2)
if err != nil || dp2 == nil {
return dp2, err
}
for dp2 != nil {
dp.Timestamp = dp2.Timestamp
dp.Duration = dp2.Duration
dp.Data = dp2.Data
dp2, err = p.Next(dp2)
if err != nil {
return nil, err
}
}
return dp, nil
}
// Append returns a pipe with the given PipeElement added
func (p *Pipe) Append(e *PipeElement) {
plen := len(p.Arr)
if plen > 0 {
e.Input(NewBuffer(p.Arr[plen-1]))
}
p.Arr = append(p.Arr, e)
}
func (p *Pipe) Copy() *Pipe {
p2 := NewPipe()
for i := range p.Arr {
p2.Append(p.Arr[i].Copy())
}
return p2
}
func (p *Pipe) Join(p2 *Pipe) {
plen := len(p.Arr)
if plen > 0 {
p2.InputIterator(p.Arr[plen-1])
}
p.Arr = append(p.Arr, p2.Arr...)
}
func (p *Pipe) OneToOne() bool {
for _, e := range p.Arr {
if !e.Iter.OneToOne() {
return false
}
}
return true
}
func (p *Pipe) String() string {
if len(p.Arr) == 0 {
return ""
}
s := p.Arr[0].String()
for i := 1; i < len(p.Arr); i++ {
s += ":" + p.Arr[i].String()
}
return s
}
func (p *Pipe) IsBasic() bool {
for i := range p.Arr {
switch p.Arr[i].Iter.(type) {
case *ConstIterator, peekIterator:
// Do nothing
case *Basic, *ArgBasic:
if !p.Arr[i].IsBasic() {
return false
}
default:
return false
}
}
return true
}
func (p *Pipe) Simplify() *Pipe {
arr2 := make([]*PipeElement, 0, len(p.Arr))
// Check which ArgBasics can be turned into consts
for i := 0; i < len(p.Arr); i++ {
switch p.Arr[i].Iter.(type) {
case *ArgBasic:
v, err := p.Arr[i].GetConst(nil)
if err == nil {
pe, err := NewPipeElement(NewConstTransform(v), nil)
if err != nil {
panic(err)
}
p.Arr[i] = pe
}
default:
// Nothing
}
}
// Remove redundant transforms from the pipe
for i := 0; i < len(p.Arr); i++ {
switch v := p.Arr[i].Iter.(type) {
case *ConstIterator:
// A const replaces all previous basic transforms
j := len(arr2) - 1
for ; j >= 0; j-- {
switch v := arr2[j].Iter.(type) {
case *ConstIterator:
case *Basic, *ArgBasic:
if !arr2[j].IsBasic() {
break
}
case peekIterator:
if v.Peek != 0 {
// Need to stop here, since must include the peek
break
}
default:
break
}
}
arr2 = append(arr2[:j+1], p.Arr[i])
case peekIterator:
// If not peeking, can remove it entirely
if v.Peek != 0 {
arr2 = append(arr2, p.Arr[i])
}
default:
arr2 = append(arr2, p.Arr[i])
}
}
p.Arr = make([]*PipeElement, 0, len(arr2))
// Second pass: propagate constants forward where possible. No longer
// need to worry about peeks anymore
for i := 0; i < len(arr2); i++ {
switch v := arr2[i].Iter.(type) {
case *ConstIterator:
cv := v.Value
for i++; i < len(arr2); i++ {
v, err := arr2[i].GetConst(cv)
if err != nil {
break
}
cv = v
}
pe, err := NewPipeElement(NewConstTransform(cv), nil)
if err != nil {
panic(err)
}
p.Append(pe)
if i < len(arr2) {
i--
}
default:
p.Append(arr2[i])
}
}
if len(p.Arr) == 0 {
// If all elements were removed, add a basic 0 peek iterator back
pe, err := NewPipeElement(&Transform{
Name: "d",
Constructor: func(transform *Transform, consts []interface{}, pipes []*Pipe) (TransformIterator, error) {
return peekIterator{0}, nil
},
}, nil)
if err != nil {
panic(err)
}
p.Arr = []*PipeElement{pe}
}
return p
}
func (p *Pipe) GetConst() (interface{}, error) {
// assumes that was already simplified
if len(p.Arr) == 1 {
ci, ok := p.Arr[0].Iter.(*ConstIterator)
if ok {
return ci.Value, nil
}
}
return nil, fmt.Errorf("Cannot use %s as constant value", p.String())
}
func (p *Pipe) MarshalJSON() ([]byte, error) {
return json.Marshal(p.String())
}
func (p *Pipe) UnmarshalJSON(b []byte) error {
var av string
err := json.Unmarshal(b, &av)
if err != nil {
return err
}
p2, err := Parse(av)
if err != nil {
return err
}
p.Arr = p2.Arr
return nil
}
type chanIterator struct {
Receiver chan *Datapoint
}
func (ci chanIterator) Next(out *Datapoint) (*Datapoint, error) {
dp, ok := <-ci.Receiver
if !ok {
return nil, errors.New("Input channel closed")
}
if dp == nil {
return nil, nil
}
out.Timestamp = dp.Timestamp
out.Duration = dp.Duration
out.Data = dp.Data
return out, nil
}
type ChanResult struct {
DP *Datapoint
Err error
}
type ChannelPipe struct {
Sender chan *Datapoint
Receiver chan ChanResult
}
func (cp *ChannelPipe) Close() {
close(cp.Sender)
}
func NewChannelPipe(p *Pipe) *ChannelPipe {
sender := make(chan *Datapoint)
receiver := make(chan ChanResult)
cp := &ChannelPipe{sender, receiver}
go func() {
p.InputIterator(chanIterator{sender})
for {
dp, err := p.Next(&Datapoint{}) // Make sure to create new datapoint each time
receiver <- ChanResult{dp, err}
if dp == nil || err != nil {
break
}
}
close(receiver)
}()
return cp
}
// Parse parses the given transform, and returns the corresponding pipe
func Parse(script string) (*Pipe, error) {
lexer := parserLex{input: script}
if parserParse(&lexer) != 0 {
if lexer.errorString != "" {
return nil, fmt.Errorf("'%s': %s", script, lexer.errorString)
}
return nil, fmt.Errorf("'%s': Unknown error", script)
}
lexer.output.Simplify()
return lexer.output, nil
}
func MustPipe(t *Transform, args []*Pipe) *Pipe {
p, err := NewElementPipe(t, args)
if err != nil {
panic(err)
}
return p
}
func MustParse(script string) *Pipe {
p, err := Parse(script)
if err != nil {
panic(err)
}
return p
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,341
|
Alice has had a nice spring this year. She has sleep-overs nearly every week-end, and play-dates most weeks. She has a nice varied group of friends. She continues to enjoy her music lessons with Ms. Tracy, and we've been spending hours at a time outside swinging from the tree, usually having conversations about anything and everything. She's pushing to be more independent and usually walks home after school now. She has also been practicing on her bike and has a small but growing range of freedom for exploring and practicing.
Alice still maintains her fairly recent appreciation of Olive Garden and prefers it to any other restaurant, so we go there more frequently than in the past. Although its P. Terry burgers if she gets anything she wants... plus the chocolate shake of course.
Also... don't want to ever forget the budding chemistry interest. Alice, like many others, is really loving playing with slime this spring. She has been looking up and experimenting with numerous online recipes. She keeps most of the results in old yogurt containers in a stack in the play room with creative and interesting names.
Also... she just finished her second after school coding club session (daddy was proud, certificate for office wall!, but Alice mostly goes for the social hang-out I think, whatever coding activities they do she seems to do easily).
Its really been a wonderful year being a parental unit. Good times.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,516
|
234791 Sport news spain,newspaper,pwfree,The Olive Press /g,">").replace(n,t.Em.replacer);return o="
Home Sport Jorge Masvidal: Fighting from the backstreets of Miami to the UFC Octagon
Jorge Masvidal: Fighting from the backstreets of Miami to the UFC Octagon
21 Oct, 2021 @ 09:44
Jorge 'Gamebred' Masvidal has become one of the UFC's most hardened veterans and popular fighters, but his journey hasn't been an easy one. Now 36, the Miami native is enjoying something of an Indian summer as a professional fighter, but it took a lot of hard work and plenty of setbacks to get there.
While most UFC fighters come from solid collegiate or amateur martial arts backgrounds, Masvidal quite literally came from the streets. From a young age, roughly 14, Masvidal became involved in mutual street fights across Miami.
While it might sound barbaric at first, these were actually organised events that didn't strictly break state law in Florida. These events were organised by the late Kimbo Slice, an MMA fighter and boxer himself who rose to prominence in the early 2000s.
These events involved one-on-one fights with no one else getting involved and no bystanders coming to any harm. Kimbo gained notoriety and popularity when he began to upload videos of these street fights to the internet, and they garnered millions of views.
One of the young men involved in some of those videos was Masvidal, who made a name for himself as a talented and scrappy fighter on the bare-knuckle scene.
He was training in various martial arts, including boxing and wrestling, and by the age of 18 he decided to make the move into professional fighting, though still continued to take part in non-sanctioned bouts on the streets.
It wasn't until he met trainer Paulino Hernandez that Masvidal moved away from street fighting and committed fully to professional mixed-martial arts. Hernandez remains a part of Masvidal's training team to this day as his striking coach.
From 2003 to 2009, Masvidal fought under various regional MMA promotions like HOOKnSHOOT and BodogFIGHT, finding plenty of success with wins over the likes of Joe Lauzon and Yves Edwards.
This caught the attention of those in charge of Bellator, who signed Masvidal in 2009. His results there were mixed and he was released by the organisation in 2010, and a year later he moved to Strikeforce.
It was there that things took a turn as he notched together some impressive wins, and when Strikeforce was dissolved in 2013, Masvidal was absorbed into the UFC. He had finally made it into the biggest MMA promotion in the world.
Even back then, Masvidal stood out for his impressive boxing and exciting style, even when he was losing. For several years he found himself treading water, experiencing wins and losses in almost equal measure.
In 2016, he began a run of three wins over Ross Pearson, Jake Ellenberger and Donald Cerrone that looked like it could become the foundation of a potential title run, however subsequent losses to Demian Maia and Stephen Thompson dashed those hopes – but only momentarily.
By 2019, it seemed as though Masvidal had been resigned to the role of gatekeeper; not quite good enough to crack the top rankings himself, but anyone hoping to get there would still have to go through him.
That was certainly the plan for Darren Till, who was booked against Masvidal in what many thought would be a showcase fight for the Liverpudlian. Masvidal had other plans and, after weathering a turbulent first round, starched Till in the second, silencing the home crowd and setting in motion a startling renaissance in his career.
Later that year he was matched with Ben Askren, who had made a lot of noise since being traded to the UFC from ONE Championship. There was plenty of needle between the two during the build-up, but no one – except perhaps Masvidal – could foresee what was about to happen.
As the siren sounded to begin the fight, Masvidal sprinted across the cage, leapt into the air and detonated a brutal flying knee on Askren's chin, rendering him unconscious. It took all of five seconds and remains the fastest knockout in UFC history.
Suddenly, Masvidal was a legitimate star. He had fought his way from the streets to produce one of the greatest knockouts the sport has ever seen, and all with that patented Miami swagger in his step.
Finally, in November of that year, he would headline a UFC pay-per-view. His opponent was another fan favorite; Nate Diaz. There was a mutual respect between the two as they vied for the "Baddest Motherf*****" title, which was walked to the ring by none other than Dwayne 'The Rock' Johnson.
Masvidal dropped Diaz and won via doctor stoppage when a cut on Nate's face was deemed too severe for him to continue.
In 2020, Masvidal received his first UFC title shot when he agreed to face welterweight champion Kamaru Usman on a week's notice. Jorge turned in a respectable performance but was soundly outpointed over five rounds.
Earlier this year, Masvidal was granted a rematch with a full training camp, but ended up being knocked out in two rounds. For those betting on UFC with DraftKings, they would have found Masvidal a significant underdog for both fights with Usman, who is considered the best UFC fighter on the planet.
With no fight scheduled, attention has turned to Masvidal's next steps. There are murmurings of a fight with Leon Edwards, with whom Jorge has previous after a backstage scuffle a few years ago.
Whatever step he takes, Masvidal has earned the large audience he now commands and it will be another fascinating move in the welterweight division.
pwfree
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DO YOU HAVE NEWS FOR US at Spain's most popular English newspaper - the Olive Press? Contact us now via email: newsdesk@theolivepress.es or call 951 273 575. To contact the newsdesk out of regular office hours please call +34 665 798 618.
Bird lovers encouraged to visit Fuente la Piedra and listen to spectacular trumpeting of cranes before they migrate North at the end of February
2 bedroom Villa for sale in Denia with pool – € 265,000
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Madrid bigamist married two women within a few months in Spain residency scam
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 5,953
|
{"url":"https:\/\/semiengineering.com\/silicon-thermo-optic-switches-with-graphene-heaters-operating-at-mid-infrared-waveband\/","text":"# Silicon Thermo-Optic Switches with Graphene Heaters Operating at Mid-Infrared Waveband\n\nResearchers demonstrate thermo-optic (TO) switches operating at 2 \u03bcm by integrating graphene onto silicon-on-insulator (SOI) structures. The thermal and optical properties of graphene make it a good heater material platform.\n\nAbstract\n\u201cThe mid-infrared (MIR, 2\u201320 \u03bcm) waveband is of great interest for integrated photonics in many applications such as on-chip spectroscopic chemical sensing, and optical communication. Thermo-optic switches are essential to large-scale integrated photonic circuits at MIR wavebands. However, current technologies require a thick cladding layer, high driving voltages or may introduce high losses in MIR wavelengths, limiting the performance. This paper has demonstrated thermo-optic (TO) switches operating at 2 \u03bcm by integrating graphene onto silicon-on-insulator (SOI) structures. The remarkable thermal and optical properties of graphene make it an excellent heater material platform. The lower loss of graphene at MIR wavelength can reduce the required cladding thickness for the thermo-optics phase shifter from micrometers to tens of nanometers, resulting in a lower driving voltage and power consumption. The modulation efficiency of the microring resonator (MRR) switch was 0.11 nm\/mW. The power consumption for 8-dB extinction ratio was 5.18 mW (0.8 V modulation voltage), and the rise\/fall time was 3.72\/3.96 \u03bcs. Furthermore, we demonstrated a 2 \u00d7 2 Mach-Zehnder interferometer (MZI) TO switch with a high extinction ratio of more than 27 dB and a switching rise\/fall time of 4.92\/4.97 \u03bcs. A comprehensive analysis of the device performance affected by the device structure and the graphene Fermi level was also performed. The theoretical figure of merit (2.644 mW\u22121\/\u03bcs\u22121) of graphene heaters is three orders of magnitude higher than that of metal heaters. Such results indicate graphene is an exceptional nanomaterial for future MIR optical interconnects.\u201d\n\nFind the open source technical paper link here.\u00a0 Published 03-2022.\n\nZhong, C.; Zhang, Z.; Ma, H.; Wei, M.; Ye, Y.; Wu, J.; Tang, B.; Zhang, P.; Liu, R.; Li, J.; Li, L.; Hu, X.; Liu, K.; Lin, H. Silicon Thermo-Optic Switches with Graphene Heaters Operating at Mid-Infrared Waveband. Nanomaterials 2022, 12, 1083. https:\/\/doi.org\/10.3390\/nano12071083.\n\nVisit Semiconductor Engineering\u2019s Technical Paper library here and discover many more chip industry academic papers.\n\nTags:","date":"2022-05-17 23:44:18","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8545790910720825, \"perplexity\": 11154.095084097358}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662520936.24\/warc\/CC-MAIN-20220517225809-20220518015809-00339.warc.gz\"}"}
| null | null |
Q: Iphone stuck on 'Connect to iTunes" screen, I don't have a computer with iTunes available I am stuck at work, and i tried to install ios 8.4 to my iPhone 5s, now it's stuck on the "Connect to iTunes" screen, well I don't have a computer with iTunes on it. My laptop is my work computer and it's and ASUS, and I can't install iTunes on any of the desktops here because it's not allowed. I desperately need my phone for work right now. I've tried resetting it, locking it, everything.
A: All you can really do is see if you can use one of your friend's computers to use iTunes on it. There's absolutely nothing you can do to the iPhone to get rid of that Connect To iTunes screen, short of connecting it to iTunes.
A: Same problem happened to iOS6, and iOS7 update, and try the steps in this article. https://support.apple.com/en-us/HT201263
If you don't have iTunes, take your phone to a computer that does. You can get your settings entered and restore from iCloud backup (if you have one) once ios is restored.
A: What worked for me after crawling around the internet was to install the latest version of iTunes then running it in Administrator mode. Meaning right clicking the iTunes program and click on "Run as Administrator."
What gave me the idea was after running a diagnostic test, it mentioned "User is not an administrator." So I said, hmmm maybe this is what is causing this. That did it and it worked and now my phone is updating and out of recovery mode.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,689
|
package at.alladin.rmbt.mapServer;
import java.awt.Color;
import java.awt.Graphics2D;
import java.awt.RenderingHints;
import java.awt.geom.Path2D;
import java.io.ByteArrayOutputStream;
import java.sql.Connection;
import java.sql.PreparedStatement;
import java.sql.ResultSet;
import java.sql.SQLException;
import java.util.ArrayList;
import java.util.List;
import javax.imageio.ImageIO;
import org.postgis.Geometry;
import org.postgis.MultiPolygon;
import org.postgis.PGgeometry;
import org.postgis.Point;
import org.postgis.Polygon;
import org.restlet.data.Form;
import at.alladin.rmbt.mapServer.MapServerOptions.MapOption;
import at.alladin.rmbt.mapServer.MapServerOptions.SQLFilter;
import at.alladin.rmbt.mapServer.parameters.ShapeTileParameters;
import at.alladin.rmbt.mapServer.parameters.TileParameters.Path;
import com.google.common.base.Strings;
public class ShapeTiles extends TileRestlet<ShapeTileParameters>
{
private static class GeometryColor
{
final Geometry geometry;
final Color color;
public GeometryColor(final Geometry geometry, final Color color)
{
this.geometry = geometry;
this.color = color;
}
}
@Override
protected ShapeTileParameters getTileParameters(Path path, Form params)
{
return new ShapeTileParameters(path, params);
}
@Override
protected byte[] generateTile(final ShapeTileParameters params, final int tileSizeIdx, final int zoom, final DBox box,
final MapOption mo, final List<SQLFilter> filters, final float quantile)
{
Connection con = null;
PreparedStatement ps = null;
ResultSet rs = null;
double _transparency = params.getTransparency();
try
{
con = DbConnection.getConnection();
final StringBuilder whereSQL = new StringBuilder(mo.sqlFilter);
for (final SQLFilter sf : filters)
whereSQL.append(" AND ").append(sf.where);
/*
* old zip code
*
final String sql = String.format(
"WITH box AS"
+ " (SELECT ST_SetSRID(ST_MakeBox2D(ST_Point(?,?),"
+ " ST_Point(?,?)), 900913) AS box)"
+ " SELECT"
+ " ST_SnapToGrid(ST_intersection(p.the_geom, box.box), ?,?,?,?) AS geom,"
+ " count(\"%1$s\") count,"
+ " quantile(\"%1$s\",?) val"
+ " FROM box, plz2001 p"
+ " JOIN v_test2 t ON t.zip_code=p.plz_4"
+ " WHERE" + " %2$s"
+ " AND p.the_geom && box.box"
+ " AND ST_intersects(p.the_geom, box.box)"
+ " GROUP BY p.the_geom, box.box", mo.valueColumnLog, whereSQL);
*/
final String sql = String.format(
"WITH box AS"
+ " (SELECT ST_SetSRID(ST_MakeBox2D(ST_Point(?,?),"
+ " ST_Point(?,?)), 900913) AS box)"
+ " SELECT"
+ " ST_SnapToGrid(ST_intersection(p.the_geom, box.box), ?,?,?,?) AS geom,"
+ " count(\"%1$s\") count,"
+ " quantile(\"%1$s\",?) val"
+ " FROM box, kategorisierte_gemeinden p"
+ " JOIN v_test2 t ON t.gkz=p.gemeinde_i"
+ " WHERE" + " %2$s"
+ " AND p.the_geom && box.box"
+ " AND ST_intersects(p.the_geom, box.box)"
+ " GROUP BY p.the_geom, box.box", mo.valueColumnLog, whereSQL);
ps = con.prepareStatement(sql);
int idx = 1;
/* makeBox2D */
final double margin = box.res * 1;
ps.setDouble(idx++, box.x1 - margin);
ps.setDouble(idx++, box.y1 - margin);
ps.setDouble(idx++, box.x2 + margin);
ps.setDouble(idx++, box.y2 + margin);
/* snapToGrid */
ps.setDouble(idx++, box.x1);
ps.setDouble(idx++, box.y1);
ps.setDouble(idx++, box.res);
ps.setDouble(idx++, box.res);
ps.setFloat(idx++, quantile);
for (final SQLFilter sf : filters)
idx = sf.fillParams(idx, ps);
rs = ps.executeQuery();
if (rs == null)
throw new IllegalArgumentException();
final List<GeometryColor> geoms = new ArrayList<>();
while (rs.next())
{
final String geomStr = rs.getString("geom");
if (! Strings.isNullOrEmpty(geomStr))
{
final Geometry geom = PGgeometry.geomFromString(geomStr);
final long count = rs.getLong("count");
final double val = rs.getDouble("val");
final int colorInt = valueToColor(mo.colorsSorted, mo.intervalsSorted, val);
double transparency = ((double)count / 20d) * _transparency;
if (transparency > _transparency)
transparency = _transparency;
final int alpha = (int) Math.round(transparency * 255) << 24;
final Color color = new Color(colorInt | alpha, true);
geoms.add(new GeometryColor(geom, color));
}
}
if (geoms.isEmpty())
return null;
final Image img = images[tileSizeIdx].get();
final Graphics2D g = img.g;
g.setBackground(new Color(0, 0, 0, 0));
g.clearRect(0, 0, img.width, img.height);
// g.setComposite(AlphaComposite.Src);
g.setRenderingHint(RenderingHints.KEY_ANTIALIASING, RenderingHints.VALUE_ANTIALIAS_ON);
final Path2D.Double path = new Path2D.Double();
for (final GeometryColor geomColor : geoms)
{
final Geometry geom = geomColor.geometry;
final Polygon[] polys;
if (geom instanceof MultiPolygon)
polys = ((MultiPolygon) geom).getPolygons();
else if (geom instanceof Polygon)
polys = new Polygon[] { (Polygon) geom };
else
polys = new Polygon[] {};
for (final Polygon poly : polys)
for (int i = 0; i < poly.numRings(); i++)
{
final Point[] points = poly.getRing(i).getPoints();
path.reset();
boolean initial = true;
for (final Point point : points)
{
final double relX = (point.x - box.x1) / box.res;
final double relY = TILE_SIZES[tileSizeIdx] - (point.y - box.y1) / box.res;
if (initial)
{
initial = false;
path.moveTo(relX, relY);
}
path.lineTo(relX, relY);
}
g.setPaint(geomColor.color);
g.fill(path);
}
}
final ByteArrayOutputStream baos = new ByteArrayOutputStream();
ImageIO.write(img.bi, "png", baos);
final byte[] data = baos.toByteArray();
return data;
}
catch (final Exception e)
{
e.printStackTrace();
throw new IllegalStateException(e);
}
finally
{
try
{
if (rs != null)
rs.close();
if (ps != null)
ps.close();
if (con != null)
con.close();
}
catch (final SQLException e)
{
e.printStackTrace();
throw new IllegalStateException(e);
}
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,873
|
Nointel est une commune française située dans le département du Val-d'Oise en région d'Île-de-France.
Ses habitants sont les Nointellois.
Géographie
Description
Nointel est un bourg périurbain de la vallée de l'Oise dans le Val-d'Oise situé sur le flanc occidental de la forêt de Carnelle.
Il est aisément accessible par l'autoroute A16 (France) et les anciennes routes nationales RN 1 et RN 322 (actuelles RD 301 et 922), et se trouve à au nord de Paris, aunord-est de Pontoise et à au sud-est de Beauvais.
Nointel est desservie par la gare de Nointel - Mours située sur la ligne d'Épinay - Villetaneuse au Tréport - Mers, et desservie par les trains de la ligne H du Transilien de la relation Paris Nord - Persan - Beaumont par Montsoult.
Communes limitrophes
Hydrographie
Un lac de s'étend sur les communes de Beaumont-sur-Oise, Nointel et Mours. Propriété de la Fédération française d'études et de sports sous-marins, il est dangereux et plusieurs noyades y ont eu lieu. Son accès est interdit.
Urbanisme
Typologie
Nointel est une commune urbaine, car elle fait partie des communes denses ou de densité intermédiaire, au sens de la grille communale de densité de l'Insee.
Par ailleurs la commune fait partie de l'aire d'attraction de Paris, dont elle est une commune de la couronne. Cette aire regroupe .
Toponymie
La localité s'appelait autrefois Nucistella en 1153, Noiatellum en 1222.
. Son nom sera définitif au .
Histoire
Son histoire remonte au . Sa seigneurie était constituée de quatre fiefs qui sont réunis par Yves comte de Beaumont.
Jean de Turmenyes, chevalier de l'Ordre du roi et garde du Trésor Royal, fait ériger le château de Nointel entre le .
Politique et administration
Rattachements administratifs et électoraux
Rattachements administratifs
Antérieurement à la loi du 10 juillet 1964, la commune faisait partie du département de Seine-et-Oise. La réorganisation de la région parisienne en 1964 fit que la commune appartient désormais au département du Val-d'Oise et à son arrondissement de Pontoise après un transfert administratif effectif au .
Elle faisait partie de 1801 à 1967 du canton de L'Isle-Adam de Seine-et-Oise. Lors de la mise en place du Val-d'Oise, la ville intègre le canton de Beaumont-sur-Oise. Dans le cadre du redécoupage cantonal de 2014 en France, cette circonscription administrative territoriale a disparu, et le canton n'est plus qu'une circonscription électorale.
La commune fait partie du tribunal judiciaire ainsi que de celle du tribunal de commerce de Pontoise.
Rattachements électoraux
Pour les élections départementales, la commune est membre depuis 2014 du canton de L'Isle-Adam
Pour l'élection des députés, elle fait partie de la première circonscription du Val-d'Oise.
Intercommunalité
Nointel est membre de la communauté de communes du Haut Val-d'Oise, un établissement public de coopération intercommunale (EPCI) à fiscalité propre créé fin 2004 et auquel la commune a transféré un certain nombre de ses compétences, dans les conditions déterminées par le code général des collectivités territoriales.
Dans le cadre des dispositions de la loi portant nouvelle organisation territoriale de la République du 7 août 2015, qui prévoit que les établissements publics de coopération intercommunale (EPCI) à fiscalité propre doivent avoir un minimum de , cette intercommunalité a fusionné avec sa voisine pour former, le , la communauté de communes du Haut Pays du Montreuillois dont est désormais membre la commune.
Liste des maires
Équipements et services publics
Enseignement
Nointel est situé dans l'académie de Versailles.
La ville dispose de l'école élémentaire Jean II de Turmeynies
Démographie
Culture locale et patrimoine
Lieux et monuments
L'ensemble du domaine du château de Nointel est inscrit monument historique en 1992, y inclus les éléments architecturaux en dehors du périmètre du parc, à savoir les bornes devant l'entrée principale est, l'obélisque, les anciens communs et les anciennes écuries. La colonne et la Vierge avec leurs perspectives en sont toutefois exclus, tout comme l'orangerie. Le parc du château a été classé au titre des sites naturels par arrêté du , sur la base de la loi du 2 mai 1930 relative à la protection des monuments naturels et des sites de caractère artistique, historique, scientifique, légendaire ou pittoresque. Les éléments protégé sont les suivants :
Le château de Nointel fut élevé en 1680 pour Jean de Turmenyes, garde du Trésor royal de Louis XIV, par un architecte élève de Jules Hardouin-Mansart. Transformé en rendez-vous de chasse par le prince de Conti en 1749, le domaine est ensuite acquis par le financier Bergeret, qui y invita Fragonard. Le château fut remanié à la fin du par son propriétaire, M. Ribault, qui fait surélever le bâtiment d'un étage et transformer le parc en jardin à l'anglaise. Propriété du prince Murat à partir de 1982, le palais fut transformé en Musée d'art contemporain par ce grand collectionneur qui y exposa sa collection personnelle avant d'être contraint, en 1987, pour des raisons financières, de revendre la bâtisse et son contenu. Le château abrite aujourd'hui un complexe hôtelier voué à l'organisation de séminaires.
Le parc du château est considéré comme l'un des plus grands parcs à l'anglaise de la région parisienne. Il possède de nombreux aménagements paysagers et architecturaux. Sont à citer notamment un grand escalier de quatre-vingts marches attribué à François Mansart ; un petit pavillon octogonal près du bassin du Mississipi de 1720 alimenté par des eaux captées en forêt de Carnelle ; des balustrades et statues sans doute de Coysevox ; vingt bassins et fontaines créés par la famille de Turmenyes. Les grilles d'entrée vers l'ouest (avenue de Verdun) et vers le sud (rue des Bohémies), ainsi que les pavillons de garde et les bornes situées sur le parterre sud et devant les écuries font également partie du patrimoine protégé.
La propriété « Le Domaine » au sud-est du château est issue de la transformation d'anciennes dépendances du château. Elles datent encore du petit château, lieu de résidence des seigneurs de Nointel avant la construction du château actuel, et remontent au . Dès l'achèvement du nouveau château, ces dépendances servaient à loger les domestiques et ont été maintes fois remaniées, avant de prendre l'aspect d'un manoir cossu du style néo-Renaissance en 1866. L'ancien colombier à l'est est encore bien identifiable, et les anciennes écuries subsistent au nord, le long de l'impasse Saint-Denis.
L'obélisque qui fait face au château au sud-est (rue Alain-Bernier) fut dressée en ce lieu en 1932, après qu'il avait été offert en cadeau à Paul Béjot, le propriétaire du château d'alors. Il commémore la victoire de Napoléon à Wagram et a été érigé en 1809 dans le parc du château de Castille à Uzès.
On peut également signaler :
L'église Saint-Denis, avenue de Verdun : Elle remonte aux , mais fut largement remaniée entre 1879 et 1892 par Paul Béjot, maire de 1868 à 1897 et propriétaire du château. Depuis, l'église se présente dans un style néogothique. Elle conserve des fonts baptismaux sculptés du .
Une meule de pressoir, impasse Saint-Denis, visible dans un enclos privé à proximité de l'église : Elle date du et provient de Normandie. La meule était actionnée par la force animale et servait à écraser des fruits (des pommes en général) placés sur de la paille.
L'ancienne orangerie du château, rue de l'Orangerie : Construite en 1871 dans un style classique, elle remplaça une orangerie plus ancienne. Situé sur un terrain appelé le clos Servoisier, elle est proche de la perspective de la colonne et de l'actuelle mairie, qui fut le logement du jardinier-chef du château. Le potager du château se trouvait à côté de l'orangerie. Après une période d'abandon, l'orangerie a été réhabilitée et sert aujourd'hui d'atelier et d'habitation à un architecte et sa famille.
La statue de la Vierge Marie, à l'extrémité ouest de l'allée de Verdun qui part de l'église, près de la voie ferrée à l'ouest du village : Connue comme « Notre-Dame-des-Moissons », elle fut offerte par la famille Béjot le lors du baptême de leur fille Amélie. La Vierge regarde vers l'église et le village. Il lui manque une main, vandalisée et refaite deux fois.
La grande colonne décorative, à l'extrémité occidentale de l'allée-perspective partant de l'entrée ouest du château, rue des Bohémies : Cette colonne provient d'Uzès, tout comme l'obélisque.
Lavoir.
Personnalités liées à la commune
Honoré de Balzac séjourna au château.
Voir aussi
Articles connexes
Liste des communes du Val-d'Oise
Liens externes
Site de la mairie
.
.
.
Notes et références
Notes
Références
Nointel
Aire urbaine de Paris
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,149
|
using System.Collections.Generic;
using System.Text.Json;
using Azure.Core;
namespace Azure.ResourceManager.MachineLearning.Models
{
public partial class EnvironmentVersionProperties : IUtf8JsonSerializable
{
void IUtf8JsonSerializable.Write(Utf8JsonWriter writer)
{
writer.WriteStartObject();
if (Optional.IsDefined(AutoRebuild))
{
writer.WritePropertyName("autoRebuild");
writer.WriteStringValue(AutoRebuild.Value.ToString());
}
if (Optional.IsDefined(Build))
{
writer.WritePropertyName("build");
writer.WriteObjectValue(Build);
}
if (Optional.IsDefined(CondaFile))
{
writer.WritePropertyName("condaFile");
writer.WriteStringValue(CondaFile);
}
if (Optional.IsDefined(Image))
{
writer.WritePropertyName("image");
writer.WriteStringValue(Image);
}
if (Optional.IsDefined(InferenceConfig))
{
writer.WritePropertyName("inferenceConfig");
writer.WriteObjectValue(InferenceConfig);
}
if (Optional.IsDefined(OSType))
{
writer.WritePropertyName("osType");
writer.WriteStringValue(OSType.Value.ToString());
}
if (Optional.IsDefined(IsAnonymous))
{
writer.WritePropertyName("isAnonymous");
writer.WriteBooleanValue(IsAnonymous.Value);
}
if (Optional.IsDefined(IsArchived))
{
writer.WritePropertyName("isArchived");
writer.WriteBooleanValue(IsArchived.Value);
}
if (Optional.IsDefined(Description))
{
if (Description != null)
{
writer.WritePropertyName("description");
writer.WriteStringValue(Description);
}
else
{
writer.WriteNull("description");
}
}
if (Optional.IsCollectionDefined(Properties))
{
if (Properties != null)
{
writer.WritePropertyName("properties");
writer.WriteStartObject();
foreach (var item in Properties)
{
writer.WritePropertyName(item.Key);
writer.WriteStringValue(item.Value);
}
writer.WriteEndObject();
}
else
{
writer.WriteNull("properties");
}
}
if (Optional.IsCollectionDefined(Tags))
{
if (Tags != null)
{
writer.WritePropertyName("tags");
writer.WriteStartObject();
foreach (var item in Tags)
{
writer.WritePropertyName(item.Key);
writer.WriteStringValue(item.Value);
}
writer.WriteEndObject();
}
else
{
writer.WriteNull("tags");
}
}
writer.WriteEndObject();
}
internal static EnvironmentVersionProperties DeserializeEnvironmentVersionProperties(JsonElement element)
{
Optional<AutoRebuildSetting> autoRebuild = default;
Optional<BuildContext> build = default;
Optional<string> condaFile = default;
Optional<EnvironmentType> environmentType = default;
Optional<string> image = default;
Optional<InferenceContainerProperties> inferenceConfig = default;
Optional<OperatingSystemType> osType = default;
Optional<bool> isAnonymous = default;
Optional<bool> isArchived = default;
Optional<string> description = default;
Optional<IDictionary<string, string>> properties = default;
Optional<IDictionary<string, string>> tags = default;
foreach (var property in element.EnumerateObject())
{
if (property.NameEquals("autoRebuild"))
{
if (property.Value.ValueKind == JsonValueKind.Null)
{
property.ThrowNonNullablePropertyIsNull();
continue;
}
autoRebuild = new AutoRebuildSetting(property.Value.GetString());
continue;
}
if (property.NameEquals("build"))
{
if (property.Value.ValueKind == JsonValueKind.Null)
{
property.ThrowNonNullablePropertyIsNull();
continue;
}
build = BuildContext.DeserializeBuildContext(property.Value);
continue;
}
if (property.NameEquals("condaFile"))
{
condaFile = property.Value.GetString();
continue;
}
if (property.NameEquals("environmentType"))
{
if (property.Value.ValueKind == JsonValueKind.Null)
{
property.ThrowNonNullablePropertyIsNull();
continue;
}
environmentType = new EnvironmentType(property.Value.GetString());
continue;
}
if (property.NameEquals("image"))
{
image = property.Value.GetString();
continue;
}
if (property.NameEquals("inferenceConfig"))
{
if (property.Value.ValueKind == JsonValueKind.Null)
{
property.ThrowNonNullablePropertyIsNull();
continue;
}
inferenceConfig = InferenceContainerProperties.DeserializeInferenceContainerProperties(property.Value);
continue;
}
if (property.NameEquals("osType"))
{
if (property.Value.ValueKind == JsonValueKind.Null)
{
property.ThrowNonNullablePropertyIsNull();
continue;
}
osType = new OperatingSystemType(property.Value.GetString());
continue;
}
if (property.NameEquals("isAnonymous"))
{
if (property.Value.ValueKind == JsonValueKind.Null)
{
property.ThrowNonNullablePropertyIsNull();
continue;
}
isAnonymous = property.Value.GetBoolean();
continue;
}
if (property.NameEquals("isArchived"))
{
if (property.Value.ValueKind == JsonValueKind.Null)
{
property.ThrowNonNullablePropertyIsNull();
continue;
}
isArchived = property.Value.GetBoolean();
continue;
}
if (property.NameEquals("description"))
{
if (property.Value.ValueKind == JsonValueKind.Null)
{
description = null;
continue;
}
description = property.Value.GetString();
continue;
}
if (property.NameEquals("properties"))
{
if (property.Value.ValueKind == JsonValueKind.Null)
{
properties = null;
continue;
}
Dictionary<string, string> dictionary = new Dictionary<string, string>();
foreach (var property0 in property.Value.EnumerateObject())
{
if (property0.Value.ValueKind == JsonValueKind.Null)
{
dictionary.Add(property0.Name, null);
}
else
{
dictionary.Add(property0.Name, property0.Value.GetString());
}
}
properties = dictionary;
continue;
}
if (property.NameEquals("tags"))
{
if (property.Value.ValueKind == JsonValueKind.Null)
{
tags = null;
continue;
}
Dictionary<string, string> dictionary = new Dictionary<string, string>();
foreach (var property0 in property.Value.EnumerateObject())
{
if (property0.Value.ValueKind == JsonValueKind.Null)
{
dictionary.Add(property0.Name, null);
}
else
{
dictionary.Add(property0.Name, property0.Value.GetString());
}
}
tags = dictionary;
continue;
}
}
return new EnvironmentVersionProperties(description.Value, Optional.ToDictionary(properties), Optional.ToDictionary(tags), Optional.ToNullable(isAnonymous), Optional.ToNullable(isArchived), Optional.ToNullable(autoRebuild), build.Value, condaFile.Value, Optional.ToNullable(environmentType), image.Value, inferenceConfig.Value, Optional.ToNullable(osType));
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,034
|
// TypeScript Version: 2.0
/// <reference types="@stdlib/types"/>
import { Iterator as Iter, IterableIterator } from '@stdlib/types/iter';
// Define a union type representing both iterable and non-iterable iterators:
type Iterator = Iter | IterableIterator;
/**
* Returns an iterator which iteratively computes the angle in the plane (in radians) between the positive x-axis and the ray from `(0,0)` to the point `(x,y)`.
*
* ## Notes
*
* - If provided a numeric value as an iterator argument, the value is broadcast as an **infinite** iterator which **always** returns the provided value.
* - If an iterated value is non-numeric (including `NaN`), the returned iterator returns `NaN`. If non-numeric iterated values are possible, you are advised to provide an iterator which type checks and handles non-numeric values accordingly.
* - The length of the returned iterator is equal to the length of the shortest provided iterator. In other words, the returned iterator ends once **one** of the provided iterators ends.
* - If an environment supports `Symbol.iterator` and all provided iterators are iterable, the returned iterator is iterable.
*
* @param y - input iterator
* @param x - input iterator
* @returns iterator
*
* @example
* var uniform = require( `@stdlib/random/iter/uniform` );
*
* var iter = iterAtan2( uniform( -2.0, 2.0 ), uniform( -2.0, 2.0 ) );
*
* var r = iter.next().value;
* // returns <number>
*
* r = iter.next().value;
* // returns <number>
*
* r = iter.next().value;
* // returns <number>
*
* // ...
*/
declare function iterAtan2( y: Iterator | number, x: Iterator | number ): Iterator; // tslint:disable-line:max-line-length
// EXPORTS //
export = iterAtan2;
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 4,371
|
using System;
using System.Collections.Generic;
using System.Linq;
using System.Reflection;
namespace Fixie.Samples.MbUnitStyle
{
public class CustomConvention : Convention
{
public CustomConvention()
{
Classes
.HasOrInherits<TestFixtureAttribute>();
Methods
.HasOrInherits<TestAttribute>();
ClassExecution
.CreateInstancePerClass()
.SortCases((caseA, caseB) => String.Compare(caseA.Name, caseB.Name, StringComparison.Ordinal));
FixtureExecution
.Wrap<FixtureSetUpTearDown>();
CaseExecution
.Wrap<SupportExpectedExceptions>()
.Wrap<SetUpTearDown>();
Parameters
.Add<RowAttributeParameterSource>()
.Add<ColumnAttributeParameterSource>();
}
class RowAttributeParameterSource : ParameterSource
{
public IEnumerable<object[]> GetParameters(MethodInfo method)
{
return method.GetCustomAttributes<RowAttribute>(true).Select(input => input.Parameters);
}
}
class ColumnAttributeParameterSource : ParameterSource
{
public IEnumerable<object[]> GetParameters(MethodInfo method)
{
return CartesianProduct(Columns(method));
}
static IEnumerable<object[]> Columns(MethodInfo method)
{
ParameterInfo[] parameters = method.GetParameters();
if (parameters.Length == 0)
return null;
if (parameters[0].GetCustomAttributes<ColumnAttribute>(true).Any() == false)
return null;
return parameters
.Select(parameter =>
parameter.GetCustomAttributes<ColumnAttribute>(true).Single().Parameters);
}
static IEnumerable<object[]> CartesianProduct(IEnumerable<object[]> sequences)
{
if (sequences == null)
return new object[][] { };
//See http://blogs.msdn.com/b/ericlippert/archive/2010/06/28/computing-a-cartesian-product-with-linq.aspx
IEnumerable<object[]> emptyProduct = new[] { new object[] { } };
return sequences.Aggregate(
emptyProduct,
(accumulator, sequence) =>
from accseq in accumulator
from item in sequence
select accseq.Concat(new[] { item }).ToArray());
}
}
}
class SupportExpectedExceptions : CaseBehavior
{
public void Execute(Case @case, Action next)
{
next();
var attribute = @case.Method.GetCustomAttributes<ExpectedExceptionAttribute>(false).SingleOrDefault();
if (attribute == null)
return;
if (@case.Exceptions.Count > 1)
return;
var exception = @case.Exceptions.SingleOrDefault();
if (exception == null)
throw new Exception("Expected exception of type " + attribute.ExpectedException + ".");
if (exception.GetType() != attribute.ExpectedException)
{
@case.ClearExceptions();
throw new Exception("Expected exception of type " + attribute.ExpectedException + " but an exception of type " + exception.GetType() + " was thrown.", exception);
}
if (attribute.ExpectedMessage != null && exception.Message != attribute.ExpectedMessage)
{
@case.ClearExceptions();
throw new Exception("Expected exception message '" + attribute.ExpectedMessage + "'" + " but was '" + exception.Message + "'.", exception);
}
@case.ClearExceptions();
}
}
class SetUpTearDown : CaseBehavior
{
public void Execute(Case @case, Action next)
{
@case.Class.InvokeAll<SetUpAttribute>(@case.Fixture.Instance);
next();
@case.Class.InvokeAll<TearDownAttribute>(@case.Fixture.Instance);
}
}
class FixtureSetUpTearDown : FixtureBehavior
{
public void Execute(Fixture fixture, Action next)
{
fixture.Class.Type.InvokeAll<FixtureSetUpAttribute>(fixture.Instance);
next();
fixture.Class.Type.InvokeAll<FixtureTearDownAttribute>(fixture.Instance);
}
}
public static class BehaviorBuilderExtensions
{
public static void InvokeAll<TAttribute>(this Type type, object instance)
where TAttribute : Attribute
{
foreach (var method in Has<TAttribute>(type))
{
try
{
method.Invoke(instance, null);
}
catch (TargetInvocationException exception)
{
throw new PreservedException(exception.InnerException);
}
}
}
static IEnumerable<MethodInfo> Has<TAttribute>(Type type) where TAttribute : Attribute
{
return type.GetMethods(BindingFlags.Public | BindingFlags.Instance)
.Where(x => x.HasOrInherits<TAttribute>());
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,523
|
sort_order: 140
tei_id: p.idm239139599632
annotation_count: 0
images:
small-thumbnail: http://example.com/books/emory:7st9j/pages/readux-rsk:pgct3/mini-thumbnail/
json: http://example.com/books/emory:7st9j/pages/readux-rsk:pgct3/info/
full: http://example.com/books/emory:7st9j/pages/readux-rsk:pgct3/fs/
page: http://example.com/books/emory:7st9j/pages/readux-rsk:pgct3/single-page/
thumbnail: http://example.com/books/emory:7st9j/pages/readux-rsk:pgct3/thumbnail/
title: Page 140
number: 140
permalink: /pages/140/
---
<div class="ocr-line ocrtext" style="left:17.39%;top:6.28%;width:1.74%;height:0.86%;text-align:left;font-size:8.6px" data-vhfontsize="0.86">
<span>{% raw %}'J{% endraw %}</span>
</div>
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<span>{% raw %}6 THE BUSINESS OF PLEASURE.{% endraw %}</span>
</div>
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<span>{% raw %}restricted our military ambition Avithin proper limits ; but our{% endraw %}</span>
</div>
<div class="ocr-line ocrtext" style="left:16.99%;top:11.35%;width:66.55%;height:1.94%;text-align:left;font-size:19.35px" data-vhfontsize="1.94">
<span>{% raw %}corps increased so tremendously, so many fresh recruits{% endraw %}</span>
</div>
<div class="ocr-line ocrtext" style="left:16.99%;top:13.76%;width:66.49%;height:1.81%;text-align:left;font-size:18.06px" data-vhfontsize="1.81">
<span>{% raw %}came flocking to our standard, that we Avere obliged to form{% endraw %}</span>
</div>
<div class="ocr-line ocrtext" style="left:16.93%;top:16.04%;width:66.55%;height:1.85%;text-align:left;font-size:18.49px" data-vhfontsize="1.85">
<span>{% raw %}a second company, who, in their turn, elected their officers,{% endraw %}</span>
</div>
<div class="ocr-line ocrtext" style="left:16.99%;top:18.37%;width:66.61%;height:1.94%;text-align:left;font-size:19.35px" data-vhfontsize="1.94">
<span>{% raw %}and Avho chose for their captain a gentleman who, from his{% endraw %}</span>
</div>
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<span>{% raw %}punctuality, exactitude, and strict attention to business,{% endraw %}</span>
</div>
<div class="ocr-line ocrtext" style="left:16.99%;top:23.1%;width:66.61%;height:1.89%;text-align:left;font-size:18.92px" data-vhfontsize="1.89">
<span>{% raw %}seems intended by Nature to supply the place of the late{% endraw %}</span>
</div>
<div class="ocr-line ocrtext" style="left:17.04%;top:25.46%;width:66.61%;height:1.94%;text-align:left;font-size:19.35px" data-vhfontsize="1.94">
<span>{% raw %}Duke of Wellington in these dominions. He Avas elected{% endraw %}</span>
</div>
<div class="ocr-line ocrtext" style="left:16.99%;top:27.83%;width:66.61%;height:1.94%;text-align:left;font-size:19.35px" data-vhfontsize="1.94">
<span>{% raw %}because he was a pleasant, strong, active young felloAv, a good{% endraw %}</span>
</div>
<div class="ocr-line ocrtext" style="left:16.99%;top:30.19%;width:66.61%;height:1.89%;text-align:left;font-size:18.92px" data-vhfontsize="1.89">
<span>{% raw %}cricketer and oarsman, and such a maniac for dancing that{% endraw %}</span>
</div>
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<span>{% raw %}venience, Avhen the senior sergeant of the regiment{% endraw %}</span>
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<span>{% raw %}Piper, Avith a jolly round rubicund face, a merry black eye,{% endraw %}</span>
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|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,529
|
\section{Introduction}
{{\it Graphane}}\footnote{In order to avoid the confusion with the word `{\g}', the word `{\gh}' is
written in italics throughout this paper.} {\it i.e.} the hydrogenated {\g}, is a recent
addition to the family of novel carbon materials known for their exotic properties.
\cite{sofo:153401, boukhvalov, casolo} Recently {\gh} is experimentally realized by Elias
{\it et al} \cite{elias01302009} who demonstrated that the process of hydrogenation is
reversible. This observation makes {\gh} a suitable candidate for the hydrogen-storage
material. Equally interesting is the possibility of a direct observation of
metal-insulator transition in two dimensional systems \cite{fuhrer}. as a function of
hydrogen coverage. {{\it Graphane}} is an insulator with a reported theoretical band gap of $\sim$
3.5 eV using density functional theory (DFT) \cite{sofo:153401}.
It is well known that a single sheet of {\g} is susceptible to a variety of disorders
like topological defects, impurity states, ripples, cracks etc. Pereira {\it et al}
\cite{pereira2008, pereira} have studied different models of local disorders in {\g} and
have investigated their electronic structure within tight binding method. They have
observed a significant changes in the low energy spectrum of {\g} viz., localized zero
modes, strong resonances, gap and pseudogap behavior etc depending upon the type of
disorder. Their results also indicate that by and large disorder significantly modifies
the states near the Fermi level. Yazyev and Helm \cite{yazyev} have studied defect
induced magnetism in {\g} using DFT. Their work shows that
the adsorption of hydrogen or the creation of defect on {\g} sheet lead to the local
magnetic moment. For an extensive survey of studies of disorders in {\g} we refer the
reader to a recent review by Castro Neto {\it et al} \cite{netormp}. We wish to point out
that most of the novel properties of {\g} arise due to the nature of the
density of states (DOS) near the Fermi level and these states are sensitive to the presence of defects.
Quite clearly a detailed study of defect induced states is warranted not only for {\g} but
also for {\gh}. So far there are no reports of systematic investigations of the effects
of such defects on the properties of {\gh}. In the present work we focus on the
electronic structure of {\gh} with topological defects created by the removal of one and
two carbon atoms. Such defects are experimentally realized using high-energy ion
beams as demonstrated by Jin {\it et al} \cite{jin:205501} by creating a stable carbon
chain from {\g} sheet.
The present work is based on spin density functional theory (SDFT), which is known to
underestimate the band gap. A recent calculation by Leb\`egue {\it et al} \cite{lebegue}
based on GW approximation estimated the band gap of {\gh} to be 5.4 eV. The authors have
also shown that the removal of a single hydrogen atom produces mid gap states.
The paper is organized as follows. We present relevant computational details in section
\ref{sec:compdet}. In section \ref{sec:prisine}, the results for pristine {\gh} is
summarized for the purpose of comparison. The main results of the electronic structure
calculations on the single and the double vacancy defects are presented in sections
\ref{sec:single} and \ref{sec:double} respectively. Finally the conclusions are presented
in section \ref{sec:concl}.
\section{Computational Details \label{sec:compdet}}
All the calculations have been performed on a monolayer {\gh}, having geometry as
described by Sofo {\it et al} \cite{sofo:153401}. We have used plane wave based DFT as
implemented in Quantum Espresso\footnote{http://www.quantum-espresso.org/} and VASP
\cite{vasp} using the generalized gradient approximation (GGA) \cite{PBE,PBEerr} for
exchange-correlation potential. For single vacancy study the primitive {\gh} cell is
repeated 5 times in $X$ and $Y$ directions while for that of double vacancies the
repetitions are of 7 units. When creating the vacancy, the carbon and the attached hydrogen
are removed. After removing two (four) atoms to form single (double) defect(s) there are
total of 98 (192) atoms on the {\gh} plane. The vertical axis ($Z$) of the cell is kept
as large as 10 {\AA} to avoid the interactions between the {\gh} sheets. The energy and
force thresholds are kept at 10$^{-6}$ eV and 10$^{-5}$ eV/{\AA} respectively. It may be
mentioned that even though the unit cell is large it was found to be necessary to use 5
$\times$ 5 Monkhorst-Pack K-grid for acceptable convergence in energy during the
optimization and the self consistency. After optimization we have used 11 $\times$ 11
Monkhorst-Pack K-grid for final calculations of DOS and other quantities.
\section{Results and discussion}
\subsection{Pristine {{\it Graphane}} \label{sec:prisine}}
Before we present the results of the vacancy studies, it is instructive to summarize the
properties of a pure {\gh}. Our results on pristine {\gh} are consistent with earlier
reports \cite{sofo:153401, boukhvalov}. {{\it Graphane}} is known to have two distinct conformations
depending upon the position of hydrogen atoms with respect to {\g} plane. In the {\it
chair} conformer the hydrogen atoms are attached to carbon atoms in alternating manner to
both the sides of the plane while in the {\it boat} conformer the pairs of hydrogen atoms
are attached in alternating manner \cite{sofo:153401}. Out of these two the chair
conformer is energetically more favorable, hence in the present work we have studied the
chair conformer only. It is interesting to note that in {\g} the $K$ point of Brillouin
zone is degenerate (no gap in DOS) however for {\gh} the minimum gap is observed at
$\Gamma$ point ($\sim$ 3.5 eV)
while $K$ point develops rather large gap $\sim$ 12 eV.
We now discuss the DOS of pristine {\gh} which is shown in Figure
\ref{fig:puregraphanedos}. The Fermi energy is taken at 0 and is marked by solid vertical
line. The largest peak seen (at $\sim$ -3 eV) is due to the peculiar $sp^3$-like bonding
between carbon and hydrogen. The states at the top of the valance band ($\sim$ -1 eV)
are mainly comprises of $p$ states forming to the $\sigma$ bonds among the carbon atoms.
Unlike {\g} there are no $\pi$ bonds in {\gh}.
\begin{figure}
\begin{center}
\includegraphics[width=8cm]{fig1.eps}
\end{center}
\caption{Spin polarized total DOS for pristine {\gh}.
The upper and lower halves of the graph represent up and down spins respectively. The
Fermi energy is taken at 0 and marked by a solid vertical line. The system is
non-magnetic with a band gap of $\sim 3.5 $ eV. \label{fig:puregraphanedos}
}
\end{figure}
\subsection{Single vacancy \label{sec:single}}
\begin{figure}
\center{
\includegraphics[width=7cm]{fig2.eps}
}
\caption{Optimized structure of {\gh} with a single vacancy. Carbon atoms are shown in
cyan color while hydrogen atoms are in white. The distorted nature of the sublattice
surrounding the vacancy is clearly evident (marked by the triangle). Pristine
sublattice is an equilateral triangle with $a=2.52$ \AA. \label{fig:GraphaneSingle}}
\end{figure}
We begin our discussion by presenting the study of a single vacancy and its effects on the
geometry, electronic structure of {\gh}. A single vacancy is created by removing one
carbon atom along with the attached hydrogen from pure {\gh}. The fully relaxed structure
is shown in figure \ref{fig:GraphaneSingle}. The immediate effect of the removal of atoms
is seen on the sublattice surrounding the vacancy (indicated by the triangle in the
figure). In an ideal {\gh} the carbon sublattice is an equilateral triangle with length of
the side $a=2.52$ \AA, however in the presence of vacancy the sublattice deforms
substantially. Due to reduced coordination number, three carbon atoms on the triangle are
pushed away from each other and remain as a part of the hexagon. The distortion of the
triangle is asymmetric and the symmetry breaking is due to Jahn-Teller effect. The
deformation seen here is qualitatively different than seen in the case of {\g} where two
of the carbon atoms move close to each other to form a $\sigma$ bond \cite{yazyev}. We do
not see formation of bonds between the carbon atoms. Thus vacancy leads three dangling
bonds on the triangle. As a consequence, the system becomes magnetic with magnetic moment
1 $\mu_B$. As we shall see these unpaired electrons have an interesting consequences
on the DOS.
\begin{figure}
\center{
\includegraphics[width=8cm]{fig3.eps}
}
\caption{DOS of {\gh} with single vacancy displaying the induced states in the gap.
Fermi energy is at zero and shown by the vertical line. The induced states are partially
occupied for up electrons. \label{fig:singledos}}
\end{figure}
Figure \ref{fig:singledos} shows the spin polarized DOS displaying some remarkable
features. The DOS with single vacancy is substantially different from that of pristine
{\gh} \cite{sofo:153401} especially near the Fermi energy (marked by a vertical line in
the figure). Clearly the vacancy has induced the {\it mid gap} states which are partially
occupied. In particular the induced spin up states are partially occupied while the
induced spin down states are completely empty. The appearance of mid gap states can be
attributed to the unpaired electrons from three carbon atoms due to which the states are
pushed up from valance band. To ascertain this observation we have hydrogenated three
available dangling bonds. Our results show that the increasing hydrogen concentration
(from one to three) steadily reduces the density of induced states (figure not shown).
Finally with complete hydrogenation, induced states vanish and the band gap reduces to
$\sim 3$ eV.
\begin{figure}
\center{
\includegraphics[width=8cm]{fig4.eps}
}
\caption{The site projected density of states (PDOS) on one of the carbon atom
surrounding the vacancy. It is evident that most of the contribution to the induced
states come from the carbon atoms on the sublattice.
\label{fig:pdos}}
\end{figure}
That the induced states are due to the unpaired electrons can be further seen from
examination of the site projected DOS. Figure \ref{fig:pdos} shows the site projected DOS
(PDOS) on one of the carbon atoms of the triangle. It can be seen that almost all the
contribution to the mid gap states come from the three carbon atoms. In fact the
contribution from carbon atoms except the three on the sublattice is negligible.
Interestingly the hydrogen atoms associated with carbon atom on the triangle do not give
any significant contribution. Furthermore the states turned out to be dominantly
$p$-like.
\begin{figure}
\center{
\includegraphics[width=6cm]{fig5.eps}
}
\caption{Isosurface of the charge density of the occupied state just below the Fermi
level for the case of single vacancy. The state is dominantly $p$-like and is highly localized.
\label{fig:fermiorbital}}
\end{figure}
Although there are states near the Fermi energy, these states need not necessarily
conduct. Figure \ref{fig:fermiorbital} shows the isosurface (at one fifth of the maximum
value) of the state just below Fermi level. This state is induced due to the vacancy and
can be seen to be localized. A careful examination of the isosurfaces at the different
values reveals that the induced state extends up to three to four nearest neighbouring
carbon site. Thus the conduction mechanism through such orbitals may possibly be by
hopping.
\subsection{Double Defect \label{sec:double}}
The results of a single vacancy gives enough impetus to study the effects of multiple
vacancies, especially on the mid gap states. Indeed, as we shall demonstrate, the nature
and the placement of induced states are sensitive to the vacancy-vacancy interaction.
This aspect brings in an interesting possibility of controlling the effective band gap via
creation of vacancy defects.
For the purpose of this study we have used a larger supercell consisting of 192 atoms.
Several possible scenarios emerge depending upon the separation of the two vacancies. We
have examined four structures obtained by removing 1) the closest carbon pair (separation,
$D$ = 1.51 \AA), 2) the nearest carbon atoms of the same sublattice ($D$ = 2.52 \AA), 3) a
pair from different sublattice ($D$ = 3.87 \AA) and 4) two carbon atoms having large
separation ($D$ = 10.71 \AA). In each of the cases we have fully relaxed the structure
and have examined the energetics. We have found that the total energy is the lowest when
the vacancies are at the closest distance (case 1). The total energies with respect to the
energy of the lowest energy system are shown in Table \ref{tab:deltae}.
\begin{table}
\centering
\begin{tabular}{c c c c c}
\hline
Separation $D$ & 1.51 \AA & 2.52 \AA & 3.87 \AA & 10.71 \AA\\
\hline
$\Delta_E$ (eV/atom)& 0.0 & 0.0906 & 0.0185 & 0.0338 \\
\hline
\end{tabular}
\caption{Energy difference ($\Delta_E$) with respect to lowest energy system ({\it i.
e.} vacancies in vicinity, $D=1.52$ \AA). $D$ indicate the separation between the
vacancies.
\label{tab:deltae}}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=5cm]{fig6a.eps} \\
(a)\\
\includegraphics[width=7cm]{fig6b.eps}\\
(b)
\end{center}
\caption{(a) Optimized structure of {\gh} with two vacancies for the case 1 (see text).
The rearrangement of the carbon atoms around the vacancies can be clearly seen.
(b) Spin polarized DOS corresponding to structure in (a). The DOS do not show any
induced mid gap states. Due to the rearrangement of the atoms the band gap is reduced by 0.5
eV. }
\label{fig:closedouble}
\end{figure}
Figure \ref{fig:closedouble} shows the optimized structure and DOS for the case 1.
Remarkably, the carbon atoms surrounding the vacancies rearrange to form two new $\sigma$
bonds leading to the formation of 5-8-5 ringed structure. This peculiar structure does
not leave any unpaired electrons (dangling bonds) unlike the case for single vacancy. This
is consistent with the observation that there are no mid gap states in the DOS as can be
seen from figure \ref{fig:closedouble} (b). In this case the band gap is reduced by 0.5 eV
with respect to pristine {\gh}.
In order to ascertain the formation of bonds we have examined the charge densities of
relevant states. Figure \ref{fig:doubledoscloseopti} (a) and \ref{fig:doubledoscloseopti}
(b) show the charged densities as isosurfaces for two states: one at the top of the
valance band and other at the bottom of the conduction band. The isosurfaces are shown at
one tenth of their maxima. The charge density of the state in figure
\ref{fig:doubledoscloseopti} (a) which corresponds to the valance band (occupied), clearly
depicts the formation of $\sigma$ bonds among the carbon atoms. It is interesting to note
that the bands at the bottom of the conduction band (figure
\ref{fig:doubledoscloseopti}(b)) are highly delocalized with almost no charge around the
vacancy site.
\begin{figure}
\begin{center}
\includegraphics[width=4cm]{fig7a.eps}\\
(a)\\~\\
\includegraphics[width=5cm]{fig7b.eps}\\
(b)
\end{center}
\caption{
(a) The charge density of the state at the top of valance band. Two $\sigma$ bonds
formed by rearrangement of carbon atoms are clearly seen.
(b) Highly delocalized state at the bottom of the conduction band. Both the
states resemble those of pristine {\gh} except for the modifications around the defect
site. \label{fig:doubledoscloseopti}}
\end{figure}
\begin{figure*}
\centering{\includegraphics[width=7cm]{fig8a.eps}\\
\centerline{(a): 2.52 \AA.} ~\\~\\
\includegraphics[width=7cm]{fig8b.eps} \\
\centerline{(b): 3.87 \AA.} ~\\~\\
\includegraphics[width=7cm]{fig8c.eps}\\
\centerline{(c): 10.71 \AA}}
\caption{The DOS of {\gh} with two vacancies for the cases (2), (3) and (4) (see text).
For each of the cases the nature of the induced states are different.
(a) The induced states have appeared near the conduction band for the vacancy
separation of 2.52 \AA.
(b) For the vacancy separation of 3.87 {\AA} the induced states are broadened,
effectively reducing the gap from valance band.
(c) A peculiar magnetic system with 2 $\mu_B$ is seen for the vacancy separation as large as 10.71
\AA. The induced states are near conduction band and the states on the top of valance band
are substantially deformed. \label{fig:doubledos}}
\end{figure*}
A different scenario emerges when the vacancies are separated from each other. Now there
are induced states in the gap. Figure \ref{fig:doubledos} shows the DOS for the three
cases with the vacancy separation of (a) 2.52 {\AA}, (b) 3.87 {\AA} and (c) 10.71 {\AA}
respectively. Quite clearly in the case of well separated vacancies, the atoms are not
able to rearrange so as to form any bonds. This leaves six unpaired electrons. These
unpaired electrons give rise to the mid gap states typically just above the valance band
and just below the conduction band. Interestingly their placement is sensitive to the
separation between the vacancies. For example, figure \ref{fig:doubledos} (a) shows the
DOS for the vacancy separation $D=$ 2.52 {\AA}. The induced states are seen at about 0.5
eV below the conduction band along with the additional modifications at the edge of the
valance band. As the separation increases to 3.87 {\AA} (figure \ref{fig:doubledos} (b))
the induced states are broadened with a width of about 1 eV and split in to two distinct
peaks. In most of the cases the unpaired electrons tend to cancel the spin giving rise to
a non-magnetic states. However for a large separation {\it i. e.} in the limit of weakly
interacting vacancies, both the vacancies carry a magnetic moment of 1 $\mu_B$. Figure
\ref{fig:doubledos} (c) shows the DOS for system with vacancy separation 10.71 {\AA},
depicting the spin polarized features.
The induced states are seen to be localized both in the case of single as well as double
vacancies. Figure \ref{fig:doubleorbital} depicts the charge density isosurfaces
corresponding two typical induced states for the case 2 ($D=3.87$ \AA). Figure
\ref{fig:doubleorbital} (a) corresponds to the state at the top of the valance band which
is occupied while figure \ref{fig:doubleorbital} (b) corresponds to the unoccupied state
below the conduction band. Both the charge densities are plotted at a lower value of
isosurfaces (one tenth of their maxima). Clearly the states are dominantly $p$-like and
are localized. At this value of isosurface occupied states (figure \ref{fig:doubleorbital}
(a)) is seen to have a weak overlap among the carbon atoms, however at the higher values
the lobes are disconnected. On the other hand the charge density of the unoccupied state
(figure \ref{fig:doubleorbital} (b)) is highly localized even at one tenth of the maxima.
\begin{figure*}
\begin{center}
\includegraphics[width=7cm]{fig9a.eps}
\centerline{(a)} ~\\~\\
\includegraphics[width=7cm]{fig9b.eps}\\
(b)
\end{center}
\caption{The charge densities corresponding to innduced states of {\gh} with two
vacancies
separated from each other (case 2). (a) The state at the top of the valance band. (b)
The state below the conduction band. The isosurface is shown at one tenth of its
maximum.
\label{fig:doubleorbital}}
\end{figure*}
Apart from the four cases discussed above, we have also calculated the electronic
structure and DOS for a few more vacancy separations. The results indicate that the
sublattice plays no role in appearance on mid gap states.
\section{Conclusions \label{sec:concl}}
We have carried out an {\it ab initio} investigation of the electronic structure of {\gh}
with single and double vacancy defects. For both the cases, we have analysed the fully
optimized structure, and have examined the DOS and charge densities as a function of
separation of the vacancies. Our calculations show that the most stable structure is
obtained when the vacancies are adjacent to each other, is accompanied by the reduction
of band gap. In this case there are no induced mid gap states. However, separated
vacancies induce mid gap states and interestingly their position and width are sensitive
to the vacancy separation. Examination of the charge densities of the induced states show
that these states are localized. Our calculation brings out the possibility of
manipulating the band gap and the nature of the mid gap states with the aid of
vacancy defects in {\gh}.
\begin{acknowledgement}
B.S.P. would like to acknowledge CSIR, Govt. of India for financial support (No:
9/137(0458)/2008-EMR-I). It is a pleasure to acknowledge Center for Development of
Advanced Computing for computational resources. Some of the figures are generated by
using VMD software \cite{vmd}.
\end{acknowledgement}
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Magnitude 5.3 earthquake rattles, but too weak to cause tsunami
Epicenter of the quake
(Los Angeles Times)
By Rong-Gong Lin IIStaff Writer
The magnitude 5.3 earthquake that rattled Southern California was the strongest in the region in several years, but it did not produce any tsunami warnings.
The quake was far too weak to generate a tsunami, said Chris Popham, lead oceanographer for the National Tsunami Warning Center in Palmer, Alaska. Generally, earthquakes of magnitude 7 or above are those that cause concern, he said.
"You need to be moving hundreds of miles of water to get a significant tsunami," said seismologist Lucy Jones. But Thursday's earthquake, by virtue of its modest magnitude, only ruptured a length of the fault of just two miles, Jones said.
Additionally, the earthquake occurred on a strike-slip fault, in which the faults are generally moving in a horizontal direction.
The worst tsunamis tend to occur on a different type of system, known as a subduction zone earthquake, in which faults produce a great deal of vertical motion. Subduction zone earthquakes are the kind that produced massive, deadly flooding in Japan in 2011 and happened off the coast of Oregon and Washington state in 1700.
There are, however, scenarios in which earthquakes in Southern California do cause significant tsunamis.
A study published in 2015 by U.S. Geological Survey and UC Riverside scientists found that tsunami wave heights could approach 20 feet in the Ventura Harbor and Channel Islands Beach area near Oxnard. That study focused on on a hypothetical scenario in which a magnitude 7.7 earthquake begins nine miles under the Earth's surface, under the mountains northeast of Santa Barbara.
Caltech seismologist Egill Hauksson said an offshore earthquake in 1812 caused a tsunami of perhaps 3 feet in the Ventura area. But because recorded history in California is so short, scientists don't have too much documented information of how big tsunamis have been.
3:40 p.m.: This article was updated with a quote from seismologist Lucy Jones.
This article was originally published at 2:10 p.m.
Rong-Gong Lin II
Rong-Gong Lin II is a metro reporter, specializing in covering statewide earthquake safety issues. The Bay Area native is a graduate of UC Berkeley and started at the Los Angeles Times in 2004.
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I have spent many years crafting my athletic life based on avoiding logistics, equipment, and planning as much as possible. This bikini competition is really throwing my normal MO off, because there's tons of stuff to do to get ready for it. I won't go into detail because you can find that other places on the internet if you're curious, but just to give you an idea – yesterday I had to order a silk robe online so that when I get my spray tan it has the greatest chance of not rubbing off all over everything, and so that I have something to keep myself decent with backstage. There are so many layers of ridiculous right there (considering that I'm doing this as a sport… kind of), I won't even try to break them down for you.
Contest prep is also insanely time-consuming. The workouts don't take that long, and now that I've figured out a regular food prep plan, that doesn't take much time either. But I never realized how much time I would have to dedicate to staring at food on Pinterest and looking at my abs in the mirror to make sure that they haven't disappeared and that this is worth it.
This week my diet has consisted of that meal, protein shakes, cream of wheat, and egg whites. Really that's a fairly close description of my diet for the past five months if you change fish to chicken and cream of wheat to oatmeal, but I've been way better at it in the past week than usual. I'll let you know in just a little over a week if it was all worth it. I'm pretty sure it will be, as long as I don't trip and fall while I'm on stage in my stripper heels (did I say mention this sport is ridiculous?).
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Toronto International Film Festival Review – Battle of the Sexes (2017)
September 11, 2017 by Ricky Church 1 Comment
Battle of the Sexes, 2017.
Directed by Valerie Faris and Jonathan Dayton.
Starring Emma Stone, Steve Carell, Andrea Riseborough, Sarah Silverman, Bill Pullman, Alan Cumming, Elisabeth Shue, Austin Stowell, and Natalie Morales.
The true story of the 1973 tennis match between World number one Billie Jean King and ex-champ and serial hustler Bobby Riggs.
Every now and then a film comes out about the past that is just as socially relevant today as it was then. This year's Battle of the Sexes is that film as it dives deep into issues that are still prevalent today, especially with what we hear in the news so much lately.
Emma Stone portrays tennis idol and social activist Billie Jean King as she preps for her famous match against Steve Carell's Bobby Riggs in what was dubbed the 'battle of the sexes' in Kings' fight for equal pay for female tennis players. Directors Valerie Faris and Jonathan Dayton direct a compelling, fun and thoughtful film with great performances all round from its cast.
Stone's portrayal as King is the film's main draw. She captures the icon's personality and really delivers the emotion through her performance. She conveys the character's complexities as King struggled through her sexual identity during this period on top of the stress of beginning her fight for female athletes. Stone also impresses with the role's physical aspects; though she's not always the one playing tennis (faraway shots are most likely a double) she shows she's quite capable of getting fit for a physically demanding role such as this.
One might say that Carell plays his usual, almost slapstick self here as Riggs sensationalizes the upcoming match and leans heavily on the 'man vs. woman' debate. While that may be true to an extent, he also imbues a lot of heart into Riggs' character, giving him some nuance and showing that he's playing a persona for the crowd (at least to a certain extent). Carell is a nice foil for Stone, though they surprisingly don't share much screentime together.
The supporting cast play their roles well. Andrea Riseborough shares nice chemistry with Stone as her lover Marilyn. She helps King find her true self and recognize the struggle that is going on within her. It's easy to believe these two share a connection with each other. Austin Stowell plays King's husband Larry, but though he too has some good chemistry with Stone, he's not featured very much in the film. Looking back, this is understandable since King is discovering she's a lesbian, but you also don't get as much insight into King's relationship with Larry because of that and how he eventually accepts her newfound sexuality.
Other prominent supporting roles are Sarah Silverman and Bill Pullman. Silverman plays King's manager and founder of World Tennis magazine, injecting the film with some more humour that fits with Silverman's personality. Pullman, meanwhile, is Jack Kramer, another famous tennis player and head of a tennis organization. Kramer is the more serious antagonist of the film. While Riggs is just putting on a show and doing the match primarily for monetary reasons, Kramer is the one advocating against Kings' beliefs in equality. Pullman's not entirely sleazy, but gets across the despicable nature of Kramer's qualities.
The story moves along at a nice pace. It never feels like it's stalled, or at least not for too long, and the tennis sequences are edited in a fairly exciting manner. I'm not a huge tennis fan, but I could feel the tension and stakes during the matches. The script was also fairly strong, knowing exactly when to inject a joke or make focus on the drama in a scene.
What was also impressive was how it never felt dumbed down, explaining things to the audience. For instance, the scene where Larry discovers Kings' preferences is done silently between Stone and Stowell. Not a word is said to address the issue, instead relying on their body language to express the awkwardness and hurt the characters feel.
As I said before, the film is very socially relevant. Some may find it beats the audience over the head with its themes, but I think they worked well by mixing its themes with the character development for Billie Jean and Riggs. Stone plays these themes earnestly and its important to be earnest in a film like this, especially with some of the controversies we see today. Overall, Battle of the Sexes delivers on the acting, characters, themes and direction, making it a solid win.
Flickering Myth Rating – Film: ★ ★ ★ ★ / Movie: ★ ★ ★ ★
Ricky Church
Filed Under: Movies, Reviews, Ricky Church Tagged With: Alan Cumming, Andrea Riseborough, Austin Stowell, battle of the sexes, Bill Pullman, Elisabeth Shue, Emma Stone, jonathan dayton, Natalie Morales, Sarah Silverman, Steve Carell, Toronto International Film Festival, valerie faris
This could have been a good review but the lack of editing was appalling. I stopped reading halfway through because the numerous spelling mistakes, typos and grammatical errors were so off-putting and would have been easy to catch if the person uploading the piece had given it a even cursory read before publishing it. Please sort it out, I thought this was a professional website not a personal blog.
First trailer for stripper heist movie Hustlers starring Jennifer Lopez
Movie Review – My Friend the Polish Girl (2018)
The tragic story behind the death of Brandon Lee on the set of The Crow
Netflix releases over 30 images from Mindhunter season 2
UPDATE: Vanessa Kirby comments on rumours linking her to Catwoman role in The Batman
Daisy Ridley addresses "vicious" social media reactions to Star War: The Last Jedi
David Leitch hasn't heard from Disney about Deadpool 3
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{"url":"https:\/\/wikimili.com\/en\/Potential_game","text":"# Potential game\n\nLast updated\n\nIn game theory, a game is said to be a potential game if the incentive of all players to change their strategy can be expressed using a single global function called the potential function. The concept originated in a 1996 paper by Dov Monderer and Lloyd Shapley. [1]\n\n## Contents\n\nThe properties of several types of potential games have since been studied. Games can be either ordinal or cardinal potential games. In cardinal games, the difference in individual payoffs for each player from individually changing one's strategy, other things equal, has to have the same value as the difference in values for the potential function. In ordinal games, only the signs of the differences have to be the same.\n\nThe potential function is a useful tool to analyze equilibrium properties of games, since the incentives of all players are mapped into one function, and the set of pure Nash equilibria can be found by locating the local optima of the potential function. Convergence and finite-time convergence of an iterated game towards a Nash equilibrium can also be understood by studying the potential function.\n\nPotential games can be studied as repeated games with state so that every round played has a direct consequence on game's state in the next round [2] . This approach has applications in distributed control such as distributed resource allocation, where players without a central correlation mechanism can cooperate to achieve a globally optimal resource distribution.\n\n## Definition\n\nWe will define some notation required for the definition. Let ${\\displaystyle N}$ be the number of players, ${\\displaystyle A}$ the set of action profiles over the action sets ${\\displaystyle A_{i}}$ of each player and ${\\displaystyle u}$ be the payoff function.\n\nA game ${\\displaystyle G=(N,A=A_{1}\\times \\ldots \\times A_{N},u:A\\rightarrow \\mathbb {R} ^{N})}$ is:\n\n\u2022 an exact potential game if there is a function ${\\displaystyle \\Phi :A\\rightarrow \\mathbb {R} }$ such that ${\\displaystyle \\forall {a_{-i}\\in A_{-i}},\\ \\forall {a'_{i},\\ a''_{i}\\in A_{i}}}$,\n${\\displaystyle \\Phi (a'_{i},a_{-i})-\\Phi (a''_{i},a_{-i})=u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i})}$\nThat is: when player ${\\displaystyle i}$ switches from action ${\\displaystyle a'}$ to action ${\\displaystyle a''}$, the change in the potential equals the change in the utility of that player.\n\u2022 a weighted potential game if there is a function ${\\displaystyle \\Phi :A\\rightarrow \\mathbb {R} }$ and a vector ${\\displaystyle w\\in \\mathbb {R} _{++}^{N}}$ such that ${\\displaystyle \\forall {a_{-i}\\in A_{-i}},\\ \\forall {a'_{i},\\ a''_{i}\\in A_{i}}}$,\n${\\displaystyle \\Phi (a'_{i},a_{-i})-\\Phi (a''_{i},a_{-i})=w_{i}(u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i}))}$\n\u2022 an ordinal potential game if there is a function ${\\displaystyle \\Phi :A\\rightarrow \\mathbb {R} }$ such that ${\\displaystyle \\forall {a_{-i}\\in A_{-i}},\\ \\forall {a'_{i},\\ a''_{i}\\in A_{i}}}$,\n${\\displaystyle u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i})>0\\Leftrightarrow \\Phi (a'_{i},a_{-i})-\\Phi (a''_{i},a_{-i})>0}$\n\u2022 a generalized ordinal potential game if there is a function ${\\displaystyle \\Phi :A\\rightarrow \\mathbb {R} }$ such that ${\\displaystyle \\forall {a_{-i}\\in A_{-i}},\\ \\forall {a'_{i},\\ a''_{i}\\in A_{i}}}$,\n${\\displaystyle u_{i}(a'_{i},a_{-i})-u_{i}(a''_{i},a_{-i})>0\\Rightarrow \\Phi (a'_{i},a_{-i})-\\Phi (a''_{i},a_{-i})>0}$\n\u2022 a best-response potential game if there is a function ${\\displaystyle \\Phi :A\\rightarrow \\mathbb {R} }$ such that ${\\displaystyle \\forall i\\in N,\\ \\forall {a_{-i}\\in A_{-i}}}$,\n${\\displaystyle b_{i}(a_{-i})=\\arg \\max _{a_{i}\\in A_{i}}\\Phi (a_{i},a_{-i})}$\n\nwhere ${\\displaystyle b_{i}(a_{-i})}$ is the best action for player ${\\displaystyle i}$ given ${\\displaystyle a_{-i}}$.\n\n## A simple example\n\nIn a 2-player, 2-strategy game with externalities, individual players' payoffs are given by the function ui(si, sj) = bisi + wsisj, where si is players i's strategy, sj is the opponent's strategy, and w is a positive externality from choosing the same strategy. The strategy choices are +1 and 1, as seen in the payoff matrix in Figure 1.\n\nThis game has a potential function P(s1, s2) = b1s1 + b2s2 + ws1s2.\n\nIf player 1 moves from 1 to +1, the payoff difference is \u0394u1 = u1(+1, s2) \u2013 u1(\u20131, s2) = 2 b1 + 2 ws2.\n\nThe change in potential is \u0394P = P(+1, s2) \u2013 P(\u20131, s2) = (b1 + b2s2 + ws2) \u2013 (\u2013b1 + b2s2ws2) = 2 b1 + 2 ws2 = \u0394u1.\n\nThe solution for player 2 is equivalent. Using numerical values b1\u00a0=\u00a02, b2\u00a0=\u00a01, w\u00a0=\u00a03, this example transforms into a simple battle of the sexes, as shown in Figure 2. The game has two pure Nash equilibria, (+1,\u00a0+1) and (1,\u00a01). These are also the local maxima of the potential function (Figure 3). The only stochastically stable equilibrium is (+1,\u00a0+1), the global maximum of the potential function.\n\n +1 \u20131 +1 +b1+w, +b2+w +b1\u2013w, \u2013b2\u2013w \u20131 \u2013b1\u2013w, +b2\u2013w \u2013b1+w, \u2013b2+w Fig. 1: Potential game example\n +1 \u20131 +1 5, 2 \u20131, \u20132 \u20131 \u20135, \u20134 1, 4 Fig. 2: Battle of the sexes(payoffs)\n +1 \u20131 +1 4 0 \u20131 \u20136 2 Fig. 3: Battle of the sexes(potentials)\n\nA 2-player, 2-strategy game cannot be a potential game unless\n\n${\\displaystyle [u_{1}(+1,-1)+u_{1}(-1,+1)]-[u_{1}(+1,+1)+u_{1}(-1,-1)]=[u_{2}(+1,-1)+u_{2}(-1,+1)]-[u_{2}(+1,+1)+u_{2}(-1,-1)]}$\n\n## Related Research Articles\n\nThe proof of G\u00f6del's completeness theorem given by Kurt G\u00f6del in his doctoral dissertation of 1929 is not easy to read today; it uses concepts and formalisms that are no longer used and terminology that is often obscure. The version given below attempts to represent all the steps in the proof and all the important ideas faithfully, while restating the proof in the modern language of mathematical logic. This outline should not be considered a rigorous proof of the theorem.\n\nIn game theory, the Nash equilibrium, named after the mathematician John Forbes Nash Jr., is a proposed solution of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.\n\nIn set theory, the axiom schema of replacement is a schema of axioms in Zermelo\u2013Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite sets in ZF.\n\nIn the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. It was first used by Paul Cohen in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo\u2013Fraenkel set theory.\n\nIn set theory, Zermelo\u2013Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo\u2013Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo\u2013Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for \"choice\", and ZF refers to the axioms of Zermelo\u2013Fraenkel set theory with the axiom of choice excluded.\n\nIn mathematical analysis, a function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the y-axis, neglecting the contribution of motion along x-axis, traveled by a point moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function, but can be every intersection of the graph itself with a hyperplane parallel to a fixed x-axis and to the y-axis.\n\nIn game theory, a cooperative game is a game with competition between groups of players (\"coalitions\") due to the possibility of external enforcement of cooperative behavior. Those are opposed to non-cooperative games in which there is either no possibility to forge alliances or all agreements need to be self-enforcing.\n\nMechanism design is a field in economics and game theory that takes an objectives-first approach to designing economic mechanisms or incentives, toward desired objectives, in strategic settings, where players act rationally. Because it starts at the end of the game, then goes backwards, it is also called reverse game theory. It has broad applications, from economics and politics to networked-systems.\n\nGame theory is the branch of mathematics in which games are studied: that is, models describing human behaviour. This is a glossary of some terms of the subject.\n\nIn game theory, a Bayesian game is a game in which players have incomplete information about the other players. For example, a player may not know the exact payoff functions of the other players, but instead have beliefs about these payoff functions. These beliefs are represented by a probability distribution over the possible payoff functions.\n\nIn game theory, normal form is a description of a game. Unlike extensive form, normal-form representations are not graphical per se, but rather represent the game by way of a matrix. While this approach can be of greater use in identifying strictly dominated strategies and Nash equilibria, some information is lost as compared to extensive-form representations. The normal-form representation of a game includes all perceptible and conceivable strategies, and their corresponding payoffs, for each player.\n\nIndependence-friendly logic is an extension of classical first-order logic (FOL) by means of slashed quantifiers of the form and . The intended reading of is \"there is a which is functionally independent from the variables in \". IF logic allows one to express more general patterns of dependence between variables than those which are implicit in first-order logic. This greater level of generality leads to an actual increase in expressive power; the set of IF sentences can characterize the same classes of structures as existential second-order logic. For example, it can express branching quantifier sentences, such as the formula which expresses infinity in the empty signature; this cannot be done in FOL. Therefore, first-order logic cannot, in general, express this pattern of dependency, in which depends only on and , and depends only on and . IF logic is more general than branching quantifiers, for example in that it can express dependencies that are not transitive, such as in the quantifier prefix .\n\nIn game theory, a symmetric game is a game where the payoffs for playing a particular strategy depend only on the other strategies employed, not on who is playing them. If one can change the identities of the players without changing the payoff to the strategies, then a game is symmetric. Symmetry can come in different varieties. Ordinally symmetric games are games that are symmetric with respect to the ordinal structure of the payoffs. A game is quantitatively symmetric if and only if it is symmetric with respect to the exact payoffs. A partnership game is a symmetric game where both players receive identical payoffs for any strategy set. That is, the payoff for playing strategy a against strategy b receives the same payoff as playing strategy b against strategy a.\n\nIn game theory, a correlated equilibrium is a solution concept that is more general than the well known Nash equilibrium. It was first discussed by mathematician Robert Aumann in 1974. The idea is that each player chooses their action according to their observation of the value of the same public signal. A strategy assigns an action to every possible observation a player can make. If no player would want to deviate from the recommended strategy, the distribution is called a correlated equilibrium.\n\nIn mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics.\n\nConstructive set theory is an approach to mathematical constructivism following the program of axiomatic set theory. The same first-order language with and of classical set theory is usually used, so this is not to be confused with a constructive types approach. On the other hand, some constructive theories are indeed motivated by their interpretability in type theories.\n\nIn computability theory, a semicomputable function is a partial function that can be approximated either from above or from below by a computable function.\n\nCongestion games are a class of games in game theory first proposed by American economist Robert W. Rosenthal in 1973. In a congestion game the payoff of each player depends on the resources it chooses and the number of players choosing the same resource. Congestion games are a special case of potential games. Rosenthal proved that any congestion game is a potential game and Monderer and Shapley (1996) proved the converse: for any potential game, there is a congestion game with the same potential function.\n\nIn game theory, an aggregative game is a game in which every player\u2019s payoff is a function of the player\u2019s own strategy and the aggregate of all players\u2019 strategies. The concept was first proposed by Nobel laureate Reinhard Selten in 1970 who considered the case where the aggregate is the sum of the players' strategies.\n\nBounded arithmetic is a collective name for a family of weak subtheories of Peano arithmetic. Such theories are typically obtained by requiring that quantifiers be bounded in the induction axiom or equivalent postulates. The main purpose is to characterize one or another class of computational complexity in the sense that a function is provably total if and only if it belongs to a given complexity class. Further, theories of bounded arithmetic present uniform counterparts to standard propositional proof systems such as Frege system and are, in particular, useful for constructing polynomial-size proofs in these systems. The characterization of standard complexity classes and correspondence to propositional proof systems allows to interpret theories of bounded arithmetic as formal systems capturing various levels of feasible reasoning.\n\n## References\n\n1. Monderer, Dov; Shapley, Lloyd (1996). \"Potential Games\". Games and Economic Behavior. 14: 124\u2013143. doi:10.1006\/game.1996.0044.\n2. 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Tools of Change is soliciting nominations for its 2013 Landmark behavior change case studies in two topic areas - (1) home / building energy conservation and (2) sustainable transportation. If you know of anyone working on a particularly effective or innovative approach for changing energy or transportation behaviours, please consider nominating them - or yourself. All nominations must include measured impact results.
Designation as a "Landmark" (best practice) case study through this peer selection process recognizes behavior change programs and approaches considered to be among the most successful, innovative, replicable and adaptable in the world. Designated programs gain exposure, credibility and free, on-line program case study materials, which may make it easier for them to maintain or increase program funding.
Nominations are screened by Tools of Change staff and then the most promising are rated by peer selection panels based on a standard scoring grid. Designated programs are captured in detail and presented to other social change practitioners through webinars, transcripts and video recordings of the webinars, and written case studies. Program organizers get a Landmark designation logo for use on websites and in electronic newsletters, providing click-through access to the program's case study materials. Individuals nominating Landmark case studies are also suitably acknowledged.
The nomination form, which can be downloaded from www.toolsofchange.com/en/landmark/, must be submitted by May 31, 2013. Designations will be announced by October 2013, and case study webinars will be presented between January and May 2014.
Help us spread the news. If you know of others who might be interested, forward this to them. If you are a media writer or editor, tell this story.
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\section{Introduction}
The standard approach for training word embeddings is to rely on monolingual corpora, which means in particular that a separate embedding model is learned for each language.
There is an increasing interest, however, in learning cross-lingual word embeddings, where words from different languages are mapped onto a single space. Such representations are attractive, for instance, for dealing with the multilingual nature of text on the Web, but also as a vehicle for transferring knowledge (e.g., labelled training data) from resource-rich languages such as English to other languages \cite{ruder2017survey}.
\blfootnote{Authors marked with an asterisk (*) contributed equally.}
Initially, the main obstacle to learning such cross-lingual embeddings was the need for large multilingual parallel corpora \cite{klementiev2012inducing,ap2014autoencoder,luong2015bilingual}.
This limitation, however, was alleviated by the development of methods that only need comparable data (e.g., Wikipedia corpora in different languages) as the main source of supervision \cite{vulic2015bilingual}. In a complementary direction, it has recently been shown that high-quality cross-lingual embeddings can be obtained by aligning two independently learned monolingual embedding spaces. This strategy is appealing, as it means that one only needs access to monolingual corpora and a bilingual dictionary as supervision signal.
Surprisingly, perhaps, it turns out that dictionaries with less than 100 word pairs are sufficient to obtain good alignments \cite{artetxe-labaka-agirre:2017:Long}. In fact, recent works have shown that cross-lingual embeddings can even be learned without any user-provided dictionary \cite{conneau2018word,artetxe:acl2018,xu2018crosslingual}.
Despite the promising results reported in the literature, it remains unclear under which conditions the aforementioned methods succeed. For example, \newcite{artetxe-labaka-agirre:2017:Long} and \newcite{conneau2018word} achieved promising results in the word translation task (i.e., bilingual lexicon induction), but their experiments relied on the availability of high-quality monolingual source corpora, namely Wikipedia, which is also the case in a more recent analysis on cross-lingual embeddings performance \cite{glavas2019evaluate}.
In fact, there exists a significant number of settings which have been largely ignored, and which might challenge models that excel in idealized environments. For instance, \newcite{ahmad2018near} found that for dissimilar languages with different word orderings than English, cross-lingual transfer is still challenging. Similarly, it remains unclear how well existing methods would perform on language pairs with significant differences in morphology (e.g., English-Finnish, the latter being an agglutinative language) or with different alphabets (e.g., English-Farsi or English-Russian). Moreover, settings with different kinds of corpora (e.g. noisy user-generated) have not been fully explored. This means, among others, that it is not clear how current cross-lingual embedding models would behave for transferring knowledge in environments such as social media centred tasks, given that such tasks usually benefit from embeddings that have been trained on social media corpora \cite{tang2014learning,godin2015multimedia,Yang2018}.
In this work, we broaden the empirical evaluation of state-of-the-art techniques for learning cross-lingual embeddings, by using several types of training corpora, various amounts of supervision, languages from different families and different alignment strategies in three different tasks. The results obtained cast some doubt on the view that high-quality cross-lingual embeddings can always be learned without much supervision.
\section{Related Work}
\noindent Cross-lingual embeddings have become increasingly popular in the past few years \cite{smith2017offline,artetxe-labaka-agirre:2017:Long,artetxe2018generalizing,conneau2018word}.
Recent efforts have focused on reducing the need for large amounts of resources (e.g., parallel corpora), which could be difficult to obtain for most languages and language pairs.
However, the evaluation of these approaches has tended to be somewhat limited, often using only one type of training corpus, including only similar languages, and considering only one evaluation task.
The most similar work to ours is that of \newcite{sogaard2018limitations}, which included an in-depth analysis of two of the factors that we also considered, namely language family and corpus type, but they only considered a single model, i.e., \texttt{MUSE} \cite{conneau2018word}.
Moreover, they studied each factor in isolation. In our case the analysis is also extended to more languages (covering up to 5 language pairs), systems (two unsupervised, two supervised, and a postprocessing technique), evaluation tasks (cross-lingual word similarity), and the impact of external bilingual dictionaries.
Another similar contribution is the analysis by \newcite{vulic2016role}, where the impact of bilingual dictionaries on cross-lingual alignments was examined. However, they only considered closely-related languages using the same alphabet and one type of corpus (i.e., Wikipedia).
Also, given the publication date, this analysis does not account for the important developments in cross-lingual embeddings from recent years, such as the methods we cover in this paper. Other empirical comparisons focused mostly on the need for different degrees of supervision, such as \cite{upadhyay2016cross}, which has been extended in a more recent survey by \newcite{ruder2017survey}.
In this paper, we complement those studies by analyzing and discussing empirical findings of the most recent state-of-the-art unsupervised and semi-supervised methods in a broader experimental setting, more in line with the recent concurrent analysis of \newcite{glavas2019evaluate}. The main differences between this empirical evaluation and the contributions of our work lie in the scope of the survey, since: (1) they only consider Wikipedia data for training; (2) they do not consider postprocessing techniques such as \texttt{Meemi}~\cite{doval:meemiemnlp2018}, which we found to improve the performance of cross-lingual models, especially in the case of distant languages and non-comparable corpora; (3) in our analysis we also consider additional settings with scarce training data such as small seed dictionaries and automatically-constructed dictionaries; and (4) we include a more exhaustive intrinsic evaluation (including cross-lingual semantic similarity).
\section{Learning Cross-lingual Word Embeddings}
\label{methods}
The focus of our evaluation is on methods
that start off with monolingual embedding models and then integrate these in a shared cross-lingual space.
Hence, given two monolingual corpora, a word vector space is first learned independently for each language. This can be achieved with common word embedding models, e.g., Word2Vec \cite{Mikolovetal:2013}, GloVe \cite{pennington2014glove} or FastText \cite{bojanowski2017enriching}. Second, a linear alignment strategy is used to map the monolingual embeddings to a common bilingual vector space (Section \ref{alignments}).
In some cases, a third transformation is applied to already aligned embeddings so the word vectors from both languages are refined and further re-positioned (Section \ref{postprocessing}).
Regardless of the overall methodology, however, these linear transformations are all learned based on a bilingual dictionary. This dictionary may be manually curated or, in some cases, automatically generated as part of the alignment process.
\subsection{Alignment methods}
\label{alignments}
In this paper we analyze two well-known orthogonal models for aligning monolingual embedding models: the corresponding version of \texttt{VecMap} and \texttt{MUSE}, plus an unsupervised non-orthogonal variant of the former. Basically, both methods use a linear transformation learned through an iterative procedure in which a seed bilingual dictionary is iteratively refined.
They can be used
with an empty initial seed dictionary, in which case the alignment process is fully unsupervised.
\texttt{VecMap} \cite{artetxe-labaka-agirre:2017:Long} uses an orthogonal transformation over normalized word embeddings.
Its semi-supervised two-step procedure is specifically designed to avoid the need for a large seed dictionary. For instance, in the original paper, a seed dictionary with 25 word pairs was used.
This seed dictionary is then augmented by applying the learned transformation to new words from the source language. The process is repeated until some convergence criterion is met.
The unsupervised variant \cite{artetxe:acl2018} obtains the initial seed dictionary automatically by exploiting the similarity distribution of words, and then applies the same method followed by a refinement step that re-weights the embeddings based on the cross-correlation of their components, which makes it the only non-orthogonal method tested in this work.
\texttt{MUSE} \cite{conneau2018word} obtains its transformation matrix in a similar way.
In this case, the seed dictionary is used as-is (supervised setting) or obtained in a fully automatically way through an adversarial learning method (unsupervised setting).
\subsection{Limitations and postprocessing}
\label{postprocessing}
By restricting transformations to orthogonal linear mappings, \texttt{VecMap} and \texttt{MUSE} rely on the assumption that the monolingual embeddings spaces are approximately isomorphic \cite{barone2016towards}. However, it has been argued that this assumption is overly restrictive, as the isomorphism assumption is not always satisfied \cite{sogaard2018limitations,yuva2018generalizing}. For this reason, it has been proposed to go beyond orthogonal transformations by modifying the internal structure of the monolingual spaces, either by giving more weight to highly correlated embedding components, as is the case for the unsupervised variant of \texttt{VecMap} in this work \cite{artetxe2018generalizing}, or by complementing the orthogonal transformation with other forms of post-processing. As an example of this latter strategy, \newcite{doval:meemiemnlp2018} fine-tune the initial alignment by learning an unconstrained linear transformation which aims to map each word vector onto the average of that vector and the corresponding word vector from the other language.
\section{Variables}
Our main aim is to explore how the choice of corpora (Section \ref{corpora}), supervision signals (Section \ref{supervision}) and languages (Section \ref{languages}) impacts the performance of cross-lingual word embedding models. In Section \ref{others} we also list some other variables which were not directly studied in this paper.
\subsection{Monolingual corpora}
\label{corpora}
It is reasonable to assume that accurate word-level alignments will be easier to obtain from corpora from similar domains with similar vocabularies and register.
Wikipedia has been the mainstream monolingual source in cross-lingual word embedding training so far \cite{artetxe-labaka-agirre:2017:Long,conneau2018word}. It provides a particularly reliable bilingual signal because of the highly comparative nature of Wikipedia corpora from different languages. As we will see, this makes finding high-quality alignments considerably easier.
In our analysis we use three different types of corpora:
Wikipedia\footnote{All Wikipedia text dumps were downloaded from the Polyglot project \cite{polyglot:2013:ACL-CoNLL}: \url{https://sites.google.com/site/rmyeid/projects/polyglot}} (as a prototypical example of comparable monolingual corpora), Web corpora from different sources\footnote{The sources of the web-corpora are: UMBC \cite{han2013umbc}, 1-billion \cite{cardellinoSBWCE}, itWaC and sdeWaC \cite{baroni2009wacky}, Hamshahri \cite{aleahmad2009hamshahri}, and Common Crawl downloaded from \url{http://www.statmt.org/wmt16/translation-task.html}.} (as a prototypical example of non-comparable but generally high-quality corpora) and social media\footnote{Social media corpora are based on Twitter, at different dates between 2015 and 2018 \cite{camacho2020learning}. Monolingual embeddings were downloaded at \url{https://github.com/pedrada88/crossembeddings-twitter}} (as a prototypical example of noisy text). Statistics of these corpora are provided in Table \ref{tab:corpora}.\footnote{Due to some restrictions, we were not able to compile a reliable Twitter corpus for Russian.}
\begin{table}
\begin{center}
{
\setlength{\tabcolsep}{5.0pt}
\scalebox{0.95}{
\begin{tabular}{llcrrrcc}
\toprule
\bf Domain & \bf Corpus & \bf Language & \bf Size & \bf Words \\
\midrule
\multirow{6}{*}{Wikipedia} & Wiki\textsubscript{en} & English & 1.7B & 12.0M \\
& Wiki\textsubscript{es} & Spanish & 407M & 3.4M \\
& Wiki\textsubscript{it} & Italian & 338M & 3.3M \\
& Wiki\textsubscript{de} & German & 605M & 7.4M \\
& Wiki\textsubscript{fi} & Finnish & 6
M & 2.8M \\
& Wiki\textsubscript{ru} & Russian & 313M & 5.4M \\
& Wiki\textsubscript{fa} & Farsi & 48M & 1.0M \\
\midrule
\multirow{6}{*}{\shortstack{Web\\corpora}} & UMBC & English &
3.5B &
8.1M \\
& 1-billion & Spanish &
1.9B & 5.5M \\
& itWaC & Italian & 1.3B & 4.2M \\
& sdeWaC & German & 438M &
1.5M\\
& Comm-crawl & Finnish & 2.8B &
1.8M\\
& Comm-crawl & Russian & 1.1B & 18.8M \\
& Hamshahri & Farsi & 167M &
0.8M \\
\midrule
\multirow{6}{*}{\shortstack{Social\\media}} & Twitter\textsubscript{en} & English & 29
M & 5.5M \\
& Twitter\textsubscript{es} & Spanish & 14
M & 3.3M \\
& Twitter\textsubscript{it} & Italian & 6
M & 1.6M \\
& Twitter\textsubscript{de} & German & 11
M & 2.3M \\
& Twitter\textsubscript{fi} & Finnish & 2
M & 1.7M \\
& Twitter\textsubscript{fa} & Farsi & 9
M & 1.0M \\
\bottomrule
\end{tabular}
}
}
\end{center}
\caption{\label{tab:corpora} Statistics of the corpora used to train monolingual word embeddings: size (measured in total number of tokens) and words (number of unique tokens).}
\end{table}
\subsection{Bilingual supervision}
\label{supervision}
Early approaches for learning bilingual embeddings relied on large parallel corpora \cite{klementiev2012inducing,luong2015bilingual}, which limited their applicability. More recent approaches instead rely on (often small) bilingual dictionaries as the only source of bilingual supervision.
In fact, some methods remove the need for a user-supplied bilingual dictionary altogether \cite{conneau2018word,artetxe:acl2018,hoshen-wolf-2018-non,xu2018crosslingual}, relying instead on synthetic dictionaries that are obtained fully automatically.
In our experiments we consider a wide range of signals, including no supervision as well as automatically generated dictionaries of identical words. In the latter case, we rely on the assumption that words that occur in both of the monolingual corpora tend to have the same meaning. While this may seem naive, this strategy has been reported in the literature to perform well in practice \cite{smith2017offline,sogaard2018limitations}.
\subsection{Languages}
\label{languages}
In most previous work, the evaluation of cross-lingual embeddings has been limited to a small set of closely-related languages. For instance, \newcite{smith2017offline} evaluated their model on the English-Italian pair only, while the evaluation of \newcite{artetxe-labaka-agirre:2017:Long} was performed on three languages, all of which share the same alphabet. Moreover, as the considered language pairs vary from one study to another, the relative performance of different methods for particular types of languages remains unclear.
More recently, however, \newcite{sogaard2018limitations} have extended the usual evaluation framework by covering additional Eastern European languages.
We similarly expand the range of languages by considering: Spanish (ES), Italian (IT), German (DE), Finnish (FI), Farsi (FA) and Russian (RU). In all cases we use English (EN) as source language. This set of languages represents not only the usual family of Indo-European languages (all of them except Finnish), but also agglutinative languages (German, Farsi and Finnish, the latter being non-Indo-European), as well as languages with different alphabets (Farsi and Russian).
\begin{table*}[!t]
\renewcommand{\arraystretch}{1.15}
\resizebox{\textwidth}{!}{
\begin{tabular}{llrrrrrrrrrrrrrrrrrrr}
\hline
\multicolumn{21}{c}{\textbf{\texttt{Wikipedia}}} \\ \hline
\multicolumn{1}{l|}{\multirow{2}{*}{\textbf{Sup.}}} & \multicolumn{1}{l|}{\multirow{2}{*}{\textbf{Model}}} & \multicolumn{3}{c|}{\textbf{Spanish}} & \multicolumn{3}{c|}{\textbf{Italian}} & \multicolumn{3}{c|}{\textbf{German}} & \multicolumn{3}{c|}{\textbf{Finnish}} & \multicolumn{3}{c|}{\textbf{Farsi}} & \multicolumn{3}{c|}{\textbf{Russian}} &
\multicolumn{1}{c}{\textbf{Avg}} \\ \cline{3-21}
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{c}{$P@5$} \\ \hline
\multicolumn{1}{l|}{\multirow{2}{*}{\textbf{\texttt{Unsup}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & \textbf{39.6} & 66.1 & \multicolumn{1}{r|}{72.3} & 42.7 & 65.7 & \multicolumn{1}{r|}{71.6} & \textbf{28.6} & \textbf{48.3} & \multicolumn{1}{r|}{\textbf{54.8}} & 19.6 & 40.4 & \multicolumn{1}{r|}{48.3} & 20.5 & 37.0 & \multicolumn{1}{r|}{42.8} & 19.5 & 45.3 & \multicolumn{1}{r|}{54.5} & 50.5 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & 39.3 & 64.7 & \multicolumn{1}{r|}{71.3} & 41.6 & 63.2 & \multicolumn{1}{r|}{69.9} & 28.3 & 46.5 & \multicolumn{1}{r|}{53.3} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 14.9 & 36.0 & \multicolumn{1}{r|}{46.5} & 35.1 \\
\hline
\multicolumn{1}{l|}{\multirow{2}{*}{\textbf{\texttt{Ident}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & \textbf{39.5} & 66.0 & \multicolumn{1}{r|}{72.4} & 42.7 & 65.8 & \multicolumn{1}{r|}{71.7} & \textbf{28.6} & \textbf{48.3} & \multicolumn{1}{r|}{54.7} & 21.6 & 43.7 & \multicolumn{1}{r|}{51.6} & 23.4 & 40.3 & \multicolumn{1}{r|}{46.1} & 19.6 & 46.0 & \multicolumn{1}{r|}{55.1} & 51.7 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & 35.9 & 60.6 & \multicolumn{1}{r|}{67.3} & 37.8 & 60.4 & \multicolumn{1}{r|}{68.5} & 24.8 & 41.9 & \multicolumn{1}{r|}{49.5} & 13.4 & 25.5 & \multicolumn{1}{r|}{32.0} & 6.7 & 16.6 & \multicolumn{1}{r|}{21.3} & 7.8 & 19.9 & \multicolumn{1}{r|}{26.1} & 37.5 \\
\hline
\multicolumn{1}{l|}{\multirow{4}{*}{\textbf{\texttt{8K}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & \textbf{39.6} & 66.2 & \multicolumn{1}{r|}{72.3} & \textbf{42.6} & 65.9 & \multicolumn{1}{r|}{71.8} & \textbf{28.6} & \textbf{48.3} & \multicolumn{1}{r|}{\textbf{54.8}} & 22.4 & 44.5 & \multicolumn{1}{r|}{52.5} & 22.8 & 39.7 & \multicolumn{1}{r|}{46.2} & 20.0 & 46.3 & \multicolumn{1}{r|}{55.6} & 51.8 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & 39.1 & 65.4 & \multicolumn{1}{r|}{72.3} & 41.1 & 63.3 & \multicolumn{1}{r|}{70.1} & 27.6 & 45.9 & \multicolumn{1}{r|}{53.2} & 19.5 & 40.4 & \multicolumn{1}{r|}{49.5} & 19.7 & 35.4 & \multicolumn{1}{r|}{42} & 21.3 & 43.7 & \multicolumn{1}{r|}{52.9} & 49.0 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Meemi}\textsubscript{VM}} & 39.3 & \textbf{67.4} & \multicolumn{1}{r|}{\textbf{73.7}} & 41.6 & 66.5 & \multicolumn{1}{r|}{72.5} & 28 & 47.8 & \multicolumn{1}{r|}{\textbf{54.8}} & \textbf{23.8} & \textbf{48.7} & \multicolumn{1}{r|}{\textbf{57.0}} & \textbf{23.4} & \textbf{41.7} & \multicolumn{1}{r|}{\textbf{47.7}} & 23.0 & 49.3 & \multicolumn{1}{r|}{58.3} & \textbf{53.4} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Meemi}\textsubscript{MS}} & 39.3 & \textbf{67.4} & \multicolumn{1}{r|}{\textbf{73.7}} & 41.3 & \textbf{66.8} & \multicolumn{1}{r|}{\textbf{72.8}} & 27.1 & 46.3 & \multicolumn{1}{r|}{53.9} & 21.7 & 45.0 & \multicolumn{1}{r|}{53.6} & 20.7 & 38.6 & \multicolumn{1}{r|}{45.1} & \textbf{24.4} & \textbf{50.3} & \multicolumn{1}{r|}{\textbf{59.3}} & 52.4 \\
\hline
\multicolumn{21}{c}{\textbf{\texttt{Web corpora}}} \\ \hline
\multicolumn{1}{l|}{\multirow{2}{*}{\textbf{Sup.}}} & \multicolumn{1}{l|}{\multirow{2}{*}{\textbf{Model}}} & \multicolumn{3}{c|}{\textbf{Spanish}} & \multicolumn{3}{c|}{\textbf{Italian}} & \multicolumn{3}{c|}{\textbf{German}} & \multicolumn{3}{c|}{\textbf{Finnish}} & \multicolumn{3}{c|}{\textbf{Farsi}} & \multicolumn{3}{c|}{\textbf{Russian}} &
\multicolumn{1}{c}{\textbf{Avg}} \\ \cline{3-21}
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{c}{$P@5$} \\ \hline
\multicolumn{1}{l|}{\multirow{2}{*}{\textbf{\texttt{Unsup}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & \textbf{34.8} & 60.6 & \multicolumn{1}{r|}{67.0} & 31.4 & 53.7 & \multicolumn{1}{r|}{60.7} & 23.2 & 42.7 & \multicolumn{1}{r|}{50.2} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 19.7 & 34.6 & \multicolumn{1}{r|}{40.4} & 13.8 & 30.9 & \multicolumn{1}{r|}{38.6} & 37.1 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & 31.4 & 51.2 & \multicolumn{1}{r|}{57.7} & 31.4 & 51.2 & \multicolumn{1}{r|}{57.7} & 20.8 & 38.7 & \multicolumn{1}{r|}{46.6} & 17.7 & 35.7 & \multicolumn{1}{r|}{42.8} & 18.1 & 32.8 & \multicolumn{1}{r|}{37.8} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 34.9 \\
\hline
\multicolumn{1}{l|}{\multirow{2}{*}{\textbf{\texttt{Ident}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & 34.7 & 60.4 & \multicolumn{1}{r|}{67.0} & 31.4 & 54.0 & \multicolumn{1}{r|}{60.7} & 23.1 & 42.9 & \multicolumn{1}{r|}{50.5} & 18.6 & 41.6 & \multicolumn{1}{r|}{49.3} & 20.0 & 35.3 & \multicolumn{1}{r|}{40.3} & 14.1 & 31.2 & \multicolumn{1}{r|}{38.8} & 44.2 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & 26.1 & 46.7 & \multicolumn{1}{r|}{53.8} & 24.7 & 45.1 & \multicolumn{1}{r|}{52.4} & 17.4 & 32.8 & \multicolumn{1}{r|}{40.5} & 12.6 & 26.0 & \multicolumn{1}{r|}{33.8} & 3.0 & 8.3 & \multicolumn{1}{r|}{5.8} & 0.1 & 0.2 & \multicolumn{1}{r|}{0.2} & 26.5 \\
\hline
\multicolumn{1}{l|}{\multirow{4}{*}{\textbf{\texttt{8K}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & 34.6 & 60.6 & \multicolumn{1}{r|}{66.9} & 31.9 & 54.2 & \multicolumn{1}{r|}{60.4} & 23.1 & 42.7 & \multicolumn{1}{r|}{50.5} & 18.9 & 40.9 & \multicolumn{1}{r|}{48.8} & 19.6 & 35.8 & \multicolumn{1}{r|}{41.4} & 14.6 & 31.7 & \multicolumn{1}{r|}{39.6} & 44.3 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & 32.5 & 58.2 & \multicolumn{1}{r|}{65.9} & 32.5 & 56.0 & \multicolumn{1}{r|}{63.2} & 22.4 & 40.9 & \multicolumn{1}{r|}{48.9} & 20.0 & 40.1 & \multicolumn{1}{r|}{48.3} & 17.4 & 31.6 & \multicolumn{1}{r|}{37.6} & 15.5 & 35.6 & \multicolumn{1}{r|}{44.1} & 43.7 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Meemi}\textsubscript{VM}} & 34.5 & \textbf{61.6} & \multicolumn{1}{r|}{67.9} & 33.6 & 58.3 & \multicolumn{1}{r|}{\textbf{65.6}} & \textbf{23.7} & \textbf{45.4} & \multicolumn{1}{r|}{\textbf{53.2}} & 22.3 & \textbf{46.7} & \multicolumn{1}{r|}{\textbf{55.0}} & \textbf{21.7} & \textbf{39.0} & \multicolumn{1}{r|}{\textbf{43.8}} & 18.2 & 40.0 & \multicolumn{1}{r|}{47.5} & \textbf{48.9} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Meemi}\textsubscript{MS}} & 33.9 & 60.7 & \multicolumn{1}{r|}{\textbf{68.4}} & \textbf{33.8} & \textbf{58.4} & \multicolumn{1}{r|}{\textbf{65.6}} & \textbf{23.7} & 45.3 & \multicolumn{1}{r|}{52.3} & \textbf{23.0} & 46.1 & \multicolumn{1}{r|}{54.0} & 19.3 & 36.0 & \multicolumn{1}{r|}{41.7} & \textbf{18.7} & \textbf{40.5} & \multicolumn{1}{r|}{\textbf{49.7}} & 47.8 \\
\hline
\multicolumn{18}{c}{\textbf{\texttt{Social media}}} \\ \cline{1-18}
\multicolumn{1}{l|}{\multirow{2}{*}{\textbf{Sup.}}} & \multicolumn{1}{l|}{\multirow{2}{*}{\textbf{Model}}} & \multicolumn{3}{c|}{\textbf{Spanish}} & \multicolumn{3}{c|}{\textbf{Italian}} & \multicolumn{3}{c|}{\textbf{German}} & \multicolumn{3}{c|}{\textbf{Finnish}} & \multicolumn{3}{c|}{\textbf{Farsi}} &
\multicolumn{1}{c}{\textbf{Avg}} \\ \cline{3-18}
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{c}{$P@5$} \\ \cline{1-18}
\multicolumn{1}{l|}{\multirow{2}{*}{\textbf{\texttt{Unsup}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & \multicolumn{1}{r}{8.1} & \multicolumn{1}{r}{16.4} & \multicolumn{1}{r|}{20.4} & \multicolumn{1}{r}{8.8} & \multicolumn{1}{r}{17.0} & \multicolumn{1}{r|}{22.3} & \multicolumn{1}{r}{0.1} & \multicolumn{1}{r}{0.4} & \multicolumn{1}{r|}{0.5} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.0} & \multicolumn{1}{r}{6.8} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.0} & \multicolumn{1}{r}{7.3} & \multicolumn{1}{r}{14.5} & \multicolumn{1}{r|}{18.3} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.1} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.1} & \multicolumn{1}{r|}{0.1} & \multicolumn{1}{r}{2.9} \\
\cline{1-18}
\multicolumn{1}{l|}{\multirow{2}{*}{\textbf{\texttt{Ident}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & \multicolumn{1}{r}{8.1} & \multicolumn{1}{r}{16.4} & \multicolumn{1}{r|}{20.4} & \multicolumn{1}{r}{8.8} & \multicolumn{1}{r}{17.0} & \multicolumn{1}{r|}{22.3} & \multicolumn{1}{r}{0.1} & \multicolumn{1}{r}{0.4} & \multicolumn{1}{r|}{0.5} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.0} & \multicolumn{1}{r}{8.2} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.0} & \multicolumn{1}{r}{7.3} & \multicolumn{1}{r}{14.5} & \multicolumn{1}{r|}{18.3} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.1} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.1} & \multicolumn{1}{r|}{0.1} & \multicolumn{1}{r}{3.7} \\
\cline{1-18}
\multicolumn{1}{l|}{\multirow{4}{*}{\textbf{\texttt{8K}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & \multicolumn{1}{r}{8.7} & \multicolumn{1}{r}{16.6} & \multicolumn{1}{r|}{21.6} & \multicolumn{1}{r}{8.9} & \multicolumn{1}{r}{17.3} & \multicolumn{1}{r|}{22.4} & \multicolumn{1}{r}{3.2} & \multicolumn{1}{r}{6.8} & \multicolumn{1}{r|}{9.5} & \multicolumn{1}{r}{0.2} & \multicolumn{1}{r}{0.8} & \multicolumn{1}{r|}{1.2} & \multicolumn{1}{r}{0.4} & \multicolumn{1}{r}{1.6} & \multicolumn{1}{r|}{2.0} & \multicolumn{1}{r}{8.6} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & \multicolumn{1}{r}{8.1} & \multicolumn{1}{r}{17.6} & \multicolumn{1}{r|}{22.7} & \multicolumn{1}{r}{8} & \multicolumn{1}{r}{16.4} & \multicolumn{1}{r|}{21.1} & \multicolumn{1}{r}{2.2} & \multicolumn{1}{r}{6.0} & \multicolumn{1}{r|}{8.4} & \multicolumn{1}{r}{0.6} & \multicolumn{1}{r}{2.2} & \multicolumn{1}{r|}{3.2} & \multicolumn{1}{r}{1.2} & \multicolumn{1}{r}{4.5} & \multicolumn{1}{r|}{6.3} & \multicolumn{1}{r}{9.3} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Meemi}\textsubscript{VM}} & \multicolumn{1}{r}{\textbf{9.8}} & \multicolumn{1}{r}{\textbf{21.3}} & \multicolumn{1}{r|}{\textbf{26.9}} & \multicolumn{1}{r}{\textbf{10.6}} & \multicolumn{1}{r}{\textbf{20.0}} & \multicolumn{1}{r|}{\textbf{25.6}} & \multicolumn{1}{r}{\textbf{3.7}} & \multicolumn{1}{r}{\textbf{9.6}} & \multicolumn{1}{r|}{\textbf{13.2}} & \multicolumn{1}{r}{1.3} & \multicolumn{1}{r}{3.6} & \multicolumn{1}{r|}{5.5} & \multicolumn{1}{r}{\textbf{1.8}} & \multicolumn{1}{r}{5.1} & \multicolumn{1}{r|}{7.0} & \multicolumn{1}{r}{\textbf{12.1}} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Meemi}\textsubscript{MS}} & \multicolumn{1}{r}{9.5} & \multicolumn{1}{r}{20.5} & \multicolumn{1}{r|}{26.3} & \multicolumn{1}{r}{9.5} & \multicolumn{1}{r}{19.1} & \multicolumn{1}{r|}{24.5} & \multicolumn{1}{r}{3.0} & \multicolumn{1}{r}{7.6} & \multicolumn{1}{r|}{11.1} & \multicolumn{1}{r}{\textbf{1.5}} & \multicolumn{1}{r}{\textbf{4.3}} & \multicolumn{1}{r|}{\textbf{6.4}} & \multicolumn{1}{r}{1.6} & \multicolumn{1}{r}{\textbf{5.3}} & \multicolumn{1}{r|}{\textbf{8.1}} & \multicolumn{1}{r}{11.4} \\
\cline{1-18}
\end{tabular}
}
\caption{Bilingual dictionary induction results using English as source language. Performance measured by $P@k$. Overall average $P@5$ is shown in the last column.}
\label{tab:dictionary-r}
\end{table*}
\subsection{Other variables}
\label{others}
It is worth mentioning that there are several other external factors that may affect the quality of cross-lingual embeddings, beyond the ones considered in this study.
For instance, in our experiments we use FastText \cite{bojanowski2017enriching}, since morphological information might be useful for agglutinative languages as noted by its authors, with default values and dimensionality\footnote{300 dimensions in the case of Wikipedia and web corpora, and 100 dimensions in the smaller social media corpora.}, but the impact of other word embedding models such as Word2Vec \cite{Mikolovetal:2013} or GloVe \cite{pennington2014glove} could also be analyzed, in the line of \newcite{sogaard2018limitations}.
Likewise, all cross-lingual models and post-processing technique we evaluate are used \textit{as is}, with their default configurations.
\section{Evaluation}
We use two standard tasks for evaluating cross-lingual word embeddings: bilingual dictionary induction (Section \ref{induction}) and cross-lingual word similarity (Section \ref{similarity}). In addition, we also consider a downstream application: cross-lingual natural language inference (Section \ref{sec:xli}).
The systems we compare are two well-known cross-lingual embedding methods which can be used in unsupervised and semi-supervised settings, namely the orthogonal version of \texttt{VecMap}\footnote{\url{https://github.com/artetxem/vecmap}} \cite{artetxe:acl2018} and \texttt{MUSE}\footnote{\url{https://github.com/facebookresearch/MUSE}} \cite{conneau2018word}.
As seed dictionaries we consider three samples of varying sizes, considering 8K, 1K and 100 word pairs, to test the robustness of the models regarding the amount of supervision available.\footnote{These dictionaries were obtained by splitting the training dictionaries provided by \newcite{conneau2018word}} For the sake of clarity, in this section we only present results for the largest dictionary (i.e., with 8K word pairs). The results for all the other dictionary sizes are included in the appendix (these results are also considered in the analysis in Section \ref{analysis}).
Additionally, we also leverage synthetic dictionaries, consisting of identical words that are found in the corpora for both languages.
Lastly, using those same bilingual dictionaries, we apply the postprocessing proposed in \newcite{doval:meemiemnlp2018}\footnote{\url{https://github.com/yeraidm/meemi}} to refine the cross-lingual embeddings obtained by \texttt{VecMap} and \texttt{MUSE}. We will refer to these postprocessed vectors as \texttt{Meemi}\textsubscript{VM} and \texttt{Meemi}\textsubscript{MS}, respectively.
\subsection{Bilingual dictionary induction}
\label{induction}
This task consists in automatically obtaining the word translations in a target language for words in a source language.
To obtain the translation candidates, we use the standard cosine distance measure, selecting the nearest neighbors from the target language to the source word in the cross-lingual embedding space.
The performance is measured with precision at $k$ ($P@k$), that is, the proportion of test instances where the correct translation candidate for a given source word was among the $k$ highest ranked candidates.
Table \ref{tab:dictionary-r} summarizes the results obtained by all comparison systems on the test dictionaries published by \newcite{conneau2018word}. Note that the test dictionaries do not overlap with the dictionaries used for training.
\begin{table}[!t]
\resizebox{\columnwidth}{!}{
\begin{tabular}{clrrrrr}
\hline
\multicolumn{7}{c}{\texttt{\textbf{Wikipedia}}} \\ \hline \hline
\multicolumn{1}{c|}{\multirow{1}{*}{\textbf{Sup.}}} & \multicolumn{1}{c|}{\multirow{1}{*}{\textbf{Model}}} & \multicolumn{1}{c|}{\textbf{EN-ES}} & \multicolumn{1}{c|}{\textbf{EN-IT}} & \multicolumn{1}{c|}{\textbf{EN-DE}} & \multicolumn{1}{c||}{\textbf{EN-FA}} & \multicolumn{1}{c}{\textbf{Avg}} \\ \hline
\multicolumn{1}{c|}{\multirow{2}{*}{\textbf{\texttt{Unsup}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & \multicolumn{1}{r|}{72.1} & \multicolumn{1}{r|}{70.6} & \multicolumn{1}{r|}{69.3} & \multicolumn{1}{r||}{61.3} & 68.3 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & \multicolumn{1}{r|}{72.6} & \multicolumn{1}{r|}{71.2} & \multicolumn{1}{r|}{68.9} & \multicolumn{1}{r||}{6.5} & 54.8 \\
\hline
\multicolumn{1}{c|}{\multirow{2}{*}{\textbf{\texttt{Ident}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & \multicolumn{1}{r|}{71.8} & \multicolumn{1}{r|}{70.6} & \multicolumn{1}{r|}{69.3} & \multicolumn{1}{r||}{61.9} & 68.4 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & \multicolumn{1}{r|}{71.9} & \multicolumn{1}{r|}{70.5} & \multicolumn{1}{r|}{68.4} & \multicolumn{1}{r||}{51.3} & 65.5 \\
\hline
\multicolumn{1}{c|}{\multirow{4}{*}{\textbf{\texttt{8K}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & \multicolumn{1}{r|}{71.8} & \multicolumn{1}{r|}{70.6} & \multicolumn{1}{r|}{69.3} & \multicolumn{1}{r||}{61.7} & 68.4 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & \multicolumn{1}{r|}{72.6} & \multicolumn{1}{r|}{70.9} & \multicolumn{1}{r|}{68.9} & \multicolumn{1}{r||}{58.7} & 67.8 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Meemi}\textsubscript{VM}} & \multicolumn{1}{r|}{71.9} & \multicolumn{1}{r|}{70.9} & \multicolumn{1}{r|}{\textbf{70.3}} & \multicolumn{1}{r||}{\textbf{63.4}} & 69.1 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Meemi}\textsubscript{MS}} & \multicolumn{1}{r|}{\textbf{72.9}} & \multicolumn{1}{r|}{\textbf{71.9}} & \multicolumn{1}{r|}{70.1} & \multicolumn{1}{r||}{62.0} & \textbf{69.2} \\ \hline \hline
\multicolumn{7}{c}{\texttt{\textbf{Web corpora}}} \\ \hline \hline
\multicolumn{1}{c|}{\multirow{1}{*}{\textbf{Sup.}}} & \multicolumn{1}{l|}{\multirow{1}{*}{\textbf{Model}}} & \multicolumn{1}{c|}{\textbf{EN-ES}} & \multicolumn{1}{c|}{\textbf{EN-IT}} & \multicolumn{1}{c|}{\textbf{EN-DE}} & \multicolumn{1}{c||}{\textbf{EN-FA}} & \multicolumn{1}{c}{\textbf{Avg}} \\ \hline
\multicolumn{1}{c|}{\multirow{2}{*}{\textbf{\texttt{Unsup}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & \multicolumn{1}{r|}{70.5} & \multicolumn{1}{r|}{68.8} & \multicolumn{1}{r|}{70.4} & \multicolumn{1}{r||}{33.4} & 60.8 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} &\multicolumn{1}{r|}{71.6} & \multicolumn{1}{r|}{69.4} & \multicolumn{1}{r|}{70.0} & \multicolumn{1}{r||}{23.8} & 58.7 \\
\hline
\multicolumn{1}{c|}{\multirow{2}{*}{\textbf{\texttt{Ident}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & \multicolumn{1}{r|}{70.6} & \multicolumn{1}{r|}{68.8} & \multicolumn{1}{r|}{70.4} & \multicolumn{1}{r||}{33.0} & 60.7 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & \multicolumn{1}{r|}{70.1} & \multicolumn{1}{r|}{67.5} & \multicolumn{1}{r|}{69.7} & \multicolumn{1}{r||}{14.5} & 55.5 \\
\hline
\multicolumn{1}{c|}{\multirow{4}{*}{\textbf{\texttt{8K}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & \multicolumn{1}{r|}{70.6} & \multicolumn{1}{r|}{68.8} & \multicolumn{1}{r|}{70.4} & \multicolumn{1}{r||}{\textbf{33.5}} & 60.8 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & \multicolumn{1}{r|}{71.9} & \multicolumn{1}{r|}{70.4} & \multicolumn{1}{r|}{70.2} & \multicolumn{1}{r||}{23.9} & 59.1 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Meemi}\textsubscript{VM}} & \multicolumn{1}{r|}{70.9} & \multicolumn{1}{r|}{70.0} & \multicolumn{1}{r|}{71.8} & \multicolumn{1}{r||}{\textbf{39.0}} & \textbf{62.9} \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Meemi}\textsubscript{MS}} & \multicolumn{1}{r|}{\textbf{72.3}} & \multicolumn{1}{r|}{\textbf{71.1}} & \multicolumn{1}{r|}{\textbf{72.1}} & \multicolumn{1}{r||}{33.0} & 62.1 \\ \hline \hline
\multicolumn{7}{c}{\texttt{\textbf{Social media}}} \\ \hline \hline
\multicolumn{1}{c|}{\multirow{1}{*}{\textbf{Sup.}}} & \multicolumn{1}{l|}{\multirow{1}{*}{\textbf{Model}}} & \multicolumn{1}{c|}{\textbf{EN-ES}} & \multicolumn{1}{c|}{\textbf{EN-IT}} & \multicolumn{1}{c|}{\textbf{EN-DE}} & \multicolumn{1}{c||}{\textbf{EN-FA}} & \multicolumn{1}{c}{\textbf{Avg}} \\ \hline
\multicolumn{1}{c|}{\multirow{2}{*}{\textbf{\texttt{Unsup}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & \multicolumn{1}{r|}{46.9} & \multicolumn{1}{r|}{51.5} & \multicolumn{1}{r|}{31.2} & \multicolumn{1}{r||}{2.4} & 33.0 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & \multicolumn{1}{r|}{10.9} & \multicolumn{1}{r|}{49.7} & \multicolumn{1}{r|}{13.0} & \multicolumn{1}{r||}{4.7} & 19.6 \\
\hline
\multicolumn{1}{c|}{\multirow{2}{*}{\textbf{\texttt{Ident}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & \multicolumn{1}{r|}{47.1} & \multicolumn{1}{r|}{51.9} & \multicolumn{1}{r|}{50.3} & \multicolumn{1}{r||}{26.5} & 44.0 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & \multicolumn{1}{r|}{47.7} & \multicolumn{1}{r|}{49.8} & \multicolumn{1}{r|}{46.8} & \multicolumn{1}{r||}{32.4} & 44.2 \\
\hline
\multicolumn{1}{c|}{\multirow{4}{*}{\textbf{\texttt{8K}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & \multicolumn{1}{r|}{47.4} & \multicolumn{1}{r|}{51.8} & \multicolumn{1}{r|}{49.5} & \multicolumn{1}{r||}{30.3} & 44.8 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & \multicolumn{1}{r|}{47.6} & \multicolumn{1}{r|}{49.3} & \multicolumn{1}{r|}{48.6} & \multicolumn{1}{r||}{42.2} & 46.9 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Meemi}\textsubscript{VM}} & \multicolumn{1}{r|}{50.1} & \multicolumn{1}{r|}{\textbf{53.6}} & \multicolumn{1}{r|}{\textbf{53.8}} & \multicolumn{1}{r||}{43.1} & 50.2 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Meemi}\textsubscript{MS}} & \multicolumn{1}{r|}{\textbf{50.4}} & \multicolumn{1}{r|}{52.5} & \multicolumn{1}{r|}{52.0} & \multicolumn{1}{r||}{\textbf{46.6}} & \textbf{50.4} \\ \hline
\end{tabular}
}
\caption{Spearman correlation performance of various cross-lingual word embedding models in the cross-lingual word similarity task.}
\label{tab:sim-r}
\end{table}
\subsection{Cross-lingual semantic word similarity}
\label{similarity}
Given a pair of words from two different languages, the task of cross-lingual semantic word similarity consists in measuring to what extent both words are semantically similar. For the evaluation we make use of the cross-lingual word similarity datasets of the SemEval 2017 task \cite{semeval2017similarity}. In this dataset each word from one language is paired with another word from the other language.
This evaluation task has been found to correlate better with downstream performance than other intrinsic benchmarks \cite{bakarov2018limitations}.
The results are reported in terms of the Pearson and Spearman correlation with respect to human similarity judgments.
The cross-lingual word similarity results for all the systems are displayed in Table \ref{tab:sim-r}. The languages available for this dataset are English, Spanish, Italian, German and Farsi, hence Finnish and Russian were not evaluated in this task.
\subsection{Cross-lingual natural language inference}
\label{sec:xli}
The task of natural language inference (NLI) consists in detecting entailment, contradiction and neutral relations between pairs of sentences. We test a zero-shot cross-lingual transfer setting where a system is trained with English corpora and is then evaluated on a different language.
It is important to highlight that in this evaluation our main aim is to compare the quality of the cross-lingual word embeddings, and not to develop a state-of-the-art NLI system. Therefore, since this is a downstream task evaluated at the sentence level (and not at the word level as in dictionary induction and semantic word similarity), we develop a simple bag-of-words approach where a sentence embedding is obtained by word vector averaging.
We then train a linear classifier\footnote{The codebase for these experiments is that of SentEval~\cite{conneau2018senteval}} to obtain the predicted label for each pair of sentences: entailment, contradiction or neutral. We use the full MultiNLI~\cite{williams2018broad} English corpus for training and the Spanish, German and Russian test sets from XNLI~\cite{conneau2018xnli} for testing.
Accuracy results are shown in Table~\ref{tab:xli}.\footnote{For this task we focused on the better performing embeddings learned from Wikipedia and web corpora.}
\begin{table}[!t]
\renewcommand{\arraystretch}{1.02}
\resizebox{\columnwidth}{!}{
\begin{tabular}{clr|r|r||r}
\hline
\multicolumn{6}{c}{\texttt{\textbf{Wikipedia}}} \\ \hline \hline
\multicolumn{1}{c|}{\textbf{Sup.}} & \multicolumn{1}{c|}{\textbf{Model}} & \multicolumn{1}{c|}{\textbf{EN-ES}} & \multicolumn{1}{c|}{\textbf{EN-DE}} & \multicolumn{1}{c||}{\textbf{EN-RU}} &
\multicolumn{1}{c}{\textbf{Avg}} \\
\hline
\multicolumn{1}{c|}{\multirow{2}{*}{\textbf{\texttt{Unsup}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & \textbf{49.6} & 46.3 & \textbf{34.1} & 43.3\\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & 48.4 & 47.4 & 33.3 & 43.0\\
\hline
\multicolumn{1}{c|}{\multirow{2}{*}{\textbf{\texttt{Ident}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & 43.0 & 42.9 & 33.2 & 39.7\\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & 39.5 & 35.8 & 33.3 & 36.2\\
\hline
\multicolumn{1}{c|}{\multirow{4}{*}{\textbf{\texttt{8K}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & 49.2 & 46.7 & 33.4 & 43.1 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & 47.7 & 47.1 & 33.1 & 42.6\\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Meemi}\textsubscript{VM}} & 49.5 & \textbf{47.6} & 33.8 & \textbf{43.6}\\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Meemi}\textsubscript{MS}} & 44.2 & 46.7 & 33.3 & 41.4\\ \hline \hline
\multicolumn{6}{c}{\texttt{\textbf{Web corpora}}} \\ \hline \hline
\multicolumn{1}{c|}{\textbf{Sup.}} & \multicolumn{1}{l|}{\textbf{Model}} & \multicolumn{1}{c|}{\textbf{EN-ES}} & \multicolumn{1}{c|}{\textbf{EN-DE}} & \multicolumn{1}{c||}{\textbf{EN-RU}} &
\multicolumn{1}{c}{\textbf{Avg}} \\ \hline
\multicolumn{1}{c|}{\multirow{2}{*}{\textbf{\texttt{Unsup}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & \textbf{48.5} & 47.9 & 33.4 & 43.3\\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & 47.7 & 47.1 & 33.6 & 42.8\\
\hline
\multicolumn{1}{c|}{\multirow{2}{*}{\textbf{\texttt{Ident}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & 45.5 & 44.4 & 33.4 & 41.1\\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & 35.2 & 36.6 & 33.3 & 35.0\\
\hline
\multicolumn{1}{c|}{\multirow{4}{*}{\textbf{\texttt{8K}}}} & \multicolumn{1}{l|}{\textbf{VecMap}} & 48.4 & 47.5 & 33.2 & 43.0 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{MUSE}} & 47.3 & \textbf{48.6} & 33.1 & 43.0 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Meemi}\textsubscript{VM}} & 47.8 & \textbf{48.6} & \textbf{33.8} & \textbf{43.4} \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Meemi}\textsubscript{MS}} & 47.3 & 48.2 & 33.2 & 42.9 \\ \hline \hline
\end{tabular}
}
\caption{Accuracy in the cross-lingual natural language inference task (XNLI) using different cross-lingual word embedding models.}
\label{tab:xli}
\end{table}
\section{Analysis}
\label{analysis}
\begin{figure}
\includegraphics[width=\columnwidth]{vecmap_unsup.png
\caption{P@1 performance of the unsupervised version of \texttt{VecMap} on dictionary induction across corpus types and language pairs.}
\label{fig:vecmap_unsup}
\end{figure}
\paragraph{Supervision signals.}
Unsurprisingly, the best alignments of monolingual spaces tend to be obtained with the largest bilingual dictionaries.
The unsupervised variants of \texttt{VecMap} (see Figure~\ref{fig:vecmap_unsup}) and \texttt{MUSE} attain competitive performance in most cases, especially for comparable corpora where alignments are easier to obtain. However, they struggle in the case of noisy social media corpora and unrelated languages (e.g.\ both \texttt{VecMap} and \texttt{MUSE} obtain inferior results, close to 0, on both Finnish and Farsi), which challenges conclusions from previous work \cite{conneau2018word,artetxe:acl2018}.
Overall, the results obtained when using social media are clearly inferior, suggesting that there is still room for improvement when it comes to dealing with noisy corpora, regardless of the supervision.
\begin{figure*}[!t]
\hspace{-16px}
\includegraphics[width=0.72\columnwidth,height=3.6cm]{wikipedia-vecmap_vs_muse.png
\hspace{0.05cm}
\includegraphics[width=0.72\columnwidth,height=3.6cm]{umbc-vecmap_vs_muse.png
\includegraphics[width=0.72\columnwidth,height=3.6cm]{twitter-vecmap_vs_muse.png
\caption{Comparison between the dictionary induction performance ($P@1$) of \texttt{VecMap} (blue) and \texttt{MUSE} (red) in English-Italian on Wikipedia (left), web corpora (middle) and social media text (right). The figure clearly shows how \texttt{VecMap} produces similar results irrespective of the seed supervision, while the results of \texttt{MUSE} fluctuate depending on the size of the seed dictionary (with its unsupervised variant being better than using a small dictionary).}
\label{fig:supervision}
\end{figure*}
\paragraph{\texttt{VecMap} vs. \texttt{MUSE}.}
One of the main differences between these two models relates to their robustness. The results of \texttt{VecMap} are largely stable across the different types of the supervision. In fact, the best performance for Spanish and Russian on the XNLI task is even obtained in its unsupervised mode. In contrast, \texttt{MUSE} does not perform well with small dictionaries. Figure \ref{fig:supervision} illustrates this trend.
In addition, \texttt{MUSE} also suffers from some stability issues, as it does not always converge to the optimal solution, which confirms findings from previous work \cite{artetxe:acl2018,sogaard2018limitations,hartmann2018unsupervised}.\footnote{This feature was not explicitly tested in this work, as in our experiments models were run until convergence.} In terms of overall results, when given a sufficiently large dictionary training data, the performance of both methods is comparable, which is perhaps unsurprising as they both rely on the solution of the orthogonal Procrustes problem to learn the transformation between the monolingual spaces.
\paragraph{Impact of corpora.}
As can be observed throughout all the experiments, the more comparable and less noisy the monolingual data is, the better the bilingual alignments.
For instance, \texttt{VecMap} goes from an average of $31.2$\% in $P@1$ on Wikipedia down to $4.3$\% on social media, considering all language pairs.
In word similarity, we observe an analogous performance drop, from $68.4$\% to $44.8$\% in Spearman correlation.
Additionally, in Figure~\ref{fig:vecmap_unsup} we can observe the negative influence of noisy corpora and distant languages on the performance of the unsupervised version of \texttt{VecMap} on dictionary induction.
In terms of error analysis, unsurprisingly we find that the low performance of the models trained on Twitter data is largely due to the noise and the informal nature of the conversation topics.
For instance, for the word \emph{discover}, instead of \emph{descubren} (one of the correct Spanish translations obtained by the models trained on Wikipedia), the translation given by VecMap corresponds to a misspelling of the correct translation: \emph{descubr}.
As another example, \emph{timeline} is not translated to \emph{cronolog\'ia} in Spanish, but to \emph{instas}, which refers to the social network Instagram. This is clearly due to the specific use of the word \emph{timeline} on Twitter.
\paragraph{Distant languages.}
As expected, the more different the languages are, the harder it is to obtain a reliable alignment of the monolingual spaces. This is particularly noticeable in the case of Farsi, Russian and Finnish (and German to a lesser extent).
For instance, in the bilingual dictionary task, while most models are over 30.0\% in $P@1$ (excluding social media text which causes performance drops in all languages), for Finnish, Farsi and Russian the results are below 20\% in most cases. A similar tendency can be observed for Farsi on the word similarity task, where the differences are even more pronounced. In addition to its idiosyncrasies (Farsi is considered agglutinative and has a noun compounding formation similar to German), the fact that it uses a different alphabet may explain this large performance gap, noting that FastText takes subword units into account.
Finally, while the poor performance could be partially explained by the small size of the monolingual training corpora for some languages, it is interesting to see notable performance differences in cases where a distant language has similar or even greater amounts of training data available; e.g., Italian and Russian on Wikipedia, or Italian, Finnish or Russian on web data.
\paragraph{Distant supervision.} As far as the synthetic dictionary of identical words is concerned, \texttt{MUSE} seems to have more difficulties coping with its noisy nature than \texttt{VecMap} (obtaining an average of $16.9$\% versus $23.6$\% in $P@1$ overall in dictionary induction in the Wikipedia and web corpora domains).
In fact, using \texttt{MUSE} in its unsupervised setting or with a small dictionary generally provides better results.
However, on social media using the dictionary of identical words appears to help \texttt{MUSE} considerably in the word similarity task compared to the unsupervised setting, going from $19.6$\% to $44.2$\% on average in Spearman correlation overall. This can be attributed to the multilinguality of social media data, where phenomena like code-switching often occur. On the other hand, the consistency of the \texttt{VecMap} semi-supervised algorithm is highlighted again, as using the identical dictionary in this case yields similar results to using external bilingual dictionaries.
\paragraph{Postprocessing.}
As explained in Section \ref{postprocessing}, for our analysis we experimented with \texttt{Meemi} \cite{doval:meemiemnlp2018}, a recent postprocessing technique which can be applied to any cross-lingual embedding space. There are two main conclusions regarding this technique. First, a clean and relatively big bilingual dictionary is needed in order to get improvements over the base methods \texttt{VecMap} and \texttt{MUSE} (for instance, $+1.2$\% $P@1$ and $+3.1$\% Spearman correlation scores on social media on average, using the 8K dictionary), with the performance otherwise ending up significantly lower. In general, the best overall results are achieved when using this postprocessing technique in combination with the largest dictionary (i.e., 8K pairs). Table \ref{tab:simweb} shows the performance gains or drops by using \texttt{Meemi} in the cross-lingual word similarity task, clearly showing the need for a reasonably large dictionary. This performance variability depending on the size of the dictionary was not addressed in the original paper. Second, \texttt{Meemi} appears to be particularly useful when the monolingual corpora are not comparable, as shown by the larger improvements attained on web-based data.
\begin{table}[!t]
\resizebox{\columnwidth}{!}{
\begin{tabular}{|ll|r|r|r|}
\cline{3-5}
\multicolumn{2}{l}{} & \multicolumn{1}{|c|}{\textbf{100}} & \multicolumn{1}{c|}{\textbf{1K}} & \multicolumn{1}{c|}{\textbf{8K}} \\ \hline
\multicolumn{1}{|l|}{\multirow{4}{*}{\textbf{VecMap}}} & \textbf{EN-ES} & -70.4 & -2.0 & \textbf{+0.3} \\ \cline{2-5}
\multicolumn{1}{|l|}{} & \textbf{EN-IT} & -68.8 & -1.0 & \textbf{+1.2} \\ \cline{2-5}
\multicolumn{1}{|l|}{} & \textbf{EN-DE} & -70.5 & -0.3 & \textbf{+1.4} \\ \cline{2-5}
\multicolumn{1}{|l|}{} & \textbf{EN-FA} & -30.5 & -32.8 &\textbf{+5.5} \\ \hline
\multicolumn{1}{|l|}{\multirow{4}{*}{\textbf{MUSE}}} & \textbf{EN-ES} & -71.5 & -1.8 & \textbf{+1.7} \\ \cline{2-5}
\multicolumn{1}{|l|}{} & \textbf{EN-IT} & -65.1 & -0.8 & \textbf{+0.7} \\ \cline{2-5}
\multicolumn{1}{|l|}{} & \textbf{EN-DE} & -68.8 & -0.1 & \textbf{+1.9} \\ \cline{2-5}
\multicolumn{1}{|l|}{} & \textbf{EN-FA} & -7.4 & -22.3 & \textbf{+9.1} \\ \hline
\end{tabular}
}
\caption{Absolute improvement (in percentage points) by applying the postprocessing (\texttt{Meemi}) over the two base models VecMap and MUSE on the cross-lingual word similarity task using web corpora.}
\label{tab:simweb}
\end{table}
\paragraph{Evaluation tasks.}
The performance variability in bilingual dictionary induction, cross-lingual word similarity and cross-lingual inference seems to be very similar across the board, with the main difference being the lower variability in results in the cross-lingual NLI task (which can be expected given that it is a downstream task where additional factors are also involved). The factors with the greatest impact on performance, namely monolingual corpora and language pairs, are clearly reflected in both cases, with analogous drops when going from training on Wikipedia to social media, and also when testing on Finnish, Farsi or Russian. To test our intuition, we computed Pearson correlation values from all overlapping results between task pairs. In this case, similarity and dictionary induction attain the highest correlation ($r=0.78$), with cross-lingual NLI and dictionary induction also attaining a high correlation score ($r=0.73$). The lowest correlation score corresponds to cross-lingual similarity and NLI, with a lower figure of $r=0.28$. Despite being positive, this relatively low correlation may suggest that dictionary induction would be a better proxy to test cross-lingual embedding performance in downstream tasks. We should note, however, that these correlation figures are only indicative and particular to the methods tested in our analysis and therefore should not be taken as the global correlation between tasks.
\section{Conclusions}
We have presented an extensive evaluation of state-of-the-art cross-lingual embedding models in a wide variety of experimental settings.
The variables explored in this paper were: the choice of training corpus, the type of supervision signal (including different types of bilingual dictionaries), and the language pairs considered. Likewise, the evaluation procedure included two standard benchmarks for cross-lingual embedding evaluation, namely bilingual dictionary induction and cross-lingual word similarity, as well as cross-lingual natural language inference as an extrinsic task. The set of languages considered included not only related languages such as English, Spanish, Italian and German, but also languages from different families such as Finnish, Farsi and Russian.
Our analysis highlights a particularly marked variability in the performance of the considered methods concerning (1) the monolingual training corpora used (e.g., between comparable corpora such as Wikipedia and non-comparable or noisy user-generated corpora) and (2) language pairs (distant language pairs still constitute a major challenge). We may also conclude that bilingual supervision signals constitute a key component for most models in non-ideal settings (i.e., non-comparable corpora or distant languages).
In general, our analysis and the results show that supervised cross-lingual word embedding learning is more robust than purely unsupervised cross-lingual learning, challenging claims from previous works on this regard \cite{conneau2018word,artetxe:acl2018,chen2018unsupervised,hoshen-wolf-2018-non,xu2018crosslingual} and in line with a concurring analysis showing a similar trend \cite{vulic2019we}.
As future work, it would be interesting to analyze multilingual embeddings that involve more than two languages, along the lines of recent multilingual approaches \cite{chen2018unsupervised,heyman-etal-2019-learning,doval2019meemi}.
\section{Acknowledgments}
Steven Schockaert is supported by ERC Starting Grant 637277. Yerai Doval has been supported by the Spanish Ministry of Economy, Industry and Competitiveness (MINECO) through the ANSWER-ASAP project (TIN2017-85160-C2-2-R); by the Spanish State Secretariat for Research, Development and Innovation (which belongs to MINECO) and the European Social Fund (ESF) through a FPI fellowship (BES-2015-073768) associated to TELEPARES project (FFI2014-51978-C2-1-R); and by the Xunta de Galicia through TELGALICIA research network (ED431D 2017/12).
\section*{Appendix: Additional Experimental Results}
\label{sec:appendix}
This appendix contains additional experimental results not included in the main body of the paper. In particular, it contains supplementary results for the dictionary induction (Table \ref{tab:dictionary}) and cross-lingual word similarity (Table \ref{tab:sim}) tasks, using all sources of supervision: no supervision, dictionary of identical words, and dictionaries containing 100, 1K and 8K translation pairs. The methods included in these tables are explained in the paper (Section \ref{methods}).
\begin{table*}[!t]
\renewcommand{\arraystretch}{1.2}
\resizebox{\textwidth}{!}{
\begin{tabular}{llrrrrrrrrrrrrrrr}
\hline
\multicolumn{17}{c}{\textbf{\texttt{Wikipedia}}} \\ \hline
\multicolumn{1}{l|}{\multirow{2}{*}{\textbf{Model}}} & \multicolumn{1}{l|}{\multirow{2}{*}{\textbf{Supervision}}} & \multicolumn{3}{c}{\textbf{English-Spanish}} & \multicolumn{3}{c|}{\textbf{English-Italian}} & \multicolumn{3}{c|}{\textbf{English-German}} & \multicolumn{3}{c|}{\textbf{English-Finnish}} & \multicolumn{3}{c}{\textbf{English-Farsi}} \\ \cline{3-17}
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l}{$P@10$} \\ \hline
\multicolumn{1}{l|}{\multirow{5}{*}{\textbf{\texttt{VecMap}}}} & \multicolumn{1}{l|}{\textbf{8K}} & \textbf{39.6} & 66.2 & \multicolumn{1}{r|}{72.3} & 42.6 & 65.9 & \multicolumn{1}{r|}{71.8} & 28.6 & \textbf{48.3} & \multicolumn{1}{r|}{\textbf{54.8}} & 22.4 & 44.5 & \multicolumn{1}{r|}{52.5} & 22.8 & 39.7 & 46.2 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{1K}} & \textbf{39.6} & 66.2 & \multicolumn{1}{r|}{72.3} & 42.6 & 65.7 & \multicolumn{1}{r|}{71.6} & \textbf{28.7} & \textbf{48.3} & \multicolumn{1}{r|}{54.7} & 22.2 & 43.9 & \multicolumn{1}{r|}{51.7} & 23.2 & 40.2 & 46.1 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{100}} & \textbf{39.6} & 66.2 & \multicolumn{1}{r|}{72.4} & \textbf{42.9} & 65.7 & \multicolumn{1}{r|}{71.6} & 28.6 & \textbf{48.3} & \multicolumn{1}{r|}{\textbf{54.8}} & 21.6 & 43.4 & \multicolumn{1}{r|}{51.7} & 22.7 & \textbf{40.6} & \textbf{46.4} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & 39.5 & 66.0 & \multicolumn{1}{r|}{72.4} & 42.7 & 65.8 & \multicolumn{1}{r|}{71.7} & 28.6 & \textbf{48.3} & \multicolumn{1}{r|}{54.7} & 21.6 & 43.7 & \multicolumn{1}{r|}{51.6} & \textbf{23.4} & 40.3 & 46.1 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Unsupervised}} & \textbf{39.6} & 66.1 & \multicolumn{1}{r|}{72.3} & 42.7 & 65.7 & \multicolumn{1}{r|}{71.6} & 28.6 & \textbf{48.3} & \multicolumn{1}{r|}{\textbf{54.8}} & 19.6 & 40.4 & \multicolumn{1}{r|}{48.3} & 20.5 & 37.0 & 42.8 \\ \hline
\multicolumn{1}{l|}{\multirow{5}{*}{\textbf{\texttt{MUSE}}}} & \multicolumn{1}{l|}{\textbf{8K}} & 39.1 & 65.4 & \multicolumn{1}{r|}{72.3} & 41.1 & 63.3 & \multicolumn{1}{r|}{70.1} & 27.6 & 45.9 & \multicolumn{1}{r|}{53.2} & 19.5 & 40.4 & \multicolumn{1}{r|}{49.5} & 19.7 & 35.4 & 42 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{1K}} & 39.2 & 65.4 & \multicolumn{1}{r|}{72.1} & 41.1 & 63.3 & \multicolumn{1}{r|}{70.1} & 27.6 & 46.0 & \multicolumn{1}{r|}{53.1} & 18.1 & 36.8 & \multicolumn{1}{r|}{44.9} & 19.8 & 35.3 & 41.5 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{100}} & 24.8 & 47.5 & \multicolumn{1}{r|}{54.6} & 20.9 & 39.2 & \multicolumn{1}{r|}{48.1} & 0.8 & 3.4 & \multicolumn{1}{r|}{5.2} & 0.3 & 1.3 & \multicolumn{1}{r|}{2.2} & 6.2 & 16.1 & 22.8 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & 35.9 & 60.6 & \multicolumn{1}{r|}{67.3} & 37.8 & 60.4 & \multicolumn{1}{r|}{68.5} & 24.8 & 41.9 & \multicolumn{1}{r|}{49.5} & 13.4 & 25.5 & \multicolumn{1}{r|}{32} & 6.7 & 16.6 & 21.3 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Unsupervised}} & 39.3 & 64.7 & \multicolumn{1}{r|}{71.3} & 41.6 & 63.2 & \multicolumn{1}{r|}{69.9} & 28.3 & 46.5 & \multicolumn{1}{r|}{53.3} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 0.0 & 0.0 & 0.0 \\ \hline
\multicolumn{1}{l|}{\multirow{4}{*}{\textbf{\textbf{Meemi}\textsubscript{VM}}}} & \multicolumn{1}{l|}{\textbf{8K}} & 39.3 & \textbf{67.4} & \multicolumn{1}{r|}{\textbf{73.7}} & 41.6 & 66.5 & \multicolumn{1}{r|}{72.5} & 28 & 47.8 & \multicolumn{1}{r|}{\textbf{54.8}} & \textbf{23.8} & \textbf{48.7} & \multicolumn{1}{r|}{\textbf{57.0}} & 0.0 & 0.0 & 0.0 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{1K}} & 35.5 & 63.7 & \multicolumn{1}{r|}{69.4} & 38.6 & 64.0 & \multicolumn{1}{r|}{70.1} & 23.1 & 42.5 & \multicolumn{1}{r|}{49.9} & 17.8 & 40.1 & \multicolumn{1}{r|}{48.6} & 0.0 & 0.0 & 0.0 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{100}} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 0.0 & 0.0 & 0.0 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & 38.7 & 63.7 & \multicolumn{1}{r|}{70.1} & 40.6 & 64.1 & \multicolumn{1}{r|}{70.2} & 27.5 & 46.6 & \multicolumn{1}{r|}{53.1} & 19.3 & 37.7 & \multicolumn{1}{r|}{45.5} & 7.1 & 14.3 & 17.9 \\ \hline
\multicolumn{1}{l|}{\multirow{4}{*}{\textbf{\textbf{Meemi}\textsubscript{MS}}}} & \multicolumn{1}{l|}{\textbf{8K}} & 39.3 & \textbf{67.4} & \multicolumn{1}{r|}{\textbf{73.7}} & 41.3 & \textbf{66.8} & \multicolumn{1}{r|}{\textbf{72.8}} & 27.1 & 46.3 & \multicolumn{1}{r|}{53.9} & 21.7 & 45.0 & \multicolumn{1}{r|}{53.6} & 0.0 & 0.0 & 0.0 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{1K}} & 35.4 & 63.1 & \multicolumn{1}{r|}{69.3} & 38.2 & 63.6 & \multicolumn{1}{r|}{70.2} & 22.4 & 40.4 & \multicolumn{1}{r|}{47.9} & 14.7 & 33.6 & \multicolumn{1}{r|}{41.8} & 0.0 & 0.0 & 0.0 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{100}} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 0.0 & 0.1 & \multicolumn{1}{r|}{0.1} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 0.0 & 0.0 & 0.0 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & 35.4 & 58.9 & \multicolumn{1}{r|}{65.4} & 37.0 & 59.0 & \multicolumn{1}{r|}{65.9} & 24.0 & 40.0 & \multicolumn{1}{r|}{47.0} & 13.0 & 25.5 & \multicolumn{1}{r|}{32.0} & 2.5 & 6.2 & 8.3 \\ \hline
\multicolumn{17}{c}{\textbf{\texttt{Web corpora}}} \\ \hline
\multicolumn{1}{l|}{\multirow{2}{*}{\textbf{Model}}} & \multicolumn{1}{l|}{\multirow{2}{*}{\textbf{Supervision}}} & \multicolumn{3}{c}{\textbf{English-Spanish}} & \multicolumn{3}{c|}{\textbf{English-Italian}} & \multicolumn{3}{c|}{\textbf{English-German}} & \multicolumn{3}{c|}{\textbf{English-Finnish}} & \multicolumn{3}{c}{\textbf{English-Farsi}} \\ \cline{3-17}
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l|}{$P@10$} & \multicolumn{1}{l}{$P@1$} & \multicolumn{1}{l}{$P@5$} & \multicolumn{1}{l}{$P@10$} \\ \hline
\multicolumn{1}{l|}{\multirow{5}{*}{\textbf{\texttt{VecMap}}}} & \multicolumn{1}{l|}{\textbf{8K}} & 34.6 & 60.6 & \multicolumn{1}{r|}{66.9} & 31.9 & 54.2 & \multicolumn{1}{r|}{60.4} & 23.1 & 42.7 & \multicolumn{1}{r|}{50.5} & 18.9 & 40.9 & \multicolumn{1}{r|}{48.8} & 19.6 & \textbf{35.8} & \textbf{41.4} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{1K}} & 34.6 & 60.5 & \multicolumn{1}{r|}{67.0} & 32.0 & 54.0 & \multicolumn{1}{r|}{60.5} & 23.1 & 42.7 & \multicolumn{1}{r|}{50.3} & 19.4 & 42.0 & \multicolumn{1}{r|}{49.4} & 19.5 & 35.6 & 41.2 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{100}} & \textbf{38.5} & 61.2 & \multicolumn{1}{r|}{67.5} & 32.0 & 54.2 & \multicolumn{1}{r|}{60.5} & 23.0 & 43.0 & \multicolumn{1}{r|}{50.2} & 19.3 & 41.6 & \multicolumn{1}{r|}{49.6} & 19.7 & 35.5 & 41.3 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & 34.7 & 60.4 & \multicolumn{1}{r|}{67.0} & 31.4 & 54.0 & \multicolumn{1}{r|}{60.7} & 23.1 & 42.9 & \multicolumn{1}{r|}{50.5} & 18.6 & 41.6 & \multicolumn{1}{r|}{49.3} & \textbf{20.0} & 35.3 & 40.3 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Unsupervised}} & 34.8 & 60.6 & \multicolumn{1}{r|}{67.0} & 31.4 & 53.7 & \multicolumn{1}{r|}{60.7} & 23.2 & 42.7 & \multicolumn{1}{r|}{50.2} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 19.7 & 34.6 & 40.4 \\ \hline
\multicolumn{1}{l|}{\multirow{5}{*}{\textbf{\texttt{MUSE}}}} & \multicolumn{1}{l|}{\textbf{8K}} & 32.5 & 58.2 & \multicolumn{1}{r|}{65.9} & 32.5 & 56.0 & \multicolumn{1}{r|}{63.2} & 22.4 & 40.9 & \multicolumn{1}{r|}{48.9} & 20.0 & 40.1 & \multicolumn{1}{r|}{48.3} & 17.4 & 31.6 & 37.6 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{1K}} & 32.9 & 56.8 & \multicolumn{1}{r|}{64.2} & 31.5 & 52.7 & \multicolumn{1}{r|}{60.6} & 22.1 & 41.0 & \multicolumn{1}{r|}{48.2} & 18.4 & 39.1 & \multicolumn{1}{r|}{47.7} & 16.6 & 31.1 & 36.4 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{100}} & 32.3 & 56.1 & \multicolumn{1}{r|}{63.9} & 27.3 & 48.0 & \multicolumn{1}{r|}{55.3} & 17.8 & 35.0 & \multicolumn{1}{r|}{41.6} & 2.7 & 7.9 & \multicolumn{1}{r|}{11.1} & 0.0 & 0.5 & 0.7 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & 26.1 & 46.7 & \multicolumn{1}{r|}{53.8} & 24.7 & 45.1 & \multicolumn{1}{r|}{52.4} & 17.4 & 32.8 & \multicolumn{1}{r|}{40.5} & 12.6 & 26.0 & \multicolumn{1}{r|}{33.8} & 3.0 & 8.3 & 5.8 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Unsupervised}} & 31.4 & 51.2 & \multicolumn{1}{r|}{57.7} & 31.4 & 51.2 & \multicolumn{1}{r|}{57.7} & 20.8 & 38.7 & \multicolumn{1}{r|}{46.6} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 18.1 & 32.8 & 37.8 \\ \hline
\multicolumn{1}{l|}{\multirow{4}{*}{\textbf{\begin{tabular}[c]{@{}l@{}}\textbf{Meemi}\textsubscript{VM}\end{tabular}}}} & \multicolumn{1}{l|}{\textbf{8K}} & 34.5 & \textbf{61.6} & \multicolumn{1}{r|}{67.9} & 33.6 & 58.3 & \multicolumn{1}{r|}{\textbf{65.6}} & \textbf{23.7} & \textbf{45.4} & \multicolumn{1}{r|}{\textbf{53.2}} & 22.3 & \textbf{46.7} & \multicolumn{1}{r|}{\textbf{55.0}} & 0.0 & 0.0 & 0.0 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{1K}} & 30.2 & 55.0 & \multicolumn{1}{r|}{62.7} & 30.7 & 54.0 & \multicolumn{1}{r|}{61.1} & 19.4 & 38.9 & \multicolumn{1}{r|}{45.9} & 18.2 & 39.9 & \multicolumn{1}{r|}{48.1} & 0.0 & 0.0 & 0.0 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{100}} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 0.0 & 0.0 & 0.0 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & 34.1 & 58.3 & \multicolumn{1}{r|}{64.8} & 31.6 & 54.6 & \multicolumn{1}{r|}{62.5} & 22.5 & 42.0 & \multicolumn{1}{r|}{49.0} & 21.1 & 43.2 & \multicolumn{1}{r|}{51.3} & 11.2 & 23.9 & 28.6 \\ \hline
\multicolumn{1}{l|}{\multirow{4}{*}{\textbf{\begin{tabular}[c]{@{}l@{}}\textbf{Meemi}\textsubscript{MS}\end{tabular}}}} & \multicolumn{1}{l|}{\textbf{8K}} & 33.9 & 60.7 & \multicolumn{1}{r|}{\textbf{68.4}} & \textbf{33.8} & \textbf{58.4} & \multicolumn{1}{r|}{\textbf{65.6}} & \textbf{23.7} & 45.3 & \multicolumn{1}{r|}{52.3} & \textbf{23.0} & 46.1 & \multicolumn{1}{r|}{54.0} & 0.0 & 0.0 & 0.0 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{1K}} & 29.1 & 54.6 & \multicolumn{1}{r|}{62.3} & 29.9 & 52.7 & \multicolumn{1}{r|}{60.3} & 18.3 & 37.0 & \multicolumn{1}{r|}{44.1} & 17.2 & 37.4 & \multicolumn{1}{r|}{45.6} & 0.0 & 0.0 & 0.0 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{100}} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 0.0 & 0.0 & \multicolumn{1}{r|}{0.0} & 0.0 & 0.0 & 0.0 \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & 24.3 & 43.0 & \multicolumn{1}{r|}{49.8} & 21.8 & 41.1 & \multicolumn{1}{r|}{48.9} & 16.4 & 30.7 & \multicolumn{1}{r|}{37.3} & 13.1 & 26.1 & \multicolumn{1}{r|}{33.9} & 2.0 & 4.2 & 5.8 \\ \hline
\multicolumn{17}{c}{\textbf{\texttt{Social media}}} \\ \hline \hline
\multicolumn{1}{l|}{\multirow{2}{*}{\textbf{Model}}} & \multicolumn{1}{l|}{\multirow{2}{*}{\textbf{Supervision}}} & \multicolumn{3}{c}{\textbf{English-Spanish}} & \multicolumn{3}{c|}{\textbf{English-Italian}} & \multicolumn{3}{c|}{\textbf{English-German}} & \multicolumn{3}{c|}{\textbf{English-Finnish}} & \multicolumn{3}{c|}{\textbf{English-Farsi}} \\ \cline{3-17}
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{} & $P@1$ & $P@5$ & \multicolumn{1}{l|}{$P@10$} & $P@1$ & $P@5$ & \multicolumn{1}{l|}{$P@10$} & $P@1$ & $P@5$ & \multicolumn{1}{l|}{$P@10$} & $P@1$ & $P@5$ & \multicolumn{1}{l|}{$P@10$} & $P@1$ & $P@5$ & $P@10$ \\ \hline
\multicolumn{1}{l|}{\multirow{5}{*}{\textbf{\texttt{VecMap}}}} & \multicolumn{1}{l|}{\textbf{8K}} & \multicolumn{1}{r}{8.7} & \multicolumn{1}{r}{16.6} & \multicolumn{1}{r|}{21.6} & \multicolumn{1}{r}{8.9} & \multicolumn{1}{r}{17.3} & \multicolumn{1}{r|}{22.4} & \multicolumn{1}{r}{3.2} & \multicolumn{1}{r}{6.8} & \multicolumn{1}{r|}{9.5} & \multicolumn{1}{r}{0.2} & \multicolumn{1}{r}{0.8} & \multicolumn{1}{r|}{1.2} & \multicolumn{1}{r}{0.4} & \multicolumn{1}{r}{1.6} & \multicolumn{1}{r}{2.0} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{1K}} & \multicolumn{1}{r}{8.3} & \multicolumn{1}{r}{17.0} & \multicolumn{1}{r|}{21.3} & \multicolumn{1}{r}{8.9} & \multicolumn{1}{r}{17.5} & \multicolumn{1}{r|}{22.0} & \multicolumn{1}{r}{2.9} & \multicolumn{1}{r}{6.5} & \multicolumn{1}{r|}{9.3} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.4} & \multicolumn{1}{r|}{1.1} & \multicolumn{1}{r}{0.3} & \multicolumn{1}{r}{1.0} & \multicolumn{1}{r}{1.4} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{100}} & \multicolumn{1}{r}{7.9} & \multicolumn{1}{r}{15.9} & \multicolumn{1}{r|}{20.2} & \multicolumn{1}{r}{8.8} & \multicolumn{1}{r}{17.6} & \multicolumn{1}{r|}{22.3} & \multicolumn{1}{r}{2.8} & \multicolumn{1}{r}{6.0} & \multicolumn{1}{r|}{8.6} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.1} & \multicolumn{1}{r}{0.1} & \multicolumn{1}{r}{0.3} & \multicolumn{1}{r}{0.4} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & \multicolumn{1}{r}{8.5} & \multicolumn{1}{r}{16.9} & \multicolumn{1}{r|}{21.6} & \multicolumn{1}{r}{9.1} & \multicolumn{1}{r}{16.8} & \multicolumn{1}{r|}{21.8} & \multicolumn{1}{r}{2.6} & \multicolumn{1}{r}{6.7} & \multicolumn{1}{r|}{9.6} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.0} & \multicolumn{1}{r}{0.2} & \multicolumn{1}{r}{0.5} & \multicolumn{1}{r}{1.1} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Unsupervised}} & \multicolumn{1}{r}{8.1} & \multicolumn{1}{r}{16.4} & \multicolumn{1}{r|}{20.4} & \multicolumn{1}{r}{8.8} & \multicolumn{1}{r}{17.0} & \multicolumn{1}{r|}{22.3} & \multicolumn{1}{r}{0.1} & \multicolumn{1}{r}{0.4} & \multicolumn{1}{r|}{0.5} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} \\ \hline
\multicolumn{1}{l|}{\multirow{5}{*}{\textbf{\texttt{MUSE}}}} & \multicolumn{1}{l|}{\textbf{8K}} & \multicolumn{1}{r}{8.1} & \multicolumn{1}{r}{17.6} & \multicolumn{1}{r|}{22.7} & \multicolumn{1}{r}{8.0} & \multicolumn{1}{r}{16.4} & \multicolumn{1}{r|}{21.1} & \multicolumn{1}{r}{2.2} & \multicolumn{1}{r}{6.0} & \multicolumn{1}{r|}{8.4} & \multicolumn{1}{r}{0.6} & \multicolumn{1}{r}{2.2} & \multicolumn{1}{r|}{3.2} & \multicolumn{1}{r}{\textbf{1.2}} & \multicolumn{1}{r}{\textbf{4.5}} & \multicolumn{1}{r}{\textbf{6.3}} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{1K}} & \multicolumn{1}{r}{7.2} & \multicolumn{1}{r}{15.9} & \multicolumn{1}{r|}{20.5} & \multicolumn{1}{r}{7.3} & \multicolumn{1}{r}{14.6} & \multicolumn{1}{r|}{18.4} & \multicolumn{1}{r}{0.9} & \multicolumn{1}{r}{3.0} & \multicolumn{1}{r|}{4.5} & \multicolumn{1}{r}{0.6} & \multicolumn{1}{r}{1.5} & \multicolumn{1}{r|}{2.1} & \multicolumn{1}{r}{0.9} & \multicolumn{1}{r}{2.1} & \multicolumn{1}{r}{3.4} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{100}} & \multicolumn{1}{r}{0.4} & \multicolumn{1}{r}{1.1} & \multicolumn{1}{r|}{1.9} & \multicolumn{1}{r}{0.3} & \multicolumn{1}{r}{1.1} & \multicolumn{1}{r|}{1.8} & \multicolumn{1}{r}{0.1} & \multicolumn{1}{r}{0.3} & \multicolumn{1}{r|}{0.6} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.2} & \multicolumn{1}{r|}{0.4} & \multicolumn{1}{r}{0.1} & \multicolumn{1}{r}{0.3} & \multicolumn{1}{r}{0.4} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & \multicolumn{1}{r}{2.5} & \multicolumn{1}{r}{5.2} & \multicolumn{1}{r|}{7.1} & \multicolumn{1}{r}{3.9} & \multicolumn{1}{r}{10.1} & \multicolumn{1}{r|}{13.7} & \multicolumn{1}{r}{1.1} & \multicolumn{1}{r}{2.6} & \multicolumn{1}{r|}{3.7} & \multicolumn{1}{r}{0.1} & \multicolumn{1}{r}{0.1} & \multicolumn{1}{r|}{0.2} & \multicolumn{1}{r}{0.1} & \multicolumn{1}{r}{0.3} & \multicolumn{1}{r}{0.8} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Unsupervised}} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.0} & \multicolumn{1}{r}{7.3} & \multicolumn{1}{r}{14.5} & \multicolumn{1}{r|}{18.3} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.1} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.1} & \multicolumn{1}{r}{0.1} \\ \hline
\multicolumn{1}{l|}{\multirow{4}{*}{\textbf{\textbf{Meemi}\textsubscript{VM}}}} & \multicolumn{1}{l|}{\textbf{8K}} & \multicolumn{1}{r}{\textbf{9.8}} & \multicolumn{1}{r}{\textbf{21.3}} & \multicolumn{1}{r|}{\textbf{26.9}} & \multicolumn{1}{r}{\textbf{10.6}} & \multicolumn{1}{r}{\textbf{20.0}} & \multicolumn{1}{r|}{\textbf{25.6}} & \multicolumn{1}{r}{\textbf{3.7}} & \multicolumn{1}{r}{\textbf{9.6}} & \multicolumn{1}{r|}{\textbf{13.2}} & \multicolumn{1}{r}{1.3} & \multicolumn{1}{r}{3.6} & \multicolumn{1}{r|}{5.5} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.1} & \multicolumn{1}{r}{0.1} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{1K}} & \multicolumn{1}{r}{8.3} & \multicolumn{1}{r}{17.7} & \multicolumn{1}{r|}{22.6} & \multicolumn{1}{r}{8.6} & \multicolumn{1}{r}{18.2} & \multicolumn{1}{r|}{23.6} & \multicolumn{1}{r}{3.0} & \multicolumn{1}{r}{7.5} & \multicolumn{1}{r|}{10.6} & \multicolumn{1}{r}{0.5} & \multicolumn{1}{r}{2.4} & \multicolumn{1}{r|}{3.7} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{100}} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & \multicolumn{1}{r}{3.8} & \multicolumn{1}{r}{9.1} & \multicolumn{1}{r|}{11.8} & \multicolumn{1}{r}{6.6} & \multicolumn{1}{r}{14.2} & \multicolumn{1}{r|}{18.2} & \multicolumn{1}{r}{2.0} & \multicolumn{1}{r}{4.1} & \multicolumn{1}{r|}{5.9} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.1} & \multicolumn{1}{r|}{0.2} & \multicolumn{1}{r}{0.1} & \multicolumn{1}{r}{0.3} & \multicolumn{1}{r}{0.5} \\ \hline
\multicolumn{1}{l|}{\multirow{4}{*}{\textbf{\textbf{Meemi}\textsubscript{MS}}}} & \multicolumn{1}{l|}{\textbf{8K}} & \multicolumn{1}{r}{9.5} & \multicolumn{1}{r}{20.5} & \multicolumn{1}{r|}{26.3} & \multicolumn{1}{r}{9.5} & \multicolumn{1}{r}{19.1} & \multicolumn{1}{r|}{24.5} & \multicolumn{1}{r}{3.0} & \multicolumn{1}{r}{7.6} & \multicolumn{1}{r|}{11.1} & \multicolumn{1}{r}{\textbf{1.5}} & \multicolumn{1}{r}{\textbf{4.3}} & \multicolumn{1}{r|}{\textbf{6.4}} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.1} & \multicolumn{1}{r}{0.2} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{1K}} & \multicolumn{1}{r}{7.6} & \multicolumn{1}{r}{16.9} & \multicolumn{1}{r|}{22.3} & \multicolumn{1}{r}{7.8} & \multicolumn{1}{r}{15.9} & \multicolumn{1}{r|}{21} & \multicolumn{1}{r}{1.7} & \multicolumn{1}{r}{4.1} & \multicolumn{1}{r|}{6.2} & \multicolumn{1}{r}{0.8} & \multicolumn{1}{r}{2.3} & \multicolumn{1}{r|}{3.7} & \multicolumn{1}{r}{0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{100}} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r|}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.0} \\
\multicolumn{1}{l|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & \multicolumn{1}{r}{2.9} & \multicolumn{1}{r}{5.3} & \multicolumn{1}{r|}{6.7} & \multicolumn{1}{r}{3.5} & \multicolumn{1}{r}{9.9} & \multicolumn{1}{r|}{13.2} & \multicolumn{1}{r}{1.2} & \multicolumn{1}{r}{2.8} & \multicolumn{1}{r|}{3.6} & \multicolumn{1}{r}{0.2} & \multicolumn{1}{r}{0.3} & \multicolumn{1}{r|}{0.3} & \multicolumn{1}{r}{0.0} & \multicolumn{1}{r}{0.1} & \multicolumn{1}{r}{0.2} \\ \hline
\end{tabular}
}
\caption{Bilingual dictionary induction results in the test sets of \protect\newcite{conneau2018word}.}
\label{tab:dictionary}
\end{table*}
\begin{table*}[!t]
\renewcommand{\arraystretch}{0.96}
\resizebox{\textwidth}{!}{
\begin{tabular}{clrrrrrrrr}
\hline
\multicolumn{10}{c}{\texttt{\textbf{Wikipedia}}} \\ \hline \hline
\multicolumn{1}{c|}{\multirow{2}{*}{\textbf{Model}}} & \multicolumn{1}{c|}{\multirow{2}{*}{\textbf{Dictionary}}} & \multicolumn{2}{c}{\textbf{English-Spanish}} & \multicolumn{2}{c|}{\textbf{English-Italian}} & \multicolumn{2}{c|}{\textbf{English-German}} & \multicolumn{2}{c|}{\textbf{English-Farsi}} \\ \cline{3-10}
\multicolumn{1}{c|}{} & \multicolumn{1}{c|}{} & \multicolumn{1}{c}{\textbf{Pearson}} & \multicolumn{1}{c|}{\textbf{Spearman}} & \multicolumn{1}{c}{\textbf{Pearson}} & \multicolumn{1}{c|}{\textbf{Spearman}} & \multicolumn{1}{c}{\textbf{Pearson}} & \multicolumn{1}{c|}{\textbf{Spearman}} & \multicolumn{1}{c}{\textbf{Pearson}} & \multicolumn{1}{c}{\textbf{Spearman}} \\ \hline
\multicolumn{1}{c|}{\multirow{5}{*}{\textbf{\texttt{VecMap}}}} & \multicolumn{1}{l|}{\textbf{8K}} & 72.1 & \multicolumn{1}{r|}{71.8} & 71.2 & \multicolumn{1}{r|}{70.6} & 70.0 & \multicolumn{1}{r|}{69.3} & 63.7 & 61.7 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{1K}} & 72.1 & \multicolumn{1}{r|}{71.8} & 71.2 & \multicolumn{1}{r|}{70.6} & 70.0 & \multicolumn{1}{r|}{69.3} & 63.9 & 61.9 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{100}} & 72.1 & \multicolumn{1}{r|}{71.8} & 71.2 & \multicolumn{1}{r|}{70.6} & 70.0 & \multicolumn{1}{r|}{69.3} & 63.9 & 62.0 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & 72.1 & \multicolumn{1}{r|}{71.8} & 71.2 & \multicolumn{1}{r|}{70.6} & 70.0 & \multicolumn{1}{r|}{69.3} & 63.8 & 61.9 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Unsupervised}} & 72.1 & \multicolumn{1}{r|}{71.8} & 71.2 & \multicolumn{1}{r|}{70.6} & 70.0 & \multicolumn{1}{r|}{69.3} & 63.4 & 61.3 \\ \hline
\multicolumn{1}{c|}{\multirow{5}{*}{\textbf{\texttt{MUSE}}}} & \multicolumn{1}{l|}{\textbf{8K}} & 72.0 & \multicolumn{1}{r|}{72.6} & 70.7 & \multicolumn{1}{r|}{70.9} & 68.8 & \multicolumn{1}{r|}{68.9} & 59.2 & 58.7 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{1K}} & 71.9 & \multicolumn{1}{r|}{72.4} & 70.6 & \multicolumn{1}{r|}{70.7} & 68.6 & \multicolumn{1}{r|}{68.7} & 58.7 & 58.4 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{100}} & 65.1 & \multicolumn{1}{r|}{66.3} & 63.0 & \multicolumn{1}{r|}{63.6} & 44.7 & \multicolumn{1}{r|}{49.9} & 47.7 & 52.1 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & 71.0 & \multicolumn{1}{r|}{71.9} & 69.9 & \multicolumn{1}{r|}{70.5} & 68.1 & \multicolumn{1}{r|}{68.4} & 47.9 & 51.3 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Unsupervised}} & 72.2 & \multicolumn{1}{r|}{72.6} & 71.0 & \multicolumn{1}{r|}{71.2} & 68.7 & \multicolumn{1}{r|}{68.9} & 8.0 & 6.5 \\ \hline
\multicolumn{1}{c|}{\multirow{4}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}\textbf{Meemi}\textsubscript{VM}\end{tabular}}}} & \multicolumn{1}{l|}{\textbf{8K}} & 72.5 & \multicolumn{1}{r|}{71.9} & 71.8 & \multicolumn{1}{r|}{70.9} & \textbf{70.9} & \multicolumn{1}{r|}{\textbf{70.3}} & \textbf{65.1} & \textbf{63.4} \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{1K}} & 70.1 & \multicolumn{1}{r|}{69.6} & 69.7 & \multicolumn{1}{r|}{69.0} & 67.6 & \multicolumn{1}{r|}{66.8} & 5.5 & 5.8 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{100}} & 0.0 & \multicolumn{1}{r|}{0.0} & 5.1 & \multicolumn{1}{r|}{5.0} & 4.1 & \multicolumn{1}{r|}{3.3} & 6.8 & 6.5 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & 71.0 & \multicolumn{1}{r|}{70.4} & 69.4 & \multicolumn{1}{r|}{68.7} & 69.3 & \multicolumn{1}{r|}{68.6} & 56.1 & 54.1 \\ \hline
\multicolumn{1}{c|}{\multirow{4}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}\textbf{Meemi}\textsubscript{MS}\end{tabular}}}} & \multicolumn{1}{l|}{\textbf{8K}} & \textbf{73.1} & \multicolumn{1}{r|}{\textbf{72.9}} & \textbf{72.4} & \multicolumn{1}{r|}{\textbf{71.9}} & 70.7 & \multicolumn{1}{r|}{70.1} & 64.1 & 62.0 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{1K}} & 70.4 & \multicolumn{1}{r|}{70.3} & 69.6 & \multicolumn{1}{r|}{69.6} & 66.7 & \multicolumn{1}{r|}{66.5} & 5.0 & 4.2 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{100}} & 2.7 & \multicolumn{1}{r|}{1.4} & 0.0 & \multicolumn{1}{r|}{0.2} & 6.3 & \multicolumn{1}{r|}{6.0} & 0.0 & 0.0 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & 70.6 & \multicolumn{1}{r|}{70.9} & 68.7 & \multicolumn{1}{r|}{68.8} & 68.0 & \multicolumn{1}{r|}{67.7} & 46.6 & 48.2 \\ \hline
\hline
\multicolumn{10}{c}{\texttt{\textbf{Web corpora}}} \\ \hline \hline
\multicolumn{1}{c|}{\multirow{2}{*}{\textbf{Model}}} & \multicolumn{1}{l|}{\multirow{2}{*}{\textbf{Dictionary}}} & \multicolumn{2}{c|}{\textbf{English-Spanish}} & \multicolumn{2}{c|}{\textbf{English-Italian}} & \multicolumn{2}{c|}{\textbf{English-German}} & \multicolumn{2}{c}{\textbf{English-Farsi}} \\ \cline{3-10}
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{} & \multicolumn{1}{l}{\textbf{Pearson}} & \multicolumn{1}{l|}{\textbf{Spearman}} & \multicolumn{1}{l}{\textbf{Pearson}} & \multicolumn{1}{l|}{\textbf{Spearman}} & \multicolumn{1}{l}{\textbf{Pearson}} & \multicolumn{1}{l|}{\textbf{Spearman}} & \multicolumn{1}{l}{\textbf{Pearson}} & \multicolumn{1}{l}{\textbf{Spearman}} \\ \hline
\multicolumn{1}{c|}{\multirow{5}{*}{\textbf{\texttt{VecMap}}}} & \multicolumn{1}{l|}{\textbf{8K}} & 71.0 & \multicolumn{1}{r|}{70.6} & 69.2 & \multicolumn{1}{r|}{68.8} & 70.9 & \multicolumn{1}{r|}{70.4} & 35.9 & 33.5 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{1K}} & 71.0 & \multicolumn{1}{r|}{70.6} & 69.3 & \multicolumn{1}{r|}{68.8} & 70.9 & \multicolumn{1}{r|}{70.4} & 35.9 & 33.5 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{100}} & 71.0 & \multicolumn{1}{r|}{70.4} & 69.2 & \multicolumn{1}{r|}{68.8} & 71.0 & \multicolumn{1}{r|}{70.5} & 35.9 & 33.5 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & 71.0 & \multicolumn{1}{r|}{70.6} & 69.3 & \multicolumn{1}{r|}{68.8} & 70.9 & \multicolumn{1}{r|}{70.4} & 35.9 & 33.0 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Unsupervised}} & 71.1 & \multicolumn{1}{r|}{70.5} & 69.2 & \multicolumn{1}{r|}{68.8} & 70.9 & \multicolumn{1}{r|}{70.4} & 35.7 & 33.4 \\ \hline
\multicolumn{1}{c|}{\multirow{5}{*}{\textbf{\texttt{MUSE}}}} & \multicolumn{1}{l|}{\textbf{8K}} & 71.9 & \multicolumn{1}{r|}{71.9} & 70.4 & \multicolumn{1}{r|}{70.4} & 70.5 & \multicolumn{1}{r|}{70.2} & 29.7 & 23.9 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{1K}} & 71.6 & \multicolumn{1}{r|}{71.5} & 69.5 & \multicolumn{1}{r|}{69.4} & 70.3 & \multicolumn{1}{r|}{70.0} & 28.3 & 22.3 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{100}} & 71.7 & \multicolumn{1}{r|}{71.6} & 67.4 & \multicolumn{1}{r|}{67.4} & 68.5 & \multicolumn{1}{r|}{68.8} & 6.3 & 7.4 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & 69.9 & \multicolumn{1}{r|}{70.1} & 67.3 & \multicolumn{1}{r|}{67.5} & 70.1 & \multicolumn{1}{r|}{69.7} & 17.5 & 14.5 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Unsupervised}} & 71.7 &\multicolumn{1}{r|}{71.6} & 69.4 & \multicolumn{1}{r|}{69.4} & 70.3 & \multicolumn{1}{r|}{70.0} & 29.6 & 23.8 \\ \hline
\multicolumn{1}{c|}{\multirow{4}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}\textbf{Meemi}\textsubscript{VM}\end{tabular}}}} & \multicolumn{1}{l|}{\textbf{8K}} & 71.5 & \multicolumn{1}{r|}{70.9} & 70.4 & \multicolumn{1}{r|}{70.0} & 72.3 & \multicolumn{1}{r|}{71.8} & \textbf{40.2} & \textbf{39.0} \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{1K}} & 69.1 & \multicolumn{1}{r|}{68.6} & 68.2 & \multicolumn{1}{r|}{67.8} & 70.9 & \multicolumn{1}{r|}{70.2} & 1.2 & 0.7 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{100}} & 0.0 & \multicolumn{1}{r|}{0.0} & 0.0 & \multicolumn{1}{r|}{0.0} & 0.0 & \multicolumn{1}{r|}{0.0} & 3.2 & 3.0 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & 70.1 & \multicolumn{1}{r|}{69.5} & 69.2 & \multicolumn{1}{r|}{68.4} & 71.2 & \multicolumn{1}{r|}{70.6} & 31.6 & 28.5 \\ \hline
\multicolumn{1}{c|}{\multirow{4}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}\textbf{Meemi}\textsubscript{MS}\end{tabular}}}} & \multicolumn{1}{l|}{\textbf{8K}} & \textbf{72.5} & \multicolumn{1}{r|}{\textbf{72.3}} & \textbf{71.5} & \multicolumn{1}{r|}{\textbf{71.1}} & \textbf{72.5} & \multicolumn{1}{r|}{\textbf{72.1}} & 36.4 & 33.0 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{1K}} & 70.0 & \multicolumn{1}{r|}{69.7} & 68.9 & \multicolumn{1}{r|}{68.6} & 70.3 & \multicolumn{1}{r|}{69.9} & 0.0 & 0.0 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{100}} & 1.7 & \multicolumn{1}{r|}{0.1} & 1.6 & \multicolumn{1}{r|}{2.3} & 0.0 & \multicolumn{1}{r|}{0.0} & 0.3 & 0.0 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & 69.2 & \multicolumn{1}{r|}{69.1} & 67.4 & \multicolumn{1}{r|}{66.9} & 70.1 & \multicolumn{1}{r|}{69.4} & 17.3 & 14.5 \\ \hline \hline
\multicolumn{10}{c}{\texttt{\textbf{Social media}}} \\ \hline \hline
\multicolumn{1}{c|}{\multirow{2}{*}{\textbf{Model}}} & \multicolumn{1}{l|}{\multirow{2}{*}{\textbf{Dictionary}}} & \multicolumn{2}{c|}{\textbf{English-Spanish}} & \multicolumn{2}{c|}{\textbf{English-Italian}} & \multicolumn{2}{c|}{\textbf{English-German}} & \multicolumn{2}{c}{\textbf{English-Farsi}} \\ \cline{3-10}
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{} & \multicolumn{1}{l}{\textbf{Pearson}} & \multicolumn{1}{l|}{\textbf{Spearman}} & \multicolumn{1}{l}{\textbf{Pearson}} & \multicolumn{1}{l|}{\textbf{Spearman}} & \multicolumn{1}{l}{\textbf{Pearson}} & \multicolumn{1}{l|}{\textbf{Spearman}} & \multicolumn{1}{l}{\textbf{Pearson}} & \multicolumn{1}{l}{\textbf{Spearman}} \\ \hline
\multicolumn{1}{c|}{\multirow{5}{*}{\textbf{\texttt{VecMap}}}} & \multicolumn{1}{l|}{\textbf{8K}} & 48.5 & \multicolumn{1}{r|}{47.4} & 53.9 & \multicolumn{1}{r|}{51.8} & 51.3 & \multicolumn{1}{r|}{49.5} & 31.1 & 30.3 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{1K}} & 48.7 & \multicolumn{1}{r|}{47.7} & 53.7 & \multicolumn{1}{r|}{51.6} & 51.7 & \multicolumn{1}{r|}{50.2} & 30.3 & 29.6 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{100}} & 49.8 & \multicolumn{1}{r|}{48.9} & 54.0 & \multicolumn{1}{r|}{51.7} & 51.0 & \multicolumn{1}{r|}{49.5} & 25.7 & 25.4 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & 48.4 & \multicolumn{1}{r|}{47.1} & 54.2 & \multicolumn{1}{r|}{51.9} & 51.8 & \multicolumn{1}{r|}{50.3} & 27.8 & 26.5 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Unsupervised}} & 48.0 & \multicolumn{1}{r|}{46.9} & 53.8 & \multicolumn{1}{r|}{51.5} & 30.1 & \multicolumn{1}{r|}{31.2} & 4.2 & 2.4 \\ \hline
\multicolumn{1}{c|}{\multirow{5}{*}{\textbf{\texttt{MUSE}}}} & \multicolumn{1}{l|}{\textbf{8K}} & 48.8 & \multicolumn{1}{r|}{47.6} & 51.0 & \multicolumn{1}{r|}{49.3} & 48.5 & \multicolumn{1}{r|}{48.6} & 43.3 & 42.2 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{1K}} & 46.6 & \multicolumn{1}{r|}{45.5} & 49.7 & \multicolumn{1}{r|}{47.8} & 44.8 & \multicolumn{1}{r|}{45.7} & 38.7 & 38.9 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{100}} & 35.8 & \multicolumn{1}{r|}{36.9} & 29.6 & \multicolumn{1}{r|}{31.3} & 30.7 & \multicolumn{1}{r|}{34.0} & 20.8 & 21.3 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & 48.1 & \multicolumn{1}{r|}{47.7} & 50.1 & \multicolumn{1}{r|}{49.8} & 45.6 & \multicolumn{1}{r|}{46.8} & 30.5 & 32.4 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Unsupervised}} & 9.9 & \multicolumn{1}{r|}{10.9} & 50.7 & \multicolumn{1}{r|}{49.7} & 12.4 & \multicolumn{1}{r|}{13.0} & 6.9 & 4.7 \\ \hline
\multicolumn{1}{c|}{\multirow{4}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}\textbf{Meemi}\textsubscript{VM}\end{tabular}}}} & \multicolumn{1}{l|}{\textbf{8K}} & 51.2 & \multicolumn{1}{r|}{50.1} & \textbf{56.1} & \multicolumn{1}{r|}{\textbf{53.6}} & \textbf{55.0} & \multicolumn{1}{r|}{\textbf{53.8}} & 45.2 & 43.1 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{1K}} & 49.7 & \multicolumn{1}{r|}{48.6} & 55.4 & \multicolumn{1}{r|}{52.8} & 52.8 & \multicolumn{1}{r|}{51.3} & 3.8 & 3.8 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{100}} & 2.4 & \multicolumn{1}{r|}{2.0} & 2.1 & \multicolumn{1}{r|}{2.8} & 0.0 & \multicolumn{1}{r|}{0.0} & 5.6 & 5.1 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & 51.7 & \multicolumn{1}{r|}{50.1} & 56.2 & \multicolumn{1}{r|}{53.4} & 52.7 & \multicolumn{1}{r|}{51.3} & 29.9 & 28.6 \\ \hline
\multicolumn{1}{c|}{\multirow{4}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}\textbf{Meemi}\textsubscript{MS}\end{tabular}}}} & \multicolumn{1}{l|}{\textbf{8K}} & \textbf{51.8} & \multicolumn{1}{r|}{\textbf{50.4}} & 54.8 & \multicolumn{1}{r|}{52.5} & 53.1 & \multicolumn{1}{r|}{52.0} & \textbf{48.9} & \textbf{46.6} \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{1K}} & 49.5 & \multicolumn{1}{r|}{47.9} & 53.1 & \multicolumn{1}{r|}{50.7} & 48.4 & \multicolumn{1}{r|}{47.2} & 0.0 & 0.0 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{100}} & 5.2 & \multicolumn{1}{r|}{5.4} & 0.0 & \multicolumn{1}{r|}{0.0} & 6.2 & \multicolumn{1}{r|}{6.8} & 0.0 & 0.0 \\
\multicolumn{1}{c|}{} & \multicolumn{1}{l|}{\textbf{Identical}} & 49.6 & \multicolumn{1}{r|}{48.2} & 53.1 & \multicolumn{1}{r|}{51.4} & 48.4 & \multicolumn{1}{r|}{47.7} & 30.5 & 30.6
\end{tabular}
}
\caption{Cross-lingual word similarity results in the SemEval-17 dataset \protect\cite{semeval2017similarity}.}
\label{tab:sim}
\end{table*}
\section{Bibliographical References}
\bibliographystyle{lrec}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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import React from 'react';
import { isEqual } from 'lodash';
import { HelpField } from 'core/help/HelpField';
export interface IPlatformHealthOverrideProps {
interestingHealthProviderNames: string[];
onChange: (healthProviderNames: string[]) => void;
platformHealthType: string;
showHelpDetails?: boolean;
}
export class PlatformHealthOverride extends React.Component<IPlatformHealthOverrideProps> {
private clicked = (event: React.ChangeEvent<HTMLInputElement>) => {
const interestingHealthProviderNames = event.target.checked ? [this.props.platformHealthType] : null;
this.props.onChange(interestingHealthProviderNames);
};
public render() {
return (
<div className="checkbox">
<label>
<input
type="checkbox"
checked={isEqual(this.props.interestingHealthProviderNames, [this.props.platformHealthType])}
onChange={this.clicked}
/>
Consider only {this.props.platformHealthType} health
</label>{' '}
<HelpField id="application.platformHealthOnly" expand={this.props.showHelpDetails} />
</div>
);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
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{"url":"http:\/\/blogs.scienceforums.net\/ajb\/?paged=3","text":"## Seminar: A first look at N-manifolds\n\nFebruary 13th, 2014 by ajb\n I will be giving a talk at the Geometry and Differential Equations Seminar at IMPAN (Warsaw) on Wednesday 26th February 2014. The title is \u201cA first look at N-manifolds\u201d.\n\nAbstract\nIn this talk I will introduce the concept of an N-manifold as refinement of the notion of a supermanifold in which the structure sheaf carries an additional grading, called weight, that takes values in the natural numbers. I will provide several motivating examples which largely come for the theory of jets, before discussing some generalities.\n\n## New exhibition at Jodrell Bank near Manchester\n\nFebruary 9th, 2014 by ajb\n Big Telescopes, Big Science is a brand new exhibition which will be unveiled in February at Jodrell Bank visitors centre. the exhibition will include hands-on activities showing how telescopes work and how it is possible to use many smaller telescopes to act as one large telescope.\n\nThere will also be running family science shows as part of the half term activities.\n\n## Son of LHC!\n\nFebruary 7th, 2014 by ajb\n CERN is putting plans in place to build a successor to the Large Hadron Collider (LHC). Possible options for the next generation of colliders will be discussed at the University of Geneva next week.\n\nThere are plans for a massive circular collider \u2013 with a circumference of 80\u2013100 km \u2013 that would accelerate protons to energies of about 100 TeV! The LHC has a 27 km circumference and can collide protons with energies up to about 7 TeV.\n\n## The Brighton Science Festival 2014\n\nJanuary 28th, 2014 by ajb\n\nThe Brighton Science Festival 2014 is going to be held from the 6th February to the 2nd March at various locations throughout the city of Brighton, East Sussex. A list of the events can be found here.\n\nIf you are in Brighton or nearby over the above dates I am sure you could find something to interest you.\n\n## Scientists get the Hollywood treatment!\n\nJanuary 27th, 2014 by ajb\n Monday night on the Science Channel, the Breakthrough Prizes are having their USA television debut!\n\nPreview and clips at http:\/\/tinyurl.com\/peo97q3\n\nHosted by Kevin Spacey, the show hands out seven \\$3 million prizes, and honors physicists Joseph Polchinski, Andrew Strominger, Cumrun Vafa, John H. Schwarz and Michael B. Green, as well as leading medical researchers James P. Allison, Mahlon R. DeLong, Michael N. Hall, Robert Langer, Richard Lifton and Alexander Varshavsky.\n\nHelping to celebrate the scientists are celebrities Conan O\u2019Brien, Anna Kendrick, Glenn Close, Rob Lowe, Michael C. Hall, as well as tech leaders Mark Zuckerberg, Larry Page, Sergey Brin, Anne Wojcicki, Jimmy Wales and Yuri Milner.\n\nThe show is at 9:00ET\/PT, 8:00CT, 7:00MT.\n\n## On curves and jets of curves and on supermanifolds\n\nJanuary 22nd, 2014 by ajb\n I placed a preprint on the arXiv entitled \u201cOn curves and jets of curves and on supermanifolds\u201d [1]. The abstract can be found below\n\nAbstract\nIn this note we examine a natural concept of a curve on a supermanifold and the subsequent notion of the jet of a curve. We then tackle the question of geometrically defining the higher order tangent bundles of a supermanifold. Finally we make a quick comparison with the notion of a curve presented here are other common notions found in the literature.\n\nQuick overview\nThe k-th order tangent bundle on a classical manifold can be defined as equivalence classes of curves at the points of the manifold that agree up to velocity, acceleration, rate of change of acceleration on so on up to k-th order. The tangent bundle which is much more widely known, is the \u201c1st order tangent bundle\u201d. These higher order tangent bundles have found applications in higher derivative Lagrangian mechanics for example and appear in the work of Grabowski & Rotkiewicz [2].\n\nThe main question posed and answered in this preprint is \u201ccan we define the k-th order tangent bundles of a supermanifold is such a kinematic way?\u201d\n\nThe problems with simply generalising the classical notions directly are two fold:\n\n1. Supermanifolds do not consist solely of topological point, thus point-wise constructions need some careful handling.\n2. The naive definition of a curve as a morphism of supermanifolds $\\mathbb{R} \\rightarrow M$ totally misses the odd dimensions of the supermanifold $M$.\n\nTo resolve these issues we look at a \u201csuperised\u201d version of curves defined via the internal Homs, a notion from category theory. In sort we consider curves that are paramaterised by all supermanifolds, however all the constructions should be functorial in this parametrisation. This allows us to define what I call S-curves that can \u201cfeel\u201d both the even and odd dimensions of a supermanifold. Moreover, this allows us to think in terms of S-points and parallel the classical constructions of jets of curves rather closely.\n\nThis allow us to define the k-th order tangent bundle of a supermanifold at first as a generalised supermanifold (a functor from the opposite category of supermanifolds to sets ) and then we show that this is representable, that is a genuine supermanifold. Moreover, locally everything looks the same as the classical higher order tangent bundles.\n\nReferences\n[1] Andrew James Bruce, On curves and jets of curves and on supermanifolds, arXiv:1401.5267 [math-ph].\n[2] J. Grabowski & M. Rotkiewicz, Graded bundles and homogeneity structures, J. geom. Phys. 62 (2012), 21-36.\n\n## My Thursday Colloquium talk\n\nDecember 30th, 2013 by ajb\n On the 9th of January 2014 I will be giving a talk at the Algebra and Geometry of Modern Physics seminar here in Warsaw.\n\nTitle \u201cWhat the functor is a superfield? \u201d\n\nAbstract Physicists are usually quite happy to formally manipulate the mathematical objects that they encounter without really understanding the structures they are dealing with. It is then the job of the mathematician to try to make sense of the physicists manipulations and give proper meaning to the structures. (Confusing physicists is not the job of mathematicians, however mathematicians are good at it!) In this talk we will uncover the structure of Grassmann odd fields as used in physics. For example such fields appear in quasi-classical theories of fermions and in the BV-BRST quantisation of gauge theories. To understand the structures here we need to jump into the theory of supermanifolds. However we find that supermanifolds are not quite enough! We need to deploy some tools from category theory and end up thinking in terms of functors! :-O\n\nI will try to post my notes here at some point after the talk\u2026\n\n## My phone number is in pi!\n\nDecember 22nd, 2013 by ajb\n I found a website that allows you to search the number $\\pi$ for stings of numbers. The Pi Searcher can search for any string of digits (up to length 120) in the first 200 million digits of $\\pi$.\n\nThe number $\\pi$ is transcendental meaning it cannot be written as any combination of rational numbers and their n-th roots.\n\nPhone numbers here in Poland are 7 digits and it turns out that there is a\u00a0 99.995% chance that the Pi Searcher can find it, Of course, if it can\u2019t find it then it may well be still in $\\pi$ somewhere,\u00a0 which is related to the fact that we think it is a normal number.\n\nAnyway my home phone number here in Poland appears 10 times in the first 200 million digits! My IOP number (6 digits) appears 186 times!\n\nTry it for yourself at the link below.\n\n## Could Newton have \u201cchosen\u201d something else?\n\nDecember 14th, 2013 by ajb\n\nIntroduction\n\n We all know Newton\u2019s second law \u201cF=ma\u201d. In words, we only have acceleration when an external force is applied. But could he have some how chosen not acceleration but say something higher order ?\n\nAcceleration is the rate of change of velocity, which itself is the rate of change of position. Thus acceleration is second order in derivatives with respect to time. This is not just a feature of Newtonian mechanics but is rather general and found throughout the fundamental laws of nature.\n\nThere are some rather general results using the Hamiltonian formalism that tells us that theories with equations of motion that are higher order than two are unstable. In particular the energy is not bounded from below and this can lead to problems classically and quantum mechanically. This is the famous Ostrogradski instability of (non-degenerate) Lagrangian theories with higher order derivative terms.\n\nNote that such theories are still of interest as effective theories, but they cannot be seen as fundamental.\n\nSo the question is can we understand simply why Newton could not really have picked anything of higher order in \u201cF=ma\u201d?\n\nNotation: I will use Newton\u2019s dot notation for the first and second order derivative with respect to time. For the n-th order derivative (n>2) I will use $x^{(n)}(t)$.\n\nOrder three\n\nLet us just pick a different form of Newton\u2019s law as\n\n$F = N x^{(3)} (t)$,\n\nthat is let us suppose the force is proportional to the third order derivative of the position. (That would be the rate of change of acceleration). Here N is some property of the particle analogous to mass.\n\nNow let us think about the motion of a free particle. So set the force term equal to zero and see what happens. Solving our \u201chigher order\u201d Newton\u2019s law with no force is simple. We have\n\n$x(t) = c_{3} t^{2} + c_{2}t + c_{1}$.\n\nThe constants here are set by our initial conditions;\n\n$x(0) = c_{1}$, $\\dot{x}(0) = c_{2}$ and $\\ddot{x}(0) = 2 c_{3}$.\n\nSo what do we notice? The velocity as a function of time is given by\n\n$v(t) := \\dot{x}(t) = \\ddot{x}(0)t + \\dot{x}(0)$.\n\nThis means that even if we set the initial velocity to zero the isolated free particle will speed up! Remember this is without any forces acting on the particle.\n\nNewton\u2019s first law says (in part) that \u201ca particle at rest will remain at rest unless acted upon by an external force\u201d. This higher order form of the second law is inconsistent with the first.\n\nWorse than this, there are no forces here and so no potentials. The particle just speeds up all by itself and so clearly we lose conservation of energy. The particle can gain kinetic energy, defined as usual, but at no loss of potential energy! We would have to abandon our usual notion of conservation of energy in simple mechanics!\n\nHigher order again\nThe same arguments work for higher order terms. The particle will just speed up by itself in violation of the first law and conservation of energy.\n\nModification of Newton\u2019s law\n\nLet us consider a slightly different situation in which \u201cF = ma\u201d becomes\n\n$F = M \\ddot{x}(t) + N x^{(3)}(t)$.\n\nThat is we will add a higher order term to Newton\u2019s law. Again, let is consider the case with no force term. We want to solve the equations of motion\n\n$\\ddot{x}(t) + \\frac{N}{M} x^{(3)}(t) =0$.\n\nHere we assume that M is not equal to zero and is positive. For now we make no assumption at all about N.\n\nOne can directly solve the equations of motion\n\n$x(t) = \\left( \\frac{N}{M}\\right)^{2} c_{1} e ^{-(M\/N) t} + c_{2}t + c_{3}$.\n\nAgain we notice that the velocity is not constant and so we do not have conservation of energy in this situation either. But let us have a look at the particles trajectory for different ratios M\/N.\n\nIn the above we have set M\/N = 2 and all the constants to 1. The purple line is what we expect from F=ma. Note that we have a quick decay to the classical case. This itself signifies we do not have conservation of energy as we have no mechanism for the loss in kinetic energy, it just happens!\n\nIn the above we have set M\/N =1. This case is very similar to the previous case.\n\nNow in the above we have set M\/N = 0.5. Again this is very similar to the previous cases.\n\nWe have a decay so that after some period of time everything looks the same as the standard Newtonian case. However, we still have to violate the first law to achieve this.\n\nNow what happens if we let N be negative?\n\nHere we see we have a runaway situation in which the particle just keeps on speeding up! Even if initially the trajectory is very close to the standard one after some time it just blows-up. Again this is in violation of the first law and conservation of energy.\n\nLower order\n\nWell if we had $F \\propto x$ then when there are no forces we simply have $x =0$. Everything not in motion would have to sit at x=0. Meaning we cannot have any extended objects that are not in motion. This cannot be consistent with our Universe.\n\nWhat about $F \\propto \\dot{x}$? Again let us set the forces to zero and we see that the solution is just $x(t) = x_{0}$, some constant. However, this does not sit comfortably with our notion of relativity. Different inertial observers will not agree on the value of $x_{0}$. Thus if we don\u2019t want to introduce absolute space we cannot allow this lower order form.\n\nConclusion\nSo as Newton wanted his first law to be true, have a good notion of statics and did not want to introduce absolute space he could have only have picked \u201cF = ma\u201d.\n\n## Physics World 2013 Breakthrough of the Year\n\nDecember 13th, 2013 by ajb\n\nThe Physics World award for the 2013 Breakthrough of the Year goes to \u201cthe IceCube South Pole Neutrino Observatory for making the first observations of cosmic neutrinos\u201d. 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War on the Children (tags)
"In 10- to 59-year-olds, there is a 15-fold higher risk of death from corona vaccination compared with corona infection" (6). https://pathologie-konferenz.de/ The broad majority is always mistaken - or allows itself to be misled. Children need love and closeness, not coercion!
The Danger in the Shadows (tags)
While everyone is distracted with Corona measures and coverage, the danger of nuclear war is increasing. The new "Krefeld Appeal" for Peace of November 2021 wants to draw attention to this danger and at the same time put public pressure on the government to stop the path it is on.
The Rich's Deal (tags)
The richest percent of the world's population accounts for fifteen percent of global emissions, which is twice as much as the entire poorer half of the world's inhabitants together are responsible for CO2. Once again, the rich countries benefit from the deal negotiated in Glasgow.
Promoting Sustainable Economics (tags)
The protagonists of neoliberal globalization seek to fight the devastations – for which they bear responsibility – by the same means that cause the devastations. Now is the time to seize the opportunity and build up economic policy pressure to enforce the development of an economic order that is sustainably oriented...
Who is Saxony's worst fucker beyond Sachsen-Paule? (tags)
In eight years the trained butcher Michael Berndt from Lausitz (Saxony) traveled through a total of 100 countries and indulged in unrestrained erotic adventures with local women.
The oppressed majority (tags)
The war on the virus is a crucial backdrop for destabilization and fear, before which vast amounts of public funds are siphoned off in the name of saving lives, saving the environment, and saving people's livelihoods - all of which are attacked by the arbitrariness of this very capitalist rule.
Rescue Plan of the Biden Administration (tags)
Thanks to the rescue plan, the US economy is expected to grow faster than at any time since 1984. The rest of the world will also benefit. Families with children will receive tax rebates. 70% of Americans are in favor of the plan.
Germany Axis Alert Puzzle (tags)
Since Sig Sauer was spotted in India more information about the Axis appear in Internet which analyses who were involved inside SS at 2nd World War. I just quoted some "random" rare puzzle pieces to inform...
Shrink or Die? (tags)
Shrinkage means the voluntary downsizing of the national economy, including a reduction in gross domestic product. "Voluntary" here indicates a preference for voluntaristic solutions - though not as individualistic and unplanned where well-to-do opt out of the consumerist market model.
The Fascism Construction Manual (tags)
Immediately after the proclamation of this pandemic, many elements of absolute all-around totalitarianism were put into effect in most countries of the West, even in countries that had previously been robust democracies. All of this happened very quickly and comprehensively.
Dissonance cognitive circlusive (tags)
De l'engrammation totalitaire...
Asia's Future (tags)
"Europe derived great power and huge profits from colonizing Asia, Asia has grown amazingly because of American and European outsourcing, and now the United States and Europe are propped up by infusions of Asian investment and talent. This is the true nature of the global system, not multipolar competition."
Corona in global retreat (tags)
The number of reported cases in the US has fallen massively since the beginning of January-by around 60 percent. Experts consider the Corona measures in the US to be less stringent than in many European countries.
"We must look at the side-effects" (tags)
"Restricting basic rights is no small matter, and it is the duty of politics to restore their exercise as soon as the infection situation permits." (Lockdown consequences: daycare centers and schools to open) German President Frank-Walter Steinmeier
Open Letter: Write off debt, win the future! (tags)
We are fortunate to have a creditor that does not have to fear losing its money: the ECB (European Central Bank). Our proposal is therefore very simple: let us conclude an agreement between the European states and the ECB. The ECB writes off the debt it holds (or converts it into perpetual interest-free debt).
The Cure is the Sickness, 2001 (tags)
"Good policy" in the sense of the Washington Consensus can be paraphrased with these ten principles. The key to development successes lies only in a correct policy mix. Development backlogs, continuing poverty and so forth are referred back everywhere to "bad policy".
Resistance against Agropoly (tags)
Food sovereignty remains the most important political program of the strongest international movement to date for a transformation of the food system: away from the agropoly of a few corporations. Tina Goethe is a sociologist and responsible for the area of the right to food at the Swiss organization Bread for All.
How should our economy grow? (tags)
If, on the other hand, the inequality distribution is reduced via increased spending on education, positive growth effects may result. In addition, too much inequality can destabilize the political system through social unrest, creating great uncertainty for investors, for example with regard to the protection of property rights.
For a borderless pragmatism. Outline of a Progressive Migration Policy (tags)
Immigration leads to stronger economic growth, which in turn creates jobs, Without migration, we would all be sitting in Olduvia in East Africa, as we were 100,000 years ago. About 3% of the world's population, about 215 million people, live in a state other than the one in which they were born. Migration has always existed.
Big Pharma profits at any price (tags)
"Almost all innovations today come from public research," says Marcia Angell. This often results in spin-offs that are then taken over by large pharmaceutical companies. "The pharmaceutical companies also locate themselves locally around the universities so that they can only cross the street to buy a patent.
New corona measures. We need "Sweden light," instead of "lockdown light" (tags)
COVID-19 is currently responsible for 1,2% of deaths in Sweden, but probably gets 99% of the attention. We need to maintain some perspective. Sebastian Rushworth, MD
Hunger is created (tags)
Today we produce and waste more food - although often produced in an unsustainable way - than is actually necessary to feed all of humanity. The fact that people are starving should therefore no longer exist. Hunger is created by other, full people.
When fear reigns (tags)
What was thought to be temporary has remained until today. Snowden warns of a similar process in the context of a real crisis. Surveillance is accepted The virus is dangerous. But the destruction of our rights is fatal because it is permanent, Snowden believes.
Blame the Covid-19 failure on your governments, not on China (tags)
The anti-Chinese narrative is based on false statements and serves to distract people from the failure of neoliberalism. Build hospitals and expand health care! Cut military spending in half!
LA Coronavirus COVID-19 Resources (tags)
Some links to local resources.
How Singapore has taken control of the virus and avoided panic (tags)
The city-state has learned a lot from the sars crisis and is not letting the coronavirus get it down. Not all recipes are transferable to other countries, but some are. And unusual measures are taken to ensure the cohesion of society.
Social Inequality and Universal Basic Income (tags)
The welfare state in Germany is not as efficient as in the Nordic countries. In Germany, more than twice as many persons are at risk of income poverty - particularly children and single parents. We have to make a decision: entrepreneurial paradise or a little less faith in the market.
Poor Despite Work (tags)
Vancouver B.C. has 26 community centers that give an identity to working and nonworking persons. Three hours of free computer use, libraries, gyms, theaters, game rooms and $4 casserole dinners are among the many amenities.
Pope Francis: Mauritius Reject "Idolatrous Economic Model" (tags)
Reject your "idolatrous economic model," were the words Pope Francis addressed Mauritius' political leaders with on Monday.
From Punitive Tariffs and Trade Conflicts to Economic War (tags)
Trump's punitive attacks on several trading partners, above all China, are changing suddenly into a destruction of the multilateral structures of the world market. With "America first," the president wants to restore the past hegemony. The wrecking ball is not the only model!
Fighting the Causes of Flight (tags)
The refugee debate is back but only the symptoms are in the spotlight. Refugees are fought instead of the causes of flight. The political failure is as dreadful as the forgetfulness of the media and politics. More than 70 million persons are fleeing from war, persecution and conflicts.
Poor Countries Lose Billions from Tax Avoidance, Reveal "Mauritius Leaks" (tags)
The International Consortium of Investigative Journalists (ICIJ) released evidence that laws in the country of Mauritius help corporations avoid taxes globally, including on the continent of Africa.
April 16 2019 Inmigration Rules do not Hold Water (tags)
people have a right to organize for sanctuary laws.There are reasons for this.What we do not like, is the funneling of tens of thousands of people who have been apprehended by border patrol and then delivered like a ups to these sanctuary zones.
Needed: a Jewish Honor Guard on US Campuses (tags)
Anti-Semitism Is Growing Worldwide, Jewish Students On College Campuses Are Getting Attacked, Jews Need To Learn To Defend Themselves. Jewish Knowledge means Jewish Pride, Jewish Self-Defense means Jewish Security!
Public Good or Private Wealth (tags)
The deep and growing chasm between the rich and the poor is not an accident. This gulf is the result of political decisions. Many rich countries have lowered or abolished property taxes. Governments reduce spending on education and health care.
On the Refugee Debate (tags)
The rulers want to divert from the real causes of the refugee wave and from their own misdeeds because the cause of the misery is an exploitation crisis of capital and the greed of mammoth world corporations. Only the prospect for a future in their country can stop refugees.
When Banana Ruled (documentary) (tags)
Documentary traces the sordid history of United Fruit/Chiquita.
Crisis Regulation in Global Capitalism (tags)
The globalization euphoria waned with the 1997 Asian crisis. Hundreds of billions were needed to bailout bankrupt banks. The US with its high solvent demand stabilized the world economy for a long time. The US could become indebted in its own currency.
Black Ops Which Involved Us In War (tags)
War profiteer networks, CIA and Pentagon hawks, and egodriven presidents have caused many immoral wars and illegal invasions of other countries.
Globalization: hope on the horizon despite Fake Liberals. (tags)
I consider the Fake Liberals as the greatest threat to global freedom and western civilization. They promote globalization, using racism as a political weapon labeling opposition as racist, fascist or white nationalists. Globalization has Corporate and right wing support creating the swamp which is being opposed.
The Beam in the White House (tags)
Iran is the "greatest supporter of terror," declared president Donald Trump. With that, he justified his peace-endangering cancellation of the nuclear agreement with Iran. Reducing foreign policy to scapegoating is Hitlerian.
Trade War and Zero Deficit Myth (tags)
State debts are often necessary and are not automatically bad economically. Investments are prerequisites of future prosperity. When private parties spend too little, the state must step into the breach.
Bioeconomical resilience (tags)
Bioeconomical resilience
The Shortwave Report 05/11/18 Listen Globally! (tags)
A weekly 30 minute review of international news and opinion, recorded from a shortwave radio and the internet. With times, frequencies, and websites for listening at home. 3 files- Highest quality broadcast, regular broadcast, and slow-modem streaming. Sputnik Radio, NHK Japan, Radio Havana Cuba, and Spanish National Radio.
Downshiftism (tags)
Downshiftism
The Struggle against Aggressive Tax Avoidance (tags)
Experts estimate a fifth of world wealth is hidden from the tax authorities of states and considerable parts of international corporate profits are not taxed. Google and Amazon parked billions in shadow financial centers. Tax avoidance makes a functioning state impossible
5 Questions on Tax Justice (tags)
Tax justice must be created and tax avoidance stopped to fight inequality and poverty. Apple paid 0.0005% tax on its profits in Ireland - merely 50 euros on a million euros profits. Corporations dodge their just contribution as part of society through tax avoidance.
Trump - Hard as Steel (tags)
There is outrage over the nationalist economic policy of "America First." The parallels to the 1929 system crisis are obvious. The unstable house of cards could now finally collapse. Credit-financed demand is created for a goods-producing system that suffocates in its own productivity.
What the State Can Do (tags)
The power of ideology is so great it easily creates a distorted perception of facts. Where money is spent is crucial, not the size of the public sector. The Internet, the touch screen, semi-conductors and the GPS were developed with state risk capital. Tax havens cause revenue shortfalls.
US Executions Violate The 14th, 8th, and 1st Amendments To The Constitution (tags)
Only 16% of US states still perform the barbarism of prisoner murder. Less than 20% of the world's countries do. The US is among the top 5 execution countries.
In Praise of Taxes by Nicola Liebert (tags)
Our tax system p[romotes private wealth and impoverishes the community. Even if no one pays them gladly, most are aware taxes int he interest of the community are sensible. "Taxes are what we pay for a civilized society" (Oliver Wendell Holmes, former Supreme Court justice).
The Unification Church is not a Church and is Seeking New People to Abuse (tags)
The Unification Church has been involved in brainwashing many people. It is a landowner and owner of hotels and newspapers in some countries and is very right wing. The leaders come from South Korea but many live in the U.S.A. and operate a small group Committee for
Urban Official Report that several Latin American Countries Sendimg Pro Dictator to U.S. (tags)
Noy much is known about this, but right wing Haitians were reported moving to Florida and other states.There are other countries who have right wing or left wing officials who have to qualify to move.
Left-Wing Secession (tags)
Secession is the withdrawal of a state or place from a larger organization such as a country. In 2018 there might be a vote in California asking people if they want to secede from the United States. There are irreconcilable differences between the left and the right. Secession is a way to settle those differences without one side ruling over the other side or annihilating the other side.
After the Hurricanes (tags)
Hurricane Harvey, followed quickly by Irma, left in its wake upended lives and enormous property damage, estimated by some at $150-180 billion. But the pummeling that America received also raise deep questions about its economic system and politics.
IN National and International News Teavana is Putting over 300 Stores Out of Business (tags)
Teavana owned by Starbucks is putting the "out of business " sign on over 300 locations in the USA. Starbucks office says it was due to low sales, profits and customer traffic.
What Happens to Immigration in Communist Countries? (tags)
A few countries have experienced communist led "revolutions" sincethe Russian upheaval in 1918. Cuba also went through a war as did China. None of these countries saw many people immigrate to their borders.
Time on our Side by Anna Coote (tags)
In her opening essay, Juliet Schor argues that countries in the rich world cannot continue to grow their economies if they are to reduce greenhouse gas emissions to sustainable levels. We must face up to a future with little or no economic growth.
Rigged reform - Oxfam Media Briefing, April 12, 2017 (tags)
US companies are dodging billions in taxes but the proposed reforms will make things worse.
KLM refuses seven passengers following Trump's muslim ban/KLM accomplice to discrimination (tags)
Airline company KLM is accomplice to discrimination, Islamophobia and deportation of refugees. Concerning discrimination and Islamophobia, it is one of the executioners of Trump's policy, by refusing passengers from the seven banned muslim countries. It's complicity with deportation of refugees consists of flying deported refugees for the Dutch government. Both are crimes and KLM's full responsibility, because they have a choice to refuse.
JEWISH GROUPS ACROSS THE SPECTRUM UNITE IN CONDEMNATION OF TRUMP'S REFUGEE BAN (tags)
By Yair Rosenberg January 30, 2017 • 9:11 AM Facebook Twitter Pinterest Google Plus Pinterest On Friday, President Donald Trump issued an executive order barring all immigrants and visa holders from entering the U.S. from seven Muslim-majority countries for 90 days. The order included green card holders and also banned all refugees for 120 days. Given the Jewish historical experience with being both refugees and subjected to religious loyalty tests, this directive unsurprisingly evoked a strong reaction from much of the American Jewish community. As so—contrary to some earlier claims—Jewish groups across the spectrum mobilized in opposition to the executive order, in a rare display of unity. The American Jewish Committee was one of the first out of the gate, offering a detailed rebuttal to the policy: "For more than 110 years, AJC has been a consistent voice for a fair, non-discriminatory, and generous U.S. refugee and immigration policy," AJC CEO David Harris said. "We are all related to those fortunate enough to have been admitted to this country – in my case, my mother, father, wife, and daughters-in-law, and we believe that other deserving individuals merit the same opportunities to be considered for permanent entry." Assuring security in the implementation of immigration laws and procedures is essential, but lost in the current discussion over the suspension of refugee admissions and the issuance of all visas to citizens of seven Muslim-majority countries this week – as in the rhetoric of the presidential campaign – are several facts: • Refugees from Syria, Iraq and other states in violent upheaval are already laboriously and intrusively vetted by U.S. immigration authorities, assisted by U.S. intelligence agencies, in cooperation with other nations' intelligence services. For those approved, it generally takes 18 to 24 months to gain U.S. admission; • Refugees from Central America are, almost exclusively, women and children who are at risk of violence, sexual assault, and even death if they are returned to their home countries. They deserve compassion and a fair hearing. • The terrorist threat attributed to refugees is a cruel and distracting fiction, especially when viewed against the actual incidence of mass violence committed with chilling frequency – in schools, churches, shopping malls and other venues – against Americans by Americans. In the 14 years ending in October 2015, a period in which 784,000 refugees were resettled in the United States, there were exactly three arrests for planning terrorist activities (none of which occurred); • Since the 1951 UN Refugee Convention, the United States and the international community as a whole have recognized an obligation to assist refugees. For America to close its eyes – and its borders – to even painstakingly vetted refugees contravenes that international treaty. President Trump, of course, is authorized to assert the sovereign right of the U.S. to assure the integrity of America's borders and the effective enforcement of the country's immigration and asylum laws… However, blanket suspensions of visas and refugee admission would suggest guilt by association – targeted primarily at Muslims fleeing violence and oppression. AJC regards such actions, contrary to international perceptions of a compassionate America and reinforcing anti-Muslim stereotypes, as both unjust and unwarranted. As the full scope of the executive order became clearer in airports across the country, the AJC later told reporters, "We are outraged by the new US refugee policy, its draconian face. This is not the American way." The Anti-Defamation League was similarly scathing. "History will look back on this order as a sad moment in American History – the time when the president turned his back on people fleeing for their lives," said ADL executive director Jonathan Greenblatt. "This will effectively shut America's doors to the most vulnerable people in the world who seek refuge from unspeakable pain and suffering." "For the Sunni family whose son languishes in prison in Iran because of his faith, for the former Army translator in Iraq who has been threatened because of his service,
Low Taxes is an Act of Desperation (tags)
A low business tax can be counter-productive if a deadbeat infrastructure arises through lower state revenues. Lowering business tax rates leads first to enormous budget holes. Employees will bear this burden at the end.
Secession (tags)
Secession is the departure of a group or community from a larger organization such as a state, province, country, or empire. Talk of secession has increased since Trump won the presidential election. Secession is the correct response to the election of Trump.
Ethics and the Profit System: Global Change of Values Necessary (tags)
Profit-making is not profit-maximizing. From the magic word profit-mongering is derived the absurd unequal distribution of wealth, cheating of consumers, the exploitation of workers and ultimately the abolition of democracy. An invisible hand makes business profit into common good
Botswana Minister supports highest level of protection for African lions (tags)
Tsekedi Khama appealed for countries to support CITES (Convention on International Trade in Endangered Species) Proposal 4
Negotiators Struggle for Agreement on Debt and Tax Issues at UN Development Summit (tags)
A United Nations global development summit begins its fourth day of difficult negotiations in Nairobi, Kenya.
Who Owns the Sky? The Climate in the Globalization Trap (tags)
The bad news is that there are more inconvenient truths. Climate change is present, not future. Globalization and protection of the atmosphere do not go together. Climate policy has a past. Economic profit dominance always has a priority over climate protection.
38 Arguments Against TTIP (tags)
TTIP (Transatlantic Trade & Investment Partnership) agreement continues an old thinking instead of everyone sharing int he profits of a few. The private arbitration courts enable corporations to sue states and could have a chilling effect on labor and environmental protection.
TTIP and CETA Must Be Prevented (tags)
Under TTIP, TPP and TiSA, corporations can sue states for lost profits. The German Judges Union calls additional protection of foreign investors "nonsense" that undermines democracy and the constitutional state.
Politics in the Age of Acceleration and Sharing (tags)
Why do international crises always come over us so suddenly, seemingly without any signal? This happens because we consistently ignore all the signs. We want to know everything in real time-and miss the big connections. Whoever sows weapons reaps refugees.
Hillary Clinton Has Helped Kill A Lot Of Animals (tags)
Hillary Clinton is aligned to the world's biggest animal killers
Brain Disorders Chemical and Structural Differences that manifest in late teens or 20's (tags)
Mental Illnesses have never been understood well even though they are a minority of health cases in all the worlds ethnic groups. Over the last 100 years physiology , anatomy, bio chem and other forms of gentle probing have illustrated worlds inside our head with MRI PET
Economic Multicutural Strangulation in US 2016 (tags)
We welcome immigrants. However certain subgroups are getting the lions share of visas both temporary and resident as well as the jobs. This economic cancer must be turned around before more locals are threatened.
Argentina Negotiates Debt Dispute with "Holdout" Hedge Funds (tags)
Argentina begins negotiations with groups that sued the country for more than $1 billion after it's 2001 default.
Pope Francis' Jubilee Year Begins (tags)
Pope Francis began the Catholic Church's "Jubilee Year of Mercy" at St. Peter's Square with a public mass and ceremony attended by 50,000 people. Pope Francis opened a "Holy Door" at St. Peter's Basilica to officially launch the year-long celebration with its themes of mercy and solidarity for the poor.
Paris Attacks - Bourgeois Terrorism (tags)
statement on Paris attacks from Internationalist Communist Perspective
Capitalism is the Problem (tags)
Prosperity in large part is based on the ecological and social exploitation of poor countries in Africa and Asia. Today 20% of humanity consume 80% of global resources. The poor are played off against each other while the richest hide billions in tax havens.
10 Shocking Realities of the TPP (tags)
Now we see why the US Trade Representative and President Obama wanted to keep the TPP secret for four years after it was ratified. It if had not been for a very aggressive fight against fast track trade authority in which hundreds of thousands of people participated, we would not see the text.
Time for a New Detente Policy (tags)
The past subordination under the US policy oriented to regime change and chaos in the Middle East must be ended. A diplomatic offensive to end the war in Syria on the path of negotiation with Russia is necessary. Since March 1999, NATO was expanded with a dozen countries.
A Tidbit about anti-Vietnam War [US] History Coffee with Soldiers (tags)
The Vietnam War was a very unneccessary, long and monumentally grotesque with over a million deaths in sevral countries and over 5 countries providing material, aid or troops for "our side".Anti-war demonstrators met with US troops at coffee houses near US bases in the US
Latest NYT Big Lie: Russia Responsible for Syrian Refugee "Surge" (tags)
Baghdad OK's Russia Bombing ISIS in Iraq (tags)
Economist: "TTIP is Doomed" (tags)
Resistance rages across the two parties. Two hegemons negotiate who are not accustomed to making compromises. The US and the EU are like two proud gorillas. The lawyer branch will put up a fierce resistance to losing that niche.
Putin Straight Talk (tags)
Tax Justice for Social Justice (tags)
Every year states lose billions in tax revenues through the aggressive tax avoidance practices of international corporations. These companies profit from public services whose financing is left to citizens. 1 trillion euros are lost every year from European public budgets.
Solidarity in Moscow Against US-Supported Terrorism (tags)
Putin: The Best Hope for World Peace (tags)
US and Russia: Geopolitical Opposites (tags)
US Imperial Wars Responsible for Refugee Crisis (tags)
Pentagon Updating War Plans Against Russia (tags)
Iceland Boycotts Israel (tags)
Obama Talks Peace, Wages Endless Wars of Aggression (tags)
Putin v. Obama on Syria (tags)
America Created and Supports the Islamic State (tags)
The Russians Are Coming, Again (tags)
Unwanted Refugees: EU Countries Block Borders (tags)
Something to chew on about people coming here [usa] (tags)
A handful of nations are the majority of legal visa recipients to USA cities where the majority of legal immigrants find jobs and housing. Other countries cannot be given legal visas.Money transfers are a major prop up for some countries with money transfers being the top 5
US Media Ignore US Responsibility for Refugee Crisis (tags)
US Phony War on ISIS (tags)
US-Led Western Nations Plan Escalated War on Syria (tags)
Israel's War on Refugees (tags)
False Reports of Russian Military Intervention in Syria (tags)
Vladimir Putin Straight Talk (tags)
Veto-Proof Senate Margin Achieved for Iran Nuclear Deal (tags)
908 Environmentalists Have Been Murdered in Brazil, Many Elsewhere As Well (tags)
Chico Mendes, Santos Rodrigues, Ohio nun Dorothy Stang are some of the nearly 1000 environmentalists murdered in Brazil by cattle ranchers, loggers, and other financial interests.
US/Israeli Attempted Color Revolution in Lebanon? (tags)
Sergey Lavrov's Message to Washington (tags)
Lavrov
Everything Western Media Report About Russia Is a Big Fat Lie (tags)
Iran to Get Upgraded Russian S-300 Missiles (tags)
fbi/cia influence over NATO et al (tags)
The fbi/cia seek to control all NATO countries, etc.
Pedagooogy 3000 World Tour for a New Education and Peace Alliance - Press Release (tags)
Conference to Help Teachers, Parents Learn New Tools to Educate Today's Children, Relieve Stress, Increase Participation, Lower School Violence
Known Cures For Cancer (tags)
Because of the trillions of dollars in the cancer industrial complex internationally (drug companies, hospitals, corporate, government and university research labs), many cancer cures are being suppressed.
Syria's Refugee Crisis Just Got Worse (tags)
Iran Nuclear Deal Aftermath: Greater Regional Destabilization? (tags)
Pope Francis Calls for Global Bankruptcy Process (tags)
Pope Francis called for an international bankruptcy process in a news conference as he left Latin America on Monday. According to the Associated Press, when asked about the Greek debt crisis, Francis stated, "if a company can declare bankruptcy, why can't a country do so and we go to the aid of others?" Francis offered further comments noting that too many countries are struggling with high debts and he suggested a United Nations bankruptcy proposal could be the solution.
Choice for Greeks: Rebel, Leave or Starve (tags)
Minors Abused in Adult US Prisons (tags)
Obama Declares Permanent War on Humanity (tags)
Ukraine: Rogue State Deadbeat (tags)
Trans-Pacific Partnership Guarantee (tags)
Former Science Ed of London Times: Smallpox Vaccine Triggered Aids (tags)
As Governor Brown, Dianne Feinstein, and Barbara Boxer have worked to violate the 1st amendment which guarantees religious freedom and to violate the health of children, it's time to examine again the data which indicates the WHO smallpox vaccine spread AIDS around the world.
Russia Vows Continued Support for Embattled Syria (tags)
BDS Effect on Israel's Economy (tags)
EU Bashes "Russian Propaganda" (tags)
TTIP: Trifling Economic Advantages at the Cost of Democracy and the Public Interest (tags)
TTIP represents a crisis for democracy and the constitutional state. Richard Trumka of the AFL-CIO describes the TTIP as "secret tribunals" where only "foreign investors" can sue. Frank Bsirske of the German Verdi service union said the TTIP is a "black box" transferring sovereignty.
Thousands in Britain Rally Against Austerity (tags)
Thousands in St. Petersburg Defy US Sanctions (tags)
Uncivilized Western Civilization (tags)
TTIP - Why the World Should Beware, 61 pp (tags)
Radical free trade policies did not begin with the birth of NAFTA, nor with the founding of the World Trade Organization (WTO) in 1995 – both had been in place in over 90 developing and transitional economies for over a decade through structural adjustment programes imposed by the World Bank and the IMF. (Walden Bello)
US Escalates Confrontation with Russia (tags)
Leaked TPP Trade Bill Healthcare Annex (tags)
The obscure legal system that lets corporations sue countries (tags)
Fast Track, TTIP, TPP and TISA should be scrapped! Richard Trumka of the AFL-CIO described the TTIP as "secret tribunals" where only "foreign investors" can sue. This article from The Guardian UK focuses on El Salvador. and Pacific Rim.
Obama Wants More Sanctions on Russia (tags)
G-7 Leaders Stay Tough on Russia (tags)
G-7 Summit: Russia Bashing Featured (tags)
Is Washington Planning Barbarossa II? (tags)
At G7 Summit Trade and Debt Policies Take Center Stage; Bishops Call on G7 to Enact Global (tags)
G7, Jubilee USA Network, Debt
Kiev Wants Foreign Troops Aiding Its War on Donbass (tags)
US-Dominated NATO Myths v. Russian Hard Truths (tags)
Jingoistic Media Scoundrels Bash China (tags)
Russian Travel Blacklist: Critics Ignore Their Own Policies (tags)
Devastated Nepal Daily Pays $600,000 on Debt (tags)
Nepal underwent another earthquake on May 12 amid recovery from the April 25 quake. It owes billions of dollars of debt to the World Bank, the Asian Development Bank, the International Monetary Fund and foreign lenders. Anti-poverty organization Jubilee USA Network states that debt relief would aid Nepal's recovery efforts.
US/Iranian Relations: Back to Square One? (tags)
Governments Negotiate UN Agreement on Global Development (tags)
The United Nations, International Monetary Fund (IMF), World Bank and finance ministries are meeting in New York to finalize an agreement for the Financing for Development Conference (FfD) to be held in Addis Ababa, Ethiopia, in July
World Bank and IMF Announce $1.1 Billion in Debt Relief for Chad (tags)
The International Monetary Fund (IMF) and the World Bank announced $1.1 billion in debt relief for Chad. The money comes through the IMF and World Bank's two major debt relief programs: the Heavily Indebted Poor Country Initiative (HIPC) and the Multilateral Debt Relief Initiative (MDRI). Chad is the 36th country to receive HIPC relief and the first since 2012. Chad is the fourth-least developed country in the world. More than half its population lives in poverty.
Aftershocks Pummel Highly Indebted Nepal: Poor Country Spends 217 Million Annually on Debt (tags)
As Nepal experiences aftershocks from Saturday's earthquake, it may find relief in the IMF's Catastrophe Containment and Relief Trust, a fund that cancels the debt of poor countries in crisis.
A Divided Society (tags)
Average wages in the US are lower than 30 years ago. Profits and income inequality explode, not investments. The austerity policy that has always worsened unemployment and economic growth is promoted as without alternative by German chancellor Merkel.
350 McDonalds Shut Down Since January 350 More Scheduled To Close (tags)
Hundreds of millions have moved away from animal flesh. McDonald's is one of many chains feeling the change.
World Bank Announces Package of Aid and Loans to Ebola-Affected Countries (tags)
The World Bank announces $650 million in new concessional loans and grants for Liberia, Guinea, and Sierra Leone at the start of the Spring IMF and World Bank Meetings.
Iran: The Leading Proponent for a Nuclear Weapons-Free World (tags)
Britain Prepares for Nonexistent Russian Attack (tags)
Israel Joins Obama's War on Yemen (tags)
Animal Agony, Union Busting, Deforestation, Human Disease At McDonald's (tags)
Unionbusting, obesity, strokes, heart attacks, cancer, animal agony, deforestation, rainforest destruction, energy waste, crime, litter associated with the Western wrold's biggest restaurant chain
Growth Criticism and Social-Ecological Transformation (tags)
Unlike a chair, an idea can be shared by a whole people. Words can be like spears breaking frozen souls. (Kafka) Education is the great transformer (John Kenneth Galbraith). Economics as taught today is like brainwashing. (Ulrich Thielemann) Incentives to investing in the real economy are vital after the phase of credit- and bubble prosperity.
US Saber-Rattling on Russia's Borders (tags)
Democracy for Sale: Stop Fast Track and TTIP! (tags)
The investment protection clause of the TTIP grants investors special protection regarding their so-called :"legitimate expectation." Businesses can sue when they feel restricted in their business activity by new laws in the public interest. Financial services are an important sector for US investors.
Kerry Threatens Russia (tags)
World Shock At USA,s Atrocities (tags)
See a summary of the global study of my reports on the complete corruption of the government of the USA.
Feinstein And Boxer Want To Make Vaccines More Mandatory (tags)
Senators Feinstein And Boxer Want To End The Right To Refuse Vaccines For Personal Belief
Monsanto's Land Grab in Ukraine (tags)
Pharma Cartel Poisons Kids with Thimerosal Ethylmercury Vaccines (tags)
Several vaccines still contain Thimerosal and the ethylmercury compound believed to be linked with higher rates of autism in children and some adults. Unfortunately the corrupt western medical establishment under pharma cartel influences remains opposed to banning Thimerosal outright as suggested by the UN and continues to distribute these toxic vaccines in lower income countries and also in the U.S.
US Official Insults Cubans (tags)
Lunatics Run the Washington Asylum (tags)
US/UK Special Relationship: Partners in High Crimes Against Humanity (tags)
New Geo;olitical/Financial Bombshells (tags)
PBS' And Frontline's Double Standard Re Putin (tags)
Soldiers, Children Of the Poor, Are Collateral Damage For The Rich
Public Funds Brought Smart Phones (tags)
All personal achievement would have been impossible without state investment in roads, schools, hospitals, airwaves, health and food safety. The state share is completely faded out in the conventional story about business history.
West?Russia Rapprochement? (tags)
Blessed Are the Peacemakers (tags)
A weekly 30 minute review of international news and opinion, recorded from a shortwave radio and the internet. With times, frequencies, and websites for listening at home. 3 files- Highest quality broadcast, regular broadcast, and slow-modem streaming. NHK World Radio Japan, Radio Havana Cuba, Sputnik Radio, and Radio Deutsche-Welle.
Gorbachev and Lavrov on US/Russian Relations (tags)
West Wants Putin Deposed (tags)
US and Israel: Partners in High Crimes Against Peace (tags)
US House Declares Cold War on Russia (tags)
Russia Bans GMOs (tags)
Kiev Plans Full-Scale War (tags)
African Mayors Call for Debt Cancellation for Ebola-Stricken West Africa (tags)
G20 Moves Forward on Ebola Debt Relief Plan.
US-Led NATO Risks Global War (tags)
US Brings Jubilee Ebola Debt Relief Request to G20 (tags)
Treasury Secretary Lew Brings Jubilee USA Request for Debt Relief for Ebola-Stricken Countries to G20. US Seeks to Move IMF to Cancel $100 Million in Debt for Sierra Leone, Liberia and Guinea.
Obama at APEC (tags)
Putin Prioritizes Strong Sino/Russian Ties (tags)
Sino/Russian Alliance Challenges West (tags)
China/Russia
Dueling Ideologies: America v. Russia and Venezuela (tags)
ani-war
Strengthening Sino-Russian Ties (tags)
Argentina's Christina Fernandez de Kirshner's UN Address (tags)
predatory capitalism
Argentina Faces US Contempt Hearing (tags)
US Federal Judge Thomas Griesa scheduled Argentina to appear for a contempt hearing on Monday, September 29. At issue is Argentina's failure to follow a court order to only continue to pay the 92% of bondholders who restructured after the country's 2001 default if Argentina pays a group of hold-out hedge funds. Argentina organized payment to restructured bondholders via an Argentine bank to avoid paying the hedge funds. The hedge funds, popularly known as "vulture funds," are asking the judge to hold Argentina in contempt and fine the South American country $50,000 per day.
Asian Socialism Conference in the Philippines (tags)
The Asian Socialism Conference will be held in Manila from November 28 to 29, 2014. The conference will be capped by the participation of guests and delegates to the November 30 rally in Manila to commemorate Bonifacio Day, a national hero who led the armed struggle in the revolution against Spain.
Arab countries in anti Isis coalition (tags)
Five Arab countries have joined the coalition against ISIS
OECD Takes Strong Stance on Corporate Tax Avoidance (tags)
The Organization for Economic Cooperation and Development (OECD) released a major proposal for curbing corporate tax avoidance yesterday. The OECD recommends international cooperation to prevent large corporations from avoiding taxes in the countries where they do business. The OECD plan comes amidst increased calls to crack down on corporate tax avoidance.
Lavrov on Syria, Ukraine and Terrorism (tags)
The BRICS: Challengers to the Global Status Quo (tags)
Can the BRICS wrest control of the global economy from the United States and Europe, or will their internal contradictions tear them apart?
Sanction Wars Delay (tags)
Anti-Russan Sanctions Wars Continue (tags)
Russia and America: Geopolitical Opposites (tags)
Anti-Democratic Straitjackets (tags)
The TTIP Transatlantic Trade and Investment Partnership (also called TAFTA) is an undemocratic corporate investment agreement that allows corporations to sue sovereign states for lost profits. This agreement negotiated in secret since 2013 does not allow states to sue.
The Middle Class Continues Shriveling (tags)
A century ago people said a progressive income tax would never happen. Today it is reality. The top tax rate in the US between 1930 and 1982 was 82 %. That was a very interesting experiment because very high incomes were covered.
The 'Asian Century' and ASEAN integration: contradictions and challenges (tags)
Talk given by Sonny Melencio, Chairperson of Partido Lakas ng Masa-Philippines (PLM), during the Socialist Alliance Conference in Sydney, Australia on June 7, 2014
IMF Paper: Corporate Tax Avoidance Hurts Global Economy and Poor Countries (tags)
The International Monetary Fund (IMF) released a staff paper noting that corporate tax avoidance negatively impacts all economies, but hurts developing countries the most. The IMF's release comes as the G20, the Organization for Economic Cooperation and Development and United Nations bodies seek vehicles to diminish corporate tax avoidance.
Competitiveness (tags)
A term becomes a myth; the myth becomes a religion. We are all conditioned by the misguided use of terms and connections so a way of looking at things seems absurd. The German economy acts like a bulldozer rather than a locomotive.
Propaganda & censorship alert: NATO countries' media coverage of the elections in Syria (tags)
The June 3 presidential election in Syria was a massive defeat for the NATO countries which have been collectively waging a genocidal war on Syria since 2011. Apart few exceptions, this momentous event was conspicuously absent from the front pages of US, British, French, German, Italian and Turkish newspapers. Those few that reported it on the front page did so in the habitual propaganda format. Meanwhile, few others went as far as publishing anti-Syria propaganda reports on their front page without making any reference to the presidential election.
Syrians Vote (tags)
The West Marches East, Part 1 (tags)
Russia can only be said to be an "aggressive" and "imperial" power so long as one accepts the unrelenting hypocrisy of U.S. and Western leaders. After all, it was not Russia that invaded and occupied Afghanistan and Iraq, killing millions.
Irresponsible Syria Bashing (tags)
Iranian Nuclear Roulette Continues (tags)
TTIP Attacks on Wages, the Social and the Environment (tags)
The belief that growth and prosperity for all people are promoted by free trade is as old as capitalism. This is often wonderfully represented in the model worlds of economists. But reality looks different. Specific particular interests are sold as the general interest.
Glow In The Dark: Consume Radioactive Fish Oil Or Fish (tags)
It is not just Fukushima that continues to pour radiation into the ocean. Many countries are illegally dumping their nuclear waste.
Interview with Jose Maria Sison (tags)
Interview with Jose Maria Sison, Founding Chairman, Communist Party of the Philippines by New Culture Magazine, Communist Reconstruction Union of Brazil
Israel's Secret Nuclear Program (tags)
Netanyahu and Rohani in Davos (tags)
Subverting Peace (tags)
Talking Peace, Waging Wa (tags)
Rogue State Accuses Victims of Incitement (tags)
Christianity's Savior in the Middle East (tags)
With one exception, Christians throughout the countries of the Middle East are at risk or on the run, their churches burned, their property expropriated, their personal safety in peril from thugs intent on beatings, rapes, and murders
Philippines' Foreign Debt Payments Dwarf Relief Aid After Typhoon Haiyan (tags)
More than a month after Typhoon Haiyan devastated the Philippines, the country has paid approximately $900 million in debt repayments—more than twice as much as it's received in pledged aid from countries around the world to support the recovery effort.
The WTO pushes through bad deal; Developed countries and TNCs are the big winners (tags)
The 9th Ministerial Conference of the World Trade Organization (WTO) pushed through a Bali Package in the final hours, extending the Conference to December 7, but at the cost of the developing countries, the poor and the hungry.
Trans Pacific Partnership Tutorial (tags)
A couple links about the Trans Pacific Partnership, a secret agreement being pushed, undemocratically, in Congress. Stop the TPP!
Ukraine Dodged a Bullet (tags)
Philippines: They Destroy, We Suffer (tags)
Capitalism is destroying the planet. Now we suffer. The devastating horror unleashed by monster-typhoon Yolanda (Haiyan) upon the eastern and central Philippines regions is unspeakable. As of writing, estimates of the number of casualties and actual damages are tentative because many areas remain isolated and communications, power, road and port systems are down.
TPP: NAFTA on Steroids (tags)
Super typhoon Haiyan is climate wake-up call (tags)
It seems these days that whenever Mother Nature wants to send an urgent message to humankind, it sends it via the Philippines. This year the messenger was Haiyan, known in the Philippines as Yolanda.
Best And Worst Country And Corporate Recyclers (tags)
As a UN environmental report urges vegan diet for the protection of Mother Earth, some countries are doing better than others.
Israel Avoids Human Rights Scrutiny (tags)
NSA Spying on World Leaders (tags)
shadowy media (tags)
If it walks like a duck .....
The World War against the Poor (tags)
More and more money capital concentrates in the hands of the rich which they let `work for itself.' Every day $1.5 trillion circulates around the globe and changes hands. Only two to three percent of that is needed to handle the business with goods and services. The rest is speculative excess.
Anti-Monsanto Activism (tags)
International Monetary Fund Reviews International Bankruptcy Process for Sovereign Countri (tags)
International Monetary Fund Reviews International Bankruptcy Process for Sovereign Countries
IMF Secures Windfall Gold Sales Profits for Concessional Lending to Low-Income Nations (tags)
On October 10, 2013, the International Monetary Fund announced that member countries have approved their commitment to fund the Poverty Reduction and Growth Trust (PRGT) for concessional lending. Financing comes from windfall profits made from the IMF selling part of their gold reserves that currently totals $3.4 billion. Currently, the IMF provides zero-interest financing to low-income nations through this fund.
Der Spiegel Interviews Assad (tags)
New Millennium Resource Wars (tags)
Rohani in New York (tags)
Assad Pledges Full Cooperation for Peace (tags)
Norwegian Audit Finds Loans to Developing Countries Don't Follow New Lending Standards (tags)
Ractopamine Banned in 160 Countries Yet US' FDA Allows Livestock Use (tags)
The butcher industry dominated FDA has allowed ractopamine to be administered to slaughterbound sheep, cows, pigs etc. though it is banned in 160 countries
EU Shamelessly Declares Hezbollah a Terrorist Organization (tags)
Snowden and Latin America Expose Washington (tags)
One of America's genuine heroes, Major General Smedley D Butler revealed in his memoirs the true, abhorrent nature of Washington's foreign policy. Butler had led countless military operations in Central America and the Caribbean as a US Marines Corp commander in the era of «gunboat diplomacy» during the early 1900s. Years after his retirement, he spoke out candidly and ruefully of his highly decorated military service in a book entitled War is a Racket. Here is how Butler characterized with unsparing words his service for country in 1935, five years before his death:
The Ideological Mantra "Competitiveness" (tags)
The policy that Merkel and Co. forced on the crisis countries has led to social catastrophes...The competition ideology has managed to establish the notion that generalized competition is the real human condition.
Europe Cravenly Colluding with U.S. in Criminality (tags)
It is hard to know which is the more outrageous: the US government's forced landing this week of the Bolivian President Evo Morales' official jet in Europe; or the European authorities' compliance with the Americans in their act of international piracy.
Targeting Foreign Leaders: Longstanding US Policy (tags)
Obama Threatens Humanity (tags)
Quelle Surprise: NSA Spies on Europe (tags)
State Department Debunks Iranian Latin American Terrorism Links (tags)
Israel popularity 2'nd from last of 197 nations - BBC poll (tags)
A new poll commissioned by the BBC was released this week, and it shows again that isra hell is ranked second from the bottom of 197 countries, is considered a danger to world peace, and is just about the world's most negatively viewed country, and that it's support globally continues to evaporate.
Slaughter of Proletarians in Bangladesh: Capitalism is the Assassin! (tags)
On April 24 "Rana Plaza" a 9-storey building in an industrial suburb of Dhaka, capital of Bangladesh (14 million), collapsed on about 3,500 workers (their number is not known precisely) working there. The building contained five textile factories, as well as a bank and many shops. The official present death toll is 1128, the number of injured is unknown. Most of the victims are women, who constitute 80 to 90% of the workforce in the Bangladeshi textile industry.
US-Style Free Trade in Good Hands with Michael Froman (tags)
corporatism
The Tiger And The Dragon: 28 Things China Can Learn From India (tags)
Abolition of capital punishment, nonviolent diet, human rights, protection of primates, noncensorship and other things China can learn from India
Neoliberalism as a Variety of Social Darwinism: Ten Million More Unemployed (tags)
Social cuts are a danger for democracy..When fear of social descent or collapse seizes the middle class, the danger grows that exclusion-ideologies may gain acceptance like racism, nationalism or social Darwinism..The economic future is given away on the altar of an ideology that raises location competition to the leitmotif of political-economic policy.
Venezuela's Maduro Inaugurated (tags)
Mirror images and ideological straitjackets on the path from Solidarity to sellout (tags)
A particularly harsh brand of capitalism was instituted via shock therapy in Poland, against the repeatedly expressed will of Poles. Unemployment, severe wage cuts and a growth in poverty were the results, just as with today's austerity programs.
Globalized Torture (tags)
Argentina Payment Plan Would Protect Poor Hedge Fund Holdout - NML Capital - has Until Apr (tags)
Cyprus Postmortems: Part II (tags)
Another Side of Israel: The Impact of Tikkun Olam (tags)
A side of Israel we never hear about
Escalating Syria's War (tags)
AIPAC Conference Promotes War (tags)
Child Poverty Massively Increases through EU Austerity Programs (tags)
Caritas warns "lost generations" are growing up in Greece, Spain, Italy, Portugal and Ireland. The Cairitas study blames the austerity programs for the impoverishment of children. A new prioritization and an even distribution of the burdens of the debt crisis are demanded.
Israel's One of the World's Biggest Land-Grabbers (tags)
Moscow to West: Hands Off Middle East/Africa (tags)
Down with French military intervention in Mali! (tags)
Commencing on January 11, the French government launched a military intervention in Mali under the pretext of "saving" the country against an alleged imminent terrorist threat and in "response" to a request for aid by the "interim president" of the country. But for several months the present "socialist" government, in continuity with the previous Sarkozy government, has multiplied its diplomatic maneuvering, its pressuring on the Government of Mali and the material preparation of military forces to organize a military attack against insurgents in northern Mali .
Western Sponsored Aleppo Mass Murder (tags)
Israelis Want Out (tags)
UNICEF's Christmas appeals, Sir Alex Ferguson and gaping contradictions (tags)
As the United Nations Security Council passes a resolution for the deployment of foreign troops in Mali and the United States announces its plan to deploy troops to 35 African countries in 2013, UNICEF and The Independent launch a campaign "to help child soldiers in Central African Republic".
Complex Chavez Recovery (tags)
USA Government has a Christmas Project in many poor countries (tags)
If Americans think they control the Brazilian oil they have to be ready to die for it because we are willing to kill Americans to defend what is ours.
Isle of Man Enacts Legislation to Outlaw Vulture Funds (tags)
This week, the Isle of Man passed legislation that would prevent vulture funds from exploiting the debts of some of the world's poorest countries.
On Typhoon Bhopa and the Doha Conference (tags)
Typhoon Bhopa (Philippine name Pablo) have resulted to a total number of 647 casualties, with another 1,842 people injured, and 780 missing.
Obama Plans African Wars (tags)
Corruption in Israel (tags)
In show of power, financiers impose will on Argentina's Navy (tags)
Billionaire speculator buys bonds of indebted countries at pennies on the dollar and then uses every means possible to get them to pay him for the full value, regardless of the cost to those countries' people.
"IMF is dead, a walking dead" - FDC Philippines (tags)
MANILA, Philippines – As the first female chief of the International Monetary Fund (IMF) visited the country for the first time and met with Pres. Benigno S. Aquino III Friday morning, debt watchdog Freedom from Debt Coalition staged a protest near the Malacañan Palace, in solidarity with the peoples of the Eurozone.
Haaretz on Obama's Reelection (tags)
Free Internet Book: Youth and Skills - Putting Education to Work, 450pp (tags)
Around one in six of the world's population are aged between 15 and 24. These young people are disproportionately concentrated in some of the poorest countries, where their numbers are still rising. more young people in 2030 than it had in 1980.
Boston University investigates causes of Chronic Kidney Disease (CKD) in Nicaragua (tags)
Since 2009, upon a request received from Nicaragua Sugar Estates Limited (NSEL) and Chichigalpa Association for Life (ASOCHIVIDA), a research team from the Boston University School of Public Health (BU) has been investigating an epidemic of chronic kidney disease (CKD) in Nicaragua of unknown cause.
Britain 'has worse record on internet freedom than Philippines' (tags)
Britain has a worse record on internet freedom than countries such as Estonia and the Philippines, a new report suggests.
Syria on the Boil (tags)
Arab Spring and the Israeli enemy (tags)
The Arab Spring showed the world that the Palestinians are happier and in better situation than their Arab brothers who fought to liberate them from the Israelis. Now, it is time to stop the hatred and wars and start to create better living conditions for the future Arab generations.
Pressure Building for More Middle East War (tags)
Pressure Builds for Full-Scale War on Syria (tags)
Another Road for Europe (tags)
Europe is in crisis because it has been hijacked by neoliberalism and finance. Along the road to another Europe, visions of change, protest and alternatives have to be woven into a common framework. Finance should be prevented from destroying the economy.
Long Knives Target Iran (tags)
US Media War on Islam (tags)
Irrational Draghi Exuberance (tags)
Stealth Corporate Coup d'Etat (tags)
Assad's fall will renew the Arab spring (tags)
As this phase of Syria's revolution closes, the debate about the future of this part of the world should also enter a new phase. Much of the debate about Syria, in the US at least, characterised the Syrian campaign as "liberal interventionism". So people argued about that, based on what they already thought about the notion. Though fed by some of the rhetoric, especially from the US government, that framing of the argument was misconceived. As I argued earlier this year, it was – and is – a mistake to see the Syrian case as a generic case for or against liberal interventionism.
Right to Rent and Social State as Prosperity Motor (tags)
The Scandinavian countries show that more social equality can be combined with more education and economic growth and less unemployment and poverty. Humanity is rich when wealth is shared.
Bona Fide Eurozone Crisis (tags)
Anti-Syrian Blame Game (tags)
Syrian False Flag Planned? (tags)
Anti-Iranian Hysteria (tags)
Full-Scale War in Syria (tags)
Volunteering in Thailand Having a good feeling of purpose and of accomplishment (tags)
Most of the program runs throughout the whole year so you have the option to plan well in advance and also talk to your family and friends about it.
Fidel Castro: Absolved by History! (tags)
Cuban revolutionary leader Fidel Castro has been absolved by history, writes Nicolas Mottas on the 59th anniversary of the Moncada barracks attack.
Organisations Launch Call for a Global Ban on Asbestos (tags)
National and International Epidemiology Organisations Launch Call for a Global Ban on Asbestos
Syria Roulette (tags)
A medieval present of austerity, a future of feudalism (tags)
Financiers can't tolerate losses flowing from their own greed and reckless gambling. Their solution is to have the state guarantee their stratospheric profits, bonuses and salaries. Governments can only do so through extracting money from their citizenry and facilitating the upward flow of wealth within corporate structures.
Opponents of Corrupt "Trade" Agreement Stage March, Rally July 7 (tags)
About 200 opponents of the Trans-Pacific Partnership (TPP), the so-called "trade" agreement that has been negotiated in secret between government and corporate representatives, marched on the Bayfront Hilton Hotel in San Diego July 7 during the most recent round of TPP talks. The march and rally finished a week of events protesting the secret negotiations of a corporate-welfare treaty under the guise of "free trade."
The high cost of private profit in health care (tags)
The extra money, extracted from people and from government, are massive subsidies for corporate profit, and gigantic payouts for executives and financiers.
Unresolved Iran Nuclear Talks (tags)
TPP: A Trade Deal from Hell (tags)
Community Activist Join Labor Against Phony "Trade" Deal (tags)
Progressive San Diego's official opposition to the proposed Trans-Pacific Partnership (TPP) — a NAFTA-like agreement seeking to subject the U.S. and a wide range of other countries to corporate dictatorship — began July 2 with a rally outside the Bayfront Hilton Hotel, where the TPP's negotiators are meeting. Speakers from labor, environmental, women's and Queer organizations explained what the TPP means to them and why it should be stopped.
Summit Fatigue (tags)
Non-Intervention: A Foreign Policy for the American Citizen (tags)
The policy of interventionism as practiced by CIA drone attacks with 17% civilian casualites is part of the reason why recruitment for Al Queda and the Taliban is on the increase.
Remembering Bob Chapman (tags)
Austerity has never worked (tags)
It's not just about the current economic environment. History shows that slashing budgets always leads to recession [...]
Charting a path out of impasse for Greece, and other countries (tags)
Should Greece remain within the eurozone or should it return to the drachma?
Economic Dictatorship (tags)
In the crisis, there was a paradigm based on the belief in unlimited economic growth on a planet with infinite resources. This paradigm identifies happiness with wealth, well-being with accumulation of material goods and progress with consumerism.
EU Austerity Madness (tags)
Looting the Seas (tags)
ACTA 2.0 (tags)
PLM Statement on increased US military presence in the Philippines (tags)
PLM Calls for an Independent Foreign Policy and a Non-Aligned ASEAN
Heading for War on Syria (tags)
No to US and Chinese expansionism (tags)
PHILIPPINES: Akbayan Party condemns plans for increased United States military presence in the country amid burgeoning Chinese military expansionism. We believe these acts by the two superpowers will inevitably destabilize the West Philippine Sea and Southeast Asian region.
Trans Pacific Free Trade Agreement Action Wed! (tags)
SOFITEL hotel in Beverly Hills, Wednesday. NOON to 2:30
Europe's Losing Game (tags)
Lee Siu Hin:Journey to My Home--Happy Spring Festival (AKA Chinese New Year)! (tags)
Greeting from Shanghai, China! Middle of my China-US solidarity working for building bi-national activist solidarity movement.
Forecasting Economic Decline (tags)
Dancing the Apocalypso with the Microbial Gestapo (tags)
The Biological Weapons Convention meets in Geneva amidst allegations that the US has violated the treaty
Heading for More Middle East War (tags)
Crisis Conditions Grip Eurozone (tags)
Netanyahu Rejects Peace (tags)
Grim Holiday Season Tidings (tags)
"The illegitimate debts must be cancelled" (tags)
A neoliberal tax policy followed for twenty years drastically lowered the taxes paid by private corporations on their profits, above all by big business. The debts are not an incomprehensible plague but the result of a conscious and completely unjust policy.
Europe in Disarray (tags)
Wrecking Europe to Fix It (tags)
A weekly 30 minute review of international news and opinion, recorded from a shortwave radio and the internet. With times, frequencies, and websites for listening at home. 3 files- HIGHEST QUALITY BROADCAST, regular broadcast, and slow-modem streaming. Free to rebroadcast. Spanish National Radio, Radio Deutsche-Welle, NHK World Radio Japan, China Radio International, Radio Havana Cuba, and the Voice of Russia.
The Crisis, Trust and Homeowners (tags)
Capitalism cannot function any more without credit-financed debt on account of its increasing productivity. The recession will come because the indebtedness-dynamic can hardly be maintained with which the capitalist zombie-economy is kept alive in its pseudo-life.
CELAC: A Washington-Controlled OAS Alternative (tags)
Central Bank Intervention: Much Ado About Nothing (tags)
At Risk Eurozone Sovereign Credit Ratings (tags)
Eurozone Doomsayer Got it Right (tags)
Deepening Debt Contagion (tags)
Dissing Europe's Flawed Bailout Scheme (tags)
Eurozone Bailout Deal (tags)
Halloween-inspired protest greets World Bank chief in Manila (tags)
Manila, Philippines – It was an early Halloween at the World Bank office here Thursday afternoon.
Obama's Imperial Arrogance (tags)
Planned Peacekeeper Occupation of Libya (tags)
9/11: Echoes of the Big Lie (tags)
These Days of Chaos (tags)
Strikes, riots, revolutions. This is the shape of the new Class War.
9/11: Who Really Benefited? (tags)
Forget so-called conspiracy theories. Instead look at reality. Dare ask yourselves who actually seems to have benefited from the 9-11 calamity. In light of the debt ceiling debates and continuous politics as usual of Washington D.C., it is time for the American people and individual states of this federation to look at a troubling set of facts. There seems to have been ?several? beneficiaries of 9-11 that don?t exactly fit the story line we were constantly fed by the propaganda machine and mainstream media as to how to connect the dots (which we were rhetorically asked to do).
Social and Economic Inequality in Israel (tags)
Urgent Appeal: Crackdown on Malaysian Socialists (tags)
Manila -- Around 70 members of Partido Lakas ng Masa (Party of the Laboring Masses - PLM) held a brief picket in front of the Malaysian Embassy this morning to protest the arrests and detention of 30 activists belonging to the Parti Sosialis Malaysia (PSM).
IMF Financial Terrorism (tags)
Greece must deny to pay an odious debt (tags)
In times of harsh neoliberal austerity and limited national sovereingty, the denial to pay an odious and illegitimate debt is a moral, political and social need. (by Nicolas Mottas/American Chronicle)
BBC: US and UK Imperial Tool (tags)
Obama in Brazil: hearts and minds operations (tags)
For the first time since John F. Kennedy, the USA has a charismatic leader to the point of being popular in other lands, becoming a reference for the electorates of different countries
No to imperialist military intervention in Libya! (tags)
Since Saturday, March 19, a US-led military coalition, including, besides English and French Forces, participation by Canadian, Italian, Spanish, and other countries, commenced the bombardment of military installations and concentrations of troops loyal to the Gaddafi government. The stated purpose of this military intervention which has been endorsed by the Security Council of the UN and the Arab League, is to get government forces off the offensive against the rebels, in order to avoid a "massacre of the civilian population".
ILPS joins the celebration of the Centennial of the First International Toiling Womens Day (tags)
We, the International League of Peoples' Struggle, express our most militant greetings to women all over the world and join you in the celebration of the centennial of the International Women's Day. The history and current trend of militant women's struggle must be highlighted in the midst of the global depression and chaos that pervade the various regions of the world today.
WikiLeaks at the Forefront of 21st-Century Journalism (tags)
"WikiLeaks is filling a void in traditional media, as the level of distrust of the mass media is now at record highs. A recent Gallup Poll found 57 percent of Americans do not trust the media, and a Pew Poll found a record low 29 percent trust the media. There is good reason for distrust. The New York Times helped start the Iraq War by publishing the false weapons of mass destruction story. It recently misled the public about a Blackwater employee arrested in Pakistan by hiding the fact that he worked for the CIA, while reporting that Obama said he was a diplomat. Even the way the Times and the Washington Post reported on WikiLeaks documents showed reason for distrust. WikiLeaks described Iranian long-range missiles that could hit European cities but also reported that Russian intelligence refuted the claim. The Times and the Post evidently made a decision to exaggerate Iranian capability and mislead readers by excluding the Russian intelligence report. The Times admits it provides WikiLeaks documents to the government in advance and excludes material at the request of the government."
"Globalize the Struggle, Globalize the Hope!" (tags)
No to market-based business deals proposed by the governments of industrialized countries and transnational corporations, the thousands of solutions to the climate crisis are in the hands of the people.
Egypt in flames (tags)
A powerful wave of anger among the poor and unemployed Arab masses shakes the young, voracious and brutal capitalism of the North African and Middle Eastern countries supported by the old and bloody capitalism in Europe and America. Anticipation of a social wave that can only be resolved in favor of the vast majority of the population by the emergence of the proletarian class.
America Replicating Japan (tags)
stagnation here like Japan
Does Money Buy Happiness? (tags)
A new article in Science Daily says money does not by happiness.
THE POLITICAL VIOLENCE IN THE USA JUST BEGAN (tags)
The America began to bleed.
Assange: Many Top Arab Officials Are CIA Spies (tags)
"These officials are spies for the US in their countries," reported Assange, adding that there was also information proving allegations that the US regularly sends "suspects" to some nations for torture."
Commission on the Measurement of Economic Performance and Social Progress (tags)
The Commission on the measurement of economic performance and social progress has been created at the beginning of 2008 on French government's initiative.
Survival calls on Obama to support the UN Declaration on indigenous rights (tags)
To mark tomorrow's opening of the White House Tribal Nations Conference, Survival International has called on President Obama to support the UN Declaration on the rights of indigenous peoples. The United States is the only country in the world that still opposes the Declaration.
Desperate to stop the "truth telling" (tags)
"The imperialists of the world are desperate to stop the "truth telling" of wikileaks. To do so they have thrown away the cover of respect of "freedom of speech" in particular and "support of democratic principals" in general. They are naked now in their hatred of democracy. Organize, resist, do all that you can to protect these true heros from attack. Most importantly, aid them in their efforts to reveal the truths that just may "set us free."
Manning Assange Bring Regime Change in Iran Foreign Minister Gone (tags)
The keyboard is mightier than the sword.
Seven Reasons Why the Economy Can't Recover (tags)
As the recession grinds on, politicians in most industrial countries have an incentive to make exaggerated claims about the supposed coming economic recovery. Some say the recession is over. Obama is in the group that claims we're on "the road to recovery," while other nations can only spot recovery "on the horizon." Below are seven important social phenomena that point to a more realistic economic and political outlook. 1) Central Banks are Dumbfounded. The usual tricks that U.S. and European central banks use to avoid recessions are long-exhausted. Interest rates cannot get any lower. And because cheap money wasn't working, the printing press was turned up a notch, into what the U.S. federal reserve calls quantitative easing -- injecting hundreds of billions of dollars into the world economy, escalating an emerging trade war.
Flotilla Support for Gaza (tags)
coming until siege ends
"Pre-Exposure Prophylaxis": Big Pharma's Latest AIDS Drug Marketing Strategy (tags)
The latest marketing strategy of the AIDS establishment and Big Pharma to sell toxic "anti-HIV" drugs at exorbitant prices is a sinister development called "pre-exposure prophylaxis." This is the idea of selling the drugs to HIV NEGATIVE Gay and Bisexual men engaged in so-called "high-risk" sexual lifestyles, supposedly to keep them from becoming infected. But it actually appears to be the industry looking for another market in the face of the recession, which is likely to reduce government and philanthropic funding for HIV meds in the developing world and thereby derail the pharmaceutical gravy train of AIDS. Fortunately, today's Gay men have got out of the intense fear of AIDS that ruled the Queer community in the 1980's and 1990's and are therefore unlikely to fall for the siren call of "pre-exposure prophylaxis."
World Geopolitics and The Battle for the Mediterranean (tags)
"If one were to live in a city where the only form of employment was a coal mine and there was no means to leave the city then one would have no choice but to work at the coal mine. Control of labour movement is a cornerstone to the socio-economic objectives of the U.S., the E.U., the World Bank, and a league of associated international financial institutions (IFIs). By rendering work forces immobile in any given geographic locality the rights of employment choice and occupational alternatives are removed and a new form of monopoly is established — a forced acceptance of work on whole pools of individuals. Rising fuel prices are also adding to the erosion of mobility rights.??The security agenda behind controlling movements is heavily tied to economic objectives, as are the international disease scares like avian influenza (bird flu) and the swine flue that lock up human movement. Control of mobility in the oceans and international waters of the world is also part of this objective. The internationally illegal Proliferation Security Initiative (PSI) was initiated by the U.S. government, with the support of the E.U., in 2003 as part of the "Global War on Terror." The Proliferation Security Initiative is presented as a means to prevent the proliferation of weapons of mass destruction (WMDs), however it can be applied to bring about a hold over global maritime mobility. The strategy is a threat to international movement on the high seas and maritime trade. There is good reason why it is illegal under international law and the 1982 U.N. Convention on the Law of the Sea.??Industrial De-location in the European Union and the Global Economic Crisis??This process of industrial de-location has already been underway in the E.U. for years, under which industries have been relocated to Eastern Europe and other global regions. Under this neo-liberal paradigm jobs and industries can gradually be removed from wealthier E.U. states to Southern Mediterranean nations, where cheap and immobile labour forces will be awaiting."
Israeli and American Rankings on Violence and Corruption (tags)
Rankings understate reality
The IMF Dictates More Than Ever (tags)
"Many developing countries that were independent in food production at the start of the 1980s import food necessary for their populations today. The subsidized agriculture of western countries has flooded the cou9ntries of the South. Haiti imports the rice that it once cultivated.."
Ecuador, Venezuela: Danger South of the Borde (tags)
"The reason there is a bizarre attempt to pretend that this coup attempt never happened, is to hope that people won't ask who might have benefited from such an action. A quick examination of the actions of President Correa sheds considerable light on that. In 2006, working with President Hugo Chávez of Venezuela, Correa moved to increase state control over oil production in the country.[9] In 2008, he announced that Ecuador would not pay several billions of its more than $10 billion foreign debt, calling it "illegitimate."[10] In 2009, he refused to renew the lease of the U.S. military airbase in Manta, saying that "the only way the US could keep their military base in Ecuador, is if Ecuador were allowed to have one of its own in Florida."[11] In 2009, he officially brought Ecuador into ALBA – the Bolivarian Alliance for the Americas led by Venezuela, Cuba and Bolivia. When a country increases state control, challenges illegitimate foreign debt, pushes the U.S. military out of the country and joins a regional alliance with Venezuela, Bolivia and Cuba – it is clear that the forces that would benefit from a coup would be: a) corporate interests inside Ecuador; b) International Financial Institutions; and c) the United States and its allies. There is another reason why the right-wing, corporate elite and the imperialist countries might have an incentive to minimize what happened September 30. There is now a shamefully long list of recent coup attempts in Latin America and the Caribbean – four of them against members of ALBA."
Chinese Currency War a Red Herring? (tags)
In the 1930s countries tried to gain advantages by exporting more. This is happening today.
Immanuel Wallerstein's Worldview (tags)
important views
The Migrant Condition (tags)
When I was recently named chairman of the Committee on Overseas Workers' Affairs of the House of Representatives of the Philippines, I was approached by members of the press for my views on migrant labor. I remembered a speech I delivered last year at the Global Forum for Migration and Development in Athens, Greece. The thrust of the speech was that migrants' rights have to be addressed on two fronts: ending the neoliberal policies that are responsible for creating poverty in their home countries, thus forcing them to emigrate, and demanding that they are given full rights in their host countries.
'Special Relationship With Israel' (tags)
"Contrary to the long-standing, official U.S. position, fewer than half of Americans show a readiness to defend Israel even against an unprovoked attack by a neighbor. Asked whether they would favor using U.S. troops in the event that Israel were attacked by a neighbor, only 47 percent say they would favor doing so, while 50 percent say they would oppose it …
Koran Burning versus Real Patriotism (tags)
Any fanatic can burn a book, or a flag, because it doesn't require rational explanation. But anyone with the brain of a troglodyte could appreciate there is complexity to the reality of most religions and belief systems. Once again high-minded religion plays its hand in the human affairs of politics, as Jones also sees the trumped up Muslim Mosque controversy, to be built close to ground zero, as seriously provocation. But real patriotism includes having a willing to seek out the truth.
Neoliberal Militarism: From Pax Americana to Pax Transatlantica (tags)
The bomb changed everything except the way we think, Albert Einstein lamented. Nuclear weapons are destabilizing, not stabilizing. Security cannot only be defined militarily.
2 U.S. JUDICIAL SYSTEMS~1 FOR PONZI TARP BUSH JR~KARL ROVE~TOM DeLAY & 1 for little people (tags)
TOM DELAY WAITED PATIENTLY 4 HIS TARP BAILOUT PONZI PALS 2 SPRING HIM ~ TOM WA$ HEARD YELLING... FREE AT LA$T..FREE AT LA$T...
KARL ROVE~GEORGE BUSH JR TARP BAILOUT PONZI SCHEME$=U.S.POLITICO ADS THIS ELECTION CYCLE ! (tags)
THEY STOLE BILLION$ IN U.S.TAXPAYERS TARP BAILOUT PONZI SCHEME MONIES & NOW CAN EVEN BE LEGAL TO BUY POLITICAL AD$ WITH THI$ STOLEN ~ BORROWED TAXPAYERS $...
Sri Lanka seeks deal to share intelligence on migrants (tags)
Pact would help stop shiploads of Tamils from heading for Canada, high commissioner says
What "Jobless Recovery" Means in Capitalism (tags)
The phrase "jobless recovery" entered the capitalist lexicon in 2002. But since then nobody want to be associated with these phrase. Although a lot of economist tried to use this term as a byword. Still noboy want to own up to this infamous phrase. The term "jobless recovery " had been associated with the terms of momentary statistical growth and corporate profits on books of accounts and in the stock market rather than in terms of production, employment and improvement of the people's living conditions.
GOD BLESS COUNTRIES LIKE PERU WHO ARE SHOWING THIS CRAZY WORLD THE FINE ART OF HEALING .. (tags)
WE HOPE AND PRAY PERU'S JUDICIAL ACTIONS CONCERNING THEIR EARLY RELEASE OF LORI BERENSON WILL ALSO INSPIRE OTHER UNIQUE COUNTRIES IN THE INTERNATIONAL COMMUNITY TO ALSO CONSIDER OFFERING SOME OF THEIR OWN LONG TIME CAPTIVE PRISONERS NEW OPPORTUNITIES TO POSSIBLE FREEDOM............................ ~TENS OF MILLIONS OF CURRENT AND FUTURE TOURISTS TRAVELING IN THE INTERNATIONAL WORLD HAVE TO RESPECT COUNTRIES AND THEIR PEOPLE LIKE PERU WHO ARE ABLE TO DEMONSTRATE TO THE REST OF US IN OUR HOST COUNTRIES THE GIFTS OF MERCY, FORGIVENESS & THE BEAUTY OF LIFE....
POOR TURN OUT IN OAV VOTES BLAMED TO DFA (tags)
PESANTE NEWS learned that the last elections showed the extent where the Filipinos have gone through-out the world. From Italy City to Tonga, and from Tanzania to Cook island, it seems there is no part of the world that the Filipino voter has not set foot on. The report of the joint congressional committee's canvassing of overseas absentee votes has underscored the extent of the Filipino diaspora, with certificates of canvass coming from cosmopolitan cities and little known territories all over the globe. The Philippine Congress used to manual elections, counted first the votes from these far-off lands were the first ones canvassed to determine the winning President and Vice President in the Philippines' first automated general elections.Most of the overseas Filipinos voted manually, either by mail or by filling out ballots at designated areas. They had one month to cast their votes.
HOW CAN IRAN & AMERICA HAVE PEACE WITHOUT DIRECT CONTACT WITH ONE ANOTHER ??? (tags)
~ THE INTERNATIONAL WORLD NEEDS TO ENCOURAGE IRAN & AMERICA TO MEET FACE 2 FACE FOR A PEACE POW~WOW ~
IRAN & AMERICA BOTH JUST MIGHT NEED 2 LOOK INTO EACH OTHERS EYES ??? (tags)
THIS NEW AND CHANGING WORLD REQUIRES ALL COUNTRIES TO BE ABLE TO COMMUNICATE WITH EACH OTHER DIRECTLY FACE TO FACE !
The IMF's mid-life crisis (tags)
The IMF has long been opposed to capital controls and punished countries. Now the IMF is wrestling with this issue. Exchange rate stabilization should be a priority in the context of unraveling economies. The IMF will be meeting in Toronto in June 2010.
BRAZIL IN 2025: DESTRUCTION or DEVELOPMENT (tags)
Brazil commemorates... but the black clouds didn't vanish.
Imprisoning Children for Life (tags)
only America and Israel do it
Greece: Blood and tears for the proletariat! (tags)
In order to resist the policy of blood and tears which the bourgeoisie inflicts on it, the proletariat has no other possibility than the resumption of the class struggle, starting with the rejection of anti-worker measures by the government and the calls for "national solidarity", and for sacrifices "to defend the fatherland and democracy".
May 8: Alliance of Anti-Fascists (tags)
The fascist plans for world domination were stopped by the Anti-Hitler coalition in a war that cost the lives of more than 50 million people. The International Federation of Resistance Fighters protests all attempts at establishing a reactionary view of history.
Human Rights Before Trade Rules (tags)
Bread for the World shows that a fundamentally different trade and economic policy oriented in human rights is possible and very necessary. Many rich countries subsidize their farmers so their foodstuffs can be produced below the real costs and sold cheaply on the world market.
BRIC STRUGGLES AGAINST THE ANGLO-AMERICAN FINANCIAL SPECULATION (tags)
Bric x USA-UK
MILITARY AGREEMENT BRAZIL-USA (tags)
Good American is American far away.
Abandoning Capitalism (tags)
The sociologist Dr. Atilio Boron, professor of political theory at the University of Buenos Aires, is an internationally known radical political author. UNESCO awarded him the "International Jose Marti prize" on July 17, 2009
Signatures Gathered for Human Rights Education (tags)
Youth for Human Rights Florida collected approximately 1,000 petition signatures over the weekend as part of a three-day human rights "Signathon", with the purpose to get the United Nations Universal Declaration of Human Rights taught in Florida classrooms of elementary, middle and senior high schools.
Freedom of Speech at Universities (tags)
Events this week in both the USA and the UK have once again shown the double standard for free speech at universities
MAHMOUD AHMADINEJAD IS THE GUY (tags)
American hegemony ended, happily.
EU-US BIAS- TUNA FISHING BANNED (tags)
PESANTE NEWS learned today from news sources in Davao City, Philippines that at least 150,000 people who depend on the tuna industry in General Santos City could end up jobless as a result of a two-year ban on tuna fishing in Western and Central Pacific, a fishing magnate said.
'Making Cents: Life Below the Bottom Rung' (tags)
A new series of oil paintings examining the daily existence of people making a living in the worst working conditions in the global economy.
Chávez: Netherlands and U.S. Planning Military Aggression Against Venezuela from Dutch Ant (tags)
"Today each year about 9.2 million children die before reaching their fifth year and 99.9 percent of these deaths occur in poorer countries. Infant mortality is 47 deaths per thousand live births, but is only 5 per thousand in rich countries. Life expectancy on the planet is 67 years, in rich countries it is 79, while in some poor nations is only 40 years. Additionally, there are 1.1 billion people without access to drinking water, 2.6 billion without sanitation services, over 800 million illiterate and 1.02 billion hungry people, that's the global scenario. Now the cause, what is the cause? Let's talk about the cause, let's not evade responsibilities, and let's not evade the depth of this problem. The cause, undoubtedly, I return to the theme of this whole disastrous panorama, is the destructive metabolic system of capital and its embodied model: Capitalism."
Filipinos join global action in Copenhagen Vs Climate Change (tags)
As world leaders negotiate on carbon emission limits, Filipino activists join tens of thousands of activists all around the world in mobilizations dubbed "People's Assembly to Reclaim Power and Avoid Climate Crisis." "We are in solidarity with the people who fight for climate justice. This big protest action is a testament to the people's demand for quick and meaningful action by national governments and international institutions like the United Nations," said Filipino climate activist Clemente Bautista of Philippine Climate Watch Alliance. "But as expected, leaders and government of capitalist countries like the United States, Japan, and European Union are evading responsibility and sabotaging the negotiation," he said in a statement e-mailed to media outfits.
SENATOR LIEBERMAN~PLEASE DO NOT SUPPORT THESE WEALTHY ELITE BILLIONAIRE INSURANCE DEMONS (tags)
PLEASE REMEMBER SENATOR THAT OUR 45 MILLION POOR AMERICANS ACROSS THE U.S. ARE RELYING ON YOUR CARING MIND AND SOUL TO HELP PULL THIS HEALTH~CARE LEGISLATION OFF. *** PLEASE SENATOR ~ DO NOT ALLOW ANOTHER 45,000 MIDDLE~CLASS AND WORKING POOR AMERICAN men, woman and children TO DIE THIS YEAR DUE TO THEIR LACK OF AVAILABLE FINANCES FOR THEIR FAMILIES HEALTH~CARE PROTECTION !!!
LOOK OUT CRAZY WORLD ~ PERU & THE U.S. JUST MIGHT KNOW HOW 2 BRING IN THE NEW YEAR ..... (tags)
DO GREAT CHRISTIAN SOCIETIES LIKE PERU & AMERICA REALLY KNOW HOW TO FORGIVE DURING THE CHRISTMAS HOLIDAY SEASONS ???
Coffin$10, Coffee $3. (tags)
The coffins will be in big demand
Youth Organize Worldwide Human Rights Walk (tags)
Youth from every corner of the globe will walk in our first ever "International Walk for Human Rights," on United Nations Human Rights Day, December 10, 2009.
The End of Poverty? in theatres and FREE TICKETS (tags)
Sold out shows during opening weekend in NYC, beating out at the box office every film playing at City Cinemas Village East including Disney's 'A Christmas Carol' starring Jim Carrey!
Poor Nations Offer Paid Sick Leave; Why Can't U.S. Workers Receive It? (tags)
This great essay by Kelber says that almost every single country out there supports new mothers and sick workers, except the United States of America. Poor countries are doing better by their workers than our country is. This is PATHETIC.
Trouble in U.S. Ranks and Not All Powerful Nations Bow to Israel (tags)
"Sixteen American soldiers killed themselves in October, an unusually high monthly toll that has senior military officers worried about the impact of sending tens of thousands of new troops into Afghanistan. WSJ's Yochi Dreazen reports. .The Army's top generals worry that surging tens of thousands more troops into Afghanistan could increase the strain felt by many military personnel after years of repeated deployments to Iraq and Afghanistan. The October suicide figures mean that at least 134 active-duty soldiers have taken their own lives so far this year, putting the Army on pace to break last year's record of 140 active-duty suicides. The number of Army suicides has risen 37% since 2006, and last year, the suicide rate surpassed that of the U.S. population for the first time. The health of ground combat forces is emerging as an element of the Obama administration's review of its Afghanistan strategy. Conditions there have deteriorated in recent months amid lingering political instability and a worsening Taliban-led insurgency. Gen. Stanley McChrystal, the top American commander in Kabul, wants more than 40,000 new troops, in addition to the 68,000 that will be in Afghanistan by year-end, and has warned that the U.S. faces possible "mission failure" unless it adopts a new strategy and quickly deploys significant reinforcements." and "Ever since the 22 day brutal and merciless bombardment of homes, schools, hospitals and mosques in Gaza by the Israeli criminal regime, many Israeli tourists are receiving the cold shoulder in countries they visit and some do not want them at all. It is no surprise that high level Israeli government and military officials do not dare set foot in certain countries for fear of being arrested but snubbing the regular Israeli citizen is a relative new phenomena."
Is Growth Only Possible Through Inequality? (tags)
Does capitalism make the rich richer and the poor poorer instead of creating prosperity for everyone as promised? Inequality declined considerably between 1930 and 1970 in western industrial states.
The Steady Stream of Lies about Venezuela or The Killers and Liars Prepare (tags)
"The fourth big lie is saved up to the end, and follows on that grand tradition of the great human rights abusers accusing others to deflect attention from themselves. Chavez is said to be "cultivating" regimes that are characterised by "rigged elections, media censorship, the criminalisation of dissent and leaders for life". Not a touch of irony, as the death squads in Colombia murder trade unionists and civil leaders unchecked, as the Obama Administration pussyfoots around the coup regime in Honduras (which deposed a Chavez ally), as election fraud and bloody war rages in Afghanistan and as the US launches missile attacks on Pakistan. It can be tedious to document such lies. They are so common and, no sooner is one lot done than the next day's lies appear – if not in The Economist then in the Washington Post, The Australian or The Times. These monopolies count on the vulnerable, who do not have alternate sources of information, who do not read history and are able to be swayed by crude and often racist agitation. In the bigger picture, this is a delegitimising process, organised by the private media monopolies which, in their hatred for the sort of popular democracy led by Chavez (as also in the past by Allende), are preparing the ground for coups, wars of intervention and conquest. It has been done before and – while these monopolies remain unchecked – it will be done again."
Ahmadinejad speech at UN 2009/Not anti-semitic, but just criticism on Israel (tags)
Not only the Western walkout at the UN 2009 Ahmadinejad speech is a part of a repeating show, consciously the Western countries make no distinction between Ahmadinejad's just criticism on Israel and his alleged ''anti-semitism'' Since the Western countries are directly and indirectly supporting the Israeli occupation, it is a shame and disgrace, that they walkout at this very speech, which brings into light the Israeli war-crimes and the Western implicite consent.
PESANTE-USA CALLS FOR ACTION AGAINST GLOBAL WARMING THAT CAUSES DISASTERS LIKE ONDOY (tags)
The Philippine Peasant Support Network (Pesante)-USA, an environmental and a peasant advocacy group based in the united States vehemently condemns the US –Arroyo regime for its criminal negligence and lack of preparedness against the natural calamities that regularly hit the Philippines. It was evident with Typhoon Ondoy (international code name Ketsana) that dumped 334 millimeters of rain in the first six hours, the highest ever recorded rainfall in the Metropolis. The previous record was 341 millimeters over a 24 hour period 42 years ago. Ondoy submerged up to 80% of the city, and covered areas that never experienced flooding before, stranding people on rooftops and bringing death and misery to Filipinos
THE NEW LATIN AMERICA FOR NOAM CHOMSKY (tags)
Interview of Noam Chomsky
Globalization and the New Political Movements (tags)
Attac, a global justice network in many European countries, demands closing off-shore tax havens and debt cancellation for the third world. The word globalization has a function as a threat. Be flexible because capital is flexible and the state can do nothing!
The H1N1 Swine Flu Pandemic: Manipulating the Data to Justify a Worldwide Public Health Em (tags)
"There was no attempt to improve the process of data collection in terms of lab. confirmation. In fact quite the opposite. Following the level 6 Pandemic announcement, both the WHO and the CDC decided that data collection of individual confirmed and probable cases was no longer necessary to ascertain the spread of swine flu. As of July 10, one month after the announcement of the level six pandemic, the WHO discontinued the collection of confirmed cases. It does not require member countries to send in figures pertaining to confirmed or probable cases. WHO will no longer issue the global tables showing the numbers of confirmed cases for all countries. However, as part of continued efforts to document the global spread of the H1N1 pandemic, regular updates will be provided describing the situation in the newly affected countries. WHO will continue to request that these countries report the first confirmed cases and, as far as feasible, provide weekly aggregated case numbers and descriptive epidemiology of the early cases. (WHO, Briefing note, 2009)."
VIDEO: The "Safe Haven" Myth (tags)
President Obama defended the expansion of the war by calling it a "war of necessity." Harvard Prof. Stephen Walt takes on Obama's justification for escalating the Afghanistan war.
August 15, 1945 : Victory Against Japan, Freedom for Asian Countries (tags)
The Alliance –Philippines (AJLPP) marked today, August 15, a day of freedom. Fifty four years ago, in August 15, 1945 several Asian countries won their freedom when Japan surrendered on this day. Although the formal surrender came later on September 2, 1945, Asian freedom fighters in Korea, Indo-China and Indonesia was freed and won their independence on August 15, 1945 from the clutches of the Japanese Imperialism.
Cracks Emerging in NAFTA (tags)
The once-solid North American Free Trade Act (NAFTA) is starting to show its age. The 1994 trade agreement that laid the foundation for the economic/political integration of North America is encountering serious internal ruptures, threatening future "progress." The problems are numerous: fights over trade, immigration, and military cooperation are all issues that Obama recently discussed in Mexico with his NAFTA partners, Mexico and Canada. The annual meeting that usually delivers plans for additional integration was instead used to remedy these heated issues, none of which were fully solved.
Real Health Care Reform - Universal Single Payer (tags)
real health care reform
For the Iranian workers, only one solution: the class struggle ! (tags)
The brutal repression of street demonstrations that have hit Tehran and the larger cities shows the determination of the Iranian government's Ahmadinejad not to tolerate any questioning of his political power. Accused of electoral fraud, the supporters of Ayatollah Khamenei and the outgoing president have responded with arrests, beatings and shootings; the police, the Pasdaran, and Basji volunteers headed by the Pasdaran are the instruments of the current repression.
The Face of Privatization (tags)
Under the present rule of neoliberal privatization, the all-embracing principle is conquest of the world transformed into rights of private property.. After destruction of the privatization myth, we need real democratization.
Mantory Swine Flu Vaccine Alert (tags)
swine flu and all other vaccines are toxic and dangerous
FDC to G8-funded World Bank: Cancel all illegitimate debts (tags)
MANILA, Philippines. Members of the Freedom from Debt Coalition (FDC) staged a protest action in front of the World Bank office here today to demand the total and unconditional cancellation of all debts it is claiming from the Philippines and developing nations. The protest action coincided with the opening of the Group of Eight Nations' (G8) 35th Summit in Italy which sets among its agenda the development of a comprehensive response to the global economic crisis.
A Cycle of New Battles (tags)
"We are moving into a world-historical situation where all the points of social, economic and political life will be reset..A revolutionary process is only possible when a radical reform gets going.. A radical redistribution from top to bottom must be fought for.."
Michael Hudson's "Super Imperialism:" The Economic Agenda of Imperial America (tags)
How imperial America is funded
The Great Divide-Iran and Leftists (tags)
A discussion of the great divergence of opinion among leftists about the situation in Iran
America's "Bases of Empire" (tags)
imperial America's global face
Obama's Outreach to Muslims - Empty Rhetoric, Same Old Policies (tags)
business as usual continues globally
Can the US Save the World? by Simon Johnson (tags)
Simon Johnson, a former chief economist at the IMF, teaches economics at MIT.
Water is a Human Right, not a Commodity (tags)
In Bolivia there was a bloody struggle over water. Indigenous persons led the water war in Cochabamba. Water is a human right and must be withdrawn from the logic of profit maximization.
Obama and the denial of genocide (tags)
Writer-activist David Boyajian's investigative articles and commentaries have appeared in Armenian media outlets in the U.S., Europe, Middle East, and Armenia and the Newton Tab and USA Armenian Life newspapers named him among their "Top 10 Newsmakers of 2007." So, when Barack Obama paid a visit to Turkey last month, it seemed like a good time to ask Boyajian for his take on the new president's approach to the issue of the Armenian genocide.
RICH AMERICA ~ S AGENDA IS ONE OF CONTINUED JUDICIAL INJUSTICES FOR POORER CITIZENS ??? (tags)
** IT IS A VERY SOMBER DAY IN AMERICAN HISTORY WHEN FREIGHTENED AND CONCERNED AMERICAN CITIZENS HAVE TO CONTINUE TO REACH OUT FOR HELP TO OTHERS WORLDWIDE CONCERNING OUR UNJUST AMERICAN JUDICIAL SYSTEM THAT IS IN DESPERATE NEED OF C*H*A*N*G*E....AND BILLION$$ IN A FEDERAL BAILOUT !!!
UC DAVIS LAW SCHOOL ~ TRIALS IN IRAN, PA & GEORGIA ALL NEED TO BE FAIR ??? (tags)
LAWYERS FOR POOR AMERICANS AGAIN PROPOSES THAT AMERICA FORMULATE A FEDERAL TASK FORCE DESIGNED TO INVESTIGATE EVERY DEATH ROW LEGAL CASE IN AMERICA !!! OUR INEPT AND UNDERFUNDED U.S. JUDICIAL SYSTEM CAN NO LONGER BE TRUSTED TO EXECUTE THE PROPER CRIMINAL !!!
Mothers Speak About War And Terror (tags)
Humankind's uncanny ability to make major advances in one civilization while simultaneously razing others beyond recognition is a strange facet of our nature. Yet most Americans are far removed from the constant dangers and terrors of living under a war. Few of us who did not serve in Vietnam or Iraq, for instance, have a clear understanding of the devastation wrought on those countries. The gap between our reality and theirs served as the driving force behind American author Susan Galleymore's venture into the war-torn countries of the Middle East. Her new book, "Long Time Passing: Mothers Speak About War & Terror," tells the stories of the repercussions of war on the mothers, families, communities, and cultures of Iraq, Israel, the West Bank, Lebanon, Syria, Afghanistan, and the United States.
For the Resumtion of the Class Struggle! (tags)
The economic crisis which has crashed down on society has already had and for a long time will have disastrous consequences for the lives of the masses, as every worker can see in his daily life or her work. The bourgeois economists themselves admit it: they do not know what the causes of the crisis are – satisfying themselves by blaming the greed of the bankers or excessive financial deregulation – and they do not know when it will end and give way to the long-awaited "economic recovery". The current crisis is the crisis of the capitalist mode of production, incapable of stopping itself, forced to produce always more commodities, at a certain point it comes up against the limits of the market.
LOYOLA MARYMOUNT UNIV LAW SCHOOL ~ REPRESSION OF BLACK AMERICANS 2009 ~ OK ??? (tags)
UC IRVINE SCHOOL OF LAW ~ REPRESSION OF BLACK AMERICANS 2009 ~ OK ??? (tags)
May Day March for Immigrants' Rights (tags)
There are several immigrants' rights marches and protests happening on May First. Ron Gochez, of the Southern California Immigration Coalition, talks about why people should come out and march.
PEPPERDINE UNIV LAW SCHOOL ~ REPRESSION OF BLACK AMERICANS 2009 ~ OK ??? (tags)
STANFORD UNIV LAW SCHOOL ~ REPRESSION OF BLACK AMERICANS 2009 ~ OK ??? (tags)
USC LAW SCHOOL STUDENTS TAUGHT ABOUT THIS U.S. REPRESSION OF THESE 2 BLACK AMERICANS ??? (tags)
DEAR MR.PRESIDENT MAHMOUD AHMADINEJAD OF IRAN,IN THE NAME OF WORLD PEACE AND GOOD WILL,WE HOPE YOU AND YOUR COUNTRY WILL MAKE AN OFFER OF A FUTURE PRISONER EXCHANGE WITH BLACK AMERICANS TROY DAVIS OF GEORGIA AND MUMIA ABU - JAMAL OF PA IN EXCHANGE FOR THE RELEASE OF OUR AMERICAN JOURNALIST ROXANA SABERI ???
PRESIDENT HUGO CHAVEZ ~ NEEDED FOR MIDDLE EAST PEACE ??? (tags)
Remembering the past to change the future – We must hold the torture state accountable (tags)
"Get the good old syringe boys and fill it to the brim We've caught another nigger and we'll operate on him Let someone take the handle who can work it with a vim Shouting the battle cry of freedom" A U.S. Army marching song composed during the Philippine War entitled "The Water Cure" to celebrate a version of waterboarding used on Filipinos "Those who cannot remember the past are condemned to repeat it." Poet and philosopher, George Santayana
Obama's Real Plan in Latin America (tags)
At first glance Obama seems to have softened U.S. policy toward Latin America, especially when compared to his predecessor. There has been no shortage of editorials praising Obama's conciliatory approach while comparing it to FDR's "Good Neighbor" Latin American policy. It's important to remember, however, that FDR's vision of being neighborly meant that the U.S. would merely stop direct military interventions in Latin America, while reserving the right to create and prop up dictators, arm and train unpopular regional militaries, promote economic dominance through free trade and bank loans, conspire with right-wing groups, etc…
IRAN RESCUES THESE TWO BLACK AMERICANS ??? (tags)
DEAR MR.PRESIDENT MAHOUD AHMADINEJAD OF IRAN,IN THE NAME OF WORLD PEACE AND GOOD WILL,WE HOPE YOU AND YOUR COUNTRY WILL MAKE AN OFFER OF A FUTURE PRISONER EXCHANGE WITH BLACK AMERICANS TROY DAVIS OF GEORGIA AND MUMIA ABU - JAMAL OF PA IN EXCHANGE FOR THE RELEASE OF OUR AMERICAN JOURNALIST ROXANA SABERI ???
Protest sweeps Europe (tags)
France Jan 29, 2009. Public and private workers mount a one-day strike with mass support. Schools, courts and transportation shut down in several cities. This article is from: Freedom Socialist Vol30 No2 April-May 2009.
The Misery of the WTO: For a Reinvention of World Trade (tags)
The crisis of the WTO has two causes: a lack of trust and an ideological prejudice. A lack of trust exists because the countries of the North judge trade with two standards: free trade for the South and protectionism for their own economies.
Pulitzer Prize-winning investigative journalist Seymour Hersh claims U.S. operating secret (tags)
Seymour Hersh created a stir last month when he said the Bush administration ran an executive assassination ring that reported directly to Vice President Dick Cheney.
Calories on the Table, not in the Tank (tags)
For decades nearly all countries of the third world had to follow the "Washington Consensus"-a strategy of opening to the world market.. Wheat and corn land in the tank, not on the table. All of our dependence on the car appears here most perversely.
Earth Without Stocks (tags)
This struggle demands more than setting up wind turbines, buying more economical cars and rehabilitating houses. An historically unique pact is necessary between North and South, between industrial countries and threshold and developing countries.
China to take over IMF (tags)
Obama sent to beg China not to pull the bond rug out from underneath it.
Guatemala and other central American countries becoming more violent than Mexico (tags)
Two armed U.S. DEA helicopters intercepted and fired on a smugglers operating from a light aircraft in far eastern Honduras, causing the aircraft to crash killing the lone pilot
The Pork That Dares Not be Called Pork (tags)
The diversionary tactics of calling things pork and the real pork
occupation 101. Please watch. (tags)
Occupation 101 Please circulate this link to friends and Blog Posts even to other cities and countries. http://video.google.com/videoplay?docid=-2451908450811690589&hl=en
Redefining the War on Terror (tags)
"The West is no longer sure of what it's fighting for On 19-20 February NATO countries held a summit in Warsaw, Poland. It was the first such summit of the new US Administration headed by Barack Obama. In many ways, the site of the summit was fitting symbolically.."
LA Musician sings at Dublin Protest Against U2's Bono (tags)
Former LA'er Paul O'Toole plays the role of Bono™ in begging for another Corporate Tax Break
Venezuela: Why Unlimited Re-election is Bad for the Revolution (tags)
A supporter of Hugo Chavez' anti-imperialist efforts explains why he is nevertheless opposed to Venezuela's recent constitutional amendment abolishing term limits.
CAPITALIST CATASTROPHE AND PROLETARIAN STRUGGLES (tags)
Hi fellows, here is the very first draft translation in English of the first pages of our last review in French "Communisme" N°60. Sorry for the language but improvements will come later. Meanwhile, have a good reading and we are waiting for your comments and critics...
The Limits of the US as Teacher (tags)
"Ivan average consumer supports the new self-confident policy of the Kremlin and hardly worries about democracy and human rights. The West must free itself from the illusion its own values can be forced on countries. The change must come from within."
Global Social Policy (tags)
The problem is not understanding but ideology. Market fundamentalists insist government jobs are not jobs. Those working government jobs under FDR are sometimes classified as unemployed to belittle FDR's great achievement in reducing unemployment from 25% to 10%.
Will Obama Start the Next Trade War? (tags)
While campaigning to be president, Obama gave the general impression that he would act "cooperatively" to solve international problems, moving away from the Bush strategy of irresponsible "unilateralism." But Obama took an aggressive stance towards China even on the campaign trail, including the accusations of "currency manipulation," "violating intellectual property rights," and "devaluing their goods." Of course, the average American cares nothing about these types of crimes, nor should they.
Is revolution brewing in America? (tags)
As things deteriorate around the globe economically Americans are starting to realize what a bad place they are finding themselves in and that Americans for the first time since the great depression are losing jobs at an alarming rate, losing their homes, savings and the way of life they have grown accustom to
Terror Detention Facility (tags)
Terror Detention Facility
The Mexican Revolution is starting (tags)
Offe's Paradox (tags)
Klaus Offe, a left thinker, once proposed Offe's paradox, the idea that"while capitalism cannot coexist with, neither can it exist without the welfare state." Canada has done better than some countries in protecting social welfare. The welfare state is not dead; what is dead is econ security.
Filipino Workers to heads of G20: Creating a better world order is no more your business (tags)
Relegating the world's future to the hands of same leaders who created the current global economic crisis is a recipe to further disaster, according to Partido ng Manggagawa (Workers Party), a militant labor party in the country which joined hundreds of other protesters in a march to the US embassy this morning in time for the G20 Summit in Washington DC.
PHILIPPINES: Labor export policy is anti-women, anti-development (tags)
MANILA, Philippines – The country should not take pride in being the model in terms of labor migration, according to women's groups who are engaging the Second Global Forum on Migration on Development (GFMD).
Part 2- World Capitalist Crisis (tags)
We expect more of the use of cheap labour in the less developed countries like China and India in order to produce cheap raw materials and semi-manufactured consumer products for the imperialist countries and thereby maximize imperialist profits. In fact the US capitalist are hell bent on transferring high paying industries to the third world and even already cheap service industries like call centers and car assemblies and other production lines in a vain attempt to cut jobs, paying insurances and other workers benefits that the people are demanding in the face of the capitalist made crisis.
The Law of the Jungle (tags)
The current crisis of the developed capitalist system is taking place when the empire is about to change leadership in the elections to be held in a few days; it was all that was left to see.
Russia Resurgent (tags)
A history of US intervention in Russia and Russia's resurgence
Lee Siu Hin: Journey to My Home-China (Part 4) In Memory of Wenchuan Earthquake (tags)
It's been over three months since the May 12th Wenchuan earthquake--the largest in modern Chinese history for over 80 years that killing nearly 70 thousand people, 20 thousand missing and affect another tens of millions of people.
De-Acceleration or Globalization 2.0 (tags)
"The wondrous global division of labor came about in which threshold countries lend money to the US to buy Chinese or Indian products..Must cut-flowers from Africa be promoted in Europe?..With de-acceleration, globalization will catapult fewer jobs around the globe.
Stillbirth of the WTO (tags)
"The negotiations in the NTO changed into a battle of the developed countries for their mammoth corporations over the market opening of developing countries. Access to education, health care, water-and energy-supply and telecommunciations should be human rights."
Towards Women's Liberation (tags)
The cause of women's liberation and the rights of women against sexual discrimination, exploitation and violence
Hugo Chavez Spearheads the South American Revolution (tags)
How Venezuela is leading the revolutionary integration of South America
Rude Awakening in Mexico (tags)
US businesses invested $120 billion in Mexico between 1994 and 2006. Only 80,000 jobs were created per year while 730,000 Mexicans pressed on the labor market every year. While migration climbed 95% from 1980 to 1994, it soared an incredible 452% from 1994 to 2006.
Hunger, The Silent Tsunami (tags)
Rice, the main food for nearly half of the world's population, costs twice as much as in December. The economic policy of the World Bank forced on countries on account of neocolonial trade relations is responsible for the unproductivity, not only the traditional small farmer system.
"Stop the Biofuel-Madness!" (tags)
"The high oil price and the food crisis threaten the existence of over a hundred million people." (Walden Bello) "People in poor countries must starve so the rich can continue driving their cars." Growing plants for biofuels is a crime against humanity.
End G8 Domination! (tags)
Joint Statement on the 34th G8 Summit in Tōyako, Hokkaidō, Japan
Alternatives to Free Trade: Fair Trade and Beyond (tags)
The global debate around free-trade and its consequences has evolved tremendously in recent years, from tiny circles of leftist critics into a broad international protest movement. Although the movement began to bloom in response to the policies of the World Trade Organization (WTO), the biggest demonstrations have been in response to the now-popular "bi-lateral" free-trade agreements that economically powerful countries sign with poorer nations. Once one has become conscious of the problems created by free-trade agreements, whether they are international or regional, an immediate task presents itself: finding a feasible alternative. Yes, the trade policy advocated by most big business politicians is "free-trade," and yes, this policy has had devastating consequences for working and poor people worldwide, while filling the already-full bank accounts of the rich. But the issue of "free-trade" alone isn't sufficient to fully explain the vast social problems that plague so many countries.
The United States, Europe and Human Rights (tags)
The discredited way in which the European Union suspended its sanctions on Cuba on June 19 has been reported in 16 international press dispatches… Such hypocrisy is made all the more evident by the brutal European measure to expel illegal immigrants from Latin American countries.
Latin America rejects new EU law on illegal immigrants (tags)
Under the new law, all illegal immigrants living in the EU member states will have to leave the bloc within a period of seven to 30 days.
"Oil Crisis Changes Globalization" (tags)
In the poorest hundred countries, the high oil price is intensifying hunger and poverty. "These people stand at the edge of a possible disaster," Jeremy Rifkin says. The high prices for oil, gasoline and gas change the rules of globalization.
The Seven Energy-Related World Crises (tags)
The market is not total, absolute and self-healing but a tool helpful after we decide what kind of society we want. The state has a social nature and can't be only a power and security state or treasure chest for the super-rich and special interests.
Disturbing 2008 Global Peace Index Report (tags)
GPI unfairly targets Venezuela.
"A Crime against Humanity" (tags)
When you fill the tank of an ethanol car with 50 liters, you burn over 700 pounds of corn. A Zambian child could live a whole year from that. More regulation instead of more market access is vital for developing countries. A rebellion of conscience is needed.
World Bank and IMF Policies Caused Food Crisis (tags)
The IMF changed its economic direction and took a market radical neoliberal economic course at the beginning of the 1980s with the rise of monetarism. Hunger inthe world can only be overcome with long-term sustainable strategies, not with the short-term horizon of Wall Street.
Hunger in Abundance (tags)
The price explosion has nothng to do with a shortage in grain. After the quasi-collapse of the US financial markets, international ivnestors floated a considerable part of their funds in raw material- and food branches.
Mexico Exports Corn and Impoverished Farmers (tags)
Food that Mexico could have easily produced itself must be purchased for ten billion dollars every year from the US. Today Mexico can offer ruined farmers as its most important export article. Prices for corn, wheat and rice soared 40-50 percent within eight months.
Israel STILL Beating Iran War Dead Horse (tags)
Shades of Iraq ...
World Food Crisis Threatens Rich Nations (That's Us), Too (tags)
Unless we act fast for a global consensus on the price spiral, the social unrest induced by food prices in several countries will conflagrate into a price contagion, leaving no country-developed or otherwise-unscathed.
Pope arrives (tags)
And was put in his place
Bush v. Chavez - An Update (tags)
Continued Bush administration efforts to destabilize Venezuela.
The Debt Boomerang (tags)
As long as the policies of the rich North represent a mixture of cruel carrot-and-stick maneuvers, coupled with basic contempt for the South, we can expect more lethal North-South tensions, more powerful boomerangs hurled back at us.
Democracy and the WTO: Alternatives to Free Trade (tags)
Trade policies that benefit multinational corporations are undemocratic and promote exploding inequality. "The negotiations are carried out on a supra-national plane where democracy or civil society interests fall by the wayside."
US court nixes RP 'privilege' over real estate tax debts (tags)
In a legal debacle for Manila, a United States federal court judge has ruled that the Philippines cannot invoke diplomatic privilege so that it would not be compelled to pay $19 million worth of unpaid real estate taxes to the New York City government. The New York City government scored a victory over real estate tax-delinquent states, including the Philippines, India and Mongolia, after US District Judge Jed Rakoff, in a written opinion, ruled in favor of the city. Rakoff's ruling will force the three countries, which are renting out offices and business establishments housed in diplomatic buildings in New York, to settle their financial obligations to the city's government. Reports said India owed the city $39.4 million including taxes, while Mongolia has to pay $4.2 million.
United States-Mexican border is experiencing an alarming rise in drug-human and Terrorist (tags)
In addition to Federal agents, State, sheriff, and local police dept's are expected to help patrol the border areas.
Lynn Rozar Wants Straight Answers from Socialist (tags)
In a letter dated December 31, 2007, Lynn Rozar and his wife, from Greeneville, Tennessee, want to know my position on some issues concerning the average person: WITH JUST SOME STRAIGHT SIMPLE ANSWERS FOR A CHANGE.
The US Will Experience Its Big Surprise (tags)
The dollar devaluation has to do with many things, above all with the double deficit. The US must become indebted more and more. China and Dubai could go on a shopping spree and buy oil companies. The US must earn euros to pay for the oil to be imported.
Forget oil, the new global crisis is food (tags)
A new crisis is emerging, a global food catastrophe that will reach further and be more crippling than anything the world has ever seen. The credit crunch and the reverberations of soaring oil prices around the world will pale in comparison to what is about to transpire, Donald Coxe, global portfolio strategist at BMO Financial Group said at the Empire Club's 14th annual investment outlook in Toronto on Thursday.
The Cart in the Speculative Mud (tags)
Liberalizing markets leads to instability, not growth. Developing and thresholdc countries suffer grievously through US speculation. The New Year is a time to abandon myths of American exceptionalism and corporate beneficence, mending our own pockets and footprints.
Bali: A Missed Opportunity (tags)
The gap between the urgent threat of global warming and the collective will to do something about it has never been greater. The recently concluded Conference on Climate Change in Bali was a grand opportunity to act. Instead, it was another missed opportunity. Unfortunately, the United States played a very negative role, standing in the way of consensus at every turn. And unfortunately, the rest of the world thought that seducing the US into a new agreement on climate action was top priority, resulting in a Bali Roadmap that was very sketchy.
U.S. isolated on the Human Rights Issue in CUBA (tags)
The recent Cuban decision to sign the International Pact on Economic, Social and Cultural Rights and the Civil and Political Rights is a clear example of the island's conduct when it is not subjected to external pressures...
Letter from US Artists and Scholars in support of Cultural Relations with CUBA (tags)
More than 500 U.S. artists and academics demand an end to the blockade...
Ameripol: a Latin America Regional Police Force, is Being Formed (tags)
http://miami.indymedia.org/news/2007/12/9917.php
Reviewing James Petras and Henry Veltmeyer's "Multinationals on Trial" (tags)
How corporate giants plunder developing nations.
Cuba's Solidarity Miracle (tags)
Cuban cooperation with Third World moving ahead.
Global Capital Flows to the Rich (tags)
The global imbalance seems more dangerous for all other countries than for the US. A fundamental reform of the international monetary system is overdue. The US dollar is no longer the stable anchor of the world monetary system.
U.S. dollar - a "worthless piece of paper." President Mahmoud Ahmadinejad (tags)
And he is right. A $100 bill is worth the paper it's printed on and backed by nothing else. Up till 1933 the U.S. was backed in gold. You could exchange a $20 bill for an ounce of gold. And up till the 1960's the dollar was backed up by silver. You could go to the mint and trade paper dollars for silver. But now the dollar is backed by NOTHING and a dollar is worth nothing more then the paper it is printed on.
UN First Committee Passes DU Resolution in Landslide Vote (tags)
US University report: "In a group of 251 soldiers in one study group in Mississippi, all of whom had normally birthed babies prior to their participation in either of the two (Persian) Gulf Wars, sixty-seven percent of their post-war offspring were born with severe deformities."
Why World War 3 is VERY possible (tags)
Following on the heels of President George W. Bush's warning last week that those countries "interested in avoiding World War III" should align themselves with Washington's escalating threats against Iran, a series of unfolding developments point to the danger of armed violence engulfing a broad swath of the Middle East and Central Asia and, indeed, posing the threat of a new world war. Six years after the US invasion of Afghanistan and four-and-a-half years after the invasion of Iraq, the continuation and deepening of the conflicts in both of these countries is setting into motion a political chain reaction of incalculable dimensions.
Retirement under capitalism (tags)
Sometimes small statistics can have huge implications. The U.S. Bureau of Labor statistics recently reported that the age at which U.S. workers retire has risen for the first time in 100 years. According to the report, in the mid-1980s, 18 percent of people in their late 60s still had jobs; today, it is 29 percent. Millions of U.S. workers can no longer look forward to the "Golden Years" of guaranteed Social Security, Medicare health insurance and employer pensions. One in four baby boomers surveyed said they would likely never be able to retire – in other words, they will "work until they drop". For those that do want to retire, the age at which they can do so is also being systematically raised. Workers born in 1960 or later must wait until age 67, rather than 65, to get their full Social Security benefits. There is talk of raising it further to 70. The mainstream media took scant notice of this, and for good reason; it's a huge indicator of deeper processes that are destroying the standard of living of workers in the United States and internationally. A quick look into why workers are delaying retirement – if they have the "luxury" of retiring at all – points a condemning finger at the current state of international capitalism, while at the same time warning younger workers of the even worse conditions they'll certainly face under the auspices of the capitalist system.
Reviewing James Petras' "Rulers and Ruled in the US Empire" (tags)
An in-depth review of James Petras' important new book.
Israel seeks exemption from atomic rules (tags)
Israel is looking to a U.S.-India nuclear deal to expand its own ties to suppliers, quietly lobbying for an exemption to non-proliferation rules so it can legally import atomic material, according to documents made available Tuesday to The Associated Press.
CUBA Calls for End to Insensitivity... (tags)
Cuban Foreign Minister Felipe Pérez Roque spoke on Monday, September 24, at a UN high-level event on climate change in New York, United States. We offers here the full text of the Cuban diplomat's speech.
2nd Screening of Videoactivism and Independent Documentaries, Caracas January 2008 (tags)
* The collective editors of the Venezuelan journal El Libertario and the Organizacion Nelson Garrido, have issued a call out for filmmakers and video activists from all over the world to present their recent autonomous works concerning active social movements in struggle. This will be held in Caracas, Venezuela, between the 21st and the 28th of January 2008.
The Future of Work-driven Society (tags)
When less and less time needs to be spent working for a living in Western industrialised countries, this partial freedom of gainful occupation is increasingly becoming a problem for those dismissed. The vision of working less and living longer is a utopia that is also realistic.
Ron Paul: Thomas Jefferson Incarnate [on Imperialism and "blowback"] (tags)
on Imperialism and "blowback"
Vatican seeks to become carbon neutral (tags)
TISZAKESZI, Hungary: This summer the cardinals at the Vatican accepted an unusual donation from a Hungarian start-up called Klimafa: The company said it would plant trees to restore an ancient forest on a denuded island by the Tizsa River to offset the Vatican's carbon emissions.
Fight GMO/GE Biofuels with Moratorium! (tags)
Call for an immediate moratorium on EU incentives for agrofuels, EU imports of agrofuels and EU agroenergy monocultures
Global Warming and Agriculture: Impact Estimates by Country (tags)
...while there is growing recognition that global warming is a problem, little attention has been paid to the likely impact at the country level, especially in the developing world. In this new book, William Cline, a joint senior fellow at CGD and the Peterson Institute for International Economics, provides the first ever estimates of the impact on agriculture by country, with a particular focus on social and economic implications in China, India, Brazil, and the poor countries of the tropical belt in Africa and Latin America.
G4 process is undemocratic, No WTO deal in Potsdam! (tags)
Trade ministers of the G4 countries (the European Union, the United States, Brazil and India) are meeting in Potsdam, Germany, this week try to stitch together a trade deal to which each of them could agree.
Trickle Down Economics (tags)
The big brother from the North misled the South.. The slogan "More Market, Less State" is losing a few old friends in Europe.. What is changing is the discourse, the eulogy of the market and its possibilities.
Who Frees Us From Capital? (tags)
Whole continents bowed to the Washington Consensus..The World Bank must end its embittered attack on distribution policy.. Privatization awakened false hopes.. Evo Morales in Bolivia nationalized the oil and gas economy and raised the monthly minimum wage to $100.
Hanging from the brush... (tags)
"They are hanging from the brush and we took the stepladder"...
The World Bank's Customers Are Disappearing (tags)
Many developing countries that gained needed credits only on World Bank conditions are poorer today than before. The Good News is that the cartel of creditors is dissolving. Former debtors threaten to become independent.
RP, Australia sign defense pact (tags)
The Philippines and Australia signed signed a security pact yesterday that will allow the countries' militaries to hold joint counterterrorism exercises. The Status of Forces Agreement (SOFA) is the second such agreement entered into by the Philippines. President Arroyo and Australian Prime Minister John Howard witnessed the signing of the agreement by Defense Secretary Hermogenes Ebdane Jr. and Australian Minister for Defense Dr. Brendan Nelson. Bayan Muna Rep. Satur Ocampo vowed to question SOFA and said that they would press for the public disclosure of all the terms indicated in the agreement. "The VFA has already allowed the year-round presence of American troops. Now, they have the SOFA and they are saying that they are also taking other countries. How many soldiers from foreign countries would be allowed in our country?" he asked. Ocampo said Congress, particularly the committee on foreign affairs, was not informed about the provisions of the agreement.
The G-8 Meeting... (tags)
Perhaps someone with one of those new computer programs developed by Bill Gates could calculate the resources being used for war expenses at the cost of education, healthcare and culture for humanity...
Ideas cannot be killed. (tags)
More than 600,000 people have lost their lives in Iraq and more than 2 million have been forced to emigrate since the American invasion began...
Nobody wants to take the bull by the horns... (tags)
On March 28, less than two months ago, when Bush proclaimed his diabolical idea of producing fuel from food, after a meeting with the most important U.S. automobile manufacturers, I wrote my first reflection...
Pax Cultura (tags)
"In Beauty we are united! Through Beauty we pray! With Beauty we conquer!"
Reflexiones de Fidel Castro, HOY en la Mesa Redonda... (tags)
En la región de América Latina la diferencia entre los sectores de la población más favorecida y los de menos ingresos es hoy la mayor del mundo.
ILPS May First 2007 Message (tags)
The International League of Peoples' Struggle extends its most militant greetings to the workers and other toiling people of the world on the occasion of International Workers' Day. We commemorate the first May Day in 1887 when hundreds of thousands of workers struck in every major American city calling for an 8-hour workday. In Chicago, the police brutally attacked the strikers and demonstrators. A rally in Haymarket Square to protest the police brutality was also attacked. A number of labor leaders were framed on trumped up charges and hanged while others were thrown in prison. One hundred twenty years later, the workers still suffer from the barbarities of capitalism. They have to struggle to achieve their own emancipation and the emancipation of mankind.
Join the Cuban Five contingents the 17th... (tags)
Join Cuban 5 contingents in the March 17 & 18 anti-war marches...
African, Muslim members block action on Darfur (tags)
The U.N. Human Rights Council will begin a three-week session in Geneva on Monday amid expressions of frustration from rights advocates at its early performance and alarm over proposals that might weaken it further.
Former RP Defense Chief ; "Don't tell us how to handle an insurgency' (tags)
The Philippine lawmaker who was defense minister during martial law said Thursday that he had told a UN human rights investigator to keep out of Manila's internal affairs. Sen. Juan Ponce Enrile had harsh words for UN special rapporteur Philip Alston, who is probing the wave of hundreds of political murders that have taken place since President Arroyo took office in 2001.
U.S. one of worst places for kids (tags)
BERLIN -- The United States and Britain ranked at the bottom of a UN survey of child welfare in 21 wealthy countries that assessed everything from infant mortality to whether children ate dinner with their parents.
Capitalism and Corruption (tags)
Until recently, the bribes paid by German firms abroad were deducted from the corpor-ation tax as operating expenses..The neoliberal dogma doesnt need any religion since it makes capitalism into a religion.. The cult of abstract riches cannot replace morality Mergers and takeovers are central.
Border Fence- The Cost is Far Too High (tags)
A study released last December by the non-partisan Congressional Research Service says a proposed 700 mile stretch of fence, along the 2,000 mile U.S. – Mexico border, could cost an estimated $49 billion. But what the report fails to say, U.S. –built walls on the border would cost both the U.S. and Mexico trillions of dollars in economic losses over the predicted 25-year lifespan of the fence, and non-monetary damage to political and cultural relations between the two countries would also last for decades.
Global poll condemns Bush administration (tags)
"The global view of the role of the US in world affairs has dramatically deteriorated over the past year, according to a BBC World Service poll released Wednesday. The study documents mounting hostility to US foreign policy, its destabilizing effect on entire regions of the world, as well as growing awareness of the threat posed to the environmental health of the planet. "
Brazil: To whom belongs the future? (tags)
In socially mature countries , consolidated and really sovereign, the prospections of the future are made by groups of university thinkers, of governmental and enterprise institutions, as well as groups associated to national NGOS.
The Dangerous Double Standard (tags)
In the early nineties, people hoped for the irrevocable ending of fear and the great peace dividend.. The governing limit themselves to fighting symptoms. The causes of terror are not a theme for them.
Something like this could only happen in the Soviet Union!!!! (tags)
Looks like all of these people were falsely arrested and jailed in the American gulag called Guantanamo for being terrorists
The SHORTWAVE REPORT 12/8/06 ¡LISTEN GLOBALLY! (tags)
A weekly 30 minute review of news and opinion, recorded from a shortwave radio. With times and freqs for listening at home. 2 files- broadcast and slow-modem streaming. Free to rebroadcast. Netherlands, China, Cuba, and Russia.
Oil in Brazil and the world: analysis under diverse aspects. (tags)
Brazilian government must suspend, while the sector is reorganized, the rounds of licitations of areas promoted by ANP.
U.S. urges restrictions on anti-vehicle mines (tags)
Is this the same US government that illegally invaded Iraq and Afganistan??? Oh I get it they want to limit 3rd world countries from materials from gettin materials that could be used to make IEDs which are the main cause of US deaths in Iraq.
U.S. seeks better ties by aiding militaries (tags)
Who is the world worste terrorist??? From articles like this it is clear that the American government is!
For the Globalization of Sustainability (tags)
The euphoria of Rio was based on the certainty that these visions could be realized thanks to a peace dividend.. Consumption is till the first civic duty in industrial countries.
Fear of the Working Class At Home And Internationally Stirs Some Debate Among the Bankers (tags)
There is growing concern among sections of the capitalist class that the rising opposition to their global offensive and the market will get out of control.
The Rise and Fall of the UN (tags)
One's opinion of the United Nations reveals a lot about political consciousness; and because perception is usually based on experience, it's only natural that people from different countries have opposing views about the UN and its pillar institutions— the International Monetary Fund (IMF), World Bank, and the World Trade Organization (WTO). In the United States, familiarity of the UN is limited to vague notions of 'international democracy' and 'peacekeeping', words that inspire the noblest of intentions; the World Bank and IMF on the other hand are institutions that invoke little reaction among the US public. How and by whom the UN was formed, whose interests it serves, and the actual history of its 'peacekeeping' missions are all things rarely examined. It is the purpose of this essay to look at the formation and development of the UN, and in so doing, attempt to show the evolution of the capitalistic system itself, which was 'reborn' upon the back of these hardly-neutral organizations.
The Financial Crisis of the IMF (tags)
The IMF is an extremely undemocratic system. Instead of one country-one vote, the good capitalist principle one dollar-one vote was in effect from the beginning. Whoever paid most had the most votes and the greatest influence.
WTO Negotiations Collapse (tags)
Liberalization in the style of the WTO damagers the developing world..Trade can be a means for development. However the mechanism of the WTO subordinates development to free trade in the interest of corporations.
AJLPP-USA Condemns Arroyo's Brazen Puppetry to US Imperialism and Attacks against People's (tags)
Pundits and blind defenders of imperial America have it all wrong: the DPRK is simply trying to defend itself from the bellicose saber-rattling that the US imperialists have directed at them and other countries and nations asserting their national sovereignty and refusing to submit to US dictates. We should not be afraid of anti-imperialist countries like the DPRK. Rather, we should link arms with all peoples, nations and countries opposed to the world's common enemy – US imperialism
Pundits and blind defenders of imperial America have it all wrong: the DPRK is simply trying to defend itself from the bellicose saber-rattling that the US imperialists have directed at them and other countries and nations asserting their national sovereignty and refusing to submit to US dictates. We should not be afraid of anti-imperialist countries like the DPRK. Rather, we should link arms with all peoples, nations and countries opposed to the world's common enemy – US imperialism.
AJLPP Condemns Arroyo's Puppetry and anti-DPRK Attacks (tags)
AFGHANISTAN - THE OTHER LOST WAR (tags)
HOW THE US PLUNDERED AFGHANISTAN IN THE NAME OF LIBERATION
How Israel is Engineering the "Clash of Civilizations" (tags)
The trajectory of a long-running campaign that gave birth this month to the preposterous all-party British parliamentary report into anti-Semitism in the UK can be traced back to intensive lobbying by the Israeli government that began more than four years ago, in early 2002.
IMF out of the Philippines and South countries now, protesters demand (tags)
On the occasion of the Annual Meeting of the International Monetary Fund (IMF) together with the World Bank in Singapore, the Freedom from Debt Coalition (FDC) and other civil society organizations blame "the Fund" for being the single most influential factor for the worsening poverty in the Philippines and other countries in the South.
TelSUR Live Feed (tags)
TelSUR offers alternative news, viewpoints and perspectives.
ALTERNATIVES TO THE COLLAPSED WTO DOHA ROUND TALKS (tags)
ALTERNATIVES TO THE FAILED WTO WORLD TRADE MODEL.
VENEZUELA LEADING THE RACE FOR THE UN SECURITY COUNCIL SEAT (tags)
VENEZUELA AHEAD IN RACE FOR UN LATIN AMERICAN SECURITY COUNCIL SEAT
A victory for legalized gambling??? (tags)
Fast forward: Antigua and Barbuda, population 69,000, is winning. The case has become an embarrassment to Washington, one that could result in economic pain. It isn't quite over, but the world's only superpower may have to capitulate to a country whose entire population could easily fit into the Rose Bowl.
""Hail Mary" or THE CRIME OF LEBANON AND PALESTINE (tags)
"If this scenario is possible, why then would US, UK, Israeli, and other Western leaders who see what's going on, be willing to take the risk? Ahmed states what a growing number of knowledgeable observers now believe - that the Western, mainly US, so-called neoliberal imperial freewheeling "free-market" model is failing and may collapse short of a desperate "Hail Mary" military solution to try to save it even though the chance for success at best would be uncertain and in some views unlikely."
Osama Has Won (tags)
Anyone who even attempts to defend the killing of the children in Qana by Israeli air strikes is criminally insane. There can be no justification for slaughtering children, in any circumstances, anywhere.
Oil Wealth as a Curse (tags)
The world energy need will increase 50 percent by 2030 necessitating investments of billions in the energy infrastructures.. With its oil greed and balance of payments deficit, the US could endanger global peace and the stability of the world economy.
Lebanon, Iran, Syria, Sudan: coldly premeditated neocon wars (tags)
Before emotions and disinformation gain the upper hand, consider what General Wesley Clark wrote in 2003--that by late 2001, the Neocon-led Pentagon was planning a five-year campaign of wars against seven countries: Iraq, Syria, Lebanon, Libya, Iran, Somalia and Sudan. (And don't expect to hear about this from Lieberman, McCain, Pelosi, Feinstein, Howard Dean, or the Clintons.)
Global Trade Talks Break Down Over Farm Issues (tags)
Trade negoatiations aimed at reaching a new global trade agreement collapsed today, dealing a blow to the Bush administration's international economic agenda and touching off a bitter new round of recriminations between the United States and Europe over farm trade barriers. After two days of discussions, the director general of the World Trade Organization, Pascal Lamy, formally suspended the talks. American trade officials said there appeared to be little prospect of resuming the talks any time soon, probably dooming the chances of a trade accord during President Bush's remaining time in office.
Jane Harman is no friend of Israel (tags)
Jane Harman's failure to condemn Israel's military aggression against the Lebanese people is detrimental to the cause of peace in the Middle East, which is in Israel's long range interests.
Counter the G8 + 5 Climate Summit! October 3-4 in Mexico City: CLIMATE JUSTICE NOW! (tags)
On Tuesday, October 3 in Mexico City, the Energy and Environment Ministers from the "Group of Eight" (G8) industrialized countries are scheduled to begin negotiating a climate change deal to succeed the Kyoto Protocol, which expires in 2012. They will be joined by Energy and Environment Ministers from the five "emerging" countries of Brazil, India, China, South Africa and Mexico. All together, these "G8 + 5" countries represent 58% of the world's human population, 61% of oil consumption, 80% of coal consumption, and 73% of C02 emissions.
Using American tax dollars to overthrow foreign governments under the guise of "democ (tags)
Foreign governments complain that these democracy-building groups maneuver behind the scenes to help destabilize and topple their governments, including in countries where the official U.S. policy has been to work through diplomatic channels with those same governments.
Lebanon, Iran, Sudan: Three more premeditated neocon wars (tags)
Before emotions get the upper hand, let's catch our breath and consider what General Wesley Clark wrote in 2003--that as of late 2001, the Neocon-led Pentagon was planning a five-year campaign of wars against seven countries: Iraq, Syria, Lebanon, Libya, Iran, Somalia and Sudan. (And let's not forget that Sudan has both oil and uranium.)
Global Social Rights Worldwide! (tags)
Deregulation of finance markets leads to a concentration of power in the hands of a few banks and investment funds..Social rights gained by struggle are already sacrificed on the altar of the market.
The time doesn't stop - of the need of a commitment with the education. (tags)
about the need to understand that the education is the only exit to the countries if they develop
Nazis and Neo-Cons: compare and contrast (tags)
It's not uncommon to hear a left-leaning person denounce the Bush administration with a barrage of Nazi-related epithets. The slurs are typically blurted out in anger with little thought to evidence or validity. In fact, whenever analogies are made linking a person or current event to either the Nazis or Holocaust, accusatory responses of "exaggeration" and "slander" are usually justified; it is indeed immature to randomly compare anything to the machine of Nazism that took the innocent lives of so many. While taking this into full consideration, the following is nevertheless an attempt to make some unpleasant connections.
Terror suspects' move out of U.S. defended (tags)
THE WALL STREET JOURNAL CALLS HUGO CHAVEZ A THREAT TO WORLD PEACE (tags)
RESPONSE TO WALL STREET JOURNAL LIES ABOUT HUGO CHAVEZ
Europe's leaders close ranks with Bush (tags)
On Tuesday, US President George Bush touched down in Vienna for the annual summit of US and European Union (EU) leaders. After brief talks in Vienna, Bush is due to fly to Budapest on Thursday for the 50th anniversary of the 1956 Hungarian uprising. Bush's trip to Europe is the first in a round of visits over the next few weeks. Next month he will return to Europe to meet with German Chancellor Angela Merkel in Stralsund, before proceeding to Russia for a meeting with President Vladimir Putin prior to the G-8 summit in St. Petersburg, to be held July 15-17.
Columbian and Peruvian Elections Prove Stalin Was Right (tags)
US arranged elections in Columbia and Peru
Detained Philippine lawmaker files resolution vs. repressive US immigration bills (tags)
Activist Philippine lawmaker Rep. Crispin Beltran recently filed House Resolution 1263 urging the Philippine government to support the call of millions of Filipinos and migrants of other nationalities in the US and actively oppose the passage of the racist and repressive Sensenbrenner-King bill and other similar proposals.
STOP CORPORATE GLOBALIZATION: ANOTHER WORLD IS POSSIBLE! (tags)
A Statement of Unity from the OUR WORLD IS NOT FOR SALE Network:(www.ourworldisnotforsale.org)
The Petrodollar: America's Achilles Heel (tags)
Virtual Chicano is webmaster and political commentator for Chicano Forums
What we can learn from the immigrants rights march (tags)
Two days after May Day's mobilization of immigrants and their supporters, Ricardo Alarcon, President of Cuba's National Assembly, took a look at the marches and drew some conclusions. for the struggle for a better world everywhere.
More Lies from the Antis (tags)
Kimberly-Clark heir Congressman James Sensenbrenner would have you believe that Americans are losing their jobs to undocumented workers. But it is his corporation, and those like it, that are the real culprits.
Propaganda on "New Foreign Aid" – (tags)
At the height of the emotional issue on immigration and the Great American Boycott mass actions nationwide, The Los Angeles Times ran a five-part series entitled, "New Foreign Aid." The series of feature articles ran from April 16, 2006 before the May 1, and featured the Philippines, Mexico, Haiti and other countries which send thousands of immigrant workers to the United States every year.We condemn such propaganda being peddled as "objective reporting." Such allusions have no place in a society that purports to be democratic and free of racial bias. Such racist notions must be criticized and exposed roundly for what they truly are: worthless
The Price of Growth (tags)
In "1984," George Orwell warned that war would become a domestic necessity to divert the people from economic contradictions. Inequality may promote productivity but also undermines public spirit and social ochesion
The Contradictions are our Hope (tags)
Multilateral trade agreements-like the WTO-are not per se something good, especially when they only codify the rules of the rich. Until trade policy changes, massive changes of social hierarchies of power are necessary Th WTO is undemocratic and unjust.
Wage war on poverty, not immigrants (tags)
"Si se puede!" Yes we can. They marched by the hundreds of thousands in Los Angeles, by the tens of thousands in Milwaukee, in Phoenix, in New York. Across the country, Hispanics dramatically entered what has been an increasingly ugly debate about immigration in this country.
ANOTHER FACE TO THE IMMIGRATION DISCUSSION (tags)
I have been inundated lately with a flood of articles analyzing the pros and cons of the various immigration bills, the impact of immigration on our social structures, the changing face of the color of our population, etc. I think there is something seriously missing in almost the entire discussion and analysis about immigrants; and that is the fact that the majority of them are misplaced people.
Changing Course: No New Nuclear Power Plants! (tags)
Offensive wars, normalization of war, militari-zation of foreign policy, exploding inequality, privatization, political corruption and nuclear power plants as panaceas have brought us to the brink of ruin. A human-friendly society would have other priorities than capital gain.
President wins 2 billion new supporters (tags)
Unfortunately a picture has been published worldwide of President Bush playing cricket. This has instantly won him the support of over half the world's population. Please hide this picture if you find it..
Rice That Saves Lives? (tags)
This Rice Can Save Lives & Cure Blindness. Why Do People Protest It?
Support the Caravan to Cuba End the Blockade (tags)
The US blockade of Cuba causes shortages of food, medicine and other important supplies for eleven million people. The blockade is an immoral policy that uses hunger and disease as political weapons. In November 2005 the countries in the United Nations General Assembly once again voted overwhelmingly (182-4) to call for the end of the blockade. A month later Condoleza Rice responded by stating that the Bush administration intends to intensify the blockade still further and is working on a package of new measure to be announced in May 2006. US policy towards Cuba is totally isolated internationally, increasingly discredited domestically and subject to many challenges from within Congress.
One Million Strong - Bolivarian Revolution (tags)
Venezuela to build an army of one million men and women.
On the Scandinavian Model (tags)
The Scandinavian model could help us find our way to social compromise and true security through investments in education and apprenticeships. Has our wealth blinded us to alternatives and encouraged our altered state of paralysis, insecurity and fear?
Women's Rights in Latin America and the Caribbean (tags)
In every country in Latin America and the Caribbean, women suffer acute discrimination. Often, the discrimination women face is related to social prejudices regarding appropriate patterns of conduct for men and women.
LISTED: CIA Planes; EU Pacts for Lynching Muslims; CIA Front Co.s, Lawyers (tags)
CIA DDO (Jose Rodriguez), the chief spy? All are listed below. The 22 American CIA agents allegedly led by Robert Seldon Lady in the 2003 Milan kidnapping of Muslim cleric Abu Omar? The Italian secret service SISMI knew in advance of their operation, just as Mr. Lady claims. There is a document that proves, in writing, such operations involve the cooperation of EU governments. The Milan prosecutor Spataro knows this. Prime Minister Berlusconi knows this. Still, these two men are on a collision course as Spataro prepares for a show trail of the 22 "in absentia" while Berlusconi attempts to get away with denying everything.
'Energy Dissent' and the 2006 G8 Summit in Russia (tags)
Reclaim the debate! Resist the G8! In July 2006, the G8 will hold its annual Summit in St.Petersburg, Russia. Vladimir Putin says he's placing "energy security" at the top of the G8 agenda during Russia's presidency. The G8 countries consume 45% of world oil and produce 47% of global CO2 emissions. Their "energy security" is our energy grave!
The SHORTWAVE REPORT 12/23/05 ¡LISTEN GLOBALLY! (tags)
A weekly 30 minute review of news and opinion, recorded from a shortwave radio. With times and freqs for listening at home. 2 files- broadcast and slow-modem streaming. Free to rebroadcast. China, Netherlands, Spain, Cuba, and Russia.
Stop the WTO Negotiations! Save Jobs! (tags)
After ten years under the WTO, unemployment has increased everywhere in the world..Many transnational businesses try to distance themselves from responsibility for working conditions by outsourcing work.. Joy to the world, not only to the superrich!
Around the world, the Death Penalty is being Abolished (tags)
The death penalty is being abolished in country after country. In 1977 only 16 countries had abolished the death penalty for all crimes, while today the number is 86. Every western European country has abolished the death penalty. An average of three countries per year are currently abolishing the death penalty.
'Tis the season: hold a peace vigil at your Congressperson's house (tags)
Before the Iraq war gets any worse--or spreads to other countries--let's take a stand for peace, in the spirit of Cindy Sheehan, in front of our Congresspersons' and Senators' homes throughout the holiday season.
Large Demos Globally Against Bush Admin's Climate Policy (tags)
In 32 countries around the world, thousands of protesters railed against the inaction of the US Government to curb it's fossil fuel polluting of the atmosphere.
Paul Samuelson Attacks a Dogma (tags)
The best-known economist of the world doubts that shifting services to india and buying cheap goods from China bring advantages to Americans when the income of workers falls through foreign competition. The rich North will lose its monopoly profits under the conditions of globalization.
"The Days of the US Empire are Numbered" (tags)
"I love the US but I hate the American empire.. The US cou9ld escape a sudden downfall if it recognizes economic equality, appears as one country among others and abandons its military presence in the 140 countries where it presently stations troops.:"
The Future of Theology (tags)
"The social state is dismantled in favor of the globalization of industry..The democratic idea of equality is incompatible with ever-greater inequalities..If globalization produces third world conditions, theology of liberation logically becomes universal.."
CHART, TABLE: CIA Secret Prison Planes--What the MSM Won't Tell (tags)
The European Commission said last week it will investigate published reports that the CIA set up secret jails in Eastern Europe to detain high-profile terrorism suspects. The Commission says the governments of the EU's 25 member nations will be informally questioned about possible human rights violations. News media reported also that the group Human Rights Watch "claims records and other evidence point to POLAND and ROMANIA as countries that allowed their territory to be used by the CIA to jail top suspected al-Qaeda captives." We report that HRW knows that from tracing the movements of CIA planes and WE PROVIDE A LIST OF THE 28 PLANES, 8 SHELL COMPANIES, AND SEVERAL CIA-RELATED COMPANIES.
Against Hollywood and Globalization: UNESCO and Cultural Diversity (tags)
The Convention on Protection of Cultural Diversity was passed at the UNESCO General assembly over the objection of the isolated US government.
Americas Summit Marked a Change, Alarcon Affirms (tags)
Americas Summit Marked a Change, Alarcon Affirms
BOOK ON CIA's SECRET JAILS: "Operation Hotel California" by Guido Olimpio (tags)
The first book-length expose of CIA kidnappings and prisons -- two of the greatest evils in the Bush government, and the world -- by the Italian investigative journalist Guido Olimpio is here, and it's new and as current as todays news stories about secret jails the CIA runs! The book is a must-read for anyone wanting to know about the CIA's practice of "rendition": kidnapping people and transporting them to places like Egypt for torture. It's an extremely controversial practice, a threat to America's security and character, and a sweat-provoking threat to potential victims and an insult to the sovereignty of countries like Italy and Sweden. Read the book, and your reactions will range from cold sweat to boiling mad.
EU, CIA's Secret Jails:a SCOOP on CIA Prison Planes (tags)
Two countries the media declined to name...a complete list of CIA shell companies and aircraft.... News media reported Thursday that the group Human Rights Watch "claims records and other evidence point to POLAND and ROMANIA as countries that allowed their territory to be used by the CIA to hold top suspected al-Qaeda captives." We report that HRW knows that from tracing the movements of CIA planes and we provide a list of the planes.
Stephen Bezruchka: Population Health Forum (tags)
The Population Health Forum's mission is to raise awareness and initiate dialogue about the ways in which political, economic, and social inequalities interact to affect the overall health status of our society. Our goals are to promote knowledge and advocate for action in service of a healthier society.
Something we need to be concerned about, the U.S. debt to foreign countries, lets not pay (tags)
The U.S. stands a good chance of becoming a third world country in the future. I just heard on Nightline that "our" government helps pay for the war in Iraq by taking out loans from other countries like China. What a stupid thing to do.
Report Back From the 16th World Festival of Students in Venezuela (part 1) (tags)
"[The U.S. ruling elites] have a lot to lose because Venezuela is setting an example. If the people of Venezuela are able to struggle and alleviate so much poverty and illiteracy and lack of healthcare in their country, then why can't we do it here as well?" -- Muna Coobtee, delegate, World Festival of Students, 2005
Report Back From the 16th World Festival of Students in Venezuela (part 2) (tags)
"[The U.S. ruling elites] have a lot to lose because Venezuela is setting an example. If the people of Venezuela are able to struggle and alleviate so much poverty and illiteracy and lack of healthcare in their country, then why can't we do it here as well?" -- Muna Coobtee, delegate, World Festival of Students, 2005
GATS as a Political Project (tags)
The goal of GATS is the worldwide liberalization in trade and investments for the service sector.. Call 1-800-828-0498 and urge your senators to reject any budget reconciliation that includes Medicade cuts.
From Haiti to New Orleans, Yes to Justice and Relief, Not Death and Occupation (tags)
A protest against the US/UN occupation of Haiti and the massacre of Haitian people was held Sept. 30th outside the Brazilian Consulate in Beverly Hills. The LA event was part of an international day of outrage; actions took place in 40 cities in 15 countries.
"The Market Has No Heart": Paul Samuelson (tags)
A post-Katrina economics must emphasize parallel worlds, solidarity and human dignity to safeguard long-term necessities and creativity. If all countries only become more competitive, mass unemployment will result. What is public must remain public.
CAPITALISM AND SOCIALISM (tags)
Every year since 1990, the United Nations publishes its Human Development Report. It contains the most authoritative data on the state of the world. These reports are available online: hdr.undp.org/reports/view_reports.cfm?type=1. Based on those reports (referred to by year, followed by page), what does our world look like?
"Counterpoints to the Standardized World": Book Review (tags)
"The world plague globalizationis a practiced philosophy in the interest of business and exploitation without responsibility toward our living space and without humanliness.."
Liberators or Occupiers? Look Who Calls the Shots (tags)
The governments of Iraq and Afghanistan are complaining about the U.S. military's brutality and its indiscriminate attacks, but the Pentagon and the White House are just blowing them off. Hey, forget "sovereignty." We're in charge here.
Danger by Eduardo Galeano (tags)
The colonial organization of the planet would run a risk if the poor countries, comprising the overwhelming majority, could vote. They are invited to the banquet - to be eaten..II turns out that power is not as powerful as it says it is "
Join us in Washington, DC! Protest Military and Economic Violence! (tags)
Global Justice Feeder March! Saturday, September 24, 2005. 11:00 am Dupont Circle, Washington D.C.
Iran hangings: Chronicle of a manipulation (tags)
A mislead gay and lesbian movement mobilizes against Iran at the height of conflict between the West and the Teheran regime.
Bush Policy: Making the World Safe...for Nukes (tags)
By invading countries unilaterally, and by threatening countries without nukes while negotiating with, or even having friendly relations with those that have them, the Bush administration is courting nuclear disaster. -----------------
Statement of the International Marxist Tendency for the 16th WFSY (tags)
Bolivarian Revolution
Purgatory of the Market (tags)
"The new capitalism is also a totalitarian movement that cannot or will not rest until it has seized the whole world and put everything in private hands that previously was subject to state or civil control.."
Disinformation as a Strategy of Power (Part II) (tags)
"Perpetual growth is not possible in nature or anywhere else..Globalization, the free market with its absolute competition protected by the WTO agreements doesn't have anything to do with fair competition...A global network of resistance is developing.."
Disappointed Hopes in Africa (tags)
"The $40 billion promised before the summit was only a small part of Africa's total $300 billion in debts and the more than $2.4 trillion of all developing countries. The IMF, World Bank and G8 still use the debts as an instru-ment for dictating development.."
The Divided Camp (tags)
"There is no reason for the Iraq war, Buchanan said..The neoconservative approach is everything but conservative: they bomb a pseudo-democracy in the countries of the Middle East.."
ACS Summit to Foster Integration (tags)
ACS Summit to Foster Integration
July 7 bombings: Another reason to end war and occupation (tags)
Will the July 7 bombings in London be followed by greater opposition to imperialist war and occupation--which has already cost so many lives and so much suffering in so many countries?
Pew Poll - The radioactive image of the US (tags)
Anybody you know? The US and it's global War of Terror: In a group of 251 soldiers from a study in Mississippi who had all had normal babies before the 1991 Gulf War, 67 percent of their post-war babies were born with severe birth defects.*
Disarming the WTO (tags)
CAFTA, written by and for investors, extends NAFTA to Central America. The lessons from 10 years of NAFTA-job losses, food insecurity are repressed. CAFTA may be voted on soon. Call your reps toll free 1-866-340-9281
How Much is Enough? (tags)
"The world has enough for everyone's need, not for everyone's greed." The fear of the further spread of the western lifestyle and economic mode with frequent flier rebates is at the center of political debate. Life without resource-intensive lifestyles is vital.
EU and MERCOSUR Foster Negotations (tags)
EU member countries create an association with Brazil to foster commercial relations with MERCOSUR.
rEVOLUCIOnARY tRANSITION dESIGNS (tags)
The following program will typically be required of the revolutions in the Andes and throughout Latin America (the pace of adaptation and implementation may vary somewhat) :
Climate Change! We Must Intervene! (tags)
Can we take climate change activism beyond uptopian demands into the realm of meaningful strategy? The climate change battle is largely being fought-and currently lost!-in the media.
'No nukes, no war!' (tags)
NEW YORK — The worldwide movement to end nuclear weapons, energized and united with the antiwar movement, is converging here for a month of actions. Kicking it off is a May 1 mass "No Nukes! No Wars!" march. On May 2, the United Nations opens a review of the Nuclear Non-Proliferation Treaty (NPT), which aims to curb and eventually eliminate these weapons of mass destruction.
Artist Gets Creative with Fundraising (tags)
Cuts in funding for the Arts and almost non-existent resources for individual artists in the United States, has lead Santa Monica based artist/painter Barbara Kolo to seek alternative methods. The artist has opened an online store with Cafepress to help sponsor her invitation to the 2005 Florence Biennale (Biennale Internazionale Dell"Arte Contemporanea) this December in Italy.
Exodus from the System (tags)
Other causes contribte to the economic misery besides the distribution problems. There are two developmental problems. There is the shift of profit from labor to capital. There is the worldwide opening of markets causing wage pressure.
Paul Wolfowitz as Head of the World Bank (tags)
In an Internet poll, Wolfowitz is regarded as a global disaster by over 80% of respondents while less than 10% see Bush's decision as a master stroke..The political scientist Bernd Kubbig points to Wolfowitz' contradictory and sinister role in the iraq conflict.
KIM IL SUNG ON DOGMATISM, REVISIONISM, AND ANTI-US STRUGGLE (tags)
Classic Text by Comrade Kim Il Sung
Pope John Paul II: The Passing Of A Monster (tags)
From women's rights to abortion to contraception to gay rights to pedophilia, this Pope has been a champion of the most conservative, the most reactionary forces in society
Venezuela, Frustrated by the U.S., Turns to Containment (tags)
After years of vitriolic rhetoric on the part of United States leaders, the government of Venezuelan president Hugo Chavez has decided on a policy of containment of the U.S.
Farewell Italia, Viva the Coalition of the Willing (tags)
Bush's Coalition of the Willing allies are dropping like flies. Of 39 "members," 14 have already quit and two more have announced plans to pull out, and now Italy, the third largest U.S. ally in the so-called coalition, says it's leaving too. Can Britain be far behind?
Manifesto of Porto Alegre (unofficial translation) (tags)
Another devel,opment model should be realized based on an energy-saving way of life and democratic controls on natural mineral resources, particularly drinking water. Military bases of foreign countries should be closed.. Another world is possible.
Call for Direct Actions A16 (tags)
Come to DC for a week of Direct Action and prevention of "Business as Usual" and the Spring World Bank and IMF meetings
The Bondage of Terror (tags)
The change for building a common security lies in recognizing oneself in the other.. Security based on trust is a hundred times more valuable than the most expensive defensive system.
False Pride of the "Super Outlaw State" (tags)
"Although few Americans realize that their country was the first to commit genocide, ethnic cleansing and use forced death marches, these historical facts didn't escape the Third Reich and Adolph Hitler. He admired the American government for their efficiency"
The End of Globalization (tags)
The economy ought to serve humanity; humanity should not serve the econmy. Beware the neoliberal idolization of growth and competition! If growth becomes a law, ecological collapse is unavoidable. If com-petition is made absolute, mass unemployment will occur everywhere.
THIS FRIDAY (Today Feb 11) Tsunami Relief Benefit show! COME!!! (tags)
This Friday Feb 11 TSUNAMI RELIEF BENEFIT SHOW! (tags)
Tsunami Relief Benefit Show Flyer For this Fri Feb 11 !!!!!!!!!!!!! (tags)
TSUNAMI RELIEF BENEFIT SHOW THIS FRI FEB 11!! (tags)
"A Subtle Form of Hypocrisy" (tags)
Between 1980 and 2003, $1.8 trillion in interests alone flowed from the South to the North. The heavy debts of developing countries more than tripled in this time period.
The fairy tale of the Dutch tolerant society and the humane asylum-policy (tags)
In contrarily with the general allegations in American and Western newsmedia, not only the Netherlands is no open society for minorities, moreover the Dutch asylum-policy is violating gravely international humanitarian principles.
Tsunami Relief Benefit Show @ Saddleback College Fri Feb. 11 (tags)
Current Globalization Sows Hatred and Terror (tags)
The future is very nebulous. Two different possibilities exist: balance or destruction.. A world in which 24,000 persons die of starvation everyday and several hundred million children are without a school educa-tion cannot be just and is unacceptable.
Hard Landing of the Dollar (tags)
The decline of the US dollar signals the economic weakness of the superpower and the structural crisis of capitalism. Since 1985 the US has become heavily indebted fo foreign countries. This foreign indebtedness amounts to 35 percent of the GDP.
No Indymedia in Socialist Countries (tags)
CROATIA ... A LAND OF CRUELTY? (tags)
an essay expressed by Croatians... for the world to read "A small country for great holidays ... But a small country for great cruelty!"
What is Progress? (tags)
Progress is understood today as the approach to European norms and the standard of life of Europeans. The new Bolivians demonstrated their emancipation with sentences like We don't want your progress of assets accumulation. We want harmony with pachamamma, mother earth.
The Economy of the Plundered South (tags)
The panacea of export orientation that the IMF and the World Bank promoted in the last 25 years for the South has led to absoute disaster..There is an alternative, abandoning export orientation and strengthening domestic demand.
How Can We Escape This Abyss? (tags)
One hand washes the other even if it is bloodstained.. The government of one of the poorest countries paid Enron $220 million per year for energy that was not produced!.. Now Bechtel and GE aresuing the Indian government for $5.6 billion..
WAR, CAPITALISM & ELECTIONS: An Interview w STEPHEN GOWANS – Canadian Political Essayist (tags)
Read ANGIE's – regular and erudite Indybay-IMC (Indybay.org) Poster fm Canada – incisive interview with Canadian political essayist STEPHEN GOWANS. Angie has scored a previous interview with Israeli pro-Palestinian human rights activist Ran Ha Cohen. We await similar interviews with Uri Avnery and others who speak directly to issues of global justice – as well as global resistance to imperialist racism and the elite capitalists' rapaciousness that will gamble ALL our lives. We, finally, are at an especially critical world cross-roads: we can choose illumination or illusion, enlightenment or inconsequence, mutual respect or profligate death.
Fidel Castro and the Cuban Revolution (tags)
When I was very young, I remember my family taking my uncle to Logan International Airport in Boston. One side of the Sumner Tunnel had been blocked to traffic due to security situations, because Fidel Castro was due to be passing through at any moment. This was the very day that he made his famous speech at Harvard University.
Remember the Red and Blue States (tags)
Don't cry for the passage of the democratic government. Weep for the real patriots that will be swept away to the concentration camps or whose blood will cover the streets, but above all else regardless of the outcome remember the red states.
Criticism of the Social State is Irrational (tags)
"Paul A Samuelson, an important economist of the 20th century, worries about the state of the US economy. He fears falling wages..`We in the US are on the way to a plutocratic democracy. Neither Bush nor Kerry can stop outsourcing'."
The Iraq War has Increased Terrorism (tags)
"The British and American population does not believe that the war against iraq has made the world safer..Only 30 percent in the US agree with the government.. The Iraq war represents a great diversion from the global struggle against terrorism.."
The Weather Map of Globalization (tags)
"Globalization is a real process rooted in the crisis of worldwide capitalist exploitation and describing a new quality of capital export. Outsourcing particular economic areas in the global repression competition replaces the export of complete industrial products.."
The World is not a Commodity: Another World is Possible (tags)
"Attac is an element of the anti-war and peace movement because a just world is impossible without peace.. We reject the militarization of foreign policy and war actions of the German army abroad.."
"Wars are Good for the Economy": The Global Free Trade System (tags)
"The international free market was created artificially in the 18th and 19th centuries with the force of the English colonial state and did not develop naturally from the free exchange of goods as neoliberals constantly claim.."
fresca (tags)
Let the whining begin!
Palestine in Numbers (tags)
I have to add one here, number of UN members in the General Assembly -189. Number of memebers who are from predominantly muslim countries or have political ties to said countries- 135
Continental Social Alliance decides its own re foundation (tags)
Gathered in Quito, at the Americas Social Forum, the Hemispheric Council of the Continental Social Alliance has decided to remove the opposition to the FTAA from its foundational declaration, as one of its main goals. After analyzing the international scenario as well as evaluating the impact of its actions, delegates from all the American countries and the most important social organizations of the continent, decided the "fight against the FTAA" would become a "fight against free trade".
Racial Accountability (tags)
In this world, there are many races, and many people who are of mixed race. Still, how you act is always going to be construed as a part of your culture.
Wolf Guards the Henhouse (tags)
What's with the ICJ?
Why Massed Demos Don't Work in America (tags)
Self-inflicted poverty (tags)
U.S. should face court, Annan says (tags)
"UN Secretary General Kofi Annan is leading the charge agaisnt Washington's call for the UN Security Council to renew its special exemption for U.S. forces from prosecution by the UN war crimes court."
European Parliament 2004 elections and new constitution (tags)
The European Parliament elections, held June 10-13, showed a strong trend among voters against ruling parties that sided with Bush's war on Iraq, as well as those that have been imposing pro-business economic policies, as in Germany and France.
BTL:At Their Georgia Summit, G-8 Nations Fail to Cancel Debt for... (tags)
...World's Poorest Countries~ Interview with Njoki Njehu, director of the 50 Years is Enough Network, conducted by Between the Lines' Scott Harris
Demonstration in Sao Paulo against the FTAA and the WTO (tags)
Yesterday, on Sao Paulo, where the 11th UNCTAD it's having place , was made a demonstration protesting against the World Trade Organization, the Free Trade Area of the Americas and the war strategy of United States.
European Parliament 2004 elections results: Throw the Bums Out! (tags)
Anti-FTAA initiative in Paraguay (tags)
The anti-FTAA movement considers that this agreement will mean subdue mechanisms for our nations. The governments will be subdued to this mechanisms or game rules that respond to the interests of the big corporations and which will negatively affect the social and sensitive sectors of the national economy. They state that the FTAA is supposed to be a stage of negotiation with low operative costs for corporations, but at the same time it impose concepts and laws that won't allow the governments to make public policies encouraging the strength of the little and middle companies and national industries.
U.S. has the highest rate of mental illness (tags)
June 2, 2004 CHICAGO - Mental illnesses including anxiety disorders and depression are common and undertreated in many countries, both developed and developing, with the highest rate found in the United States, according to a study of 14 countries.
Immigrants Drain $30 Billion in Cash Annually (tags)
In the past nine years the cash that immigrants send from the United States back to their home countries has almost doubled, but the Bush Administration is planning to use the upcoming G-8 summit to discuss ways to increase the outward flow of cash.
Over 800 US fatalities in Iraq so far (tags)
The total US fatalities just passed the 800 mark today. The total for the whole coalition is over 900. For the month of May, there has been an average of 2.78 coalition fatalities per day. The average since the war began is 2.11 per day.
The Boy Who Cried Wolf (by Latuff) (tags)
Copyright-free artwork by Brazilian cartoonist Latuff, on behalf of brave Palestinian people and their struggle against U.S. backed Israeli terror.
IF (tags)
If AIDS is sexually transmitted how can one explain these figures:
Dangers Hang Over Latin America and the World (tags)
Havana.-
Globalization and the Sex Trade:Trafficking and the Commodification of Women and Children (tags)
In this important article, Richard Poulin, professor of sociology at the University of Ottawa (Canada), argues that the sex industry, previously considered marginal, has come to occupy a strategic and central position in the development of international capitalism.
Undocumented Workers Are Not The Enemy Of labor (tags)
I submitted the following article to my Local Union newsletter.
CoupsRUS: Haiti and Venezuela (tags)
An argument that the CIA is involved in both Haiti and Venezuela and the techniques they use to overthrow governments.
Developing countries summit in Venezuela: "To recover the South-South identity" (tags)
The developing countries encounter will take place next February 26th to 28th in Caracas, Venezuela. It will count with the presence of the presidents of Argentina, Brazil, Mexico, Peru, Chile, among others.
Bricks (tags)
It's been brought.
Bush's Immigrant Trap (tags)
Celebrating Black History All Year Long African style (tags)
Discover Africa will play host to African civic leaders and businessmen and women
Bush rewards allies in terror war with billion$ (tags)
The President's budget does not include funding for the US's military operations in Afghanistan or Iraq, expected to cost and additional 30 to 50 billion dollars.
Capitalism is Killing Free Trade (tags)
Are libertarians going to allow their ideology to do their thinking? What good does it do for libertarians to go into denial and to call me, patronizingly, names?
Arundhati Roy's Speech at the WSF (tags)
In the great cities of Europe and America, where a few years ago these things would only have been whispered, now people are openly talking about the good side of Imperialism and the need for a strong Empire to police an unruly world.
Sierra Club a Target of Anti-Immigrant Movement (tags)
I heard this on the radio and found an article on the net. Once again, the Sierra Club, the nation's largest environmental group, is finding itself split on the issue of immigration.
Lula in India (tags)
Lula in India: "we can change the world commercial geography"
Bushies in spaaaace! (tags)
Just imagine what the moon will be like in a few years. There'll be a Wal-Mart in every crater and a Disneyland at each pole. There'll be a Howard Johnson every five miles and solar powered swimming pools on every street corner. To make it seem like home, there will be vacant factories, rundown tenements and drunks sleeping in dumpsters...
Miami FTAA Protest: Cops Rampage Against Youth, Labor (tags)
What Strategy to Defeat Imperialism?
They want my fingerprints (by Latuff) (tags)
Copyright-free artwork by Brazilian cartoonist Latuff.
Iraqi Resistance Solidarity Network January 5th 2004 Updates! (tags)
1. Iraqi Resistance Report Jan. 1-4, 2004 2. Newsletter From the International League of People's Struggle, formed by Comrade Jose Maria Sison, former head of the New People's Army of the Philippines (PDF) 3.***A Call to Support the March 20th Demonstrations!!!*** 4. Join the Anti-Imperialist Camp Delegation to Iraq! 5. ANSWER stands against Colonialism in Palestine, Iraq, the world! 6.International Progress Organization: PROSECUTION OF INTERNATIONAL CRIMES IN IRAQ – LACK OF CONSTITUTIONAL BASIS FOR WAR CRIMES TRIALS 7. The True Figures of Imperialist Casualities in Iraq! 8. Examples of online zionist hatred and racism against Palestinians
The Same Question Remain (tags)
The apprehension of Saddam Hussein has done nothing to answer the critics
FTAA – No way! (tags)
In Miami last month, trade ministers from around the Americas gathered for talks on the Free Trade Area of the Americas.
Bush Translation (tags)
President Bush explains why other nations will be barred from getting the contracts to rebuild the mess we made
Editor of Kuwaiti Daily: Syria's Secret Service is Behind the 'Iraqi Resistance' (tags)
08 December, 2003 The Middle East Media Research Institute The November 1, 2003 conference of foreign ministers of the countries neighboring Iraq yielded numerous articles in the Arab press condemning Syria's pro-Saddam stance. Among the authors were editor of the Kuwaiti daily Al-Siyassa, Ahmad Al-Jarallah, and Iraqi opposition member and political commentator Dr. Riyadh Al-Amir. The following are excerpts from the articles
Opposition to Government and Climate, is there a link? (tags)
Is there a link between the level of discontent with the "Government" and the climate? I think there may be.
After the Failure of the World Trade Conference (tags)
"There is growing skepticism toward the past structure of the WTO and the procedures and results of previous world trade conferences. With the founding of the WTO in 1994, all 148 member countries were promised that there would `only be winners'."
Interview with Comrade-in-Arms, Awni al Kalemji, of the Iraqi Patriotic Alliance (tags)
Bush & the Democrats: Liars in the service of imperialist conquest (tags)
Despite all their bickering with Bush, the Democrats too can scarcely contain their glee about the unchallenged power of U.S. armies of world conquest. After all, the Democrats like the Republicans are in the pockets of the big capitalists who want their domination of the world defended at all costs.
Exodus from Growth (tags)
"Growth, prosperity, free trade, liberal world economy and international division of labor are all ideologically charged terms. They re-flect veiled interests more than a search for truth. The starting-point for an alternative econmics is an ideology-critical position."
History- 17 Years Later (tags)
A wonderful piece by John Bennett in a time of conglomorate-controlled media.
Anti-Globalize the Corporate Globalization Movement - DO IT NOW! (tags)
'Free trade' scam:'It's bad for workers everywhere' (tags)
"We'll be marching with truck drivers and electrical workers, nurses and teachers, health care workers and Teamsters, all kinds of working people," Fred Frost, president of the South Florida AFL-CIO, told the Miami Herald.
Ban sought on US Military Death Weapons (tags)
If it wasn't already bad enough for the killers in the White House now that the Afghan/Iraqui misadventure is spiralling into chaos, now the people of the planet want to stop the US Military from building and dropping cluster bombs. What's a republican war monger to do with such a disgusting life-affirming development?
Bush - A Crazed Mattoid (tags)
Insanity in individuals is something rare -- but in groups, parties, nations, and epochs it is the rule.
STATE VS ENVIRONMENT. SECRET PLOT TO UNDERMINE GLOBAL POLLUTION CONTROL REVEALED (tags)
According to data from 30-year-old government records given lately in public a secret plot was formed back then by developed countries in an attempt to limit first UN's project concerning environment and pollution control. Reason? Simply would damage their economic development…
Those who seek ''the death of the United Nations'' (tags)
While the United Nations is seen by many as merely a talk shop that gets little done, the truth lies elsewhere. For example, UNESCO has done a great deal for education and the work of UNICEF has been phenomenal.
Is Zionism a Failed Ideology? (tags)
"In Israel, Avraham Burg, former speaker of the Knesset, recently warned that if Israel wishes to preserve what little democracy it still has, it must either withdraw to its pre-1967 boundaries or grant full citizenship to the approximately 3.5 million Palestinians in the occupied territories, a step that would spell the virtual end of the Jewish state...."
LA Times: Officials Confirm Israeli Nuke Launching Subs (tags)
The story contains the first official confirmation from both American and Israeli officials that Israel can launch nuclear weapons from its submarines. The ultimate guarantee of mutually assured destruction (MAD). The acronym MAD also makes for a most appropriate word to describe the human condition.
The Cancun Summit-WTO on the edge of abyss (tags)
Abusing "Anti-Semitism" (tags)
The abuse of alleged anti-semitism is morally despicable. It took hundreds of years and millions of victims to turn anti-semitism – a specific case of racism which led historically to genocide – into a taboo. People abusing this taboo in order to support Israel's racist and genocidal policy towards the Palestinians do nothing less than desecrate the memory of those Jewish victims, whose death, from a humanistic perspective, is meaningful only inasmuch as it serves as an eternal warning to the human kind against all kinds of discrimination, racism, and genocide.
Imperialist Excuses for War (tags)
A comparison of the excuses for wars of aggression used by Nazi Germany with those used by the United States.
Impunity for US forces in Colombia (tags)
The Bush Administration has forced Colombia to sign an accord granting immunity to US citizens that commit war crimes in Colombia. Bogotá will now need approval from Washington if they wish to hand US soldiers or other officials over to the International Criminal Court for trial.
Public Servants reject WTO Tactics (tags)
Cancun Confrontation: Thousands Protest the WTO and Capitalist Globalization (tags)
Protesters Should Love WTO (tags)
Why the anti-globalization movement should try to disrupt the work of the World Trade Organization is a mystery to me. Instead, they should learn to love corporations like McDonald's.
Will NeoCons Drive The Decent From The GOP? (tags)
"We've watched while he used the tragedy of 911 and international terrorism to polarize the nation even further, vexing us with fears, while his mindless minions and yes-men from Ashcroft right down to commercial automatons like Fox's smarmy, imbecilic Sean Hannity, blaspheme the grand traditions and Constitutional institutions of this nation in order to wrestle everything from the flag to love of country to being holy relics of the GOP alone. "
Solidarity Economics is Ecology (tags)
How can there be a global movement with no platform and no economic vision and leaders who hide? If the Brazilian constitution defends the right of the starving to steal food, then surely the poor of the world have the right - under any way of thinking - to steal back control over their lives and their local economies.
"Stop GATS!" Against Privatization of Services (tags)
"The `Washington Consensus' from the 80s promised that a neoliberal policy - liberalization and deregulation - would be good for eveyrone. In fact, local structures are destroyed, human lives are annihilated through hunger and poverty.."Trans fr German
STOP CAFTA IN HOUSTON OCT 20-24 (tags)
YO BASTA!!!! STOP CAFTA Free Trade Talks are coming to Houston and the resistance needs to begin
Country's and Individual's Involved with Saudi Arabia and Terrorism. (tags)
To all countries and individuals that are in a current religious trade journey with Saudi Arabia in an effort to expand extreme fundamentalism through the training and support of jihad terrorists, and to any officials taking "enticement" money, here is some pertinent information you need to consider:
Growing World Poverty and Conflict Shows the Barbarity of Capitalism (tags)
Down with Capitalism
Where Was The Color In Seattle? Looking for reasons why the Battle was so White (tags)
"I was at the jail where a lot of protesters were being held and a big crowd of people was chanting 'This Is What Democracy Looks Like!' At first it sounded kind of nice. But then I thought: is this really what democracy looks like? Nobody here looks like me." --Jinee Kim, Bay Area youth organizer
My Two Countries: America & Iran (tags)
Artist John Phillip Nybe (a.k.a. Sam Jon Nayebi) will light up Melrose Lightspace with his latest painting & photography series inspired by his recent journey to Iran last May; "My Two Countries: America & Iran" opens July 26 - 8:00pm at Melrose Lightspace, in Los Angeles.
Sidestepping Sanctions (tags)
While the Bush administration looks the other way, U.S. companies are dodging laws that bar them from doing business with nations accused of sponsoring terrorism.
Trading on fear (tags)
Trading on fear From the start, the invasion of Iraq was seen in the US as a marketing project. Selling 'Brand America' abroad was an abject failure; but at home, it worked. Manufacturers of 4x4s, oil prospectors, the nuclear power industry, politicians keen to roll back civil liberties - all seized the moment to capitalise on the war. PR analysts Sheldon Rampton and John Stauber explain how it worked.
U.S. is 3rd on World's "Economic Freedom" Index (tags)
Hong Kong, Singapore, and the United States rank first, second and third, respectively, in the latest edition of the 'Economic Freedom of the World' annual report by the Cato Institute and more than 50 other libertarian think tanks around the world.
Who wants to be a Palestinian Refugee? (tags)
ZIon will Win!
Immigration control- atool of the bosses (tags)
US M-16s to Nepal, Nepali troops to Iraq? (tags)
US expects Nepali troops in Iraq; M-16 airlift complete. See a connection?
About the "Kill a cop for the cause!" Redding Murder Case - Pt. 2 (tags)
The man who killed a Police Officer in Red Bluff, CA claims immunity because he was incorporated.
American Oil's next target: Africa (tags)
American oil corpse like ExxonMobil are moving aggresively into African nations. There is no end to the mess they seek to create for millions of people.
''Pogroms in America'' (tags)
The oddest thing about the "Jewish lobby" is that while its ability to intimidate both the U.S. government and the American media stems from a well-deserved reputation for highly effective grassroots organizing, it does so without having a genuine constituency.
U.S. Gets War Crimes Tribunal Exemption (tags)
The final deal dented the court's underlying principle that no one should be exempt from punishment for war crimes, and it angered court supporters and human rights groups.
EU ratifies UN Biosafety Proposal (tags)
"The protocol lets countries ban imports of a genetically-modified product if they feel there is not enough scientific evidence the product is safe and requires exporters to label shipments containing genetically-altered commodties such as corn or cotton."
Selecting Regimes for Change: The U.S. is Unfair in Choosing (tags)
It seems that the U.S. government has become the expert on regime change. The problem is that they have ignored some of the worst examples of bad regimes because of valued trade status, being allied in the war on terror, and economic importance.
Survey: Israel yet to grasp concept of democracy (tags)
The general conclusion of the survey, which is dubbed the "Israeli Democracy Survey" and will be conducted every year, is that Israel is basically a democracy in form more than in substance, and that it has yet to internalize fully the concept of democracy.
Questions of Life and Death (tags)
"Many politicians lack a moral imperative today. They act as though they have no options because the economy dictates conditions. This is not true. Politicians can still make decisions in the interests of many, not in the interests of a few.." From German
Libertarians mock the Left, May Day (tags)
The giveaway, and the clue to the real motive of today's left and their hangers-on, is that all their protests are against -- they are anti-capitalism, anti-free trade, anti-using the environment for man's benefit -- but they are not for anything.
Do We Really Have Free Speech? (tags)
The following is a transcript of a speech given by 13-year-old Charlotte Aldebron in Augusta, Maine on April 19, 2003.
"The System is always Fair", the Cat says to the Mouse (tags)
"Behind closed doors, the WTO negotiated GATS, an agreement on liberalization of services. Health conglomerates all over the world can steal solvent patients. The existential necessity water could soon be much more expensive." Trans. from German
Foreign aid, foreign disaster (tags)
Be it bilateral or multilateral, dispensed directly from a government or through an international lending agency, foreign aid embodies all the failures and tragedies that have come to typify our government-run domestic poverty schemes.
Libertarian Links Free Trade to Freedom, Peace (tags)
Palmer, a senior fellow at the Cato Institute, said anti-globalization activists are missing the bigger picture. By way of analogy, Palmer referred to Homer's Odyssey saying that Homer is telling the reader that "the cyclops is a savage because he doesn't trade. He lives in the preferred world of the anti-globalization activists."
Will Pre-emptive War Make America Safer? (tags)
Another aspect of U.S. safety is threatened by pre-emptive war. The Founding Fathers realized that not all enemies are foreign. What keeps U.S. citizens safe is adherence to the U.S. Constitution. Losing any part of the Constitution is precedent for losing other parts. Congress - the American people's representative - now has lost the constitutional right to declare war.
God Bless America (tags)
Globe and Mail Editorial (Toronto) (tags)
The team leader of the trial, Dr Anatoli Kamalai, acknowledged that there was "no measurable reduction" in HIV incidence with "no hint of even a small effect." The trial showed there to be no connection between sexual activity and AIDS.
DICTATORSHIPS (tags)
THE UNITED STATES MUST STOP SUPPORTING DICTATORSHIPS
The Spoils of War (tags)
Capitalists profit from war both from the destruction of a country and from the reconstrution afterwards, and of course from the natural resources like oil.
The US citizens are also to be condemned (tags)
The US citizens are also to be condemned, this was not just George Bush, you elected him. I am norwegian, and not an angry arab.
The only way to stop this massacre (tags)
If you ask an iraqi child who just lost 2 legs and one arm plus family members " what does it feel like to be free now?" What would you think it would be the answer ?
What Is Socialism, and Why We Oppose The Invasion of Iraq (tags)
Resist the war in the streets, workplaces, schools, and in the barracks! Organize to end all imperial bloodshed through socialist revolution!
hate feeling of young in Islamic and other countries for American. (tags)
Major issue that is coming from war is hate feeling of young in Islamic and other countries for American. American administration and its policies should understand that it is one of the largest democracies and democracy has its value. In order to retain its democratic values.
Iraqi Civilians Detained! (tags)
More human rights abuse.
Humanitarian Aid - what is the US responsibility in Iraq? (tags)
Excellent analysis. Especially issue #3 where she explains that the companies getting rich on war & rebuilding - Halliburton, Bechtel, etc - should pay for humanitarian aid to Iraq through a special 50% Excess Profit/Windfall for War tax.
Gulf States May Be Next: British MP (tags)
So this is the kind of thing that goes on behind their oaths of secrecy eh?
Greenpeace: Momentum Builds for New UN Peace Resolution (tags)
Demands for a UN emergency session are on the rise! 32,015 of you have written to UN Ambassadors around the world. You've sent 29,700 E-cards to friends, colleagues, fellow students, and family members. This is an extraordinary response in a very short time, and what do we want??? MORE! Please re-post onother indymedia sites.
Launch of a worldwide boycott of the US, the UK, Australia and Turkey. (tags)
A worldwide boycott of all trade with the US, the UK Australia and Turkey, and go-slow within those countries, was launched at the blockade in Adelaide Street, Brisbane, Australia, on Friday night.
COALITION OF THE WILLING? (tags)
Some reasons why countries support the US.
Call for an International Boycott and Go-slow (tags)
A call has been issued by a small Australian peace group, for a consumer boycott of all goods and services from the United States, Britain, Australia and their allies, and for everyone in these countries who is opposed to this war, to go slow economically in every way possible, until the war is over.
Spying, Lying, Coercion and Bribery (tags)
The dirty tricks that the United States has been using to influence public opinion and obtain a vote for war in the United Nations
Gnome$ for peace- How to Stop the Attack (tags)
Boycott the dollars and bring peace to the world. The more people you can persuade to join you in becoming a Gnome for Peace, the better the chance there is of weakening the dollar and the American economy by enough to prevent or limit a war.
War soon please. (tags)
Wake up from the ditto storm. Think for yourself and do not follow the heard in the streets of LA.
THOMAS JEFFERSON CALLING: The time for revolution is now (tags)
"Prudence, indeed, will dictate that governments long established, should not be changed for light and transient causes; and, accordingly, all experience [has] shown that mankind are more disposed to suffer while evils are sufferable than to right themselves by abolishing the forms to which they are accustomed. But, when a long train of abuses and usurpations, pursuing invariably the same object, evinces a design to reduce [the people] under absolute despotism, it is their right, it is their duty, to throw off such government, and to provide new guards for their future security." Thomas Jefferson
Bush's Arab and Muslim Propaganda Chief Quits (tags)
Bush's Arab and Muslim Propaganda Chief Quits CNN.com Monday 3 March 2003 U.S. official: 'She didn't do anything that worked'
March 5th! The War on Iraq and the People's Determination to Stop It (tags)
We are in the rapids. Powerful forces in Washington are clearly determined to launch this war. And this moment demands that millions of people respond with an even more powerful determination to resist. The Revolutionary Worker salutes all those who manifest bold resistance to this war on the National Moratorium to Stop the War on March 5 and in the days ahead.
How the Propaganda System Works (tags)
Arab League Vote against war is ignored
Bush Admin dirty tricks campaign to bug UN Security Council member phones & email (tags)
A leaked memo reveals the US is bugging work & home communications of UN Security Council members. Nixon did this sort of thing, and he was forced to leave office or face impeachment.
US boycott of products from countries against war? (tags)
After a long succession of nasty diplomatic slips by US diplomats, there is now talk of a US boycott of products from countries that do not want to be part of the coalition of the willing.. Published 2/14/03 in Telepolis
EU/WTO want to privatize Everything! (tags)
Now that we're totally distracted by the "War", the corporate dictators (EU & WTO) want to dismantle and privatize the US Postal Service, our schools, our water supplies, public power and anything else they can get away with!
MIDDLE EAST (tags)
THE UNITED STATES FOREIGN POLICY TOWARDS THE MIDDLE EAST CHANGED FOREVER AFTER SEPTEMBER 11, 2001. THE POLICY NOW IS TO REPLACE DICTATORSHIPS WITH DEMOCRACIES.
Appeasement is not the Answer (tags)
Appeasement is not the answer
Diplomatic Riffs (tags)
It's funny that the president brags about having the support of Spain, Italy, Slovenia, Poland, Hungary, Lithuania, Estonia, Bulgaria and Denmark. Those countries together couldn't whip Iraq.
USDA Announces WTO-Level Ministerial Conference for Sacramento (tags)
VENEMAN ANNOUNCES AGRICULTURAL SCIENCE AND TECHNOLOGY MINISTERIAL CONFERENCE
BBC: US Senators Question Iraq Evidence (tags)
"However, the surprising mood of scepticism evident in the committee room was an indication that President Bush and Mr Powell still have work to do if they are to gain bipartisan support for their Iraq policy."
Chomsky interview about Haiti (tags)
On November 8th, 2002 Faiz Ahmad sat down with Noam Chomsky, MIT professor and world renowned critic of American foreign policy, to discuss the current situation in Haiti.
Problems! (tags)
So the Iraq is an problem, France is a problem, Germany is a problem.
Hard facts! (tags)
In effect we don't need any proofs to attack other countries, we attack them quite simply as the case may be
The Flexible Person (tags)
Hundreds of millions of people are forced to work migration between countries and continents.. People are transformed into socially uprooted vagabonds of markets..Flexibility means ..more output and more stress for less money. From German
IT'S NOT ABOUT OIL (tags)
WHY INVADE IRAQ?
The European Summit in Copenhagen (tags)
European summit
The fight for water. A fight for human rights (tags)
Water. We see it everywhere. (see related story below) Many of us take water for granted. All we have to do is open a faucet and this life-sustaining liquid just pours out into our glass to quench our thirst. But in many places it is not so simple to get that clean glass of water.
Hypocrisy that underlies HIPC (tags)
Vultures at the IMF/World Bank use accounting tricks to starve and further impoverish the poor.
Seattle Somali Community Speaks Out (tags)
In response to indefinite detentions, deportations and growing racist attacks, Somali Community leaders in Seattle are joined by the Hate Free Zone and Somali Justice Advocacy Center to bring attention to this international human rights situation.
FREE IMMIGRATION LEGAL CLINIC (tags)
Do you know if you must Register with the INS??? As per the INS new program of Alien registration
Next INS deadline Jan 10, 2003--- (tags)
January 10, 2003 is the next deadline imposed by the INS for registration, fingerprinting and interviews for so-called "visitors" from Afghanistan, Algeria, Bahrain, Eritrea, Lebanon, Morocco, North Korea, Oman, Qatar, Somalia, Tunisia, United Arab Emirates, or Yemen.
ACLU Calls Immigrant Registration Program Pretext for Mass Detentions (tags)
By January 10, 2003, citizens of 13 additional countries – Afghanistan, Algeria, Bahrain, Eritrea, Lebanon, Morocco, North Korea, Oman, Qatar, Somalia, Tunisia, United Arab Emirates and Yemen – must also submit to registration, a move that could push the detentions into the tens of thousands, the ACLU said.
INS Dragnet: Only Muslims need apply? (tags)
INS deadline (anyone even heard about it?) to register is today, December 16th, 2002. Do your best to take ABSOLUTELY no notice of the glaring ommision of Saudi Arabia...
Atrocity propaganda! (tags)
The Right to Development (tags)
"The 1994 UN Development Program proposed a world social charter with the goal of building a society where the right to food is as sacred as the right to vote..and the right to development is a fundamentaql human right Brigitte Hamm is a professor in Duisberg.
Proud and Insolent Youth Incorporated (tags)
Presidents Chance & Putin Call for Israel to Disarm (tags)
The British Empire's imposition of the Zionist state has obviously been among the worst blunders in history. As always, Jews, Muslims, Christians, Buddhists, pagans, agnostics, atheists and UFO-worshipers are welcome to live in secular nations such as the United States and Russia....
Sweatshops and Imperialism (tags)
The conditions in Third World sweatshops and the relationship between sweatshops and imperialism.
McDonald's closing 175 restaurants (tags)
McDonald's Corp. announced today that it will cease or restructure operations in seven Middle Eastern and Latin American nations, eliminate up to 600 jobs and close about 175 restaurants in 10 other countries.
CONDOLEEZZA'S NONSENSE ABOUT DEMOCRACY (tags)
Some history rather than hysteria...
Online Petition: US CITIZENS OPPOSE WAR ON IRAQ (tags)
US citizens oppose war on Iraq
An International Call From Veterans and Soldiers To Members Of The Armed Forces: (tags)
We Are Facing a Serious Choice: Perpetual War or Resistance!
World Bank and International Monetary Fund strangle economies of Third World (tags)
The World Bank's origins go back to the Bretton Woods conference of July 1944, when leaders from 43 nations assembled to start building the global economy in the aftermath of World War II. U.S. Treasury Secretary Henry Morgenthau, hoping to avoid another worldwide depression, called for creation of institutions that would create "a dynamic world community in which the peoples of every nation will be able to realize their potentialities in peace."
Capitalism, Fascism and World War 2 (tags)
The relationship of fascism and capitalism as illustrated by the events of World War 2
Africa AIDS Crisis Worsening (tags)
The HIV/AIDS pandemic is playing a key role in the worsening health crisis in Zambia and southern Africa, according to James Morris, the U.N. secretary-general's special envoy for the humanitarian crisis in six southern African countries.
Is September 11 driving you nuts? (tags)
The people of the United States should rightly remember, and so should the people of the entire world. But the events of the 11th of September 2001 were so powerful, so disturbing, that it would be catastrophic if they were remembered for the wrong reasons.
Earth summit hits poverty, pollution (tags)
JOHANNESBURG – Delegates to the third Earth Summit reached a minimal agreement on policies aimed at lifting four billion people from poverty and curbing threats to the global environment.
Global Day of Protest against Coca-Cola: O17 (tags)
In Barcelona, major AIDS groups from across the globe issued a call for actions on October 17th demanding AIDS drugs for Coca Cola's 100,000 employees and bottling plant workers in Africa. The numbers of countries and cities planning actions is growing rapidly, and the company is running scared. Here's how you can get involved in the Global Day of Protest.
World Leaders Debate Environment and Economics at Largest UN Summit Ever Held (tags)
World leaders, activists, and business elite gather in Johannesburg to decide the fate of the planet in what is shaping up to be merely another clash between globolizatiion proponents who want to impose free markets and privatization as the path to sustainable development, versus environmental and social justice activists who say the neoliberal model is inherently unsustainable.
take our Gmaize or starve (tags)
they call themselves humanitarian, but in pushing GM food on to third world countries they are opportunistically exploiting other peoples tragedies
Sachs says: 'Forget Debt, Spend on AIDS' (tags)
Columbia University economist Jeffrey Sachs says poor countries should simply stop making debt service payments and spend that money on health & education instead. As Jubilee South says, "Don't Owe, Won't Pay."
Anti-globalization (tags)
Why the Palestinians
anit-globalizatiom (tags)
Palestine (tags)
Globalization Has Helped the Poor: Study says (tags)
Contrary to anti-globalizers study says that globalization has helped the poorest in the world
See Ya @ 5 (tags)
Take the Not In our Name Pledge @ pm TODAY. Downtown.
Peace Concert in Tijuana/San Diego (tags)
WHAT:CROSS BORDER PEACE CONCERT WHEN:FRIDAY, JUNE 21 12:00 NOON- MIDNIGHT SOLSTICE CELEBRATION WHO:THE SAN DIEGO COALITION FOR PEACE AND JUSTICE and HUMANISTAS POR LA PAZ WHERE: THE US/MEXICO BORDER AT PLAYAS De TIJUANA/BORDERFIELD STATE PARK
Not In our Name Nation June 6th Protests- LA Info (tags)
a kick off to what is to be a mass demo in the fall
Bush in Paris (3) (tags)
One French demonstrator carries a mock U.S. Flag made up of the countries invaded or bombed by Washington over the years. Photo by JJ Méric.
Muslims Are Good Folks (tags)
wish more Americans had an opportunity to get to know Muslims. Then they would not be susceptible to the silly anti-Muslim propaganda that is floated by some right-wing Christians.
The objections some Muslims have to Western culture are the same as those some Christians have. They don't like the violence, the immorality, the pornography and the greed.
Bush Admin to Nix Int'l Court (tags)
Petition: US CITIZENS OPPOSE WAR ON IRAQ, OTHER COUNTRIES, AND US GOV MILITARISM (tags)
US citizens and permanent residents OPPOSE war on Iraq, other countries, and US government militarism.
The Shortwave Report 3/8/02 ¡LISTEN GLOBALLY! (tags)
A 30 minute review of news and opinions recorded from a shortwave radio. 2 files- broadcast-13.7MB & quick/streaming-3.4MB. With times and freqs for listening at home. Free to rebroadcast upon notification. Netherlands, Germany, Russia, Cuba.
Millennium Ride in LA (tags)
Round the world charity biker offers free powerpoint presenttion of his two years/29 countries on the road. He comes to your event free, presentation is an hour, appeals for voluntary donations. charities: Doctors withut Borders & Health for All. Highly entertaining, funny. Up to mid April 2002. $56,000 raised so far. Many adventures.
Third World Debt Deemed Illegal (tags)
People's Tribunal on the Debt in Porto Alegre finds $1.8 trillion in debt illegal; World Bank, IMF & other lenders guilty of genocide
Take over Argentinean embassies! (tags)
Social revolution in Argentina NOW, tomorrow the world!
The Ugly American Dollar Campaign (tags)
The United States dollar is being introduced in several Central and South American countries as a means of economic development. Even in Russia, the American dollar is more popular than the Ruble -- proving correct the old proverb: if you can't beat'em, buy 'em.
TRIPS: will the majority prevail? (tags)
NGO Statement on Ministerial Declaration on TRIPS and Public Health. From Act-Up Paris, Consumer Project on Technology, Consumers International, Health GAP Coalition, Médecins sans Fronteires, Oxfam, Tebtebba Foundation, Third World Network
Naked protest by Ex-British Tank Commander (tags)
An ex-British army tank commander, who saw service in Northern Ireland and other countries, carried out a naked protest against the war in Afghanistan today, Sunday 28th October 2001.
QATAR CHARGES U.S IS A TERRORIST (tags)
The leading newspapers in a dozen Near East countries and India and Pakistan oppose the U.S. action in Afghganistan.
QATAR CHARGES U.S. AS TERROIST (tags)
The leading newspapers in 12 Near East countries and others condemn U.S. action in Afghanistan.
Afghanistan one of the most mine-affected countries (tags)
PESHAWAR: Latest reports suggest that nearly two decades after Afghanistan plunged into civil war, it is still one of the severely mine-affected countries in the world.
U.S. WAR AGAINST 60 "TERRORISTS COUNTRIES" (tags)
U.S. War Against 60 "Terrorists Countries"
U.S. to U.N.: We May Have To Attack Other Countries (tags)
So when will we bomb Florida?
In 32 of 35 Countries People Prefer Non-Military Approach (tags)
A Gallup International poll finds that substantial majorities in 32 out of 35 countries favor a criminal justice response to the terrorist attack on America, rather than military action. Opposition to attacking civilians is even stronger. However, when asked if their country should join in US military action, supporters outnumber opponents in 14 countries. This forced-choice approach still left miliatry supporters outnumbered in 19 countries.
Al-Ahram Weekly: Caught in the Middle (tags)
A View From An Egyptian Weekly Online: Cairo is urging caution and restraint to a hotheaded Washington, warning: look before you leap, write Nevine Khalil and Soha Abdelaty. [Excerpts. For full story, click link provided.]
BBC : 60 nations on US target list (tags)
just up on BBC site
IPPN Executive Committee Statement (tags)
"The progressive movement must stand firm at this dangerous time in opposition to the scapegoating of Arab and Islamic people, in support of basic civil and democratic rights, and for a U.S. foreign policy which is about justice, fair trade, environmental sustainability and a raising of the living standards of all people, especially those in the countries of the Global South."
America: The Good Neighbor. (tags)
This, from a Canadian newspaper, no less, is worth sharing.
Why indymedia should have ongoing reports on Colombia's civil war (tags)
This is a comment I posted in response to the article "The FARC faces the empire" http://la.indymedia.org/display.php3?article_id=9478
Hanged Effigies Kick Off Expected Wave of G8 Demos (tags)
Dummies hang from a crane in Genoa July 19, 2001 as anti-globalization demonstrators gather in Genoa to launch protests during the upcoming G8 Summit. (Dylan Martinez/Reuters)
Universities and Countries Begging Companies (tags)
Science was noticeably absent at BIO2001: Most of the booths were of Western cities, states, countries et al. and universities competing with each other to give biotech corporations the sweetest offer in the form of tax breaks and office parks etc. Here we see one of Germany's booths: It compares the contributions of Martin Luther and Back to those of biotech companies. Come on! The actual biotech company booths appeared to be not staffed by scientists seriously discussing the pros and cons of their work, but by capitalists out to make as much money as possible.
Golden Rice Is Not A Substitute For Vitamin A (tags)
Golden Rice is a genetically modified food intended to reduce the Vitamin A deficiency that plagues people in developing countries, unfortunately it creates more problems than it solves.
Brutal Attack on Argentine Activist's Daughter!!! (tags)
Things we '1st world citizens' don't hear about.
States of Unrest: Resistance to IMF Policies in Poor Countries (tags)
Ignored by the corporate media fixated on "spoiled students," 50 mass demonstrations in the global south took place in at least 13 countries between protests against the WTO in Seattle in November, 1999 and the IMF/World Bank demonstrations in Prague in September, 2000.
Letter Bomb the President (tags)
Open letter from Deivid's Thang to George www. Bush
Leonard Peltier On The FTAA--Roots In Colonization, Feudalism & Imperialism (tags)
The FTAA is a continuation of the imperialism that began thousands of years ago in Europe…. Advocates of the FTAA would not dare refer to their policies as forms of colonization or feudalism…. Instead they will justify their actions in the name of "development" for the "poor" countries of Central and South America. Development? What the first peoples of the Americas need is "recovery" not development. Recovery from the very same colonization, domination, and genocide that multi-national corporations want to perpetuate for their own gains today.
U.S., WTO threaten Brazil medicine (tags)
The U.S. and the WTO threaten Brazil with trade sanctions to prevent them from distributing life-saving AIDS drugs to their people for free. They claim Brazil violates the patents of giant drug companies.
FREE ELECTIONS? (tags)
For free and fair elections in the United States!
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Let's talk about female friendships in fiction
Ditch the drama! Here are 5 of the most awesome, supportive female friendships that fiction has to offer.
In Virginia Woolf's infamous essay on females and fiction, A Room of One's Own, one of the most potent points Woolf raises is about is how negatively women's relationships have been portrayed in literature. Up until the point when she was writing in the 1920s, she argued, fiction was almost exclusively written by men and – as a result – the women depicted were almost exclusively in relation to men too. What's more, when women were shown to have relationships with other women – aside from when those women were related – these relationships were almost entirely combative, riddled with jealousy, sabotage and betrayal.
Ninety years later, women are writing more books than ever but how much has this shift changed the way that female relationships are portrayed? It would be easy to argue, not that much! More often than not, even those friendships which begin fondly still turn sour. The Girls by Emma Cline, Girls on Fire by Robin Wasserman and Dare Me by Megan Abbott are a few recent hits where women are at odds. Whilst competition and back-stabbing makes for great drama, we all know that this doesn't fully reflect the female experience in real life. So to balance the scales, we've picked five of the most fabulously awesome and supportive female friendships that literature has to offer…
Beaches by Iris Rainer Dart
Cee Cee Bloom is a loudmouthed, spunky 10-year-old from the Bronx with dreams of being a Hollywood actress. Bertie White is a timid 7-year-old from Pittsburgh who wants to be a mother. They meet by chance in Atlantic City and promise write. What follows is a friendship that spans thirty years; one that starts with simple letters between children but becomes an extraordinary, life-long bond. They reunite over the years – always at the ocean – their friendship weathering love affairs, jealousy, missed opportunities, crushing disappointment, loneliness and perceived betrayals, in bursts of emotion they've had nothing to do with but repress in their times apart. This is an energetic, warm and funny novel that will leave you reaching for the tissues.
Girls of Riyadh by Rajaa Alsanea
Girls of Riyadh is an exposé of the life and loves of four privileged twentysomething women, looking for happiness in Riyadh. It is often described as the Saudi version of Sex and The City, in which designer handbags and nose-jobs are converted like diamonds but interactions with men are much harder to engineer. It was banned in Saudi Arabia when it was published in 2005 so – of course – then became an international bestseller, but it actually gives politics a wide berth, focusing instead on how women here fall swiftly and madly in and out of love just like women everywhere else. Written as a series of emails between the women, it is a sharp-tongued, sassy and subversive story about friendship, love, lust and hypocrisy, in a society where repression feeds rather than suppresses desire.
The Girls of Slender Means by Muriel Sparks
First published in the 1960s, The Girls of Slender Means is a small novel but a firecracker, slight enough to be read in one sitting but that stands strong against the winds of time. It follows a group of young, single women living in together in post-war London, working as secretaries and clerks, bickering over dresses and suitors, and trying to pretend the war never happened. Sparks writes with piercing wit but the darkness of her subject matter is never far away. The women are impulsive, fearless and bewildered in that way that only comes with youth, but each is struggling to hide secrets far greater than their years. This is a book about the end of innocence, and – with new awareness of the world and themselves – how the dynamics of friendship and its value change.
The Joy Luck Club by Amy Tan
In 1949, four Chinese women who had recently immigrated to San Francisco start The Joy Luck Club to eat dim sum and play mah-jong. Forty years later, their meetings have become more than just an opportunity to giggle and gossip. The women are still outsiders in America, clinging to fading memories of their homeland and uncertain in their search for a new cultural identity. The novel's structure cleverly emulates a mag-jong game, with four parts and sixteen interlinking chapters that reveal the women's individual tragedies and their fraught relationships with their westernized daughters as they grapple to reconcile the present with the past. Written with tenderness, empathy and straightforward prose, this is a novel about overcoming the enormity of change in your life, and of raising friends up.
A Thousand Splendid Suns by Khaled Hosseini
Mariam and Laila – two Afghani women married to the same vile husband – ought to be enemies. Mariam is twenty years Laila's senior, bitter in her childlessness and subject to constant physical and emotional abuse from overbearing Rasheed. Laila is privileged by comparison, young, smart and fertile as she is. Yet from where their relationship begins as confrontational, it evolves into an inspirational, fiercely protective friendship, where connection is found in shared terrors and decimated aspirations, where they face their captor together and which ultimately saves their lives. This is a beautiful, brutal tale of fear, loss and what it means to be family – and the heroism that the deepest friendship can inspire.
Click here for more great bookish articles, and to find books you'll love and readers like you.
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Weimarrepubliken (: ) är en period i Tysklands historia från den 11 augusti 1919, då en demokratisk och parlamentarisk författning antogs i Weimar, till den 30 januari 1933, då Adolf Hitler utnämndes till rikskansler. Landets officiella namn var från landets enande 1871 till 1943 Tyska riket (: ).
Politiskt system
Weimarrepubliken var en parlamentarisk demokrati och en förbundsstat, dock med betydande befogenheter för den centrala makten, riksstyrelsen. Riksstyrelsen leddes av en folkvald rikspresident. Förste presidenten blev socialdemokraten Friedrich Ebert (1919–1925), dess andra och sista president blev den partipolitiskt oberoende konservative Paul von Hindenburg (1925–1934); von Hindenburg omvaldes i presidentvalet 1932 men avled 1934, varefter Hitler avskaffade presidentämbetet och antog titeln Führer ("ledare").
Regeringen i Weimarrepubliken leddes av rikskanslern som var ansvarig inför parlamentet, riksdagen, som var direktvalt med allmän och lika rösträtt för män och kvinnor. Riksdagen valdes i direkta val och saknade spärregler, vilket gynnade uppkomsten av småpartier. Regeringarna blev därför svaga, vilket ledde till regeringskriser och återkommande nyval.
1918–1919
Tiden efter det tyska kejsardömets fall och vapenstilleståndet i november 1918 och innan republikens utropande var en tid präglad av kaos och enorma sociala konflikter i Tyskland. I flera tyska städer tog arbetar- och soldatråd över styrelsen, i storstäderna utvecklades situationen till gatukamper och kuppförsök som Spartacusförbundets uppror. Dagligen begicks det mord, och stridigheter förekom mellan olika fraktioner på gatorna i de tyska storstäderna, mellan kommunister och frikårer och andra fraktioner. Vidare betydde Versaillesfreden att landet blev kraftigt skuldsatt mot utlandet och spädde på den oroliga utvecklingen. När man till sist skapade en ny, demokratisk tysk republik valde delegaterna att samla sig till kongress i kulturstaden Weimar då andra, större städer ansågs vara för oroliga för att delegaterna skulle kunna samla sig där. Republiken utropades den 11 augusti 1919.
1919–1925
Republiken hade ingen lätt start. I mångt och mycket fortsatte situationen de första åren som den hade varit i slutet av 1910-talet med hög arbetslöshet, hyperinflation, sociala oroligheter m.m. Den ekonomiska situationen förvärrades särskilt akut efter att franska och belgiska trupper gick in i Ruhr för att försäkra sig om att skadeståndet skulle komma dem tillgodo i form av varor, se även Ruhrockupationen. Hyperinflationen ledde till att alla besparingar blev mer eller mindre värdelösa, strejker och kravaller var vardagsmat i stora delar av Tyskland. Se vidare inflationen i Weimarrepubliken. Den 12 mars 1920 startades Kappkuppen i Berlin av högerextremister.
År 1924 tog man tillsammans med USA och de övriga segrarmakterna från första världskriget kontakt med varandra och gjorde upp om en ny avbetalningsplan för krigsskadeståndet som landet hade ålagts i Versaillesfreden. Detta ledde till Dawesplanen. Den ekonomiska situationen under senare delen av 1920-talet förbättrades radikalt, mycket tack vare amerikanska krediter och investeringar som flödade in i Tyskland. Tyskland tilläts gå med i Nationernas Förbund 1925.
1929–1933
Börskraschen 1929 kom snabbt att få återverkningar även i Tyskland. Den amerikanska marknaden föll samman som ett korthus, och då amerikanska investeringar stod bakom lejonparten av den tyska uppgången under slutet av 1920-talet drabbades Tyskland snabbast och hårdast av alla europeiska stater. Situationen i början av 1930-talet liknade den som hade varit i republikens början: Hög arbetslöshet, hög inflation, enorma sociala oroligheter, gatukamper och kravaller varje dag. Nyval utlystes 1930 och flera gånger under 1932.
Nationalsocialistiska tyska arbetarepartiet (NSDAP) hade fram tills börskraschen 1929 haft endast marginellt stöd, men de nya dåliga tiderna gav partiet ett rejält uppsving. Även kommunisterna drog fördel av de dåliga tiderna, medan särskilt de borgerligt konservativa och de liberala partierna tappade stöd. Vid det sista riksdagsvalet 1932 blev NSDAP största parti, men de hade bara stöd av drygt en tredjedel av väljarna. Socialdemokraterna och kommunisterna var dock tillsammans större.
Weimarrepubliken gick slutligt i graven när Hindenburg utsåg Adolf Hitler till rikskansler, det vill säga Adolf Hitlers Machtübernahme den 30 januari 1933.
Förbundsländer
År 1925 bestod Weimarrepubliken av 18 länder:
Referenser
Litteratur
Externa länkar
Lindgren, Lena (1999) "David Dyzenhaus: Legality and Legitimacy. Carl Schmitt, Hans Kelsen and Herman Heller in Weimar", Retfærd nr 85 /nr 2 1999, sid. 85-90
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Q: 401 Unauthorized error with WebDeploy using non-administrator user and IIS user on Windows Server 2016 I am getting 401 error when trying to deploy remotely.
Event log from server -
The following information was included with the event:
IISWMSVC_AUTHORIZATION_SERVER_NOT_ALLOWED
Only Windows Administrators are allowed to connect using a server connection. Other users should use the 'Connect To Site or Application' task to be able to connect.
Process:WMSvc
User=deployer
I am using following script to deploy
msdeploy.exe -verb:sync -source:package=c:\Jenkins\workspace\Myproject.zip - dest:auto,wmsvc=[server_ip],usernverame=iis_deployer,password=iis_deployer,authType=basic -setparam:name='IIS Web Application Name',value='Default Web Site\myproject\master' -allowUntrusted
It works fine if I use the admin user. But it fails for non-admin and IIS users.
I have tried to set the permission and delegated rules by following this article-
https://blogs.iis.net/krolson/using-iis-manager-accounts-for-web-deployment-tool-msdeploy-delegation
Nothing seems to work.
A: I have been struggling with this error for hours and I found the solution. In the MsDeploy.exe command line you must specify the site name in the "wmsvc" parameter. For instance:
"C:\Program Files\IIS\Microsoft Web Deploy V3\msdeploy.exe" -source:package='d:\temp\xxx.zip' -dest:auto,wmsvc="https://192.168.XXX.XXX:8172/msdeploy.axd?site=Test",userName='usr',password='pwd',includeAcls="False" -verb:sync -disableLink:AppPoolExtension -disableLink:ContentExtension -disableLink:CertificateExtension -AllowUntrusted -retryAttempts=2 -setParam:'IIS Web Application Name'='Test'
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Sunday, May 2, 1999 - Page updated at 12:00 AM
Northwest Places
The Olmsted Legacy -- The Fabled Massachusetts Landscape Firm Got To Seattle Early, And That Has Made All The Difference
By David B. Williams
The M Word Along with the rest of the planet, Pacific Northwest is dubiously eyeing the approach of a new millenium. We don't want to add to the general racket on this subject. But we have noticed that thinking about the future seems to tug at our feelings about the past. So we'll quietly honor the millennium we're about to leave with a few pieces about interesting, sometimes overlooked local history. Today's feature is the first of an occasional series. -------------------------------
CREATING AN ELABORATE park system was probably the furthest thought from their minds when members of the Denny party arrived at Alki Point on Nov. 13, 1851. All they could see through the rain were trees, more trees and water. Building a home was far more important than building a place to play. And yet, just 33 years later Seattle had its first park and 20 years after that, the city had a comprehensive plan for major parks and parkways that would rival any found in the United States.
Few cities in the world can claim such an achievement, either in breadth of vision or efficiency. Today's Seattle residents, famous for their ability to squabble for years over creating even a neighborhood traffic circle, might look for inspiration to the architects most responsible for crafting the master plan, the Olmsted family of Massachusetts.
The family is best known for designing Central Park in New York City and the Boston and Buffalo park systems, but the Olmsted Brothers firm also worked in Seattle for 38 years. The 37 parks and playgrounds the firm designed include Colman, Frink, Green Lake, Interlaken, Jefferson, Mount Baker, Seward, Volunteer and Woodland parks, Washington Park and Arboretum, as well as Lincoln Park (now known as the Bobby Morris Playfield), Hiawatha Playground and Lake Washington, Magnolia and Ravenna boulevards.
Park planners across the country recognize Seattle's Olmsted park system as one of the best-preserved and best-designed in the United States. Many eastern cities have only one or two Olmsted-designed parks; in Seattle the architects created a system of linked beaches, boulevards and natural spaces that is as distinctive as the city's setting.
ALTHOUGH THE OLMSTED Brothers are sometimes credited with starting Seattle's park system, five public parks existed when the firm's lead designer, John Charles Olmsted, arrived in 1903. The city's park movement had begun inauspiciously in 1884, when the city converted the town cemetery into a five-acre park and named it Seattle Park. Now known as Denny Park, for David and Louisa Denny, who donated the property, the land was located then on the outskirts of town. Visitors reached the park via a narrow track cut through the surrounding virgin forest.
The system grew tenfold in 1887. George Kinnear donated 14 acres at the base of Queen Anne Hill. The city also designated 40 acres on top of Capitol Hill as Lake View Park. A few years later the name was changed to City Park and in 1901 to Volunteer Park, to honor veterans of the Spanish-American War.
While the city created these parks, five privately owned parks also opened to the public. The owners of Seattle's trolley system created Madison, Madrona, and Leschi parks. Often described as "pleasure grounds," they included pavilions, horse tracks, amusement rides and dance halls. They served one purpose for the developers: promotion and sale of the surrounding land, which also happened to be owned by the trolley lines.
Another developer, William Beck, owned the fourth private park, Ravenna, which he had named for the Italian city. Beck cleared the surrounding land but left the deep ravine mostly intact. He built meandering paths through the forest of Brobdingnagian-sized Douglas fir and Western red cedar, installed benches and pavilions and developed the sulfur springs. Beck charged 25 cents to enter his park.
The biggest change to Seattle's nascent park system occurred in 1900 with the acquisition of the fifth private park, Woodland, and what would become Washington Park. The city bought Woodland Park for $100,000 from the Guy Phinney estate. With the purchase, Seattle gained more than 100 acres of mostly cleared land, along with Phinney's menagerie, which included herds of deer, elk and buffalo he had gathered to promote his own land sales around Green Lake. Seattle acquired most of Washington Park from the Puget Mill company, which later developed Broadmoor Country Club.
Public sentiment did not favor such extravagant purchases. The Argus newspaper wrote of the proposed acquisition of Woodland Park: "If this park was properly located there might be some excuse for paying the enormous price. But to pay more than it would bring to cut up into town lots for a piece of land much of which is barren wilderness, and none of which possesses the features that Seattle should demand in a park, is little short of criminal." No more large tracts of land would be acquired for another eight years.
DESPITE OPPOSITION to the purchase of Woodland Park, the Seattle Board of Park Commissioners decided in 1902 that it wanted a more elaborate park system. The board planned on hiring the best landscape architect in the country, Frederick Law Olmsted, believing his name would add an air of distinction to the growing city.
When the board contacted the landscape-architecture firm in Brookline, Mass., however, members discovered that Frederick Law Olmsted was in poor health. (He would die the following year.) His son, Frederick Jr., had joined the firm, now known as the Olmsted Brothers, but he was teaching and could not make it to Seattle. The firm wrote that its senior partner, John Charles Olmsted, was available.
The dubious board wanted to know more about this "other Olmsted." After the firm sent a letter listing his extensive park-planning work (something the firm did not normally consider necessary) the board finally hired John Charles Olmsted. What they got, apparently without realizing it, was the most experienced landscape architect in the country in 1903.
His career had begun in 1875, when he apprenticed with his stepfather, Frederick Law Olmsted Sr. The younger Olmsted's credits included design work on the Boston, Louisville and Rochester park systems, as well as numerous private estates throughout the east.
Olmsted and his assistant, Percy Jones, arrived in Seattle on April 30, 1903. They could not have timed it better. Money from the Klondike Gold Rush had helped Seattle become a more wealthy city. Furthermore, the anti-park tide was ebbing. The year before, a full-page article in the Seattle Post-Intelligencer had urged the city to acquire more land and to develop an elaborate park system. The story, "Let Us Make a Beautiful City of Seattle," ended with sparkling endorsements for parks from many of the city's leading citizens.
Besides public support, Olmsted had two important documents that helped his evaluation of Seattle.
Eleven years earlier in 1892, Superintendent of Parks Edward Otto Schwagerl had produced a city-wide plan for a park system. Schwagerl proposed two parks on Puget Sound (Fort Lawton and Alki Point) and two on Lake Washington (Sand Point and Seward Park). Parkways would link these parks with the privately owned "pleasure grounds" and with the existing public parks. Schwagerl wrote that "establishment of a fine system of Parks and Drive-ways as such is manifestly the most effective means of rendering a city a beautiful and desirable place of residence."
Later, in 1900, assistant city engineer George Cotterill had produced a plan for a 25-mile system of bicycle paths around the city. Olmsted incorporated several miles of these trails into his plan, including the paths through what would become part of Interlaken and Washington parks.
With these plans, and accompanied by a host of park commisioners, Olmsted and Jones spent the month of May surveying Seattle by horse, trolley, foot and boat. They left for Massachusetts on June 6 and just under a month later, on July 2, 1903, sent their formal report back to Seattle.
The Seattle City Council approved the Olmsted Brothers' "A Comprehensive System of Parks and Parkways" in October 1903.
Olmsted wrote that the "primary aim should be to secure and preserve for the use of the people as much as possible of these advantages of water and mountain views and of woodlands, well distributed and conveniently located." Recognizing the changing real-estate market, he urged the city to move swiftly to acquire as much land as possible, especially "all the borders of the different bodies of water."
Seattle's citizens actively supported the plan. In the eight years following the original proposal, city citizens approved bond issues totaling $3.5 million (about $57 million in 1999 dollars) for park enhancement.
One other event also made a significant impact on their plan. In March 1904, city residents voted to give the Board of Park Commissioners powers separate from the City Council. (The pre-1904 powers of the Park Board were later characterized by its successors: "After learning it was necessary to kneel to the council and play the political game to secure appropriations on the work, nearly all of these 35 commissioners resigned or retired in disgust.")
With the change, the new board would have the same independence as park boards in eastern cities. Taking advantage of this change in the power structure, Olmsted Brothers proposed to "furnish a competent and experienced park superintendent" and then pay him, too. Their man, John Thompson, remained the Seattle parks superintendent for 17 years. With new power, the board began to implement the Olmsted plan for Seattle.
The central feature of the Olmsted plan was a 20-mile-long parkway that ran from Bailey Peninsula (Seward Park) to Fort Lawton (Discovery Park). From Bailey, the pleasure drive would snake along the lake shore, climb up and wrap along the bluff that now encompasses Colman and Frink parks, dive back down to the water at Madrona Park, and eventually turn inland to Washington Park. From here, the roadway would cut north to the University of Washington campus, pass through it to Ravenna Park and the adjacent ravine (Cowen Park), and eventually parallel the brook that flowed from Green Lake. The parkway would continue through Woodland Park, turn south again to reach Queen Anne Hill, wrap around the hill's north end and through Interbay to Smith's Cove, then end with an extension along the Magnolia bluffs to Fort Lawton.
A spur road would connect Lake Washington Boulevard at Mount Baker Park to Beacon Hill Park (Jefferson). A second spur would go from Washington Park along Interlaken Boulevard of North Capitol Hill, with forks to Volunteer Park and Roanoke Park. Another boulevard would connect Kinnear Park on lower Queen Anne with Magnolia.
While the Olmsted report focused on park and boulevard development, it also promoted a new concept to Seattle - playgrounds.
According to David Streatfield, a University of Washington landscape-architecture professor, John Charles Olmsted was a pioneer in this concept.
"Olmsted believed that playgrounds were a necessity for a civilized society. Children would learn fairness and decency via sports in the playground," Streatfield says.
In the firm's 1908 report to the park commissioners, Olmsted recommended locating small parks and playgrounds, oriented toward young children and women with babies, within a half-mile of every home. He also supported additional playgrounds and outdoor gymnasiums for older boys.
THE 1903 REPORT was the beginning of a relationship between the Olmsted Brothers and Seattle that lasted until 1941. Initially, the city hired the firm to make plans for all the parks it already owned, none of which had been formally designed. The city then asked the firm to create plans for the new lands it acquired: Colman, Cowen, Frink, Green Lake, Leschi, Madrona, Ravenna, Seward and Schmitz parks. The Washington Park Arboretum, designed by John Charles Olmsted's collaborator and successor in Seattle, James Dawson, was the Olmsted Brothers' final major public project in Seattle.
One of John Charles Olmsted's main concerns in Seattle was that parks should fit into their surroundings. In the 1903 report, he wrote: "The different parks of the city should not be made to look as much like each other as possible, but on the contrary every advantage should be taken of differing conditions to give each one a distinct individuality of its own."
Today Streatfield points out Colman and Frink parks and says, "It is clear that Olmsted recognized the fragility of the environment in these ravines." He did not alter the rough terrain, wild growth and tall trees, except next to the roads. Winding roads follow the land's contours, while overpasses allow people to easily move through the parks. Many consider these parks to be the best examples of Olmsted park design in Seattle.
On the other hand, since Volunteer Park was located within a "highly finished style of city development," Olmsted proposed a formal design with large grassy areas and extensive flower beds, clearing of densely growing fir trees and construction of an observation tower. Another style emerged at Washington Park, which combined a shady brook and swampy areas surrounded by native trees with "marked open spaces, (not) of large extent, and . . . covered smoothly with grass so as to adapt it for use by large crowds."
Olmsted did not always leave the land alone, though. He recommended lowering Green Lake by 4 feet to create a "lake within a park." The city went further and eventually lowered the lake 7 feet, creating almost 100 acres of additional land
According to Jerry Arbes, board member of the National Association of Olmsted Parks and also of Friends of Seattle's Olmsted Parks, the key to appreciating what the Olmsteds did for Seattle is not to look at the separate parks but to look at the system as a whole.
"There is a sense of connection in the Olmsted plan," Arbes says. "The interlinked parks provide a diversity of experience, yet one senses a relationship between these separate elements."
ALL OF THIS was far in the future on that dreary November day in 1851. A virgin wilderness of unknown proportions spread out in all directions from the Dennys and their clan. Their vision, though, was so grand that they named the place "New York, Alki" - New York bye and bye.
Seattle's founders rightly saw no need for preserves or parks in their forested domain. Fifty years later, though, civic leaders, and most important, John Charles Olmsted, understood that the city's growth had changed the fabric of the landscape so that Seattle's citizens now needed places away from "the restraining and confining conditions of the town."
Almost 100 years later, parks have become even more central to the city's existence. Surveys by the Park Department over the past 30 years show that parks act upon us at a level we don't always understand.
When asked if they used parks, many of those surveyed initially said "No," but when probed further, the respondents realized that they walked, biked or drove through or by a park, and most often one designed by the Olmsted Brothers, almost every day. Part of the park system's appeal is that these green spaces do not feel like designed landscapes, but blend into their residential surroundings. People noticed the trees and water in the parks almost daily.
This is part of the Olmsted Brothers legacy to Seattle - that so much land was protected and that the designs were good enough to survive.
How many Seattleites had their first experience with the natural world in an Olmsted park? We are fortunate. We still have places like Schmitz Park, which has hints of the virgin forest that once covered this land; Interlaken Park, which remains the province of frogs and quiet streams or even a more formal park like Woodland, where anyone can sit quietly under an 80-foot-tall oak and forget the city for a while.
Equally important is that the Olmsteds also gave the city a philosophy that protecting natural scenery was important. John Charles Olmsted wrote in his initial report to the city that it should "secure and preserve . . . these advantages of water and mountain views." Don Harris, director of environmental programs for the Seattle Department of Parks and Recreation, says the admonishment still guides planners. He cites as an example the 1989 King County Open Spaces and Trails Bond, which provided funding to preserve 531 additional acres of green space, as an example.
"Over the years, we have created a wonderful mix of larger parks such as Gasworks and Discovery and smaller jewels like Thornton Creek and Maple School Ravine, but the key element was the Olmsted legacy," Harris says. "It has helped provide the links that sustain the system."
David B. Williams, a Seattle writer, was once an interpretive specialist at the Frederick Law Olmsted National Historic Site in Massachusetts. Harley Soltes is staff photographer for Pacific Northwest magazine.
Next week: A closer look at the quirky history of the Woodland Park Zoo, which celebrates its 100th year in 1999. ------------------------------- WANT TO KNOW MORE?
THE WATER TOWER in Volunteer Park is a good place to begin learning more about the Olmsted legacy. In 1997, the Friends of Seattle's Olmsted Parks created a multi-faceted, interpretive exhibit that details the city's association with the Olmsted Brothers. It includes photographs; maps; text; field notes; historic postcards that address the creation of Volunteer Park, Seattle's water system and its relationship to the parks; specifics on what the Olmsteds designed in Seattle and a history of the Olmsted family's design work.
The permanent exhibit is open daily from 10 a.m. to dusk. For those with disabilities, an accessible version of the Olmsted interpretive exhibit is located at the Seattle Department of Parks and Recreation. For more information contact Parks and Recreation at 206-684-4075.
On Saturday, May 22, Friends of Seattle's Olmsted Parks will its show its six-projector "Olmsted Revisited" slide show in the Asian Art Museum at Volunteer Park. Showings will be at 11 a.m., noon, 1 and 2 p.m.
The Friends of Seattle's Olmsted Parks was formed in 1983 to promote awareness, enjoyment and care of the the city's Olmsted parks and landscapes, both public and private. For more information contact the group at P.O Box 9884, Seattle, WA 98109.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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A week of Games.
It's been a funny old week for gamers.
We've lost and regained the loved/hated high street retailer Game in blink of an eye, with its future still uncertain. Half the branches have been shuttered and the remaining portfolio is been split between the best performing locations of Gamestation and Game, with the business apparently being readied for a subsequent sale – prompting the return of GameStop to these shores?
In Sheffield, for example, we've gone from six stores to two – a branch of Game in the Crystal Peaks shopping centre and a Gamestation in the larger, presumably more expensive Meadowhall shopping mall. What this means for me is that neither are now particularly near me and I'd have to make a special trip to two not-especially-pleasant shopping venues. I see online and a couple of independent retailers in my future. The Guardian's Keza McDonald has a great piece on Game, the market space now open for indy retailers and what it all means over at their Games blog.
The vultures, of course, have begun to circle…
On another note, embattled publisher THQ announced that they'll be giving me what I wanted but in a totally roundabout way. By all accounts, we will see another single-player action game in the "Warhammer 40K" universe but at the cost of their expensive and apparently too unwieldy 40K MMO, which has been shuttered. Developers Relic and Vigil are accordingly reducing headcount by over 100 staff – who would work in an industry that precarious?
Sony's new console, meanwhile, may be called Orbis rather than PS4, will eschew any attempt to be backwards compatible and will eat any pre-owned disc that you attempt to put inside it's ravenous, bio-mechanical, terrifying cyber-maw.
Or, if you want to be boring, the new console will lock games to PSN accounts, preventing trade-in and the second-user market which is even now destroying the video game business from within. But I prefer my version. It's more Cyber.
Finally, in news which isn't actually news to anybody who's picked up a controller or installed a game to their hard drive in the last decade, EA are in the running for the coveted title of Worst Company In America…
Whatever you're playing this weekend, you probably want to make sure it isn't "Resident Evil: Operation Raccoon City"…
Filed under Gaming
Tagged as EA, Game, Gamestation, Gamestop, MMO, PS4, Resident Evil: Operation Raccoon City, Sony, THQ, Warhammer 40K
Une image d'une porte
Via Reddit user Pajamazombie
See, Brussels is interesting.
I'm always fond of a spot of Art Nouveau. When Mrs Rolling Eyeballs and I were holidaying in Paris nearly ten years ago, the Musee D'Orsay had an exhibit devoted to the style and it was one of those genuinely unexpected treats which you sometimes find yourself blundering into: the museum has free admission on the first Sunday of the month, unbeknownst to us, so we got to enjoy a glorious location, in arguably the world's most beautiful city, with inspiring art and design thrown in almost as an afterthought.
I'd love to talk more, but I'm off to gaze admiringly at Alphonse Mucha images for a few minutes…
Filed under Fluffrick, Random Notes
Tagged as Alphonse Mucha, Art Nouveau, Brussels, Musee D'Orsay, New Art, Paris
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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\section{Introduction}
There is nothing unphysical about matter being localised in a region of space; matter can simply exist at one location and not another. But for it to be localised in a finite period of time is altogether different. Indeed, as the matter would exist only for that period and no other, the situation would be a direct violation of mass conservation.
In conventional quantum mechanics, this undesirable situation is avoided axiomatically by requiring matter to be represented by a quantum state vector whose norm is fixed over time. There is, however, no corresponding restriction of the state vector over space.
\noindent
The underlying time-space asymmetry here can be traced to the fact that the state, and the matter it represents, is presumed to undergo continuous translation over time (as time evolution) but there is no corresponding presumption about the state undergoing translations over space. Even in relativistic quantum field theory, where both time and space are treated equally as the coordinates of a spacetime background, a similar asymmetry holds because time evolution and conservation laws are presumed to operate differently over time and space.
Nevertheless, time and space could have an equivalent footing at a fundamental level if any asymmetry between them were to arise \textit{phenomenologically} rather than being imposed axiomatically. Such a prospect is well worth pursuing because it would help us to understand the relationship between time and space. It would require finding an underlying mechanism that, due to phenomenological conditions, affects the spatial and temporal translational degrees of freedom in different ways to the extent matter can be localised in space but not in time. This suggests that we should examine the phenomenological character of the operations associated with the translational degrees of freedom.
The generators of translations in space and time are given by the momentum and Hamiltonian operators, respectively, and with them lies a difference that sets space and time apart in the quantum regime.
In fact, the last fifty years \cite{parity,CP,T1,T2,T3,T4} has shown that Nature is not invariant to particular combinations of the discrete symmetry operations of charge conjugation (C), parity inversion (P) and time reversal (T). The violation of these discrete symmetries are observed in various particle decays independent of position in space, and so they occur over translations in time and not translations in space. In terms of the corresponding generators, this implies that the Hamiltonian violates the discrete symmetries whereas the momentum operator does not.
The discrete symmetry violations are accounted for in the Standard Model of particle physics by the Cabibbo-Kobayashi-Maskawa (CKM) matrix \cite{Cabibbo,Kobayashi-Maskawa}.
Studies of the violations have been made in relation to baryogenesis in the early universe \cite{Sakharov}, the arrow of time and irreversibility \cite{Aharony1,Aharony2,Gell-Mann,Berger}, the time operator \cite{Courbage2}, quantum entanglement and Bell inequalities \cite{Datta1,Finkelstein,Hall,Lindblad,Squires,Datta2,Corbett,Clifton,Bramon,Ancochea,Bertlmann1,Bertlmann2,Andrianov,Barnett,Samal,Bertlmann4,Genovese}, decay and decoherence \cite{Courbage1,Fassarella}, complementarity and quantum information \cite{Bertlmann3,Bertlmann5,Hiesmayr}, quantum walks \cite{Lu}, and the potential for T violation to have large scale physical effects \cite{FPhys,FPhys2}. In particular, in Ref.~\cite{FPhys} I modelled the state of the universe as a superposition of paths that zigzag through time, and showed that T violation can, in principle, affect the time evolution in a global way. Then in Ref.~\cite{FPhys2} I showed that the effect on time evolution is greater when the paths are constructed in the limit of infinitely-small steps.
Here we explore the potential impact the violations of the discrete symmetries may have for giving quantum states different representations in space and time. The aim is not to study specific instances of the violations as observed experimentally, but rather to look for possible consequences of the violations in general terms. For this, the definitions of the P and T operations given by Wigner in relation to non-relativistic quantum mechanics \cite{Wigner} are sufficient and so we shall undertake the analysis using the same theory as a basis. Many previous studies have used the same framework \cite{Aharony1,Aharony2,Gell-Mann,Berger,Courbage2,Datta1,Finkelstein,Hall,Lindblad,Squires,Datta2,Corbett,Clifton,Bramon,Ancochea,Bertlmann1,Bertlmann2,Andrianov,Barnett,Samal,Bertlmann4,Genovese,Courbage1,Fassarella,Bertlmann3,Bertlmann5,Hiesmayr,Lu}.
A relativistic analysis will be left for a future study.
We will need, however, to depart from conventional quantum mechanics in three important ways. The justification for these departures lies in the eventually recovery of the conventional formalism under appropriate conditions. The first departure is that we will not impose any equation of motion, such as the Schr\"{o}dinger equation, on states because to do so would directly build in the asymmetry between time and space mentioned above. Instead, we anticipate that an effective equation of motion will arise phenomenologically in some way. Second, we will consider states that describe the location of a material object either in space or in time. While the location in space can simply be given by a wave function, say $\psi(\vec{x})$, in the position representation, the location in time is quite unconventional as it would need to be given by a wave function, say $\phi(t)$, in what might be called the ``time representation''. Here, $|\phi(t)|^2$ gives the probability density for the object being at time $t$ just as $|\psi(\vec{x})|^2$ gives the probability density for the object being at position $\vec{x}$. Note that, in general, $\phi(t)$ violates mass conservation as it describes the object as having potential existence at the points in time where $|\phi(t)|^2\ne0$ and definitely not existing at the points where $|\phi(t)|^2=0$. This lack of mass conservation is the price we must pay to keep the formalism symmetric with respect to time and space. Nevertheless we anticipate that mass conservation will arise phenomenologically rather than being imposed on the formalism. The third departure from conventional quantum mechanics is that we need to include the P and T symmetry operations in the formalism explicitly, even in the situation where the corresponding symmetries are obeyed and the actions of P and T are redundant. This will ensure that we have a consistent formalism that operates both when the symmetries are obeyed and also when they are violated.
The particular way in which we include the P and T discrete symmetry operators is motivated by the fact that they reverse the directions of spatial and temporal translations, respectively. A one-dimensional path in space that consists of a sequence of translations that alternate in direction can, therefore, be expressed explicitly in terms of parity inversions P. The same can be said for the T operation in relation to a sequence of time translations that alternate in direction. The roles that P and T play will be greater in mathematical constructions that involve a greater number of direction reversals. Such constructions have the potential to display the effects of any discrete symmetry violation to a greater extent. Feynman's path integral formalism immediately comes to mind as one that involves a superposition of all paths that zigzag through configuration space between two states. However, this formalism is inextricably associated with dynamics and thus is tied to the space-time asymmetry mentioned above. We need to develop a different approach if we are to keep time and space on an equal footing at a fundamental level.
Our approach is as follows. We will first consider the effects of P violation on spatial localisation.
We will begin with a quantum state that represents some material object as being localised in space, and for this we will need the variance in the object's position to be finite.
We will place no other constraint on the position and so we will need a quantum state that yields the least information about position (and thus maximum entropy) for a fixed variance; the optimum pure state fitting this requirement has a Gaussian wave function in the position representation \cite{Bialynicki-Birula}. Note that a classical particle undergoing a one-dimensional Wiener process has a position probability density that is Gaussian; it too has trajectories that consist of infinitely many reversals in direction of the kind we have been considering. With this in mind, we will decompose the Gaussian quantum state into a superposition of infinitely-many paths through space where each path has infinitely-many reversals in direction. The reversals in the direction of each path will be expressed explicitly in terms of the P symmetry operation and the translations in terms of the momentum operator. We will find that the violation of P symmetry has no effect on the construction. Then, we will apply the same sum-over-paths construction to a quantum state that represents the object as being localised in time, but with the path reversals expressed explicitly in terms of the T symmetry operator and the translations in terms of the Hamiltonian. Using the same construction will ensure that \textit{the formalism is symmetric with respect to the representation of states of matter in both space and time} when the discrete symmetries hold. The situation will be found to change dramatically when T symmetry is violated. Only then will the formalism exhibit a time-space asymmetry that is consistent with conventional quantum mechanics. The important point to be made here is that the asymmetry will not be imposed on the formalism at a fundamental level, but rather it will arise \textit{phenomenologically} due to the T violation.
Given the fundamental character of the issues involved, one should not be surprised to find that to make any progress we need to pay due attention to quite subtle mathematical details. In particular, while the concept of the limit of an infinite sequence has rigorous meaning in a mathematical context, there is no \textit{a priori} reason to suppose that it automatically carries a corresponding value in a theory that is designed to underpin experimental physics. After all, the accuracy of observations made in experimental physics are always restricted by finite resources. For example, consider a theory in which the limit point $a$ of the convergent sequence $a_1$, $a_2$, $a_3$, $\ldots$ (i.e. where $a_n\to a$ as $n\to\infty$) represents an experimental parameter, and let $\epsilon$ represent the experimental accuracy of measuring $a$ for a given level of resources. The convergence of the sequence implies that there exists a natural number $N_\epsilon$ that depends on $\epsilon$ for which $|a-a_n|<\epsilon$ for all $n>N_\epsilon$, and so it is not possible to physically distinguish (using the given resources) the limit point $a$ from any of the terms $a_n$ for $n>N_\epsilon$. Under such circumstances, the set $\{a_n:n>N_\epsilon\}$ would be more representative of the physical situation than just the limit point $a$. Set representations of this kind will be important for expressing quantum states in a manner which better represents their physical implications.
The structure of the remainder of the paper is as follows. We develop a sum-over-paths construction of a quantum state that is localised in space and examine the effects of the violation of P symmetry in section \ref{sec:construction}. We apply the same construction to quantum states that are localised in time and examine the effects of the violation of T symmetry in section \ref{sec:applied to time}. Following that, in section \ref{sec:emergence}, we show how the conventional Schr\"{o}dinger equation and conservation of mass emerge as a result of coarse graining over time, and explore how the new formalism might be tested experimentally. We end with a discussion in section \ref{sec:conclusion}.
\section{\label{sec:construction}Mathematical construction of quantum states in space}
\subsection{Developing the construction}
We first need to develop the mathematical construction of quantum states that are localised in space and consist of a superposition of infinitely-many paths each of which has possibly infinitely-many reversals in direction. For this consider a simple 1-dimensional model universe composed of a single ``galaxy'' as our material object. The galaxy is representative of any spatially localised physical system with mass and could in fact be a star, a planet or just a single particle; its details are not important for this study. The location of the galaxy is described by a set of observables that represent all its spatial degrees of freedom. Imagine that at a particular time, each observable is in some localised state that is uncorrelated with respect to every other observable in the set. This will almost certainly result in the galaxy being far from its minimum energy state, however neither the energetics nor the dynamics are important in this section. Also, because the same analysis applies to each observable, we will only treat one representative observable explicitly. Let that observable be the centre of mass coordinate, which we assume to have a finite variance. As mentioned above, the best choice for a pure state under these circumstances is one described by a Gaussian wave function \cite{Bialynicki-Birula}, which we write as follows:
\begin{equation} \label{eq:spatial gaussian state}
\ket{\psi} \propto \int dx \exp(-\frac{x^2}{2\sigma^2_{\rm x}})\ket{x}_{\rm x}
\end{equation}
where $x$ and $\ket{x}_{\rm x}$ are the eigenvalue and corresponding eigenstate of $\hat X$, the operator representing the $x$ component of the centre of mass position, and $\sigma_{\rm x}$ is a width parameter.
This state can be written explicitly in terms of spatial translations as
\begin{equation} \label{eq:gaussian spatial translation}
\ket{\psi} \propto \int dx \exp(-\frac{x^2}{2\sigma^2_{\rm x}})\exp(-i\hat Px)\ket{0}_{\rm x}
\end{equation}
where operator representing the total momentum of the galaxy, $\hat P$, generates spatial translations according to
\[
\exp(-i\hat P\delta x)\ket{x}_{\rm x}=\ket{x+\delta x}_{\rm x}
\]
as illustrated in Fig.~\ref{fig:translation}(a). Here, and throughout this paper, we use units in which $\hbar=1$.
Inserting the resolution of the identity $\hat{\boldsymbol 1}=\int dp \ket{p}_{\rm p}{}_{\rm p}\bra{p}$ into \eq{eq:gaussian spatial translation} gives
\[
\ket{\psi} \propto \iint dx\,dp \exp(-\frac{x^2}{2\sigma^2_{\rm x}})\exp(-ipx)\ket{p}_{\rm p}{}_{\rm p}\ip{p|0}_{\rm x}\ ,
\]
where $\{\ket{p}_{\rm p}: \hat P\ket{p}_{\rm p}=p\ket{p}_{\rm p}\}$ is the momentum basis. Carrying out the Fourier transform with respect to $x$, yields
$\ket{\psi} \propto \exp(-\sfc{1}{2}\hat P^2\sigma^2_{\rm x})\ket{0}_{\rm x}$ and making use of the result
\begin{equation}
\exp(-A^2/2)=\lim_{N\to\infty}\cos^N(A/\sqrt{N})
\end{equation}
then leads to
\begin{equation} \label{eq:path in space}
\ket{\psi} \propto
\lim_{N\to\infty}\frac{1}{2^N}\!\left[\exp(i\frac{\hat P\sigma_{\rm x}}{\sqrt N})+\exp(-i\frac{\hat P\sigma_{\rm x}}{\sqrt N})\right]^N\kern-1mm\ket{0}_{\rm x}\ .
\end{equation}
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm]{Fig_translation.pdf}
\end{center}
\vspace{-5mm}
\caption{Sketches illustrating the translation of wave functions along (a) the $x$ axis and (b) the time axis. In (a) the wave functions represent the position eigenket $\ket{x}_{\rm x}$ and an arbitrary state $\ket{\chi}$ and the translation is by a distance $\delta x$. In (b) the wave function represents the state $\ket{f}$ and the translation is by an interval $t$.
\label{fig:translation}}
\end{figure}
Expanding the $N$-fold product in \eq{eq:path in space} gives a series of terms each of which comprise $N$ translations (or ``steps'') of $\pm\sigma_{\rm x}/\sqrt{N}$ along the $x$ axis.
For example, a term of the form
\begin{equation} \label{eq:one path}
\cdots\exp(-i\hat P a)\exp(-i\hat P a)\exp(i\hat P a)\exp(-i\hat P a)\ket{0}_{\rm x}\ ,
\end{equation}
where $a=\sigma_{\rm x}/\sqrt{N}$, describes a path on the $x$ axis from the origin $0$ through the sequence of points $a$, $0$, $a$, $2a$ and so on, as illustrated in Fig.~\ref{fig:random path}(a). Equation \eq{eq:path in space} can be viewed, therefore, as a superposition of random paths away from the origin $\ket{0}_{\rm x}$ in the limit of infinitely small steps, and shares similarities with both quantum walks \cite{quantum-walk} and Feynman's sum over paths \cite{Feynman}.
Note that here, however, the random path is traversed without reference to time, and so it should be considered to be traversed in a zero time interval. Each random path is, therefore, a generalisation of the virtual displacements in D'Alembert's principle in classical mechanics \cite{Goldstein}. For this reason each individual path shall be called a {\it random virtual path} and the superposition of a set of random virtual paths like that in \eq{eq:path in space} shall be called a {\it quantum virtual path}.
\begin{figure}
\begin{center}
\includegraphics[width=8.0cm]{Fig_ran_path.pdf}
\end{center}
\vspace{-8mm}
\caption{Binary tree diagrams representing virtual paths in (a) space and (b) time. Each edge (white dashed line) in the tree represents a virtual displacement along the black horizontal axis. The thick blue edges in (a) represents a virtual path that passes through the sequence of points $0$, $a$, $0$, $a$, $2a$ on the $x$ axis. In (b) four different virtual paths from $0$ to $2\delta t$ on the $t_{\rm c}$ axis are represented in the tree by thick edges coloured yellow, red, blue and purple.
\label{fig:random path}}
\end{figure}
Although they share similarities, a quantum virtual path is quite distinct from Feynman's sum over paths \cite{Feynman}. For example, Feynman's method is used to calculate the probability amplitude for a system to evolve from one state to another. The paths represent potential classical trajectories between the same starting and ending points and the sum gives the total probability amplitude for evolving between the points. In contrast, a quantum virtual path represents a single state. The accumulated displacement over one random virtual path, like that in \eq{eq:one path}, gives a potential classical position of the system, and the whole quantum virtual path represents the state \eq{eq:path in space} in terms of a distribution of potential classical positions. Moreover, calculating the innerproduct of two states where one (or both) is represented by a quantum virtual path would result in a Feynman-like sum over paths calculation. So in this sense, a quantum virtual path is a \textit{precursor} of Feynman's sum over paths.
The right-hand side of \eq{eq:path in space} is not the only way to decompose the state in \eq{eq:spatial gaussian state}. But what makes \eq{eq:path in space} special is that it consists of a superposition of an infinite number of continuous paths with the property that if one path is picked at random, it will effectively consist of a sequence of infinitesimal segments where each segment has an equal likelihood of representing a step in the positive or negative $x$ directions. The set of paths is unbiased with respect to direction in this sense. Another feature that sets the decomposition in \eq{eq:path in space} apart is that it comprises all possible paths. The justification of why it should is that in decomposing the state in \eq{eq:spatial gaussian state} in terms of a superposition of paths, we have no reason for leaving out particular paths or, alternatively, for including only particular paths; in the absence of such reasons all possible paths should be included.
As $N\to\infty$ the step length $\sigma_{\rm x}/\sqrt{N}$ in \eq{eq:path in space} will eventually breach the fundamental lower bound, say $\delta x_{\rm min}$, that is expected for physically distinguishable positions.
For example, there are reasons \cite{Planck} to believe that points in space are indistinguishable at the scale of the Planck length $\ell_{\rm P}\approx 1.6\times 10^{-35}$~m.
Let $N^{\rm (space)}_{\rm min}$ be the value of $N$ where the step length $\sigma_{\rm x}/\sqrt N$ becomes equal to $\delta x_{\rm min}$, i.e. $N^{\rm (space)}_{\rm min}=\sigma^2_{\rm x}/\delta x_{\rm min}^2$. This implies that the limit on the right side of \eq{eq:path in space} can be replaced by a term corresponding to any value of $N$ larger than $N^{\rm (space)}_{\rm min}$ without any physically meaningful consequences. There are an infinite number of such terms, each of which has an \textit{equal status} in representing the state of the universe.
They form the set
\begin{equation} \label{eq:set Psi}
\boldsymbol{\Uppsi}=\{\ket\psi_N: N\ge N^{\rm (space)}_{\rm min}\}
\end{equation}
where
\begin{equation} \label{eq:psi_N}
\ket\psi_N=
\frac{1}{2^N}\!\left[\hat{\mathbf{P}}^{-1}\!\exp(-i\frac{\hat P\sigma_{\rm x}}{\sqrt N})\hat{\mathbf{P}}+\exp(-i\frac{\hat P\sigma_{\rm x}}{\sqrt N})\right]^N\kern-1mm\ket{0}_{\rm x}\ .
\end{equation}
In \eq{eq:psi_N} we have written the translations explicitly in terms of the parity inversion operator $\hat{\mathbf{P}}$. It has the property that
\begin{equation} \label{eq:parity op for step}
\exp(i\hat P x')=\hat{\mathbf{P}}^{-1}\!\exp(-i\hat P x')\hat{\mathbf{P}}
\end{equation}
which expresses the fact that a translation along the $x$ axis by $-x'$ (left side of \eq{eq:parity op for step}) can be produced by first performing a parity inversion, translating by $x'$ and then reversing the parity inversion (right side).
Every element in the set $\boldsymbol{\Uppsi}$ can serve equally well as a representation of the state in \eq{eq:spatial gaussian state} as far as the physically-distinguishable spatial limit allows;
they all have equal status in this respect.
The mathematical construction represented by \eq{eq:set Psi} and \eq{eq:psi_N} is in the form of the explicit translations and discrete symmetry operations that we need for comparing the difference between quantum states in space and time.
Although being equivalent to \eq{eq:spatial gaussian state}, we shall henceforth regard \eq{eq:set Psi} and \eq{eq:psi_N} as being a \textit{more fundamental} description of the state of the galaxy due to this explicit form.
Note that the interpretation of \eq{eq:psi_N} in terms of quantum virtual paths does not hinge on the state $\ket{0}_{\rm x}$ being the eigenstate of position with zero eigenvalue. In fact any state $\ket{\chi}$ with a variance in position very much smaller than $\sigma^2_{\rm x}/2$ (and, correspondingly, a variance in total momentum very much larger than $1/2\sigma^2_{\rm x}$) could be used in its place, in which case the steps in a path represent translations of $\ket{\chi}$ along the $x$ axis, as illustrated in Fig.~\ref{fig:translation}(a), rather than steps along the $x$ axis itself. While this situation allows some ambiguity in the formalism, it does not have any effect on results provided that corresponding adjustments to $\ket{\psi}$ and $\sigma_{\rm x}$ are duly taken.
\subsection{Parity inversion invariance and its violation}
Given that the Hamiltonian does not appear explicitly in the construction of spatial states, we should not expect to find any effects of discrete symmetry violation here. In particular, \textit{regardless of whether the galaxy obeys parity inversion symmetry or not}
\begin{equation}
\hat{\mathbf{P}}^{-1}\hat P\,\hat{\mathbf{P}}=-\hat P
\end{equation}
always holds, and so \eq{eq:psi_N} can be rewritten as a binomially weighted superposition of spatially translated states, i.e.
\begin{equation} \label{eq:spatial binomial state}
\ket\psi_N
= \sum_{n=0}^N B_n \exp\left[-i(2n-N)\frac{\hat P\sigma_{\rm x}}{\sqrt N}\right]\ket{0}_{\rm x}
\end{equation}
where
\begin{equation} \label{eq:binomial coefficients}
B_n =
\frac{1}{2^N}\left({}^N_n\right)\ .
\end{equation}
In the large $N$ limit, $\ket\psi_N$ tends to the Gaussian state $\ket{\psi}$ in \eq{eq:spatial gaussian state}, i.e.
\begin{equation}
\lim\limits_{N\to\infty}\ket\psi_N \propto \int dx\ g(x,\sigma_{\rm x}) \ket{x}_{\rm x}
\end{equation}
where
\begin{equation} \label{eq:gaussian function}
g(x,\sigma_{\rm x}) = \exp(-\frac{x^2}{2\sigma^2_{\rm x}})\ .
\end{equation}
\begin{figure}
\begin{center}
\includegraphics[width=9.0cm]{Fig_TInv.pdf}
\end{center}
\vspace{-8mm}
\caption{The position representation of the state $\ket{\psi}_N$ as a function of the scaled position $x/\sigma_{\rm x}$. The dots represent the exact values $B_n$ given by \eq{eq:binomial coefficients} and the solid curves represent the Gaussian approximation $g(x,\sigma_{\rm x})$ given by \eq{eq:gaussian function}. The abscissae for the discrete coefficients $B_n$ are given by $x/\sigma_{\rm x}=(2n-N)/\sqrt{N}$ in accord with \eq{eq:spatial binomial state}. For clarity, $B_n$ and $g(x,\sigma_{\rm x})$ have been scaled to give a maximum of unity, and the green ($N=100$) and blue ($N=1000$) dots and curves have been displaced vertically by 0.2 and 0.4, respectively.
\label{fig:TInv}}
\end{figure}
Figure \ref{fig:TInv} compares the coefficients $B_n$ (shown as dots) with their large-$N$ limit $g(x,\sigma_{\rm x})$ (continuous curves) for a number of different $N$ values. The values of $N$ have been chosen purposely to exaggerate the discreteness of the state $\ket{\psi}_N$ in comparison to the limiting state $\ket{\psi}$ from \eq{eq:spatial gaussian state}. In truth, for every $\ket{\psi}_N\in \boldsymbol{\Uppsi}$ in \eq{eq:set Psi}, the values of $N$ are sufficiently large (viz. $N\ge N^{\rm (space)}_{\rm min}$) that the dots representing $B_n$ for consecutive $n$ values are physically indistinguishable, and the locus of points representing $B_n$ is essentially equivalent to the curve $g(x,\sigma_{\rm x})$ up to a proportionality constant. As a consequence, every $\ket{\psi}_N\in \boldsymbol{\Uppsi}$ is physically indistinguishable from the state $\ket{\psi}$ in \eq{eq:spatial gaussian state}.
\section{\label{sec:applied to time}Applying the construction to quantum states in time}
\subsection{Adapting the construction}
We now use our construction to explore the temporal analogy of \eq{eq:spatial gaussian state} in which the galaxy is represented in time rather than space.
We begin by recalling that the Hamiltonian $\hat H$ generates translations through time according to
\[
\exp(-i\hat H t)\ket{f}=\ket{f'}
\]
where $\ket{f}$ and $\ket{f'}$ represent states at times differing by $t$, as illustrated in Fig.~\ref{fig:translation}(b).
Next, we construct a set of states analogous to \eq{eq:set Psi} but with each state representing a superposition of random virtual paths through {\it time} as
\begin{equation} \label{eq:set Upsilon}
\boldsymbol{\Upupsilon}_\lambda =\{ \ket{\Upsilon_\lambda}_N: N\ge N^{\rm (time)}_{\rm min}\}
\end{equation}
where
\begin{equation} \label{eq:Upsilon_N}
\ket{\Upsilon_\lambda}_N \propto \frac{1}{2^N}\!\left[\hat{\mathbf{T}}^{-1}\!\exp(-i\frac{\hat H\sigma_{\rm t}}{\sqrt N})\hat{\mathbf{T}}+\exp(-i\frac{\hat H\sigma_{\rm t}}{\sqrt N})\right]^N\kern-1mm\ket{\phi}\ .
\end{equation}
Here $\lambda$ distinguishes different physical situations that will be specified later, $N^{\rm (time)}_{\rm min}=\sigma_{\rm t}^2/\delta t_{\rm min}^2$ is the value of $N$ for which the step size $\sigma_{\rm t}/\sqrt{N}$ reaches some fundamental resolution limit in time $\delta t_{\rm min}$ (e.g. taking the resolution limit as the Planck time would mean that $\delta t_{\rm min}=5.4\times 10^{-44}$~s), and $\hat{\mathbf{T}}$ is Wigner's time reversal operator \cite{Wigner}.
The state $\ket{\phi}$ plays the role of $\ket{0}_{\rm x}$ in \eq{eq:psi_N} and is assumed to be sharply defined in time and, correspondingly, to have a broad distribution in energy \cite{Pegg}. More specifically, $\ket{\phi}$ must have a variance in energy that is very much larger than $1/2\sigma_{\rm t}^2$ in analogy with the requirement for any state $\ket{\chi}$ to be used in place of $\ket{0}_{\rm x}$. Other details of $\ket{\phi}$ are not crucial for our main results.
It is perhaps worth elaborating a little on what is meant by $\ket{\phi}$ being sharply defined in time given that there are well-known difficulties associated with defining an operator to represent time \cite{Moyer,Pegg}. Fortunately, the absence of a universally accepted time operator does not prevent uncertainties in time from being physically meaningful. Rather, we can use the fact that the Hamiltonian is the generator of translations in time to probe the time uncertainty of a state. For example, $\ket{\varphi_\tau}=\exp(-i\hat H\tau)\ket{\varphi_0}$ represents the state $\ket{\varphi_0}$ translated in time by $\tau$. If the overlap $\ip{\varphi_\tau|\varphi_0}$ is negligible for all values of $\tau$ except for $|\tau|\approx 0$ then $\ket{\varphi_0}$ can be regarded as sharply defined in time, at least for the purposes needed here. A more rigorous definition of such states is given by Moyer's timeline states \cite{Moyer}. The ambiguity mentioned at the end of the previous section also occurs here regarding the choice of the state $\ket{\phi}$, and can be treated in a similar way. We will return to this point in section \ref{sec:emergence}\ref{sec:conventional qm}.
The violation of T symmetry is expressed by $\hat{\mathbf{T}}^{-1}\!\hat H\hat{\mathbf{T}}\ne\hat H$ which implies that there are two versions of the Hamiltonian \cite{FPhys,FPhys2}. We label the two versions as $\hat H_{\rm F}=\hat H$ and $\hat H_{\rm B}=\hat{\mathbf{T}}^{-1}\!\hat H\hat{\mathbf{T}}$.
In this construction, one direction of time is not physically distinguishable from the other and the subscripted labels ${\rm F}$ and ${\rm B}$ simply refer to opposite directions; nevertheless, it may be convenient to think of the labels as referring to the customary ``forwards'' and ``backwards'' directions of time.
Using these definitions together with the fact \cite{Wigner} that $\hat{\mathbf{T}}^{-1}i\hat{\mathbf{T}}=-i$ then gives
\begin{equation} \label{eq:Upsilon_N in terms of H_F H_B}
\ket{\Upsilon_\lambda}_N \propto \frac{1}{2^N}\!\left[\exp(i\hat H_{\rm B}\delta t)+\exp(-i\hat H_{\rm F}\delta t)\right]^N\kern-1mm\ket{\phi}
\end{equation}
where, for convenience, we have set
\begin{equation} \label{eq:time step}
\delta t=\frac{\sigma_{\rm t}}{\sqrt{N}}
\end{equation}
as the step in time.
\eq{eq:Upsilon_N in terms of H_F H_B} shows that $\hat H_{\rm F}$ and $\hat H_{\rm B}$ are responsible for translations in opposite directions of time. This is an important point that warrants particular emphasis: a translation in time in the opposite direction to that given by $\exp(-i\hat H_{\rm F} t)$ is not produced by its {\it inverse} $\exp(i\hat H_{\rm F} t)$ but rather by its {\it time reverse}:
\[
\exp(i\hat H_{\rm B} t)=\hat{\mathbf{T}}^{-1}\! \exp(-i\hat H_{\rm F} t)\hat{\mathbf{T}}\ .
\]
Evidently we need to associate the operators $\exp(-i\hat H_{\rm F} t)$ and $\exp(i\hat H_{\rm B} t)$ with {\it physical evolution} in different directions of time according to \eq{eq:Upsilon_N in terms of H_F H_B}. This leaves their respective inverses $\exp(i\hat H_{\rm F} t)$ and $\exp(-i\hat H_{\rm B} t)$ to be associated with the {\it mathematical operations of rewinding} that physical evolution. In short, physical time evolution is described by the former pair of operators, and not the latter.
In fact, these associated meanings follow from conventional quantum mechanics. For example, let $\ket{f(t)}$ represent the state of an arbitrary closed system at time $t$. Unitary evolution implies that
\begin{equation}
\ket{f(t)}=\exp(-i\hat h t)\ket{f(0)}
\end{equation}
where $\ket{f(0)}$ is the state at $t=0$ and $\hat h$ is the corresponding Hamiltonian. Recall that Wigner's time reversal operator $\hat{\mathbf{T}}$ reverses the direction of all momenta and spin \cite{Wigner}. Let the time-reversed states at times $0$ and $t$ be $\ket{b(0)}=\hat{\mathbf{T}}^{-1}\ket{f(0)}$ and $\ket{b(-t)}=\hat{\mathbf{T}}^{-1}\ket{f(t)}$, respectively.
Using $\hat{\mathbf{T}}\hat{\mathbf{T}}^{-1}=\hat 1$ and rearranging shows that $\ket{b(-t)}=\exp(i\hat{\mathbf{T}}^{-1}\hat h\hat{\mathbf{T}} t)\hat{\mathbf{T}}^{-1}\ket{f(0)}$, i.e.
\begin{equation}
\ket{b(-t)}=\exp(i\hat{\mathbf{T}}^{-1}\hat h\hat{\mathbf{T}} t)\ket{b(0)}\ ,
\end{equation}
and so the time-reversed state $\ket{b(-t)}$ represents the evolution from the time-reversed state $\ket{b(0)}$ according to the Hamiltonian $\hat{\mathbf{T}}^{-1}\hat h\hat{\mathbf{T}}$ for the time $-t$.
That is, evolving from the state $\ket{f(0)}$ for the time $t$ with the Hamiltonian $\hat h$ is equivalent to evolving from the time-reversed state $\ket{b(0)}$ for the time $-t$ with the Hamiltonian $\hat{\mathbf{T}}^{-1}\hat h\hat{\mathbf{T}}$. Clearly $\hat h$ generates translations in one direction of time and $\hat{\mathbf{T}}^{-1}\hat h\hat{\mathbf{T}}$ generates translation is the opposite direction, which is consistent with \eq{eq:Upsilon_N in terms of H_F H_B}.
If our model universe satisfied T symmetry, $\hat H_{\rm F}$ and $\hat H_{\rm B}$ would be commuting operators and the terms in \eq{eq:Upsilon_N in terms of H_F H_B} would be able to be manipulated algebraically in exactly the same way as those in \eq{eq:psi_N} led to \eq{eq:spatial binomial state}. Thus, for the temporal quantum virtual path to be qualitatively distinct from the spatial one, the model universe must violate T symmetry to the extent of giving a non zero commutator $[\hat H_{\rm F},\hat H_{\rm B}]$. We could model such a commutator using details of the T violation that has been observed in the decay of mesons \cite{T1,T2,T3,T4} or that has been speculated for a Higgs field \cite{Ivanov,Lee}. However, the potential repercussions of T violation will be manifest most clearly for the simplest departure from time reversal invariance. Accordingly we shall imagine that our model universe contains an unspecified T-violating mechanism that is consistent with the commutator
\begin{equation} \label{eq:commutator}
[\hat H_{\rm B},\hat H_{\rm F}]=i\lambda
\end{equation}
for real valued $\lambda$. This is the origin of the parameter $\lambda$ that appears in \eq{eq:set Upsilon} and \eq{eq:Upsilon_N}.
Eqs.~(6) and (8) of Ref.~\cite{FPhys} show that the operator on the right side of \eq{eq:Upsilon_N in terms of H_F H_B} can be expanded and reordered using the Zassenhaus formula \cite{Suzuki} as follows
\begin{eqnarray} \label{eq:Zassenhaus}
&&\hspace{-5mm}\left[\exp(i\hat H_{\rm B}\delta t)+\exp(-i\hat H_{\rm F}\delta t)\right]^N\\
&&\hspace{2mm}= \sum_{n=0}^N \exp[i\hat H_{\rm B}(N-n)\delta t]\exp(-i\hat H_{\rm F}n\delta t)\nonumber\\
&&\hspace{2mm} \times\sum\limits_{v=0}^{N-n}
\cdots \sum\limits_{\ell =0}^s \sum\limits_{k=0}^\ell \exp \left[
(v+\cdots +\ell +k)(\delta t ^{2}[\hat H_{F} ,\hat H_{B}]+\hat Q) \right]\nonumber
\end{eqnarray}
where $\hat Q$ contains terms representing higher order commutators of the form $[[\hat H_{\rm B},\hat H_{\rm F}],\cdots]$. It follows from \eq{eq:commutator} that $\hat Q=0$ here.
Substituting \eq{eq:Zassenhaus} into \eq{eq:Upsilon_N in terms of H_F H_B} and then simplifying the resulting expression using Eqs.~(B.14) and (B.15) in Appendix B of Ref.~\cite{FPhys} yields
\begin{equation}
\label{eq:Upsilon with interference fn}
\ket{\Upsilon_\lambda}_N \propto \sum_{n=0}^N I_{N-n,n}(\delta t^2\lambda)\exp[i\hat H_{\rm B}(N\!-\!n)\delta t]\exp[-i\hat H_{\rm F}n\delta t]\ket{\phi}
\end{equation}
where
\begin{equation}
\label{eq:interference function}
I_{N-n,n}({z})
=\exp[-in(N-n){z}/2]\prod_{q=1}^n \frac{\sin [(N+1-q){z}/2]}{\sin (q{z}/2)}
\end{equation}
is an \textit{interference function} that takes account of the non-commutativity of $\hat H_{\rm F}$ and $\hat H_{\rm B}$.
To relate this to what an observer in the galaxy would see, imagine that the galaxy contains a clock that is constructed from T-invariant matter. We will refer to any time shown by the clock as ``clock time'' and use the symbol $t_{\rm c}$ to represent its value. Let the state $\ket{\phi}$ represents the clock showing the time $t_{\rm c}=0$. The state
\begin{equation} \label{eq:clock state}
\exp[i\hat H_{\rm B}(N-n)\delta t]\exp[-i\hat H_{\rm F}n\delta t]\ket{\phi}
\end{equation}
represents evolution by $\exp[-i\hat H_{\rm F}n\delta t]$ in one direction of time followed by $\exp[i\hat H_{\rm B}(N-n)\delta t]$ in the opposite direction which, by convention, first increases $t_{\rm c}$ by $n\delta t$ and then decreases it by $(N-n)\delta t$, respectively. The state in \eq{eq:clock state} would therefore represent the clock showing the net clock time of
\begin{equation} \label{eq:clock time}
t_{\rm c}=(2n-N)\delta t\ ,
\end{equation}
and so the state in \eq{eq:Upsilon with interference fn} represents a weighted superposition of states over the range of net clock times from $t_{\rm c}=-N\delta t$ to $N\delta t$.
\subsection{Time reversal invariance}
It is useful to first consider the special case where the universe is invariant under time reversal. For this we set $\lambda=0$, $\hat H_{\rm F}=\hat H_{\rm B}=\hat H$ in \eq{eq:Upsilon with interference fn}. The interference function for $\lambda=0$ is the binomial coefficient $I_{N-n,n}(0)=\left({}^N_n\right)$ and so
\begin{equation} \label{eq:temporal binomial state}
\ket{\Upsilon_0}_N \propto \sum_{n=0}^N B_n\exp[-i(2n-N)\hat H\delta t]\ket{\phi}
\end{equation}
where $B_n$ is given by \eq{eq:binomial coefficients}. The coefficient $B_n$ becomes proportional to the Gaussian function $\exp[-(2n-N)^2/2N]$ for large $N$ and so
\[
\ket{\Upsilon_0}_N \srel{\propto}{\sim} \sum_{n=0}^N \exp[-(2n-N)^2/2N]\exp[-i(2n-N)\hat H\delta t]\ket{\phi}\ .
\]
Re-expressing the summation in terms of the index $m=2n-N$ and using the definition $\delta t=\sigma_{\rm t}/\sqrt{N}$ then yields
\begin{equation} \label{eq:Upsilon_N temporal gaussian}
\ket{\Upsilon_0}_N \srel{\propto}{\sim} \sum_{m\in S} \exp[-\frac{(m\delta t)^2}{2\sigma^2_{\rm t}}]\exp(-i\hat H m\delta t)\ket{\phi}
\end{equation}
where $S=\{-N,-N+2,\ldots,N\}$. We define the large-$N$ limit as
\begin{equation} \label{eq:temporal gaussian state}
\ket{\Upsilon_0} = \lim\limits_{N\to\infty} \ket{\Upsilon_0}_N \propto \int dt\ g(t,\sigma_{\rm t})\exp(-i\hat H t)\ket{\phi}
\end{equation}
where $g(t,\sigma_{\rm t})$ is given by \eq{eq:gaussian function}. Although Fig. \ref{fig:TInv} is explicitly for the spatial case, it can also be used here as a comparison of $B_n$ and $g(t,\sigma_{\rm t})$ in Eqs.~(\ref{eq:temporal binomial state}) and (\ref{eq:temporal gaussian state}) provided we interpret the horizontal axis as $t/\sigma_{\rm t}$. Likewise, for $N\ge N^{\rm (time)}_{\rm min}$ the locus of points representng $B_n$ is essentially equivalent to the curve $g(t,\sigma_{\rm t})$ up to a proportionality constant, and so every $\ket{\Upsilon_0}_N\in \boldsymbol{\Upupsilon}_0$ in \eq{eq:set Upsilon} is physically indistinguishable from the state $\ket{\Upsilon_0}$ in \eq{eq:temporal gaussian state}.
Hence, for time reversal invariance, the construction yields a state, given by \eq{eq:temporal gaussian state}, that
is a Gaussian weighted superposition of the time-translated states $\exp(-i\hat H t)\ket{\phi}$. This state represents the galaxy existing in time only for a duration of the order of $\sigma_{\rm t}$ and is analogous to \eq{eq:spatial gaussian state} which represents the centre of mass of the galaxy existing only in a spatial region with a size of the order of $\sigma_{\rm x}$.
Our construction, therefore, allows for the same kind of quantum state in time as in space, in the absence of T violation.
In other words, there is a symmetry between time and space for quantum states in this special case.
As discussed in the Introduction, this symmetry comes at the cost of the non-conservation of mass.
\subsection{Violation of time reversal invariance}
Next we examine the quite different situation of T violation where $\lambda\ne 0$ and $\hat H_{\rm F}\ne\hat H_{\rm B}$.
In that case the amplitudes for different virtual paths to the same point in time, as illustrated in Fig.~\ref{fig:random path}(b), can interfere leading to undulations in $I_{N-n,n}({z})$ as a function of $n$. To find the values of $n$ where the modulus of the interference function $I_{N-n,n}(z)$ is maximized it is sufficient to look for the position where $|I_{N-n,n}(z)|$ is unchanged for consecutive values of $n$, i.e. where $|I_{N-(n-1),n-1}(z)|=|I_{N-n,n}(z)|$. This condition reduces, on using \eq{eq:interference function} and performing some algebraic manipulation, to $|\sin[(N+1-n)z/2]|=|\sin(nz/2)|$. Note that Eqs.~(\ref{eq:Upsilon with interference fn}) and (\ref{eq:interference function}) imply $z=\delta t^2\lambda$ and given $\delta t=\sigma_{\rm t}/\sqrt{N}$ from \eq{eq:time step}, this means $z$ is inversely proportional to $N$; thus we let $z=\theta/N$ where
\[
\theta=\sigma_{\rm t}^2\lambda
\]
is the coefficient of proportionality (i.e. $\theta$ is independent of $N$). Hence we wish to know the values of $n$ that satisfy $|\sin[(N+1-n)\theta/2N]|=|\sin(n\theta/2N)|$. Writing $x=\theta(N+1)/2N$ and $y=n\theta/2N$ transforms this equation into $|\sin(x-y)|=|\sin(y)|$ which has the solutions $y=(x-\pi)/2+m\pi$ for integer $m$. Re-expressing the solutions in terms of $n$ then gives
\[
n=\frac{N+1}{2}+\frac{N(2m-1)\pi}{\theta}\ .
\]
The modulus of the interference function reaches a maximum value at this value of $n$ and one less (i.e for $n-1$). Taking the midpoint and choosing the particular values $m=0, 1$ then gives the positions of two maxima (or ``peaks'') at $n=n_\pm$ where \begin{equation} \label{eq:n for peaks}
n_{\pm}=N\left( \sfc{1}{2} \pm\sfc{\pi}{\theta}\right)\ .
\end{equation}
Substituting $n_\pm$ for $n$ in \eq{eq:clock time} gives the corresponding clock times as
\begin{equation} \label{eq:peak clock time}
\pm t_{\rm c}^{\rm (peak)}=(2n_\pm-N)\delta t=\pm\frac{2\pi\sigma_{\rm t}\sqrt{N}}{\theta}
\end{equation}
where $t_{\rm c}^{\rm (peak)}$ is defined to be positive.
The modulus of the interference function \eq{eq:interference function} is shown in Section A of the Supplementary Material to be approximately Gaussian about these maxima, which allows us to write $\ket{\Upsilon_\lambda}_N$ in \eq{eq:Upsilon with interference fn} as a superposition of two states as follows:
\begin{equation} \label{eq:Upsilon with two peaks}
\ket{\Upsilon_\lambda}_N \propto \ket{\Upsilon_\lambda^{(+)}}_N+\ket{\Upsilon_\lambda^{(-)}}_N
\end{equation}
where
\begin{equation} \label{eq:Upsilon_N in terms of f and g}
\ket{\Upsilon_{\lambda}^{(\pm)}}_N \propto \sum_{n=0}^N f_{n}^{(\pm)} g_{n}^{(\pm)}\exp[i\hat H_{\rm B}(N\!-\!n)\delta t]\exp[-i\hat H_{\rm F}n\delta t]
\ket{\phi}
\end{equation}
for $2\pi<\theta<4\pi$. Here
\begin{eqnarray}
\label{eq:f_n}
f_{n}^{(\pm)}&=&\exp\{-i[n_+n_- - (n-n_\pm)^2]\theta/2N\}\ ,\\
\label{eq:g_n}
g_{n}^{(\pm)}&=&\exp[-(n-n_{\pm})^2|\theta\tan(\theta/4)|/2N]
\end{eqnarray}
are a complex phase function and a Gaussian weighting function, respectively. Keeping in mind the definition of the clock time $t_{\rm c}$ from \eq{eq:clock time} for the state in \eq{eq:clock state}, we find that $\ket{\Upsilon_\lambda^{(\pm)}}_N$ is a Gaussian-weighted superposition of states over a range of clock times with a mean of $t_{\rm c}=\pm t_{\rm c}^{\rm (peak)}$ and a variance of $(\Delta t_{\rm c})^2 \approx 2/|\lambda\tan(\theta/4)|$. In other words, the states $\ket{\Upsilon_\lambda^{(+)}}_N$ and $\ket{\Upsilon_\lambda^{(-)}}_N$ represent the universe localised in time for a duration of the order of $\Delta t_{\rm c}$ about the mean times $t_{\rm c}= t_{\rm c}^{\rm (peak)}$ and $t_{\rm c}= -t_{\rm c}^{\rm (peak)}$, respectively.
\begin{figure}
\begin{center}
\includegraphics[width=9.0cm]{Fig_Approx.pdf}
\end{center}
\vspace{-8mm}
\caption{$|I_{N-n,n}(z)|$ plotted as a function of the scaled clock time $(t_{\rm c}-t^{\rm (peak)}_{\rm c})/\sigma_{\rm t}$ where $t_{\rm c}=(2n-N)\delta t$. The points $(|I_{N-n,n}(z)|,(t_{\rm c}-t^{\rm (peak)}_{\rm c})/\sigma_{\rm t})$ are generated parametrically by varying $n$. The dots represent the exact values from \eq{eq:interference function} and the solid curves represent the approximation given by $|f_{n}^{+}g_{n}^{+}|$ in \eq{eq:Upsilon_N in terms of f and g}. The numerical values used are $z=\theta/N$ where $\theta=2.23\pi$ and $N=100$ (red curve), $N=1000$ (green) and $N=10000$ (blue). For clarity, the functions have been scaled to give a maximum of unity, and the green ($N=1000$) and blue ($N=10000$) curves have been displaced vertically by 0.2 and 0.4, respectively.
\label{fig:approx}}
\end{figure}
The symmetry of the clock times associated with $\ket{\Upsilon_\lambda^{(+)}}_N$ and $\ket{\Upsilon_\lambda^{(-)}}_N$ about the time $t_{\rm c}=0$ reflects the symmetry of the construction \eq{eq:set Upsilon} and \eq{eq:Upsilon_N} which has no bias toward one direction of time or the other. Moreover, if the state $\ket{\phi}$ is T invariant (i.e. if $\hat{\mathbf{T}}\ket{\phi}\propto\ket{\phi}$) and we shall assume that it is, then $\hat{\mathbf{T}}\ket{\Upsilon_\lambda^{(+)}}_N\propto\ket{\Upsilon_\lambda^{(-)}}_N$ and $\hat{\mathbf{T}}\ket{\Upsilon_\lambda}_N\propto\ket{\Upsilon_\lambda}_N$. This symmetry also arises in time-symmetric cosmological and gravitational studies of the direction of time \cite{Carroll,Barbour}. As the time evolution in one component of the superposition in \eq{eq:Upsilon with two peaks} is mirrored in the other, it suffices for us to consider just $\ket{\Upsilon_\lambda^{(+)}}_N$ and its corresponding value of $t_{\rm c}^{\rm (peak)}=2\pi\sqrt{N}\sigma_{\rm t}/\theta$. Accordingly, we will call this value of $t_{\rm c}^{\rm (peak)}$ the \textit{representative clock time} and use it to label the whole state $\ket{\Upsilon_\lambda}_N$. The minimum representative clock time of a state in the set $\boldsymbol{\Upupsilon}_\lambda$ is found, using \eq{eq:peak clock time} with $N=N^{\rm (time)}_{\rm min}=\sigma_{\rm t}^2/\delta t_{\rm min}^2$ and $\theta = \sigma^2_{\rm t}\lambda$, to be
\begin{equation} \label{eq:shortest time}
t_{\rm c, min}^{\rm (peak)}=\frac{2\pi}{\lambda \delta t_{\rm min}}\ .
\end{equation}
A discussion of the values of $\lambda$ and $\delta t_{\rm min}$ in relation to $t_{\rm c, min}^{\rm (peak)}$ is given in Section C of the Supplementary Material.
Figure \ref{fig:approx} compares the coefficients $I_{N-n,n}(z)$ of the state $\ket{\Upsilon_\lambda}_N$ in \eq{eq:Upsilon with interference fn} with their Gaussian approximation $f_{n}^{+}g_{n}^{+}$ in \eq{eq:Upsilon_N in terms of f and g} near a maximum. The coefficients have been plotted as a function of $(t_{\rm c}-t_{\rm c}^{\rm (peak)})/\sigma_{\rm t}$ to centre them in the figure, where $t_{\rm c}^{\rm (peak)}$ is the position of the maxima given by \eq{eq:peak clock time}. As in Fig.~\ref{fig:TInv}, the values of $N$ have been chosen purposely to exaggerate the discreteness of the state $\ket{\Upsilon_\lambda}_N$. However, for every $\ket{\Upsilon_\lambda}_N\in \boldsymbol{\Upupsilon}_\lambda$ in \eq{eq:set Upsilon}, the values of $N$ are sufficiently large (i.e. $N\ge N^{\rm (time)}_{\rm min}$) that the locus of points representing $I_{N-n,n}(z)$ is essentially equivalent to the Gaussian approximation $f_{n}^{+}g_{n}^{+}$ up to a proportionality constant. It follows that each $\ket{\Upsilon_\lambda}_N\in \boldsymbol{\Upupsilon}_\lambda$ is physically indistinguishable from a state given by \eq{eq:Upsilon with two peaks} and \eq{eq:Upsilon_N in terms of f and g} with the same value of $N$ but where the sum over $n$ in \eq{eq:Upsilon_N in terms of f and g} is replaced with its integral equivalent.
The broad properties of the states $\ket{\Upsilon_\lambda}_N$ are illustrated in Fig.~\ref{fig:interference function} which shows $|I_{N-n,n}(z)|$ plotted as a function of the scaled clock time $t_{\rm c}/\sigma_{\rm t}$. The black curve corresponds to the time reversal invariance case where $\lambda=0$ (and so $\theta= \sigma^2_{\rm t}\lambda=0$).
All other curves correspond to the violation of time reversal invariance (i.e. $\lambda\ne 0$) and have been generated for $\theta=2.23\pi$ which gives the minimum uncertainty in energy and time (see Section B of the Supplementary Material for details). The figure illustrates how the location of the maxima at $t_{\rm c}=\pm t_{\rm c}^{\rm (peak)}$ increases with $N$ as given by \eq{eq:peak clock time}.
For clarity, $|I_{N-n,n}(z)|$ is plotted in Fig.~\ref{fig:interference function} only for a select few values of $N$ for which the peaks in the corresponding curves are widely separated. To see how close the peaks can be, consider the difference $\delta t_{\rm c}^{\rm (peak)}$ in the representative clock times $t_{\rm c}^{\rm (peak)}$ of states $\ket{\Upsilon_\lambda}_N$ with consecutive values of $N$, which is found from \eq{eq:peak clock time} to be
\[
\delta t_{\rm c}^{\rm (peak)}=\frac{2\pi\sigma_{\rm t}\sqrt{N+1}}{\theta}-\frac{2\pi\sigma_{\rm t}\sqrt{N}}{\theta}
\approx \frac{\sigma_{\rm t}\pi}{\theta\sqrt{N}}
\]
for large $N$. Noting that $N \ge N^{\rm (time)}_{\rm min}=\sigma_{\rm t}^2/\delta t_{\rm min}^2$ gives $\delta t_{\rm c}^{\rm (peak)} \le (\pi/\theta)\delta t_{\rm min}$ and as $2\pi < \theta < 4\pi$ we find
\[
\delta t_{\rm c}^{\rm (peak)} < \frac{1}{2}\delta t_{\rm min}\ .
\]
Hence, for any given time $t>t_{\rm c, min}^{\rm (peak)}$, there is a state in the set $\mathbf{\Upsilon}_\lambda$ given by \eq{eq:set Upsilon} whose representative clock time $t_{\rm c}^{\rm (peak)}$ is equal to $t$ to within the resolution limit $\delta t_{\rm min}$.
\begin{figure}
\begin{center}
\includegraphics[width=9.0cm]{Fig_Interference_Fn.pdf}
\end{center}
\vspace{-5mm}
\caption{$|I_{N-n,n}(z)|$ as a function of the scaled clock time $t_{\rm c}/\sigma_{\rm t}$ where $t_{\rm c}=(2n-N)\delta t$ for different values of $\lambda$ and $N$. As in Fig.~\ref{fig:approx}, the points $(|I_{N-n,n}(z)|,t_{\rm c}/\sigma_{\rm t})$ are generated parametrically by varying $n$. For clarity, in each case straight lines connect consecutive discrete points of $|I_{N-n,n}(z)|$ to form a continuous curve. The black curve represents the T invariant case (i.e. $\lambda=0$) and has been generated for $N=1000$. It does not visibly change with increasing values of $N$. The remaining curves represent the T violation case (i.e. $\lambda\ne 0$) for $\theta=2.23\pi$ and a range of $N$ values as follows: red curve for $N=300$ and $t_{\rm c}^{\rm (peak)}=15.5\sigma_{\rm t}$, green curve for $N=1200$ and $t_{\rm c}^{\rm (peak)}=31.1\sigma_{\rm t}$, light blue curve for $N=2600$ and $t_{\rm c}^{\rm (peak)}=45.7\sigma_{\rm t}$, and dark blue curve for $N=4600$ and $t_{\rm c}^{\rm (peak)}=60.8\sigma_{\rm t}$. All curves have been scaled to give a maximum of unity.
\label{fig:interference function}}
\end{figure}
Fig.~\ref{fig:interference function} clearly shows that the inclusion of the violation of time reversal invariance {\it dramatically} changes the set $\boldsymbol{\Upupsilon}_{\lambda}$ in \eq{eq:set Upsilon} from one containing elements that are physically equivalent (represented by the black curve), to one containing states that are diverging in time (other curves).
This striking outcome warrants careful consideration. Both sets $\boldsymbol{\Upupsilon}_{\lambda=0}$ and $\boldsymbol{\Upupsilon}_{\lambda\ne 0}$ have the same mathematical construction given by \eq{eq:set Upsilon}; the striking difference we found between them is due solely to the phenomenological Hamiltonian and whether it respects T symmetry ($\lambda=0$) or not ($\lambda\ne 0$). All the states in $\boldsymbol{\Upupsilon}_{\lambda=0}$ are physically equivalent to a unique state, $\ket{\Upsilon_0}$, which represents the galaxy as existing for one particular finite period in time. This constitutes phenomenology associated with T symmetry.
In contrast, with T violation there are infinitely-many different states in the set $\boldsymbol{\Upupsilon}_{\lambda\ne 0}$. There is no physically-based reason to suppose that any of them has special significance and so, by default, all states in $\boldsymbol{\Upupsilon}_{\lambda\ne 0}$ have equal status in representing the state of the galaxy in time. This \textit{pluralism} constitutes phenomenology associated with T violation. That different states can equally represent the galaxy is not contradictory because each state represents the galaxy at a different representative clock time. In fact, the same pluralism is assumed in conventional quantum physics, and is the root of the asymmetry between time and space discussed in the Introduction.
\subsection{Impact for quantum states in time and space}
These remarkable results manifest a fundamental difference between quantum states in time and space. All the states in the set $\boldsymbol{\Uppsi}$, irrespective of whether the discrete symmetries are obeyed or not, represent the galaxy existing only in a region of order $\sigma_{\rm x}$ near $x=0$.
Likewise, all the states in the set $\boldsymbol{\Upupsilon}_{\lambda=0}$ associated with T symmetry represent the galaxy existing only for a duration of order $\sigma_{\rm t}$ near $t_{\rm c}=0$. The fact that the states in the set $\boldsymbol{\Upupsilon}_{\lambda=0}$ don't conserve mass is testament to mass conservation not being an explicit property of the construction defined by \eq{eq:set Upsilon} and \eq{eq:Upsilon_N}.
But for a set $\boldsymbol{\Upupsilon}_{\lambda'}$ associated with T violation with $\lambda'\ne 0$, for any given time $t\ge t_{\rm c, min}^{\rm (peak)}$ we have just seen that there is a state $\ket{\Upsilon_{\lambda'}}_N\in\boldsymbol{\Upupsilon}_{\lambda'}$, whose representative clock time $t_{\rm c}^{\rm (peak)}$ is equal to $t$ to within the resolution limit $\delta t_{\rm min}$. In other words, the set $\boldsymbol{\Upupsilon}_{\lambda'}$ contains a state that represents the galaxy's existence at each corresponding moment in time. That being the case, it would not be unreasonable to regard the set as representing a \textit{history} of the universe. It follows that the set $\boldsymbol{\Upupsilon}_{\lambda'}$ represents the persistence of the mass of the galaxy over the same period of time, in so far as the Hamiltonians $\hat H_F$ and $\hat H_B$ conserve mass. This raises a subtle point regarding conservation laws; while they may be due to deep principles (such as Noether's theorem) they are not manifested in quantum mechanics unless the state persists over a period of time. The crucial point being that in conventional quantum mechanics, the persistence of the state is essentially \textit{axiomatic} and ensured by adopting a compliant dynamical equation of motion whereas here it arises \textit{phenomenologically} as a property of the set of states $\boldsymbol{\Upupsilon}_{\lambda'}$. Finally, on comparing the two sets $\boldsymbol{\Upupsilon}_{\lambda=0}$ and $\boldsymbol{\Upupsilon}_{\lambda'\ne0}$ one could even venture to say that T violation, in effect, \textit{causes} the contents of the universe to be translated or, indeed, to \textit{evolve}, over an unbounded period of time.
\section{\label{sec:emergence}Emergence of conventional quantum mechanics}
\subsection{\label{sec:conventional qm}Coarse graining over time}
The spread of the state $\ket{\Upsilon_\lambda}_N$ along the time axis, as illustrated by the plots of $|I_{N-n,n}(z)|$ in Fig.~\ref{fig:interference function}, represents a significant departure from conventional quantum mechanics for which states are interpreted as having no extension in time. Nevertheless, the conventional formalism can be recovered in the following way. Imagine that observations of the galaxy are made with a resolution in time that is much larger than the width of the Gaussian weighting function $g_{n}^{\pm}$ in \eq{eq:Upsilon_N in terms of f and g}. Under such coarse graining, the summation in \eq{eq:Upsilon_N in terms of f and g} can be replaced by the term corresponding to the maximum in $g_{n}^{\pm}$ and so, for example,
\[
\ket{\Upsilon_\lambda^{(+)}}_N \srel{\propto}{\sim} \exp[i\hat H_{\rm B}(N-n_+)\delta t]\exp(-i\hat H_{\rm F}n_+\delta t) \ket{\phi}\ .
\]
We can re-express this state in terms of its representative clock time, $t_{\rm c}^{\rm (peak)}$, which we shall shorten to $t_{\rm c}$ for brevity, as
\begin{equation} \label{eq:Upsilon_+ coarse grained}
\ket{\Upsilon_\lambda^{(+)}}_N \srel{\propto}{\sim} \exp(i\hat H_{\rm B}t_{\rm c}a_-)\exp(-i\hat H_{\rm F}t_{\rm c}a_+) \ket{\phi}
\end{equation}
where $a_\pm=n_\pm/(n_+-n_-)$ and we have used $t_{\rm c}=(2n_+-N)\delta t=(n_+-n_-)\delta t$ and $N-n_+=n_-$. At this level of coarse graining, the time step $\delta t$ is effectively zero and $t_{\rm c}$ is effectively a continuous variable.
Making use of the Baker-Campbell-Hausdorff formula \cite{Suzuki} in \eq{eq:Upsilon_+ coarse grained} yields
\begin{equation}
\ket{\Upsilon_\lambda^{(+)}}_N \srel{\propto}{\sim} \exp(\sfc{1}{2}ia_+a_-t_{\rm c}^2\lambda)\exp[-i(\hat H_{\rm F}a_+ - \hat H_{\rm B}a_-)t_{\rm c}] \ket{\phi}\ .
\end{equation}
The complex phase factor can be accommodated by transforming to a new state, $\ket{\widetilde\Upsilon(t_{\rm c})}$, as follows
\begin{equation} \label{eq:Upsilon_+ coarse grained transformed}
\ket{\widetilde\Upsilon(t_{\rm c})}= \exp(-\sfc{1}{2}ia_+a_-t_{\rm c}^2\lambda)\ket{\Upsilon_\lambda^{(+)}}_N\propto \exp[-i(\hat H_{\rm F}a_+ - \hat H_{\rm B}a_-)t_{\rm c}] \ket{\phi}\ .
\end{equation}
On taking the derivative with respect to $t_{\rm c}$ we recover Schr\"{o}dinger's equation,
\begin{equation} \label{eq:schrodingers equation}
\frac{d}{dt_{\rm c}}\ket{\widetilde\Upsilon(t_{\rm c})} \srel{\propto}{\sim}
-i(\hat H_{\rm F}a_+ - \hat H_{\rm B}a_-)\ket{\widetilde\Upsilon(t_{\rm c})}\ .
\end{equation}
Here, the coarse-grained Hamiltonian $(\hat H_{\rm F}a_+ - \hat H_{\rm B}a_-)$ is a linear combination of $\hat H_{\rm F}$ and $\hat H_{\rm B}$ owing to the fact that the quantum virtual path involves contributions from both.
Notice that the differential equation \eq{eq:schrodingers equation} does not depend on the state $\ket{\phi}$. The ambiguity associated with loosely specifying $\ket{\phi}$ as being sharply defined in time does not play a role here. It is true that different choices for $\ket{\phi}$ will lead to different states $\ket{\widetilde\Upsilon(t_{\rm c})}$, but that is no concern when the goal is to show that conventional quantum mechanics is recovered. The fact that \eq{eq:schrodingers equation} results for all allowed choices of $\ket{\phi}$ is all that is needed for this.
It is useful at this point to divide the galaxy into two non-interacting subsystems, one whose Hamiltonian $\hat H^{\rm (i)}=\hat{\mathbf{T}}^{-1}\!\hat H^{\rm (i)}\hat{\mathbf{T}}$ is T-invariant and the remainder whose Hamiltonian $\hat H_{\rm F}^{\rm (v)}=\hat{\mathbf{T}}^{-1}\!\hat H_{\rm B}^{\rm (v)}\hat{\mathbf{T}}\ne\hat H_{\rm B}^{\rm (v)}$ is T-violating; in that case we can write
\begin{equation} \label{eq:Separated Hamiltonian}
\hat H_{\rm F}=\hat H^{\rm (i)}\otimes\hat{\boldsymbol 1}^{\rm (v)}+\hat{\boldsymbol 1}^{\rm (i)}\otimes\hat H_{\rm F}^{\rm (v)}\ ,\
\hat H_{\rm B}=\hat H^{\rm (i)}\otimes\hat{\boldsymbol 1}^{\rm (v)}+\hat{\boldsymbol 1}^{\rm (i)}\otimes\hat H_{\rm B}^{\rm (v)}
\end{equation}
where the superscripts ``i'' and ``v'' label operators associated with the state space of the T-invariant and T-violating Hamiltonians, respectively, and $\hat{\boldsymbol 1}^{(\rm \cdot)}$ is an appropriate identity operator. Equation \eq{eq:schrodingers equation} can then be rewritten as
\begin{equation} \label{eq:schrodinger eqn for I and V}
\frac{d}{dt_{\rm c}}\ket{\widetilde\Upsilon(t_{\rm c})} \srel{\propto}{\sim}
-i(\hat H^{\rm (i)}\otimes\hat{\boldsymbol 1}^{\rm (v)}+\hat{\boldsymbol 1}^{\rm (i)}\otimes\hat H_{\rm phen}^{\rm (v)})\ket{\widetilde\Upsilon(t_{\rm c})}
\end{equation}
where $\hat H_{\rm phen}^{\rm (v)}=\hat H_{\rm F}^{\rm (v)}a_+ - \hat H_{\rm B}^{\rm (v)}a_-$ is the phenomenological Hamiltonian for the T-violating subsystem.
It is straightforward to show that the commutator of $\hat H_{\rm phen}^{\rm (v)}=\hat H_{\rm F}^{\rm (v)}a_+ - \hat H_{\rm B}^{\rm (v)}a_-$ with its time reversed version is
\[
[\hat H_{\rm phen}^{\rm (v)}, \hat{\mathbf{T}}^{-1}\!\hat H_{\rm phen}^{\rm (v)}\hat{\mathbf{T}}] = -i\frac{\theta}{2\pi}\lambda
\]
which is $\theta/2\pi$ times the commutator $[\hat H_{\rm F}^{\rm (v)}, \hat H_{\rm B}^{\rm (v)}]$. Thus, in principle, the commutation relation could be used to distinguish the phenomenological Hamiltonians $\hat H_{\rm phen}^{\rm (v)}$ and $ \hat{\mathbf{T}}^{-1}\!\hat H_{\rm phen}^{\rm (v)}\hat{\mathbf{T}}$ from the more elementary versions $\hat H_{\rm F}^{\rm (v)}$ and $\hat H_{\rm B}^{\rm (v)}$.
\subsection{Conventional formalism and potential experimental tests}
These results are important because they not only show how the conventional formalism of quantum mechanics is recovered, but they also show how the construction introduced here may be verified experimentally. To see this consider the following three points. First, \eq{eq:schrodinger eqn for I and V} shows that the T-invariant subsystem behaves in accord with the conventional Hamiltonian $\hat H^{\rm (i)}$ with respect to clock time $t_{\rm c}$. This means that conventional quantum mechanics is recovered for this subsystem. Second, \eq{eq:schrodinger eqn for I and V} shows that, due to the coarse graining, the role of the clock time $t_{\rm c}$ has been reduced from being a physical variable that describes the location and uncertainty of the galaxy with respect to time as illustrated in Fig.~\ref{fig:interference function}, to being simply a parameter that labels a different state in the set $\boldsymbol{\Upupsilon}_\lambda$ according to the time $t_{\rm c}=t_{\rm c}^{\rm (peak)}$ of the maximum in $g_{n}^{+}$. Indeed, its demoted role is the very reason we are able to recover Schr\"{o}dinger's equation. Third, any experiments involving T-violating matter that are performed by observers in the galaxy would give results that are consistent with \eq{eq:schrodinger eqn for I and V} and so they would provide evidence of the phenomenological Hamiltonian $\hat H_{\rm phen}^{\rm (v)}$ in the same way that experiments in our universe give evidence of the Hamiltonian associated with meson decay. Any demonstration that $\hat H_{\rm phen}^{\rm (v)}$ differs from the more elemental Hamiltonians $\hat H_{\rm F}^{\rm (v)}$ and $\hat H_{\rm B}^{\rm (v)}$ represents a ``smoking gun'' for the construction introduced here. Of course, this specific result can not be used in practice because it applies to the simple model of T violation chosen here for its clarity rather than accuracy, and also because the present knowledge of T violating Hamiltonians is based on empirical results and so it is limited to the phenomenological version of the Hamiltonians. More realistic models of the universe and T violating mechanisms may provide experimentally testable predictions, such as novel deviations from exponential decay for T violating matter or local variations in clock time. But these are beyond the scope of the present work whose aim is to show, in the clearest way possible, how T violation may underlie differences between time and space.
\section{\label{sec:conclusion}Discussion}
We began by drawing attention to the asymmetry between time and space in conventional quantum theory where states are presumed to undergo continuous translation over time whereas there is no corresponding presumption about the state necessarily undergoing translations over space. We set out to explore an alternate possibility by introducing a new quantum formalism that gives both space and time analogous quantum descriptions. In developing the formalism, we paid particular attention to subtle mathematical details that play no significant role in conventional quantum mechanics. These details involve explicitly taking into account the P and T symmetry operations, translations of states in space and time, and fundamental limits of precision. We incorporated them in a mathematical construction where quantum states are represented as a superposition of random paths in space or time. We found that with no P or T symmetry violations, quantum states had analogous representations in space and time: just as matter can be represented as existing only in a finite region of space, it can also be represented as existing only for a finite interval of time. Clearly the price we pay for this symmetry is absence of the conservation of mass. However with the violation of T symmetry, dramatic differences between the representation of quantum states in space and time arise through the quantum interference between different paths. The state (and the matter it describes) is found to persist over an unbounded range of time values. This result gives a new appreciation of conservation laws: while they may be due to deep principles they are not manifested unless the state persists over a sufficient period of time. The Schr\"{o}dinger equation of conventional quantum mechanics, where time is reduced to a classical parameter, also emerges as a result of coarse graining over time. As such, T violation is seen in the new formalism as being responsible for fundamental differences between space and time in conventional quantum mechanics.
The new formalism may also help resolve other perplexing issues associated with space and time. For example, the arrows of time indicate a preferred direction from ``past'' to ``future'' \cite{Price}, but there is no analogous preferred direction of space. The new formalism appears to offer a basis for understanding why. Indeed the set of states in time, $\boldsymbol{\Upupsilon}_\lambda$ for $\lambda\ne 0$ in \eq{eq:set Upsilon}, has a natural order over time in the following sense. First recall that our interpretation of \eq{eq:Upsilon_N in terms of H_F H_B} is that $\exp(-i\hat H_{\rm F} t)$ and $\exp(i\hat H_{\rm B} t)$ are associated with physical evolution in different directions of time, whereas the inverses $\exp(i\hat H_{\rm F} t)$ and $\exp(-i\hat H_{\rm B} t)$ are associated with the mathematical operations of rewinding that physical evolution. Within this context, the coarse-grained state $\ket{\widetilde\Upsilon(t_{\rm c})}$ in \eq{eq:Upsilon_+ coarse grained transformed} is interpreted as resulting from evolution by $t_{\rm c}a_+$ in the positive direction of time and $t_{\rm c}a_-$ in the reverse direction, giving a net evolution of $t_{\rm c}(a_+-a_-)=t_{\rm c}$ in time from the state $\ket{\phi}$. Correspondingly, the state $\ket{\widetilde\Upsilon(t'_{\rm c})}$ with $t'_{\rm c}>t_{\rm c}$ represents a \textit{more-evolved state} than $\ket{\widetilde\Upsilon(t_{\rm c})}$. In fact writing
\begin{equation} \label{eq:Upsilon evolving}
\ket{\widetilde\Upsilon(t'_{\rm c})}\propto \exp[-i(\hat H_{\rm F}a_+ - \hat H_{\rm B}a_-)\Delta t] \ket{\widetilde\Upsilon(t_{\rm c})}
\end{equation}
where $\Delta t=t'_{\rm c}-t_{\rm c}>0$ shows that $\ket{\widetilde\Upsilon(t'_{\rm c})}$ evolves from $\ket{\widetilde\Upsilon(t_{\rm c})}$. One might be tempted to argue that we could equally well regard $\ket{\widetilde\Upsilon(t_{\rm c})}$ as evolving from $\ket{\widetilde\Upsilon(t'_{\rm c})}$ because
\begin{equation} \label{eq:Upsilon rewinding}
\ket{\widetilde\Upsilon(t_{\rm c})}\propto \exp[i(\hat H_{\rm F}a_+ - \hat H_{\rm B}a_-)\Delta t]\ket{\widetilde\Upsilon(t'_{\rm c})}\ ,
\end{equation}
but doing so would be inconsistent with our interpretation of \eq{eq:Upsilon_N in terms of H_F H_B}. According to that interpretation, \eq{eq:Upsilon rewinding} represents the mathematical \textit{rewinding} of the physical evolution represented by \eq{eq:Upsilon evolving}. Note that the state $\ket{\widetilde\Upsilon(t_{\rm c})}$ is a coarsed-grained version of the component $\ket{\Upsilon_\lambda^{(+)}}_N$ of $\ket{\Upsilon_\lambda}_N$ in \eq{eq:Upsilon with two peaks}; an analogous argument also applies to the coarse-grained version of the other component $\ket{\Upsilon_\lambda^{(-)}}_N$, and thus to the whole state $\ket{\Upsilon_\lambda}_N$. Hence, the set of states $\boldsymbol{\Upupsilon}_\lambda$ for $\lambda\ne 0$ are ordered by the degree of time evolution from the state $\ket{\phi}$. This gives two preferred directions of time away from the origin of the time axis and so represents a \textit{symmetric arrow of time}. Time-symmetric arrows have also been explored by Carroll, Barbour and co-workers \cite{Carroll,Barbour}. In stark contrast, there is no analogous ordering for $\boldsymbol{\Uppsi}$ in \eq{eq:set Psi}, the set of states distributed over space. Indeed, all the states in $\boldsymbol{\Uppsi}$ are physically indistinguishable. Also the ordering of the set $\boldsymbol{\Upupsilon}_\lambda$ vanishes at $\lambda= 0$ which corresponds to T symmetry. It appears, therefore, that T violation is also responsible giving time a direction (in the sense of orientating time away from the occurrence of $\ket{\phi}$).
In addition to these conceptual results, the new formalism was also found to have potential experimentally-testable consequences. Indeed, for a subsystem associated with T violation, the formalism predicts that the experimentally-determined Hamiltonian, $\hat H_{\rm phen}^{\rm (v)}$ in \eq{eq:schrodinger eqn for I and V}, will be different to the Hamiltonians, $\hat H_{\rm F}^{\rm (v)}$ or $\hat H_{\rm B}^{\rm (v)}$ in \eq{eq:Separated Hamiltonian}, associated with conventional quantum mechanics. Further work is needed to develop feasible experiments for testing predicted departures from conventional theory like this. An experimental verification of the new formalism would have profound impact on our understanding of time.
In conclusion, the importance of Feynman's sums over paths for describing quantum phenomena is well beyond doubt \cite{Feynman}. A distinctive feature of the quantum virtual paths in the new formalism is that they explicitly take into account the violation of T symmetry. The new formalism has the advantage of giving \textit{time and space an equal footing at a fundamental level} while allowing familiar differences, such as matter being localised in space but undergoing unbounded evolution in time, to arise \textit{phenomenologically} due to the fact that T violation is a property of translations in time and not space.
As such, the violation of the discrete symmetries are seen to play a defining role in the quantum nature of time and space.
I thank D.T. Pegg, H.M. Wiseman, M.J. Hall and T. Croucher for helpful discussions.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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# Sunset Express
An Elvis Cole Novel
## Robert Crais
**Begin Reading**
Table of Contents
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In accordance with the U.S. Copyright Act of 1976, the scanning, uploading, and electronic sharing of any part of this book without the permission of the publisher constitutes unlawful piracy and theft of the author's intellectual property. If you would like to use material from the book (other than for review purposes), prior written permission must be obtained by contacting the publisher at permissions@hbgusa.com. Thank you for your support of the author's rights.
_For Leonard Isaacs,_
_who opened the door,_
_and_
_for Kate Wilhelm and Damon Knight,_
_who invited me in._
# Prologue
The sky above the San Fernando Valley that Saturday morning was a deep blue, washed clean of the dirt and chemical particulates that typically color L.A. air by a breeze that burbled out of the San Gabriel Mountains and over the flat valley floor and across the high ridge of the Santa Monica Mountains. Mulholland Drive snakes along the crest of the Santa Monicas, and, if you were walking along Mulholland as Sandra Bernson and her father were doing that morning, you would have been able to look south almost forty miles across the Los Angeles basin to the tip of the Long Beach Peninsula or north some thirty-five miles across the San Fernando Valley and through the Newhall Pass to the deep purples of the Santa Susana Mountains and the peaks surrounding Lake Castaic. It was a day of unusual clarity, the far horizons magnified as if by some rare trick of optical law that might even allow you to see into the lives of the sleeping millions in the valleys below. Sandra Bernson later said that as she watched the small private airplanes floating into and out of Van Nuys Airport in the center of the valley that morning, she imagined them to be flying carpets. On mornings like these, she also said, it was easy to believe in magic.
Sandra was a fifteen-year-old honor student at the prestigious Harvard-Westlake School, and her father, Dave Bernson, was a television writer and producer of moderate success, then working as the supervising producer of a popular series on the Fox Television network. The Bernsons lived in a contemporary home on a small private road off Mulholland Drive in Sherman Oaks, approximately one mile west of Beverly Glen, and they left their home at exactly 6:42 that morning. Both Sandra and Dave were able to tell investigators their exact departure time because it was Dave's habit to call out when their walks began so that they could time themselves. They intended to walk east along Mulholland to Warren Beatty's home approximately one mile east of Beverly Glen, where they planned to reverse course and return. Their typical walk would cover four miles round-trip and take almost exactly fifty minutes. On this particular Saturday, however, they never made it to Beatty's and they didn't complete the walk.
On this Saturday, Sandra Bernson saw the deer.
They proceeded east from their home, climbing one of Mulholland's steeper grades to a high, flat stretch of road abreast Stone Canyon Reservoir. That was Sandra's favorite part of the walk because she could see the valley to the north and the reservoir to the south, and just before they came to Beverly Glen Canyon they would reach the Stone Canyon overlook. The overlook is built into the top of a little knoll there beside Mulholland, with manicured walks and observation points and benches if you want to sit and admire what realtors like to call a 360-degree jetliner view. Sandra remembers that as she and her father reached the top of the overlook she saw the deer creeping up from the valley side of Mulholland, sniffing and listening, and she whispered to her father, "Look, Dad!"
"Mule deer. See the size of his ears? It's a buck, but he's already shed his horns. See the knobs above his eyes?"
The deer heard them. It looked in their direction, its huge ears cocked forward, and then it bounded across Mulholland and the overlook's little parking lot and disappeared. Sandra said, "I wanna see where he goes!"
She slid across the overlook's low wall and ran to the edge of the knoll just as the buck vanished near a cut in the slope that had caught a lot of dead brush and beer cans and newspapers and brown plastic garbage bags. Dave arrived at her side a moment later. Everything caught by the cut looked old and dusty and weathered as if it had been there for a very long time, except for the garbage bags. They looked shiny and new, and Sandra was using them as a landmark to point out to Dave where she had last seen the mule deer when she saw the hand sticking out of the bags. The nail polish was very red and seemed to gleam in the breathtakingly clear morning sun.
It never entered Dave's mind that the hand might be a movie prop or belong to a mannequin; the moment he saw it he knew it was real. It looked real, and it also looked dead. Dave recalls that he considered working his way down to the body, but then says that he remembered things like clues and evidence, and so he led his daughter back to Mulholland where they flagged down a passing Westec private security car. The security cop, a twenty-eight-year-old ex-Marine named Chris Bell, parked his unit and went to see for himself, then returned to his car and reported the find to the Westec offices. In less than eight minutes, two LAPD patrol units arrived on the scene. The uniforms observed the hand protruding from the plastic, but, as had Dave Bernson, decided not to venture down the slope. The uniforms relayed their observations in code by radio, then secured the area to await the arrival of the detectives.
Dave Bernson offered to wait also, but by that time Sandra had to pee really bad, so one of the uniforms drove them home. Forty minutes after Sandra Bernson and her father were returned to their home, and thirty-nine minutes after Sandra began calling her friends just as quickly as she could to tell them about this incredibly gross thing that had just happened, the first detective unit arrived on the scene.
Detective Sergeant Dan "Tommy" Tomsic and Detective-two Angela Rossi were in the first car. Tomsic was a powerfully built man who'd spent a dozen years on the street before making the transfer to detectives. He had almost thirty years on the job, and he viewed the world through suspicious, unblinking eyes. Angela Rossi was thirty-four years old, with twelve years on the job, and had been Tomsic's partner for only five weeks. Rossi spoke her mind, was entirely too confrontational, and, because of this, she had trouble keeping partners. So far Tomsic didn't seem bothered, but that was probably because he ignored her.
Eleven minutes after the first car, the senior detectives arrived on the scene. Detective Sergeant Lincoln Gibbs was a tall, thin African-American with mocha-colored skin, a profoundly receding hairline, and tortoiseshell spectacles. He looked like a college professor, which was a look he cultivated. He had twenty-eight years on the job, less than Tomsic, but more time in grade as a detective sergeant, so Linc Gibbs would be in charge. He arrived with Detective-three Pete Bishop, a twenty-two-year veteran with an M.A. in psychology and five divorces. Bishop rarely spoke, but was known to make copious notes, which he referred to often. He had a measured IQ of 178 and a drinking problem. He was currently in twelve-step.
The four detectives got the story from the uniforms and the Westec cop, then went to the edge of the overlook and stared down at the hand. Gibbs said, "Anybody been down there?"
One of the uniforms said, "No, sir. It's undisturbed."
The detectives searched the ground for anything that might present itself as evidence—scuff marks, drops of blood, footprints, that kind of thing. There were none. They could see the path that the body had followed as it slid down the slope. Scuffs on the soil, broken and bent plants, dislodged rocks. Linc followed the trail with his eyes and figured that the body had been dumped from a point just at the rear of the parking lot. The body was between twelve and fifteen yards down a damned steep slope. Someone would have to go down, and that presented certain problems. You wouldn't want to follow the same path as the body because that might disturb evidence. That meant they'd have to find another route, only everything else was steeper and the drop-off more pronounced. Linc was thinking that it might take mountaineering gear when Angela Rossi said, "I can get down there."
The three male detectives looked at her.
"I've done some rock climbing in Chatsworth and I work terrain like this all the time when I'm backpacking." She pointed out her route. "I can work my way down the slide over there, then traverse back and come up under the body. No sweat."
Dan Tomsic said, "That goddamned soil is like sand. It won't hold your weight."
"It's no sweat, Dan. Really."
Rossi looked like the athletic type, and Gibbs knew that she had run in the last two L.A. marathons. Tomsic sucked down three packs a day and Bishop had the muscle tone of Jell-O. Rossi was also fifteen years younger than the rest of them, and she wanted to go. Gibbs gave his permission, told her to take the camera, and Angela Rossi went back to the car to trade her Max Avante pumps for a busted-out pair of New Balance running shoes. She reappeared a minute later, and Gibbs, Tomsic, and the others watched as she worked her way down to the body. Tomsic frowned as he watched, but Gibbs nodded in approval—Rossi seemed graceful and confident in her movements. Tomsic was praying that she wouldn't lose her balance and break her damned neck—one slip and she'd flop ass over teapot another sixty or eighty yards down the slope.
Down below, Rossi never once entertained the notion that she might fall. She was feeling absolutely confident and more than a little jazzed that it was she who had taken the lead in recovering the body. If you took the lead you got the promotions, and Rossi made no secret that she wanted to become LAPD's first female chief of detectives. It was a goal she had aggressively pursued since her days at the academy and, though there had been what she called her Big Setback, she still hoped that she could get her career back on track and pull it off.
When Rossi reached the body, she could smell it. The sun was rising and the dark plastic was heating quickly and holding the heat. As water evaporated from the body it collected on the plastic's inner surface, and, Rossi knew, it would be humid and damp inside the bag. The victim's abdomen would swell and the gases of decay would vent. Decomposition had begun.
Linc called down to her, "Try not to move the body. Just take the snaps and peel back the bags."
Rossi used the Polaroid to fix the body's position for the record, then pulled on rubber surgical gloves and touched the wrist, checking for a pulse. She knew that there would be none, but she had to check anyway. The skin was pliant but the muscles beneath were stiff. Rigor.
Rossi couldn't see much, as yet, but the body appeared intact and double-bagged in two dark brown plastic garbage bags. The bags were secured around the body with silver duct tape, but the job appeared to have been done hastily. The bags had parted and the hand had plopped out. Angela Rossi peeled the bags apart to expose the shoulder and head of a blonde Caucasian woman who appeared to be in her early thirties. The woman was clothed in what looked like a pale blue Banana Republic T-shirt that was splattered with blood. The woman's left eye was open but her right eye was closed, and the tip of her tongue protruded between small, perfect teeth. The hair on the back and right side of her head was ropey and matted with blood. Much of the blood was dried, but there was a shiny, wet quality to much of it, also. The skull at that portion of the hair appeared depressed and dark, and brain matter and ridges of white skull were obvious. The woman's nose was straight and her features rectangular and contoured. In life, she would've been pretty. Angela Rossi had an immediate sense that the woman looked familiar.
Tomsic yelled down, "Don't pitch a goddamned tent down there. What's the deal?"
Rossi hated it when he spoke to her that way, but she clenched her jaw and took it. She'd been taking it more and smarting off less since the Big Setback. Anything to resurrect the career. She called back without looking at them. "Caucasian female. Early thirties. Blunt force trauma to the back of the head." She pushed the garbage bag back farther, exposing the victim's head and shoulders. She saw no additional injuries and wanted to peel back the bags even farther, but was concerned that the body would dislodge and tumble down the slope, possibly taking her with it. She took more pictures, then said, "The blood around the wound appears to be tacky, and it's wet in some spots. She hasn't been here long."
Bishop said, "Lividity?"
"A little, but it could be bruising."
Above her, Linc Gibbs was growing impatient with all the conversation. He didn't like Rossi perched on such a steep slope, and he wanted to call in the criminalist. He said, "What about a weapon?" Murderers almost always dumped the murder weapon with the body.
He watched Rossi lean across the body and feel around the bags. She moved out of sight twice, and each time he tasted acid because he thought she'd fallen. Another Tagamet day. He remembers that he was just getting ready to ask her what in hell was taking so long when she said, "Don't see anything, but it could be under the body or in the bag."
Gibbs nodded. "Leave it for the criminalist. Take some more pix and get back up here."
Rossi took the remainder of the roll, then worked her way back up the slope. When she reached the top, the others crowded around to see the pictures. All of the male detectives pulled out reading glasses except for Gibbs, who wore bifocals.
One of the uniformed cops said, "Hey. She looks like somebody."
Rossi said, "I thought so, too."
She didn't look like anyone to Gibbs. "You guys recognize her?"
Bishop was turning the pictures round and round, as if seeing the victim from every possible view was important. All the turning was making Tomsic nauseated. Bishop said, "Her name is Susan Martin."
The Westec cop said, "Holy Christ, you're right. Teddy Martin's wife."
All four detectives looked at him.
The Westec cop said, "They live right over here in Benedict Canyon. It's on my route." Benedict Canyon was less than one mile from the overlook.
Gibbs said, "I'll be damned."
The four detectives later testified that they thought pretty much the same thing at the same time. Teddy Martin meant money and, more important than money, political power, and that meant the case would require special handling. Dan Tomsic remembers thinking that he wished he had called in sick that day so some other asshole would've answered the call. Special cases always meant special trouble, and investigating officers almost always caught the short end of the deal. Teddy Martin was a rich boy who'd made himself even richer; a successful restaurateur and businessman who used his wealth to cultivate friends and social position and notoriety. He was always having dinner with city councilmen and movie stars, and he was always in the newspaper for giving millions of dollars to all the right causes. Tomsic knew the name because Teddy Martin had opened a new theme restaurant with a couple of movie star partners that his wife had been nagging him to take her to. He'd been foot-dragging because he knew it'd cost sixty bucks for a couple of pieces of fish just so the wife could eyeball some second-rate movie props and maybe some closet-fag actor. Tomsic hated guys like Teddy Martin, but he kept it to himself. Guys like Teddy Martin were headline grabbers and almost always phonies, but a phony with the right connections could end your career.
Pete Bishop said, "It's gonna be a headliner. We'd better call the boss."
Gibbs said, "Use your cell phone. You put it on the radio, we'll have media all over us. Tommy, see if there's anything on the wire."
Angela Rossi walked with Tomsic and Bishop back to their units. Fine soil and foxtails had worked down into her running shoes and between her toes, so she sat in the backseat of her radio car and cleaned her feet with a Handiwipe before changing back into her Max Avantes. While she sat in the car, Tomsic and Bishop stood apart from each other in the overlook's parking lot, each talking into their respective cell phones.
By the time Rossi finished cleaning her feet and had rejoined Gibbs at the top of the slope, both Tomsic and Bishop were off their phones. Tomsic said, "Nothing on the board about a Susan Martin."
Bishop said, "I called the boss and notified the coroner. Criminalists are on the way, and the boss is coming out." The boss was the detective captain who oversaw the Westside detectives. When he reached the scene, everyone knew he'd decide whether Gibbs would keep the case or it would be reassigned to someone else. Gibbs knew that because of Mr. Martin's stature, the case would almost certainly be assigned to one of the elite robbery-homicide units downtown. He had no problem with that.
Gibbs said, "Okay, we'd better notify Mr. Martin and see what he says." He looked at the Westec guy. "You know where they live?"
"Sure. I'll take you over, you want."
Gibbs started for his car. "Okay. Let's go."
Bishop was shaking his head. "We'd better stick around for the boss, Linc."
Tomsic said, "Angie and I'll go."
Angela Rossi later said that if she'd known where it was going to lead, she would have shot Tomsic right there.
Dan Tomsic and Angela Rossi followed the Westec guy east along Mulholland to Benedict, then south down through the canyon into a lush winding world of million-dollar homes and Mercedes convertibles. Most of the homes were new and modern, but the Westec guy pulled off the road in front of a Mediterranean mansion that could have been a hundred years old. A big mortar wall with an ornate iron gate protected the mansion from the street, the wall laced by delicate ivy with tiny, blood-red leaves. The wall was cracked and crumbling beneath the ivy, but you could see the cracks only if you took your time and looked between the vines. A gate phone stood to the left of the drive so you could identify yourself before being buzzed in. Tomsic figured the grounds for four or five acres, and the house beyond for maybe twenty thousand square feet. Tomsic and his wife and four children were squeezed into a twenty-two-hundred-square-foot cracker box in Simi Valley, but those were the breaks. Anyone could be a cop, but it took real talent to serve bad food in an overpriced restaurant.
They were getting out of the car when Angie said, "The gate's open."
The big wrought iron gate was open maybe nine or ten inches. You didn't live behind walls and gates and security cops, then leave the front gate open so that any stray goofball or passing psycho could come inside and make himself at home. Tomsic remembers that his first thought on seeing the open gate was that they would find a body inside.
They went to the gate and pressed the button on the call box twice, but they got no answers. Angie said, "We don't need to wait for a warrant, do we?"
Tomsic said, "Shit." He pushed at the gate and went through.
The Westec guy said, "We can't just walk in, can we?" He looked nervous. "I'll call the office and they can ring the house."
Tomsic ignored him, and Rossi followed Tomsic toward the house.
The drive was hand-laid Mexican pavers and had probably cost more than Tomsic's house, his two cars, and the quarter interest he owned in a Big Bear Lake cabin combined. The mansion itself was built of mortar and rough-hewn wooden beams and was finished with an ancient Spanish tile roof. A healthy growth of ivy covered the ground along the east side of the drive, nestling up to a couple of monstrous podocarpus trees before continuing around a four-car garage. Each car had its own door, and the whole effect was more that of a stable than a garage. A large fountain sat just off the front entry, trickling water.
Tomsic thought that it looked like the kind of house that Errol Flynn might've owned. His wife would love the place, but Tomsic knew that most of the old stars, just like most of the new stars, were perverts and scumbags, and if you knew the things that went on in places like this you wouldn't be so thrilled with being here. Normal people didn't go into the movie business. Movie people were shitbirds with serious emotional problems who kept their secret lives hidden. Just like most lawyers and all politicians. Tomsic completely believed this, probably because everything he'd seen in almost thirty years on the job confirmed it. Of course, Tomsic had never in his thirty years shared what he knew with his wife because he didn't want to rain on her parade. It was easier to let her think he was a grump.
Nothing seemed amiss. No bodies were floating in the fountain and no cars were parked crazily on the front lawn. The massive front door was closed and appeared undamaged. A large ornate knocker hung in the center of the door, but there was also a bell. Tomsic pressed the button, then used the knocker. Loud. The Westec guy came running up behind them. "Hey, take it easy. You're gonna break it." His face was white.
Angie said, "Stay back, okay? We don't know what we have here."
Tomsic glanced at Angie and shook his head. Fuckin' Westec geek, worried about losin' the account. Angie rolled her eyes.
Tomsic slammed at the door two more times without getting an answer and was starting back to the car when the door opened and Theodore "Teddy" Martin blinked out. Martin was a medium-sized man, a little shorter than average, with pale, delicate skin. He was unshaven and drawn with hollow, red-rimmed eyes. Tomsic says that he would've bet that the guy had spent most of the night blasted on coke or crystal meth. "Mr. Martin?"
Martin nodded, his head snapping up and down. He was wearing baggy gray sweatpants and no shirt. His torso was soft and undeveloped and covered with a thick growth of fine hair. He squinted against the bright morning sun. "Yeah, sure. What do you want?"
Both Tomsic and Angela Rossi later testified that Tomsic badged him and identified himself as a detective with the Los Angeles Police Department. Angela Rossi noted that Teddy Martin never looked at the badge. He kept his eyes on Tomsic and blinked harder as if something were in his eyes. Angela Rossi thought at the time that he might have allergies. Tomsic said, "Mr. Martin, does a woman named Susan Martin live here with you?"
When Tomsic asked the question, Angela Rossi says that Teddy Martin took a single sharp breath and said, "Oh, my Christ, they killed her, didn't they?"
People say the damnedest things.
Tomsic took Rossi aside, gave her his cell phone, and told her to call Gibbs and tell him to get over here. Rossi walked out to the drive and made the call. When she returned to the house, Tomsic and Teddy Martin and the Westec geek were inside, Tomsic and Martin sitting on an antique bench in the entry. Teddy Martin was blubbering like a baby. "I did everything they said. I did everything, and they said they'd let her go. Jesus Christ. Oh, Jesus, tell me this isn't happening."
Tomsic was sitting very close to Martin and his voice was soft. He could make it soft whenever he wanted to calm people. "You're saying she was kidnapped?"
Martin sucked great gulps of air as if he couldn't breathe. "Christ, yes, of course she was kidnapped." He put his face in his hands and wailed. "I did everything they said. I gave them every nickel. They said they'd let her go."
Angela Rossi said, "You gave someone money?"
Martin waved his hands, like a jumble of words were floating around him and he had to grab hold of the ones he wanted to use. "Half a million dollars. Just like they said. I did everything exactly the way they said. They promised they'd let her go. They _promised_."
Tomsic gently took Teddy's wrists and pushed his hands down. He said, "Tell me what happened, Mr. Martin. You want to tell me what happened? Can you do that?"
Martin seemed to regain control of himself and rubbed at his eyes. He said, "I came home Thursday night and she was gone. Then this guy calls and says he's got Susan and he puts her on. I think it was around eight o'clock."
Rossi distinctly remembers asking, "You spoke with her?"
"She was crying. She said she couldn't see anything and then the guy came back and he told me that if I didn't give them the five hundred thousand they'd kill her. I could hear her screaming. I could hear her crying."
Tomsic said, "Did you recognize this man's voice?"
"No. No, I asked him who he was and he said I should call him James X."
Tomsic glanced at Rossi and raised his eyebrows. "James X?"
"He said they were watching the house. He said they would know if I called the police and they would kill her. Oh, Jesus, I was so scared." Teddy Martin stood, taking deep breaths and rubbing his stomach as if it hurt. "He said I should get the money and he would call tomorrow and tell me what to do with it."
Angie said, "Tomorrow was yesterday?"
Martin nodded. "That's right. Friday. I got the money just like he said. All in hundreds. He wanted hundreds. Then I came back here and waited for his call."
Tomsic said, "You just walked into the bank and got five hundred thousand dollars?"
Teddy Martin snapped him an angry look. "Of course not. My business manager arranged it. He cashed bonds. Something like that. He wanted to know why I wanted the money and I told him not to ask."
Rossi saw Tomsic frown. Tomsic prompted Martin to continue. "Okay. So you got the money, then came back here to wait."
Martin nodded again. "I guess it was around four, something like that, when he called. He told me to put the money in a garbage bag and bring it to a parking lot just off Mulholland at the four-o-five. They have a little lot there for people who carpool. He told me that there was a dumpster, and I should put the money into the dumpster, then go home. He said they would give me exactly twelve minutes to get there, and if I was late they'd know I was working with the police and they'd kill Susan. They said I should just drop the money and leave, and that after I was gone they'd pick up the money and count it and if everything was okay they'd let Susan go. They said it probably wouldn't be until nine or ten with the counting." He sat again and started rocking. "I did everything just like they said and I've been waiting all night. I never heard from them again. I never heard from Susan. When you rang the bell I thought you were her." Teddy Martin put his face in his hands and sobbed. "I made it in the twelve minutes. I swear to God I made it. I was driving like a maniac."
Tomsic told Angie to take the cell phone again, call Gibbs, and this time tell him to have someone check the dumpster. She left, and Tomsic stayed with Martin and the Westec guy. Rossi was gone for only four or five minutes, but when she returned she looked burned around the edges. He said, "You get Gibbs?"
She didn't answer the question. Instead, she said, "Dan, may I see you, please?"
Tomsic followed her outside to the ivy alongside the expensive Mexican drive. She took out her pen, pushed aside some leaves, and exposed a ball-peen hammer clotted with blond hair and bits of pink matter. Tomsic said, "I'll be damned."
Rossi said, "I was just looking around when I saw it. The handle was sticking up out of the ivy."
Tomsic stared at the hammer for several seconds, noticing that a single black ant was crawling in the pink matter. Tomsic made the same whistling sound that he'd made at the Stone Canyon overlook when he'd seen the body. Angela Rossi then said, "He killed her, didn't he, Dan?"
Lincoln Gibbs and Pete Bishop turned into the drive as she said it. Dan Tomsic, who had a million years on the job and whose opinion as a professional cynic almost everyone valued, glanced at the mansion and said, "The sonofabitch killed her, all right, but now we have to convict him."
"Hey, we've _got_ this guy, Dan! He's _ours_!"
Dan Tomsic stared at her with the disdain he reserved for shitbirds, defense attorneys, and card-carrying members of the ACLU. He said, "It's easier to cut off your own goddamned leg than convict a rich man in this state, detective. Haven't you been around long enough to know that?"
It was the last thing that Dan Tomsic said to her that day.
Susan Martin's murder made the evening news, as did the events that followed.
I was able, months later, to piece together the events of that Saturday morning from police reports, participant interviews, court testimony, and newspaper articles, but I couldn't tell you what I was doing when I heard, or where I was or who I was with. It didn't seem important.
I did not think, nor did I have reason to believe, that Susan Martin's murder and everything that grew from it would have such a profound and permanent impact upon my life.
#
Jonathan Green came to my office on a hazy June morning with an entourage of three attorneys, a videographer, and an intense young woman lugging eight hundred pounds of sound recording equipment. The videographer shoved past the attorneys and swung his camera around my office, saying, "This is just what we need, Jonathan! It's real, it's colorful, it's _L.A_.!" He aimed his camera at me past the Mickey Mouse phone and began taping. "Pretend I'm not here."
I frowned at him, and he waved toward the lawyers. "Don't look at me. At _them_. Look at _them_."
I looked at _them_. "What is this?" I was expecting Green and an attorney named Elliot Truly, but not the others. Truly had arranged the meeting.
A man in his mid-forties wearing an immaculately tailored blue Armani suit said, "Mr. Cole? I'm Elliot Truly. This is Jonathan Green. Thanks for seeing us."
I shook hands with Truly first, then Green. Green looked exactly the way he had the two times I'd seen him on _60 Minutes,_ once when he defended an abortion rights activist accused of murder in Texas and once when he defended a wealthy textile manufacturer accused of murder in Iowa. The Texas case was popular and the Iowa case wasn't, but both were victories for the defense.
The videographer scrambled backward across the office to fit us into his frame, the woman with the sound gear hustling to stay behind the camera as they captured the moment of our first meeting. Armstrong steps onto the moon; the Arabs and the Israelis sign a peace accord; Jonathan Green meets the private detective. The woman with the sound equipment bumped into my desk and the videographer slammed against the file cabinet. The little figures of Jiminy Cricket on the cabinet fell over and the framed photo of Lucy Chenier tottered. I frowned at him again. "Be careful."
The videographer waved some more. "Don't look at me! _Not at me!_ You'll ruin the shot!"
I said, "If you break anything, I'll ruin more than the shot."
Green seemed embarrassed. "This is tiresome, Elliot. We have business here, and I'm afraid we're making a bad impression on Mr. Cole."
Truly touched my arm, trying to mitigate the bad impression. "They're from _Inside News_. They're doing a six-part documentary on Jonathan's involvement in the case."
The woman with the sound equipment nodded. "The inner workings of the Big Green Defense Machine."
I said, "Big Green Defense Machine?"
The videographer stopped taping and looked me up and down as if he found me lacking but wasn't quite sure how. Then it hit him. "Don't you have a gun?" He glanced around the office as if there might be one hanging on a wall hook.
"A gun?"
He looked at Truly. "He should be wearing a gun. One of those things under the arm." He was a small man with furry arms.
Truly frowned. "A shoulder holster?"
The woman nodded. "A hat would be nice. Hats are romantic."
I said, "Truly."
Jonathan Green's face clouded. "I apologize, Mr. Cole. They've been with us for the past week and it's becoming offensive. If it bothers you, I'll ask them to leave."
The videographer grew frantic. "Hey, forget the gun. I was just trying to make it a little more entertaining, that's all." He crouched beside the watercooler and lifted his camera. "You won't even know we're here. I promise."
Truly pursed his lips at me. My call.
I made a little shrug. "The people who come to me usually don't want a record of what we discuss."
Jonathan Green chuckled. "It may come to that, but let's hope not." He went to the French doors that open onto the little balcony, then looked at the picture of Lucy Chenier. "Very pretty. Your wife?"
"A friend."
He nodded, approving. When he nodded, the two lesser attorneys nodded, too. No one had bothered to introduce them, but they didn't seem to mind.
Jonathan Green sat in one of the leather director's chairs across from my desk and the two lesser attorneys went to the couch. Truly stayed on his feet. The videographer noticed the Pinocchio clock on the wall, then hustled around to the opposite side of my desk so that he could get both me and the clock in the frame. The Pinocchio clock has eyes that move side to side as it tocks. Photogenic. Like Green.
Jonathan Green had a firm handshake, clear eyes, and a jawline not dissimilar to Dudley Do-Right's. He was in his early sixties, with graying hair, a beach-club tan, and a voice that was rich and comforting. A minister's voice. He wasn't a handsome man, but there was a sincerity in his eyes that put you at ease. Jonathan Green was reputed to be one of the top five criminal defense attorneys in America, with a success rate in high-profile criminal defense cases of one hundred percent. Like Elliot Truly, Jonathan Green was wearing an impeccably tailored blue Armani suit. So were the lesser attorneys. Maybe they got a bulk discount. I was wearing impeccably tailored black Gap jeans, a linen aloha shirt, and white Reebok sneakers. Green said, "Did Elliot explain why we wanted to see you?"
"You represent Theodore Martin. You need investigators to help in the defense effort." Theodore "Teddy" Martin had been arrested for Susan Martin's murder and was awaiting trial. He had gone through two prior defense attorneys, hadn't been happy with them, and had recently hired Jonathan Green. All the hirings and firings had been covered big time by the local media.
Green nodded. "That's right. Mr. Cole, I've spoken at length with Teddy and I believe that he's innocent. I want your help in proving it."
I smiled. _"Moi?"_
The videographer edged in closer. I raised a finger at him. Unh-unh-unnh. He edged back.
Truly said, "We've talked to people, Mr. Cole. You've an outstanding reputation for diligence, and your integrity is above reproach."
"How about that." I glanced at the camera and wiggled my eyebrows. The videographer frowned and lowered the lens.
Jonathan Green leaned toward me, all business. "What do you know about the case?"
"I know what everybody knows. I watch the news." You couldn't read the _Times_ or watch local television without knowing the business about James X and the five hundred thousand dollars and the dumpster. I'd heard Theodore Martin's sound-bite version of it ten thousand times, but I'd also heard the DA's sound-bite version, too, that Teddy and Susan weren't getting along, that Susan had secretly consulted a divorce attorney and told a friend that she was planning a divorce, and that Teddy had offed her to keep her from walking away with half of his estimated one-hundred-twenty-million-dollar fortune. I said, "From what I hear, the police have a pretty good case."
"They believe they have, yes. But I don't think all the facts are in." Green smiled and laced his fingers across a knee. It was a warm smile, tired and knowing. "Did you know that Teddy and Susan loved to cook?"
I shook my head. That one had slipped right by me.
"Teddy arrived home early that night, and they had no engagements, so the two of them decided to cook something elaborate and fun. They spent the next couple of hours making a pepper-roasted pork tenderloin with wild cherry sauce. Teddy makes the sauce with fresh cherries, only they didn't have any, so he ran out to get some."
Truly took a step toward me and ticked points off his fingers. "We have the receipt and the cashier whom Teddy paid. That's where he was when Susan was kidnapped."
Green spread his hands. "And then there's the question of the money. What happened to the money?"
Truly ticked more fingers. "We have the bank transactions and the business manager. The manager says that Teddy was visibly shaken when he came for the money that Friday morning. He says Teddy was white as a sheet and his hands were shaking."
Green nodded. "Yet the cashier remembers that Teddy was relaxed and happy a dozen hours earlier." Green stood and went back to the balcony. The videographer followed him. At the French doors he turned back to me and spread his hands again. I wondered if he thought he was in court. "And then we have the murder weapon and the crime scene evidence."
Truly ticked more fingers. He had used up one hand and was starting on the next. "There were fingerprints on the hammer, but none of them match Teddy. There were also fingerprints on the garbage bags that Susan was in, but those don't match Teddy, either."
I said, "You think he's innocent because of that?"
Green came back to the director's chair, but this time he didn't sit. He stood behind it, resting his hands on the wooden posts that hold the back. "Mr. Cole, I don't win the number of cases that I do because I'm good. I turn down ten cases every day, cases that would bill millions of dollars, because I will not represent people I believe to be guilty."
The videographer went down to the floor for a low-angle shot, the woman with the sound equipment with him, and I heard him mumble, "Oh, man, this is great."
Green said, "I don't represent drug dealers or child molesters. I only take cases that I believe in, so that every time I walk into court I have the moral high ground."
I leaned back and put my foot on the edge of my desk. "And you believe that Teddy is innocent."
"Yes. Yes, I do." He came around to the front of the chair and tapped his chest. "In here I know he's innocent."
The videographer muttered, "This is fabulous," and scrambled around to keep Jonathan Green in the shot.
Green sat and leaned toward me, elbows on knees. "I don't yet know all the facts. I need people like you to help me with that. But I do know that we've received several calls that are disturbing."
Elliot Truly said, "Have you heard of our tip line?"
"I've seen the ads." Green's office was running television, radio, and print ads offering a reward of one hundred thousand dollars for anything leading to the capture, arrest, and conviction of James X. There was a number you could call.
Green said, "We've received over twenty-six hundred calls and there are more every day. We try to weed out the cranks as quickly as possible, but the workload is enormous."
I cleared my throat and tried to look professional. "Okay. You need help running these things down."
Green raised his eyebrows. "Yes, but there's more to it than that. Several of the callers have indicated that one of the arresting officers has a history of fabricating cases."
I stared at him. The videographer scrambled back across the office, again running into the cabinet, but this time I did not look. "Which officer?"
Truly said, "The detective who claims to have found the hammer. Angela Rossi."
I looked at Truly. "Claims?"
Jonathan Green, Elliot Truly, and the camera stared at me. No one spoke.
I looked back at Green. "Do you believe that Angela Rossi planted evidence against Teddy Martin?"
Green shifted in the chair and the camera swung back toward him. He looked uncomfortable, as if the subject bothered him. "I don't want to say that, not yet, but I believe that the possibility exists. She was the first to go down to Susan's body, and she went alone."
Truly said, "She had the opportunity to recover the murder weapon and secrete it on her person."
"A full-size ball-peen hammer."
Truly smiled. "Where there's a will."
I shook my head. "Why would she take the chance?"
Green said, "Elliot."
Truly leaned toward me, serious. "Rossi was on a fast track up the promotion ladder until she blew a homicide investigation two years ago. She failed to Mirandize a suspect who subsequently confessed, and the suspect walked. She might feel she needs a headline case to resurrect her career, and if she tampered with evidence to make this case, it may not be the first time she's done so." Truly made a little hand move at one of the lesser attorneys, and the lesser attorney slipped a manila envelope from his Gucci case and brought it to me. Truly said, "Rossi arrested a man named LeCedrick Earle five years ago for possessing counterfeit money and attempting to bribe an officer. He's currently serving a six-year sentence at Terminal Island." Terminal Island is the federal facility down in San Pedro. "Earle phoned six days ago and told us that Rossi planted the money." He gestured at the envelope. "He's been saying that he was set up since day one, and sent us a copy of his case file and the various letters of complaint to prove it."
I opened the envelope and fingered through the arrest reports, legal correspondence, and letters of complaint. Terminal Island return address, all right. I said, "All perps claim they're innocent and every cop I know has had charges brought against him. It goes with the job."
Green nodded, reasonably. "Of course, but Mr. Earle's claim seems to have a bit more merit than the others."
Truly said, "A former LAPD officer named Raymond Haig told us about the Earle case, also. Haig was Rossi's partner."
I said, "Haig was her partner at that time?"
"Yes."
"And he said that she planted the goods?"
Truly smiled again. "He wouldn't say that, but he says that he knew her and that she would do anything to further her career. He suggested that we look into it."
I said, "If Earle made the allegation, there would've been an internal police investigation."
The smaller lesser attorney said, "There was, but no charges were filed."
Green said, "Mr. Haig indicated that Detective Rossi has a history of excessive behavior."
I put the envelope down and tapped at the edge of my desk. The videographer crept back to the watercooler and focused on me. I said, "Mr. Green, you should know that my partner, Joe Pike, is a former LAPD officer."
"We're familiar with Mr. Pike."
"I work with LAPD often, and I have many friends there, and in the district attorney's office."
He leaned toward me again, very serious now, sincere. "I'm not looking for a stooge. I have plenty of those, believe me." He tried not to glance at the lesser attorneys but couldn't help himself. "I'm looking for an honest detective who won't just tell me what I want to hear. I want the truth. Without the truth, I have nothing. Do you see?"
I nodded. Maybe I could see why he was one of the world's greatest defense attorneys after all.
Truly said, "What we're discussing with you is only a small part of the larger picture. We have sixteen investigators working with us now, and we'll probably have as many as thirty, but you'll be the only investigator working on this aspect of the case."
The larger lesser attorney said, "We have fourteen attorneys on board, in addition to the investigators."
The smaller lesser attorney's head bobbed. "Not to mention eight forensic specialists and three criminalists." He seemed proud when he said it. Peace through superior firepower.
I made a whistling sound. "The best defense money can buy."
Jonathan Green stayed serious. "As I said, there's plenty of work to go around, and more work every day. Will you help us, Mr. Cole?"
I leaned back, thinking about it, and then I held up the envelope. "And what if I find out that Rossi's okay?"
"Then that's what you find. I owe it to myself and my client to exhaust every possibility. Do you see?"
I said, "Wherever it leads."
"That's exactly right."
"The moral high ground."
"My reputation rests on it."
I watched the Pinocchio clock. I looked at the picture of Lucy Chenier. I nodded. "If Rossi's clean, that's what I'll report."
"I wouldn't have it any other way."
Jonathan Green put out his hand and we shook.
#
We worked out my fee, Elliot Truly cut me a check, and the Big Green Defense Machine left me to get on with it. I stood in the door as they walked to the elevator, watching the videographer record every moment of the departure. Cindy, the woman who runs the beauty supply distribution office next door, came out of the elevator as they were getting on and saw Jonathan. She stared at him until the doors closed, and then she smiled at me. Incredulous. "Isn't he that guy? The lawyer?"
"Jonathan Green."
"I saw him on _Geraldo_. He's famous."
I held out crossed fingers. "We're like this."
Cindy opened her door, then cocked an eyebrow at me. "I always did think you were cute."
"Big time. I am nothing if not big time."
She laughed and disappeared into her office. That's Cindy.
I went back into my office, closed the door, and looked at the picture of Lucy Chenier. She was sitting in her backyard wearing shorts and hiking boots and an LSU T-shirt. I had had the picture in my office since Lucy sent it to me a little over three months ago, and I looked at it a lot. Lucy was a lawyer, too, but she hadn't been on _Geraldo_. His loss. I stared at the picture. Something about it wasn't right and, being an astute detective, I deduced that this was because the videographer had bumped the cabinet. It was not too late to rush down the stairs and shoot him, but that would probably be overreacting. Besides, he was part of the Big Green Defense Machine, and teammates shouldn't shoot each other. Jonathan Green might think me small.
I adjusted the picture, then went back to my desk and dialed Lucy's office in Baton Rouge, Louisiana. If Cindy was impressed with Jonathan Green, so might be Lucy Chenier. I am also nothing if not a show-off.
A warm southern voice said, "Ms. Chenier's office." Lucy's assistant, Mrs. Darlene Thomas.
"It's me." I'd phoned quite often in the three months since I'd been in Louisiana, and the calls were becoming more frequent.
"Hello, Mr. Cole. How are we today?"
"We're fine, Darlene. And yourself?" Small talk.
"Very well, thank you. I'm sorry, but she's in court today."
"Oh." Dejected.
Darlene said, "She'll call for her messages, though. I'll tell her that you phoned."
"Tell her that I'm lonely, Darlene."
Darlene laughed. "I'll tell her that Mr. Cole says he's lonely."
"Tell her that I miss her, Darlene. That the longing grows with every passing moment and has become a weight impossible for me to bear."
Darlene gasped. "Oh _my,_ but you do go on!"
I was grinning. Darlene did that to me. "Darlene, have I ever said that you've got a very sexy voice?"
"Get on with you, now! You stop this nonsense before I tell Ms. Chenier!"
We said our good-byes and I called Joe Pike to tell him that we were once more employed. His answering machine picked up on the first ring and beeped. He used to have a one-word message that just said, "Speak," but I guess he felt it was long-winded. Now, there was just the beep. When I asked him how people were supposed to know who they had gotten or what to do, he'd said, "Intelligence test." That Pike is something, isn't he?
I said, "This is the Lone Ranger, calling to inform you that someone has once again been foolish enough to give us money. We're working for Jonathan Green." I hung up. It might be days before I heard from him.
The envelope that Truly left contained a copy of LeCedrick Earle's arrest report as well as a formal letter of complaint written by a public defender on Earle's behalf. The arrest report was written by Officer Angela Rossi and stated that Rossi had arrested Mr. Earle at his home after Mr. Earle attempted to bribe his way out of a traffic code violation with eight hundred dollars in counterfeit one-hundred-dollar bills. The letter of complaint alleged that Rossi had planted the counterfeit money on Mr. Earle and that Mr. Earle was innocent of all wrongdoing. The arrest report said little, and the letter of complaint said even less. She said, he said. A single sheet bearing both Angela Rossi's home address and Raymond Haig's business address and phone number was the last entry in the file. A newspaper photograph of Rossi was clipped to the sheet. It was an old photo that showed an attractive woman with a lean, rectangular face and intelligent eyes. She looked determined.
I put everything back into the envelope, then called my friend Eddie Ditko at the _Examiner_. Eddie has been a reporter for about ten million years, and he answered with a voice that was maybe three weeks away from throat cancer. "Ditko."
"Is this Eddie Ditko, the world's finest reporter?"
He made a hacking sound like a cat gakking up a hairball. "Yeah, sure, it says that right here on my Pulitzer. Hold on a minute while I wipe my ass with it." That Eddie. Always with just the right thing to say.
"A guy named LeCedrick Earle was busted on a funny money beef five years ago. He claimed it was a setup by the arresting officer."
"They all claim that. It's a natural law." You see?
"The arresting officer was Angela Rossi."
"I'm hearing Notre Dame." Bells.
"Rossi put the cuffs on Teddy Martin. She found the hammer."
Eddie made the gakking sound again. "You're shitting me."
"Nope."
He wasn't saying anything. Thinking. Sniffing the words and smelling a story. "What's this to you?"
I didn't say anything.
He gave the big sigh, like I was asking for an organ donation. "What do you want?"
"Whatever you've got on the Earle arrest, and anything in your files about Rossi." Ever since the Christopher Commission the _Examiner_ kept a database on LAPD officers. The Fourth Estate's version of Big Brother.
"What's this have to do with Teddy Martin?"
I didn't say anything some more.
"Yeah, right. I'll get back to you." Then he said, "You really give me ass cramps."
He hung up without another word. Always the pleasant conversationalist.
I put everything back in the envelope, then locked the office and drove up through Hollywood and the Cahuenga Pass and into the San Fernando Valley. I left the Hollywood Freeway at Barham and drove east along the foot of the Verdugo Hills through Burbank into Glendale. Raymond Haig owned a Mr. Rubber Discount Tire franchise in an area of gas stations and falafel stands and flat single-story buildings with shops that sold secondhand clothes and wholesale electronics. A weathered Hispanic guy in a broken straw hat had set up a little churro cart outside the tire store, the churros hanging in ropes inside the glass cart. The Hispanic guy was decked out in cowboy boots and jeans and a wide leather belt with a gleaming silver buckle inlaid with the image of a Brahma bull. A vaquero. A couple of kids with skateboards were holding fistfuls of wax paper and long brown churros, and a black dog with a bandana around its neck was sitting between them, looking first at one, then the other. Hopeful.
I parked on the street in front of the churro cart, then went into the store. A young Hispanic woman with tired eyes and too much makeup was sitting behind the counter, staring at a little television. I handed her a card. "I need to see Mr. Haig. If you tell him that Elliot Truly sent me, he'll know what it's about."
She took the card and disappeared through a door leading to the service bay, and a couple of minutes later she came back with a tall guy in his late forties. Haig. He was wearing a plaid shirt and a maroon knit tie, and he had a pencil caddy in his shirt pocket. The caddy's plastic flap said _Beamis Shocks_. He came over. "You Cole?"
"That's right. Elliot Truly said that someone from his office spoke to you, and that you'd be willing to answer a few questions about Angela Rossi."
His face split with a sleek smile and he put out his hand. "You bet. Let's go in back and I'll tell you everything you need to know about that rotten bitch Rossi." Nothing like an unbiased opinion.
He led me to a small office cluttered with parts catalogs and product manuals and posters of bikinied young women posing on lug wrench displays. Enlightened. A couple of padded chairs sat opposite his desk for customers, and a Mr. Coffee with a tower of Styrofoam cups sat on a table next to the glass door. "You want a little coffee?"
"No, thanks."
Haig poured a cup for himself and brought it to his desk. There was a picture of a younger Haig in an LAPD uniform on the desk.
I said, "How long were you on the job?"
"Fifteen bullshit years." Unbiased, all right. "Best move I ever made was getting out and going into business for myself. Yes, sir." He settled in behind the desk, then picked up an unlit cigar and popped it into the side of his mouth. I took out a little pad and a Uniball pen to take notes. He said, "Rossi's the reason I left the goddamned force."
"How so?"
"I didn't want to ride with a woman."
I smiled at him. "You left because you didn't want to ride with a woman."
He pulled the cigar from his mouth and made a move with it. "Hey, you get these women in a car, they're either scared shitless and not worth a damn when things get hairy, or they're out of their minds aggressive and you never know what they're gonna do."
"And Rossi was aggressive?"
"Christ, yes. Always tryin' to be more man than a man." He had some of the coffee, then sucked at the cigar again.
I said, "You were partners when she made the LeCedrick Earle arrest?"
"Yep. That's the bust got her into plainclothes. She got a big promotion off that bust." He leaned back, and I noticed that small brown flecks of matter were scattered over the catalogs and desk and floor. I squinted at them and wondered what they were.
I said, "LeCedrick Earle claims that she planted the money, and Truly says that you agree." I felt something gritty on the arms of the chair and looked. More flecks. Sort of like brown dandruff.
Haig chewed at the cigar, then took it out and examined it. The end was soggy and frayed, and while he looked he absently spit little pieces of tobacco off his tongue. I saw a piece land on an air filter catalog. I saw another piece land on the framed photo of young LAPD Haig. Haig didn't seem to notice, or didn't care. I lifted my elbows from the chair and brushed at my arms. Yuck. Haig shook his head. "Nope. I didn't say that. I said that I wouldn't put it past the bitch."
"But you don't know?"
He shrugged and spit more tobacco. "If you read the arrest report you know I wasn't listed as an arresting officer. Rossi went back later without me. That way only one name gets credit for the collar. You see how she was?"
"She cut you out."
Another shrug. "Just her way. When it came to wearin' a uniform she was just passin' through and she made no secret of it. All she used to talk about was gettin' ahead, gettin' that gold shield. She told me she'd do anything to get that gold shield, and that's what I told Truly. I had to listen to that every goddamned day like a goddamned matrah."
"Mantra."
"Whatever."
The Hispanic woman rapped at the glass then stepped into Haig's office. She was holding a clipboard. "Warren wan's you to sign these estimates."
Haig grinned and made a little c'mere gesture. "Lemme see what you've got."
She kept her eyes down when she crossed to him, probably because Haig was making a big deal out of looking at her. A gold wedding band and a large, ornate engagement ring were on her left hand, the stone square and flat and enormous, and probably zircon. The polished gold of the rings looked warm against her brown skin. She said, "Warren says a truck is here with the new tires. He says he needs you to come see." Warren was probably Haig's assistant.
"Yeah. I'll be out in a minute."
Haig took the clipboard and flipped through a couple of pages without really looking at them. He used one hand to flip the pages and the other to feel her right hip. He scratched his name and handed back the board, still with the big grin. " _Gracias,_ babe. Lookin' good."
"Warren says he needs you about the new tires." Like Warren had been making a thing and she didn't want to mention it, but felt obligated.
Haig's grin turned brittle. "Tell Warren to hold his water. I'll come when I come. _Comprende?_ " He patted her hip again, letting his hand linger.
She took the board and walked out, Haig watching her go. He spit more tobacco, and I thought that if any of the flecks landed on me I might shoot him. Haig glanced at his watch and frowned. Warren.
I said, "Okay, Rossi was ambitious. But did she ever do anything illegal to your knowledge?"
"Not to my knowledge."
"Ever rig an arrest?"
Haig shook his head.
"Plant evidence?"
"Not with me around." Offended.
"You told Truly that you thought Rossi was capable of falsifying evidence. You said that your statement was based upon your experience as her partner. Do you really know anything, Haig, or are you just blowing smoke?"
Haig frowned. "Look, Rossi used to skirt the line all the time. She'd do anything to make a case, go through a window, pop a trunk, jump a fence. I used to say, hey, you ever heard of the search and seizure laws? You ever heard of a warrant?"
"And what would she do when you said that?"
"Look at me like I'm an asshole." He chewed at the cigar some more, then suddenly seemed to realize what he was doing and dropped it into the trash. "Christ, she made me crazy in the car, always running plates, always looking for the collar."
"Sounds like good police work."
"Try livin' with it every day." He glanced at his watch again. "I gotta get going."
"One more thing. You weren't with her when she made the Miranda violation."
"Nah. That was later. I was already off the job and she was a detective-one. Rossi the hot shot, bustin' balls like always."
"Then how do you know about it?"
"I saw her after. Bobby Driskoll's retirement up at the Revolver and Athletic Club." The Revolver and Athletic Club is the Police Academy's bar. "She was goin' on about it, sayin' how rotten it was, sayin' that she was going to do whatever it took to get her career back on track."
"Were there other people around?"
"Hell, yes. Rossi never made a secret about her ambition. 'They can't keep me down.' That's the way she talked. 'All it takes is one big bust and I'm on top again.' Like that."
"But you have no personal knowledge of her having done anything illegal?"
Haig frowned at me. "Any bitch that in-your-face is up to something."
I closed the pad and put it away. Jonathan Green probably wasn't going to like what I had to say about Haig. "Tell me something, Haig. Are you an asshole by choice?"
Haig gave me the hard cop eyes, and then the slick grin came back and he stood. "Yeah, I guess it sounds that way, but there's more to it than her attitude. You see where she lives?"
I didn't know what he meant. "No."
"Go see where she lives."
We walked out to the little showroom together. A guy who was probably Warren was standing with a black guy in a Goodyear shirt, and together they were reading what was probably a delivery manifest. They looked up when we came out and Warren said, "We got those tires."
Haig ignored him. He slipped behind the counter and I went to the door, and neither of us said anything to the other.
The Hispanic woman was behind the counter. Haig moved against her and mumbled something that the rest of us couldn't hear. She didn't look at him, and she didn't respond. She stared at the TV, as if by staring hard enough it wouldn't be happening.
I went out into the sun, thinking that maybe I should have shot him anyway.
#
The two kids with their skateboards were gone, but the dog was still sitting by the churro cart, watching the vaquero. The vaquero was still waving his churro at the passing cars and looking sad. All the way up from Zacatecas to stand on a corner and sell something that no one except a couple of kids and a dog wanted. A man who had worked with the Brahmas, no less.
I climbed into my car and opened Truly's envelope and looked at Angela Rossi's address, wondering what Haig had meant about seeing where she lived. 724 Clarion Way. I looked up Clarion Way in the Thomas Brothers Guide, found it in Marina del Rey, and thought, "Well, hell."
The Marina wraps around the ocean on a stretch of sand just south of Santa Monica. It's home to sitcom writers and music producers and people who own Carpeteria franchises, maybe, but not cops. The cheapest house in the Marina maybe goes for six hundred thousand, and even the smallest apartments would set you back fifteen hundred a month before utilities. Condos had to start at three hundred grand. Raymond Haig was probably just a raging sexist who had been shown up on the job and was working out on the person who had shown him up, but how did that explain a cop living in the Marina? Of course, there were probably ten million explanations for how Rossi might live there, but I probably wouldn't ferret them out sitting in front of a tire store in Glendale.
The churro salesman caught me staring at him and gestured with the churro, his eyes somehow embarrassed in their sadness. I climbed out of my car and paid him thirty-five cents for ten inches of fried dough that had been dusted with powdered sugar and cinnamon. He thanked me profusely, but he still seemed sad. I guess there's only so much you can do.
I went back to my car and worked my way across the valley floor, then up onto the San Diego Freeway and down through the westside of Los Angeles to the Marina. It was sunny and bright, with the sun still riding a couple of hours above the horizon. The air smelled of the sea and crisp white gulls floated and circled overhead, eyeing McDonald's and Taco Bell parking lots for fast-food leftovers. Women with ponytails raced along the wide boulevards on Rollerblades and shirtless young men pedaled hard on two-thousand-dollar mountain bikes, and everybody had great tans. Aging vaqueros selling rubber-hose churros weren't in evidence, but maybe I hadn't looked close enough.
I turned down Admiralty Way with its wide green traffic island and drove along the Silver Strand to a short cul-de-sac lined with low-density condominiums partially hidden behind tropical plantings. Clarion Way. Seven twenty-four was part of a four-unit building at the front of the curve, and even from the street I could see that the units were large and spacious and expensive. Definitely not cop digs. A gated drive led down beneath the building, and a gated walk led along the front of the units. A mail drop was built into the front gate, along with a security phone so that you could call inside to let the residents know you'd come to visit. I circled the cul-de-sac, parked across the street at the curb, and walked back to the mail drop to see if Angela Rossi's name matched the address. No names. I guess the postman was expected to know who lived where.
A thin man with thick glasses and a bulging forehead squinted out at me from behind the gate. "May I help you?"
I gave him one of my better smiles and tried to look reasonable. "Do you know if Keith's home?"
He frowned at me. "Keith?"
I nodded. "That's right. Keith Adams in seven two four. He said he'd wait for me, but no one answers."
He shook his head. "You must have the wrong address. There are only four of us in the building, and no one by that name lives here."
I dug out my wallet, drew a cash receipt from Hughes Market, and frowned at it. "It says seven twenty-four Clarion."
He was shaking his head before I finished. "Maybe there's another Clarion. I know the woman in seven twenty-four. I don't think she's home now." The woman.
"You don't think we could be talking about Keith's wife, do you?" I peered through the gate. A boy's red bike was leaning against a planter in the entry to seven two four. A plastic hamper filled with Nerfballs stood behind the bike.
He put his hands on his hips, still shaking the head. "Oh, no. It's just Angie and her kids." Angie. You see how it adds up?
I put my wallet away and scratched my head. Klem Kadiddlehopper comes to the big city. "Has she lived here long? Maybe Keith moved." Trying to find out how a cop could afford to live here. Trying to find out if she rented or stayed with a friend or had won the place in a lottery.
"Not long. She moved in two years ago."
"She own it, or does she rent?"
Now he was frowning. Suspicious. "Why don't you leave your number. Maybe the lady knows something about your friend and will call you." The detective presses his luck a tad too hard.
"That's okay. I'm pretty sure I've got Keith's number back at the office."
I thanked him for his time, went back to my car, then drove to a pay phone in a little shopping center at the mouth of the Marina where I called a realtor friend who works in Pacific Palisades. A bright woman's voice said, "Westside Realty, how may I help you?"
I tried to sound like a G-man. "Adrienne Carter, please."
"May I tell her who's calling?"
"Richard Tracy."
"Please hold."
Maybe twenty seconds later another woman's voice came on. "This is Adrienne Carter."
"I'd like to buy the Hearst Castle. Wanna handle the deal?"
Adrienne Carter laughed. "Dick Tracy. Oh, _please_."
I gave her Angela Rossi's address and asked if she could run an owner-of-record check for me. I told her it was a matter of utmost urgency and the security of the nation depended on her. She said, "I'll bet, _Dick_." I think I had started something that I was going to regret.
Forty minutes later I made the slow pull up Laurel Canyon into the mountains above Hollywood and the rustic A-frame I have there. It's woodsy where I live, and though I have neighbors, our homes are separated by mature eucalyptus and olive trees that give us shade and lend stability to the steep slopes upon which we live. I bought the place many years ago when it was in disrepair and, over time, have rebuilt and refinished it both alone and with the help of friends.
I parked in the carport, let myself in through the kitchen, and was looking in the refrigerator for something to eat when the cat door squeaked and the cat who lives with me walked in. I said, "Hey."
The cat is large and black and one ear sits kind of cocked to the side from when he was head-shot with a .22. The flat top of his head is laced with scars and his ears are shredded and lumpy. When he was younger he would often bring me bits of squirrel and bird to share, but he's older now and the gifts are not as frequent. Perhaps he's slowing, or perhaps he's just less generous. He snicked across the floor and sat by his bowl. "Naow."
"I'm hungry, too. Hang on."
I took out leftover chicken that I'd baked with garlic and rosemary, and a half can of tuna. I turned the oven to 350, wrapped the chicken and canned new potatoes together in foil, then put it in the oven to heat. I forked the tuna into the cat's bowl, then set the can next to it so he could lick the juice. He prefers the chicken, but the garlic gives him gas, so I've had to draw the line. He doesn't like me for it, but there you go.
It was eighteen minutes after seven, and I was getting ready to take a shower when the phone rang. Adrienne. I said, "Hi, Adrienne." Elvis Cole, Too Hip Detective, pretends he can read minds.
Lucy Chenier said, "Adrienne?" The Too Hip Detective steps in deep doo-doo.
"A realtor friend," I said. "I'm expecting her to call with some information I need."
"Do tell. Well, heaven forbid I should tie up your line."
I gave her Groucho. "Can't think of anyone I'd rather have tie me up, heh heh."
"Oh, you." I love it when she says 'oh, you.' And then she said, "Hi, Studly."
I felt the smile start deep in my chest and grow large like an expanding bubble, and then I was standing in my kitchen with the phone and Lucy Chenier's presence seemed to fill the house with warmth and light. I said, "I miss you, Luce."
"I miss you, too."
"Hmm."
"Hmm-mm." We often have conversations like this.
I had met Lucy Chenier three months earlier when I was working in Louisiana for an actress named Jodi Taylor. Lucy was Jodi Taylor's lawyer and I was Jodi Taylor's detective, and the attraction, as they say, was immediate. We had called each other regularly since then, and two months ago I had flown back to Louisiana to spend a long weekend with Lucy and her eight-year-old son, Ben. Three weeks after that, Lucy and I had met in Cancun for four days of snorkeling and grilled shrimp and sunburns, and it was harder still to say good-byes when she boarded her plane and I boarded mine. Thereafter, the phoning grew more frequent, and the conversation less necessary, and soon we were in a kind of comfortable/uncomfortable place where the occasional murmur on the other end of the line was enough, but not nearly enough. Over the weeks an increasing part of my day has become the anticipation of the evening's call, when I would sit in my home and Lucy would sit in hers and we would share a few minutes together linked by two thousand miles of fiber-optic satellite relays. It wasn't as nice as actually being with her, but if romance were easy, everyone would do it. I said, "You may be interested in why I am waiting for Adrienne to call."
"I'm sure I don't want to know."
"Do I detect coolness?"
"You detect indifference. They are not the same."
I said, "Ha. We'll see if you feel the same after you hear my news."
She said, "Let me guess. You've changed your name to Jerry Lee Lewis Cole?" You see what passes for humor in Louisiana?
"I'm working with Jonathan Green."
There was a moment's silence, and then Lucy Chenier said, "Is that true, or is this more of the famous Elvis Cole wit?" Not joking, now.
"Hired me today for the Big Green Defense Machine."
Lucy Chenier made a soft whistling sound, then said, "Oh, Elvis. That's wonderful." You see? Impressed. Lucy being impressed made me want to thump my hind leg on the floor and roll over so that she could scratch my belly. She said, "We used to study his cases in law school."
"How about that."
"It must be very exciting."
"He's just another client."
She said, "I have news, too." She sounded happy, like maybe she was smiling when she said it.
"Okay."
"The firm has business to take care of in Long Beach, and they're sending me out. Ben's out of school, so how would you like a couple of freeloading house guests?"
The background noise of the TV and the CNN newscasters was suddenly a million miles away. I said, "I could handle that."
"What?" I guess she hadn't heard me. I guess my voice had come out hoarse and small.
"Hold on a minute and let me check my calendar."
"You rat."
I was smiling. I was smiling so wide that my face felt tight and brittle, as if smiling any farther would make my cheeks crack. "Yes. Yes, I think that would be fine. Are you kidding? That's great."
"I thought so, too."
I said, "I'll be at the airport in an hour."
She laughed. "You can be there in an hour, but Ben and I won't be there until the day after tomorrow. I'm sorry to spring this on you, but I didn't know for sure until this afternoon."
I was too busy smiling to answer.
"I'll call tomorrow and give you the flight information."
"Hey, Luce."
"Hm?"
"I'm really happy about this."
"Me, too, Studly. Oh, you don't know."
We talked for another hour, mostly about where we would go and what we would do and how excited we were that we would see each other again. When my food was warm I sat on the kitchen floor, eating as we talked, and the cat came over and stared at me. Purring. Lucy asked about Green and the Teddy Martin case, and as I told her I listened to the soft country sounds of k.d. lang behind her, and the passing voices of Ben and his best friend as they tumbled through her home. The sounds of Lucy Chenier's life. I told her about the videographer and that Green was shorter and thinner than he looked on television, though still imposing, but after a while our conversation drifted back to us, and to how our tans from Cancun were fading and how much fun we'd had drinking blue iced cocktails and eating the fresh ceviche that the hotel chefs would make at the beach, and then after a while the conversation was over.
Lucy blew me a kiss and hung up and I lay back on the kitchen floor with the phone on my stomach, grinning at the ceiling. The cat stopped purring and came closer to stare into my face. He looked concerned. Maybe he didn't know I was grinning. Maybe he thought I was dying of some sort of hideous facial stricture. Is that possible? Death by grinning. I said, "She's coming to see us."
He hopped up onto my chest and sniffed at my chin and began to buzz again. The certainty of love.
Later, I washed the dishes and shut the lights and went up to bed. I lay there for a very long time, but sleep wouldn't come. I could only think of Lucy, and of seeing her, and as I thought the grin seemed to grow. Perhaps the grin would grow so wide that it would crash through the sides of the house and slop down across the mountain and just keep expanding until it became The Grin That Ate L.A. Of course, if that happened, the grin would eat LAX and Lucy couldn't land. Then where would I be?
At a little after two that morning, I went downstairs to the guest room and stripped the bed and put on fresh linen and then dusted and vacuumed and cleaned the guest bath. I figured I could borrow a camper's cot from Joe Pike; Ben could use the cot and Lucy could have the bed.
At sixteen minutes before four, I went out onto the deck and stared down at the lights in the canyon below. A family of coyotes who live around Franklin Reservoir were singing, and a great desert owl who lived in the eucalyptus trees made his hooting call. I breathed the cool night air and listened to the coyotes and the owl, and I thought how fine it was that so much of my being could have so suddenly become focused on an airplane's time of arrival.
I did not sleep, but I did not mind.
#
By nine o'clock the next morning I had gained some measure of control over the sappy grin and was once more feeling focused, productive, and ready to swing into investigative action. Sappy grins are fine in your personal life but somehow seem less than professional when one is representing the Big Green Defense Machine. Credibility, as they say, is everything.
By eight-forty I had shaved, showered, and phoned Terminal Island to arrange an interview with LeCedrick Earle. I was eating a breakfast of nonfat yogurt and sliced bananas when Eddie Ditko called and said, "Hold on a sec while I fire up a smoke." First thing out of his mouth.
"Top of the morning to you, too, Edward."
There was the sound of the strike and a little pause like maybe Eddie was sucking up half of the earth's pollutant supply, and then a burst of coughing that sounded wet and phlegmy. He said, "Christ, I'm passing blood."
So much for breakfast. I pushed the bowl away and said, "Are you all right?"
"Think I'm gonna drop a goddamned lung." He croaked it out between coughs.
"You want to call back?"
The coughs settled to a phlegmy wheezing. "Nah, nah, I'm fine." When he got his breathing under control, he said, "Whadda they make these things outta nowadays, fiberglass? Ya gotta rip the filters off to get any taste."
"Jesus Christ, Eddie."
Eddie Ditko said, "Listen, I made a few calls and got some stuff for you."
"Okay."
"Rossi looks like a pretty sharp gal." Gal. "Divorced. Got a couple of little boys. Her ex is some kind of middle manager at Water and Power."
"All right." I was making notes. I had been thinking that she might've married well and gotten the expensive house in the divorce, but middle managers at Water and Power aren't known for their bank accounts.
"She was top of her class at the academy and moved right up the promotion ladder once she got into uniform. She responded to more calls, worked more hours, and made more arrests than all but three other officers with her time in grade. That's probably where the marriage went."
I was still writing.
"The LeCedrick Earle bust is what led to the gold shield, and everybody kind of figured that Rossi had a shot at being the first female chief of detectives until the Miranda thing. You blow a murder-one case because you failed to Mirandize a suspect, and that's it for you. She lost a grade in rank and received a letter of censure. That pretty much killed her career."
I was nodding as I wrote. Everything he said was confirming both Haig and Truly. "What happened with the Miranda?"
"Two idiots armed with machetes robbed a Burito King in Silverlake and hacked three people to death. Rossi spotted a car matching the getaway vehicle and collared one of the suspects after a high-speed chase. She was jazzed from the pursuit and forgot to give the guy his warning before he confessed and implicated his accomplice. They hadda let both idiots walk, and Rossi took the heat for it. You see?"
"Man. Did she dispute the Miranda?"
"Nope. She blew it and she admitted it. How about that?" Like he was surprised that someone would take responsibility for their actions. "I can fax you this stuff, you want."
"Thanks, Eddie. What about Earle?"
"Another genius. Rossi tags the guy for a taillight violation and he slides across a C-note with his license, which he saw some moron do in a Dirty Harry movie. Rossi recognizes the Franklin's a fake and tells him it'll cost him a lot more than that, so he brings her back to his house where he pulls out a stash and says she can have all she wants. She says thank you very much and let's go to jail."
"That's her side of it."
Eddie laughed. "Yeah, sure. Your man LeCedrick is what we call a career-type criminal. Prior to the funny-money arrest, he'd been in and out of the system half a dozen times, mostly dope and burglary charges, including two prior associations with a guy named Waylon Mustapha. Mustapha makes his living by selling down funny money for points." Selling for points is when you discount the face value of the counterfeit money to sell it in quantity. Sort of like being a broker. "My guy at the PD says that the bills they recovered when Rossi made the collar matched up with the goods Mustapha handles."
I tapped the pen against the pad, frowning. "Just because LeCedrick was a creep most times doesn't mean he was a creep _that_ time."
Eddie laughed harder. "Keep dreaming."
I said, "You hear anything that would indicate she might be willing to fudge a case?"
"You talk to his mother?"
"Whose mother?"
"Earle's mother was in the house when Rossi made the collar. She saw the whole thing."
"Anything in the file?"
" _Nada_. IA would've talked to her, though. 'Course, whether they listened is a different matter."
"Do you have her address, Eddie?"
He did, and he gave it to me. It was the same address in Olympic Park as that listed on LeCedrick Earle's arrest report. I hung up, then phoned information for Louise Earle's number and called her. I still needed to see LeCedrick, but maybe I could see her first. Maybe she had something to offer that might bolster his version of events, or clarify it. I let the phone ring ten times but got no answer. Guess I'd have to see LeCedrick _sans_ clarity.
I hung up again, washed the dishes, then climbed into my car and made the long drive south to see LeCedrick Earle.
The harbor town of San Pedro lies on the water at the southeast point of the Palos Verdes peninsula, sixty miles south of Los Angeles. It's pretty much a straight shot down the San Diego Freeway across a rolling flat fuzz of low buildings and single-family homes, past Inglewood and Hawthorne and Gardena to Torrance, and then yet farther south on the Harbor Freeway to the water. The Port of Los Angeles is down there, with the gleaming white cruise ships that come and go and the great _Queen Mary_ that forever stays and the U. S. Federal Correctional Facility at Terminal Island.
Terminal Island is on the western side of the harbor, and the facility itself is on the outermost end of the island. The _Queen Mary_ is next door, as are the berths for the cruise ships, but neither can be seen from the prison. From the prison, you could only see open water, and the water looked very much like iron. Sort of like the bars of the cells.
I crossed a land bridge to the island and followed the signs to the prison, and pretty soon I passed through a high chain-link gate and parked at the administration building. A tall link fence topped by concertina wire surrounded the prison, which was new and modern and clean. A guard tower overlooked the grounds, but it was new and modern and clean, too. No gun ports. No swivel-mounted machine guns. No snarling guard dogs or barrel-chested yard-bulls sapping prisoners into line. All of the guards wore blue blazers and ties, and none of them carried guns. They carried walkie-talkies, instead. Modern justice.
I went inside to the reception desk, identified myself, and told the guard that I had an appointment to see LeCedrick Earle. The guard was a clean-cut guy in his early thirties. He found my name in his log, then turned it around for me. "Sign here, please. Are you armed?"
"Nope."
He flipped through a large loose-leaf book until he found Earle's name, then used his phone to tell someone that he wanted prisoner number E2847 brought out. When he was finished he smiled at me and said, "Someone will be right out for you. Wait by the sally port."
A couple of minutes later a second guard brought me through the sally port to a glass-walled interview room. A neat new table sat in the middle of the floor with four comfortable chairs around it. A second glass door was behind the table, and there was a nice gray berber carpet. The air smelled of Airwick. If it weren't for the guards peering in at you and the wire in the glass, you'd never know you were in a prison. Portrait of the Big House as corporate America.
Thirty seconds later the same guard opened the rear door and an African-American guy in his late twenties came in and squinted at me. "You that guy come about Rossi?"
The guard said, "Buzz me when you're done and I'll come get him." The guard had bored eyes and spoke to me as if Earle wasn't there and hadn't said anything.
"Sure. Thanks."
The guard left, locking the door.
LeCedrick Earle was maybe an inch shorter than me, with dark glossy skin and a shaved head. He was wearing a prison-issue orange jumpsuit and Keds. I said, "That's right. I work for an attorney named Jonathan Green."
"You a lawyer?"
"Nope. I'm a private investigator."
Earle shrugged. "I saw that ad in the paper and called. I talked to some guy say he was a lawyer."
"The ad was about information leading to the arrest of James X for the murder of Susan Martin." Truly had filled me in before he'd left the office. "You know anything about that?"
He dropped into the near chair, put his feet on the table, and crossed his arms. Showing smug. "Don't give a damn about that. I know about Rossi. I read in the paper she one of the cops arrest Teddy Martin. She put the fuck on me, I figure she maybe put the fuck on him, too."
"You don't care about the reward?"
"Fuck the reward." Giving me righteous. Giving me can-you-believe-this? "Can't a brother just wanna do his civic duty?"
"I read your arrest report, and I read the letter of complaint your lawyer filed against her. What happened with that?"
"Shit, what you think happened? They didn't do a goddamned thing. Say it's my word against hers."
"Your mother was there."
All the show and the exaggeration flicked away. His eyes darkened and his face seemed to knot. "Yes, well, she don't know nothing. Just a crazy old lady scared of the police."
I said, "Okay, so the arrest report is wrong and Rossi is lying."
"Goddamned right. Bitch set me up."
"She says that you tried to buy your way out of a traffic violation with a fake C-note."
"Bullshit. That money was real."
"You really tried to buy your way out with a C?"
"Man, I had so many outstanding warrants I was scared she was gonna run me in. _That's_ what I was tryin' to avoid."
"So what happened?"
He uncrossed his arms and leaned forward. "I pass her the note and she laughs. She says she don't come that cheap and I say it's all I got. She says I guess we gonna get locked down, then won't we? I'm gettin' the Hershey squirts cause of all the warrants, so I say I got a few hundred stashed at the house. She says let's see it, and that's when we go home."
"She followed you to your house to get more money."
"Oh, yeah. That part's true."
"Okay."
"So we get there and go inside and I got the money back in my room, not much, a few hundred, but it's real. I worked for that cash."
"Okay."
"We go back to my room to get the money and the next thing I know the gun's coming out and she's screamin' at me to get on the floor an' I'm squirtin' for real' cause I think the crazy bitch gonna shoot me and so I go down and she snaps on the cuffs and then she takes this little bag of cash from under her jacket and that's the shit."
"The funny money?"
He was nodding. "I say, what's that? I say, whatchu think you doin'? She say shut the fuck up. Oh, man, next thing I know more cars are pullin' up and she's tellin' them other cops that the flash cash is mine and now I'm in here. How you like that shit?"
I stared at LeCedrick Earle and LeCedrick Earle stared back. His eyes did not waver. I said, "Well?"
"Well what?"
"Just thinking."
"Thinkin' what?"
"Wondering about you and Waylon Mustapha."
He waved his hand. "That's just bullshit bad luck." He waved the hand some more. "Waylon grow up down the street from me. Waylon and me know each other since kindergarten and blow a little smoke together, that's all. I can't help it I know Waylon. I know guys who killed people, an' I ain't no murderer."
"The money Rossi booked into evidence matched with paper that Waylon deals."
LeCedrick crossed his arms and grinned. "Half the funny money on the street come from Waylon. She probably got it from the goddamned evidence room. She mighta even bought it from Waylon his own damn self."
"Okay." I stared at him some more.
LeCedrick Earle started to fidget. "Now what you lookin' at? You don't believe me, jus' say so, callin' me a liar." He got up and walked in a little circle.
I said, "I'm going to write down everything you've said. I'm going to check it out. I'm going to pass it along to Jonathan Green. You sure you don't want a piece of the money?"
"Fuck the money. I just wanna get out of here."
I nodded.
He jabbed a finger at me. "I'm tellin' you and God and everyone else that bitch set me up. You check it out, you see. Bet she set up this Teddy Martin, too."
I said, "Something about what you're saying bothers me, LeCedrick. You want to help me with something?"
His eyes narrowed. Suspicious. "What?"
"If she wanted to set you up, she didn't need to go to your house. All she had to do is bust you on the street and say she found the money under the front seat."
"Damn bitch is crazy! Who know how a goddamn crazy bitch think?" He threw up both hands, then came back to the table and slapped the buzzer for the guards. "Shit on this. I shoulda known you asshole muthuhfuckuhs wouldn't believe me. Fuck you and fuck her, too. I guess a brother just has to rot in here."
The guard came and took LeCedrick Earle back to his cell.
#
As I tooled north back to Los Angeles I tried to keep an open mind. Just because someone looks like a liar and acts like a liar doesn't mean that he is a liar. It doesn't even mean he's a liar when his story is full of holes. Even the truth has been known to have holes. Of course, when his story doesn't make sense it becomes a little more difficult to swallow. I could see Angela Rossi's side of it, but not LeCedrick Earle's. Rossi's report said that she followed Earle to his house because he only had the single hundred-dollar bill on his person and she knew that he could plead innocent to a knowledge of its being counterfeit; she reasoned that if he had more at home as he stated, he couldn't reasonably deny knowledge and the intent to defraud, and the arrest would stick. LeCedrick Earle said that she followed him to his home where she produced a hidden amount of counterfeit money and made the arrest. He opined that she might've done this so that there would be no witnesses, yet Mrs. Louise Earle had been there and Rossi apparently consummated the arrest. Rossi's version made sense and LeCedrick Earle's didn't.
Still, people sometimes do strange things for strange reasons, and I decided to see what Mrs. Louise Earle had to offer. I expected that she would support her son's claims, but in the doing perhaps she would add something to give them greater credence.
I opened Truly's envelope, shook out my notes, and looked up her address. It would be polite to pull off the freeway and call again to see if she was at home, but when people know you're coming they often find reasons to leave. I decided to risk it.
Forty-five minutes later I dropped off the Harbor Freeway onto Martin Luther King Boulevard, and five minutes after that I found my way to Olympic Park.
Olympic Park is a downscale residential area just north of USC and Exposition Park and the Natural History Museum, not far from downtown L.A. The Coliseum is nearby, along with the L.A. Sports Arena, and on game nights the surrounding residential streets are jammed belly to butt with parked cars and pushcarts and hawkers selling souvenirs and iced drinks.
Louise Earle lived in a stucco bungalow on Twenty-fifth Street, four blocks south of the freeway, within walking distance of USC. The houses and the yards are small and the drives are narrow, but the properties are neat and clean, and the Earle home was painted a happy yellow with about a million multicolored flowers blooming on her porch in about a million clay pots and wooden planters. Flowers hung from the eaves and filled the porch and two large wrought iron baker's racks. There were so many flowers on the porch that you had to walk along a narrow path to make your way to the door. It probably took her two hours a day just to water the things.
A six-year-old Buick Skylark was parked in the drive and an air conditioner was humming in a side window. I parked at the curb opposite her house, then went up the drive past the Buick and through the jungle of flowers to her door. The Buick's engine was still ticking. Recent arrival. A little metal plaque under the doorbell said WELCOME. I rang the bell.
The door opened and a thin woman in her early sixties looked at me. She was wearing a simple print dress in a flowered pattern and comfortable canvas shoes and her gray hair had been pulled into a bun. Neat. I said, "Mrs. Earle?"
She smiled at me. "Yes?"
I gave her my card. "Mrs. Earle, my name is Elvis Cole. I'm an investigator looking into your son's arrest. May I ask you a few questions?"
She frowned, but she might've been squinting at the sun. "Are you from the police?"
"No, ma'am. I'm private." I told her that I was working for an attorney named Jonathan Green, and though Green did not represent LeCedrick, the events of his arrest might have a bearing on another case.
She shifted in the door, uncomfortable and unsure about what I might want. "LeCedrick is at Terminal Island."
"I know. I understand that you witnessed his arrest, and I have some questions about that." Something moved in the house behind her.
"Well, I guess it would be all right." Reluctant. She glanced back into the house, then stepped aside and opened the door. "Why don't you come in so we don't let all the cool air out."
I stepped in and she closed the door.
A short, slight gentleman was standing in the living room. He had wavy marcelled hair and he was wearing a brown summer-weight suit that had probably been new twenty years ago. His hair was more gray than not, and his skin was the color of fine cocoa parchment. He was holding a small bouquet of zinnias. I made him for his late sixties, but I could've been off five years either way.
Louise Earle said, "This is my friend, Walter Lawrence. He just dropped in, and now he'll have to be leaving. Won't you, Mr. Lawrence?" She said it more to Mr. Lawrence than to me, and he didn't seem to like it very much.
Mr. Lawrence frowned, clearly disappointed. "I suppose I could come back later."
Louise Earle said, "And I suppose you could just phone later and see whether or not a person is busy before you drop around, now couldn't you?"
Mr. Lawrence ground about four inches of enamel off his teeth, but he managed a grim smile anyway. He wasn't liking this one bit. "I suppose."
She nodded approvingly, then took the flowers. "Now you just let me get these lovely flowers in some water and we'll speak later." She cradled the flowers and encouraged him toward the door.
Mr. Lawrence stood very straight when he walked, trying to get as much height as he could. He mumbled something to her that I couldn't hear, frowned at me as he passed, and then Louise Earle shut the door. A couple of heartbeats later the Skylark backed out of the drive. I said, "Ah, romance."
Louise Earle laughed, and the laugh made her fifteen years younger. "May I offer you coffee, Mr. Cole, or something cool to drink?"
"Coffee would be fine, Mrs. Earle. Thank you."
She took the flowers back to her kitchen, calling over her shoulder. "Please make yourself comfortable."
I sat on a well-worn cloth couch with a handmade slipcover and needlepoint throw pillows. An overstuffed chair made an L with the couch, and the couch and the chair were angled around an inexpensive coffee table, and all of it looked across the room at a cherrywood armoire. The armoire was open and its shelves were lined with tiny vases and knickknacks and family photographs, some of which were of LeCedrick. LeCedrick as a teenager. LeCedrick as a child. LeCedrick before choosing a life of crime. He seemed like a happy child with a bright smile. Her home was neat and cared for and smelled of the flowers.
Mrs. Earle appeared a few moments later with two cups of coffee, walking carefully so as not to slosh. She said, "That business with LeCedrick was several years ago. Why are you interested in that now?"
"I'm investigating the officer who arrested him."
"Oh, yes. I remember her." She put the cups on the table, then offered one to me. "Would you care for milk or sugar?"
"No, ma'am. Then you were present during the arrest?"
She nodded again. "Oh, yes. The police came to see me about that. They came back three or four times. Those affairs people."
"Internal Affairs?"
"Mm-hm." She sipped at her coffee. It was so hot that swirls of steam followed the contours of her face and fogged her glasses.
"You know LeCedrick is disputing the arrest."
"Of course, I know."
"LeCedrick claimed at the time of his arrest, and still claims, that Officer Rossi planted counterfeit bills in order to make the arrest."
Mrs. Earle nodded, but it was noncommittal, like she was waiting to hear more.
"Is that what you told the Internal Affairs people?"
Mrs. Louise Earle gave a deep sigh and the mask of noncommittal detachment melted away into eyes that were tired and pained. "I know he says that, and I'll tell you just what I told those affairs people."
I leaned toward her.
"You can't believe a thing that child says."
I blinked at her.
She put down the coffee and waved toward the armoire. "I was standing right there when LeCedrick and that officer came in. I saw every little thing that happened." Louise Earle closed her tired eyes, as if by closing them she could see it all again, just like she'd told the affairs people. "The officer stood right there, holding her hat and telling me about her day. I remember that she was holding her hat because I thought how polite that was, to hold her hat like that. I didn't know she'd come to arrest him."
"She didn't go back to his room?" LeCedrick had said that Rossi had gone back to his room.
"Oh, no. She just came in and stood there, talking with me the whole time. I was certainly angry when she arrested the boy, but she was very nice about it." Very nice about it. I could see Jonathan Green when I related this. I could see his color drain, his eyes bulge. I wondered if he would pass out and Truly and I would have to administer CPR.
"LeCedrick claims that she accompanied him to his room. He says that she had a bag under her jacket containing the counterfeit bills."
"It was summer. What would anyone be doing with a jacket in summer?" Louise Earle shook her head, and now there was a sadness to her. She crossed her hands in her lap. "Mr. Cole, you listen to LeCedrick and you'd think he was just the most innocent thing, but that just isn't the way it is. LeCedrick will lie at the drop of a hat, and always has."
I sighed. So much for LeCedrick Earle.
Louise Earle said, "Make no mistake about it. I love that child and it grieves me no end he's in jail, but he's said exactly the same thing every other time he's been arrested. It's always somebody else's fault. It's always the police out to get him. Like that."
I nodded. "Yes, ma'am."
"If you're lookin' for me to say that boy is innocent, I can't. If you're lookin' for me to speak against that lady officer, I can't do that, either." She looked stern when she said it.
"No, ma'am. I'm not looking for that."
"He wanted me to lie for him back then, and I wouldn't. He wanted me to cover for him, and make excuses, and I said no. I said, LeCedrick, you have to learn to stop makin' excuses, you have to learn to be a man." Her voice wavered and she stopped. She picked up the coffee, sipped, then said, "It's cost me greatly, but it's for him. Something has to shock some sense into that boy."
"Yes, ma'am."
"He hasn't spoken to me since the trial. He said he'd never speak to me again."
"I'm sorry, Mrs. Earle." I didn't know what else to say. I felt awkward and ashamed that I'd come into her life and driven off Mr. Lawrence and made her relive something that was clearly so painful.
"I tried to raise that boy right. I loved that boy as much as any mother could, and tried to show a good example, but he just went wrong." Her eyes grew pink and a single tear worked its way down her cheek. "Maybe that was where I went wrong. Maybe I held him too close and excused too much. Is it possible to love someone too much?"
I looked at her, and then I looked at the furniture and the pictures, and then back at her weary eyes and the weight they carried. "I don't think there can ever be too much love, Mrs. Earle."
She seemed to consider that, and then she put her coffee down again. "Has this helped you?"
"Yes, ma'am. It has." Jonathan Green wouldn't think so, but there you go.
She stood, and it was clear that she wanted me to leave. "If you don't mind, then, I should clip those zinnias and get them in water."
"Yes, ma'am. I'm sorry I interrupted you and Mr. Lawrence."
The tiny smile came back, though it wasn't as strong as before. "Yes, well, it'll take more than a little interruption to discourage that man."
"Men are like that, Mrs. Earle. We find something worthwhile, we stay with it."
The tired eyes crinkled and suddenly the younger self was there again. "Oh, you get on with you, now."
She walked me to the door and I went out into the sun and got on with me.
#
The early afternoon heat shimmered off the sidewalks and cars and surrounding roofs in a kind of urban illusion of life's silver lining. It was just before two on the second day of my investigation into Angela Rossi and the doors of investigative possibility were rapidly closing, and with every closed door Angela Rossi looked better and the people making claims against her looked worse. Louise Earle was credible, cogent, in full command of her faculties, and did not seem to be a person who would miss seeing a cop carrying a bag of funny money through her living room. Of course, maybe Angela Rossi was a master of misdirection and had secreted the money behind her back. She might've shouted, "Look over there!" and run to LeCedrick's room and planted the cash when Louise turned to look. Perhaps my investigative task for the afternoon should be finding out whether or not Angela Rossi was an amateur magician.
Or maybe not. Three teenaged girls with long skinny legs and halter tops came out of the house across the street and went to an ancient Volkswagen Beetle parked in their drive. They were lugging beach towels and bottles of Evian water, and everybody wore thongs. Off to the beach. Maybe I should offer to go with them and protect them from the thugs at the beach. Maybe we could discuss my findings. On the other hand, Lucy Chenier was arriving tomorrow, and maybe I should snap out of it before I found myself in really deep doo-doo. _C'est la vie_.
When I reached the sidewalk a tall, muscular black guy appeared beside my car. As he reached the car a heavy white guy in his early fifties climbed out of a blue sedan parked across the street and started toward me. The black guy was in impeccably pressed designer jeans and a tight knit shirt that showed his muscles, and the white guy was in a rumpled light gray winter-weight suit. A million degrees, and he's wearing winter weight. Cops. A woman's voice said, "Excuse me, sir. May I have a word with you?" Polite, and kind of cheery.
The cheery woman was coming toward me from the adjoining yard as if she had been standing at the corner of the house there, waiting. She was maybe five-eight, and dark the way you're dark when you spend a lot of time in the sun running and working out and playing sports. I made her for her early- to mid-thirties, but the lines around her eyes and mouth were deep. Probably from all the sun. She was wearing designer jeans like the black guy and Reebok court shoes and a loose linen top that she would probably cover with a linen sport coat if it weren't so hot. Stylish and attractive, even with the Browning 9mm clipped to her right hip. She badged me with an LAPD detective shield as she approached, still cheery with the smile, and I recognized her just before she said, "Mr. Cole, my name is Angela Rossi. The detective in the gray suit would like to ask you a few questions."
She glanced at the guy in the bad suit and I followed her look just as she knew I would, and when I did she stepped close and threw an overhand with a black leather sap, trying for the side of my head. Sucker shot. I picked up her move and tried to twist out of the way, but she was good and fast and I caught most of the sap on my right cheek with a blossom of pain. The guy in the suit yelled, "Hey!" and the black guy grunted, "Shit!" like they were surprised, too. Rossi followed the sap with a hard knee, but it caught me in the thigh instead of the groin, and then the older guy was there, wedging himself between us, forcing her away and saying, "Dammit, Rossi, you want another beef in your file? Is that what you want?"
I wobbled, but kept my feet and let the older guy move her back.
The black guy hustled up behind me and his hands went to my wrists, pulling my arms behind me. The three girls ran up onto their porch and watched from the door, one of them with her hand to her mouth. My right cheek felt like someone had popped a firecracker under the skin and my eyes were watering. I didn't want to double over, but I couldn't exactly stand up straight either. It's hard to look tough when you're thinking that maybe you'll vomit. Especially when you've been suckered with an eye-fake. Maybe Rossi was a master of misdirection after all.
Angela Rossi jabbed her finger at me, saying, "This shitbird came to my _home_! What were you doing at my _home_ , you creep?" She wasn't smiling, now. Her face was etched and drawn, and she looked as if she wanted to rip out my eyes.
The older guy pushed her hand down and shoved her farther away. "Dammit, Rossi. Step back."
The black guy locked my right arm above the elbow, walked me to a white Cressida, and pushed me down across the trunk. The skin of the car was so hot from the sun it felt like a branding iron. I said, "Are you guys really cops or is this _America's Funniest Home Videos_?"
The black guy ignored me. He went through my pockets and down my pants, and then he said, "He's clean, Tommy."
Rossi stopped all the squirming and trying to get at me. The older guy came over and badged me, too. "I'm Detective Tomsic, and you're being investigated for stalking a Los Angeles police officer. Do you understand that?"
The teenage girl with her hand to her mouth disappeared inside the house. The other two stayed on the porch, watching. A couple of faces appeared in the windows, and I said, "Hey, look, Tomsic. I think they've got a video camera."
Tomsic said, "Good. Let'm watch."
"Maybe they got the sap on tape. You think?" Saps are classified as dangerous weapons. They are illegal to carry, sort of like rocket launchers and samurai swords.
Rossi said, "What were you doing at my home?" She was breathing hard, but she was well back on the sidewalk and she probably wasn't going to hit me again.
"ID and license are in my wallet. I'm a private investigator." The black guy tossed my wallet to Tomsic.
Rossi said, "We know who you are, shitbird. Tell me why you came to my house."
"I was investigating a lead that you were living beyond your means."
"Why?"
"It's what I do. Investigate."
The third girl returned from her house to join her two friends, but Tomsic didn't seem overly concerned. He was going through the wallet like he had all the time in the world. "He's our boy, all right. California PI license. Elvis Cole." He looked at me. "You've got a license to carry here. Where's the piece?"
"Under the seat."
The black guy laughed. "You left it under the seat?"
"I was talking to a woman in her sixties. Who would I shoot?"
The black guy said, "I hear you." He went to my Corvette without having to ask which car was mine. They'd probably followed me. Rossi's neighbor had probably copied my tag number and they'd run the plates and picked me up at my house or maybe even on the way to Terminal Island.
Rossi frowned at Louise Earle's place. "You investigating the LeCedrick Earle thing?"
"Earle claims you planted the cash."
"That's bullshit."
I nodded. "I had to check it out."
She put her right hand on her right hip, just above the Browning. "Who are you working for?"
"Jonathan Green. In the matter of Teddy Martin."
Tomsic said, "Well, fuck me."
The black guy stood out of the Corvette, grinning. "You on the Martin defense? Whadda they call it, the Big Green Defense Machine?" Like he wanted to laugh.
I looked back at Rossi. "People are making accusations that may be relevant to the defense effort, and I'm checking them out. So far you look pretty good."
She looked surprised. "What accusations? Teddy Martin killed that woman."
I made a little shrug. "If you planted evidence once, the theory is that you'd plant it again. Some people called Green and told him that you've got a history of doing anything it takes to jump your career. Green hired me to see if there's anything to it."
Angela Rossi squared herself and took a step toward me. Tomsic shook his head. "Angie."
Rossi took another step closer and the black guy came back to stand with Tomsic between us. Like the two of them were scared of what she might do. She said, "Green's a shithog and so are you."
Tomsic said, "Take it easy, Angie."
Rossi shoved at Tomsic. "Hey, I don't have to take this shit! Assholes coming into my life and trying to put this on me!"
I said, "No one's trying to put anything on you. I just want the facts. No one's looking to axe you."
Rossi jabbed her finger at me, but spoke to Tomsic. "This guy's in _my life_ , Dan!"
Tomsic said, "Chill out, will you? This stuff happens. I've been investigated nine thousand times."
I said, "Look, Rossi, it's like I said. I've been through most of it and you're looking good. This is a legal investigation, and if you check out clean I'll report that to Green and that'll be the end of it."
Tomsic said, "You hear that? Clean." Like we were both on the same team, now, trying to keep her calm. Maybe Haig had been right about her being a nutcase. Tomsic was acting as if he was scared what might happen if she lost control of herself. He turned back to me. "You understand why we dropped on you, right? Your nosing around her house."
"No problem." My cheek was throbbing and the skin around my eye was starting to stretch, but there was no problem. Sure.
A black and white LAPD radio car turned onto the block and came at us with its light bar flashing, probably responding to a call from the three girls. The radio car roared in to a sliding stop with a couple of uniforms unloading even before the car stopped rocking. An Asian guy in his mid-forties was driving with a Hispanic guy in his late-twenties along for the ride. Tomsic said, "Fuckin' great. A cheering section." He nodded toward the black guy, then the uniforms. "Robert, chill out these guys, okay?"
Robert badged the uniforms and trotted over. The Asian guy had a couple of stripes on his sleeve and was built like he'd spent the last twenty years in the LAPD's weight room. His name tag read SAMURA. Robert met Samura first and spoke to him in low tones as they walked back to us. When Samura heard my name he looked at me. "You're Cole?"
"Unh-hunh."
He looked at Tomsic. "This guy works with Joe Pike."
Robert and Tomsic stared at me. So did Rossi. Robert said, "No shit?"
I spread my hands. "Somebody has to."
Tomsic's face went red and he wasn't so friendly any more, like he and I were no longer on the same team. " _The_ Joe Pike?"
"How many you know?"
His jaw worked, and he said, "The Joe Pike I know can kiss my goddamned ass." When Joe left the PD it hadn't gone well.
I smiled at him. "I'll give you his number. You can tell him yourself."
A little tick started in Tomsic's left eye. "Maybe we should march your butt in, after all. Dig around and see if you're in violation of your license."
I rolled my eyes. "Oh, please, Tomsic. Spare me."
The tick fluttered into a rapid-fire blink, but then he stepped back and looked embarrassed. Samura pretended not to notice. "We got a robbery in progress call. What's the deal?"
Tomsic filled him in, telling him about my nosing around Rossi's home, telling him about Teddy Martin and Jonathan Green and Rossi's role in the Martin arrest. Samura listened, but didn't seem particularly interested. You spend enough years on the street, you're not even interested if a nuke goes off.
When Tomsic was finished, Samura said, "Cole has a good rep. I know guys who've worked with him." He squinted at me, then took off his hat and wiped his face. It had to be a million degrees, standing in the sun. "You remember a guy named Terry Ito?"
"Sure." I'd worked with Ito four or five years back.
Samura put his hat back on and looked at Tomsic. "You don't have to sweat it. Ito thinks that this guy's the cat's ass."
I said, "Terry has a way with words, all right."
Robert said, "We didn't know who the guy was and he was poking around an officer. You know how it is."
"Sure." Samura squared his hat, then nodded toward his radio car. His partner drifted away. Samura started after him, then turned back and looked me over. "I'd never heard Terry Ito say a good thing about anybody. Terry know you work with Joe Pike?"
"Yes."
Samura cracked the world's smallest grin, then went back to his car and drove away. The three girls were still gaggled at their front door, but most of the other faces had disappeared from the windows. You've seen one crime scene, you've seen'm all.
Tomsic looked at Rossi. "Okay. We know who this guy is and what he's doing. You okay with it?"
She made a grudging shrug.
Tomsic looked back to me. "How about you? You gonna file a beef because of the sap?"
"Barely touched me."
Robert laughed. "Yeah. Look at you."
Tomsic said, "Okay, then. Everybody knows where it stands." He nudged Rossi. "We don't have to like it, we just have to know where it stands."
Rossi said, "One thing."
I looked at her.
"You're doing a job, and I can live with that. Investigate all you want, but stay the hell away from my home. If you come around my home again, I'll break you down. If you even look at my kids, I'll kill you on the spot."
Tomsic said, "Jesus Christ, Angie, knock that shit off. Sayin' shit like that is what gets you in deep."
She raised a neutral hand. "Just laying it out."
I said, "You're looking good, Rossi. Don't sweat it."
"Yeah, sure." She stared at me for another couple of seconds, but she didn't look relaxed and she didn't look as if she believed it was over. She was breathing hard, and the crinkled skin around her eyes was jumping and fluttering as if tiny butterflies were trapped there, trying to get out. Then something that looked like it might've been a smile flickered at the corners of her mouth and she said, "Tell Joe that Rossi says hi."
Angela Rossi turned away without another word, crossed the street, and slid into the passenger side of Tomsic's dark blue G-ride. Tomsic joined her, and Robert got into a tan Explorer. In a couple of minutes they were gone. Even the three girls were gone, vanished in their Volkswagen for a belated trip to the beach.
I stood there for a time, alone except for the dull ache in the side of my face, and then I got into my car and drove to my office.
#
I stopped at a 7-Eleven to buy ice for my eye. A Pakistani gentleman was behind the counter, watching a miniature TV. He was watching an episode of _COPS_ , and he viewed me with suspicion as I paid.
I told him what the ice was for and asked if I could use the bathroom to look at myself, but he said that the bathroom was for employees only. I asked if he had a little mirror that I could borrow, but he said no again. He sneaked a look toward the door as if he wanted me to leave, as if whatever wraith of urban violence had assaulted me might suddenly be visited upon him and his store. Guess I couldn't blame the guy. You look at enough episodes of _COPS_ , and pretty soon you're thinking that life is a war zone.
I thanked him for the ice, then went out to the car and looked at my eye in the rearview mirror. A neat little mouse was riding high on my right cheek and was already starting to color. Great. I wrapped a handful of ice in my handkerchief and drove back to my office with one hand. Nothing like bucking rush-hour traffic with a faceful of ice.
It was just after five when I reached my building and turned down the ramp into the building's garage. A line of cars was on its way out, but most of the garage was already empty. Cindy's Mazda was missing, and so were the cars belonging to the people who worked at the insurance company across the hall from my office. I left my car in its spot, walked up to the lobby, then took the elevator to my floor. Lights off, doors locked, empty. Empty was good. Maybe if Los Angeles had been empty I would've been able to spot two carloads of cops tailing me around half the city.
I let myself into my office, popped on the lights, and found Joe Pike sitting at my desk. I said, "You could've turned on the lights, Joe. We're not broke."
Pike cocked his head to the side, looking at my eye. "Is that a pimple?"
"Ha-ha." That Pike is a riot. A real comedian, that guy.
Joe Pike is six foot one, with long ropey muscles, dark hair cut short, and bright red arrows tattooed on the outside of each deltoid. He got the tattoos in a faraway place long before it was stylish for rock stars and TV actors and Gen X rave queens to flash skin art. The arrows point forward, and are not a fashion statement. They are a statement of being. Pike was wearing a gray sweatshirt with the sleeves cut off and Levi's and dark pilot's glasses. Even at night he wears the glasses. For all I know he sleeps in them.
I went to a little mirror I have on the wall and looked at the eye. The side of my face hurt like hell, but the ice was working; the swelling had stopped. "Your friend Angela Rossi hit me with a six-ounce sap. Suckered me with an eye move."
"I know."
I looked at him. "How do you know?"
He got up, took two Falstaffs from the little fridge, and handed one to me. If you listened as hard as you could, you still wouldn't hear him move. "Angie called and told me. She wanted to know what we were doing."
"She called you."
He popped the tab on his Falstaff and had some. "I've been here a while. Lucy called. I didn't know she was coming out."
"Tomorrow."
"I left her flight information on your desk." Pike took his beer to the couch. "Why are we working for Theodore Martin?"
"We're not. We're working for Jonathan Green." I told him about Haig and his allegations that Rossi would fabricate evidence to boost her career. I told him about LeCedrick Earl and his allegations that Rossi had done just that. "Green hired us to look into the allegations. I told him that we would report what we found, even if it hurt his case. He said okay."
"Lawyers are lizard people." Life is simple for Pike.
"Lucy's a lawyer."
Pike's head shifted a quarter of an inch. "Not Lucy."
I said again, "Angela Rossi called you."
He stared at me with impenetrable black lenses. Two months before I'd had canvas Roman shades installed on the French doors to cut the western exposure in the afternoon, and when the shades were down the office filled with a beautiful gold light. They were down now, and Pike was bathed in the light. It made his dark glasses glow. "We worked Rampart Division together. She was coming on when I was going out." Pike had spent three years riding in a radio car for LAPD. "I knew Haig. Haig was an asshole. I knew Rossi, too. I didn't ride in a car with her, but she seemed like a straight shooter."
"Okay."
"That what you found?"
I took my ice and my Falstaff and went to my desk. I saw the notepaper with Lucy's flight information. Pike's printing was meticulously neat, but so small it was almost impossible to read. "She's aggressive, ambitious, and no one likes her much, but there's no evidence that she dumped LeCedrick Earle or anyone else. Haig comes across like a crank, and Earle's own mother said that her son is a liar."
Pike nodded.
"The only thing that doesn't fit is her house. Two years ago she bought a condo in the Marina that had to go for four hundred thousand dollars. I've got a call in to Adrienne Martin."
"Forget the house. Her mother left her an apartment building in Long Beach. When Rossi sold it she had to roll the cash into another property or get hit with the capital gains."
I stared at him.
"We were close."
"I see."
"Very close." Still hidden behind the black lenses.
I stared at him some more, and then I nodded. "I guess that's it, then. No crime, no graft, no corruption. Jonathan won't like it, but there it is." There hadn't been much to check and it hadn't taken long, but it rarely does when everything is aboveboard.
"She's a sharp cop, Elvis. It's a tough game for a woman, tougher still if the woman is better than the boys and lets them know it."
I smiled at him. "She doesn't seem like the retiring type."
He canted his head a couple of degrees. "She had a real shot at being the first female chief of detectives. She still might, even with the Miranda beef."
"High praise coming from you."
Pike shrugged.
I said, "Joe, are you soft on this woman?"
Pike finished his beer, then got up and placed the empty carefully into the wastebasket. "I admire her, Elvis. In much the same way I admire you."
I didn't know what to say to that, so I said, "Since you admire me so much, I've got a favor to ask."
He waited.
"Lucy and Ben are coming, and I've got the two-seater. Can I borrow your Jeep to pick them up?"
Pike stood motionless. The Jeep was in immaculate condition, and Pike kept it flawless. You could shave in the fender. You could eat off the engine block.
I said, "I'll wash it before I give it back. If someone dents it I'll shoot them."
Pike's head swiveled one-half a degree. I think he was stricken. "Why don't I come with you to pick them up?"
"Joe." It was like pulling teeth.
He still wasn't happy about it, but he finally nodded. Once.
I said, "I'll draft the report on Rossi tonight. I'll call Truly and tell him that I'm going to turn it in tomorrow, and he'll probably want to see me. You want to go along?"
Pike said, "No." Lizard people.
"Just thought I'd ask."
Pike went to the door, then looked back at me, and gestured to his right eye. "That's going to look nice for Lucy."
"Thanks, Joe."
"Good to see Angie hasn't lost her touch." His mouth twitched a single time and he left. Pike never smiles or laughs, but sometimes you'll get the twitch. Mr. Hilarity.
I had the rest of my beer, then phoned Elliot Truly. When Truly came on the line, I said, "I've concluded the investigation into Angela Rossi. I'm going to write the report tonight."
He didn't say anything for a second. "So soon?"
"I'm fast, Truly. Cases solved in no time flat or your money back."
Truly said, "Well, hell." Like he was disappointed it hadn't taken longer, like he was maybe thinking that I had given the job short shrift. "What did you find?"
"She's clean. Earle is a liar and Haig is a crank with a grudge. There's absolutely no evidence that Rossi's ever been anything other than a good cop."
Another silence. "You'd better come in. Jonathan will want to talk about it." You see?
"I have guests coming in from out of town at five tomorrow evening."
I could hear him fumbling with something. "We're going to have a staff meeting here tomorrow morning at nine. Can you make that?"
"I'll be there."
It took less than twenty minutes to write the report, and then I drove home listening to k. d. lang. k. d. lang was Lucy's favorite, and as I drove I found that I was thinking less about Jonathan Green and Angela Rossi, and more about Lucy Chenier. I thought that I might clean the house and make a shopping list. The house was already clean and it was too late to shop, but that didn't matter. My work was done and Lucy was coming, and what could be better than that? Anticipation is everything.
When I got home, Pike's Jeep was waiting in the drive, freshly washed, immaculate and gleaming. I found a note under the windshield that said, _Give my love to Lucy, and please drive carefully_.
That Pike is something, isn't he?
#
At twenty minutes before nine the next morning I worked my way down the mountain along Laurel Canyon to Sunset, then turned west toward Jonathan Green's office.
Most prominent attorneys in Los Angeles will black-jack their mothers to find office space in Beverly Hills or Century City, both of which are considered prestige addresses for the legal community. Jonathan Green's office was on Sunset Boulevard in an ornate four-story Spanish office building across from the Mondrian Hotel. I guess if you're Jonathan Green, any place you happen to be is a prestige address.
The building was older, with an established landscape of royal palms and bougainvillea, and state-of-the-art security equipment discreetly hidden from public view. A tasteful sign built into the front of the building simply said THE LAW OFFICES OF JONATHAN GREEN. The parking garage was gated, and the gate wouldn't open until a gentleman wearing a red blazer strolled out to my car and asked my name. He was exceedingly polite and possessed a bulge in the line of his jacket beneath his left arm. The bulge, like the sign and the security equipment, was also discreet.
I left my car in the garage, then followed the guard's directions past a Spanish tile fountain in the lobby to the elevators, and then to the top floor. Another blazered gentleman smiled at me in the lobby, and a third just happened to be on the elevator. Both were polite and both, like the guard in the parking garage, had the corded necks of men who spent a lot of their time honing confrontational skills. Corded necks are a dead giveaway.
When the elevator opened, Elliot Truly was waiting for me. I guess the parking guard must've called. I said, "Some security."
He stared at my eye.
"Cut myself shaving."
Truly realized he was staring and looked away. "Yes, well, I guess that happens."
I followed him past the floor receptionist and along a glass hall. "Why all the spooks?"
"Many of Jonathan's cases are unpopular, as you might imagine. You'd be surprised at the number of people who don't believe that defendants are entitled to the best possible defense."
"No kidding."
Men and women in business suits hurried in both directions, some carrying files, others long yellow legal pads, still others small Styrofoam cups of what I took to be coffee. Nine in the morning, and everyone looked tense. I guess tension is a way of life when you're trying to give people the best possible defense. Especially at five hundred dollars an hour.
I said, "Are all of these people working for Teddy Martin?"
"Oh, no. The firm is involved in over two hundred active cases."
"Mm."
"Jonathan only involves himself in the more, ah, trying cases." He gave me a sly smile.
I nodded.
He looked at me. " 'Trying.' "
"I got it."
Truly looked disappointed. "Oh." Lawyer humor.
We turned down another hall and then into a conference room about the size of Rhode Island. A breakfast buffet had been set up at one end of the room with coffee and mineral water and enough lox and bagels to sink the _Lexington_. Six men and three women were crowded around the buffet, talking in soft whispers. Everyone had coffee, but no one was eating. Probably too tense. Truly said, "Would you like something to eat?"
"Just coffee." Elvis Cole, at one with the team.
"Let me introduce you. Jonathan will be along in a moment."
We got the coffee, and Elliot Truly introduced me. Everyone in the room was an attorney except me. While the introductions were under way, yet more attorneys arrived. I stopped counting at fourteen. The large lesser attorney came in, followed by the small lesser attorney, both of whom were wearing beige linen Armani suits. So was Elliot Truly. I said, "Beige."
Truly said, "Pardon me?"
"Nothing." Jonathan Green would be wearing beige, too. You could bet your house on it.
Thirty seconds later Jonathan Green came in wearing a beige linen Armani. You see? I said, "Shucks."
Truly glanced at me and whispered, "What?" Now that Jonathan was here I guess we would whisper.
"No videographer. I was hoping for more air time."
Truly blinked at me, then seemed to get it. "Oh, right. Ha-ha." Ha-ha. We're just a riot at nine A.M.
Another man came in behind Jonathan. He was a little shorter than me, but his arms were as long as backhoe shovels and his shoulders so wide they looked like they had been built of steel frame girders. The arms and the shoulders didn't go with the rest of him, as if they had once belonged to King Kong or Mighty Joe Young or some other large mammal, and now this guy was using them. He was carrying a manila envelope.
Green smiled when he saw me and offered his hand. "Thank you for coming. This is Stan Kerris, our chief of security. Stan, this is Mr. Cole." Stan Kerris was the guy with the shoulders. He had a monstrously high forehead, sort of like a Klingon's, and eyes that looked at you but gave you nothing, like windows to an empty room.
Truly said, "Let's get started."
Jonathan Green took his seat at the head of the table with Stan Kerris sitting next to him. The two lesser attorneys elbowed each other to sit nearby. Like the lesser attorneys, everyone else tried to jockey as close to Jonathan Green as possible. Truly sat next to me. When everyone was down, Green crossed his legs and smiled at me. "So. Elliot tells me that you've found no corroborating evidence to Mr. Earle's claims."
"That's right."
"And the same for Mr. Haig?" He raised his eyebrows in a question.
"That's right. I spoke with Haig and with Earle, then with Earle's mother. I did a cursory background check on Earle, and reviewed the Internal Affairs investigation into the funny money bust. IA found that Rossi made a quality bust."
Truly was shaking his head. "What does that mean? Of course, they would say that."
"No, Mr. Truly. They wouldn't. LAPD takes these things seriously." I looked at Green. "I concur."
Green laced his fingers across a knee and settled back. "Please tell us why."
At least seven of the assembled attorneys copied what I said. I started with Raymond Haig and worked my way through Eddie Ditko and Rossi's condo and my interviews with both LeCedrick Earle and Louise Earle. I told them about LeCedrick's past record, including his close association with Waylon Mustapha, and I described in detail how Louise Earle's version of events matched with Rossi's police report. I spoke for close to twenty minutes, and for twenty minutes pens scratched on legal pads and Jonathan Green sat unmoving. His eyes narrowed a couple of times, but mostly he watched me as if he could absorb the details without effort and assimilate them. Or maybe he was just bored.
When I finished Kerris said, "Anything we can use in the Miranda?"
"What do you mean, use?"
Truly smiled. "Was there anything in her action indicative of malice aforethought or a willingness to commit an illegal act?"
I took the reports that Eddie Ditko had faxed me from my file and passed them to Truly. I told them about the guys with the machetes. I described what had happened at the Burrito King. "They let both these guys walk and Rossi took the heat for it. I don't think there was much forethought to blowing out her career at the end of a high-speed chase because of an adrenaline rush."
Truly smiled again and shrugged at Kerris. "Guess not."
Jonathan Green said, "You're sure about these things?"
"Yes, sir. There is no evidence that this woman has ever done anything illegal or even improper other than the Miranda beef, and she stood up for that one. She wouldn't have had to set up LeCedrick Earle. He's a career criminal."
Green nodded. "Then you don't believe that she could've planted the hammer on Theodore's property?"
"No, sir."
"We should abandon this as a legal theory?"
"That would be my opinion, yes, sir."
Jonathan Green nodded again, then stared at the far wall for what seemed like several minutes. No one moved, and no one spoke. All of the other attorneys stared at Jonathan as if he might suddenly utter some dictum and they would have to act on it. Apprehensive.
I looked at my watch. It was nine forty-two, and the staring continued. Maybe Jonathan Green had lapsed into a trance and no one knew it. Maybe he would continue to stare all day and I'd still be sitting here when Lucy and Ben landed at LAX. I drummed my fingers on the table and Elliot Truly looked horrified. I guess it just wasn't done.
Jonathan Green suddenly spread his hands, then placed them on the table and leaned forward. "Well, that's that. Better to know now than embarrass ourselves in court. You've done an outstanding job, Mr. Cole. Thank you."
The other attorneys breathed as one and broke into large smiles, saying what an outstanding job I'd done.
Green swiveled toward Truly and said, "It was one theory, and there's still plenty of ground to cover. We'll just have to roll up our sleeves and try harder." Green swiveled back to me and leaned forward again, absolutely serious. "I remain convinced of Teddy's innocence, and I'm determined to work all the harder to prove it."
The fourteen other attorneys around the big table nodded, and I guess I could understand why. Green seemed to bring it out in you. I wanted to nod, too.
Jonathan Green said, "Mr. Cole, I know you were hired for this specific part of our investigation, but it's very important to me that people of your caliber work with the team."
Elliot Truly said, "Here, here." Really.
Green gestured toward Kerris. "We've been absolutely overwhelmed with people calling our hotline, haven't we, Stan?"
Kerris nodded, but the nod conveyed nothing, sort of like his eyes. "We've gotten several hundred calls from people claiming to have information about the kidnapping. We can dismiss some based on the phone interview, but most have to be checked. We're dividing these things up among our investigators."
Green said, "Stan, give him the envelope, please."
Kerris pushed the envelope down along the table to me. I opened it. Eight single-sheet interview forms were inside.
Jonathan said, "Each sheet contains the name, phone number, and address of a person claiming to have information about Susan Martin's murder. If you could see your way clear to staying with us on this and checking these people out, we would appreciate it."
I looked at the sheets. I slipped them back into the envelope. "I have guests coming into town."
Truly shrugged. "There isn't a rush with this, Cole. Sure, sooner is better than later, but you know the justice system."
"Okay."
Green broke into a wide smile. "Well, that's just great. That's fabulous."
The assembled attorneys told me how great it was.
I glanced at my watch, thinking I could knock off three or four interviews before Lucy's plane. The more I finished before Lucy's plane, the more time I'd have for her.
Truly said, "We don't know anything about these people. As Stan said, our screeners were able to rule out the obvious cranks, but you never know. We want you to use your best judgment to determine if they have anything of merit to offer."
"Judgment. Okay." I looked at my watch again. "I've got it."
Truly spread his hands. "And when you're done with those, of course, there's more."
The lesser attorneys chuckled and someone said, "A _lot_ more." Even Jonathan Green chuckled at that one.
Green stood and everyone stood with him, and I was hoping I hadn't been too obvious with all the watch-glancing. Jonathan came around the table and offered his hand again, and this time when we shook he held it. He said, "I want you to know that I appreciate the good, fast work you've done, Mr. Cole. It's important to me, and it's important to Teddy, also. I spoke with him yesterday and told him that you're on the team. You're going to like Teddy, Mr. Cole. Everyone does."
"I'll look forward."
"Good hunting." He tried to let go of my hand, but this time I held onto him, not realizing that I had. In that instant he smiled warmly and I let go.
Jonathan Green swept out in a wave, Kerris beside him and the lesser attorneys in his wake, jostling each other to better their positions.
#
It was a little before ten when I followed the trail of security men down to my car, then zipped to the Virgin Megastore, bought the new k.d. lang and a collection of Louisiana hits called _Cajun Party,_ then sat in the Megastore's parking structure and went through the envelope of hotline tipsters. I had almost seven hours until Lucy's plane; plenty of time for the world's fastest detective to do his marketing and work his way through a significant number of interviews, especially if he attacked his investigatorial responsibilities in a methodical and professional manner.
I organized the twenty statement forms by location and decided to start with those people who were closest and work outward.
I went back into the Virgin, got change from a pretty young woman with a pin through her nose, then found a pay phone on Sunset Boulevard to arrange the interviews. A homeless man with a shopping cart filled with neatly folded cardboard squares was seated beneath the phone, but he graciously moved aside when I told him I needed to make some calls. He said, "Please feel free. It is, after all, a public instrument." He was wearing spats.
I fed in a quarter and dialed Mr. C. Bertrand Rujillio, who lived less than five minutes away. A man with a soft, raspy voice answered on the fourth ring and said, "Who is this?"
"My name is Cole, for the law firm of Jonathan Green. I'm calling for Mr. C. Bertrand Rujillio, please."
There was a pause, and then the rasp came back. "Do you have the money?"
"Is this Mr. Rujillio?"
Another pause, softer. "The money?"
"If you mean the reward, that won't be paid unless the information you provide leads to the arrest and conviction of Ms. Martin's murderer." Truly said that the phone bank operators had explained all this. Truly said I wouldn't have to worry about it. "I need to take your statement, Mr. Rujillio. Can we arrange that?"
The pause again, and this time the line went dead. I stared at the phone for a couple of seconds, then hung up and scratched C. Bertrand Rujillio's name off the list.
The homeless man said, "No luck?"
I shook my head.
Of the next three calls, two reached answering machines and one went unanswered. Nobody home. I said, "Damn."
The homeless man said, "Four out of four is poor luck."
"It can't last forever."
"Will you have many more calls?"
"A couple."
He sighed and looked away.
Two more calls and two more answering machines and all the nearby people were done. So much for efficiency. So much for my plan of starting in close and working out. I said, "Well, hell."
The homeless man said, "Tell me about it."
I looked at him. "I had a plan, but no one's home."
He made a sympathetic shrug, then spread his hands. "Flexibility, my friend. Flexibility is the key to all happiness. Remember that."
I told him that I would and shuffled through the witness forms and decided to hell with starting close. I called Floyd M. Thomas in Chatsworth. Chatsworth was a good forty minutes away. Floyd M. Thomas answered on the third ring in a fast, nervous voice and told me that he had been expecting my call and that he would be happy to see me. I hung up. The homeless man said, "You see? When we force events we corrupt them. Your flexibility allowed events to unfold in a way that pleases you. We know this as synchronicity."
"You're a very wise man. Thank you."
He spread his hands. "To possess great wisdom obliges one to share it. Enjoy."
I drove to Chatsworth.
Floyd Thomas lived in a studio apartment on the second floor of a ten-unit garden apartment just off Nordhoff. Scaffolding was rigged around the front and sides of the place, and Hispanic men in baggy pants were chipping away cracked stucco. Earthquake repairs. Thomas himself was a thin, hunched man in his early fifties who opened his door only wide enough to peer out at me with one eye. When he opened the door a cloud of moist heat oozed out around him like a fog. I slipped in a card. "Elvis Cole. I called you about the Martin murder."
He looked at the card without taking it. "Oh, yes. Floyd Thomas saw that. Floyd Thomas saw exactly what happened." Floyd Thomas. Don't you love it when they speak of themselves in the third person?
"That's great, Mr. Thomas. I'll need to take your statement."
He unlocked four chains and opened the door just wide enough for me to enter. If it was in the high nineties outside, Thomas's apartment must've been a hundred ten with at least three industrial-strength humidifiers pumping out jets of water vapor. Stacks of newspapers and magazines and periodicals sprouted around the room like some out-of-control toadstool jungle, and everything smelled of mildew and body odor. I said, "Hot in here."
"Floyd Thomas chills easily." Sweat leaked down out of his scalp and along the contours of his face and made his thin shirt cling to his skin. Thirty seconds inside his apartment, and I was beginning to sweat, too.
"So what did you see, Mr. Thomas?" I dug out the form and prepared to take notes.
He said, "We were over the Encino Reservoir. They were in a long black convertible. A Mercury, I think."
I looked at him without writing. "Over the Encino Reservoir?"
He nodded. "That's right. I saw them with a woman in their car, and I'm sure it was her. She was struggling." His eyes shifted side to side as he spoke.
I put down the pen. "How were you over the reservoir?"
His eyes narrowed and he looked suspicious. "They'd taken me up in the orb to adjust the chips."
"The orb?" I said. "The chips?"
He pulled back his upper lips so that his gums were exposed. "They force chips into my gums that no one can see. They won't even show up on X-rays." He made a tiny laugh. Hee-hee. Like that.
I said, "You believe you saw Susan Martin in a black Mercury convertible when you were up in the orb."
He nodded again. "There were three men in black and they had the woman. Black suits, black ties, black hats, dark glasses. She had seen the orb and the men in black had to make sure she was silenced. They work for the government, don't you know."
"Of course."
"When will I get the reward?"
"We'll let you know, Mr. Thomas."
I thanked Floyd Thomas for his time, then drove to a nearby 7-Eleven and made five more calls, which resulted in three more interviews. Mr. Walter S. Warren of Van Nuys was a retired general contractor who was convinced that his younger brother, Phil, was behind the kidnapping. He revealed that Phil had once eaten in Teddy Martin's Santa Monica restaurant, had cracked a tooth while enjoying the steak tartare, and had promised to "get that prick" for what had happened to his tooth. Ms. Victoria Bonell, also of Van Nuys, was an extremely thin woman who shared her ranch-style home with seven pug dogs and nine million fleas. Ms. Bonell described an elaborate scenario in which "lipstick lesbians" and "power dykes" were behind Susan Martin's murder, information she had overheard while having her hair colored at a place called Rosa's. I dutifully noted these things, then went to see Mrs. Lewis P. Reese of Sherman Oaks, who offered me tea and finger cakes, and who clearly knew nothing of Teddy Martin, Susan Martin, or the kidnapping. She was elderly and lonely, and I stayed twenty minutes longer than necessary, chatting about her dead husband. The detective does his good turn.
I left Mrs. Reese at twenty minutes after two, bloated on tea cakes, itching from fleas, and smelling of Floyd C. Thomas's pod-person environment. I thought that if I was going to make any more calls maybe they should be to Jonathan. Maybe I should ask him if he really wanted to spend his money having me interview these people?
I stopped at a Ralph's market, bought Tide, Downy Fabric Softener, two Long Island ducklings, enough salad ingredients for a family of nine, and was home by ten minutes after three. The airline told me that Lucy's flight was expected to arrive on time. I put the ducks into a large pot, covered them with water to thaw, and put the pot in the refrigerator. I showered, shaved, put on fresh clothes, and made a last-minute check of the house. Spotless. Pristine. Free from embarrassing dust bunnies.
I took Pike's Jeep, pushed back down the hill and made my way to LAX, arriving at the gate twenty-eight minutes early. I took a seat across from an older woman with brittle white hair and pleasant eyes. I nodded hello and she nodded back. She said, "I'll bet she's very pretty."
"Who?"
"The one you're waiting for. You should see the smile on your face." Know-it-all.
The gate grew crowded and, with the growing crowd, I began to feel anxious and goofy. Then the plane was down and my heart was hammering and it was hard to breath. I said, "Snap out of it, dummy. Try to get a grip."
The older woman laughed, and a man holding a two-year-old moved away.
I saw Lucy first, emerging from the jetway behind three elderly gentlemen, and I wanted to yell, "Hey, Luce!" and jump up and down.
Lucy Chenier is five feet five, with amber green eyes and auburn hair rich with golden highlights from all the time she spends in the sun. She was wearing black shorts and a white long-sleeved shirt with the sleeves rolled and white Reebok tennis shoes, and she was carrying a gray canvas shoulder bag that probably weighed nine thousand pounds and her Gucci briefcase. When she saw me she tried to wave but her hands were full with the bags. Ben yelled, "Hey, there's Elvis!" and then I shouldered past two Marines and Lucy was hugging me and I was hugging her back, and then she stepped away and said, "Oh, your poor eye!"
"You look so good, Luce. You don't know."
We gave each other a long kiss, and then I hugged Ben, too. Ben Chenier had grown maybe four inches in the three months since I'd last seen him. "You're taller."
He beamed. "Four six and a quarter. I'm getting close to five feet."
"Wow."
I took the shoulder bag and we moved with the flow of arrivals down to baggage claim, Lucy and I holding hands and Ben ranging ahead of us, burning off eight-year-old-boy energy. Lucy's hand felt dry and warm and natural in mine, and as we moved along the white-tiled corridors they told me about their flight (uneventful) and how Ben was spending his summer (a week at Camp Avondale with his Cub Scout pack) and about Lucy's business in Long Beach (amicably renegotiating a six-year-old divorce settlement involving complex corporate holdings). As we talked there was a growing feeling that these were not just two people with whom I would spend time, but two people I was allowing into my life. It was a thought that made me smile, and Lucy said, "What?"
"Just thinking how glad I am that you guys are here."
She squeezed my hand.
When their luggage arrived we loaded it into the Jeep and followed LaTijera out of the airport northeast up through the city. It was rush hour, and the going was slow, but going slow didn't seem to matter. Ben said, "We're going to your house?"
"That's right. I live in the hills above West Hollywood."
"Where are we gonna sleep?"
Lucy and I traded a smile. "I've got a guest room. There's a bed for your mom, and a camper's cot for you."
"What's your house like?"
Lucy said, "You'll see when we get there, Ben."
I smiled at him in the rearview. "It's perched on the side of a mountain and it's surrounded by trees. A friend said that it reminds her of a tree house."
Ben said, "Cool."
Lucy raised an eyebrow and looked at me. "What friend?"
I said, "That was years ago."
"Mm-hmm."
We made great time through the Slauson Pass, then climbed north through the Fairfax District past CBS and finally up Laurel Canyon and into the mountains, and then we were home. The summer sun was still high in the west as we turned into the carport and got out, and Lucy said, "Oh, this is just wonderful!" You could smell the eucalyptus and the pine and, high above us, the two red-tailed hawks who lived in the canyon floated on rising thermals. I said, "You guys hungry?"
Ben said, "Yeah!"
Lucy said, "Starving, but I want to take a bath first."
I showed them in through the kitchen and led them past the entry and across the living room and, as we walked, I watched Lucy's eyes flick over the kitchen counters and the refrigerator with its Spider-Man magnets and the bar built into the dining room wall and the stone hearth in the living room and the bookcases and pictures; trying to take in as much of my life in those few seconds as she could. She caught me watching her and gave me a smile of approval. "I like."
I showed them their room and bath, then brought them out onto the deck. Ben said, "Oh, wow," and raced around the handrail, looking down. It's about a twenty-foot drop.
Lucy said, "Elvis, it's beautiful."
"This canyon merges with Nichols Canyon, which opens out into the basin. The little bit of city you see is part of Hollywood. Tomorrow morning we'll take the road below us down to the Budget Rent-a-Car."
She turned back to the house and lowered her voice. "And where does the master sleep?"
I grinned and pulled her close. "The stairs off the living room lead to the master's quarters."
She pushed away, then leaned against the rail and crossed her arms. It was a pretty good pose. "Perhaps a bit later I'll get a chance to inspect the premises."
I shrugged, but even pretending to be disinterested was somehow impossible. My voice came out hoarse and broken. "If you're good, perhaps I'll let you."
She let a smile curl out from under the world's longest eyelashes and lowered her voice still more and let the southern accent come thick. "Oh, Studly, Ah intend to be very, very bad."
The air seemed to spark with a kind of electric heat and then Ben raced back from the side of the house. "Elvis, can I go down the hill?"
"Up to your mom, pal."
Lucy looked over the rail. "Is it safe?"
"Sure. It's a gentle slope. The people who live over there have a couple of boys, and they play all along the ridges."
Lucy didn't look convinced, but you could tell she was going to give in. "Well, okay, but stay close to the house."
Ben ran around the side of the house again, and this time we could hear him crashing down through the dried grass and into the trees. Lucy looked at me and I looked back, but now she was giving me serious. "So. Are you going to tell me about the eye, or do I have to keep wondering?"
"A police officer named Angela Rossi popped me with a sap."
Lucy sighed and shook her head. "Other women date doctors or businessmen. I have to fall for someone who gets into street fights."
"It wasn't much of a fight. She suckered me." I told her about what Green had hired me to do, and how I had done it, and how I had come to get the eye.
Lucy listened, interested more in the parts about Jonathan Green, and frowning when I told her how Rossi had eye-faked me. "She caught you off guard. You underestimated her because she was a woman."
"If I said that it would be taking something away from her. I didn't underestimate her; she was just good enough to sucker me with an eye-fake."
Lucy gave me one of her gentle smiles, then touched the mouse. "You're such a sweetie."
I nodded.
She came close and went up on her toes and kissed it. "I need to make some calls about tomorrow, and I want to take that bath. May I use your phone?"
"Sure." I brushed at her hair, then stroked her upper arms. "You don't have to ask, okay? Whatever you want to do while you're here, just do it. Ben, too."
She went up on her toes and kissed me again. "Keep an eye on Ben?"
"The good eye or the bad eye?"
"Funny."
While Lucy was making her calls I fired the grill, then split the ducks and rubbed them with lemon juice and garlic and pepper. Lucy phoned two attorneys to arrange her next day's meeting, and then she called Jodi Taylor. Jodi was filming her series, _Songbird,_ and had invited Ben to spend the day with her on the set. When Lucy was off the phone and in the bath I checked on Ben and, when the coals were right, put the four duck halves on the grill and covered them. I was back in the kitchen working on tarragon rice and salad when the cat door clacked and the cat walked in. He froze in the center of the kitchen floor and growled.
I said, "Knock that off."
He moved through the kitchen, stopping every couple of steps, his cat nose working and the growl soft in his chest. I said, "We're going to have guests for a few days, and if you bite or scratch either one of them it will go hard for you."
His eyes narrowed and he looked at me. I said, "I mean it."
He sprinted back through his door. There are some things you just can't talk to him about.
I checked on Ben again, then finished with the salad and set the table and put on the new k. d. lang. Lucy reappeared in fresh shorts and wet, slicked-back hair, wrapping her arms around me from behind and sharing her warmth. She said, "Everything is just perfect."
"Not yet," I said. "But soon."
We called in Ben and ate, and little by little we moved through the evening, talking about and planning our coming days, Lucy and I gently touching as we talked, each touch a way of sharing something larger than a simple tactile experience, and after a while even the excitement of the adventure couldn't keep Ben going and Lucy finally whispered, "He's sleeping."
"Need help getting him to bed?"
"No. I'll get him on his feet and he'll walk."
When their door was closed I shut all the lights save one, then went upstairs and took off my clothes. The house was still, and I thought that I could smell her the way, I supposed, the cat had. But maybe that was my imagination.
I lay in the dark for what seemed forever, and then I heard the door below open and the sound of her on the stairs, and I thought how very lucky I was that she had come, and that I was the one whom she had come to see.
#
The sun was bright and hot on the sheets, and I woke smelling coffee and hearing _Bewitched_ on the television, Elizabeth Montgomery saying, "But Darren is a wonderful man, Mother," and Agnes Moorehead saying, "That's the problem, dear. He's a _man,_ and you deserve so much more."
When I went downstairs, Lucy and Ben were up and dressed, Ben on the couch watching television, and Lucy at the dining room table, sipping coffee. She was wearing a pale yellow pants suit and her Gucci briefcase was open, with papers spread on the table beside her. Preparing for business. I said, "Hey. There are people in my house."
Lucy smiled. "We tried to be quiet."
"You were. I didn't hear a thing." She held out her hand, fingers spread, and I laced my fingers through hers.
She said, "Mm."
I wiggled my eyebrows, then made a shifty look back toward the stairs. "Mm-mm."
Lucy took back her hand. "No time, my dear. Jodi's going to pick up Ben on her way in to the studio, then you have to take me to the Budget office. She should be here soon."
"Great." We were grinning at each other with great loopy grins that probably looked silly. "Did you sleep all right?"
Lucy managed a straight face. "Very well, thank you. And yourself?"
I pretended to stifle a yawn. "A little restless. I feel drained this morning."
Lucy raised her eyebrows. "Imagine that. Perhaps you need more rest."
Ben looked at us from the couch, confused. "You don't look tired to me."
Lucy and I grinned, and Ben looked even more confused. "What did I say?"
Lucy said, "I got directions to my meeting, so all we need to do is pick up the car. You shower and dress, and I'll make breakfast. Deal?"
"Deal."
I did and she did, and we were finishing coffee and toasted banana bread and scrambled eggs when Jodi Taylor's black-on-black Beemer tooled up and stopped across the drive. I pushed open the kitchen door and gave her a kiss as she entered. "What, no limo for the star?"
Jodi Taylor tugged at my shirt and said, "I'll buy a stretch if you'll come for a ride, handsome." Then she winked at Lucy and said, "Oops, sorry. I see he's already taken."
I gave her the eyebrows. "Taken, yes, but perhaps available for rent."
Lucy said, "In that case she should buy a hearse. Better to lay out the body."
Jodi laughed. "Grr-owl. These southern belles are _very_ territorial."
"Possessive," Lucy said. "The word is possessive."
Lucy and Jodi hugged, and Ben ran in from the living room. Like Lucy, Jodi Taylor was from Louisiana, though, unlike Lucy, you couldn't hear it in her voice. She was maybe an inch taller than Lucy, with hazel eyes and dusky red hair and a kind of natural beauty that made her accessible and real to thirty million people every week. Supermarket beauty, they called it. The kind and quality of beauty that let you believe that you might bump into her in the market, buying Pampers or Diet Coke. _Songbird_ had been renewed for a second full season, and Jodi Taylor had just begun production on the new episodes. She was happy and confident in returning to work, and was at ease with herself in a way that she hadn't been three months ago. Lucy said, "Jodi, you look wonderful."
Jodi smiled shyly. "Thanks to you two."
I had seen Jodi from time to time in the three months since I'd helped her, but Lucy hadn't, and they chatted and worked out the details of Ben's day while I cleared the table, loaded the dishwasher, then went upstairs to gather together my file of tipsters. I considered bringing along a can of bug repellent for the day's assignment, but decided against it. Too hard to force the can into my holster.
When I went back downstairs, Jodi and Lucy were standing together, grinning. Jodi said, "You're working for Jonathan Green? My, my." Impressed.
I spread my hands. "He's just another client, ladies." Mr. Modest.
Lucy put her hands on her hips. "No, he's not. He's Jonathan Green."
I spread my hands again. They're carrying on like this, and I'm battling fleas and talking to people who think they've got chips in their gums.
Lucy made her voice low and breathy. "He positively _dominates_ a court room. And his presence is so _commanding_."
Jodi Taylor slinked over to me and toyed with my collar. "Could you arrange a personal introduction?"
Lucy said, "Would he autograph my law school diploma? Would he do that for lil' ol' me?"
Jodi purred, "I've got something else he could autograph."
Girl humor.
Jodi and Ben finally left for the studio, and then I brought Lucy down to the Budget office, working our way along the back canyon road in silence. Lucy was staring out of the car, and I thought that she might be watching the alien scenery and the strange mountain houses, but she wasn't. She said, "What I said about possessive. I was joking." Her voice was soft, and when she said it she didn't look at me.
"Sure."
Her hands were in her lap and her briefcase was on the floor beneath her legs. She said, "Elvis?"
"Hm?"
Another pause. Longer. "Do you see anyone else?"
I looked at her, but she still wasn't looking at me. I went back to the road.
Lucy said, "I mean, it's none of my business. We've never talked about other people."
I nodded. I looked at her again, but she still was focused outside. "I went out twice in the month after I came back from Louisiana. Once with a woman I'd seen several times before, and once with a waitress I met in the Valley, and both times went poorly."
"Oh." She didn't sound disappointed.
"I was with them, but I was thinking of you. Then you and I started talking about going to Cancun. I haven't been out with anyone since then. I don't want to go out with anyone else." I was looking more at her than the road, which isn't smart in the hills.
Lucy Chenier looked at me, then nodded once and turned back to the window.
I said, "Have you been seeing anyone?"
She shook her head. "No."
I thought about it and what it meant. "Good."
Without looking at me, she put out her hand. I took it. We drove like that the rest of the way to the Budget office, where I dropped her off and began another exciting day in the employ of the Big Green Defense Machine.
#
After I dropped Lucy off I stopped at a diner on Hollywood Boulevard and made more calls. Of the remaining names on my list, two were in El Monte, one in San Marino, and one was in Pasadena, all of which were on the eastern rim of the Los Angeles sprawl.
I called a Mr. James Lester first. A woman answered, sounding young and whiny, and told me that he was sleeping. She said that he didn't have to go in until noon, so he always slept late. I told her that I would be in their area later, and how about I call back then. She said, "Mister, I don't give a rat's ass what you do." Nothing like starting off your work day with a bang.
No one was home on my next call, and then I phoned Ms. Mary Mason of San Marino. A woman with a low, breathy voice answered on the third ring. She identified herself as Mistress Maggie Mason and told me that Mary was her sister. When I told her why I was calling she said that Mary would be available shortly and gave me directions to their home. One for three.
Mary Mason lived on Winston Drive in a stately well-kept home set back from the street. It was an older place, built of heavy stone and stucco. I rang the bell three times, knocked twice, and was just getting ready to leave when the door opened and a tall, statuesque woman in a black leather teddy, net stockings, and six-inch platform shoes stepped out. A twined cobra was tattooed on her right thigh. She said, "May I help you?" She had long black hair pulled back tight against her head.
"Are you Mary Mason?"
She smiled nicely. It was a friendly smile, relaxed and personable. "No, I'm her sister, Maggie. I spoke with you earlier."
"Ah."
"Come in and I'll get Mary."
The living room was tastefully decorated with minimalist Italian furniture, a spherical saltwater aquarium, and custom bookshelves lining three walls. The bookshelves were African teak and must've cost a fortune. Maggie Mason said, "Wait right here and I'll get her." She was bright and cheery, not unlike a Girl Scout troop leader from Nebraska.
I waited. The house was so quiet that I could hear neither street noise nor passing cars nor the sound of Maggie Mason getting her sister. I looked at the books. Short fiction by Raymond Carver and Joan Didion. Asian philosophy by T'sun T'su and Koji Toyoda. Crime novels by James Ellroy and Jim Thompson. Science fiction by Olaf Stapledon and Jack Finney. Eclectic and impressive. I had finished reading the titles on one wall and was starting on a second when Mary and Maggie Mason returned. Twins. Both were tall, but where Maggie was dressed in the teddy and the fishnets, Mary wore a smartly tailored business suit and conservative low-heeled pumps. Her face was very white and her lips were liquid red and her black hair was cropped short and oiled to severe perfection. I said, "Mary Mason?"
Mary Mason sat next to the aquarium, crossed one gleaming leg over the other, and said, "Four payments. I want the first payment now, another when there's an arrest, the third on arraignment, and the final on the first day of the trial. That's the only way I'll do business."
I said, "Business?"
Her sister smiled politely. "If you'll excuse me, I have something to take care of." She left without waiting for either of us to respond.
Mary Mason leaned toward me. "I hear things." She arched her eyebrows, which, like the rest of her, were perfect. "I know the identity of James X. I can help Teddy Martin."
I gave her the same news that I'd given Floyd Thomas, that there would be no money until a conviction.
Mary Mason said, "Bullshit." When she said it, a muffled _crack_ came from the back of the house.
I looked past her. "What was that?"
Mary Mason leaned closer and put her hand on my knee. "Pay something as a sign of good faith. Five thousand dollars, and I'll give you a physical description. How about that?" There was another dull _crack_ and then a whimpering sound.
I looked past her again. "I can't do that, Ms. Mason."
She squeezed the knee. "Three thousand, then. Teddy Martin can afford it." She ran her tongue along glistening lips, and then a man in the rear of the house moaned something about being called a dog. The voice was muffled and far away, and I thought that maybe I'd heard him wrong. Then the man howled.
"Thanks for your time, Ms. Mason." I walked out, wondering if it were too late to change professions.
It was twenty-eight minutes after ten when I left the Mason twins and dropped south out of San Marino to San Gabriel. I pulled into a strip mall, made two more calls, and on each of the calls got an answering machine. That meant I was back to James Lester, who may or may not be awake. I called his number again anyway, and this time a man answered. I said, "Mr. Lester?"
A woman was shouting in the background. Lester shouted back at her, "Just shut the fuck up, goddammit," and then he came on the line. "Yeah?"
"Mr. James Lester?"
"Who wants to know?" One of those.
I told him who I was and what I wanted.
"You're the guy from the lawyer, right?"
"That's right."
"Okay, sure. C'mon over."
I went over.
El Monte, California, is a mostly industrial area north of the Puente Hills and south of Santa Anita, with small working-class neighborhoods to the south and west. James and Jonna Lester lived in a poorly kept bungalow on a narrow street just west of the San Gabriel River in an area of postwar low-income housing. The lawn was patchy and yellow from lack of water, as if the Lesters had given up against the desert and the desert was reclaiming their yard. Everything looked dusty and old, as if there were no future here, only a past.
I left my car on the street, walked up across the dead yard, and a guy I took to be James Lester opened the door. He was average-sized in dark gray cotton work pants, dirty white socks, and a dingy undershirt. His hair was cut short on the sides and on top, but had been left long and shaggy in back, and he looked at me with a squint. He was thin, with knobby, grease-embedded hands and pale skin sporting Bic-pen tattoos on his arms and shoulders and chest. Work farm stuff. I made him for thirty, but he could've been younger. He said, "You're the guy who called. You're from the lawyer, right?" A quarter to eleven in the morning and he smelled of beer.
"That's right."
I followed him into a poorly furnished living room that wasn't in any better shape than the yard. Stacks of magazines and newspapers and comic books were piled around on the furniture, and no one had dusted since 1942. A tattered poster of the Silver Surfer was thumbtacked to the wall, four darts growing out of the Silver Surfer's chest. Lester dropped into a battered, over-stuffed chair and pulled on a work boot. An open can of Hamm's was on the floor by the boots. "I gotta get ready for work. You wanna brewscalero?"
"Pass."
"Your loss, dude. I can't get going without it."
A barefoot woman with a swollen, discolored lip came out of the kitchen carrying a sandwich in a paper towel. She was wearing baggy shorts and a loose top and her skin was very white, as if she didn't get out in the sun much. She dropped the sandwich on a little table next to the chair as if she didn't give a damn whether he ate it or not. She looked sixteen, but she was probably older.
I smiled and said, "I believe we spoke earlier."
She said, "Well, whoop-de-doo."
James Lester pulled hard at his bootlaces. "I need another brewscalero, Jonna. Go get it."
Jonna Lester shot a hard look at her husband's back, then stomped back into the kitchen. Pouty.
James said, "She don't do nothing but run around with her friends all day while I'm bustin' my ass. That's why it's such a sty in here. That's why it's a goddamned shithole." They didn't have air conditioning. A couple of ancient electric fans blew hot air around the room, one of the fans making a slow, monotonous _chinging_ sound. Jonna Lester came back with a fresh Hamm's, put it down next to the sandwich, then stomped out again. I hadn't been in their house for thirty seconds and already my neck was starting to ache.
I said, "I'm here to follow up the call you made about Susan Martin's kidnapping and murder."
Lester finished tying the first boot, then started on the second. "Sure. That guy I spoke to on the phone, he said someone would come talk to me about it. That's you, I guess."
"I guess." Mr. Lucky.
He looked over and grinned when he saw my eye. "Hey, you and Jonna kinda match, doncha?" He laughed after he said it, huh-yuk, huh-yuk, huh-yuk. Like Jughead.
I stared at him.
James Lester killed what was left of the first Hamm's, then popped the tab on the second. "I think I met the guys who did it."
"Okay."
He took another pull on the Hamm's, then had some of the sandwich. When he bit into the sandwich he jumped up and opened the sandwich as if he'd just bitten into a turd. "Goddammit, Jonna, what in hell is this?"
"That's your potted meat!" Yelling from the kitchen.
"Where's the fuckin' mayonnaise?"
"We're out. I gotta get some."
"Where's the _little pickles_?" Now he was whining worse than her.
"I'm gonna go get some, all right?" Screaming, now. _"Do you think I'm your fuckin' slave?"_
His face went sullen and his breathing grew loud. He had more of the Hamm's. He had more of the Hamm's again. My neck was hurting so bad I thought it would go into spasm.
"Tell me what you know, James."
He stayed with the loud breathing a little longer, then closed the sandwich and took another bite. You'd think it was killing him, having to eat his sandwich without the mayonnaise and the little pickles.
I said, "James."
He went on with his mouth full. "A week before it's on the news about her gettin' killed I stop in this place for a couple of brewscaleros. There's these two guys, one of the guys, he was wearing a Shell station shirt had the name 'Steve' sewn over the pocket."
"Okay." I wrote _Shell station_ on my notepad. I wrote _Steve_.
"We were talkin' about how shitty it was, havin' to work for a livin', and this guy, he gives me the big wink and says he's got her whipped. I'm all, whaddaya mean you got'r whipped? He goes, hey, a guy with the 'nads could snatch one of these rich Beverly Hills bitches and score enough fast cash to retire in style."
I said, "Steve said that?"
"Unh-hunh." He stuffed the rest of the sandwich in his mouth and washed it down. "I tell'm that sounds like a fast track to the gas chamber to me, but he goes, all you need is a layout of the house and a slick way in and out, stuff like that." He swallowed hard and let out a gassy belch.
"The other guy say anything?"
"Nope. Just sat there drinkin'."
"What'd they look like?"
"Steve was kinda tall and skinny, with light hair. I'm not sure about the other guy. Shorter. Darker."
A phone rang in the kitchen and we could hear Jonna Lester answer. James's face clouded and he yelled, "That better not be one'a your cunt friends!"
She yelled back, "Fuck you!"
I said, "James."
He turned the cloud my way.
" 'Cunt' is an ugly word."
He squinted at me as if he wasn't sure what I'd said, and then he shook his head. "All she does is yack with her friends. All she does is run around the mall while I'm bustin' my ass." Like that should explain it.
I said, "Steve and the dark guy say anything else?"
He sucked at his teeth, getting rid of the last bits of the sandwich. "I hadda pee so I went to the head. When I come back they was gone."
I stared at him, thinking about it. Seven interviews so far, and his was the only one that seemed to be worth checking out. It would probably add up to nothing, but you never know until you know. "You remember the bar?"
"Sure. It was a place called the Hangar over on Mission Boulevard. I go there sometimes."
I wrote it down. _The Hangar_.
"Last thing the guy says before I go to the head, he says he knows just who to grab, too. He says she's a one-way ticket to Easy Street."
"Steve said that?"
"Yeah. Steve."
"He say a name?"
"Unh-unh."
Jonna Lester reappeared wearing strap sandals and carrying a small purse. She'd made up her face, but the lip still looked puffy. He said, "Where the fuck do you think you're going?"
She pouted the lips at him, giving him attitude. "I gotta go to the store. I got things to buy."
"You think you're gonna run around with your cunt friends while I'm bustin' my ass? You think you're gonna spend my dough in some fuckin' mall?"
_"We're outta mayonnaise. We're outta those little pickles."_
He jumped up and grabbed her right arm. "You're gonna stay here and clean this fuckin' rathole, that's what you're gonna do!"
I stood.
She tried to twist away from him, screaming, "You piece of shit! I'm not your fuckin' slave!" She pounded at him with her left fist, pretty good shots that nailed him on the head and face and chest until he was able to grab her left arm, too.
"James." The ache in my neck had moved up to my scalp. Never a good sign.
She said, "You're hurtin' me, you asshole!"
"James. Leave go of her."
James Lester said, "Fuck you. This is my house. This is my wife. She's gonna do what I say or I'll give'r a fat lip!"
I held up my right index finger. "Watch the finger, James. I want to show you something."
His eyes went to the finger, like maybe it was a trick, only he couldn't figure out what the trick might be.
"Are you watching my finger?"
"Suck my ass." She was watching my right finger, too.
I hit him flush on the nose with a left.
He yelled, "Ow!" and grabbed at his face with both hands. He stumbled back and tripped over the little side table. Jonna Lester leaned over him, wiggled her butt, and yelled, "Ha-ha, _asshole_!" Some wife.
James Lester was on his back, eyes watering, blinking at me. He said, "You piece of shit. You wait'll I get up!"
I put my notes in the manila envelope, then went to the door. Lucy was probably in the midst of her negotiation right now. Ben was probably watching Jodi Taylor shoot a scene right now. The world was turning on its axis right now.
I said, "Thanks for the statement, James. If anything comes of it we'll be in touch about the reward."
"You better not jew me out of that reward! I'm gonna call the cops, you hear? I'm gonna have you _arrested_!"
I left them to their lives and walked out into the sun. You want to do the right thing, but sometimes there is no right thing to be done.
Another day, another moron. And to think, some people have to work for a living.
#
The Hangar was a small, bright hole-in-the-fence-type bar wedged between a place that sold balsa-wood rocket kits and another place that repaired appliances. They were doing a pretty good lunch business when I got there, selling chili tacos and grilled sausages to people swilling down schooners of beer. Both of the bartenders were women in their fifties, and neither of them knew a blond guy named Steve who worked for Shell. I didn't expect that they would, but you never know. The older of the two women called me 'sweetie.' The younger of the two didn't like it very much. Jealous.
I bought a grilled sausage with kraut, a schooner of Miller, and asked if they'd mind letting me use their phone book. The older one didn't, but the younger one warned me not to walk out with it. I assured them that I wouldn't. The younger one told me to be careful not to spill anything on it. The older one asked the younger one why she always had to make such a big thing, and the younger one said what if I ruined it? I assured them that I'd buy them a new phone book if I ruined the loaner. The older one said, "Oh, don't you give it another thought, sweetie," and the younger one went down to the far end of the bar and sulked.
Half the schooner later I had addresses for the nine Shell service stations located in the El Monte/Baldwin Park/West Covina area. I finished the sausage, thanked the older one for her help, and made the round of the Shell stations. At each stop I spoke to the manager or assistant manager, identified myself, and asked if a tall blond guy named Steve had worked there anytime in the past six months. At the first four stations I visited, the answer was no, but at the fifth station the manager said, "You mean Pritzik?"
"Who's Pritzik?"
"We had a fellow named Steve Pritzik." The manager was a Persian gentleman named Mr. Pavlavi. He was short and round and stood in the shade of his maintenance center with his arms crossed. His maintenance center, like the rest of his service station, was polished and gleaming.
I said, "Was he tall?"
"Oh, yes. Very tall."
"Was he blond?"
"Oh, yes. Very blond."
I said, "Mr. Pavlavi, is he employed here now?" Just because a tall blond guy named Steve worked here didn't mean it was the _same_ tall blond Steve. Maybe it was just a coincidence.
Pavlavi frowned. "Not in a very long while. He quit, you know. One day here, the next day not, never to return." He sighed as if such things are the stuff of life, to be expected and therefore no great cause for anxiety or resentment.
"About how long ago was that?"
"Well," he said. "Let us see."
He led me into the air-conditioned office and took a ledger from his desk. The ledger was filled with page after page of handwriting that, like the service station, was immaculate. "Pritzik was last here exactly one hundred two days ago."
"Hm." Steve Pritzik had last been in four days before Susan Martin's murder.
"I owe him forty-eight dollars and sixteen cents, but he has not been in to collect. I will keep it for exactly one year, then give it to charity."
"Mr. Pavlavi, would you have an address on Pritzik?" He did, and he gave it to me.
Steve Pritzik lived in one of a cluster of six small duplex cottages in an older neighborhood at the base of the Puente Hills, not far from the Pomona Freeway. The duplexes were single-story stucco and clapboard buildings stepping up the side of the hill and overgrown with original planting fruit trees and ivy and climbing roses.
I parked at the curb, then made my way up broken cement steps, looking for Pritzik's address. The steps were narrow, and the heavy growth of ivy and roses made them feel still more narrow. Pritzik's apartment was the western half of the third duplex up from the street. Each side of the cottage had its own little porch, separated by a couple of ancient orange trees and a trellis of roses. The eastern porch was neat and clean and decorated by a small cactus garden. Pritzik's porch was dirty and unadorned, and his mailbox was heavy with letters and flyers. I rang the bell and could hear it inside, but no one answered. I listened harder. Nothing. I went to the mailbox and fingered through gas and phone and electric bills. They weren't addressed to Steve Pritzik; they were addressed to a Mr. Elton Richards. Hmm. I walked around the orange trees and up onto the adjoining porch and rang the bell. You could hear music inside. Alanis Morissette.
A woman in her late twenties opened the door. "Yes?" She had long dark hair and great floppy bangs and she was wearing cutoff jeans under an oversized man's T-shirt. The T-shirt was blotched with small smears of color. So were her hands.
I gave her my card and introduced myself. "I'm trying to find a guy named Steve Pritzik. I think he lives or used to live next door."
She read the card and grinned. "Are you really a private eye?"
"Pretty amazing, huh?"
She grinned wider and nodded. "Cool."
"You know Pritzik?"
She offered the card back, but I raised a hand, telling her to keep it. "I don't think so. Elton lives next door."
"Is Elton tall and blond?"
"Oh, no. He's short and kinda dark." Ah. She rolled her eyes. "He's such a creep. He's always hitting on me, so I try to avoid him."
"I was just over there, and it looks like Elton hasn't been around." I told her about the mail.
She pushed her hands in her pockets. "You know, now that I think about it, I haven't seen him in a while. I haven't heard his TV or anything."
"You think he might've moved?"
"I don't know."
"Can you give me a guess how long he's been gone?"
She scrunched her face, thinking. "Couple of months, maybe."
"Between three and four months?"
She waffled her hand. "He's just such a creep I try to duck him. Sorry."
I said, "You ever see a tall blond guy hanging around with him?"
She frowned.
"Maybe four months ago."
She was swaying with Alanis, then she kind of cocked her head. "You know, I think maybe there was a guy like that. Elton had such scuzzy friends." She nodded, then, starting to see it. "Yeah. There was this blond guy." She nodded harder, the image pulling into focus. "Oh, yuck, what an asshole. He sees me on the street and follows me up the walk one day. He asks me if I want to go inside and fuck, just like that. Oh, yuck. I think he worked at a gas station or something."
I nodded.
"All of Elton's friends were like that. Real lowlifes." She suddenly put out her hand. "I'm Tyler, by the way."
"Hi, Tyler." We shook, and I gave her the big smile. "Can I ask you something?"
"Sure." She smiled back, anxious to hear what I was going to ask. Alanis was really tearing it up inside.
"I'm thinking about popping Elton's door and sneaking in to look around. You wouldn't call the police if I did that, would you?"
Her smile grew wider as I said it. "No way! Could I come, too?"
I shook my head. "Then if we're caught, we're both in trouble, you see?"
She looked disappointed. Behind her, Alanis stopped singing and Tyler pulled a hand out of her pocket long enough to brush at the bangs. They were pretty incredible. "You really know how to pick locks and stuff?"
"I'm a full-service professional, Tyler."
She stared at me for a few seconds and then she crossed her arms. She looked out from under her bangs at me. "And just what kind of service do you provide?"
"I've got a girlfriend. Sorry."
Tyler stared at me from under the bangs for another couple of seconds, then uncrossed her arms and looked at my card again. "Yeah, well. If I ever need anything detected, maybe I'll call."
"How about the cops?"
Tyler made a zipping move across her lips.
I gave her the big smile again, then went next door, slipped the lock, and let myself into Elton Richards' half of the house. It was dim from the drawn shades, and I flipped the light switch but the lights stayed dark. I guess the power company had killed the juice. I said, "Mr. Richards?"
No answer. Next door, I could hear Alanis start again, faint and far away.
The house smelled musty. A ratty couch was against the wall under a Green Day poster, fronted by a coffee table made of a couple of 2 by 10 planks lying on cinderblocks and cornered by someone's second-hand lawn chair. A black streamline phone waited on the planks. A pretty good Hitachi electronics stack was against the opposite wall, and a beat-up Zenith television with a coat hanger antenna was on the floor, and everything was covered with a light patina of undisturbed dust.
I crossed into the kitchen and turned on the tap. No water. I went back to the living room, used my handkerchief, and lifted the phone. No tone. I guess Elton Richards had ignored his bills long enough for the power and water and phone companies to turn everything off. Say, about four months.
I stood in the living room by the phone and thought about it. James Lester had met a short dark man and a tall blond man named Steve in a bar about a week before Susan Martin's kidnapping and murder. Steve speaks of snatching a rich woman as a means of attaining the better things in life, and maybe the two are connected, but maybe not. Four months after the fact, I identify a possible Steve and trace him to this address which, in fact, is apparently owned by a shorter, darker man named Elton Richards. Maybe they are the same two men, but maybe not. Maybe tall blond guys named Steve just naturally have short dark friends.
Two small bedrooms bracketed the bath. I searched each thoroughly, looking for receipts or ticket stubs or anything else that might provide a clue as to when and where Elton Richards and Steve Pritzik went. There was nothing. I went into the bathroom and checked behind and beneath the toilet and in the water tank. I pulled the medicine cabinet out of the wall. I checked in the little wooden cabinet beneath the lavatory. _Nada_. I went back into the living room and pulled the cushions off the couch and found a single 9 by 12 manila envelope. It was the kind of envelope you get in the mail from those sweepstakes companies declaring that you've just won ten million dollars, and it was addressed to Mr. Elton Richards. The end of the envelope had been scissored open, then retaped. I pushed my car keys under the tape, opened the envelope, and looked inside. Then I sat down.
I took deep rhythmic breaths, flooding my blood with oxygen and forcing myself to calm. Pranayamic breathing, they call it.
I looked in the envelope again, then tilted it so that the contents spilled out onto the couch. Inside there were seven separate photographs of Susan Martin and Teddy Martin, and two hand-drawn maps. One map was the floorplan of a very large house. The other was a street map showing the layout of someone's neighborhood and a house on Benedict Canyon Road. It was Teddy Martin's neighborhood, and it was Teddy Martin's house.
#
I went to my car for the new Canon Auto Focus I keep in the glove box. I made sure I had film and that the flash worked, and then I took a pair of disposable plastic gloves and went back into the house. I put on the gloves, then photographed everything as I had found it, making sure I had clear shots of the handdrawn maps as well as the photos. When I was done, I left everything lying on the couch, then went next door and asked Tyler if I could use her phone.
I called Truly first, who listened quietly until I was finished, then said, "I'll notify Jonathan and we'll get there as quickly as we can. Don't let anyone else in the residence." He cupped the phone, and I could hear muffled voices. Then he came back. "We'll notify the police, too. Cooperate with them when they arrive, but keep an eye on them. Watch that they don't destroy the evidence."
"Truly, they won't do anything like that."
He said, "Ha."
When I hung up, Tyler was leaning against the back of her couch, arms crossed, a long paintbrush in one hand. Her home smelled of fresh jasmine tea and acrylic paint, and was decorated with oversized sunflower sculptures that she'd made from cardboard and wire. "You really think that this creep next door had something to do with Susan Martin's murder?"
"Maybe."
"I thought her husband did it. That restaurant guy."
"You never know."
"They said on TV that he did."
"That's TV."
She shook her head. "L.A. is so perverted."
The first black and white arrived eighteen minutes later. The senior officer was a guy named Hernandez, and his partner was a younger African-American woman named Flutey. I went out to meet them carrying a glass of Tyler's jasmine iced tea. Hernandez said, "You Cole?"
"Yep." I told him what we had.
He nodded. "Okay. Flutey, get the tape from the car and let's seal it, okay? I'll check inside and around back."
Flutey went for the tape, and Hernandez looked at me. "Where you gonna be?"
"I'll hang around out here unless you want company."
Tyler called from the porch. "Would you and the other officer like some iced tea?"
Hernandez smiled at her. "That'd be real nice, miss. Thank you." Tyler ducked back inside. Hernandez stared after her. Portrait of the crime scene as a social occasion.
Two detectives from the L.A. County Sheriff's Office arrived, followed almost immediately by a criminalist van. The lead detective was a heavyset guy with thinning hair named Don Phillips. A DA's car came next, off-loading a thin woman named Sherman, a bald guy named Stu Miller, and an intense African-American guy in dark glasses named Warren Bidwell. Sherman was the Assistant Deputy DA charged with prosecuting the Teddy Martin case. Miller and Bidwell worked for her.
All three of them slipped under the tape and went into Richards's duplex, then Miller and Sherman slipped out again and came over to me. Tyler gave them a bright smile and pushed aside her bangs. "Would either of you like iced tea?"
Sherman said, "No." She squinted at me. "I'm Anna Sherman from the district attorney's office and this is Stu Miller. Would you come inside, please?"
"Sure."
Tyler said, "Can I come, too?"
Anna Sherman said, "No."
I shrugged at Tyler and followed them.
Inside, Sherman said, "Okay. Walk me through what happened."
I told them about getting the address from Pavlavi and finding the duplex deserted and popping the lock to let myself in. I told them about finding the envelope under the couch cushions and opening the envelope. Sherman stopped me. "You touched the envelope?"
"That's right."
The criminalist said, "What about the contents?"
I shook my head. "Edges only. When I saw what I had I slid the stuff out onto the couch. I used my knuckles to separate the pages first time through. When I photographed the material I was wearing gloves."
Bidwell was glowering so hard his body was making little jerks and lurches and I wondered if he knew he was doing it. He said, "I want those photographs."
I shook my head. "I don't think so."
Bidwell lurched harder. "You don't? Are you a sworn officer? You have a search warrant or any authority to break into a private residence?"
I looked at Sherman. "You want me to continue or should I call my lawyer?"
Sherman closed her eyes and shook her head. "Not now, Warren."
The yard and the walk outside grew crowded with cops and media people and rubberneckers from the neighborhood drawn by gathering news vans. Between questions I watched the on-air television talent fan out among the cops. A woman I'd seen a thousand times on the local NBC affiliate was talking with her camera operator when the camera operator saw me standing in the window and pointed me out. The reporter said something and the operator trained his camera on me. The reporter ducked past Flutey and hurried over to the window. She was all frosted hair and intelligent eyes. "Are you the detective who found the kidnappers?"
I gave her Bill Dana. "My name José Jimenez."
She waved her camera operator closer. "Look, we know that two men named Elton Richards and Steve Pritzik lived here and we'd like an on-camera statement." The camera operator held the camera over his head, trying to scan the room.
Don Phillips saw the camera coming through the window and said, "Jesus Christ!" He pushed in front of me, then leaned out the window and yelled at a uniformed sergeant. "Clear the area, for Christ's sake. Seal it off from the street back." The sergeant hustled away, and Phillips looked at me. "Are you trying to be cute?"
I spread my hands. "Trying has nothing to do with it."
The uniforms were pushing the press and gawkers along the walk when a ripple spread up from the street and across the crowd as if someone had amped a jolt of electricity through the air. Heads turned and voices rose, and the TV people surged toward the street. Phillips said, "Now what?"
Jonathan Green and Elliot Truly and the videographer from _Inside News_ were working their way through the crowd. The videographer's sound tech was trying her best to move people out of their way, but it was hard going until Hernandez and Flutey and a couple of other uniforms lent a hand. Anna Sherman came to the window, then gathered Bidwell and Miller for a whispered conference. When Green and the others pushed their way through the front door past the uniformed sergeant, Phillips said, "Where in hell do you think you're going?"
Anna Sherman came over and smiled tightly. "Let them pass, detective." She offered her hand. "Hello, Mr. Green."
"Ms. Sherman." Jonathan Green smiled at me. "Congratulations, son. I think you've made my day." The videographer bumped into Phillips as he tried to get the shot, and Phillips shoved him away. Hard. The videographer said, "Hey."
Anna Sherman said, "Detective Phillips, this is Jonathan Green. Mr. Green represents Theodore Martin."
Phillips said, "How about that."
Jonathan and Truly went to the couch and leaned over the papers without touching them. Phillips said, "Don't touch anything. We haven't printed them yet."
Truly was grinning wildly and shaking his head. "This is wonderful. Would you look at this? This is absolutely fabulous." He grinned at me and then he grinned at Sherman, only Sherman didn't return it.
Green said, "Mr. Cole, are these the same documents you found when you entered this residence?" He said it loudly so that everyone in the room could hear.
"Yes."
Green motioned to the videographer. "Would you get a close up of this, please?"
The videographer almost tripped over himself getting there. Bidwell said, "Who _is_ this dork?"
Truly said, "They're from _Inside News_. They're doing a documentary on Jonathan."
Bidwell said, "Oh, for God's sake," and shook his head.
As the videographer panned the evidence, Jonathan looked back at me. "There are no new documents, and none of the documents you found are now missing?"
"Of course not."
The videographer panned up to Jonathan, and Jonathan said, "Mr. Cole photographed the documents found in this envelope before the police were summoned. That photographic record constitutes an accurate accounting of exactly what was here before the police took possession of the evidence. We intend to compare those photographs with these to see if the evidence has been tampered with."
Phillips went red. "Hey, what the fuck?"
Anna Sherman told him to shut up. She said that if Phillips couldn't control himself he should go outside.
Phillips said, "I know what he's saying and I don't like it. I run a clean house, goddammit." He was purple.
Sherman said something to Bidwell and Bidwell led Phillips out.
They had me go through it again, Jonathan Green and Elliot Truly asking questions and the videographer and the sound tech recording me. Anna Sherman listened with her arms crossed, occasionally digging her heel into the floor and rocking her foot, and, like Green and Truly, occasionally asking more questions. Bidwell and Phillips came back, but this time Phillips kept his mouth shut and glowered at us from the corner. When I was done, Jonathan Green looked at Sherman again and said, "We'll want these documents preserved, and we'll want to examine them as soon as practicable. We'll want the results of your fingerprint analysis, and then, of course, we'll want to do our own."
Anna Sherman's jaw was tight. "Of course."
"Do you have anything more for Mr. Cole?"
The criminalist said, "I asked Cole for permission to take his prints. He said okay."
Green nodded. "Please do it now in our presence." The criminalist broke out his fingerprint kit and had me sit on one of the dinette chairs. He took my prints quickly and professionally, then gave me a Handiwipe to clean off the ink. The videographer recorded every moment. I said, "Don't you ever run out of tape?"
The sound tech laughed.
Green walked back to the couch, again examined the papers without touching them, then looked back at Sherman. "You realize what we have here, don't you, Anna?" The patient father.
Anna Sherman did not respond. The pouty daughter.
Jonathan Green smiled. "If you don't, Ms. Sherman, I'm sure the district attorney will. Tell him I'll expect his call soon, if you would."
Her jaw flexed.
Green said, "I think we can go, Elliot. Mr. Cole's had a long and fruitful day. I expect he wants to go home."
Phillips coughed loudly from his corner of the room, but the cough soundly suspiciously like, "Fuck you."
I followed them out. The street at the end of the walk was jammed with media people and broadcast vans and uniformed cops trying to clear a path. Hernandez and Flutey flanked Jonathan and we crossed under the tape, and the media people surged around us, pushing their cameras and microphones at Jonathan and shouting their questions. There were so many broadcast vans that it looked as if we were in a forest of transmitters, each spindly stack pointing at the same invisible satellite 22,500 miles above in geosynchronous orbit, like so many coyotes crying at the moon. I said, "This is nuts."
Truly yelled in my ear so that I could hear him. "It hasn't even begun."
The woman with the frosted hair jammed her microphone past Hernandez and shouted, "Jonathan, can you tell us what was found?"
"I'm sorry. That information should come from the district attorney's office."
She yelled, "Is it true that a plan of Teddy Martin's house was found?"
"I'm sorry." We were working our way toward Jonathan Green's Rolls-Royce.
A short man who himself had been an attorney before becoming a broadcast journalist shouted, "Jonathan? Is it true that evidence found in the house exculpates Theodore Martin in the murder of his wife?"
Jonathan smiled benignly. "I've seen the evidence that Mr. Cole found, and I'll be in consultation with the district attorney's office sometime in the next few days. Now, if you'll excuse me."
More questions exploded at us from a dozen directions, and they were all about Mr. Cole.
I didn't think Jonathan was going to answer, but he stopped and put his hand on my shoulder and said, "This is Mr. Elvis Cole of the Elvis Cole Detective Agency, and I believe that his discovery is going to be the breakthrough that we need. I can't tell you how proud I am of this young man, and how impressed I am with the caliber of his work."
I said, "Gee."
The microphones shifted toward me as one and the questions came so fast and loud that the words blended into white noise. I was pretty sure that no one heard me say, "Gee." I may have said it twice.
Green said, "All we can say at this time is that we received a tip through our hotline, and Mr. Cole followed it to this conclusion." He squeezed my shoulder again as if I were his son and I'd just made Eagle Scout. "What we have here is the result of good, solid detective work, and I suspect that when all is said and done Mr. Cole will be the hero of this little drama."
Truly added, "And Teddy Martin the victim."
Jonathan slid into his Rolls-Royce, and then Truly and the two uniforms walked me to my car. The press stayed with us, jostling and shoving and keeping up with the questions. We had to push a fat guy and two women away from my car to get the door open. Flutey lost her hat. Truly said, "Screen your calls. If anyone gets through to you, refer them to our office. Jonathan is the only one who deals with the press. Do you have a problem with that?"
"No."
"It should die down in a few days."
"And if it doesn't?"
Truly shrugged. "Enjoy the idolatry. You earned it, my friend. You really came through for us."
A tall thin guy from one of the national networks yelled, "Hey, Sherlock Holmes! Are you really that good or did you just get lucky?"
I said, "Some idol."
Truly laughed and I climbed into my car and drove away. Slowly. I almost ran over a cameraman.
#
I pulled into the carport at two minutes after six that evening. The TV was on, and Lucy and Ben were at the dining table, Lucy still in her business suit and Ben wearing a _Songbird_ T-shirt. The cat was nowhere to be found, but that was probably just as well. If he'd been home, Lucy and Ben would probably need stitches.
Lucy smiled when she saw me and said, "It's the world's greatest detective. Congratulations, Sherlock."
Ben jumped up and clapped. "We saw you on television!"
I said, "How do you know it was me? Maybe it was an imposter."
Lucy crossed her arms and considered me. "Now that you mention it, the man on television was devastatingly handsome and darkly mysterious."
I said, "Oh. That was me."
Lucy was beaming. "We just turned on the news and there you were. You and Jonathan Green. Was it exciting?"
"Being with Jonathan?"
"No, silly! They said you made some kind of break-through that might turn the case around. Jonathan said that you were the finest investigator he's ever worked with."
I tried to look blasé and stifled a yawn. "Oh, that."
She punched me in the arm. "Be serious."
I gave her a kiss. "There were so many reporters I thought I'd have to shoot my way out." I gave her a second kiss and then a nuzzle. "Enough about me. How was your day?"
"It was good. We'll meet again the day after tomorrow, then perhaps once more, so there's plenty of time to play." She was surrounded by tourist brochures and tour books with Post-Its and a list of things to see and do.
I looked at her list. They wanted to see my office and visit both Disneyland and Universal Studios and take in a Dodgers game and eat a hot dog at Pink's on LaBrea in Hollywood. They wanted to ride the roller coasters at Magic Mountain and go to Malibu and spend a day at the beach. They wanted to see the Venice boardwalk and Beverly Hills and Rodeo Drive. They wanted to see Griffith Observatory, where James Dean had his famous knife fight in _Rebel Without a Cause,_ and the Hollywood sign. They wanted to see Ronald Colman's house. I said, "Ronald Colman?"
Lucy said, "Of course, silly. We can't miss that." She was marking yet more things as I watched. She would finger through the tour books and refer to notes and frown as she juggled alternatives and weighed options and planned the Great L.A. Adventure. She glanced at me, then went back to the Frommer's, then came back to me again. She said, "What are you smiling at?"
"How I Spent My Summer Vacation."
She closed the Frommer's on a finger and looked miserable. "There's so much."
"Too much. You're never going to do all that in the few days that you have."
She put down the Frommer's. "What are your suggestions?"
"Visit more often."
She smiled and patted my hand. "What are your suggestions for _now_?"
"For now, how about dinner at Spago? For tomorrow, how about the Universal tour with lunch at the Universal City Walk, then either Beverly Hills and Rodeo Drive or Malibu and dinner at the beach?"
She looked longingly at the Frommer's. "Couldn't we squeeze in Ronald Colman?"
I leaned close and lowered my voice so that Ben couldn't hear. "We could, but that would fill the forty-five minutes I've alloted for lovemaking." I stepped back and spread my hands. "Your call."
She frowned and drummed the table. "We didn't need forty-five minutes last time." Everyone's a comedian.
She shrugged and frowned like it was the trade-off of the century. "Okay. Forget Ronald Colman."
Ben said, "Hey! You're on TV again!"
Lucy grabbed my hand. "Oh, look!"
I looked as the local anchor said that there had been a "surprise development today" in the Theodore Martin murder investigation that might "derail the prosecution's case." They cut to a clip of Elton Richards's duplex and the frosty-haired remote reporter took over. You could see me talking with Hernandez and Flutey in the background. Ben and Lucy both yelled, _"There you are!"_
The reporter told us that a private investigator working for the Big Green Defense Machine had followed a tip to evidence that implicated two El Monte men in the kidnapping/murder of Susan Martin. She referred to notes and said, "We've learned that the two men are Stephen Pritzik and Elton Richards, both of whom have lengthy criminal records." The image shifted to grainy mug shots of Pritzik and Richards. Pritzik looked narrow and mean; Richards looked stupid. Lucy said, "Oh, those guys are choice."
The reporter said, "Sources close to the investigation tell us that the evidence found here today provides a direct link between these men and Susan Martin's kidnapping." They cut to a clip of me and Jonathan Green standing by Jonathan's Rolls-Royce, Jonathan with his hand on my shoulder, saying that I had found the breakthrough that the defense has needed. Both Lucy and Ben cheered again when Jonathan said it, and Lucy hooked her finger in my belt loop. I thought that I looked like a turnip head.
The anchor reappeared, said that the two men were being sought for questioning, then shifted to a story about sweatshops in East L.A.
I said, "Shucks. He didn't put in the part where Jonathan said I would be the hero of the case."
Lucy tugged at my belt loop. "So what did you find?"
I told her about the map and the pictures. Lucy wasn't smiling now. She looked grave, and then she shook her head. "Wow."
I nodded.
"Do you think you'll be able to come out and play with us tomorrow?"
"I'll call Jonathan in the morning. I'll have to follow up on Pritzik and Richards, but it shouldn't take all day. Maybe just half a day."
We stared at each other.
She held out her hand, and I took it. She said, "It's okay, Studly. I understand."
"I'll grab a shower and we'll eat."
I phoned Spago for a reservation, then showered, changed, and when I came back down she was grinning. I said, "What?"
Grinning wider. "Nothing."
"What?"
"Just a little surprise. Let's go."
It was almost eight by the time we made it down the mountain, and the sky was deep purple edging into darkness. The Sunset Strip was alive with middle-aged hipsters driving Porsches to show off for women twenty years younger and goateed Val Dudes chasing the Christian Slater look and young women sporting navel rings set fire by the neon. The sidewalks outside of clubs like the Viper Room and the House of Blues and the Roxy were jammed with people, some of whom wanted to get in but most of whom were content to make the concrete scene out front, laughing and goofing and tossing back test-tube shooters of red dye number six vodka. Ben said, "Mom, look! There's a man with a bone through his nose."
I said, "Welcome to Planet Los Angeles."
Lucy shook her head and smiled. "Well, it isn't Baton Rouge, is it?"
"Wait'll you see Melrose."
"It's fun, though. Sort of like Mardi Gras three hundred sixty-five nights a year."
"Yeah," I said. "L.A. is okay that way."
She turned back to me. Serious. "Do you like living here?"
"If I didn't, I wouldn't."
She stared at me for a moment, then she nodded and turned back toward the window. "Yes, I guess you wouldn't."
We pulled into Spago and let the valet have the car. Lucy suggested that I wear my Groucho Marx nose as a disguise to prevent adoring fans from mobbing me, but I pointed out that then everyone might think I was Groucho Marx and I would be mobbed anyway. I decided to risk going as myself.
We went upstairs for very nice Caesar salads and duck sausage pizza and a pretty good merlot. Johnny Depp was there with several friends, and so were three of the cast members of _Beverly Hills 90210_. No one stared at me and no one asked for my autograph and no one took my picture. Everyone was looking at Johnny Depp. Even the _90210_ people. Disappointing, but maybe the people who go to Spago don't watch the news.
Lucy said, "Perhaps you should've worn the Groucho after all."
"Perhaps."
She patted my arm. "Don't feel bad, sweetie. They would recognize you if all these _faux_ celebrities weren't here."
"Yeah." I sneered. "Johnny Depp."
Throughout the meal Lucy would grin with the knowledge of her secret, and I would ask, "What?" and she would say, "You'll see." Then everyone stopped looking at Johnny Depp and turned toward the door. Lucy grinned wider, and I looked, too.
Joe came over, gliding across the floor as the room parted for him. Tall men in sleeveless sweatshirts and dark glasses and brilliant red tattoos tend to stand out in Spago. Even Johnny Depp was looking.
Lucy stood to greet him. "Hi, Joseph."
Joe kissed Lucy's cheek, hugged her, then shook Ben's hand. "You ready, sport?"
"Yeah!"
I said, "What's going on?"
Joe swiveled my way, and you could tell he was amused. You could see that he was positively dying, even though his face showed nothing. "Peter Nelsen's down in the car. Peter and I are taking Big Ben to a screening of Peter's new movie." Peter Alan Nelsen is the third most successful movie director in the world. Once he was a client, but now he's a friend.
I looked at Lucy.
Joe said, "We won't be back until late."
I looked back at Joe.
"Very late."
Lucy gave Ben a hug. "You guys have fun."
Joe's mouth gave the twitch, and then he and Ben were gone.
I looked back at Lucy, and she said, "Alone time is very important."
"You called him while I was in the shower?"
"Uh-huh."
"I knew there was something I liked about you."
She sipped her wine. "I think that there's something you like very much about me."
We enjoyed a slow, noisy dessert with Lucy and I playing footsie under the table. We spoke in greater detail about her day and mine, and I told her about the Mason twins and the Pug Woman and the man who claimed to have seen Susan Martin's kidnapping from the Orb. I didn't often speak of my work, but it seemed natural with Lucy, and as we talked we laughed and goofed about Orb people and stroked each other's hands and arms and fingers. The time passed with the slow, warm feel of dripping honey, and finally Lucy wrapped my feet with hers and said, "Maybe we should go."
We left Spago at 10:35, and when we got home Lucy went into her room and I put Janis Ian on the CD player, then poured a merlot that would go nicely with the one we'd had at Spago.
When Lucy came out she had changed into shorts and a cropped T-shirt that said _Tank Girl_ and silver evening slippers with four-inch heels. The lights were low and Janis was singing. Lucy did a slow pirouette. "Is the world's greatest detective tired after his long and successful day?"
I watched the pirouette. I watched the way the warm light caught her back and hair and the long, smooth line of her legs and the sexy counterpoint of the formal evening slippers to the shorts and T-shirt. "He was, but now he feels a growing revitalization."
"Ah. Is that what's growing?"
"One way to look at it." I held out the merlot and Lucy took a sip. I said, "Nice shoes."
Lucy brushed against me, swaying to the music. The merlot left a sweet, rich taste in my mouth that I liked a lot. She said, "You'll probably be on the news again at eleven. Shall we turn on the TV and see?"
I shook my head. "Seeing me once is plenty. Besides, something else is already turned on."
"Ah."
"I think I'm ready for the rest of my surprise."
She took my hand and tugged me toward the big glass doors. Outside, the sky was clear, and would be filled with stars.
I said, "The deck?"
She let her hair fall across one eye. "I thought your middle name was Adventure."
"So it is."
I followed her out, and what I found there tasted better than any wine, and was more beautiful than the stars.
#
Lucy and Ben woke giggling and excited and filled with plans. As they readied for their day, I phoned Elliot Truly at his office. "It's Cole. Has there been word on Pritzik or Richards?"
"Not yet, but there'll be something soon." He sounded distracted.
"I was thinking that I'd go back to Richards's place and talk to some of the neighbors, but the police will be there and they won't like it. Maybe Jonathan could talk to Sherman and smooth the way in the spirit of cooperation."
Truly didn't say anything for a moment. "Why do you want to go back there?"
"To try to get a lead on Pritzik and Richards."
"Forget it. We're talking with the cops. We've got Kerris on it. Take the day off and relax." I could hear voices behind him.
"It's a cold trail already, Elliot. We shouldn't let it get any colder."
"Look, you just said that the entire area will be swarming with cops. We've got a staff meeting here in a couple of minutes to try to figure out what to do next. Jonathan's trying to get a meeting with the DA."
"What does that have to do with finding Pritzik and Richards?"
"Take the day off, enjoy yourself, and I'll get back to you."
"You want me to do nothing."
"Yeah. What could be better than that?" He hung up, and I stared at the phone.
Lucy said, "What's wrong?"
I looked at the phone some more and then I put it down. "Not a thing. I get to spend the day with you. What could be better than that?"
Their excitement was contagious. We made a fast breakfast of sliced fruit and cottage cheese and toast, and then we dressed in shorts and light shirts and baseball caps for the always popular Ralph Cramden look. I considered bringing my Dan Wesson .38 caliber revolver, but thought it unsightly strapped over my flowered shirt. Besides, blue steel wheel guns aren't exactly requisite tourist attire in southern California. Florida maybe, but not yet California.
It was early, so we decided to see my office first, then head for the Universal tour. We took Lucy's rented Taurus down Laurel Canyon, then along Sunset toward the office. The sky was free of haze and smog, and more blue than white because of it. A great V of gulls floated above West Hollywood, heading toward the sea, and the streets were busy with cars sporting out-of-state license plates and people with camcorders and young Middle Eastern guys selling maps to the stars' homes. Summer had come to the City of Angels.
When we turned onto Santa Monica, one block up from my office, we saw two television news vans parked at the curb in front of my building. I said, "Uh-oh."
Lucy said, "Do you think they're here for you?"
"I don't know." Maybe they were here for Cindy. Maybe they wanted to do a story about hot new beauty supply products.
"You don't want to speak to them?"
"Jonathan's the only person on the team who talks to the press."
I pulled past the vans. A very attractive young Asian-American woman was on the sidewalk talking to a guy holding a Minicam, and a guy who looked like a surfer in a sport coat was smoking with a scruffy woman in a work shirt. I pulled to the curb on the next block, asked Lucy for her cell phone, and called Cindy's office. Cindy answered on the first ring and said, "Wow, are you ever the big deal."
"Have they been upstairs?"
"All morning. They knock on your door and when you don't answer they come to me or the insurance people and ask about your hours." The insurance people had the office across the hall. "That was a great picture in the paper."
"I'm in the paper?"
"You haven't seen it?"
"Uh-uh." Mr. With-it. Mr. Hip L.A. Private Eye with his fingers on the pulse.
"Oh, man, you look so cool. And I saw you on TV, too. I saw you _twice_." Even Cindy was excited.
"Is anyone upstairs now?"
"Yeah. There's a guy sitting in the hall. I think he's from a radio station."
I thanked her and handed back Lucy's phone. Lucy was looking at me. "They're upstairs?"
I nodded. "You mind if we don't go up? I'll show you guys my office another time."
She patted my leg and put away her phone. "Another time is fine, Studly. I want to see my man in the paper."
We stopped at a Sav-on drugstore where we bought the _Times_ , the _Examiner_ , and the _Daily News_ , then stood in the parking lot, reading. Elton Richards, Steve Pritzik, and the discovery I'd made in Richards's duplex were front page news in all three papers. A picture of me with Jonathan Green was on page one of both the _Examiner_ and the _Daily News_ and on page three of the _Times_. Guess the people at the _Times_ had higher standards. Lucy said, "Oh, Elvis. This is so exciting."
I said, "Um."
"Aren't you proud?"
"It's kinda neat, I guess." I held up the paper next to my face and frowned. "Do I look like Moe Howard?"
Lucy compared me to the picture, then nodded. "Yes. Yes, I think you do."
A round man with thick glasses and a nervous tic walked past, staring. He went to a brown Cressida, still staring, then called out, "Hey, are you that guy?"
I folded the paper and tossed it in the car.
"I read about what you did. I saw you on the news. That was good work."
I gave a little wave. "Thanks."
He said, "These cops here in L.A. suck, don't they?"
I frowned at him. "Some of my best friends are cops."
He made a nasal, braying laugh, then climbed into his car and drove away.
I opened the door for Lucy and we drove east across West Hollywood and Hollywood, and then up through the Cahuenga Pass to Universal Studios. We parked in one of the big parking structures with about twelve million other tourists, then followed along with what seemed an endless stream of people to the ticket kiosks and then into yet more lines that led to the trams. It made me feel like a lemming.
We rode the trams around the Universal back lot and took goofy pictures of ourselves posing with giant toothpaste tubes and rode little cars past screeching dinosaurs and gargantuan gorillas, and then Lucy said, "I feel the urge to spend."
I looked at her. "Spend?"
Ben made as if he was horrified. "Not that, Mom! Not that! Try to control it!"
Lucy's eyes narrowed in concentration and her gaze went blank. "The shopping gene is beyond all control. Souvenirs. _I must have souvenirs!_ "
It was horrible to behold. Lucy bought; I carried. Three T-shirts, two sweatshirts, and a snow-shaker paperweight later, we had exhausted the selection in the upper park and trekked down to CityWalk in search of more booty. The CityWalk is a large, open-air mall with shops, bookstores, restaurants, and other fine places to spend your money. Some people have described the CityWalk as an urban version of Disney's Main Street U.S.A., but I've always thought of it as a G-rated take on _Blade Runner_. Only without the rain.
It was just before noon when we got there, and, like the park above, the CityWalk was thick with tour groups from Asia and visitors from around the country. We walked the length of the CityWalk, browsing in the shops and watching the people, Lucy and I holding hands while Ben ranged around us. It felt good to be not working and good to be with Lucy. I said, "Do you think you can rein in your spending spree long enough to eat?"
She looked at me the way the cat does when I take his bowl before he's finished.
"I may not be able to carry this stuff much longer without an infusion of calories."
"You'll manage."
"We may have to hire porters."
"It's only money."
"We may have to stop spending."
She made a big sigh and rolled her eyes. "Modern men are such wimps."
I leaned close to her ear. "That's not what you said last night on the deck."
Lucy laughed and hugged my arm tight, biting my shoulder through the shirt. "O.K., Studly, your wish is my command. Where would you like to eat?"
"You said that last night, too."
She dug her thumb in my ribs and said, "Shh! Ben!"
"He didn't hear. C'mon. There's a Puck's ahead. We can eat there."
"Puck's! Oh, goody!"
We went to Wolfgang Puck's and stood in line for a table. Everyone around us was from Iowa or Canada or Japan, and no one seemed to have seen the news or read the paper or, if they had, didn't care. There was plenty of outdoor seating, and the people at the tables were enjoying salads and sandwiches.
We worked our way up the line to a pretty blonde hostess who told us that it would be just another minute when I caught an overweight guy staring at me. He was sitting at one of the tables, eating shredded chicken salad and reading a _Times_. He looked from me to the paper, then back to me. He stopped a passing waitress, showed her the paper, then they both looked at me. I turned so that I was facing the opposite direction. Lucy said, "Those people are looking at you."
"Great."
"I think they recognize you."
"I know."
"He's pointing at you."
The Korean couple behind us looked at me, too. I guess they saw the pointing. I smiled and nodded at them, and they smiled back.
Lucy said, "Ohmigod, he's showing the paper to the people at the next table."
I touched the hostess's arm. "Do you think you could find us a table, please. Inside or out. First available."
"Let me check." She disappeared into the restaurant.
Lucy said, "Maybe we should run for it."
"Very funny."
"We could leave. I don't mind."
"No. You want Puck's, we're going to eat at Puck's."
An older couple behind the Korean people craned around to see what all the looking and pointing was about. The woman looked from me to the people with the newspaper, then back to me. She said something to her husband and he shrugged. I turned the other way, and now the heavy man with the newspaper was locked in conversation with a table of six people, all of whom were twisted around in their seats to see me. I said, "This is nuts."
Lucy was smiling.
I said, "This isn't funny."
The woman behind the Korean couple said, "Excuse me. Are you somebody?"
I said, "No."
She smiled at me. "You're an actor, aren't you? You're on that show."
Lucy began one of those silent laughs where your face goes red and you're trying not to but can't help yourself.
I said, "I'm not. Really."
"Then why is everybody looking at you?"
"It's a long story."
The woman gave me huffy. "Well, it's not very friendly of you, if you ask me, snubbing your public like this."
Lucy leaned toward the woman. "He can be just horrible, can't he? I talk to him about it all the time."
I stared at her.
The woman said, "Well, you should. It's so unkind."
Lucy gave me a little push. "Why don't you give her an autograph."
I stared harder. "You're some kind of riot, you know that?"
Lucy nodded. Brightly.
The woman said, "Oh, that would be just so nice." She gestured to her husband. "Merle, we have a pen, don't we?" She shoved a pen and a souvenir napkin from Jodi Maroni's sausage kitchen at me to sign. The Korean couple were talking in Korean to each other, the man searching frantically through a shoulder bag.
I took the napkin and leaned close to Lucy. "I'm going to get you for this."
She turned away so no one could see her breaking up. "Oh, I really, really hope you do."
Ben said, "Mom? Why are these people looking at Elvis?"
The older woman's eyes grew large. "You're _Elvis_?"
The Korean woman held out an autograph book and the Korean man began taking pictures. Two teenaged girls who were seated behind the party of six saw me signing the Jodi Maroni napkin and came over, and then two younger guys from the table of six followed. A tall, thin man across the restaurant stood up at his table and aimed his video camera at me. His wife stood with him. A Hispanic couple passing on the CityWalk stopped to see what was going on, and then three young women who looked like they'd come up to the CityWalk on their lunch hour stopped, too. A woman with very loose upper arms pointed at me and told her friend, "Oh, I just love his movies, don't you?" She said it loudly.
The heavy man with the newspaper who had started it got up and walked away. Lucy and Ben were walking away, too. Quickly. Off to ruin someone else's life, no doubt.
The crowd grew. I signed twenty-two autographs in four minutes, and they were the longest four minutes of my life. I finally begged off by announcing that as much as I enjoyed meeting them, the President required my counsel and so I must leave. When I said it the woman with the loose arms said, "I didn't know he was in politics, too!"
When I finally found Lucy and Ben they were well along the CityWalk, grinning and walking fast away from me.
I said, "Lucille Chenier, you can run but you can't hide." I said it loud enough for them to hear.
Lucy and Ben laughed, and then they ran.
#
After another $182.64 in souvenirs, postcards, and gifts, Lucy called Baton Rouge to check her messages. I was hoping that there might be word on Pritzik or Richards, so I phoned my office, also. Sixteen messages were waiting for me. Of the sixteen, seven were from newspeople asking for interviews and five were from friends who had seen me on the news. Of the remaining four calls, two were hang-ups and two were from Elliot Truly. On the first hang-up a woman's voice said, "Oh, shit," and on the second the same voice said, "Just eat me!" The voice was muffled and irritated. Truly's secretary left the first message from his office, asking me to return the call. Truly himself left the second message, saying, "Cole? Cole, if you're there, pick up. This is important." I guess Truly was irritated, too. Maybe I bring it out in people.
I returned Truly's call. When he came on the line he said, "Thank Christ! I've been trying to reach you all day. Where have you been?" He sounded frantic.
"You told me to take the day off, remember?"
"Yeah, well, we don't want you to do that anymore. Channel Eight wants to interview you on the evening news and Jonathan thinks it would be a good idea."
I said, "Go on television?"
"It's maybe three minutes on the four o'clock newscast, and Jonathan wants you to do it."
"Truly, I made plans. I've got guests from out of town."
"Look, the team talked about this today and we want the press to have access to you. Either we're going to control the media or the district attorney's office will, and we'd rather it be us. Openness is important. Honesty is everything. That's all we have going for us."
I was sorry that I had returned his call.
"They want to know how some guy all by himself beat the entire LAPD at their own game."
"I didn't beat anybody. I followed a tip and got lucky." Lucy had finished her call and was looking at me.
"Right. That's why you scored the breakthrough while eight thousand blue suits were sopping up coffee and donuts."
"I didn't beat anyone, Truly." He was getting on my nerves with that.
"All you have to do is sit there and be likable. People like you; you're a likable guy. That's all they care about. It's TV."
I cupped the receiver and told Lucy, "They want me to give a television interview this afternoon, and it'll interfere with going to Beverly Hills."
Lucy smiled and rubbed my arm. "If you have to you have to. We'll do Beverly Hills after."
"It'll cut into your shopping time. Are you sure?"
She smiled again. "We'll come watch you get interviewed. It'll be fun."
Truly said, "What did you say?"
"Relax, Elliot. I'll do it."
Truly said, "It's almost three now and they want you at Channel Eight by four-thirty. Grab a pencil and let me tell you where to go."
Truly gave me the directions. Lucy, Ben, and I drove home, changed, then made our way back down the mountain to Channel Eight's broadcast studio just east of Western in Hollywood. KROK-TV. _Personal News from Us to You—We take it personally!_
We parked in the lot beside the building, then walked in the front entrance to a receptionist seated in a bullet-proof glass booth. The lobby was walled off from the rest of the building with more heavy glass, and there was a big door next to the receptionist that she would have to buzz open to let you enter. I wondered if anyone had ever tried to shoot their way in. _Put me on the news or die!_ You never know.
I told her who I was and why I was there, and a few minutes later a woman in her early forties appeared and opened the door from the inside. She said, "Hi. I'm Kara Sykes, the news director. Are you Mr. Cole?"
"That's right. This is Lucy Chenier and her son, Ben. They're with me." I was holding Lucy's hand.
Kara Sykes held the door. "That's fine. You'll go on in a few minutes, so we don't have much time. Please come this way."
We followed her down a long hall, then through a newsroom filled with desks and production people and onto the news set. A man and a woman were seated at the anchor desk, facing cameras fitted with TelePrompTers. A floor director was standing between the cameras with his hand touching the TelePrompTer that the man was reading from. There were places at the anchor desk for a sportscaster and a weatherperson, but those seats were empty. The set was built so that the anchors were seated with their backs to the newsroom so the audience could see that the Channel Eight news team was bringing them personal news personally. Kara whispered, "Lyle Stodge and Marcy Bernside are the five o'clock anchors. Lyle is going to interview you."
"Okay." Lyle Stodge was a rugged-looking guy in his early fifties, just going gray at the temples. Marcy Bernside was a profoundly attractive woman in her late thirties with dark hair, expressive eyes, and a wholesome, girl-next-door smile.
Kara said, "Have you done a live interview before?"
"No."
"It's no big deal. Just speak directly to Lyle. Don't look at the camera."
"Okay."
"I spoke with Jonathan, so I know how important this is. Everyone here is on your side."
"My side?"
"Just relax and enjoy it. You're the man of the hour."
Lucy squeezed my hand and whispered, "I guess they heard how you were mobbed at Universal."
Lucy's a riot, isn't she?
Lyle finished reading a story about illegal Taiwanese aliens found working in a sweatshop in Gardena, and Marcy began reading a story about Pritzik and Richards. She said that the police and the FBI had expanded their search into seven states, and that there was a growing though unofficial belief that Pritzik was, in fact, James X.
The floor director raised his hand, made a circling gesture, and Marcy Bernside said that Channel Eight's Personal News Team would return in just one minute. The director raised both hands, then announced, "In commercial. We're clear."
Marcy Bernside shouted, " _Fuck!_ Who blew the feed to my _fucking_ ear phone?" She twisted around to glare at the newsroom. "Come on, Stuart. What're you assholes doing back there? Jesus _Christ_!" So much for wholesome.
Kara pulled my arm and said, "Showtime."
She hustled me to the anchor desk and had me sit in the sportscaster's vacant seat while the camera operators repositioned for a two-shot of me and Lyle. I could see Lyle's lines frozen on the TelePrompTer, waiting for the commercial to end. The floor director clipped a tiny microphone inside the lapel of my sport coat, then ran the wire under my jacket and plugged it into a larger cable that had been lying on the floor. Kara introduced me to Lyle Stodge who said, "I'm glad that you could join us. You're quite a guy."
I said, "Will anyone notice if I make faces at the camera?"
Lyle Stodge shuffled loose yellow legal sheets. "Don't worry about anything. I've done this ten thousand times, and I can make anyone look good. Even you." I looked at Lucy and Lucy laughed. I looked back at Lyle Stodge and he winked. Another comedian.
A makeup person was adjusting Marcy Bernside's hair. Marcy was singing to herself and moving to the song as if she were alone in her home. She was singing the Z.Z. Topp song, _Legs_. Nervous energy.
The floor director said, "Ten seconds." He raised his hand above Lyle's camera. Lyle straightened his jacket and leaned toward the camera. The makeup person left the set. Lyle said, "Would you stop with the goddamned singing, for Christ's sake?"
Marcy Bernside gave him the finger and kept singing.
"Three, two, one—" The floor director touched the TelePrompTer and Lyle's script scrolled upward. Lyle made his patented crinkly-eyed smile at the camera. "As we reported at the top of the hour, a private investigator working for the Big Green Defense Machine has made a startling discovery that may shed new light on the Theodore Martin murder investigation. He joins us now in a Channel Eight Personal News exclusive, bringing you the people who _make the news_." Lyle turned the pleased smile toward me. "Mr. Elvis Cole, thank you for joining us in a Channel Eight Personal News exclusive."
"Thanks, Lyle. It's good to be here." Mr. Sincerity.
Lyle laced his fingers and leaned toward me, getting down to serious journalistic business. "How is it that one man working alone was able to uncover these things when the entire Los Angeles police department working for three months couldn't?"
"I followed a tip that Jonathan Green's office received on the hotline. If LAPD would've gotten the tip, they would've made the same discoveries, and probably sooner."
Lyle chuckled good-naturedly. "Sounds like you're being modest to me." The chuckle vanished and Lyle turned serious again, cocking an eyebrow to let everyone know just how serious he was. "Tell us, was it dangerous?"
"It's just meeting people and asking questions, Lyle. It's no more dangerous than crossing the street."
Lyle made the chuckle again, then twisted around to smile at Marcy Bernside. "Marce, I'll tell you, I've never met the real McCoy who liked to blow his own horn, have you?"
Marcy Bernside said, "Never, Lyle. Real men let their deeds speak for themselves."
Lyle twisted back to me. "Theodore Martin has proclaimed his innocence from the beginning. Many people are now saying that your discovery proves him right."
"It's another piece of the puzzle."
Lyle leaned toward me, serious and professional. "Many people are also saying that the LAPD botched this investigation, and now they're unwilling to admit their mistake."
"LAPD is the finest police force in the nation, Lyle."
Lyle nodded as if I'd just laid out the Unified Field Theory. "Well, sir, we've checked into _your_ background and learned that _you_ certainly have an excellent reputation, even among members of the police department and the district attorney's office."
"Those guys. Did they really say that?"
Lyle nodded gravely. "Personal News Eight is told that this isn't your first high-profile case. Apparently, you've worked in a confidential capacity for some very high-profile celebrities."
"I never discuss my clients, Lyle. That's why it's called 'confidential.' "
Lyle squinted approvingly. "A man of integrity." He gave an encouraging smile. "Most of us see private eyes on television or in movies but never get a chance to meet the real thing. Tell me, is it as exciting as it seems?"
"No."
Lyle laughed. They paid him seven hundred thousand dollars a year for that laugh, and I wondered if he practiced it. "Looks like you're a truthful man, as well. How does it feel to be compared to that famous, fictional Los Angeles detective, Raymond Marlowe?"
Marcy said, "Philip Marlowe."
Lyle looked confused and twisted to look at her again. I guess she'd said her bit and he hadn't expected her to speak again. "What was that, Marce?"
"Raymond _Chandler_ created _Philip_ Marlowe."
Lyle laughed again, but this time the laugh was strained. Guess you weren't supposed to correct the anchor while you were on the air. He twisted back to the camera and said, "Well, it looks as if Los Angeles has found its very own Sherlock Homes, and, unfortunately, that's all the time we have for this segment." Lyle Stodge offered his hand to me, and we shook as if he had just awarded me the Congressional Medal of Honor. "Mr. Cole, it's been my privilege to meet you. Congratulations, and thank you for taking the time to talk with us."
"Thanks, Lyle. It's been personal."
The floor director raised both hands. "In promo. We're clear."
Lyle Stodge glared at Marcy Bernside. "You fucking cunt! Don't you ever do that to me again on air!"
Marcy Bernside gave him the finger again. "It's Holmes, moron. Sherlock _Holmes_. With an _L_."
"Oh, yeah, right. Sure."
Kara Sykes unclipped my lapel mike and helped me off the set. No one gave me a second glance.
We followed Kara Sykes back to the lobby, then left the building and walked to the car. Lucy hugged my arm. "That was almost as much fun as Beverly Hills."
"Un."
She stepped back and looked at me. She cocked her head. "Are you okay, Studly?"
I said, "Luce?"
"Mm?"
"If Truly wants me to do another of these, I'm going to shoot him to death. Will you represent me?"
She smiled sweetly. "Oh, you know that I will, hon. You shoot him all you like."
"Thanks, Luce."
#
Lucy, Ben, and I spent the next two days seeing Disneyland and Malibu and the Griffith Observatory. We saw Ronald Colman's house. We shopped in Beverly Hills. I called Jonathan's office twice each day, asking to speak with either Jonathan or Truly, but neither was ever available. Busy, they said. In meetings. No one returned my calls.
I stayed away from my office because of the press. The answering machine was flooded with so many interview requests that I deleted them without playing them. The eat-me lady called back twice.
Elliot Truly's assistant phoned to arrange three more television interviews and two appearances on local talk radio. It's important to Jonathan, she said. We need our side of it known, she said. I asked her about Pritzik and Richards. I said that I wanted to know what was going on. She said that she would talk to Jonathan and get back to me. She didn't.
News reports questioning LAPD's investigative techniques appeared with greater frequency. A summer marine layer moved in, filling the morning sky with an oppressive layer of dark clouds. Sometimes they burned off by noon, but not always.
On the morning of the third day, Peter Alan Nelsen took Ben to spend the day on the set of his new movie and Lucy was dressing for her second meeting when the phone rang and Elliot Truly said, "We're meeting with Teddy Martin at ten this morning in the Men's Central Jail. Teddy wants to meet you, and Jonathan would like you there. Can you make it?"
I said, "What in hell is going on, Truly? How come no one returns my calls?"
"You're not the only investigator we have on this, Cole. We've been swamped. Jonathan's working sixteen hours a day."
"I'm an investigator. I investigate. If you don't want me to investigate anymore, fine." I was feeling sullen and petulant. Mr. Maturity.
Truly said, "Look, talk about it with Jonathan at the jail. One other thing. Jonathan's having a get-together at his home tonight, people who've been behind Teddy through this thing, some press people, like that. Jonathan personally asked me to invite you. You can bring a date if you want."
I cupped the phone and looked at Lucy. She was standing in the kitchen, dressed and Guccied and ready for business, eating peach yogurt. "Would you like to go to a party at Jonathan Green's house tonight?"
Lucy blinked at me and the spoon froze between cup and mouth. "Are you serious?"
"Truly just asked."
She shook her head, the spoon forgotten. "I don't have anything to wear to meet Jonathan Green."
I uncupped the phone. "Forget it, Truly. We can't make it."
The yogurt cup hit the floor and Lucy grabbed my arm. "I didn't say that! I'll get something!"
"My mistake, Truly. We'll be there."
Truly said, "Great. I'll see you at the jail. Ten o'clock."
I smiled at Lucy. "How about that? You'll get to meet Jonathan."
Her eyes were glazed and distant. "Ohmigod, what am I going to wear?"
"Wear what you have on. You look great."
She shook her head. "You don't understand. I'm going to meet Jonathan Green."
I said, "You've got time. Go to your meeting, then go into Beverly Hills. You'll find something."
Lucy looked miserable. "I wouldn't know where to go. It could take _days_."
"Call Jodi. Jodi can tell you."
Lucy's eyes widened and she latched onto my arm again. "That's right. Jodi can save me!" I guess these things are relative.
Lucy set about arranging her salvation, and I drove down to my office. I hadn't been there in three days and wanted to check my mail and return calls. There weren't any news vans parked at the curb. Maybe my fifteen minutes of fame was over. Live in hope.
I locked the door in the outer office, then answered mail. Most of the mail was bills, but even World Famous Private Eyes have to pay their Visa charges. When the bills were done I was getting ready to return calls when the phone rang and I answered, "Elvis Cole Detective Agency. Please leave a message at the sound of the beep. Beep." The detective as Natural Born Wit.
There was a pause, and then a muffled woman's voice said, "You're not a machine." The eat-me lady.
"Who is this?"
"That weevil-dicked fuck James Lester is fulla shit. You find out about Stuart Langolier in Santa Barbara." She was speaking through cloth, but I'd heard the voice before.
"El-ay-gee-oh..." Spelling it. "No, wait... Capital el-ay- _en_ -gee-oh-el-eye-ee-are."
I said, "Jonna?"
There was another pause, and then Jonna Lester hung up. I listened to the dial tone for several seconds, then called an investigator friend of mine named Toni Abatemarco who works at a large agency in Santa Barbara. Toni had worked as an investigator since the day she was old enough to get the license and had hammered out twelve-hour days for years, building her small agency into one of Santa Barbara's finest. Then she met a guy, fell head over heels, and decided that she wanted a small herd of children. She sold the small agency to a larger outfit, had four little girls, and now worked three days a week for the organization that had bought her. She loved investigating, she loved being a mom, and the little girls often accompanied her to the office. They would probably grow up to be investigators, also.
I gave Toni the name, asked her to see what she could find, and then I went to jail.
The Men's Central Jail is an anonymous building behind Central Station, less than ten minutes from the Criminal Courts Building in downtown L.A. I parked in a neat, modern underground parking structure, then walked up steps to a very nice plaza. Nicely dressed people were sipping _lattes_ and strolling about the plaza, and no one seemed to mind that the plaza adjoined a place housing felons and gangbangers and the wild men of an otherwise civil society. Perhaps because this is L.A. and the jail is so nice. There's a fountain in the plaza, and it's very nice, too.
Truly was waiting for me in the jail lobby. "Jonathan and the others are in with Teddy. Come on. I've checked us in."
"I'm carrying a gun."
"Okay. Sure." Like Terminal Island, you can't bring guns into the interview room or the holding areas.
We crossed the lobby past the deputies at the information desk to the gun locker, then went through the metal detector and flashed our IDs at the security gate. The guard there sits behind bulletproof glass and controls the metal doors that let you into or out of the interview area. He's the last guard that you'll see in the jail who has guns. He has shotguns, pistols, tasers, and CS gas. Preparation is everything.
The guard threw switches and the metal door crawled to the side. We stepped through into a room like a gray air lock, and then the door closed. When the door behind us was closed, the door in front of us opened and we stepped through into a large room sporting two long tables lined by metal stools. The tables were narrow and dark, sort of like public-school cafeteria tables, only with low vertical partitions running lengthwise down their centers. Inmates in orange jumpsuits sat on stools along the inside of each table, staring across at the attorneys who sat opposite them. The vertical partition was supposed to make it hard for illegal contraband or weapons to be passed from one to another. Sometimes it worked. Another deputy sat behind glass in the far corner, keeping track of who came and who left and making sure that no one was stabbed to death. Sometimes that worked, too.
Everyday dirtbags had to sit in the big room at the long tables and talk about their cases with no privacy, but high-profile defendants like Teddy Martin rated a private interview room. I followed Truly along a short hall, then into a room that was not dissimilar to the one in which I had seen LeCedrick Earle at Terminal Island, only older and uglier and smelling of urine.
Jonathan Green said, "Here he is now."
The interview room was small and crowded. Stan Kerris, Green's chief of security, was leaning against the glass with his Fred Munster arms crossed. Jonathan Green was seated at a worktable with one of the lesser attorneys and Teddy Martin. I had never met Teddy Martin before, but I knew him from his picture. Teddy Martin had a round, boyish face, a steeply receding hairline, and pale, soft skin. Theodore Martin looked like someone's younger brother grown older; a kind of nonguy who just happened to have built six family-owned hot dog stands into an empire. Truly said, "Elvis Cole, this is Teddy Martin. Teddy, the man."
Teddy Martin came around the table and offered his hand. He said, "I don't know what to say except thank you." His eyes were wide and kind of frantic. "I did not kill my wife. I loved her, Cole. I tried to save her, do you see? They're blaming this thing on me, and it feels like you're the first one who's done anything to help me."
"I'm glad we could finally meet." He gripped my hand with both of his and pumped hard, as if hanging onto me was the most important thing in his life.
Green said, "Theodore."
Teddy Martin seemed to realize what he was doing and flushed. "Sorry." He let go and went back to the table.
I said, "Why did you have me come down here?"
Green patted Teddy on the shoulder, much the way that he had patted me. "Twofold. Teddy very much wanted to meet you, and I've arranged a press conference to take place in the plaza. The core of the team will be there, and I'd like you to be there, too."
I looked at Kerris. The empty eyes were unimpressed. "Press conferences are fine, Jonathan, but what about the investigation? I've called you guys five times, and nobody returns my calls."
Jonathan Green's face stiffened ever so slightly, as if he wasn't used to being questioned and didn't like it.
Truly said, "We're swamped. I told you."
Jonathan waved his hand, cutting off Truly. "What would you like to do?"
"Follow up Pritzik and Richards. Run down more hotline tips."
Kerris shifted against the glass. "I've got other people on Pritzik and Richards. I can give you all the hotline tips you want."
Jonathan made the hand wave again. "Let's not waste Mr. Cole's time with that." He left Teddy and sat on the edge of the table.
I said, "The police and the feds are looking for Pritzik and Richards. We can launch a collaborative effort with them. The cops aren't our enemy."
Jonathan spread his hands. "If you want to work with the police, fine. If it helps us free Teddy any sooner, that's all to the good."
I looked from Jonathan to Kerris to Truly. They were staring at me. The lesser attorney was staring at me, too. I said, "There's something else. A woman I believe to be Jonna Lester called me. She said that James Lester was lying. She said that I should check into someone named Stuart Langolier."
Jonathan nodded. "By all means." He looked at his watch. "We really should be going now. Can't keep our friends in the press waiting."
We said our good-byes to Theodore Martin and walked out. Jonathan walked beside me. When we were out the door and down the hall, Jonathan said, "A proper criminal defense effort is an enormous managerial task, akin to staging the Normandy invasion or launching the Gulf War. All the pieces will come together. Trust me on that."
I nodded.
"Elliot tells me you'll be joining our little soiree this evening."
"That's right. Thanks for inviting me."
"I understand you have a lady friend."
"She's an attorney, also. She's excited about meeting you."
"Well, who can blame her?" Jonathan made a little laugh. "Ha-ha." I glanced at Truly and Truly was nodding. Serious.
Jonathan said, "We'll discuss the team's progress and direction. I want you to be a part of that meeting. I don't want you to feel left out."
I said, "You don't have to handle me, Jonathan."
"I know that, son. I respect you."
I recovered my gun, then we stepped out into the plaza and a wall of people and cameras and microphones surged forward and enveloped us. I thought that maybe this wasn't the jail anymore and maybe I wasn't me. Maybe I'd stepped through Calvin and Hobbes's transmogrifier and I was no longer a detective and Green was no longer a lawyer. Maybe we had just discovered life on Titan. Maybe we had found the cure for AIDS and were about to tell the world. Why else would so many people be here shouting questions?
Jonathan went to the microphones. "We're not here to answer questions, but I want to make a short statement." He spoke in his normal voice, and the crowd shushed itself to hear him.
Jonathan's expression turned somber, and then he looked at me and again rested his hand on my shoulder. He said, "As you all know, three days ago Mr. Cole found important evidence that both the police department and the district attorney's office failed to uncover, evidence that we believe supports our client's claim of innocence. Both the police department and the district attorney's office promised to evaluate this evidence, and act on it, but they have not." He let go of my shoulder, and the somber expression turned fierce. "We demand that the police stop their foot-dragging and issue immediate arrest warrants for Stephen Pritzik and Elton Richards. Concurrently with this, and in consideration of the state's weakened case, I hereby request that the district attorney stop this injustice, admit the failure of his investigation, and dismiss all charges against Theodore Martin. In lieu of that, we have filed a motion with the bench to set bail so that Mr. Martin might be released."
Reporters in the back were tossing out questions as the reporters in front pushed their microphones even closer.
Jonathan's voice grew, and the fierce expression became outraged. He grabbed my shoulder again, and all the grabbing was making me uncomfortable. "The tyranny of evil men cannot be hidden from the light of truth! We have not only uncovered evidence of a specific crime, but also of gross incompetence, negligence, and a police department all too willing to obfuscate the truth in an attempt to hide their own shortcomings." Still cameras were clicking and videocameras were panning, and they seemed to be panning toward me.
I said, "What's he talking about?"
Truly nudged me. "Jonathan knows what he's doing."
Green bellowed, "We do not rest. We continue to investigate. And, ladies and gentlemen, we are about to blow the lid off the evil and the desire for personal gain that underlies this tragic and wrongful prosecution!"
Jonathan abruptly turned away from the microphones, and a wall of sound came from the press. They surged around us and shouted their questions, and just as abruptly Kerris and maybe a dozen of his security guys appeared from nowhere and surrounded us in a kind of flying wedge. Truly was smiling. I grabbed his jacket and shouted to make myself heard. "What's he talking about, Truly? What just happened here?"
Truly laughed. "The truth happened, Cole. Don't worry about it. We'll see you at the party."
Kerris's people worked us across the plaza and down to the parking structure. I moved with the crush of bodies the way a leaf is carried by the wind, a part of an unseen world, yet not.
#
I drove back to the house feeling hollow and uncertain and spent the rest of the afternoon waiting for Lucy to return from her shopping excursion with Jodi Taylor.
Darlene called at ten minutes after three and said, "Good afternoon, Mr. Cole. How are we today?"
"We're fine, Darlene. And yourself?" I wondered if she had seen the press conference.
"Would Ms. Chenier be about?" I guess not.
"I expect her return shortly, Darlene. May I take a message?"
Darlene hesitated, and seemed confused. I have never known Darlene to sound confused. "Oh, no message. Please ask her to call."
"I don't expect her for another hour or so, Darlene, and it's already after five, your time. Is tomorrow okay?" Baton Rouge was two hours ahead of us.
"She could call me at home."
"Is everything all right, Darlene?"
"Everything is fine, Mr. Cole. Please have a good evening."
We hung up, and maybe five minutes later the cat door clacked and I heard him in the kitchen. I got up from the couch and found him standing just inside his door, motionless, tiny nose twitching as he tested the air. I said, "It's just us."
He stared at me for maybe forty seconds, then crept to the living room and tested the air again. I said, "How about some tuna?" He hadn't been home in almost four days, and I had missed him.
I opened a small can of Bumble Bee Fancy White, sat on the floor, and put it down beside me. He loves Bumble Bee Fancy White. It's his most favorite thing. That and field mice. "Well?"
You could see him catch the scent. You could see his eyes widen and his nose shift gears and his ears perk. He looked at the can, took two steps toward me, then squinted back toward the living room. He made his little growl.
"Lucy and Ben aren't here, but they will be. You'd best get used to it and get over this attitude you have."
He stopped the growling and came over but did not touch the tuna. I stroked his back, but he did not purr. "I know, buddy. I feel a little bit disrupted, too."
He head-bumped me, then trotted out of the kitchen and up the stairs, heading for the safety of my loft, moving fast in case Lucy or Ben was lying in wait. I had to shake my head, but at least he was home. You take your progress where you find it.
I checked my office messages at 3:45. Thirteen more interview requests were jamming the machine, but there was also a message from Toni Abatemarco, saying she had something on Stuart Langolier. I called her back and said, "What's the word, Toni?"
"I'm showing seven arrests over a five-year period, starting when he was sixteen for grand theft, auto. We've got a couple more GTAs, one count of fencing stolen auto parts, and an armed robbery. Real working-class doofball stuff."
"That's it?" I was thinking about Jonna Lester. I was wondering what Stuart Langolier had to do with James Lester.
"His most recent arrest was eight years ago. Nothing after that. I can fax this stuff to you if you want."
"Sure." I gave her the number. "Is there a James Lester listed as an accomplice or a known associate?"
"Hang on and lemme see." I waited. "Nope. I don't see one."
I thought about it some more. "How about a phone number or address listed for Langolier?" I thought I might call him. I thought I might ask him why Jonna Lester had brought him into this.
"There is, but it's eight years old, so I double-checked with information. There is no Stuart Langolier listed or unlisted in Santa Barbara, or anywhere in Ventura county."
"How about an attorney?" His docket sheet would list his attorney of record. I could call the attorney and see if they had a current address.
She said, "Sure. He had a public defender named Elliot Truly."
I was poised to write it down, but I didn't. I said, "Stuart Langolier was represented by a public defender named Elliot Truly."
"That's right. You want his number?"
"No, babe. I think I have it." I thanked Toni for the good work, told her to say hi to her husband, Frank, and then I hung up.
I stood in my kitchen, staring at the canyon through the glass doors for a time, and then I dialed Truly's number. "Mr. Truly's office."
"This is Elvis Cole. Is Truly in?"
"I'm sorry. Could I take a message?"
"How about Jonathan?"
"I'm afraid they can't be disturbed."
I hung up again.
I showered and changed and was just getting ready to run down to my office when Lucy got back. I wanted to check the fax. I wanted to have the facts with me when I confronted Truly at the party and asked him what in hell was going on. Lucy came in flushed and excited and beaming, carrying a shopping bag with shoes and a long plastic dress bag. She said, "I want to show you! It's absolutely gorgeous and they took up the hem right there while we waited and it's just perfect!"
Her smile made me smile. "You would look perfect in anything."
"Yes, but I'll look even better in this."
I reached to peek into the bag, but she held it away. "Don't peek. I want you to see me with it on."
"How about I see you without it on, then with it on, so I can decide which way I like you better. Sort of like before and after."
She smiled. "If you're as smart as I think you are, you'll rave about me both ways."
I pulled her close. "I'll rave, but smart has nothing to do with it."
She kissed my nose. "I'm having such fun."
"Me, too, Luce. I'm glad you guys are here."
We kissed again, and then I told her that Darlene had called and said that Lucy should phone her at home.
Lucy frowned. "She said to call her at home?"
"Unh-hunh. I asked if there was a problem, but she said no."
Now Lucy wasn't smiling. She seemed somehow distant and distracted.
"Lucy?"
She smiled again, but now it was forced. She stepped back. "I'd better call Darlene and see. Why don't you go along to the office and I'll show you the dress when you get back?"
"You sure?"
She was already moving toward her room. "I'm sure it's business and it could take a while. I'll model the dress when you get back."
She disappeared into the guest room and closed the door.
I said, "Okay."
The marine layer had burned off but it was bright and hot as I drove down to my office. We get these inversion layers, and the air stops moving and grows milky from the exhaust of five million cars. A thin haze was forming to the east. I was surprised that Jonathan Green would allow an inversion layer on a day when he was going to have a party. Might cast a pall on the entire affair.
I parked in my spot, walked up the four flights to my floor, and saw that my door was open. I stepped in and found Dan Tomsic sitting on the couch. He looked large and heavy, and his eyes were closed. I glanced at the fax. Something had printed out in its basket. I looked back at Tomsic. "I could've sworn that I locked the door."
Tomsic opened his eyes but didn't move. His arms were spread along the back of the couch, and he appeared neither surprised nor concerned. "You did, but what's that to a couple of guys like us?"
I stared at him.
"I'm trying to figure you out, Cole. I ask around and everyone says that you're solid, but now there's this shit with Pritzik and Richards, and the double-dealing with Rossi."
I shook my head. "What are you talking about?"
"The press conference. You and Green looked real sweet standing out there on the plaza. A couple of liars."
"I don't know what you're talking about, and I don't know what Jonathan was talking about, either. All I know is that no one seems to be doing very much about Pritzik and Richards."
Tomsic frowned, like maybe he was confused, and then the frown became a nasty smile. "You don't know, do you?"
"Know what, Tomsic?"
"Pritzik and Richards are dead. They died together in an auto accident three weeks ago in Tempe, Arizona."
"So what's the big secret? All you had to do was let us know."
Now the smile dropped away like a gold digger's interest. "We didn't find out until last night. We called Green's office and notified him at five minutes after nine this morning."
I stared at him. I opened my mouth, then closed it.
Tomsic stood and walked past me to the door. "That's some asshole you're working for. He knew that they were dead even when he was making a big speech about how we weren't doing enough to find them. Foot-dragging, he said. Covering up."
I said, "Were either Pritzik or Richards ever represented by Elliot Truly?"
Tomsic squinted at me. "How in hell should I know?"
I glanced at the fax again.
Tomsic came very close to me. "Shitting on the department is one thing, but Rossi's personal. You said she was clear. You said she was out of it."
"She is, Dan."
"That's not what Green's saying on the news. They're saying she planted the hammer. They're saying she set him up and that they've got proof. You call that being out of it?"
I didn't know what to say.
Tomsic turned back to the door, then raised a single finger, like a teacher instructing a pupil. "My first name is for my friends. You don't rate." He lowered the finger. "Jonathan Green is willing to destroy a good detective's life to save a piece of shit murderer. That makes him a piece of shit, and you're a piece of shit, too."
"Don't mince words, Dan. Tell me what you really think."
Dan Tomsic kept the flat cop eyes on me for another lifetime, and then he left.
My heart was hammering and my head felt swollen. I collected the pages from the fax, then turned on the little Sony TV and found the four o'clock news. The frosty-haired reporter was saying that Pritzik and Richards had plowed into a culvert, saying that they had been drinking, saying we might never know if Pritzik had in fact been James X.
The chiseled male anchor came on, and they cut to a live shot of Jonathan on the sidewalk outside his office. Jonathan and Truly and the lesser attorneys were accusing Angela Rossi of planting the murder weapon, and they were demanding a full investigation, not only of Rossi but of the LAPD command that was protecting her. Jonathan said that his team had uncovered proof that Rossi had tampered with evidence on other occasions, and then Stan Kerris brought out Mrs. Louise Earle. When I saw Mrs. Earle I leaned forward and the swollen feeling spread to my neck and my shoulders. Jonathan introduced her, saying that she had come forward through the efforts of Elvis Cole. He reminded everyone that Elvis Cole was the fine young detective who had made the breakthrough about Pritzik and Richards. He said that what Mrs. Earle was about to say was even more shocking. The camera closed on Mrs. Louise Earle, and she said that Detective Angela Rossi had planted counterfeit money on her son, LeCedrick, and then arrested him. She said that Rossi had threatened to have him killed in prison if she said anything. Mrs. Earle was crying when she said it, and Jonathan Green put his arm around her shoulders to comfort her.
I watched the news for another ten minutes and then I turned off the television. I said, "What in hell is going on here?"
No one answered.
I took a deep breath, let it out, then leaned back in my chair and wondered if I could feel any more out of the loop. I could, and in about twelve seconds I did.
I paged through the faxes until I came to Stuart Langolier's D-55 booking page from the Ventura County Sheriff's office. The booking page showed Stuart Paul Langolier's fingerprints in two rows of five along the bottom of the page, and his front- and side-view mug shots above the prints. The fax quality was poor and the prints had come through mostly as black smudges, but the mug shots were clear enough.
It was eight years ago and the hairstyle was different, but Stuart Langolier wasn't just Stuart Langolier. He was also James Lester, one-time client of Elliot Truly.
I gathered together the faxes, locked my office, and went home to pick up Lucy.
It was going to be a hell of a party.
#
It was just after six when I got back to the house. I let myself in through the kitchen and saw Lucy on the deck. She was standing at the rail, and she was wearing a white silk slip dress with spaghetti straps that left her shoulders and back bare. The silk was without embroidery or detail, and seemed to glow in the lowering sun.
I said, "Simple. Elegant. Utterly devastating."
She turned and smiled, but the smile seemed strained. "Ben called. Peter's going to bring him home after dinner."
"Great."
"You were gone a long time."
"Angela Rossi's partner was waiting for me. Have you seen the news?"
"No."
I turned on the local station, but now they were talking about a fruit fly infestation in Orange County. I changed channels twice, but other things were happening in the world. "They've got a woman I interviewed saying that Rossi framed her son."
"Congratulations." She didn't understand.
"That isn't what she told me. Rossi didn't frame anyone. I cleared her, and that's what I reported to Jonathan."
"I'm sure it's just a misunderstanding. These things happen." She said it, but it was as if she wasn't really there.
I turned off the television and looked at her. "Is everything okay with Darlene?"
"Of course." She glanced away, then made a little shrug. "Just something at the office."
I looked closer. "You sure?"
Lucy stiffened ever so slightly. "Shouldn't you get ready, or are we not going?"
"Luce, he made it sound like I uncovered this woman. He made it sound like I turned up something that implicates Angela Rossi." I said it carefully.
"Perhaps you're just being sensitive." Cool.
I took a step back and went upstairs and put on a jacket and tie. The cat watched me from the closet. Hiding. I said, "Don't say a word."
He didn't.
I folded the fax from Santa Barbara and put it into my inside jacket pocket, and then we went out to the car. I said, "Would you like the top up or down?" Thinking of her hair.
"It doesn't matter."
I left the top down.
I said, "If there's a problem, I wish we could talk about it."
She looked out the window. "Please don't start one of those conversations."
I nodded.
Lucy relaxed as we moved along Mulholland and down Coldwater, and by the time we gave the car to a valet she was smiling again and holding my hand. She said, "There're so many people."
Jonathan Green lived in an expensive home on a corner lot just north of Sunset in Coldwater Canyon. It was an older, established area of great red pines and curving drives and ranch-style estates that looked not unlike the Ponderosa. A small army of valets was trotting along the walks, and the curbs were already lined with cars and limousines and an awful lot of people who looked as if they'd just stepped out of the Academy Players Directory.
Jonathan's front entry was open, and, as we approached, we could see that his home was crowded. I said, "Prepare to be stared at."
She glanced at me. "Why?"
"You'll be the most beautiful woman there."
She hooked her arm through mine.
"In the most beautiful dress."
She squeezed my arm. I'm such a charmer.
A news crew from Channel Eight had lights set up on Jonathan's front lawn and was interviewing a well-known figure who had starred in a hit television series in the early seventies, and who now ran a major studio. Lucy said, "Isn't he somebody?"
"Yep." He was well known for his efforts as an active fund-raiser for private social programs and had received humanitarian-of-the-year awards twice, in large part because Teddy Martin had contributed heavily to his causes. He was less well known for the violent, hair-trigger temper that he has frequently shown toward the young men whom he supplies with heroin.
As we passed, he was telling the reporter, "I've known from the beginning that Teddy is innocent, and this proves it. Teddy has been a force for good in our community for years. He's stood by us, and now it's our turn to stand by him. I can't understand why the district attorney has this vendetta." Other reporters were spread through the crowd, interviewing other supporters.
The entry was wide and long and opened onto a great room that flowed outside through a line of French doors. The floors were Spanish tile and the decor was western, with plenty of rich woods and bookshelves and oil paintings of cattle and horses. An original Russell hung over a great stone fireplace. Behind the French doors were a pool and a pool house and, still farther back, a tennis court. Maybe a half dozen of Kerris's security people were standing around, trying to be unobtrusive and not having a lot of luck at it. The grounds were lush and dramatically lit, and waiters and waitresses moved through the crowd, offering wine and canapés. Maybe three hundred people were drifting through the house and around the pool. Lucy said, "This is beautiful."
I nodded. "Crime pays."
"Oh. There's Jonathan."
Green was near the fireplace, talking with a couple of men in dark suits and a together-looking woman in her late fifties. One of the men was tall and thin, with little round spectacles and a great forehead and bulging Adam's apple. Intense. As we approached, he said, "LAPD has an entrenched white male racist attitude that is impervious to change. I'm telling you that the time is right to simply abolish them."
The together woman said, "That's a non-issue, Willis. Angela Rossi is a white female."
Willis jabbed the air. Agitated. "And as such must subjugate herself to the dominant white male racist attitudes that surround her. Don't you see that?"
The together woman said, "But LAPD is over fifty percent women and minority now, and the percentage is increasing."
Willis's eyes bulged. "But is it increasing fast enough to save us? My God, we're living in a virtual police state! If it could happen to Teddy, it could happen to any of us!"
Jonathan saw me and offered his hand, looking not altogether unhappy to shut Willis off. "Everyone, I'd like to introduce Elvis Cole, an integral member of the team."
Willis's eyes lit up and he grabbed my hand. "Great to meet you. You're the one who nailed that fascist bitch."
The together woman drew a deep breath and Lucy said, "Please don't refer to any women by that word in my presence." She said it politely.
Willis stepped back and held up his hands. "Oh, hey, I apologize. Really. But these cops have just gone over the line, and I'm so frustrated."
The together woman said, "You're such a hog."
Jonathan introduced us. The woman was Tracy Mannos, the station manager from Channel Eight. Willis was a writer for a local alternative weekly, the _L. A. Freak_.
When Green was finished introducing me, I introduced Lucy. She said, "It's a pleasure, Mr. Green."
He smiled warmly and took her hand. "Please call me Jonathan. I understand that you're an attorney."
She nodded. "I practice civil law, but your cases have been inspirational. Especially the Williams case in nineteen seventy-two." He was still holding her hand.
"That's a lovely accent. Where are you from?" He patted her hand.
"Louisiana."
"Well, perhaps we'll have the pleasure of working together some time."
He patted her hand again, and I said, "Jonathan, I'd like to see you."
As I said it, Kerris appeared behind Jonathan and whispered something. Jonathan stared at me as Kerris spoke, and then Jonathan nodded at me. "I have to see the others for a moment. Why don't you come along?"
I left Lucy with Tracy Mannos and followed Jonathan through his house to an office that was the size of my living room. Elliot Truly was there, along with the larger of the lesser attorneys and two men who looked vaguely familiar. One of them was tall and hard and African-American. When Kerris closed the door, I said, "Jonathan, I saw the statement you made this afternoon. What's going on with Louise Earle?"
Jonathan spread his hands. "I'm sorry. I don't know what you mean."
"She's changed her story. She didn't implicate Rossi when I talked to her."
Kerris said, "Guess you got it wrong." He had drifted to the wall behind Jonathan so he could lean. Every time I saw him he was leaning. Guess it wore the guy out carrying all those shoulders and arms.
"I cleared Rossi, and now you're attacking her. You made it sound like I'm behind it."
No one said anything for a moment, and then Jonathan spread his hands. "Angela Rossi found the murder weapon when she went down the slope to Susan's body. She hid it on her person, then planted it on Teddy's property in order to frame him for Susan's murder. She was hoping that if she was credited with solving such a high-profile case, her career would be resuscitated." He smiled at me. "It's as simple as that."
I looked at Truly and Kerris and the two other guys. "That's nuts."
Kerris crossed his arms. He was so wide that maybe he was twins who didn't quite separate. "What's your problem? Everyone thinks you're a hero."
I stared at him. "What's going on?"
Jonathan shook his head.
"How'd you get Mrs. Earle to change her story?"
Jonathan smiled the way you smile when you're incredulous. "Excuse me. Are you accusing me of tampering with evidence?"
Kerris said, "Good thing for us that I double-checked your work. Here everyone thinks that you're some kind of top-dog investigator, and the truth is you suck."
Jonathan frowned. "Please, Stan. There's no need to be insulting."
Kerris kept the empty eyes my way. "He sucks. I'm with the woman five minutes and she breaks down, telling me she's terrified, telling me she's wanted someone to help her for damn near six years because those cops framed her son, then threatened her into keeping her mouth shut."
Everyone was so still that they might have been a fresco. Elliot Truly had a kind of idiot half-grin. He glanced away when I looked at him. I said, "James Lester is a fraud."
Truly was shaking his head before I finished. "That's not true. I should've said something about him when you mentioned Langolier at the jail, but I didn't know how Jonathan wanted to handle it."
Jonathan glared at him.
I took the fax from my pocket and tossed it at Jonathan. "James Lester is an alias. James Lester is a convicted felon named Stuart Langolier. Truly knew him."
Jonathan didn't touch the fax. "This is my fault. You're used to working on small cases and this is a large case, and I should've briefed you on our meetings. Then you wouldn't think we're keeping things from you."
Truly shrugged and looked apologetic. "Look, I didn't realize that Lester was Langolier until I saw his picture in the paper, okay? As soon as I knew I notified Jonathan. We called the district attorney's office and filed a brief about it this afternoon."
Kerris said, "There's a reason they call it 'coincidence.' "
I said, "No secrets?"
Jonathan shook his head. "I'm sorry that I've left you out of the loop. An effort like this is such a large undertaking."
"Like the Gulf War."
"That's right. There are no secrets here."
I said, "What about the lies here? You knew Pritzik and Richards were dead when you attacked the police this morning."
Jonathan frowned as if I were a child he had once thought backward but now stubborn. "I'm disappointed, Elvis. Clearly, you don't understand a team effort, or my obligations as a defense attorney."
Truly shook his head. "What a spoilsport. This case has made you a celebrity."
I said, "Spoilsport?"
Kerris said, "How about 'prick'?"
I looked at him, and Kerris shifted away from the wall. Jonathan said, "No, Stan."
I smiled at him. "Kerris, anytime you want to go for it, I'm available."
Jonathan said, "No, Stan."
Kerris settled against the wall again, and still the empty eyes did not move. The black guy was grinning at me. So was the other idiot.
I looked back at Jonathan Green. "You're right, Jonathan. I don't appreciate it. I quit."
Jonathan said, "I'm sorry to hear that, but under the circumstances I understand."
Kerris said, "You want I should walk him out?"
I said, "Kerris, if you walk me out you won't make it to the door."
Kerris said, "Oo."
I walked out of the office and slammed the door and stood in the crowded living room until my pulse slowed and my ears stopped ringing. The room was so crowded and so noisy that no one heard the door slam. Foiled again. I wandered around for twenty minutes before I found Lucy and Tracy Mannos talking by the pool. Willis and the other guy were nowhere in sight. Just as well for Willis. I said, "Excuse me." My face felt tight, and obvious. "Luce, could I see you please?"
Tracy Mannos handed Lucy a card. "It's been fun, Ms. Chenier. Call me when you get the chance."
Lucy smiled at her, then Tracy Mannos walked away. Lucy said, "Interesting woman."
"I'm glad you're having a good time."
She looked at me. "What's wrong?"
"I am no longer a member of the Big Green Defense Machine. It would probably be appropriate for us to leave."
Lucy stared at me. "What happened?"
"I quit."
We got the car from the valets and found our way back to Coldwater and climbed the mountain to Mulholland. "I'm sorry that we have to leave this way. I know you were excited about meeting Jonathan."
"I don't care about Jonathan. Are you all right?"
I told her about Truly and Lester. I told her again about Mrs. Earle, and about Jonathan making the misleading statement about Pritzik and Richards. I said, "I don't get it. The guy's Jonathan Green. He's an All-World attorney. What does he think he's doing, behaving in this manner?"
She looked at me. "He probably thinks he's doing his job."
I shook my head.
"It's his job to attack the prosecution's case. That's how he creates reasonable doubt."
"Is it his job to lie?"
"No, but you're assuming it's a lie. Reasonable people can disagree and have opposing interpretations of the facts. It's Jonathan's job to present an interpretation that's favorable to his client. It would be malpractice for him to do otherwise." When she said it she was stiff and testy, and it felt like we were having a confrontation.
I said, "What's wrong?"
"Nothing's wrong."
"Are you mad that we had to leave the party?"
"Why are you staying with this? I told you that nothing is wrong."
"Fine."
"Fine."
I turned on k. d. lang. k. d. sang, but I'm not sure either Lucy or I listened. Neither of us spoke.
Peter Alan Nelsen's black Range Rover was parked off the road across from my house, waiting. I said, "Looks like they're home."
Lucy still didn't speak.
We parked and went inside. Peter and Ben were on the couch watching a laser disc of _When Worlds Collide_. The house smelled of popcorn. Peter yelled, "Hide the babes, Ben! It's the police!" Peter always yells things like that.
Ben said, "Hi, Mom. You shoulda seen the neat stuff on Peter's set!"
"You can tell me in the morning, sweetie."
Lucy walked across the living room and into the guest room and shut the door. Ben and Peter looked at me. I said, "I guess she's tired."
Peter said, "Oh, yeah. Looks that way to me."
I frowned at him, and then I stalked up to the loft.
Another fun evening in Tinsel City.
#
Sometime before sunrise the cat's door made its sound, then, a few minutes later, made its sound again. Come and gone.
When the eastern sky was lit gold and the great glass steeple opposite my loft was filled with copper I pulled on gym shorts and slipped down the stairs. The door to the guest room was closed. I went out onto the deck and breathed the cool morning air and did twelve sun salutes from the hatha yoga as the finches and the sparrows and two mockingbirds watched. The canyon was still and quiet and just beginning to fill with light. I did one hundred push-ups and one hundred sit-ups, enjoying the rhythm of the count and the feeling of accomplishment that came with the exertion and the sweat.
The cat climbed onto the deck and watched me from the corner of the house. He didn't look happy.
I worked through the stronger asanas, starting with the half locust, then the full, and then the scorpion and the peacock. The air warmed and the sweat began to flow more freely, and then I saw Ben standing in the glass doors, his face thoughtful. I said, "You're up early."
He nodded. Upset about last night, maybe.
"Come on out."
Ben came out. He was wearing baggy pajama bottoms and a white T-shirt. When he came out the cat lowered his ears and growled. Ben said, "He doesn't like me."
"It's not you. He doesn't like anyone."
"He likes you."
I nodded. "Yeah. He likes me and Joe, pretty much, but he doesn't care for other people. I've never known why."
The low gutter of his growl spiraled up into his war cry and I grew worried he might charge. I'd seen him charge, and it wasn't pretty. I said, "Knock it off." Loud.
The growling stopped.
"That's better."
His ears stayed down, but at least he didn't leave.
Ben crossed the deck to the rail, keeping one eye on the cat, and looked out at the canyon. He put his weight on the rail, then leaned out. He said, "Hawks."
Two red-tailed hawks were gliding low over the canyon. "They're red-tails. They nest up the canyon."
He bounced on the rail. "I think I heard coyotes last night. Was that coyotes?"
"Yep. A family lives by the reservoir."
He bounced faster, then edged along the rail and bounced more. Nervous. I guess he hadn't come out just to look at the hawks. "Your mom and I are going to work things out, Ben. It's okay."
The bouncing stopped and he gave me the same eyes that he'd given me when we'd first met, eight-going-on-nine and taking care of his mom. "She was crying."
I drew a deep breath. I squinted at the canyon, then looked back at him. "Is she crying now?"
He shook his head. "I think she's sleeping."
"She's upset about something, but I'm not sure what."
The bouncing was over, but he still looked uncomfortable.
"She say anything?"
He looked down at the deck, and seemed even more uncomfortable.
"She seemed okay until Darlene called." I watched him. "After Darlene, she seemed kind of upset."
Ben looked at the cat. The cat's ears were up now, and he seemed calm. Ben said, "She's fighting with my dad." Fighting.
"Ah."
"My dad didn't want us staying here. He said we should be in a hotel."
"I see." The hawks reappeared, higher now, following the air back toward their nest. The female had something in her talons. "Are you okay with this, Ben?"
He shrugged without looking at me.
I went to the rail and leaned next to him. "It's tough when your parents are fighting. You get caught in the middle and no matter what you do, you always feel like you're letting one of them down."
Ben said, "She really likes you."
"I really like her. I like you, too. I'm glad you guys are here."
He didn't seem moved by that, but there you go.
I took a breath and went to the center of the deck and worked through a simple kata from the tae kwon do called the Crane. You do a lot of bending and your arms pinwheel a lot and you spin, but it isn't difficult. Ben watched me. I did the Crane slowly, moving from one end of the deck to the other, and taking great care in my movements, sort of like with the tai chi. When I reached the end of the deck, I turned and did it again, back to the other side, only much faster, moving at three-quarter speed. Ben said, "What's that?"
"Ballet."
Ben grinned. "Nunh-unh." He stopped leaning over the rail and crossed his arms. "Is that karate?"
"Korean karate. It's called tae kwon do." I went through it again. Left to right, right to left.
He said, "They do that on _Power Rangers_. They beat up monsters."
"Well, it's a fighting skill, but only if you look at it that way. That's a choice you make. You could also choose to look at it as a way to make yourself stronger and more flexible and healthy. It's also fun." I did it again and watched him watch me. "Want me to show you how?"
He came over and I showed him. I modeled the postures and adjusted his position and walked him through the moves. "Don't try to hurry. Slow is better."
"Okay."
We did the Crane. After the Crane I showed him the Tiger. Ben took off his T-shirt and tossed it aside. Sweating. We worked through the katas together as the sun floated up from the eastern ridge and the air warmed, and then I saw Lucy watching us from the door. I smiled. "Morning."
"Hi."
Ben said, "Look at this, Mom! This is called the Crane. It's a tae kwon do kata. Watch."
Ben worked his way through, and as he did, Lucy put her hand to the glass, fingers spread, and I put my hand to hers. She said, "Joe's on the phone."
Ben said, "Mom, you're not watching!"
I went in and found the phone on the counter. "Now what?"
Pike said, "Put on Channel Five."
I put it on and went back to the phone. The morning anchor was recapping yesterday's report on Green's accusations, and again ran the clip where Green made it look like I had been the one who turned up Mrs. LeCedrick Earle. I said, "We quit last night. We're no longer working for the Big Green Defense Machine."
Pike grunted. "Keep watching."
The anchor said that LAPD had announced a full investigation into Angela Rossi. The anchor said that Rossi had been suspended pending the outcome. I felt a drop-away feeling in my stomach and said, "Oh, man."
Pike said, "I tried calling her, but the phone's off the hook."
"How about I pick you up?"
He hung up without answering. Lucy had come inside, and Ben was still on the deck. I said, "We've got to go see about Rossi."
Lucy nodded. "I thought you might. I've got the meeting later in Long Beach. I'll take Ben."
"Sure."
She started away, then turned back. "I liked seeing you together with him."
I smiled, but I didn't say anything. I wanted to ask what was going on with her former husband, but I didn't want to press her. I wanted to be supportive, but sometimes support can be oppressive. Maybe it would work itself out. Maybe, too, it was none of my business. I decided to give her some room. Giving them room is often the better part of valor, especially when you're trying not to make things worse.
I showered and dressed, and then I drove down to Culver City and found Joe waiting at the curb. Pike slid into the right front seat and closed the door without a word. He buckled the seat belt and still didn't say anything. I guess he was angry, too.
It was a few minutes after nine when we drove to the beach, then turned south to the Marina and slowed at the mouth of Angela Rossi's cul-de-sac. We would've turned onto her street, but we couldn't because of the news vans jamming the cul-de-sac and spilling out onto Admiralty Way. Knots of reporters and camera people were clustered on the sidewalks and in the street, and a couple of women who were probably Rossi's neighbors were arguing with a short, stocky guy in a sport coat. Apparently, his van was blocking their drive. Apparently, they wanted the reporters to lay off Rossi and get out of their neighborhood. Pike said, "Look at this crap."
We parked across Admiralty and walked back. A beefy reporter sitting in a Blazer did a double take when we passed, then hurried after us, asking if he could have a word. He reached Pike first and Pike seemed to give a lurch, and then the reporter sat down on the street hard, going "Omph!"
Pike didn't lose a step. "No comment."
I guess some interviews are harder than others.
We walked past the reporters to the front gate. The thin man with the glasses and an older woman were telling an attractive red-haired reporter that they weren't going to let her in, when the thin man recognized me and shook his finger at me. "It's you. You lied to me when you were here. You weren't looking for anyone named Keith!"
I said, "Would you please tell Detective Rossi that Joe Pike and I would like to see her?"
The red-haired reporter turned and yelled for her camera operator to hurry up. She yelled that she wanted a shot of this.
The thin man kept shaking his finger. "You're a prick. You should be ashamed of yourself."
Joe Pike stepped to the gate and murmured something that I couldn't hear. Pike didn't seem threatening now. He seemed gentle and calming. The woman went to Rossi's front door. I guess she was the thin man's wife.
The red-haired reporter's camera operator hustled up behind us and began taping. The reporter asked if I had any additional information implicating or incriminating Angela Rossi. She asked if I was here to get a statement from Rossi or to follow up a line of inquiry. I kept my back to her. I stared at the hamper filled with Nerf balls. I stared at the red bike.
The thin man's wife came back and let us through the gate. The red-haired reporter tried to push through, but the wife shoved her back, yelling, "Don't you dare!" The thin man wasn't happy that I was coming in.
Joe Pike rapped at the door once, then opened it, and we stepped through into Angela Rossi's life.
It was a nice place, roomy and spacious, though the furnishings weren't expensive, just a sofa and love seat arranged in an L, and a BarcaLounger. I guess she'd put all of the money into buying the place and hadn't had a lot left over for furniture. A woman and a man were standing behind the love seat, and another woman was sitting on the couch, and two little boys were sitting on the floor, the smaller sitting in the larger's lap. I guess the boys belonged to Rossi. I guess the adults were friends or family come to lend support. Off-duty cops, maybe, but maybe not. Everyone in the room was looking at me. Even the boys.
Angela Rossi was standing by the sofa with her arms crossed. Her cheeks looked hollow and her eyes were dark and haunted. I said, "I wanted to tell you that I didn't have anything to do with this. I told Green that you were clean. He told me that he bought it. I don't know what happened."
"Okay. Thanks." Like she was numb.
Joe said, "Angie."
She shook her head. "I didn't do those things. I didn't frame that guy."
Joe said, "I know."
Angela Rossi looked confused. "I don't know why she's lying. She seemed like such a nice woman."
I said, "We'll talk to her. We'll get this straightened out."
Angela Rossi said, "It won't matter. I'm done with the job."
Joe stiffened and shook his head once. "Don't say that, Angela. You're not."
"So what kind of career will I have when it's over?" She walked past us to the window and peeked out. "I can't believe that all these people have nothing better to do." She looked back. "Can you?"
All of them kept staring. I wanted to say something, but I didn't know what to say. My eye still hurt where she'd hit me, and I was thinking that maybe she ought to hit me again. "I'm sorry."
"Forget it." She shrugged, no big deal.
Joe said, "We'll help you fight it."
"Nothing to help. I've decided to resign."
Joe leaned forward. His dark lenses seemed to blaze. "Don't resign. You're too good to resign."
She said, "Oh, Joe."
Pike was leaning so far forward he seemed to sway.
"They've taken everything away, but that's okay. I just have to survive this, and I know I can." She smiled when she said it, as if she were at peace with all of this.
Joe said, "What's wrong with you?" His voice was so soft I could barely hear him.
Angela Rossi's left eye began to flutter, then grew wet, and I had the sense that if she were fine china there would be a webwork of spider-silk cracks spreading beneath her surface. She held up her right hand and said, "Please go now."
Pike nodded, and I started to say something else, but then she turned the hand to me, and I nodded, too.
#
We left Angela Rossi's and walked out to the car. The whiny reporter who had once been a lawyer saw us first and ran toward us, shouting, "They've come out! They've come out!" The rest of the reporters stayed back, shifting their feet and keeping their distance. Pike raised a palm at the whiny reporter, and he stopped, too. I guess word had spread, or maybe it was in our faces.
We drove slowly, neither of us speaking, and worked our way out of the Marina, up through Venice, and along the beach. It was automatic driving, going through the motions without conscious thought or direction, movement without destination or design. Pike hunkered low in the passenger's seat, his face dark in the bright sun, his dark lenses somehow molten and angry. It is not good to see Joe Pike angry. Better to see a male lion charge at close quarters. Better to hear someone scream, _"Incoming!"_
I said, "Where do you want to go?"
His head swiveled sideways maybe half an inch.
"How about we just drive?"
His head moved up, then down. Maybe half an inch.
"Okay. We'll drive."
We followed Ocean Avenue up through Venice and along the bluff above the beach, Pike as still as an undisturbed lake. We stopped for a light on Ocean Park, and I watched the joggers and bikers and smiling young women with deep tans who dotted the bike paths along the bluff. Everyone was smiling. Happy people having a great time on a beautiful day. What could be better than that? Of course, they could be happy because they hadn't just come from Angela Rossi's house. It's always easy to smile when you haven't helped destroy an innocent person's life.
The light turned green, and a red Toyota pickup filled with surfers and surfboards blew their horn behind us. The driver yelled for us to get out of the way, and Joe Pike floated up out of his seat and twisted around, and when he did the honking stopped and the Toyota jammed into reverse and sped away at high speed. Backwards.
I said, "Well. I guess we'd better talk about this before we kill somebody."
Pike frowned. His arms were knotted and tight, and the veins in his forearms were large. His dark glasses caught the bright sun, looking hot enough to sear flesh. The red arrow tattoos on his deltoids were as bright as arterial blood. I wondered if the idiots in the Toyota knew how close they'd come.
I said, "It isn't just Angela Rossi, is it?"
Pike's head moved from side to side one time.
"You don't like the cops we know thinking that we're part of this. You don't like people thinking that you and I believe this garbage or had a part in destroying an innocent woman's life."
Pike's head moved again. Just a bit. Just the smallest of moves.
"But that's the way it looks."
Pike's jaw rippled with tension.
We went to a Thai place a few blocks up from the beach. It was still shy of noon when we parked at the curb and went in. Early. It's tiny place with beat-up Formica tables, and it was empty except for two women sitting at the single window table. The young guy who greeted us said we could sit where we liked. An older woman who was probably his grandmother was sitting at the table nearest the kitchen, snapping the stems off an enormous pile of snow peas and watching a miniature Hitachi television. She smiled and nodded, and I smiled back. I have never been in their restaurant when she was not snapping peas. We took a table near her, ordered two Thai beers, squid pad thai, vegetable fried rice, and seafood curry. The little woman was watching the midday news as she worked. Something about the Middle East.
The beer came and I said, "Joe, I'm thinking that there is something larger here than an attorney's zealous defense of his client." The master of understatement.
Pike cocked his head toward me.
I told him about the connection between James Lester and Elliot Truly, and about Lester's record. "Lester could be for real, and his tie to Truly could be a coincidence, but maybe it isn't. Pritzik and Richards were killed before Lester called the hotline."
"Are you thinking he knew that?"
"Say he knew them better than he let on. Say he knew that they had gone to Arizona and were dead, and figured that they would be the perfect crash-test dummies to take the heat for Susan Martin's murder. Lester may have done a little homework and planted the evidence himself to take a shot at the reward."
"Or Truly might have helped him."
I nodded. "Just thinking out loud."
"Because you have no proof." The veins in his arms weren't as prominent, and his tattoos had lost their glow. The danger of thermonuclear meltdown was passing.
I shook my head. "No. Lester could be on the level, even though he's a creep."
"What about the woman?"
"Louise Earle is different. Kerris went to see her, and now she's changed her story. I don't buy that she was lying to me, and I don't buy that Rossi held a gun to her head and made her lie six years ago. Rossi wouldn't have done that, and Louise Earle wouldn't have lied about it."
"If she wasn't lying then, she's lying now."
"Yes. But why?" The waiter brought our food, and the smells of mint and garlic and curry were strong. He set out the dishes and said, "We make spicy. Like always."
"Great."
When the waiter was gone, Pike said, "Because the law is war, and to defeat the prosecution Green must do two things. He must float a viable theory for what happened to Susan Martin, and he must discredit the prosecution's theory."
"Okay."
"Lester gives him the alternative theory. The business with Rossi gives him a way to discredit the prosecution's evidence."
"If Rossi framed LeCedrick Earle, she's also framing Teddy Martin."
Pike nodded. "Yes." Pike twisted toward the Hitachi and said, "Listen."
Jonathan Green was on the noon news. The lead story was Elliot Truly's connection to James Lester, also known as Stuart Langolier. Green was announcing that James Lester had revealed to a defense investigator that he had once been known as Stuart Langolier and, under that name, had once been represented by Elliot Truly. Green said that it was his understanding that Mr. Truly had no recollection of Mr. Lester as a client and added that the defense team had immediately notified the district attorney's office to mitigate the appearance of a conflict and to allow them the opportunity for a complete investigation. I said, "He's doing just what he said."
Pike grunted. "Covering his ass."
The little woman noticed that we were watching the TV and turned the Hitachi so that it would be easier for us to see.
The news anchor shifted the story to the charges against Angela Rossi and cut to the same tape of Louise Earle that I'd seen last night, Mrs. Earle crying as she charged that Angela Rossi had framed her son, saying that the police had made her lie before, saying that they had threatened her. The tears looked real. Her pain looked real. Jonathan Green was standing next to her. Elliot Truly was standing behind them. Everyone looked oh-so-concerned.
Pike turned away. "I can't look at this."
I stared at the Hitachi. I watched Green and I watched Louise Earle, and it just didn't make sense. "If what we're thinking about Lester and Louise Earle is true, why would a guy like Jonathan Green risk who he is and what he does?"
"Because he's an asshole." The world according to Pike.
I said, "Lizard people."
Pike's glasses gleamed. "We can talk about this forever, but the only way we're going to find out what's going on with these people is to ask them."
The young waiter was watching us. He didn't like it that we hadn't touched the food, and he looked concerned. He said something to the little woman. She frowned at us and seemed to share his concern.
The waiter came over and wanted to know if anything was wrong. Pike looked at him and stood. "Probably. But if there is we'll fix it."
We picked up the Santa Monica Freeway and drove to Louise Earle's home in Olympic Park. We knocked twice, and rang the bell three times, but she didn't answer. Pike said, "I'll look in back."
Pike disappeared around the side of the house. The day was bright, and the same three girls were across the street, whiling away their summer on their porch. I waved and they waved back. Getting to be old friends. Pike reappeared from the opposite side. "She's not home."
"Then let's see Lester."
We climbed back onto the freeway and worked our way east past Pasadena to La Puente and James Lester's house.
Lester's home was unchanged from the last time I was there. The yard was still dead, the Fairlane was still rusted, and everything was still covered with fine gray sand. We parked at the curb and walked across the gray soil to the house. The front door was open, and music was coming from the house. The George Baker Selection doing "Little Green Bag." When we got closer, Pike said, "Smell it?"
"Yep." The sweet rope smell of hashish was coming from the house.
When we reached the door we didn't have to knock. Jonna Lester was sitting on the couch, sucking hard on a glass pipe, the little electric fans arcing back and forth as they scattered her hash smoke. She was wearing a Michigan State University T-shirt and short-shorts and the clear plastic clogs. Her left eye was red and blue and swollen almost closed, and the bottoms of the clogs were crudded with something dark, as if she'd stepped through mud. She smiled stupidly when she saw me and waved the pipe at her eye. "Helps with the pain. You wanna smoke a bowl?"
I opened the screen door and we went in. There was another smell in the room, just beneath the dope. I tilted her face to better see the eye. "James do this?"
She pulled away from me and waved the pipe again. "It'll be the last time, yessireebob." She took another pull on the pipe.
"We need to see him."
Jonna Lester giggled. "He's in the bathroom. It's his favorite room in the house. He always said that." She giggled again.
"Would you tell him we want to see him, please?" The other smell felt wet and old, like melons that had gone soft with age.
Jonna Lester sank back on the couch. "This is such a cool song."
Joe Pike walked over to the radio and turned it off. Jonna Lester screwed up her face and said, "Hey!"
I called, "James?"
Jonna Lester pushed to her feet and angrily waved toward the back of the house. "He's back there, you wanna see the sonofabitch so bad. C'mon, I'll show ya."
Pike and I looked at each other, and then Pike took out his .357 Python and held it down along his leg. We followed her out of the living room and across a square little hall to the bathroom. It was an old bathroom, built sometime back in the fifties, with a buckled linoleum floor and corroded fixtures and a brittle glass shower door, the kind that can hurt you bad if you fall through it. Jonna Lester stopped in the door and waved the hash pipe. "Here he is. Talk to the sonofabitch all you want."
I said, "Oh, man."
James Lester was lying through the broken shower door, half in the tub and half out, impaled on half a dozen jagged glass spikes. His head was almost severed, and the walls and the tub and the buckled linoleum were sprayed with gouts of dark red blood that looked not unlike wings raised toward heaven.
We had wanted to ask James Lester about Pritzik and Richards and the fabrication of evidence, but now he wasn't around to answer our questions. Neither were Pritzik and Richards.
Funny how that works. Isn't it?
#
I got as close to the body as I could without stepping in blood. Jonna Lester's footprints were already on the linoleum from an earlier visit, but there didn't seem to be any other marks or tracks or signs of passage. There was a single small window at the far end of the bathroom above the toilet, open for the air. The window's screen was dirty and torn, but was hooked from the inside and appeared undisturbed. Metallic black flies bumped against the screen, drawn by the blood. I said, "Did you touch anything?"
She said, "Yee-uck! I ain't touchin' that mess."
"Your footprints are on the floor. There's dried blood on your shoes."
Jonna Lester took another pull on the hash pipe. The hash nut must've gone out, because she frowned at the pipe and poked the bowl. "I hadda turn off the water." One of the black flies worked its way through the screen and droned low across the slick floor. You could see its reflection in the blood.
"The water in the sink was running?"
"Yeah."
James Lester was wearing pants and the work boots, but no shirt. Both legs and one arm were crumpled in a kind of K on the floor, with the other arm and the upper half of his body hanging through the glass into the tub. There was water on the linoleum around the base of the sink where it had spilled over and mixed with James's blood. A bar of soap and a Bic razor and a can of Edge shaving cream were on the sink, which was splashed with water, like maybe he had been getting ready for work and turned and slipped and gone head first through the glass. I said, "What happened, Jonna?"
She shook her head. "I spent the night with my friend Dorrie, and he was like this when I came home. I guess he fell." She made a big deal out of showing me her eye. "The prick did this to me yesterday. You see what he did?" She shook her head and her lips went _wubba-wubba-wubba_ like a cartoon character. "Oh, man, doesn't that smell just make you wanna vomit?"
She went back into the living room, and we followed her. She tried stoking the pipe again, and I pulled it away from her. "Hey, whatcha doin'?!"
"He's dead, Jonna. A material witness in a murder case who stands to collect a hundred thousand dollar reward doesn't just fall through a shower door."
Jonna Lester slapped at me and tried to push me away. "We had this big fight yesterday and I hadda get outta here! I don't know what happened!"
"Was he expecting anyone?"
"I don't know!"
"Did he mention anyone to you, like maybe he was concerned?"
She put her hands over her ears. "I don't know I don't know I don't know!" Shouting.
I stepped back, breathing hard, and let her calm. I looked at Pike and Pike shrugged. I took a breath, let it out, then sat next to her. I said, "Okay, Jonna, what did you guys fight about?" Calm.
"We fought because he's an asshole!"
"Was it because you blew the whistle to me about James being Stuart Langolier?"
She froze for a moment, and then she squinted at me. Suspicious. "I don't know what you're talking about."
"C'mon, Jonna. I recognized your voice. Why'd you tip me about James's real name?"
She slumped back on the couch and stuck out her lower lip. Sulking. "James Lester was his real name. He changed it legally to get a fresh start when he gave up his life of crime."
I said, "Jonna."
"I did it to fuck him." Her voice was soft and petulant.
"Why?"
" 'Cause he was gonna cut me out. I know it."
"How do you know it?"
" 'Cause he said that when he got the big payday he was gonna blow me off and get a Bud Lite girl." Her eyes were welling in a delicate balance at the edge of tears. The point of her chin trembled.
Pike walked away. He has little tolerance for the vagaries of the human condition.
I said, "Jonna? What else do you know?"
"What do you mean?" She rubbed at her eyes. When she touched the bruised eye she winced.
"He may not have been telling the truth. He might have made up the story about meeting Pritzik in a bar. I think maybe he planted the things I found in order to collect the reward, or someone else planted them and James was in on it."
She shrugged, even more sulky. "I dunno."
"Did he know Pritzik and Richards? Did he tell you how he was going to set this up?"
She suddenly sat up, loud and animated. "Hey, I'm still gonna get the reward money, ain't I? I mean, I get it now that he's dead, right?"
Pike said, "Forget the reward. You'll be lucky if you don't go to jail." Pike, the Intimidator.
Jonna Lester's eyes filled again and this time the tears leaked down her cheeks. "Well, that's no fair." No fair.
I said, "Tell me about Pritzik and Richards."
She shook her head. "I don't think he knew them. I mean, he _coulda,_ but I don't think so."
"Why not?"
Shrug. " 'Cause he didn't have any friends. Just this guy from the video store and Clarence at the transmission shop. Clarence is a Mexican."
I glanced at Pike, but Pike was staring out the front door. Intimidating the neighborhood. I said, "Maybe he mentioned a buddy who worked at a Shell Station or an ex-con he would have drinks with."
She shook her head. "He just went out with Clarence. I know 'cause I followed him."
"You followed him." The detective using advanced interrogation techniques.
She made the kind-of shrug again. "When he started all that talk about gettin' a Bud Lite girl I got worried he might be doin' more than drinkin' when he went out."
"And all he ever did was go out with Clarence?"
Her head bobbed. "Uh-huh."
"How many times did you follow him?"
"Eight or nine." She thought about it. "Maybe ten."
I described Pritzik and Richards. "You ever see him with guys like that?"
Another head shake. "Nuh-uh. James and Clarence would just sit there and drink, and sometimes play video games." Another big fly cruised through the room, this time passing between us before heading toward the bathroom. Jonna Lester watched it, realized where it was going, and made a face. "Oh, yuck."
Pike followed the fly and closed the bathroom door.
I walked over to the front door, stared out at the hot earth, then went back to James Lester's chair and sat. Maybe James hadn't known Pritzik and Richards. It was still possible that he had, but if he hadn't then he wouldn't have been able to fake the evidence. He wouldn't have known they were dead. He wouldn't have known where to plant it. Maybe James had been telling the truth. Of course, maybe his dive through the shower glass was an accident, too.
Jonna Lester got the hey-waitaminute-! look again, then frowned as if she was trying to see shadows within shadows and not having a lot of luck with it. She wiggled her finger in the air and said, "I take it back! There was another guy I saw him with."
I stared at her.
"This time that I followed him, he went to the Mayfair Market over here and talked to this guy."
Pike crossed his arms and looked at me. Well, well.
"A guy in the Mayfair?"
"A guy in the parking lot. I thought he was going to the store, but he just parked there in the lot and went over to this other car. James just kinda squatted by the driver and talked through the window, and then this guy gives him a bag and James left."
"The man in the car gave him a bag?"
"Mm-mm. Like a Mayfair bag. Brown paper."
"When did this happen?"
Her lips made a tight line. Her eyebrows jumped up and down. Time sense distorted by all the hash. "A long time ago. Two or three weeks."
I looked at Pike again, and Pike's mouth twitched. It could've been after Pritzik and Richards were killed and before James Lester phoned the hotline. Maybe we were getting somewhere.
I said, "What did the guy look like, Jonna?"
"Like a guy. I was behind them and he didn't get out."
Pike said, "What kind of car was it?"
"I don't know anything about cars. It was little."
"What color?"
She frowned. "Dark blue. No, waitaminute. I think it was black. A little black car." She was nodding like she could see it.
I said, "Did James ever mention someone named Elliot Truly to you?"
She shook her head. "Who's that?"
"Truly was James's lawyer in San Diego."
She shook her head again. "Nuh-uh."
I looked around their living room. I dug through the comic books and monster truck magazines, and looked under the couch. I finally found four days' worth of the _Los Angeles Times_ at the bottom of a plastic trash can in the kitchen. I found the one with my picture and brought it out to her. You could see Elliot Truly clearly behind me and Jonathan Green. I pointed at Truly. "Was this the man in the car?"
Jonna Lester shook her head. "Oh, no. He didn't look anything like that."
I pointed at Green. "Him?"
"Oh, no. Not him, either."
I glanced at Pike and Pike shrugged. He said, "Could've been anybody about anything. Doesn't have to relate to this. Maybe he was buying the hash."
Jonna Lester's pout had come back, and now it was rimmed with petulance. "Look, I've been trying to help, haven't I? All those news people said it looked like we were gonna get the reward, and I think we still should. I mean, even though he's dead he's still due the reward, and that means I should have it, right?"
I stared at her.
"Well, it's only right. You're only guessing that he made it up, and even if he did you can't prove it. I don't think he made it up at all. I think he was telling the truth, even if he was a lyin' no good sonofabitch."
I said, "Jonna, in about two minutes you're going to call the police. Do yourself a favor and don't tell them how much you should get the cash."
The pout edged over into full-blown petulance. "Well, why not?"
Pike said, "Because with all the remorse you're showing, they'll think you killed him for the money. You don't want them to think that, do you?"
Jonna Lester slapped hard at the couch, then threw the glass pipe to the floor. She stamped both feet. Mad. "Life really sucks."
"That's true," I said. "But think of it this way."
She squinted at me, and I glanced toward the bathroom.
"Death sucks worse."
#
Jonna Lester dialed 911, identified herself, and told them that she'd found her husband dead of an apparent bathtub accident. Jonna related the facts as I outlined them, and the operator said that the paramedics were on their way.
I made Jonna dump her hash down the disposal and spray Lysol to kill the smell. Flushing it down the toilet would've been better, but I didn't want anyone in the bathroom. Evidence. I had her wash her mouth with bourbon; if she acted goofy or giggly, they'd smell the booze and figure her for a drunk. The paramedics arrived first, then the police. A uniformed sergeant named Belflower shook his head when we told him who James Lester was and said, "Hell of a thing, ain't it? Guy stands to collect a hundred grand and he gets his neck slit from slipping on a bar of Ivory."
I said, "You think?"
He frowned at me. "You don't?"
We stared at each other until he went out to his squad car and called the detectives. Pike and I stayed until the police were satisfied that Jonna Lester had found the body on her own and that we had stumbled in later, and then they said we could go.
We stopped at an Arco station two blocks away where I used the pay phone to call a friend of mine who works at the Medical Examiner's office. I told him that James might've had help falling through the glass, and I asked if he might share his findings after the autopsy. He said that such a thing might be possible if I was able to share four first-base-side tickets to a Dodgers game. I said, "I don't have first-base-side tickets to the Dodgers."
My friend didn't say anything.
"But maybe I can find some."
My friend hung up, promising to call.
I dropped Pike off, and it was twenty minutes before seven when I arrived home.
Lucy's rental was wedged on the far left side of the carport, silent and cool in the deepening air. The far ridge was rimmed with copper and bronze, and honeysuckle was just beginning to lace in and around the musky scent of the eucalyptus. I stood at the edge of the carport and breathed deep. I could smell the grease and the oil and the road scents of my Stingray mixing with the smells of the mountain. I could feel the heat of its engine, and hear the dings and pops of the cooling metal. The house was quiet. A horned owl glided across the road and down along the slope, disappearing past the edge of my home. Insects swirled over the canyon, erased by the dark blur of bats. I stood there, enjoying the cooling air and the night creatures just beginning to stir and twilight in the mountains. Home is the detective, home for the night. Sandbagged, unemployed, and feeling more than a little suspicious.
I let myself in through the kitchen. Lucy was on the couch in the living room, reading _Los Angeles Magazine_. Ben was on the deck, sitting cross-legged in one of the deck chairs, reading Robert A. Heinlein's _Have Spacesuit, Will Travel_. There wasn't much light, and he would have to come in soon. I said, "Another strange day in Oz, Lucille."
Lucy closed the magazine on a finger and smiled, but the smile was small and uncertain. "We got back around four."
"Sorry I'm so late."
"It's okay." She made a little shrug, and in that moment I wondered how much of the tension from last night was still with us.
"Are you two starving?"
Lucy made the uncertain smile again as if she recognized the tension and was trying to soften it. "I made Ben a snack a little while ago, but we could eat."
"How about I make spaghetti?"
"Oh, that would be nice."
I went into the kitchen, popped open a Falstaff, and took a package of venison sausage from the freezer. I filled a large pot with enough water for the spaghetti, dropped in the sausage, then put on the heat. I heard the glass doors slide open and Ben yelled hi. I yelled hi back. I heard Lucy tell Ben that dinner would be ready soon and that he should take a bath. I heard the guest room door close and water run. The sounds of other people in my house.
I drank most of the Falstaff, then examined the cat's tray. Crumbs of dry food speckled the paper towel around his food bowl and a hair floated in his water. He'd probably slipped down the stairs during the day when no one was home, eaten, then made his escape. I tossed the old food and water, put out fresh, and wished that he was here.
I finished the Falstaff, then opened a bottle of pinot grigio, poured two glasses, and brought one to Lucy. She was still reading the magazine, so I put the wine on the table near her. I said, "I meant to get home sooner, but Rossi's in pretty bad shape, and the day just sort of grew from there." I didn't tell her about James Lester. Lester would bring us back to Green, and I didn't want to go there. "I was hoping that we'd have more time together."
Lucy's face grew sad and she covered my hand with hers. "Oh, Studly, I know you can't be with us every moment. It's okay."
"It doesn't seem okay."
Lucy stared past me and the sadness grew deeper. She wet the corner of her mouth as if she were going to say something, then shook her head as if changing her mind. "There's a lot going on right now, Elvis, but it doesn't have anything to do with us."
"Can we talk about it?"
She wet the corner of her mouth again, but she still didn't look back at me. She was staring at a point in midspace as if there was a third presence in the room, floating in space and demanding the weight of her attention. "I'd really rather not. Not now."
I nodded. "Okay. Up to you."
She looked back at me and made the little smile again, and now it was clearly forced. "Let me help you cook. Would that be okay?"
"Sure."
We went into the kitchen and collected things for the spaghetti sauce and talked about her day. We chopped mushrooms and onions and green peppers, and opened cans of tomatoes and jars of oregano and basil, and talked as we did it, but the talking was empty and forced, the way it might be if there was a distance between us and we had to shout to make ourselves heard. I asked how her meetings had gone and she said fine. I asked if she was finished with the negotiation, and she said that a final meeting tomorrow would do it. Ben came in and parked on one of the counter stools, but he seemed to sense the tension and said little. After a time, he went into the living room and turned on my Macintosh and went on-line.
We had just put the spaghetti in boiling water and were setting the table when the doorbell rang. I said, "If it's a reporter, I'm going to shoot him."
It was Joe Pike and Angela Rossi. Rossi looked ragged and uncertain, and there were great hollow smudges beneath her eyes. Lucy stared soundlessly from the kitchen, and Rossi glanced from her to me. "I hope you don't mind."
"Of course not." I introduced them.
Angela Rossi glanced at Lucy again, and in that moment there was something very female in the room, as if Rossi somehow sensed the tension and felt that she was not so much invading my space but Lucy's. She said, "I'm sorry." To Lucy, not to me.
Lucy said, "We were going to eat soon. Would you like to join us?" She was holding the sauce spoon over the pan, frozen in mid-stir.
Rossi said, "No. Thank you. I can't stay very long." She smiled at Ben. "I have children."
"Of course." Lucy put the sauce spoon on the counter, then excused herself and took Ben out onto the deck.
We watched the glass doors slide shut, and Rossi looked even more uncomfortable. "Looks like I've come at a bad time."
"Forget it."
Pike moved behind her. He hadn't yet spoken, and probably wouldn't.
Angela Rossi looked at the floor, then looked at me, as if her energy reserves were so depleted she had to conserve what little remained. She said, "Joe told me about Lester. He told me what you've been trying to do."
I nodded.
"I lost it this morning and I want to apologize. You're caught in this, too, just like me."
"Yes, but it's worse for you."
"Maybe." She looked at the floor again, then looked back. "I want you to know that I didn't lie to you. I want you to know that everything I told you was the truth. LeCedrick Earle is lying, and so is his mother. I didn't do those things."
"I believe you, detective."
When I said it her breath gave and her eyes filled and her face collapsed, but in that same instant she caught herself and rebuilt the calm cop exterior: her breathing steadied, her eyes dried, her face calmed. It wasn't easy to recreate herself that way, but I imagined that she'd had plenty of practice over the years and that, as with every other professional police officer that I'd known, it had become a necessary survival skill. She had allowed a window to her heart to open, then had slammed it shut the way you take a covered pan off the fire when it begins to boil over, removing the heat so that you don't lose the contents. "I'm suspended. I've been ordered to stay away from all official police business or activities pending an IA investigation. The district attorney's office is also investigating me."
"I know."
"The people I work with, there's only so much they can do."
I knew that, too. If Tomsic or the others did anything to find out what was going on, they'd be pounded for obstructing justice and probably accused of trying to cover up Rossi's alleged crimes.
She looked at Joe. "You guys offered to help. Joe said that the offer still stands."
"Of course." I glanced at Lucy on the deck. She and Ben were at the rail. Ben was pointing at something far down the canyon and yakking, but Lucy seemed neither to hear him nor to see. As if the other presence were out there, too, and drawing her attention. I felt my own eyes fill, but, like Angela Rossi, I also knew the tricks of survival. "We're not going to walk away, Angie. We're not going to leave you hanging."
Angela Rossi looked at me for a time, first in one eye and then the other, and then she glanced again at Lucy and Ben. "I'm sorry I intruded."
"Don't worry about it."
She put out her hand. We shook, and then Angela Rossi left my home.
Joe Pike stood in the entry, staring out onto the deck, as if he, too, could somehow sense the tension. Maybe I should just put up a huge sign: DOMESTIC PROBLEM. I said, "What?"
Pike stared a moment longer, then turned and followed Angela Rossi, leaving me in the shadows.
I went back into the kitchen, stirred the sauce, then turned off the heat. The spaghetti was limp and swollen. I poured it in the colander, rinsed it, and let it drain.
I could see Lucy and Ben in the light at the rail, haloed by a swirl of flying insects, Lucy still there but not there, Ben now quiet. The cat door made its _clack-clack_ behind me, and the cat crept in. He moved cautiously, pausing between steps, sniffing the air. I smiled at him. "It's okay, bud. They're outside."
He blinked at me, but you could tell he was suspicious. He crept to the dining area, still testing the air, then came back and stood by my feet. I broke off a piece of the venison sausage, sucked off the tomato sauce, then blew on it until it was cool. I offered it to him, and as he ate it I stroked him. His fur was flecked with dust and bits of plant matter, and felt cool from the night air. White hairs were beginning to show through the black, and I wondered how old he was. We had been together a long time.
When he was finished he looked up at me, and I smiled. I picked him up and held him close, and after a time he purred. I said, "Life is complicated, isn't it?"
He licked my cheek, then bit my jaw, but he didn't bite hard.
After a time he hopped down and made his way through the house. He moved slowly, staring toward the deck for a very long time before finally bolting up the stairs and into my bedroom.
I told Lucy and Ben that dinner was ready. We ate, and not long after that we doused the lights and went to bed.
Since Lucy did not come upstairs that night, the cat slept well.
#
The next day Lucy and Ben planned to spend the morning in Beverly Hills, then make the drive to Long Beach for what Lucy hoped would be the final meeting of her negotiation. They were leaving the day after tomorrow.
We made banana pancakes and eggs and coffee, and ate together, but Lucy still seemed pained and distracted as she readied to leave. I found that I was thinking more about her and less about me, but neither of us seemed to be making much progress toward a resolution. Of course, maybe this was because we had so far successfully avoided talking, and maybe the time for talk-avoidance had passed. The ducking of communication rarely leads to a resolution. I said, "What time do you guys expect to be home?"
"Sixish." Lucy was replacing her files in her briefcase. "I don't expect that anything will hang us up in Long Beach."
"Good. I'm going to take us someplace special for dinner."
She smiled at me. The soft smile. "Where?"
"Surprise."
We held each other's gaze for the first time that morning, and then Lucy put out her hand. Her skin was warm and soft, and touching her made me tingle. "A surprise would be nice."
"Leave everything to me." Elvis Cole, Master of the Universe, turns on the ol' charm.
They left the house at ten minutes before nine, and then I phoned my friend at the coroner's office. The autopsy of James Lester had been completed, and when I asked after the cause of death, he said, "The guy took a header through the glass, and he was still alive when he made the fall. You want to know just what was severed and how?"
"Not necessary. Was there an indication that he might've had help going through the glass?"
"You mean, like, did someone beat the hell out of him first, then push him through?"
"That's one way to put it." I could hear papers rustling in the background, and laughter. Someone sharing a big joke to start the day at the morgue.
"Nah. No sign of blunt-force trauma. No bruising, cuts, or scrapes that would indicate a physical altercation."
"Hm." So maybe it wasn't murder. Maybe James Lester was just clumsy.
"But we did find one thing that was odd." Maybe James Lester wasn't just clumsy after all. "There's a pattern of subcutaneous capillary rupture over the carotid area on his neck."
"That sounds like bruising."
"It's not the kind of bruising you'd ever see, and it wasn't caused by impact trauma."
"So no one hit him."
"You see stuff like this when someone vomits or has a coughing fit. Coughing can do stuff like that. You'd be surprised what coughing can do." These medical examiners.
I was thinking about the carotid artery, and I was trying to imagine a type of force that might rupture microcapillaries without creating an impact bruise. "Are you saying that he was strangled?"
"Nah. Bruising would be severe."
"Could he have been strangled in a way to avoid the bruising?"
He thought about it. "I guess he could've been strangled with something soft, like a towel, or maybe choked out, like with a police choke hold. That might show a rupture pattern like this."
"So he could've been choked out, then tossed through the glass."
"Hey, you're saying it, I'm not. We're just speculating."
"But it's possible."
"It's possible the guy swallowed wrong, started coughing, then lost his balance and went through the glass."
I didn't say anything.
"But, yeah, he could've been choked out, too."
I hung up, then called Mrs. Louise Earle. Her answering machine answered, and I said, "Mrs. Earle, this is Elvis Cole. If you're there, would you pick up, please? We need to talk." I was hoping to catch her before she started her day. I was hoping to convince her to see me.
No one picked up.
"Mrs. Earle, if Angela Rossi or any other police officer threatened you, I wish you would've told me. I'd sure like to hear about it, now."
Still no answer.
I hung up, then once more made the drive to Olympic Park. If I couldn't get her on the phone, I would try to see her in person. If she wasn't home, I would wait. What better way for an unemployed detective to fill his day?
The streets were still heavy with morning traffic, and the day was bright and hot, but a marine cloud cover had rolled across the basin that made the light seem sourceless and somehow disorienting and had charged the air with a kind of vague dampness. It was as if the sun had vanished and the landscape was lit by a weird kind of indirect lighting that made Los Angeles take on a 1950s tract-home fluorescent reality.
I parked two houses down from Louise Earle's, walked back, and rang her bell exactly as I had done yesterday. Still no answer. I stepped through the dozens of plants and peeked through the gap between the curtains of the same front window. What I could see of the room appeared unchanged from yesterday. Hmm. It was twenty-five minutes after nine, and I stood at the edge of Louise Earle's porch and wondered what I should do. The neighborhood looked calm and ordinary; maybe Louise Earle had simply run to the market and would soon be back. Of course, even it she wasn't back soon, it didn't matter a whole hell of a lot. Such are the joys of unemployment.
I went out to my car, put up the roof to cut the sun, and waited. It was hot, and, as the sun rose, it grew hotter. Sweat leaked out of my hairline, and my shirt stuck to my chest and back. A couple of Hispanic kids pedaled by on mountain bikes, both kids sucking on Big Gulps. A thin brown dog trotted behind them, the dog's tongue hanging from its mouth. The dog looked hot, too, and was probably wishing one of the kids would drop his drink. A Carrier Air Conditioning van pulled into a drive on the next block. Probably making an emergency call. An elderly man came down the sidewalk a few minutes later, covering his head with a _Daily News_ the way you would if it was raining and you were trying to stay dry.
Two of the three girls showed up in their Volkswagen Beetle, pulled into their friend's drive, and honked. Guess it was too hot to go to the door. The third girl came out with her bag and an orange beach towel and jumped into the Beetle. As they drove away, they waved, and I waved back. Guess the third girl had noticed me when she was watching for her friends. People came and went, and when they did they raced between air-conditioned cars and air-conditioned homes at a dead run. No one stayed in the heat any longer than they had to, except, of course, for displaced private eyes working on a slow case of dehydration.
Louise Earle still had not returned two hours and twenty-one minutes later, when a very thin white woman wearing an enormous sun hat emerged from the house next door and crossed her yard to Louise Earle's porch. I made her for her late seventies, but she might've been older. She rang the bell, then peered through Louise Earle's window just as I had done. She tromped around to the side of the house, came back with a watering can, and began watering the plants. I got out of the car and went up to her. "Pardon me, ma'am, but Mrs. Earle doesn't wish to be disturbed." The detective resorts to subterfuge.
She stopped the watering and squinted at me. "And who are you?"
I showed her my license. You show them a license and everything looks official. "The news people were bothering her, so I've been hired to keep them away."
She made a little sniff and continued with the watering. Guess she didn't give too much of a damn whether I was official or not. "Well, my name is Mrs. Eleanor Harris and I can assure you that Louise Earle does not consider me a bother. We've been friends for forty years."
I nodded, trying to seem understanding. "Then you must've seen how awful the news people were."
The stern look softened and she resumed the watering. "Aren't they always, though. You watch the way these people on television act and you wonder how they can live with themselves. All that smug attitude." She made a little shiver. "That Geraldo Rivera. That horrible little man on Channel Two. Ugh." She shook her head in disgust and the stern look came back. "You should've been here yesterday. Yesterday is when people were trying to bother her."
"They were?"
She squinted harder. "You know, one of them looked an awful lot like you."
"I came by yesterday to introduce myself, but she wasn't home. I came with my partner, a tall man with dark glasses."
The squint relaxed, and she nodded. "Well, you and your partner weren't the only ones. There were others. One of them even tried to get into her house."
I looked at her. "Who tried to get into her house, Mrs. Harris?"
"Some man." Great. "I remember him because he came three different times. You and your friend came the once. All the different press people came the once."
"What did he look like?"
She made a waving motion. "He was pretty big. You'd better watch out."
"Big." I put my hand a couple of inches above my head. "Like this?"
"Well, not tall, so much. But wide. Much wider than you." She gave me a just-between-you-and-me look. "His arms were so long he looked like a monkey." Kerris.
"And he was here three times."
She was nodding. "The first time was before you and your friend, then he came back in the afternoon and once more at dusk. When he was here in the afternoon he tried the door and he went around back. He was back there for quite a while, and for all I know he got in. For all I know he did all manner of horrible things in there." She made the little shudder again, equating all manner of horrible things with Geraldo Rivera and the little man on Channel Two. "It's a good thing Louise went away."
"No one told me that she'd gone away."
Mrs. Harris continued with the watering. "Well, no one told me, either, and that is highly unusual. We've been friends for forty years and I always water her plants when she's away. We watch out for each other. Older people have to."
I looked more closely at the plants. Some of the leaves were wilting and the soil was dry and beginning to crack. "Do you know where I can find her?"
Mrs. Harris continued with the watering and did not answer.
I said, "Mrs. Harris, I can't keep people away from her if I'm here and she's somewhere else. Do you see?"
The water can wavered, and then Mrs. Harris looked around at the drying plants and seemed lost. She shook her head. "She always calls when she goes away. Why wouldn't she call?"
I waited.
Mrs. Harris said, "I saw her leave and it just wasn't like her, let me tell you. It was the day before yesterday, the evening after all those horrible people were here, and she just walked away."
I thought about it. "Could she have gone to visit Mr. Lawrence?"
"Not walking. Mr. Lawrence would always come in the car."
"Do you know where Mr. Lawrence lives?" I thought I might drive over.
"I'm afraid I don't. I saw her from the window, dressed very nicely and carrying her bag, walking right up this street, and in all this heat, too." She made her lips into a thin, wrinkled line. She was holding the can with both hands, and both hands were twisting on the handle. "I came out and called after her. I said, 'Louise, it's too hot for all of that, you'll catch a stroke,' but I guess she didn't hear." The thin lips were pressing together. Worried. "People our age are very sensitive to this heat."
"Yes, ma'am. And she didn't call."
Mrs. Harris looked at me with wet, frightened eyes. "You don't think she's mad at me, do you? We've been friends for forty years, and I just don't know what I'd do if she was mad at me."
"No, ma'am. I don't think she's mad." I was wondering why she might be in such a hurry that she would just walk away.
"But why wouldn't she call? I always water her plants."
"I don't know, Mrs. Harris. Maybe she was just trying to get away from the press. You know how horrible they are."
Her eyes brightened a bit, drawing a little hope. "Yes. Yes, I'm sure that must be it."
"I'm sure she'll be back soon."
The ancient eyes finally smiled, and she turned back to the plants.
"When you find her, you'll keep them away from her, won't you? It must be awful, having people like that around."
"Yes, ma'am. I'll take good care."
I helped Mrs. Harris water the remainder of Louise Earle's plants, and then I went back to my car, wondering why Kerris had come three times, and wondering if his coming around had had anything to do with her going away. If he had come here three times, that meant he very much wanted to see Louise Earle. Three times was a pattern, and if the pattern maintained, he might return again today. Of course, he might not, but I still didn't have a whole lot else to do.
I went back to my car, drove four blocks to a 7-Eleven, bought two large bottles of chilled Evian water, then drove back to Louise Earle's, parked on the next block behind the Carrier van so that Eleanor Harris couldn't watch me, and continued to wait.
Exactly twelve minutes after I pulled up behind the van and turned off my car, Stan Kerris returned, but did not stop. He was driving a Mercedes SL300, and this time he slowly cruised the block, peering at Louise Earle's house, maybe hoping to see if she was home. I copied his tag number, then pulled out the little Canon and took four quick snaps just as he turned the corner.
The Mercedes was small and black, and I was hoping that Jonna Lester would recognize it.
#
I drove south to a Fast-Foto in a minimall on Jefferson Boulevard about six blocks west of USC. A Persian kid was alone in the place, working at the photo processing machine. He said, "I'll be with you in a moment."
"I don't have a moment. I'll pay you twenty bucks if you stop what you're doing and take care of me now."
He eyed me like maybe I was pulling his leg, but he got up and came to the counter. I put the film on the counter between us. "There are only four exposures on the roll. I've got to make a call. If they're done when I get back, you get the twenty."
He wet his lips. "What size?"
"Whatever's fastest."
I used a pay phone in the parking lot to call Angela Rossi at home. She didn't answer her phone; her machine got it. Screening. "Detective Rossi, it's Elvis Cole. I think I might have something."
She picked up before I finished saying it. She sounded tired, but then she probably hadn't slept last night.
I told her where I was and what I was doing and what I had seen. I said, "Do you want a piece of it?"
"Yes." She said it without hesitation and without fear, the way someone would say it when they were still in the game.
"I have to show the pictures to Jonna Lester, first. Call Joe. Have Tomsic call Anna Sherman in the DA's office. If this is going where I think, everything will begin to happen very quickly."
"I'll be ready."
"I'll bet you will."
I hung up, then called Jonna Lester. She answered on the second ring, and I told her that I was on my way to see her.
She said, "But me and Dorrie was just goin' to the mall!"
"Go to the mall after. This is important, Jonna. _Please_." The detective stoops to begging.
"Oh, all right." Long and drawn out and whiny. "Dorrie wants to meet you. I told her you were really cute." Then she giggled.
I hung up and closed my eyes, thinking that only twenty-four hours ago she'd found her husband impaled on glass. Man. I called the information operator last and asked if they had a listing for Mr. Walter Lawrence. They did not.
The Persian kid was waiting at the counter when I went back inside. He had the four shots waiting, too. Fast-Foto, all right. He said, "That's all you wanted?" You could see the Mercedes clearly in three of the four pictures. You could see Kerris clearly enough to recognize him.
"That's all."
I paid him for the developing, gave him the extra twenty, then drove hard to the freeway and made my way across town to Jonna Lester. She and her friend, Dorrie, were waiting for me in a cloud of hash smoke so thick that I tried not to breathe. Jonna Lester giggled. "Y'see. I tol' you he was cute."
Dorrie giggled, too.
Dorrie looked so much like Jonna that they might've been clones. Same shorts, same top, same clear plastic clogs and dark blue nail polish. Same gum. Dorrie sat on the couch and grinned at me with wide, vacant eyes while I showed the pictures to Jonna. I said, "Have you ever seen this car?"
She nodded and popped her gum. "Oh, yeah. That's the guy James went to see." She didn't even have to think about it.
"The man at the Mayfair?"
"Uh-huh."
"The man who gave James a large paper bag?"
"Yup."
Dorrie said, "You wanna get high an' fuck?"
I went to the phone without asking and called Angela Rossi, who answered on the first ring. "A man named Stan Kerris met with James Lester twenty-three days ago, eight days before Lester phoned the hotline. Stan Kerris works for Jonathan Green. I think we can build a case that these guys have fabricated evidence and set you up."
Angela Rossi said, "That sonofabitch."
"Yes."
#
We agreed to meet on the second floor of Greenblatt's Delicatessen at the eastern end of the Sunset Strip at three that afternoon.
Angela Rossi was pacing in the parking lot behind Greenblatt's when I pulled up at two minutes before three. Rossi was wearing black Levi's and a blue cotton T-shirt and metallic blue Persol sunglasses. She was pacing with her arms crossed and her head down, and when she stopped to wait for me, she scuffed at the fine gravel on the tarmac with her shoes. I said, "Didn't you think I'd show?"
Rossi shook her head. "Too wired to sit. I think I'm going to vomit."
"Is Sherman here?"
"Yeah. She's not happy about it, and she's not happy about me being here."
I followed Rossi in past the deli counter and up the stairs to the dining room. This late in the afternoon Greenblatt's was mostly empty. Earlier, the upstairs dining room had probably been filled with wannabe television writers and ninety-year-old regulars and Sunset Strip habitués, but not now. Now, the only civilians were a couple of young guys with mushroom cuts and an African-American woman sitting alone with _People_ magazine. Everybody else was cops.
Linc Gibbs, Pete Bishop, Dan Tomsic, and Anna Sherman were sitting at a table as far from everyone else as possible. Gibbs had coffee, and Bishop and Tomsic had iced tea. Anna Sherman didn't have anything, and she was seated with her back to the restaurant, probably because she was concerned about being recognized. Tomsic said, "Here they are."
Gibbs and Bishop turned, but Anna Sherman didn't. I hadn't met Gibbs and Bishop before. Tomsic introduced us, but before he was finished, Anna Sherman said, "I want to make it clear that the only reason I'm here is because Linc and I have a history, and he's asked me to listen. I make no claims that anything said here is off the record. Is that clear?"
Tomsic scowled. "It's great you're on the right side in this."
Linc said, "Dan."
Tomsic crossed his arms and leaned back, his mouth a hard slash. Nothing like having everyone work to the same end.
Linc Gibbs hooked a thumb toward me. "As I understand it, we're here to discuss possible criminal wrongdoing on the part of the attorneys involved in Teddy Martin's defense. Is that it?"
"Yes. I believe that Jonathan Green or agents working on his behalf fabricated the James Lester evidence. I believe that Lester was in on it. I suspect that they also coerced Louise Earle into changing her story, but that's only a suspicion. I haven't been able to locate Mrs. Earle to ask her about it."
Anna Sherman pooched her lips into a knot. She was leaning forward on her elbows, arms crossed.
Gibbs said, "I thought you were working for these people."
"I quit yesterday."
Anna Sherman raised her eyebrows, saying let's hear it.
I said, "James Lester's real name was Stuart Langolier. Eight years ago, he was represented on a grand theft beef in Santa Barbara by Elliot Truly. That's prior association."
Sherman looked impatient. "Green's office notified us about that. It's even been on the news."
"James Lester's original call to Green's hotline was logged eleven days ago. Eighteen days ago, Jonna Lester followed James to a Mayfair Market where she saw him meet this man." I handed her the three snapshots that I'd taken of the black Mercedes. She looked at them.
Linc Gibbs frowned. "Looks familiar."
"Stan Kerris. He's the chief investigator for Green's office. She saw Kerris and Lester speak, then Kerris passed a shopping bag to Lester, who drove away."
Tomsic said, "Man."
Anna Sherman glanced at me, and Pete Bishop made a tiny smile. Gibbs held out a hand, and Sherman passed him the first of the three pictures, then the second. She stared at the third. "Jonna Lester identified him?"
"Yes. Green hired me to check out the allegations against Detective Rossi, then run down a series of tips he'd received via the reward hotline, one of which was from James Lester. I checked out Louise Earle and the allegations, and Rossi came up clean. I reported that to Jonathan Green, and he seemed to accept it."
Sherman chewed at the inside of her cheek as if she was thinking about leaving.
I tapped the photo she was still holding. "I took these photographs this morning outside Louise Earle's home. A neighbor saw Kerris visit Louise Earle's home three times yesterday, and I saw him there today. When I spoke with Mrs. Earle a week ago, everything she told me confirmed Rossi's version of her son's arrest and the subsequent LAPD investigation. Now she's suddenly changed her story and Kerris is living on her porch. First Lester, now Louise Earle. I think there's a connection."
Sherman passed the final photograph to Lincoln Gibbs and began ticking her right index fingernail on the table. "All right. What else do you have?"
"When the James Lester story hit the news, I wanted to stay after Pritzik and Richards, which would've been the natural thing to do, but Green had me work a dog and pony with the press. I now believe that it was a media manipulation to make Louise Earle's changing her story more credible to the public."
Bishop said, "I thought you were the guy who got her to change her story."
I shook my head. "That's part of the big lie. I saw her one time, and at that time everything she said confirmed Rossi's story. Three days later Stan Kerris pays her a visit and everything changes, and the next thing I know Green holds a press conference and says that I've turned up evidence to prove Rossi rotten. The wonder boy who showed up the cops and found James Lester now ferrets out the truth from the intimidated mother. You see?"
Anna Sherman continued ticking the nail. She stared at the table and made her mouth the small knot again. Then she looked up and shook her head. "All of you must be out of your minds."
Tomsic threw up his hands. "What does _that_ mean?"
The two kids with the mushroom cuts and the African-American woman looked over, and Lincoln Gibbs zapped Tomsic with a look that must've come from the days before he started affecting the professor image, flashing street eyes, mess-with-me-and-I'll-choke-your-eyes out.
Tomsic settled back.
Sherman said, "It means that if my office or the LAPD launched an investigation into Jonathan Green at this time based on this kind of bullshit evidence it would be a public relations nightmare."
Gibbs said, "This is worth something, Anna. You know it is. You can't just ignore it."
She leaned toward him, ticking off the points. "I spoke with Jonna Lester and I know her to be a hash head. Jonna Lester doesn't know if it was eighteen days ago or twenty-eight or just eight, which is exactly what Stan Kerris would say _if_ he admitted to having met James Lester, which he almost certainly won't." She ticked another point. "Then, if he did admit to such a meeting, he would say that it was a preliminary interview conducted prior to Mr. Cole's being assigned the follow-up, and, in case you've forgotten, Mr. Lester isn't around to dispute that statement."
I said, "Did you review Lester's autopsy report?"
"There was no sign of foul play."
"That isn't quite correct. Someone who was good could've choked out Lester, then put him through the glass."
Sherman's nostrils flared and she closed her eyes. "Could have."
"I know you can't go to court with that one, Sherman, but it fits with the theory. You really think Lester just happened to cut his own throat?"
Tomsic said, "Subpoena Lester's phone records, and pull Green's records, too. See who was calling who and when they were talking."
Sherman made a hissing sound.
Angela Rossi said, "No one's forgotten about Pritzik and Richards, either."
Anna Sherman shook her head. "You people are talking about accusing an attorney of Jonathan Green's stature of fabricating evidence without any substantive proof to back it up. With even less proof, you want me to accuse him of murder. Ask yourself this: why would Jonathan Green risk his career and his reputation and his freedom to falsify evidence for one client? The press is going to ask that, and you don't have an answer because it doesn't make sense." She glanced from cop to cop, finally coming back to me. It was exactly what I had asked Pike. "All you have are some unseemly coincidences and the testimony of a hash head. Jonathan Green will charge us with harassment, and he will bring us before the state bar, and I, for one, am tired of getting my ass handed to me in the _L. A. Times_ every day."
I said, "Is that it?"
She nodded.
I looked at Rossi. I looked back at Anna Sherman. I said, "Getting our asses kicked in the press is how we define truth in the American legal system?"
Anna Sherman stood. "My boss is being pressured to drop the charges against Teddy Martin. I've been fighting him on it because I want to see this through, but I don't think he has the balls. I think he'll give in because he has arrived at his own personal definition of justice. He defines it as political survival." Anna Sherman didn't say anything more for a time, then she looked directly at Rossi. "I'm sorry, but this meeting is now over." She tucked her purse under her arm and walked out.
Tomsic slapped the Formica hard, and Bishop made a soft whistling sound through his teeth. Angela Rossi had pushed her fists between her legs onto her chair and gently rocked. Finally, Bishop said, "So where are we, Linc?"
Lincoln Gibbs took a breath. "You heard her. The district attorney's office is not interested in pursuing this investigation."
Tomsic said, "That's bullshit." He jabbed the finger at me. "Cole's onto something! These bastards are over the line!"
Gibbs made his voice harder. "They will not pursue this line, Sergeant. That's the end of it."
Tomsic wasn't letting it go. Now he was waving both hands. "So Green can do whatever? He can murder people? He can rob banks? We just say, oh, we'll look bad if we do something?"
Lincoln Gibbs's nostrils were wide and hard and you could hear him breathe. But then the breathing calmed and he looked at Rossi. Sad. "Sometimes we have the worst job in the world. High-priced, sleazebag shysters make millions getting off murderers and dope dealers and the dregs of this society, but they are wrong, and we are right. And if we have to take some bullets along the way, then we take'm." He reached across the table and squeezed Angie's arm. "Goes with the job."
Tomsic said, "That's bullshit."
Linc Gibbs nodded. "Of course it's bullshit, Sergeant, but it's where we are." He looked at me. "Thanks, Cole. It didn't pan out, but we owe you for the effort."
Bishop got up, then Gibbs. Gibbs told Tomsic to come with them, and he told Angela Rossi he thought she should probably go home. He told her not to worry. He said that they weren't going to let it go, and that they would keep digging into the LeCedrick Earle thing and that they wouldn't abandon her. She nodded and got up and went with them, but she looked abandoned to me. Of course, maybe it was just my imagination.
I sat alone at the empty table for another three minutes, wondering what to do next and having no great surges of inspiration. I think I was feeling abandoned, too, but I probably wasn't feeling as abandoned as Angela Rossi.
I went down the stairs and out the back of Greenblatt's
to my car. Anna Sherman was sitting in the passenger seat, waiting for me. A bead of sweat worked its way down along her temple and her cheek. She said, "It's hotter than hell out here."
I stared at her. "Yes. It is."
She ran her fingers along the dash. She tapped the shift lever. "This is a classic, isn't it?"
"Yeah."
"It's a Corvette?"
"Yeah. A Stingray." I looked where she was looking. I touched where she was touching. "I wanted one when I was a kid, and a few years ago I had the opportunity to buy this one, so I did. I couldn't afford it, but I bought it anyway."
She nodded. "I should do something like that. Something crazy." She ran her fingers along the console. "When was it made?"
"Nineteen sixty-six."
"God. I was ten years old." She looked older.
I wanted to start the engine and turn on the air conditioner, but I didn't.
Anna Sherman said, "Three months ago an attorney named Lucas Worley was arrested in a drug sting in Santa Monica. He wasn't the target. He just happened to be there." She tapped my glove box. "I put his address in here."
I waited.
"Worley has a heroin problem. He'll buy a kilo every now and then, then cut it and sell it to his friends to cover his costs. Worley was a junior litigator in Green's office."
I smiled. "Was."
"Green had the case handled to minimize bad publicity for his firm, so Worley was able to cop a first offense probation plea."
"Is Worley still with the firm?"
She shook her head. "Resigned. I guess that was part of the deal." She finally looked at me. It was the first time she'd looked at me since I got into the car. "Worley was a tort litigator. That means he worked in Green's contract department. He would've had access to retainer agreements and to the contracts that Green had with his clients."
"Is he employed?"
She made a little dismissive shrug. "Probably dealing full time, but I don't know."
"So you think there's something to this."
She touched the dash again, watching her fingers move along the gentle lines. "You always follow the money." She shook her head and made a little smile. "I've been doing this for twelve years. I've prosecuted hundreds of cases, and I have learned that people do crime for only two reasons: sex and money. There are no other motives."
"What about power and revenge?"
"That's just sex and money under aliases." The tiny smile again. "If you're right, and if Jonathan Green is willing to break the law, then he's doing it for sex or money."
I was starting to like Anna Sherman. I was starting to like her just fine. "Do you think Worley will cooperate?"
She shrugged. "Lucas Worley is a piece of shit. He sells dope because he likes it. He likes the people, he likes the scene. He says that it's a step up from practicing law." She looked tired. "Maybe he's right."
I said, "Hey."
She looked at me.
"I'll tell you what I told Rossi. Don't give up. The good part of the system outweighs the bad. We just have to fix the bad."
Anna Sherman got out of the car, closed the door, then looked in at me. She said, "This conversation never happened. If you say it did, and if you say I gave you Worley, I'll deny it and sue you for slander. Is that clear?"
"Clear."
She walked away without another word. I opened the glove box and found a plain white sheet of notepaper with Lucas Worley's address written in anonymous block letters.
#
I stopped for roses. I bought a dozen red long-stems, plus a single daisy, then went to a wine shop I know for a bottle of Dom Perignon and an ounce of Beluga caviar. While the clerk was bagging the champagne I used their phone to make a reservation at Musso & Frank for eight o'clock. When I was off the phone, the clerk grinned at me. "Special date?"
"Very special."
He laughed. "Are there any other kind?" Cynic.
I drove home hard, hoping that I would get there before Lucy and Ben. I did. I put the flowers in the refrigerator and the Dom Perignon and three flute glasses in the freezer. The Dom Perignon was cold, but I wanted it colder. I hard-boiled an egg, minced an onion, then minced the egg. I put the egg, the onion, and some capers in three little Japanese serving plates, covered them with Saran Wrap, then arranged the plates on a matching tray with the caviar and put the tray in the refrigerator next to the flowers. I put out Carr's Table Water Crackers, then phoned Joe Pike and told him about Lucas Worley. Pike said, "You think he might know something?"
"I think he might, or, if he doesn't, he might be able to help us find someone who does."
"How do you want to play it?"
I told him.
Joe was silent for a time, then said, "How about we bring in Ray Depente? Ray would be effective on a guy like Worley."
"You think?"
"Ray could get a corpse to talk."
I told him that would be fine. I told him that I would meet them outside Worley's place early tomorrow, and when I was done, Joe said, "Is it going any better with Lucy?"
"Not yet, but soon. I'm about to turn on the charm."
"Why don't you try working it out, instead." Mr. Sensitive.
I hung up, then ran upstairs to finish getting ready. I shaved, showered, put on a jacket and tie, then ran downstairs and took the Dom Perignon out of the freezer. I wanted it cold, not frozen.
When Lucy and Ben pulled into the carport I was waiting at the door when they came through with shopping bags from Saks and Bottega Veneta and Giorgio and Pierre Deux. Lucy looked tired until she saw me, and then she looked surprised. I held out the flowers. "My God, you're beautiful."
Ben smiled so wide I thought his face would turn inside out.
Lucy looked at the flowers. She glanced at me and then the flowers again, and then back to me. Her hands were still full of shopping bags. "Oh, a daisy."
I put the shopping bags on the dining room table, then opened the Dom Perignon. I poured apple juice for Ben. "We have champagne. We have caviar. Then we will have dinner at Musso & Frank."
She said, "The restaurant in Hollywood?"
"Dashiell Hammett fell in love with Lillian Hellman there." I gave her a glass of the Dom Perignon. "It was a love that changed their lives, and endured for as long as they lived."
Lucy seemed embarrassed. "You're being so nice."
I said, "Ben. Would you give your mother and me a moment alone, please?"
Ben giggled. "You want me to amscray?"
"Yes, Ben, I want you to amscray."
Ben amscrayed into the living room. When the TV came on and Agent Mulder started talking about something that ate five human livers every thirty years, I took the flowers from Lucy and put them aside. I put aside her champagne glass, too, and held her upper arms and looked into her eyes. "You have two more nights in Los Angeles. I want those nights to be easy for you. It's okay with me if you'd like to move to a hotel."
Lucy stared at me for ten heartbeats, then shook her head. "I'm exactly where I want to be."
"I know that you're having trouble with your ex-husband. I know that he has a problem with you and Ben staying here. I want you to know that I'll support you in anything you want to do."
Lucy sighed, and glanced toward the living room. "Ben."
"Don't blame Ben. I am a detective, Lucille. I know all and see all."
"Darlene."
"Does it matter?"
She sighed again, then leaned forward to rest her forehead against my chest. "Oh, Studly, there is so much going on right now. I'm sorry."
I put my arms around her and held her. "You don't have anything to be sorry for."
She looked up and her eyes were rimmed red and wet. "I feel like I've ruined our time together."
"You haven't."
"I've let him intrude, and that's not fair to you or to me. I didn't tell you, and that is not the quality of honesty that I want in our relationship."
"You were trying to protect me."
She stepped back and looked into my eyes as if she were searching for something faraway and hard to see, something that she feared might change even as she saw it. "There's so much going on right now. You just don't know." She took a breath, then let it out. "I really need to talk about this."
"Then let's talk."
She took my hand and led me out onto the deck into the cooling night air, with the last breath of day fading in the west. She held my right hand in both of hers and said, "There are things you need to know."
"I don't need to know anything about you, Lucille."
"I'm not going to tell you deep dark secrets about myself. I don't _have_ any secrets."
"Shucks." Trying to lighten the moment with a little humor.
Lucy frowned and looked away. "These are things I need to say as much to help me get them straight as for you to be aware of what's going on. Do you see that?"
"Okay."
She looked back. "There are things happening between me and my ex-husband that I should've told you about, but didn't."
I nodded, letting her talk.
"Not because they're secret or because I wanted to keep anything from you, but because I resent the intrusion and did not want these things to impact upon our time together. I did not want _him_ to share this time with us." The other presence. "But I let him get to me, and he has intruded and that is not fair to either me or to you and I apologize."
I started to tell her that she didn't have to apologize, but she raised a hand, stopping me.
I sighed. "Okay. I accept."
"I'm not asking for advice. I'm an adult, I'm an attorney, and I will handle this. Okay?"
I nodded.
"I mean, God, I'm paid to advise other people, am I not?"
I nodded again. Getting a lot of nod practice tonight.
She said, "Richard has moved back to Baton Rouge." Richard was her ex-husband. He'd been living in Shreveport for the past three years, and, in the time that I'd known Lucy, she'd mentioned him exactly twice. He, too, was an attorney. "I've encouraged Ben to develop a relationship with his father, but Richard has taken it beyond that. He phones me at my office; he drops around my house unannounced; he invites himself to outings that I've planned with Ben; he's resurrected his friendship with a lot of the people at my firm. He has systematically reinserted himself into my life, and I do not like it."
"You feel invaded."
She made a brief, flickering smile. "Studly, I feel like Normandy Beach."
I said, "Joe likes you. Joe would probably fly down and have a talk with him."
The smile flickered again and, for just a moment, Lucy laughed. The tension was easing. "Perhaps it will come to that." The laugh and the smile faded then, and she said, "When he found out that Ben and I were going to stay here, with you, instead of a hotel, he became abusive. He criticized my judgment and told me that I was setting a bad example for Ben and demanded that I leave Ben with him."
I said, "Luce?"
She looked at me.
I opened my mouth but did not speak. My mouth felt dry and there was a kind of faraway ringing and my fingers and legs suddenly went cold. There are those times when intellect fails us. There are those moments when the modern man fades to a shadow and something from the brain stem reasserts itself, and in that moment the joking is gone and we frighten ourselves with our dark potential. I said, quite normally, quite conversationally, "What do you mean, abusive? Did he touch you?"
She shook her head, and then she placed both palms on my chest. "Oh, no. No, Elvis. And if he had I promise you fully that I would've had him arrested so fast he would've had whiplash."
I nodded again, but now the nods weren't funny. My fingers and legs began to tingle with returning blood.
She said, "I thought it was past, but it isn't. That's why Darlene called. He's been phoning the office and leaving messages on my machine at home, and then I got upset even more that I had let him get me upset in the first place. Do you see?"
My breathing had evened out and the ringing was gone. I nodded. "He pushed your buttons."
"Yes."
"He exerted a kind of power over you that you thought was behind you."
She said, "I'm so sorry you thought it was you, or that you had something to do with this. Oh, sweetie, it wasn't you at all. It was me."
"It's okay, Luce. It's really okay."
She rubbed my chest again and stared up at me because there was more. "Everything is complicated because I haven't been happy at the firm or with where I am in my life, and I don't know what I'm going to do about it."
I looked at her, and my heart began to thud.
"It started before I met you. It started even before Richard moved back."
I looked at her some more, and the night air was suddenly sparkling with a kind of expectant electricity.
"I don't know if I want to stay at the firm. I don't even know if I want to stay in Baton Rouge." She shook her head, glancing past me at Ben, glancing out at the warm house lights in the canyon. She finally looked back at me. "Do you know what I'm saying?"
"Would you consider coming out here?" My heart was thudding so loudly I wondered if the people across the canyon could hear it.
"I don't know." She took a deep breath and rubbed my chest again. "I guess I just needed to tell you that I don't know." She tried to make a joke. "Damn, and I thought I was too young for menopause."
I nodded.
"I'm feeling kind of stupid right now. It just seemed important to tell you."
I touched her lips. I kissed her, with the center of my heart. "I love you, Lucille. Rotten ex-husband or no. Long distance relationship or no. Do you know that?"
Her eyes grew wet again, and she ran her hand along the line of my shoulder. She touched my tie. "You look so nice."
I smiled.
"You went to so much trouble with the champagne and caviar."
I said, "Would you like to go eat? We still have time." They would hold the reservation. I was sure I could talk them into holding the reservation.
She took a breath, then let it out and carefully looked up at me. "What I would like to do is stay home with my two guys. What I would like is to order a pizza and drink your wonderful champagne and play Clue."
I grinned. "You want to play Clue?"
She was suddenly very serious. "I just want to be with you, Elvis. I just want to relax and enjoy being here. Do you know?"
I kissed her fingers. "I know."
I took off my jacket and tie, and we ordered Domino's pizza. We made a large Italian salad with pepperocinis and garbanzo beans and fresh garlic while we waited for the pie. When the pizza came, we drank the Dom Perignon and ate the pizza between bites of Beluga caviar mixed with capers and minced onion, and played Clue. There was a smile on Lucy's face that did not leave, and made the room feel light and warm and explosive with energy. Ben laughed so hard that he blew soda through his nose.
It was as if the other presence was no longer with us, as if by exposing the other it vanished the way a shadow will when exposed to light.
We played until very late, and when Ben went to bed, Lucy and I finished the last of the champagne, and then she followed me upstairs into a night filled with warmth and love and laughter.
#
The next morning I left the house as the eastern sky bloomed with the onrushing sun and drove to Lucas Worley's condominium on a one-way street just off Gretna Green Way in Brentwood. Gretna Green is a connecting street between Sunset Boulevard and San Vicente, lined with apartment houses and condominium complexes and some very nice single-family homes, but in the dim time just before sunrise the traffic was sparse and the neighborhood still. It was a wonderful time of the day for lurking.
Worley's condo was set between the street and a service alley in a lush green setting. They were nice condos, large and airy and stylishly ideal for former on-the-rise young attorneys turned dope dealers. I slow-cruised the street first, then turned down the alley and idled past the rear. Each condominium had a double carport at its back protected by an overhead wrought-iron door, and Worley's was filled with a gunmetal blue Porsche 911 sporting a vanity plate. The vanity plate read EZLIVN. Guess the loss of his day job hadn't inhibited his lifestyle.
When I reached the end of the alley, Joe Pike and Ray Depente materialized out of the murk and drifted silently to my car. Ray was wearing a black suit over a white shirt with a thin black bow tie. I said, "When did you go Muslim?"
Ray looked at himself and smiled. "Joe said you wanted scary. You tell me anything a white boy's more scared of than a Muslim with a hard-on?"
Ray Depente was an inch taller than Joe, but slimmer, with mocha skin and gray-flecked hair and the ramrod-straight bearing of a career Marine, which he had been. For the better part of twenty-two years Ray Depente had taught unarmed combat at Camp Pendleton, in Oceanside, California, before retiring to open a karate school in South Central Los Angeles. Now, he taught children the art of self-respect for ten cents a lesson, and instructed Hollywood actors how to look tough on screen for five hundred dollars an hour. The one paid for the other.
Ray extended his hand and we shook as he said, "Haven't seen you in a while, my friend. Better get your butt down to my place before you get out of shape."
"Too many tough guys down there, Ray. Some actor might beat me up."
Ray smiled wider. "Way I hear things been going for you, I guess it could happen." The smile fell away. "We got a plan for Mr. Dope Dealer, or are we just gonna stand around in the dark waitin' to be discovered?" The eastern sky was cooling from pink to violet to blue. Traffic was picking up out on Gretna, and we could hear garbage trucks and cars pulling out of driveways as people left for work. Pretty soon housekeepers would be trudging past to their day work.
Joe tilted his head toward the Porsche."Worley's been inside since eight-thirty last night."
"Is he alone?"
"Yes."
I said, "He's got to leave sooner or later. When he leaves we'll go in the house and find his stash. We find the stash, we'll have some leverage."
Ray said, "What if he doesn't have a stash?"
I shrugged. "Then we'll live with him until he scores."
Ray stared at the Porsche. "Joe said this guy was a lawyer."
"Yep. Until he got caught with the dope."
Ray looked at the nice car and the nice condo and shook his head. "Asshole."
Joe and Ray vanished back into the thinning shadows, and I pulled out of the alley and down the little street to Gretna Green. I parked beneath a Moroccan gum-ball tree with an easy eyes-forward view of Lucas Worley's street and waited while the air slowly filled with a mist of brightening light and early morning commuter traffic increased and the city began its day.
At twelve minutes after nine that morning the 911 nosed out onto Gretna and turned south, heading for San Vicente. Worley was a pudgy guy with tight curly hair cut short and close-set eyes and a stud in his left ear. He was wearing a tattered dark gray sweatshirt with no sleeves, and his arms were thin and hairy. Probably just running out for coffee.
I left the Corvette, trotted across Gretna and down along the little street to Worley's condo, where Pike and Ray were waiting at the front door. Pike already had the door open.
Lucas Worley's condominium was all high-angled ceilings and stark white walls and rented furniture of the too low, too wide, and too ugly variety. A fabric and plastic ficus sat in the L of two full-sized sofas, and a big-screen TV filled one wall. A stack of stereo equipment ran along the adjoining wall with what looked to be a couple of thousand CDs scattered over the floor and the furniture and on top of the big screen. I guess neatness wasn't one of Lucas Worley's strengths. Framed movie posters from _Easy Rider_ and _To Live and Die in L.A_. hung above the fireplace opposite mediocre lithographs of Jimi Hendrix and Madonna, and the effect was sort of like a nebbish's fantasy of how a high-end life-in-the-fast-lane hipster would live. He even had a lava lamp. Ray said, "Would you look at this?"
A framed Harvard Law School diploma was leaning against the lava lamp.
Ray was shaking his head. Incredulous. "The kids I work with down in South Central bust their asses just to get a high school diploma so they can get away from this shit, and here this fool is with a goddamned ticket from Harvard Law."
I said, "He won't be gone long, Ray. We've got to find the stash."
Ray moved away from the diploma. He glanced back at it twice and sighed as if he'd seen something so incomprehensible that understanding would forever be denied.
I started for the stairs. "I'll take the second floor. You guys search down here."
Pike said, "Don't bother. It's in the tree." Pike was circling the ficus.
I stopped at the base of the stairs. "What do you mean, it's in the tree? How would you know that?"
"Because it's where a lightweight would put it." Pike grabbed the ficus and yanked it up hard. The ficus came out of its pot, and there was the dope stash. Like Pike had sensed it.
Ray and I stared at each other. We stared at Pike. Ray said, "Nawwww."
Pike made a little shrug.
Ray said, "You're pulling our legs. You saw him foolin' in there through the window last night."
Pike angled the flat lenses at Ray. "You think?"
You never know with Pike.
The ficus had covered two Baggies of white powder, one Baggie of brown powder, a metric scale, and assorted drug-sales paraphernalia. I told Joe and Ray what I wanted them to do, and when, and then they left. I stayed. I took the dope out of the planter and put it in a neat pile on the coffee table, then replaced the ficus, looked through the scattered CDs until I found something that I liked, put it in the changer, turned on the music, and sat on the couch to wait. The Police. _Reggatta De Blanc_.
Forty-two minutes later, keys worked the lock, the door swung open, and Lucas Worley came halfway through the door before seeing me. He was carrying a newspaper and a Starbucks cup. He looked surprised, but he hadn't yet seen the dope on the table. "What the fuck is this? Who are you?"
"Come inside and close the door, Luke. Can I call you Luke? Or is it Lucas? Lucas seems pretentious." He was a little bit taller than he had looked in the car. His eyes were bright and sharp, and he spoke quickly. You could tell he was used to talking. You could tell he was used to saying bright things and having them appreciated, and you could tell that he thought he was brighter than he really was. Probably where the smugness came from.
He said, "Maybe I'm confused. Isn't this my house? Isn't that my sofa? The only thing that doesn't seem to belong here is you." Showing attitude.
"Look at me, Luke. Do you recognize me?"
"Sure. On television. You're the detective who's working with Jonathan." He closed the door. He was moving slowly. Wary, but trying to be oh-so-cool about it. "How's Jonathan?"
I smiled at him. "Funny you should ask, Luke. Jonathan is why I'm here."
That's when he saw the Baggies. He stared at them for most of an eternity, and then he said, "What's that?" Like he'd never seen them before.
"Here's the deal, Luke. You used to work in Jonathan's contracts department, and I want to know everything there is to know about Jonathan and his relationship to Teddy Martin. You're going to tell me what you know, and then you're going to get me into his office so that I can see for myself. Are we on the same page with that?"
He shook his head as if I'd spoken Somali. "Are you high? I don't know you. Get out of here."
I leaned back and spread my arms along the back of the couch so that my jacket would open and he could see the Dan Wesson.
"Look, I'm not doing anything for you. I'm going to call Jonathan right now. I'm going to tell him what's going on."
"Oh, you'll go along, Luke. Trust me." I pointed at the Baggies with my foot. "You've been a bad boy."
He smiled like he'd decided exactly how he was going to play it out and he knew he could beat me because he was smarter than me. "Is this how you're going to get me to do what you want? You're going to call the police? You figure you can have me bounced for violating probation?"
I shook my head. "No way, Luke. We don't need the police."
He smiled wider and moved past me, going to the phone. "Tell you what. I'll call them for you." He picked up the phone and waved it, showing me just how in control he thought he was. "Because when they get here and pull us in, I promise you that I can beat this nine ways from Monday in court." Waving the phone at the dope. "That's not mine. You're here, you planted it, and you're trying to extort me to screw Jonathan because of the Martin case. Man, Jonathan will have a field day with that one. I can see it now."
I looked disappointed. "You didn't listen, Luke. I'm not going to call the cops. I've already made my call."
Worley frowned and looked uncertain. "Who'd you call?"
Someone knocked at the door.
Lucas Worley suddenly didn't look so sure of himself.
"Don't you think you should get that?"
He didn't look at the door. "Who is it?"
Someone knocked again.
I said, "I kinda figured that you wouldn't cooperate, and that if I tried setting you up with the police that you'd find a way to beat it, so I called a guy I know named Gerald DiVega. You know DiVega?"
His mouth formed into a little O, like the name was ringing a bell but he couldn't quite be sure of it.
I went to the door. "Gerald DiVega sells drugs to westside hipsters like yourself. For many years he sold drugs on the streets, like so many other gentlemen of free enterprise, but in the past few years he's chosen to cultivate a more upscale clientele: movie and TV people, music people, lawyers and doctors, the very same people you're selling to with your little pissant business." I opened the door and Ray and Joe stepped in. They were both wearing sunglasses and looking somber. Ray reached under his jacket and drew out a Colt. 45 Government model. Joe Pike took out his Python. I said, "This is Mr. X and this is Mr. Y. Mr. D sent them because he doesn't like you cutting in on his clientele."
Ray Depente said, "This the muthuhfuckuh?" He took a black tube from his jacket pocket and screwed it onto the muzzle of the .45 as he said it.
"That's him."
Lucas Worley's eyes went wide, and he took one step back. "Hey. What is this? What's going on?" Smug was gone. Arrogance had vanished.
Ray and Joe crossed the room like two large, sinuous sharks gliding toward a blood spoor. Ray moved between Worley and the stairs, and Joe moved in from the other side and grabbed Worley's throat hard and rode him down on the couch. When Joe grabbed him, Lucas made a gurgling sound. I said, "I guess you should've called the cops when you had the chance, Luke."
Ray waved the .45 at me. "You can split now, you want. Mr. DiVega says thanks."
"Can't I stay?"
Ray shrugged like it was nothing to him. "Either way."
Lucas Worley's eyes were bulging and his face changed from red to purple. He was clawing at Joe's one hand with both of his, but it was like a child trying to bend steel bars.
Ray jacked a round into the .45, then put the muzzle of the suppressor against Worley's cheek and held out his other hand to shield himself from the blood-splatter that would surely follow and Lucas Worley thrashed and moaned and his bowels and bladder went loose at the same time. Guess the real world wasn't seeming like _Easy Rider_ anymore. Guess it wasn't like a movie or a television program. Not much glamor in messing your shorts.
I said, "You guys, don't shoot him, yet."
Lucas Worley's eyes rolled toward me.
I walked over and squatted by him to look into the rolling eyes. I said, "I helped Mr. DiVega out a couple of years back, and he owes me. He knows that I want something from you, and he's willing to play this however I want. You see?"
Lucas Worley was trying to shake his head, trying to say he wasn't trying to cut in on anyone's trade and wouldn't do it anymore if only they'd let him live. Of course, since Joe was strangling him, we couldn't quite make out the words.
"These gentlemen have orders to kill you unless I tell them not to."
Ray said, "Kill yo' ass dead." I frowned at Ray over the top of Worley's head, and Ray shrugged. Overacting.
I said, "So what's it going to be, Luke? You going to help me out with Jonathan Green, or do I walk out the door and make these guys happy?"
Lucas Worley gurgled some more.
I said, "I didn't understand you, Luke."
Joe released some of the pressure, and Lucas Worley croaked, "Anything. I'll do anything."
Ray Depente pressed the gun in harder and looked angry. "Shit. You mean we don't get to kill the little muthuhfuckuh?"
"Not yet. But maybe later."
Ray squinted down at the rolling eyes, then withdrew the gun and stepped back. Joe let Worley go and also stepped away. Ray said, "You got a pass this time, dipshit. But Mr. DiVega be on your ass now, you understand?"
Lucas Worley was frozen on the couch like a squirrel in front of an onrushing car.
Ray said, "You just retired from the dope dealin' business, didn't you?"
Worley nodded.
Ray said, "You're giving Mr. DiVega your word, and you know what will happen if you break your word, don't you?"
Worley nodded again. I think he was too terrified to speak.
Ray looked at the framed Harvard Law School diploma and shook his head. "Dumb muthuhfuckuh. You oughta be ashamed of yourself."
He put away the .45, then he and Joe Pike walked over to the bar and made themselves a drink.
I said, "I told you that you'd see it my way, Luke. Now go wash off and change your clothes. We've got some work to do."
#
When Lucas Worley was in the shower I looked at Ray Depente. " 'Kill yo' ass dead'?"
"I thought it was very effective."
Joe Pike shook his head. "Samuel L. Jackson."
Ray frowned. "Since when did you become Sir Laurence Olivier?" Pike's mouth twitched, and he went over to browse through Worley's CDs.
By the time we got Worley out of the shower and dressed and sitting in the living room, it was two-forty that afternoon. Joe and Ray were back in character, Joe standing behind the couch like an ominous shadow, Ray watching ESPN on the big-screen. I said, "Luke, do you have a gun here in your house?"
He was sitting on the couch with his hands in his lap and his hair wet and spikey. He still looked scared, but now he wasn't looking panicked. "Yeah. Up in the nightstand."
Joe drifted up the stairs.
"That the only one, Luke? You wouldn't have any surprises tucked away, would you?"
He shook his head, eyes jumping with the certain knowledge that surprises would get him killed. "That's all. I swear."
"Are you expecting anyone?"
"No."
"No one dropping around to pick up a little smack? No girlfriends? No repairmen?"
"No. Honest." A dope-dealing ex-attorney saying _honest_.
"Okay. I am now going to tell you exactly what I want, and you're going to tell me how to do it. Okay?"
He looked worried. "If I can."
Ray whirled away from the big-screen, loud and angry and snapping, "What did you say?"
Lucas Worley jumped as if he'd been slapped. "I'll tell you how. Sure. Whatever you want."
Ray's eyes narrowed, and he turned back to the big-screen, mumbling.
Joe Pike came back down the stairs with a pistol. "Glock nine."
"Anything else?"
"Nope." He sat by Ray.
I said, "Okay, Luke. Here's my problem. I suspect that your mentor, Mr. Green, is suborning testimony. I think he may even be involved in murder, only I can't figure out why a man in his position and of his stature would risk his ass by so doing. Do you understand that?"
Worley wasn't just looking at me; he was watching my lips move, careful to get every word. He blinked when he realized that I'd quit speaking, then shook his head. "Of course, he wouldn't. That's dumb."
"That's what everyone says."
"It's true. If he's caught he'd be throwing away his career."
I smiled at him. "Sort of like you."
Lucas Worley swallowed, then shrugged. Like he was embarrassed. "Yeah, but I was just a lawyer, and I never liked it much. He's Jonathan Green. He _loves_ it."
"Well, you're going to help me find out if it's dumb or not. Would Jonathan enter into a verbal agreement with a client?"
Worley grinned. "You've got to be kidding."
"Okay, so everything would be written."
"Absolutely. But no one is going to admit to a crime on paper. You're not going to find a paper that says 'I will do murder for X dollars.' " He was smiling at the thought of it. "Such a contract isn't enforceable, anyway. You couldn't sue somebody because they didn't perform an illegal act. You'd be incriminating yourself in conspiracy."
"So Jonathan wouldn't put anything in writing that he couldn't support in a civil action."
"Not a chance. No lawyer would." He spread his hands. "Look, you're not going to find anything incriminating there. I promise you. Jonathan isn't that stupid."
"That's not your concern. Your job is to get me access to all the contracts between Teddy and Jonathan. That is the sum total of your value to me." I nodded toward Joe and Ray. "You know that much, don't you?"
The worried look came back. "Hey, I said that I would. We can't just walk in there in the middle of the day. There're people."
"When do the people go home?"
"The office closes at six, but some of them stay late. Christ, we used to work until ten, eleven at night. Sometimes later."
Joe said, "How many people?"
"A few. It's a big office."
I said, "But most of the people go at six?"
"Yeah. There shouldn't be more than eight or nine there later than that."
"You have a card key to get in?"
"Oh, yeah. I kept it."
"How about the elevator to Jonathan's floor?"
"The card key accesses the parking garage, the elevator, everything."
I thought about it. "How long would it take you to get into the files?"
Lucas Worley stared at me about six seconds too long. "I dunno. It could take a while."
Ray Depente pushed up from his seat and drew out the. 45 and stalked over like he'd just hit the red line on the biggest bunch of bullshit he'd ever heard. "I'm killin' this fuckwad right goddamned now! Weasely muthuhfuckin' bullshit, take a while my ass!"
Worley threw himself to the side and covered his head, screaming, _"Twominutes! Icandoitintwominutes, sweartochrist! It'sallondiskandIcangeteverycontractinthegoddamnedoffice!"_
Ray stood over him, breathing hard and pointing the big .45. Across the room I could see Pike shake his head as he flipped through a magazine. _Modern Living_. Ray smirked and went back to his seat.
I said, "That's better, Luke. I think you and I are going to work this out just fine."
We had Worley describe the layout of the contracts department, and how we could get in and get out, and then we settled in for the afternoon. Pike left for a time, then returned with a small blue gym bag.
We listened to Lucas Worley's CD collection until five forty-five that evening, and then the four of us wedged into Worley's Porsche and drove to Green's building on Sunset. We bypassed the public parking entrance and used Worley's card key to access tenant parking. It was fourteen minutes after six when we worked our way beneath the building, and Worley said, "You see all these cars? There're still plenty of people working."
We found an empty spot as far in the back as possible, pulled in, cut the engine, and waited. Secretaries and office workers and blue-coated security people and attorneys of one stripe or another trickled out of the elevators and, little by little, the offices above us emptied. By seven-forty the trickle had dried and there were only six cars left, every one of which Worley recognized. He said, "The 420 belongs to Deke Kelly and the white Jag belongs to Sharon Lewis. They both work in Contracts. The little Stanza over there works in Contracts, also; I forget the kid's name. He was new. Sharon's assistant."
Pike said, "Contracts is on the third floor."
"That's right. Just like I said." We'd had him describe it five times. He'd even drawn a little map.
I said, "And Jonathan is on the fourth."
Worley nodded. "Yeah, but we won't have to go up there. All we have to do is go to Contracts. They have everything in their computers."
"What if Jonathan wanted something kept secret?"
Worley shook his head. "We can still access it from Contracts. The whole office is on the same computer net. Jesus, I should know. I helped design the system."
I looked at Pike and Pike shrugged. "Whenever."
Worley looked worried. "But what about the people up there?"
"What about them?"
Worley was looking even more worried. "You aren't going to kill them, are you?"
Ray glared at him. "That up to you. You get outta line, we be killin' people now till next Tuesday."
Pike looked at me and I rolled my eyes. Jesus, what a ham.
I pushed Worley out of the car and we walked in a tight group to the elevator, Pike with the gym bag, Ray with a hand on Worley's shoulder. Our footsteps were loud and gritty. "You said two minutes, and that's all you're going to have, Luke. Don't mess up."
Lucas Worley didn't answer. His eyes were blinking fast, and he kept wetting and rewetting his lips. Fear.
We got into the elevator and rode up to the third floor. If the doors opened and someone we recognized got on, I planned to say that I had come to see Truly and Jonathan and brazen it out, but when the doors opened on the third floor, the reception area was empty. The cleaning crews wouldn't be in until nine. The door to Contracts was on the left side of reception, opposite a pair of restrooms. Joe checked the men's room and Ray the women's. They both reappeared, shaking their heads. Clear. Pike opened the gym bag and pulled out a single gray cylinder. Worley said, "What's that?"
I pushed him toward Ray without answering. "Okay, Luke. Here we go." Ray pulled him to the men's room.
I pulled the fire alarm at the same time that Joe Pike used Worley's card key to open the door to Contracts, then yanked the fuse on the smoke canister and tossed it through the door. He held the door long enough to yell, "We have a fire in the building! Please use the main stairs and go to the street!" The main stairs fed into the ground floor lobby and were off the reception area. There were utility stairs in the rear of the Contracts department that would lead down to parking. That's how we planned to get out.
Joe let the door close, and then he and I followed Ray and Worley into the bathroom and pressed against the door. We heard voices and curses and a woman's nervous laughter, and then I said, "That's it, Luke. Showtime."
I dragged Worley out, and we used the card key to open the door again as Pike turned off the alarm. I pushed Worley through white smoke and said, "That's a minute, forty-five. The clock is running."
Joe and Ray scrambled in behind us, Joe taking a dousing blanket from the gym bag and pulling on heavy gloves to recover the smoke canister. Leave no evidence. They stayed at the door and Worley led me into an office. He said, "This used to be mine. Sharon must've taken it." A Macintosh computer was up and running on the desk, as if she'd been in the middle of something when we pulled the alarm. I said, "Ninety seconds. They'll be asking each other what happened. They'll be wondering why the alarm stopped and wondering if they should come take a look."
Worley closed the files that were on-screen and opened others. A case log heading that read MARTIN, THEODORE appeared on the screen along with a list of topics. He grinned and slapped the desk. "Y'see. Fuckin' magic. It's all right here." Like we were on the same team, now. Like he'd forgotten that we'd had to put a gun to his head.
"Print it and open Green's personal file."
Worley frowned. "Whaddaya mean, personal file?"
"Letters, bills, work product, anything that has his name on it." I went to the door and looked at Pike. The canister was out, but a heavy mist of white was spreading through the office as the smoke settled. I said, "C'mon, Luke. Sixty seconds."
Worley frowned harder. "Faster if I disk it." He could tell I didn't know what he was talking about. "I'll just dupe it onto a disk. It's faster than printing."
"Do it."
Ray stepped into the door. "We've got voices on the other side of the door."
Worley slapped in a disk. He punched buttons.
I said, "You'd better not be screwing with me, Luke."
"Jesus Christ, I'm almost done." His eyes were big again. "Okay, _now_! We've got it! That's everything!"
He ejected the disk, and we hurried through the smoke in the outer office to the rear stairs and took them down to the parking level. I was sweating hard, thinking we might meet a blue coat or a maintenance man taking the back way up, but we didn't. Luck.
We crossed the parking garage and got into the Porsche and drove back to Lucas Worley's condominium. It was dark when we got there. No one had thrown up a road block to stop us, and a phalanx of police cars hadn't chased us in hot pursuit. I'd never seen a phalanx before, but I was happy to avoid the experience. I said, "You did okay, Luke, but there's one other thing."
He looked at me. The four of us were still in the Porsche, sitting there in his carport.
I said, "You're going to keep your mouth shut about this. You're not going to tell your buddies. You're not going to brag to your girlfriend. We clear on that?" I was pretty sure that he would, eventually, but I wanted some time.
Ray said, "DiVega still wants this fuckuh dead."
I ignored Ray. "We together on this, Luke?"
Lucas Worley's head bobbed. "I won't breathe a word. I swear to Christ."
I held up the disk. "I'm going to check this stuff, and if it isn't complete, or if I figure you've screwed me, I'm going to call DiVega. We together on that, too?"
Luke flicked from Ray to Joe to Ray again. Ray was glaring at him. "Man, I copied _everything_. If it was there, you've got it. I _swear_."
Ray said, "DiVega said we should do what you say, but I know he don't like it."
I looked at him, making a big deal out of the look so that Worley would see. "Tell Mr. DiVega that we're even now. Tell him I said thanks."
Ray turned back to Luck and punched him once in the forehead, lightly.
Worley said, _"Ow!"_
Ray said, "You ever buy any more dope, we'll hear about it. You ever sell dope again, we'll be back. What happened here won't matter a damn. You understand, Mr. Harvard Law?"
Worley's head snapped up and down like it was on a spring. "Hey, I'm retired. You tell Mr. DiVega. I swear."
Ray and Joe and I climbed out of the Porsche, left Lucas Worley sitting in his carport, and walked out to the street and back to our cars. Ray said, "Is this guy DiVega for real?"
"Nope. I made it up."
Ray nodded. "I was trying to scare the little dip. Maybe wake him up."
"I know."
"That little sonofabitch will be dealing again inside the month."
"You can bet on it."
Ray thought about it. "If this fool goes back to dealing he's gonna meet a real Mr. DiVega sooner or later."
"They always do." We stopped at my car and shook hands. "Thanks, Ray. I appreciate the help."
Ray was staring back toward Worley's condominium, looking more than a little sad. "Think of the waste. Goddamned Harvard."
"Yep."
Ray Depente took a deep breath, let it out, and then walked on to his car. I guess he just couldn't understand how someone could turn his back on so much opportunity. I guess he'd be thinking about it most of the night.
Pike and I watched him leave, and then we drove back to my house.
#
We drove directly to my home, me in my car, Joe following in his Jeep, anxious to see if we had anything that Anna Sherman could use. It was eight-twenty when we arrived, and Lucy and Ben were snuggled together on my couch, watching what looked to be a Discovery Channel program about African plains game. The cat was watching the TV, too, but from the edge of the loft. He still didn't like Lucy and Ben much, but at least he wasn't growling.
Ben said, "They're home! Hi, Joe."
Joe said, "Hey, bud. You want to show me how to boot up this Macintosh?"
"Sure." Ben jumped up and the two of them went to the Mac. The cat stopped watching the television and started watching Joe. He began kneading his paws, but he still did not come down.
Lucy held up her hand, and I took it. She said, "I'm still not going to ask where you've been or what you've been doing."
I kissed her nose. "Damnedest thing. Joe and I found a computer disk on the street. We suspect that it contains contracts and business agreements between Jonathan Green and Theodore Martin." I held it up and showed her.
Lucy closed her eyes and slumped back miserably on the couch. "God. For sure I don't want to know."
"Of course, we won't know where it leads until we review what's here, and it would probably help to have an attorney decipher the stuff."
Lucy buried her face in her hands. "I'll be disbarred. I'll go to jail."
Joe said, "We're ready."
I went over to the Mac. "Yeah, you're right, Luce. Better stay over there out of the way."
Lucy jumped up and hurried around the couch to join us. "Oh, hell. It won't hurt to peek over your shoulder."
We fed the disk into the computer and opened the files. The list of available documents pertaining to Teddy Martin's representation was lengthy. Lucy leaned past me and tapped her nail on the screen. She had put on her reading glasses. "Most of this probably has to do with billing. You want the retainer agreement."
I looked at Lucy. "I thought you wanted no part in this."
She took a half-step back and showed her palms. "You're right. Forget I said anything."
I turned back to the screen.
Lucy said, "But you still want the retainer agreement."
Ben went back to the couch. We found the retainer agreement files and opened them. There were three documents, the original agreement plus two amendments. The original agreement called for a flat fee of five hundred thousand dollars for Green to represent Teddy from the date of the agreement through final appeal, plus all expenses and costs related to the defense. The five hundred thousand was to be deposited into an escrow account of Jonathan Green's choosing and dispensed in equal parts between signing, pretrial hearing start date, pretrial finish date, main trial start date, and main trial finish date, with the ongoing balance payable on demand should the case be dismissed for any reason. I looked at Lucy and she shrugged. "Looks pretty ordinary."
Pike's face was dark. "Five hundred grand. Ordinary."
I said, "Yeah. But these guys work for it."
Lucy knuckled me in the ribs, and then we opened the amendments. Lucy made a soft, whistling sound, and said, "I guess the price of justice went up."
The first amendment transferred the functional ownership of the entirety of Theodore Martin's business holdings, known corporately as Teddy Jay Enterprises, Inc., as well as Theodore Martin's personal property, into twenty-six different escrow accounts under the control of the Law Offices of Jonathan Green. The list of property and assets went on for pages and included fourteen specific restaurants, the real property associated with same, Teddy's Benedict Canyon mansion, plus homes, apartment buildings, and commercial property in Palm Springs, Honolulu, Denver, and Dallas. Approximate values had been given to each holding, and the total valuation was listed as one hundred twenty million dollars. I said, "Is this legal?"
Lucy scrolled through the document, lips parted, the screen reflected in her glasses. "Free enterprise, Studly. It looks like the parties renegotiated Green's fee for services, and who cares if it's akin to hyenas feeding on the bones of the dead?"
I looked back at the screen and shook my head. There were retirement accounts and bonds and stock portfolios. "Jesus Christ, Green's getting _everything_."
She continued scrolling. "Appears so." Then her breath caught and the scrolling stopped. "This is odd."
"What?"
She touched the final paragraph of the amendment. "These things are in escrow, but they're payable to Green only in the event that the charges against Teddy are dropped, or that he is acquitted." She shook her head. "This just isn't done. No attorney would predicate payment on the outcome of a case."
Pike said, "This one did."
I nodded. "Sex and money. A hundred twenty million is an awful lot of motivation."
Pike leaned back, and the left corner of his mouth twitched. "Enough to use James Lester to plant phony evidence, and enough to convince Louise Earle to change her story so that the press and the public doubt Angela Rossi's honesty."
I frowned. "I can see it with Lester, but you're not going to buy Mrs. Earle. They had to threaten her in some way, and I'm wondering if maybe they've increased the threat."
Lucy stepped away from the Mac and took off her glasses. "I agree that you could argue motivation now, but there is nothing illegal about this agreement. It's simply unusual. It could also be argued that Jonathan is willing to take the chance on an outcome-based payment because the funds are so large. The very thing that makes it unusual also makes it reasonable."
"You don't think Anna Sherman would be interested?"
Lucy spread her hands. "I'm sure that she would be interested, but what could she do? The California Bar certainly has no grounds for an investigation, and, unless there were some corroborating grounds for an investigation, neither does she." She gestured at the Mac. "Besides, she couldn't show this to anyone. It was illegally obtained."
I said, "Hey, we found it."
Lucy put on her glasses again and leaned past me to the keyboard. "Let's see the final amendment."
The final amendment was less than a page. It simply deleted four personal accounts and a vacation home in Brazil from the second amendment and contained an order releasing the accounts and home from escrow, returning them to Teddy Martin's control. Lucy said, "Mm."
"What?"
She shook her head and took off her glasses again. I guess, "Mm," meant nothing.
She said, "I'm sorry. It's still a stretch."
I looked at Pike, but Pike only shrugged.
I scrolled back through the original contract, then through the amendments. I considered the dates. "Okay, how about this. The first agreement is legitimate. Teddy hires the best lawyer he can, and that's Green. He's thinking that if anyone can get him off, it's Jonathan."
Lucy pulled over one of the kitchen stools and sat."Okay."
"But as the blood evidence comes in from the police and FBI laboratories, and the investigation proceeds, things aren't looking so good. Maybe Jonathan goes to him and says that they should negotiate a plea. Teddy freaks. He's a spoiled, arrogant, egomaniac and he can't imagine not beating this thing. I don't know who mentions it first, maybe Jonathan, maybe Teddy, but someone floats the notion that there has to be a way to beat this thing, and if such a way were found it would be worth everything that Teddy Martin owns. One of them says it, and the other thinks about it, and then they agree. Maybe the actual plan is never discussed. Maybe the words are never spoken, but they both know what they're talking about and the amendment is drawn, and then things begin to happen. Truly suggests James Lester; Kerris contacts Lester; Lester calls the tip line; I get put on the job. You see?"
Joe shifted in his seat. "Reality begins changing."
Lucy crossed her arms and leaned forward. "Are you saying that Jonathan stays away from it?"
"Sure. He's got Truly. He's got Kerris. He's hidden by layers of people. Jonathan Green's experience is that he has the ability to face twelve people and persuade them to accept the facts as he describes them. More often than not, the reality he constructs is false, but his entire experience is that he is able to convince a jury that this false interpretation is real."
Lucy sighed. "That's what makes a great defense attorney."
"And Jonathan Green is one of the best. He's very good at it, he's very careful, and he leaves no direct evidence to link him to any crime."
Lucy was nodding. "But if what you're saying is true, and he created Lester as a witness, why would he have him killed? Lester was the one link who tied Pritzik and Richard to Susan's kidnapping and could testify to that end."
Joe said, "Green knew that we'd begun to suspect him of manufacturing evidence. Maybe he decided to eliminate Lester because he was scared that Lester would give him up."
I shrugged. "Or maybe Lester realized what he had. Maybe he went back to Green and threatened to spill the beans. Maybe that's what he was talking about when he told Jonna about a big payoff coming in. Maybe he wasn't talking about the hundred thousand dollar reward, maybe he was talking about whatever he could get by extorting Jonathan Green, only when he made the move and tried to put a gun to Jonathan's head, Jonathan took care of the problem."
Lucy didn't look convinced. "Or maybe he just slipped on a bar of soap." She frowned at the look that I gave her. "Hey, bad luck happens."
I stared at her some more, and then I looked back at the Macintosh. Nothing on the screen had changed. Nothing had presented itself that irrefutably linked Jonathan Green to any wrongdoing. "That's what makes this guy so good, I guess. Everything can be explained. None of it leads anywhere else."
Joe said, "No. It all leads back to the money, and Green doesn't get the money unless Teddy beats the rap."
Lucy was staring at the computer again, the temple of her glasses against her teeth. She said, "Unless they aren't planning to get to trial."
I shook my head. "There's no way that the district attorney will drop these charges."
Lucy reopened the final amendment, the one that released accounts and property back to Teddy, and put on the glasses again. "A house in Brazil. A little less than ten million dollars in various holdings." She stepped back and took off the glasses. "We have no extradition with Brazil. Why would Green release the money and the house? Teddy had already agreed to them as part of his fee."
Pike said, "Bail. They're pushing hard for bail."
Lucy was nodding, clicking at her teeth again with her glasses. "I'll bet he's going to run. If he was willing to give up everything he owns to beat the charge, he's willing to leave it behind. Do you see?"
"Sure." Maybe I should just sit with Ben and watch television. Let Lucy and Joe figure it out.
Lucy said, "Maybe they've amended their agreement again, only this time not on paper. Maybe now it's payable on bail."
I was nodding, too. Mr. Getting-on-Board. Mr. Getting-with-the-Program. "Why wouldn't it be on paper?"
Pike said, "Because payment on bail would indicate a foreknowledge of flight."
I stared at him.
Lucy said, "Joe's right. You two are in the picture and you're making trouble. Lester was a problem, and that's more trouble. Maybe Teddy and Jonathan are getting so pressed that they're willing to take the chance on each other."
I was grinning. "So once Teddy has the money, he arranges a funds transfer to Brazil while he's still in jail. Jonathan doesn't have anything to do with it. Then, if he's granted bail, he jumps. Teddy will have his freedom, and Green can deny any knowledge of Teddy's proposed flight."
Lucy nodded. "That would work. Plus, any communication between the two is privileged and not admissible in court."
Pike said, "Ain't justice grand."
I said, "Sonofagun," and held up my hand and Lucy gave me a high five. It felt like we'd done something.
But then Joe said, "And there's nothing we can do about it."
I blinked at him. "Man, are you ever Mr. Wet Blanket."
Joe watched me for a moment, then stood and went to Lucy. He towered over her. "You're going tomorrow."
"That's right. In the morning."
Pike looked at me, but he spoke to Lucy. "He's going to miss you. He's done nothing but pine since he got back from Louisiana."
I said, "Pine?"
Lucy smiled. "I like pining."
Joe frowned at me. "You must be out of your mind, talking about this stuff when it's her last night." He turned back to Lucy. "I'm going to miss you, too."
Lucy stood on her toes and gave him a quick kiss on the lips. "Joe, thank you."
Joe said, "Hey, Ben."
Ben rolled over the back of the couch and grinned at him. "Bye, Joe. I hope you come visit."
Joe pointed at him, then glanced again at Lucy and walked to the door. The cat saw that Joe was leaving, hurried down the stairs, and slipped out with him. Soul mates.
When Joe was gone Lucy wrapped her arms around me. "He's so nice."
"Nice isn't a word often used to describe him."
"He cares a very great deal for you."
"Joe's okay."
She said, "I care about you, too."
"I know." I put my arms around her then and hugged her. I lifted her off the floor and my heart filled, and in a strange moment I felt as if I were fading into a shadow and if I did not hang onto her tight enough I would disappear. I said, "Want to do something wild?" I think I whispered it.
"Yes."
"Want to do something crazy?" I said it louder.
"Oh, God, I can't wait."
Ben said, "Hey, can I do it, too?"
And I said, "You bet, bud."
I put her down, and then the three of us made hot cocoa and sat in the cool night air on my deck and talked about our time together as the coyotes sang.
We talked until very late, and then Lucy put Ben to bed, and she and I sat up still longer, no longer talking, now simply holding each other in the safety of my home, pretending that tomorrow would not come.
#
I brought Lucy and Ben to LAX at just after nine the next morning. We returned her car to the rental agency, then sat together at the departure gate until the plane boarded, and then I stood with them in line until they entered the jetway and I could go no farther. I watched them until an efficient young woman in a neat airline uniform told me that I was blocking the door and asked me to move. I went to the great glass windows and watched the plane, hoping to see Lucy or Ben in one of the ports, but didn't. I guess they were seated on the other side. We had spent the morning speaking of innocuous things: _It's certainly cloudy this morning, isn't it? Yes, but it will burn off by ten. Oh, darn, I forgot to phone the airline and order the fruit plate_. I guess it was a way of minimizing our separation. I guess it was a way of somehow pretending that her getting on an airplane and both of us going back to our lives wasn't somehow painful and confusing.
When the little tractor pushed the airplane away from the dock and out to the taxiway, I said, "Damn."
An older gentleman was standing next to me. He was stooped and balding, with a thin cotton shirt and baggy old-man pants pulled too high and a walking stick. He said, "It's never easy."
I nodded.
He said, "Your wife and son?"
"My friends."
"With me, it was my grandkids." He shook his head. "They come out twice a year from Cleveland. I put them on the plane, I always think that this could be the last time. The plane could crash. I could drop dead."
I stared at him.
"I'm not a young man anymore. Death is everywhere."
I walked away. Too bad you couldn't get a restraining order against negativity.
Joe picked me up outside the terminal and we drove directly to Louise Earle's. We parked at the mouth of her drive, again went up to her door, and once more rang the bell and knocked. If we knocked much more we'd probably wear a groove in the wood. I was hoping that she might've returned home, but the drapes were still pulled and the house was still dark, and there was no sign that she had come back, then left again. While we were standing there, Mrs. Harris came out of her house and made a nervous wave at us. Pike said, "Looks worried."
"Yeah."
We walked over to her. I could see that her face was pinched and frightened, and that she was cupping one hand with the other, over and over. She said, "That man came back this morning. I thought it was the milkman, they came so early."
"They."
"There were three men. They were walking all around Louise's house. They walked around the side. They went in the back."
Pike looked at me, and I showed her the photograph of Kerris. "Was this one of them?"
She squinted at the picture and then she nodded. "Oh, yes. That's the one who was here before." She bustled to the edge of the porch, wringing her hands, flustered by the dark thoughts. "They were in her house. The lights came on and I could see them moving."
"Did you see them leave?"
She nodded.
"Did Mrs. Earle leave with them?"
She looked at me with large, frightened eyes. "What do you mean by that? What are you saying to me?"
"Did she leave with them?"
Mrs. Eleanor Harris shook her head. Just once. Imperceptibly.
I said, "Had Mrs. Earle come home?"
She was looking at her friend's house, wringing her hands, shifting in a kind of encompassing agitation.
"Was Mrs. Earle at home?"
She looked back at me with big eyes. "I don't know. I don't think so, but she may have been."
Pike and I trotted around the side of Louise Earle's house and into her backyard. I felt washed in a cold air, the hair along the back and sides of my head prickling, and scared of what we might find. Pike said, "The door."
Louise Earle's back door had been forced. We slipped out our guns and went in and moved through the house. It was a small home, just the kitchen and the dining room and the living room and two small bedrooms and a single bath. Papers had been pulled from drawers and furniture shoved out of place and closet doors left open, as if someone had searched the place more out of frustration than with a specific goal. I was worried that we might find Mrs. Earle, and that she might be dead, but there was nothing. I guess she hadn't come home, after all. Pike said, "First Lester, now her. Green's tying off the loose ends to protect himself."
"If she got scared, then she ran. If she ran, she might've bought tickets and they might show up on her credit cards. Also, she might've called a guy named Walter Lawrence."
Pike said, "I'll take the bedroom. You start in the kitchen."
We went through her house quickly and without speaking. She had two phones, one in her kitchen and one in her bedroom. The kitchen phone was an older dial-operated wall mount with a little corkboard next to it filled with notes and clippings and Prayers-for-the-Day and messages that she'd written to herself and probably not needed for years. I looked through them all, then checked the Post-its on her refrigerator door, and then I went through the papers that Kerris's people had left on the floor. I was looking for a personal phone book or notes or anything that might help me find Walter Lawrence or point to where she might've gone, but if there had been anything like that Kerris and his people had taken it. When I finished in the kitchen I went back through into the bedroom. Pike was working in the closet. He said, "Credit card bills by the phone."
I sat on the edge of the bed by the phone and looked at what he'd found. There were five past Visa and MasterCard bills, three Visas and two MasterCards. Charges were minimal, and nothing on the bills gave any indication of where Louise Earle might've gone, but then I didn't expect them to. Tickets purchased within the past few days would not yet have been billed to her, but I didn't expect that, either. I picked up her phone, called the toll free number on back of the Visa bill, and said, "Hi. I'm calling for my mom, Mrs. Louise Earle." I gave them the credit card number that showed on the bill and the billing address. "She charged a plane ticket yesterday, and we need to cancel, please."
The Visa woman said, "Let me punch up her account." She was very pleasant when she said it.
"Thanks. That'd be great."
Maybe three seconds later, she said, "I'm sorry, sir, but we're not showing an airline charge."
"Gosh, she told me she'd bought the tickets. She always flies United."
"I'm sorry, sir."
I said, "You know, maybe it wasn't an airline. Are you showing a bus or a train?"
"No, sir. I'm not."
I made a big deal out of sighing. "I'm terribly sorry. She told me about this trip and I got concerned. She's a bit older, now...." I let it trail off.
The Visa woman said, "I know how that is." Understanding.
I thanked her for her time, and then I called MasterCard and went through it again, and again I learned that Louise Earle had bought no tickets. Of course, she might've paid cash, but since I couldn't know that, it wasn't worth worrying about. Like most other things in life.
When I hung up from MasterCard, Pike was waiting."Looks to be some missing clothes. No toothbrush."
"Great."
"She has to be somewhere."
I picked up the phone again, called my friend at Pacific Bell, gave her Louise Earle's phone number, and asked for every call that Louise Earle had made in the past five days. Her records would show only toll calls, so if she'd phoned someone the next street over I'd never know it. But, like paying cash for airline tickets, it wasn't worth worrying about.
My friend read off twelve numbers that I dutifully copied, nine of which were in local area codes (310, 213, or 818), and three of which were long distance. The long distance calls were all to the same number, the first two of which were collect calls that she'd accepted the charges on. The third time she'd dialed the number direct. I thanked her for the help, then hung up and started dialing. Minimum-wage detective work.
I called each number and got two answers out of the first five calls, one from a pharmacy and one from an elderly woman. I hung up on the pharmacy and asked the elderly woman if she knew where I could find Mrs. Earle. She didn't. The sixth number was long distance. The phone rang twice, and a male voice said, "Federal Correctional Facility, Terminal Island."
I didn't speak.
The voice said, "Hello?"
I told him I was sorry, then hung up and looked at Pike. "LeCedrick."
Pike said, "She probably didn't go to stay with him." Everyone's a comedian.
"She didn't call LeCedrick. LeCedrick called her. LeCedrick calls, and she changes her story. She wouldn't do it six years ago, but she does it now. What do you think he told her?"
Pike shrugged.
I tapped the phone, thinking about it, and then I called Angela Rossi at her home. Her machine answered, but again she picked up when she heard that it was me. I said, "At six this morning, Kerris and two other guys broke into Louise Earle's house, looking for her. They searched the place, and I don't know if they got a line on her or not."
"Why are you telling me this?"
"Because LeCedrick Earle might know where she's gone. When I spoke with Louise she told me that she hadn't spoken to LeCedrick since he was sent up. She said he wouldn't speak to her. But four days ago he called her twice. Three days ago she changed her story. She called him the day before yesterday. That's the day she disappeared. He might know where she's gone. Do you see?"
Angela Rossi didn't say anything.
"I saw him before, but the last time he agreed to see me. I'm pretty sure he won't this time, and I need a badge to get in without his permission. Maybe you could talk to Tomsic. Maybe he could get me in."
Angela Rossi said, "Pick me up."
"You're suspended, Rossi. You don't have a badge."
"I'll get one, goddammit. Pick me up and we'll go see him. I'll get it set up before you get here."
She hung up before I could say anything else.
#
Angela Rossi was waiting at the mouth of her cul-de-sac, looking professional in a dark blue business suit that'd she'd probably worn to work every other week for the past three years. She swayed back and forth the way cops do when they're anxious. It's an unconscious habit they pick up in their uniform days when they have to stand in a place for long hours with nothing to occupy themselves except their baton. It's called the nightstick rock.
We stopped at the curb, and she climbed into the back seat. She said, "It's set up. The guards think we're coming to interview him about a past association. That's what he thinks, too."
Pike said, "Did you get a badge?"
"Don't worry about it." Protecting someone, saying if you don't know you can't tell.
Pike pulled back in traffic without waiting for her to buckle in. I said, "You could give us the badge, then you wouldn't have to come in. Less chance of anyone finding out that you're violating your suspension."
She neither answered nor looked at me. Her mouth was set and her eyes empty. Cop eyes. Just another day on the job walking the razor's edge.
We picked up the San Diego Freeway and headed south, and once more I was passing Inglewood and Hawthorne and Gardena and Torrance. Angela Rossi sat behind me in silence, hands in her lap, gazing out the window without seeing, dressed in her cop clothes, carrying a cop's badge, going on a cop's mission. She had given her all to it for a great long while, and I wondered if she was thinking that it might now be at an end. I wondered if she was thinking that the dream of being the first female chief of detectives had been a silly one. I wondered if she had regrets.
Forty minutes later we crossed the land bridge onto Terminal Island and passed through the gate, and then we were at the administration building. We parked, took off our guns, and then Angela Rossi and I went in. I said, "You okay?"
Rossi said, "Keep your mouth shut and try to look like an officer. I'll do the talking."
Yes, ma'am.
We went through the front door and up to the reception desk. I was worried that the reception guard would be the same guy, but he wasn't. This guy was paging through _Saltwater Fisherman_ magazine, but looked up when we approached. He said, "May I help you?" He was a young guy, tall and athletic and looking as if he'd just mustered out of the military. He was wearing the blue blazer and tie.
Rossi showed the badge. "West L.A. robbery/homicide. I called to see an inmate named LeCedrick Earle."
The receptionist jotted down the badge number, then said, "Sure. Hold on." He flipped through the loose-leaf book until he found Earle's name, then told someone on the phone that he wanted prisoner number E2847 in the interview room. When he hung up he said, "Guns?"
Rossi said, "Left'm in the car."
"Great. Someone will be right out for you. Wait by the sally port."
Rossi said, "Would it be a problem to check your logs for the visitors that Mr. Earle has had over the past two weeks?"
"No sweat." He turned to a computer and typed something. "We enter the log into the computer at the end of each day for the record. You want a hard copy?"
"Yes."
It took maybe sixteen seconds, and then a laser printer spit out a single sheet. Modern crime fighting at its finest. He said, "Here you go."
Rossi took it and we looked at it as we went to the sally port. The only visitors that LeCedrick Earle had had in the past two weeks were Elliot Truly and Stan Kerris. How about that?
A second guy in a blue blazer opened the sally port for us and said, "This way, please."
We followed him through and turned right. He was a couple of years younger than Rossi and he looked her over. "You guys down from L.A.?"
Rossi said, "That's right."
"What kind of case?" Rossi was trying to ignore him, but the guard was giving her the grin.
"Don't know yet."
The guard grinned wider. "How long are you going to be down here? Maybe we could get together for a drink."
Rossi never looked at him. "Do yourself a favor, sport. I just tested positive for chlamydia."
The guard's grin faltered and he moved a half-step away. Talk about a conversation stopper.
He brought us to the same interview room that I had used before and opened the door. He stood kind of bent to the side so that Rossi wouldn't brush against him when she went by. "I've got to lock you in. Your guy will be here in a minute."
Rossi said, "Thanks."
He locked the door behind us and we were alone. I nodded at her. "Chlamydia. Nice."
Rossi shrugged. I guess it was something she'd had to do ten thousand times.
We had been there less than thirty seconds when the rear door opened and a third guard led in LeCedrick Earle. His eyes widened when he recognized us, and he shook his head at the guard. "Forget this shit. I don't wanna see'm."
The guard shoved LeCedrick toward the table without acknowledging him and said, "Just punch the buzzer when you're finished."
LeCedrick Earle said, "Hey, fuck this shit. Take me back to my cell."
Rossi said, "Thanks, officer."
The guard closed the door and locked it, and Rossi smiled. "It's my favorite perp. How're you doing, LeCedrick?"
LeCedrick Earle glowered at us and stood with his back to the door, as far from us as possible. He said, "I don't have anything to say to you." He wiggled a finger at me. "I said everything I had to say to you before. I ain't gotta see you without my lawyer."
I said, "Stan Kerris is trying to kill your mother."
He blinked twice, and then he laughed. "Oh, that's right. You drove all the way down here for that?" He laughed some more.
Rossi said, "Jonathan Green's scam is falling apart, LeCedrick. He's falsified evidence and suborned testimony, and now he's scared that it's coming out. We believe that he ordered the death of a man named James Lester, and we believe that he's after your mother, too. If he is, then he'll probably come after you as well."
"Bullshit. You just talkin' trash." He wiggled the finger at Rossi. "You just worried cause your ass is in a crack. You know I'm gonna get your ass for puttin' me in here." He went to the near chair, plopped down, and put up his feet. "I ain't sayin' nothing without my lawyer."
"You want Mr. Green?"
LeCedrick smiled wide. "I think you'll find that he represents me in all matters criminal and civil. Especially in the civil case where we whack your ass for every nickel in your pension fund for planting bullshit evidence on me."
I stepped past Rossi and slapped LeCedrick's feet from the table. He said, "Hey!"
I said, "We've got to get past that right now, LeCedrick." He tried to get up but I dug my thumb under his jawline beneath his right ear. He said, "Ow!" and tried to wiggle away, but I stayed with him.
Rossi pulled at me from behind. "Stop it. We can't do that."
I didn't stop it. I said, "You didn't call the hotline about this, they called you. That's the way it started, isn't it?"
He grabbed at my hand, but he couldn't pry it away.
Rossi said, "Stop it, dammit. That's over the line."
"Kerris and Truly came to see you and convinced you to speak with your mother, didn't they?"
He was finally listening.
"What did they say, LeCedrick? You hadn't spoken to the woman in years, but you called her and convinced her to change her story. They offer you money? They say they could get you an early release?"
He stopped trying to pull at my hand, and I relaxed the pressure. Rossi said, "Jesus Christ, they could arrest us for this."
I said, "Think about it, LeCedrick. Jonathan and Truly and all those guys went to see her and probably told her what to say and how to say it, and that means she could testify against them."
Now he was squinting at me, hearing the truth of it, even though it was masked by his suspicions.
"I uncovered a connection between Lester and Green, and two days later Lester went through his shower door and damn near cut off his head. You see that in the papers?"
He nodded.
"The day after that I went to your mother's house to ask why she changed her story, and she was missing. You know Mrs. Harris next door? Mrs. Harris told me that Kerris had cruised your mother's house three times, that he'd walked around the place and tried to get in."
He said, "Mrs. Harris?"
"At six this morning Kerris and two other guys went back to her house and turned the place upside down. Why would they do that, LeCedrick?"
Now he was shaking his head. "This all bullshit."
"Would Mrs. Eleanor Harris bullshit you? You grew up next door to her. Would she bullshit you?"
He made a little head-shake. One so tiny that it was hard to see. "Lady 'bout raised me. Like a second mama."
Rossi pushed the buzzer, and when the guard came she asked if we could have a phone. He said no problem, brought one, and when he was gone again I turned it toward LeCedrick Earle and said, "Call her. I've got the number, if you need it."
He stared at the phone.
"We have to find your mother, LeCedrick. If we don't find her before Kerris, he'll kill her. Do you see?"
He wet his lips.
Rossi said, "Goddammit, you piece of shit, call the woman."
LeCedrick Earle snatched up the phone and punched the number without asking for it, and spoke with Mrs. Eleanor Harris. When she answered his manner changed, and he hunched over the phone and spoke in a voice that was surprisingly young and considerate. I guess the lessons we learn when we're small stay with us, even as we harden with the years. They spoke for several minutes, and then LeCedrick Earle put down the phone and kept his eyes on it, as if the phone had taken on an importance that dwarfed everything else in the room. He crossed his arms and started rocking. He said, "Why they do that? Why they go there so early?"
Rossi said, "They want to kill her. And after they kill her, they will almost certainly arrange to have you killed, and then no one can implicate them in the manufacture of false evidence. Do you see that?"
He didn't say anything.
I said, "She left the house with a bag. She has a gentleman friend named Mr. Lawrence."
LeCedrick Earle nodded dumbly. "That old man been chasin' her for years."
"Would she go there?"
"Sure, she'd go there. She ain't got nobody else."
I felt something loosen in my chest. I felt like I could breathe again. "Okay, LeCedrick. That's great. Just great. Do you know where he lives?"
LeCedrick Earle slumped back in the chair with an emptiness that made him seem lost and forever alone. His eyes filled with tears, and the tears spilled down across his cheeks and dripped on his shirt. He said, "I can't believe this shit. I just can't believe it."
Rossi said, "What?"
He rubbed at the tears, then blew his nose. "I must be the stupidest muthuhfuckuh ever been born. That woman ain't never done nuthin' but what she try to do right, and this what she gets for it. A fool for a son. A goddamned stupid fool." He was sobbing.
Rossi said, "Goddammit, LeCedrick, what?"
LeCedrick Earle blinked through the tears at us. "Your man Kerris called me 'bout an hour ago and asked about old Mr. Lawrence, too. He say they need to get her story straight. He say they want her to do a news conference, and I told him where she was. I told him how to get there and now they gonna kill my momma. Ain't I a fool? Ain't I God's own stupid muthuhfuckin' fool?"
I was pressing the buzzer even before he was finished, and Angela Rossi was shaking him until he told us the address, and then we were running out to the Jeep. It was almost certain that Louise Earle was dead, but neither of us was yet willing to give up on her.
Maybe we were God's own fools, too.
#
Pike pushed the Jeep hard out of the parking lot and through the gate and across the land bridge. Angela Rossi used her cell phone to call Tomsic as we were climbing back onto the freeway. She told him about Kerris, and that Louise Earle was probably staying with a Mr. Walter Lawrence in Baldwin Hills. They spoke for about six minutes, and then Rossi turned off her phone. "He's on the way."
I said, "You sure you want to go to the scene?"
"Of course."
Pike glanced at her in the rearview. "It gets back to the brass that you're involved, it's over for you."
Rossi took her Browning from under the seat and clipped it onto her waistband. "I'm going."
We scorched up the Harbor Freeway to the San Diego, the speedometer pegged at a hundred ten, Pike gliding the Jeep between and around traffic that seemed frozen in space. We drove as much on the shoulder as the main road, and several times Pike stood on the brakes, bringing us to screaming, sliding stops before he would once more stomp the accelerator to rocket around lane-changers or people merging off an entrance ramp. I said, "We can't help anybody if we're piled up on the side of the road."
Pike went faster.
Hawthorne slipped past, then Inglewood, and then we were off the freeway and climbing through the southern edge of Baldwin Hills along clean, wide residential streets lined with spacious postwar houses. Baldwin Hills is at the southwestern edge of South Central Los Angeles, where it was developed in the late forties as a homesite for the affluent African-American doctors and dentists and lawyers who served the South-Central community. At one time it was called the black Beverly Hills, though in recent years the community has diversified with upwardly mobile Hispanic, Asian, and Anglo families. Rossi's phone beeped, and she answered, mumbling for maybe ten seconds before ending the call. "Dan just got off the freeway. They're three minutes behind us, and he's got a black-and-white behind him."
We used Pike's Thomas Brothers map to find our way through the streets, watching for turns and scoping the area. Mothers were pushing strollers and children were playing with dogs and everyone was enjoying a fine summer day. I said, "We're almost there."
We were two blocks from Walter Lawrence's home when a tan Aerostar van passed us going fast in the opposite direction and Pike said, "That's Kerris. Three others on board."
Rossi and I twisted around, trying to see. "Louise Earle's in the back. Looks like Lawrence and someone else, too." Louise Earle looked scared.
Rossi said, "The other guy is probably one of Kerris's security people."
Pike jerked the Jeep into a drive and did a fast reversal. I said, "Did they make us?"
Pike shook his head. The Aerostar turned a far corner, but it hadn't increased its speed, and its driving seemed even. We went after them, Pike hanging back. In cases like this there are always two choices: You can let them know that you're there, or you can hide from them. If they know that you're there they might get nutty and start shooting. As long as they're not shooting, you're better off. Louise Earle and Walter Lawrence would be better off, too. Rossi unfastened her seat belt and leaned forward between me and Joe, better to see. "Don't crowd them, Joseph. Let's give them room."
Pike pursed his lips. "I know, Angela." Nothing like a backseat driver in a pursuit situation.
Rossi got on her phone again and told Tomsic where we were and what we were doing. She didn't cut the circuit this time, but kept up a running flow of information so that Tomsic knew where we were at all times. I said, "Can he get in front of them?"
"No. He's west of the hills and behind us. He's calling in more black-and-whites."
I glanced at Rossi, but she seemed impassive. The brass would know now, for sure.
We followed the van down out of the residential area onto Stocker Boulevard, then started climbing again almost at once, leaving the residential area behind as we wound our way through dry, undeveloped hills dotted with oil pumpers and radio towers. I had hoped that they would turn into the city, but they didn't. They were heading into a barren place away from prying eyes.
We followed them deeper into the hills, staying well back, catching only glimpses of their dust trail so that we wouldn't be seen, and as the peaks rose around us Rossi's cell phone connection became garbled and our link to Tomsic was broken. She tossed the phone aside."I lost him."
Pike said, "He knows about where we are."
"About."
Maybe a half mile ahead of us the van turned up the side of a hill along a gravel service road, making its way toward two great radio towers. We could see the towers, and what was probably a maintenance shed at their bases, and another car parked there. I said, "They're going to kill them. They couldn't kill them at the house with so many people on the street, but they're going to do them here."
Rossi craned her head out the window. "If we take the road up after them, they'll see us coming a mile away."
Pike slapped the Jeep into four-wheel drive, and we left the road, heading first down into a gully, then up. We lost sight of the towers and the van, but we watched the ridgeline and followed the slope of hills and we did what we could until we came to an elevated pipeline that we could not cross. Pike said, "Looks like we're on foot."
Pike and I were wearing running shoes, but Angela Rossi was wearing dress flats. I said, "Going to be a hard run."
She said, "Fuck it."
She threw her jacket into the backseat, took her Browning from its holster, then kicked off her shoes and set out at a jog. Barefoot. The ground was rough and bristling with stiff dried grass and foxtails and must have hurt, but she gave no sign.
The hill was steep and the going was slow. The soil was loose and brittle, and the desiccated grass did not help bind it together. Our feet sunk deep and every step caused a minor landslide, but halfway up the hill we saw the tops of the towers, and pretty soon after that the roof of the shed. We went down to our hands and knees and eased our way to the ridge. The Aerostar was parked next to a bronze Jaguar. Kerris was already out of the van and moving toward the shed. He'd left the van's driver side door open. The same black security guard I'd seen at Green's party came out of the shed. The van's side door slid open and a younger guy with a very short crew cut pushed out. Walter Lawrence climbed out after him, but I guess he wasn't moving fast enough because the crew cut took his arm and yanked, and Mr. Lawrence stumbled sideways to fall in a little cloud of dust. The black guy ignored all of that and opened the Jag's trunk to lift out two shovels and a large roll of plastic. Rossi said, "They're going to execute these people."
Pike said, "Yes."
I edged higher on the ridge. "They'll bring them inside the building. Maybe we can work our way around to the backside of the slope and come up behind the building without being seen." I didn't think Kerris would just shoot them in the open, even out here in the middle of nowhere.
Pike started backwards with Rossi behind him when the crew cut leaned into the van and said something to Louise Earle. I guess she didn't want to get out, because he reached in and pulled. He had her by the upper arm and it must've hurt. She tried swatting at him like you might a fly, but it did no good. That's when Walter Lawrence scrambled up out of the dust and grabbed the crew cut's jacket and tried pulling him away. Defending his woman. The crew cut guy put a hand on Walter Lawrence's face and pushed. Walter Lawrence flailed backwards and fell again, landing flat on his back, and the crew cut guy took a steel Smith & Wesson 9mm from beneath his left arm, pointed it at Walter Lawrence, and fired one shot.
The shot sounded hollow and faraway, and Mrs. Earle screamed just as Elliot Truly stepped out of the maintenance shed.
#
Pike worked the Python out of his waist holster and pushed it in front of him, lining up on the crew cut.
Rossi said, "We're too far."
"If they point a gun at her, Joe." Ignoring Rossi.
"I'm on it."
Rossi said, "Can he make this shot?"
We were more than a hundred yards from them. It was a very long shot for a four-inch barrel, but Pike could brace his hand on the ground, and he was the finest pistol shot I've ever seen.
Truly waved his arms, raising hell with Kerris and the guy with the crew cut, and the guy with the crew cut put away his gun. Truly did some more waving, then went back into the maintenance shed. Kerris raised hell with the crew cut too, then he and the black guy lifted Mrs. Earle by the arms and dragged her past Walter Lawrence's body to the shed. The crew cut went over to the shovels and plastic, and he didn't look happy about it.
I said, "We don't have much time."
We crabbed back down beneath the ridgeline and trotted around the side of the hill until we had the maintenance shed between us and the van. The shed was at the base of the north tower, and its structure formed a kind of latticework around the shed and would provide cover between the shed and the Jaguar. We moved fast, but with every passing second I was frightened that we'd hear the second shot. I guess we could've just started yelling and let them know we were here, but they had already committed murder; Mrs. Earle would probably catch the first shot.
When the shed was between us and the van, we crept up the hill to the rear of the base of the north radio tower. I said, "Rossi and I will take the shed. You take the guy at the van."
Pike slipped away to the edge of the shed, then disappeared among the girders at the base of the radio tower.
I looked at Rossi. "You ready?"
She nodded. Her stockings were shredded, her feet torn and bleeding and clotted with dirt and little bits of brown grass. Her nice suit pants were ripped.
The maintenance shed was a squat cinder block and corrugated metal building built against the base of the north tower. Inside, there would be tools and parts and paint for maintaining the towers and adjusting the repeater antennas. There were no windows, but doors were built into the front and back. Truly had probably been here for a while and had opened the doors for the air. The door nearest the cars was wide and tall so you could move oversized parts and equipment in and out, but the rear door, the door by the tower, was a people door.
Rossi and I slipped up to the side of the shed, then crept toward the door. We listened, but all we could hear was Mrs. Earle crying. I touched Rossi, then pointed to myself, then the door, telling her that I was going to risk a look. She nodded. I went down onto my hands and knees, edged forward, and peeked inside. Mrs. Earle was on the floor, tied, and Kerris and Truly were standing together just inside the far door. Truly looked nervous, like he didn't want to be there. The black guy wasn't inside; he'd probably gone back to help the crew cut with the shovels. I was still looking at them when the guy with the crew cut walked past the side of the shed with the shovel and the plastic and a sour expression and saw us. He did a classic double take, said, "Hey!" then dropped the shovels and plastic to claw for his gun when I shot him two times in the chest. I said, "Get Mrs. Earle."
Rossi rolled past into the door with me behind her when we heard three shots from the front of the shed. Kerris grabbed Truly and pushed him in the way and fired fast four times. Rossi said, "Shit."
Truly was looking confused and Mrs. Earle was staring at us with wide, frightened eyes, and I was scared that if I tried to hit Kerris I would hit her. I fired high and Kerris fell back, scrambling through the door, firing as he went. Truly turned to run after him, and when he did he turned square into Kerris and was kicked backwards by one of the rounds, and then Kerris was gone. There was shouting out front, Kerris and the black guy, and more firing. The black guy was yelling, "I'm hit! Oh, Holy Jesus, I'm hit!"
Rossi went to Mrs. Earle and I went to Truly. Truly was trying to get up and not having a good time of it. The bullet had hit him maybe three inches to the right of his sternum, and a flower of red was blooming on his shirt. He said, "I think I've been shot."
Rossi was untying Mrs. Earle. I said, "Are you all right?"
Mrs. Earle was still crying. "They shot Mr. Lawrence."
Rossi helped her up, telling her that she had to stand, that she had to move to the side, out of the way, telling her that everything was going to be okay. The lies you tell someone when you need them to cooperate because their life depends on it. Truly said, "Am I going to die?"
I tore off my shirt and bundled it and pressed it to his chest. "I don't know."
I pulled off his belt and wrapped it around his chest and the shirt and buckled it tight. He said, "Oh, God, that hurts."
There were more shots by the cars and running footsteps, and then Joe Pike slipped through the door. Maybe six shots slammed into the door and the walls and through the open doorway. Maybe seven. Pike said, "Those Glocks are something."
Rossi duck-walked over. "What's the deal out front?"
Joe said, "The black guy's punched out. Kerris is behind the Jaguar. I don't know about the crew cut."
Rossi nodded toward the rear. "Forget him."
I said, "Can we get to Kerris?"
Pike made a little shrug. "He's got a clean field of fire at us. We could go back the way we came, maybe, and work our way around." He glanced at Truly. "Take about twenty minutes to work around behind the Jag."
I turned Truly's face so that he looked at me. "You hear that, Elliot? You're bleeding and we're pinned down in here and Kerris is doing the pinning."
Truly opened his mouth, then closed it. He blinked at me, then shook his head. "Kerris kidnapped these people. He shot that old man. I didn't know anything about it."
Rossi said, "Bullshit."
I shook Truly's face. "Stop lying, you idiot. Stop worrying about incriminating yourself, and start worrying about dying."
He shook his head. His eyes filled with tears, and the tears tumbled out and ran down into his hair.
I said, "It's you and Kerris and the black guy and the guy with the crew cut. Is there anyone else up here?"
He shook his head again. "No." A whisper.
"Is anyone else supposed to come up here?"
The crying grew worse and became a cough. When he coughed, pink spittle blew out across his chin and the chest wound made a wheezing sound.
I said, "Tell Kerris to give it up. If Kerris gives it up, we can get you to a hospital."
Truly's face wrinkled from the pain and he yelled, "Kerris! Kerris, it's over. I need a doctor!" It wasn't much of a yell.
Kerris didn't answer.
Elliot Truly yelled, "Goddammit, Kerris, enough of this, would you, please?! I'm dying! I've got to get to a doctor!" He coughed again, and this time a great red bubble floated up from his mouth.
Rossi duck-walked over. She said, "You're fucked, Elliot. Your man outside is in for murder and he's looking to save himself. He's got to kill us and this woman to do that, and he doesn't give a damn if you live or die."
Truly moaned. "Oh, God."
Rossi leaned closer to him. "Maybe you'll make it, but maybe you won't. We still might get Kerris, though, and the sonofabitch who put you into this spot. Give him up, Elliot. Tell us what we want to hear."
Truly squeezed his eyes shut, but still the tears came out. "It was Jonathan."
Rossi smiled. It was small, and it was personal.
I said, "Everything that's happened, it's so Jonathan can take over Teddy's companies, isn't it?"
Truly tried to nod, but it didn't look like much. "Not at first. At first, Jonathan was just going to defend him, like anyone else."
"But Teddy got scared."
Truly coughed, and more bubbles came up. "Oh, God, it hurts. God, it hurts so bad."
I said, "Did Teddy kill his wife?"
Truly wet his lips to answer, and made his lips red. "Yes. He denied it at first, but Jonathan knew. You can always tell. You know when they did it."
Rossi frowned at me and nodded. You see?
Truly said, "Then he just admitted it. I'm not sure why, but he did, just out of the blue one night when we were going over his story. Jonathan and I were alone with him and he started to cry and he admitted that he killed her. That changed everything. Jonathan advised him to negotiate a plea, but Teddy wouldn't do that. He was scared of going to prison, and he begged Jonathan not to quit the case. He said that he'd do anything rather to go to prison."
"Even give away everything he owned."
Another nod. "That was Jonathan's price."
Rossi said, "All that stuff about Pritzik and Richards. That was bullshit?"
"Jonathan and Kerris and I put it together. Jonathan had the idea of a straw man, and Kerris came up with Pritzik and Richards, and I knew Lester. We just put it together." He started coughing again, and this time a great gout of blood bubbled up and he moaned. I put my hands on the compress and leaned on it. He said, "I don't want to die. Oh, God, please Jesus, I don't want to die. Please save me."
I wiped the blood off his face and forced open his eyes and said, "You're a piece of shit, Truly, but I'm going to save you, do you hear? Just hang on, and I will get you to a hospital. Do you hear me?"
He nodded. "Uh-hunh."
"Don't die on me, you sonofabitch."
He moaned, and his eyes rolled back.
I checked on Mrs. Earle, and made sure that she was behind as much metal as possible, and then Rossi and I went over to Pike. Pike was peering through a split in the door jamb. "He got a shotgun from the van. He's talking on his cell phone."
"Great. Probably calling for reinforcements."
Pike glanced at Rossi. "Be real nice if Tomsic happened to find us about now."
Rossi shrugged. "Let's all hold our breaths."
I edged past Pike and looked through the split. Kerris was behind the Jaguar with the shotgun. The black guy was lying on his side between the Jag and the van, and Mr. Lawrence was on his back a few yards behind him. The black guy was probably unconscious, but he might've been dead. I yelled, "Come on, Kerris. There's three of us and one of you. Don't be stupid."
The shotgun boomed twice, slamming buckshot into the corrugated metal about eighteen inches above my head. Mrs. Earle made a kind of moaning wail, and Rossi dived across the doorway, popping off caps to force Kerris down.
Pike looked at me. "I don't think he's scared of the odds."
"Guess not."
Rossi edged toward the door and stopped just shy of the jamb, squinting out into the sun. She said, "Hey, the old man's still alive."
Mrs. Earle stopped wailing. "Walter?"
I went back to the split and saw Walter Lawrence slowly roll onto his belly, then push up to his knees before falling onto his face.
Mrs. Earle started for the door, but Pike pulled her down. "Stay back, ma'am. Please."
"But Walter needs help." She said it loudly, and Pike put his hand over her mouth.
"Don't draw attention to him. If Kerris sees him he's a dead man."
Her eyes were wide, but she nodded.
Walter Lawrence pushed up again, then looked around as if what he was seeing was new and strange. He saw the guy in the red knit shirt about ten feet in front of him and he saw the guy's pistol, a nice blue metal automatic, lying in the dust. He looked past the guy in the knit shirt and almost certainly saw Kerris hiding behind the Jaguar, pointing the shotgun at us. Walter Lawrence was behind Kerris, and since Kerris was looking at us, he wasn't looking at Walter Lawrence. Mr. Walter Lawrence began crawling for the pistol. I said, "Rossi."
"I see him."
I watched through the split jamb, and could see the hills and the pumpers and the rough service roads below, and as I watched a dark sedan appeared on the road between the pumpers, heading our way, kicking up a great gray rooster tail of dust. Rossi saw it, too. I said, "Is that Tomsic?"
She ejected her Browning's clip, checked the number of bullets left, then put it back in her gun. "I can't tell."
I glanced at Pike and Pike shrugged. Guess it didn't matter to him. Guess he figured the more the merrier.
Walter Lawrence crept toward the gun like a drunken infant, weaving on his hands and knees, bloody shirt hanging loose and sodden between his arms. He reached the pistol and sat heavily, but he did not touch the gun. As if simply reaching it had taken all of his energy. Rossi said, "In a couple of seconds we're going to be able to hear the car. If Kerris looks that way, the old man's dead."
I looked at Pike and Pike nodded. I took a breath and peered out the split again. Kerris had taken up a position behind the Jaguar's front end. You could see about a quarter of his face behind the left front tire. The tire was probably a steel-belted Pirelli. Might be able to shoot through it, but it wasn't much of a target. "Kerris? Truly's dying. He needs a doctor."
"It's the cost of doing business." All heart.
I stood. "Listen, Kerris! Maybe we can work something out." I sprinted past the open door to the other side of the shed. When I flashed past the door, the shotgun boomed again, but the buckshot hit the wall behind me.
Pike said, "Lucky."
I yelled, "I didn't sign on to this job to get killed, and neither did Pike. You want the old lady, we just want to go home. You hear what I'm saying?" I hopped past the door in the opposite direction. Kerris fired twice more, once behind me through the doorway and once high through the wall. Maybe I could just keep running back and forth until he ran out of ammo.
Kerris said, "Bullshit, Cole. I checked out you and your partner, remember? You aren't built that way."
Another boom, and this time the number four slammed through the wall just over Joe's back.
I crawled across him to the split again and looked out. Walter Lawrence had once more focused on the gun. He leaned forward from the waist, picked it up, then held it as if he had never held a gun before in his life. Maybe he hadn't. He cupped it in both hands and pointed it at Kerris, but the gun wavered wildly. He lowered the gun. I yelled, "I'm serious, Kerris. What's all this to me?"
"If you're so goddamned serious, throw out your guns and come out."
"Forget that."
"Then let's wait it out."
The car was close, now, and if I strained I thought that I might hear it. Walter Lawrence raised the gun again. Rossi said, "That's Tomsic!"
I yelled, "Okay, Kerris. Let's talk."
I stepped into the door, and as I did Mr. Walter Lawrence pulled the trigger. There was a loud _BANG_ and his shot slammed into the Jaguar's rear fender and Kerris jumped back from the wheel, yelling, "Sonofabitch!"
Walter Lawrence fired again, and again the shot went wide, and Kerris swung the shotgun toward him but as he did Angela Rossi shouted, "No!" and she and Joe Pike and I launched out the door, firing as fast as we could.
Kerris brought the shotgun back, pulling the trigger _BOOM-BOOM-BOOM-BOOM_ as our bullets caught him and lifted him, and then slammed him into the gray earth, and then the noise was gone and it was over and there was only the sound of Louise Earle crying.
#
Mr. Walter Lawrence fell onto his back and kept trying to right himself the way a turtle might, clawing at the air with his arms and legs. I took the gun from him and told him to lie still, but he wouldn't until Louise Earle hurried out from the shed and made him.
Linc Gibbs and Dan Tomsic pulled up in a cloud of dry gray dust, then ran over with their guns out. Tomsic said, "Who's this?"
"One of the good guys. Get an ambulance, for Christ's sake. We've got another wounded in the shed."
Linc Gibbs made the call while Tomsic ran for the first aid kit that every cop keeps in his trunk. The crew cut had put one high into the left side of Mr. Lawrence's chest. His shirt and jacket were soaked red, and he felt cold to the touch. The blood loss was extreme. When Tomsic came with the kit, we put a compress bandage over the wound and held it in place. Mrs. Earle held it. While Tomsic was working with the bandage he glanced at Angela Rossi. "You okay, Slick?"
She made an uncertain smile. "Yeah."
When Mr. Lawrence was bandaged we ran into the shed, but Elliot Truly was dead. Tomsic looked close at Truly as if he wanted to be sure of what he was seeing. "Is this who I think it is?"
"Unh-huh."
"Sonofabitch."
Gibbs had them send a medivac helicopter, and while we waited, we secured the scene. There wasn't much to secure. Both the guy with the crew cut and the guy in the knit shirt were dead. Kerris was dead, too. Tomsic said, "Do all of these guys work for Green?"
"Kerris was his chief investigator. I think these other two worked for Kerris. I saw the black guy at Green's home."
Tomsic shook his head and stared at the bodies. "Man, you really wrack'm up."
I frowned at him. "Do you have a spare shirt in your car?" My shirt was still a bloody wad on Elliot Truly's chest.
"Think I might have something." Most cops keep a spare shirt for just such occasions.
He had a plain blue cotton dress shirt still in its original plastic bag stowed in his trunk. It had probably been there for years. "Thanks, Tomsic." When I put it on, it was like wearing a tent. Two sizes too big.
The medivac chopper came in from the north and settled to a rest well away from the radio towers. Two paramedics hustled out with a stretcher and loaded Mr. Lawrence into the helicopter's bay. They told us that they were going to lift him to Martin Luther King, Jr. Hospital, which would be a five-minute flight, and Mrs. Earle wanted to go. They refused to take her until Angela Rossi volunteered to go with her. Lincoln Gibbs told Rossi that we would pick her up at the hospital.
When the helicopter had lifted away and disappeared over the hills, Gibbs looked at me and Pike, and said, "Well?" The first of the black-and-whites was just now kicking up dust on the roads below.
"Green's people got to LeCedrick Earle. They offered him money and an early out from prison if he could get his mother to change her story. He hadn't spoken to her in six years, but he called and told her that the guards and the other prisoners were beating him because she was defending the police. Green's people went to her also, and helped convince her that it was real, and that the only way they could save LeCedrick was if she changed her story so that they could get him away from the guards."
Gibbs nodded. "Figured it had to be something like that. Figured she wouldn't do it for money."
Tomsic said, "Will she say that on the record?"
"Yes. And we've got something else, too."
They looked at me.
"Truly made a dying declaration that Teddy Martin admitted murdering his wife, and that Jonathan Green conspired with Truly and Kerris to fabricate false evidence against Pritzik and Richards."
Tomsic smiled, and Lincoln Gibbs made a little whistle. Gibbs said, "Truly said that to you?"
"Pike and Rossi heard it, too. Mrs. Earle might've heard it, but I'm not certain that she did."
Gibbs went back to his car and spoke on his cell phone for a time. As the black-and-whites rolled up, Tomsic met them and told them to hang around. There wasn't anything for them to do until the detectives who would handle the scene arrived. Gibbs came back in a few minutes and said, "Is that your Jeep on the other side of the hills?"
Pike said, "Mine."
"Okay. We'll pick up Rossi and Mrs. Earle at MLK and go see Sherman."
I spread my arms. "Like this?"
Tomsic was already walking to his car. "The shirt looks great on you. What's your beef?"
"It looks like I'm wearing a tent."
Pike's mouth twitched.
I said, "Hey, Gibbs."
He looked back.
"How about I pick up Mrs. Earle? It might be easier for her."
He stared at me for a short moment, and then he nodded. "We'll meet you at Sherman's."
A black-and-white brought us to Pike's Jeep, and we drove directly to the MLK emergency trauma center. Mr. Lawrence was in surgery, and Rossi and Mrs. Earle were in the waiting room. I sat next to Mrs. Earle and took her hands. "We need to go see the district attorney. We need to tell her what we know about all of this. Do you see?"
She looked at me with clear eyes that were free of doubt or equivocation. "Of course. I knew that we would."
The four of us drove to Anna Sherman's office in Pike's Jeep. Mrs. Earle rode with her hands in her lap and her head up. I guess she was thinking about LeCedrick. We did not listen to the radio during this time, and perhaps we should have. Things might've worked out differently if we had.
It was just after three that afternoon when Louise Earle, Angela Rossi, and I were shown into Anna Sherman's office. The bald prosecutor, Warren Bidwell, was there, along with another man I hadn't seen before, and Gibbs and Tomsic.
Sherman greeted us, smiling politely at Louise Earle and giving me a kind of curious neutrality, as if the meeting in Greenblatt's parking lot had never happened. I guess that they had told her what to expect.
Sherman offered coffee, which everyone declined, and as we took our seats she passed close to me and whispered, "Great shirt."
I guess that they'd told her about the shirt, too.
Anna Sherman asked Mrs. Earle if she would mind being recorded, and if she would like to have an attorney present.
Mrs. Earle said, "Am I going to be arrested?"
Anna Sherman smiled and shook her head. "No, ma'am, but it's your right, and some people feel more comfortable."
Mrs. Earle raised her hands. "Oh, Lord, no. I don't care for all those lawyers."
Tomsic grinned big time at that one. Even Bidwell smiled. Sherman said, "Do you mind if we record?"
"You can record whatever you want. I don't care who hears what I have to say." Her jaw worked, and for a moment she looked as if she was going to cry again. "You know, those things I said about LeCedrick and the officer weren't true." She looked at Angela. "I want to apologize for that."
Angela Rossi said, "It's okay."
Mrs. Earle said, "No, it is not. I am so ashamed that I don't know what to do." She looked back at Sherman. "They said that the most horrible things were happening to my boy. They said that he would surely die in that place unless I helped get him out of there."
Anna Sherman turned on the recorder. "Who is 'they,' Mrs. Earle?"
Mrs. Louise Earle went through her part of it first, telling how she received the first phone call that she'd had from LeCedrick in six years, how he'd pleaded with her that his life was in danger there in the prison, that he'd called again, crying this time, begging her to help and saying that he'd hired an attorney named Elliot Truly who wanted to come speak with her. She told us how Truly and Kerris had come to the house, confirming the horror stories that LeCedrick had claimed, and convincing her that the fastest way to get LeCedrick moved from harm's way was to claim that the police had framed him those six years ago, just as LeCedrick had always said. She said that Truly helped her work out what to say.
Anna Sherman took notes on a yellow legal pad even though the recorder was running. Bidwell was taking notes, too. Sherman said, "Did Jonathan Green take part in any of these conversations?"
"No, ma'am."
Bidwell said, "I saw you and Green together at a news conference."
"That's right. When Mr. Truly said it was time to say my piece, he drove me over to meet Mr. Green."
"Did you and Mr. Green talk about what you were going to say?"
Louise Earle frowned. "I don't think so." She frowned harder, trying to remember. "I guess we didn't. I guess he knew from Mr. Truly. He just said to say it to the newspeople as direct and as honest as I could."
Gibbs leaned forward. "He said for you to be direct and honest?"
Sherman shook her head. "Green's smart." She drew a line across her pad. "Okay. Let's hear what you have."
I told them how Rossi and I had gone to see LeCedrick, and what we had learned from him, and how Kerris and his people had gotten to Mrs. Earle first and how we had followed them to the pumping fields west of Baldwin Hills, and what happened there. I told them what Truly had said as he lay dying. I said, "Truly confirmed everything that Mrs. Earle and LeCedrick said. He tied in Jonathan Green, and stated that it was Green who directed the fabricating of phony evidence implicating Pritzik and Richards."
Bidwell put down his pad. "Why would Green do that?"
I handed him the hard copy printout of the contracts between Jonathan Green and Theodore Martin. "These are copies of confidential retainer agreements between Green and Teddy Martin. They have an amended agreement that gives Jonathan Green ownership and control of most of Teddy's businesses." Anna Sherman stared at me without emotion as I said it.
Bidwell flipped through the sheets, frowning. "How in hell did you come by these?"
I shrugged. "You just find things sometimes."
Sherman smiled, still without emotion.
Bidwell passed the pages to her. "Inadmissable."
Anna Sherman took the pages but didn't look at them. The neutral smile stayed. She said, "You have a dying declaration from Elliot Truly implicating Jonathan Green in the falsification of evidence."
I nodded. "We do."
"Who heard it besides you?"
Rossi said, "I did. So did Joe Pike."
Sherman looked at Louise Earle. "Did you hear it, Mrs. Earle?"
Louise Earle looked uncertain. "I don't think so. They put me behind all this metal. There was shooting, and I thought Mr. Lawrence was dead."
Anna Sherman patted her hand. "That's all right."
Bidwell said, "So what we've got is a dying declaration witnessed by three people who have an interest in attacking Jonathan Green."
Rossi said, "What in hell does that mean?" She stood. "We're giving it to you on a plate, and you're saying it's not enough?"
Bidwell crossed his arms and rocked.
Anna Sherman looked at the third guy. He hadn't said anything, and now he was staring at her. She stood and said, "It's not the best, but I want to move on this. I am confident that these people are telling the truth, and that Jonathan Green is guilty of these crimes."
Rossi said, "Truly said something else, too."
Everyone looked at her.
"He said that Theodore Martin admitted killing his wife."
Anna Sherman smiled again, and Bidwell leaned forward.
"That's why the agreement was amended. Teddy said that he'd pay anything for Green to save him, and Green went for everything. Teddy put almost all of his personal and corporate holdings into escrow as payment to Green."
Bidwell snatched up the pages and flipped through them again.
I said, "There's also a second amendment that releases several million dollars in holdings back to Teddy Martin. I figure it's because Teddy thought he could get bail, and if he got it he was planning to skip."
Rossi said, "Truly confirmed that."
Anna Sherman leaned forward just like Bidwell now, but she wasn't smiling anymore. "Truly said Teddy was planning to skip?"
Rossi and I answered at the same time. "Yes."
Bidwell ran out of the room. The third guy angrily slapped his hands and said, "Sonofabitch!"
I said, "What?"
Anna Sherman slumped back in her chair and looked terribly tired. "Theodore Martin was granted bail this morning at ten o'clock."
#
Theodore "Teddy" Martin was granted bail in the amount of five hundred thousand dollars at ten that morning under a nine-nine-five motion made by Jonathan Green on the defendant's behalf in the Los Angeles Superior Court. The nine-nine-five was granted, according to the presiding judge, due to the revelation of "evidence consistent with innocence." Namely, the evidence found by one Elvis Cole linking Pritzik and Richards to the kidnapping of Susan Martin. The same evidence that Elliot Truly declared to have falsified as he bled to death in a maintenance shed in the Baldwin Hills.
Lincoln Gibbs and Anna Sherman got on the phones in a mad scramble to ascertain Teddy's whereabouts. Calls were made to Green's office, Teddy's business manager, and Teddy's home. Radio cars were sent to all three locations. Both Green's office and the business manager denied any knowledge of Teddy's whereabouts, and there was no answer at his home. The radio car reported that his home appeared empty, and that a Hispanic housekeeper had responded to their knock and said that "Mr. Teddy" was not and had not been home. Sherman grew so angry that she slammed her phone and cursed, and Mrs. Earle said, "What's going on?"
I said, "Teddy jumped bail."
Sherman snapped, "We don't know that."
I picked up the amended retainer agreement and flipped to the list of Teddy's personal and corporate possessions. Teddy Jay Enterprises owned a Cessna Citation jet aircraft. It was listed among the properties transferred to Jonathan Green's control, but what does that matter when you're running for your life? Stealing jets isn't much when you compare it to killing people.
Anna Sherman was yelling into the phone at someone in Jonathan Green's office when I held the amendment in front of her with my finger pointing to the jet. She saw the listing, then said, "Call you back," and hung up. "Where does he keep it?"
"I don't know."
Sherman called Green's business manager again and demanded to know where Teddy housed the jet. She was yelling, and the business manager probably got his nose out of joint because of it, and he probably made the mistake of asking if she had a court order. Sherman went ballistic. Her face turned purple and a webwork of veins stood out on her forehead, and Gibbs said, "Lord, Anna. You'll have a stroke."
Anna Sherman shouted into the phone that if the business manager didn't cooperate she would have him arrested within the hour for accessory after the fact and conspiracy. It worked. The business manager told her, and Anna Sherman repeated the information as he gave it. "Van Nuys airport. Skyway Aviation." She also repeated a phone number, which Dan Tomsic copied.
Gibbs, Tomsic, Rossi, and I watched Anna Sherman dial Skyway, identify herself, and ask to speak with whoever was in charge. Mrs. Earle was watching, too, but you could tell that it wasn't as important to her. Bidwell was arranging a ride back to the hospital for her. The Skyway manager came on the line, and Anna Sherman identified herself again. She asked as to the status of Theodore Martin's Citation jet, then asked several follow-up questions. We knew the answers from her expression. Lincoln Gibbs yelled, "That sonofabitch," and kicked the couch. Tomsic sat and put his face in his hands, as if he'd played a long, close game and given it everything and lost in the end. After maybe six minutes Anna Sherman hung up and looked at us with an ashen, strained face. "Theodore Martin boarded his airplane at approximately eleven-forty this morning, and the jet departed at exactly eleven-fifty-five. His pilot filed a typical IFR flight plan to Rio de Janeiro." Anna Sherman sat in her chair with her hands in her lap and put her head back. "He's gone."
Mrs. Louise Earle looked as if she was about to cry. "Did I do something wrong?"
Angela Rossi stared at her for a moment, then put her arm around Mrs. Earle's shoulders. "No, ma'am. No, you didn't. He just left. It happens all the time."
Sherman took a deep breath, then sat forward and picked up the phone again. Only this time there wasn't any urgency to it. "I'll notify the FBI and ask them to speak to the State Department. He's still in the air. Maybe we can work something out with the Brazilians."
Bidwell said, "We don't have reciprocal extradition with Brazil."
Sherman snapped, "Maybe we can work something out."
I said, "You going to do anything about Green?"
Anna Sherman stared at me for maybe six seconds, then she put down the phone. "Oh, yes. Yes, I'm definitely going to do something about Mr. Green."
Bidwell said, "You want to file an arrest warrant?"
Anna Sherman was looking at Angela Rossi. "Yes, we'll file an arrest warrant. I saw Judge Kelton downstairs. Look him up and have it signed." Arrest warrants had to be signed by a judge.
Bidwell started toward the door. "I'll call Green's office and set it up. How much time do you want to give him to turn himself in?" Often in cases like this, the attorney is notified that a warrant has been issued and is allowed to turn himself in.
Anna Sherman shook her head, still looking at Angela Rossi. "To hell with that. We're going to go over there and arrest his ass."
Angela Rossi smiled. So did everyone else.
I said, "You guys mind if I tag along?"
Lincoln Gibbs was pacing now. Grinning and anxious to take action, sort of like a leopard sensing that a hunt was on. "No sweat."
Rossi wanted to come, too, but Lincoln Gibbs told her no. She was still suspended, and an administrative action could be taken against her for violating her suspension.
Sherman and Bidwell made their calls and drafted their documents, and one hour and ten minutes later they were ready to pay a visit to Mr. Jonathan Green, Attorney to the Stars. Mrs. Louise Earle had already been returned to the hospital. Rossi walked out with us, but in the lobby she had to go one way and we another. A radio car was going to take her home.
Rossi put out her hand and we shook. "I want to thank you."
"No problem."
"No, I mean it."
"I understand."
"I'll call Joe."
I said, "So long, Rossi."
We smiled at each other and then she walked away.
Gibbs and Tomsic and I crowded into Anna Sherman's car and drove to Jonathan Green's office on Sunset Boulevard. A couple of uniforms in a radio car followed us. A district attorney almost never accompanied the police on an arrest, but then neither did freelance private eyes. I guess this was just too good to pass up.
We double-parked in front of his building, jamming up the west-bound flow on Sunset, and walked in past the receptionist and the security guys in their blazers. A blond security guy with a red face tried to make a deal about stopping us to see the warrant, but Dan Tomsic said, "You've gotta be kidding," and motioned at the uniforms to walk the guy out of the way.
We took the elevator up to the fourth floor and Sherman said, "You've been here before. Which way?"
I showed them to Green's office. Green had not been notified of Elliot Truly's death, nor of the deaths of his other people, nor had it yet hit the news. As we walked through the halls, lawyers and legal assistants and secretaries and clerks appeared in their doors. Jonathan Green's secretary stood as we approached, and I said, "Knock knock knock, Chicken Delight!"
She looked at Anna Sherman. "May I help you?"
Anna Sherman said, "No." We trooped past the secretary and through the door and into Green's office. Green and the two lesser attorneys and the videographer and three people I'd never seen before were seated around his conference table with their jackets off and their sleeves rolled. The videographer and his sound tech were seated in the background, the camera on the floor, sipping coffee and talking between themselves. Guess there's only so much you can do with endless footage of lawyers sitting around tables. Jonathan Green looked at us without a great deal of surprise and said, "Doors are made for knocking."
I said, "Not bad. I was kinda hoping you'd say, 'What's the meaning of this?' "
Anna Sherman smiled sweetly. "Sorry for the intrusion, Jonathan. But we're here to arrest you on the charges of tampering with evidence, obstruction of justice, conspiracy to commit murder, and murder."
The videographer's eyes got big and his jaw dropped. I waved at his camera. "Better turn it on. You don't want to miss this." The videographer jumped across the sound tech for his camera, spilling both his coffee and hers.
Anna Sherman turned to Lincoln Gibbs. "Lieutenant, please inform Mr. Green of his rights and take him into custody."
Lt. Lincoln Gibbs handed the warrant forms to Jonathan, then recited his rights. Jonathan didn't interrupt, and didn't bother to examine the forms. He sat with a kind of half-smile, as if he had anticipated these events. Maybe he had. When Gibbs finished with the rights, he said, "Would you stand, sir? I have to handcuff you." Polite.
Jonathan submitted without complaint. He said, "Anna, this is the most flagrant case of judicial manipulation I've ever seen. I'll have you before the bar for this."
Anna Sherman said, "Teddy Martin has jumped bail and is on his way to Brazil. Elliot Truly, Stan Kerris, and two other men in your employ are dead. Elliot Truly supplied a dying declaration implicating you in the manufacture of false evidence, as well as the murder of James Lester and the Kidnapping of Louise Earle."
Jonathan Green said, "That's absurd. I don't know what you're talking about." He angled his face toward the camera when he said it.
She said, "That's why we have trials, Jonathan. To determine the facts."
Lincoln Gibbs took Green by the arm and guided him to the door. Jonathan Green turned back just long enough to say, "We won't get to trial, Anna." He smiled when he said it, and his smile was confident and without fear. "I guess you believe you have reason to do this, but for the life of me I can't imagine what it might be." He angled toward the video camera again. "I look forward to seeing your proof, and I hope for your sake that this isn't some ugly form of harassment."
Gibbs and Tomsic escorted him out, the videographer scurrying ahead of them to capture every moment of the arrest and departure.
I stood with Sherman, watching them go, and wondered at Jonathan Green's lack of concern. I was thinking that maybe he was crazy, or arrogant, or brimming with the fatal flaw of hubris, but you never know.
Maybe he was just used to winning.
#
Theodore Martin's flight from the country was covered throughout the evening by every one of the local Los Angeles television stations, effectively eliminating regular programming. Live news remote teams assaulted Skyway Aviation, Angela Rossi's home, Jonathan Green's office, and spokespeople for both the LAPD and the District Attorney's office. Angela Rossi did not return home that night, and so was unavailable for comment. She picked up her boys and spent the night with a friend. The Skyway people were available, however, and were more than a little surprised by the army of microwave vans and news teams who invaded their otherwise quiet world.
The Skyway employees who were interviewed included the flight operations manager, a young female flight dispatcher, and an even younger male line attendant. The line attendant was a seventeen-year-old kid name Billy Galovich who washed the planes, pumped them full of jet fuel, and pushed them in and out of a hangar with a little tractor. The sum total of his involvement in Teddy Martin's escape was that he had towed out Teddy's Citation, fueled it, then greeted the pilot, a very nice Hispanic man who introduced himself as Mr. Garcia. I counted fourteen interviews with Billy Galovich that evening, and then I stopped counting.
The flight dispatcher's claim to fame was that she had taken the call from Teddy Martin, who personally ordered that his Citation be readied for flight. The dispatcher's name was Shannon Denleigh, and she related that Mr. Martin told her that his pilot would be a man named Mr. Roberto Garcia, and that Mr. Garcia would be along directly. She said that she informed the flight operations manager, a Mr. Dale Ellison, of the call and then she left the premises to have her nails done. I stopped counting her interviews at sixteen. Dale Ellison related that Mr. Garcia arrived moments later, preflighted the Citation, and filed his flight plan. He said that Mr. Garcia was an amiable, friendly man who identified himself as a flight officer with Air Argentina who picked up corporate charters to earn extra money. I didn't bother to count the number of times that Dale Ellison was interviewed, but it was plenty.
Reports of Jonathan Green's arrest and the charges against him were interspersed with the coverage of Teddy's flight, but when the newspeople discovered that the Citation was still in the air, the real show began. Reporters and cameras descended upon the FAA and the various Flight Operations Centers between Los Angeles and Rio. The Citation's path was charted, and its progress was depicted on a global map. It was kind of like watching the beginning of _Casablanca_. Every network put a little clock in the corner of their picture, counting down the time until the Citation landed. Crime and show business had merged.
Foreign bureau reporters flocked to the Rio de Janeiro airport, and Teddy Martin's landing was covered live even though it was after midnight in Rio and you really couldn't see anything. The Citation taxied to a private flight service facility for corporate jets where it was met by Brazilian authorities and a small army of newspeople. A spokesman for the Brazilian authorities said that Mr. Martin would be questioned as to his plans, but thereafter would be free to go. Teddy Martin pushed through the cameras with his face covered, ignoring the shouting reporters. He reached the flight service facility's door, then apparently changed his mind and paused to make a short statement. Teddy Martin said, "Please don't interpret my flight from California as indicative of guilt. I promise you, I swear to you all, that I did not murder my wife. I loved her. I left because I am convinced that I could not and would not get a fair and just hearing. I do not know why they are doing this to me." He disappeared into the building and must have slipped out by some prearranged and secret manner because he was not seen again.
I went to bed at twenty minutes after one that night, and still the networks were on the air, rehashing the landing, replaying the interviews, offering taped "live" coverage of something that was no more alive than a nightmare.
#
The phone rang several times throughout the night. I stopped answering and let the machine get the calls after I realized that they were reporters, looking for yet another comment. I finally unplugged the phone.
I slept late the next morning and woke to a quiet house. The cat was sleeping on the foot of my bed and the finches were waiting on the deck rail and no one was trying to shoot me, which was good, but for the first time in many days I felt the emptiness of Lucy's absence, which wasn't.
My involvement with Angela Rossi and Louise Earle and the events in their lives seemed to be at an end or, if not ended, then certainly diminished. Anna Sherman wanted to interview me in greater detail, but she would speak to Rossi first, then Gibbs and Tomsic. It might be days before we could get together.
I got out of bed, took a shower, then ate a bowl of granola and cottage cheese and sliced peaches. I drank a glass of nonfat milk. I phoned Martin Luther King Hospital, asked about Mr. Lawrence, and was told that he was doing well even though he was listed in critical condition. The nurse remembered me and told me that Mrs. Earle was still there, asleep in the waiting room. She had been there throughout the night. I called a florist I know and sent flowers, addressing them to Mrs. Earle as well as to Mr. Lawrence. I hoped that they would brighten her day.
At twenty minutes after eleven my phone rang again, and this time I answered. Life in the fast lane. Joe Pike said, "Are you looking at this?"
"What?"
"Turn on your television."
I did.
Jonathan Green was surrounded by reporters on the steps of the Superior Court Building. The network legal analyst was saying that Green had been arraigned at ten A.M., had posted minimal bail, and was now about to make a statement. The two lesser attorneys were behind him, as was an older, gray-haired African-American attorney named Edwin Foss. Foss was a criminal defense attorney of Green's stature who had made his reputation defending a transient who had shot four people to death while robbing an AM-PM Minimart. The murders had been caught on videotape, but Foss had still managed to gain an acquittal. I guess he had convinced the jury that it was reasonable to doubt what they had seen.
Edwin Foss whispered in Jonathan's ear, then Jonathan stepped to the microphones and made his statement. His tone was somber and apologetic, and Foss kept a hand on Jonathan's shoulder as he spoke. Guidance. Green said, "No one is more surprised by Theodore Martin's actions than me. I have believed in his innocence from the beginning, and I still believe him to be an innocent man. I believed then, and believe now, that the evidence against Theodore Martin was planted by unscrupulous officers involved in the investigation. Teddy, if you can hear these words, I urge you to return. Justice will prevail."
Pike said, "You think Teddy's tuned in, down there in Rio?"
"Shh."
Green said, "I pledge my full cooperation to those investigating the charges that have been made against me. I will aid in uncovering whatever wrongdoing has occurred, if any, and in the prosecution of anyone in my employ who has conspired to breech the canon of ethics by which I have lived my life. I state now, publicly and for the record, that I have behaved honorably and within the law. I have done no wrong."
Green's attorney again whispered something in Green's ear and gently pulled him away from the microphones. The reporters shouted questions, but Green's attorney waved them off and said that there would be no questions.
I turned off the television and said, "This guy is something. He's already doctoring the spin."
Pike didn't respond.
"You don't think he can beat this, do you?"
There was a pause, then Pike hung up. Guess he didn't have an answer. Or maybe he didn't want to think that it was possible.
I made an early lunch for myself, then brought the phone out onto the deck and called Lucy Chenier at her office. She had heard about Jonathan's arrest and Teddy's flight on the national news, but she didn't seem particularly anxious to hear the inside dirt. When I described the events beneath the radio towers, she told me that she was late for a meeting. Great. Anna Sherman called later that afternoon and asked me to come to her office the following day to make a statement. I did, and spent three hours in the Criminal Courts Building being interviewed by Sherman, Bidwell, and three LAPD detectives whom I had not previously met. Pike came in as I was leaving. Sherman told me that Mrs. Earle had been interviewed the day before.
Two days after my interview, Mr. Walter Lawrence was taken off the critical list. His prognosis was excellent. I went to see him and brought more flowers. Mrs. Earle was still there and told me that she planned to visit LeCedrick. It would be the first time that she'd seen him in the six years that he had been at Terminal Island. I offered to drive her.
Teddy's flight and Green's arrest stayed in the headlines. "Teddy Sightings" were a regular feature in the tabloids, which reported on various occasions that Teddy was now living in a palatial Brazilian mansion that had been built by a famous Nazi war criminal, that Teddy had been seen in the company of Princess Diana, and that Teddy was gone for good because he had been abducted by short gray aliens with large heads. The California State Bar Association announced that it was launching an investigation into Jonathan Green's conduct independent of that by the Los Angeles Police Department and the District Attorney's office. Green said that he welcomed the opportunity to clear his name and would cooperate fully.
Jonathan Green and his attorney appeared regularly on local television news, local radio talk shows, and in the L.A. _Times_. Reports from "unnamed sources" began surfacing that Elliot Truly had made a secret deal with Teddy, unknown to Mr. Green. Leaks "close to the prosecution" were quoted as saying that computer files found at Elliot Truly's home confirmed such an agreement. Other sources leaked that Truly had had several meetings with Teddy while Teddy was in jail to which Mr. Green was not privy. Carefully worded public opinion polls charted a swing in the belief of Jonathan Green's involvement from "absolutely" to "probably" to "uncertain."
Eleven days after the events beneath the radio tower, the LAPD Internal Affairs Division announced that it had completed its investigation of Detective Angela Rossi and had found there to be no evidence either in the LeCedrick Earle matter (LeCedrick Earle himself had recanted his claims against her) or that she had manufactured or planted evidence against Theodore Martin. The story was given two inches on page nineteen of the _Times_ , and the same public opinion polls indicated that seventy-three percent of the public still believed that she was a corrupt cop who had framed LeCedrick Earle (even though he now denied it) and who had "probably" mishandled evidence against Teddy Martin. She was returned to active duty with her partner, Dan Tomsic.
I listened to the news and followed the investigations with a growing sense of unease. Jonathan Green signed a two-million-dollar contract with a major book publisher to publish his version of the story. He appeared on _Larry King Live_ and _Rivera Live_ , and each time he presented himself as a victim. I was offered many jobs, but I declined them. The press still called, though with less frequency, and I avoided them. I listened to talk radio and gained weight, as if I felt a hunger that I couldn't satisfy.
The days grew warm again, and I decided to refinish the deck. It had been almost eight years since I'd last stained and sealed the deck, and the wood was showing its age. Joe offered his help, and we spent the core of each day sanding and staining and sealing. We listened to music as we worked, but from time to time we turned to the news. Twenty-three days after the events beneath the radio tower, the California Bar quietly closed its investigation, saying that all evidence pointed to wrongdoing by Elliot Truly and not by Jonathan Green. Twenty-five days after the tower, the District Attorney's office dropped all charges against Jonathan Green save one count of tampering with evidence. I was on a ladder beneath the deck when we heard the news, and Pike said, "He's getting away with it."
I went inside and called Anna Sherman, who said, "It's the best we can do." Her voice was faraway and sounded lost.
I said, "This is crap. You _know_ he was behind it."
"Of course."
"He set up Truly just like he set up Rossi and Pritzik and Richards. He ordered Lester's murder. They were going to kill Louise Earle. He did his best to destroy the life and career of a police officer who did nothing worse than do her job."
She didn't say anything for a time, and then she said, "He knows how to play the game, Elvis. What can I tell you?" Then she hung up.
Twenty-eight days after the towers, Pike and I finished sealing the deck. It was slick and gleaming and smelled of marine-grade varnish. After the varnish had cured, we put the deck chairs and the Weber and the little table back and sat in the sun drinking cold Falstaff. We sat for awhile, and then Pike said, "Say something."
I looked at him.
"You haven't said anything for three days. You've said next to nothing for almost two weeks."
"Guess I'm getting like you."
I smiled at him, but he didn't smile back.
I finished my Falstaff, crimped the can, then put it carefully onto the shining deck. Little rings of condensation beaded on the thick varnish. I said, "I'm not sure that I want to do this anymore."
"Be an investigator?"
I nodded.
"What do you want to do?"
I shrugged.
"You want to stop being what you've been for almost fifteen years because Jonathan Green is getting away with murder?" He frowned when he said it. Like maybe he was disappointed.
I spread my hands. "I guess that's it. Elvis Cole, sore loser."
Pike shook his head.
I went inside, brought out two fresh Falstaffs, and gave him one. I said, "What would you say if I told you that I was thinking about moving to Louisiana to be closer to Lucy?"
Pike sipped some of the Falstaff, then gazed out at the canyon, then wet his lips and nodded. "I'd say that I'd miss you."
I nodded.
"I'd say that if that's what you needed to do, that I would help any way that I could."
I nodded again.
"You talk to her about it?"
"Not yet."
Pike shook his head. "You're something."
Four hours later Pike was gone and I was cooking a very nice puttanesca sauce when I decided to call Lucy Chenier. I was most of the way through a bottle of California merlot. In the course of my life I've been shot, sapped, slugged, stabbed with a broken beer bottle, and I've faced down any number of thugs and miscreants, but talking to Lucy about moving to Louisiana seemed to require fortification. She answered on the third ring, and I said, "Guess who?"
"Have you been drinking?" Don't you hate smart women?
"Absolutely not." Giving her affronted. Giving her shocked. Then I said, "Well, maybe a little."
She sighed. "I heard on the news that the charges against Green were reduced. How's Angela?"
"Not great, but not bad, either. The public still thinks that she's rotten, but IA cleared her."
"How nice for her children."
"Green kept himself insulated so that there was always plausible deniability."
"What about Truly's dying declaration?" I had told her about Truly weeks ago. "That's legitimate evidence."
"It is, but since it was witnessed only by me and Angela and Joe, the powers that be view it as questionable. Because I resigned from Green's employ, and because he accused Rossi, the powers that be feel that a jury would discount our version of events."
She didn't say anything for a time, and then she said, "Well, in this case the powers that be are probably right."
I nodded, but she probably couldn't see it. "I don't believe Truly had a secret agreement with Teddy Martin. Green fabricated that, just as he fabricated the business about Pritzik and Richards."
"I'm sure you're right."
"Truly was telling the truth."
"I'm sure of that, too."
I didn't say anything. I was staring at the bubbles rising in the sauce and my shoulders felt tight and I was wishing that I hadn't drunk all the wine.
Lucy said, "It hurts, doesn't it?"
I moved my tongue, trying to scrub away the wine's taste. "Oh, God, yes."
"You try so hard to make things right, and here's this man, and he's oozing through the system in a way that keeps things wrong."
"He is defiling justice." Defiling. That was probably the merlot talking.
She said, "Oh, Studly." I could see her smile. "The law is not about justice. You know that."
I finished the merlot and turned off the sauce. It was thick with chunks of tomatoes and black olives and raisins. I had cooked it without being hungry. Maybe I just wanted to give myself something worthwhile to do. "Of course I know, but it should be."
Lucy said, "The law is an adversarial contest that defines justice as staying within the rules and seeing the game to its conclusion. Justice is reaching a conclusion. It has very little to do with right and wrong. The law gives us order. Only men and women can give us what you want to call justice."
I took a deep breath and let it out. "God, Lucille, I wish you were here."
"I know." Her voice was soft and hard to hear. Then she said, "You're still the World's Greatest Detective, honey pie. They can't take that away from you."
It made me smile.
Neither of us spoke for a time, and then Lucy said, "Do you remember Tracy Mannos at Channel Eight? We met her at Green's party."
"Sure. The program manager."
"She called me last week. She arranged for the network affiliate here in Baton Rouge to shoot a test tape of me, and after she saw it she offered me a job as an on-air legal commentator."
I said, "In Baton Rouge?"
"No, Elvis. Out there. In Los Angeles."
I couldn't say anything. The merlot seemed to be rushing through my ears.
Lucy said, "It's more money, and we would be closer to you, but it's such a big move." You could hear her uncertainty.
I said, "You'd come to Los Angeles?"
"There's so much to think about. There's Ben. There's my house and my friends. I'm not sure what to do about Richard."
"Please say yes." It came out hoarse.
She didn't say anything for a time. "I don't know just yet. I need to think about it."
"I told Joe that I was thinking about moving to Baton Rouge."
Another pause. "Are you?"
"Yes."
"Would you?"
"Yes."
"Why?"
"You know why, Lucille. I love you."
She didn't speak for another moment, and when she did her voice seemed lighter, somehow more at ease. "I need to think."
"Call me tomorrow."
"I may not know tomorrow."
"Call me anyway."
She said, "I love you, Studly. Always remember that."
Lucille Chenier hung up, and I lay on my kitchen floor and smiled at the ceiling, and not very much later I knew that I had found the last and final way to bring Jonathan Green to justice.
Or, at least, a close approximation.
#
I called Eddie Ditko first. He came over that night, coughing and wheezing, but happy to eat spaghetti with the puttanesca sauce and listen to my account of the events in the maintenance shed while he recorded my every word. He grinned a lot while I talked and said that he could guarantee a bottom half of the front-page position for the story. He said, "Man, the shit's gonna hit the fan when this comes out."
"That's the idea."
When Eddie was gone, I called Tracy Mannos, who put me in touch with Lyle Stodge at twenty minutes after ten. Lyle and Marcy anchored the eleven P.M. newscast as well as the five. Lyle was only too happy to talk to me, and only too happy to accept my offer of an interview. He said, "We've been hoping to get you for a comment on all of this! Can you make the eleven o'clock?"
"Nope."
"How about tomorrow at five?"
"I'll be there." The five o'clock newscast had the larger audience.
I phoned every person who had interviewed me in print or on radio or television, or who had wanted to interview me. I spent most of the night and part of the next morning on the phone, and everybody was happy to talk to me. I called both Peter Alan Nelsen and Jodi Taylor, and asked if they could put me in touch with any of the major network and cable news people, and of course they could. Even _Daily Variety_ wanted an interview. Everybody wanted to know if I had been duped by Theodore Martin, and everybody wanted to know what had happened in the maintenance shed, and everyone still considered me the hero of the defense effort, just the way Jonathan had hoped when he had staged the news conferences with his hand on my shoulder. I told them that I would be happy to tell them exactly what happened, especially if we were on the air live.
By three the following afternoon, I had completed eleven interviews and had provided each interviewer with a copy of Green's amended retainer agreement with Theodore Martin. Seven other interviews were scheduled, and more would be forthcoming. I had copies for them, too.
At twelve minutes after three, I parked in a red zone outside Jonathan Green's Sunset Boulevard building and went inside. I shoved past the receptionist and ran up the stairs and barged past the army of clerks and assistants and minions. There was a noticeable absence of blue-blazered security guards, but I guess those few who hadn't been killed in Baldwin Hills had been fired. All the better for Green to separate himself from Kerris.
The _Inside News_ videographer and his sound technician were talking to a slim woman by the coffee machine when I went past. The videographer's eyes went wide when he saw me, and the sound tech dropped her coffee. The videographer said, "What are you doing here?"
I grabbed him by the arm and pulled him along. "Do you have tape in that thing?"
"Sure."
"You're going to love this."
The sound tech scrambled after us.
Jonathan Green's office occupied the entire east end of the fourth floor. An efficient-looking woman in her early forties tried to tell me that I couldn't go in, but I ducked around her and hit the door, only the door wouldn't open. The woman said, "You stop that! You stop that before I call the police!"
The sound tech said, "You have to buzz it open."
I said, "Where?"
The sound tech hurried to the woman's desk and pressed the buzzer. The sound tech was grinning.
I kicked open the door and stormed in and found Jonathan Green on the phone. The two lesser attorneys were with him, along with a younger man with a notepad. Somebody's secretary. The smaller of the lesser attorneys fell over a chair trying to get out of my way. Green said, "I'm calling the police!"
I pulled the phone out of his hands and tossed it aside. I said, "Here's the bad news, Jonathan: You've become my hobby. I know what Truly knew, and I am telling it to anyone who will listen."
Green maneuvered to keep his desk between us. His face had grown white. "The police are on their way! I'm warning you!"
I threw a copy of the retainer agreement at him. "I'm also passing out copies of this. The _Examiner_ is going to print it in this evening's edition."
Green looked at it without touching it and shook his head. "This means nothing. For all anyone knows you wrote it yourself. It isn't admissible."
"Not in a court of law, Jonathan. But we're going to try you in the court of public opinion." I shoved his desk, and Jonathan jumped backwards. "I will hound you, and I will not stop. I will tell everyone that it was you who falsified the evidence, and you who ordered James Lester killed, and you who attempted to take the life of Louise Earle." I started around the end of the desk, and Jonathan scrambled in the opposite direction.
"You can't do that! I'll get a restraining order!"
"What's that to a tough guy like me?"
"No one will believe you!"
"Sure they will, Jonathan. I am the World's Greatest Detective, remember? Above reproach. Trustworthy."
Jonathan glared at the lesser attorneys and yelled, "Don't just stand there! Do something!"
The larger lesser attorney ran out the door.
"I will keep this alive until the DA can finally build a case or until you are driven out of business. I will haunt you like a bad dream. I will come to your house and follow you into restaurants and send videotapes of my interviews to your clients."
He drew himself up into a vision of outrage. "We have laws against that, you idiot! That's libel! That's slander! You won't get away with it!"
I looked at the videographer. "Are you getting this?"
The videographer was all smiles. "Hell, yes! What an ending!"
I jumped across the desk and punched Jonathan Green hard in the mouth one time. He floundered backwards and went over his chair and landed on his ass. The smaller lesser attorney shouted, "Oh, my God," and then he ran, too.
Jonathan Green said, "You hit me! You actually laid hands on me!" He felt his mouth, then looked at his red fingers and started crying. "You broke my teeth!"
I walked over to Jonathan Green, looked down at him, and said, "So sue me."
And then I walked out.
# Acknowledgments
The author would like to thank Bruce J. Kelton, former Assistant United States Attorney and a managing director of the investigative firm Kroll Associates, for sharing his knowledge of the law and the criminal justice system. Additional thanks go to Det. John Petievich, whose counsel on matters relating to the Los Angeles Police Department in this novel and others has been invaluable. Any errors contained herein are solely the responsibility of the author.
Special thanks go to the world's greatest editor, Leslie Wells.
The author would also like to thank Patricia Crais, Lauren Crais, Robert Miller, Lisa Kitei, Carol Perfumo, Samantha Miller, Brian De Fiore, Marcy Goot, Chris Murphy, Kim Dower, and Jennifer Lang for their support, superior talents, and great efforts on the author's behalf.
# About the Author
Jonathan Exley
Robert Crais is the author of many novels, including the _New York Times_ bestsellers _The Last Detective_ , _Hostage_ , and _L.A. Requiem_.
www.robertcrais.com
# Also by Robert Crais
Suspect
Taken
The Sentry
The First Rule
Chasing Darkness
The Watchman
The Two Minute Rule
The Forgotten Man
The Last Detective
Hostage
Demolition Angel
L.A. Requiem
Indigo Slam
Voodoo River
Free Fall
Lullaby Town
Stalking the Angel
The Monkey's Raincoat
# Praise for _Voodoo River_
"Crais tells a super-active tale, with plenty of local color, including a massive old turtle who lives in a catfish breeding pond. The larger issues are raised, rather than tidily concluded (not that they often are), but they give a resonance to this story."
—Charles Champlin, _Los Angeles Times Book Review_
"Elvis Cole has never been wittier, more passionate or more violently committed to his clients. Crais makes not a misstep here."
— _Publishers Weekly_ (starred review)
"What a terrific book. With this fifth mystery featuring the private eye with the unlikely moniker of Elvis Cole, author Robert Crais has firmly established himself at the top of the mystery field. He should be mentioned in the same breath as Robert B. Parker, Tony Hillerman, Sue Grafton and James Lee Burke."
—John B. Clutterbuck, _Houston Chronicle_
"With _Voodoo River,_ Elvis and his creator have finally made it into mystery's big time, right up there with Tony Hillerman, Robert B. Parker, Ed McBain, et al."
—Ed Kelly, _Buffalo News_
"Like the thick, rich mix in a Louisiana gumbo, Elvis Cole's latest case has enough spice and flavor to make you wish for more even before you've finished this helping... You'll love this book."
—Bruce Southworth, _Bookpage_
"Elvis Cole, in his fifth appearance, is in fine form. He's tough, witty, and clever as ever, and his partner Pike exudes danger like no one in suspense fiction this side of Spenser's Hawk."
—Wes Lukowsky, _Booklist_
"Elvis Cole is the King of Smart-Aleck Detectives, and _Voodoo River_ is another jewel in his throne. All the things you love about this genre—snappy dialogue, crisp action and vivid characters—are here in spades. Crais is clearly the undisputed leader in the field; _Voodoo River_ is a book deserving of much attention."
—Don Crouch, _Mostly Murder_
"Look for great dialogue, scary action, a twisty plot, and lots of quality time with an irresistible private eye."
—Mary Cannon, _Alfred Hitchcock's Mystery Magazine_
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# Contents
1. Cover
2. Title Page
3. Welcome
4. Dedication
5. Prologue
6. Chapter 1
7. Chapter 2
8. Chapter 3
9. Chapter 4
10. Chapter 5
11. Chapter 6
12. Chapter 7
13. Chapter 8
14. Chapter 9
15. Chapter 10
16. Chapter 11
17. Chapter 12
18. Chapter 13
19. Chapter 14
20. Chapter 15
21. Chapter 16
22. Chapter 17
23. Chapter 18
24. Chapter 19
25. Chapter 20
26. Chapter 21
27. Chapter 22
28. Chapter 23
29. Chapter 24
30. Chapter 25
31. Chapter 26
32. Chapter 27
33. Chapter 28
34. Chapter 29
35. Chapter 30
36. Chapter 31
37. Chapter 32
38. Chapter 33
39. Chapter 34
40. Chapter 35
41. Chapter 36
42. Chapter 37
43. Chapter 38
44. Chapter 39
45. Acknowledgments
46. About the Author
47. Also by Robert Crais
48. Praise for _Voodoo River_
49. Newsletters
50. Copyright
# Navigation
1. Begin Reading
2. Table of Contents
# Copyright
Copyright © 1996 by Robert Crais
Cover copyright © 2014 by Hachette Book Group, Inc.
All rights reserved. In accordance with the U.S. Copyright Act of 1976, the scanning, uploading, and electronic sharing of any part of this book without the permission of the publisher constitutes unlawful piracy and theft of the author's intellectual property. If you would like to use material from the book (other than for review purposes), prior written permission must be obtained by contacting the publisher at permissions@hbgusa.com. Thank you for your support of the author's rights.
Hyperion
Hachette Book Group
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First ebook edition: February 2014
Hyperion is a division of Hachette Book Group, Inc.
The publisher is not responsible for websites (or their content) that are not owned by the publisher.
ISBN 978-0-316-37541-2
E3
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"redpajama_set_name": "RedPajamaBook"
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\section{Introduction}
\input{01-intro}
\section{Related Works}
\noindent\textbf{Mesh-based representation and rendering.}
In computer vision and graphics, polygon mesh has been widely used in 3D scene modeling and rendering~\cite{izadi2011kinectfusion,akenine2019real,liu2019soft}.
Traditional methods utilize multi-view geometry and numerical theories to reconstruct surface meshes of a captured scene~\cite{colmap,XuT19,kazhdan2006poisson,waechter2014let}.
Recently, more attention has been paid to neural network based scene reconstruction~\cite{murez2020atlas,sun2021neuralrecon} and texture learning~\cite{thies2019deferred,gao2020deferred,xiang2021neutex}.
However, existing mesh-based rendering pipelines usually require UV-mapping to build correspondences between meshes vertices and texture maps, which limits the applicability from representing scenes with complex topology and delicate structure.
Another line of methods uses MVS based mesh as a geometry proxy for image feature aggregation~\cite{riegler2020free,riegler2021stable}, but requires nearby source images to be warped back to the mesh surface and is not feasible for high-level editing operations.
Instead of storing textures in a flat 2D map or warping-based view synthesis, our method directly encodes appearance information on 3D vertices, and is more flexible in representing complex objects whose UV-maps are difficult to be unwrapped.
\noindent\textbf{Neural rendering.}
Given a set of image captures, neural rendering methods~\cite{dellaert2020neural,IDR} aim to render photo-realistic images of novel views.
NeRF~\cite{nerf} takes advantages of volume rendering to boost rendering quality, which inspires a lot of works, including surface reconstruction~\cite{unisurf,neus,volsdf}, human modeling~\cite{neural_actor,peng2021neural}, pose estimation~\cite{yen2021inerf},
scene understanding~\cite{yang2022_nr_in_a_room} and relighting~\cite{srinivasan2021nerv,zhang2021nerfactor,boss2021nerd,neural_outdoor_rerender}, \emph{etc}\onedot} \def\vs{\emph{vs}\onedot{}.
To further increase network capacity and reduce computation, many works propose to decompose scene into local representations, such as multiple tiny networks~\cite{reiser2021kilonerf}, point clouds~\cite{ost2021neural} and voxels~\cite{nsvf,yu2021plenoxels}.
Although these works explicitly encode scenes in a 3D spatial structure, they are not designed to be easily manipulated as polygon meshes, thus not capable of high-level applications like geometry and texture editing.
\noindent\textbf{Neural scene editing.}
Scene editing is a popular topic in computer vision and photography.
Early methods mainly focus on editing a single static view by inserting~\cite{karsch2011rendering}, compositing~\cite{perez2003poisson}, moving~\cite{kholgade20143d,shetty2018adversarial} objects or changing lighting~\cite{luo2020niid} for an existing photograph.
With the development of neural rendering, many works start to edit scenes with movable~\cite{zhang2021editable,yang2021learning,guo2020object} and deformable~\cite{nerf_editing} objects, changeable colors, shapes~\cite{xie2021fig,liu2021editing} and textures~\cite{xiang2021neutex}.
However, existing methods are either limited to object-level rigid transformation~\cite{zhang2021editable,yang2021learning,guo2020object}, not generalize to out-of-distribution categories~\cite{deng2021deformed,liu2021editing,oechsle2019texture,sun2022fenerf,mvsnerf,wang2016unsupervised}, restricts its representation to simple shapes~\cite{xiang2021neutex} or orthographic projection~\cite{rematas2020neural}, or does not support fine-grained texture editing~\cite{liu2021editing,IDR,zhang2021nerfactor,srinivasan2021nerv,niemeyer2021giraffe,nerf_editing}.
By contrast, we pick up triangle mesh as a scaffold to encode the scene, since the mesh can be edited conveniently and intuitively in mature industry software, and the region of interest on the mesh can be precisely selected by vertices.
Built upon this, our method delivers the capability of non-rigid geometry editing and fine-grained texture editing.
\begin{figure*}[t!]
\centering
\includegraphics[width=1.0\linewidth, trim={0 0 0 0}, clip]{figures/framework.pdf}
\caption{
\textbf{Overview.}
We encode neural implicit field on a mesh-based scaffold, where each vertex possesses a geometry and texture code ${\bm{l}}^{g}, {\bm{l}}^{t}$, and a sign indicator $\textbf{n}$ for computing projected distance $h$.
For a query point $\bm{x}$ along a casted camera ray, we retrieve interpolated codes and signed distances from the nearby mesh vertices, and forward to the geometry/radiance decoder to obtain SDF value $s$ and color $\textbf{c}$.
}
\label{fig:framework}
\end{figure*}
\section{Method}
We introduce NeuMesh, a novel scene representation that encodes neural implicit field at a mesh-based scaffold.
As demonstrated in Fig.~\ref{fig:framework}, instead of learning the entire scene as a whole in a coordinate-based network, we leverage 3D mesh structure by decomposing the scene into a set of local-vertex-bounded implicit fields (Sec.~\ref{ssec:repr}), where each vertex stores geometry and texture information of its neighboring local space.
Motivated by previous works~\cite{nerf,neus,reiser2021kilonerf}, we adopt the volume rendering technique to render pixels, and employ a distillation and fine-tuning training scheme to encode the neural implicit field into the mesh surface (Sec.~\ref{ssec:learn}).
During the rendering stage, we retrieve interpolated codes and learnable signed distances
(\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, projected distances to the mesh vertices, which complements spatial distinguishability) from the mesh, and use two separated MLPs to decode geometry (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, SDF values) and radiance color.
In this way, the scene representation is locally aligned to the mesh, and the geometry and color are encoded in two separated latent spaces, which naturally derives the approaches of mesh-guided geometry deforming and designatable texture editing (Sec.~\ref{ssec:edit}).
\subsection{Neural Mesh-based Implicit Field}
\label{ssec:repr}
\noindent\textbf{Mesh-based representation.}
As illustrated in Fig.~\ref{fig:framework}, we use a mesh-based scaffold to model the neural implicit field.
First, we reconstruct the target object using out-of-box NeuS~\cite{neus} and marching cubes~\cite{lorensen1987marching}, which yields a triangle mesh with about 50K$\sim$150K vertices.
Then, for each vertex $\textbf{v}$ on the mesh, we store a set of learnable parameters, including a geometry code $\bm{l}^{g}$, a texture code $\bm{l}^{t}$ and a sign indicator $\textbf{n}$ ($\textbf{n}$ helps to identify relative position, and will be introduced later).
In a typically volume rendering process that sample points $\textbf{x}$ along the ray, we first find $K$ nearest vertices $\{\textbf{v}_k | k = 1,2,...,K\}$ for each point $\textbf{x}$, and perform spatial interpolation to obtain the interpolated codes $\tilde{\bm{l}}^{g}(\textbf{x})$, $\tilde{\bm{l}}^{t}(\textbf{x})$ and signed distances $\tilde{h}(\textbf{x})$.
Specifically, we adopt inverse distance weighting based interpolation~\cite{qi2017pointnet}, as:
\begin{equation}
\label{eq:weight}
\tilde{\bm{l}}(\textbf{x})=\frac{\sum^K_{k=1}w_k \bm{l}_k}{\sum^K_{k=1}w_k}, \;\; w_k=\frac{1}{||\textbf{v}_k-\textbf{x}||}.
\end{equation}
Then, we forward all these variables to geometry decoder $F_{G}$ and radiance decoder $F_{R}$ to obtain the SDF value $s = F_{G}\left(\tilde{\bm{l}}^g, \tilde{h}\right)$ and color $\bm{c} =F_{R}\left(\tilde{\bm{l}}^t,\tilde{h},\textbf{d},\nabla_\textbf{x} s\right)$ at point $\textbf{x}$,
where $\textbf{d}$ is the viewing direction, $\nabla_\textbf{x} s$ is the gradient of the SDF w.r.t query position.
Different from the previous methods~\cite{IDR,neus,volsdf},
we replace the global coordinate $\textbf{x}$ with locally retrieved codes $\tilde{\bm{l}}^g, \tilde{\bm{l}}^t$ and sign distances $\tilde{h}$, where $\tilde{h}$ complements spatial distinguishability without hurting the locality of the representation.
Note that we also apply positional encoding $\gamma(\cdot)$~\cite{nerf} to the interpolated codes, distance and direction before feeding them into the MLP, but we omit it in the equations for brevity.
Following the formulation of NeuS~\cite{neus} and quadrature rules~\cite{nerf}, we render the pixel $\hat{C}(\bm{r})$ with points $\{\mathbf{x}_i|i=1,...,N\}$ along the ray $\bm{r}$ as:
\begin{equation}
\hat{C}(\bm{r}) = \sum_{i=1}^{N} T_i \alpha_i {\mathbf{c}}_i, \; T_i = \prod_{j=1}^{i-1}(1-{\alpha}_j), \;
{\alpha}_j = \max \left (\frac{\Phi_s(s_{i}) - \Phi_s(s_{i+1})}{\Phi_s(s_i)}, 0 \right),
\end{equation}
where $T$ is accumulated transmittance, $\Phi_s$ is the cumulative distribution of logistic distribution, and $\alpha$ is opacity derived from adjacent SDF.
\noindent\textbf{Learnable sign indicator for interpolated signed distance.}
To complement the spatial distinguishability of the network query along the direction perpendicular to the surface (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, inside or outside the mesh),
we introduce a \textit{learnable} sign indicator $\textbf{n}_k$ for each vertex $\textbf{v}_k$ that aids at computing interpolated signed distances for spatial query points.
Indeed, the sign indicator plays a similar role as vertex normal (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, initialized with vertex normal), but is continuously adjusted during the training process to best fit the target loss.
The computation of interpolated signed distance $\tilde{h}(\textbf{x})$ is defined as:
\begin{equation}
\begin{split}
\tilde{h}(\textbf{x})=\frac{\sum^K_{k=1}w_k h_k}{\sum^K_{k=1}w_k}, \; h_k = \textbf{p}_k \cdot \frac{\omega^n \textbf{n}_k+ \omega^p_k\textbf{p}_k}{\omega^n + \omega^p_k}, \;\textbf{p}_k=\textbf{x} - \textbf{v}_k,
\end{split}
\end{equation}
where $w_k$ is inverse distance weighting as defined in Eq.~\eqref{eq:weight},
$\omega^n$ and $\omega^p_k$ controls the influence between sign indicator and point-to-vertex vector $\textbf{p}_k$, and we empirically set $\omega^n=0.1, \omega^p_k=||\textbf{p}_k||$.
Intuitively, when the sample points are far from the surface, $\tilde{h}(\textbf{x})$ is numerically close to the point-to-surface distance; otherwise, when the sample points are getting close to the surface, $\tilde{h}(\textbf{x})$ would be gradually perturbed by learnable sign indicators.
\subsection{Optimizing Mesh-based Implicit Field}
\label{ssec:learn}
\noindent\textbf{Distillation and fine-tuning.}
We observe that training NeuMesh from scratch leads to artifacts and converges to sub-optimal results (see Fig.~\ref{fig:ablation_vis} and Tab.~\ref{tab:ablation}).
Inspired by Reiser \emph{et al}\onedot~\cite{reiser2021kilonerf}, we apply a distillation and fine-tuning training scheme, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, we supervise NeuMesh simultaneously with the output from a coordinate-based teacher model (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, NeuS), and also the images.
For a batched training rays $\bm{r}\in R$, we defined the distillation loss $L_d$ and photometric fine-tuning loss $L_f$ as:
\begin{equation}
\mathcal{L}_{\text{d}} = \sum_{\bm{r}\in R}\sum_{i\in N}||{s}_i - s^t_i|| + ||{\bm{c}}_i-\bm{c}^t_i|| ,\;\;
\mathcal{L}_{\text{f}} =\sum_{\bm{r}\in R}||\hat{C}(\bm{r})-C(\bm{r})||^2_2
\end{equation}
where ${s}_i^t$ and $\bm{c}_i^t$ are the SDF value and color from the teacher model, and $C(\bm{r})$ is the ground-truth pixel color from images.
By leveraging distillation and fine-tuning, we smoothly transfer a pure MLP-based neural implicit model into a flexible and editable mesh-based representation, and the final model even produces better appearance details, as shown in our experiment (Sec.~\ref{ssec:compare_quality}).
\noindent\textbf{Regularization.}
As introduced in Sec.~\ref{ssec:repr}, we dynamically adjust a set of per-vertex sign indicators during the training process.
To ensure a smooth convergence, we empirically apply a regularization to the sign indicator by slightly encouraging them being close to pre-computed vertex normal $\bm{n}^t$, as: $\mathcal{L}_{\text{rs}}= \sum_{k} ||\bm{n}_k-\bm{n}^t_k||_2^2$.
Besides, as suggested by Gropp \emph{et al}\onedot~\cite{igr}, we add an Eikonal loss to regularize the norm of the spatial gradients to 1, as: $\mathcal{L}_{\text{re}}=\sum _{k}||\left\lVert\nabla_{\textbf{x}_k} s_k\right\rVert-1||^2_2$.
The final loss is then defined as:
\begin{equation}
\mathcal{L}_{\text{total}} =
\lambda_{\text{d}} \mathcal{L}_{\text{d}}
+ \lambda_{\text{f}} \mathcal{L}_{\text{f}}
+ \lambda_{\text{rs}} \mathcal{L}_{\text{rs}}
+ \lambda_{\text{re}} \mathcal{L}_{\text{re}},
\end{equation}
where we set $\lambda_{\text{d}}=1.0$, $\lambda_{\text{f}}=1.0$, $\lambda_{\text{rs}}=0.01$ and $\lambda_{\text{re}}=0.01$.
\subsection{Mesh-guided Geometry Editing}
\label{ssec:edit}
\begin{figure}[t!]
\centering
\includegraphics[width=1.0\linewidth, trim={0 0 0 0}, clip]{figures/method_geo_edit.pdf}
\caption{
\textbf{Mesh-guided Geometry editing.}
By simply deforming the corresponding mesh, the change will synchronously take effect on the implicit field, and the rendered object will also be deformed accordingly.
}
\label{fig:geo_edit_process}
\end{figure}
In NeuMesh, since the neural implicit field has been tightly aligned to the mesh surface, any manipulation on mesh vertices would directly take effect on the field and the volume rendering results.
Therefore, to perform geometry editing with a NeuMesh-based scene, users are only required to edit the corresponding mesh, which can be easily accomplished by interactively moving a few vertices with out-of-box mesh deforming methods (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, as-rigid-as-possible, or ARAP~\cite{arap}), or 3D modeling software like Blender.
We show an example of the geometry editing in Fig.~\ref{fig:geo_edit_process}, where we first deform the microphone by bending its head and lifting the wire on the corresponding mesh.
Then, to maintain the local consistency of signed distance (Sec~\ref{ssec:repr}), for each transformed vertex, we also compute a relative rotation of the surface normal and compensate the rotation according to its sign indicator (Sec.~\ref{ssec:repr}).
Without any fine-tuning, the microphone's implicit field has been deformed in the meantime, and we can easily render the deformed view (see Fig.~\ref{fig:geo_edit_process}).
Please refer to the supplementary materials for more details.
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\linewidth, trim={0 0 0 0}, clip]{figures/method_tex_edit.pdf}
\caption[]{
\textbf{Designatable Texture editing.} By exchanging texture codes (and decoders), our representation delivers various texture editing pipelines on neural implicit field.
\footnotemark[1]
}
\label{fig:tex_edit_process}
\end{figure}
\footnotetext[1]{Icon credit: Flaticon~\cite{flaticon}}
\subsection{Designatable texture editing}
Until then, texture editing on the neural implicit model is still an open problem.
Previous methods tend to replace the entire materials by swapping the appearance branch~\cite{IDR,zhang2021nerfactor,srinivasan2021nerv}, changing a uniformed color~\cite{liu2021editing}, or learning an editable UV mapping for simple and plump shapes~\cite{xiang2021neutex}.
However, in real texturing of 3D modeling software, artists are used to working with a mesh-based workflow, which allows them to select a partial region of an object and modify it with arbitrary colors and material properties.
We propose to mimic such pipelines by introducing a designatable texture editing, where the selection of mesh vertices can be used to precisely guide the texture editing on the region of interest.
The core step of our texture editing is that we update the latent texture code $\bm{l}^t$ (`material properties') and the binding decoder $F_R$ (`rendering palette') at the selected region.
As shown in Fig.~\ref{fig:tex_edit_process}, we deliver three ways of texture editing:
\noindent\textbf{1) Texture swapping} by exchanging textures between two objects through 3D geometry (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, swapping textures of two apples in Fig.~\ref{fig:tex_edit_process}).
Users are first asked to mark out the source and target object on the mesh, which can be done by mature 3D model software, or point-based instance segmentation~\cite{wang2018sgpn}.
Then, given a putative point matches with interactive annotation~\cite{zhou2018open3d}, we perform non-rigid 3D alignment to the source and target object with Umeyama~\cite{umeyama} and ARAP~\cite{arap}.
Finally, we transfer texture codes by assigning each target vertex with code interpolated from nearby source vertices.
\noindent\textbf{2) Texture filling} by filling a targeting object area with repeated textures from a pre-captured model (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, assigning part of Buddha with two furry materials from a teddy bear as shown in Fig.~\ref{fig:tex_edit_process}).
In real applications, artists might want to try out some materials from a daily captured scene or pre-built material library, or want to fill some areas (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, floor and walls) with uniformed materials.
Therefore, we build a compatible workflow for the standard texturing operation, where we first construct a UV map for the user-interest areas, and
then repeatedly fill the mapped vertices (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, chest and cloth of Buddha) with template textures from a pre-captured model (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, gray and brown hairs from the teddy bear).
\noindent\textbf{3) Texture painting} from a single 2D view to the 3D field.
Users paint an arbitrary pattern or put some text on a captured image, and we can transfer these paintings into the 3D neural implicit field and freely preview in rendered novel views (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, painting ECCV logo on a skull in Fig.~\ref{fig:tex_edit_process}).
Compared to NeuTex~\cite{xiang2021neutex} that might be difficult to edit on the desired position due to distorted UV-mapping, our method delivers a more natural editing way, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, what you draw and see is what you get.
However, it is not trivial to precisely control the painting transferring with only one image, since the overfitting of a single image might lead to appearance drifting at unconstrained views, which inevitably introduces artifacts in rendered novel views.
To tackle this issue, we propose a \textit{spatial-aware optimization mechanism}.
Specifically, we first shoot rays through the painted pixels to obtain the surface points and find the affected texture codes of nearby vertices around the points.
During the fine-tuning stage, we optimize by minimizing photometric loss of rendered pixels and painted pixels, and only backward gradient of these codes while detaching the others.
Besides, to improve the training efficiency and the painting consistency across views, we restrict training rays inside a slightly dilated paint mask, and also augment with random viewing directions at the input to the radiance decoder.
Please refer to the supplementary materials for more details.
\section{Experiments}
\subsection{Datasets}
\label{ssec:dataset}
We evaluate our method on the real captured DTU~\cite{dtu} dataset and NeRF 360$^{\circ}$ Synthetic dataset.
For the DTU dataset, we follow the setting of IDR~\cite{IDR} by using 15 scenes with images of $1600\times 1200$ resolution and foreground masks for experiments.
To facilitate the metric evaluation for both rendering and mesh quality, we randomly select 10\% images as test split and use the remaining images for training.
For NeRF 360$^{\circ}$ Synthetic dataset, we follow the official split and choose 4 representative scenes for evaluation, including thin structures (Mic), complex shapes (Lego) and rich textures (Chair and Hotdog).
\begin{figure}[!t]
\centering
\includegraphics[width=1.0\linewidth, trim={0 0 0 0}, clip]{figures/render_quality.pdf}
\caption{
We show rendering examples of NeuTex~\cite{xiang2021neutex}, NeuS~\cite{neus} and our method on the DTU dataset and the NeRF 360$^\circ$ Synthetic dataset.
}
\label{fig:render_quality}
\end{figure}
\input{tables/tab_render}
\subsection{Comparison of Rendering and Mesh Quality}
\label{ssec:compare_quality}
We first compare the rendering and mesh reconstruction quality of our representation with the baseline method NeuS~\cite{neus} and the SOTA texture-editable implicit neural rendering method NeuTex~\cite{xiang2021neutex}.
Following previous works~\cite{nerf,neus,IDR}, we use PSNR, SSIM and LPIPS to measure the rendering quality, and use Chamfer distance to measure the reconstructed mesh quality.
Please note that for mesh quality comparison, we use a subset (training split) of images, while NeuS takes all images for training in their paper, so the result is slightly different.
As demonstrated in Fig.~\ref{fig:render_quality} and Tab.~\ref{tab:render}, our method is comparable or even better than NeuS and NeuTex on rendering quality.
To achieve texture editing, NeuTex attempts to encode all the textures in a single continuous UV space, which works for plump objects (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, plush toys or Buddah as shown in its paper) but struggles to reconstruct objects with complex shapes (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, scissors in
DTU Scan 37 and gears in NeRF-Synthetic Lego as shown in Fig.~\ref{fig:render_quality}).
We consider that because NeuTex tries to memorize all textures in the single continuous UV-map by using a simplified Atlas-Net~\cite{groueix2018papier} (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, one atlas), which limits its representation of complex shapes.
For NeuS, as it pursues better mesh reconstruction quality than novel view synthesis, the details of rendered images are slightly blurred and smoothed.
By contrast, our representation not only delivers the capability of geometry and texture editing, but also shows clear appearance detail (see Fig.~\ref{fig:render_quality}) and maintains mesh quality on par with NeuS (see Tab.~\ref{tab:mesh}).
\begin{figure}[t!]
\centering
\includegraphics[width=1.0\linewidth, trim={0 0 0 0}, clip]{figures/geo_edit_example.pdf}
\caption{
We show examples of mesh-guided geometry editing in (a) and also compare with na\"ive field warping solution in (b).
}
\label{fig:geo_edit}
\end{figure}
\input{tables/tab_mesh}
\subsection{Experiment on Geometry Editing}
\label{ssec:expr_geo_edit}
We now show the result of mesh-guided geometry editing in Fig.~\ref{fig:geo_edit} (a), where we simply deform meshes with Blender, and the rendered objects are deformed simultaneously.
Since an implicit field can be trivially deformed with non-rigid warping, we also compare our editing with a na\"ive field warping solution that is directly applied to NeuS, which bends the query points from the deformed space to the original space by computing interpolated warping with the offsets from 3 nearest vertices of the extracted mesh.
As shown in Fig.~\ref{fig:geo_edit} (b), the object boundary of the field warping results is much jaggier than ours, which proves the necessity of our mesh-based representation on this task.
\begin{figure}[!htbp]
\centering
\includegraphics[width=1.0\linewidth, trim={0 0 0 0}, clip]{figures/tex_edit_example.pdf}
\caption{
We show texture editing examples on the DTU dataset and the NeRF 360$^\circ$ Synthetic dataset.
}
\label{fig:tex_edit}
\end{figure}
\subsection{Experiment on Texture Editing}
\label{ssec:expr_tex_edit}
To the best of our knowledge, prior to our work, only NeuTex~\cite{xiang2021neutex} supports texture editing of the neural implicit field by painting on 2D UV texture.
However, due to the distorted UV mapping, we find NeuTex hard to perform all the editing operations like ours, so we only compare it on the texture painting task.
\noindent\textbf{Texture swapping.}
We present 2 examples of texture swapping in Fig.~\ref{fig:tex_edit} (a), where the textures of the snowman's body and the packaging of cans have been seamlessly swapped, and even the details (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, texts on the cans) have been clearly transferred into the target object, while the geometry is kept unchanged.
This demonstrates that our representation successfully disentangles geometry and texture in two spaces, and the disentangled texture representation can be seamlessly integrated into new shapes.
\noindent\textbf{Texture filling.}
We show 3 examples of texture filling in Fig.~\ref{fig:tex_edit} (b), in which the targeting areas are repeatedly filled with template texture code and decoder from previously captured source models.
It is worth noting that even though the source template only covers a small area of texture codes, we can still observe view-dependent effects (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, golden metal naturally exhibits specular reflections at different views).
\noindent\textbf{Texture painting.}
In Fig.~\ref{fig:tex_edit} (c), we exhibit 4 examples of texture painting, and also conduct similar editing with NeuTex~\cite{xiang2021neutex} by painting on the unwrapped UV-map.
For NeuTex, as the learned texture mapping is somehow irregular and distorted (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, the head of the teddy bear is separated in UV-map), we find it hard to paint at the desired location.
In the second row (painting on bricks), we have to adjust the painting position back and forth to get a reasonable editing result.
Besides, due to the mapping issue explained in Sec.~\ref{ssec:compare_quality}, NeuTex cannot picture a clear result when editing on complex shapes (Lego in the second row).
On the contrary, our method offers a user-friendly editing pipeline by directly painting on 2D images and then transferring the painting into the 3D implicit field.
\subsection{Ablation Studies}
\label{ssec:ablation}
\begin{figure}[t!]
\centering
\includegraphics[width=1.0\linewidth, trim={0 0 0 0}, clip]{figures/ablation_vis.pdf}
\caption{
We show the rendering results with different settings in (a) and also show the effectiveness of spatial-aware optimization for texture painting in (b).
}
\label{fig:ablation_vis}
\end{figure}
\input{tables/tab_ablation}
\noindent\textbf{Learnable sign indicator.}
We first inspect the effectiveness of the proposed learnable sign indicator in each vertex.
Specifically, we set sign indicators as constant vertex normal without adjusting during the training process and evaluate the model both qualitatively and quantitatively.
As demonstrated in Fig.~\ref{fig:ablation_vis} (a) and Tab.~\ref{tab:ablation} (first row), online adjusting sign indicators consistently improves the image quality.
By the way, we notice that the PSNR improvement on real data (DTU Scan 69) is more significant than the synthetic one (Lego). We consider that the mesh quality (and vertex normal) of real data is worse than the synthetic data due to sensor noises, which degrades the rendering quality, while the learnable sign indicator helps to mitigate this issue.
\noindent\textbf{Distillation and fine-tuning training scheme.}
We then study the necessity of distillation and fine-tuning training scheme by ablating one of them during model training.
As shown in Fig.~\ref{fig:ablation_vis} (a) and Tab.~\ref{tab:ablation} (second and third row),
by enabling distillation only, the rendered image is blurry than the full model ones.
When using fine-tuning without distillation, the rendering result ends up with noticeable artifacts.
These results suggest that both distillation and fine-tuning are indispensable when training our mesh-based representation.
\noindent\textbf{Spatial-aware optimization in texture painting.}
We also evaluate the proposed spatial-aware optimization in the texture painting task and visualize the comparison in Fig.~\ref{fig:ablation_vis} (b).
It is clear that when na\"ively optimizing painting with a single image, the model will overfit to the specific viewpoint, and the change to the texture codes might break the appearance consistency, which results in visual artifacts when rendering the implicit field from the side view.
By introducing a spatial-aware optimization mechanism, we successfully avoid such artifacts and obtain the modified field while maintaining other parts untouched.
\section{Conclusion}
We have proposed a novel mesh-based neural representation, which supports high-fidelity volume rendering, and flexible geometry and texture editing.
Specifically, we encode the neural implicit field into a mesh scaffold, where each mesh vertex possesses learnable geometry and texture code for its neighboring local space.
One limitation of our method is that we do not model fine-grained lighting effects such as shadowing and specular reflection of a certain lighting environment, which can be improved by introducing material and lighting estimation in future works.
Besides, due to the reliance on mesh scaffold, we cannot represent objects that fails during reconstruction (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, smoke or liquid).
\vspace*{\baselineskip}
\noindent\textbf{Acknowledgment.} This work was partially supported by NSF of China~(No. 61932003, No. 62102356).
\clearpage
\bibliographystyle{splncs04}
\section{Model Architecture}
\label{sec:model_arch}
The detailed model architecture is shown in Fig.\ref{fig:supp_architecture}.
To begin with, we first extract a triangle mesh with marching cubes~\cite{lorensen1987marching} from NeuS's~\cite{neus} SDF field, where we set the voxel resolution as 256 and the spatial range as $[-1, 1]$.
Then, for each query point $\mathbf{x}$, we find $K$ nearest vertices (with $K=8$ in our experiments) and obtain the interpolated geometry code (32 dimensions), texture code (32 dimensions) and learnable signed distance (scalar) from these vertices.
Before feeding into the network, we apply positional encoding to the signed distances (with 8 frequencies), interpolated codes (with 2 frequencies) and viewing directions (with 4 frequencies).
The geometry decoder and the radiance decoder are constructed with a MLP of 3 / 4 hidden layers and 256 hidden sizes, and we use SoftPlus / ReLU activation, respectively.
During the rendering stage, we first sample 64 coarse points along the ray and adopt a progressive up-sampling strategy from Wang \emph{et al}\onedot ~\cite{neus} to guide the sampling of 64 fine points, which yields 128 samples for each ray.
Besides, to accelerate rendering and training, we pre-compute near-far bound for each ray by counting the minimum and maximum distances of ray-to-mesh intersections.
\begin{figure}[h]
\centering
\includegraphics[width=1.0\linewidth, trim={0 0 0 0}, clip]{figures/supp_architecture_tight.pdf}
\caption{
\textbf{The model architecture of NeuMesh.}
}
\label{fig:supp_architecture}
\end{figure}
\section{Implementation Details}
\label{sec:impl}
\subsection{Training Details}
\label{ssec:train}
As introduced in our main paper, we adopt a distillation and fine-tuning training scheme.
Practically, for each object, we first train a teacher model (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, NeuS~\cite{neus}).
Then, we optimize codes and decoders with output from the teacher model and the images.
During the training process, we use a batch size of 512 rays on a single Nvidia RTX3090-24G GPU, where the queried color and SDF value for each sample point will also be supervised with the output from the teacher model (a.k.a distillation loss in Sec.~3.2 Eq.(4)).
We adopt the Adam optimizer with an initial learning rate of 0.0005 and a cosine annealing scheduler with 5000 warm-up steps.
The training process takes about 16 hours for each model.
Besides, to train on the DTU dataset that contains unbounded background, we follow previous works~\cite{IDR,neus} by taking foreground masks into the supervision with a binary cross-entropy loss.
\begin{figure}[t!]
\centering
\includegraphics[width=1.0\linewidth, trim={0 0 0 0}, clip]{figures/supp_render_quality.pdf}
\caption{
\textbf{
We show more comparison of rendering quality on the DTU dataset and the NeRF 360$^\circ$ Synthetic dataset.
} Our rendering results show better appearance details than NeuS and NeuTex (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, the roof at DTU Scan 37, and the metal grids at NeRF-Synthetic Mic).
}
\label{fig:supp_render_compare}
\end{figure}
\subsection{Details of Geometry Editing}
\label{ssec:detail_geo_edit}
With our mesh-based representation, deforming a neural implicit field is as simple as deforming the corresponding mesh scaffold.
The only thing to note is to keep the local consistency of the learnable signed distances (Sec.~3.1), \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the interpolated signed distance of the locally deformed or rotated region should keep the same.
To achieve this goal, we simply compensate the rotation of the surface normal to the learnable signed indicator $\tilde{\bm{h}}(\textbf{x})$, as:
$\tilde{\bm{h}}'(\textbf{x})=\tilde{\bm{h}}(\textbf{x}) + \delta h_x$,
where $\delta h_x$ is the relative difference of vertex normal (averaged from the nearby surface normal) from the original mesh to the deformed mesh, and $\tilde{\bm{h}}'(\textbf{x})$ is the compensated signed indicator.
\subsection{Details of Texture Editing}
\label{ssec:detail_tex_edit}
Since our representation disentangles textures into locally bounded texture codes saved on mesh vertices, texture editing for a neural implicit field can be accomplished by updating or optimizing texture codes (and the binding encoders) for the region of interest.
\noindent\textbf{Texture swapping.}
We can easily swap textures of two areas by swapping texture codes on the surface, as long as we find the correspondence from the source area's vertices to the target area's vertices.
To this end, we provide a solution to perform texture swapping on two areas that can be reasonably aligned but with slightly different shapes (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, two apples in Fig.4 (a) from the main paper).
In practice, we first choose source and target areas by selecting mesh vertices with Blender, and annotate 4$\sim$9 point correspondences with our scripts between these two areas.
Note that this can also be automated with point cloud or image segmentation tools when deploying to user-friendly applications.
Then, we perform non-rigid mesh alignment by solving scaled transformation with Umeyama~\cite{umeyama} between point correspondences, and then feed the point residual to ARAP~\cite{arap}, so as to deform the source area to the target area.
Finally, we update the texture codes on the target area by assigning interpolated code (with inverse distance weighting) from 4 nearest deformed source vertices.
\noindent\textbf{Texture filling.}
By leveraging NeuMesh, our model supports filling of the user-selected area on a neural implicit field with a texture template (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, furry hair or golden metal in Fig.~7 (c)) from a pre-captured object model.
First, we need to obtain the target UV-map of the selected area, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, utilizing Blender to unwrap the UV-map of the selected vertices.
Then, we select a texture template from a pre-trained NeuMesh model, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, a small squared patch with $\sim$10 vertices, and repeatedly fill the target UV-map with the template in a sliding-window manner.
Practically, we assign texture codes in the target vertices with interpolated codes from the template and also bind the radiance decoder to the one from the pre-trained model, as the target texture code and the template texture code do not share the same latent radiance space.
Besides, to make a smooth transition near the area boundary (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, naturally transiting from the edited appearance to the original appearance), for each query point that has texture codes/decoders from different sources, we fuse the color contribution from different decoders with inverse distance weighting.
\noindent\textbf{Texture painting.}
As introduced in our main paper (Sec.3.4), we propose a spatial-aware optimization to precisely transfer the painting from 2D image to 3D field, while keeping geometry and appearance of other parts unchanged.
In detail, we first shoot probing rays from the painted pixels to the mesh scaffold, and find the affected texture codes by collecting the vertices of the hit faces.
During optimization, we adopt Adam optimizer with the fixed learning rate of 0.01, and only allow these codes to be changed.
The whole texture painting optimization takes about $\sim$1 hours with 8000 iterations.
\section{More Discussions}
\label{sec:supp_discussion}
\noindent\textbf{Using neural implicit representation instead of traditional textured mesh.}
Neural implicit representation merits easy-reconstruction with photo-realistic volumetric rendering and view-dependent effects (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, shiny golden materials) on both real-world and synthetic data, and flexibility to accomplish some fine-grained editing demands (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, material editing or appearance variations) on the real-world scene with latent space operations.
While the rendering quality of the textured mesh is bounded by the MVS reconstruction and texturing.
It is not feasible for the textured 3D mesh to achieve such effects (see Fig.~\ref{fig:supp_traditional_mesh_edit}) without BRDF material properties and lighting estimation.
\begin{figure}[t!]
\centering
\includegraphics[width=0.95\linewidth, trim={0 0 0 0}, clip]{figures/traditional_mesh_edit.pdf}
\label{fig:supp_traditional_mesh_edit}
\caption{We show the comparison between textured mesh editing and our NeuMesh editing on a statue.
This proves that direct editing texture meshes with template patterns without lighting and material property estimation cannot provide satisfactory results.
}
\end{figure}
\noindent
\textbf{Using learnable signed distance.}
Unlike voxel-based~\cite{nsvf,yu2021plenoxels} or point-cloud-based~\cite{ost2021neural} methods that possess spatially scattered features, we only learn a set of `single layer' features on mesh surfaces as we want to build a surface-aligned implicit field.
Therefore, a bare code interpolation is not sufficient to coordinate the query relative position for our mesh-based representation, (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the inner and outer point queries along the direction perpendicular to the surface still lack spatial distinguishability).
One plausible solution is to use a physically computed signed distance to the surface as Liu \emph{et al}\onedot~\cite{neural_actor} does, but it is not applicable for general object meshes because the geometry is not always well-defined (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, watertight or even predefined skinning weights) as a human-body model (SMPL)~\cite{loper2015smpl}, which confuses ray-to-mesh intersection counting and the sign of the distance might be unexpectedly reversed.
Therefore, we propose to use a learnable sign indicator to compute interpolated signed distances for spatial query points, as described in Sec.~3.1.
\noindent
\textbf{Using distillation instead of training from scratch.}
As explained in Sec.~3.2, we exploit the teacher NeuS model with distillation and fine-tuning training scheme instead of training from scratch.
The teacher NeuS model serves two purposes:
\textbf{1)} it provides an SDF field where we could extract a mesh scaffold;
\textbf{2)} the locally embedded geometry and appearance features in our model facilitate region-based editing but may lead the training to fall into a local minimum (as shown in our ablation studies), and the use of distillation helps to alleviate such training issue.
\noindent
\textbf{Texture swapping with different geometry/topology.}
Our method can be applied to objects with a moderate geometry difference (see Fig.~\ref{fig:swap_diff_geo}).
If there is a significant topology difference between two objects, we suggest using texture filling (Sec.~3.4 (2)) that swaps textures in UV spaces regardless of object geometries.
\begin{figure}[t!]
\centering
\includegraphics[width=1.0\linewidth, trim={0 0 0 0}, clip]{figures/texture_swap_diff_geo.pdf}
\label{fig:swap_diff_geo}
\caption{Texture swapping with different geometry.
}
\end{figure}
\noindent\textbf{Limitations for real-world applications.}
Currently, our rendering speed (about 30s for each view) is bounded by the intensive network queries and nearest neighboring searching operations.
When deploying to real-world applications, we might consider accelerating the inference speed to fulfill the real-time rendering demand
with recently proposed coefficient caching techniques~\cite{yu2021plenoxels,garbin2021fastnerf}
, multiresolution hash encoding~\cite{muller2022instant} or MVS priors~\cite{mvsnerf}.
Besides, we rely on 3D modeling software to select vertices for the region of interest, which can be replaced with some semantic annotation approaches~\cite{wang2018sgpn} to facilitate broaden users.
\noindent\textbf{Relation to point-based methods.}
From a high-level perspective, both ours and point-based methods can be regarded as building upon local feature-based representations, while the main differences include:
\textbf{1)} Our model encodes features on mesh vertices, so we can easily deform objects with a mesh proxy or modify textures through a UV space.
Point-based methods use scattered point features, so it is non-trivial to perform mesh-based editing like ours, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, each point that is projected at the pixel (NPBG) or lying nearby ray samples (Point-NeRF) would contribute to the appearance, making it hard to distinguish which point features should be edited.
\textbf{2)} We embed surface normal (similar to IDR/NeuS) to realize view-dependent effects of texture filling, which cannot be directly inherited by point-based methods.
\section{More Experiment Results}
\label{sec:expr}
\begin{figure}[t!]
\centering
\includegraphics[width=1.0\linewidth, trim={0 0 0 0}, clip]{figures/supp_other_quality.pdf}
\caption{
\textbf{
We present visual comparison to alternative designs.}
}
\label{fig:supp_ablation}
\end{figure}
\input{tables/tab_supp_ablation}
\noindent\textbf{Rendering quality comparison.}
We present more results of rendering quality comparison in Fig.~\ref{fig:supp_render_compare}.
It is clear that our method renders more details than other competitors, especially when reconstructing with complex shapes and textures (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, the roof at DTU Scan 37, and the metal grids at NeRF-Synthetic Mic in Fig~\ref{fig:supp_render_compare}).
\noindent\textbf{Rendering quality with varying mesh vertex numbers.}
We analyze the impact of varying mesh vertex numbers on rendering quality.
Specifically, we train on DTU Scan 114 with 3 sets of mesh vertices (10K, 50K, and 100K).
As shown in Fig.~\ref{fig:ablation_vertex_num}, the metric quality of rendered images are slightly affected when decreasing vertex numbers, but still outperform NeuS even with only 10K vertices, which demonstrate the robustness and advantages of our representation.
\begin{figure}[t!]
\centering
\includegraphics[width=1.0\linewidth, trim={0 0 0 0}, clip]{figures/ablation_vertex_num.pdf}
\caption{
We analyze the impact of vertex numbers on the rendering quality by training with 10K / 50K / 100K vertices.
}
\label{fig:ablation_vertex_num}
\end{figure}
\noindent\textbf{Learnable signed distance.}
We report the training results without learnable signed distance as network input in Fig.~\ref{fig:supp_ablation} (first column) and Tab.~\ref{tab:supp_ablation} (first row), which proves the necessity of this design in our mesh-based representation, as it complements spatial distinguishability on the direction perpendicular to the surface (Sec.~3.1).
\noindent\textbf{Mesh-based representation vs. uniform grid-based representation.}
We first compare our `single layer' mesh-based representation with a uniform grid-based representation (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, similar to NSVF~\cite{nsvf} or Plenoxel~\cite{yu2021plenoxels}).
Specifically, we thicken the mesh vertices to uniform grids, so the interpolated codes can be fully aware of the spatial coordinates, and the signed distance can be omitted.
Note that this also loses some flexibility for fine-grained editing.
As shown in Fig.~\ref{fig:supp_ablation} (second column) and Tab.~\ref{tab:supp_ablation} (second row), even with only a single slice of spatial features, our method shows on par visual quality with these uniform grid-based representation, but enables the functionalities of geometry and texture editing.
\noindent\textbf{Learnable signed distance vs. relative position.}
We then compare the encoding of our learnable signed distance with an alternative design, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, relative position encoding from PointNet~\cite{qi2017pointnet}.
Specifically, for each query point, we first concatenate codes (from nearby vertices) and relative coordinate offsets (from query to vertex), and encode with a shallow MLP (with 2 hidden layers and 64 hidden sizes).
Then, we use the same inverse distance weighting to obtain the final interpolated embedding for the query.
As demonstrated in Fig.~\ref{fig:supp_ablation} (third column) and Tab.~\ref{tab:supp_ablation} (third row), our learnable signed distance encoding shows better rendering quality when incorporated with such `single layer' surface features and is a better choice for mesh-based representation.
\begin{figure}[t!]
\centering
\includegraphics[width=0.8\linewidth, trim={0 0 0 0}, clip]{figures/supp_edit_comparsion.pdf}
\caption{
\textbf{We show the comparison of our texture editing to the field warping.}
}
\label{fig:supp_edit_comparsion}
\end{figure}
\noindent\textbf{Our texture editing vs. radiance field warping.}
One possible workaround of texture editing is to warp the radiance field from the original space to the aligned space according to the non-rigid mesh alignment (Sec.~3.4).
So, we compare our code updating based texture editing with such na\"ive radiance field warping on DTU Scan 63.
As shown in Fig.~\ref{fig:supp_edit_comparsion},
the rendered apple of the na\"ive approach contains noticeable artifacts, while our editing result is visually much more natural.
We believe that this is mainly due to the fact that the warped texture field might not be compatible with the geometry (SDF field), which leads to spatial misalignment (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, SDF field is close to the surface while radiance field is not) during the volume rendering process and produces erroneous color.
In contrast, since our method exchanges textures through code swapping, the edited texture field is tightly fit to the geometry, which yields a better rendering quality.
\begin{figure}[t!]
\centering
\includegraphics[width=1.0\linewidth, trim={0 0 0 0}, clip]{figures/hybrid_edit.pdf}
\caption{
We show examples of hybrid object editing by combining multiple editing operations.
}
\label{fig:hybrid_edit}
\end{figure}
\noindent\textbf{Hybrid object editing.}
\noindent
To demonstrate the editing flexibility of our method, we show examples of hybrid object editing in Fig.~\ref{fig:hybrid_edit} by combining geometry/texture editing operations, which sheds light on integrating our representation into modern 3D modeling workflow.
\begin{figure}[t!]
\centering
\includegraphics[width=1.0\linewidth, trim={0 0 0 0}, clip]{figures/edit_large_scene_vertical.pdf}
\caption{Texture editing of large-scale scenes on the Tanks\&Temple~\cite{Knapitsch2017} dataset.}
\label{fig:supp_large_scale}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=1.0\linewidth, trim={0 0 0 0}, clip]{figures/uneven_triangle.pdf}
\caption{Rendering quality under sparse and uneven triangulation.}
\label{fig:supp_uneven_triangle}
\end{figure}
\noindent
\textbf{Large-scale scenes.}
The modeling ability of our method depends mainly on the teacher SDF model.
As long as the scaffold mesh is available, our method can be freely scaled-up thanks to the locally embedded features.
For large scenes with complicated backgrounds, we can adopt NeRF++~\cite{nerf++}'s parameterization to handle unbounded backgrounds, or use pre-computed segmentation masks like in NSVF and IDR.
Here we show two texture editing examples (see Fig.~\ref{fig:supp_large_scale}) on the Tanks\&Temple dataset~\cite{Knapitsch2017} with foreground segmentation provided by NSVF~\cite{nsvf}.
\noindent
\textbf{Influence of triangle quality.}
Our method can still deliver reasonable rendering quality with locally sparse/uneven triangulation (see Fig.~\ref{fig:supp_uneven_triangle}).
In fact, as the mesh scaffold is created based on the SDF from teacher NeuS, we can handily guarantee a uniformed distribution of vertices with off-the-shelf mesh regularization algorithms (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, isotropic remeshing by Botsch \emph{et al}\onedot~\cite{botsch2004remeshing}).
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 1,190
|
21/07/2014 · you will need a piece of printer paper to make the paper crane steps of how to fold a paper crane 1-take the corner of the sheet and fold it down to make …... When you google "origami," one of the first products that appears is a paper crane. I didn't want to give you just a plain crane, but also one of many variations that you can make with cranes – twin cranes.
I once folded 1,000 miniature paper cranes for a wedding (which is a symbol of good luck) and so folding this shape is ingrained on my brain! Today I thought I'd share how to make an origami paper crane … how to make a 3d manatee by folding the paper in half on both diagonals.paper "peace" cranes, these instructions will make a flapping bird. Try playing Origami Flapping Bird 1.
Want to spice up your origami cranes? MIT's High-Low Tech Group made two electronic origami cranes that flap their wings when you squeeze their tails.
When you google "origami," one of the first products that appears is a paper crane. I didn't want to give you just a plain crane, but also one of many variations that you can make with cranes – twin cranes.
How to Fold a Paper Crane The origami crane is perfect as a gift or gift tag as a decoration, or as the first step to making asenbazuru. The cranes are delicate, but surprisingly easy (and fun) to fold, so don't hesitate to give this craft a try.
This is a step by step video on how to make the origami paper crane that flaps its wings when you pull its tail!
|
{
"redpajama_set_name": "RedPajamaC4"
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| 2,751
|
{"url":"https:\/\/pt3mathematics.blog.onlinetuition.com.my\/2017\/01\/42-statistics-ii-pt3-focus-practice.html","text":"# 12.1.1 Measures of Central Tendencies\n\n \n \n \n12.1 Measures of Central Tendencies\n \n\n \n12.1.1 Pie Charts\n \n1. \u00a0 A pie chart is a graphic representation of data using sectors of a circle. The size of each sector shows the percentage of each category of data it represents.\n \n\n \nExample:\n \n \n \n \n\n \n2. \u00a0 The percentage of data represented by each sector can be calculated by using the formula below.\n \n \n $\\text{Percentage of data}=\\frac{\\text{angle of sector}}{{360}^{o}}\u00d7100%$ \n \n\n \n \n3. \u00a0 To construct a pie chart, the angle of each sector can be calculated by using the formulae below.\n \n \n $\\text{Angle of sector}=\\frac{\\text{data of each category}}{\\text{overall data}}\u00d7{360}^{o}$ \n \n\n \n \n4. \u00a0 Data can also be represented by pictograms, bar charts and line graphs.\n \n \n5. \u00a0 The choice of any these representations depends very much on the suitability of the data and the aim we wish to achieve in representing the data.","date":"2021-05-06 18:47:55","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 2, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9191640615463257, \"perplexity\": 614.4673242337044}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243988759.29\/warc\/CC-MAIN-20210506175146-20210506205146-00632.warc.gz\"}"}
| null | null |
Douglas Robert Brown (born 29 October 1969) is a Scottish former cricketer and former head coach of the United Arab Emirates national team. Brown represented the Scottish national team as an all-rounder at One Day International (ODI) and Twenty20 International level, having earlier played One Day International cricket for England in 1997 and 1998. He played English county cricket for Warwickshire.
Playing career
As a youngster Brown attended Alloa Academy, represented Clackmannan County Cricket Club and played football at under-18 level for Scotland, and played for Feltham Rugby Football club as a competent fullback, choosing to focus on cricket shortly afterwards.
He first came to the attention of Warwickshire County Cricket Club in 1992 whilst playing for Scotland, he signed for the county and made his debut in the same year. He went on to spend his entire career with the county, for whom he has scored over 12500 runs and taken over 850 wickets at first-class and list A cricket. At this point he also played for the north Birmingham club Aston Unity CC.
Highlights of Brown's Warwickshire career included his contributions to their successes of the mid-1990s. In 1995 he helped Warwickshire to win the NatWest Trophy,
and also scored 506 runs, and took 37 wickets at 27.32, as the county won the County Championship. In 1997 he scored 374 runs and took 17 wickets as Warwickshire won the AXA Life League, and in the same year took his career best first-class bowling figures, 8 for 89 when playing for a First Class Counties XI against Pakistan 'A'.
In 1997 Brown was named in England's one-day squad for four nations Champions Trophy at Sharjah. He made his debut on 11 December in England's narrow 7-run victory over their hosts, in this game Brown took no wickets and scored 6 runs. However, it was England's next game, on 13 December against the West Indies, that Brown gave his greatest performance as an England player.
He dismissed Philo Wallace with the opening ball of the game before going on to dismiss his Warwickshire teammate Brian Lara, then at the height of his powers, leg before wicket two balls later. This opening allowed England to take an easy victory. This game aside, Brown failed to excel at one-day cricket, and the 1998 tour of the West Indies was to prove his last.
He played for Scotland in the 2005 ICC Trophy, taking 11 wickets in the tournament, and scoring 59 runs in the final, to help Scotland win the competition. As a result, Scotland gained temporary One-day International status, and Brown has since played ODI cricket for Scotland in the 2006 European Cricket Championships, 2007 World Cricket League and 2007 Cricket World Cup.
In a twenty20 fixture between Warwickshire and Northamptonshire Steelbacks in July 2007, Brown suffered a severe Achilles tendon injury which ultimately ended his career.
Coaching career
During the 2003 Cricket World Cup, Brown coached the Namibian team. He also coached them in South African domestic one-day cricket in the 2002/03 season, playing one game as an overseas player. Outside of cricket Brown works as a PE teacher.
At the end of the 2007 season, Brown retired from cricket to take up a coaching role at Warwickshire County Cricket Club. He was sacked despite having had success with the team by winning won two trophies, the T20 Blast in 2014 and 2016 Royal London One-Day Cup.
In January 2017, Brown was appointed head coach of the United Arab Emirates national team, succeeding interim coach Owais Shah. Brown moved on from UAE Cricket in 2020 and is now actively involved with the PSL (Pakistan Super League) and RedBear Consulting in the UAE.
See also
List of cricketers who have played for more than one international team
References
External links
1969 births
Living people
Directors of Cricket
Scottish cricketers
England One Day International cricketers
Warwickshire cricketers
Wellington cricketers
Dual international cricketers
Scotland One Day International cricketers
Scotland Twenty20 International cricketers
Cricketers at the 2007 Cricket World Cup
Scottish cricket coaches
Scottish cricket commentators
BBC sports presenters and reporters
People educated at Alloa Academy
Scottish footballers
Scotland youth international footballers
Marylebone Cricket Club cricketers
Namibia cricketers
Coaches of the Namibia national cricket team
British expatriates in Namibia
Association footballers not categorized by position
Scottish expatriate sportspeople in the United Arab Emirates
Coaches of the United Arab Emirates national cricket team
First-Class Counties Select XI cricketers
Sportspeople from Clackmannanshire
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,498
|
require 'spec_helper'
module Bullet
module Notification
describe CounterCache do
subject { CounterCache.new(Post, [:comments, :votes]) }
its(:body) { should == " Post => [:comments, :votes]" }
its(:title) { should == "Need Counter Cache" }
end
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,731
|
{"url":"https:\/\/socratic.org\/questions\/what-is-the-reaction-of-a-benedict-s-reagent-with-a-aldehyde-b-ketone","text":"# What is the reaction of a benedict's reagent with: a.) Aldehyde? b.) Ketone?\n\nThen teach the underlying concepts\nDon't copy without citing sources\npreview\n?\n\n#### Explanation\n\nExplain in detail...\n\n#### Explanation:\n\nI want someone to double check my answer\n\n2\nOct 28, 2017\n\nAn aldehyde produces a brick red precipitate; a ketone does not react.\n\n#### Explanation:\n\nBenedict's solution\n\nBenedict's solution contains copper(II) ions in a sodium carbonate solution.\n\nThe copper(II) ions are complexed with citrate ions to prevent the formation of a precipitate of copper(II) carbonate.\n\nBenedict's test\n\nA few drops of the aldehyde or ketone are added to the Benedict solution, and the mixture is warmed gently for a few minutes.\n\nWith a ketone, there will be no change in the blue solution.\n\nIn a positive test for an aldehyde, the blue solution produces a dark red precipitate of copper(I) oxide.\n\nThe equations\n\nAn aldehyde reduces the complexed $\\text{Cu\"^\"2+}$ ions to $\\text{Cu\"_2\"O}$ and is in turn oxidized to a carboxylate ion.\n\nThe equations for the reaction are\n\n$\\text{RCHO + 3OH\"^\"-\" \u2192 \"RCOO\"^\"-\" + 2\"H\"_2\"O\" + 2\"e\"^\"-}$\n$\\underline{2 \\text{Cu\"^\"2+\" + 2\"OH\"^\"-\" + 2\"e\"^\"-\" \u2192 \"Cu\"_2\"O(s)\" + \"H\"_2\"O} \\textcolor{w h i t e}{m m m m m m m m m}}$\n$\\text{RCHO +\"underbrace(2\"Cu\"^\"2+\")_color(red)(\"blue complex\") + 5\"OH\"^\"-\" \u2192 \"RCOO\"^\"-\" + underbrace(\"Cu\"_2\"O(s)\")_color(red)(\"brick red ppt\") + \"3H\"_2\"O}$\n\n\u2022 11 minutes ago\n\u2022 17 minutes ago\n\u2022 18 minutes ago\n\u2022 20 minutes ago\n\u2022 12 seconds ago\n\u2022 27 seconds ago\n\u2022 4 minutes ago\n\u2022 5 minutes ago\n\u2022 6 minutes ago\n\u2022 8 minutes ago\n\u2022 11 minutes ago\n\u2022 17 minutes ago\n\u2022 18 minutes ago\n\u2022 20 minutes ago","date":"2018-02-21 05:31:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 5, \"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\": 0.6366069316864014, \"perplexity\": 13040.093656347266}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-09\/segments\/1518891813431.5\/warc\/CC-MAIN-20180221044156-20180221064156-00318.warc.gz\"}"}
| null | null |
Once again, giant Yule goat burns in Sweden
Giant Swedish Yule goat set on fire yet again, soot-covered suspect arrested
STOCKHOLM (AP) -- In what's become a Christmas tradition to some Swedes, a giant decorative goat made of straw was set ablaze early Sunday and police arrested a 25-year-old man suspected of arson.
The straw goat is a beloved Christmas symbol in the city of Gavle, in central Sweden. However, it's also become a tradition of sorts to burn it down.
This year's edition lasted nearly a month on a downtown square before going up in flames.
Police said they arrested a suspect wearing a balaclava and clothes reeking of lighter fluid. His face was covered in soot and his hair damaged by fire.
Police said he would be questioned once he sobered up.
The goat is an ancient Scandinavian Yuletide tradition that preceded Santa Claus as the bringer of gifts.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 2,213
|
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