text
stringlengths 14
5.77M
| meta
dict | __index_level_0__
int64 0
9.97k
⌀ |
|---|---|---|
Why Trader Joe's Corn Wheat Tortillas?
They're chewy and have flavor. A lot of the times, tortillas don't have any flavor.
It's fresh and made with fresh ingredients & it's actually spicy.
You can also get Casa Sanchez chips and do nachos!
Heat up tortillas on a pan on medium heat until soft on both sides. Don't let them get hard- it will taste horrible.
Chop avocado into 4 slices and add 1 to each taco.
Add 1/8 cup of black beans, 1/4 chopped tomato, and 1/2 tablespoon of salsa to each taco.
Garnish with chopped cilantro and chopped onion.
Squeeze some lime juice on top of the taco for a tangy taste.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,875
|
{"url":"https:\/\/stacks.math.columbia.edu\/tag\/0ACW","text":"Lemma 66.22.2. Let $S$ be a scheme. Let $f : X \\to Y$ be a morphism of algebraic spaces over $S$ which are decent and have finitely many irreducible components. If $f$ is birational then $f$ is dominant.\n\nProof. Follows immediately from the definitions. See Morphisms of Spaces, Definition 65.18.1. $\\square$\n\nIn your comment you can use Markdown and LaTeX style mathematics (enclose it like $\\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).","date":"2020-08-08 00:07:57","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\": 2, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9964032769203186, \"perplexity\": 308.7598775552153}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"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-2020-34\/segments\/1596439737233.51\/warc\/CC-MAIN-20200807231820-20200808021820-00452.warc.gz\"}"}
| null | null |
\section{Introduction}
A variety of formation channels have been proposed for binary black holes (BBHs;see, e.g., \citealt{mapelli2021review} for a recent review): BBH mergers can be the outcome of isolated binary evolution via common envelope \citep{tutukov1973,bethe1998,portegieszwart1998,belczynski2002,belczynski2008,belczynski2016,eldridge2016,dvorkin2016,dvorkin2018,stevenson2017,mapelli2017,mapelli2019,kruckow2018,spera2019,tanikawa2020,belczynski2020,klencki2021,olejak2021}, stable mass transfer \citep{giacobbo2018,neijssel2019,bavera2020,gallegosgarcia2021} or chemically homogeneous evolution \citep{marchant2016,mandel2016,demink2016,dubuisson2020,riley2021}. Alternatively, BBHs can form dynamically in triples \citep[e.g.,][]{antonini2017,silsbee2017,arcasedda2018b,fragione2020,vigna2021}, young star clusters (YSCs, \citealt[][]{banerjee2010,mapelli2016,banerjee2017,banerjee2020,dicarlo2019,dicarlo2020a,kumamoto2019,kumamoto2020}), globular clusters (GCs, \citealt[][]{portegieszwart2000,tanikawa2013,samsing2014,rodriguez2016,askar2017,fragionekocsis2018,choksi2019,hong2018,kamlah2021}), and nuclear star clusters (NSCs, \citealt[][]{antonini2016,petrovich2017,antonini2019,arcasedda2020,arcasedda2020b,fragione2020b}). Furthermore, gas torques in AGN discs trigger the formation of BBHs and speed up their mergers \citep[e.g.,][]{bartos2017,stone2017,mckernan2018,yang2019,tagawa2020,ishibashi2020}. Finally, primordial black holes (BHs), born from gravitational collapses in the early Universe, might also pair up and merge via gravitational wave (GW) emission \citep[e.g.,][]{carr1974,carr2016, sasaki2016,alihaimoud2017,clesse2017,deluca2021}.
One of the key signatures of the dynamical scenario is the formation of massive BHs via hierarchical merger chains \citep{miller2002,giersz2015,fishbach2017,gerosa2017,rodriguez2019,arcasedda2021b,mapelli2021,gerosa2021review}: the remnant of a BBH merger is a single object at birth, but, if it is inside a dense stellar environment, it may pair up dynamically with other BHs and merge again. The merger remnant has a distinctive feature, which is a large spin magnitude $\chi\sim{0.7}$, mostly inherited from pre-merger orbital angular momentum \citep{jimenez-forteza2017,gerosa2021review}. The efficiency of hierarchical mergers is hampered by relativistic kicks, that the merger remnant suffers at birth because of radiation
of linear momentum through beamed GW emission \citep{fitchett1983,favata2004,campanelli2007,lousto2011}.
The magnitude of the relativistic kick is generally comparable to (or higher than) the escape velocity of a massive star cluster, and can lead to the ejection of the merger remnant, interrupting the hierarchical chain \citep{holley-bockelmann2008,moody2009}.
Advanced LIGO \citep{LIGOdetector} and Virgo \citep{VIRGOdetector} observed more than 50 BBH mergers to date \citep{abbottO3a,abbottGWTC-2.1}. Population analyses on these BBHs moderately support the co-existence of multiple formation channels
(\citealt{abbottO3popandrate,callister2021,zevin2021,wong2021,bouffanais2021}, but see \citealt{roulet2021} for a different result). Moreover, GW190521 \citep{abbottGW190521,abbottGW190521astro}, and possibly GW190403\_051519 and GW190426\_190642 \citep{abbottGWTC-2.1} challenge current models of massive star evolution, hosting BHs in the pair-instability mass gap \citep{belczynski2016pair,woosley2017,spera2017,marchant2019,stevenson2019,farmer2019,mapelli2020,farrell2020,vink2021,costa2021,tanikawa2020}. The properties of these events are easier to explain with a dynamical scenario \citep{fragione2020,anagnostou2020,kimball2020,mapelli2021,arcasedda2021,liu2021,spera2019,dicarlo2019,dicarlo2020a,dicarlo2020b,gerosa2021,kremer2020,renzo2020b,gonzalez2021} than with binary evolution \citep{belczynski2020b}.
Several studies performed a multi-channel analysis,
trying to constrain the relative contribution of each formation scenario to the observed BBH population \citep[e.g.,][]{zevin2017,stevenson2017,mandel2018,bouffanais2019,bouffanais2021,zevin2021,wong2021}.
Models of different formation channels are generally produced with different numerical techniques:
For example, BBHs in NSCs
are generally studied with semi-analytical models \citep[e.g.,][]{antonini2016,arcasedda2020b}, BBHs in GCs are often modelled with hybrid Monte Carlo simulations \citep[e.g.,][]{rodriguez2016,askar2017}, BBHs in YSCs with direct N-body simulations \citep[e.g.,][]{banerjee2010,ziosi2014,fujii2014} and isolated BBHs with population-synthesis simulations \citep[e.g.,][]{belczynski2016,mapelli2017,eldridge2016}, run with different codes and assumptions.
Comparing catalogs of BBHs simulated with different codes might
lead to biased results: the differences among the considered BBH catalogs might be due to the adopted numerical approach rather than to the intrinsic physical differences among channels \citep[e.g.,][]{belczynski2021}. For example, if the assumed initial BH mass function is different for different models, the results of the multi-channel comparison will be conditioned by this discrepancy in the initial conditions. The only way to avoid such bias is to simulate different dynamical channels with the same numerical code, starting from the same underlying initial conditions (e.g., the same BH mass function).
At present, this cannot be done with direct N-body simulations, because of their high computational cost, especially for the most massive and long-lived star clusters \citep{wang2020}.
The purpose of this work is to compare the merger rate and other BBH properties (mass and spin distribution) we obtain for different channels, by adopting the same numerical code for all the considered scenarios.
We use the semi-analytic dynamical code {\sc fastcluster} \citep{mapelli2021}, which can handle isolated BBHs and dynamical BBHs in YSCs, GCs and NSCs within the same numerical framework. {\sc fastcluster} overcomes the numerical challenge of simulating BBHs in massive and long-lived star clusters by integrating the effect of dynamical hardening and GW emission with a fast semi-analytic approach, calibrated on direct N-body models.
Finally, we derive the mixing fraction of each channel, by running Bayesian hierarchical inference on the public data of the second GW transient catalog (GWTC-2, \citealt{abbottO3a}).
\section{Methods}
\subsection{Isolated BBHs}
Isolated BBHs
form and evolve in the field;
they are not perturbed by dynamical interactions. To generate masses, delay times\footnote{The delay time is the time between the formation of a binary star and the merger of the final BBH.} and spin orientations of isolated BBHs, we use the population-synthesis code {\sc mobse} \citep{giacobbo2018,giacobbo2018b}.
Dimensionless spin magnitudes $\chi$ are randomly drawn from a Maxwellian distribution truncated to $\chi=1$, with root mean square $\sigma_{\chi}=0.1$ (fiducial case) or $\sigma_{\chi}=0.01$ (low-spin case).
{\sc mobse} is an upgraded and custom version of {\sc bse} \citep{hurley2002}. It implements up-to-date models for stellar winds \citep{vink2001,graefener2008,chen2015}, core-collapse supernovae \citep[SNe,][]{fryer2012}, pair-instability SNe \citep{mapelli2020} and SN kicks \citep{giacobbo2020}. For more details, we refer to \cite{giacobbo2018} and \cite{giacobbo2018b}. BHs with mass up to $\approx{65}$ M$_\odot$ can form from metal-poor stars in {\sc mobse}, but only BHs with mass up to $\approx{45}$ M$_\odot$ merge within a Hubble time in isolated binary systems (see, e.g., Figure~11 of \citealt{giacobbo2018b}). The main reason of this difference is that tight isolated binary stars, which are the progenitors of isolated BBH mergers, evolve via mass transfer or common envelope. These are dissipative processes and lead to the complete removal of stellar envelopes, leaving behind naked He cores. The maximum mass of a BH that forms from a naked He core is $\approx{45}$ M$_\odot$ in {\sc mobse} models.
\subsection{First generation (1g) BBHs in star clusters}
\label{sec:firstgen}
To generate catalogs of BBH mergers in dynamical environments (YSCs, GCs and NSCs), we use the semi-analytic code {\sc fastcluster} \citep{mapelli2021}.
Here below, we summarize the main features of this code and refer to \cite{mapelli2021} for more details.
{\sc fastcluster} takes into account two classes of BBHs: original and dynamical BBHs. The former originate from binary stars that are already present in the initial conditions (hereafter, original binaries), while the latter are dynamically assembled. Both original and dynamical BBHs evolve inside their parent star cluster and are affected by dynamical encounters.
A dynamical BBH forms in a timescale
\begin{equation}\label{eq:tdyn}
t_{\rm dyn}=\max{\left[t_{\rm SN},\,{}t_{\rm DF}+\min{(t_{\rm 3bb},t_{\rm 12})}\right]},
\end{equation}
where $t_{\rm SN}$ is the time of the core-collapse SN explosion or direct collapse, $t_{\rm DF}$ is the dynamical friction timescale \citep{chandrasekhar1943}, $t_{\rm 3bb}$ is the timescale for dynamical formation of a BBH via three-body encounters \citep{goodman1993,lee1995} and $t_{\rm 12}$ is the timescale for dynamical formation of a BBH via exchange into an existing binary star \citep{millerlauburg2009}. For the aforementioned timescales, we use the following approximations:
\begin{eqnarray}
t_{\rm DF}=\frac{3}{4\left(2\,{}\pi{}\right)^{1/2}\,{}G^2\ln{\Lambda{}}}\,{}\frac{\sigma^3}{m_{\rm BH}\,{}\rho{}},\nonumber\\
t_{\rm 3bb}=125\,{}{\rm Myr}\,{}\left(\frac{10^6\,{}{\rm M}_\odot\,{}{\rm pc}^{-3}}{\rho_{\rm c}}\right)^2\,{}\left(\zeta{}^{-1}\,{}\frac{\sigma_{\rm 1D}}{30\,{}{\rm km}\,{}{\rm s}^{-1}}\right)^9\,{}\left(\frac{20\,{}{\rm M}_\odot}{m_{\rm BH}}\right)^5,\nonumber\\
t_{\rm 12}=3\,{}{\rm Gyr}\,{}\left(\frac{0.01}{f_{\rm bin}}\right)\,{}\left(\frac{10^6\,{}{\rm M}_\odot\,{}{\rm pc}^{-3}}{\rho_{\rm c}}\right)\,{}\left(\frac{\sigma}{50\,{}{\rm km}\,{}{\rm s}^{-1}}\right) \nonumber\\
\,{}\left(\frac{12\,{}{\rm M}_\odot}{m_{\rm BH}+2\,{}m_\ast}\right)\,{}\left(\frac{1\,{}{\rm AU}}{a_{\rm hard}}\right),
\end{eqnarray}
where $G$ is the gravity constant, $m_{\rm BH}$ is the mass of the BH, $\sigma{}$ is the 3D velocity dispersion, $\rho{}$ is the mass density at the half-mass radius, $\ln{}\Lambda{}\sim{}10$ is the Coulomb logarithm,
$\rho_{\rm c}$ is the central density of the star cluster, $\sigma{}_{\rm 1D}=\sigma{}/\sqrt{3}$ is the one-dimensional velocity dispersion at the half-mass radius (assuming an isotropic distribution of stellar velocities) and $\zeta{}\leq{1}$ accounts for deviations from equipartition of a BH subsystem (here we assume that there is equipartition, \citealt{spitzer1969}).
Furthermore, $f_{\rm bin}$ is the binary fraction, $m_\ast$ is the average mass of a star in the cluster and $a_{\rm hard}=G\,{}m_\ast/\sigma{}^2$ is the minimum semi-major axis of a hard binary system. Equation~\ref{eq:tdyn} indicates that a dynamical BBH forms only after the primary BH had enough time to sink to the cluster core by dynamical friction and acquire a companion via either three-body or exchange interactions.
The masses of both original and dynamical BBHs are generated from the population-synthesis code {\sc mobse} \citep{giacobbo2018,giacobbo2018b}. {\sc fastcluster} can take any other possible initial conditions for BH masses. However, this choice ensures that the underlying BH mass spectrum is the same for isolated, original and dynamical BBHs. The main difference between original and dynamical BBHs is that the masses of original BBHs are taken from isolated BBH simulations (they are the same as isolated BBHs), while the masses of dynamical BBHs are extracted from the distribution of single BHs. The secondary component mass of a dynamical BBH is extracted from a distribution $p(m_2)\propto{}(m_1+m_2)^4$, where $m_1$ and $m_2$ are the primary and secondary component, respectively \citep{oleary2016}.
Consistently with isolated BBHs, BH spin magnitudes are randomly sampled from a Maxwellian distribution with root mean square $\sigma_{\chi}=0.1$ (fiducial case) or $\sigma_{\chi}=0.01$ (low-spin case). We randomly draw spin directions isotropic over the sphere, because dynamics resets any spin alignments.
The semi-major axis $a$ and the eccentricity $e$ at the time of BBH formation are calculated with {\sc mobse} in the case of original BBHs and are drawn from the following probability distributions in the case of dynamical BBHs \citep{heggie1975}:
\begin{eqnarray}
p(a)\propto{}a^{-1}\quad{}\quad{}a\in[1,\,{}10^3]\,{}{\rm R}_\odot\nonumber\\
p(e)=2\,{}e\quad{}\quad{}e\in[0,\,{}1).
\end{eqnarray}
At the beginning of the integration, we check if a (dynamical or original) BBH is hard, i.e. if its binding energy $E_{\rm b}$ satisfies the following relationship \citep{heggie1975}
\begin{equation}
E_{\rm b}=\frac{G\,{}m_1\,{}m_2}{2\,{}a}\geq{}\frac{1}{2}m_\ast{}\,{}\sigma^2.
\end{equation}
If the binary is hard, we integrate its orbital evolution. Otherwise, we assume it breaks via dynamical encounters.
\begin{figure}
\begin{center}
\includegraphics[width = 0.53 \textwidth]{FASTCLUSTER_scheme2.png}
\end{center}
\caption{Flow chart of {\sc fastcluster}. \label{fig:FASTCLUSTER_scheme}}
\end{figure}
\subsection{Orbital evolution}\label{eq:orbev}
When a BBH is hard and is inside its parent star cluster, the evolution of its semi-major axis $a$ and eccentricity $e$ can be described as \citep{mapelli2021review}:
\begin{eqnarray}\label{eq:mapelli2018}
\frac{{\rm d}a}{{\rm d}t}=-2\,{}\pi{}\,{}\xi{}\,{}\frac{G\,{}\rho{}_{\rm c}}{\sigma}\,{}a^2-\frac{64}{5}\,{} \frac{G^3 \,{} m_1 \,{} m_2 \,{} (m_1+m_2)}{c^5 \,{} a^3\,{} (1-e^2)^{7/2}}\,{}f_1(e) \nonumber\\
\frac{{\rm d}e}{{\rm d}t}=2\,{}\pi{}\,{}\xi{}\,{}\kappa{}\,{}\frac{G\,{}\rho{}_{\rm c}}{\sigma}\,{}a-\frac{304}{15}\,{} e \frac{ G^3 \,{} m_1 \,{} m_2 \,{} (m_1+m_2)}{c^5 \,{}a^4 \,{} (1-e^2)^{5/2}}\,{}f_2(e),\nonumber\\
\end{eqnarray}
where $c$ is the speed of light and \citep{peters1964}
\begin{eqnarray}
f_1(e)=\left(1+\frac{73}{24}\,{}e^2+\frac{37}{96}\,{} e^4\right) \nonumber\\
f_2(e)=\left(1+\frac{121}{304} \,{} e^2\right).
\end{eqnarray}
In eqs.~\ref{eq:mapelli2018}, $\xi$ and $\kappa$ are two dimensionless parameters, calibrated with direct N-body simulations \citep{hills1983,quinlan1996,miller2002,sesana2006}. Here, we assume $\xi=3$ \citep{quinlan1996} and $\kappa=0.1$ \citep{sesana2006}. Equations~\ref{eq:mapelli2018} are composed of two terms. The first ones ($\frac{{\rm d}a}{{\rm d}t}\propto{}-a^{2}$ and $\frac{{\rm d}e}{{\rm d}t}\propto{}a$) describe the dynamical hardening and the evolution of eccentricity via Newtonian dynamical scatterings; the second ones ($\frac{{\rm d}a}{{\rm d}t}\propto{}-a^{-3}$ and $\frac{{\rm d}e}{{\rm d}t}\propto{}-a^{-4}$) describe hardening and circularization via GW emission \citep{peters1964}.
{\sc fastcluster} integrates eqs.~\ref{eq:mapelli2018} until the BBH is ejected from the cluster, or it merges, or the star cluster dies by evaporation, or we reach the Hubble time (which one of these cases happens first). If the BBH is ejected from the cluster, {\sc fastcluster} integrates only the second terms of eqs.~\ref{eq:mapelli2018} (hardening and circularization by GW emission) until either the BBH merges in the field or a Hubble time has elapsed.
A binary is assumed to be ejected from the cluster when $a_{\rm ej}>a_{\rm GW}$ \citep{baibhav2020} with
\begin{eqnarray}\label{eq:aej}
a_{\rm ej}=\frac{2\,{}\xi{}\,{}m_\ast{}^2}{(m_1+m_2)^3}\,{}\frac{G\,{}m_1\,{}m_2}{v_{\rm esc}^2}\nonumber{}\\
a_{\rm GW}=\left[\frac{32\,{}G^2}{5\,{}\pi{}\,{}\xi{}\,{}c^5}\,{}\frac{\sigma{}\,{}m_1\,{}m_2\,{}(m_1+m_2)}{\rho{}_{\rm c}\,{}(1-e^2)^{7/2}}\,{}f_1(e)\right]^{1/5}.
\end{eqnarray}
The former of the two eqs.~\ref{eq:aej} describes the semi-major axis below which the BBH is ejected by dynamical recoil, while the latter describes the maximum semi-major axis for the regime of efficient orbital decay via GW emission.
\subsection{Nth generation (Ng) dynamical BBHs}
If the BBH merges in less than a Hubble time, we estimate the mass and spin of the merger remnant using the fitting formulas by \cite{jimenez-forteza2017}. If the BBH merges inside its parent star cluster, we also calculate the relativistic kick magnitude $v_{\rm K}$ using the fit by \cite{lousto2012}. We assume that the merger remnant remains inside its parent cluster if the relativistic kick magnitude $v_{\rm K}<v_{\rm esc}$, where $v_{\rm esc}$ is the escape velocity from the star cluster. Otherwise, the merger remnant is ejected from the parent cluster and cannot participate in any further hierarchical mergers.
Even when the merger remnant remains inside its parent cluster, the kick sends it far away from the cluster core. The BH must sink back to the core via dynamical friction before it can acquire new companions via three-body encounters or exchanges. We then calculate the timescale $t_{\rm Ng}$ for the merger remnant to pair up dynamically with a new companion BH as
\begin{equation}\label{eq:tNg}
t_{\rm Ng}=t_{\rm merg}+t_{\rm DF}+\min{(t_{\rm 3bb},\,{}t_{\rm 12})}.
\end{equation}
In the above equation, $t_{\rm merg}=t_{\rm dyn}+t_{\rm GW}$ is the delay time of the first generation (1g) BBH (where $t_{\rm dyn}$ is defined in eq.~\ref{eq:tdyn}, while $t_{\rm GW}$ is the time elapsed from the formation of the BBH to its merger, according to eqs.~\ref{eq:mapelli2018}). If $t_{\rm Ng}$ is shorter than the Hubble time, we start the loop again by integrating the second generation (2g) BBH with Eqs~\ref{eq:mapelli2018}. We iterate the hierarchical merger chain until the merger remnant is ejected from the cluster, or the cluster evaporates, or we reach the Hubble time. Figure~\ref{fig:FASTCLUSTER_scheme} is a flow chart of {\sc fastcluster}.
\subsection{Properties of star clusters}\label{sec:clusters}
We consider three different flavours
of star clusters: NSCs, GCs and YSCs. Each star cluster is uniquely defined by its lifetime $t_{\rm SC}$, total mass $M_{\rm tot}$, binary fraction $f_{\rm bin}$ and half-mass density $\rho{}$. We assume $t_{\rm SC}=13.6$, 13.6 and 1 Gyr for NSCs, GCs \citep{gratton1997,gratton2003,vandenberg2013} and YSCs \citep{portegieszwart2010}, respectively. Furthermore, we assume $f_{\rm bin}=0.01$, 0.1 and 1 in NSCs \citep{antonini2016}, GCs \citep{jibregman2015} and YSCs \citep{sana2012}, respectively. We draw the total masses from a log-normal distribution with mean $\langle{}\log_{10}{M_{\rm tot}/{\rm M}_\odot}\rangle{}=6.18,\,{}5.6$ and 4.3 for NSCs, GCs and YSCs, respectively. We assume a fiducial standard deviation $\sigma_{\rm M}=0.4$ for all star cluster flavours
We draw the density at the half-mass radius from a log-normal distribution with mean $\langle{}\log_{10}{\rho{}/({\rm M}_\odot\,{}{\rm pc}^{-3})}\rangle{}=5,$ 3.7 and 3.3 for NSCs, GCs and YSCs, respectively. We assume a fiducial standard deviation $\sigma_\rho=0.4$ for all star cluster flavours. The values of $M_{\rm tot}$ and $\rho{}$ are inferred from the observations reported in \cite{neumayer2020} for NSCs and GCs (see also \citealt{harris1996,georgiev2016}) and from \cite{portegieszwart2010} for YSCs. For each star cluster, we assume a core density $\rho_{\rm c}=20\,{}\rho$. We derive the escape velocity from $M_{\rm tot}$ and $\rho$ \citep{georgiev2009a,georgiev2009b,fragione2020} using the following relationship
\begin{equation}\label{eq:vesc}
v_{\rm esc}=40\,{}{\rm km}\,{}{\rm s}^{-1}\,{}\left(\frac{M_{\rm tot}}{10^5\,{}{\rm M}_\odot}\right)^{1/3}\,{}\left(\frac{\rho}{10^5\,{}{\rm M}_\odot\,{}{\rm pc}^{-3}}\right)^{1/6}.
\end{equation}
Equation~\ref{eq:vesc} results in a distribution of escape velocities fairly consistent with the observational sample reported in Figure~1 of \cite{antonini2016} for GCs and NSCs. In the initial conditions, we generate each star cluster by randomly drawing a value of $M_{\rm TOT}$ and $\rho$ from the aforementioned distributions. We simulate only one BBH per each randomly drawn star cluster, in order to better sample the parameter space of BBHs and possible host clusters. Here, we do not consider NSCs that host a supermassive BH. In such clusters, most of the binaries inside the influence radius of the supermassive BH are soft.
We refer to \cite{arcasedda2020b} for a detailed treatment of this case. We assume, for the sake of simplicity, that the star cluster properties do not evolve in time. We will add the evolution of the star cluster in a follow-up study.
\subsection{BBH merger rate}\label{sec:MRD}
The BBH merger rate per each channel $i$ can be estimated as
\begin{eqnarray}\label{eq:cosmorate}
\mathcal{R}_i(z) = \frac{\rm d\quad{}\quad{}}{{\rm d}t(z)}\int_{z_{\rm max}}^{z}\psi_i(z')\,{}\frac{{\rm d}t(z')}{{\rm d}z'}\,{}{\rm d}z' \,{} \nonumber \\
\int_{Z_{\rm min}(z')}^{Z_{\rm max}(z')}\eta{}_i(Z)\,{}\mathcal{F}_i(z',z, Z)\,{}{\rm d}Z,
\end{eqnarray}
where $t(z)$ is the look-back time at redshift $z$, $\psi_i(z')$ is the formation rate density at redshift $z'$ for the $i-$th channel, where $i=$~NSCs, GCs, YSCs or field, $Z_{\rm min}(z')$ and $Z_{\rm max}(z')$ are the minimum and maximum metallicity of stars formed at redshift $z'$, $\eta{}_i(Z)$ is the merger efficiency at metallicity $Z$, and $\mathcal{F}_i(z', z, Z)$ is the fraction of BBHs belonging to a given channel $i$ that form at redshift $z'$ from stars with metallicity $Z$ and merge at redshift $z$, normalized to all BBHs belonging to the same channel $i$ that form from stars with metallicity $Z$. To calculate the look-back time we take the cosmological parameters ($H_{0}$, $\Omega_{\rm M}$ and $\Omega_{\Lambda}$) from \cite{planck2016}.
\subsubsection{Formation rate density}
In our fiducial model, we define $\psi_i(z)$ as follows. For the formation rate of GCs as a function of redshift we assume a Gaussian distribution
\begin{equation}\label{eq:GCs}
\psi_{\rm GC}(z)=\mathcal{B}_{\rm GC}\,{}\exp{\left[-(z-z_{\rm GC})^2/(2\,{}\sigma_{\rm GC}^2)\right]},
\end{equation}
where, in the fiducial model, $z_{\rm GC}=3.2$ is the redshift where the formation rate of GCs is maximum, $\sigma_{\rm GC}=1.5$ is the standard deviation of the distribution and $\mathcal{B}_{\rm GC}$ is the normalization factor. This distribution is reminiscent of the one estimated by \cite{el-badry2019} (see also \citealt{rodriguezloeb2018}). In particular, the fiducial normalization we adopt, $\mathcal{B}_{\rm GC}=2\times{}10^{-4}\,{}{\rm M}_{\odot}\,{}{\rm Mpc}^{-3}\,{}{\rm yr}^{-1}$, is consistent with both \cite{el-badry2019} and \cite{reina-campos2019}. The peak redshift $z_{\rm GC}=3.2$ is not taken from \cite{el-badry2019}, who report $z_{\rm GC}=4$, but rather is calibrated on the distribution of the ages of Galactic GCs, which peaks at $z=3.2$ \citep{gratton1997,gratton2003,vandenberg2013}.
In Section~\ref{sec:uncertainties}, we will discuss the impact of these parameters on the merger rate. If we assume that none of our GCs dies by evaporation, eq.~\ref{eq:GCs} yields a density of GCs in the local Universe $n_{\rm GC}\approx{4}$ Mpc$^{-3}$. This is higher than the observed value ($n_{\rm GC}\approx{2.5}$ Mpc$^{-3}$, \citealt{portegieszwart2000}), but our estimate of $n_{\rm GC}$ must be regarded as an upper limit because we assume that all GCs, even the least massive, survive to redshift zero. Fig.~\ref{fig:SFR} shows the formation rate density as a function of redshift for the four channels considered here.
The uncertainty on the formation rate of NSCs is even higher. According to several models \citep{tremaine1975,capuzzo1993,capuzzo2008,antonini2012}, NSCs form from the merger of GCs sinking to the centre of their host galaxies by dynamical friction. Thus,
for NSCs we adopt the same functional form as for GC formation history, but we reduce the normalization:
\begin{equation}\label{eq:NSCs}
\psi_{\rm NSC}(z)=\mathcal{B}_{\rm NSC}\,{}\exp{\left[-(z-z_{\rm NSC})^2/(2\,{}\sigma_{\rm NSC}^2)\right]},
\end{equation}
where, in the fiducial model, $z_{\rm NSC}=3.2$ and $\sigma_{\rm NSC}=1.5$ for analogy with GCs. This formalism is subject to large uncertainties, because of the scarce observational constraints. In Section~\ref{sec:uncertainties}, we will comment on these uncertainties. In our fiducial model, the normalization of eq.~\ref{eq:NSCs} is $\mathcal{B}_{\rm NSC}=10^{-5}\,{}{\rm M}_{\odot}\,{}{\rm Mpc}^{-3}\,{}{\rm yr}^{-1}$, and was chosen so that we obtain a NSC density in the local Universe comparable with the observed one. If we assume that all NSCs survive to redshift zero (which is reasonable for NSCs) and integrate eq.~\ref{eq:NSCs} over cosmic time, we find a current density of NSCs $n_{\rm NSC}\approx{0.06}$ Mpc$^{-3}$. For comparison, if we take the density of galaxies with stellar mass $>10^7$ M$_\odot$ from observations \citep{conselice2016} and assume that all such galaxies have a NSC, we expect a current NSC density $n_{\rm NSC}\approx{0.05-0.1}$ Mpc$^{-3}$, which compares nicely with our estimate.
Modelling the redshift evolution of YSCs is a somewhat easier task, because YSCs are expected to trace the total cosmic star formation rate density \citep{lada2003, portegieszwart2010}. Hence, we assume
\begin{equation}\label{eq:YSCs}
\psi_{\rm YSC}(z)=\mathcal{B}_{\rm YSC}(z)\,{}\psi{}(z),
\end{equation}
where
\begin{equation}\label{eq:madau}
\psi{}(z)=0.01\,{}\frac{(1+z)^{2.6}}{1+[(1+z)/3.2]^{6.2}}~\text{M}_\odot\,{}\text{Mpc}^{-3}\,{}\text{yr}^{-1}
\end{equation}
is the fit to the total cosmic star formation rate density by \cite{madau2017} and $\mathcal{B}_{\rm YSC}(z)$ is the fraction of the cosmic star formation rate density that happens in YSCs. In our fiducial model, we adopt
\begin{equation}
\mathcal{B}_{\rm YSC}(z)=\min{\left(0.1,\,{}1-\frac{\psi_{\rm NSC}(z)}{\psi{}(z)}-\frac{\psi_{\rm GC}(z)}{\psi{}(z)}\right)}.
\end{equation}
In the above equation, we assume that YSCs represent $\sim{10}$\% of the total cosmic star formation rate, as a reasonable guess from \cite{kruijssen2014} and impose that the total star formation rate density in our model at a given redshift cannot be higher than $\psi{}(z)$.
Finally, the star formation rate in the field will be equal to the remaining portion of the total cosmic star formation rate density:
\begin{equation}\label{eq:field}
\psi_{\rm iso}(z)=\mathcal{B}_{\rm iso}(z)\,{}\psi{}(z),
\end{equation}
where
\begin{equation}
\mathcal{B}_{\rm iso}(z)=1-\frac{\psi_{\rm NSC}(z)}{\psi{}(z)}-\frac{\psi_{\rm GC}(z)}{\psi{}(z)}-\frac{\psi_{\rm YSC}(z)}{\psi{}(z)}.
\end{equation}
The star formation rate in the field is dominant over the other channels in our fiducial model, as shown in Fig.~\ref{fig:SFR}.
\begin{figure}
\begin{center}
\includegraphics[width = 0.45 \textwidth]{SFR.pdf}
\end{center}
\caption{Star formation rate density as a function of redshift for isolated stars (orange dot-dashed line), YSCs (magenta short-dashed line), GCs (violet solid line) and NSCs (blue long-dashed line). \label{fig:SFR}}
\end{figure}
\subsubsection{Merger efficiency}
The merger efficiency is the total number of BBHs of a given population that merge within a Hubble time divided by the total initial stellar mass of that population \citep{giacobbo2018,klencki2018}. For isolated BBHs, this is simply
\begin{equation}\label{eq:eta_field}
\eta_{\rm field}(Z) = \frac{\mathcal{N}_{\text{TOT}}(Z)}{M_\ast{}(Z)},
\end{equation}
where $N_{\rm TOT}(Z)$ is the number of BBH mergers for a given metallicity $Z$ and $M_\ast(Z)$ is the total initial stellar mass of the population, assuming a Kroupa mass function between 0.1 and 150 M$_\odot$ \citep{kroupa2001}.
For dynamical and original BBHs, the calculation of $\eta{}(Z)$ is less straightforward, because {\sc fastcluster} does not integrate the entire BH population of a star cluster, but only a sub-set, in order to sample the parameter space more efficiently (see Section~\ref{sec:clusters}). We thus estimate the merger efficiency in star clusters as
\begin{equation}\label{eq:eta_dyn}
\eta_{\rm SC}(Z) = \frac{\mathcal{N}_{\rm merg,\,{}sim }(Z)}{\mathcal{N}_{\rm sim}(Z)}\,{}\frac{\mathcal{N}_{\rm BH}(Z)}{M_\ast{}(Z)},
\end{equation}
where $\mathcal{N}_{\rm merg,\,{}sim }(Z)$ is the number of BHs simulated with {\sc fastcluster} that merge within a Hubble time for a given metallicity $Z$, $\mathcal{N}_{\rm sim}(Z)$ is the number of BHs simulated with {\sc fastcluster} for a given metallicity $Z$, $\mathcal{N}_{\rm BH}$ is the total number of BHs associated with a given metallicity (including the BHs we did not simulate with {\sc fastcluster}) and $M_\ast{}(Z)$ is the total initial stellar mass for a given metallicity $Z$. $\mathcal{N}_{\rm merg,\,{}sim }(Z)$ and $\mathcal{N}_{\rm sim }(Z)$ are directly extracted from the simulations. We calculate $M_\ast{}(Z)=\sum{}M_{\rm TOT}(Z)$, i.e. the sum of the initial total mass of all simulated star clusters with a given $Z$. We derive $\mathcal{N}_{\rm BH}(Z)$ as the number of BHs we expect from a stellar population following a Kroupa mass function between 0.1 and 150~M$_\odot$, assuming that all stars with zero-age main sequence mass $\ge{}20$~M$_\odot$ are BH progenitors \citep{heger2003}. In our definition, $\mathcal{N}_{\rm merg,\,{}sim }(Z)$ includes even Nth generation (Ng) mergers, while $\mathcal{N}_{\rm sim }(Z)$ counts only 1g BHs. Hence, the ratio $\mathcal{N}_{\rm merg,\,{}sim }(Z)/\mathcal{N}_{\rm sim}(Z)$ can be $>1$ if hierarchical mergers are extremely efficient.
\begin{table}[h]
\begin{center}
\caption{Model properties.\label{tab:table1}}
\begin{tabular}{lcccccc}
\toprule
Model & Channel & SN model & $\alpha$ & $\sigma_{\chi}$ & $\sigma_{\rm Z}$ & $f_{\rm orig}$\\
\midrule
A02 & Isolated & delayed & 1 & 0.1 & 0.2 & 1\\
A02 & YSC & delayed & 1 & 0.1 & 0.2 & 0.6\\
A02 & GC & delayed & 1 & 0.1 & 0.2 & 0.1\\
A02 & NSC & delayed & 1 & 0.1 & 0.2 & 0.01\\
A03 & Isolated & delayed & 1 & 0.1 & 0.3 & 1 \\
A03 & YSC & delayed & 1 & 0.1 & 0.3 & 0.6 \\
A03 & GC & delayed & 1 & 0.1 & 0.3 & 0.1 \\
A03 & NSC & delayed & 1 & 0.1 & 0.3 & 0.01\\
A04 & Isolated & delayed & 1 & 0.1 & 0.4 & 1 \\
A04 & YSC & delayed & 1 & 0.1 & 0.4 & 0.6\\
A04 & GC & delayed & 1 & 0.1 & 0.4 & 0.1 \\
A04 & NSC & delayed & 1 & 0.1 & 0.4 & 0.01 \vspace{0.1cm}\\
B02 & Isolated & rapid & 1 & 0.1 & 0.2 & 1 \\
B02 & YSC & rapid & 1 & 0.1 & 0.2 & 0.6 \\
B02 & GC & rapid & 1 & 0.1 & 0.2 & 0.1 \\
B02 & NSC & rapid & 1 & 0.1 & 0.2 & 0.01\\
B03 & Isolated & rapid & 1 & 0.1 & 0.3 & 1\\
B03 & YSC & rapid & 1 & 0.1 & 0.3 & 0.6\\
B03 & GC & rapid & 1 & 0.1 & 0.3 & 0.1\\
B03 & NSC & rapid & 1 & 0.1 & 0.3 & 0.01\\
B04 & Isolated & rapid & 1 & 0.1 & 0.4 & 1 \\
B04 & YSC & rapid & 1 & 0.1 & 0.4 & 0.6\\
B04 & GC & rapid & 1 & 0.1 & 0.4 & 0.1\\
B04 & NSC & rapid & 1 & 0.1 & 0.4 & 0.01\vspace{0.1cm}\\
C02 & Isolated & delayed & 1 & 0.01 & 0.2 &1\\
C02 & YSC & delayed & 1 & 0.01 & 0.2 &0.6\\
C02 & GC & delayed & 1 & 0.01 & 0.2 &0.1\\
C02 & NSC & delayed & 1 & 0.01 & 0.2 &0.01\\
C03 & Isolated & delayed & 1 & 0.01 & 0.3 &1\\
C03 & YSC & delayed & 1 & 0.01 & 0.3 &0.6\\
C03 & GC & delayed & 1 & 0.01 & 0.3 &0.1\\
C03 & NSC & delayed & 1 & 0.01 & 0.3 &0.01\\
C04 & Isolated & delayed & 1 & 0.01 & 0.4 &1\\
C04 & YSC & delayed & 1 & 0.01 & 0.4 &0.6\\
C04 & GC & delayed & 1 & 0.01 & 0.4 &0.1\\
C04 & NSC & delayed & 1 & 0.01 & 0.4 &0.01\vspace{0.1cm}\\
D02 & Isolated & delayed & 5 & 0.1 & 0.2 &1\\
D02 & YSC & delayed & 5 & 0.1 & 0.2 &0.6\\
D02 & GC & delayed & 5 & 0.1 & 0.2 &0.1\\
D02 & NSC & delayed & 5 & 0.1 & 0.2 &0.01\\
D03 & Isolated & delayed & 5 & 0.1 & 0.3 &1\\
D03 & YSC & delayed & 5 & 0.1 & 0.3 &0.6\\
D03 & GC & delayed & 5 & 0.1 & 0.3 &0.1\\
D03 & NSC & delayed & 5 & 0.1 & 0.3 &0.01\\
D04 & Isolated & delayed & 5 & 0.1 & 0.4 &1\\
D04 & YSC & delayed & 5 & 0.1 & 0.4 &0.6\\
D04 & GC & delayed & 5 & 0.1 & 0.4 &0.1\\
D04 & NSC & delayed & 5 & 0.1 & 0.4 &0.01\\
\bottomrule
\end{tabular}
\end{center}
\footnotesize{Column 1: Name of the model, composed of a letter (A, B, C and D) followed by a number indicating the metallicity spread (02, 03 and 04 indicate $\sigma_{\rm Z}=0.2,$ 0.3 and 0.4, respectively) ; Column 2: formation channel (isolated, YSC, GC or NSC); Column 3: core-collapse SN model (delayed or rapid); Column 4: parameter $\alpha$ of common envelope for isolated binaries and original binaries; Column 5: spin parameter $\sigma_{\chi}=0.1$ or 0.01; Column 6: metallicity spread $\sigma_{\rm Z}=0.2, $ 0.3, 0.4; Column 7 ($f_{\rm orig}$): original BBH fraction (in the isolated channel every binary is original).}
\end{table}
\subsubsection{Metallicity evolution}
For the metallicity evolution, we adopt a formalism similar to the one described by \cite{bouffanais2021}, namely we use the fit to the mass-weighted metallicity evolution given by \cite{madau2017}:
\begin{equation}\label{eq:met}
\log{\langle{}Z/{\rm Z}_\odot\rangle{}}=0.153-0.074\,{}z^{1.34}
\end{equation}
To describe the spread around the mass-weighted metallicity, we assume that metallicities are distributed according to a log-normal distribution:
\begin{equation}
\label{eq:pdf}
p(z', Z) = \frac{1}{\sqrt{2 \pi\,{}\sigma_{\rm Z}^2}}\,{} \exp\left\{{-\,{} \frac{\left[\log{(Z(z')/{\rm Z}_\odot)} - {\langle{}\log{Z(z')/Z_\odot}\rangle{}}\right]^2}{2\,{}\sigma_{\rm Z}^2}}\right\},
\end{equation}
where
\begin{equation}
\langle{}\log{Z(z')/Z_\odot}\rangle{}=\log{\langle{}Z(z')/Z_\odot\rangle{}}-\frac{{\ln(10)}\,{}\sigma_{\rm Z}^2}{2}.
\label{eq:average_Z}
\end{equation}
The standard deviation $\sigma{}_Z$ is highly uncertain. Here, we probe different values of $\sigma{}_Z=0.2,$ 0.3 and 0.4. Equation~\ref{eq:pdf} allows us to estimate
the term $\mathcal{F}_i(z',z,Z)$ of eq.~\ref{eq:cosmorate}:
\begin{equation}\label{eq:Fz}
\mathcal{F}_i(z',z,Z)=\frac{\mathcal{N}_i(z',z,Z)}{\mathcal{N}_{\text{TOT\,{}i}}(Z)}\,{}p(z', Z),
\end{equation}
where $\mathcal{N}_i(z',z,Z)$ is the total number of BBHs of channel $i$ that form at redshift $z'$ with metallicity $Z$ and merge at redshift $z$, while $\mathcal{N}_{\text{TOT,\,{}i}}(Z)$ is the total number of BBH mergers of channel $i$ with progenitor's metallicity $Z$. We use the same metallicity formalism for all the considered channels.
\subsubsection{Fraction of original and dynamical BBHs}\label{sec:fractions}
With {\sc fastcluster}, we evaluate original BBHs (i.e., BBHs that form from a binary star but then evolve dynamically in a star cluster) and dynamical BBHs (i.e., BBHs that form via three-body encounters or exchanges), separately. In order to estimate the total BBH merger rate, we need to know the percentage of original and dynamical BBHs. Ideally, the mixing fraction between original and dynamical BBHs can be obtained by running Bayesian inference on GWTC-2. However, this would significantly increase the number of dimensions of our multi-channel analysis (see the next section); hence, we prefer to assume some physically motivated guess for the fraction of original BBHs.
In NSCs, the fraction of original binaries surviving dynamical interactions is expected to be of the order of $\sim{0.01}$, because most binary systems are soft in such extreme environment \citep{antonini2016}. Hence, we assume that the fraction of original BBHs in NSCs is $\sim{0.01}$, analogous to the total surviving binary fraction.
We also assume that the fraction of original BBHs in GCs is $\sim{0.1}$, corresponding to the typical binary fraction measured in the core of GCs, with large fluctuations from cluster to cluster \citep{sollima2007,milone2012}.
For YSCs we use the recent results by \cite{dicarlo2020b} and \cite{rastello2021}. Based on direct N-body simulations of YSCs, they find that the percentage of original BBH mergers is $\approx{60}$\%, with large fluctuations depending on metallicity. In Section~\ref{sec:uncertainties}, we will comment on the impact of these assumptions about the original BBH merger fraction. Finally, in the isolated BBH channel, each BBH is original by definition. The only difference between isolated BBHs and original binaries in YSCs/GCs/NSCs is that the latter are perturbed by dynamical encounters, while the former are unperturbed.
\subsection{Description of runs}
For the isolated BBH channel, we ran $1.44\times{}10^8$
massive isolated binary systems with {\sc mobse}, considering twelve different metallicities ($Z=[0.0002,$ 0.0004, 0.0008, 0.0012, 0.0016, 0.002, 0.004, 0.006, 0.008, 0.012, 0.016, $0.02]$), two different SN models (rapid and delayed model, from \citealt{fryer2012}) and two
values for the parameter $\alpha{}$ of common envelope ($\alpha=1,$ 5).
The zero-age main-sequence masses of the primary component of each binary star are distributed according to a Kroupa \citep{kroupa2001} initial mass function in the range $[5,\,{}150]\,{}{\rm M}_\odot$. The orbital periods, eccentricities and mass ratios of binaries are drawn from \cite{sana2012}. In particular, we derive the mass ratio $q$ as $\mathcal{F}(q) \propto q^{-0.1}$ with $q\in [0.1,\,{}1]$, the orbital period $P$ from $\mathcal{F}(\Pi) \propto \Pi^{-0.55}$ with $\Pi = \log{(P/\text{day})} \in [0.15,\,{} 5.5]$ and the eccentricity $e$ from $\mathcal{F}(e) \propto e^{-0.42}~~\text{with}~~ 0\leq e \leq 0.9$.
For the dynamical channels, we ran 288 different realizations of our models with {\sc fastcluster}, half of them for original binaries and the other half for dynamical binaries. Each of these 288 realizations consists of $10^6$ BBH systems. We consider three
families of star clusters (NSCs, GCs and YSCs), twelve
metallicities (the same as for the isolated BBHs), two
values of the spin magnitude parameter ($\sigma_\chi=0.01$ and 0.1), two
core-collapse SN models (rapid and delayed model, from \citealt{fryer2012}) and two
values for the parameter $\alpha{}$ of common envelope ($\alpha=1,$ 5). The properties of the star clusters are the same as described in Section~\ref{sec:clusters}.
For each of the isolated and dynamical models, we ran the {\sc cosmo${\mathcal{R}}$ate} code, in order to derive the merger rate of each specific channel. We considered three
values of the metallicity spread $\sigma{}_{\rm Z}=0.2$, 0.3 and 0.4. Table~\ref{tab:table1} summarizes the details of each resulting model. Each model presented in Table~\ref{tab:table1} includes the 12 simulated progenitor metallicities, mixed according to the formalism of {\sc cosmo${\mathcal{R}}$ate} (Section~\ref{sec:MRD}). Furthermore, each star cluster model in Table~\ref{tab:table1} includes both dynamical and original BBHs, mixed according to the fractions described in Section~\ref{sec:fractions}. In Section~\ref{sec:uncertainties}, we will consider additional models with respect to the ones summarized in Table~\ref{tab:table1}, to discuss the main uncertainties related to the formation rate of each channel, to the proportion between original and dynamical BBHs and to the properties of the considered star clusters.
\begin{table}
\begin{center}
\caption{BBH merger rate density at redshift $z=0$.
\label{tab:table2}}
\begin{tabular}{lccc}
\toprule
Model & Channel & $\mathcal{R}(0)$ & $\mathcal{R}_{\rm Ng}(0)$\\
\midrule
A02 & Isolated & 5.14 & -- \\
A02 & YSC & 2.35 & 0.07 \\
A02 & GC & 3.64 & 0.82 \\
A02 & NSC & 1.31 & 0.47\\
A03 & Isolated & 17.53 & -- \\
A03 & YSC & 4.40 & 0.11\\
A03 & GC & 4.58 & 0.98\\
A03 & NSC & 1.41 & 0.51 \\
A04 & Isolated & 60.67 & -- \\
A04 & YSC & 8.72 & 0.16\\
A04 & GC & 5.59 & 1.14\\
A04 & NSC & 1.50 & 0.54
\vspace{0.1cm}\\
B02 & Isolated & 7.41 & -- \\
B02 & YSC & 3.10 & 0.08\\
B02 & GC & 5.62 & 1.27\\
B02 & NSC & 2.08 & 0.76 \\
B03 & Isolated & 24.66 & -- \\
B03 & YSC & 6.07 & 0.14\\
B03 & GC & 6.97 & 1.50\\
B03 & NSC & 2.15 & 0.79\\
B04 & Isolated & 77.75 & -- \\
B04 & YSC & 11.93 & 0.23\\
B04 & GC & 8.47 & 1.74\\
B04 & NSC & 2.22 & 0.81
\vspace{0.1cm}\\
C02 & Isolated & 5.14 & --\\
C02 & YSC & 2.74 & 0.37\\
C02 & GC & 4.58 & 1.75\\
C02 & NSC & 1.50 & 0.66\\
C03 & Isolated & 17.53 & --\\
C03 & YSC &4.98 & 0.61\\
C03 & GC & 5.74 & 2.14\\
C03 & NSC & 1.62 & 0.71\\
C04 & Isolated & 60.67 & -- \\
C04 & YSC & 9.55 & 0.94\\
C04 & GC & 6.98 & 2.53\\
C04 & NSC & 1.72 & 0.76
\vspace{0.1cm}\\
D02 & Isolated & 4.41 & --\\
D02 & YSC & 2.38 & 0.08\\
D02 & GC & 3.73 & 0.83\\
D02 & NSC & 1.32 & 0.47 \\
D03 & Isolated & 13.23 & --\\
D03 & YSC & 4.39 & 0.11\\
D03 & GC & 4.74 & 1.00\\
D03 & NSC & 1.43 & 0.51\\
D04 & Isolated & 45.85 & --\\
D04 & YSC & 8.43 & 0.17\\
D04 & GC & 5.82 & 1.17\\
D04 & NSC & 1.52 & 0.54\\
\bottomrule
\end{tabular}
\end{center}
\footnotesize{Column 1: Model name; column 2: formation channel; column 3, $\mathcal{R}(0)$: merger rate density of BBHs at $z=0$ in units of Gpc$^{-3}$ yr$^{-1}$; column 4, $\mathcal{R}_{\rm Ng}(0)$: merger rate density of Nth generation (Ng) BBHs with N$>1$ at $z=0$, in units of Gpc$^{-3}$ yr$^{-1}$.}
\end{table}
\begin{figure*}
\begin{center}
\includegraphics[width = 0.9 \textwidth]{rate.pdf}
\end{center}
\caption{BBH merger rate density $\mathcal{R}(z)$ as a function of redshift, in the comoving frame, for all the models listed in Table~\ref{tab:table1}. From left to right, the upper row shows models A02, A03 and A04, the second row models B02, B03 and B04, the third row models C02, C03 and C04 and the lower row models D02, D03 and D04. In all the panels, yellow dot-dashed line: isolated BBHs; light-blue short-dashed line: BBHs in YSCs; blue solid line: BBHs in GCs; dark-blue long-dashed line: BBHs in NSCs; black solid line: total merger rate density. \label{fig:rate}}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width = 0.9 \textwidth]{rate_Ng.pdf}
\end{center}
\caption{Merger rate density of Nth generation (Ng) BBHs $\mathcal{R}_{\rm Ng}(z)$ as a function of redshift, in the comoving frame, for all the models listed in Table~\ref{tab:table1}. The order of the panels is the same as in Fig.~\ref{fig:rate}. In all the panels, light-blue short-dashed line: BBHs in YSCs; blue solid line: BBHs in GCs; dark-blue long-dashed line: BBHs in NSCs; red solid line: total merger rate density of Ng BBHs. \label{fig:rate_ng}}
\end{figure*}
\subsection{Bayesian inference and mixing fractions}
To compare our models against GW events in the first (O1), second (O2) and in the first part of the third observing run (O3a) of the LIGO--Virgo collaboration (LVC), we use a hierarchical Bayesian approach. Given a number $N_{\rm obs}$ of GW observations, $\mathcal{H}=\lbrace h^{k} \rbrace_{k=1}^{N_{\rm obs}}$, described by an ensemble of parameters $\theta$, the posterior distribution of the hyper-parameters $\lambda{}$ associated with the models is described as an in-homogeneous Poisson distribution \citep{loredo2004,mandel2018}
\begin{eqnarray}\label{eq:post_hier_model}
p(\lambda{}, N_\lambda | \mathcal{H}) = \text{e}^{-\mu_{\lambda}}\,{} \pi(\lambda{}, N_\lambda{}) \prod_{k=1}^{N_{\rm obs}} N_{\lambda} \int_{\theta} \mathcal{L}^{k}(h^k | \theta) \,{}p(\theta | \lambda )\,{}{\rm d}\theta{},
\end{eqnarray}
where $\theta$ are the GW parameters, $N_{\lambda}$ is the number of events predicted by the astrophysical model, $\mu_{\lambda}$ is the predicted number of detections associated with the model and the GW detector, $\pi{}(\lambda{},N_\lambda{})$ is the prior distribution on $\lambda$ and $N_\lambda$, and $\mathcal{L}^{k}(\lbrace h\rbrace^k | \theta)$ is the likelihood of the $k-$th detection. The predicted number of detections is given by $\mu{}(\lambda{})=N_\lambda\,{}\beta{}(\lambda{})$, where
\begin{equation}\label{eq:beta}
\beta{}(\lambda{})=\int_\theta p(\theta{}|\lambda{})\,{}p_{\rm det}(\theta{})\,{}{\rm d}\theta
\end{equation}
is the detection efficiency of the model. In eq.~\ref{eq:beta}, $p_{\rm det}(\theta{})$ is the probability of detecting a source with parameters $\theta$ and can be inferred by computing the optimal signal-to-noise ratio and comparing it to a detection threshold, as described, e.g., in \cite{bouffanais2021}. The values for the event's log-likelihood are derived from the posterior and prior samples released by the LVC, such that the integral in equation~\ref{eq:post_hier_model} is approximated with a Monte Carlo approach as
\begin{equation}\label{eq:approx_integral_likeli}
\mathcal{I}^{k} = \int_{\theta}\mathcal{L}^{k}(h^k | \theta) \,{}p(\theta | \lambda )\,{}{\rm d}\theta{}\sim{}\frac{1}{N_s^k}\,{}\sum_{i=1}^{N_s^k}\frac{p(\theta^k_i | \lambda{})}{\pi^k(\theta_i^k)},
\end{equation}
where $\theta_i^k$ is the $i-$th posterior sample for the $k-$th detection and $N_s^k$ is the total number of posterior samples for the $k-$th detection. Both the model and prior distributions are estimated with Gaussian kernel density estimation.
In our analysis, we further marginalise eq.~\ref{eq:post_hier_model} over $N_{\lambda}$ using a prior $\pi(N_{\lambda}) \sim 1 / N_{\lambda}$ \citep{fishbach2018}, which yields the following expression
\begin{eqnarray}\label{eq:post_hier_model_marg}
p(\lambda| \mathcal{H}) \sim \pi(\lambda) \prod_{k=1}^{N_{\rm obs}} \dfrac{\mathcal{I}^k}{\beta(\lambda)} ,
\end{eqnarray}
where the integral can be approximated in the same way as in eq.~\ref{eq:approx_integral_likeli} and $\beta(\lambda)$ is given by eq.~\ref{eq:beta}. We make this choice to neglect the information coming from the number of sources predicted by the model when estimating the posterior distribution. By doing this assumption, our analysis is not affected by the large uncertainties on the rates (see, e.g., Section~\ref{sec:uncertainties}). More details on this procedure are described in \cite{mandel2018} and \cite{bouffanais2021}.
In our analysis, our model distribution is the sum of the contributions from multiple channels (isolated BBHs, dynamical BBHs in YSCs, GCs and NSCs) weighted by mixing fraction hyper-parameters as
\begin{eqnarray}\label{eq:mixfrac}
p(\theta{}|f_{\rm iso},\,{}f_{\rm YSC},\,{}f_{\rm GC},\,{}f_{\rm NSC},\lambda{})=f_{\rm iso}\,{}p(\theta{}|{\rm iso}, \lambda{})\nonumber\\
+f_{\rm YSC}\,{}p(\theta{}|{\rm YSC},{}\lambda{})
+f_{\rm GC}\,{}p(\theta{}|{\rm GC},{}\lambda{})+f_{\rm NSC}\,{}p(\theta{}|{\rm NSC},{}\lambda{}),
\end{eqnarray}
where $f_{\rm iso}$, $f_{\rm YSC}$, $f_{\rm GC}$ and $f_{\rm NSC}$ are the mixing fractions of BBHs from isolated binary stars, YSCs, GCs and NSCs, defined so that $f_{\rm iso} + f_{\rm YSC} + f_{\rm GC} + f_{\rm NSC} = 1$. Based on this definition, the mixing fraction for each channel approximately is the fraction of merger events associated with that specific channel.
In our analysis, we do not consider all GWTC-2 event candidates \citep{abbottO3a} but only the 45 BBHs analyzed in \cite{abbottO3popandrate}, which represent a sub-sample with false alarm rate $<1$ yr$^{-1}$.
For these 45 BBHs, we use the GWTC-2 posterior samples for $\theta=\{\mathcal{M},\,{}q,\,{}\chi_{\rm eff},z\}$, where $\mathcal{M}=(m_1\,{}m_2)^{3/5}(m_1+m_2)^{-1/5}$ is the chirp mass, $q=m_2/m_1$ is the mass ratio, and $\chi_{\rm eff}$ is the effective spin:
\begin{equation}\label{eq:chieff}
\chi_{\rm eff}= \frac{(m_1\,{}\vec{\chi}_1+m_2\,{}\vec{\chi}_2)}{m_1+m_2}\cdot{}\frac{\vec{L}}{L},
\end{equation}
where $\vec{L}$ is the BBH orbital angular momentum, while
$\vec{\chi}_1$ and $\vec{\chi}_2$
are the dimensionless spin vectors. We used a Metropolis-Hastings algorithm to generate samples from the posterior of eq. \ref{eq:post_hier_model_marg}. We ran chains of $10^{7}$ iterations for each set of hyper-parameters, and then trimmed the chains using auto-correlation length.
\section{Results}\label{sec:results}
\begin{figure*}
\begin{center}
\includegraphics[width = 0.8 \textwidth]{masses2_sigma03.pdf}
\end{center}
\caption{Distribution of primary BH masses in BBH mergers. From top to bottom: NSCs, GCs, YSCs and isolated BBHs. Left-hand column: model A03 (exploiting the delayed SN model); right-hand column: model B03 (with the rapid SN model). Unfilled histograms: all BBH mergers; filled histograms: Ng BBH mergers (with N$>1$). Blue, purple, pink and orange histograms: BBHs merging at $z=0,$ 1, 2 and 4, respectively. We show the same number of simulated BBHs per each channel and per each redshift. \label{fig:masses}}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width = 0.9 \textwidth]{mix_mass_lin_z0z1.pdf}
\end{center}
\caption{Probability distribution function of primary BH masses ($m_1$) of BBHs merging at redshift $z=0$ (blue), $z=1$ (purple) and $z=2$ (pink). In each panel, we have put together different channels (isolated, YSC, GC and NSC) based on their merger rate, to obtain a synthetic Universe. We truncate the plots at 150 M$_\odot$ to improve the readability of this Figure, but there are several BHs with even higher masses (see Fig.~\ref{fig:masses}). The order of the panels is the same as in Fig.~\ref{fig:rate}. The black solid (dashed) line is the median value of the {\sc power law + peak} model applied to GWTC-2 BBHs excluding (including) GW190814 \protect{\citep{abbottO3popandrate}}. The shaded gray areas are the corresponding 90\% credible intervals. We arbitrarly rescaled the {\sc power law + peak} model on the $y$ axis. \label{fig:mix_mass}}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width = 0.75 \textwidth]{spins2.pdf}
\end{center}
\caption{Distribution of effective spins ($\chi_{\rm eff}$, blue) and precessing spins ($\chi_{\rm p}$, red) in BBH mergers at redshift $z=0$. From top to bottom: NSCs, GCs, YSCs and isolated BBHs. Left-hand column: model A03 ($\sigma_{\chi}=0.1$); right-hand column: model C03 ($\sigma{}_\chi{}=0.01$). Unfilled histograms: all BBH mergers; filled histograms: Ng BBH mergers (with N$>1$). We show the same number of simulated BBHs per each channel. \label{fig:spins}}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width = 0.9 \textwidth]{mix_spin.pdf}
\end{center}
\caption{Probability distribution function of effective ($\chi_{\rm eff}$, blue dashed line) and precessing spin ($\chi_{\rm p}$, red solid line) of BBHs merging at $z=0$. In each panel, we have put together different channels (isolated, YSC, GC and NSC) based on their merger rate, to obtain a synthetic Universe, as already done in Fig.~\ref{fig:mix_mass}. The order of the panels is the same as in Fig.~\ref{fig:rate}. \label{fig:mix_spin}}
\end{figure*}
\subsection{Multi-channel rates}
Figure~\ref{fig:rate} shows the BBH merger rate density as a function of redshift, while Table~\ref{tab:table2} shows the BBH merger rate density at $z=0$, for all of our models. The BBH merger rate density evolution of NSCs and GCs are only weakly affected by the metallicity spread $\sigma_{\rm Z}$, because BBHs efficiently pair up and harden in these massive star clusters, regardless of progenitor's metallicity. In contrast, the merger rate density of isolated BBHs is dramatically affected by progenitor's metallicity. As already discussed in previous work \citep[e.g.,][]{chruslinska2019,santoliquido2021,mandel2021}, there is about one order of magnitude difference in the BBH merger rate if we assume $\sigma_{\rm Z}=0.2$ or $0.4$. The behaviour of YSCs is intermediate between the field and GCs/NSCs.
The impact of the core-collapse SN model is nearly the same for all considered channels: the local BBH merger rate is $\approx{40-60\%}$ higher if we assume the rapid instead of the delayed SN model. This happens because the minimum mass of 1g BHs is higher in the rapid ($m_{\rm min}=5$ M$_\odot$) than in the delayed model ($m_{\rm min}=3$ M$_\odot$), leading to a shorter GW decay time. Actually, the difference between the two SN models tends to be slightly higher for GC and NSC BBHs than for isolated BBHs, because larger BH masses favour their retention inside the parent star cluster after the SN kick.
The merger rate density of isolated BBHs is $\approx{16-32}$\% higher if we assume $\alpha_{\rm CE}=1$ than if we assume $\alpha_{\rm CE}=5$, as already discussed by \cite{giacobbo2020}. In contrast, the BBH merger rate density of YSCs, GCs and NSCs is almost unaffected by the choice of the common envelope parameter.
The merger rate density of isolated BBHs and YSC BBHs extends to higher redshift with respect to GC BBHs and NSC BBHs, but this is probably a mere effect of the extrapolation of the fitting formula for the SFR density to redshift $z\gtrsim{10}$, where we do not have measurements \citep{madau2017}. Furthermore, we do not model population III stars in this work (see, e.g. \citealt{ng2021} for an accurate modeling of BBHs from metal-free stars).
The total local merger rate density (i.e., the sum of the merger rate densities of the four channels) is within the 90\% credible interval of the value inferred by the LVC [$\mathcal{R}(0)=19.3_{-9}^{+15}$ Gpc$^{-3}$ yr$^{-1}$ if GW190814 is not considered a BBH and if we allow the merger rate to evolve with redshift, \citealt{abbottO3popandrate}] for the models A02, A03, B02, C02, C03, D02 and D03, while it is too high in the other models. In particular, the models with $\sigma_{\rm Z}=0.4$ (A04, B04, C04 and D04) always produce a total local merger rate $\mathcal{R}>60$ Gpc$^{-3}$ yr$^{-1}$, which is a factor of $>3$ higher than the median value inferred by the LVC.
Figure~\ref{fig:rate_ng} shows the merger rate density evolution of Ng BHs only, with N$>1$. The fourth column of Table~\ref{tab:table2} reports the local merger rate density of Ng BBHs. GCs give the main contribution to the merger rate of Ng BBHs in all our models [$\mathcal{R}_{\rm Ng}(0)\approx{0.8-2.5}$ Gpc$^{-3}$ yr$^{-1}$], followed by NSCs [$\mathcal{R}_{\rm Ng}(0)\approx{0.5-1}$ Gpc$^{-3}$ yr$^{-1}$] and YSCs [$\mathcal{R}_{\rm Ng}(0)\approx{0.1-0.9}$ Gpc$^{-3}$ yr$^{-1}$]. In models A02--A04, the merger rate of Ng BHs is $\approx{36}\%$, $\approx{22}\%$ and $\approx{2-3}\%$ of the total merger rate in NSCs, GCs and YSCs, respectively.
Models C02--CO4 have higher values of $\mathcal{R}_{\rm Ng}$ with respect to the other models, because lower spin magnitudes are associated with lower relativistic kicks and hence favour the merger of Ng BHs. The spin magnitude parameter $\sigma_{\chi}$ has a stronger impact on the rate of Ng BBH mergers in YSCs than in GCs and especially NSCs. For example, the local Ng BBH rate in YSCs is a factor of $\approx{6}$ higher in model C03 ($\sigma_{\chi}=0.01$) with respect to model A03 ($\sigma_{\chi}=0.1$). In the case of GCs and NSCs, the difference between the C03 and A03 models is equal to a factor of $\approx{2}$ and $\approx{1.4}$, respectively. This trend is a consequence of the different escape velocities of YSCs, GCs and NSCs: in our models, NSCs have escape velocities of the order of 100 km s$^{-1}$, hence they retain a large fraction of the merger remnants even if $\sigma_{\chi}=0.1$; in contrast, GCs and especially YSCs have lower escape velocities and lose most of their BH merger remnants if $\sigma_{\chi}=0.1$. If we lower $\sigma_{\chi}$ to 0.01, even YSCs can efficiently retain their BH merger remnants. Accounting for all these uncertainties, the total merger rate of Ng BBHs in the local Universe ranges from $\approx{1}$ Gpc$^{-3}$ yr$^{-1}$ (D02) to $\approx{4}$ Gpc$^{-3}$ yr$^{-1}$ (C04).
\subsection{BBH mass}
Figure~\ref{fig:masses} shows the distribution of the primary BH masses in the four considered channels at different redshifts, for models A03
(delayed SN model) and B03
(rapid SN model). The overall primary BH mass distribution strongly depends on the core-collapse SN model by construction: while the delayed SN model allows the formation of BHs with mass as low as 3 M$_\odot$, the rapid SN model prevents the formation of BHs with mass $<5$ M$_\odot$.
The mass function of primary BHs in YSCs is similar to the distribution of primary BHs in isolated binary systems, but while the latter has a sharp truncation at $\approx{50}$ M$_\odot$, the former has a tail up to $\approx{200}$ M$_\odot$ because of Ng systems.
The contribution of dynamically formed BBHs and Ng BBHs is more important for NSCs and GCs, which are the most dynamically active systems. However, BHs with mass $>100$ M$_\odot$ are extremely rare even in GCs and NSCs. As already discussed in \cite{mapelli2021}, NSCs are the channel with the largest number of low-mass primary BHs. This happens because single BHs that receive a SN kick higher than the escape velocity leave their parent star cluster and cannot pair up dynamically. Since the natal kick in our models is higher for less massive BHs, this strongly suppresses the formation of light BBHs in YSCs and GCs, which have relatively low escape velocity, while NSCs are able to retain even the least massive BHs.
The mass distribution of isolated BBHs and YSC BBHs does not show any strong dependence on the merger redshift. In contrast, the mass distribution of BBHs in GCs and especially NSCs shows a relevant trend: low-mass BBH mergers are more common at low redshift than at high redshift. This is a consequence of dynamics: more massive BHs
dynamically pair up on a shorter timescale than lighter BHs (eq.~\ref{eq:tdyn}). Moreover, the timescale for GW decay is shorter for more massive systems than for lighter ones (eqs.~\ref{eq:mapelli2018}).
Figure~\ref{fig:mix_mass} shows a realization of the primary BH mass distribution we obtain by putting together BBHs from various channels according to their merger rate. In other words, this is the entire population of BBH mergers at $z=0,$ 1 and 2 in our synthetic Universe, without including observation biases. The most notable difference is between the rapid and delayed model, the former displaying a stronger peak at primary mass $m_1\approx{10}$ M$_\odot$ with respect to the latter. The tail of high-mass BHs ($\geq{}50$ M$_\odot$) is more populated in models C02--C04 with respect to the other models, because the low-spin models have a higher percentage of Ng BBHs.
Going from models with $\sigma_{\rm Z}=0.2$ to models with $\sigma_{\rm Z}=0.4$, the contribution of primary
BHs with mass $m_{\rm BH}\sim{20}$ M$_\odot$ becomes more and more important, because the isolated BBH channel (which has the largest population of BHs with mass $m_{\rm BH}\sim{20}$ M$_\odot$, Fig.~\ref{fig:masses}) is associated with a higher merger rate for larger values of $\sigma_{\rm Z}$.
In Figure~\ref{fig:mix_mass}, we also visually compare our synthetic populations with the {\sc power law + peak} model from Fig.~8 of \cite{abbottO3popandrate}. Our populations
match the {\sc power law + peak} model, the main difference being the number of primary BHs with mass $\sim{20}$~M$_\odot$, which is higher in our models, especially if we adopt the delayed model and $\sigma_{\rm Z}=0.4$. At the high-mass end, our low spin models C02--C04 better match the {\sc power law + peak} model than the other runs, but all of our synthetic populations are within the 90\% credible interval of the phenomenological model by \cite{abbottO3popandrate}.
Figure~\ref{fig:mix_mass} compares the population of BBHs at redshift 0, 1 and 2. The fraction of BBHs with primary mass $\ge{}30$ M$_\odot$ increases with redshift, because of the
contribution of GCs and NSCs to the overall BBH population (Fig.~\ref{fig:masses}).
This dependence on redshift is more evident in models with $\sigma_{\rm Z}=0.2$ and $\alpha=5$, in which the contribution of isolated BBHs is quenched. From GWTC-2 data, there is no clear evidence that the mass of BBH mergers evolves with redshift \citep{abbottO3a,abbottO3popandrate}, but some recent analysis suggests a possible weak trend under several assumptions \citep{fishbach2021a,fishbach2021b}. In our models, we also predict a weak trend, driven by BBHs in GCs and NSCs.
\subsection{BBH spins}
Figure~\ref{fig:spins} shows the distribution of effective ($\chi_{\rm eff}$, eq.~\ref{eq:chieff}) and precessing spins ($\chi_{\rm p}$) for our BBH mergers at redshift $z=0$. We calculated $\chi_{\rm p}$ according to the following definition:
\begin{eqnarray}\label{eq:chip}
\chi_{\rm p}=\max{\left[\chi_{\rm 1\perp},\,{}\frac{q\,{}(4\,{}q+3)}{4+3\,{}q}\,{}\chi_{\rm 2\perp}\right]},
\end{eqnarray}
where $\chi_{1\perp{}}$ and $\chi{}_{2\perp{}}$ are the components of the dimensionless spin vectors ($\vec{\chi}_1$ and $\vec{\chi}_2$) perpendicular to the orbital angular momentum.
The distribution of $\chi_{\rm eff}$ and $\chi_{\rm p}$ is nearly independent of redshift, but this is not surprising, because we derive the magnitudes of 1g BBHs from a toy model which does not depend on either redshift or mass.
In the dynamical channels $\chi_{\rm eff}$ is symmetric around zero, because we assume isotropic spin orientation, while the isolated channel has a strong preference for positive $\chi_{\rm eff}$ because of angular momentum alignment during mass transfer and tidal evolution. Ng mergers extend the distribution of $\chi_{\rm eff}$ to very low and very high values with respect to 1g mergers.
The distribution of $\chi_{\rm p}$ for isolated BBHs has a strong peak at zero, because of the preferential alignment, while the distribution of $\chi_{\rm p}$ for dynamical BBHs has two peaks. The position of the primary peak depends on the choice of $\sigma_{\chi}$, while the secondary peak is at $\chi_{\rm p}\approx{0.7}$ and is completely determined by Ng BBHs.
Fig.~\ref{fig:mix_spin} shows the distribution of $\chi_{\rm eff}$ and $\chi_{\rm p}$ we obtain by putting together BBHs formed via different channels according to their merger rate at $z=0$. In the fiducial spin case ($\sigma_{\chi}=0.1$), the distribution of $\chi_{\rm eff}$ becomes more asymmetric if we assume a larger value of $\sigma_{\rm Z}$, because the contribution of isolated BBHs to the total merger population increases for larger metallicity spreads. In the low-spin case ($\sigma_{\chi}=0.01$), this dependence on $\sigma_{\rm Z}$ is not visible because all 1g BBHs have vanishingly small spins.
In the fiducial spin case ($\sigma_{\chi}=0.1$), the total distribution of $\chi_{\rm p}$ even shows three peaks: a first narrow peak at zero because of isolated BBHs, a second broader peak at $\chi_{\rm p}\sim{0.1-0.2}$ because of 1g dynamical BBHs and a third peak at $\chi_{\rm p}\sim{0.7}$ because of Ng mergers. The peak at $\chi_{\rm p}\sim{0.1-0.2}$ is an effect of our choice of $\sigma_\chi$. In the low-spin case ($\sigma_{\chi}=0.01$) there are only two peaks: one at zero (due to both isolated BBHs and 1g dynamical BBHs) and the other at 0.7 (Ng BBHs).
\subsection{Mixing fractions}\label{sec:bayes}
\begin{figure*}
\begin{center}
\includegraphics[width = 0.9 \textwidth]{Summary_newdata_NoRate.pdf}
\end{center}
\caption{Posterior probability distribution of the mixing fractions $f_i$ (with $i=$ iso, YSC, GC and NSC, eq.~\ref{eq:mixfrac}) for all our models. The order of the panels is the same as in Fig.~\ref{fig:rate}. Yellow dot-dashed line: isolated BBHs; light-blue short-dashed line: BBHs in YSCs; blue solid line: BBHs in GCs; dark-blue long-dashed line: BBHs in NSCs. To produce this Figure we used the posterior samples from GWTC-2 \citep{abbottO3a}.
\label{fig:mixing_frac}}
\end{figure*}
Figure~\ref{fig:mixing_frac} shows the posterior distribution of the mixing fractions $f_i$ (with $i=$ iso, YSC, GC and NSC), defined in eq.~\ref{eq:mixfrac}. These values are obtained taking into account the detection efficiency (eq.~\ref{eq:beta})
and marginalizing eq.~\ref{eq:approx_integral_likeli} over $N_\lambda$ (eq.~\ref{eq:post_hier_model_marg}).
Table~\ref{tab:table3} shows the median and 90\% credible interval of the mixing fractions. The mixing fractions wildly depend on the details of each model: small differences between one model and another result in large differences in terms of $f_i$. There are still too many uncertainties about astrophysical models to claim we know the relative impact of each channel onto the global merger population.
However, there is a common feature of all our models: GWTC-2 data moderately support the co-existence of multiple channels: in each of our models, the mixing fraction is significantly larger than zero for at least two of the four considered channels. Hence, multiple formation channels likely are at work, to produce the population of BBH mergers we observe with GWs. In particular, the contribution of either isolated BBHs or BBHs in YSCs is needed to explain the low-mass portion of the BH mass function (Figs.~\ref{fig:masses} and \ref{fig:mix_mass}), while the contribution of BBHs in GCs or NSCs is fundamental to reproduce the high-mass tail ($m_1\geq{}50$ M$_\odot$).
Isolated BBHs are associated with higher mixing fractions in models with very low spins (C02--C04) possibly because the observed population does not favour a strongly asymmetric $\chi_{\rm eff}$ distribution with support for large positive values \citep{abbottO3popandrate}.
The metallicity spread also has a large effect on the mixing fractions. Fig.~\ref{fig:mixing_frac} does not account for the predicted number of detections. Hence, the impact of $\sigma_{\rm Z}$ on our mixing fractions is rather connected with BH mass and redshift distribution than with rates. A larger metallicity spread increases the percentage of isolated BBHs born from metal-poor stars that merge in the low-redshift Universe. Since these tend to be more massive than BBHs from metal-rich stars, the mass function of isolated BBHs tends to be more top-heavy when $\sigma_{\rm Z}$ is large, hence more similar to the one of dynamical BBHs. As a consequence, $f_{\rm iso}$ tends to be larger.
Figure \ref{fig:match_model} shows the values of $\mathcal{I}^{k}$ (defined in eq.~\ref{eq:approx_integral_likeli}) for model A03.
The other models yield similar results. The integral $\mathcal{I}^{k}$
gives an idea of how well our models are able to
match the posterior distributions of a GW event. Figure \ref{fig:match_model} only shows the 10 BBHs with the largest chirp mass from GWTC-2 \citep{abbottO3a}. The isolated channel struggles to explain the five most massive events, which have $\mathcal{M}\ge{}40$~M$_\odot$. In the case of GW190521, we find $\ln (\mathcal{I}^{k} )\approx{-453}$, even if we do not include $\chi_{\rm p}$ among the considered parameters. While a strongly negative value of $\mathcal{I}^{k}$ for a single event does not significantly affect the mixing fractions, significantly negative values for at least five BBHs (over the 45 events we included in our sample) have some impact on $f_{\rm i}$. In model A03, the mixing fraction of the isolated channel increases from $f_{\rm iso}=0.07_{-0.07}^{+0.17}$ to $0.10_{-0.09}^{+0.20}$ if we recalculate it after removing the five events with the largest chirp mass (the reported uncertainty is the 90\% credible interval). Correspondingly, the mixing fraction of GCs decreases from $f_{\rm GC}=0.28_{-0.23}^{+0.33}$ to $0.21_{-0.18}^{+0.32}$ when we remove these five events, while $f_{\rm YSC}$ and $f_{\rm NSC}$ remain nearly unchanged.
\begin{figure}
\begin{center}
\includegraphics[width = 0.45 \textwidth]{Massive_events_likelihood.pdf}
\end{center}
\caption{Value of $\mathcal{I}^{k}$ (eq.~\ref{eq:approx_integral_likeli}) for the 10 GWTC-2 events with the largest chirp mass $\mathcal{M}$. The GW events on the $x$ axis are ordered by decreasing median value of $\mathcal{M}$. Yellow squares: isolated BBHs; light-blue open circles: BBHs in YSCs; blue stars: BBHs in GCs; dark-blue triangles: BBHs in NSCs.
\label{fig:match_model}}
\end{figure}
\begin{table}
\begin{center}
\caption{Median values of the mixing fractions.\label{tab:table3}}
\begin{tabular}{lcccc}
\toprule
Model & $f_{\rm iso}$ & $f_{\rm YSC}$ &$ f_{\rm GC}$ & $f_{\rm NSC}$\\
\midrule
A02 & $0.17_{-0.13}^{+0.19}$ & $0.28_{-0.23}^{+0.26}$ & $0.32_{-0.27}^{+0.34}$ & $0.17_{-0.15}^{+0.25}$ \\
A03 & $0.07_{-0.07}^{+0.17}$ & $0.45_{-0.27}^{+0.25}$ & $0.28_{-0.23}^{+0.33}$ & $0.15_{-0.13}^{+0.23}$ \\
A04 & $0.17_{-0.15}^{+0.24}$ & $0.20_{-0.18}^{+0.29}$ & $0.43_{-0.28}^{+0.30}$ & $0.13_{-0.11}^{+0.23}$ \\
B02 & $0.05_{-0.04}^{+0.13}$ & $0.33_{-0.27}^{+0.29}$ & $0.42_{-0.32}^{+0.32}$ & $0.17_{-0.13}^{+0.20}$ \\
B03 & $0.08_{-0.07}^{+0.18}$ & $0.54_{-0.26}^{+0.22}$ & $0.17_{-0.13}^{+0.26}$ & $0.16_{-0.14}^{+0.22}$ \\
B04 & $0.15_{-0.13}^{+0.24}$ & $0.38_{-0.28}^{+0.27}$ & $0.19_{-0.17}^{+0.33}$ & $0.21_{-0.15}^{+0.22}$ \\
C02 & $0.26_{-0.23}^{+0.30}$ & $0.03_{-0.03}^{+0.06}$ & $0.54_{-0.27}^{+0.26}$ & $0.13_{-0.11}^{+0.22}$ \\
C03 & $0.56_{-0.23}^{+0.19}$ & $0.02_{-0.01}^{+0.04}$ & $0.16_{-0.13}^{+0.21}$ & $0.24_{-0.19}^{+0.24}$ \\
C04 & $0.66_{-0.22}^{+0.15}$ & $0.02_{-0.02}^{+0.05}$ & $0.18_{-0.12}^{+0.19}$ & $0.12_{-0.10}^{+0.18}$ \\
D02 & $0.08_{-0.07}^{+0.15}$ & $0.47_{-0.31}^{+0.23}$ & $0.23_{-0.19}^{+0.33}$ & $0.20_{-0.14}^{+0.20}$ \\
D03 & $0.07_{-0.06}^{+0.15}$ & $0.23_{-0.20}^{+0.29}$ & $0.57_{-0.31}^{+0.26}$ & $0.09_{-0.08}^{+0.21}$ \\
D04 & $0.12_{-0.10}^{+0.18}$ & $0.13_{-0.12}^{+0.26}$ & $0.51_{-0.31}^{+0.26}$ & $0.19_{-0.14}^{+0.24}$ \\
\bottomrule
\end{tabular}
\end{center}
\footnotesize{This Table shows the median values and 90\% intervals of the mixing fractions shown in Fig.~\ref{fig:mixing_frac}.}
\end{table}
\section{Discussion: main sources of uncertainty and further caveats}\label{sec:uncertainties}
The formation rate density of star clusters is extremely uncertain. Here, we discuss what happens if we consider different assumptions within the observational uncertainties. For GCs, we
start from model A03
and change the normalization $\mathcal{B}_{\rm GC}$, the position of the peak $z_{\rm GC}$ and the spread of the distribution $\sigma_{\rm GC}$. A change of the normalization of the GC formation rate causes the same change of the value of the BBH merger rate density: if we increase (reduce) the normalization by a factor of two from $\mathcal{B}_{\rm GC}=2\times{}10^{-4}$ to $4\times{}10^{-4}$ ($10^{-4}$) M$_\odot$ Mpc$^{-3}$ yr$^{-2}$, we obtain a factor of two higher (lower) merger rate density at each redshift, as shown in Fig.~\ref{fig:GCrate}.
If we change the peak redshift from $z_{\rm GC}=3.2$, as inferred from Galactic GCs, to $z_{\rm GC}=4$, as suggested by the models of \cite{el-badry2019}, the BBH merger rate density also shifts: the maximum value of $\mathcal{R}(z)$ is at redshift $z=3.55$ ($z=2.75$) when $z_{\rm GC}=4$ ($z_{\rm GC}=3.2$). This shift of the peak has a strong impact on the local merger rate density, which decreases from $\mathcal{R}(0)\approx{5}$ Gpc$^{-3}$ yr$^{-1}$ to $\mathcal{R}(0)\approx{2}$ Gpc$^{-3}$ yr$^{-1}$ if we change $z_{\rm GC}$ from 3.2 to 4. Finally,
the standard deviation $\sigma_{\rm GC}$
has an even larger impact on the local merger rate: $\mathcal{R}(0)$ drops from $\approx{5}$ Gpc$^{-3}$ yr$^{-1}$ to $\approx{0.3}$ Gpc$^{-3}$ yr$^{-1}$ if we change $\sigma_{\rm GC}$ from 1.5 to 0.5. However, $\sigma_{\rm GC}=0.5$ is an extreme value when compared with other models (e.g., \citealt{el-badry2019,reina-campos2019}).
The formation history of NSCs is even more uncertain. We assumed that $\psi{}_{\rm NSC}(z)$ is a Gaussian function for analogy with GCs, but the shape of NSC formation history is essentially unconstrained \citep{neumayer2020}. In Figure~\ref{fig:NSCrate}, we assume that $\psi{}_{\rm NSC}(z)$ scales with the global star formation rate density $\psi{}(z)$ \citep{madau2017} as
$\psi{}_{\rm NSC}(z)=\mathcal{B}_{\rm NSC}\,{}\,{}\psi(z)$,
where $\mathcal{B}_{\rm NSC}=10^{-5},$ 10$^{-4}$ and 10$^{-3}$ in the three cases shown in Fig.~\ref{fig:NSCrate}. The case with $\mathcal{B}_{\rm NSC}=10^{-4}$ has a similar behaviour to our fiducial model A03 at redshift $z<2$. The models with $\mathcal{B}_{\rm NSC}=10^{-3}$ and $\mathcal{B}_{\rm NSC}=10^{-5}$ give a local merger rate density a factor of 10 higher and a factor of 10 lower than model A03, respectively. The case with $\mathcal{B}_{\rm NSC}=10^{-3}$ is a strong upper limit, because it gives a local density of NSCs $n_{\rm NSC}\sim{0.6}$ Mpc$^{-3}$, i.e. larger than the number of galaxies which can host such NSCs.
\begin{figure}
\begin{center}
\includegraphics[width=0.45 \textwidth]{GC_rate_comparison.pdf}
\end{center}
\caption{Merger rate density of BBHs in GCs, as a function of redshift. Different lines show the uncertainties connected with the formation rate history of GCs. Blue solid line: model A03 for GCs. Violet dotted line: same as A03 but with normalization $\mathcal{B}_{\rm GC}=10^{-4}$ in units of M$_\odot$
Mpc$^{-3}$ yr$^{-1}$. Magenta long-dashed line: same as A03 but with normalization $\mathcal{B}_{\rm GC}=4\times{}10^{-4}$ M$_\odot$
Mpc$^{-3}$ yr$^{-1}$. Pink dash-dotted line: same as A03 but with peak redshift $z_{\rm GC}=4$. Orange short-dashed line: same as A03 but with standard deviation $\sigma_{\rm GC}=0.5$. \label{fig:GCrate}}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.45 \textwidth]{NSC_rate_comparison.pdf}
\end{center}
\caption{ Merger rate density of BBHs in NSCs, as a function of redshift. Different lines show the uncertainties connected with the formation rate history of NSCs. Blue solid line: model A03 for GCs. Magenta long-dashed line: same as A03 but with NSC formation rate density $\psi_{\rm NSC}(z)=10^{-3}\,{}\psi{}(z)$ ($\psi{}(z)$ is defined in eq.~\ref{eq:madau}). Pink dash-dotted line: same as A03 but with $\psi_{\rm NSC}(z)=10^{-4}\,{}\psi{}(z)$. Orange short-dashed line: same as A03 but with standard deviation but with $\psi_{\rm NSC}(z)=10^{-5}\,{}\psi{}(z)$.
\label{fig:NSCrate}}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.45 \textwidth]{YSC_rate_comparison.pdf}
\end{center}
\caption{Merger rate density of BBHs in YSCs (thick lines) and in the field (thin lines), as a function of redshift. Different lines show the uncertainties connected with the formation rate history of YSCs. Blue thick (thin) solid line: model A03 for GCs (isolated BBHs). Magenta long-dashed line: same as A03 but with YSC formation rate density $\psi_{\rm YSC}(z)=0.3\,{}\psi{}(z)$. Orange dash-dotted line: same as A03 but with $\psi_{\rm YSC}(z)=0.7\,{}\psi{}(z)$.
\label{fig:YSCrate}}
\end{figure}
In the case of YSCs, the main uncertainty concerns which fraction of the cosmic star formation rate happens in YSCs versus the field. In our fiducial model, we adopted a conservative assumption that only $\sim{10}$\% of the cosmic star formation rate happens in YSCs, as suggested by recent studies (e.g., \citealt{kruijssen2014,ward2020}). In Figure~\ref{fig:YSCrate}, we consider two more optimistic assumptions in which $\sim{30}$\% and $\sim{70}$\% of the cosmic star formation rate happen in YSCs \citep{lada2003,portegieszwart2010}. As expected, the merger rate density of BBHs in YSCs scales accordingly, while the merger rate density of isolated BBHs decreases by the corresponding amount.
Another source of uncertainty is the fraction of original versus dynamical BBHs. While the fraction of original BBHs is deemed to be very low in both GCs and NSCs (hence their population properties are mostly driven by dynamical BBHs, \citealt{antonini2016,rodriguez2016}), the percentage of original BBHs in YSCs is more uncertain. Here, we have assumed they are 60\% of all BBH mergers, based on the results of \cite{rastello2021}. However, \cite{rastello2021} also show that the percentage of original BBHs strongly fluctuates from a cluster to another and possibly depends on both YSC mass and metallicity. Figure~\ref{fig:YSCeta} shows the merger efficiency in the field and in YSCs (defined in eqs. \ref{eq:eta_field} and \ref{eq:eta_dyn}). The merger efficiency of original BBHs in YSCs is very similar to the one of isolated BBHs, while the merger efficiency of dynamical BBHs in YSCs has a
much less steep dependence on metallicity. Hence, if we assume a higher percentage of dynamical BBHs in YSCs, we end up with a higher local merger rate density in YSCs and with a milder dependence of the YSC merger rate on metallicity spread.
One of the main approximations of our approach is that we do not model stellar and binary evolution together with dynamics. This approximation is well motivated for GCs and NSCs, which have two-body relaxation timescales of several Gyrs \citep{binney1987}, but is less good for YSCs, which have two-body relaxation timescales of several ten Myrs. \cite{dicarlo2020b} showed that most dynamical exchanges leading to the formation of merging BBHs involve their stellar progenitors, before they collapse to BHs. Moreover, hierarchical BBH mergers are rare in YSCs, but runaway collisions seem to be more efficient in producing massive BHs in these environments \citep{mapelli2016,rizzuto2020,dicarlo2021}. Thus, our results likely underestimate the presence of massive BBHs ($m_1+m_2>100$ M$_\odot$) in YSCs. We will include a
treatment of stellar/binary evolution
in {\sc fastcluster} in future work.
The properties of our star clusters do not evolve with time. On the one hand, star clusters lose mass by stellar evolution and dynamical ejection and expand by two-body relaxation. This leads to lower star cluster mass and density, possibly quenching the formation of hierarchical mergers \citep[e.g.,][]{antonini2020a, antonini2020b}. On the other hand, by assuming no evolution with time, we do not account for core collapse episodes and gravothermal oscillations, which lead to a dramatic temporary increase of the central density, possibly boosting BBH formation and hierarchical mergers \citep[e.g.,][]{breen2013}. NSCs might even acquire mass during their life by fresh star formation \citep[e.g.,][]{mapelli2012} and by accreting GCs \citep[e.g.,][]{capuzzo2008,antonini2012}. These processes might lead to a higher efficiency of hierarchical mergers in NSCs. The overall effect of including star cluster evolution in our model is thus quite difficult to predict and might be significantly different for YSCs, GCs and NSCs. We will add a formalism for star cluster evolution in a follow-up study. Furthermore, we neglect the impact of additional formation channels, such as BBHs in AGN discs and field triples.
The approach of {\sc fastcluster} is very flexible, and we can add more channels in the future.
Comparing to previous studies, which use more sophisticated and computationally expensive simulations, we find similar results. For example, our local merger rate density in GCs is consistent with the one found by \cite{rodriguezloeb2018}, even if our values $\left[\mathcal{R}(0)\approx{4-8}\,{}{\rm Gpc}^{-3}\,{}{\rm yr}^{-1}\right]$ are rather on the lower side of their range $\left[\mathcal{R}(0)\approx{4-18}\,{} {\rm Gpc}^{-3}\,{} {\rm yr}^{-1}\right]$. The difference is easily explained by the fact that we do not model GW captures, which require direct N-body integration with post-Newtonian terms \citep{samsing2018,zevin2019,kremer2020b}. Moreover, we use a different mass function and spin distribution. To confirm the good performance of {\sc fastcluster}, we also find that the maximum merger rate density (at $z\sim{2.8}$) is about six times higher than the local merger rate density, in perfect agreement with \cite{rodriguezloeb2018}. Finally, our percentages of Ng to 1g BBH mergers in GCs are comparable to the ones derived by several authors with different approaches \citep{rodriguez2019,zevin2019,kimball2020a,kimball2020,doctor2020}. For more details on this comparison, see the Discussion in \cite{mapelli2021}.
The main result of our mixing fraction analysis is that at least two formation channels need to be at work to produce the population of GWTC-2. This result is in agreement with previous work \citep{abbottO3popandrate,zevin2021,bouffanais2021,wong2021}. Taking advantage of {\sc fastcluster} flexibility and speed, we probed a larger parameter space than previously done (including different metallicity spreads, different core-collapse SN models and a large number of stellar metallicities). This analysis shows that the mixing fraction of each channel varies wildly from one model to another, being extremely sensitive to the metallicity spread $\sigma_{\rm Z}$, spin parameter $\sigma_{\chi}$, common envelope parameter and core-collapse SN model. Hence, we must be very cautious when drawing conclusions from a mixing-fraction analysis: the relevant parameter space and the uncertainties of current models are still utterly large.
\begin{figure}
\begin{center}
\includegraphics[width=0.45 \textwidth]{eta_YSC.pdf}
\end{center}
\caption{Merger efficiency $\eta$ of BBHs as a function of metallicity $Z$, in model A03. Blue open circles: dynamical BBHs in YSCs. Red stars: original BBHs in YSCs. Black open squares: isolated BBHs.
\label{fig:YSCeta}}
\end{figure}
\section{Conclusions}\label{sec:summary}
We interfaced our semi-analytic codes {\sc fastcluster} \citep{mapelli2021} and {\sc cosmo$\mathcal{R}$ate} \citep{santoliquido2021}. {\sc fastcluster} dynamically pairs up binary black holes (BBHs) in dense star clusters,
and integrates their orbital evolution via three-body hardening and gravitational-wave (GW) decay. With {\sc fastcluster} we can study the dynamical formation of BBHs in very different star clusters, from the least massive young star clusters (YSCs) to the most massive globular clusters (GCs) and nuclear star clusters (NSCs). Furthermore, {\sc fastcluster} includes a treatment for hierarchical mergers.
{\sc cosmo$\mathcal{R}$ate} calculates the BBH merger rate evolution, by using catalogues of BBH mergers simulated with {\sc fastcluster} and by coupling them with the cosmic star formation rate and metallicity evolution. Here, we included the mass formation rate of NSCs, GCs and YSCs in {\sc cosmo$\mathcal{R}$ate}.
We use {\sc fastcluster} + {\sc cosmo$\mathcal{R}$ate} to study four BBH formation channels: isolated BBHs and dynamical BBHs in NSCs, GCs and YSCs. This technique allows us to model different BBH formation channels with the same code, starting from the same
BH mass function. Our approach prevents any systematic bias which arises from comparing outputs of different codes, that assume different stellar evolution models and BH mass function. We consider a large range of progenitor's metallicities (twelve values of $Z\in[0.0002,\,{}0.02]$), three values of the metallicity spread ($\sigma_{\rm Z}=0.2,$ 0.3 and 0.4), two models of core-collapse SN (delayed and rapid), two values of the common envelope parameter ($\alpha=1$, 5) and two models for the dimensionless spin $\chi$ (two truncated Maxwellian distributions with $\sigma_{\chi}=0.01$ and 0.1).
We find a local BBH merger rate density $\mathcal{R}(0)\sim{4-8}$ Gpc$^{-3}$ yr$^{-1}$ in GCs. The BBH merger rate density in GCs increases up to redshift $z\sim{2.5-2.8}$, reaching values $\sim{6}$ times higher than the local merger rate density. The local merger rate density of BBHs in NSCs spans $\mathcal{R}(0)\sim{1-2}$ Gpc$^{-3}$ yr$^{-1}$. The rate associated with NSCs also peaks at $z\sim{2.5-2.8}$, reaching values $\sim{4-5}$ times higher than at $z=0$ (Fig.~\ref{fig:rate}).
The merger rate density of BBHs in both GCs and NSCs is very mildly affected by stellar metallicity, while the merger rate of isolated BBHs changes wildly with the metallicity spread $\sigma{}_{\rm Z}$. BBHs in YSCs behave in an intermediate way between isolated BBHs and dynamical BBHs in GCs/NSCs. Enforcing or not the lower BH mass gap affects the merger rate density of all channels, from isolated BBHs to dynamical BBHs: the rapid core-collapse SN model (which prevents the formation of BHs with mass $<5$ M$_\odot$) produces a higher merger rate by $\sim{40-60}\%$ with respect to the delayed model (where we can have BHs with mass $3-5$ M$_\odot$). This happens because a higher minimum BH mass results in shorter delay times.
Our star cluster models grow a population of Nth generation (Ng) mergers. The local merger rate density of Ng BBHs is $\sim{0.8-2.5}$, $\sim{0.5-1.0}$, and $\sim{0.1-0.9}$ Gpc$^{-3}$ yr$^{-1}$ in GCs, NSCs and YSCs, respectively (Fig.~\ref{fig:rate_ng}). The total merger rate density of Ng BBHs in the local Universe, obtained by summing up these three channels, ranges from $\sim{1}$ to $\sim{4}$ Gpc$^{-3}$ yr$^{-1}$ and is mostly sensitive to the spin parameter: we find higher (lower) values for our low-spin model with $\sigma_\chi=0.01$ (fiducial model with $\sigma_\chi=0.1$).
The primary BH mass function has a high-mass tail, extending up to several hundred M$_\odot$ in the three dynamical channels, because of hierarchical mergers. The primary BH mass function evolves with redshift in both GCs and NSCs: lower mass BH mergers become less and less common at high redshift ($z\geq{}1$), because they are associated with longer delay times (Fig.~\ref{fig:masses}). In contrast, the primary BH mass function does not significantly evolve with redshift in isolated BBHs, in agreement with previous studies \citep{mapelli2019,santoliquido2020}. This happens because binary evolution processes (e.g., common envelope) generate tight systems of low-mass BHs with short delay time \citep[see, e.g.,][]{mapelli2019}. This difference has exciting implications for third-generation ground-based GW detectors:
if Einstein Telescope and Cosmic Explorer will find a heavier BH mass function at higher redshift, this will be a signature that most BBH mergers have a dynamical origin;
vice versa, isolated BBHs dominate the observed population if the mass function does not evolve with redshift.
The resulting primary BH mass function we obtain by combining our four channels according to their merger rate is similar to the {\sc power law + peak} model used by the LIGO--Virgo--KAGRA collaboration \citep{abbottO3popandrate}. The main difference is that our models predict more BHs with mass $\sim{20}$ M$_\odot$ with respect to the {\sc power law + peak} model (Fig.~\ref{fig:mix_mass}). In our mass function, low-mass BHs are
mostly given by isolated BBHs, YSCS and NSCs, while the high-mass tail ($\geq{}50$ M$_\odot$) is mostly due to Ng BHs in GCs and NSCs.
The distribution of effective ($\chi_{\rm eff}$) and precessing ($\chi_{\rm p}$) spins we obtain by combining our four channels
strongly depend on $\sigma_\chi$. For $\sigma_\chi=0.01$ (low-spin models), 1g BBHs have vanishingly small values of $\chi_{\rm eff}$. Hence, the effective spin distribution has a sharp peak at zero, surrounded by two symmetric broad wings due to Ng BBHs. The distribution of $\chi_{\rm p}$ has two peaks: a primary peak, very narrow, at $\chi_{\rm p}=0$ and a secondary peak at $\chi_{\rm p}\approx{0.7}$, because of Ng BBHs. In contrast, for $\sigma_\chi=0.1$, the distribution of effective spins becomes asymmetric: it peaks at $\chi_{\rm eff}\approx{0.2}$, because of isolated BBHs. In this case, the distribution of precessing spins has three peaks: a sharp primary peak at $\chi_{\rm p}=0$ because of isolated BBHs, a broader secondary peak at $\chi_{\rm p}=0.1-0.2$, because of 1g dynamical BBHs, and a third, lower peak at $\chi_{\rm p}\sim{0.7}$ because of Ng BBHs (Fig.~\ref{fig:mix_spin}).
We calculated the posterior probability distribution of the mixing fractions associated with our four channels, by running a Bayesian hierarchical analysis with posterior samples from GWTC-2 \citep{abbottO3a}. The resulting mixing fractions indicate that at least two formation channels
are likely at work to produce the observed BBH population (Fig.~\ref{fig:mixing_frac}). However, the mixing fraction of each channel varies wildly from a model to another, being extremely sensitive to the metallicity spread $\sigma_{\rm Z}$, the spin parameter $\sigma_{\chi}$, the common envelope parameter $\alpha$ and the core-collapse SN model.
Our models still suffer from large uncertainties (e.g. on the formation history of star clusters), but {\sc fastcluster} and {\sc cosmo$\mathcal{R}$ate} are extremely flexible and fast tool, and we can use them to probe the relevant parameter space.
\section*{Acknowledgements}
MM, YB and FS acknowledge financial support from the European Research Council for the ERC Consolidator grant DEMOBLACK, under contract no. 770017. MCA and MM acknowledge financial support from the Austrian National Science Foundation through FWF stand-alone grant P31154-N27. MAS acknowledges financial support from the Alexander von Humboldt Foundation for the research program ``The evolution of black holes from stellar to galactic scales'', the Volkswagen Foundation Trilateral Partnership through project No. I/97778, and the Deutsche Forschungsgemeinschaft (DFG) -- Project-ID 138713538 -- SFB 881.
\section*{Data availability}
The data underlying this article will be shared on reasonable request to the corresponding authors.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 76
|
{"url":"https:\/\/web2.0calc.com\/questions\/trig-conversion","text":"+0\n\n# Trig Conversion\n\n0\n39\n4\n+992\n\nHi friends,\n\nI hopefully have only 2 more questions for today. Both these questions come from the same problem. Please explain to me..please...\n\nThe 2nd step in the sum goes like this:\n\n$${Sin135Cosx - Cos135Sinx} \\over{Sinx}$$\n\nbecomes:\n\n$${Sin45Cosx + Cos45Sinx \\over Sinx}$$\n\nHow can this be?...sorry if I'm wasting someone's time with this one......I tried to make the font a little bigger...unsuccessful.\n\nMar 3, 2023\n\n#1\n+118455\n+1\n\nSeems like you need to study up on the unit circle.\n\nHere is a play page.\n\nIts great if you already understand what you are looking at but maybe not so great if you don't.\n\nhttps:\/\/www.mathsisfun.com\/geometry\/unit-circle.html\n\nThis video looks really good.\u00a0 \u00a0I have only watch about half but she sweemed to be explaining it really well.\n\nhttps:\/\/online.clickview.com.au\/exchange\/channels\/3010412\/maths-with-heather-davis\/playlists\/3014610\/trigonometry\/videos\/928592\/lesson-12-angles-of-any-magnitude\n\nsummarizing.\n\nIf an angle is drawn on the unit circle as shown in the video, the hypotenuse of any triangle will be 1.\n\nSo\n\ncos of the angle is the x value on the circumference of the unit circle\u00a0 \u00a0 \u00a0So cos will be positive in the 1st and 4th quadrants\n\nsin of the angle is the y value on the circumference of the unit circle\u00a0 \u00a0 \u00a0 So sin will be positive in the\u00a0 1st and 2nd quadrants\n\ntan of the angle is the y value divided by the x value on the circumference of the unit circle\u00a0 \u00a0 \u00a0 So tan will be positive in the\u00a0 1st and 3rd quadrants\n\nA sentance to help you with this is\u00a0 \u00a0 \u00a0ALL\u00a0 \u00a0 Stations\u00a0 \u00a0To\u00a0 \u00a0 Central\n\nAll trig values are positive for angles less than 90 degrees, that is angles in the 1st quadrant\n\nSin (but not cos or tan) will be positive for all angles in the 2nd quadrant,\u00a0 that is angles between 90 and 180 degrees\n\nTan is positive in the 3rd quadrant\n\nCos is positive in the 4th quadrant.\n\nMar 4, 2023\n#2\n+118455\n+1\n\nso\n\nsin135 = +sin (180-135)\u00a0 \u00a0 Note:\u00a0\u00a0it\u00a0is in the secend quadrant fo it is positive\n\nsin 135 = sin 45\n\nMaybe you can try the rest yourself.\n\nMar 4, 2023\n#3\n+992\n+1\n\nmy my my ...goodness!!!..off course!\n\ngosh, you know what Melody...I have looked at for example Sin 140, as Sin 140 DEG,\u00a0and therefore could not understand how on earth 135\u00a0DEGREES could be the same as 45\u00a0DEGREES\n\nbut yes, I see my error. Getting the actual values of the different Sin values, equate to the same answer....ALL RIGHTIE!!!...\n\nJust a last thing...The different quads where the trig functions are either positive or negative, I fully understand, I use the phrase\u00a0CAST, starting with quad 4...C=Cos, A=All, S=Sin and T=Tan...(obviously..)\n\nThank you Melody for going through the trouble of finding those urls, I have not watched them, since I understand all of this, I just made a stupid error for mistaking the value for an angle...\n\nI'm sending a BIG BUNCH of flowers!!\n\njuriemagic \u00a0Mar 4, 2023\nedited by juriemagic \u00a0Mar 4, 2023\n#4\n+118455\n0\n\nI am glad I could help :)\n\nMelody \u00a0Mar 4, 2023","date":"2023-03-23 18:01:41","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\": 0.8387308120727539, \"perplexity\": 2135.2017683360286}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296945182.12\/warc\/CC-MAIN-20230323163125-20230323193125-00046.warc.gz\"}"}
| null | null |
\section{Introduction}
Increasing vehicle population worldwide has led to traffic congestion, delays and unpredictable travel times. Existing adaptive traffic control systems (ATCS) are widely used to select traffic signal plans based on a real-time model of the traffic. Currently traffic engineers design signal plans and hand-tune the necessary parameters for different traffic situations to ensure optimal flow of traffic. It is, however, difficult to manually design signal plans for a large number of traffic scenarios, especially in the event of traffic anomalies such as accidents, road construction and traffic violations such as illegal parking/stopping. Traditional control algorithms work under predefined rules and suffer from high inductive bias. Typically, such bias is supplied by domain experts and require manual intervention.
Our goal is to control traffic signals in an adaptive manner by providing a variable green signal time, based on instantaneous traffic flow without the need for hand engineered signal plans. We use the asynchronous property of A3C\cite{mnih2016asynchronous} algorithm to build single and multi-agent reinforcement learning systems. These algorithms learn the local and global optimal behavior of multiple intersections such that overall traffic flow is maximized. A global reward function is defined to incorporate cooperation among multiple agents to optimize global policies.
The remainder of the paper is organized into the following sections: Section \ref{refs} presents related work. Section \ref{setting} introduces the simulation environment along with state space, action space and reward functions. In Section \ref{algorithm} we discuss the different algorithms evaluated for single and multi-agent network architectures. Section \ref{result} describes in detail the results and conclusions.
\section{Related Work} \label{refs}
The work done in \cite{liang2018deep} provides insights to a basic state encoding of the traffic signal. Studies such as \cite{genders2016using} and \cite{gao2017adaptive} use Deep Q-Networks with experience replay\cite{adam2011experience} and target network\cite{lillicrap2015continuous} to address sampling\cite{zhai2016deep} and non-stationary target\cite{foerster2016learning} issues respectively. When dealing with a large and continuous state space, methods like Monte Carlo\cite{guo2014deep}, DQN\cite{mnih2013playing}, DDQN\cite{tsividis2017human} are not efficient and face convergence and sampling problems \cite{pan2018organizing} \cite{arulkumaran2017deep}. Thus, in this work, we use Asynchronous Advantage Actor Critic-A3C \cite{mnih2016asynchronous}, a variant of actor-critic algorithms that uses the advantage function \cite{peters2008natural} to address convergence issues, in addition to two separate actor and critic networks to handle sampling and non-stationary target problems.
\cite{tan1993multi} and \cite{littman1994markov} explain the dynamics of multiple agents working together and independently in a cooperative environment to maximize their rewards. The team work by the players in \cite{OpenAI_dota} allow for collaboration among agents only by introducing a global reward function to induce team spirit among the operating players without an external linkage. \cite{zhang2018fully} proposes the use of actor-critic method in a multi-agent setup.
We also use the asynchronous property of A3C to deploy a single model onto multiple intersections, thereby reducing training time.
\section{Setting} \label{setting}
In this section we describe the simulation environment, state space, action space and reward functions used to train the RL (reinforcement learning) agent.
\subsection{Environment}
We use Aimsun Next\cite{barcelo2005microscopic}\cite{casas2010traffic} to simulate realistic traffic conditions and train RL agents. The real traffic measurements (number of vehicles and traffic density) calculated by the approach described in \cite{yeshwanth2017estimation} is used to enhance the simulations. The collected data follows \textit{Weibull distribution}\cite{lai2003modified}. Fig.\ref{fig:aimsun rendering} shows a sample simulation in Aimsun Next.
\begin{figure}
\centering
\subfloat{\includegraphics[width=1\textwidth]{Aimsun_1.eps}}
\caption{Aimsun 2D and 3D simulation of traffic from the city of Bengaluru, India. Different composition of vehicles, including 2-wheelers and 3-wheelers are considered to create realistic simulations for accurate results.}
\label{fig:aimsun rendering}
\end{figure}
\begin{figure}
\centering
\subfloat{\includegraphics[width=.8\textwidth]{network_.eps}}
\caption{Policy and Value Network used in A3C for both single and multi-agent setup. For the single agent the input is a single queue and for multi-agent setup it is a matrix, both containing density and phase information of single and multiple intersections respectively. Single agent utilises a Linear Layer + LSTM network while a multi-agent setup uses a Conv-LSTM architecture for its input. }
\label{fig:a3c network used for single and multi}
\end{figure}
\subsection{State Space}
The state space of an agent controlling its intersection is \textbf{[encoded density, encoded phase]} where encoded density is the density of different approaches towards the intersection and encoded phase contains the intersection phase information. Density is defined as \textit{average number of vehicles per kilometer across all approaches of an intersection} and intersection phase information \textit{is the information encoding that represents sections of an intersection that are active (green) at an instance of time}. State encoding along with one-hot encoded phase information for a single agent is explained in Table \ref{Phase encoding}.
The densities are normalized with the highest density among the approaches using eq.\ref{eq:density} and eq.\ref{eq:normalised}. This is done to help the model learn which section (or approach) has maximum amount of traffic and accordingly allocate necessary green time to it.
\begin{figure}
\centering
\subfloat[][Delay Graph]{\includegraphics[width=.5\textwidth]{normalised_delay.eps}}
\subfloat[][Density Graph]{\includegraphics[width=00.5\textwidth]{normalised_density.eps}}
\caption{ Comparison graphs for neighbouring intersection state space representation in a multi-agent scenario against Fixed Signal Timing (FST) of 60, 90 and 120 seconds. Delay and density on y-axis are the average delay (seconds per kilometer) and average density (vehicles per kilometer) across the four intersections respectively. X-axis is the simulation time in seconds.}
\label{fig:state graphs}
\end{figure}
\begin{table}
\caption{Single Agent State Space Encoding}
\label{Phase encoding}
\centering
\begin{tabular}{llllll}
\toprule
\cmidrule(r){1-2}
Phase & Section Upper & Section Right & Section Lower & Section Left
& Encoded Phase \\
\midrule
1 & Green &Red &Red &Red &[1,0,0,0] \\
2 &Red &Green &Red &Red & [0,1,0,0] \\
3 &Red &Red &Green &Red & [0,0,1,0] \\
4 &Red &Red &Red &Green & [0,0,0,1] \\
\bottomrule
\\
& & & & &Encoded Density \\
\midrule
Density &$D_1$ &$D_2$ &$D_3$ &$D_4$ &[$D_1$,$D_2$,$D_3$,$D_4$]\\
\bottomrule
\end{tabular}
\end{table}
\begin{equation}
D_{max} = max(D_1 ,D_2 ,D_3 ,D_4)
\label{eq:density}
\end{equation}
\begin{equation}
Encoded Density= \frac{[D_1,D_2,D_3,D_4]}{D_{max}}
\label{eq:normalised}
\end{equation}
\begin{figure}
\centering
\subfloat{\includegraphics[width=.5\textwidth]{State_space_4_junctions_edited.eps}}
\caption{Intersection representation in Aimsun of a small part of the city of Bengaluru in India for Neighbouring State Representation where each intersection is represented with an alphabet. }
\label{fig:Neighbouring intersections in Aimsun}
\end{figure}
Figure \ref{fig:Neighbouring intersections in Aimsun} shows the road network considered to train RL agents in our experiments. In the figure, intersections D, B and E are neighbours of C. Their densities and phase information are considered as the state space for intersection C. We represent the state space comprising of density and phase encoding of the neighboring intersections as a matrix shown below for training multi-agent networks.
$$ C= \left(\begin{matrix}C_{d1}&C_{d2}&C_{d3}&C_{d4}&C_{p1}&C_{p2}&C_{p3}&C_{p4}\\B_{d1}&B_{d2}&B_{d3}&B_{d4}&B_{p1}&B_{p2}&B_{p3}&B_{p4}\\E_{d1}&E_{d2}&E_{d3}&E_{d4}&E_{p1}&E_{p2}&E_{p3}&E_{p4}\\D_{d1}&D_{d2}&D_{d3}&D_{d4}&D_{p1}&D_{p2}&D_{p3}&D_{p4}\end{matrix}\right) $$
where $C_{d1} - C_{d4}$ represents the densities and $C_{p1} - C_{p4}$ represents the encoded phase information of the respective intersection. Similarly for intersection D, density and phase of intersections A and E are encoded in its state space.
Fig.\ref{fig:state graphs} depicts the impact of using density and phase information of neighbouring intersections as the state space on average delay time\footnote{Delay time for a section in the intersection is calculated as average delay time per vehicle per kilometre. In Aimsun, it is the difference between the expected travel time (the time it would take to traverse the system under ideal conditions) and the travel time. For a section, it is calculated as the average of all vehicles and then converted into time per kilometre and does not include the time spent in a virtual queue.} and average density of intersections C, D, E and F as referred to in Fig. \ref{fig:Neighbouring intersections in Aimsun}.
\subsection{Action Space}
A discrete action space is considered with a fixed range from 20 to 60 seconds of green signal time in increments of +5. The model selects four different actions in the form of four different green signal times for each approach of the intersection. The action space is shown below-
Green signal time - [20, 25, 30, 35, 40, 45, 50, 55, 60]
We experimented with various action spaces such as green signal times from 0 to 60 seconds with increments of 5 and 10 seconds respectively and 20 to 120 seconds with increments of 20, yet none of them were effective in reducing the traffic congestion. The sum of the four green times obtained is always less than or equal to a certain threshold of the intersection which can be set manually. In the experiments performed, 240 seconds is used as the threshold.
\subsection{Reward Function}
The task of designing reward functions is important in solving a problem using reinforcement learning and therefore comprehensive testing for multiple reward functions is necessary. For our agents to not be solely governed by the reward functions\cite{bush2005modeling}\cite{glascher2010states}, we chose density (also a parameter in state-space) as our reward function parameter. The reward is estimated by considering densities of two different time stamps $D_{T}$ and $D_{T+1}$. We experimented with the following reward functions-
\begin{itemize}
\setlength\itemsep{.12em}
\item \textbf{Product of Densities-} The product of densities over a particular time $T_{t}$ is calculated given in eq. \ref{eq:product}.
\begin{align}
D_{t} = \prod_{i=1}^{4} D^{i}_{t}
\label{eq:product}
\end{align}
where $D^{i}$ is the density of the i$^{th}$ section of the intersection and $i$ ranges from 1 to 4
\item \textbf{Sum of Densities-} The sum of densities of each of the four sections of the intersection is calculated in eq.\ref{eq:sum}
\begin{align}
D_{t} = \sum_{i=1}^{4} D^{i}_{t}
\label{eq:sum}
\end{align}
\item \textbf{Sum of Squares of Densities-} The sum of squares of density of each section is calculated in eq. \ref{eq:square}
\begin{align}
D_{t} = \sum_{i=1}^{4} {D^{i}_{t}}^{2}
\label{eq:square}
\end{align}
\end{itemize}
The final reward is calculated with the difference across two time-stamps as mentioned in equation \ref{eq:reward}. If the obtained reward is positive, a +1 is supplied, else if the reward is negative, a value of -1 is supplied. The clipping of rewards eliminate the possibility of sparse rewards, making the agent end goal oriented which in turn enhances the learning\cite{henderson2018deep}\cite{mnih2015human}.
\begin{equation}
Reward = D_{t}-D_{t+1}
\label{eq:reward}
\end{equation}
Among the reward functions mentioned above, the product of densities resulted in reducing the overall delay at the intersection. This is primarily because in a dense/chaotic environment, such as on Indian roads, the traffic is stochastic in nature. During peak hours the traffic densities are nearly the same and the product of densities will amplify the effect. The product as a reward function accommodates even the minute fluctuations in densities across four approaches in contrast to the sum and sum of squares of densities.
\section{Adaptive Signal Control using Single and Multi-agent architectures} \label{algorithm}
In the following section we elaborate on the techniques evaluated and showcase their results using average delay time and average density (in case of single agent) of intersections by taking FST (Fixed Signal Timing) as our baseline. In this method a fixed measure of green signal time is assigned to every approach of the intersection during each cycle in a simple round-robin fashion. The fixed signal timing of 60, 90 and 120 seconds is chosen for baseline evaluation of our RL based model since they are most frequently used as the traffic signal timings in Indian cities.
\ref{single agent} discusses single agent operating on an intersection. The remaining sections describe multi-agent behaviour to optimize traffic flow across multiple intersections.
\subsection{Single Agent - Single intersection}\label{single agent}
We evaluated a single agent controlling a single intersection by using real traffic data. The aim is to reduce the overall delay time at this intersection. The hyperparameters that yielded the best results in terms of lowest delay per vehicle at the intersection and the overall average density of intersection are given below -
\begin{itemize}
\item Network - LSTM
\item State Space - [Density | Phase] for each intersection
\item Action Space - Green time: 20 seconds to 60 seconds with increments of 5 seconds
\item Reward Function - Product of densities
\end{itemize}
\subsection{Independent RL (InRL)}
We utilize the notion of Independent RL wherein agents are trained collectively in the same environment and while doing so the agents formulate their own strategies of cooperation, without actually communicating with each other. This produces self competitive behaviour among the agents to maximize their individual cumulative reward while working together with other agents without any direct communication in place. The setup was imitated individually on four different agents operating across four different intersections which include A, B, C and D as shown in Fig.\ref{fig:Neighbouring intersections in Aimsun}.
\begin{figure}
\centering
\subfloat{\includegraphics[width=.8\textwidth]{A3C.eps}}
\caption{Schematic Representation of A3C being used for multiple intersections. Each intersection is an asynchronous worker running in its individual environment. Each worker in the figure outputs: Current State, Reward, Action and Next State where State is Density and Action is the Green Signal Time.}
\label{fig:a3c multi intersection}
\end{figure}
\subsection{Asynchronous Instances as intersections}
We now employ the asynchronous attribute of A3C where multiple instances of the environment are used for training, by deploying four different traffic intersections as four distinctive environments as illustrated in Fig.\ref{fig:a3c multi intersection}. Instead of having different agents for four different intersections, only one agent trained on the experience buffer of all four intersections decides the signal timings at their respective intersections.
\subsection{Coordination via Global Reward Function}
As we already know the performance of RL agents is governed by the reward function, we propose a model of global reward function for multi-agent setting. This global reward function fuses the agent's individual reward and also takes into account a global reward which is the average of rewards of all agents and is given in eq.\ref{eq:gloabl_reward}
\begin{align}
Reward_{Global} = \frac{\sum_{i=1}^{4}{Reward_i}}{4}
\label{eq:gloabl_reward}
\end{align}
\begin{align}
Reward_{Final} = 0.5* (Reward_{Global} + Reward_{Individual intersection})
\label{eq:ireward}
\end{align}
The individual agent then takes the average value of the two reward functions - its own and global reward. The final reward is clipped to +1 if positive and -1 if negative. Currently we consider equal weight for both global as well as individual reward and intend to experiment with different ratios in the future.
\begin{figure}
\centering
\subfloat[][Delay Time comparison]{\includegraphics[width=.5\textwidth]{single_delay.eps}}
\subfloat[][Density comparison]{\includegraphics[width=.5\textwidth]{single_density.eps}}
\caption{ Plots of single agent showing comparisons between FST 60, 120 and our RL agent. On y-axis is average delay time of the intersection in seconds per kilometer in (a) and average density of the intersection in vehicle per kilometer in (b). Both figures have simulation time in seconds on x-axis.}
\label{fig:single agent graphs}
\end{figure}
\begin{figure}
\centering
\subfloat[][A1:Async A3C\\ B1:Async A3C+Coordination]{\includegraphics[width=.5\textwidth]{asyn_delay.eps}}
\subfloat[][A2:InRL\\ B2:InRL+Coordination]{\includegraphics[width=.5\textwidth]{inrl_delay.eps}}
\caption{Delay graphs with coordination i.e global reward function introduced in A3C and Independent RL (InRL) on four intersections. Y-axis is average delay time of the four intersections in seconds per kilometer and x-axis represents the simulation time in seconds. }
\label{fig:Multi agent graphs}
\end{figure}
\section{Results and Conclusions} \label{result}
We describe the results of single and multi-agent RL experiments in this section.
Fig.\ref{fig:single agent graphs} compares the average delay time and average density achieved by a single agent with FST 60, FST 90 and FST 120 at an intersection. The RL agent reduces the average density and delay by 33\% compared to FST 60 and 66.67\% compared to FST 120 respectively. This suggests that the drop in delay is because the RL agent, unlike fixed signal timing, considers real-time traffic density and dynamically adapts to the changing conditions.
The results of experiments with the multi-agent setup, shown in Fig.\ref{fig:Multi agent graphs} lead to similar conclusions where RL performs better than FST in reducing the average delay time. It is seen in Fig.\ref{fig:Multi agent graphs}(a) that although the Async A3C algorithm is successful in lowering the delay at intersections initially, it starts performing similar to FST 60 after ~\textasciitilde1500 simulation seconds. With the introduction of the global reward function, the delay across intersections dropped by 38\%. The results confirm that coordination among agents along with Async A3C results in lower congestion across neighbouring intersections.
Fig.\ref{fig:Multi agent graphs}(b) shows a drop in average delay in a multi-agent setup when agents are trained independent of state information of other agents. InRL achieved better results than Async A3C because each agent focused on reducing the delay at its intersection and learnt strategies to coordinate with each other to optimize globally. It is observed that while using this technique the green time at intersections synced in a manner to achieve a coordinated green. On introducing a global reward function in the setup, a drop in delay is seen initially and at ~\textasciitilde1500 simulation seconds the average delay is similar to FST 60.
In this paper, we evaluated multiple methods to optimize green signal time at single intersections and across multiple intersections using deep reinforcement learning. Using A3C \cite{mnih2016asynchronous} algorithm and multi-agent state space we introduced information sharing and coordination between intersections, further reducing the delay time. The experiments demonstrated that training different agents independently in a multi-agent setting led to self competitive behaviour among them, thus working better than fixed signal timing. Finally, introducing a global reward function in both Async A3C and InRL methods induced team work and cooperation among agents and their respective intersections for a smoother traffic flow. Our RL agents consider real-time traffic densities and show better performance than fixed signal timing in chaotic traffic conditions. Directions for future work include introducing traffic anomalies and lane prioritization.
\medskip
\section{Acknowledgements}
The authors acknowledge the support provided by Professor Shalabh Bhatnagar and Jayanth Prakash Kulkarni, Department of Computer Science and Automation (CSA), Indian Institute of Science Bengaluru. Their expertise and recommendations assisted the research conducted on adaptive signal control with Reinforcement Learning techniques, especially the use of Asynchronous Advantage Actor Critic (A3C) algorithm and Aimsun simulation environment.
\bibliographystyle{unsrt}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,864
|
Stefan P. Müller (* 1. Juli 1978 in Olten; heimatberechtigt in Luzern) ist ein Schweizer Politiker (SVP), Nidwaldner Landrat und Gemeinderat von Emmetten.
Leben und Politik
Müller arbeitet als Kundenberater in einer Grossbank in Luzern. Er wohnt in Emmetten, ist verheiratet und hat drei Kinder.
Seine erste politische Tätigkeit übernahm Müller als Mitglied der Finanzkommission der Gemeinde Emmetten. Diese Aufgabe legte er nieder, als er im Mai 2016 in den Gemeinderat von Emmetten gewählt wurde. Im folgenden Jahr übernahm er die Präsidentschaft der SVP-Ortspartei. Seit Juni 2018 vertritt er die SVP im Nidwaldner Landrat. Sein oberstes Ziel ist nach eigenen Angaben «die Werte und Kulturen von Emmetten und Nidwalden pflegen und aufrecht erhalten».
Weblinks
Stefan P. Müller auf der Website des Kantons Nidwalden
Stefan P. Müller auf der Website der Gemeinde Emmetten
Stefan P. Müller auf der Website der SVP Nidwalden
Landrat (Nidwalden, Person)
Politiker (21. Jahrhundert)
SVP-Mitglied
Schweizer
Geboren 1978
Mann
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,532
|
Q: XSD Cannot generate classes but works on similar file I converted a lot of xsd to C# in the past, but today I'm facing a new error message, for me: "cannot generate classes because no top-level elements with complex type were found."
I have this problem on 2 files. I read a lot of posts about this, and they helped me to solve at least 1 of the 2 problems I have.
The file I fixed was:
<?xml version="1.0" encoding="utf-8"?>
<xs:schema xmlns:mg="urn:crif-messagegateway:2006-08-23" xmlns:xs="http://www.w3.org/2001/XMLSchema" targetNamespace="urn:crif-messagegateway:2006-08-23" elementFormDefault="qualified" attributeFormDefault="unqualified">
<xs:element name="MGRequest" type="xs:string"/>
<xs:element name="MGResponse" type="xs:string"/>
</xs:schema>
and I edited it to:
<?xml version="1.0" encoding="UTF-8"?>
<xs:schema xmlns:mg="urn:crif-messagegateway:2006-08-23" xmlns:xs="http://www.w3.org/2001/XMLSchema" targetNamespace="urn:crif-messagegateway:2006-08-23" elementFormDefault="qualified" attributeFormDefault="unqualified">
<xs:element name="MGRequest">
<xs:complexType>
<xs:simpleContent>
<xs:extension base="xs:string" />
</xs:simpleContent>
</xs:complexType>
</xs:element>
<xs:element name="MGResponse">
<xs:complexType>
<xs:simpleContent>
<xs:extension base="xs:string" />
</xs:simpleContent>
</xs:complexType>
</xs:element>
</xs:schema>
Now, I'm trying also to convert the following xsd file (quite similar to the previous fixed), but it xsd.exe throws the error "cannot generate classes because no top-level elements with complex type were found.". Which is the problem? What are the differences between the working file above?
<?xml version="1.0" encoding="UTF-8"?>
<xs:schema xmlns:mg="urn:crif-messagegateway:2006-08-23" xmlns:xs="http://www.w3.org/2001/XMLSchema" targetNamespace="urn:crif-messagegateway:2006-08-23" elementFormDefault="qualified" attributeFormDefault="unqualified">
<xs:element name="MGRequest">
<xs:complexType>
<xs:sequence>
<xs:any namespace="##other"/>
</xs:sequence>
</xs:complexType>
</xs:element>
<xs:element name="MGResponse">
<xs:complexType>
<xs:sequence>
<xs:any namespace="##other"/>
</xs:sequence>
</xs:complexType>
</xs:element>
</xs:schema>
I tried also with xsd2code, but what I get is just an empty class.
A: I guess it doesn't see the need to create a class to wrap a primitive type, if you run it through Liquid XML Objects you get this
namespace LiquidTechnologies.GeneratedLx.Mg
{
#region Elements
/// <summary>A class representing the root XSD element MGRequest@urn:crif-messagegateway:2006-08-23</summary>
/// <XsdPath>schema:schema.xsd/element:MGRequest</XsdPath>
/// <XsdFile>file://sandbox/schema.xsd</XsdFile>
/// <XsdLocation>3:5-3:52</XsdLocation>
[LxSimpleElementDefinition("MGRequest", "urn:crif-messagegateway:2006-08-23", ElementScopeType.GlobalElement)]
public partial class MGRequestElm
{
/// <summary>Holds the <see cref="System.String" /> (xs:http://www.w3.org/2001/XMLSchema:string) value of the element</summary>
/// <XsdPath>schema:schema.xsd/element:MGRequest</XsdPath>
/// <XsdFile>file://sandbox/schema.xsd</XsdFile>
/// <XsdLocation>3:5-3:52</XsdLocation>
[LxValue(LxValueType.Value, XsdType.XsdString)]
public System.String Value { get; set; } = "";
}
/// <summary>A class representing the root XSD element MGResponse@urn:crif-messagegateway:2006-08-23</summary>
/// <XsdPath>schema:schema.xsd/element:MGResponse</XsdPath>
/// <XsdFile>file://sandbox/schema.xsd</XsdFile>
/// <XsdLocation>4:5-4:53</XsdLocation>
[LxSimpleElementDefinition("MGResponse", "urn:crif-messagegateway:2006-08-23", ElementScopeType.GlobalElement)]
public partial class MGResponseElm
{
/// <summary>Holds the <see cref="System.String" /> (xs:http://www.w3.org/2001/XMLSchema:string) value of the element</summary>
/// <XsdPath>schema:schema.xsd/element:MGResponse</XsdPath>
/// <XsdFile>file://sandbox/schema.xsd</XsdFile>
/// <XsdLocation>4:5-4:53</XsdLocation>
[LxValue(LxValueType.Value, XsdType.XsdString)]
public System.String Value { get; set; } = "";
}
#endregion
}
Liquid XML Objects is free for small xsd's.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,640
|
export ICE_JAR="/Users/bkonowitz/code/MGM/images/mgm-ice/1.0.0/bin/java/lib/mgm-ice.jar"
export ICE_MAIN="/Users/bkonowitz/code/MGM/mgm-ice/main"
export ICE_WEBAPP="/Users/bkonowitz/servers/apache-tomcat-7.0.23/webapps/mgm-ice"
export RAVEN_BUILDIR="/Users/bkonowitz/code/MGM"
function depice() {
case $1 in
-jar)
echo "copying mgm-ice.jar"
cp $ICE_JAR $ICE_WEBAPP/WEB-INF/lib
tom-down
sleep 3
tom-up
tft
;;
-pages)
echo "copying pages..."
cp -r $ICE_MAIN/web/pages $ICE_WEBAPP
;;
-scripts)
echo "copying scripts..."
cp -r $ICE_MAIN/web/scripts $ICE_WEBAPP
;;
-web)
echo "copying pages..."
cp -r $ICE_MAIN/web/pages $ICE_WEBAPP
echo "copying scripts..."
cp -r $ICE_MAIN/web/scripts $ICE_WEBAPP
;;
-build)
echo "setting environment ..."
pushd $RAVEN_BUILDIR
pwd
. ./setenv.sh
echo 'building....'
./build_component.sh -target rebuild mgm-ice
popd
echo 'out of build dir...'
;&
-all)
echo "copying jar..."
cp -r $ICE_MAIN/web/pages $ICE_WEBAPP
echo "copying scripts..."
cp -r $ICE_MAIN/web/scripts $ICE_WEBAPP
echo "copying pages..."
cp $ICE_JAR $ICE_WEBAPP/WEB-INF/lib
tom-down
sleep 3
tom-up
tft
;;
*)
echo "Unrecognized operation. Valid operations are: -jar -pages -scripts -web -all"
;;
esac
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,517
|
« Million drivers face losing license under EU diabetes directive
FEMA'S use of term 'federal family' for government expands under Obama »
Investigators Probe White House Role in Massive Energy Loan
[Republicans in Congress] expressed concern that so much federal money was headed to a company whose key investor was George Kaiser, an Oklahoma billionaire who raised more than $50,000 for Obama's 2008 presidential campaign.
President Barack Obama during a tour of the Solyndra solar panel company May 26, 2010 in Fremont, California. (Paul Chinn/Pool via Getty Images) Click to enlarge
Matthew Mosk and Ronnie Greene
ABC News and iWATCH News
House investigators said they have uncovered evidence that White House officials became personally involved in an Energy Department review of a hot-button $535 million loan guarantee to the now-failed California solar company Solyndra.
The allegation surfaced in a letter House Energy Committee Chairman Fred Upton (R-Mich.) sent to the White House Thursday night, saying he planned to accelerate efforts to understand an investment deal that may have left taxpayers out half a billion dollars.
"We have learned from our investigation that White House officials monitored Solyndra's application and communicated with [Department of Energy] and Office of Management and Budget officials during the course of their review," the letter says.
Thursday's letter, which calls on the White House to turn over correspondence between administration officials, Solyndra and its investors, presents the most pointed suggestion that the White House had direct involvement in the financing.
"How did this company, without maybe the best economic plan, all of a sudden get to the head of the line?" Upton told ABC News in an interview this week. "We want to know who made this decision … and we're not going to stop until we get those answers."…
READ: Solyndra Collapse a 'Waste' of Half a Billion By Obama, GOP Critics Say
READ at CPI: White House Had Role in Federal Benefit for Failed Solar Company, House Investigators Say
The complete article, with videos, is at ABC News.
H/T Legal Insurrection where Professor Jacobson compares Solyndra to the Teapot Dome Scandal of the 1920s.
Update: Obama Bundler George Kaiser Made Multiple Visits to White House in Months Prior to $535 Million Loan Guarantee to Solyndra
September 3rd, 2011 | Tags: 'green' technology, "green jobs", bankruptcy, campaign contributions, cronyism, Energy Secretary Stephen Chu, George Kaiser, George Kaiser Family Foundation, green energy, House Energy and Commerce Committee, Office of Management and Budget, Rep. Fred Upton, Solyndra Inc., The Center for Public Integrity | Category: Barack Obama, Department of Energy, Environment, Oklahoma
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 330
|
{"url":"https:\/\/docs.exabyte.io\/entities-general\/actions\/set-entity-index\/","text":"In order to change the order of entities inside an ordered set, navigate to the set first and then use Set Index option in the actions drop-down menu. The user is presented with the following dialog to set a new index.","date":"2019-08-22 00:23:49","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\": 0.4957598149776459, \"perplexity\": 722.8259308375556}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027316555.4\/warc\/CC-MAIN-20190822000659-20190822022659-00117.warc.gz\"}"}
| null | null |
\section{Introduction}
One of the big surprises of the RHIC experimental program has been
the large elliptic flow \cite{Ackermann:2000tr,Voloshin:2008dg},
which, contrary to SPS energies, agreed more or less with predictions
from ideal hydrodynamics \cite{Teaney:2000cw,Kolb:2003dz}. This
surprising agreement let to the conjecture that the matter at RHIC
is a strongly coupled, nearly perfect fluid, with very small shear
viscosity. Indeed using the AdS-CFT correspondence, is was shown that
a large class of strongly coupled theories seem to have a universal
minimal shear viscosity of $\eta/s=1/4\pi$ \cite{Son:2004iv}.
Meanwhile, more refined calculations
based on relativistic viscous hydrodynamic \cite{Luzum,Song} seem to indicate
that a
finite but small value for the shear viscosity is required in order
to reproduce the $p_{t}$ dependence of the measured transverse flow.
On the theoretical side very little is known about the shear viscosity
of high temperature QCD. Perturbative calculations lead to a considerably
larger value than the conjectured lower bound. Extracting a value
for the shear viscosity from Lattice QCD (LQCD), on the other hand,
requires analytic continuation to real time, thus leading to substantial
uncertainties \cite{meyer}. It was also found that the elliptic flow of the
observed
hadrons scales with the number of quarks \cite{Voloshin:2008dg}
as predicted by a quark coalescence picture of
hadronization \cite{Voloshin:2002wa}.
This in turn implies that the scaled $v_{2}$ may be interpreted as
that of the quarks prior to hadronization, and we will use this interpretation
in the following where we always refer to the elliptic anisotropy of
quarks.
In this contribution we want to entertain an entirely different view
and interpretation of the observed elliptic flow. First, we note that
Lattice QCD results \cite{Cheng:2008zh} suggest a
quasi-particle picture, at least for the quarks. Both flavor-off-diagonal
susceptibilities \cite{Koch:2005vg} and higher order baryon number
susceptibilities \cite{Ejiri:2005wq} are consistent with vanishing
correlations for temperatures right above the transition, $T\gtrsim1.2T_{c}$.
Estimating the strength of correlations in the gluon sector is not
so straightforward, due to the lack of any additional quantum numbers,
such as flavor, which one can use to study correlations. Therefore,
let us conjecture, that gluons behave like quasi-particles as well.
In this case we have a single-particle description of the QGP right
above $T_{c}$.
Next we need to address the equation of state above $T_{c}$, where LQCD
finds the pressure to be considerably $(\approx15\%)$ below that of
a free gas of massless quarks and gluons. Since LQCD calculations
are carried out in the grand-canonical ensemble, i.e. at fixed chemical
potential rather than particle number, a reduction of the pressure
in a single-particle picture implies a \emph{repulsive}, density dependent
single-particle potential, i.e,
\begin{equation}
p\sim\int d^{3}p\,\exp\left[-\frac{E_{0}+U}{T}\right]<\int
d^{3}p\,\exp\left[-\frac{E_{0}}{T}\right]\sim p_{0}
\end{equation}
for $U>0$. Here $p_{0}$ denotes the pressure of a free, non-interacting
gas of partons.
The presence of a \emph{repulsive} single-particle potential has interesting
consequences, especially for the elliptic anisotropy, $v_{2}.$ Given
the almond shaped initial distribution of matter in the transverse
plane in a semi-central heavy-ion collision, the (negative) gradient
of the potential, and thus the force, is larger in the in-plane
than in the out-of-plane direction. As a consequence the momentum
kick due to the repulsive single-particle potential is larger in plane
than out of plane, resulting in a deformation of the momentum distribution
in qualitative agreement with the observed elliptic anisotropy, $v_{2}(p_{t})$
(see also \cite{cassing}).
We note that this effect does \emph{not} require a short mean free
path or, equivalently, a small viscosity.
Besides the positive $v_{2}$ the single-particle dynamics
leads to two additional, qualitative predictions. First, the elliptic
anisotropy should vanish for large transverse momenta, since the
additional momentum kick due to the potential becomes negligible.
Thus, contrary to ideal hydrodynamics we predict a maximum of the transverse
momentum dependent elliptic anisotropy, $v_{2}(p_{t})$, which is observed
in experiment. At large $p_t$, of course, $v_{2}$ will be dominated by
the attenuation of fast partons in the matter, and it needs to be
determined at what momentum this transition will take place \cite{Liao:2009zg}.
Second, since for a given temperature the momentum distribution for
heavy quarks is titled towards higher momenta, $T\sim\frac{p^{2}}{m}$,
the resulting elliptic anisotropy for heavy quarks should be considerably
smaller than that for light quarks.
\section{Schematic Model}
In order to study the qualitative features of the proposed single-particle
dynamics in a transparent fashion let us start with a simple
schematic model. If we ignore collisions among the partons, the dynamics
of the phasespace distribution follows as Vlasov equation. To allow
for an analytical treatment of the expansion, we assume a Gaussian
distribution in configuration space and non-relativistic kinematics,
resulting in a Gaussian momentum space distribution,
\begin{equation}
f(x,y,v_{x,}v_{y},t=0)=\frac{N}{2\pi^{2}\sigma_{x}\sigma_{y}(T/m)}\exp(\frac{-x^
{2}}{\sigma_{x}^{2}})\exp(-\frac{mv_{x}^{2}}{2T})\,\exp(\frac{-y^{2}}{\sigma_{y}
^{2}})\exp(-\frac{mv_{y}^{2}}{2T}).
\end{equation}
Here, $v_{x}$, $v_{y}$ are the velocities and $\sigma_{x}$, $\sigma_{y}$
denote the widths of the distribution in the transverse $x$, $y$
direction, respectively. Assuming, for simplicity, that the single
particle potential $U$ is proportional to the density of the light
degrees of freedom, \[
U\left(\vec{x},t\right)=g\,\rho\left(\vec{x},t\right)=g\int d\vec{v}\, f(\vec{x},\vec{v},t),\]
the Vlasov equation leads to the following expression for the velocity
space distribution, $n\left(\vec{v}\right)$,
\begin{equation}
\frac{\partial}{\partial t}n(\vec{v})=\frac{g}{m}\int
d^{2}x\,\vec{\nabla}_{v}f(\vec{x},\vec{v},t)\vec{\nabla}_{x}\rho(\vec{x},
t)\label{eq:transport}
\end{equation}
which can be solved analytically under the assumption that
the time dependence of the density follows free streaming, i.e. we
ignore the effect of the potential on the density distribution. A
fully consistent solution will require a numerical treatment, which
will be briefly discussed below. Since the heavy quarks are rare, we ignore
their contribution to the potential, and propagate them in the potential
generated by the light degrees of freedom.
To leading order in the initial spatial eccentricity
$\epsilon\equiv\frac{\sigma_{y}^{2}-\sigma_{x}^{2}}{\sigma_{x}^{2}+\sigma_{y}^{2
}}$
we then obtain the following result for the elliptic anisotropy of
the light quarks
\begin{equation}
v_{2,light}\left(u\right)=\frac{1}{2}\epsilon\frac{U_{0}}{T}\frac{1}{u^{2}}\left
[1-\exp\left(-u^{2}\right)\left(u^{2}+1\right)\right]
\end{equation}
which only depends on the kinetic energy $u=\frac{mv^{2}}{2T}$. Here,
$U_{0}=U(\vec{r}=0,t=0)$ is the initial strength of the potential
at the center. We note that $v_{2}$ is (a) proportional to the initial
spatial eccentricity, (b) proportional to the initial transverse density
(via $U_{0}$) and (c) a function of the kinetic energy only. This
is precisely what is seen in experiment.
At a given velocity the heavy quark elliptic anisotropy, $v_{2,heavy}$,
is related to that of the light quarks by $v_{2,heavy}(v)=\frac{1+\gamma}{2}\,
v_{2.light}(v)$.
Both, $v_{2,light}$ and $v_{2,heavy}$ are plotted as a function of the kinetic
energy in the left panel of Fig.\ref{fig:supress} for an
initial temperature of $T=300\,{\rm MeV}$. As expected $v_{2}$ exhibits
a maximum, the position of which depends on the choice of initial
temperature and is shifted to larger values of the kinetic energy
for heavy quarks. As a result the momentum averaged, or integrated
anisotropy, $\bar{v}_{2}$, for the heavy quarks is much smaller. This
is shown in left panel of Fig.\ref{fig:supress}, where we plot the
ratio of heavy over light quark $v_{2}$ as a function of the ratio
of the quark masses, $\gamma=m_{light}/M_{heavy}$ . Thus, a unique
prediction of the single-particle dynamics is that the integrate $v_{2}$
for heavy quarks, specifically the $J/\Psi$ should be considerably
smaller than that for the pions, contrary to hydrodynamics where they should be
about equal. \vspace{.2cm}
\begin{figure}[ht]
\centering
\includegraphics[width=0.4\textwidth]{schem_v2_comp.eps}
\hspace{.5cm}
\includegraphics[width=0.4\textwidth]{supress_2.eps}
\caption{Left panel: Schematic model results for the elliptic anisotropy as
a function of the kinetic energy, $v_{2}(E_{kinetic})$ for light
quarks (full line) and heavy quarks (dashed line). The dotted line represents
the kinetic energy spectrum for both heavy and light quarks. Right
panel: Ratio of integrated $v_{2}$ for heavy over light quarks as
a function of the ratio of heavy and light quark mass.}
\label{fig:supress}
\end{figure}
\section{More realistic (transport) model}
A realistic treatment of the proposed single-particle dynamics requires
relativistic kinematics as well as a fully consistent
treatment of the Vlasov (Boltzmann) equation. In order to ensure energy
momentum conservation it is best to start from an effective energy
functional which is tuned to reproduce the Lattice equation of
state and from which one derives the single-particle Vlasov equation.
The results from such an exercise (for details see \cite{Koch_soon}) is shown in
Fig. \ref{Flo:transport}. Again we see the same features as in the
schematic model, and we find that the intergrated $v_{2}$ for heavy
quarks is considerably smaller than that of light quarks.
\begin{figure}
\begin{centering}
\includegraphics[width=0.4\textwidth]{v2_transp.eps}
\par\end{centering}
\caption{Result for elliptic anisotropy from transport calculation without
parton
scattering.}
\label{Flo:transport}
\end{figure}
\section{Conclusions}
In this contribution we have shown that the qualitative features of
the observed elliptic anisotropy can be obtained from single-particle
dynamics motivated by recent Lattice QCD results. In
this description
there is no need for a very short mean free path or, equivalently,
small viscosities. This model can be tested in experiment by measuring
the elliptic anisotropy of heavy quarkonium, which is predicted to
be considerably smaller than that of pions, in contradistinction
to the predictions from hydrodynamics.
\section*{Acknowledgments}
This work is supported by the Director, Office of Energy Research,
Office of High Energy and Nuclear Physics, Divisions of Nuclear Physics,
of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,552
|
\section{Introduction}
Asymptotically flat spacetimes present a surprisingly rich symmetry structure, with infinite dimensional symmetry enhancements appearing when we consider gauge transformations that act non-trivially on the asymptotic data. As elucidated by Strominger~\cite{Strominger:2017zoo} these give rise to non-trivial Ward identities manifested as soft theorems for $\mathcal{S}$-matrix elements.
However in repeating the standard derivation of the `Ward identity $=$ soft theorem' equivalence, one can't help but take pause at the number of steps needed to demonstrate such an elegant final result. Nominally we would proceed roughly as follows
\begin{enumerate}
\item Take a Cauchy slice of my asymptotically flat spacetime.
\item Push it down to past null infinity to define my $in$-state.
\item Push it up to future null infinity to define my $out$-state.
\item Evaluate the canonical charges on each slice.
\item Use an antipodal matching condition (across infinitely time-separated spheres!) at $\mathcal{I}^+_-$ and $\mathcal{I}^-_+$ to equate these two expressions.
\item Use the constraint equations and integrate by parts to write this charge as a flux.
\item Insert this as an operator equation in $\mathcal{S}$-matrix elements and see that
\begin{equation}
\langle out|(Q^+_{S}+Q^+_H)\mathcal{S}-\mathcal{S}(Q^-_{S}+Q^-_H)|in\rangle=0
\end{equation}
arises from the soft theorem by plugging the perturbative mode expansion into $Q_S$, which is linear in the gauge field.
\end{enumerate}
This summary does not include the additional historical step where we would ignore massive contributions in the early derivations~\cite{He:2014laa,Kapec:2014opa,He:2014cra}, to make our lives easier when integrating by parts along $u$ and $v$ in step 6.
These Ward identities play a crucial role in motivating the celestial holographic dictionary, whereby $\mathcal{S}$-matrix elements in a boost basis transform like correlators in a CFT living on the celestial sphere~\cite{Pasterski:2016qvg,Pasterski:2017kqt,Pasterski:2017ylz}. It thus strongly behooves us to have a clean bulk interpretation of the manipulations we are doing in 2D. The aim of this note is to show that we can skip to what is roughly step 6 at least for the purposes of celestial Ward identities if we combine the Noether's second theorem-based approach of~\cite{Avery:2015rga}, with an extension of the left/right Hilbert space picture of~\cite{Crawley:2021ivb} into the bulk, and the extrapolate dictionary of~\cite{Pasterski:2021dqe}.
The punchline of our story is that for an appropriate choice of 3-surface $\Sigma_C$ the celestial charge $\mathcal{J}_C$ in the radially quantized CCFT is related to the surface charge of the corresponding gauge theory in the bulk
\begin{equation}\label{result}
\boxed{~~\mathcal{J}_C(\lambda)=\int_{\partial\Sigma_C}\star k(\lambda).~~}
\end{equation}
Namely we can jump directly from the canonical charges in the bulk to the symmetry generators in the boundary in a manner similar to what one would do in AdS/CFT. The price we pay is that the surface $\Sigma_C$ is not a Cauchy slice, but this is not surprising considering that radial evolution is Rindler time evolution which is spacelike outside the Rindler wedge. In terms of the standard presentation, the bulk operator corresponding to the left hand side is normally constructed from the soft charge. By deriving~\eqref{result} we get an immediate route to it. As we will explore in more detail below this should help explain why the soft and hard splitting of the flux operators separately obey the symmetry algebra relations.
\section{Surface Charges for Gauge Symmetries}
In this section we will follow the conventions of~\cite{Avery:2015rga}. We will focus on the electromagnetic case for simplicity and use units where $e=1$.
Recall that in gauge theory we can write the canonical charges for a gauge transformation $\lambda$ in terms of a 2-form $k(\lambda)$. For a Cauchy slice $\Sigma$ we have
\begin{equation}
Q(\lambda)=\int_{\partial\Sigma} \star k .
\end{equation}
For the case of electromagnetism the charge generating the gauge transformation
\begin{equation}
A_\mu\mapsto A_\mu+\partial_\mu\lambda
\end{equation}
corresponds to
\begin{equation}\label{klF}
k=\lambda F,~~~F=\frac{1}{2}F_{\mu\nu}dx^\mu\wedge dx^\mu
\end{equation}
so that on a constant time Cauchy slice this reduces to
\begin{equation}
Q(\lambda)=\lim_{r\rightarrow\infty} \int_{S^2} d^2z\sqrt{\gamma}~\lambda(r^2 F_{ru})
\end{equation}
where we use the coordinates
\begin{equation}
X^\mu=(u+r,r\frac{z+{\bar z}}{1+z{\bar z}},ir\frac{{\bar z}-z}{1+z{\bar z}},r\frac{1-z{\bar z}}{1+z{\bar z}}).
\end{equation}
Large gauge transformations are precisely those for which $\lambda$ has non-compact support so that it survives this large $r$ limit. In the case where $\lambda$ is constant we get back to the global $U(1)$ symmetry generator, but in the context of studying asymptotic symmetries we allow $\lambda(z,{\bar z})$.
On the equations of motion we have
\begin{equation}\label{ktoj}
\partial_\nu k^{\nu\mu}\overset{w}{=}{\cal j}^\mu~~~~ \Rightarrow ~~~~\int_{\partial R} \star k=\int_R \star {\cal j}
\end{equation}
where ${\cal j}$ is the Noether current one gets from path integral Ward identities. For our electromagnetic example
\begin{equation}\label{jem}
{\cal j}_\mu=F_{\nu\mu}\partial^\nu\lambda+\lambda J^M_\mu
\end{equation}
where the last term is the matter current. The right hand equality in~\eqref{ktoj} is the starting point for talking about charge conservation. Namely, if we consider two Cauchy slices with the same boundary, the charges would be the same.
The essence of the asymptotic symmetry Ward identities is to write
\begin{equation}
\partial M=\Sigma_1\cup\Sigma_2
\end{equation}
and try to take $\Sigma_1\mapsto \mathcal I^-$ and $\Sigma_2\mapsto \mathcal I^+$. The series of steps needed in our outline is to justify that there is no additional contribution near spatial infinity $i^0$. Namely
\begin{equation}
\partial \mathcal{I}^{\pm}= \mathcal{I}^{\pm}_+\cup \mathcal{I}^{\pm}_-
\end{equation}
and until we have the antipodal matching condition it is not obvious that $k|_{\mathcal{I}^+_-}=k|_{\mathcal{I}^-_+}$. Once we do have this matching condition, the Ward identity has a very nice interpretation in terms of a relationship between time integrals of radiation called memory effects and the external scattering states~\cite{Pasterski:2015zua}. By~\eqref{jem} the charge at future null infinity is
\begin{equation}\label{Qp}
Q^+=\lim\limits_{r\rightarrow\infty} r^2\int du\sqrt{\gamma} d^2 z n^\mu(F_{\nu\mu}\partial^\nu\lambda+\lambda J^M_\mu)
\end{equation}
where the normal vector of null infinity is
\begin{equation}
n^\mu\partial_\mu =\partial_u-\frac{1}{2}\partial_r.
\end{equation}
For the standard radiative phase space only the $F_{uA}$ term contributes. Moreover, each of these terms is $\mathcal{O}(1)$ in the large-$r$ limit since raising the sphere metric gives us an $r^{-2}$ in the first term. The first term in~\eqref{Qp} is the soft charge $Q^+_S$ while the second term is the massless contribution to the hard charge $Q^+_H$. Step 6 in our procedure is precisely the fact that the weak equality in~\eqref{ktoj} amounts to using the constraint equations. For the full Ward identity we need to repeat this procedure for past null infinity and take into account additional contributions near $i^{\pm}$ for the massive scatterers~\cite{Campiglia:2015qka,Kapec:2015ena}. A nice recent review can be found in~\cite{Miller:2021hty}. This procedure generalizes to any other asymptotic gauge symmetry. The appearance of a term in the charge that is linear in the gauge field is symptomatic of the fact that the majority of these asymptotic symmetries are spontaneously broken by our choice of vacuum. The Ward identities are manifested in $\mathcal{S}$-matrix elements as soft theorems precisely because the term linear in the gauge field reduces to a $u$-integral that picks out a zero frequency mode of the gauge field~\cite{He:2014cra}.
\section{Celestial Symmetries}
The main motivation for the celestial CFT program is that these Ward identities for 4D asymptotic symmetries can be naturally recast as 2D ward identities for a CFT living on the celestial sphere. This is most striking for the case of the subleading soft graviton~\cite{Cachazo:2014fwa,Kapec:2014opa}, from which we get a candidate 2D stress tensor~\cite{Kapec:2016jld} if we go to a basis of boost-eigenstates. For the U(1) gauge theory case at hand we can define the following current~\cite{Strominger:2013lka,He:2015zea,Nande:2017dba}
\begin{equation}\label{jisqs}
j=j^++j^-,~~~~j^+=Q_S^+\left(\lambda=\frac{1}{z-w}\right)
\end{equation}
where $j^-$ is a corresponding antipodally-related $\mathcal{I}^-$ contribution. Because of a helicity redundancy~\cite{He:2014cra} (corresponding to a shadow relation for $\Delta=1$ photons~\cite{ss}) the $F_{uz}$ and $F_{u{\bar z}}$ contributions to~\eqref{Qp} are equal, and we can just as well write
\begin{equation}\label{jfuz}
j^+=-{4\pi}\int du F_{uz}.
\end{equation}
Viewing $\mathcal{S}$-matrix elements as celestial correlators, Weinberg's soft photon theorem~\cite{Weinberg:1965nx} turns into the relation
\begin{equation}\label{jzward}
\langle j(z)\mathcal{O}_1(z_1,{\bar z}_1)...\mathcal{O}_n(z_n,{\bar z}_n)\rangle=\sum_i\frac{Q_i}{z-z_i}\langle\mathcal{O}_1(z_1,{\bar z}_1)...\mathcal{O}_n(z_n,{\bar z}_n)\rangle.
\end{equation}
We can generate an arbitrary holomorphic $U(1)$ transformation $\lambda$ using the operator
\begin{equation}\label{Jcvj}
\mathcal{J}_C(\lambda)=\oint_C\frac{dz}{2\pi i }\lambda j
\end{equation}
so that
\begin{equation}
\langle \mathcal{J}_C(z)\mathcal{O}_1(z_1,{\bar z}_1)...\mathcal{O}_n(z_n,{\bar z}_n)\rangle=\sum_{i\in C}{Q_i}\lambda(z_i)\langle\mathcal{O}_1(z_1,{\bar z}_1)...\mathcal{O}_n(z_n,{\bar z}_n)\rangle.
\end{equation}
In particular we get the global transformations when we take $\lambda=1$ and the contour $C$ to enclose all of the charged scattering states. In that case the left hand side vanishes, which we can see by shrinking the contour in the direction where there are no operator insertions. We see this from point of view of the soft charge by the fact that $Q_S$ vanishes when $\partial_\mu\lambda=0$.
We emphasize that we want $\lambda$ holomorphic (not just meromorphic) within the contour $C$. As pointed out in~\cite{He:2015zea} if $\lambda$ has poles within this contour then we will get extra insertions of $j$ coming from the residues. It is in this sense that $\mathcal{J}$ is more properly thought of as the full charge rather than just the soft contribution. Within $\mathcal{S}$-matrix insertions this was appreciated in~\cite{He:2015zea}, however their starting point was at the level of the soft theorems.
In what follows we will be able to give an extra dimension (pun intended!) to the statements about contours dividing in and out particles we see in~\cite{He:2015zea} and the cover art of~\cite{Strominger:2017zoo}.
\section{Slicing the Sphere and the Bulk}
Here we will show that from the canonical charge perspective it is clear that the quantity $\mathcal{J}_C$ is the object generating gauge transformations on the operators within the contour. The essence of what we will be doing in this section is to consider what happens when we do the alternate splitting
\begin{equation}
\partial M=\Sigma_1\cup\Sigma_2 = \Sigma_L\cup \Sigma_R
\end{equation}
where the contour $C$ that divides the left from the right patch of the celestial sphere determines the division between left and right patches of our boundary. The downside is that these left and right surfaces are not Cauchy slices. However, this splitting is actually natural from the point of view of celestial CFT.
The main reason for going to the celestial basis is so that we can better take advantage of the asymptotic symmetries that should constrain scattering. The structure of the celestial OPEs derived from collinear limits of scattering~\cite{Fan:2019emx,Fotopoulos:2019tpe,Pate:2019lpp,Fotopoulos:2019vac} strongly suggest that we can treat them as if we are doing radial quantization. In doing so one finds an even more surprisingly rich symmetry structure~\cite{Guevara:2021abz,Strominger:2021lvk,Himwich:2021dau}.
Recent attempts to formalize the out states and the state-operator correspondence in this language can be found in~\cite{Fan:2021isc,Crawley:2021ivb}. Here we will be particularly interested in the left/right splitting of states suggested in~\cite{Crawley:2021ivb}.
\begin{figure}[ht]
\centering
\includegraphics[trim=150 100 100 50,clip, width=15cm]{splitscri.pdf}
\caption{Splitting the spacetime across the hyperplane $X^3=0$ splits the celestial sphere along the equator. By the extrapolate dictionary, we can prepare massless scattering states by inserting operators along generators of null infinity. The north patch of the celestial sphere corresponds to the regions shaded in purple, related across spatial infinity by an antipodal matching.}
\label{slice}
\end{figure}
Radial evolution in celestial CFT corresponds to Rindler evolution in the bulk.\footnote{In~\cite{Pasterski:2022lsl} we examine of some implications for CCFT that naturally arise when considering the perspective of a Rindler observer. Here the direction of the particle's acceleration sets the foliation of both the sphere and the bulk.} If we slice our spacetime along the hypersurface $X^3=0$, the celestial sphere will get cut along its equator $|z|=1$. This is illustrated by the blue surface in figure~\ref{slice}. In a reference frame boosted along the $X^3$ direction this slice will tilt towards one or the other Rindler horizons and the corresponding locus on the celestial sphere will shrink. Global conformal transformations of the Riemann sphere map circles to circles. Global conformal transformations of the celestial sphere are induced by Lorentz transformations in the bulk. Under such Lorentz transformations we can go from our $X^3=0$ hyperplane to any other hyperplane with spacelike normal. As such we expect to be able to map any circle on the celestial sphere to a hyperplane with spacelike normal in the bulk.\footnote{We've seen the special role of `celestial circles' popping up in Mellin-transformed momentum space when examining the 4-pt kinematics for massless scattering~\cite{Pasterski:2017ylz}. We will show that this hyperplane trick has a much more general utility when studying constraints from translation invariance on celestial amplitudes in~\cite{MP}.} While more intricate contours $C$ will not be boost images of this canonical contour, one can consider propagating into the bulk the deformation one would do on the celestial sphere to get from a circle to $C$.
What is nice about this hypersurface is that it respects our ability to quotient along the generators of null infinity. When we go down by one codimension to a Cauchy slice, we have a boundary that is the full sphere at infinity. While this is a codimension-1 cut of the boundary if we want to look at $u$-evolution, we have to do a lot of gymnastics to construct an object that looks like a radial quantization `time'-slice. Namely the steps outlined in the introduction. Instead our hypersurface $\Sigma_C$ extends along the $u$-direction and is also codimension-1 on the celestial sphere.
Now the boundary of this slice is still in the asymptotic region and that is all that we will need to evaluate the charges $Q^C$. Recalling that
\begin{equation}
\star(dx^\mu \wedge dx^\nu)=\sqrt{-g}g^{\mu\kappa}g^{\nu \lambda}\epsilon_{\kappa\lambda\rho\sigma}\frac{1}{2!}dx^\rho\wedge dx^\sigma
\end{equation}
and using the radiative falloffs
\begin{equation}
F_{ur}\sim\mathcal{O}(r^{-2}),~~~F_{z{\bar z}}\sim\mathcal{O}(1),~~~F_{uA}\sim\mathcal{O}(1),~~~F_{rA}\sim\mathcal{O}(r^{-2})
\end{equation}
the integral of the 2-form~\eqref{klF} evaluates to
\begin{equation}\label{QFC}
Q^C(\lambda)=\int_{\partial \Sigma_C}\star (\lambda F)= -i \int du \oint_C dx^A \lambda \epsilon_{AC}\gamma^{CB}F_{ B u} + (\mathcal{I}^-~\rm{contribution})
\end{equation}
(where $\epsilon_{AC}$ contains a factor of $\sqrt{\gamma}$) which we recognize as a contour integral of the memory operator, and which is determined by Weinberg's soft photon theorem. While in the standard presentation the `weak' equality we need to go from from the charge as a surface integral to the flux presentation involves a careful choice of constraint equations, here it is just an application of Stokes' theorem changing the contour integral to an area integral
\begin{equation}
\partial^A (\lambda F_{Au})=(\partial^A \lambda) F_{Au}+ \lambda \partial^A F_{Au}.
\end{equation}
The first term is soft charge while the second term can be evaluated by projecting Maxwell's equations to the boundary
\begin{equation}
\nabla^\nu F_{\nu\mu}=j_\mu.
\end{equation}
Indeed if we are careful about this we actually get both the massless and massive contributions to the hard charge in one fell swoop! Namely~\cite{Himwich:2019dug}
\begin{equation}
\gamma^{z{\bar z}}(\partial_{\bar z} F_{zu}+\partial_z F_{{\bar z} u})=r^2j_u+(\partial_u-\partial_r)(r^2F_{ru})
\end{equation}
so that the surviving term in the large-$r$ limit involves an area integral of
\begin{equation}
\int du J^M_u=\int r^2j_u+(r^2F_{ru})\Big|_{\mathcal{I}^+_\pm}.
\end{equation}
We don't need to add $i^\pm$ in by hand. To get the full Ward identity for a scattering process we can consider closing the contour to the left or right for the two ways we can cap off the contour on the celestial sphere.
Our claim is that the charge for the 2D CCFT, defined on a contour $C$, is equal to the canonical charge evaluated on a hypersurface $\Sigma_C$ that lifts this contour into the bulk
\begin{equation}\label{JCl}
\mathcal{J}_C(\lambda)=\int_{\partial\Sigma_C}\star k(\lambda).
\end{equation}
We get the necessary antipodal part (which distinguishes memory effects from a scattering process from plane waves passing through) by nature of the cut through the bulk. While we don't appear to directly need the antipodal matching to get to the Ward identity, it does come in handy if we want to avoid the bulk. Namely, as illustrated by the purple surface in figure~\eqref{slice}, we can hug the boundary and jump antipodally across spacelike infinity if we want to consider the flux expressions for the charges. In the case where we think of this surface as hugging the boundary the full Ward identity of in + out is trivially equal to left + right because we have just split the same boundary $S^3$ differently.
While~\eqref{JCl} is the full charge for this hypersurface, we can isolate a soft contribution by considering a non-constant $\lambda$ and picking a contour that avoids any hard particles. Combining~\eqref{jfuz} and~\eqref{Jcvj}, if we pick a point $w$ where there are no other operator insertions and use Cauchy's integral theorem we see that
\begin{equation}\label{Jtoj}
\mathcal{J}_{C_w}\left(\lambda=\frac{1}{z-w}\right)=\oint_{C_w}\frac{dz}{2\pi i}\frac{j(z)}{z-w}=j(w).
\end{equation}
Because the soft charge~\eqref{jisqs} only reduced to~\eqref{jfuz} due to the shadow relation, we need to be careful about restricting to a finite region. However, as we will discuss below, once we introduce the magnetic dual charge we can indeed isolate the single helicity soft contribution in this manner. From the point of view of the bulk, the hypersurface $\Sigma_{C_w}$ corresponding to a small circle around the point $w$ should be a highly boosted image of the $X^3=0$ cut such that it approaches the Rindler horizon for an observer accelerating towards $w$. From the point of view of the extrapolate dictionary we can nominally do this for the massless charge case. Going to boost eigenstates should help us localize the massive charge contributions towards their respective reference directions.
By contrast only the hard part of the charge survives whenever $\lambda$ is holomorphic inside the contour. As a cute check, the case $\lambda=1$ is indeed an application of Gauss's law for a weird hypersurface
\begin{equation}\label{kqc}
\int_{\partial\Sigma_C}\star k(\lambda)=\sum_{i\in C}Q_i.
\end{equation}
Namely, rather than the constant `ordinary-time' slices we are used to this is a constant Rindler time cut of the bulk (but in the region outside the Rindler wedge). This makes sense when we consider how the hypersurface cuts the worldlines of the particles. The in versus out charges are weighted by a relative sign coming from the orientation of our slice when pushed towards the boundary. From this perspective, the soft factor can be viewed as a Greens function solving~\eqref{kqc}.
\section{Discussion}
We will close with some comments about why this perspective is useful. First, it is very satisfying to see that the charges in the dual theory correspond to the same symmetries in the bulk. This is closer to how we talk about the relation between bulk and boundary charges in AdS/CFT. While the choice of hypersurface is unnatural from the point of view of canonical gauge theory, it is natural from the point of view of the celestial dictionary since radial evolution on the sphere corresponds to Rindler evolution in the bulk. We would like to view the fact that this is clearly not a Cauchy slice as a positive in that it should help inform how to handle issues that might arise in formulating CCFT as a radially quantized theory.
Second, we see that we are able to work directly with the field strength in our discussion of the large gauge charge. We only needed the gauge field $A$ to give meaning to the transformation $\lambda$. If we work in dual variables there is a corresponding magnetic charge and soft theorem~\cite{Strominger:2015bla}. For the electromagnetic case this is literally
\begin{equation}
\int_{\partial\Sigma} \star k\mapsto \int_{\partial\Sigma} k.
\end{equation}
For a spacelike Cauchy slice the integrand of the surface charge changes from the electric to the magnetic field
\begin{equation}
r^2 F_{ru}\mapsto iF_{z{\bar z}}
\end{equation}
for the dual charge. At null infinity the Hodge dual acts diagonally on the $F_{uz}$ and $F_{u{\bar z}}$ components since these correspond to self dual and anti-self dual solutions, respectively. The curl in~\eqref{QFC} gets replaced by a divergence and we can pick out the single helicity photons by taking the combinations
\begin{equation}
Q_{(A)SD}=\frac{1}{2}(Q\mp i\widetilde{Q}).
\end{equation}
This alleviates the need to use the shadow relation for the leading soft photon to select out the positive helicity soft theorem.
Finally, recent investigations~\cite{Donnay:2021wrk} have shown that one can split the BMS fluxes into a hard and a soft part which each obey the expected symmetry algebra provided you use the appropriate bracket~\cite{Barnich:2011mi,Campiglia:2020qvc,Compere:2020lrt}. These involve non-trivial quadratic corrections to the soft generators which one might have anticipated from loop corrections~\cite{He:2017fsb} or from the point of view of the descendancy relations for the celestial diamonds~\cite{Pasterski:2021fjn,Pasterski:2021dqe}. See also~\cite{Freidel:2021qpz,Freidel:2021dfs}. The picture where we go from the full charge to the soft charge using a reference point~\eqref{Jtoj} away from any hard particles might help explain why we expect such a splitting. Getting the soft charge in this manner is clearly sensitive to any discussions of boundary terms in the definition of the charge, as well as the (in)ability to isolate hard particles to distinct points on the celestial sphere~\cite{Freidel:2021ytz}.
In sum, we see that taking `radial = Rindler' evolution seriously gives us a nice way to jump directly to the celestial Ward identities from the bulk. The perspective brings us closer the standard holographic relation between between bulk and boundary symmetry generators, informs the sensitivity of these Ward identities to our choice of boundary terms in the charge, and warns us of issues with radial time ordering that we might run into when we consider the fully interacting theory.
\section*{Acknowledgements}
Many thanks to Laurent Freidel, Sebastian Mizera, and Herman Verlinde for interesting discussions. My research is supported by the Sam B. Treiman Fellowship at the Princeton Center for Theoretical Science.
\bibliographystyle{utphys}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,981
|
\section{Introduction}\label{sec:intro}
\begin{figure}[t]
\centering
\includegraphics[width=1.0\textwidth]{figs/fig1.pdf}
\captionsetup{font=small}
\caption{
Overview of the method and
latent space representation.
We start from an original image $I_o$ that can be edited $t(\cdot)$ in various ways: its feature extraction $f(t(I_o))$ spawns the shaded region in the embedding space.
The edited versions should be recoverable by nearest neighbor search on quantized representations.
In the regular (non-active) case, $f(I_o)$ is quantized by the index as \includegraphics[width=0.5em]{figs/assets/c1.pdf}.
When the image is edited, $t(I_o)$ switches cells and the closest neighbor returned by the index is the wrong one \includegraphics[width=0.5em]{figs/assets/c2.pdf}.
In active indexing: $I_o$ is modified in an imperceptible way to generate $I^\star$ such that $f(I^\star)$ is further away from the boundary.
When edited copies $f(t(I^\star))$ are queried, retrieval errors are significantly reduced.
\label{fig:fig1}}
\end{figure}
The traceability of images on a media sharing platform is a challenge:
they are widely used, easily edited and disseminated both inside and outside the platform.
In this paper, we tackle the corresponding task of Image Copy Detection (ICD), \textit{i.e}.\@ finding whether an image already exists in the database; and if so, give back its identifier.
ICD methods power reverse search engines, photography service providers checking copyrights, or media platforms moderating and tracking down malicious content (\textit{e.g}.\@ Microsoft's \cite{photodna} or Apple's \cite{neuralhash}).
Image identification systems have to be robust to identify images that are edited (cropping, colorimetric change, JPEG compression \ldots) after their release~\citep{douze2021disc, wang2022benchmark}.
The common approach for content-based image retrieval reduces images to high dimensional vectors, referred to as \emph{representations}.
Early representations used for retrieval were hand-crafted features such as color histograms~\citep{swain1991color}, GIST~\citep{oliva2001modeling}, or Fisher Vectors~\citep{perronnin2010large}.
As of now, a large body of work on self-supervised learning focuses on producing discriminative representations with deep neural networks, which has inspired recent ICD systems.
In fact, \emph{all} submissions to the NeurIPS2021 Image Similarity challenge~\citep{papakipos2022results} exploit neural networks.
They are trained to provide invariance to potential image transformations, akin to data augmentation in self-supervised learning.
Scalability is another key requirement of image similarity search: searching must be fast on large-scale databases, which exhaustive vector comparisons cannot do.
In practice, ICD engines leverage approximate neighbor search algorithms, that trade search accuracy against scalability.
Approximate similarity search algorithms speed up the search by \emph{not} computing the exact distance between all representations in the dataset~\citep{johnson2019faiss, guo2020scann}.
First they lower the number of scored items by partitioning the representation space, and evaluate the distances of only a few subsets.
Second, they reduce the computational cost of similarity evaluation with quantization or binarization.
These mechanisms make indexing methods subject to the curse of dimensionality.
In particular, in high-dimensional spaces, vector representations lie close to boundaries of the partition~\citep{bohm2001searching}.
Since edited versions of an original image have noisy vector representations, they sometimes fall into different subsets or are not quantized the same way by the index.
All in all, it makes approximate similarity search very sensitive to perturbations of the edited image representations, which causes images to evade detection.
In this paper, we introduce a method that improves similarity search on large databases, provided that the platform or photo provider can modify the images before their release (see Fig.~\ref{fig:fig1}).
We put the popular saying ``attack is the best form of defense'' into practice by applying image perturbations and drawing inspiration from adversarial attacks.
Indeed, representations produced with neural networks are subject to \emph{adversarial examples}~\citep{szegedy2013intriguing}:
small perturbations of the input image can lead to very different vector representations, making it possible to
create adversarial queries that fool image retrieval systems~\citep{liu2019whos, tolias2019targeted,dolhansky2020adversarial}.
In contrast, we modify an image to make it \emph{more} indexing friendly.
With minimal changes in the image domain, the image representation is pushed towards the center of the indexing partition, rising the odds that edited versions will remain in the same subset.
This property is obtained by minimizing an indexation loss by gradient descent back to the image pixels, like for adversarial examples.
For indexing structures based on product quantization~\citep{jegou2010pq}, this strategy amounts to pushing the representation closer to its quantized codeword, in which case the indexation loss is simply measured by the reconstruction error.
Since the image quality is an important constraint here, the perturbation is shaped by perceptual filters to remain invisible to the human eye.
Our contributions are:
\begin{itemize}[leftmargin=1cm,itemsep=0cm,topsep=-0.1cm]
\item a new approach to improve ICD and retrieval, when images can be changed before release;
\item an adversarial image optimization scheme that adds minimal perceptual perturbations to images in order to reduce reconstruction errors, and improve vector representation for indexing;
\item experimental evidence that the method significantly improves index performance.
\end{itemize}
\section{Related Work} \label{sec:related}
\paragraph{Image watermarking}
hides a message in a host image, such that it can be reliably decoded even if the host image is edited.
Early methods directly embed the watermark signal in the spatial or transform domain like DCT or DWT~\citep{cox2007digital}.
Recently, deep-learning based methods jointly train an encoder and a decoder to learn how to watermark images~\citep{zhu2018hidden,ahmadi2020redmark,zhang2020udh}.
Watermarking is an alternative technology for ICD.
Our method bridges indexing and watermarking, where the image is modified before publication.
Regarding retrieval performance, active indexing is more robust than watermarking.
Indeed, the embedded signal reinforces the structure naturally present in the original image, whereas watermarking has to hide a large secret keyed signal independent of the original feature.
App. \ref{sec:watermarking} provides a more thorough discussion and experimental results comparing indexing and watermarking.
\paragraph{Active fingerprint} is more related to our work.
As far as we know, this concept was invented by Voloshynovskiy \textit{et al}.\@ ~\cite{voloshynovskiy2012active}.
They consider that the image $I\in \mathbb{R}^N$ is mapped to $x \in \mathbb{R}^N$ by an invertible transform $W$ such that $WW^\top$.
The binary fingerprint is obtained by taking the sign of the projections of $x$ against a set of vectors $b_1,., b_L \in \mathbb{R}^N$ (à la LSH).
Then, they change $x$ to strengthen the amplitude of these projections so that their signs become more robust to noise.
They recover $I^\star$ with $W^\top$.
This scheme is applied to image patches in~\citep{7533094} where the performance is measured as a bit error rate after JPEG compression.
Our paper adapts this idea from fingerprinting to indexing, with modern deep learning representations and state-of-the-art indexing techniques.
The range of transformations is also much broader and includes geometric transforms.
\section{Preliminaries: Representation Learning and Indexing}
For the sake of simplicity, the exposure focuses on image representations from SSCD networks~\citep{pizzi2022sscd} and the indexing technique IVF-PQ~\citep{jegou2010pq}, since both are typically used for ICD.
Extensions to other methods can be found in Sec.~\ref{sec:generalization}.
\subsection{Deep descriptor learning}
Metric embedding learning aims to learn a mapping $f: \mathbb{R}^{c\times h\times w} \to \mathbb{R}^d$, such that measuring the similarity between images $I$ and $I'$ amounts to computing the distance $\norm{f(I) - f(I')}$.
In recent works, $f$ is typically a neural network trained with self-supervision on raw data to learn metrically meaningful representations.
Methods include contrastive learning~\citep{chen2020simclr}, self-distillation~\citep{grill2020bootstrap, caron2021dino}, or masking random patches of images~\citep{he2022masked, assran2022masked}.
In particular, SSCD~\citep{pizzi2022sscd} is a training method specialized for ICD.
It employs the contrastive self-supervised method SimCLR~\citep{chen2020simclr} and entropy regularization~\citep{sablayrolles2018catalyser} to improve the distribution of the representations.
\subsection{Indexing}
Given a dataset $\mathcal X = \{x_i\}_{i=1}^{n}\subset \mathbb{R}^d$ of $d$-dimensional vector representations extracted from $n$ images and a query vector $x_q$, we consider the indexing task that addresses the problem:
\begin{align}
x^* := \mathop{\mathrm{argmin}}_{x \in \mathcal X}
\; \norm{x - x_q}.
\end{align}
This exact nearest neighbor search is not tractable over large-scale databases.
Approximate search algorithms lower the amount of scored items thanks to space partitioning and/or accelerate the computations of distances thanks to quantization and pre-computation.
\paragraph{Space partitioning and cell-probe algorithms.}
As a first approximation,
nearest neighbors are sought only within a fraction of $\mathcal{X}$:
at indexing time, $\mathcal{X}$ is partitioned into $\mathcal X = \bigcup_{i=1}^{b} \mathcal{X}_i$.
At search time, an algorithm $Q: \mathbb{R}^d \to \{1,..,b\}^{k'}$ determines a subset of ${k'}$ buckets in which to search, such that ${k'}=|Q(x_q)| \ll b$, yielding the approximation:
\begin{align}
\mathop{\mathrm{argmin}}_{x \in \mathcal X} \; \norm{x-x_q}
\approx
\mathop{\mathrm{argmin}}_{x \in \bigcup_{i\in Q(x_q)} \mathcal{X}_i} \; \norm{x-x_q}.
\end{align}
A well known partition is the KD-tree~\citep{bentley1975kdtree} that divides the space along predetermined directions.
Subsequently, locality sensitive hashing (LSH)~\citep{indyk1998lsh, gionis1999lsh} and derivative~\citep{datar2004lsh,pauleve2010locality} employ various hash functions for bucket assignment, which implicitly partitions the space.
We focus on the popular clustering and Inverted Files methods~\citep{sivic2003video}, herein denoted by IVF.
They employ a codebook $\mathcal{C} = \{c_i\}_{i=1}^{k}\subset\mathbb{R}^d$ of $k$ centroids (also called ``visual words'' in a local descriptor context), for instance learned with k-means over a training set of representations.
Then, $Q$ associates $x$ to its nearest centroid $q_\mathrm{c}(x)$ such that the induced partition is the set of the $k$ Voronoï cells.
When indexing $x$, the IVF stores $x$ in the bucket associated with $c_i=q_\mathrm{c}(x)$.
When querying $x_q$, IVF searches only the ${k'}$ buckets associated to centroids $c_i$ nearest to $x_q$.
\paragraph{Efficient metric computation and product quantization.}
Another approximation comes from compressed-domain distance estimation.
Vector Quantization (VQ) maps a representation $x \in \mathbb{R}^d$ to a codeword $q_\mathrm{f}(x) \in \mathcal{C} = \{C_i\}_{i=1}^{K}$.
The function $q_\mathrm{f}$ is often referred to a \emph{quantizer} and $C_i$ as a \emph{reproduction value}.
The vector $x$ is then stored as an integer in $\{1, .., K\}$ corresponding to $q_\mathrm{f}(x)$.
The distance between $x$ and query $x_q$ is approximated by $\norm{q_\mathrm{f}(x) - x_q}$, which is an ``asymmetric'' distance computation (ADC) because the query is not compressed.
This leads to:
\begin{align}
\mathop{\mathrm{argmin}}_{x \in \mathcal X} \; \norm{x-x_q}
\approx
\mathop{\mathrm{argmin}}_{x \in \mathcal X} \; \norm{q_\mathrm{f}(x)- x_q} .
\end{align}
Binary quantizers ({a.k.a}.\@ sketches, \cite{charikar2002similarity} lead to efficient computations but inaccurate distance estimates~\citep{weiss2008spectral}.
Product Quantization (PQ)~\citep{jegou2010pq} or derivatives \cite{ge2013optimized} offer better estimates.
In PQ, a vector $x\in \mathbb{R}^d$ is split into $m$ subvectors in $\mathbb{R}^{d/m}$: $x=(x^1, \ldots, x^m)$.
The product quantizer then quantizes the subvectors: $q_\mathrm{f}: x \mapsto (q^1(x^1), \ldots, q^m(x^m))$.
If each subquantizer $q^j$ has $K_s$ reproduction values, the resulting quantizer $q_\mathrm{f}$ has a high $K=(K_s)^m$.
The squared distance estimate is decomposed as:
\begin{align}
\norm{q_\mathrm{f}(x)-x_q}^2 = \sum_{j=1}^m \norm{q^j(x^j)-x_q^j}^2.
\end{align}
This is efficient since $x$ is stored by the index as $q_\mathrm{f} (x)$ which has $m\log_2 K_s$ bits, and since summands can be precomputed without requiring decompression at search time.
\section{Active Indexing}\label{section:method}
Active indexing takes as input an image $I_o$, adds the image representation to the index and outputs an activated image $I^\star$ with better traceability properties for the index.
It makes the feature representation produced by the neural network more compliant with the indexing structure.
The activated image is the one that is disseminated on the platform, therefore the alteration must not degrade the perceived quality of the image.
Images are activated by an optimization on their pixels. The general optimization problem reads:
\begin{align}
I^\star := \mathop{\mathrm{argmin}}_{I \in \mathcal{C}(I_o)}
\; \mathcal{L}\left(I;I_o\right),
\label{eq:active_image}
\end{align}
where $\mathcal{L}$ is an indexation loss dependent on the indexing structure, $\mathcal{C}(I_o)$ is the set of images perceptually close to $I_o$.
Algorithm~\ref{alg:1} and Figure~\ref{fig:fig1} provide an overview of active indexing.
\begin{wrapfigure}{R}{0.45\textwidth}
\vspace{-0.7cm}
\resizebox{1.0\linewidth}{!}{
\begin{minipage}{0.5\textwidth}
\begin{algorithm}[H]
\caption{Active indexing for IVF-PQ}
\label{alg:1}
\begin{algorithmic}
\State \textbf{Input}: $I_o$: original image; $f$: feature extractor;
\State Add $x_o = f(I_o)$ to Index, get $q(x_o)$;
\State Initialize $\delta_0 = 0_{(c\times h\times w)}$;
\For{$t = 0, ..., N-1$}
\State $I_t \gets I_o + \alpha \,.\, H_{\mathrm{JND}}(I_o) \odot \mathrm{tanh}(\delta_t)$
\State $x_{t}\gets f(I_{t})$
\State $\mathcal{L} \gets \mathcal L _{\mathrm{f}} (x_{t}, q(x_o)) + \lambda \mathcal L _{\mathrm{i}} (\delta_t)$
\State $\delta_{t+1} \gets \delta_t + \eta \times \mathrm{Adam}(\mathcal{L})$
\EndFor
\State \textbf{Output}: $I^\star=I_N$ activated image
\end{algorithmic}
\end{algorithm}
\end{minipage}
}
\end{wrapfigure}
\subsection{Image optimization dedicated to IVF-PQ (``activation'')}
The indexing structure IVF-PQ involves a coarse quantizer $q_\mathrm{c}$ built with k-means clustering for space partitioning, and a fine product quantizer $q_\mathrm{f}$ on the residual vectors, such that a vector $x \in \mathbb{R}^d$ is approximated by $q(x) = q_\mathrm{c}(x) + q_\mathrm{f}\left( x-q_\mathrm{c}(x) \right)$.
We solve the optimization problem~\eqref{eq:active_image} by iterative gradient descent, back-propagating through the neural network back to the image.
The method is classically used in adversarial example generation~\citep{szegedy2013intriguing, carlini2017c&w} and watermarking~\citep{vukotic2020classification, fernandez2022sslwatermarking}.
Given an original image $I_o$, the loss is an aggregation of the following objectives:
\begin{align} \label{eq:objective}
& \mathcal L _{\mathrm{f}} (x,q(x_o)) = \norm{x - q(x_o)}^2
\textrm{\qquad with } x_o = f(I_o) ,\, x = f(I) \\
& \mathcal L _{\mathrm{i}} (I,I_o) = \norm{I - I_o}^2.
\end{align}
$ \mathcal L _{\mathrm{i}} $ is a regularization on the image distortion.
$ \mathcal L _{\mathrm{f}} $ is the indexation loss that operates on the representation space.
$ \mathcal L _{\mathrm{f}} $ is the Euclidean distance between $x$ and the target $q(x_o)$ and its goal is to push the image feature towards $q(x_o)$.
With IVF-PQ as index, the representation of the activated image gets closer to the quantized version of the original representation, but also closer to the coarse centroid.
Finally, the losses are combined as
$\mathcal{L}(I;I_o) = \mathcal L _{\mathrm{f}} (x,q(x_o)) + \lambda \mathcal L _{\mathrm{i}} (I,I_o)$.
\subsection{Perceptual attenuation}
It is common to optimize a perturbation $\delta$ added to the image, rather than the image itself.
The adversarial example literature often considers perceptual constraints in the form of an $\ell_p$-norm bound applied on $\delta$ (\cite{madry2017towards} use $\norm{\delta}_\infty < \varepsilon = 8/255$).
Although a smaller $\varepsilon$ makes the perturbation less visible, this constraint is not optimal for the human visual system (HVS), \textit{e.g}.\@ perturbations are more noticeable on flat than on textured areas of the image (see App.~\ref{subsec:linf}).
We employ a handcrafted perceptual attenuation model
based on a Just Noticeable Difference (JND) map~\citep{wu2017enhanced}, that adjusts the perturbation intensity according to luminance and contrast masking.
Given an image $I$, the JND map $H_{\mathrm{JND}}(I)\in \mathbb{R}^{c\times h \times w}$ models the minimum difference perceivable by the HVS at each pixel and additionally rescales the perturbation channel-wise since the human eye is more sensible to red and green than blue color shift (see App.~\ref{sec:perceptual} for details).
The relation that links the image $I$ sent to $f$, $\delta$ being optimized and the original $I_o$, reads:
\begin{align}
I = I_o + \alpha \,.\, H_{\mathrm{JND}}(I_o) \odot \mathrm{tanh}(\delta),
\label{eq:scaling}
\end{align}
with $\alpha$ a global scaling parameter that controls the strength of the perturbation and $\odot$ the pointwise multiplication.
Coupled with the regularization $ \mathcal L _{\mathrm{i}} $~\eqref{eq:objective}, it enforces that the activated image is perceptually similar, \textit{i.e}.\@ $I^\star\in \mathcal{C}(I_o)$ as required in~\eqref{eq:active_image}.
\subsection{Impact on the indexing performance}
Figure~\ref{fig:fig1} illustrates that the representation of the activated image gets closer to the reproduction value $q(f(I_o))$, and farther away from the Voronoï boundary.
This is expected to make image similarity search more robust because
(1) it decreases the probability that $x=f(t(I_o))$ ``falls'' outside the bucket; and
(2) it lowers the distance between $x$ and $q(x)$, improving the PQ distance estimate.
Besides, by design, the representation stored by the index is invariant to the activation.
Formally stated, consider two images $I$, $J$, and one activated version $J^\star$ together with their representations $x,y,y^\star$.
When querying $x=f(I)$, the distance estimate is $\norm{q(y^\star)- x} = \norm{q(y)- x}$, so the index is oblivious to the change $J\rightarrow J^\star$.
This means that the structure can index passive and activated images at the same time.
Retrieval of activated images is more accurate but the performance on passive images does not change.
This compatibility property makes it possible to select only a subset of images to activate, but also to activate already-indexed images at any time.
\section{Experimental Results}
\subsection{Experimental setup}\label{sec:experimental}
\paragraph{Dataset.}
We use DISC21~\citep{douze2021disc} a dataset dedicated to ICD.
It includes 1M reference images and 50k query images, 10k of which are true copies from reference images.
A disjoint 1M-image set with same distribution as the reference images is given for training.
Images resolutions range from 200$\times$200 to 1024$\times$1024 pixels (most of the images are around 1024$\times$768 pixels).
The queries used in our experiments are \emph{not} the queries in DISC21, since we need to control the image transformations in our experiments, and most transformations of DISC21 were done manually so they are not reproducible.
Our queries are transformations of images \emph{after active indexing}.
These transformations range from simple attacks like rotation to more realistic social network transformations which created the original DISC21 queries (see App. \ref{subsec:dataset}).
\paragraph{Metrics.}
For retrieval, our main metric is Recall $1$@$1$ ($R$@$1$\@ for simplicity), which corresponds to the proportion of positive queries where the top-1 retrieved results is the reference image.
For copy detection, we use the same metric as the NeurIPS Image Similarity Challenge~\citep{douze2021disc}.
We retrieve the $k=10$ most similar database features for every query; and we declare a pair is a match if the distance is lower than a threshold $\tau$.
To evaluate detection efficiency, we use the 10k matching queries above-mentioned together with 40k negative queries (\textit{i.e}.\@ not included in the database).
We use precision and recall, as well as the area under the precision-recall curve, which is equivalent to the micro average precision (\emph{$\mu$AP}).
While $R$@$1$\@ only measures ranking quality of the index, $\mu$AP takes into account the confidence of a match.
As for image quality metric, we use the Peak Signal-to-Noise Ratio (PSNR) which is defined as $10\log_{10} \left( 255^2 / \mathrm{MSE}(I, I')^2 \right)$, as well as SSIM~\citep{wang2004ssim} and the norm $\norm{I-I'}_\infty$.
\paragraph{Implementation details.}\label{par:details}
The evaluation procedure is: (1) we train an index on the 1M training images, (2) index the 1M reference images, (3) activate (or not) 10k images from this reference set.
(4) At search time, we use the index to get closest neighbors (and their distances) of transformed versions from a query set made of the 10k images.
Unless stated otherwise, we use a IVF4096,PQ8x8 index (IVF quantizer with 4096 centroids, and PQ with 8 subquantizers of $2^8$ centroids), and use only one probe on IVF search for shortlist selection ($k'=1$).
Compared to a realistic setting, we voluntarily use an indexing method that severely degrades learned representations to showcase and analyze the effect of the active indexing.
For feature extraction, we use an SSCD model with a ResNet50 trunk~\citep{he2016resnet}.
It takes image resized to 288$\times$288 and generates normalized representations in $\mathbb{R}^{512}$.
Optimization~\eqref{eq:active_image} is done with the Adam optimizer~\citep{kingma2014adam}, the learning rate is set to $1$, the number of iterations to $N=10$ and the regularization to $\lambda=1$.
In~\eqref{eq:scaling}, the distortion scaling is set to $\alpha=3$ (leading to an average PNSR around $43$~dB).
In this setup, activating 128 images takes around 6s ($\approx$ 40ms/image) with a 32GB GPU.
It can be sped-up at the cost of some accuracy (see App.~\ref{sec:speedup}).
\subsection{Active vs. Passive}\label{sec:act_vs_passive}
This section compares retrieval performance of active and passive indexing.
We evaluate $R$@1 when different transformations are applied to the 10k reference images before search.
The ``Passive'' lines of Tab.~\ref{tab:act_vs_pas_retrieval} show how the IVF-PQ degrades the recall.
This is expected, but the IVF-PQ also accelerates search 500$\times$ and the index is 256$\times$ more compact, which is necessary for large-scale applications.
Edited images are retrieved more often when they were activated for the index:
increase of up to $+60$ $R$@$1$\@ for strong brightness and contrast changes, close to results of the brute-force search.
We also notice that the performance of the active IVF-PQ$^{k'=1}$ is approximately the same as the one of the passive IVF-PQ$^{k'=16}$, meaning that the search can be made more efficient at equal performance.
For the IVF-PQ$^\dagger$ that does less approximation in the search (but is slower and takes more memory), retrieval on activated images is also improved, though to a lesser extent.
\begin{table}[t]
\centering
\captionsetup{font=small}
\caption{
Comparison of the index performance between activated and passive images.
The search is done on a 1M image set and $R$@1 is averaged over 10k query images submitted to different transformations before search.
\textbf{Random}: randomly apply 1 to 4 transformations.
\textbf{Avg.}: average on the transformations presented in the table (details in App. \ref{subsec:transformations}).
\textbf{No index}: exhaustive brute-force nearest neighbor search. \textbf{IVF-PQ}: \textsc{IVF4096,PQ8x8} index with $k'$=1 (16 for \textbf{IVF-PQ}$^{16}$).
\textbf{IVF-PQ}$^\dagger$: \textsc{IVF512,PQ32x8} with $k'=32$.
}
\label{tab:act_vs_pas_retrieval}
\vspace{-0.3cm}
\resizebox{1.0\linewidth}{!}{
\begingroup
\setlength{\tabcolsep}{4pt}
\def1.1{1.1}
\begin{tabular}{ l |l| l| c| *{15}{p{0.04\textwidth}}}
\multicolumn{1}{c}{} & \multicolumn{1}{c}{\rot{Search (ms)}} & \multicolumn{1}{c}{\rot{Bytes/vector}} & \multicolumn{1}{c}{\rot{Activated}} & \rot{Identity} & \rot{Contr. 0.5} & \rot{Contr. 2.0} & \rot{Bright. 0.5} & \rot{Bright. 2.0} & \rot{Hue 0.2} & \rot{Blur 2.0} & \rot{JPEG 50} & \rot{Rot. 25} & \rot{Rot. 90} & \rot{Crop 0.5} & \rot{Resi. 0.5} & \rot{Meme} & \rot{Random} & \rot{Avg.} \\ \midrule
No index & 252 & 2048 & \ding{55} & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 0.90 & 0.99 \\ \midrule
& & & \ding{55} & 1.00 & 0.73 & 0.39 & 0.73 & 0.28 & 0.62 & 0.48 & 0.72 & 0.07 & 0.14 & 0.14 & 0.72 & 0.14 & 0.13 & 0.45 \\
\rowcolor{apricot!30} \cellcolor{white!0} \multirow{-2}{*}{IVF-PQ} & \multirow{-2}{*}{0.38} \cellcolor{white!0} & \multirow{-2}{*}{8} \cellcolor{white!0}
& \checkmark & 1.00 & 1.00 & 0.96 & 1.00 & 0.92 & 1.00 & 0.96 & 0.99 & 0.10 & 0.50 & 0.29 & 1.00 & 0.43 & 0.32 & 0.75 \\
\midrule
& & & \ding{55} & 1.00 & 1.00 & 0.90 & 1.00 & 0.78 & 0.99 & 0.95 & 0.99 & 0.35 & 0.57 & 0.57 & 1.00 & 0.56 & 0.39 & 0.79 \\
\rowcolor{apricot!30} \cellcolor{white!0} \multirow{-2}{*}{IVF-PQ$^{16}$} & \multirow{-2}{*}{0.42} \cellcolor{white!0} & \multirow{-2}{*}{8} \cellcolor{white!0} &
\checkmark & 1.00 & 1.00 & 1.00 & 1.00 & 0.98 & 1.00 & 1.00 & 1.00 & 0.43 & 0.88 & 0.75 & 1.00 & 0.84 & 0.50 & 0.88 \\
\midrule
& & & \ding{55} & 1.00 & 1.00 & 0.99 & 1.00 & 0.95 & 1.00 & 0.99 & 1.00 & 0.72 & 0.87 & 0.88 & 1.00 & 0.87 & 0.61 & 0.92 \\
\rowcolor{apricot!30} \cellcolor{white!0} \multirow{-2}{*}{IVF-PQ$^\dagger$} & \multirow{-2}{*}{1.9} \cellcolor{white!0} & \multirow{-2}{*}{32} \cellcolor{white!0} &
\checkmark & 1.00 & 1.00 & 0.99 & 1.00 & 0.98 & 1.00 & 1.00 & 1.00 & 0.75 & 0.92 & 0.91 & 1.00 & 0.92 & 0.63 & 0.94 \\
\bottomrule
\end{tabular}
\endgroup
}
\vspace{-0.3cm}
\end{table}
As for copy detection, Figure~\ref{fig:prc} gives the precision-recall curves obtained for a sliding value of $\tau$, and corresponding $\mu$AP.
Again, we observe a significant increase ($\times 2$) in $\mu$AP with active indexing.
Note that the detection performance is much weaker than the brute-force search even in the active case because of the strong approximation made by space partitioning (more details in Sec.~\ref{sec:space_partitioning}).
Example of activated images are given in Fig.~\ref{fig:qualitative_short} (more in App. \ref{sec:more_qualitative}), while the qualitative image metrics are as follows: PSNR$=43.8\pm 2.2$~dB, SSIM$=0.98 \pm 0.01$, and $\norm{I-I'}_\infty=14.5 \pm 1.2$.
These results are computed on 10k images, the $\pm$ indicates the standard deviation.
\subsection{Image quality trade-off}
For a fixed index and neural extractor, the performance of active indexing mainly depends on the scaling $\alpha$ that controls the activated image quality.
In Fig. \ref{fig:psnr}, we repeat the previous experiment for different values of $\alpha$ and plot the $\mu$AP against the average PSNR.
As expected, lower PSNR implies better $\mu$AP. For instance, at PSNR 30~dB, the $\mu$AP is augmented threefold compared to the passive case.
Indeed, for strong perturbations the objective function of \eqref{eq:objective} can be further lowered, reducing even more the gap between representations and their quantized counterparts.
\begin{figure}[b]
\begin{minipage}{0.4\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{figs/exp/psnr_loss_muap.pdf}
\captionsetup{font=small}
\caption{PSNR trade-off. As the PSNR decreases, the {\color{orange}$\mu$AP\@ (orange)} gets better, because the {\color{blue} distance (blue)} between activated representations $x$ and $q(x)$ decreases.}
\label{fig:psnr}
\end{minipage}\hfill
\begin{minipage}{0.55\textwidth}
\centering
\includegraphics[width=0.95\textwidth]{figs/exp/qualitative_short.jpg}
\captionsetup{font=small}
\caption[Caption]{Activated images. \emph{Left:} reference from DISC (\href{http://www.flickr.com/photos/11805179@N04/1816673349/}{R000643.jpg} and \href{http://www.flickr.com/photos/36449457@N00/8585081884/}{R000761.jpg}), \emph{middle:} activated image, right: pixel-wise difference.}
\label{fig:qualitative_short}
\end{minipage}
\end{figure}
\subsection{Generalization}\label{sec:generalization}
\paragraph{Generalization to other neural feature extractors.}
We first reproduce the experiment of Sec.~\ref{par:details} with different extractors, that cover distinct training methods and architectures.
Among them, we evaluate a ResNext101~\citep{xie2017aggregated} trained with SSCD~\citep{pizzi2022sscd}, a larger network than the ResNet50 used in our main experiments ;
the winner of the descriptor track of the NeurIPS ISC, \textsc{Lyakaap}-dt1~\citep{yokoo2021isc}, that uses an EfficientNetv2 architecture~\citep{tan2021efficientnetv2} ; networks from DINO~\citep{caron2021dino}, either based on ResNet50 or ViT~\citep{dosovitskiy2020vit}, like the ViT-S model~\citep{touvron2021training}.
Table~\ref{tab:extractors} presents the $R$@$1$\@ obtained on 10k activated images when applying different transformations before search.
The $R$@$1$\@ is better for activated images for all transformations and all neural networks. The average improvement on all transformations ranges from $+12\%$ for DINO ViT-s to $+30\%$ for SSCD ResNet50.
\begin{table}[t]
\centering
\captionsetup{font=small}
\caption{
$R$@$1$\@ for different transformations before search.
We use our method to activate images for indexing with IVF-PQ, with different neural networks used as feature extractors.
}
\label{tab:extractors}
\vspace{-0.3cm}
\resizebox{1.0\linewidth}{!}{
\begingroup
\setlength{\tabcolsep}{4pt}
\def1.1{1.1}
\begin{tabular}{ l| l| c| *{15}{p{0.04\textwidth}}}
\multicolumn{1}{c}{\rot{Name}} & \multicolumn{1}{c}{\rot{Arch.}} & \multicolumn{1}{l}{\rot{Activated}} & \rot{Identity} & \rot{Contr. 0.5} & \rot{Contr. 2.0} & \rot{Bright. 0.5} & \rot{Bright. 2.0} & \rot{Hue 0.2} & \rot{Blur 2.0} & \rot{JPEG 50} & \rot{Rot. 25} & \rot{Rot. 90} & \rot{Crop 0.5} & \rot{Resi. 0.5} & \rot{Meme} & \rot{Random} & \rot{Avg.} \\ \midrule
& & \ding{55} & 1.00 & 0.73 & 0.39 & 0.73 & 0.28 & 0.62 & 0.48 & 0.72 & 0.07 & 0.14 & 0.14 & 0.72 & 0.14 & 0.13 & 0.45 \\
\rowcolor{apricot!30} \cellcolor{white!0} & \multirow{-2}{*}{ResNet50} \cellcolor{white!0}& \checkmark & 1.00 & 1.00 & 0.96 & 1.00 & 0.92 & 1.00 & 0.96 & 0.99 & 0.10 & 0.50 & 0.29 & 1.00 & 0.43 & 0.32 & 0.75 \\ \cmidrule{2-18}
& & \ding{55} & 1.00 & 0.88 & 0.68 & 0.88 & 0.57 & 0.84 & 0.46 & 0.79 & 0.46 & 0.63 & 0.53 & 0.80 & 0.48 & 0.28 & 0.66 \\
\rowcolor{apricot!30} \cellcolor{white!0} \multirow{-4}{*}{SSCD} & \multirow{-2}{*}{ResNext101} \cellcolor{white!0} & \checkmark & 1.00 & 1.00 & 0.96 & 1.00 & 0.90 & 0.99 & 0.77 & 0.97 & 0.53 & 0.85 & 0.64 & 1.00 & 0.74 & 0.37 & 0.84 \\ \midrule
& & \ding{55} & 1.00 & 0.66 & 0.65 & 0.65 & 0.52 & 0.71 & 0.52 & 0.82 & 0.07 & 0.20 & 0.51 & 0.84 & 0.62 & 0.18 & 0.57 \\
\rowcolor{apricot!30} \cellcolor{white!0} & \multirow{-2}{*}{ResNet50} \cellcolor{white!0} & \checkmark & 1.00 & 0.99 & 0.88 & 0.99 & 0.75 & 0.93 & 0.72 & 0.94 & 0.08 & 0.25 & 0.57 & 0.99 & 0.82 & 0.23 & 0.72 \\ \cmidrule{2-18}
& & \ding{55} & 1.00 & 0.89 & 0.71 & 0.86 & 0.64 & 0.75 & 0.74 & 0.90 & 0.14 & 0.18 & 0.57 & 0.88 & 0.61 & 0.25 & 0.65 \\
\rowcolor{apricot!30} \cellcolor{white!0} \multirow{-4}{*}{DINO} & \multirow{-2}{*}{ViT-s} \cellcolor{white!0} & \checkmark & 1.00 & 0.99 & 0.94 & 0.99 & 0.92 & 0.98 & 0.89 & 0.99 & 0.15 & 0.28 & 0.63 & 0.99 & 0.77 & 0.32 & 0.77 \\ \midrule
& & \ding{55} & 1.00 & 0.25 & 0.08 & 0.16 & 0.01 & 0.51 & 0.54 & 0.84 & 0.18 & 0.16 & 0.23 & 0.79 & 0.16 & 0.18 & 0.36 \\
\rowcolor{apricot!30} \cellcolor{white!0} \multirow{-2}{*}{ISC-dt1} & \multirow{-2}{*}{EffNetv2} \cellcolor{white!0} & \checkmark & 1.00 & 0.57 & 0.16 & 0.33 & 0.01 & 0.88 & 0.79 & 0.97 & 0.20 & 0.24 & 0.29 & 0.97 & 0.26 & 0.26 & 0.49 \\
\bottomrule
\end{tabular}
\endgroup
}
\end{table}
\vspace*{-0.1cm}
\paragraph{Generalization to other indexes.}
The method easily generalizes to other types of indexing structures, the only difference being in the indexation loss $ \mathcal L _{\mathrm{f}} $~\eqref{eq:objective}.
We present some of them below:
\begin{itemize}[leftmargin=0.5cm,itemsep=0cm,topsep=-0.1cm]
\item \textbf{PQ and OPQ}.\quad
In PQ~\citep{jegou2010pq}, a vector $x \in \mathbb{R}^d$ is approximated by $q_\mathrm{f}(x)$.
$ \mathcal L _{\mathrm{f}} $ reads $\norm{x-q_\mathrm{f}(x_o)}$.
In OPQ~\citep{ge2013optimized}, vectors are rotated by matrix $R$ before codeword assignment, such that $RR^\top = I$.
$ \mathcal L _{\mathrm{f}} $ becomes $\norm{x-R^\topq_\mathrm{f}(Rx_o)}$.
\item \textbf{IVF.} \quad
Here, we only do space partitioning.
Employing $ \mathcal L _{\mathrm{f}} = \norm{x- q_\mathrm{c} (x_o)}$ (``pushing towards the cluster centroid'') decreases the odds of $x$ falling in the wrong cell (see Sec.~\ref{sec:space_partitioning}).
In this case, an issue can be that similar representations are all pushed together to a same centroid, which makes them less discriminate.
Empirically, we found that this does not happen because perceptual constraint in the image domain prevents features from getting too close.
\item \textbf{LSH.} \quad
Locality Sensitive Hashing maps $x\in \mathbb{R}^d$ to a binary hash $b(x)\in \mathbb{R}^L$.
It is commonly done with projections against a set of vectors, which give for $j \in [1,..,L]$, $b_j(x) = \mathrm{sign} (w_j^\top x)$.
The objective $ \mathcal L _{\mathrm{f}} = -1/L \sum_{j} \mathrm{sign}(b(x_o))\cdot w_j^\top x$, allows to push $x$ along the LSH directions and to improve the robustness of the hash.
\end{itemize}\vspace*{0.2cm}
Table~\ref{tab:indexes} presents the $R$@$1$\@ and $\mu$AP\@ obtained on the 50k query set.
Again, results are always better in the active scenario.
We remark that active indexing has more impact on space partitioning techniques: the improvement for IVF is higher than with PQ and the LSH binary sketches. As to be expected, the impact is smaller when the indexing method is more accurate.
\begin{table}[h]
\centering
\resizebox{0.6\linewidth}{!}{
\begingroup
\setlength{\tabcolsep}{3pt}
\begin{tabular}{ c|c |cc|cc}
\toprule
\multirow{2}{*}{Index} & \multirow{2}{*}{Search time} & \multicolumn{2}{c|}{$R$@$1$\@ avg.} & \multicolumn{2}{c}{$\mu$AP\@ } \\
& & Passive & Activated & Passive & Activated \\ \midrule
IVF 1024 & 0.32 ms & 0.47 & \textbf{0.83} & 0.16 & \textbf{0.43} \\
OPQ 8x8 & 5.71 ms & 0.92 & \textbf{0.94} & 0.48 & \textbf{0.55} \\
PCA64, LSH & 0.99 ms & 0.72 & \textbf{0.83} & 0.25 & \textbf{0.39} \\
\bottomrule
\end{tabular}
\endgroup
}
\captionsetup{font=small}
\caption{\makebox{$R$@$1$\@ averaged on transformations presented in Tab.~\ref{tab:act_vs_pas_retrieval} and $\mu$AP\@ for different indexing structures}}
\label{tab:indexes}
\end{table}
\section{Analyses}\label{sec:analyses}
We provide insights on the method for IVF-PQ, considering the effects of quantization and space partitioning.
For an image $I$ whose representation is $x=f(I)\in\mathbb{R}^d$,
$\hat{x}$ denotes the representation of a transformed version: $\hat{x} = f(t(I))\in\mathbb{R}^d$, and $x^\star$ the representation of the activated image $I^\star$.
For details on the images and the implementation used in the experimental validations, see Sec. \ref{sec:experimental}.
\begin{figure}[b!]
\begin{minipage}{0.45\textwidth}
\centering
\vspace{3pt}
\includegraphics[width=1.05\linewidth, trim={0 0.8em 0 0em},clip]{figs/exp/prc.pdf}
\captionsetup{font=small}
\caption{Precision-Recall curve for ICD with 50k queries and 1M reference images (more details for the experimental setup in Sec. \ref{sec:experimental}). $p_\mathrm{f}^{\mathrm{ivf}}$ is the probability of failure of the IVF (Sec. \ref{sec:space_partitioning}).
}
\label{fig:prc}
\end{minipage}\hfill
\begin{minipage}{0.51\textwidth}
\centering
\includegraphics[width=0.8\textwidth, trim={0 0.15cm 0 0.23cm}, clip]{figs/exp/distances.pdf}
\captionsetup{font=small}
\caption{
Distance estimates histograms (sec. \ref{sec:quantization}).
With active indexing, $\|x- q(x)\|^2$ is reduced ({\color{blue}$\leftarrow$}), inducing a shift ({\color{orange}$\leftarrow$}) in the distribution of $\|\hat{x}- q(x)\|^2$, where $t(I)$ a hue-shifted version of $I$.
$y$ is a random query.
}
\label{fig:dists}
\end{minipage}
\end{figure}
\subsection{Product quantization: impact on distance estimate}\label{sec:quantization}
We start by analyzing the distance estimate considered by the index:
\begin{equation}
\|\hat{x}-q(x)\|^2 = \|x-q(x)\|^2 + \|\hat{x}-x\|^2
+ 2 (x-q(x))^\top (\hat{x}-x).
\label{eq:distance}
\end{equation}
The activation aims to reduce the first term, \textit{i.e}.\@ the quantization error $\|x- q(x)\|^2$, which in turn reduces $\|\hat{x}- q(x)\|^2$.
Figure~\ref{fig:dists} shows in blue the empirical distributions of $\|x- q(x)\|^2$ (passive) and $\|x^\star- q(x)\|^2$ (activated).
As expected the latter has a lower mean, but also a stronger variance.
The variation of the following factors may explain this: \emph{i)} the strength of the perturbation (due to the HVS modeled by $H_{\mathrm{JND}}$ in~\eqref{eq:scaling}), \emph{ii)} the sensitivity of the feature extractor $\|\nabla_x f(x)\|$ (some features are easier to push than others), \emph{iii)} the shapes and sizes of the Voronoï cells of PQ.
The second term of \eqref{eq:distance} models the impact of the image transformation in the feature space.
Comparing the orange and blue distributions in Fig.~\ref{fig:dists}, we see that it has a positive mean, but the shift is bigger for activated images.
We can assume that the third term has null expectation for two reasons: \emph{i)} the noise $\hat{x}-x$ is independent of $q(x)$ and centered around 0, \emph{ii)} in the high definition regime, quantification noise $x-q(x)$ is independent of $x$ and centered on 0.
Thus, this term only increases the variance.
Since $x^\star-q(x)$ has smaller norm, this increase is smaller for activated images.
All in all, $\|\hat{x}^\star-q(x)\|^2$ has a lower mean but a stronger variance than its passive counterpart $\|\hat{x}-q(x)\|^2$.
Nevertheless, the decrease of the mean is so large that it compensates the larger variance.
The orange distribution in active indexing is further away from the green distribution for negative pairs, \textit{i.e}.\@ the distance between an indexed vector $q(x)$ and an independent query $y$.
\subsection{Space partitioning: impact on the IVF probability of failure}\label{sec:space_partitioning}
We denote by $p_\mathrm{f} := \mathbb{P}(q_\mathrm{c}(x) \neq q_\mathrm{c} (\hat{x}) )$ the probability that $\hat{x}$ is assigned to a wrong bucket by IVF assignment $q_\mathrm{c}$.
In the single-probe search ($k'=1$), the recall (probability that a pair is detected when it is a true match, for a given threshold $\tau$ on the distance) is upper-bounded by $1 - p_\mathrm{f}$:
\begin{align}
R_\tau
= \mathbb{P} \left(\{ q_\mathrm{c}(\hat{x}) = q_\mathrm{c}(x) \} \cap \{ \| \hat{x}-q(x)\| < \tau \} \right)
\leq \mathbb{P} \left(\{ q_\mathrm{c}(\hat{x}) = q_\mathrm{c}(x) \} \right)
=1 - p_\mathrm{f}.
\end{align}
In other terms, even with a high threshold $\tau \rightarrow \infty$ (and low precision), the detection misses representations that ought to be matched, with probability $p_\mathrm{f}$. It explains the sharp drop at recall $R=0.13$ in Fig.~\ref{fig:prc}.
This is why it is crucial to decrease $p_\mathrm{f}$.
The effect of active indexing is to reduce $\|\hat{x}-q_\mathrm{c}(x)\|$
therefore reducing $p_\mathrm{f}$ and increasing the upper-bound for $R$:
the drop shifts towards $R=0.32$.
This explanation suggests that pushing $x$ towards $q_\mathrm{c}(x)$ decreases even more efficiently $p_\mathrm{f}$.
This makes the IVF more robust to transformation but this may jeopardize the PQ search because features of activated images are packed altogether.
In a way, our strategy, which pushes $x$ towards $q(x)$, dispatches the improvement over the IVF and the PQ search.
\section{Conclusion \& Discussion}
We introduce a way to improve ICD in large-scale settings, when images can be changed before release.
It leverages an optimization scheme, similar to adversarial examples, that modifies images so that (1) their representations are better suited for indexing, (2) the perturbation is invisible to the human eye.
We provide grounded analyses on the method and show that it significantly improves retrieval performance of activated images, on a number of neural extractors and indexing structures.
Activating images takes time (in the order of 10~ms/image) but one advantage is that the database may contain both active and passive images: active indexing does not spoil the performance of passive indexing and vice-versa. This is good for legacy compliance and also opens the door to flexible digital asset management strategies (actively indexing images of particular importance).
The method has several limitations.
First, it is not agnostic to the indexing structure and extractor that are used by the similarity search.
Second, an adversary could break the indexing system in several ways.
In a black-box setting (no knowledge of the indexing structure and neural network extractor), adversarial purification~\citep{shi2021online} could get rid of the perturbation that activated the image.
In a semi-white-box setting (knowledge of the feature extractor), targeted mismatch attacks against passive indexing like ~\cite{tolias2019targeted} may also work.
Adversarial training ~\citep{madry2017towards} could be a defense.
For instance, it is interesting to know if adversarial training prevents active indexing, or if the perceptual perturbation that is used in our method is still able to push features in the latent space of a robust and defended neural network.
\newpage
\subsection*{Ethics Statement}
\paragraph{Societal impact statement.}
Content tracing is a double-edged sword.
On the one hand, it allows media platforms to more accurately track malicious content (pornographic, terrorist, violent images, \textit{e.g}.\@ Apple's NeuralHash and Microsoft's PhotoDNA) and to protect copyright (\textit{e.g}.\@ Youtube's Content ID).
On the other hand it can be used as a means of societal and political censorship, to restrict free speech of specific communities.
However, we still believe that research needs to be advanced to improve global moderation in the internet.
We also believe that advantages that a better copy detection could bring are more numerous than its drawbacks.
\paragraph{Environmental impact statement.}
We roughly estimated that the total GPU-days used for running all our experiments to $200$, or $\approx 5000$ GPU-hours.
Experiments were conducted using a private infrastructure and we estimate total emissions to be in the order of a ton CO$_2$eq.
Estimations were conducted using the \href{https://mlco2.github.io/impact#compute}{MachineLearning Impact calculator} presented in \cite{lacoste2019quantifying}.
We do not consider in this approximation: memory storage, CPU-hours, production cost of GPUs/ CPUs, etc. as well as the environmental cost of training the neural networks used as feature extractors.
Although the cost of the experiments and the method is high, it could possibly allow a reduction of the computations needed in large data-centers thanks to improved performance of indexing structures.
\subsection*{Reproducibility Statement}
\emph{The implementation will be made available.}
Models used for feature extraction (\href{https://github.com/facebookresearch/sscd-copy-detection/}{SSCD}, \href{https://github.com/facebookresearch/dino}{DINO}, \href{https://github.com/lyakaap/ISC21-Descriptor-Track-1st}{ISC-dt1}) can be downloaded in their respective repositories.
It builds upon the open-source Pytorch~\citep{paszke2019pytorch} and FAISS~\citep{johnson2019faiss} libraries.
The main dataset used in the experiments (DISC21) can be freely downloaded on its webpage \href{https://ai.facebook.com/datasets/disc21-dataset/}{https://ai.facebook.com/datasets/disc21-dataset/}.
Dataset processing is described in App. \ref{subsec:dataset}.
\section{Details on the Perceptual Attenuation Model}\label{sec:perceptual}
\subsection{Just Noticeable Difference map}
\begin{figure}[b]
\centering
\includegraphics[width= 0.23\textwidth]{figs/qualitative/ref.jpg}
\hspace{0.1\textwidth}
\captionsetup{font=small}
\includegraphics[width= 0.23\textwidth]{figs/qualitative/heatmap.png}
\caption[Caption]{A reference image $I$ from DISC21 (\href{http://www.flickr.com/photos/61368956@N00/5060849004/}{R002815.jpg}), and the associated perceptual heatmap $H_{\mathrm{JND}}(I)$.}
\label{fig:heatmap}
\vspace*{-0.5cm}
\end{figure}
The maximum change that the human visual system (HVS) cannot perceive is sometimes referred to as the just noticeable difference (JND)~\cite{krueger1989reconciling}.
It is used in many applications, such as image/video watemarking, compression, quality assessment (JND is also used in audio).
JND models in pixel domain directly calculate the JND at each pixel location (\textit{i.e}.\@ how much pixel difference is perceivable by the HVS). The JND map that we use is based on the work of \cite{chou1995perceptually}.
We use this model for its simplicity, its efficiency and its good qualitative results.
More complex HVS models could also be used if even higher imperceptibility is needed (\cite{watson1993dct, yang2005just, zhang2008just, jiang2022jnd} to cite a few).
The JND map takes into account two characteristics of the HVS, namely the luminance adaptation (LA) and the contrast masking (CM) phenomena. We follow the same notations as \cite{wu2017enhanced}.
The CM map $\mathcal{M}_C$ is a function of the image gradient magnitude $\mathcal{C}_l$ (the Sobel filter of the image):
\begin{equation}
\mathcal{M}_C(x) = 0.115 \times
\frac{\alpha \cdot \mathcal{C}_l(x)^{2.4}}
{ \mathcal{C}_l(x)^{2} + \beta^2}
\textrm{\quad , with \,}
\mathcal{C}_l = \sqrt{ \nabla_x I(x)^2 + \nabla_y I(x)^2},
\end{equation}
where $x$ is the pixel location, $I(x)$ the image intensity, $\alpha = 16$, and $\beta = 26$.
It is an increasing function of $\mathcal{C}_l$, meaning that the stronger the gradient is at $x$, the more the image is masking a local perturbation, and the higher the noticeable pixel difference is.
LA takes into account the fact that the HVS presents different sensitivity to background luminance (\textit{e.g}.\@ it is less sensible in dark backgrounds).
It is modeled as:
\begin{align}
\mathcal{L}_A (x) =
\begin{cases}
\displaystyle 17 \times \left( 1-\sqrt{\frac{B(x)}{127}} \right) & \textrm{\quad if\,} B(x)<127 \\
\displaystyle \frac{3 \times \left( B(x) - 127 \right)}{128} +3 & \textrm{\quad if\,}B(x)\geq 127,
\end{cases}
\end{align}
where $B(x)$ is the background luminance, which is calculated as the mean luminance value of a local patch centered on $x$.
Finally, both effects are combined with a nonlinear additivity model:
\begin{equation}
H_{\mathrm{JND}} = \mathcal{L}_A + \mathcal{M}_C - C \cdot \min \{ \mathcal{L}_A, \mathcal{M}_C \},
\end{equation}
where $C$ is set to $0.3$ and determines the overlapping effect.
For color images, the final RGB heatmap is $H_{\mathrm{JND}} = [\alpha_R H, \alpha_G H, \alpha_B H]$, where $(\alpha_{R}, \alpha_{G}, \alpha_{B})$ are inversely proportional to the mixing coefficients for the luminance: $(\alpha_{R}, \alpha_{G}, \alpha_{B}) = 0.072 / (0.299, 0.587, 0.114)$.
\subsection{Comparison with $\ell_\infty$ Constraint Embedding}\label{subsec:linf}
Figure~\ref{fig:linf_vs_perc} shows the same image activated using either the $\ell_\infty$ constraint (commonly used in the adversarial attack literature) or our perceptual constraint based on the JND model explained above.
Even with very small $\varepsilon$ ($4$ over 255 in the example bellow), the perturbation is visible especially in the flat regions of the images, such as the sea or sky.
\cite{laidlaw2021perceptual} also show that the $\ell_\infty$ is not a good perceptual constraint.
They use the LPIPS loss~\citep{zhang2018unreasonable} as a surrogate for the HVS to develop more imperceptible adversarial attacks.
Although a similar approach could be used here, we found that at this small level of image distortion the LPIPS did not capture CM and LA as well as the handcrafted perceptual models present in the compression and watermarking literature.
\begin{figure}[h]
\centering
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width= 0.48\textwidth]{figs/qualitative/linf4_psnr36,4.png}
\includegraphics[width= 0.48\textwidth]{figs/qualitative/diff_linf.png}
\captionsetup{font=small}
\caption{$\ell_\infty=4$, $\mathrm{PSNR}=36.4$~dB, $\mathrm{SSIM}=0.91$}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width= 0.48\textwidth]{figs/qualitative/linf24_psnr34,4.png}
\includegraphics[width= 0.48\textwidth]{figs/qualitative/diff_perc.png}
\captionsetup{font=small}
\caption{$\ell_\infty=23$, $\mathrm{PSNR}=34.4$~dB, $\mathrm{SSIM}=0.94$}
\end{subfigure}
\captionsetup{font=small}
\caption{
Activated images, either with (a) the $\ell_\infty \leq 4$ constraint or with (b) our perceptual model (best viewed on screen).
We give the corresponding measures between the original and the protected image, as well as the pixel-wise difference.
The perturbation on the right is much less perceptible thanks to the perceptual model, even though its $\ell_\infty$ distance with the original image is much higher.
}
\label{fig:linf_vs_perc}
\end{figure}
\section{More Experiments Details}
\subsection{Dataset}\label{subsec:dataset}
The dataset DISC 2021 was designed for the Image Similarity Challenge~\citep{douze2021disc} and can be downloaded in the dataset webpage: \href{https://ai.facebook.com/datasets/disc21-dataset/}{https://ai.facebook.com/datasets/disc21-dataset/}.
We want to test performance on edited versions of activated images but in DISC query set transformations are already applied to images. Therefore the query set cannot be used as it is.
We create a first test set ``Ref10k'' by selecting the 10k images from the reference set that were originally used to generate the queries (the ``dev queries'' from the downloadable version).
We also re-create a query set ``Query50k''.
To be as close as possible, we use the same images that were used for generating queries in DISC.
Edited images are generated using the AugLy library~\citep{papakipos2022augly}, following the guidelines given in the ``Automatic Transformations" section of the DISC paper.
Therefore, the main difference between the query set used in our experiments and the original one is that ours do not have manual augmentations.
\subsection{Transformations seen at test time}\label{subsec:transformations}
They cover both spatial transformations (crops, rotation, etc.), pixel-value transformations (contrast, hue, jpeg, etc.) and ``everyday life'' transformations with the AugLy augmentations.
All transformations are illustrated in Fig.~\ref{fig:all_transformations}.
The parameters for all transformations are the ones of the torchvision library~\citep{marcel2010torchvision}, except for the crop and resize that represent area ratios. For the Gaussian blur transformation we use alternatively $\sigma$, the scaling factor in the exponential, or the kernel size $k_b$ (in torchvision $k_b = (\sigma-0.35)/0.15$).
The ``Random'' transformation is the one used to develop the 50k query set.
A series of simple 1-4 AugLy transformations are picked at random, with skewed probability for a higher number.
Among the possible transformations, there are pixel-level, geometric ones, as well as embedding the image as a screenshot of a social network GUI.
\begin{table}[t]
\centering
\captionsetup{font=small}
\caption{Illustration of all transformations evaluated in Tab.~\ref{tab:act_vs_pas_retrieval}.}
\label{fig:all_transformations}
\begin{tabular}{*{5}{l}}
Identity & Contrast 0.5 & Contrast 2.0 & Brightness 0.5 & Brightness 2.0 \\
\begin{minipage}{.16\linewidth}\includegraphics[width=\linewidth]{figs/attacks/none.jpg}\end{minipage} &
\begin{minipage}{.16\linewidth}\includegraphics[width=\linewidth]{figs/attacks/contrast1.jpg}\end{minipage} &
\begin{minipage}{.16\linewidth}\includegraphics[width=\linewidth]{figs/attacks/contrast2.jpg}\end{minipage} &
\begin{minipage}{.16\linewidth}\includegraphics[width=\linewidth]{figs/attacks/brightness1.jpg}\end{minipage} &
\begin{minipage}{.16\linewidth}\includegraphics[width=\linewidth]{figs/attacks/brightness2.jpg}\end{minipage}
\\ \\
Hue 0.2 & Blur 2.0 & JPEG 50 & Rotation 25 & Rotation 90 \\
\begin{minipage}{.16\linewidth}\includegraphics[width=\linewidth]{figs/attacks/hue.jpg}\end{minipage} &
\begin{minipage}{.16\linewidth}\includegraphics[width=\linewidth]{figs/attacks/blur.jpg}\end{minipage} &
\begin{minipage}{.16\linewidth}\includegraphics[width=\linewidth]{figs/attacks/jpeg.jpg}\end{minipage} &
\begin{minipage}{.16\linewidth}\includegraphics[width=\linewidth]{figs/attacks/rotation1.jpg}\end{minipage} &
\begin{minipage}{.16\linewidth}\includegraphics[width=\linewidth]{figs/attacks/rotation2.jpg}\end{minipage}
\\ \\
Crop 0.5 & Resize 0.5 & Meme & Random \\
\begin{minipage}{.16\linewidth}\includegraphics[width=\linewidth]{figs/attacks/centercrop.jpg}\end{minipage} &
\begin{minipage}{.16\linewidth}\includegraphics[width=\linewidth]{figs/attacks/resize.jpg}\end{minipage} &
\begin{minipage}{.12\linewidth}\includegraphics[width=\linewidth]{figs/attacks/meme.jpg}\end{minipage} &
\begin{minipage}{.16\linewidth}\includegraphics[width=1.7\linewidth]{figs/attacks/auto.jpg}\end{minipage} &
\\
\end{tabular}
\end{table}
\begin{figure}[b!]
\centering
\includegraphics[width=0.8\linewidth,trim={0 0.4cm 0 0.2cm}, clip]{figs/exp/transformations.pdf}
\vspace*{-0.5cm}
\captionsetup{font=small}
\caption{
Average $R$@1 over 10k images indexed with IVF-PQ.
}
\label{fig:tranformations}
\end{figure}
\section{More Experimental Results}
\subsection{Detailed metrics on different image transformations}
On Fig.~\ref{fig:tranformations}, we evaluate the average $R$@$1$\@ over the 10k images from the reference dataset.
The experimental setup is the same as for Tab.~\ref{tab:act_vs_pas_retrieval} but a higher number of transformation parameters are evaluated.
As expected, the higher the strength of the transformation, the lower the retrieval performance is.
The decrease in performance is significantly reduced with activated images.
\vspace*{-0.3cm}
\subsection{Additional ablations}
\paragraph{Speeding-up the optimization.}\label{sec:speedup}
In our experiments, the optimization is done using 10 iterations of gradient descent, which takes approximately 40ms/image.
If the indexation time is important (often, this is not the case and only the search time is), it can be reduced at the cost of some accuracy.
We activated 10k reference images, with the same IVF-PQ indexed presented in Sec.~\ref{sec:act_vs_passive} with only one step of gradient descent with a higher learning rate.
Activation times are computed on average.
The $R$@$1$\@ results in Tab.~\ref{tab:speedup} indicate that the speed-up in the image optimization has a small cost in retrieval accuracy.
Specifically, it reduces the $R$@$1$\@ for unedited images.
The reason is that the learning rate is too high: it can cause the representation to be pushed too far and to leave the indexing cell.
This is why a higher number number of steps and a lower learning rate are used in practice.
If activation time is a bottleneck, it can however be useful to use less optimization steps.
\begin{table}[h]
\centering
\captionsetup{font=small}
\caption{$R$@$1$\@ for different transformations applied before search, with either 1 step at learning rate 10, or 10 steps at learning rate 1.}
\label{tab:speedup}
\vspace*{-0.5cm}
\resizebox{0.95\linewidth}{!}{
\begingroup
\setlength{\tabcolsep}{4pt}
\def1.1{1.1}
\begin{tabular}{ l|l| *{15}{c}}
\multicolumn{1}{c}{} & \multicolumn{1}{l}{\rot{Activation}} & \rot{Identity} & \rot{Contr. 0.5} & \rot{Contr. 2.0} & \rot{Bright. 0.5} & \rot{Bright. 2.0} & \rot{Hue 0.2} & \rot{Blur 2.0} & \rot{JPEG 50} & \rot{Rot. 25} & \rot{Rot. 90} & \rot{Crop 0.5} & \rot{Resi. 0.5} & \rot{Meme} & \rot{Random} & \rot{Avg.} \\ \midrule
Passive & - & 1.00 & 0.73 & 0.39 & 0.73 & 0.28 & 0.62 & 0.48 & 0.72 & 0.07 & 0.14 & 0.14 & 0.72 & 0.14 & 0.13 & 0.45 \\
\rowcolor{apricot!30} lr=1 - 10 steps & 39.8 ms/img & 1.00 & 1.00 & 0.96 & 1.00 & 0.92 & 1.00 & 0.96 & 0.99 & 0.10 & 0.50 & 0.29 & 1.00 & 0.43 & 0.32 & 0.75 \\
lr=10 - 1 step & 4.3 ms/img & 0.99 & 0.99 & 0.92 & 0.99 & 0.84 & 0.99 & 0.95 & 0.99 & 0.10 & 0.39 & 0.25 & 0.99 & 0.36 & 0.27 & 0.72 \\
\bottomrule
\end{tabular}
\endgroup
}
\vspace*{-0.3cm}
\end{table}
\paragraph{Data augmentation at indexing time and EoT.}
Expectation over Transformations~\citep{athalye2018eot} was originally designed to create adversarial attacks robust to a set of image transformations.
We follow a similar approach to improve robustness of the marked image against a set of augmentations $\mathcal{T}$.
At each optimization step, we randomly sample $A$ augmentations $\{t_i\}_{i=1}^A$ in $\mathcal{T}$ and consider the average loss:
$ \mathcal L _{\mathrm{f}} = \sum_{i=1}^{A} \mathcal{L}(I,t_i;I_o) /A $.
In our experiments, $\mathcal{T}$ encompasses rotations, Gaussian blurs, color jitters and a differentiable approximation of the JPEG compression~\cite{shin2017jpeg}.
$A$ is set to $8$ and we always take the un-augmented image in the chosen set of augmentations.
We activated 10k reference images, with the same IVF-PQ as Sec.~\ref{sec:act_vs_passive} with or without using EoT.
Table \ref{tab:eot} shows the average $R$@$1$\@ performance over the images submitted to different transformations before search.
EoT brings a small improvement, specifically on transformations where base performance is low (\textit{e.g}.\@ rotation or crops here).
However, it comes at a higher computational cost since each gradient descent iteration needs $A$ passes through the network, and since fewer images can be jointly activated due to GPU memory limitations (we need to store and back-propagate through $A$ transformations for every image).
If the time needed to index or activate an image is not a bottleneck, using EoT can therefore be useful.
Otherwise, it is not worth the computational cost.
\begin{table}[h]
\centering
\captionsetup{font=small}
\caption{$R$@$1$\@ for different transformations applied before search, with or without EoT when activating the images.}
\label{tab:eot}
\vspace*{-0.5cm}
\resizebox{0.95\linewidth}{!}{
\begingroup
\setlength{\tabcolsep}{4pt}
\def1.1{1.1}
\begin{tabular}{ l |l| *{15}{c}}
\multicolumn{1}{c}{} & \multicolumn{1}{l}{\rot{Activation}} & \rot{Identity} & \rot{Contr. 0.5} & \rot{Contr. 2.0} & \rot{Bright. 0.5} & \rot{Bright. 2.0} & \rot{Hue 0.2} & \rot{Blur 2.0} & \rot{JPEG 50} & \rot{Rot. 25} & \rot{Rot. 90} & \rot{Crop 0.5} & \rot{Resi. 0.5} & \rot{Meme} & \rot{Random} & \rot{Avg.} \\ \midrule
Without EOT & 40 ms & 1.00 & 1.00 & 0.96 & 1.00 & 0.92 & 1.00 & 0.96 & 0.99 & 0.10 & 0.50 & 0.29 & 1.00 & 0.43 & 0.32 & 0.75 \\
\rowcolor{apricot!30}
With EOT & 870 ms & 1.00 & 1.00 & 0.95 & 1.00 & 0.92 & 1.00 & 0.95 & 0.99 & 0.14 & 0.64 & 0.33 & 1.00 & 0.45 & 0.33 & 0.76 \\
\bottomrule
\end{tabular}
\endgroup
}
\vspace*{-0.8cm}
\end{table}
\pagebreak
\section{Active Indexing vs. Watermarking}\label{sec:watermarking}
\paragraph{Discussion.}
Watermarking and active indexing both modify images for tracing and authentication, however there are significant differences between them.
Watermarking embeds arbitrary information into the image. The information can be a message, a copyright, a user ID, etc.
In contrast, active indexing modifies it to improve the efficiency of the search engine.
Watermarking also focuses on the control over the False Positive Rate of copyright detection, \textit{i.e}.\@ a bound on the probability that a random image has the same message as the watermarked one (up to a certain distance).
Although watermarking considers different settings than indexing methods, it could also be leveraged to facilitate the re-identification of near-duplicate images. In this supplemental section, we consider it to address a use-case similar to the one we address in this paper with our active indexing approach.
In this scenario, the watermark encoder embeds binary identifiers into database images.
The decoded identifier is then directly mapped to the image (as the index of a list of images).
\paragraph{Experimental setup.}
In the rest of the section, we compare active indexing against recent watermarking techniques based on deep learning.
\begin{itemize}[leftmargin=0.5cm,itemsep=0cm,topsep=-0.1cm]
\item For indexing, we use the same setting as in Sec.~\ref{sec:experimental} (IVF-PQ index with 1M reference images).
When searching for an image, we look up the closest neighbor with the help of the index.
\item For watermarking, we encode $20$-bit messages into images, which allows to represent $2^{10}\approx 10^6$ images (the number of reference images).
When searching for an image, we use the watermark decoder to get back an identifier and the corresponding image in the database.
\end{itemize}
Like before, we use $R$@$1$\@ as evaluation metric.
For indexing, it corresponds to the accuracy of the top-1 search result.
For watermarking, the $R$@$1$\@ also corresponds to the word accuracy of the decoding, that is the proportion of images where the message is perfectly decoded.
Indeed, with $20$-bit encoding almost all messages have an associated image in the reference set, so an error on a single bit causes a mis-identification (there is no error correction\footnote{In order to provide error correction capabilities, one needs longer messages. This makes it more difficult to insert bits: in our experiments, with 64 bits we observe a drastic increase of the watermarking bit error rate. }).
We use two state-of-the-art watermarking methods based on deep learning:
SSL Watermarking~\citep{fernandez2022sslwatermarking}, which also uses an adversarial-like optimization to embed messages, and HiDDeN~\citep{zhu2018hidden}, which encodes and decodes messages thanks to Conv-BN-ReLU networks.
The only difference with the original methods is that their perturbation $\delta$ is modulated by the handcrafted perceptual attenuation model presented in App.~\ref{sec:perceptual}.
This approximately gives the same image quality, thereby allowing for a direct comparison between active indexing and watermarking.
\paragraph{Results.}
Tab.~\ref{tab:watermarking} compares the $R$@$1$\@ when different transformations are applied before search or decoding.
Our active indexing method is overall the best by a large margin.
For some transformations, watermarking methods are not as effective as passive indexing, yet for some others, like crops for HiDDeN, the watermarks are more robust.
\begin{table}[h]
\centering
\captionsetup{font=small}
\caption{$R$@$1$\@ for different transformations applied before search, when using either watermarking or active indexing. Results are averaged on 1k images. Best result is in \textbf{bold} and second best in \textit{italic}. }
\label{tab:watermarking}
\resizebox{0.99\linewidth}{!}{
\begingroup
\setlength{\tabcolsep}{4pt}
\def1.1{1.1}
\begin{tabular}{ p{4.0cm}| *{14}{c}c}
\multicolumn{1}{c}{} & \rot{Identity} & \rot{Contr. 0.5} & \rot{Contr. 2.0} & \rot{Bright. 0.5} & \rot{Bright. 2.0} & \rot{Hue 0.2} & \rot{Blur 2.0} & \rot{JPEG 50} & \rot{Rot. 25} & \rot{Rot. 90} & \rot{Crop 0.5} & \rot{Resi. 0.5} & \rot{Meme} & \rot{Random} & \rot{Avg.} \\ \midrule
Passive indexing & \bf 1.00 & 0.73 & 0.39 & 0.73 & 0.28 & 0.62 & 0.48 & \it 0.72 & \it 0.07 & 0.14 & 0.14 & \it 0.72 & 0.14 & 0.13 & 0.45 \\
Active indexing (ours) & \bf 1.00 & \bf 1.00 & \bf 0.96 & \bf 1.00 & \bf 0.92 & \bf 1.00 & \bf 0.96 & \bf 0.99 & \bf 0.10 & \bf 0.50 & \it 0.29 & \bf 1.00 & 0.43 & \bf 0.32 & \bf 0.75 \\ \midrule
\parbox{4.0cm}{SSL Watermarking \citep{fernandez2022sslwatermarking}} & \bf 1.00 & \it 0.98 & \it 0.53 & \it 0.98 & \it 0.63 & \it 0.85 & 0.13 & 0.00 & 0.00 & \it 0.15 & 0.11 & 0.00 & \it 0.46 & 0.07 & 0.42 \\ \midrule
\parbox{3.5cm}{HiDDeN\footnote{} \citep{zhu2018hidden}} & \it 0.94 & 0.87 & 0.36 & 0.85 & 0.55 & 0.00 & \it 0.81 & 0.00 & 0.00 & 0.00 & \bf 0.92 & 0.44 & \bf 0.77 & \it 0.16 & \it 0.48 \\
\bottomrule
\end{tabular}
\endgroup
}
\end{table}
\footnotetext{Our implementation.
As reported in other papers from the literature, results of the original paper are hard to reproduce.
Therefore to make it work better, our model is trained on higher resolution images (224$\times$224), with a payload of $20$-bits, instead of 30 bits embedded into 128$\times$128.
Afterwards, the same network is used on images of arbitrary resolutions, to predict the image distortion which is later rescaled as in Eq.~\eqref{eq:scaling}.
In this setting the watermark can not always be inserted (6\% failure).}
\newpage
\section{More Qualitative Results}\label{sec:more_qualitative}
Figure~\ref{fig:more_qualitative2} gives more examples of activated images from the DISC dataset, using the same parameters as in Sec.~\ref{sec:act_vs_passive}.
The perturbation is very hard to notice (if not invisible), even in flat areas of the images because the perceptual model focuses on textures.
We also see that the perturbation forms a regular pattern.
This is due to the image (bilinear) resize that happens before feature extraction.
Figure~\ref{fig:more_qualitative} gives example of an image activated at several values of perturbation strength $\alpha$ of Eq.~\eqref{eq:scaling}
(for instance, for $\alpha=20$ the image has PSNR $27$dB and for $\alpha=1$ the image has PSNR $49$dB).
The higher the $\alpha$, the more visible the perturbation induced by the activation is.
Nevertheless, even with low PSNR values ($<35$dB), it is hard to notice if an image is activated or not.
\begin{figure}[h]
\centering
\includegraphics[width=1.0\textwidth]{figs/qualitative/psnr/psnr.pdf}
\captionsetup{font=small}
\caption{Example of one activated image at different levels of $\alpha$.}
\label{fig:more_qualitative}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=0.9\textwidth, trim={0 0.4cm 0 0.4cm}, clip]{figs/qualitative/diffs/qual_0.jpg}
\includegraphics[width=0.9\textwidth, trim={0 0.4cm 0 0.4cm}, clip]{figs/qualitative/diffs/qual_2.jpg}
\includegraphics[width=0.9\textwidth, trim={0 0.4cm 0 0.4cm}, clip]{figs/qualitative/diffs/qual_3.jpg}
\includegraphics[width=0.9\textwidth, trim={0 0.4cm 0 0.4cm}, clip]{figs/qualitative/diffs/qual_4.jpg}
\includegraphics[width=0.9\textwidth, trim={0 0.4cm 0 0.4cm}, clip]{figs/qualitative/diffs/qual_1.jpg}
\captionsetup{font=small}
\caption{Example of activated images for $\alpha=3.0$. (Left) original images, (Middle) activated images, (Right) pixel-wise difference. Images are
\href{http://www.flickr.com/photos/87738888@N03/8039927199/}{R000005.jpg},
\href{http://www.flickr.com/photos/79653482@N00/7766743380/}{R000045.jpg},
\href{http://www.flickr.com/photos/12420018@N03/5452964454/}{R000076.jpg},
\href{http://www.flickr.com/photos/56594044@N06/5933951717/}{R000172.jpg} and
\href{http://www.flickr.com/photos/31369133@N04/5570867046/}{R000396.jpg}.
}
\label{fig:more_qualitative2}
\end{figure}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,925
|
US8082255B1 - Branding digital content - Google Patents
Branding digital content Download PDF
melded
Edward J. Carlson, Jr.
Steve Jernigan
Robert Hubbard
Ferol Vernon
eMinor Inc
2009-11-20 Application filed by eMinor Inc filed Critical eMinor Inc
2011-02-03 Assigned to eMinor Incorporated reassignment eMinor Incorporated ASSIGNMENT OF ASSIGNORS INTEREST (SEE DOCUMENT FOR DETAILS). Assignors: CARLSON, EDWARD J., JR., HUBBARD, ROBERT, JERNIGAN, STEVE, VERNON, FEROL, WATTS, ANDREW
230000000007 visual effect Effects 0 abstract claims description 72
238000000034 methods Methods 0 abstract claims description 64
238000009826 distribution Methods 0 claims description 8
230000002452 interceptive Effects 0 claims description 8
230000001960 triggered Effects 0 claims description 7
239000000470 constituents Substances 0 abstract 1
230000015654 memory Effects 0 claims 1
238000007906 compression Methods 0 description 1
230000000630 rising Effects 0 description 1
239000011435 rock Substances 0 description 1
230000001755 vocal Effects 0 description 1
G06Q—DATA PROCESSING SYSTEMS OR METHODS, SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL, SUPERVISORY OR FORECASTING PURPOSES; SYSTEMS OR METHODS SPECIALLY ADAPTED FOR ADMINISTRATIVE, COMMERCIAL, FINANCIAL, MANAGERIAL, SUPERVISORY OR FORECASTING PURPOSES, NOT OTHERWISE PROVIDED FOR
G06Q10/00—Administration; Management
G06Q10/08—Logistics, e.g. warehousing, loading, distribution or shipping; Inventory or stock management, e.g. order filling, procurement or balancing against orders
G06Q10/087—Inventory or stock management, e.g. order filling, procurement, balancing against orders
G06Q30/00—Commerce, e.g. shopping or e-commerce
G06Q30/06—Buying, selling or leasing transactions
G06Q30/0601—Electronic shopping
G06Q30/0621—Item configuration or customization
Y10S707/00—Data processing: database and file management or data structures
Y10S707/912—Applications of a database
Y10S707/913—Multimedia
Y10S707/964—Database arrangement
Y10S707/966—Distributed
A process for creating a melded visual image to accompany a delivery of digital content. The melded image including at least a digital image associated with the digital content and a first placement image. The first placement image selected for delivery with the digital image in accordance with the execution of one or more rules concerning the digital content or the end user receiving the digital content. The melded image may contain constituent parts that have been altered such as changing the aspect ratio or transparency so that the digital image and the placement image can be displayed on a display screen associated with at least one class of player device that may play the digital content. The placement image may be part of an advertising campaign.
This application claims benefit of and incorporates by references U.S. Provisional Application No. 61/116,868 filed Nov. 21, 2008.
This disclosure relates generally to advertising and to processes for distributing digital content including distributions over a network such as a site that provides music downloads.
Process of Music Downloads
One likely use of the present disclosure is in connection with music download web sites. Thus, it is useful to review the concept of music downloads. One way of acquiring music downloads is via a download button on the website that appears in proximity to the digital content, or via social network applications or widgets that contain the digital content. To acquire the digital content through this route the user must activate a download button. This may be described as a user-pull operation as the user pulls the specific download.
An alternative to a user-pull operation is a push operation where a user is designated to receive certain types of content (such as a selection of the week) based on particular criteria and then material is delivered to the end user.
The examples provided below to illustrate implementations of the teachings of the present invention focus on digital content for music, but nothing in this disclosure is necessarily limited to the use with music files as opposed to other digital content. Likewise, nothing should be interpreted as limiting this disclosure to audio files as opposed to audio/visual content (such as movies, television programs, or applications such as computer games), or pure visual content such as electronic books, electronic newspapers, or representations of works of art.
Aspects of the teachings contained within this disclosure are addressed in the claims submitted with this application upon filing. Rather than adding redundant restatements of the contents of the claims, these claims should be considered incorporated by reference into this summary.
This summary is meant to provide an introduction to the concepts that are disclosed within the specification without being an exhaustive list of the many teachings and variations upon those teachings that are provided in the extended discussion within this disclosure. Thus, the contents of this summary should not be used to limit the scope of the claims that follow.
Inventive concepts are illustrated in a series of examples, some examples showing more than one inventive concept. Individual inventive concepts can be implemented without implementing all details provided in a particular example. It is not necessary to provide examples of every possible combination of the inventive concepts provided below as one of skill in the art will recognize that inventive concepts illustrated in various examples can be combined together in order to address a specific application.
Other systems, methods, features and advantages of the disclosed teachings will be or will become apparent to one with skill in the art upon examination of the following figures and detailed description. It is intended that all such additional systems, methods, features and advantages be included within the scope of and be protected by the accompanying claims.
BRIEF DESCRIPTION OF THE FIGURES
The invention can be better understood with reference to the following figures. The components in the figures are not necessarily to scale, emphasis instead being placed upon illustrating the principles of the invention. Moreover, in the figures, like reference numerals designate corresponding parts throughout the different views.
FIG. 1 is a flow chart with a high level representation of one implementation of the process for delivering melded visual content along with delivered digital content.
FIG. 2 is a graphic representation of one implementation of the process for delivering melded visual content along with delivered digital content.
FIG. 3 is a representation of one implementation of a process to create a melded image with a placement image.
Process of Providing Melded Image with Delivery of Digital Content
FIG. 1 is a flow chart with a high level representation of one implementation of the process for delivering melded visual content along with delivered digital content. FIG. 2 is a graphic representation of one implementation of the process for delivering melded visual content along with delivered digital content.
Delivered digital content includes audio, visual, or audio-visual content such as music, books, or videos. It includes digital content of other sorts including games, software applications, or applets. The scope of the disclosed method is not to be limited by the type of digital content. The melded visual content could be provided within the delivered digital content as part of a single file. For example the melded image could be part of Meta data that is accessible by a device that uses the digital content. In other instances, the melded image may be in a separate file sent with a set of files provided as the digital content. As digital content is not closely linked to the confines of a single file, it is perhaps easier to reference the delivery and playing of digital content or a data set than to describe it as the delivery and playing of a "file".
STEP 1006—The rights holder 204 places the digital content 208 onto the web site. The process of creating and delivering a melded image does not rely on a particular process for providing the digital content 208 to the web site. The digital content could be uploaded to a web site or the digital content could be provided by physical media or provided to the web site by an application interface that does not use a web browser. While a web site is a useful example that helps convey the concept, the present disclosure could be used with some other repository of digital content that provides digital content to recipients. An on-demand delivery service for downloadable content via any type of communication form could implement teachings of the present disclosure.
The rights holder may be an individual that has recorded a song or produced a video, authored, or otherwise created the digital content. The rights holder may simply be the current "Rights Holder" of such content as the rights may have passed one or more times and ended up with the current rights holder. The rights holder may be a record label, Disney, a publisher or some other entity that owns the rights to the digital content.
The concept of required rights for various types of digital content varies: A) over time, B) across digital content classes, and C) across countries. In some instances, many or all people may have sufficient rights to make the digital content available through the web site. In those instances, people with sufficient rights to place the content on the web site are "rights holders" even if they do not have unique rights. This may be the situation where the digital content is in the public domain or in a system where peer to peer file sharing is accepted.
The digital content may include components that may be selected and delivered separately. For example, if the digital content is a music album, then it may be possible to obtain individual songs. Alternatively, it may be possible to obtain separate tracks, for example the vocals separately from the music or the lead guitar separately from the rest of the music. The variety of ways that the digital content may be subdivided and distributed are not at the core of this disclosure but the disclosure is intended to include any parsing of digital content into segments that may be delivered to a recipient. The format of the digital content is not central to this disclosure and will evolve over time as new formats for communicating digital content are developed. As an example for music, the music digital content may be in the form of an MP3. (MP3 is the file extension for a compressed audio file according to the conventions of MPEG Audio Layer-3 format)
STEP 1012—The rights holder places visual art work onto the web site to be associated with this digital content as base art work which can be called the base image 212. While it is possible that a rights holder may place a separate image for each sub-component of the digital content (such as a separate image for each song), it is more likely that the rights holder will have one image as base art work for the entire album. In some instances, the rights holder may not have album art work for this particular digital content and may simply use a general image of the artist that performed the digital content. Thus, the base image 212 may be provided to the site before, with, or after delivery of the digital content 208. There may be a number of different items of digital content 208 that become associated with the same base image 212.
STEP 1018—Depending on the scheme used for creating the melded image 424, it may be necessary to alter the aspect ratio of the base image 212 to allow room in the melded image 424 for a placement image (308, 318, or 328) provided by an advertiser (304, 314, or 324).
For example, as shown in FIG. 3, if the base image 212 has a 1 to 1 aspect ratio and the space allocated in the melded image 424 calls for the selected placement image 404 to have the bottom 25% of a 1 to 1 aspect ratio image, then the base image needs to be modified to a revised image 216 with a 4 to 3 aspect ratio. The combination of a 4 to 3 revised image 216 and a 4 to 1 selected placement image 404 produces the 1 to 1 melded image 424. The revised image may be created through stretching or compressing the base image, cropping a portion of the base image, or a combination of both.
In the event that the base image 212 is in the desired aspect ratio for creating the melded image 424, then the revision process does not do anything and the revised image 216 is the same as the base image 212. A variation of this concept is for the rights holder to provide a set of base images with different aspect ratios, (perhaps through selective cropping of the image) so that either no additional change is needed to adjust the aspect ratio or the changes made by stretching/compression are less dramatic as a base image close to the desired aspect ratio was selected as the most appropriate base image.
Step 1024—Load placement image. The placement image (308, 318, or 328) is what an advertiser (304, 314, or 324) wants to put before the digital content recipient 450. The placement image (308, 318, or 328) becomes part of the melded image 424. A digital content site is apt to receive digital content 208 with associated base images 212 on an ongoing basis. Likewise the digital content site is apt to receive placement images (308, 318, or 324) on an ongoing basis. The order of when a particular base image arrives and a particular placement image arrives is not critical as long as a base image is present so that a revised image may be created and combined with a selected placement image 404 to form a melded image 424.
Step 1030—Combine the selected placement image 404 with the revised image 216 to form the melded image 424 as shown in FIGS. 2 and 3.
Step 1036—Provide the melded image 424 and digital content 208 to a recipient 450. The recipient 450 may receive the digital content 208 and melded image 424 via a download, via streaming, via receipt of digital media, or via an analogous process. The recipient may be designated to receive a particular piece of digital content in a number of ways such as A) using various menu or search tools, B) downloading this digital content as part of a set of related digital content (such as all the songs on a album), or C) signing up for a subscription that causes certain types of digital content to be delivered in the future. How the recipient selects the digital content is not part of the focus of the present disclosure.
Typically, the movement of the digital content 208 to the recipient 450 would be over a communications network such as the Internet or a telephone network, but it could be through other communication networks or through the delivery of physical media such as a disc with a set of digital content on the physical media.
Step 1042—Recipient 450 moves digital content 208 and melded image 424 to a player device 454. In some instances the act of providing the digital content to the recipient is the same as the act of moving the digital content to the player device as the digital content is delivered directly to the player device (such as an IPOD® brand electronic player device or a cellular telephone adapted to receive and use digital content). In other instances, the digital content would be received by an intermediate device such as a laptop computer and communicated to a player device. In yet another instance, the player device may be the laptop computer. The communication to the player device may involve wired communication, wireless, communication, or the movement of physical media containing the digital content 208 and the melded image 424.
Step 1048—Recipient 450 uses the player device 454 to play the digital content 208 and a display screen 458 associated with the player device 454 displays the melded image 424 including the selected placement image 404. For digital content 208 that is exclusively audio content, the melded image 424 may be displayed during the entire time the digital content is played. For digital content 208 that is a mix of audio and visual content or just visual content, the melded image 424 may be displayed only when seeking a title to play (like a box for a rental movie), immediately before the digital content 208 is played, or during pauses of the digital content 208. These examples are provided to show the variety of options for when the melded content 424 is displayed. One of skill in the art will recognize that the actual display of the melded content 424 being driven by the way that the player device 454 handles a representative graphic image for particular digital content. The player device 454 that displays visual content as part of the digital content 208 may have a separate visual screen for this use, may use a portion of the screen that displays the visual content that is used for the melded image, or may display the visual content for selected times before or after the display of the visual content.
Note that if the recipient 450 makes a copy of the digital content 208 and distributes that copy to either a second player device belonging to the recipient 450 or to a third party, the melded image 424 may be passed with the digital content 208. Thus there may be secondary distribution of the selected placement image 404 within the melded image 424 even if the digital content 208 is redistributed contrary to rights held or reserved by the rights holder 204.
Selection of Placement Image
1) Placement Based On Digital Content & Placement Price
The selection of the placement image may be done at least in part based upon a request from an advertiser to place a placement image into the melded image for a particular piece of digital content. The advertiser ABC might authorize distribution of 10,000 placement images with delivery of digital content "A" for a placement price of 5 money units per delivery. Advertiser DEF may authorize distribution of 1,000 placement images of the same digital content "A" for a placement price of 8 money units per delivery. Assuming that both of these advertisers placed these authorizations before the digital content was available for delivery and no other advertiser makes an authorization for payment:
the first 1,000 requests for delivery would contain a melded image with the placement image from advertiser DEF;
the next 10,000 deliveries of this digital content would have the melded image containing the placement image from advertiser ABC; and
after 11,000 deliveries, the digital content may be provided with the base image rather than the melded image or the digital content may optionally be provided with a melded image that includes a placement image from the digital content site to advertise itself.
The concept of offering a placement price for a particular piece of digital content could be extended to all digital content from a given album or from a particular artist or group. The concept could be further extended to all digital content within a particular micro-genre such as Celtic instrumental music or a particular style of rap music but only if performed by a female artist.
2) Placement Based Upon Demographics
An implementation may be created that selects the placement image based upon the demographics of the digital content recipient. The demographics may be based upon a combination of sources. The sources may include:
information that the digital content recipient provided to the site when registering with the site;
information about the digital content recipient obtained from other information repositories; and
information obtained about the digital content recipient through interactions with a network such as:
the site from which the digital content recipient entered the present site,
the server location for the digital content recipient,
the frequency of downloads by this digital content recipient; and
the types of digital content sought by this digital content recipient including a categorization of the digital content for this content request.
Thus a given digital content recipient, Pat, may have a series of demographic attributes associated with Pat. For sake of this example, assume that Pat has 26 attributes which we will call A, B, C, . . . X, Y, Z. If advertiser BM1 offers a placement price of 5 money units per placement for 1,000 placements for digital content recipients with demographic qualities A, B, C, and D, then Pat would be a qualified digital content recipient for this placement. However, if some other advertiser BM2 has authorized a placement price of 7.5 money units for 2,000 placements to digital content recipient with demographic qualities A, E, I, O, U, and Y, the system would insert the placement for BM2 into Pat's digital content's melded image if the 2,000 placement authorization had not been exhausted.
The demographic based system works particularly well for sites with many instances of digital content that may not be familiar to particular advertisers. This allows the advertisers to place their images onto melded images for digital content deliveries for the artists that are rising from obscurity to popularity before the popularity has reached the point where the advertisers are vying to offer placement prices for that particular artist.
An example of a set of demographics for selecting a digital content recipient is an advertiser seeking to reach a music fan that lives in the United States, is 17 to 24 years old, is seeking to download digital content from "rock" genre and is a female.
3) Placement Based Upon a Blended Model
In a blended model, when digital content recipient Pat seeks to download digital content A, then the highest available placement price is selected based upon either the demographics or the digital content specific placements. In this example, the highest price for placement for this download is from Advertiser ABC as 8 money units for delivery of a placement image with delivery of digital content A is more than 7.5 money units offered by BM2 for a delivery of the placement image to someone with demographic qualities A, E, I, O, U, and Y.
Reports to Advertisers
Advertisers may receive reports that indicate how many placements were made in a given time period and information about the individual placements such as:
digital content;
demographic profile factors of interest to the advertiser; and
date and time of the download.
Alternative Implementations
The examples given above are intended to provide insight on how the present disclosure may be used. The examples are not exhaustive and are not intended to recite every possible use of the present disclosure. Certain variations of the implementation options are discussed below.
Location of the Placement Image
In order to provide a concrete example for a drawing, FIG. 3 shows the selected placement image 404 being placed adjacent and below the revised image 216. The teachings of the present disclosure apply equally to placement images placed above or to either side of the revised image 216.
Overlap of Placement Image onto the Revised Image
The example uses a revised image 216 and a selected placement image 404 that are placed adjacent to one another and do not overlap one another. This is not necessary for the practice of the present invention. Thus the selected placement image 404 may be placed over the revised image 216 to partially cover some of the revised image 216. The selected placement image 404 placed over the revised image may have some level of transparency so that the colors of the revised image 216 are visible through the revised image but certain text or graphics of the placement image are opaque or less transparent than other portions of the selected placement image.
Aspect Ratio of the Melded Image
While a 1 to 1 aspect ratio for the melded image may be particularly convenient for use by a wide range of player devices, the present disclosure does not require any particular aspect ratio for the melded image. Consequently, the present disclosure does not require specific aspect ratios for the revised image or the placement image.
Variable Aspect Ratio for Revised Images, Placement Images, or Melded Images
While for the sake of simplicity, it may be desirable to have the same process for creating all melded images, this is not required. Thus not all revised images 216 need to have a particular aspect ratio and not all selected placement images 404 of another particular aspect ratio in order to combine the two images to form every time a melded image 424 of a particular aspect ratio. For example, the aspect ratio for the revised image for a particular base image does not have to be constant, as it may be 4 to 3, or slightly different aspect ratios such as 5 to 4 or 3 to 2. Having the ability to make revised images of different aspect ratios allows the different revised images to be used with placement images of different aspect ratios so that the melded image has a particular desired aspect ratio.
Likewise if the desired aspect ratio for the melded image varies from situation to situation, the aspect ratio of the revised image and possibly of the placement image may be modified to achieve that desired aspect ratio for the melded image used in that specific instance of a delivery of that digital content.
A variation in the optimal aspect ratio may arise from providing digital content to be used with different player devices with different aspect ratios in their displays.
Relationship between Advertiser and Placement Image
For the sake of simplicity, FIG. 2 has a set of three advertisers each providing one placement image. It is likely that some, perhaps most advertisers, will have a number of different placement images on the site at any given time. Sometimes the placement images will be targeted towards different end users and thus have different rules for selecting which digital content deliveries are eligible to receive a particular placement image. Sometimes a particular campaign may have several placement images that are used for the same group of end users and the web site will rotate through the set. Within the scope of the teachings contained herein, the placement image provided with the digital content may actually be a series of several placement images that are all delivered with a single delivery of digital content but are displayed at different times in different versions of the melded image when the end user is interacting with the digital content via the player device. Thus, the melded image may not be a static image but is digital information sufficient for the player device to create the melded image.
As an example of the use of the present disclosure was for music downloads, the term download is used repeatedly in this disclosure. The way the digital content is delivered is not central to the use of the disclosed concepts. The delivery of digital content could be triggered by an end-user pull—a request from the end user to deliver specific digital content, or a content push where digital content is pushed to various end users in accordance with a previously determined rule.
The digital content could be delivered by any type of carrier media including systems currently associated with the Internet network, telephone networks, and broadcast networks such as radio or television. In some contexts, the type of delivery may be described as streaming rather than a download. The present disclosure could be extended to digital content that is provided via tangible media such as a disc (including Compact Disc, DVD, or other type of disc) or a portable drive such as a flash drive or SD card.
The device that receives the digital content could be a portable personal media player such as an IPOD® brand electronic player device, MP3 player, or other device. The device that receives the digital content could be a laptop or other computer including one adapted to be a media server. The device that receives the digital content could be a device that communicates with a network to send and receive data such as a telephone device, Blackberry® device, or analogous device. The player device could be a device that specializes in the receipt and display of electronic books.
The device that receives digital content could be a device that receives broadcast content such as a radio in a home or automobile (including one adapted to receive XM radio or analogous transmissions), or a television device.
The device may receive the digital content through a communication port. The communication port may be adapted for use with a wired communication protocol. The communication port may include components such as an antenna to allow the receipt of wireless communications and may allow the transmission from the device of wireless communication.
The device that receives digital content could be one that is not normally connected to a communication network for receiving broadcast information but can play digital media such as a camera, video camera, digital picture frame, video player, game station device, or other analogous device.
If the digital content is intended for use with an external network such as the Internet or a telephone network, the placement image and thus the melded image may include interactive content. One example is clickable content that provides an instruction to the device currently holding the melded image to link the recipient to a particular location on that network. For example, the clickable content may cause the player device to display a web page from a particular address. The clickable content may cause identifying information about the recipient or the device to be registered such as registration to a newsletter or a fan club. The range of activities that could be enabled by the passing of clickable content in the placement image portion of the melded image is not limited by these few examples.
The interactive content does not need to respond to a "click." The interactive content could respond to any type of user input. Examples of user input include actions such as:
tapping a touch screen,
hovering an x-y input cursor over a portion of the placement image, and even
inaction as in some cases the failure of a user to proceed with making a selection may result in a time-out that automatically provides the user with an option to make a different selection or routes the user to an address where the user may receive additional help.
Distribution of Software
It is also important to note that although the present disclosure has been described in the context of a fully functional computer system, those skilled in the art will appreciate that the mechanisms of the present disclosure are capable of being distributed as a program product or a portion of a suite of programs. This distribution may be done in a variety of forms. The inventiveness of the present disclosure is present in a set of computer instructions adapted to implement some or all of the innovations described above regardless of how this set of instructions is conveyed. A set of computer instructions is a set of instructions adapted for use by a computer in achieving some or all of the advantages set forth above and is distinguishable from a paper such as this disclosure that describes the attributes of an implementation without providing anything that can be processed by computer components available in 2008 to ultimately be executed by a computer.
One of skill in the art will recognize that the set of computer instructions may be stored on one or more mass storage memory devices that are accessible by a particular computer system to implement some or all of the innovations described above. The set of computer instructions may be conveyed in one of many types of signal bearing media. Signal bearing media carrying instructions to be executed by one or more computer programming languages may be conveyed in different formats including, without limitation, program instructions in high level programming languages or in machine code.
The signal bearing media may be located on traditional articles of manufacture that are any one of a variety of recordable type media such as floppy disks or compact discs (including write once and re-recordable media). In this instance the recordable type media receives a written set of computer instructions which can subsequently be read by an input device. The recordable type media may then be shipped from one place to another such as shipped to a customer and then the customer may access the computer instructions written into the recordable type media.
A separate category of signal bearing media not currently considered a traditional article of manufacture under the United States patent laws is a paper printout carrying the sequence of computer instructions in at least one computer software language. One of skill in the art will recognize that an appropriate scanner may read paper through such routes as bar code readers, optical character recognition (OCR) of text, or via detection of holes in paper cards or paper tape.
The signal bearing media may be any of the many transmission type media such as analog or digital communication links as the software may be conveyed to a purchaser without the shipment of permanent tangible media but through a transitory propagating signal such as a series of Internet protocol packets.
To the extent that the relevant patent laws allow issuance of claims covering each of these three types of signal bearing media, (recordable media, paper printout, and transmission type media), then it is the intent to include such signal bearing media within the scope of relevant claims.
One of skill in the art will recognize that some of the alternative implementations set forth above are not universally mutually exclusive and that in some cases additional implementations can be created that employ aspects of two or more of the variations described above. Likewise, the present disclosure is not limited to the specific examples or particular embodiments provided to promote understanding of the various teachings of the present disclosure.
1. A process for creating a melded visual image to accompany a delivery of digital content, the process comprising:
storing a first collection of digital content in a first memory device accessible to a first computer system;
storing a first image data set containing at least a first digital image associated with the first collection of digital content in a second memory device accessible to the first computer system;
storing a second collection of digital content in a third memory device accessible to the first computer system;
storing a second image data set containing at least a second digital image associated with the second collection of digital content in a fourth memory device accessible to the first computer system;
storing a first digital representation of a first placement image associated with a first rule for selecting digital content deliveries as eligible to include the first placement image;
recognizing that a planned delivery of the first collection of digital content is eligible under the first rule to include the first placement image;
delivering to a first communication network, the first collection of digital content with a first melded data set containing a digital representation of a first melded visual image comprising a combination of the first digital image and the first placement image, the first melded data set delivered in a first format which allows at least one class of player devices capable of using the first collection of digital content to display the first melded visual image in association with the first collection of digital content;
recognizing that a planned delivery of the second collection of digital content is eligible under the first rule to include the first placement image; and
delivering to a second communication network, the second collection of digital content with a second melded data set containing a digital representation of a second melded visual image comprising a combination of the second digital image and the first placement image, the second melded data set delivered in a second format that may be the same as the first format and which allows at least one class of player devices capable of using the second collection of digital content to display the second melded visual image in association with the second collection of digital content.
2. The process of claim 1 wherein the first melded visual image contains a representation of the first digital image adjacent to a representation of the first placement image.
3. The process of claim 2 wherein a digital representation of the first digital image within the first melded image does not contain all of the first digital image.
4. The process of claim 1 wherein the representation of the first digital image within the first melded image use an aspect ratio that is different from the first digital image.
5. The process of claim 1 wherein the first melded visual image contains a representation of the first digital image under a representation of the first placement image.
6. The process of claim 5 wherein the first placement image is at least partially transparent so that the first digital image is visible through the first placement image.
7. The process of claim 1 wherein a representation of the first placement image within the first melded visual image has an aspect ratio that is different from the first placement image.
8. The process of claim 1 further comprising delivering the first cluster of digital content a second time with a representation for the first digital image but without the first placement image.
9. The process of claim 1 wherein output from a player device of the first cluster of digital content includes audio output.
10. The process of claim 1 wherein output from a player device of the first cluster of digital content includes a display of visual content.
11. The process of claim 10 wherein the output from a player device of the first cluster of digital content includes both audio output and the display of visual content.
12. The process of claim 1 wherein a microprocessor-based computer adjusts an aspect ratio of the first digital image for use in the first melded visual image.
13. The process of claim 1 wherein the planned delivery of the first collection of digital content is triggered by an end user communicating across a communication network requesting delivery of the first collection of digital content.
14. The process of claim 13 wherein the first rule is triggered at least in part by an attribute of the first collection of digital content.
15. The process of claim 13 wherein the first rule is triggered at least in part by an attribute of the end user.
16. The process of claim 1 wherein the first melded data set includes interactive content associated with a portion of the first melded visual image so that an input device associated with the at least one class of player devices may trigger an action via the interactive content.
17. The process of claim 1 wherein at least two of the first memory device, second memory device, third memory device, and fourth memory device are implemented in a common memory device.
18. The process of claim 1 wherein the first communication network is also the second communication network.
19. A process for creating a melded visual image to accompany a delivery of digital content, the process comprising:
recognizing that a first planned delivery of the first collection of digital content is eligible under the first rule to include the first placement image;
storing a second digital representation of a second placement image associated with a second rule for selecting digital content deliveries as eligible to include the second placement image;
recognizing that a second planned delivery of the first collection of digital content is eligible under the second rule to include the second placement image; and
delivering to a second communication network, the first collection of digital content with a second melded data set containing a digital representation of a second melded visual image comprising a combination of the first digital image and the second placement image, the second melded data set delivered in a second format that may be the same as the first format and which allows at least one class of player devices capable of using the first collection of digital content to display the second melded visual image in association with the first collection of digital content.
20. The process of claim 19 wherein the first melded visual image contains a representation of the first digital image adjacent to a representation of the first placement image.
21. The process of claim 20 wherein a digital representation of the first digital image within the first melded image does not contain all of the first digital image.
22. The process of claim 19 wherein a digital representation of the first digital image within the first melded visual image uses an aspect ratio that is different from the first digital image.
23. The process of claim 19 wherein the first melded visual image contains a representation of the first digital image under a representation of the first placement image.
24. The process of claim 23 wherein the first placement image is at least partially transparent so that the first digital image is visible through the first placement image.
25. The process of claim 19 wherein a representation of the first placement image within the first melded visual image has an aspect ratio that is different from the first placement image.
26. The process of claim 19 further comprising delivering the first collection of digital content a second time with a representation of the first digital image but without the first placement image.
27. The process of claim 19 wherein output from a player device of the first cluster of digital content includes audio output.
28. The process of claim 19 wherein output from a player device of the first cluster of digital content includes a display of visual content.
30. The process of claim 19 wherein a microprocessor-based computer adjusts an aspect ratio of the first digital image for use in the first melded visual image.
31. The process of claim 19 wherein the first planned delivery is triggered by an end user communicating across a communication network requesting delivery of the first collection of digital content.
34. The process of claim 19 wherein the first melded data set includes interactive content associated with a portion of the first melded visual image so that an input device associated with the at least one class of player devices may trigger an action via the interactive content.
35. The process of claim 19 further comprising:
recognizing that a third planned delivery of the second collection of digital content is eligible under the first rule to include the first placement image; and
delivering to a third communication network, the second collection of digital content with a third melded data set containing a digital representation of a third melded visual image comprising a combination of the second digital image and the first placement image, the third melded data set delivered in a third format that may be the same as the first format and which allows at least one class of player devices capable of using the second collection of digital content to display the third melded visual image in association with the second collection of digital content.
36. The process of claim 35 wherein at least two of the first memory device, second memory device, third memory device, and fourth memory device are implemented in a common memory device.
37. The process of claim 35 wherein at least two of the first communication network, the second communication network, and the third communication network are implemented on a common communication network.
38. A player device for playing a selected cluster of digital content from a set of at least two clusters of digital content, the player device comprising:
at least one memory for storing digital information;
at least one display used at least in part to display visual content associated with but separate from the selected cluster of digital content;
at least one component for receiving a cluster of digital content and visual content associated but separate from the cluster of digital content;
a first cluster of digital content with a first melded visual image associated but separate from the first cluster of digital content wherein the first melded visual image comprises a representation of a first visual image associated with the first cluster of digital content melded with a first placement image provided by a third party unconnected to creation or distribution of the first cluster of digital content; and
a second cluster of digital content with second visual content associated but separate from the second cluster of digital content wherein the second melded visual image comprises a representation of a second visual image associated with the second cluster of digital content melded with the first placement image provided by the third party unconnected to creation or distribution of the second cluster of digital content such that the player device contains at least the first visual content and the second visual content each containing a representation of the first placement image provided by the third party.
39. The player device of claim 38 wherein the first melded visual image is displayed by the player device to simultaneously display a representation of the first visual image associated with the first cluster of digital content melded with the first placement image provided by the third party unconnected to the creation or distribution of the first cluster of digital content.
40. A process for altering a visual image displayed by a digital player device, the process comprising:
recognizing that a first planned delivery of the first collection of digital content is eligible under the first rule to include the first placement image; and
recognizing that a second planned delivery of the second collection of digital content is eligible under the first rule to include the first placement image; and
loading onto a player device the first collection of digital content with the first melded data set containing the first digital representation of the first melded visual image comprising the combination of the first digital image and the first placement image; and
interacting with the player device to select the first collection of digital content and thus cause display of the first melded visual image.
43. The process of claim 40 wherein the first communication network is also the second communication network.
delivering to a third communication network, the second collection of digital content with a third melded data set containing a digital representation of a third melded visual image comprising a combination of the second digital image and the first placement image, the third melded data set delivered in a third format that may be the same as the first format which allows at least one class of player devices capable of using the second collection of digital content to display the third melded visual image in association with the second collection of digital content.
loading onto a player device the first collection of digital content with the first melded data set containing the digital representation of the first melded visual image comprising the combination of the first digital image and the first placement image; and
US12/623,065 2008-11-21 2009-11-20 Branding digital content Active 2030-08-13 US8082255B1 (en)
US12/623,065 US8082255B1 (en) 2008-11-21 2009-11-20 Branding digital content
US12/623,065 Active 2030-08-13 US8082255B1 (en) 2008-11-21 2009-11-20 Branding digital content
WO2014115136A1 (en) * 2013-01-28 2014-07-31 Sanderling Management Limited Dynamic promotional layout management and distribution rules
US9825898B2 (en) 2014-06-13 2017-11-21 Snap Inc. Prioritization of messages within a message collection
US9843720B1 (en) 2014-11-12 2017-12-12 Snap Inc. User interface for accessing media at a geographic location
US20170374003A1 (en) 2014-10-02 2017-12-28 Snapchat, Inc. Ephemeral gallery of ephemeral messages
US9881094B2 (en) 2015-05-05 2018-01-30 Snap Inc. Systems and methods for automated local story generation and curation
US10080102B1 (en) 2014-01-12 2018-09-18 Investment Asset Holdings Llc Location-based messaging
US10102680B2 (en) 2015-10-30 2018-10-16 Snap Inc. Image based tracking in augmented reality systems
US10123166B2 (en) 2015-01-26 2018-11-06 Snap Inc. Content request by location
US10154192B1 (en) 2014-07-07 2018-12-11 Snap Inc. Apparatus and method for supplying content aware photo filters
US10157449B1 (en) 2015-01-09 2018-12-18 Snap Inc. Geo-location-based image filters
US10165402B1 (en) 2016-06-28 2018-12-25 Snap Inc. System to track engagement of media items
US10203855B2 (en) 2016-12-09 2019-02-12 Snap Inc. Customized user-controlled media overlays
US10219111B1 (en) 2018-04-18 2019-02-26 Snap Inc. Visitation tracking system
US10223397B1 (en) 2015-03-13 2019-03-05 Snap Inc. Social graph based co-location of network users
US10319149B1 (en) 2017-02-17 2019-06-11 Snap Inc. Augmented reality anamorphosis system
US10327096B1 (en) 2018-03-06 2019-06-18 Snap Inc. Geo-fence selection system
US10334307B2 (en) 2011-07-12 2019-06-25 Snap Inc. Methods and systems of providing visual content editing functions
US10348662B2 (en) 2016-07-19 2019-07-09 Snap Inc. Generating customized electronic messaging graphics
US10354425B2 (en) 2015-12-18 2019-07-16 Snap Inc. Method and system for providing context relevant media augmentation
US10387730B1 (en) 2017-04-20 2019-08-20 Snap Inc. Augmented reality typography personalization system
US10387514B1 (en) 2016-06-30 2019-08-20 Snap Inc. Automated content curation and communication
US10423983B2 (en) 2014-09-16 2019-09-24 Snap Inc. Determining targeting information based on a predictive targeting model
US10430838B1 (en) 2016-06-28 2019-10-01 Snap Inc. Methods and systems for generation, curation, and presentation of media collections with automated advertising
US10474321B2 (en) 2015-11-30 2019-11-12 Snap Inc. Network resource location linking and visual content sharing
US10499191B1 (en) 2017-10-09 2019-12-03 Snap Inc. Context sensitive presentation of content
US10523625B1 (en) 2017-03-09 2019-12-31 Snap Inc. Restricted group content collection
US20010044779A1 (en) * 2000-04-04 2001-11-22 Shin Iima Transmission apparatus and method, reception apparatus and method, management apparatus and method, charging apparatus and method, providing apparatus and method, and recording medium
US6338044B1 (en) * 1999-03-17 2002-01-08 Loudeye Technologies, Inc. Personal digital content system
US20030028432A1 (en) * 2001-08-01 2003-02-06 Vidius Inc. Method for the customization of commercial product placement advertisements in digital media
US20030191816A1 (en) * 2000-01-11 2003-10-09 Spoovy, Llc System and method for creating and delivering customized multimedia communications
US20040268413A1 (en) * 2003-05-29 2004-12-30 Reid Duane M. System for presentation of multimedia content
WO2006000967A1 (en) * 2004-06-22 2006-01-05 Koninklijke Philips Electronics N.V. System for transaction of digital content
US7188342B2 (en) * 2001-04-20 2007-03-06 Microsoft Corporation Server controlled branding of client software deployed over computer networks
US20090012873A1 (en) * 2007-07-05 2009-01-08 Mike Hamling Systems for managing digital media distribution
US20090053992A1 (en) * 2007-07-05 2009-02-26 Butler Jon F Systems and methods for ordering and delivering digital content
"Be My Patron." Obtained Nov. 17, 2009 from a link at the end of the "Thoughts on Music-Podington Bear" material described above (C-02). May be from on or before Feb. 23, 2007 (2 pgs).
"Thoughts on Music-Podington Bear" material obtained from www.podingtonbear.com/thoughtsonmusic.htm on Nov. 17, 2009 and believed to be the unabridged material referenced in the Feb. 23, 2007 posting. May be from on or before Feb. 23, 2007 (5 pgs).
"Be My Patron." Obtained Nov. 17, 2009 from a link at the end of the "Thoughts on Music—Podington Bear" material described above (C-02). May be from on or before Feb. 23, 2007 (2 pgs).
"Thoughts on Music—Podington Bear" material obtained from www.podingtonbear.com/thoughtsonmusic.htm on Nov. 17, 2009 and believed to be the unabridged material referenced in the Feb. 23, 2007 posting. May be from on or before Feb. 23, 2007 (5 pgs).
Datta et al., Content-Based Image Retrieval-Approaches and Trends of the New Age, 2005 ACM, pp. 253-262. *
Datta et al., Content-Based Image Retrieval—Approaches and Trends of the New Age, 2005 ACM, pp. 253-262. *
Faloutsos et al, Efficient and Effective Querying by Image Content, 1994 Kluwer Academic Publishers, pp. 231-262. *
Kosch et al., Smooth-A Distributed Multimedia Database System, 2001 VLDB, pp. 1-2. *
Kosch et al., Smooth—A Distributed Multimedia Database System, 2001 VLDB, pp. 1-2. *
www.archive.org entry printed Nov. 17, 2009 (9 pgs total) showing: a) search results for www.podingtonbear.com (1 pg); b) entry that purports to be Mar. 3, 2007 version of the www.podingtonbear.com website (2 pgs); and c) material purported to be a Feb. 23, 2007 posting on the www.podingtonbear.com website with title "Thoughts on Music" (6 pgs).
US9355412B2 (en) 2013-01-28 2016-05-31 Sanderling Management Limited Dynamic promotional layout management and distribution rules
US10108986B2 (en) 2013-01-28 2018-10-23 Sanderling Management Limited Dynamic promotional layout management and distribution rules
US10182311B2 (en) 2014-06-13 2019-01-15 Snap Inc. Prioritization of messages within a message collection
US10200813B1 (en) 2014-06-13 2019-02-05 Snap Inc. Geo-location based event gallery
US10476830B2 (en) 2014-10-02 2019-11-12 Snap Inc. Ephemeral gallery of ephemeral messages
US10380720B1 (en) 2015-01-09 2019-08-13 Snap Inc. Location-based image filters
US10524087B1 (en) 2018-04-06 2019-12-31 Snap Inc. Message destination list mechanism
KR101645763B1 (en) 2016-08-04 Server apparatus, electronic apparatus, electronic book providing system, electronic book providing method, electronic book displaying method, and recording medium
KR101195621B1 (en) 2012-10-30 Virtual group shopping mall
JP5250100B2 (en) 2013-07-31 Programming, distribution and consumption of media content
CN1190080C (en) 2005-02-16 TV appaaratus of storage broadcasting information display, distributing device, and information distribution method
US7478163B2 (en) 2009-01-13 Method and apparatus for presenting multimedia content and for facilitating third party representation of an object
US8315950B2 (en) 2012-11-20 Powerfully simple digital media player and methods for use therewith
US9003288B2 (en) 2015-04-07 System and method for displaying contextual advertisements with media
JP4903047B2 (en) 2012-03-21 Method and apparatus for organizing and reproducing data
US9760911B2 (en) 2017-09-12 Non-expanding interactive advertisement
CN101699505B (en) 2016-02-17 A kind of network media system
KR101334821B1 (en) 2013-12-09 Bid-based delivery of advertising promotions on internet-connected media players
US8375019B2 (en) 2013-02-12 Methods and systems of content mobilization, mobile search, and video editing through a web interface
US20120272127A1 (en) 2012-10-25 System and Method for Structured News Release Generation and Distribution
TWI528824B (en) 2016-04-01 Method and computer program product for sharing media item
US20130047123A1 (en) 2013-02-21 Method for presenting user-defined menu of digital content choices, organized as ring of icons surrounding preview pane
US20080086689A1 (en) 2008-04-10 Multimedia content production, publication, and player apparatus, system and method
US9762967B2 (en) 2017-09-12 System and method for presenting content with time based metadata
US8069414B2 (en) 2011-11-29 Embedded video player
US20110289445A1 (en) 2011-11-24 Virtual media shelf
US20100153831A1 (en) 2010-06-17 System and method for overlay advertising and purchasing utilizing on-line video or streaming media
US10210529B2 (en) 2019-02-19 Systems and methods for advertising on remote locations
US20100023863A1 (en) 2010-01-28 System and method for dynamic generation of video content
US8412021B2 (en) 2013-04-02 Video player user interface
US20090070673A1 (en) 2009-03-12 System and method for presenting multimedia content and application interface
TWI498835B (en) 2015-09-01 Systems and methods for providing textual and visual interactive advertisements in videos
Owner name: EMINOR INCORPORATED, NORTH CAROLINA
Free format text: ASSIGNMENT OF ASSIGNORS INTEREST;ASSIGNORS:CARLSON, EDWARD J., JR.;WATTS, ANDREW;JERNIGAN, STEVE;AND OTHERS;REEL/FRAME:025738/0731
Free format text: PAYMENT OF MAINTENANCE FEE, 8TH YR, SMALL ENTITY (ORIGINAL EVENT CODE: M2552); ENTITY STATUS OF PATENT OWNER: SMALL ENTITY
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,807
|
package com.datafueled.trace.core.attributes
import scala.collection.mutable.HashMap
trait HasAttributes {
private val attributes = HashMap[Symbol, Attribute[_]]()
def getAttribute(attributeName: Attribute[_]) : Option[Attribute[_]] = {
attributes.get(attributeName.name)
}
def getAttributeValue[A <: Attribute[_]](implicit type2attr: (() => A)) : Option[A#ValueType] = {
val a = getAttribute(type2attr())
if (a.isDefined) {
Some(a.get.value)
} else {
None
}
}
def getAttributeValue[A <: Attribute[_]](defaultValue: A#ValueType)(implicit type2attr: (() => A)) : A#ValueType = {
val a = getAttribute(type2attr())
if (a.isDefined) {
a.get.value
} else {
defaultValue
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,859
|
Q: Java - Not getting right items using array and Streams to parse a file Am trying to parse the /etc/group file on a macOS Mojave 10.14.3 operating system using Java 1.8's stream feature.
The full set of lines inside my /etc/group file are as follows:
nobody:*:-2:
nogroup:*:-1:
wheel:*:0:root
daemon:*:1:root
kmem:*:2:root
sys:*:3:root
tty:*:4:root
operator:*:5:root
mail:*:6:_teamsserver
bin:*:7:
procview:*:8:root
procmod:*:9:root
owner:*:10:
everyone:*:12:
_taskgated:*:13:_taskgated
group:*:16:
staff:*:20:root
_networkd:*:24:
_installassistant:*:25:
_lp:*:26:
_postfix:*:27:
_postdrop:*:28:
certusers:*:29:root,_jabber,_postfix,_cyrus,_calendar,_dovecot
_keytabusers:*:30:_calendar,_jabber,_postfix
_scsd:*:31:
_ces:*:32:
_appstore:*:33:_appstore
utmp:*:45:
authedusers:*:50:
interactusers:*:51:
netusers:*:52:
consoleusers:*:53:
_mcxalr:*:54:
_appleevents:*:55:
_geod:*:56:
_devdocs:*:59:
_sandbox:*:60:
localaccounts:*:61:
netaccounts:*:62:
_mdnsresponder:*:65:
_uucp:*:66:
_ard:*:67:
dialer:*:68:
network:*:69:
_www:*:70:_devicemgr,_teamsserver
_eppc:*:71:_eppc
_cvs:*:72:
_svn:*:73:
_mysql:*:74:
_sshd:*:75:
_qtss:*:76:
_mailman:*:78:
_appserverusr:*:79:
admin:*:80:root
_appserveradm:*:81:
_clamav:*:82:
_amavisd:*:83:
_jabber:*:84:
_appowner:*:87:
_windowserver:*:88:
_spotlight:*:89:
accessibility:*:90:
_tokend:*:91:
_securityagent:*:92:
_calendar:*:93:_teamsserver
_teamsserver:*:94:_devicemgr
_update_sharing:*:95:
_installer:*:96:
_atsserver:*:97:
_lpadmin:*:98:
_unknown:*:99:
_lpoperator:*:100:
_softwareupdate:*:200:_softwareupdate
_guest:*:201:
_coreaudiod:*:202:
_screensaver:*:203:
_developer:*:204:
_locationd:*:205:
_detachedsig:*:207:_locationd
_trustevaluationagent:*:208:
_odchpass:*:209:_teamsserver
_timezone:*:210:
_lda:*:211:
_cvms:*:212:
_usbmuxd:*:213:
_postgres:*:216:_devicemgr,_calendar,_teamsserver,_xserverdocs
_devicemgr:*:220:
_webauthserver:*:221:_teamsserver,_devicemgr
_netbios:*:222:
_warmd:*:224:_warmd
_dovenull:*:227:
_netstatistics:*:228:
_assetcache:*:235:
_coremediaiod:*:236:
_launchservicesd:*:239:
_iconservices:*:240:
_distnote:*:241:
_nsurlsessiond:*:242:
_nsurlstoraged:*:243:
_displaypolicyd:*:244:
_astris:*:245:
_gamecontrollerd:*:247:
_mbsetupuser:*:248:
_ondemand:*:249:
_analyticsusers:*:250:_analyticsd,_networkd,_timed,_reportmemoryexception
_xserverdocs:*:251:
_wwwproxy:*:252:
_mobileasset:*:253:
_findmydevice:*:254:
_datadetectors:*:257:
_captiveagent:*:258:
_ctkd:*:259:
_applepay:*:260:
_hidd:*:261:
_cmiodalassistants:*:262:
_analyticsd:*:263:_analyticsd
_webdeveloper:*:264:
_fpsd:*:265:_fpsd
_timed:*:266:
_reportmemoryexception:*:269:_reportmemoryexception
com.apple.access_ftp:*:395:
com.apple.access_disabled:*:396:
com.apple.access_sessionkey:*:397:
com.apple.access_screensharing:*:398:
com.apple.access_ssh:*:399:
Group.java:
public class Group {
private String gid;
private String name;
private String members;
public Group(String line) {
String[] items = line.split(":");
if (items.length <= 3) {
this.name = items[0];
System.out.print("Name: " + name + ", ");
this.gid = items[2];
System.out.print(" gid: " + gid + ", ");
}
else if (items.length >=3 || this.name != null || "".equals(this.name)){
this.members = items[3];
System.out.println("Members: " + members);
}
}
// Omitted getters & setters for brevity
}
Created this GroupsParser:
public class GroupParser {
public static List<Group> getAllGroups(String line) {
List<Group> groups = null;
try (Stream<String> stream = Files.lines(Paths.get(line))) {
groups = stream.filter(s -> s.charAt(0) != '#').map(Group::new)
.collect(Collectors.toCollection(ArrayList::new));
}
catch (Exception e) {
e.printStackTrace();
}
return groups;
}
public static void main(String[] args) {
List<Group> groups = GroupParser.getAllGroups("/etc/group");
for (Group group : groups) {
System.out.println("Group: " + group.getName() + ", gid: " + group.getGid() + ", members: " + group.getMembers());
}
}
}
Received the following output (which is obviously out of order and missing some items):
Group: nobody, gid: -2, members: null
Group: nogroup, gid: -1, members: null
Group: null, gid: null, members: root
Group: null, gid: null, members: root
Group: null, gid: null, members: root
Group: null, gid: null, members: root
Group: null, gid: null, members: root
Group: null, gid: null, members: root
Group: null, gid: null, members: _teamsserver
Group: bin, gid: 7, members: null
Group: null, gid: null, members: root
Group: null, gid: null, members: root
Group: owner, gid: 10, members: null
Group: everyone, gid: 12, members: null
Group: null, gid: null, members: _taskgated
Group: group, gid: 16, members: null
Group: null, gid: null, members: root
Group: _networkd, gid: 24, members: null
Group: _installassistant, gid: 25, members: null
Group: _lp, gid: 26, members: null
Group: _postfix, gid: 27, members: null
Group: _postdrop, gid: 28, members: null
Group: null, gid: null, members: root,_jabber,_postfix,_cyrus,_calendar,_dovecot
Group: null, gid: null, members: _calendar,_jabber,_postfix
Group: _scsd, gid: 31, members: null
Group: _ces, gid: 32, members: null
Group: null, gid: null, members: _appstore
Group: utmp, gid: 45, members: null
Group: authedusers, gid: 50, members: null
Group: interactusers, gid: 51, members: null
Group: netusers, gid: 52, members: null
Group: consoleusers, gid: 53, members: null
Group: _mcxalr, gid: 54, members: null
Group: _appleevents, gid: 55, members: null
Group: _geod, gid: 56, members: null
Group: _devdocs, gid: 59, members: null
Group: _sandbox, gid: 60, members: null
Group: localaccounts, gid: 61, members: null
Group: netaccounts, gid: 62, members: null
Group: _mdnsresponder, gid: 65, members: null
Group: _uucp, gid: 66, members: null
Group: _ard, gid: 67, members: null
Group: dialer, gid: 68, members: null
Group: network, gid: 69, members: null
Group: null, gid: null, members: _devicemgr,_teamsserver
Group: null, gid: null, members: _eppc
Group: _cvs, gid: 72, members: null
Group: _svn, gid: 73, members: null
Group: _mysql, gid: 74, members: null
Group: _sshd, gid: 75, members: null
Group: _qtss, gid: 76, members: null
Group: _mailman, gid: 78, members: null
Group: _appserverusr, gid: 79, members: null
Group: null, gid: null, members: root
Group: _appserveradm, gid: 81, members: null
Group: _clamav, gid: 82, members: null
Group: _amavisd, gid: 83, members: null
Group: _jabber, gid: 84, members: null
Group: _appowner, gid: 87, members: null
Group: _windowserver, gid: 88, members: null
Group: _spotlight, gid: 89, members: null
Group: accessibility, gid: 90, members: null
Group: _tokend, gid: 91, members: null
Group: _securityagent, gid: 92, members: null
Group: null, gid: null, members: _teamsserver
Group: null, gid: null, members: _devicemgr
Group: _update_sharing, gid: 95, members: null
Group: _installer, gid: 96, members: null
Group: _atsserver, gid: 97, members: null
Group: _lpadmin, gid: 98, members: null
Group: _unknown, gid: 99, members: null
Group: _lpoperator, gid: 100, members: null
Group: null, gid: null, members: _softwareupdate
Group: _guest, gid: 201, members: null
Group: _coreaudiod, gid: 202, members: null
Group: _screensaver, gid: 203, members: null
Group: _developer, gid: 204, members: null
Group: _locationd, gid: 205, members: null
Group: null, gid: null, members: _locationd
Group: _trustevaluationagent, gid: 208, members: null
Group: null, gid: null, members: _teamsserver
Group: _timezone, gid: 210, members: null
Group: _lda, gid: 211, members: null
Group: _cvms, gid: 212, members: null
Group: _usbmuxd, gid: 213, members: null
Group: null, gid: null, members: _devicemgr,_calendar,_teamsserver,_xserverdocs
Group: _devicemgr, gid: 220, members: null
Group: null, gid: null, members: _teamsserver,_devicemgr
Group: _netbios, gid: 222, members: null
Group: null, gid: null, members: _warmd
Group: _dovenull, gid: 227, members: null
Group: _netstatistics, gid: 228, members: null
Group: _assetcache, gid: 235, members: null
Group: _coremediaiod, gid: 236, members: null
Group: _launchservicesd, gid: 239, members: null
Group: _iconservices, gid: 240, members: null
Group: _distnote, gid: 241, members: null
Group: _nsurlsessiond, gid: 242, members: null
Group: _nsurlstoraged, gid: 243, members: null
Group: _displaypolicyd, gid: 244, members: null
Group: _astris, gid: 245, members: null
Group: _gamecontrollerd, gid: 247, members: null
Group: _mbsetupuser, gid: 248, members: null
Group: _ondemand, gid: 249, members: null
Group: null, gid: null, members: _analyticsd,_networkd,_timed,_reportmemoryexception
Group: _xserverdocs, gid: 251, members: null
Group: _wwwproxy, gid: 252, members: null
Group: _mobileasset, gid: 253, members: null
Group: _findmydevice, gid: 254, members: null
Group: _datadetectors, gid: 257, members: null
Group: _captiveagent, gid: 258, members: null
Group: _ctkd, gid: 259, members: null
Group: _applepay, gid: 260, members: null
Group: _hidd, gid: 261, members: null
Group: _cmiodalassistants, gid: 262, members: null
Group: null, gid: null, members: _analyticsd
Group: _webdeveloper, gid: 264, members: null
Group: null, gid: null, members: _fpsd
Group: _timed, gid: 266, members: null
Group: null, gid: null, members: _reportmemoryexception
Group: com.apple.access_ftp, gid: 395, members: null
Group: com.apple.access_disabled, gid: 396, members: null
Group: com.apple.access_sessionkey, gid: 397, members: null
Group: com.apple.access_screensharing, gid: 398, members: null
Group: com.apple.access_ssh, gid: 399, members: null
Group: nobody, gid: -2, members: null
Group: nogroup, gid: -1, members: null
Group: null, gid: null, members: root
Group: null, gid: null, members: root
Group: null, gid: null, members: root
Group: null, gid: null, members: root
Group: null, gid: null, members: root
Group: null, gid: null, members: root
Group: null, gid: null, members: _teamsserver
Group: bin, gid: 7, members: null
Group: null, gid: null, members: root
Group: null, gid: null, members: root
Group: owner, gid: 10, members: null
Group: everyone, gid: 12, members: null
Group: null, gid: null, members: _taskgated
Group: group, gid: 16, members: null
Group: null, gid: null, members: root
Group: _networkd, gid: 24, members: null
Group: _installassistant, gid: 25, members: null
Group: _lp, gid: 26, members: null
Group: _postfix, gid: 27, members: null
Group: _postdrop, gid: 28, members: null
Group: null, gid: null, members: root,_jabber,_postfix,_cyrus,_calendar,_dovecot
Group: null, gid: null, members: _calendar,_jabber,_postfix
Group: _scsd, gid: 31, members: null
Group: _ces, gid: 32, members: null
Group: null, gid: null, members: _appstore
Group: utmp, gid: 45, members: null
Group: authedusers, gid: 50, members: null
Group: interactusers, gid: 51, members: null
Group: netusers, gid: 52, members: null
Group: consoleusers, gid: 53, members: null
Group: _mcxalr, gid: 54, members: null
Group: _appleevents, gid: 55, members: null
Group: _geod, gid: 56, members: null
Group: _devdocs, gid: 59, members: null
Group: _sandbox, gid: 60, members: null
Group: localaccounts, gid: 61, members: null
Group: netaccounts, gid: 62, members: null
Group: _mdnsresponder, gid: 65, members: null
Group: _uucp, gid: 66, members: null
Group: _ard, gid: 67, members: null
Group: dialer, gid: 68, members: null
Group: network, gid: 69, members: null
Group: null, gid: null, members: _devicemgr,_teamsserver
Group: null, gid: null, members: _eppc
Group: _cvs, gid: 72, members: null
Group: _svn, gid: 73, members: null
Group: _mysql, gid: 74, members: null
Group: _sshd, gid: 75, members: null
Group: _qtss, gid: 76, members: null
Group: _mailman, gid: 78, members: null
Group: _appserverusr, gid: 79, members: null
Group: null, gid: null, members: root
Group: _appserveradm, gid: 81, members: null
Group: _clamav, gid: 82, members: null
Group: _amavisd, gid: 83, members: null
Group: _jabber, gid: 84, members: null
Group: _appowner, gid: 87, members: null
Group: _windowserver, gid: 88, members: null
Group: _spotlight, gid: 89, members: null
Group: accessibility, gid: 90, members: null
Group: _tokend, gid: 91, members: null
Group: _securityagent, gid: 92, members: null
Group: null, gid: null, members: _teamsserver
Group: null, gid: null, members: _devicemgr
Group: _update_sharing, gid: 95, members: null
Group: _installer, gid: 96, members: null
Group: _atsserver, gid: 97, members: null
Group: _lpadmin, gid: 98, members: null
Group: _unknown, gid: 99, members: null
Group: _lpoperator, gid: 100, members: null
Group: null, gid: null, members: _softwareupdate
Group: _guest, gid: 201, members: null
Group: _coreaudiod, gid: 202, members: null
Group: _screensaver, gid: 203, members: null
Group: _developer, gid: 204, members: null
Group: _locationd, gid: 205, members: null
Group: null, gid: null, members: _locationd
Group: _trustevaluationagent, gid: 208, members: null
Group: null, gid: null, members: _teamsserver
Group: _timezone, gid: 210, members: null
Group: _lda, gid: 211, members: null
Group: _cvms, gid: 212, members: null
Group: _usbmuxd, gid: 213, members: null
Group: null, gid: null, members: _devicemgr,_calendar,_teamsserver,_xserverdocs
Group: _devicemgr, gid: 220, members: null
Group: null, gid: null, members: _teamsserver,_devicemgr
Group: _netbios, gid: 222, members: null
Group: null, gid: null, members: _warmd
Group: _dovenull, gid: 227, members: null
Group: _netstatistics, gid: 228, members: null
Group: _assetcache, gid: 235, members: null
Group: _coremediaiod, gid: 236, members: null
Group: _launchservicesd, gid: 239, members: null
Group: _iconservices, gid: 240, members: null
Group: _distnote, gid: 241, members: null
Group: _nsurlsessiond, gid: 242, members: null
Group: _nsurlstoraged, gid: 243, members: null
Group: _displaypolicyd, gid: 244, members: null
Group: _astris, gid: 245, members: null
Group: _gamecontrollerd, gid: 247, members: null
Group: _mbsetupuser, gid: 248, members: null
Group: _ondemand, gid: 249, members: null
Group: null, gid: null, members: _analyticsd,_networkd,_timed,_reportmemoryexception
Group: _xserverdocs, gid: 251, members: null
Group: _wwwproxy, gid: 252, members: null
Group: _mobileasset, gid: 253, members: null
Group: _findmydevice, gid: 254, members: null
Group: _datadetectors, gid: 257, members: null
Group: _captiveagent, gid: 258, members: null
Group: _ctkd, gid: 259, members: null
Group: _applepay, gid: 260, members: null
Group: _hidd, gid: 261, members: null
Group: _cmiodalassistants, gid: 262, members: null
Group: null, gid: null, members: _analyticsd
Group: _webdeveloper, gid: 264, members: null
Group: null, gid: null, members: _fpsd
Group: _timed, gid: 266, members: null
Group: null, gid: null, members: _reportmemoryexception
Group: com.apple.access_ftp, gid: 395, members: null
Group: com.apple.access_disabled, gid: 396, members: null
Group: com.apple.access_sessionkey, gid: 397, members: null
Group: com.apple.access_screensharing, gid: 398, members: null
Group: com.apple.access_ssh, gid: 399, members: null
Question(s):
*
*This output doesn't seem ordered
Group: nobody, gid: -2, members: null
Group: nogroup, gid: -1, members: null
Why am I getting this, in between nogroup and wheel?
*Is there a better way to extract these items? Why can't I see kmem or wheel anywhere?
*Is there a way to check / fix this via Streams?
A: The core of the issue lies in the constructor you are using in mapping as .map(Group::new). The logic of the constructor implemented currently is a reason for a few questions arising. Trying to explain how below:
public Group(String line) {
String[] items = line.split(":"); // you split up the string into contents based on ':' separator
in the condition to check if the contents are split into less than or equal to 3 parts, you've also made an assumption here that the size of array would at least be 3, which is what you intend to access when doing items[2]. Notice though, if the items.length could have been 2, the if condition would have successfully allowed you to end up getting an AIOOBE for this kind of access still.
if (items.length <= 3) {
this.name = items[0];
System.out.print("Name: " + name + ", ");
this.gid = items[2];
System.out.print(" gid: " + gid + ", ");
}
when the content of the string could be split into more than 3 parts, you tend not to initialize the name, gid for that Group and hence you've got those null s in your output.
else {
this.members = items[3];
System.out.println("Members: " + members);
}
}
Is there a better way to extract these items? Why can't I see kmem or
wheel anywhere?
The section related to handling null values by initializing them would have solved for it.
To avoid NPE as well, you might want to initialize groups
to an empty list;
List<Group> groups = new ArrayList<>();
A simpler implementation based on an assumption that the string would always have 3 or more components when split using : could be:
public Group(String line) {
String[] items = line.split(":");
this.name = items[0];
this.gid = Integer.parseInt(items[2]);
if (items.length > 3){
this.members = items[3];
}
}
A: Not sure if I'm understanding your question correctly. You just want to parse the file line by line, without any groupBy/merge action? items.length > 3 is unnecessary with Splitter from Google Guava or my library abacus-common. Here is the simple sample solution:
Splitter splitter = Splitter.with(':').trim(true);
Splitter memSplitter = Splitter.with(',').trim(true).omitEmptyStrings(true);
ExceptionalStream.lines(new File("./tmp.txt")) // Or StreamEx.of(IOUtil.readLines(new File("./tmp.txt")))
.filter(s -> s.charAt(0) != '#')
.map(s -> splitter.splitToArray(s))
// .map(a -> Tuple.of(a[0], a[2], memSplitter.splitToArray(a[3])))
// TODO whatever you need/want.
.forEach(group -> N.println(group));
A: From my observations:
*
*This output doesn't seem ordered
Change the code ArrayList::new to LinkedList::new to maintain ordering.
*Is there a better way to extract these items? Why can't I see kmem or wheel anywhere?
The code if (items.length <= 3) { is the culprit. The else case does not assign values for fields other than members
3.Is there a way to check / fix this via Streams?
I think the fixes are done. Better ways - there will always.
A: Here below is the code which would help you :
class Group {
private String gid;
private String name;
private String members;
public Group(String line) {
String[] consolidated = line.split(":");
if (consolidated != null && consolidated.length > 0) {
this.name = doesIndexExists(consolidated, 0) ? consolidated[0] : "";
this.gid = doesIndexExists(consolidated, 2) ? consolidated[2] : "";
this.members = consolidated.length < 4 ? "" : doesIndexExists(consolidated, 3) ? consolidated[3] : "";
}
}
public String getGid() {
return gid;
}
public void setGid(String gid) {
this.gid = gid;
}
public String getName() {
return name;
}
public String getMembers() {
return members;
}
public boolean doesIndexExists(String[] inputArray, int index) {
return inputArray.length > index && inputArray[index] != null;
}
public void setName(String name) {
this.name = name;
}
public void setMembers(String members) {
this.members = members;
}
@Override
public String toString() {
return "Group [gid=" + gid + ", name=" + name + ", members=" + members + "]";
}
}
import java.io.IOException;
import java.nio.file.Files;
import java.nio.file.Paths;
import java.util.ArrayList;
import java.util.List;
import java.util.stream.Collectors;
public class GroupParser {
public static final String FILE_PATH = "YOURFILEPATH";
public static boolean isNotEmpty(String inputContent) {
return (inputContent != null && !"".equalsIgnoreCase(inputContent));
}
public static List<Group> getAllGroups(String line) throws IOException {
final List<Group> groupList = new ArrayList<>();
Files.readAllLines(Paths.get(line)).stream()
.filter(lineInstance -> isNotEmpty(lineInstance) && !lineInstance.startsWith("#")).collect(Collectors.toList())
.forEach(parsedInputObject -> groupList.add(new Group(parsedInputObject)));
return groupList;
}
public static void main(String[] args) throws IOException {
List<Group> groups = GroupParser.getAllGroups(FILE_PATH);
for (Group group : groups) {
System.out.println("Group: " + group.getName() + ", gid: " + group.getGid() + ", members: " + group.getMembers());
}
}
}
A: May I suggest a simpler fix than checking length of array or bringing in extra libraries: just add a few of the separators to guarantee sufficient number of array items:
String[] items = (line + "::::").split(":");
And String.split(String, int) could then save some negligible amount of time by not parsing more than the columns you want, if you want [up]tight code.
Well, that is all.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,053
|
Review: Disenchanted! The Musical
Disenchanted the Musical is a project that began in 2009 and made its way to off-Broadway theaters in 2012. Not to be confused with the upcoming Disney+ sequel to Enchanted, this stage show is a parody of the Disney Princess brand that is in no way sanctioned by Disney, as proven by the public domain-approved character autographs on the poster. The show contains far too many Disney-specific references to be an interpretative musical about fairy tales in the vein of Stephen Sondheim's Into the Woods. Disenchanted! is far less sophisticated than Into the Woods with a focus on low-brow humor and an overall lack of plot, which explains why it never could have been a contender for the elite world of Broadway. It must instead settle for off-Broadway and low-budget local productions like the one I streamed over the weekend. The show's childish humor about cartoon princesses might imply that it is meant for very young audiences, but the light swearing and song about boobs imply otherwise, which makes me wonder what sort of audience it is intended for.
I saw an ad for Disenchanted! on Facebook a few years ago, and it peaked my interest enough to research some songs and clips, but not enough to make travel plans to see it it at some faraway venue. Times have changed since then. Nowadays, entertainment is created with the intent that people will be able to watch it from home. When I saw a new ad for a small theater in Michigan that was offering livestream tickets, I decided to sign up for the Fenton Village Players' performance so I could finally see what all the fuss was about. The production I streamed over the weekend was extremely low budget with no costumes or sets and poor audio quality. The only actress who tried to dress up for her role was Cinderella, who wore a nice dress. Between the performance I watched, the song clips from the official website, and other videos I had seen from the show online, I was able to get the overall gist of the musical. For the most part, Disenchanted! is about a group of Disney Princesses led by Snow White who sing about how terrible their lives are.
Now that this show is over a decade old, I think the idea behind it is too outdated to be enjoyable. Poking fun at princesses was a big fad when the Disney Princess brand was first introduced in the early 2000s with movies like Shrek and Enchanted that recoded the DNA of what it means to be a fairy tale princess, paving the way for the modern era in which Disney redefined it unironically. It no longer makes sense to sing about being tired of the ballgown-wearing damsel in distress waiting for a prince to rescue her because modern princesses wear pants and rescue their own kingdoms. Most of the jokes in this show are tired and overused, which mitigates the humor. Those of us who still dream of romance and ballgowns have trouble relating to princesses who dislike them, and quite frankly, the bitter attitudes behind the princesses of Disenchanted! are uncharacteristic of the girls they are imitating. The only two princesses whose songs I found charming and believable were Pocahontas, who sang about how disappointed she was by the many historical inaccuracies of her movie and Tiana, who sang about how pleased she was to be the first black Disney Princess.
As a narrative structure, Disenchanted! incorporates a series of vignettes in which Snow White summons various princesses from her mirror. There is very little dialogue outside of these introductions, and most songs have little to do with the one that preceded them. The only running gag throughout the show is Sleeping Beauty falling asleep whenever it is her turn to sing until the finale in which she sings the song "Perfect" with the other princesses. A narcoleptic "Sleeping Beauty" is a gat that has been done to death even though she never spontaneously falls asleep in her fairy tale outside of the initial curse. The final message of the show is that the princesses aren't as perfect as the movies make them out to be, but they're still perfect just the way they are. Did they really need a two-hour long play just to tell us that? As a lover of princesses, I found very little to enjoy about this spiteful look at beloved Disney classics. Some of the gags were quite a stretch, such as Ariel complaining about having to shave her new legs. It must have been difficult for the show's writers to come up with something for such famously optimistic characters to complain about.
Overall, Disenchanted! The Musical is an outdated low-brow satire made for people who dislike the Disney Princess brand as it was when it started in the early 2000s. It no longer applies to the new kickass princesses that Disney has been promoting over the past few years, and the jokes have been done to death in other movies and shows. The minor swearing and sexual references are inappropriate for kids, but the potty humor and trite gags are uninspiring for adults. The only audience that I think could possibly enjoy this show is college kids who have been drinking heavily. In that respect, it might be a good contender for a Vegas casino. Aside from that, it has little to offer to older fans of Disney Princess movies, as they are unlikely to agree with its bitter sentiments about these beloved characters.
ariel belle broadway cinderella disenchanted disney princesses jasmine live events musicals parodies plays pocahontas reviews sleeping beauty snow white tiana
PrincessContent said…
Hi there! It's been a while. :)
Sad to hear that this musical was a disappointment. And I must say I got a bit upset to hear that the streamed version was that low-budged! I mean, even if it's streamed, people are paying to see it so they should have good camera work and the actresses should be in costume! Very unprofessional…
I did look up Pocahontas' song though and I agree that it's a charming song. A dark joke that, compared to the other out-dated princess jokes, is still relevant and will be in till the day we have reached more and better native American representation in the media.
It's always good to hear from you! :)
Ordinarily, I would agree that the stream should have been better quality, but it was performed by a community theater in Michigan, an area that is not very well-known for the arts. Plus, their website said that most people at the theater are volunteers. The tickets were pretty cheap, so I just have to take it for what it is. It's amazing they were able to do a show at all when the pandemic is still happening.
Aha, ok! Then I immediately take back my harsh criticism!
Review: The Scarred Prince
Review: Throne of Elves
New Trailer Takes a Deep Dive into Raya and the La...
Review: Poisoned
Fate on Netflix Is the Anti-Winx Saga!
Review: Stalks of Gold
Review: Disenchantment - Part 3
Review: The Opal Crown
Review: Bridgerton
Review: Tangled in Time - The Burning Queen
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,793
|
Михаил Александрович Волковойнов (, Вольск — 9 мая 1933, Москва) — советский лётчик-испытатель, участник первой мировой войны, Гражданской войны, Заслуженный лётчик СССР (17 июля 1925 года).
Биография
Родился а в городе Вольск ныне Саратовской области. В 1914 году окончил реальную школу.
В армии с 1914 года, подпоручик. В 1916 году окончил Гатчинскую авиашколу.
Участник первой мировой войны: в 1916—1917 годах — старший унтер-офицер 4-го армейского авиаотряда. Был контужен. Награждён Георгиевскими крестами 4 и 3 степени.
В мае 1918 года, будучи в Сибири (по другим данным — в Поволжье), был мобилизован и назначен командиром Чешского авиаотряда, участвовал в Тобольской операции белых, где был ранен в руку. В декабре 1919 года в составе 10 военлётов своего отряда перелетел к красным. В 1919—1921 годах участвовал в гражданской войне на стороне красных. В марте 1921 года участвовал в подавлении Кронштадтского восстания.
С февраля 1923 года — на лётно-испытательной работе в НИИ ВВС (до 1926 — НОА). 14 — 20 августа 1924 года на самолёте Р-1 совершил перелёт Москва — Севастополь — Москва. 10 июня — 13 июля 1925 года на самолёте Р-1 участвовал в групповом перелёте Москва — Пекин в качестве начальника лётной части. 30 августа — 2 сентября 1925 года на самолёте Р-1 участвовал в перелёте Москва — Пекин — Токио. Из-за плохих метеоусловий совершил вынужденную посадку, не долетев до Токио.
С 193? — лётчик-испытатель ЦАГИ, заместитель начальника ОЭЛИД (Отдел эксплуатации, лётных испытаний и доводок) ЦАГИ.
Погиб 9 мая 1933 года при проведении испытаний самолёта Р-5 с поворотными стойками на штопор.
Жил в Москве. Похоронен в колумбарии № 1 (в церкви) Донского кладбища (Москва).
Награждён двумя орденами Красного Знамени.
Примечания
Литература
Шелест И. «Лечу за мечтой». — М.: Молодая Гвардия, 1989.
Громов М. М. На земле и в небе. — Жуковский: Печатный двор, 1999.
Шавров В. Б. История конструкций самолетов в СССР до 1938 г. — 4-е изд. — Машиностроение, 1994.
Волковойнов М. На самолете из Москвы в Японию через Пекин. — М. Авиоиздательство, 1926.
Ссылки
Волковойнов Михаил Александрович
Лётчики-испытатели СССР
Заслуженные лётчики СССР
Погибшие в авиакатастрофах в РСФСР
Лётчики Гражданской войны в России (белые)
Лётчики Гражданской войны в России (красные)
Погибшие при испытании боевой техники
Похороненные на Донском кладбище
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,383
|
\section{Introduction}
The main goal of this work is to study in detail
under which conditions the right-handed (RH) neutrinos present
in a general type I seesaw scenario \cite{seesaw} can give a direct sizable
contribution to the neutrinoless double beta ($0\nu\beta\beta$)
decay rate, i.e., a contribution in the range of sensitivity of the current
and upcoming $0\nu\beta\beta$ decay experiments,
once all the relevant constraints are included in the analysis.
In~\cite{Blennow:2010th,LopezPavon:2012zg}, it was shown that
a sizable sterile neutrino
contribution to the $0\nu\beta\beta$ decay can be achieved
if the heavy neutrino spectrum is
hierarchical, with at least one RH neutrino with mass
$M$ below $100$ MeV and the other state(s) above
this scale. However, this spectrum is disfavoured
by cosmological observations since the region
$M\in[1$ eV, $100$ MeV$]$ is excluded
by BBN and CMB data~\cite{Hernandez:2013lza,Hernandez:2014fha}.
In~\cite{Ibarra:2010xw,Ibarra:2011xn,Mitra:2011qr}
the possibility of having a relevant contribution
from heavy RH neutrinos up to the TeV
scale was explored.~\footnote{The interplay between the light and
heavy Majorana neutrino contributions in $0\nu\beta\beta$ decay was investigated phenomenologically
first in \cite{HPR83}.}~It was found that indeed RH neutrinos as heavy as
$100$ GeV$\textendash$10 TeV could, in principle, give a
sizable and observable contribution to the $0\nu\beta\beta$ decay rate.
In~\cite{Mitra:2011qr} the role of the fine-tuning and one-loop effects were discussed,
concluding that for RH neutrino masses above $10$ GeV a relatively high level of fine-tuning
would be required. In~\cite{LopezPavon:2012zg} a more detailed study of
the one-loop effects was performed and it was found that indeed
they are significant and can play a very important role in the type I seesaw scenario.
The lepton number violation introduced through the RH neutrino Majorana mass term,
required to obtain a sizable effect in the $0\nu\beta\beta$ decay rate, naturally
appears at one-loop level in the light neutrino sector.
If fine-tuning is not invoked, the light neutrino mass constraints on
the one-loop corrections make it very difficult to obtain a
significant (RH) heavy Majorana neutrino contribution
in the $0\nu\beta\beta$ decay
effective Majorana mass, i.e., to have $|m_{\beta\beta}^{\text{heavy}}|\gtrsim 0.01$ eV,
$m_{\beta\beta}^{\text{heavy}}$ being the heavy
Majorana neutrino contribution under discussion.
We will show, in particular, that the scenario in which RH neutrinos
with a mass $M\gtrsim 1$ GeV can give a sizable contribution
to the $0\nu\beta\beta$ decay rate necessarily involves a fine-tuned
cancellation between the tree-level and one-loop light neutrino contributions.
More specifically, in this work we re-analyse the conditions under which
the heavy Majorana neutrinos with masses $M >100$ MeV
of the type I seesaw scenario
can give a significant direct contribution to
the $0\nu\beta\beta$ decay effective Majorana mass,
i.e., a contribution in the range of sensitivity of the current
and upcoming $0\nu\beta\beta$ decay experiments.
We show that for $M \gtrsim$ a few GeV
this requires a relatively large active-sterile neutrino mixing
(charged current couplings of the heavy Majorana neutrinos).
We clarify which seesaw realisations can provide
the requisite mixing. We discuss the impact of the
one-loop corrections in the different type I seesaw
realisations considered. We analyse also numerically
the problem of the sizable heavy Majorana neutrino contribution
to the $0\nu\beta\beta$ decay effective Majorana
mass, by studying the full parameter space, including the
relevant one-loop corrections and the bounds on the
active-sterile neutrino mixing from direct searches, charged lepton flavour violation
and non-unitarity~\cite{Antusch:2006vwa,Antusch:2008tz,Atre:2009rg,Alonso:2012ji,Antusch:2014woa,Drewes:2015iva,Dinh:2012bp,Cely:2012bz}. We
quantify, in particular, the level of fine-tuning required in order to have a
sizable heavy neutrino contribution to the $0\nu\beta\beta$
decay rate. In order to do the analysis and
generate the right pattern
for the light neutrino masses and mixing,
we have constructed a modification of the Casas-Ibarra
parametrization~\cite{Casas:2001sr},
which takes into account the impact of the
one-loop corrections.
The paper is organized as follows: in section~\ref{sec1} we derive under which conditions it is possible to obtain a sizable active-sterile neutrino mixing,
which can strongly affect the effective Majorana neutrino mass, $m_{\beta\beta}$. In section~\ref{sec3} we study the impact on $m_{\beta\beta}$ of the one-loop
corrections to the light neutrino masses and present our modified Casas-Ibarra parametrization which takes into account the one-loop effects. In
section~\ref{BetaBeta} we perform the numerical analysis and quantify the level of fine-tuning necessary to have a dominant contribution in $m_{\beta\beta}$ from the exchange of the heavy (sterile) neutrinos. Finally, we summarise our results in the concluding section.
\mathversion{bold}
\section{Large Active-Sterile Neutrino
Mixing and $0\nu\beta\beta$ Decay}
\mathversion{normal}
\label{sec1}
We consider the most general type I seesaw scenario \cite{seesaw}
with $n\geq2$ RH neutrino fields $\nu_{sR}$ ($s=1,\ldots,n$).
After the spontaneous breaking of the electroweak (EW) symmetry the
full neutrino mass Lagrangian is
\begin{equation}
\mathcal{L}_{\nu}\;=\; -\, \overline{\nu_{\ell L}}\,(m_{D})_{\ell s}\, \nu_{sR} -
\frac1 2\, \overline{\nu^{c}_{sL}}\,(M_{R})_{st}\,\nu_{tR}\;+\;{\rm h.c.}
\label{typeI}
\end{equation}
where $\ell=e,\mu,\tau$ and $\nu^{c}_{sL}\equiv C\, \overline{\nu_{sR}}^T$,
$C$ being the charge
conjugation matrix. $M_{R} = (M_{R})^T$ is the Majorana mass matrix of the
RH neutrinos and $m_{D}$ is the $3\times n$ neutrino Dirac mass matrix.
The full mass matrix derived from Lagrangian (\ref{typeI}) is therefore
\begin{equation}
\mathcal{M} \equiv \begin{pmatrix} \mathbf{O} & m_D \\
m_D^T & M_R \end{pmatrix} = U^* \,\text{diag}\left(m_i,M_k\right)U^\dagger,
\label{Mnu}
\end{equation}
where $m_i$ ($i=1,2,3$) and $M_k$ ($k=1,\ldots,n$) are the
light and heavy Majorana neutrino masses, respectively.
We define $\mathbf{O}$ as a $3\times3$ matrix with all elements
equal to zero.
The full neutrino mass $\mathcal{M}$
is diagonalised by a $(3+n)\times (3+n)$ unitary matrix $U$,
through a well known rotation between the neutrino flavour and
mass eigenstates. We give below the relation between
the left-handed (LH) components of the corresponding fields
($\nu_{\ell L}$, $\nu^c_{sL}$ and $\chi_{iL}$, $N_{kL}$):
\begin{equation}
\begin{pmatrix}
\nu_{\ell L} \\ \nu^c_{s L} \end{pmatrix} = U \begin{pmatrix}
\chi_{iL} \\ N_{k L} \end{pmatrix}.
\end{equation}
Taking into account that the active block of $U$ is unitary
to a very good approximation, the
complete mixing matrix can be expanded as
\footnote{In the following we work in the basis in which the
charged lepton mass matrix is diagonal.}
\begin{equation}
U =\begin{pmatrix} 1-\theta\theta^\dagger/2 & \theta \\
-\theta^\dagger & 1-\theta^\dagger \theta/2 \end{pmatrix}
\begin{pmatrix} U_{\text{PMNS}} & 0\\
0 & V \end{pmatrix} +\mathcal{O}\left( \theta^3 \right)
= \begin{pmatrix} U_{\text{PMNS}} & \theta V\\
-\theta^\dagger U_{\text{PMNS}} & V \end{pmatrix}
+\mathcal{O}\left( \theta^2 \right)
\,,
\label{U}
\end{equation}
where $\theta$ is a $3\times n$ matrix with ``small'' entries,
which characterises the mixing between the active and
the sterile neutrinos, $U_{\text{PMNS}}$ is the PMNS neutrino mixing matrix \cite{BPont57,MNS62}
and $V$ is a $n\times n$ unitary matrix.
The quantity $(\theta\,V)_{\ell k}$, $\ell=e,\mu,\tau$, $k=1,\ldots,n$,
is the coupling of the heavy
Majorana neutrino $N_k$ to the charged lepton $\ell$ in the weak
charged lepton current, and to the flavour neutrino $\nu_\ell$
in the weak neutral lepton current.
From the diagonalization of the complete neutrino mass matrix $\mathcal{M}$,
at leading order in $\theta$ we have \cite{Ibarra:2010xw}
\begin{eqnarray}
\theta^* \,M_R\, \theta^\dagger
&\approx&
- U_{\text{PMNS}}^*\,\hat m \,U_{\text{PMNS}}^\dagger\,,
\label{constraints1}\\
\theta^* \,M_R &\approx& m_D\,,
\label{constraints2}\\
M_R &\approx& V^*\,\hat M \,V^\dagger \,,
\label{constraints3}
\end{eqnarray}
where
\begin{equation}
\label{diagmasses}
\hat m \;\equiv\; \text{diag}(m_1,m_2,m_3)\,,\quad\quad
\hat M \;\equiv\; \text{diag}(M_1,\ldots,M_n)\,.
\end{equation}
It follows from Eqs. (\ref{constraints1}) and (\ref{constraints3}) that
\begin{equation}
(\theta\,V)^* \,\hat M\, (\theta\,V)^\dagger
\approx
- U_{\text{PMNS}}^*\,\hat m \,U_{\text{PMNS}}^\dagger\,.
\label{constraints4}
\end{equation}
In terms of the seesaw parameters we have for
the active-sterile neutrino mixing:
\begin{equation}
\theta^* \approx m_D \,M_R^{-1}.
\label{theta}
\end{equation}
Using Eqs.~(\ref{constraints1}) and (\ref{theta}), we recover the
usual type I seesaw relation for the (tree-level)
light neutrino mass matrix, namely
\begin{equation}
\label{mvtreetypeI}
m_{\nu}^{\rm tree}\;=\; -m_D\, M_R^{-1} \,m_D^T\;
\equiv\; -\theta^* \,M_R\, \theta^\dagger \;
=\; -(\theta\,V)^* \, \hat M\, (\theta\,V)^\dagger\,
=\, U_{\text{PMNS}}^*\,\hat m \,U_{\text{PMNS}}^\dagger\,.
\end{equation}
The effective Majorana neutrino mass, $m_{\beta\beta}$, which enters in
the $0\nu\beta\beta$ decay amplitude, receives, in general, two
different contributions,
corresponding to the exchanges of the light and heavy virtual
Majorana neutrinos:
\begin{equation}
m_{\beta\beta} \;=\; m_{\beta\beta}^{\rm light}\,+\,m_{\beta\beta}^{\rm heavy} \,,\label{mefftot}
\end{equation}
with
\begin{equation}
m_{\beta\beta}^{\rm light} \;=\;
\sum\limits_{i=1}^3\,(U_{\rm PMNS})_{ei}^2\,m_i
\;=\; - \sum\limits_{k}\,(\theta\,V)^2_{ek}\,M_k\,,
\label{mefflight}
\end{equation}
where we have used Eq.~(\ref{constraints4}), which holds at tree-level in the type I seesaw models.
A good estimate for the contribution due to the heavy Majorana
neutrino exchange for $M_k\gg 100$ MeV is
\cite{Blennow:2010th}
\begin{equation}
m_{\beta\beta}^{\rm heavy} \approx
- \sum_{k} (\theta V)_{ek}^2\, f(A)\,
(M_{a}/M_{k})^{2}\,M_k \,,
\label{meeh1}
\end{equation}
where $M_{a}\approx 0.9$ GeV and $f(A)$
depends on the decaying isotope considered.
For, e.g., $^{48}$Ca,
$^{76}$Ge, $^{82}$Se, $^{130}$Te and $^{136}$Xe,
the function $f(A)$ takes the values $f(A)\approx$
0.033, 0.079, 0.073, 0.085 and 0.068, respectively.
Using Eq. (\ref{meeh1}),
it is easy to estimate the minimum mixing $(\theta V)_{\rm min}$
required in order to have a contribution
at the aimed sensitivity of the next generation of $0\nu\beta\beta$
decay experiments, that is $|m_{\beta\beta}^{\rm heavy}|\gtrsim 10^{-2}$ eV.
In Fig.~\ref{estimate} we compare this
estimate for $(\theta V)_{\text{min}}$ for the $^{76}$Ge isotope,
$(\theta V)^2_{\text{min}} \simeq 1.6\times 10^{-10} M~\text{GeV}^{-1}$ (dashed line),
with the naive seesaw scaling suggested by Eq.~(\ref{mvtreetypeI}),
$(\theta V)^2_{\text{naive}}=\sqrt{\Delta m^2_{\text{atm}}}/M
\simeq 5\times 10^{-11}~\text{GeV}/M$ (solid line)
as a function of the RH neutrino mass scale $M$
(expressed in units of GeV).
From Fig.~\ref{estimate} it is clear that for RH neutrino masses
larger than $\sim1$~GeV a considerable enhancement with respect to the naive
seesaw scaling of $\theta V$ is required in order to have a
sizable RH neutrino contribution. Obviously, this enhancement
increases with the mass of the RH neutrinos.
We notice that in the region
$M\approx500$ MeV$\textendash 1$ GeV,
the naively estimated mixing, $(\theta V)^2_{\text{naive}}$,
is in the right ballpark.
Similar conclusions are valid for
$(\theta V)^2_{\text{min}}$ and $(\theta V)^2_{\text{naive}}$
in the cases of $0\nu\beta\beta$ decay of other
isotopes ($^{48}$Ca, $^{82}$Se, $^{130}$Te, $^{136}$Xe, etc.).
\begin{figure}[t!]
\includegraphics[width=0.6\textwidth,angle=0]{figures/estimation2.pdf}
\caption{
\label{estimate}
{\small \textbf{Active-sterile neutrino mixing.} The dashed line stands for
an estimate of the minimum $(\theta V)^2$
required in order to have
$|m_{\beta\beta}^{\rm heavy}| > 10^{-2}$ eV in the case of
$0\nu\beta\beta$ decay of $^{76}$Ge.
The solid line corresponds to the naive seesaw scaling of
$(\theta V)^2$ (see the text for further details).}}
\end{figure}
\subsection{Casas-Ibarra Parametrization and
Large Active-Sterile Neutrino Mixing}
In order to understand under which conditions an enhancement
with respect to the naive scaling of the active-sterile mixing
(or equivalently, of the charged current couplings of the
heavy Majorana neutrinos $(\theta V)_{\ell k}$)
can be expected, we employ the Casas-Ibarra parametrization of
$\theta V$~\cite{Casas:2001sr}.
In this parametrization
the light neutrino masses and the angles and phases of the PMNS
matrix are input parameters, in such a
way that the correct light neutrino mixing pattern is always recovered.
The Casas-Ibarra parametrization is obtained
rewriting Eq.~(\ref{constraints1}) as
\begin{equation}
\left(\pm i \,\hat m^{-1/2} \,U_{\text{PMNS}}^\dagger \,\theta V \,\hat{M}^{1/2}\right)\,\left(\pm i \,\hat m^{-1/2}\, U_{\text{PMNS}}^\dagger \,\theta V\, \hat M^{1/2}\right)^T
\equiv R\, R^T=1\,,
\end{equation}
where $R$ is a general $3 \times n$ complex
matrix which parametrizes the new physics
degrees of freedom associated to the sterile neutrino sector.
Using this parametrization, $\theta V$ can be written as
\begin{equation}
\theta V =
\mp\, i\, U_{\text{PMNS}}\, \hat m^{1/2}\, R\, \hat M^{-1/2}\,.
\label{thetaR}
\end{equation}
The matrix $V$ can be set to the unit matrix if one works in the basis
in which the Majorana sub-matrix $M_R$ is diagonal.~\footnote{
An extension of this parametrization to all orders in the seesaw
expansion can be found in~\cite{Donini:2012tt,Blennow:2011vn}.}
Naively, from Eq.~(\ref{mvtreetypeI})
one may conclude that
$\theta V\approx\mathcal{O}\left(\sqrt{\frac{\hat m}{\hat M}}\right)$,
i.e., that the mixing (or coupling) $\theta V$ is expected to be suppressed by
the heavy neutrino mass scale. However, having a larger mixing
is perfectly possible
due to an enhancement factor contained in the matrix $R$
\cite{Ibarra:2010xw,Ibarra:2011xn}. Obviously, such enhancement can only
be in agreement with the light
neutrino spectrum if there is a non-trivial suppression/cancellation
in the l.h.s.~of Eq.~(\ref{constraints4}). This
extra suppression is related to particular textures of
the neutrino mass matrix, which can be motivated,
for instance, introducing an extra $U(1)$ global symmetry
in the Lagrangian, as it is the case in the
so called ``inverse'' and ``direct'' seesaw models
~\cite{Mohapatra:1986bd,Branco:1988ex}.
In these models the indicated
global symmetry can be identified with that
corresponding to the conservation of
a non-standard lepton charge (see further).
In the following we will focus on the minimal seesaw scenario
with $n=2$ RH sterile neutrinos~\footnote{In the present article we will use the term
``heavy Majorana neutrinos'' for Majorana neutrinos having masses
exceeding approximately 100 MeV.}~(see, e.g., \cite{3X2Models}) giving rise
to two heavy Majorana mass-eigenstate neutrinos,
which predicts one massless and
two massive light active neutrinos. For the light neutrino mass spectrum with
normal hierarchy (NH) and inverted hierarchy (IH) we have
\begin{eqnarray}
\label{measuredmassesNH}
&& m_1=0 \,, \quad \quad m_2= \sqrt{\Delta m^2_{21}} \,,
\quad \quad m_3= \sqrt{\Delta m^2_{31}} \, ,
\quad\quad\mbox{(NH)}\\
&& m_1= \sqrt{ |\Delta m^2_{32}| - \Delta m^2_{21} } \,, \quad\quad m_2= \sqrt{|\Delta m^2_{32}| } \,,
\quad\quad m_3= 0\quad \quad\mbox{(IH)} \,.
\label{measuredmassesIH}
\end{eqnarray}
The current best fit values obtained from the global fit analysis in
\cite{nufit} are
\begin{eqnarray}
\label{dmbf}
&\Delta m^2_{21} = 7.50 \times 10^{-5} \; \mathrm{eV}^2 \; ,&\\
\nonumber
&\Delta m^2_{31} = 2.457 \times 10^{-3} \;\; \mathrm{eV}^2 \;\; \mbox{(NH)} \;\;\;
\mbox{and} \;\;\;
\Delta m^2_{32} =-2.449 \times 10^{-3} \;\mathrm{eV}^2 \;\; \mbox{(IH)} \; .
&
\end{eqnarray}
In this minimal seesaw scenario, the two (tree-level)
non-zero light neutrino masses $m_2^{\rm tree}$ and $m_3^{\rm tree}$
($m_1^{\rm tree}$) in the case of NH (IH) neutrino mass spectrum
satisfy the relation:
\begin{equation}
\label{treeproduct}
m_{2}^{\rm tree}\,m_{3(1)}^{\rm tree}\;
\equiv\; - \det[M_{R}^{-1}]\det[m_{D}^{T}m_{D}]\,,~~~{\rm NH~(IH)}\,,
\end{equation}
which is basis independent.
In the considered case the $R$-matrix, which enters into Eq. (\ref{thetaR}),
can be parametrized~as~\cite{Ibarra:2011xn}
\begin{eqnarray}
R &= &\begin{pmatrix} 0 & 0 \\
\cos\left(\theta_{45}+i\gamma\right) & -\sin\left(\theta_{45}+i\gamma\right) \\
\sin\left(\theta_{45}+i\gamma\right) & \cos\left(\theta_{45}+i\gamma\right) \end{pmatrix}\,,\quad\quad\text{for \:\:NH}\,,
\label{RNO}\\
R &= &\begin{pmatrix}
\cos\left(\theta_{45}+i\gamma\right) & -\sin\left(\theta_{45}+i\gamma\right) \\
\sin\left(\theta_{45}+i\gamma\right) & \cos\left(\theta_{45}+i\gamma\right) \\
0 & 0 \end{pmatrix}\,,\quad\quad\text{for \:\:IH}\,,
\label{RIO}\
\end{eqnarray}
where $\theta_{45}$ and $\gamma$ are real
parameters.
If $R$ were real, i.e., $\gamma=0$, there is no way to obtain
any enhancement of the couplings/mixings $\theta V$ of interest
since $R$ would essentially be a
real orthogonal matrix. However,
for $\gamma\neq0$ and $e^{\pm\gamma}\gg 1$
an enhancement of $\theta V$ is possible:
\begin{eqnarray}
|\cos\left(\theta_{45}+i\gamma\right)|^2 &=& \cos^2\theta_{45}+\sinh^2\gamma\gg 1\Leftrightarrow e^{\pm\gamma}\gg1\,,
\nonumber\\
|\sin\left(\theta_{45}+i\gamma\right)|^2 &=& \sin^2\theta_{45}+\sinh^2\gamma\gg 1\Leftrightarrow e^{\pm\gamma}\gg1\,.
\end{eqnarray}
In fact, for $e^{\pm\gamma}\gg 1$ the expression of $R$
in the NH case
reduces to
\begin{equation}
R \approx e^{-i\,\theta_{45}}\,\frac{e^{\pm\gamma}}{2}\,
\begin{pmatrix} 0&0 \\ 1 & \pm i \\
\mp i & 1 \end{pmatrix}\,,~~~{\rm NH}\,.
\label{RissNH}
\end{equation}
Similarly, one can derive from (\ref{RIO}) the same limit of $R$ for
the IH neutrino mass spectrum:
\begin{equation}
R \approx e^{-i\,\theta_{45}}\,\frac{e^{\pm\gamma}}{2}\,
\begin{pmatrix} 1 & \pm i \\
\mp i & 1
\\ 0 & 0
\end{pmatrix}\,,~~~{\rm IH}\,.
\label{RissIH}
\end{equation}
Notice that the Casas-Ibarra parameter $\gamma$ in (\ref{RissNH}) and (\ref{RissIH}) can be related to the maximum eigenvalue $y$ \cite{Ibarra:2011xn} of the Dirac mass matrix $m_D$ in Eq.~(\ref{Mnu}), that is
\begin{eqnarray}
y^2\,v^2 & = & 2\, \text{max}\left\{ \text{eig}\left( m_D\, m_D^\dagger \right) \right\}\;=\;\frac12 e^{\pm \gamma}\,M_1\left(m_2+m_3 \right)\left(2+z\right)\,,\quad \quad \text{NH}\,,\\
y^2\,v^2 & = & 2\, \text{max}\left\{ \text{eig}\left( m_D\, m_D^\dagger \right) \right\}\;=\;\frac12 e^{\pm \gamma}\,M_1\left(m_1+m_2 \right)\left(2+z\right)\,,\quad \quad \text{IH}\,,
\end{eqnarray}
where $z$ denotes the relative mass splitting of the two heavy Majorana neutrino masses, $z=(M_2-M_1)/M_1$, and $v=246$ GeV is the EW symmetry breaking scale.
Introducing the expression (\ref{RissNH}) (or (\ref{RissIH}))
in Eq.~(\ref{thetaR})
one obtains \cite{del Aguila:2006dx,Gavela:2009cd, Ibarra:2010xw,Ibarra:2011xn}
\begin{equation}
\frac{\left(\theta V\right)_{\ell 1}}{\left(\theta V\right)_{\ell 2}} \approx\pm i\,\sqrt{\frac{M_2}{M_1}}.
\label{condition}
\end{equation}
Then, in terms of $y$ the active-sterile neutrino mixing in Eq.~(\ref{thetaR})
takes the form \cite{Ibarra:2011xn}
\begin{eqnarray}
\label{mixing-vs-y}
\left|\left(\theta V\right)_{\ell 1} \right|^{2}&=&
\frac{1}{2\,(2+z)}\frac{y^{2} v^{2}}{M_{1}^{2}}\frac{m_{3}}{m_{2}+m_{3}}
\left|U_{\ell 3}+i\sqrt{m_{2}/m_{3}}\,U_{\ell 2} \right|^{2}\,,
\quad{\rm NH}\,,\\
\left|\left(\theta V\right)_{\ell 1} \right|^{2}&=&
\frac{1}{2\,(2+z)}\frac{y^{2} v^{2}}{M_{1}^{2}}\frac{m_{2}}{m_{1}+m_{2}}
\left|U_{\ell 2}+i\sqrt{m_{1}/m_{2}}\,U_{\ell 1} \right|^{2}\,,
\quad {\rm IH}\,.
\label{mixing-vs-yIH}
\end{eqnarray}
All in all, the previous relations imply that in the basis
in which the RH neutrino Majorana mass term is diagonal,
the neutrino Yukawa couplings, or equivalently
$(m_D)_{\ell 1}$ and $(m_D)_{\ell 2}$,
should satisfy the following relation:
\begin{equation}
\frac{\left(m_D\right)_{\ell 1}}{\left(m_D\right)_{\ell 2}}\approx
\pm i\,\sqrt{\frac{M_1}{M_2}}
\label{conditionMD}
\end{equation}
Any texture of the neutrino mass matrix which satisfies
this condition gives rise to relatively large couplings $\theta V$
with the right suppression/cancellation in the light (flavour) neutrino
mass matrix, which allows to recover the correct light neutrino mass
spectrum at tree-level.
The relatively large $\theta V$
thus generated can
saturate the present bounds even
in the case in which the heavy Majorana neutrino
spectrum is hierarchical.
Using Eqs.~(\ref{meeh1}) and (\ref{condition}),
one can easily estimate the contribution
to the $0\nu\beta\beta$ decay effective Majorana mass due to the
exchange of the heavy Majorana neutrinos in the large coupling/mixing
case of interest \cite{Ibarra:2010xw}:
\begin{equation}
m_{\beta\beta}^{\rm heavy} \approx
- \, (\theta V)_{e1}^2\, f(A)\,\frac{M^2_a}{M_1}\,
\left\{1-\left(\frac{M_1}{M_1+\Delta M}\right)^2\right\}\,,
\label{meeh2}
\end{equation}
with~\footnote{
Note that $(\theta V)_{e1}^2$ depends, in particular,
on the phase $\theta_{45}$. This implies that
$m_{\beta\beta}$ will also depend on $\theta_{45}$~\cite{Ibarra:2011xn}.}~$\Delta M =M_2-M_1$.
Clearly, if $\Delta M \ll M_1$ the contribution will be proportional
to $\Delta M$, while in the limit $\Delta M \gg M_1$ the dependence
on $\Delta M$ is subleading since the lightest
RH neutrino dominates the contribution.
The interplay between the light and heavy Majorana neutrino exchange
contributions in the effective Majorana mass,
$m_{\beta\beta} = m_{\beta\beta}^{\rm light} + m_{\beta\beta}^{\rm heavy}$,
in the scheme under discussion in which
Eq. (\ref{condition}) holds and $m_{\beta\beta}^{\rm heavy}$ is
given by Eq. (\ref{meeh2}), was investigated in detail
in \cite{Ibarra:2011xn} in the case when the two heavy Majorana
neutrinos form a pseudo-Dirac pair,
$0 < \Delta M =M_2-M_1 \ll M_1,M_2$, and have masses in the interval
$\sim (50 - 1000)$ GeV. It was found that there exists a
relatively large region of the allowed parameter space of the scheme
in which the heavy Majorana neutrino contribution
can change drastically the predictions based on the light Majorana
neutrino exchange contribution. More specifically, it was found
that \cite{Ibarra:2011xn}:
i) $|m_{\beta\beta}|$ in the case of NH spectrum
can have values in the interval
$0.01~{\rm eV}\lesssim |m_{\beta\beta}| \lesssim 0.1$ eV, i.e.,
in the range of sensitivity of the current
GERDA \cite{GERDA}, EXO \cite{EXO200}, Kamland-Zen \cite{KamlandZen} and CUORE \cite{CUORE0} experiments
and of a few other experiments under preparation
(Majorana \cite{Abgrall:2013rze}, SNO+ \cite{Hartnell:2012qd}, AMORE \cite{Bhang:2012gn}, etc.). We recall that in the case of
$0\nu\beta\beta$ decay generated only by
light Majorana neutrino exchange we have
(see, e.g., \cite{bb0nuNH2008,PDG2014})
$|m_{\beta\beta}| = |m_{\beta\beta}^{\rm light}| \lesssim 0.005$ eV;\\
ii) $|m_{\beta\beta}|$ in the case of IH spectrum
can be strongly suppressed due to partial, or even total,
cancellation between $m_{\beta\beta}^{\rm light}$ and
$m_{\beta\beta}^{\rm heavy}$ in $m_{\beta\beta}$ (see also \cite{Pascoli:2013fiz}).
Since the magnitude of $m_{\beta\beta}^{\rm heavy}$, as it follows from
Eq.~(\ref{meeh2}), depends on the atomic number $A$ of the
decaying nucleus \cite{HPR83},
the cancellation between $m_{\beta\beta}^{\rm light}$ and
$m_{\beta\beta}^{\rm heavy}$ in $m_{\beta\beta}$ can take place
for a given nucleus (say, e.g., for $^{48}$Ca)
but will not hold for other
nuclei ($^{76}$Ge, $^{82}$Se, $^{130}$Te, $^{136}$Xe, etc.).
If the $0\nu\beta\beta$ decay is due only to the
light Majorana neutrino exchange we have
in the case of IH spectrum, as is well known \cite{bb0nuNHIH}
(see also, e.g., \cite{PDG2014}),
$ 0.013~{\rm eV} \lesssim |m_{\beta\beta}| =
|m_{\beta\beta}^{\rm light}| \lesssim 0.050$ eV.
On the other hand, in \cite{Ibarra:2011xn} the role of the one-loop corrections was not studied. In
\cite{LopezPavon:2012zg} it was shown that the one-loop corrections to the light neutrino
masses generated in the scheme under discussion turn out to be very relevant. Essentially,
a sizable heavy contribution to the $0\nu\beta\beta$ decay for heavy masses in the
range $\sim (50 - 1000)$ GeV generates at the same time a very large one-loop correction
to the light neutrino masses. In this work we analyse in detail the role of the
one-loop effects showing that similar conclusions to the ones drawn in \cite{Ibarra:2011xn}
will be obtained. However, we will also show that the price one has to pay in order to
have a significant impact of the heavy neutrinos in the $0\nu\beta\beta$ decay is the
requirement of a highly fine-tuned cancellation between the tree-level and one-loop contributions
to the light neutrino masses.
\subsection{Comparison with Extended and
Inverse Seesaw Scenarios}
As an application of the previous results,
we consider the effect of heavy RH neutrinos on the
$0\nu\beta\beta$ decay amplitude in the case of two
different realisations of the type I seesaw scenario,
which predict a large active-sterile neutrino mixing $\theta V$, that is the
well known extended seesaw (ESS)~\cite{Kang:2006sn} and
inverse/direct seesaw (ISS)~\cite{Mohapatra:1986bd, Branco:1988ex} models.
In particular, we will clarify how the large mixing realisations
described in the previous section in terms of the Casas-Ibarra
parametrization match with the ISS and ESS scenarios.
In order to understand the predictions in these classes of models
it is useful to adopt the following parametrization of the generic
mass terms in the seesaw Lagrangian (\ref{typeI}), namely
\begin{eqnarray}
\mathcal{M}\;\equiv\; \left( \begin{array}{cc} \mathbf{O} & m_D \\ m_D^T
& M_R
\end{array} \right)\; =\; \left( \begin{array}{ccc} \mathbf{O} & \mathbf{Y}_1 \,v/\sqrt{2}& \epsilon \,\mathbf{Y}_2 \,v/\sqrt{2} \\ \mathbf{Y}_1^T v/\sqrt{2}
& \mu' & \Lambda \\ \epsilon \,\mathbf{Y}_2^T \,v/\sqrt{2}
& \Lambda & \mu
\end{array} \right)\,,
\label{Mnu2}
\end{eqnarray}
where
$\mathbf{Y}_i\equiv(y_{i e},y_{i \mu}, y_{i \tau})^T$, for $i=1,2$. This parametrization is completely general and, in principle,
$\epsilon$, $\mu$, $\mu'$ and $\Lambda$
can take any value.~\footnote{In the following we will assume for
simplicity that all the parameters introduced in Eq.~(\ref{Mnu2}) are real.}~However,
$\epsilon$, $\mu$ and $\mu'$ can be interpreted as lepton
number violating couplings and, therefore, in principle they take arbitrarily
small values,
because in this case there is an approximate global
symmetry of the seesaw Lagrangian corresponding to the
conservation of the lepton charge $L' = L_e + L_{\mu} + L_{\tau} + L_1 - L_2$,
where $L_1$ and $L_2$ are the charges carried by the
RH neutrino fields $\nu_{1R}$ and $\nu_{2R}$, respectively.
In the limit of $\epsilon = \mu = \mu' = 0$, the conservation of $L'$
is exact. In this case the neutrino sector consists of three massless neutrinos
and one massive Dirac fermion, which can be inferred, in particular,
directly from the expression of the charge $L'$ in terms of the charges
$L_{\ell}$ and $L_{1,2}$ \cite{Leung:1983ti,Bilenky:1987ty}. The exact conservation of $L'$
corresponds to the case in which condition (\ref{condition}) is exactly fulfilled and
the RH neutrino splitting satisfies: $\Delta M =M_2-M_1 \rightarrow 0$.
In terms of the new parameters, the exact (tree-level) expression
of the light neutrino mass matrix given in (\ref{mvtreetypeI})
is proportional to $\mu$ and $\epsilon$, that is
\begin{equation}
\label{treemass}
m_{\nu}^{\rm tree}\;=\;\frac{v^2}{2\,(\Lambda^2-\mu'\mu)}
\left(\mu \,\mathbf{Y}_1\,\mathbf{Y}_{1}^T\,+\, \epsilon^2\,\mu' \,\mathbf{Y}_2\, \mathbf{Y}_{2}^T\,-\,\Lambda\,\epsilon\, (\mathbf{Y}_2\, \mathbf{Y}_{1}^T+\mathbf{Y}_1\, \mathbf{Y}_{2}^T)\right)\,,
\end{equation}
and thus if $\mu=\epsilon=0$ there is a complete
cancellation at tree-level for the light neutrino masses.
As we will see in the next section, if $\mu'$ is different from zero,
at least one neutrino mass can be generated at one-loop, even for
$\mu=\epsilon=0$~\cite{LopezPavon:2012zg}. Furthermore, from the
diagonalization of (\ref{treemass}), we obtain for the product of the smallest ($m_l^\text{tree}$) and
the largest ($m_h^{\rm tree}$) light neutrino masses:
\begin{eqnarray}
&& \left| m_{l}^{\rm tree}\,m_{h}^{\rm tree} \right| \;=\;\left| \det \left[M_R^{-1}\right]\,\det\left[m_D^T\,m_D\right]\right|\;=\nonumber\\\\
&& \frac{v^{4}\,\epsilon^{2}\left| y_{2 e}^{2}\,(y_{1 \mu}^{2}+y_{1 \tau}^{2})+
y_{1 e}^{2}(y_{2 \mu}^{2}+y_{2 \tau}^{2}) -2\, y_{1 e} y_{2 e} (y_{1 \mu} y_{2\mu}+y_{1 \tau}y_{2 \tau})+(y_{2 \mu}\, y_{1 \tau}-y_{1 \mu}y_{2\tau})^{2} \right| }{4|\Lambda^{2}-\mu\,\mu^{\prime}|}\,.
\nonumber
\end{eqnarray}
From this relation it follows that in order to have
two massive active neutrinos at tree-level, i.e.,
$m_{l,h}^{\rm tree}\neq 0$, $i)$ an explicit breaking of the
lepton charge conservation via the neutrino Yukawa
couplings is necessary, that is the parameter $\epsilon$ must always be
different from zero;
$ii)$ the vectors of neutrino Yukawa couplings $\mathbf{Y}_{1}$ and $\mathbf{Y}_{2}$
cannot be proportional.
Accordingly, the two seesaw limits of Eq.~(\ref{treemass}) which give rise to
large active-sterile neutrino mixing $\theta V$ and generate
sufficiently small active
neutrino masses are:
\begin{itemize}
\item i) $\mu'\gg \Lambda,\,y_{1 \alpha}\,v \gg \mu,\,\epsilon\, y_{2 \alpha}\, v$ (\textbf{ESS limit}).
This limit matches the so-called extended seesaw~\cite{Kang:2006sn}
models and corresponds to a hierarchical spectrum for the heavy neutrinos:
\begin{eqnarray}
M_1 &\approx& (\Lambda^2/\mu'-\mu)\,,\quad\;\;
(\theta V)_{\ell 1}\;\approx\; i\frac{v}{\sqrt{2}\,M_1}
\left [ y_{1\ell}\,\frac{\Lambda}{\mu' - \mu}
- \epsilon \, y_{2\ell} \left (1 - \frac{\Lambda^2}{2(\mu' - \mu)^2}\right)
\right]\,,\nonumber\\&&
\label{ESS}\\
M_2 &\approx&\mu'\,+\,\Lambda^2/\mu'\,,\quad\;\;
(\theta V)_{\ell 2}\;\approx\;
\frac{v}{\sqrt{2}\, M_2}
\left [ y_{1\ell}\, \left( 1 - \frac{\Lambda^2}{2(\mu' - \mu)^2}\right )
+\, \epsilon \, y_{2\ell} \, \frac{\Lambda}{\mu' - \mu}\,
\right ]\,,\nonumber\\ &&
\label{ESSb}
\end{eqnarray}
where we also show the corresponding mixing with the
active neutrinos. Then, the approximate tree-level contribution
to the $0\nu\beta\beta$ decay effective Majorana
mass due to the exchange of the light and the heavy neutrinos is
\begin{eqnarray}
\label{ESSbb0nul}
m^{\text{light}}_{\beta\beta} &\approx& \frac{v^2}{2\left(\Lambda^2/\mu'-\mu\right)}\left(\frac{\mu}{\mu'}\,y_{1e}^2\, -\,2\,\epsilon\frac{\Lambda}{\mu'}\,y_{1e}\,y_{2e}
\right)\,,\\
m^{\text{heavy}}_{\beta\beta} &\approx&
f(A)\,\frac{v^2\, M^2_a}{2\left(\Lambda^2/\mu'-\mu\right)^3}
\left(\frac{\Lambda^2}{\mu'^2}\,y_{1e}^2\,
-\,2\,\epsilon\,\frac{\Lambda}{\mu'}\,y_{1e}\,y_{2e}\right)\,,
\label{ESSbb0nuh}
\end{eqnarray}
respectively.
The dominant term in $m^{\text{heavy}}_{\beta\beta}$
is due to the exchange of the lighter of the two heavy Majorana
neutrinos $N_1$, the exchange of $N_2$ giving a subleading
(and negligible in the leading approximation we employed)
correction.
Notice that,
if $\Lambda^2/\mu'\gg\mu$, $m^{\text{light}}_{\beta\beta}$
becomes independent of $\mu'$ while
$m^{\text{heavy}}_{\beta\beta}$ is proportional to $\mu'$:
\begin{eqnarray}
\label{mbbESS1}
m^{\text{light}}_{\beta\beta} &\approx& \frac{v^2}{2\,\Lambda^2}\left(\mu \,y_{1e}^2 \,-\,2\,\epsilon\,\Lambda\, y_{1e}\,y_{2e}\right)\,,
\\
m^{\text{heavy}}_{\beta\beta} &\approx&
f(A)\,\frac{\mu' \, v^2\, M^2_a}{2\,\Lambda^4}
\left(\,y_{1e}^2 \,
-\,2\,\epsilon\,\frac{\mu'}{\Lambda}\,y_{1e}\,y_{2e}\right)\,.
\label{mbbESS2}
\end{eqnarray}
\item ii) $\Lambda\gg y_{1\alpha}\,v \gg \mu',\mu,\,\epsilon\, y_{2\alpha}\, v$
(\textbf{ISS limit}). This limit corresponds to a minimal realisation with
only two RH neutrinos of the so-called inverse or direct
seesaw models~\cite{Gavela:2009cd}.
In this case the heavy neutrino spectrum is quasi-degenerate,
forming a quasi-Dirac pair \cite{Wolfenstein:1981kw, Petcov:1982ya}
\begin{eqnarray}
M_1 &\approx&\Lambda -\frac{\mu+\mu'}{2}\,,\quad\;\;
(\theta V)_{\ell 1}\,\approx\;
i\frac{v}{2M_1}\,
\left[y_{1\ell}\left(1 + \frac{\mu - \mu'}{4\Lambda}\right)
-\,\epsilon \,y_{2 \ell}\left(1 - \frac{\mu - \mu'}{4\Lambda}\right)
\right]\,,\nonumber\\ &&
\label{ISS}\\
M_2 &\approx& \Lambda\,+\,\frac{\mu\,+\,\mu'}{2}\,,\quad
(\theta V)_{\ell 2} \;\approx\;
\frac{v}{2M_2}\,\left[y_{1 \ell}\left(1 - \frac{\mu - \mu'}{4\Lambda}\right)
+\,\epsilon \,y_{2 \ell}\left(1 + \frac{\mu - \mu'}{4\Lambda}\right)
\right]\,,\nonumber\\ &&
\label{ISSb}
\end{eqnarray}
In this limit the light and heavy contributions to
the $0\nu\beta\beta$ decay rate are given by:
\begin{eqnarray}
\label{mbbISSl}
m^{\text{light}}_{\beta\beta} &\approx&
\frac{v^2}{2\,\Lambda^2}\left(\mu\, y_{1e}^2
- 2\,\epsilon\,\Lambda \,y_{1e}\,y_{2e}\right)\,,\\
m^{\text{heavy}}_{\beta\beta} &\approx&
f(A)\,\frac{v^2\, M^2_a}{2\,\Lambda^4}
\left((2\,\mu\,+\,\mu')\, y_{1e}^2 \,
-\,2\,\epsilon\,\Lambda\, y_{1e}\,y_{2e}\right)\,.
\label{mbbISS}
\end{eqnarray}
Both of them are proportional to the
small lepton number violating parameters, as it should be.
Notice that the expression
of $m^{\text{light}}_{\beta\beta}$ above is exactly the same
as the one given in Eq.~(\ref{mbbESS1}).
\end{itemize}
On one hand, it follows from
Eqs.~(\ref{mbbESS1}), (\ref{mbbESS2}), (\ref{mbbISSl}) and
(\ref{mbbISS}) that a relatively large contribution to the
$0\nu\beta\beta$ decay rate due to the heavy Majorana neutrino exchange
might be possible at tree-level without affecting the smallness of the
light neutrino masses since in the limits considered here
$m^{\text{heavy}}_{\beta\beta}\propto \mu'$,
while $m^{\text{light}}_{\beta\beta}$ is independent of $\mu'$.
On the other hand, Eqs.~(\ref{ESS}-\ref{ESSb}) and (\ref{ISS}-\ref{ISSb})
confirm that the condition to obtain relatively large mixings,
Eq.~(\ref{condition}), is fulfilled at leading order,
that is in the Casas-Ibarra
parametrization the $R$-matrix corresponding to these two cases
is similar to the textures reported in Eqs. (\ref{RissNH}) and
(\ref{RissIH}).
Finally, we note that in the case of the ISS model, the
smallness of the light neutrino masses comes
from the existence of an approximate
symmetry corresponding to the conservation of the
lepton charge $L'$.
In contrast, in the ESS limit, the conservation of
the lepton charge $L'$
is strongly violated through the $\mu'$ coupling.
This means that, in principle,
the one-loop corrections to the neutrino masses
can be expected to be more important
in the ESS limit than in the ISS one since
in the ESS case there is no symmetry protecting the light neutrino masses
from getting relatively large corrections~\cite{LopezPavon:2012zg}.
\section{One-loop Corrections to the Neutrino
Mass Matrix}\label{sec3}
We turn now to the computation of the one-loop corrections to
the light neutrino mass matrix and the effective Majorana neutrino
mass associated to $0\nu\beta\beta$ decay amplitude.
At one-loop the neutrino self-energy $\Sigma(p)$ provides the
dominant finite correction to $m_\nu$~\cite{Pilaftsis:1991ug,Grimus:2002nk,AristizabalSierra:2011mn,LopezPavon:2012zg, Dev:2012sg},
which depends on the square of the neutrino Yukawa couplings,
as in the tree-level contribution (\ref{mvtreetypeI}), and is
further suppressed by the one-loop factor $1/(16\,\pi^{2})$.
In a generic basis, with the
Dirac and Majorana mass terms defined in Lagrangian (\ref{typeI}), we obtain:
\begin{equation}
\mathcal{M}\; = \; \begin{pmatrix} m_{\nu}^{1-\text{loop}} & m_D \\
m_D^T & M_R \end{pmatrix} = U^* \,\text{diag}\left(m_i, M_k\right)U^\dagger\,,
\label{Mnuloop}
\end{equation}
where the new Majorana mass term generated at one-loop is in this case
\begin{equation}
m_{\nu}^{\rm 1-loop}\;=\;
\frac{1}{(4 \,\pi\, v)^2}\,m_{D}\,\left(M_{R}^{-1}\,F(M_{R}M_{R}^{\dagger})+F(M_{R}^{\dagger}M_{R})\,M_{R}^{-1} \right) \,m_{D}^{ T}\,.
\label{mv1loopB}
\end{equation}
The loop function $F(x)$ is defined as
\begin{equation}
F(x)\equiv \frac{x}{2}\,\left(\,3\log(x/M_{Z}^{2})\,(x/M_{Z}^{2}-1)^{-1}+\log(x/M_{H}^{2})\,(x/M_{H}^{2}-1)^{-1}\,\right)\,,
\label{1loopfunc}
\end{equation}
$M_H$ and $M_Z$ denoting the Higgs and the $Z$ boson mass, respectively. Hence,
the overall light neutrino mass matrix, $m_\nu$, is given by
the sum of the tree-level (\ref{mvtreetypeI}) and
one-loop (\ref{mv1loopB}) terms, which
in the basis of charged lepton mass eigenstates satisfies the relation
\begin{equation}
m_{\nu}=m_{\nu}^{\rm tree}+m_{\nu}^{\rm 1-loop}\;=\; U_{\rm PMNS}^*\,\text{diag}(m_1,m_2,m_3)\,U_{\rm PMNS}^\dagger\,.
\label{mvtot}
\end{equation}
The finite radiative correction given in (\ref{mv1loopB}) is in general
subdominant in the case of RH neutrinos with a high mass scale $M\gg v$,
but it may be sizable and comparable to the tree-level term
in seesaw scenarios where
the lepton number violating scale is taken below the TeV range.
It is therefore interesting to analyse in greater detail the dependence
of the light neutrino masses on the additional
finite one-loop contribution, Eq.~(\ref{mv1loopB}).
In the basis in which the RH neutrino mass is diagonal,
the one-loop correction of interest has the following form:
\begin{eqnarray}
(m^{\rm 1-loop}_{\nu})_{\ell\ell'}
&=&\, \frac{1 }{(4\, \pi\, v)^2}\,
(\theta V)^*_{\ell k}\, M^3_k\,
\left(\frac{3 \log( M_k^{2}/M_{Z}^{2})}{M_k^{2}/M_{Z}^{2}-1}
+\frac{\log(M_k^{2}/M_{H}^{2})}{M_k^{2}/M_{H}^{2}-1} \right)\,
(\theta V)^{\dagger}_{k\ell'}\,,
\label{mv1loopcorr0}
\end{eqnarray}
where we have used Eqs.~(\ref{constraints2}) and (\ref{constraints3}).
The contribution of the one-loop correction
under discussion to the effective Majorana neutrino mass
$m^{\text{light}}_{\beta\beta}$, generated by the light Majorana
neutrino exchange, as can be shown, is given by
\begin{equation}\label{mbb1loopA}
m^{\rm 1-loop}_{\beta\beta} \;=\; (m^{\rm 1-loop}_{\nu})^*_{ee}\,.
\end{equation}
\subsection{The Scheme with Two RH Neutrinos}
In the phenomenologically interesting scheme with two RH neutrinos,
for each non-zero eigenvalue $m_k$ of Eq.~(\ref{mvtot}),
we have the exact relation
\begin{eqnarray}
0 &=& \det\left[\, m_k\,\mathbf{1_{3\times 3}}\,+\,m_{D}\,M_{R}^{-1}\,\left(\,\mathbf{1_{2\times 2}}-\mathcal{H}(M_{R})\,\right)\,m_{D}^{T} \,\right]\nonumber\\
&=& m_k\, \text{\rm det}\left[\, m_k\, \mathbf{1_{2\times 2}} \,+\, M_{R}^{-1}\left(\, \mathbf{1_{2\times 2}}- \mathcal{H}(M_{R}) \,\right)\,m_{D}^{T}m_{D}\,\right]\,,
\label{id1}
\end{eqnarray}
where the second equality follows form the Sylvester's determinant theorem
and we have introduced the function~\footnote{The definition given in Eq.~(\ref{Hfun}) is by construction basis
independent.}
\begin{equation}
\mathcal{H}(M_{R})\;\equiv\;\frac{1}{(4\,\pi\,v)^2}\, \left(F(M_{R}M_{R}^{\dagger})+M_{R}\,F(M_{R}^{\dagger}M_{R})\,M_{R}^{-1}\right)\,.
\label{Hfun}
\end{equation}
Using (\ref{id1}) and (\ref{treeproduct}), we get the identity
\begin{eqnarray}
\det\left[ \, \mathbf{1_{2\times 2}}- \mathcal{H}(M_{R}) \,\right]\, \left| m_{l}^{\rm tree}\,m_{h}^{\rm tree} \right| \;=\;m_l\,m_h\,,
\label{id2}
\end{eqnarray}
where $m_l$ ($m_h$) is the smaller (larger) non-zero active neutrino
mass, whose experimental value in the cases of NH and IH neutrino mass spectrum
is given in Eqs.~(\ref{measuredmassesNH}) and (\ref{measuredmassesIH}),
respectively.~\footnote{In the convention we are using
$m_{l}^{\rm tree} m_{h}^{\rm tree} =m_{2}^{\rm tree} m_{3}^{\rm tree}$
( $m_{l}^{\rm tree} m_{h}^{\rm tree} =m_{1}^{\rm tree} m_{2}^{\rm tree}$)
and $m_l m_h = m_2 m_{3}$ ($m_l m_h = m_1 m_{2}$) in the NH (IH) case.}~Therefore,
the determinant on the left hand side of
Eq.~(\ref{id2}) provides a measurement of the deviation of
the tree-level mass eigenvalues from the
observed neutrino masses. Notice that, this is a positive
quantity smaller than one
in the scenarios considered here.
As a consequence of Eq.~(\ref{id2}), one has that in
the case $m_l^{\rm tree}=0$, i.e. if two of the active neutrinos are
massless at tree-level, it is not possible to generate at one-loop
level two non-zero light (active) neutrino masses in the spectrum.
In other words, in such a scenario both the solar and atmospheric
neutrino oscillation mass differences cannot be radiatively generated.
As it is not difficult to show,
in the minimal scenario with only two
heavy Majorana neutrinos, in which
condition (\ref{condition}) is exactly fulfilled,
the one-loop contribution to the light neutrino mass
matrix goes to zero in the limit $\Delta M =M_2-M_1 \rightarrow 0$.
Indeed, from Eqs.~(\ref{condition}) and (\ref{mbb1loopA}) we find:
\begin{eqnarray}
m^{\rm 1-loop}_{\beta\beta}
&=&\, \frac{1}{(4\, \pi\, v)^2}\,
(\theta V)_{e 1}^2\, M^3_1\,
\left\{\left [\left ( \frac{3 \log( M_1^{2}/M_{Z}^{2})}{M_1^{2}/M_{Z}^{2}-1}
+\frac{\log(M_1^{2}/M_{H}^{2})}{M_1^{2}/M_{H}^{2}-1} \right) -
\left ( M^2_1 \rightarrow M^2_2\right )\right ] \right.
\nonumber \\ [0.30cm]
&-&\, \left. z(2+z)\,
\left(\frac{3 \log( M_2^{2}/M_{Z}^{2})}{M_2^{2}/M_{Z}^{2}-1}
+\frac{\log(M_2^{2}/M_{H}^{2})}{M_2^{2}/M_{H}^{2}-1} \right)
\right \}\,,
\label{mv1loopcorr1}
\end{eqnarray}
where $z \equiv \Delta M/M_1$, i.e., $M_2 = (1 + z) M_1$.
Note that Eq.~(\ref{mv1loopcorr1}) is valid for
arbitrary values of $z$ and $M_1$.
In the case of $M^2_1,M^2_2 \ll M^2_{Z},M^2_{H}$ we get:
\begin{eqnarray}
m^{\rm 1-loop}_{\beta\beta}
&=&\, \frac{(\theta V)^2_{e 1}}{(4\, \pi\, v)^2}\, M^3_1\,
\left [8\,(1+z)^2 \log (1 + z) + z(2+z)\,\left(3 \log( M_1^{2}/M_{Z}^{2})
+ \log(M_1^{2}/M_{H}^{2}) \right)
\right ]\,.\nonumber\\
\label{mv1loopcorr2}
\end{eqnarray}
If, in addition, $z \ll1$, this expression further simplifies to:
\begin{eqnarray}
m^{\rm 1-loop}_{\beta\beta}
&=&\, \frac{(\theta V)^2_{e 1}}{(4\, \pi\, v)^2}\, M^3_1\, z(2+z)\,
\left [4(1+z)^2 + 3 \log( M_1^{2}/M_{Z}^{2})
+ \log(M_1^{2}/M_{H}^{2}) \right ]\,.
\label{mv1loopcorr3}
\end{eqnarray}
In the opposite limit, namely, $M^2_1,M^2_2 \gg M^2_{Z},M^2_{H}$,
$m^{\rm 1-loop}_{\beta\beta} $ takes also a rather simple form
for $z \ll 1$. In this case, to leading order in $z \ll 1$,
we obtain:
\begin{eqnarray}
m^{\rm 1-loop}_{\beta\beta}
&=&\,-\, 2\, z\,\frac{1}{(4\, \pi\, v)^2}\,
(\theta V)^2_{e 1}\, M_1\,\left (3\, M^2_{Z} + M^2_{H} \right )\,.
\label{mv1loopcorr4}
\end{eqnarray}
Thus, in the scheme considered here, in which
condition (\ref{condition}) is fulfilled,
the magnitude of the one-loop correction
to $m^{\text{light}}_{\beta\beta}$
of interest, $m^{\rm 1-loop}_{\beta\beta} $,
exhibits a strong dependence on $z$.
This dependence is particularly important
in the case when the two heavy Majorana
neutrinos form a pseudo-Dirac pair,
$0 < \Delta M \ll M_1,M_2$, or $z\ll 1$.
In this case the ratio of the
one-loop correction to
the $0\nu\beta\beta$ decay amplitude and
the heavy Majorana neutrino exchange
contribution given in Eq.~(\ref{meeh2}),
$|m^{\rm 1-loop}_{\beta\beta} /m_{\beta\beta}^{\rm heavy}|$, practically
depends only on the mass $M_1$.
As it is not difficult to show,
for $f(A) = 0.79~(0.033)$, i.e., for $^{76}$Ge ($^{48}$Ca),
we have $|m^{\rm 1-loop}_{\beta\beta}/m_{\beta\beta}^{\rm heavy}| \approx 1$
at $M_1 \approx 15~(9.7)$ GeV.
For $M_1 > 15~(9.7)$ GeV
($M_1 < 15~(9.7)$ GeV), $|m^{\rm 1-loop}_{\beta\beta} |$
is bigger (smaller) than $|m_{\beta\beta}^{\rm heavy}|$.
\begin{figure}[t!]
\includegraphics[width=0.6\textwidth,angle=0]{figures/JLPplotSP090615.pdf}
\caption{
\label{1loopandh}
{\small
The contributions to
the $0\nu\beta\beta$ decay effective Majorana mass
due to the one-loop correction to the light neutrino mass
matrix (dashed line) and due to the heavy Majorana neutrino
exchange (solid line),
$|m^{\rm 1-loop}_{\beta\beta}|$ and $|m_{\beta\beta}^{\rm heavy}|$
(Eqs.~(\ref{mv1loopcorr1}) and (\ref{meeh2})),
as functions of the heavy Majorana neutrino mass $M_1$,
for $\Delta M = 10^{-2}$ GeV, $|(\theta V)_{e1}|^2 = 10^{-3}$
and $f(A) = 0.079$ (i.e., for $^{76}$Ge).
The range of values the effective Majorana neutrino mass can take
in the case of light Majorana neutrino exchange and
IH spectrum is also shown (the band in red color).
See the text for further details.
}}
\end{figure}
This is illustrated in Fig.~\ref{1loopandh}, which shows the dependence
of $|m^{\rm 1-loop}_{\beta\beta} |$ and $|m_{\beta\beta}^{\rm heavy}|$
on $M_1 > 0.5$ GeV for $\Delta M = 10^{-2}$ GeV in the scheme in which
condition~(\ref{condition}) is exactly fulfilled and
fixing the active-sterile mixing to
the reference value of $|(\theta V)^2_{e 1}|= 10^{-3}$.
In this
plot the Higgs mass has been set to $M_H=125$ GeV.
Note, however, that given the values of $M_Z = 90$ GeV and
$M_H = 125$ GeV, for $M_1 = 15~(9.7)$ GeV,
the factor $(4(1+z)^2 + 3 \log( M_1^{2}/M_{Z}^{2})
+ \log(M_1^{2}/M_{H}^{2}))$ in Eq. (\ref{mv1loopcorr3})
for $m^{\rm 1-loop}_{\beta\beta}$
is negative. Thus, at $M_1 = 15~(9.7)$ GeV,
we have $m^{\rm 1-loop}_{\beta\beta}/m_{\beta\beta}^{\rm heavy} > 0$
(see Eq. (\ref{meeh2})),
and therefore a cancellation, or even a partial
compensation, between the two terms
$m^{\rm 1-loop}_{\beta\beta} $ and
$m_{\beta\beta}^{\rm heavy}$ in the
$0\nu\beta\beta$ decay amplitude is impossible.
As it should be clear from Fig.~\ref{1loopandh} and Eqs.~(\ref{mv1loopcorr2}-\ref{mv1loopcorr4}),
$|m^{\rm 1-loop}_{\beta\beta}|$ grows rapidly with the
increase of $M_1$. However, the dependence of
$|m^{\rm 1-loop}_{\beta\beta} |$ on $z$ when
$z << 1$ makes it possible, in principle,
for $|m^{\rm 1-loop}_{\beta\beta} |$ to have values
in the range of sensitivity of the current and
next generation of $0\nu\beta\beta$ decay experiments,
i.e., to have $|m^{\rm 1-loop}_{\beta\beta} | \sim (0.01 - 0.10)$ eV
even for, e.g., $M_1 = 10^{3}$ GeV and
the maximal value of $|(\theta V)^2_{e 1}| = 10^{-3}$
allowed by the current data.
This requires, however, exceedingly small values of
$z$, which lead to a subleading heavy neutrino contribution.
Indeed, using the quoted values of $M_1$ and $|(\theta V)^2_{e 1}|$,
and taking into account that $v = 246$ GeV,
it is not difficult to find from Eq.~(\ref{mv1loopcorr4}) that
we can have $|m^{\rm 1-loop}_{\beta\beta} | \approx 0.01~(0.10)$ eV
for $z \approx 6\times 10^{-10}~(6\times 10^{-9})$. Such a small value of $z$ suggests a severe fine-tuning,
but it can also be understood in the context of the
ISS scenario as a technically naturally small value of the lepton number
violating parameters of this model.
In the analyses which follow we will not assume that
Eq.~(\ref{condition}) relating $(\theta V)_{e 1}$ and $(\theta V)_{e 2}$
is satisfied. We will use only the phenomenological constraint
on $(\theta V)_{e 1}$ and $(\theta V)_{e 2}$
\cite{Antusch:2006vwa,Antusch:2008tz,Atre:2009rg,Alonso:2012ji,Antusch:2014woa,Drewes:2015iva,Dinh:2012bp,Cely:2012bz}. Notice,
however, that for values of the Casas-Ibarra parameter $|\gamma| \gtrsim 6$ (see Eqs. (\ref{thetaR}),
(\ref{RissNH}) and (\ref{RissIH})), the relation given in Eq.~(\ref{condition}) is effectively satisfied.
\subsection{One-loop Generalisation of the Casas-Ibarra Parametrization}
\label{CIloop}
In order to make sure that we generate the correct light neutrino
mixing pattern, it is useful to generalise the
Casas-Ibarra parametrization introduced in the previous section
including the one-loop correction to the neutrino mass matrix.
Taking into account the expression (\ref{mv1loopcorr0}) for
$(m^{\rm 1-loop}_{\nu})_{\ell\ell'}$
in the basis in which the RH neutrino mass is diagonal,
Eq.~(\ref{mvtot}) takes the explicit form:
\begin{eqnarray}
(m_{\nu})_{\ell\ell'}&=&-\,(m_{D}\,V)_{\ell k} \left[ M^{-1}_k
-\frac{1}{(4\, \pi\, v)^2}\,M_k\,
\left(\frac{3 \log( M_k^{2}/M_{Z}^{2})}{M_k^{2}/M_{Z}^{2}-1}
+\frac{\log( M_k^{2}/M_{H}^{2})}{ M_k^{2}/M_{H}^{2}-1} \right)\right]\,
(V^{T}\,m_{D}^{ T})_{k\ell'}\,
\nonumber\\
&\equiv&
-\,(m_{D}\,V)_{\ell k}\,\Delta_k^{-1}\,(V^{T}\,m_{D}^{T})_{k \ell'}\;
=\;(U_{\rm PMNS}^*\,\text{diag}(m_1,m_2,m_3)\,U_{\rm PMNS}^\dagger)_{\ell \ell'}\,.
\label{mv1loopcorr}
\end{eqnarray}
Hence, in analogy to the tree-level contribution,
we have now
\begin{equation}
\left(\pm i \,\hat m^{-1/2} \,U_{\text{PMNS}}^\dagger \,\theta V \,\hat{M}\,\Delta^{-1/2}\right)\,\left(\pm i \,\hat m^{-1/2}\, U_{\text{PMNS}}^\dagger \,\theta V\,\hat{M}\, \Delta^{-1/2}\right)^T
\equiv R\, R^T=1\,.\label{CIloop1}
\end{equation}
Thus, we get the following expression for the
heavy Majorana neutrino couplings in the weak charged current, or
equivalently, for the active-sterile
neutrino mixing, at one-loop order:
\begin{equation}
\theta V = \mp i\, U_{\text{PMNS}}\, \hat m^{1/2}\, R\, \Delta^{1/2}\,\hat{M}^{-1}\,.
\label{thetaR2}
\end{equation}
In the numerical analysis reported in section~\ref{BetaBeta} we will make use
of this parametrization of $\theta V$, with $R$ given
in (\ref{RNO}) and (\ref{RIO}), in order to include the one-loop corrections
to the light neutrino masses and at the same time ensure that all
the neutrino mixing parameters match with their experimental values.
\begin{figure}[t!]
\begin{tabular}{cc}
\includegraphics[width=0.5\textwidth,angle=0]{figures/comparisonIHm2.pdf} &
\includegraphics[width=0.5\textwidth,angle=0]{figures/comparisonNHm2.pdf}
\end{tabular}
\caption{
\label{mbbMax}
{\small
Maximum value of the contribution to the $0\nu\beta\beta$ decay effective Majorana mass due to the heavy
Majorana neutrino exchange $|m_{\beta\beta}^{\rm heavy}|$ (solid thick line) for $^{76}$Ge and $\Delta M =10^{-2}$ GeV in the
IH (left panel) and NH (right panel) case, including the following constraints: $|m_{\beta\beta}^{\rm heavy}|\leq 0.5$ eV
and $|(\theta V)_{e 1}|^2 + |(\theta V)_{e 2}|^2 \leq 2\times10^{-3}$. The corresponding values of the contributions to the $0\nu\beta\beta$ decay
effective Majorana mass due to the tree-level (dashed line) and one-loop correction (dotted line)
to the light neutrino mass matrix, $|m^{\rm tree}_{\beta\beta}|$ and $|m^{\rm 1-loop}_{\beta\beta}|$, are also shown. The range of
values the effective Majorana mass can take in the case of light Majorana neutrino exchange and IH (NH) spectrum is shown in the red (blue) band. See the text for further details.
}}
\end{figure}
In Fig.~\ref{mbbMax} we illustrate the interplay between the contributions to
the $0\nu\beta\beta$ decay effective Majorana neutrino mass due to the heavy Majorana neutrino
exchange, $|m_{\beta\beta}^{\rm heavy}|$, the tree-level light neutrino masses, $|m^{\rm tree}_{\beta\beta}|=|(m^{\rm tree}_{\nu})^*_{ee}|$, and the one-loop correction to
the light neutrino mass matrix, $|m^{\rm 1-loop}_{\beta\beta}|=|(m^{\rm 1-loop}_{\nu})^*_{ee}|$, using the generalised Casas-Ibarra parametrization derived
above. In particular, we have maximised $|m_{\beta\beta}^{\rm heavy}|$ over the free parameters of the model ($\theta_{45}$, $\gamma$ and the Dirac and Majorana
phases of the PMNS matrix), in order to show the maximum heavy neutrino contribution to the process (solid thick line)
as a function of $M_1$ for $\Delta M =10^{-2}$ GeV and fixing the already measured PMNS parameters and neutrino squared mass differences to the best fit values
given in \cite{nufit}. The Higgs mass has been set to $M_H=125$ GeV. In the plot we show the corresponding value of the separate contributions associated to the tree-level (dashed line) and one-loop correction (dotted line) to the light neutrino mass matrix. We also impose the following constraints: $|m_{\beta\beta}^{\rm heavy}|\leq 0.5$ eV and $|(\theta V)_{e 1}|^2 + |(\theta V)_{e 2}|^2\leq 2\times10^{-3}$.
From Fig.~\ref{mbbMax} we conclude that for $M_1\lesssim 1$ GeV the one-loop correction is subleading for $\Delta M =10^{-2}$ GeV, being the tree-level contribution
the one responsible for the light neutrino mass generation. At the same time, in that region the heavy neutrino contribution to the $0\nu\beta\beta$ decay effective Majorana neutrino mass
can be sizable and larger than the one from light neutrino exchange. According to the estimate given in Fig.~\ref{estimate}, for $M_1\lesssim 1$ GeV there is no
need of any enhancement of the active-sterile mixing with respect to the naive seesaw scaling in order to obtain a sizable $|m_{\beta\beta}^{\rm heavy}|$.
However, around $M_1\sim 2$ GeV, the one-loop correction starts to be of the same size as the value of the light neutrino contribution dictated by neutrino oscillation
data. Indeed, this correction increases with $M_1$ in such a way that in order to stabilise the light neutrino mass and mixing, a fine-tuned cancellation
between the tree-level and one-loop correction is required. This is reflected in the fact that for $M_1\gtrsim 5$ GeV the dotted and dashed lines merge.
Therefore, as it is shown in Fig.~\ref{mbbMax}, for $5$ GeV $\lesssim M_1\lesssim 1$ TeV a sizable $|m_{\beta\beta}^{\rm heavy}|$ can in principle be realised, but a fine-tuned cancellation between the tree-level and one-loop
contributions to the light neutrino masses is also necessary.
Note that the bound $|m_{\beta\beta}^{\rm heavy}|\leq 0.5$ eV imposed by us can be saturated for $M_1\lesssim 100$ GeV. At
$M_1 = 10$ GeV, for instance, we have $|m_{\beta\beta}^{\rm heavy}| = 0.5$ eV for $|(\theta V)_{e1}|^2 + |(\theta V)_{e2}|^2 \simeq 0.8\times 10^{-4}$,
where we have used $f(A) = 0.079$ corresponding to $^{76}$Ge. For $M_1 \gtrsim 100$ GeV the
maximum value of $|m_{\beta\beta}^{\rm heavy}|$ decreases with $M_1$ since an active-sterile mixing $|(\theta V)_{ei}|^2$ bigger than $2\times10^{-3}$
would be required in order to saturate the bound.
It is interesting that the solid line and the blue and red bands
in Fig.~\ref{mbbMax} intersect around $M_1\sim 10^3$ GeV.
This implies that in the case of NH neutrino mass spectrum,
the effective Majorana neutrino mass $|m_{\beta\beta}|$ can be larger at
$0.1~{\rm GeV}\lesssim M_1\lesssim 10^3$ GeV
than that predicted in the case of the light neutrino exchange
mechanism. In particular, it can be in the range of sensitivity
of the experiments aiming to probe the range of
values of the effective Majorana mass corresponding to the
IH and quasi-degenerate (QD) light neutrino mass spectra
(see, e.g., \cite{PDG2014}).
In the case of the IH light neutrino mass spectrum,
the indicated result implies that at
$M_1\lesssim 10^3$ GeV there can be, in principle,
a significant interplay between the
light and heavy Majorana neutrino exchange contributions in
the effective Majorana mass, as discussed in detail in
\cite{Ibarra:2011xn} and summarised by us
at the end of subsection II.A
(see the paragraph before the last in subsection II.A).
More specifically, due to this interplay
of the light and heavy Majorana neutrino contributions,
$|m_{\beta\beta}|$ can be larger (smaller)
than that predicted in the case of the exchange of light neutrinos
with IH mass spectrum and
$|m_{\beta\beta}|$ will exhibit a dependence on the
atomic number $A$ of the decaying nucleus.
It should be mentioned that,
given the already high level of fine-tuning
required for the cancellation between the tree-level and one-loop light
neutrino contributions in $m_{\beta\beta}$,
an additional cancellation between the light and heavy
Majorana neutrino contributions would suggest further
fine-tuning.
The main features of Fig.~\ref{mbbMax} also appear
for larger splittings $\Delta M$.
In particular, the necessity of fine-tuned cancellation
between the tree-level and one-loop
correction to the light neutrino mass matrix is present also
in this case.
The level of the fine-tuning required increases
with $M_1$, as we will show in Section \ref{BetaBeta}.
\subsection{Radiative Corrections to the ESS and ISS Scenarios}
In this section, we compute the one-loop contribution to the
effective Majorana neutrino mass in the ESS and ISS limits of
the seesaw Lagrangian (\ref{typeI}) with two RH neutrinos.
Accordingly, we apply the parametrization of the Dirac and Majorana
mass matrices reported in Eq.~(\ref{Mnu2}) to the general
expression given in Eq.~(\ref{mv1loopB}).
The exact result of the one-loop contribution in terms of
the parameters introduced in (\ref{Mnu2}) is reported in Appendix~\ref{App}.
For the ESS scenario we have at leading order in $\Lambda/\mu'$
\begin{eqnarray}
m^{\rm 1-loop}_{\beta\beta}&\approx&\frac{\mu'}{2}\frac{y_{1e}^2}{\left(4\,\pi\right)^2}
\left(\frac{3\ln\left(\mu'^2/M_Z^2\right)}{\mu'^2/M^2_Z-1}+\frac{\ln\left(\mu'^2/M^2_H\right)}{\mu'^2/M^2_H-1}
\right)\,.
\label{meff1loopESS}
\end{eqnarray}
Notice that for $\mu'\gg M_H, M_Z$, this expression
reduces to
\begin{equation} m^{\rm 1-loop}_{\beta\beta}\;\approx\;\frac{y_{1e}^2}{\left(4\,\pi\right)^2}
\left(\frac{3\,M_Z^2}{2\,\mu'}\ln\left(\mu'^2/M_Z^2\right)+\frac{M^2_H}{2\,\mu'}\ln\left(\mu'^2/M^2_H\right)\right)\,.
\end{equation}
Therefore, when $\mu'\gg M_H, M_Z$, since the lepton number violating scale
$\mu'$ is introduced at high energies, the one-loop contribution
to the light neutrino masses appears to be suppressed as $1/\mu'$, as expected.
In the ISS realisation, i.e. for $\epsilon\,v,\,\mu,\,\mu'\ll \Lambda$,
we obtain
\begin{eqnarray}
m^{\rm 1-loop}_{\beta\beta}&\approx&\frac{1}{\left(4\,\pi\right)^2}\left(\epsilon \,\Lambda\,y_{1e}\,y_{2e}\,-\,\frac{\mu}{2}\,y_{1e}^2\right)
\left(\frac{3\ln\left(\Lambda^2/M_Z^2\right)}{\Lambda^2/M^2_Z-1}+\frac{\ln\left(\Lambda/M^2_H\right)}{\Lambda^2/M^2_H-1}\right)
\label{meff1loopISS}\\
&-&\frac{\mu+\mu'}{2}\frac{y_{1e}^2}{(4\,\pi)^2}
\left(\frac{4M_H^2M_Z^2-\Lambda^2\left(M_H^2+3M_Z^2\right)}{\left(\Lambda^2-M^2_Z\right)\left(\Lambda^2-M^2_H\right)}
+\frac{\ln\left(\Lambda^2/M_H^2\right)}{\left(\Lambda^2/M_H^2-1\right)^2}
+\frac{3\ln\left(\Lambda^2/M_Z^2\right)}{\left(\Lambda^2/M_Z^2-1\right)^2}\right)\,.
\nonumber
\end{eqnarray}
It is remarkable that in the ESS limit with $\mu'\lesssim M_H, M_Z$ and in the ISS limit the one-loop correction
to the light neutrino masses has a contribution proportional to $\mu'$. This dependence on $\mu'$ is very relevant since at one-loop
the light neutrino contribution to the $0\nu\beta\beta$ decay amplitude does depend directly on $\mu'$, as for the heavy contribution in (\ref{mbbESS2})
and (\ref{mbbISS}). This makes much more difficult to obtain a dominant contribution from the RH neutrinos in this limit, unless a fine-tuning of
the seesaw parameters is introduced to guarantee the smallness of the neutrino masses as it was indeed already shown in Fig. \ref{mbbMax}.
\mathversion{bold}
\section{Large heavy neutrino contribution to $0\nu\beta\beta$ decay}
\mathversion{normal}\label{BetaBeta}
In this section, we will address in more detail the question if
the RH neutrinos can eventually give a sizable contribution to
the $0\nu\beta\beta$ decay rate. As we have already mentioned,
cosmological constraints close the mass window of
$M<100$ MeV~\cite{Hernandez:2013lza,Hernandez:2014fha}
and thus only if the RH neutrino masses are larger
than $100$ MeV, a direct contribution to the process of interest
can be expected.
Following the notation in Ref.~\cite{Blennow:2010th},
the $0\nu\beta\beta$ decay rate can be written as
\begin{equation}
\frac{\Gamma_{0\nu\beta\beta}}{\ln2}=
G_{01}\left|\sum_{j}U_{ej}^{2}\frac{m_j}{m_e}\mathcal{M}^{0\nu\beta\beta}(m_j)\right|^{2},
\label{decayrate}
\end{equation}
where $G_{01}$ is a well-known kinematic factor, $U$ is the unitary
matrix given in Eq. (\ref{U}) which diagonalizes the
complete neutrino mass matrix,
$m_{j}$ are the corresponding eigenvalues, i.e.,
the neutrino masses (light and heavy), and
$\mathcal{M}^{0\nu\beta\beta}$ are the Nuclear Matrix
Elements (NMEs) associated with the process. Notice that
the NMEs depend on the mass of the neutrino mediating the
process since the dependence on the
neutrino propagator is already included in the NMEs computation.
The sum should be made over all the neutrino masses,
including the heavy ones. In the following
we will use the NMEs data provided in \cite{Blennow:2010th}.
In particular, we will consider the NMEs computed for the $^{76}$Ge.
However, we have checked that the conclusions of our
analysis do not significantly change considering a different nucleus.
We will use the modified Casas-Ibarra parametrization of the
active-sterile neutrino mixing given in Eq.~(\ref{thetaR2}),
to compute the full effective Majorana neutrino mass
$m_{\beta\beta}$, which is given by the sum of the
contributions from the exchange of the light and heavy Majorana neutrinos.
In this way, we include in the computation the effect of the
one-loop correction to the light neutrino masses, reproducing
at the same time the correct neutrino oscillation parameters.
We will also take into account the relevant bounds on the active-sterile
mixing which come from direct searches, charged lepton flavour violation
and non-unitarity constraints
~\cite{Antusch:2006vwa,Antusch:2008tz,Atre:2009rg,Alonso:2012ji,Antusch:2014woa,Drewes:2015iva,Dinh:2012bp,Cely:2012bz}.
Notice that the inclusion of such bounds guarantees the perturbativity of the neutrino
Yukawa couplings for any value of RH neutrino masses considered in this paper.
\begin{figure}[t!]
\begin{tabular}{ccc}
\includegraphics[width=0.33\textwidth,angle=0]{figures/100GeV_NH.pdf} &
\includegraphics[width=0.33\textwidth,angle=0]{figures/TeV_NH.pdf}&
\includegraphics[width=0.33\textwidth,angle=0]{figures/10TeV_NH.pdf}
\end{tabular}
\begin{tabular}{ccc}
\includegraphics[width=0.33\textwidth,angle=0]{figures/100GeV_IH_h.pdf} &
\includegraphics[width=0.33\textwidth,angle=0]{figures/TeV_IH_h.pdf}&
\includegraphics[width=0.33\textwidth,angle=0]{figures/10TeV_IH_h.pdf}
\end{tabular}
\caption{\label{fig2}
{\small \mathversion{bold}
\textbf{Neutrinoless double beta decay ($M_1\geq 100~\text{GeV}$).}
\mathversion{normal} The blue shaded areas in
the top panels (down panels)
represent the region of the parameter space in which we have
$10^{-2}~\text{eV}<|m_{\beta\beta}^{\rm light}\,+\,m_{\beta\beta}^{\rm heavy}|<0.5~\text{eV}$
($10^{-2}~\text{eV}<|m_{\beta\beta}^{\rm heavy}|<0.5~\text{eV}$)
in the case of NH (IH) neutrino mass spectrum
with the active-sterile mixing (or couplings) $(\theta V)_{\ell k}$
satisfying the bounds form direct searches, charged lepton flavour violation and non-unitarity constraints.
The black solid line stands for different values of the parameter
$\alpha \equiv |m^{\rm 1-loop}_{\beta\beta}|/|m_{\beta\beta}^{\rm light}|$,
which quantifies the fine-tuning required in order to achieve
the cancellation between the one-loop and tree-level
contributions to the light neutrino masses.
In the region to the right of the red dashed line
the ratio between the leading order and the next to
leading order contributions to the light
neutrino masses in the seesaw expansion is smaller than $10$.
The gray region to the right of the dotted line corresponds to
$y_{1e}^2\, m^{\rm 1-loop}_{\beta\beta}>
16\,\pi^2\,m_{\beta\beta}^{\rm light}$. The blue dashed line corresponds
to $|m_{\beta\beta}^{\rm heavy}|=0.05$ eV.
The measured neutrino oscillation parameters are fixed to the central
values reported in~\cite{nufit}.}}
\end{figure}
In the top panels (down panels) of Figs.~\ref{fig2} and \ref{fig3}, the blue shaded area corresponds to the region of the parameter space
in which $10^{-2}~\text{eV}<|m_{\beta\beta}^{\rm light}\,+\,m_{\beta\beta}^{\rm heavy}|<0.5~\text{eV}$ ($10^{-2}~\text{eV}<|m_{\beta\beta}^{\rm heavy}|<0.5~\text{eV}$),
projected on the $\gamma -\Delta M$ plane for NH (IH) and several values of $M_1$. In these plots we have fixed the already measured PMNS parameters
and neutrino oscillation mass differences to the best fit values given in \cite{nufit}.
The relevant Majorana and Dirac CP violation phases
in the PMNS matrix have been set to zero, but we have checked
that there is no significant impact on the results when other values
are considered.
The Casas-Ibarra parameter $\theta_{45}$ is also
set to zero. It is irrelevant when the heavy Majorana neutrino
exchange contribution is dominant (subdominant) in $m_{\beta\beta}$,
but can play an important role in the interplay of the light and heavy
Majorana neutrino exchange contributions when these two contributions are
comparable in size \cite{Ibarra:2011xn}.
The Higgs mass has been fixed to $M_H=125$ GeV. The solid black line stands for different values, stated in the plots, of the $\alpha$ parameter defined as
\begin{equation}
\alpha \equiv |m^{\rm 1-loop}_{\beta\beta}|/|m_{\beta\beta}^{\rm light}|\,,
\end{equation}
where $m_{\beta\beta}^{\rm light}=m^{\rm tree}_{\beta\beta}+m^{\rm 1-loop}_{\beta\beta}$ is the full (tree-level plus one-loop)
contribution to $m_{\beta\beta}$ given by the light neutrinos.
Therefore, $\alpha$ quantifies
the level of fine-tuning in the cancellation between $m^{\rm tree}_{\beta\beta}$ and $m^{\rm 1-loop}_{\beta\beta}$ described
in section \ref{CIloop} and required in order to keep the light neutrino masses and mixing to the observational values. Notice that
the level of fine-tuning increases with $\alpha$. The region to the right of the black solid line corresponds to values of $\alpha$ larger than those
stated in the plots.
In the red shaded area of Figs.~\ref{fig2} and \ref{fig3}, the ratio between the leading and next to leading order contributions to the light
neutrino masses in the seesaw expansion is smaller than $10$. The next to leading order contribution is given by~\cite{Grimus:2000vj}:
\begin{equation}
\delta m_\nu =
-\frac{1}{2}\left(m^{\rm tree}_\nu + m^{\rm loop}_\nu\right)\,
(\theta\,V) (\theta V)^\dagger\; -\;
\frac{1}{2}\,(\theta V)^*\,(\theta V)^T\,
\left(m^{\rm tree}_\nu+m^{\rm loop}_\nu\right)\,.
\end{equation}
From this expression, one can conclude that a cancellation
between the one-loop and tree-level contributions
to the light neutrino masses remains
at next to leading order in the seesaw expansion.
This is in agreement with Figs.~\ref{fig2}
and~\ref{fig3}, which show that the next to leading
order contribution is always negligible in the range of parameters of interest.
Ignoring for the time being
the impact of the two-loop corrections,
which will be commented below, two main conclusions can be extracted from
Figs.~\ref{fig2} and \ref{fig3}. First, we have proved that a
sizable and dominant heavy neutrino contribution to the
$0\nu\beta\beta$ decay is possible for RH neutrino masses as
heavy as $10$ TeV, satisfying at the same time the relevant
constraints and keeping
under control the light neutrino mass and mixing pattern.
Second, and not less important, it is shown that this possibility
can only take place if a highly
fine-tuned cancellation between the tree-level and one-loop light
neutrino masses is at work. The level of fine-tuning ranges
from $\alpha=10^{4}$ to $10^{9}$,
for heavy masses between $M_1=100$ GeV and $M_1=10$ TeV.
On the other hand, the level of fine-tuning is smaller
for lighter masses, being in the case of $M_1=100$ MeV smaller than
$\alpha = 2$. In addition, we have checked that
for $M_1\gtrsim 10$ TeV a heavy contribution to
$m_{\beta\beta}$ in the range of sensitivity
of the next-generation of experiments,
$|m_{\beta\beta}|\gtrsim 0.01$ eV,
cannot be expected.
Figs.~\ref{fig2} and \ref{fig3} also show that in the limit
$\Delta M \gg M_1$ the sizable heavy neutrino contribution
corresponding to the blue region becomes
independent of $\Delta M$, according with the ESS
limit -- see Eq.~(\ref{meeh2}).
However, in the ISS limit $\Delta M \ll M_1$
this is not the case and, according to Eq.~(\ref{meeh2}),
the smaller the heavy splitting $\Delta M$, the larger is the value of $\gamma$.
Notice that in the IH case we have plotted only
$m_{\beta\beta}^{\rm heavy}$ because $|m_{\beta\beta}^{\rm light}|$
is already in the planned range of sensitivity
of the next generation of $0\nu\beta\beta$ decay experiments.
In this case for
$M_1\lesssim 10^3$ GeV and $\Delta M << M_{1,2}$, there can be, in principle,
a significant interplay between the light and heavy Majorana neutrino exchange contributions in
the effective Majorana mass, as discussed at tree level in detail in \cite{Ibarra:2011xn} and summarised by us
at the end of subsection II.A (see the paragraph before the last in subsection II.A).
More specifically, due to this interplay of the light and heavy Majorana neutrino contributions,
$|m_{\beta\beta}|$ can be larger (smaller) than that predicted in the case of the exchange of light neutrinos
with IH mass spectrum and $|m_{\beta\beta}|$ will exhibit a dependence on the
atomic number $A$ of the decaying nucleus. This can happen roughly in the region located to the left of the
blue dashed line corresponding to $|m_{\beta\beta}^{\rm heavy}|=0.05$ eV inside the blue areas in Figs. \ref{fig2}
and \ref{fig3}.
In the NH case, the light neutrino contribution
is smaller than $10^{-2}$ eV and
therefore any sizable effect to the process is due to the heavy neutrinos. This is why in the NH case we plot
the total contribution $m_{\beta\beta}$, including light and heavy neutrinos.
\begin{figure}[t!]
\begin{tabular}{ccc}
\includegraphics[width=0.33\textwidth,angle=0]{figures/15GeV_NH.pdf} &
\includegraphics[width=0.33\textwidth,angle=0]{figures/GeV_NH.pdf}&
\includegraphics[width=0.33\textwidth,angle=0]{figures/100MeV_NH.pdf}
\end{tabular}
\begin{tabular}{ccc}
\includegraphics[width=0.33\textwidth,angle=0]{figures/15GeV_IH_h.pdf} &
\includegraphics[width=0.33\textwidth,angle=0]{figures/GeV_IH_h.pdf}&
\includegraphics[width=0.33\textwidth,angle=0]{figures/100MeV_IH_h.pdf}
\end{tabular}
\caption{\label{fig3}{\small\mathversion{bold}
\textbf{Neutrinoless double beta decay ($M_1< 100~\text{GeV}$).} \mathversion{normal}
The same conventions as in Fig.~\ref{fig2}, but for different choices of $M_1$.}}
\end{figure}
It follows from Fig. 4 that for $M_1\geq 100$ GeV the regions of interest
(the blue shaded areas) correspond to $\gamma \gtrsim 6$.
For such values of $\gamma$, as it is not difficult to show, we have for the
NH and IH neutrino mass spectra:
\begin{eqnarray}
&M_1& |(\theta V)_{e1}|^2 \approx M_2 |(\theta V)_{e2}|^2
\nonumber \\[0.30cm]
&\approx& \frac{e^{2\gamma}}{4}\,
\left |U_{e2}\sqrt{m_2} -\,i\, U_{e3}\sqrt{m_3} \right |^2\,,~~{\rm NH}
\label{thVgammaNH1}
\\ [0.30cm]
&\approx& \frac{e^{2\gamma}}{4}\,
\left |U_{e1}\sqrt{m_1} -\,i\, U_{e2}\sqrt{m_2} \right |^2\,.~~{\rm IH}
\label{thVgammaIH1}
\end{eqnarray}
Taking into account that Fig. 4 is obtained by setting to zero
the phase $\theta_{45}$ and the Dirac and Majorana phases in the PMNS matrix
and by using the best fit values of the neutrino oscillation parameters,
Eqs. (\ref{thVgammaNH1}) and (\ref{thVgammaIH1}) imply the following relations
between $|(\theta V)_{e1 (e2)}|^2$ and the parameter $\gamma$:
\begin{equation}
M_1 |(\theta V)_{e1}|^2 \approx M_2 |(\theta V)_{e2}|^2
\approx e^{2\gamma}\, 0.94~(12.4)\times 10^{-3}~{\rm eV}\,,~~{\rm NH~(IH)}\,.
\label{thVgammaNHIH2}
\end{equation}
In view of the high level of fine-tuning required in order to have a
cancellation between the tree-level and one-loop light neutrino masses,
the obvious question arising here is what is the role of
the two-loop corrections. Can the two-loop corrections spoil
this fine-tuned cancellation? In order to answer this question,
we estimate the impact of the two-loop contributions.
Since we are studying the case in which heavy neutrinos can
give a sizable contribution to the $0\nu\beta\beta$ decay, which means
relatively large
Yukawa couplings, we expect the diagram with two Higgs bosons
in the loop to be the leading two-loop contribution to the
light neutrino mass matrix.
The contribution of this diagram can be roughly estimated as
\begin{equation}
m^{\rm 2-loop}_{\beta\beta} \sim
\frac{y_{1e}^2}{(4\,\pi)^2}\,m^{\rm 1-loop}_{\beta\beta}\,,
\end{equation}
where $m^{\rm 1-loop}_{\beta\beta}$ is the one-loop contribution
in $m_{\beta\beta}^{\rm light}$. This estimate of the impact of the two
loop corrections is also shown in Figs.~\ref{fig2} and \ref{fig3}, where
the gray area to the right of the dotted line corresponds
to the region of the parameter space with
$y_{1e}^2\, m^{\rm 1-loop}_{\beta\beta}> 16\,\pi^2\,m_{\beta\beta}^{\rm light}$.
This region of the parameter space is excluded since the
two-loop correction, which would dominate the light neutrino masses,
would be larger than the value dictated by neutrino oscillation data.
Notice that this would essentially exclude the possibility of having a
large sterile neutrino contribution for $M_1 \gtrsim 1$ TeV,
as can be seen in Figs.~\ref{fig2} and \ref{fig3}.
For $M_1 \lesssim 100$ GeV the impact of the two-loop correction is
basically negligible.
\section{Conclusions}
We have performed a systematic analysis of the radiative corrections
to the light neutrino masses arising in low
scale type I seesaw scenarios, where the RH (sterile) neutrino masses
vary in the interval $100~\text{MeV}\lesssim M \lesssim 10~\text{TeV}$.
Within this range of masses a significant enhancement of the
neutrinoless double beta ($0\nu\beta\beta$) decay rate
in several isotopes - at the level of
sensitivity of the present and next generation experiments searching for
this rare process - is possible, due to the new physics contribution
in the decay amplitude given by the exchange of the
virtual heavy sterile neutrinos. Notice that one of the most clear signatures of a significant
heavy sterile Majorana neutrino contribution to the
$0\nu\beta\beta$ decay amplitude is the dependence of the
$0\nu\beta\beta$ decay effective Majorana mass,
$|m_{\beta\beta}|$, on the atomic number $A$ of the
decaying nucleus \cite{HPR83}.
The requirement of a sizable contribution of heavy neutrinos
with masses $\gtrsim 1$ GeV
to the $0\nu\beta\beta$ decay implies strong cancellations
between the tree-level and one-loop expressions in the
light neutrino mass matrix $m_\nu$ originated from the seesaw mechanism.
We show that such a cancellation can always be achieved while
being consistent with neutrino
oscillation data and low energy constraints from direct searches,
charged lepton flavour violation and non-unitarity by
using a generalisation of the Casas-Ibarra parametrization of
the neutrino Yukawa matrix, which can be derived from
Eqs.~(\ref{CIloop1}) and (\ref{thetaR2}).
We clarify the connection between this parametrization and
the lepton number breaking terms in the seesaw Lagrangian,
as usually defined in extended as well as inverse/direct
seesaw UV completions of
the Standard Model. Then, we numerically quantify the level of
fine-tuning between
the tree-level and one-loop parts of $m_\nu$ in the case the
heavy neutrino contribution $m_{\beta\beta}^{\text{heavy}}$
to the effective Majorana neutrino mass - which enters in the $0\nu\beta\beta$ decay amplitude -
is sizable, namely $|m_{\beta\beta}^{\text{heavy}}|\gtrsim 0.01$ eV.
The main results of our analysis
are summarised in Figs.~\ref{fig2} and \ref{fig3},
where we show that a fine-tuning of one part in $10^{4}$ ($10^{5}$)
for RH neutrino masses $\sim 100$ ($1000$) GeV is unavoidable
in order to have an observable effect in $0\nu\beta\beta$ decay experiments.
Furthermore, we conclude that for seesaw scales $M$ larger than few TeV,
two-loop effects in the generation of the light neutrino masses cannot
be neglected, thus
excluding the possibility of having a large $|m_{\beta\beta}^{\text{heavy}}|$.
Conversely, in the low mass regime, $M\lesssim 1$ GeV, the level
of fine-tuning in the seesaw parameter space is very mild and
the sterile neutrino contribution can easily
exceed the current limits on the effective Majorana neutrino mass.
Finally, we can conclude on the basis of the results obtained
in the present analysis that $0\nu\beta\beta$ sets the strongest constraints on lepton number
violation in low scale type I seesaw extensions of the Standard Model.
In particular, this implies a strong suppression of processes
which involve the production at colliders (LHC included)
of RH neutrinos and their decays with two like-sign charged leptons
in the final state (see, e.g., \cite{delAguila:2008cj, Ibarra:2010xw}).
\acknowledgments
The work of J.L.P. and S.T.P. was supported in part by the European Union FP7
ITN INVISIBLES (Marie Curie Actions, PITN-GA-2011-289442-INVISIBLES),
by the INFN program on Theoretical Astroparticle Physics (TASP)
and by the research grant 2012CPPYP7
({\it Theoretical Astroparticle Physics})
under the program PRIN 2012 funded by the Italian
Ministry of Education, University and Research (MIUR).
S.T.P. acknowledges partial support
from the World Premier International Research Center
Initiative (WPI Initiative), MEXT, Japan.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 3,383
|
{"url":"https:\/\/brilliant.org\/discussions\/thread\/calvin-which-solution-would-you-feature-8\/","text":"\u00d7\n\n[Calvin] Which solution would you feature (8)?\n\nPrevious Discussion\n\nBelow, we present a problem from the 2\/4 Algebra and Number Theory set, along with 3 student submitted solution. You may vote up for the solutions that you think should be featured, and should vote down for those solutions that you think are wrong (voting is anonymous!). Also, feel free to make remarks about these solutions, especially since threading of comments has been introduced :).\n\nPolynomial powered by 2 A polynomial $$f(x)$$ has degree $$8$$ and $$f(i)=2^i$$ for $$i=0,1,2,3,4,5,6,7,8$$. Find $$f(9).$$\n\nYou may try the problem by clicking on the above link.\n\nAll solutions may have LaTeX edits to make the math appear properly. The exposition is presented as is, and has not been edited.\n\nAbout 50% of those who answered this problem got it correct. Most guessed $$2^9 = 512$$, which is incorrect. The formula $$f(n) = 2^n$$ is an exponential, not a polynomial.\n\n[I am aware of solutions using the Lagrange Interpolation Formula, though none of them bothered to write it up properly. Please do not post any other solutions.]\n\n$\\mbox{Remarks from Calvin}$\n\nSolution A - This is a typical example of solutions obtained through lucky guessing. In this case, he happened to sum up the various values and got the answer of 511, so claimed that must be the solution. No justification for the steps have been given. While it can be considered the final step of Solution B, there is no indication that the method of differences is part of the thought process. Such a solution is marked incomplete at best.\n\nSolution B - This is a great solution, for those who understand the method of differences for a polynomial. It is simple, direct, and works if you are given consecutive values of the polynomial. I like that he provided enough detail that I can tell he has the correct solution at one quick glance. This solution is presented by Ryan.\n\nFor those who do not know the method of differences, do a quick search on the internet or read the comments.\n\nSolution C - Not every pattern can be proved by Mathematical Induction. In this case, because the polynomial is hard to generalize, the induction method will not work nicely. While that is not to say it is impossible, you will have to provide details of how to do the actual induction, and not merely \"let's do another one to make sure\".\n\nSolution Karan - Calculus works on incremental changes, and not such discrete large step changes. As Peter points out, we know that exponential growth is much faster than polynomial growth, so there is no polynomial function with can take on values $$f(n) = 2^n$$ for all positive integers $$n$$. Try proving this fact, it can be slightly tricky!\n\nSolution Titas - Clearly ignored my request to not state a solution using the Lagrange Interpolation Formula. Yes we know that it works, but the calculations can be tedious and are certainly non-enlightening. Here's a direct approach:\n\nWe know that binomial coefficients can be interpreted as polynomial. For example, $${ x \\choose 3} = \\frac {x (x-1)(x-2} { 3!}$$. I claim that the polynomial is equal to\n\n${x \\choose 0} + { x \\choose 1} + { x \\choose 2} + {x \\choose 3} + {x \\choose 4} + {x \\choose 5} + { x \\choose 6 } + { x \\choose 7 } + { x \\choose 8}.$\n\nThis follows since the above is a degree 8 polynomial that agrees on the 9 given values. Hence\n\n$f(9) = {9 \\choose 0} + { 9 \\choose 1} + { 9 \\choose 2} + {9 \\choose 3} + {9 \\choose 4} + {9 \\choose 5} + { 9 \\choose 6 } + { 9 \\choose 7 } + { 9 \\choose 8} = (1+1)^9 - 1 = 511.$\n\nNote by Calvin Lin\n4\u00a0years, 6\u00a0months ago\n\nSort by:\n\nSolution B - We can construct a finite difference table: $\\begin{array}{ccccccccccccccccc} f(0)&&f(1)&&f(2)&&f(3)&&f(4)&&f(5)&&f(6)&&f(7)&&f(8)\\\\\\hline 1&&2&&4&&8&&16&&32&&64&&128&&256\\\\ &1&&2&&4&&8&&16&&32&&64&&128\\\\ &&1&&2&&4&&8&&16&&32&&64\\\\ &&&1&&2&&4&&8&&16&&32\\\\ &&&&1&&2&&4&&8&&16\\\\ &&&&&1&&2&&4&&8\\\\ &&&&&&1&&2&&4\\\\ &&&&&&&1&&2\\\\ &&&&&&&&1 \\end{array}$ This last row has to be constant, so we can extend the finite difference table: $\\begin{array}{ccccccccccccccccccc} f(0)&&f(1)&&f(2)&&f(3)&&f(4)&&f(5)&&f(6)&&f(7)&&f(8)&&f(9)\\\\\\hline 1&&2&&4&&8&&16&&32&&64&&128&&256&&511\\\\ &1&&2&&4&&8&&16&&32&&64&&128&&255\\\\ &&1&&2&&4&&8&&16&&32&&64&&127\\\\ &&&1&&2&&4&&8&&16&&32&&63\\\\ &&&&1&&2&&4&&8&&16&&31\\\\ &&&&&1&&2&&4&&8&&15\\\\ &&&&&&1&&2&&4&&7\\\\ &&&&&&&1&&2&&3\\\\ &&&&&&&&1&&1 \\end{array}$ Therefore, $$f(9)=\\boxed{511}$$. Staff \u00b7 4\u00a0years, 6\u00a0months ago\n\nThis is clearly the clearest solution (no pun intended). It shows a nice visual representation of this problem and leaves out nothing for the reader to guess. \u00b7 4\u00a0years, 6\u00a0months ago\n\nvery good ...after knowing the solution,everything in the universe looks nice and leaves out nothing for the reader to guess. \u00b7 4\u00a0years, 6\u00a0months ago\n\nWhy does the last row has to be constant? \u00b7 4\u00a0years, 6\u00a0months ago\n\nIt is given that the polynomial is of order 8, and there are eight rows in the difference table that are the differences.\n\nn rows of differences means that the polynomial is of order n, so eight rows is that maximum an 8th degree polynomial can have. \u00b7 4\u00a0years, 6\u00a0months ago\n\nI've used exactly this method, extending the finite difference table. The table allows to find the general term and can be easily extended forward or backward indefinitely. \u00b7 4\u00a0years, 6\u00a0months ago\n\nReason for keeping the last row as constant??? \u00b7 4\u00a0years, 6\u00a0months ago\n\nIt is given that the polynomial is of order 8, and there are eight rows in the difference table that are the differences.\n\nn rows of differences means that the polynomial is of order n, so eight rows is that maximum an 8th degree polynomial can have. \u00b7 4\u00a0years, 6\u00a0months ago\n\ni consider the polynomial is f(x) = a.x(x-1)(x-2)......(x-7) + b.x(x-1)(x-2)....(x-6) + c.x(x-1)(x-2)...(x-5) + ........f.x(x-1)(x-2) + g.x(x-1) + h.x + i . this is a polynomial of degree 8. now putting x =0 we get i=2^0 = 1. again putting x=1 we get h+1=2^1 or, h=1 . again putting x= 2,3,....8 we can easily find out the value of a,b,...g. we get i = 1, h =1\/1 =1, g=1\/12 =1\/2 ,f =1\/123 =1\/6 ,......., a = 1\/123.....*8 . now putting x=9 we get f(x)= 511 . \u00b7 4\u00a0years, 6\u00a0months ago\n\nRemarks have been added. Congrats to Ryan! Staff \u00b7 4\u00a0years, 6\u00a0months ago\n\nLagrange interpolation would be tedious in this case ...... your first method is mathematically solid and more feasible in my opinion \u00b7 4\u00a0years, 6\u00a0months ago\n\nDo we only discuss one question per week for the featured solution? \u00b7 4\u00a0years, 6\u00a0months ago\n\nYes. You can look up previous discussions through the tag featured solutions.\n\nIt is difficult to get a question where people submit such varying viewpoints, or solutions in which they seem correct but are actually being wrong. Bear in mind that only students who have submitted the correct numerical answer may be chosen to submit a solution, so this already removes solutions which are just completely wrong. Staff \u00b7 4\u00a0years, 6\u00a0months ago\n\nBut How do you do $${0 \\choose 8}$$ for $$f(0)$$? Yours is a good solution, summing up rows of Pascal's triangle, but wouldn't it involve evaluating $$-8!$$? \u00b7 4\u00a0years, 5\u00a0months ago\n\nHow do you justify the claim?\n\ni did this using lagrange interpolation, evaluating $$f(9) = \\sum_{i=0}^{8} \\bigg(2^i \\times \\prod_{j=0, j \\neq i}^{8} \\frac{9-j}{i-j} \\bigg)$$\n\nI noticed that this was equal to $$\\sum_{i=0}^{8} (-2)^i \\times \\frac{9}{9-i} \\times {8 \\choose i}$$\n\nHow do you convert this into what you claim? \u00b7 4\u00a0years, 6\u00a0months ago\n\nI'm intentionally avoiding Lagrange Interpolation to calculate the polynomial, which merely provides the formula. I do not care what the coefficients are, and they can get pretty ugly.\n\nMy claim is to consider the polynomial $$g(x) = \\sum_{i=0}^8 { x \\choose i}$$. [Do you understand how this is a polynomial?] Then, to establish that this is the degree 8 polynomial that we want, we see that $$g(x)$$ has degree 8, and $$g(n) = f(n)$$ for $$n = 0$$ to 8 (pretty obvious), so it agrees on 9 values, hence $$g(x) = f(x)$$. [This follows because $$g(x) - f(x)$$ is a degree at most 8 polynomial with 9 roots, hence must be the zero polynomial.] Staff \u00b7 4\u00a0years, 6\u00a0months ago\n\nI would say that since 2^n - 2^(n-1) = 2^(n-1), its fairly obvious that the sequences of differences will be the same, and since we were given 9 terms and told that the polynomial is of the 8th degree, then it follows that the 8th difference sequence will be equal to one 1 (given the number of terms) So instead of following by 2, it would follow by 1 so the next term will be (2^n) -1. In other words I see no need to construct the table! \u00b7 4\u00a0years, 6\u00a0months ago\n\nlet the polynomial be ax^8+ bx^7+..........gx^2+ hx+ k We insert x=0 and find the value of k as 1 since 2^0=1( for 0 to 8 the fuction is 2^x) Next,we differentiate the above polynomial on both sides. Hence, we get 8ax^7+ 7bx^6.......+2gx+h= 2^x ln2. Now put x=0 on both LHS and RHS and we will get the value of h. If we go on doing this we get the constants as (ln2)^n\/n!....When we insert them back into the above equation and put x=9,we do not get 511..Can anyone tell me what is wrong with this solution? Thanks. \u00b7 4\u00a0years, 6\u00a0months ago\n\nThe polynomial is only equal to $$2^x$$ on the integers from 0 to 8, not on the interval $$[0,8]$$. (As a matter of fact, that would impossible for any polynomial.) \u00b7 4\u00a0years, 6\u00a0months ago\n\nIn solution B why the last rowto be kept constant \u00bf\u00bf\u00bf\u00bf\u00bf\u00bf\u00bf\u00bf?????????? No proper solution \u00b7 4\u00a0years, 6\u00a0months ago\n\nIf $$g(x)$$ is any non-constant polynomial, then the degree of the polynomial $$h(x)=g(x+1)-g(x)$$ will be one less than the degree of $$g(x)$$. \u00b7 4\u00a0years, 6\u00a0months ago\n\ni just want to ask why differentiation of e^4x which is 4*e^4x greater than it as differentiation is breaking down in smaller parts \u00b7 4\u00a0years, 6\u00a0months ago\n\nsir the second explanation of taking a polynomial is like the way i thought but can't we keep the same polynomial to get the values and predict what is taking place....... \u00b7 4\u00a0years, 6\u00a0months ago\n\nI don't think that u can. That process will be very lengthy. Look at the simplicity yet usefullness of solution B. \u00b7 4\u00a0years, 6\u00a0months ago\n\nSolution C - A polynomial of the 8th degree sounds scary, so we should start with simpler cases. To begin with, look at the polynomial of 1st degree (a line) given that $$f(i) = 2^{i}$$ for i=0,1. This obviously generates a line with equation $$y=x+1$$, and f(2) in this case (since we're looking for f(n+1) where n is the degree of the polynomial) = 3. Next, we examine the case with a polynomial of degree 2, or a quadratic, given that $$f(i) = 2^{i}$$ for i=0,1,2. Solving the system of 3 equations with 3 variables, we find that $$f(x) = 0.5x^{2}+0.5x+1$$, and f(3) = 7. We're beginning to see a pattern here, of f(n+1), where the polynomial has degree of n, has a value of $$2^{n+1}-1$$, but let's do another one to make sure. Again, we have a polynomial, this time of degree 3 and having $$f(i) = 2^{i}$$ for i=0,1,2,3. Solving the system of 4 variables and 4 equations, we get that $$f(x) = x^{3}\/6+5x\/6+1$$, and f(4) = 15, which is equal to $$2^{4}-1$$. We now hypothesize that f(9) in the original problem follows our formula, thus $$f(9) = 2^{9}-1 = 512-1 = 511$$, giving us the final answer of 511.\n\n[Note: Induction was listed as the Key Technique.] Staff \u00b7 4\u00a0years, 6\u00a0months ago\n\n\"We now hypothesize that f(9) in the original problem follows our formula\"\n\nProof?? \u00b7 4\u00a0years, 6\u00a0months ago\n\nInduction? \u00b7 4\u00a0years, 6\u00a0months ago\n\nI'm having trouble seeing how induction would work here, since we're adding a variable to each of the original equations AND adding another equation to the system as the degree of the polynomial increases. \u00b7 4\u00a0years, 6\u00a0months ago\n\nSir how can we say that this formula works for 4,5,6,7,8 degree polynomial , obviously checking like this will be very lengthy, so i think this doesn't prove that f(9)=511 \u00b7 4\u00a0years, 6\u00a0months ago\n\nyou cannot rely upon the pattern as you need to check for other degrees as well. we cannot say that watching the pattern gives us a definite answer. you have to check it for other degrees as well including degree 8. so, this is a bad solution. \u00b7 4\u00a0years, 6\u00a0months ago\n\nHow you are sure that the pattern working for ,2,3,4 will also work on 9 also You should state the reason or you should prove that f(x) of n degree will give f(n+1)=2^(n+1)-1 \u00b7 4\u00a0years, 6\u00a0months ago\n\nSolution A - $$f(9)=\\sum(2^i)$$ for $$i=0$$ to $$8$$. Hence, we have $$f(9) = 2^9 -1=511$$. Staff \u00b7 4\u00a0years, 6\u00a0months ago\n\nWhy? \u00b7 4\u00a0years, 6\u00a0months ago\n\nIt does follow from the finite difference table in Solution B, but other than that I have no idea. \u00b7 4\u00a0years, 6\u00a0months ago\n\nhow can you say so as this is a different solution and has nothing to do with solution- B \u00b7 4\u00a0years, 6\u00a0months ago","date":"2017-08-20 11:34:49","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.861153781414032, \"perplexity\": 588.142541478767}, \"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-2017-34\/segments\/1502886106465.71\/warc\/CC-MAIN-20170820112115-20170820132115-00071.warc.gz\"}"}
| null | null |
{"url":"https:\/\/tex.stackexchange.com\/questions\/150569\/get-math-width-in-math-unit-mu","text":"# get math width in math unit (mu)\n\nIs there any command to get the width of math in math mode and give the result in math unit of the used font?\nI am using `\\settowidth{\\myl}{\\$M\\$}\\the\\myl` but the result is in pt and in would like to get the value in mu directly.\n\n\u2022 The TeXBook mentions \"There are 18 mu to an em...\" \u2013\u00a0Werner Dec 18 '13 at 14:46\n\u2022 what do you need it for? `\\kern` does not need math units and works just fine in math mode \u2013\u00a0daleif Dec 18 '13 at 14:50\n\nMath units are transient things that make sense in the math list before TeX has committed that part of the list to being in text or script\/script size. At the end of a math expression TeX converts a math list into a horizontal list, raising and lowering boxes as needed, and assigning characters from the 16 font families. So there is no command to do this. You could measure the width in `pt` and divide by `\\fontdimen6\\textfont2` (em of the main math font) and then multiply by 18 to get something you might think of as the length in math units, but it doesn't really relate to anything that TeX uses, as the width of 1mu changes at different points within the expression.","date":"2021-01-19 06:11:53","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\": 0.9099403023719788, \"perplexity\": 736.9482798096954}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610703517966.39\/warc\/CC-MAIN-20210119042046-20210119072046-00056.warc.gz\"}"}
| null | null |
from PySide import QtCore, QtGui
class Add_Groups(object):
def __init__(self):
super(Add_Groups, self).__init__()
# self.list_groups.addItems(list_groups)
def setupUi(self, Form_Groups):
Form_Groups.setObjectName("Form_Groups")
Form_Groups.resize(356, 265)
self.gridLayout_2 = QtGui.QGridLayout(Form_Groups)
self.gridLayout_2.setObjectName("gridLayout_2")
self.gridLayout = QtGui.QGridLayout()
self.gridLayout.setObjectName("gridLayout")
self.list_groups = QtGui.QListWidget(Form_Groups)
self.list_groups.setObjectName("list_groups")
self.gridLayout.addWidget(self.list_groups, 0, 0, 1, 2)
self.btn_OK = QtGui.QPushButton(Form_Groups)
self.btn_OK.setObjectName("btn_OK")
self.gridLayout.addWidget(self.btn_OK, 1, 0, 1, 1)
self.btn_cancel = QtGui.QPushButton(Form_Groups)
self.btn_cancel.setObjectName("btn_cancel")
self.gridLayout.addWidget(self.btn_cancel, 1, 1, 1, 1)
self.gridLayout_2.addLayout(self.gridLayout, 0, 0, 1, 1)
self.retranslateUi(Form_Groups)
QtCore.QMetaObject.connectSlotsByName(Form_Groups)
# Bindings
self.btn_cancel.clicked.connect(self.Cancel)
self.btn_OK.clicked.connect(self.Add)
def retranslateUi(self, Form_Groups):
Form_Groups.setWindowTitle(QtGui.QApplication.translate("Form_Groups", "Add Groups", None, QtGui.QApplication.UnicodeUTF8))
self.btn_OK.setText(QtGui.QApplication.translate("Form_Groups", "Add", None, QtGui.QApplication.UnicodeUTF8))
self.btn_cancel.setText(QtGui.QApplication.translate("Form_Groups", "Close", None, QtGui.QApplication.UnicodeUTF8))
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,332
|
NGC 693 é uma galáxia espiral (S0-a) localizada na direcção da constelação de Pisces. Possui uma declinação de +06° 08' 41" e uma ascensão recta de 1 horas, 50 minutos e 30,9 segundos.
A galáxia NGC 693 foi descoberta em 25 de Dezembro de 1790 por William Herschel.
Ver também
Astronomia extragaláctica
Lista de galáxias
Lista de objectos NGC
New General Catalogue
Ligações externas
NGC 0693
Constelação de Pisces
Galáxias espirais
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 88
|
\section*{Introduction}
In the last years a few different approaches to quantum
statistics which generalize the usual boson or fermion
statistics has been intensively developed by several
authors. The so--called $q$--statistics and corresponding
$q$--relations have been studied by Greenberg \cite{owg,gre},
Mohapatra \cite{moh}, Fivel \cite{fi} and many others, see
\cite{zag,mepe,bs2} for example. The deformation of
commutation relations for bosons and fermions corresponding
to quantum groups $SU_q(2)$ has been given by Pusz and
Woronowicz \cite{P,PW}. The $q$--relations corresponding
to superparticles has been considered by Chaichian,
Kulisch and Lukierski \cite{ckl}. Quantum deformations
have been also studied by Vokos \cite{V},
Fairle and Zachos \cite{FZ} and many others.
Note that there is also an approach to particle systems
with some nonstandard statistics in low dimensional spaces
based on the notion of the braid group $B_n$ \cite{Wu,I}.
In this approach the configuration space for the system of
$n$--identical particles moving on a manifold ${\cal M}$ is given
by the formula
$$
Q_{n}({\cal M}) = {\left({\cal M}^{\times n} - D\right) }/{S_{n}},
$$
where $D$ is the subcomplex of the Cartesian product
${\cal M}^{\times n}$ on which two or more particles occupy
the same position and $S_{n}$ is the symmetric group.
The group $\pi_{1}\left( Q_{n}(M)\right)\equiv B_{n} (M)$
is known as the $n$--string braid group on ${\cal M}$. Note that
the group $\Sigma_n ({\cal M})$ is a subgroup of $B_n({\cal M})$ and
is an extension of the symmetric group $S_n$ describing the
interchange process of two arbitrary indistinguishable
particles. It is obvious that the statistics of the given
system of particles is determined by the group $\Sigma_n$
\cite{Wu,I}. The mathematical formalism related to the braid
group statistics has been developed intensively by Majid, see
\cite{SM,Maj,SMa,bm,sma,qm,ssm} for example.
It is interesting that all commutation relations for particles
equipped with arbitrary statistics can be described as
representations of the so-called quantum Weyl or Wick algebra
${\cal W}$. Such algebraic formalism has been considered by J\"orgensen,
Schmith and Werner \cite{jswe} and further developed by
the author in the series of papers
\cite{WM3,WM4,WM6,WM8,wm7,mad,mco,qweyl,wmq}.
and also by Ralowski \cite{jswe,RM,m10,ral}.
Similar approach has been also considered
by others authors, see \cite{sci,twy,mep} and \cite{mphi,mira}.
An interesting approach to quantum statistics has been also given
in \cite{fios,melme}. A proposal for the general algebraic formalism
for description of particle systems equipped with an arbitrary
generalized statistics based on the concept of monoidal categories
with duality has been given by the author in \cite{castat}.
The physical interpretation for this formalism was shortly
indicated. A few examples of applications for this formalism
are considered in \cite{top,sin}.
The generalization of the concept of quantum statistics
is motivated by many different applications in quantum field
theory and statistical physics. Some problems in condensed
matter physics, magnetism or quantum optics lead to the
study of particle systems obeying nonstandard statistics.
It is interesting that in the last years new and highly
organized structures of matter has been discovered.
For example in fractional quantum Hall effect a system
with well defined internal order has appears \cite{zee}.
Another interesting structures appear in the so called
$\frac{1}{2}$ electronic magnetotransport
anomaly \cite{jai,dst}, high temperature superconductors
or laser excitations of electrons. In these cases
certain anomalous behaviour of electron have appear.
An another example is given by the so--called Lutinger
liquid \cite{hal1}. The concept of statistical--spin
liquids has been studied by Byczuk and Spalek \cite{bys}.
It is interesting that half of the available
single--particle states are removed by the statistical
interaction between the particles with opposite spins.
The study of highly organized structures leads to
the investigation of correlated systems
of interacting particles. The essential problem in
such study is to transform the system of interacting
particles into an effective model convenient for
the description of ordered structures.
A system with generalized statistics seems to be
one of the best candidate for such model.
Hence there is an interest in the development of
formalism related to the particle systems
with unusual statistics and the possible
physical applications.
In this paper we would like to discuss some problems
of possibility of application of systems with generalized
statistics for the further description of ordered structures.
All our considerations are based on the assumption that
the quantum statistics of charged particles is determined
by some specific interactions.
\section*{Fundamental Assumptions}
The starting point for our discussion
is a system of charged particles interacting with certain
quantum field. The proper physical nature of the system is
not essential for our considerations.
The fundamental assumption is that the problem of
interacting particles can be reduced to the study
of a system consisting $n$ charged particles and
$N$--species of quanta of the field. In this way we
can restrict our attention to study of such system.
It is natural to expect that some new excited states
of the system have appear as a result of certain specific
interaction. The existence of new ordered structures
depends on the existence of such additional excitations.
Hence we can restrict our attention to the study
of possibility of appearance for these excitations.
For the description of such possible excited states
we use the concept of dressed particles.
We assume that every charged particle is equipped with
ability to absorb quanta of the external field. A system
which contains a particle and certain number of quanta as
a result of interaction with the external field is said to
be dressed particle. A particle without quantum is called
undressed or a quasihole. The particle dressed with two quanta
of certain species is understand as a system of two new objects
called quasiparticles. A quasiparticle is in fact the charged
particle dressed with a single quantum. Two quasiparticles
are said to be identical if they are dressed with quanta
of the same species. In the opposite case when the particle
is equipped with two different species of quanta then we have
different quasiparticles. We describe excited states as
composition of quasiparticles and quasiholes.
It is interesting that quasiparticles and quasiholes have
also their own statistics.
We give the following assumption for the algebraic
description of excitation spectrum of single dressed
particle.
\paragraph*{Assumption 0: The ground state.}
There is a state $|0> = {\bf 1}$ called the ground one.
There is also the conjugate ground state $<0| \equiv 1^{\ast}$.
This is the state of the system before intersection.
\paragraph*{Assumption 1: Elementary states.}
There is an ordered (finite) set of single quasiparticle states
\begin{equation}
S := \{x^i : i = 1,\ldots, N<\infty\}.
\end{equation}
These states are said to be elementary (simple). They
represent elementary excitations of the system.
We assume that the set $S$ of elementary states forms
a basis for a finite linear space $E$ over a field
of complex numbers $\mbox{\numb C}$.
\paragraph*{Assumption 2: Elementary conjugate states.}
There is also a corresponding set of single quasihole states
\begin{equation}
S^{\ast} := \{x^{\ast i} : i = N, N-1,\ldots , 1\}.
\end{equation}
These states are said to be conjugated. The set $S^{\ast}$ of
conjugate states forms a basis for the complex conjugate space
$E^{\ast}$. The pairing $(.|.) : E^{\ast} \otimes E \longrightarrow \mbox{\numb C}$
is given by
\begin{equation}
(x^{\ast i}|x^j) := \delta^{ij}.
\end{equation}
\paragraph*{Assumption 3: Composite states.}
There is a set of projectors
\begin{equation}
\Pi_n : E^{\otimes n} \longrightarrow E^{\otimes n}
\end{equation}
such that we have a $n$--multilinear mapping
\begin{equation}
\odot_n : E^{\times n} \longrightarrow E^{\otimes n}.
\end{equation}
defined by the following formula
\begin{equation}
x^{i_1} \odot \cdots \odot x^{i_n} :=
\Pi_{n}(x^{i_1} \otimes \cdots \otimes x^{i_n}).
\end{equation}
The set of $n$--multiquasiparticle states is denoted by
$P^n (S)$. All such states are result of composition
(or clustering) of elementary ones. These states are
also called composite states of order $n$. They represent
additional excitations charged particle under interaction.
In this way for multiquasiparticle states we have the
following set of states
\begin{equation}
P^n (S) := \{x^{\sigma} \equiv x^{i_1} \odot \cdots \odot x^{i_n}
: \sigma = (i_1 ,\ldots,i_n) \in I\},
\end{equation}
Here $I$ is a set of sequences of indices such that the
above set of states forms a basis for a linear space
${\cal A}^n$. We have
\begin{equation}
{\cal A}^n = Im(\Pi_n).
\end{equation}
Obviously we have where ${\cal A}^0 \equiv {\bf 1}\mbox{\numb C}$,
${\cal A}^1 \equiv E$ and ${\cal A}^n \subset E^{\otimes n}$.
\paragraph*{Assumption 4: Composite conjugated states.}
We also have a set of projectors
\begin{equation}
\Pi^{\ast}_n : E^{\ast\otimes n} \longrightarrow E^{\ast\otimes n}
\end{equation}
and the corresponding set of composite
conjugated states of length $n$
\begin{equation}
P^{n}(S^{\ast}) := \{x^{\ast \sigma} \equiv x^{\ast i_n}
\odot \cdots \odot x^{\ast i_1} :
\sigma = (i_1 ,\ldots,i_n) \in I \}.
\end{equation}
The set $P^n (S^{\ast})$ of composite conjugated states of
length $n$ forms a basis for a linear space ${\cal A}^{\ast n}$.
\paragraph*{Assumption 5: Algebra of states.}
The set of all composite states of arbitrary length is denoted
by $P(S)$. For this set of states we have the following
linear space
\begin{equation}
{\cal A} := \bigoplus\limits_{n} \ {\cal A}^n .
\end{equation}
If the formula
\begin{equation}
m(s \otimes t)s \equiv s \odot t := \Pi_{m+n}(\tl s, \otimes \tl t,)
\label{mul}
\end{equation}
for $s=\Pi_m (\tl s,), t=\Pi_n (\tl t,)$,
$\tl s, \in E^{\otimes n}, \tl t, \in E^{\otimes m}$,
defines an associative multiplication in ${\cal A}$, then we say
that we have an algebra of states. This algebra represents
excitation spectrum for single dressed particle.
\paragraph*{Assumption 6: Algebra of conjugated states.}
The set of composite conjugated states of arbitrary length
is denoted $by P(S^{\ast})$. We have here a linear space
\begin{equation}
{\cal A}^{\ast} := \bigoplus\limits_{n} \ {\cal A}^{\ast n} ,
\end{equation}
If $m$ is the multiplication in ${\cal A}$, then the
multiplication in ${\cal A}^{\ast}$ corresponds to
the opposite multiplication in ${\cal A}$
\begin{equation}
m^{op}(t^{\ast} \otimes s^{\ast}) = (m(s \otimes t))^{\ast}.
\end{equation}
\section*{Creation and Annihilation Operators}
We define creation operators for our model as multiplication
in the algebra ${\cal A}$
\begin{equation}
a^+_{s} t := s \odot t, \quad \mbox{for} \quad s, t \in {\cal A},
\end{equation}
where the multiplication is given by $\ref{mul}$.
For the ground state and annihilation operators we assume that
\begin{equation}
\langle 0|0 \rangle = 0, \quad a_{s^{\ast}} |0\rangle = 0
\quad \mbox{for} \quad s^{\ast} \in {\cal A}^{\ast}.
\end{equation}
The proper definition of action of annihilation
operators on the whole algebra ${\cal A}$ is a problem.
For the pairing
$\pg -, -, n, : {\cal A}^{\ast n} \otimes {\cal A}^n \longrightarrow \mbox{\numb C}$
we assume in addition that we have the following
formulae
\begin{equation}
\pg 0, 0, 0, := 0,\quad \pg i, j, 1, :=
(x^{\ast i}|x^j) = \delta^{ij},\\
\pg s, t, n, := \pg \tilde{s}, P_n \tilde{t}, n, _0
\quad\mbox{for}\quad n\geq2
\end{equation}
where $\tl s, , \tl t, \in E^{\otimes n}$,
$P_n : E^{\otimes n}\longrightarrow E^{\otimes n}$ is an additional linear
operator and
\begin{equation}
\langle i_1 \cdots i_n |j_1\cdots j_n \rangle^n_0 :=
\pg i_1, j_1, 1, \cdots \pg j_n, j_n, 1, .
\end{equation}
Observe that we need two sets $\Pi := \{\Pi_n\}$ and
$P := \{P_n\}$ of operators and the action
\begin{equation}
a: s^{\ast} \otimes t \in {\cal A}^{\ast k} \otimes {\cal A}^n \longrightarrow
a_{s^{\ast}} t \in {\cal A}^{n-k}.
\label{act}
\end{equation}
of annihilation operators for the algebraic description of
our system. In this way the triple $\{\Pi , P, a\}$, where
$\Pi$ and $P$ are set of linear operators and $a$ is the action
of annihilation operators, is the initial data for our model.
The problem is to find and classify all triples of initial
data which lead to the well--defined models. The general
solution for this problem is not known for us. Hence we
must restrict our attention for some examples.
\paragraph*{Definition:}
If operators $P$ and $\Pi$ and the action $a$ of
annihilation operators are given in such a way that
there is unique, nondegenerate, positive definite
scalar product, creation operators are adjoint to
annihilation ones and vice versa, then we say that
we have a well--defined system with generalized
statistics.
\paragraph*{Example 1:}
We assume here that $\Pi_n \equiv P_n \equiv id_{E^{\otimes n}}$.
This means that the algebra of states ${\cal A}$ is identical with
the full tensor algebra $TE$ over the space $E$, and the second
algebra ${\cal A}^{\ast}$ is identical with the tensor algebra
$TE^{\ast}$. The action (\ref{act}) of annihilation operators
is given by the formula
\begin{equation}
a_{x^{\ast i_k} \otimes \cdots \otimes x^{\ast i_1}}
(x^{j_1} \otimes \cdots \otimes x^{j_n})
:= \delta_{i_1}^{j_1} \cdots \delta_{i_k}^{j_k} \
x^{j_{n-k+1}} \otimes \cdots \otimes x^{j_n}.
\end{equation}
For the scalar product we have the equation
\begin{equation}
\langle i_n \cdots i_1 |j_1\cdots j_n \rangle^n
:= \delta^{i_1 j_1}\cdots \delta^{i_n j_n}
\end{equation}
It is easy to see that we have the relation
and
\begin{equation}
a_{x^{\ast i}} a_{x_j} := \delta_i^j {\bf 1}.
\end{equation}
In this way we obtain the most simple example of
well--defined system with generalized statistics.
The corresponding statistics is the so--called
infinite (Bolzman) statistics \cite{owg,gre}.
\paragraph*{Example 2:}
For this example we assume that $\Pi_n \equiv id_{E^{\otimes n}}$.
This means that ${\cal A} \equiv TE$ and ${\cal A}^{\ast} \equiv TE^{\ast}$.
For the scalar product and for the action of annihilation operators
we assume that there is a linear and invertible operator
$T : E^{\ast} \otimes E \longrightarrow E \otimes E^{\ast}$ defined by its
matrix elements
\begin{equation}
T(x^{\ast i}\otimes x^j) = \Sigma_{k,\ast l} \
T^{\ast ij}_{k\ast l} \ x^k \otimes x^{\ast l},
\label{teo}
\end{equation}
such that we have
\begin{equation}
(T^{\ast ij}_{k\ast l})^{\ast} =
\overline{T}^{\ast ji}_{l\ast k},
\mbox{i.e.} T^{\ast} = \overline{T}^t,
\end{equation}
and $(T^t)^{\ast ij}_{k\ast l} = T^{\ast ji}_{l\ast k}$.
Note that this operator not need to be linear, one can
also consider the case of nonlinear one. We also
assume that the operator $T^{\ast}$ act to the left,
i.e. we have the relation
\begin{equation}
(x^{\ast j} \otimes x^{i})T^{\ast} = \Sigma_{l,\ast k} \
(x^{l} \otimes x^{\ast k}) \ \overline{T}^{\ast ji}_{l\ast k},
\end{equation}
and
\begin{equation}
(T(x^{\ast i} \otimes x^j))^{\ast} \equiv (x^{\ast j}
\otimes x^{i}) T^{\ast}.
\end{equation}
The operator $T$ given by the formula (\ref{teo})
is said to be {\it a twist} or {\it a cross} operator.
The operator $T$ describes the cross statistics
of quasiparticles and quasiholes.
The set $P$ of projectors is defined by induction
\begin{equation}
P_{n+1} := (id \otimes P_n) \circ R_{n+1},
\end{equation}
where $P_1 \equiv id$ and the operator $R_n$ is given by
the formula
\begin{equation}
R_n := id + \tilde{T}^{(1)} + \tilde{T}^{(1)} \tilde{T}^{(2)}
+ \cdots + \tilde{T}^{(1)}\dots\tilde{T}^{(n-1)} ,
\end{equation}
where
$\tilde{T}^{(i)} := id_E \otimes \cdots
\otimes \tilde{T} \otimes \cdots \otimes id_E$,
$\tilde{T}$ on the $i$--th place, and
\begin{equation}
(\tilde{T})^{ij}_{kl} = T^{\ast ki}_{l\ast j}.
\end{equation}
If the operator $\tilde{T}$ is a bounded operator acting
on some Hilbert space such that we have the following
Yang-Baxter equation on $E\otimes E \otimes E$
\begin{equation}
(\tl T, \otimes id_E )\circ (id_E \otimes \tl T, )\circ
(\tl T, \otimes id_E ) = (id_E \otimes \tl T, ) \circ
(\tl T, \otimes id_E )\circ (id_E \otimes \tl T, ),
\end{equation}
and $||\tl T,|| \leq 1$,
then according to Bo$\dot{z}$ejko and Speicher \cite{bs2}
there is a positive definite scalar product
\begin{equation}
\pg s, t, n, _T := \pg s, P_n t, n, _0
\label{csca}
\end{equation}
for $s, t \in {\cal A}^n \equiv E^{\otimes n}$. Note that the
existence of nontrivial kernel of operator
$P_2 \equiv R_1 \equiv id_{E\otimes E} + \tilde{T}$
is essential for the nondegeneracy of the scalar product
\cite{jswe}.
One can see that if this kernel is trivial, then we
obtain well--defined system with generalized statistics
\cite{m10,ral}.
\paragraph*{Example 3:}
If the kernel of $P_2$ is nontrivial, then the scalar
product (\ref{csca}) is degenerate. Hence we must remove
this degeneracy by factoring the mentioned above scalar
product by the kernel. We assume that there is an ideal
$I \subset TE$ generated by a subspace
$I_{2} \subset ker P_2 \subset E \otimes E$ such that
\begin{equation}
a_{s^{\ast}} I \subset I
\end{equation}
for every $s^{\ast} \in {\cal A}^{\ast}$, and for the
corresponding ideal $I^{\ast} \subset E^{\ast} \otimes E^{\ast}$
we have
\begin{equation}
a_{s^{\ast}} t = 0
\end{equation}
for every $t \in TE$ and $s^{\ast} \in I^{\ast}$.
The above ideal $I$ is said to be Wick ideal \cite{jswe}.
We have here the following formulae
\begin{equation}
{\cal A} := TE/I, \quad {\cal A}^{\ast} := TE^{\ast}/I^{\ast}
\end{equation}
for our algebras. The projection $\Pi$ is the quotient map
\begin{equation}
\Pi : \tl s, \in TE \longrightarrow s \in TE/I \equiv {\cal A}
\end{equation}
For the scalar product we have here the following relation
\begin{equation}
\langle s|t\rangle_{B,T} := \langle\tilde{s}|\tilde{t}\rangle_T
\end{equation}
for $s = P_m (\tilde{s})$ and $t = P_n (\tilde{t})$.
One can define here the action of annihilation operators
in such a way that we obtain well--defined system with
generalized statistics \cite{ral}.
\paragraph*{Example 4:}
If a linear and invertible operator $B: E \otimes E \longrightarrow E \otimes E$
defined by its matrix elements
\begin{equation}
B(x^i \otimes x^j) := B^{ij}_{kl} (x^k \otimes x^l)
\end{equation}
is given such that we have the following conditions
\begin{equation}
\begin{array}{l}
B^{(1)} B^{(2)} B^{(1)} = B^{(2)} B^{(1)} B^{(2)},\\
B^{(1)}T\2nT^{(1)} = T\2nT^{(1)} B^{(2)},\\
(id_{E \otimes E} + \tilde{T})(id_{E \otimes E} - B) = 0,
\label{cd}
\ea
\end{equation}
then one can prove that there is well defined action
of annihilation operators and scalar product. In this
case we need two operators $T$ and $B$ satisfying the
above consistency conditions for the model with
generalized statistics \cite{RM,m10,ral}.
\paragraph*{Example 5:}
If $B = \frac{1}{\mu} \tl T,$, where $\mu$ is a parameter,
then the third condition (\ref{cd}) is equivalent to the well
known Hecke condition for $\tl T,$ and we obtain the
well--known relations for Hecke symmetry and quantum groups
\cite{P,PW,Ke}.
\section*{Physical applications}
Let us consider the system equipped with generalized
statistics and described by two operators $T$ and
$B$ like in Example 4. We assume here in addition that a
linear and Hermitian operator $S : E\otimes E\longrightarrow E\otimes E$ such that
\begin{equation}
S^{(1)} S^{(2)} S^{(1)} = S^{(2)} S^{(1)} S^{(2)}, \quad \mbox{and} \quad
S^2 = id_{E\otimes E}.
\end{equation}
is given. If we have the following relation
\begin{equation}
\tl T, \equiv B \equiv S,
\end{equation}
then it is easy to see that the conditions (\ref{cd})
are satisfied and we have well--defined system with
generalized statistics.
Let us assume for simplicity that the operator $S$ is
diagonal and is given by the following equation
\begin{equation}
S(x^i \otimes x^j) = \epsilon^{ij} x^j \otimes x^i,
\end{equation}
for $i, j = 1, \ldots, N$, where $\epsilon^{ij} \in \mbox{\numb C}$,
and $\epsilon^{ij}\epsilon^{ji} = 1$. In the general case we have
\begin{equation}
\epsilon_{ij} = (-1)^{\Sigma_{ij}} q^{\Omega_{ij}},
\label{comf}
\end{equation}
where $\Sigma := (\Sigma_{ij})$ and $\Omega := (\Omega_{ij})$ are
integer--valued matrices such that $\Sigma_{ij} = \Sigma_{ji}$
and $\Omega_{ij} = - \Omega_{ji}$, $q \in \mbox{\numb C} \setminus \{0\}$ is
a parameter \cite{zoz}. The algebra ${\cal A}$ is here a quadratic
algebra generated by relations
\begin{equation}
x^i \odot x^j = \epsilon^{ij} x^j \odot x^i,\quad\mbox{and}
\quad (x^i)^2 = 0 \quad\mbox{if}\quad\epsilon^{ii} = -1
\end{equation}
We also assume that $\epsilon^{ii} = -1$ for every
$i = 1, \ldots, N$. In this case the algebra ${\cal A}$
is denoted by $\Lambda_{\epsilon}(N)$. One can see that this
is a $G$--graded $\epsilon$--commutative algebra \cite{sch}.
Now let us study the algebra $\Lambda_{\epsilon}(2)$, where
$\epsilon^{ii} = -1$ for $i=1, 2$, and $\epsilon^{ij} = 1$ for $i\neq j$,
in more details. In this case our algebra is generated by $x^1$
and $x^2$ such that we have
\begin{equation}
x^1 \odot x^2 = x^2 \odot x^1, \quad (x^1)^2 = (x^2)^2 = 0
\end{equation}
Note that the algebra $\Lambda_{\epsilon}(2)$ is an example of
the so--called $Z_2 \oplus Z_2$--graded commutative colour
Lie superalgebra \cite{luri}. Such algebra can be transformed
into the usual grassmann algebra $\Lambda_2$ generated by
$\Theta^1$ and $\Theta^2$ such that we have the anticommutation
relation
\begin{equation}
\Theta^1 \ \Theta^2 = - \Theta^2 \ \Theta^1,
\label{thr}
\end{equation}
and $(\Theta^1)^2 = (\Theta^1)^2 = 0$.
In order to do such transformation we use the Clifford algebra
$C_2$ generated by $e^1, e^2$ such that we have the relations
\begin{equation}
e^i \ e^j + e^j \ e^i = 2 \delta^{ij}\quad \mbox{for} \quad
i, j = 1, 2.
\end{equation}
For generators $x^1$, and $x^2$ of the algebra $\Lambda_{\epsilon}(2)$
the transformation is given by
\begin{equation}
\Theta^1 := x^1 \otimes e^1, \quad\mbox{and}\quad \Theta^2 := x^2 \otimes e^2.
\end{equation}
It is interesting that the algebra $\Lambda_{\epsilon}(2)$ can be
represented by one grassmann variable $\Theta$, $\Theta^2 = 0$
\begin{equation}
x^1 = (\Theta, 1),\quad x^2 = (1, \Theta).
\label{sqa}
\end{equation}
For the product $x^1 \odot x^2$ we obtain
\begin{equation}
x^1 \odot x^2 = (\Theta, \Theta).
\label{haf}
\end{equation}
In physical interpretation generators $\Theta^1$ and $\Theta^2$ of
the algebra $\Lambda_2$ represents two fermions. They anticommute
and according to the Pauli exclusion principle we can not put
them into one energy level. Observe that the corresponding
generators $x^1$ and $x^2$ of the algebra $\Lambda_{\epsilon}(2)$
commute, their squares disappear and they describe two different
quasiparticles. This means that these quasiparticles behave
partially like bosons, we can put them simultaneously into one
energy levels. This also means that single fermion can be
transform under certain interactions into a system of two
different quasiparticles.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,215
|
El Chicago Fire Soccer Club és un club de futbol professional de la ciutat de Chicago, situat al suburbi de Bridgeview, Illinois, als Estats Units. Equip de la Major League Soccer des de 1997, té la seu al nou estadi específic de futbol Toyota Park, que té una capacitat de 20.000 espectadors, juga amb samarreta i pantalons de color vermell.
Història
El club va ser fundat el 8 d'octubre de 1997 en el 126è aniversari del gran incendi de Chicago de 1871. El 1998, en la seva primera temporada, va guanyar l'MLS Cup, així com la US Open Cup, aquest darrer títol també el guanyà els anys 2000, 2003 i 2006. També va guanyar l'Escut dels seguidors de l'MLS al millor equip de la lliga regular el 2003.
El club té un acord de cooperació amb el CA Morelia mexicà des del 2001. Competeix amb el FC Dallas des del mateix any per la Brimstone Cup, que es decideix en funció dels partits que disputen ambdós durant la lliga regular.
Inicialment disputava els seus partits al Soldier Field, l'estadi dels Chicago Bears de l'NFL. Actualment juga al Toyota Park, un estadi específic de futbol. Per l'equip han passat destacats jugadors entre els quals destaquen: Frank Klopas, Eric Wynalda, Hristo Stoítxkov, Tomasz Frankowski o Cuauhtémoc Blanco.
Palmarès
MLS Cup (1): 1998
MLS Supporters' Shield (1): 2003
US Open Cup (4): 1998, 2000, 2003, 2006
Estadis
Soldier Field (1998–2001, 2004–2005)
Cardinal Stadium (2002–2003)
Toyota Park (2006—)
Entrenadors
Bob Bradley (1998–2002)
Dave Sarachan (2003—2007)
Juan Carlos Osorio (2007)
Denis Hamlett (2008–)
Jugadors destacats
Hristo Stoítxkov
Gonzalo Segares
Paulo Wanchope
Lubos Kubik
Damani Ralph
Dema Kovalenko
Cuauhtémoc Blanco
Jorge Campos
Tomasz Frankowski
Roman Kosecki
Piotr Nowak
Jerzy Podbrozny
Chris Armas
DaMarcus Beasley
Carlos Bocanegra
CJ Brown
Frank Klopas
Justin Mapp
Ante Razov
Chris Rolfe
Zach Thornton
Josh Wolff
Eric Wynalda
Enllaços externs
Web oficial
Història
Web de l'Independent Supporters' Association
Fòrum
Chicago Fire Soccer
Equips de l'MLS
Clubs de futbol estatunidencs
Clubs esportius de Chicago
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,603
|
require 'em_test_helper'
require 'socket'
class TestSetSockOpt < Test::Unit::TestCase
if EM.respond_to? :set_sock_opt
def setup
assert(!EM.reactor_running?)
end
def teardown
assert(!EM.reactor_running?)
end
#-------------------------------------
def test_set_sock_opt
omit_if(windows?)
test = self
EM.run do
EM.connect 'google.com', 80, Module.new {
define_method :post_init do
val = set_sock_opt Socket::SOL_SOCKET, Socket::SO_BROADCAST, true
test.assert_equal 0, val
EM.stop
end
}
end
end
else
warn "EM.set_sock_opt not implemented, skipping tests in #{__FILE__}"
# Because some rubies will complain if a TestCase class has no tests
def test_em_set_sock_opt_unsupported
assert true
end
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,071
|
#
# About the Book
Rescuing endangered species, piloting choppers and coming nose-to-snout with some of Australia's deadliest creatures is all in a day's work for Matt Wright. With his mates by his side, Matt ventures into the outback and beyond, managing to get into (and, remarkably, out of) some insanely nail-biting situations.
This new collection of adventures moves from his home in the Northern Territory to the jungles of Borneo to the rivers of the Congo. Follow Matt as he tracks down a monster croc in the Congo, relocates fifty saltwater crocodiles over state borders in the space of a few days, rescues elephants and orangutans (and two giant snakes) in Borneo and spends time at home with his pet crocodile Tripod, and gain some behind-the scenes insights into the making of some of Outback Wrangler's most intense moments.
Told with wit, candour and a hit of adrenaline that makes you feel like you're riding shotgun in Matt's chopper, Outback Adventures is a ripping collection of unforgettable experiences from a remarkable Australian.
#
#
Introduction
1 Two rascals
2 Training the crew
3 The wild west
4 Under investigation
5 Operation Congo
6 Unparalleled wildlife
7 My mate Tripod
8 Hiccups and headaches
9 Island adventure
10 Colossal croc
11 Outback Wrangler, season three
12 My wife
13 A bloke's dream
14 Caught off guard
15 Success
Picture Section
#
I was raised in Papua New Guinea and the Australian outback, where living off the land and connecting with animals became second nature to me. From an early age I was drawn to animals most people would run from, and developed a love of deadly snakes, spiders and sharks in particular. My fascination for reptiles grew and grew, and I became obsessed not only with snakes but alligators and crocodiles too. By the time I turned 10, much to my mum's disgust, I had acquired my very own prized collection of one of Australia's deadliest animals, the king brown snake. I kept three of these beauties in a bucket that sat at the end of my bed while I slept. I'd take the snakes everywhere with me, and always seemed to be in trouble at home and school for sharing my 'pets' with my classmates and teachers. My mischief and love of wildlife developed as I grew into my teens, and if anything involved an adventure or rogue animal, I was there!
As a young adult, I was keen for my adventures to continue so I said yes to everything, upskilling myself and taking on whatever work was going. I've done plenty of jobs in my time – toilet cleaner, cattle station ringer, drill rig worker, Australian Army soldier, helicopter instructor and crocodile egg collector. Everything I learned along the way has helped develop my passion and skills for my career today as a helicopter pilot, wildlife conservationist and tourism operator.
For my work, I get to travel the world, capturing and relocating a diverse range of animals from crocodiles to wild buffalo and even polar bears. It's an honour to be able to interact so intimately with these animals, and I always have the preservation of wildlife front of mind. My focus is to remove and relocate problem animals rather than kill them, and I feel pretty bloody lucky to be where I am and to do what I do.
I love nothing more than sharing my slice of paradise in northern Australia with everyone else. My knowledge of helicopters, boats and airboats means that I've been able to create some awesome tourism experiences for people to come and spend a day in my life. But I'm not exactly a shrewd businessman, so there are some good-value 'What not to do' stories in this book about my troubles (and my successes) in setting up a new company.
I guess you could say that my lifestyle in Australia's Top End is pretty unique and it's something that the National Geographic Channel picked up on. In 2012, we premiered Outback Wrangler, a TV show that follows my life in the Northern Territory, catching and relocating the biggest saltwater crocodiles in the world. Luckily for me, people enjoyed my story and watching the show, and we're now coming into its fourth season.
As you can imagine, there are a lot of hairy moments and touch-and-go encounters that come with my line of work. This book gives you a few of my most memorable 'outback adventures' over the years from Australia to Malaysia and the Congo. This dip-in dip-out selection of experiences complements my life story, which I told in my first book, The Outback Wrangler.
#
In 2004, our egg-collecting missions were pretty perilous. This was before the days of slings and rings, when we didn't have the luxury of a 100-foot line, harness and helmet to get us safely and conveniently onto a crocodile nest, so we had to use other means, which led to some shitty situations.
In areas where vegetation or uneven terrain prevents a helicopter from making a full skid landing, you've got to make a hover exit. And during egg collecting, this means jumping out of the chopper into the air, and often landing right next to a crocodile. Or, other times, getting dropped off as close as possible and walking my way through tall timber to get to a nest. Sometimes, as close as possible was a long way away. (The worst technique of them all was the provisional 'chain and stick' sling I invented, but I'll get to that later in the chapter.
In the early days, it was just me and my good mate Jimmy who did the collecting. We were footloose and fancy-free and did whatever we had to do to get in and out of a nest without being chewed. We would pack ourselves but loved the adventure. Without the proper equipment and gear, overheating and exhaustion was commonplace.
One particular day on the Victoria River in the Northern Territory is ingrained in my memory forever. It was the peak of the wet season and the heat was unbearable – it was 42 degrees Celsius and the humidity was worse than a steam room. Jimmy was collecting from a patch of nests in open country and I got the gig of taking care of a creek bed that ran through thick jungle. It was my first time in this sort of setting and I didn't know what to expect. My boss had thrown me straight in the deep end with very little direction and I was too scared to speak up back then.
'All right, Matty, you go look now. Follow the water and don't get lost,' he said to me. And that was it. He landed his rattling R22 chopper at the top of an escarpment and I jumped out with my two blue banana crates and stick – crates to carry the eggs and a stick for defence. A radio would have been good but I wasn't going to ask for one.
I was fairly fit, so bouncing through the spinifex and rocky country into the valley was easy enough but the heat was pretty vicious, and not only was the sun burning from above but the rocks beneath my feet were so hot you could fry an egg on one. With every step, I could feel the warmth pulsating through the soles of my shoes. The heat was a good reminder to slow down and save energy, and it wasn't long until I got into the jungle.
It was cooler in there, under the canopy of trees, but while I'd lost the hot breeze I'd gained the humidity. And I'd been so focused on preparing myself for collecting the eggs along the river that I hadn't even thought about the challenges of doing it in the dense jungle. It was inhospitable. I struggled to move through it in a straight line; actually, I struggled to move much at all. I stepped one way and got hung up in vines, and stepped another way and got hooked on spikey bushes, retreating only to trip on fallen dead wood. I moved two steps forward, one step back and ten steps to the side to find myself where I'd began: on swampy ground and amongst razor-sharp grass.
I eventually got through a small opening. I stepped up onto a big log and jumped forwards into the air, only to drop right into a big spider web. Now, I can deal with crocs and snakes any day but spiders are not my favourite animals, and I had just managed to completely cocoon myself in this web with a golden orb-weaving spider the size of a dinner plate on my chest. I lost it, dropped my gear and started dancing around in circles, pulling the spider and sticky web off my body.
'Farrrrk!' I yelled, hearing it echo out.
I got the rest of the gunk off and stood back to look at what was left of the web. Its size was unbelievable: it would have been 10 by 10 metres, easy. It was thick with insects and even a couple of small finches were tangled up in there.
Spider webs are pretty remarkable creations. They say that if the threads of a spider web were as thick as rope, the web could stop a Boeing 747 aircraft. The sun peeped through the trees and shone down on the web, making it shimmer. Despite the big-arse spider, it was beautiful.
I grabbed my crates and stick, and kept moving. The path became easier to navigate and soon I found the creek. I trudged down the right bank of the creek through the jungle, looking out for a nest.
I was stopped in my tracks again when a golden tree snake fell at my feet. It made me jump a little but I kept walking. Then a few steps further, another three snakes tumbled out of the trees. One landed on my shoulder, so I put my crates down and picked up the long, thin snake, admiring its brilliant yellow colour. I was careful with it because their fangs carry poison – nothing fatal, but that was the last thing I needed. As I moved, they continued to fall like gumnuts and then slide off to safer ground. I must have startled them.
I checked my watch. I'd already been gone for 35 minutes and I hadn't even collected a nest. I'd spotted most of the nests from the air earlier, so I knew there were about six of them around. I could hear the chopper buzzing above me, assuring me that I wasn't alone.
And then I saw it. Sitting perfectly on the bank behind some small trees was a pile of scratched-up razor grass, sand and paperbark – my first nest.
Carefully approaching its vicinity, I kept an eye on the nest and an eye out for the croc.
'Where are you, girl, where are you?' I whispered.
You never know if the croc is hiding off to one side ready to ambush, so it's always a big relief when you can spot the croc before you set foot on the nest. I stepped carefully and got down low to look under the surrounding foliage, keeping one crate close, like a shield at the ready. In the other hand I held my stick firm, ready to jam into any unexpected teeth. My senses were heightened, tuning in to the slightest of movements. In moments like that, a pin dropping sounds like a hammer slamming.
I did one last scout of the area. The only place I could think she might be was the creek. The water was crystal clear and looked about six feet deep. A scaly tail poked out from underneath where I was standing. She must have been wedged in an underwater ledge. Perfect, I thought. Leaving her there, I quietly stepped back to the nest, keeping the crate in between the water and me. I opened the nest up, pulled out the eggs and marked them one by one with a tradie pencil. It was a great nest – I collected 50-odd eggs and there were only three infertile ones. When you pick up an egg you look for a thin white banding around its circumference, which tells you that the egg is fertile. If there is no band and the egg is a lighter colour, almost translucent, you know it's ruined. I finished packing the nest and kept trucking.
I already had one half-full crate, but it was still early days and the heat was getting worse. My light-blue shirt had turned a deep blue and my pants were wet with sweat. I hoped the chopper would fly over again so I could feel the cool down-draft from its blades. I stopped at a small pool and cupped my hands together in the water for a drink. The creek was idyllic, with delicate green ferns and sandstone rocks rippled with different colours of the earth. Minnows and a couple of barramundi darted around, an archer fish spurted water into the air to drown an insect, and a pair of turtles cruised along happily. It was a little oasis of life.
I collected the second, third and fourth nests without too much trouble, making sure I packed the crates with paper-bark and nesting material (not mud or sand) to keep them as light as possible. By this point, I had half a crate left to fill. Each time I arrived at a new nest, I would see the croc from the previous nest pass me by in the creek, heading towards the main waterway to look for a feed. It was like a domino effect of crocs, so I had to keep a keen eye out for more than just one.
By the time I got to the second-last nest I was all hot and bothered. On approach, I could see a 10-foot female sitting on top of the nest with her jaw open. Phew – at least I knew where she was. I put down my one full crate and moved towards the nest with my stick and the other, half-full crate. This time I was protecting the eggs as well – I needed to defend myself without destroying my cargo. As she came at me, I gave her my stick to chew on and, after a quick tussle, she spun around and took off into the stream. That was easy, I thought. I was having a good run, despite the heat. I sat down on the nest with the crate in front of me and scratched it open.
All of a sudden an almighty commotion erupted behind me. It sounded like a T-Rex was thundering through the trees. I spun around and saw the gaping jaws of a 16-foot male crocodile lining me up. It was overwhelming, being sized up by this dinosaur of an animal and knowing I had nowhere to run to, but I had to make a move. I jammed the half-full crate of eggs into the croc's mouth and he slammed his jaws down, crushing the crate and tearing it out of my hands. He swung the crate around violently, knocking me in the chest and pushing me to the ground so I was belly-up in front of him. It all happened so quickly, but he was clearly more interested in getting to the water than munching on me because he bolted right over the top of me into the river. There I was, pushed into the mud with claw marks all over me and a torn shirt and ego. I got lucky. I climbed to my feet and took a minute to gather myself. I tried to salvage the crate and pick up any eggs that weren't destroyed in the chaos, but I was buggered and couldn't move properly. I was in shock.
The heat was making me delusional, I got vertigo each time I went from squatting to standing up. I couldn't remember ever feeling this hot, not even when I worked out bush on the drill rigs in blistering 45-degree heat. I sat back down on the ground to finish off the nest. I pulled out an egg and as I went to mark it, it exploded in my hand, squirting a milky green fluid. Some landed on my lip, making me gag; it was putrid. I grabbed a second egg and it popped again. The whole nest was rotten.
It must have gone underwater, I thought.
I was frustrated – I'd just lost some eggs in a battle to save my own arse and so I could collect from this nest and, after all that, the eggs were completely ruined. I had one crate full of eggs and another mangled crate with a few sad eggs. I felt a little defeated and didn't bother trying to salvage any more. I got to my feet and picked up the crates, staggering around. They weighed at least 30 kilos each. I had a fair way to go and they weren't getting any lighter. I just wanted to collapse, but I knew I had to find the croc at the last nest before I could rest.
I found the female croc straight away and she wasn't hard to move off. After a few pokes with the stick she bolted, but I couldn't get it together. My energy tank was empty. Carrying on and clearing out to the next spot seemed impossible. I took off my clothes, ripping another big hole in my shirt as it clung to my sweaty skin. I crawled on all fours along the bank, looking for somewhere out of the way to lie down. My body was tingling and I felt like I'd been hit over the head with a hammer. Then I started convulsing, and vomited everywhere. I rolled over into the shallow water and vomited again and again, and then completely lost control and shat myself. I lay there in my own shit and vomit, with stars spinning above me. I've seen dogs overheat and die before and I thought that was what was next for me. Even trying to cool down in the water was counterproductive because it was warm – and I didn't want to go into one of the main pools because I would be croc food.
So I stayed on the bank in the shade and rested. I could hear other crocs as they slopped down past me in the stream. I just lay there silently with my crate and stick, waiting until my nausea subsided. Finally, I mustered enough energy to move to a clearing where the chopper could find me. To my relief, 30 minutes later it came and landed at the bottom of the creek. I hopped into the chopper with the crates on my lap.
'Jesus, mate, is that all you got?' my boss admonished me.
I shrugged. 'Yeah, sorry, mate. It was getting a bit hot out there.'
I was done for the day, that's for sure.
I suffered from heat stroke for three days after our Vic River expedition, so I was stuck in bed at the cattle station where Jimmy and I were living. I used that time to take it easy and brainstorm different ways of reducing the stress on our bodies while we were out collecting. I'd heard stories of some fellas slinging under a chopper and I liked the sound of it.
Jimmy came into my room that night and told me there had been torrential rain, and nests were going underwater on the Moyle. The Moyle River flows down near Peppimenarti, which is located about 300 km southwest of Darwin and is notorious for its big crocs.
'Big wet out there, Matty. It'll be a new challenge for us collecting tomorrow,' Jimmy said.
'Yeah, we'll be right. Hey, I've been thinking and I reckon we should try out a new technique. We should get a thick chain, like a pull-chain you use on a boat, with a big D-shackle on the end, and hook it underneath the chopper. Then we can put a plank of wood through the shackle and sit on the plank. That way, we can get slung under the chopper and dropped onto nests so we don't get as knackered walking in the heat. What do you reckon?' I asked Jimmy.
'Oh yeah, but things could go wrong pretty quickly doing that.' Jimmy wasn't exactly sold on the idea.
'Not if we only fly above land and never go higher than 10 or so feet. What's the worst that can happen – we fall and break an arm? I'd rather that than overheating and dying any day,' I said, convincingly.
Jimmy paused and gave it a few seconds of thought. 'Righto, let's give it a crack then.'
I am fully aware of how illegal and downright dangerous this technique is, and nowadays we would be in a bit of strife for doing something like that, and rightly so. Back then I was naïve, and collecting croc eggs was uncharted territory, so we made up our own rules.
We got going at first light. Jimmy and I flew the R44 to a clearing on the edge of the Moyle and we set up our base camp, where we would return to pack eggs and refuel. My boss went for a quick scout of the area in the R22 to see how many nests there were. He landed back and told us there were about 18. It was going to be a big day.
'Righto, Matty, want to give this harebrained idea of yours a go?' my boss asked me. I was excited to give this new plan a go so I wouldn't get so hot all the time, but nervous about the prospect of falling off mid-air and dropping down hard to the ground.
I pulled out the plank of wood and 20-foot-long chain from under the seat of the chopper. We hooked it up and off we went. My boss was flying the little R22 to save fuel, with me dangling on the chain beneath. These sorts of choppers are designed for a one-man band, not a two-man circus. Nevertheless, I was having a ball. I was sitting on the wood with the chain between my legs, holding onto the chain and my gear, while I skimmed along the top of the cane grass. We weren't trying to be silly, we were trying to be efficient. And it was quite literally a breeze – I was staying cool and I didn't have to face the jungle and risk suffering another heatstroke.
But at one point we moved from about 12 feet from the ground to 20 feet because of a patch of trees and the fear of god was put in me.
'Please never go that high again, that was no good,' I said to my boss when we landed. I realised in that moment slinging like this could in fact go pear-shaped pretty quickly.
We'd collected about 12 nests when we stopped for some tinned spam for lunch. We had a handful of nests left, but I still wasn't feeling 100 per cent after overheating on the Vic, so I decided to stay back for a bit. It was Jimmy's turn to sling in. From where I was standing, I could see the small opening in the grass where the next nest was located. Off Jimmy went on the plank.
He looked back at me with an 'Are you sure about this?' expression, and I yelled back, 'It's as safe as house, what could go wrong?!'
The water level was high, meaning collecting was much more dangerous because the crocs have full advantage. I could see the chopper buzzing around in circles and lowering down to let him off the chain. But each time the bottom of the chain emerged from the top of the cane grass, Jimmy was still attached to it.
'What the hell is taking him so long?' I said under my breath.
He lowered in again and then the uproar started – Jimmy screaming, 'Noooo! Stop!' He yo-yoed up. 'No, no, noooo! Please stop, please stop! I'm not going in, not this one!' I could just make out what he was saying through the noise of the chopper.
He was throwing a cracking tantrum at the top of his lungs. My boss couldn't hear him properly and dangled him back down; it was his fifth attempt by this point. I saw his figure rise up again, but this time Jimmy had dropped his stick and crate and was monkeying up the chain towards the chopper. It was hilarious to watch. My boss saw what he was doing and headed back to meet me in the clearing. He carefully lowered Jimmy to the ground and parked the chopper next to us. I'd never seen Jimmy so wound up – he was huffing and puffing, crying a little, and scarlet-red in the face.
It wasn't the right reaction, but I burst out laughing. 'Mate, breathe, calm down. What the hell went wrong?'
'Fuck that nest, Matty – that female is nuts.' He was shaking his head, pacing two steps one way then two steps back the other way. 'I came in the first time and the croc swung her tail at me and it clipped me, then each time she wouldn't go away, she just wanted to eat me, I swear! And when I saw the other dead shit around her nest I didn't fancy ending up the same way.'
'All right, come on, you're up,' my boss said to me. 'I'll lower you in and hover there for a moment so you can get a close look, and if you don't like what you see then I'll bring you back in.'
I turned to Jimmy who was still distressed. 'You wait here and drink up some water for now. But if I go in, I want you to come with me.' He nodded at me.
I got the plank of wood ready to slide into the D-shackle but noticed it had nearly worn in half. I found a piece of solid wood from a nearby tree – nothing old or rotten, I wanted hard green timber – and hacked off a decent log with my knife, slid it into the D-shackle and twitched it on with a bit of telecom rope.
As I got lowered down, I saw what the problem was and I could understand Jimmy's reaction. Everything was dead. There were about 30 chewed-up turtles, a half-eaten six-foot crocodile, and mangled fish everywhere. It smelt absolutely foul, and the nest was thick with maggots and bugs. It was a proper haven for a killer croc.
Holy hell, I don't want to come face to face with whatever has chewed all that shit, I thought. But I wasn't going to back off, because I'd just made fun of Jimmy and I was confident that with the two of us we'd be able to collect the nest. So I got dropped in and a few minutes later Jimmy joined me. We started collecting the eggs together, but I couldn't shake off a growing sense of dread. I had a bad feeling about this nest.
'I'll stand guard and you collect,' I said to Jimmy. I had my crate and two sticks in my hands, ready to go.
Jimmy was on edge, looking over his shoulder every five seconds or so. The worst thing about this nest was that it was surrounded by deep water where the croc could be hiding, ready to launch at any time.
And sure enough, out of nowhere she propelled herself onto the island, grabbing one of my sticks and thrashing around, tearing it to pieces. I hoped she would've given up then, but it was clear she wasn't going anywhere.
'Hurry up, let's get this done!' I yelled out.
She went at us again, lunging at Jimmy. I pulled him towards me, and we both fell in the water, gripping at the edge of the island bank. My heart was pounding and Jimmy was white in the face.
I still had the crate in my hand, so I fed it to her in the water and she grabbed a hold of it, scattering the eggs everywhere. We backed ourselves up onto the little island, right on top of the animal remains. I could feel the maggots wriggling around me. We scrambled to our feet and she came out with us. I fed her the second stick and she death-rolled, so I gave her another couple of good hits and pokes until she finally retreated. But I knew she wouldn't have gone far.
Jimmy hastily packed the crates with the eggs that weren't damaged and slung out first. I stayed back with the stick and a crate at the ready until the chopper came for me. As I landed in the clearing, just two feet from the ground, the plank I was sitting on snapped. Jimmy was already there, sitting on the ground, defeated.
'Well, I think we've established two things – we might give that nest a miss next time and we're not using a stick and chain again,' I said.
'Yeah, and we need to invest in a harness and make our slinging a bit safer,' Jimmy added.
It's crazy to think how far our slinging has come since then. We were extremely lucky that nothing ever went terribly wrong in the early days of collecting, given our cowboy approaches. Today, under the leadership of Mick Burns, we have developed equipment and techniques that have made slinging into crocodile nests a hell of a lot more secure. We work closely with the Civil Aviation Safety Authority (CASA) and local Darwin engineers, and have designed the safest sling operation for external human load in the world. We now use a dual-hook set-up on the belly of an R44 helicopter, with two lines in use at all times, both equipped with manual release function and electric buttons that operate on separate circuits. The lines have lightning protection in the form of an electromagnetic shield wire, so the hooks won't release if lightning strikes. The hooks and lines are tested and replaced every 12 months, and we use harnesses that are certified to Australian safety standards, which are also replaced every 12 months. We wear a helmet, and carry a knife and emergency locating transmitter at all times. How's that for safety!
And when there are tricky nests on islands we surround ourselves with a big ring as a barricade. The ring is made up of shade-cloth material, which is wrapped around a series of metal rings. The ring sits surrounding you while you're on top of a nest and comes up about 1.5 metres high and has about the same diameter. It's more of a bluff for the croc rather than legitimate protection, but it does the trick most of the time and hangs off our harnesses as we fly through the sky from nest to nest. You couldn't pay me a million dollars to go back to using a stick and chain, never in my wildest dreams would I do that again.
#
It has always been a precarious task to train new crew in how to work safely with, and in close proximity to, dangerous animals. Whether it's someone making a cameo appearance on my TV show Outback Wrangler, a helicopter pilot or green cameraman, anyone getting close to wildlife must have an understanding of animal behaviour and how to act around them, especially under pressure.
People often ask me, 'How do I learn to work with crocodiles?' Or, 'What do I do if I come face to face with a croc?' It's mostly true that knowing what to do in those situations is instinctual, something that is innately built into a person, but a lot of it comes from experience and, like anything, heaps of practice. I've had crocs chew and smash up cameras and tail-whip people off their feet. I've seen camera crew thrown off horses, rolled by bulls and trampled by buffalo. When you're heading out bush for these jobs, everyone has to understand the dangers surrounding the work. Otherwise you can lose a man pretty quickly, which isn't something you want to happen.
In order to study a croc – or any animal, for that matter – you need to understand how the animal's physiology and anatomy works with its behaviour. Lots of external and internal factors prompt different reactions from an animal. For example, sounds, smells, territorial threats, weather and stuff going on inside the animal – like feelings of hunger and fear – all play a part. These elements will impact the way an animal responds to you, and the more time you spend with a particular species the more you realise that most of them act very similarly. However, it is a mistake to think that animals of a certain species will always act the same. That's an assumption, which leads to complacency, and complacency can lead to things going wrong very fast.
Another thing to keep in mind is how genes and the environment have interacted over time to inform the behaviour of an animal. Genes relate to evolutionary responses of previous populations, and the certain behaviour patterns they have undertaken to survive. So, with crocodiles, a species that's been around for over 250 million years, one thing's for sure: they know how to kill and survive. And this is something I never take for granted.
When I'm in a croc's territory I not only home in on the animal itself, but I assess everything around me. Situational awareness is critical. I scope out nearby trees as a potential barrier to hide behind, look for high ground so I can stand above the croc, and test the ground to see if it's wet, dry or slippery. I check all paths available to the croc and work out which route they are most likely to take if they try to escape. Nine times out of ten, a croc will take the track with the least resistance, which is the easiest for them to navigate. (Although this only applies to a croc who is out in the open and on the defensive. You would see something very different from a female croc who is on a nest and on the attack, in which case you need to throw anything you have – crate, stick or pole – into her mouth and hope she chews on that instead of you.)
The conditions in the outback are always unforgiving, so there is a real skill in consciously zoning out the foul stuff – the stifling heat, the swarms of mosquitos, the smell of rotten pig. They are all things that, if you pay them too much attention, will affect your focus.
This job has so many moving parts to learn about and be aware of that it's hard to know where to start when a newbie comes on board. I'm a big fan of baptism by fire (within reason) because it shows you if someone is cut out for the work or not. Although often, if a person is really inexperienced, you've got to go back to basics first.
In the early days, before I started filming Outback Wrangler, I had a job catching a few crocs out at Marrakai cattle station towards Corroboree in the Territory. A young Kiwi bloke by the name of Ross was working for me at the time. He was a great guy but had a lot to learn. He had come up from New Zealand and had no idea about crocs, so I had to train him in everything from the get-go. I eased him into it step by step. I started with small crocs, showing him how to work around them, taking it steady and not giving him too much to take on.
On one particular day, I was slinging in a handful of traps that we were using to catch and relocate some local problem crocs. I hooked them onto the chopper and flew them back to the Toyota land cruiser, where Ross was waiting with our boss, the station manager. The plan was for me to pick up the traps from the paddock then drop them off, one by one, to the Toyota. Once that was done I would pull each croc out of the trap with a head rope, tape them up and load them on the back of the Toyota, ready to be relocated. My boss was in a shitty mood that day; he was time-poor and was pushing Ross beyond his capabilities in order to get the job done as quickly as possible.
I was flying back with the second trap when I looked down and saw the first croc out of the trap. Ross was sitting on its back while my boss was pulling on the head rope, holding it tight while Ross gaffer-taped its snout shut.
This isn't a good look, Ross has never done this before, I thought.
It was only a six-foot-long croc, but the next trap had a big croc in it and I knew there was no way Ross could handle it.
'Leave the next one, guys,' I radioed down to them. 'I'll be done with everything in 10 minutes, so I'll come back and give you a hand with the rest. The croc I'm bringing in now is huge, so I need to do it. Just wait for me.'
I was worried. I knew Ross hadn't been trained in how to jump on the back of a croc. He was very impressionable, naïve, eager to please and well out of his depth: a bad combination around my reckless boss.
I repeated myself over the radio after I dropped the trap. 'See, fellas, this one's about 12 feet so please leave it until I'm back.'
I was flying over to the paddock for the next one when I heard my boss cry on the radio, 'Matt, Matt, Matt! Quick, come quickly, we're in trouble! Ross has been bitten, he's been bitten!'
'Oh shit! You idiots!' I flew back immediately but I didn't want to land too close in case the chopper spooked the crocs. As I came in I could see my boss holding the rope and Ross sitting on the croc, but not much else. I landed a couple of hundred metres away and ran over but was still confused about what was going on because when I got there, everything appeared to be calm. Ross was on top of the croc, the hessian was over the croc's eyes and my boss was pulling the rope tight.
'Matty, grab the tyre leavers from the Toyota!' my boss yelled.
I snatched them and ran up to the croc and finally saw what I was dealing with.
'Oh Jesus, you poor bastard.'
Ross was sitting there, white in the face, tears streaming down his cheeks, with his hands locked into place on either side of the croc's jaws. He had jumped on the back of the croc, thrown the hessian over the croc's eyes and misjudged. Both hands had overshot the eyes and slipped into the croc's mouth as it shut, engulfing and then crushing his fingers. He was stuck. And, knowing that crocs have jaws that can slam down at 3700 pounds per square inch of bite force, I could only imagine the pain he was in.
My boss and Ross were helpless. If my boss took the pressure off the rope, the croc would have death-rolled and torn off both of Ross's arms, so he had to stay put until I levered the croc's mouth open. I got in behind Ross and opened up its mouth, which stunk like no one's business and was festering with maggots from being in the trap with a rotten pig. A croc is like a snake, though; if you touch it under pressure, it restricts or clamps down even tighter. As I attempted to pry wide the jaws of the croc, poor Ross began wailing, letting out the most harrowing shriek. Finally, I managed to open its mouth enough for Ross to pull his fingers out.
'Get out of here,' I told him.
Ross sprang off the croc and I jumped on in his place, putting on the hessian in the right spot and taping the animal's snout shut. Then I got up and had a look at Ross's hands, and I kid you not, it looked as though every one of his fingers had fallen in between the croc's teeth. Sure, his fingers were crushed but there was not one puncture wound or any blood drawn. Still to this day, he is the luckiest bloke I've ever come across. Bitten by a croc without a mark, it was unbelievable.
I did my block at my boss then turned to Ross.
'If you're not comfortable doing something, you've got to speak up and say so! That was just plain stupid, and unfortunately, I can't afford to work with you anymore, mate – I'm sorry.'
Ross looked relieved, not upset, and I think he was quietly happy to get the hell out of there. To be honest, I think he was done with crocs altogether after that encounter.
I've moved a lot of people along over the years who haven't been the right fit for catching crocs, and many of them have included film crew. Being behind the lens is just as challenging as being in front of it, and you need someone who doesn't crack under pressure. The first person I trained for filming Outback Wrangler was my cameraman, Mark Priest. He was a fit, older bloke, very capable and a bit of a silver fox. He was one of the first people to go diving, snorkelling and filming with our world's great sharks. He's filmed broadnose sevengills feeding, a tiger shark killing and eating a grey reef shark, hundreds of spinner sharks in one place and up-close encounters with great white sharks, all captured outside of a cage. He was still alive to tell these tales when I came across him, so I figured he had to be all right.
The more time I spent with Mark the more I realised how special he was. Always looking for the next best adventure, his zest for life was contagious and the best bit was, he was egoless. His fascination for wildlife and the beauty in nature would make me stop to smell the roses more than I ever had. I took a lot of things for granted before Mark came along, but he reminded me of the simple beauty in the reflection of the sun on the water, or a unique marking on a croc. The passion he brought with him made me feel like I was doing everything again for the first time.
The first croc I had to relocate alongside Mark was a nine-foot female. The tricky bit was that she was in water. In order to get the footage, Mark had to partly submerge himself underwater without making big movements, which is virtually impossible if you're filming. If a croc sees movement in its periphery it will change its course immediately towards that direction. I've been in situations where I've been one metre away from a croc with a pig leg in my hand, and someone 10 metres away, off to its side, has moved, and the croc has gone straight towards the other person. They are incredibly reactive animals, so I made sure I spelt the basics out to Mark over and over again, just to be certain.
'Mate, if you see the croc swimming around don't move abruptly or do anything to disrupt her environment. If you leave her be then she doesn't have a reason to be aggressive,' I said. 'If she looks like she's moving in on you, that's a whole new story. You need to jump out of the water immediately.'
Mark nodded. 'Got it, Matty, all good.'
There really is a difference between a croc swimming in your direction and one coming to attack. If you don't give them a reason to attack, most of the time they are more focused on either swimming away from you or moving about their waterhole as usual. Mark and I moved onto the high ground above the waterhole, sussing out our location and chatting some more.
'Always be mindful to not muddy up the waterhole when you're wandering around it, otherwise we will have no idea where she is and then we'll be stuck,' I warned him.
'Yep, I'll tread lightly.'
It was a freshwater spring and the water came to above knee height, so we were still able to move quick enough and stay above water most of the time while keeping the croc in sight.
I moved into the water and turned to see Mark already kneeling down behind me, working out his angles. He wasn't anxious or uptight at all, he seemed very composed. I slowly waded through the water, making sure I didn't disturb the ground too much, turning over my shoulder every now and then to see how Mark was doing. It was only the two of us; we didn't have the luxury of a spotter or extra helping hands back then. I turned again to see that Mark had got right down on the bottom and was lying there. I stopped in my tracks and stood still. He was underwater for at least 30 seconds making absolutely no movement at all, aside from the odd small bubble. I watched as the croc swam right into Mark's view and imagined the epic footage he was capturing. Once the croc had swum a distance away, he slowly rose from the water, with no gasping or desperation for air. It was incredible.
'You've got good lungs, mate,' I said to him.
He smiled and gave me a thumbs up. As I approached the croc, who was lurking in the corner, Mark came in closer. I stepped out of the water and onto the bank so I had more height. Mark stopped stirring as I went to harpoon the croc. I locked onto it and Mark moved straight in on the action.
'Whoa! Back it off, back it off!' I said.
Mark had got a little too close for comfort and the croc's death roll almost smacked him as he launched in the air out of the way. He had focused too much on getting the shot that he forgot about his proximity to the croc, and boy, was he reminded of it quickly. I was holding the rope tight as the croc tired itself on the other end, swimming and rolling about. Mark cleaned off his lenses and got back in position.
'You've got to wait to move in on the croc until it has tired itself out,' I said. 'You usually give them about five or so minutes of death-rolling, then they calm down and that's your time to move in for the close-ups. If you go too soon you'll be cactus.'
'Righto.'
It was Mark's first mishap but he was definitely the man for the job. He learnt quickly and became more proficient as time went on. His only downside was being too eager. He liked to get up close to capture the perfect moment and sometimes I had to pull him up for getting a bit too intimate. In the earlier days, I exposed Mark to three of the most dangerous crocodile encounters – a water relocation, catching an 18-foot monster and collecting croc eggs.
The croc relocation was a very close call for Mark and me. Catching a croc that size brings with it many added dangers. The most concerning is that it has the force to buckle and potentially open a steel trap and manoeuvre itself out. We had this big-arse panel trap, double the size of a normal one at about 10 by 10 metres and about three metres high. It was extremely sturdy, made of high-quality steel sheets from the local cattle yard and reinforced with narrower steel panelling just to be sure.
We'd caught the croc in the trap and it was hiding itself underwater. Half of the pen covered elevated dry land and the other half sloped down into the water, so I had a nice height advantage over the croc. I was inside the trap mucking around with the rope, trying to detach it from a pig leg then retie the rope as a noose. Mark was following me around, asking me questions and getting some stuff to camera. My mate Mick Jacobi was there too, standing on the railing outside of the trap, keeping an eye out.
'So now I'm going to use this rope to throw over the croc's snout and secure him,' I said to the camera. I picked up the pig leg. 'And I'll use this to entice him out of the wa–'
'Fucking move!' Mick yelled over the top of me.
I registered the horror on Mark's face as he stumbled backwards and I swung around to see the croc in mid-air, at least eight feet off the ground, flying towards us.
I instinctively chucked the leg in its gob as it landed at my feet. It bolted it down within seconds. I felt Mick behind me, tapping me on the shoulder. Any sudden movements and the croc would launch – Mick knew this all too well. He slowly picked up another leg and pole for defence. I reached my hands up, keeping my eyes on the croc, and Mick put them in my hands from behind. I was armed and ready. Mark stood in my shadow slightly to my left.
I had trapped us both in. I'm okay if it's just a croc and me in that sort of situation because I can work things out pretty quickly, but I had Mark to worry about and I could see out of the corner of my eye that he was still filming. For a cameraman, this sort of situation would send their adrenaline through the roof and a lot of people would flip out, but he jumped back, stayed calm and kept going.
'Don't move. Even if I move, I need you to stay dead still, Mark,' I instructed. I didn't need to say anything to Mick, he was sitting frozen at the top of the railing.
'Okay, I'm going to feed him this second pig leg and use the pole as backup. As soon as I chuck the leg, Mark, you need to scale that trap and get out.'
'Roger,' he said.
'Move!' Mick screamed again from above.
I darted to the right, jabbing the pole towards the croc for him to chew on. I fell onto Mark in the process and knocked him over, causing the camera to topple into the water. But he left it behind and scaled the railing in a flash while I chucked the second pig leg to the croc. It let go of the pole and snapped it up and boy, did I move quickly out of the trap. The croc launched again once I was on safer ground, spearing straight into the trap and bending it outwards.
'Back, back, back, both of you get back behind the car! This croc is nuts!' I yelled.
Mick and Mark sprinted towards the car. There was a large tree to the left of me, which I planned on climbing if the croc somehow managed to get out of the trap. It continued to launch itself repeatedly, ploughing the trap with unimaginable force.
'This is one crazy fucking crocodile,' I remarked.
It was no wonder the station owner needed it off his property, it would eat a person in a heartbeat. It hadn't stopped throwing itself against the trap, so I decided to leave it there overnight to give it time to calm down and so I could bring back some more manpower.
We caught and relocated the croc quite easily the next day and amazingly Mark was able to fish his camera out of the water and download the footage. I've still got the video on an old hard drive at home and it sends shivers down my spine. The footage turns to shit each time the croc makes an attack but you get a sense of how close we came to disaster that day. Mark did an amazing job keeping the camera rolling, but there are times like that where you have no other choice than to screw up the filming because it's a life or death situation.
Mark moved on to other gigs soon after that, paving the way for my current cameraman, Ash Dunn, to learn the ropes. Ash had been working on Whale Wars back in 2007 with the Discovery Channel. It was a reality show that followed the Sea Shepherd's campaign against Japanese whaling off the coast of Antarctica. Ash was faced with life-threatening situations every day while capturing the Sea Shepherd's attempts to disable the Japanese vessels and the relentless Japanese retaliations, which were often extremely violent. So he knew how to film under pressure. While Mark filmed a little bit of egg-collecting, it was Ash who got involved with filming in the thick of collecting season.
Training people and the crew in egg collecting takes even more work than relocating a croc. You've got to explain the basics of choppers, slinging, being on the nest and looking out for a croc. It is such a unique environment. You're in waist-deep water, and the croc is aggressive and always has the upper hand. It's not the time and place to be pushing the boundaries.
Ash joined me in the very early days of Outback Wrangler when I was trying to get a show of some sort off the ground. He shot a sizzle reel with me for Discovery, which is a promo that outlines the concept for a new TV show. Before we went out I took Ash and the rest of the crew over the safety aspects of slinging in and out of croc nests: how to check the rigging; hook and unhook; the communication signals; the line entries in and out of choppers. And when it came to the crocs, my only advice was: 'Stay close to me and listen to EXACTLY what I have to say.' I could have gone into more detail, but female crocs on nests are unpredictable at the best of times so I didn't want to plant one concrete idea about how to act because it's different every time.
I slung into the first nest and Ash followed me in. We were on a bit of high ground but there was water all around us. Ash is a strong, tall lad with big hooves, so he's a little heavy-footed. His footsteps shook the ground causing vibrations, which is an open invitation for a croc to come say hi.
I stopped in my tracks and turned to Ash. 'Before we go any further you've got to quieten down, mate. You need to step a lot lighter. I sound like I have a bloody elephant in tow.'
'Yeah, sure thing.'
We reached the outskirts of the nest and I helped him get into the right position to film me collecting the eggs. Then I got cracking and signalled him to back up a bit so he could get more perspective. But instead of slowly shuffling backwards, he crab-walked quickly, lost his footing, and fell into the water. It was no surprise that a female appeared instantly, within a metre of him.
For god's sake, I thought. Ash was lying in the shallow water with a 12-foot female at his feet. I was certain he was about to get his foot chomped on. It was really serious.
'Don't fucking move,' I said to him softly.
I stamped my foot repeatedly and managed to get the croc's attention. She came at me and I fed her my crate while Ash staggered to his feet and moved behind a nearby tree, panting profusely.
'This is no bullshit,' Ash said to me, as we returned to safer ground. 'I thought a lot of this stuff was half-staged but that was very fucking real.'
'Yup, there's no room for error out here.'
I called my mate Craig in, who was collecting a nest nearby. 'Okay we've got about 15 metres to walk until we get to the next nest, and Craig is going to get slung right onto it with us. Keep your wits about you because the chopper's noise means you won't be able to hear crocs approaching.'
'Gotcha.'
The line appeared with Craig on the end and he was lowered into the nest. Ash was standing beside me looking left, right, left, right, and rotating around on the spot making sure there were no more surprises. It was amusing to watch but I was happy to see that he at least had his guard up now.
Craig was coming down to land, about 10 feet from the ground, when a croc propelled herself out of a deep wallow and into the air, missing his foot by inches. She disappeared back into the water, but we knew she was still there and would soon have another crack. The chopper pilot didn't see the commotion, and kept lowering Craig.
'Ash, you have to stay here. Don't move,' I told him, then headed over to help Craig.
I got around to where Craig was but didn't realise that Ash had ignored my instructions and followed me to film. Craig landed in front of me and I heard a sudden scream from behind. I spun around and there was Ash on his back with the croc crunching down on his camera. I rammed her with my stick and she retreated, dropping the camera and leaving the crushed lens behind.
'My bad, Matt, sorry.'
'You can't do that, mate. I need to be in control of the situation as much as possible, I can't have you going rogue on me. The only thing I will say is, well done for giving her the camera, otherwise you'd be limbless.'
Not long after Ash's first run-in we had another calamity releasing a handful of crocs in a clear patch of water near a reef. It was a breathtaking day: the water was calm and pristine and a massive storm was brewing in the distance, creating a dramatic backdrop that was perfect for filming. Ash was knee-deep in water, lugging around an old-school underwater camera that weighed around 40 kilos, which is no walk in the park. He was squatting down in a channel about five metres in front of me, holding the camera half-in half-out of the water.
'Righto, Ash, I'm going to release this eight-footer right here and I'm going to walk up the channel behind it,' I told him. 'And you can get the shot as we come past you. She'll swim past you, but you know the drill: when she comes don't flinch, just keep the camera pointing forward and she'll swim into the shot. Keep absolutely still, otherwise she will have a crack.'
He nodded. 'Yeah, yeah, Matty. I got this, no worries at all.'
'Okay, here we go. Are you sure you're ready?'
'Yeah.'
'What did I say, Ash?'
'Don't move, you said don't move.'
'So what are you going to do?'
'Not move,' he confirmed.
'Okay, perfect. Here we go.'
I released the croc and walked behind her as planned. We were a metre out from Ash and . . . I couldn't believe it. He decided to move the camera to follow the crocodile. As soon as he tried to spin the camera, the croc launched out sideways and grabbed it, dragging it through the water, then letting go and swimming off. Ash just stood there, speechless.
'What happened?' I asked him.
'I was just trying to get the . . .'
'No, Ash, what happened?'
'I moved.'
'No shit you moved!' I was livid. 'You gotta listen to what I have to say, there's a reason for it!'
'Yeah, Matty, I get it, sorry. I'm not having a good start to things, am I?'
Ash was right. He got off to a bad start but 13 years later he has done his training, knows the drill inside out and has well and truly earnt his stripes. He listens a lot more these days, but most of the time he runs his own show and knows exactly what to do and not do.
We all get caught off guard. No matter how many times I tell the experienced boys to be careful around the croc's tail, they still get wiped off their feet. It happens to me too. Last year, I was acting blasé around a 14-foot male that I had tied up under a tree. I touched its tail and it exploded with everything it had. Its tail flipped me upside down six feet in the air, and I came crashing on my head with my legs vertical. It felt like I had snapped my neck. Luckily for me it didn't get caught on camera because I know the guys would have stitched me up and put it on the show.
I'm very lucky that I now have a very strong behind–the-scenes crew on Outback Wrangler, after years of people coming and going. I have a small team consisting of Ash, Jamie, Jez and Row who know the landscape, the local people and how to act. None of them are full of themselves and none have been chewed on.
One of the biggest challenges I had was trying to teach them general practical skills like how to open and close a gate, hook up a trailer, drive on a dirt road without rolling a motorcar, and station etiquette. Those little things go such a long way in the bush, and a lot of people have come out over the years who thought they were above being kind to people on the communities or above having to roll up their sleeves to get the job done. They didn't last long.
Then there's my on-camera crew for Outback Wrangler. Willow started with me when we were catching crocs on La Belle Station in 2011. He loved adventure and stuck by my side, venturing out with me, collecting eggs and learning the ropes. Because he's a bigger lad he's always had to be mindful that he's not as agile as the rest of us, but he does a good job of holding his own. Training Willow in the little things early on has helped hugely with filming nowadays because he knows all of the smaller skills and details that make a big difference. For example, spotlighting at night. You can't just shine a light and hope for the best, he knows that you've got to keep the torch level at eye line because it's at that level that you can see the reflection of the croc's eyes, otherwise you miss them. He knew how to drive a boat before Outback Wrangler, but I trained him in how to safely drive a boat in waters with a high population of crocs. Now he knows how to do it in stealth mode at the right speed and the right time.
My other main man is Jono. Like me, Jono has always had an innate sense of animals and adventure, so when he came up north to film from South Australia he didn't need much training. But he did learn a couple of things the hard way on his first day in the swamps. We had to relocate a 15-foot croc caught in a small channel. Because the croc was trapped, we had the advantage, and it was a good opportunity to train Jono in how to catch a big animal. I sat down with him to give him a rundown before we got into things.
'Okay, mate, when this croc launches out, his mouth is going to be open and he's going to head towards you, so you have to throw that rope as soon as he shows himself. When you get it over his mouth don't go pulling on the rope too hard because you'll get terrible rope burn.'
'Roger,' he said.
I was a bit worried that he hadn't asked me any questions, but nevertheless I had faith in him. He was wide-eyed and eager, preparing himself on the bank with utter concentration. He had never caught a croc before in his life but I knew he'd be able to handle himself.
I gave the croc a little nudge under the ribs with my stick and he launched out mouth-open, just as I'd predicted. Jono dropped the noose over the croc's snout as accurate as anything. His smile took up his whole face; he was loving every minute of it. His excitement took over and he pulled the rope as hard as he could.
The croc retaliated and pulled him into the water, jerking violently on the rope. Jono pulled back with his own force but then yelled out in pain. All of the skin on the palms of his hands and his fingers came off completely. He had to let go and I took over until we had the animal secured. It was the worst rope burn I've ever seen, it looked like he'd just put his hands on a BBQ hotplate. They were blistered and red raw. All it took was the croc to give one quick flick of its head and the rope ripped through Jono's hands like a knife cutting deep into butter. He was very sore after that, but it was a good learning curve for him and he learnt very quickly to resist the pull of a big croc, and to never grip the rope tightly. Even now, Jono watches other people and reminds them not to try to overpower a big crocodile. He knows to let the croc move around where it pleases, tiring itself out on its own accord. And it's only then, once it's exhausted, that you move in on it.
In this sort of work, if you're not trained properly or focused on the job you'll get hurt pretty quickly. It's quite simple at the end of the day: you just need to do what you know is the right thing to do. Don't exceed your training and conditions without instruction, and don't go along with peer pressure. There are a lot of times I've stuffed up good scenes because I've stopped addressing the camera mid-sentence to yell at someone who's in danger, but I wouldn't have it any other way because that caution and safety is why I have never been chewed or lost a man while out on the job.
#
As the leader of the team tasked with biggest croc capture in the world, I had an enormous job on my hands. Within the space of two short days, me and my team of six had to catch 50 saltwater crocodiles and relocate them across Australian state borders. And these weren't your average-sized crocodiles: they all ranged from 12 to 17 feet long. In fact, five of them came in around the 17-foot mark, making them some of the largest in Australia. About 30 of them were males and the rest were large females. They were big and had even bigger personalities to go with them.
The Wyndham Crocodile Farm and tourism site in WA was set up in the late eighties and was a popular tourist attraction for more than two decades. It fell on hard times in 2012 and had to close its doors two years later, leading to the most remarkable bulk sell-off of crocodiles in Australian history. The farm's animals were also some of the oldest crocodiles ever held in captivity, so the owner wanted to avoid destroying them at all costs and was keen to see them moved on to another Australian reptile park or farm.
That's where, Mick Burns, the owner of NT Crocodile Park and local tourist attraction Crocosaurus Cove came into the picture. Mick bought the crocs off the owners, and contracted me to help move the 50 crocs over 900 kilometres from Wyndham in Western Australia to Darwin in the Northern Territory. I was aware of the farm's notorious reputation for having troublesome crocs who had been exiled from their communities for causing problems and unrest. Two of the culprits went by the names of Oombi and Devil. Oombi was named after Oombulgurri, an Aboriginal community where he left his mark after eating more than 25 dogs before he was captured and sent to the farm. Devil built his reputation once he was in the farm by breaking out of his pen to mate with a nearby female and eating one of his penmates in the process.
Willow, as always, was the logistics man for this job, organising the excavators, trailers, trucks, slings and other machinery and equipment. Mick arranged the special permits, which allowed us to transport the crocs across borders from WA to the NT, and as soon as that was sorted, we were ready to boot. Willow and Craig hit the road and I flew over to Wyndham with Jono and Mick. From Darwin, we tracked over the spectacular salt flats of Port Keats down through the start of Kimberley country, where the mouths of the Fitzmaurice and Victoria rivers meet, then over the scattered boab trees and rocky outcrops of Kununurra, finally landing at the croc farm in Wyndham. We met up with the last additions to the team: my mates, father and son combo Nick and Seb Robinson. But before we got started, I had one thing to discuss with everyone.
'Okay, fellas, we had slim pickings for accommodation so we're being put up at the old Wyndham hospital,' I said. 'It's not the most inviting spot but we'll make do.'
Mick had organised the hospital accommodation with Paul, the owner of the croc farm, who had left the hospital lights on for us, as well as some instructions for us to navigate the joint.
We arrived on dark at this historical-looking hospital. Some of the lights were already turned on, and cobwebs creeped all over the ceilings. I followed the instructions to get to our rooms.
'Ya kidding. We're actually sleeping on the old hospital beds?' Seb asked.
'It looks that way,' I said.
The door opened and Willow and Craig came walking in.
'I think I'd rather camp out in the swamps than here. This place is spooky as,' Willow said. Everyone looked like they were thinking the same thing.
I lay back on the bed to test it out and it collapsed inwards. 'It's going to be an interesting night's sleep, that's for sure. Let's hit the sack, fellas.'
I slept a total of two hours that night, and that's being generous. It was hands-down the worst sleep I've ever had. There was a constant creaking from a door swinging on its hinges, the tin roof clanged in the wind, the windows rattled, and it sounded like there was nonstop activity down the corridors. I could hear every one of the lads tossing and turning; no one was resting easy.
I rolled over and checked my watch. It was 2 a.m.
Jono, who was next to me, must have heard me. 'This place is deadset haunted, mate,' he whispered.
Then Craig chimed in: 'I can hear footsteps every 10 or so minutes.'
'No shit!' Willow called out.
'God knows what the other noises are, but this place is the absolute pits,' Seb said.
'Why are you scared, Sebby? You're sleeping with your dad!' I yelled across the room.
The room burst out laughing. Everyone was wide awake and equally disturbed by the place. But I was a little concerned that no one was getting any sleep the night before the biggest relocation of our lives.
'We've got four hours left of shut-eye, so let's make the most of it,' I told them. 'Shove ya pillow over your eyes, count sheep or do whatever works to get some rest. I'll find us somewhere else for tomorrow night.'
It was game on at 6 a.m., and time to catch those monster crocs. We'd brought along a road train with three trailers on the back, which we typically used to cart wild bulls. These were going to be our means of transport for the crocs. We rolled massive hay bales out onto the bases of the trailers to make it more comfortable for the crocodiles, and then we were ready for the real work.
The owner of the farm, Paul, guided the way. 'We'll start with this pen here, lads. The biggest croc we have at the farm is in this pen, but he's in there by himself and he's also the most placid so it's probably a good one to warm up with.'
'Roger. How big?' I asked.
'Just over 17 feet and weighing in at almost a tonne.'
'Is the croc you're talking about called Axel?' Nick asked.
'Yeah, mate, have you heard the stories about him?'
'Yeah, they're pretty outrageous, tell the rest of the lads,' Nick said.
'Well, before my time, one of the old workers at the croc farm used to stick his seven-year-old daughter on the back of Axel and he would just sit there without a worry in the world.'
'Ya kidding? That's just plain stupid,' I commented.
'Yeah, pretty nuts,' Paul agreed. 'But I guess it illustrates just how docile this big fella is.'
I couldn't believe what I'd heard. I don't care how docile a croc is, it's still an apex predator and anything can happen. 'Fellas, don't go into this pen with your guard down, okay?'
Everyone heard me. 'Righto.'
I went in alone to start with, stick in hand. Axel was out of the water and sitting on a concrete slab. I poked him . . . nothing. I poked him again . . . still nothing. He just sat there like a statue. I half-questioned whether the thing was even alive. I went over and crouched alongside him. He opened his jaws a bit, confirming he was still in the land of the living, but there were no big movements. It was crazy, I'd never seen anything like it.
'Jesus, lads, I don't know if I want to give this animal the drugs. It's like he's already had them!' I walked towards his tail and administered a sedative, which relaxes the croc's muscles. I gave him a small dose, just to be safe, before we attempted to move him. After 20 minutes, we picked him up and put him straight on the forklift and into the truck parked in the shade. I hadn't even needed a head rope, harpoon or stick for defence. He was as placid as anything.
We had more machinery than we knew what to do with – forklifts of all sizes, telehandlers, a range of excavators, a front-end loader, motor vehicles, a road train, choppers, ropes and slings for days. And enough men to run a cattle station, because we weren't there just to catch and move the crocs, we also had to knock down the infrastructure at the farm to ensure we had enough space to move around in. There were fences, steps, railings and massive concrete walls to get rid of. It was like a demolition party with some prehistoric monsters involved too – a bloke's perfect idea of fun, really.
I was excited by the adventure to come, but I was also very saddened by the whole operation. So much history was being lost because of the WA government's uninformed policy on collecting crocodile eggs from the wild. The policy only allowed the farm to collect 500 eggs a year, which sent them broke. It was sad to see a local business shut down and a good tourism attraction lost.
Axel's pen was right next to the road train, so he was an easy move. But before we touched any more crocs we had to knock down some of the main walls. Willow got going on an excavator, pulling out the concrete that surrounded the premises. We nickname Willow 'The Operator', and for good reason.
Once the big stuff was done it was time to target the remaining individual pens. Paul told us that on average there were about four to five crocs in each pen, and every pen had at least one territorial male that was totally nuts. And a few of the larger, more aggressive crocs had a pen to themselves.
We separated into small groups of two and three and got cracking, starting from the pen that was furthest away and working our way back towards the truck. A couple of the farm workers went around before we got to our pens, partially draining out the water with a fire pump to make our jobs a little easier. Mick and I paired up, and Paul worked his way around the groups letting everyone know what to expect in each pen.
'Okay, this one is a pen with a single male, about 15 feet long,' Paul said, when he reached Mick and me.
We opened the door and quietly snuck in, with our crates and sticks in tow. The rotting smell from the patch of dark muddy water saturated the air. As I shut the door behind me I saw a sign a sign saying BEWARE OF THE DEVIL.
'Ah, shit,' I said. 'Here we go, Mick, we're just about to meet the infamous Devil. Strap yourself in.'
'Of course we're the ones who get this guy,' Mick said, sarcastically. 'Here's an idea, mate – Craig and Jono are next to us, we'll tell them this one is really placid and I'll rip the sign down, then we'll move onto the next one and leave them to it,' he continued, with a big smirk plastered across his face.
We both cracked up laughing – 'If only we were that mean!' – and then we stopped talking shit and got into it. Tiptoeing about, we sussed out our surroundings. It was only a small pen, so there wasn't much room.
We stepped cautiously down to the water's edge and, without warning, the croc sprang out from the water, making straight for Mick. Mick hurled his crate at him and the croc wasn't impressed, chomping down a couple of times and spitting it straight out. I chucked my crate as backup, but he didn't have a real go at that either.
I wasn't in the right position to help, so it was up to Mick and his stick to fend him off. Mick spun back to the croc but because the water had just been pumped out of the pen, the wet concrete surface was like a slippery slide. In what seemed like slow motion, Mick's stick went into the croc's mouth, then, still holding onto the stick, Mick slid three metres to the right of the crocodile, half pushing the animal back into the water. But as he slid down around to the croc's side, he lost grip of the stick and landed in the shallow water, on top of the croc's tail.
'Fuck, Mick!' I yelled.
I was at the croc's snout, so I poked and prodded it like mad, drawing the attention away from Mick, who was scrambling to his feet. Mick ran up behind me and Devil retreated into what was left of his water.
I turned to Mick, who was covered in the tar-like black mud. The smell coming off him was unbearable, which was no surprise because the muddy water was laden with old faeces and dead shit that had been decomposing and fermenting for years. It's hard to explain unless you've been around dead things before, but think of the worst stench you've ever experienced, then times that by 10. And a decomposed animal is the worst of them all. As a carcass breaks down in the heat, it begins to bloat up and smell, and then as it starts to decay, it deflates right down like a melted candle. The fluids and fat just melt out, spreading everywhere. That's what the mud was made up of and that's what was all over Mick. I was gagging, and Mick was gagging at his own stench.
We exited the pen and ran straight to the pressure hose, where I sprayed him down immediately. We were well aware of the risks associated with working in the mud. It was about more than just washing away the smell – the mud was disease-ridden. The crocs were one danger, but equally as life-threatening was a disease called melioidosis (more commonly known as Nightcliff Gardeners Disease). The disease can be fatal and is found in tropical areas throughout the world, particularly in northern Australia. The bacteria that causes the disease enters your body through small cuts or can be transmitted from breathing in contaminated air. In endemic areas, like the Kimberley region, people are warned to avoid contact with soil, mud and surface water. When we'd moved Axel out of his pen earlier that day we'd all been walking on eggshells trying to avoid contact with the muddy water, and now here was Mick covered head to toe in it. The farm was a haven for crocs, not humans. I made sure Mick was thoroughly hosed off before we returned for round two with our cranky mate.
Sure enough, as we walked back into the pen, there was Devil, waiting. But this time we were more prepared, and there was no messing around. The moment he pounced I was going to go at him with a head rope. It's easier to noose a croc when they're on the attack because their mouth is open and they're coming straight at you, but you also can't afford to miss. He lunged at me, so I threw the rope, lobbing it over his snout. I pulled it tight and Mick jumped on the rope behind me, pulling hard to keep the tension. But Devil kept coming, moving forward with force.
I shuffled backwards, attempting to push him away from me with the stick in my other hand but he flung his head towards me instead, knocking the stick straight into my chest.
'Arghhhh!' I cried, falling hard onto the concrete and hitting my head. I started seeing stars. By this point, Craig had heard the commotion and had hurried in to help. Mick pulled me to my feet with his free hand, and Craig, Mick and I tugged hard on the rope. We had to be careful with the rough concrete floor beneath us because I didn't want to hurt the croc. I'm always focused on going steady and trying to pull the animal out without banging it up. The last thing I wanted was for the croc to do a death roll and hurt his osteoderms (scales and plates) or crush his brisket. I taped him up with hessian to calm him down and injected the relaxant.
After a period of grace, we lifted Devil and placed him on three connected pallets, before pulling him out and getting the forklift to transport him into the back of the truck with Axel. The 20-tonne excavators were already following in our tracks, bringing down the bricks and concrete pieces. They also knocked down part of the neighbouring pen so we could move in to access the next crocs. We worked our way from pen to pen, wrestling, noosing and relocating the rest of them.
It wasn't like catching a croc in the wild, it was a completely different ball game. Each animal posed its own unique challenge. When you're in their environment, the landscape is less forgiving; there is grass, mud or water, and they are less likely to stress or hurt themselves. Plus, you have space to move or run to if the animal chucks a hissy fit. In a small pen there is limited space and therefore small room for error but we managed to work it out, getting more efficient as we pulled them out one by one. If the forklift was busy helping someone else out, then we would tie the crocs up to nearby trees until we had the equipment to move them to the truck.
I was driving the forklift carrying a big 16-footer, moving it to the truck, when the police turned up. Oh dear, what have we done? This can't be good, I thought.
'G'day, fellas, how you going?' they asked.
'Good,' I said, giving a short answer. From past experience, a copper showing up without an invitation usually resulted in me in the paddy wagon. I didn't think we were doing anything wrong, we'd got all the right permits sorted but I just know a mountain can be made out of a molehill in these sorts of situations. They walked up right beside the forklift and examined my croc. Here we go, I thought.
'We heard you were in town, mate, wondering if we can get a pic with you and the croc?' one of them asked.
I felt a rush of relief as my worry subsided and I smiled back at them. I never thought I'd be so pleased to hear someone ask for a photo with me, I thought.
'Oh yeah, of course no worries, lads,' I said, and hopped out to stand beside them.
We became mates in minutes. 'Shit, I thought I was in strife and was heading to the lock-up!' I said.
The coppers laughed. 'Nah, mate, we just like your show!' one of them said.
I was relieved. After the first visit, word got around and more locals came down to take pictures of this one-off occasion. So many large crocs lying side-by-side in the one place was a very impressive sight.
Willow joined Mick and me for our last pen of the day, which was a big one. We had to pull five out in one hit and were slowly losing daylight. The first few crocs were easy because they came up out of the water, showing themselves with minimal aggression. They were females and surprisingly easy to handle, especially between the three of us. We poked around in the water and found the large male we were told about, but we couldn't seem to find the last female.
'Maybe there were only four in this pen after all,' I pondered. 'Let's focus on this male then.'
The croc in question was 15 foot but he was a big boy, with the girth you would expect to see on an 18-foot crocodile. He'd showed himself initially but went to ground after further pokes and prods. That's always something you have to be careful of when catching a croc – too many pokes and prods can tire them out, and when they're tired they hide away at the bottom and are near impossible to get out.
I couldn't see him, but I knew he was down towards the middle of the remaining patch of slosh. And then I had an idea.
'This muddy water is dense enough that we should be able to make a bit of a bridge for me to walk across with sheets of tin,' I said.
The guys agreed and went about collecting sticks, pieces of timber and tin sheets. We chucked them out onto the mud and created a pathway so I could get into the middle of the water. The plan was to reach where I imagined the croc to be, poke him with my stick and bring him to the surface. I knew there was the possibility of falling in from the 'bridge', but I preferred to take that chance rather than wading through disease-ridden mud and potentially stepping on the croc in the process.
I walked across the precarious planks and pieces of material, using my stick as support, stepping slow and steady. I was poking and prodding, poking and prodding but nothing was showing up. I submerged my stick next to me and squatted down to realign my balance and that's when I felt myself connect with the croc. It must have been lying right underneath me and it reared up with so much power the makeshift boardwalk went flying into pieces. The croc's tail cleaned me up, lifting me and flinging me to the back of the pen. I swallowed a big gulp of mud in the process, and felt it fill up my ears and trousers. Luckily for me, there was a small concrete ledge that skirted the back of the pen that allowed me to tiptoe back around to the front where Mick and Willow were. I spat out the mud until my guts gave way and, I couldn't take it, I vomited. I looked as if I'd been dipped in chocolate, but it certainly didn't taste like that.
I attempted to wipe the putrid mud from around my eyes and mouth where possible, but with it already all over my hands, I didn't do a very good job. The odour coming off me was terrible, but I wanted to get the croc out sooner rather than later and be done with it.
'There's no time to hose me off and it's not going to do me any good now that I've already swallowed the shit,' I said to Mick and Willow.
The mud was moving around like a mole was burrowing under the soil, so we could see where the croc was. I waded through slowly, like it was quicksand. The movement stopped.
'Ah shit, I hope he's not at my feet,' I said to the boys. 'I'm backing out. Let's build the bridge again, I don't fancy stepping into his jaws.'
So we went again, rebuilding the dodgy platform and I stepped out, this time with a head rope and my stick. I took the first two steps and, in a stroke of luck, he rose to my left. I took the opportunity and gave him a nudge, and he presented his jaws and I noosed him!
'Pull!' I called, as my platform sank and I found myself in knee-deep. The guys pulled him out, meanwhile I prayed his tail didn't swipe and knock me over along the way.
'Matty, move!' Mick yelled. Suddenly, to my left, another croc came at me and had a go at my stick. There WAS one more female!
The boys were battling by themselves, managing to only just keep the croc away from the water. I waded out of the crap and lent a hand, managing to pull the male out relatively easily. We tied him to a nearby tree before the drugs and forklift arrived.
I watched as Craig helped Willow and Mick get the last female out of the hole and we were done for the day. I joined the croc in getting a hose-down. Each time someone got mud on them they cleaned it off, but it was just as important to make sure the crocs were totally hosed down, so they didn't carry any diseases back to the Territory with them. We drove to the local motel and slept off our weariness, ready for day two.
There were 10 crocs in two pens left to clear out. We got into it early because we already had the 40 other crocs waiting in the back of the truck. Nick was topping up the sedative drugs every four hours to keep them calm. The truck was parked up in the shade and the crocs were all positioned comfortably with extra layers of hay padding and shade sails pulled over the top, but we still wanted to get the job finished sooner rather than later.
We moved the first out pretty quickly, because we had the entire team helping, which made everything easier. Well, that is, until we came across another cranky male in a pond. All of us were around this pond, all of us had a head rope and all of us had a pole to rouse him up. But every time the croc came up we kept missing! I decided to change tack.
'All right, that's it! I'm going to bring the little excavator in here, park it on the edge, extend the arm and scoop this croc out from a vantage point.'
The guys weren't that keen on this, but I thought it was a good idea, so I ran back and hopped in the excavator and drove it down a small walkway, over an embankment and to the edge of the pond. I edged it down the bank just a bit and soon realised it was a lot steeper than I had anticipated. The excavator began sliding and then started tipping, so I extended the arm with the bucket to stop the machine rolling over. That worked for a second but then the machine was moving sideways and it kept going and going, and I couldn't get the bucket around quick enough. The soil shifted from beneath me and both the machine and I slipped straight into the middle of the pond with the bloody crocodile.
I was sitting in the excavator half submerged in the mud while all of the guys standing around the pond slapped their sticks like crazy, in an attempt to keep the attention away from me. I watched as the croc swirled around in the mud. I was ready to jump on the excavator roof if I needed to.
'Boys, quickly, go grab the chains and the big excavator and help get me out of here!' I yelled.
Craig dropped his stick, ran off and then came straight back, flinging me some chains. I jumped on the roof, hooked up a couple of D-shackles and chucked the chains back to the guys with the big excavator, which had just arrived. I was tempted to try my hand at making it from the excavator to the bank on foot, but I knew how unforgiving the mud was to move through, so I sat myself back in the excavator cab, waiting for the bigger machine to pull me out. Each time the big excavator pulled I looked on at the chains in fear that they would snap, and I'd end up in even deeper shit with the croc. I knew I was slowly sinking, but I still had a small height advantage in the cab. Every so often I saw the croc do a bog lap around me, so I took the opportunity to see if I could coax it up to the surface with a little stick I had with me.
I tried to swipe him with the stick as he went past and the cheeky bastard showed himself all right, launching up and almost ending up in the cab with me! A commotion erupted from the guys as the croc came up for a second crack. His jaw knocked the bottom of the excavator as I finally became airborne, getting lifted to the bank. The croc was so intrigued by the sound of the excavator engine that he followed me up to the bank and the job was made easy. Nick got a head rope on him instantly and Craig, Willow, Seb and Mick pulled the rope off to one side until they got the croc under control.
I didn't say much immediately after that debacle because I wasn't the most liked person that morning, but the guys got over it pretty quickly. We relocated a couple more of them and, in true croc-catching style, finished the operation off with a real pearler.
In the last pen, there was a number of small ponds scattered around the place. We knew there was a male in one of them, but the question was which one. It was hard to pick because this particular enclosure had a heap of two-to three-foot crocs bounding around, so all of the ponds appeared to be pumping with life. (We weren't allowed to catch any of the little ones; they had come from eggs that had accidentally not been collected at the right time, and were going elsewhere.)
Willow, Jono and I did the rounds of the ponds, scoping out where the guy would most likely be hiding. Willow kneeled down at the water's edge of one of the ponds and you wouldn't read about it! The 14-foot croc we were after torpedoed straight towards Willow. He slipped over in the mud and fell flat on his back. But the croc didn't want to attack, it just wanted to get out of the way. Flying at him, it went straight over the top of Willow, using his body as a platform to slide across – over his back, over the top of his shoulders – and pushing off to get a bit more propulsion, slapping him in the face with its tail on the way out.
'Good thing we know where he is,' I said, pointing to the new pond the croc had just entered, but Willow wasn't impressed. We moved along to the pond where there were 10 little crocs darting about the place. They looked like tadpoles in comparison to their big crocodile counterparts that we'd been catching. Still, the two-footers aren't to be taken lightly either, their bite can rip apart muscles in an instant and leave behind a nasty infectious disease, so we had to be careful.
The croc reappeared at the surface of the pond, calmly keeping its distance.
'At least this one isn't playing hide and seek,' Jono said.
I chucked a stone into the pond to see if it would get a reaction and the croc snapped, swiping its tail and cleaning up one of the little crocs along the way. The large croc propelled the fella through the air, and it beelined towards Jono. It hit him square in the chest and knocked him back in the mud, arse over tit. The little croc scurried back into the pond. It was entertaining for us all, but not for Jono, who was in a foul mood and joined Willow for a hose-down. The operation went on for another hour. We secured the last male and one more female, and then we were done.
The local media showed up with the intention of doing a story about the big relocation, but I cut things short because I wanted the crocs moved out ASAP. I pulled out a big fire hose, which was looped up on a spiel attached to a nearby wall, and used it to give the crocs one big spray-down. I added more hay and shading to the inside of the trailer and made sure each croc was safely secured. Then Jono, Mick and I flew to Kununurra, where we waited until the road train arrived and the crocs cleared the border. The customs fella was a bit confused when the cattle truck revealed a mob of crocs, but he signed off on the permits and across the border they went. The boys and I camped the night in Kununurra, leaving at first light, and met the road train at Darwin Crocodile Park at 9 o'clock the next morning.
Unloading the crocs was a hell of a lot easier than catching them, because they were still groggy and emerged from the truck with hardly any resistance. We made ramps so that the crocs could comfortably slide off the backs of the trailers, and once they were off we stretchered them out to their new pens. To this day they are all living happily in Darwin and some of them, one being Axel, have been moved to Crocosaurus Cove, where they live as display and education animals at the popular Darwin tourist attraction. Their new cushy homes are a far cry from their wild west digs.
#
I had five good mates – Matt, Danny, Ben, Changa and AJ – come up to Darwin from down south, all of them eager to get amongst the Territory life and landscape. They were city slickers who had never been up north before and were keen to hook a few barra, chase a couple of hogs and rip about in the chopper. I had planned the ultimate lads' trip.
The southerns stepped out of Darwin Airport with their rosy-cheeked pale faces, bucket hats and fresh Mountain Designs outfits, still with the crease lines from being in the packet. Their shirts certainly hadn't got a workout before, neither had their scuff-free Blundstone boots.
We loaded up the troupe and left Darwin Airport en route to my hangar, where my two choppers were waiting for us to boot out.
'When we get there, I will give you a full safety brief on the choppers and how to operate firearms,' I told them. 'But before we go out for some fun I have to stop off at the shack and check on my genny to make sure she's all good to go for when we get home tonight.'
'No worries, mate. Who's Jenny? Is she your housekeeper?' asked Changa. I was confused, but quickly caught on to what he meant.
'Jesus, old son, genny as in the generator that gives the shack power, not a sheila. There's no housekeeper out here, mate, you're going to have to do your own washing and cleaning.'
The troupe erupted with laughter, and the lads started hanging shit on Changa. 'Looks like Changa's buying the first round at the pub tonight!'
After dropping our gear at the shack and checking out the genny I introduced the boys to Tripod, my 17-foot pet crocodile, which they couldn't quite comprehend either.
'Wow, Matty, that's one hell of a statue, it looks so real.'
'Ohhhh, ya shitting me, aren't ya?!'
They think he's made of concrete, I thought to myself.
I decided to play along. 'All right, lads, let's get a picture.'
I got the guys close enough to Tripod and asked Crab, my other chopper pilot, to get a picture of us. Just as everyone was smiling for the camera I flicked a rock next to Tripod's head and he exploded with an almighty snap, slamming his jaws down and splashing water over everyone. I've never seen a bunch of blokes scream like that before, running in the opposite direction.
'Righty-o, you lot, into the choppers,' I said, once they'd calmed down. 'It's time to give Tripod a feed and get some bait for my croc traps.'
The guys piled into the choppers and we flew over a 25,000-hectare cattle station, located south of Dundee in the Northern Territory, and a prime location for catching feral pigs. We saw mobs of pigs running beneath us, like ants scurrying around on a hot day.
'There's over 20 million of these guys spread across Australia, and they're a big problem around here,' I explained to Matt, Danny and Ben, who were in my chopper. 'They're feral pests and they muck up the land, spreading weeds and destroying the habitat of our native animals, so I don't mind using a few for croc bait.'
The NT and other state governments spend millions of dollars every year trying to manage the feral pig population through baited trapping, mustering and aerial culling. So when it comes time to set my croc traps, feral pigs are the most accessible and sustainable bait there is.
Crab was flying Changa and AJ in the second chopper, following me closely behind. We stopped a couple of times so the boys could get their eyes in with a few pot shots at termite mounds. After everyone had become more acquainted with their guns, we got back into the air and found a good mob of pigs in a swamp adjacent to the river.
'Oi, Crab, I'm going to land here and I want you to drop your lads off next to me, fly back up and scoot around those pigs. There's a couple of good boars I want to get,' I said over the radio as I landed my chopper. 'Give me a moment to shut down and get the boys in position. Just make sure you keep those pigs bailed up in the cane grass and try not to bring too many at us at once. I don't want to get hit up the nuts here.'
'Roger,' Crab radioed back.
Pigs are pretty easy to muster. Like with cattle, you drop in behind with the chopper, pushing them to clear open ground. You stay low in the trees and drive them out using the downwash of the chopper. They move slowly through the swamps but as soon as they're out into the open, they bolt.
I got the guys lined up in a formation where they wouldn't be at risk of shooting themselves or the chopper. In the past, I've had guests accidentally shoot my helicopter, blow its skids off or almost shoot themselves in the foot. And sometimes the gun jams, there's a misfire, or in the heat of the moment they just get completely overwhelmed and the hog gets the upper hand. These past experiences make me very savvy when it comes to safety precautions.
I called Crab on my handheld radio: 'You're up, mate!'
Crab began mustering the first group of hogs out of the waterhole and towards us. The plan was for the guys to hit them before the hogs got across the plain to the river. I made sure the group of five blokes was shooting in a northerly direction, away from the river or anywhere there could be fishermen. My palms were getting clammy just watching the guys, so I could only imagine the sweat building on their hands as they saw the chopper in the background and the cane grass moving about with the pigs within. It could be an angry boar, a group of piglets or even a raging buffalo. Everyone was fidgeting in their stance.
'Take your time, focus, and be calm and steady with your shot,' I said with a raised, stern voice. 'One good shot outweighs 10 bad ones.'
The first group of pigs emerged from the cane grass, coming at us, grunting in the distance. There were mostly sows and smaller pigs, which was a good start, but I could see a big boar at the back.
'Hold your fire, hold your fire. Wait, wait!' I yelled out to the guys.
Once the pigs got within 20 metres of the boys, I called, 'Fire!'
And away they shot. Cracks went out and the smell of gunpowder filled the air. I saw two pigs fall, while the big boar continued to trail at the back. A couple more cracks of the rifles and the boar fell too.
'No time to rest, guys, reload and get ready. The next mob is coming,' I advised. Before I could finish my sentence Crab had another four big boars beelining straight towards us.
'Bend those pigs, Crab!' I shouted on the radio.
It was too late. The boars had lined us up and I screamed again, 'Fire, fire, fire!' Another two went down, but the other two were charging. I grabbed Changa's rifle and quickly jammed a round, letting the closest one have it. He fell, and I quickly reloaded and took a shot from my hip, dropping the second pig just as it reached my feet.
'Wow. Well, that was exciting, wasn't it, boys?'
Crab landed and we sorted the six pigs. The boys were having a big barney about who shot the biggest one. It was comical. We hooked the pigs up to the chopper and Crab slung them back to the traps. It was happy days. Or so I thought . . .
It just so happened that a local fishing guide – let's call him Scott – had seen my choppers flying around that day. Scott was running charters in the area and from all accounts wasn't too happy with the competition from my new adventure tourism business I had opened not far from where he operated. Apparently Scott was on the floodplains of the river the same morning we were out shooting and heard the guns going off. Coincidentally he also saw choppers that he presumed were mine, even though he had never met me or seen my choppers before. He just saw a chopper and thought, That's Matt Wright, and made up a story to go with it.
I got a call the next day and I was ropable.
'Matt, it's Darren here from the NT police department. We've got a report from a bloke called Scott who said you were flying around recklessly in a chopper and shooting at his fishing charter boats with an automatic rifle.'
'Ya kidding me, right?' I snapped back at the cops.
'Well, were you out flying in that area?' Darren asked.
'Yeah, I was, actually, but I wasn't shooting at no fishing charter. If I was, I wouldn't miss. What a tosser!'
I couldn't believe it. I didn't see anyone else out there and the supposed 'fully automatic rifle' he was talking about was probably the sound of our five rifles shooting simultaneously at the pigs. I made certain that we were on private property and away from people. It was just another case of tall poppy syndrome that seems to be rife in Australia.
I understand the cops were doing their due diligence but, unfortunately for me, this petty complaint led to a long and serious investigation that started from that afternoon and continued on for the next few years. But I didn't let the lads know what had happened and made sure they enjoyed the rest of their trip to the NT.
The saying 'when it rains it pours', I swear, was written with me in mind. Not only was I under investigation for a shooting incident that didn't happen, but I also had lawyers going in all sorts of directions with a CASA court case into my flying on Outback Wrangler and an ex-girlfriend trying to take me to court for a stack of money after a 12-month relationship. The investigation into the pig-shooting incident had dragged on for just over three years. I'd been in and out of the cop shop, answering silly questions and giving evidence. Finally, on Tuesday, 11 November 2014, I was going to pick up two coppers from Sandpalms resort in Bynoe Harbour, fly them to the area where the reports were made and show them the guns we were shooting with. The cops needed to do this one last time before the case was closed and I was really looking forward to laying it all to rest. But the day was far from restful – there were coppers, choppers, chainsaws and a monster croc. It was an absolute clusterfuck.
This final visit from the cops was in the middle of filming for season two of Outback Wrangler. Jono, Willow, me and my other good mate Craig were all camping out at a large cattle station, along with the camera crew. The lads and I had already pulled out and relocated four crocs from the area, but there was one massive croc left to go. And it was this croc that was causing the most damage to cattle. I thought it would be a quick enough job and that I'd be out of there by lunchtime, ready to meet the coppers at Sandpalms.
'All right, fellas, hop on this off-roader Polaris and head down to the landing, sort out the airboat and grab the gear we need for the swamp. I'll get the chopper ready just in case we need it,' I said to the guys.
I flew the chopper and landed it near the water so I was ready to boot out of there when the time came. We didn't know what to expect because we'd never done a relocation using an airboat. It would have been an easy job if we'd been able to access the waterhole by foot, pull the croc out and put it straight onto a trailer. But this particular croc was in an area completely encased by jungle, so we had to cut our way through dense foliage to get the airboat in there. We had two chainsaws, duct tape, rope, a harpoon and stick at the ready. I was driving, and Willow and Jono were on the chainsaws. Craig and our cameraman, Ash, as well as our second cameraman, producer and sound guy followed us on foot along the bank as we tracked the airboat through the thick paperbark swamps at a snail's pace. Every so often we'd make a little break, but we'd soon be stopped again by the paperbarks and floating logs. It was quite obvious that no person had ever been here before.
I made a sharp turn on the airboat and it got stuck on some floating mat, which is a dense, free-floating weed. The airboat hitched up on an angle, tipping Jono and me overboard, along with the two chainsaws.
'Shit! Move quickly and scoop up those chainsaws,' Willow yelled from the boat.
I was wide-eyed and couldn't believe what had just happened.
Jono on the other hand thought it was all a bit of fun. 'At least we've got the chainsaws to fend off the crocs,' he joked.
I forced a laugh. We managed to grab the chainsaws and waded towards the airboat. The rest of the lads were on the bank at this point, telling us to get a move on, knowing full well what lay beneath the water. The floating mat made it difficult to move quickly. It felt like I was trying to walk against a strong current of water, and I was second-guessing whether I was actually making any progress. After a very long five minutes of unease we finally made it onto the boat unscathed.
It was the beginning of the wet season in the Territory so it was 39 degrees Celsius with 90 per cent humidity. It was 11 a.m., not even the peak of the heat, and everyone was already giving me daggers. By this point we had spent three hours cutting down trees and got as far as 300 metres, with at least a kilometre to go. We made progress with another 10-metre stretch until the blade broke on Jono's chainsaw. At the same time, I heard Willow cursing next to me – he'd got his chainsaw completely stuck in the trunk of a 10-inch paperbark and it wasn't coming out. What a fucking disaster. Willow was sweating bullets, Jono's humour had dried up, I was quiet and it was the first time I'd ever seen Craig crack it. We were all exhausted and grumpy.
'Is this worth it? What happens if we get there and the croc's not even in the waterhole?' Willow whined.
He had a point, but I'd promised Paul, the station's owner, that I'd get the croc out for him and now was the only opportunity we had. My gut told me the croc was there and that we'd find him, and besides, a bit of hard work never hurt anyone. I knew I had that meeting soon and I'd have to get on the bank and bolt back to my chopper on foot to get there in time. Jono persisted with the stuck chainsaw, using a small pocketknife to hack it out of the tree. We sat there exhausted and watched and waited until finally the chainsaw came free – it was 12.30 p.m. by this point.
'All right, lads, you keep trucking. I've got to head over to Sandpalms. I'll be gone for about an hour but hopefully by the time I'm back you'll be in the waterhole and I'll come and find you.' No one said a word as I jumped onto the bank and returned to the chopper. They were pissed off and I don't blame them.
I flew over to Sandpalms. I knew I needed to return bearing gifts so I could earn some brownie points with the guys back at the swamp, so I borrowed an esky from Sandpalms and packed the bottom of it with cold beers, followed by a layer of ice, then Powerade, bottled water and some more ice. Then I picked up the coppers. They were great people and were keen to get everything closed off, so after a quick fly around and debrief they were ready to head home.
'Guys, you don't mind if I just fly over a swamp here on the way back and check in on my boys?' I asked the two cops. 'They're on an airboat trying to catch a croc.'
'Of course not, mate. Would be interesting to have a look, anyway,' the sheila cop said.
We flew in low to the waterhole and I couldn't believe it! The guys had made it through the jungle, and there was Jono, on the edge of the airboat with a stick, poking the water as it erupted with a huge crocodile. Behind Jono was the rest of the crew, violently waving me down.
'I'm sorry, guys,' I said to the police, 'but I'm going to have to land here and make my way to the waterhole to catch this croc. You're welcome to follow.'
I landed the chopper and started running through the trees along the bank with the esky in hand and the cops close on my tail. They sounded like a marching band, with their tasers, batons and guns banging and clanging as they ran.
'Make sure you don't get your gear hung up on the vines and taser yourself,' I yelled. I finally saw the lads through the clearing, and they were beaming from ear to ear. Boy, they looked absolutely cooked, but their energy levels had sure as hell lifted. Jono had got the croc on the end of the harpoon, and was waiting for my call. The croc was situated between heaps of floating mat and fallen melaleuca trees with six feet of water beneath. It had nowhere to go, but it was going to be one hell of a battle to get him out and onto the airboat.
'Take a break, lads, the croc's not going anywhere. He'll just lie here, he's got nowhere to go.'
The lads sighed with relief as I chucked them all a Powerade and a bottle of water. I was back in the good books. The condensation on the bottles and the sweat dripping off of their heads would have made for a great TV ad.
But then soon it was back to work. Craig made his way over near the floating mat, taking hold of one side of the croc, while Jono and I perched up on a fallen tree that was sticking out across the water. We had to leverage the croc's head up to get a rope around its snout, but it was proving to be difficult.
And then, without warning, the croc flung up and grabbed the tree branch sticking out between Jono and me. It wobbled and we lost our footing, falling onto the floating mat and sinking into the water. We hoisted ourselves back onto the tree quick-smart with the croc in sight. With all this action going on, it was our cameraman Ash who had the hardest job of us all – bouncing around between the airboat, the landing, the floating mat and the tree, trying to get the best shot, at the same time as making sure he didn't lose his camera or an arm. Every now and then I spotted the coppers in my periphery. I don't think either of them knew what to do, so they just stood there frozen in shock, with wide eyes and gaping mouths.
Finally, we managed to get the head rope on the croc when it came up for a breath of air. We got its head up the side of the airboat and, luckily for us, the croc's mouth was already bound up with weeds from death-rolling underwater, which made taping its mouth shut heaps easier. Craig and I jumped into the waterhole with the croc while Jono and Willow pulled the rope, but we had to get some behind-the-scenes help from the Outback Wrangler crew. The lads put down their tools and gave us a hand to lift the croc up onto the deck of the airboat. The job was done and it was time to head home with the croc on the airboat, but not before the coppers got a selfie with this monster croc.
#
On the edge of the Democratic Republic of the Congo and Tanzania border is Lake Tanganyika, an immense body of water with outflowing rivers full of wildlife. In 2010, South African Johannes Hendrik Coetzee or 'Hendri', a hero in the world of whitewater exploration, set out on what would be his last ever paddling expedition. His in-depth knowledge of the area resulted in people saying he was the man who possessed the 'key to the Congo'. Hendri guided two American expedition paddlers, Ben Stookesberry and Chris Korbulic, on an epic kayaking adventure through central Africa. It is because of my friendship with Chris that I know the story I am about to tell you.
The trio started in Uganda and paddled through a series of rapids along the Ruzizi River and then water-taxied across Lake Tanganyika to the outflowing Lukuga River. The region has seen decades of bloody battles where millions of lives were taken and thousands of deceased bodies were disposed of in the waterways. This dark history means the large crocs in the river acquired a taste for human flesh and, given the abundance of food, they grew to incredible sizes. One croc, whom the locals call Gustave, is reported to be an incredible 20 feet long and has eaten in excess of 300 people. And Gustave isn't alone; he has lots of scaly mates who are 16-to 18-foot alpha predators with a liking for humans.
Hendri, Chris and Ben journeyed down the Lukuga River, stopping off at villages along the way. At each stop, travellers and locals warned them of a man-eating crocodile hanging out in the waters where they were headed. This sort of story wasn't anything special for someone like Hendri – he knew about the risks involved with the crocodiles in the Congo but accepted that it was just part of the adventure.
They were travelling in November, the end of the dry season, when the water was at its lowest and the heat was at its highest. Hippos wallowed around, posing their own danger, and the crocs were starting to become even more territorial as their nesting season approached.
The boys came up to a section in the river that Hendri called the 'supermarket'. It was an expression he used to describe where the dead water pooled at the end of a fast-flowing rapid. This was where crocs would hang out, waiting for something to get caught up to feast on. When they reached the 'supermarket', Hendri told Chris and Ben to keep paddling until the crocs were gone and the river had widened out. Ben was in the lead, followed closely by Hendri, with Chris lagging behind a little. Two crocs launched from the banks thrashing into the water and Hendri called out to Chris: 'Come on, mate, catch up and get in front of me, keep moving and keep the pace up.' Chris got up alongside the left of Hendri and, as he overtook him, he remembers hearing Hendri say in a weird, matter-of-fact way, 'Oh my god.' As Chris turned his head to see what was going on, he saw an image that would stay with him for the rest of his life.
A croc's head, the same size as Hendri's torso, emerged from the water and grabbed Hendri, slowly pulling him upside down into the murky water. It dragged him underwater so laxly that there was barely a splash.
Chris was in so much shock that he completely froze for a moment, but then snapped out of it, turned his kayak and powered over to Hendri's kayak, which was spinning around in circles. Feeling utterly helpless and shocked about what had just happened, Chris took a few jabs under water, trying to hit the animal. Ben frantically made his way over to Chris to give him a hand, but the chaos stopped and the kayak appeared empty.
Hendri was gone.
Ben and Chris paddled for their lives and got out of the water as soon as they saw a small clearing on the riverbank. They were dwarfed by the tall cane grass and could see nothing but jungle in front of them. I can only imagine how helpless they would have felt at that moment. Their safety net had just been torn open, they were alone in the middle of nowhere and the one person who knew how to conquer the Congo was gone. They had no choice but to get back in their kayaks and continue to race downstream. They got very lucky – only a few more clicks along the river, they came to a bridge. It was the only bridge they had seen in days, and situated alongside it, was a small grass hut village called Kabeya Maji.
Chris and Ben paddled their kayaks up to a makeshift washing bay and ran up the banks yelling out to the locals, 'Crocodile! Crocodile!' They communicated in basic French with the help of hand gestures, explaining what had happened, with Chris raising his arms and clapping them down repeatedly to mimic the jaws of a croc. A few of the men from the village started taking their shirts off to go for a swim and investigate.
'No, no, no, just wait, just wait,' Ben urged them to stay. He begged the village chief for a motorised boat to take them back upstream to look for Hendri, but they quickly learnt that the village had no boats. The locals had stopped using water travel after several of them had been pulled overboard and eaten by a big croc in the area. They told Ben and Chris that eight people had been taken in six years. The guys realised in that moment that they had no hope of finding their mate's body.
The guys sat on the bank helplessly as Chris told Ben what he had witnessed.
'I saw the whole thing happen. The croc came out of the water, it was massive.'
'It got him by the shoulder?'
'Yeah, just there.' Chris signalled to his left shoulder.
They watched as Hendri's kayak drifted downstream towards them. The village people recovered the boat, but sadly Hendri's remains were never found.
It wasn't easy for Chris and Ben to return home to America; they had to stay in the Congo and wait around for a hearing, conducted by the United Nations, into Hendri's death. The guys spoke, in front of the tribunal, reliving the trauma and describing the incident, blow by blow. They had to stay in the country for another week until the case was finally cleared and they were allowed to leave. Chris and Ben flew out and spent a week in Jinja, Uganda, with Hendri's friends before going home.
In August 2011, while I was doing some flying work in the Yukon region in Canada, I got a call from a guy at the National Geographic Channel who explained what had happened to Hendri.
'Matt, we're wondering if you'd be interested in heading to the Congo in three months to do a show on catching and relocating this man-eating crocodile?' He spoke with a hybrid accent; not quite American, not quite English, so I figured he must be from the global headquarters. I wasn't really thinking about the trip or the show, I was in shock at what I'd just heard about Hendri.
'Sure, I would be really interested in the project, but what do Hendri's mates and family think?' I was keen to do the job but I wanted to speak with Chris and Ben and get their perspectives before committing.
The next day I found myself on Skype talking to Chris and Ben, and listening to a raw rendition of events. Ben led the conversation, and there was no sensationalism or drama in what he said, only real emotion. I felt his sadness and connected with his calmness and sincerity.
'Hendri spent a lot of time in the Congo and it was a dream of his to build a well in Kabeya Maji so people can access fresh water without having to go to the riverbank and risk being eaten. We want to go back and build the well as well as catch and relocate the crocodile, but we need your help with the croc part.'
I was sold on the trip and confirmed my involvement with Nat Geo that day.
I couldn't shake the thought of Hendri being engulfed in the jaws of a croc and how painful and overwhelming his last moments would have been. Chris sent through photos, videos and diaries from the trip, including footage of Hendri being taken that Ben had captured on a GoPro strapped to his head. I also viewed the intense footage from the United Nations hearing – the guys had really been through the thick of it. I spent hours on the phone to our African film crew and a personal guide, also known as a fixer, on the ground in the Congo, explaining the ins and outs of catching a croc. I emailed the fixer pictures and descriptions of the traps and harpoons I needed so we could hit the ground running. I started tracking the weather and researching the region. I worked out that November was the perfect time to travel because it would be hot with low water, which meant crocodiles would be out on the banks, not deep in the water. Chris and Ben agreed, and were familiar with the conditions at that time of year because it had almost been 12 months since they were in the Congo when Hendri was taken.
Come November, everything was in place – visas were sorted, flights were booked and I had my yellow fever shot. I was ready to fly out when I received an unexpected call from Nat Geo informing me the trip had been cancelled. I was aware of the political unrest building in the Congo but I thought that was part and parcel of visiting that part of the world, so I didn't think it would affect our trip. It was the lead-up to the December 2011 presidential and parliamentary elections, and the riots were building every day. People were getting killed in political rallies and that was only the beginning. Nat Geo knew that if the local people didn't accept who won it was going to get worse, and it did.
I was disappointed we couldn't go, but seeing what unfolded in the Congo, I was glad to be home in Australia for the time being. I flew back to Darwin and started work collecting crocodile eggs, and kept in contact with the producer from Nat Geo. She was adamant that the trip would still go ahead, but we just had to wait until the country was more stable and the weather more favourable. Our fixer on the ground in the Congo advised that February would be an appropriate time to visit, with kind weather and a calmer political situation.
I was sceptical from the get-go because the seasons in the Congo line up with what we have in Darwin, and February is the peak of the wet season; meaning high water and thunderstorms, which is the shittiest conditions for spotting a croc, let alone catching one. But Nat Geo trusted their fixer because they'd used him before. He knew the country and the people in power, so he could handle the logistics and politics without too many headaches. When I questioned the fixer about the time of year and the weather, he insisted it would be hot and dry with low water and crocodiles everywhere, so we went with his intel and booked our second attempt. I brought Mick Burns along with me – he's an experienced croc wrangler and I needed another set of hands.
Mick and I, along with our crew, landed in Johannesburg, South Africa, to discover that the fixer hadn't fixed anything. It turned out the visas hadn't been sorted and there was no one to meet us. The rest of the crew were in the same boat so our producer, assistant producer, cameraman, second cameraman, soundy, Chris, Ben, me and Mick were all stuck in the customs office. After a few phone calls, we managed to get cleared but missed our connecting flight. There wasn't another flight for quite some time, so we had to hang around in Johannesburg for a week.
The airport motel was less than ideal, with ceilings covered in mould that resembled moss, grumpy staff and corridors that echoed voices, footsteps and doors. We were well and truly ready to get out by the seventh day. Nat Geo arranged a charter flight for our crew to fly direct to the town of Kalemie, located next to the mouth of Lakuga River. We were eager to board the plane and arrived safely at the other end, where we finally met up with our fixer in person.
I'm impatient at the best of times and value time more than anything – I absolutely hate wasting it. And after all the delays we'd experienced, I was recharged and ready to hook in.
'So, mate, lets grab the gear and equipment and hit the ground running, hey?'
Our fixer stared blankly back at me. 'I haven't had the chance to get your equipment sorted yet, so we'll do that over the next few days.' He wasn't asking me, he was telling me.
I wanted to react, but I didn't want to rock the already rocky boat. 'I guess that's our only option then.'
I couldn't believe it. We were meant to have harpoons, ropes and steel for traps, and we didn't have a fucking thing. He could sense how pissed off I was, so he disappeared and came back half an hour later with what he said was steel. I raised my knee and pulled down on the steel panel. It looked like steel and kind of felt like steel, but it sure wasn't steel. It bent like a plastic fork and snapped. I didn't have to say anything, I just turned to the fixer.
He seemed nervous. 'There's heaps of other stuff, mate, don't worry. Leave it with me.'
But I wasn't about to let him waste any more time. 'I don't know about that idea, how about WE have a look around.'
We set off together and found some sturdy old gates, bamboo and chicken wire in a shonky little hardware store. I still needed a harpoon pole, so the fixer organised for two of the locals to cut down a tree and make four poles. When I saw how eager they were to do the work and make some extra money, I couldn't say no. I shot myself in the foot a bit because it took them another two days to whittle down the wood for the harpoons, which meant more waiting, but it was worth it. All they'd used was a small knife and sandpaper, but I'd never seen anything like it. The poles were completely smooth and as straight as a die. Our boat and trailer arrived soon after, and then we packed up our three banged-up four-wheel drives, ready for our nine-hour journey through open savannah country to Kabeya Maji.
It was a wet and bumpy ride along winding roads – or what passed for roads – in the Congo. They were muddy and hard to make out. I felt like I was back home in Darwin during a wet season monsoon; we were engulfed by grey and the rain was pelting down. The rain subsided for a couple of hours, so I got on the back of the truck and took in the landscape – the striking blue skies and towering cumulus clouds were pretty special. I could taste the heavy damp air and felt the humidity sticking to my skin. We took the opportunity to do some introductory pieces to camera for the show.
'The boys have been in a very tough situation. There was nothing they could have done to save Hendri – you've got absolutely no chance when you're up against something of that size deep in the water,' I explained. 'It will be good for the guys to go back to where everything happened and see with a clear head how impossible it would have been to do anything for Hendri once he was taken.'
I got in the four-wheel drive with everyone and we drove off. We stopped a couple of times for a hot beer and a quick stretch, then kept on trucking. The rain started to pour down again just as we reached the Lakuga River crossing. Our fixer told us that we would need to put our cars onto a ferry, which would take us to the other side of the river.
The 'ferry' wasn't what any of us expected. As I watched it approach I couldn't believe my eyes. It was comprised of four large canoes strapped together with a steel sheet laid across the top, creating a platform. There was a man on the front and back of each canoe using large bamboo pole to leverage the ferry across the river.
By this point, the rain was building and I couldn't see further than two feet in front of me. 'I don't know about this, I can't see how this dinky thing is going to transport a car full of gear and all of us.'
I love adventure but this just seemed silly to me – the current in the river looked strong, and I had a good idea of how many crocs would be waiting, ready for a feed if we fell in.
'There's no other option to get us over to the other side, I've done this lots of times before and haven't ever had a problem.' The fixer seemed confident, but I was frustrated that I had to put our fate into his hands when I already didn't trust him.
The car drove over a steel ramp that connected the riverbank to the do-it-yourself ferry and we got on. There were holes and leaks in all of the canoes.
I tried to make light of the situation: 'Okay, I'll plug my finger into this hole and we'll be right to go, hey?!'
A couple of nervous laughs came from the camera crew.
'All right, Chris, are you ready?' I called out to Chris, who was at the front of the boat. He had been given a bucket to bail out water from the most damaged canoe so that we wouldn't sink as we moved across the river.
'Yup, let's do this!' he shouted back.
The moody current jerked the ferry in different directions but we finally made it downstream and hit into the bank on the other side of the river. Everyone looked relieved.
We travelled for another hour and made it across the bridge and into the village of Kabeya Maji. Silhouettes of wispy clouds spread across the big sun, which had sunk halfway behind the horizon. Ben and Chris were both pretty emotional as we crossed the bridge near where their friend had been taken.
'It's nice to arrive at Hendri's favourite part of the day, sunset. He always stopped to appreciate the magic in it,' Ben said.
Chris nodded. 'After spending seven weeks with him I knew he was someone inspiring, who I would call a very good friend.'
The unknown is the greatest thing about adventure but unfortunately in the case of what happened to Hendri, Ben and Chris had no way of knowing that their expedition was going to go horribly wrong. Waking up on my first morning in Kabeya Maji, I was filled with that same excitement of not knowing what would be in store for me during my next month in the Congo. I was aware of the potential danger, but confident in my own abilities.
I woke up at 4.30 with butterflies in my belly. But before we went anywhere, I had to slaughter a goat for bait. I'd bought a goat from the local people for a lot of money (it worked out to be about 100 Australian dollars, which is out of control!), went to a quiet area and put the goat down with a sharp knife. I came back to camp with fresh meat for dinner and bait for the traps. We got going at dawn, and our first stop was the river's 'supermarket' stretch, where Hendri had disappeared.
Ben and Chris came out with Mick and me to show us exactly where it had happened. The camera crew followed along as the boys retold the story to camera.
'When you put kayaks in the water in the Congo, you know you're paddling into sketchy territory and, equally so, the locals who fetch water along the riverbanks know they're entering the kill zone,' Ben told me. He was right – locals will continue to go about their day-to-day business and will continue to get chewed unless the croc is removed. The river is such an integral part of their lives and culture that they'd rather run the risk of being eaten than give up their lifestyle.
We reached the 'supermarket', where there was an island created by the swollen river.
'That's where we'll look at setting a big trap.' I pointed to a cleared-out space on the island bank. 'I'm sure we could lure him there with bait.'
Our driver banked our boat and the cameraman and soundy followed us onto the island. The guys were being too complacent, turning their backs to the water. It was putting me on edge. I was used to being on crocodile nests or in the swamps with guys who knew about crocodiles, but this time it was just Mick and me and four guys who were completely green.
'Just stay away from the edge of the water!' I warned. 'If it's not a croc it will be a hippo, and you'll be breakfast.' I turned to Mick. 'Pass me the bait and we'll wedge it up here.' I pointed to a tree situated about five or so metres from the water.
Mick lugged the fresh goat meat out of the boat and I clambered up the tree. He passed me the bait and I hung it in the branches so that the blood and guts dripped all over the ground. Then it was time to set the motion-sensor cameras. I strapped them in, tested them and tilted them up and down until I was satisfied with their position.
It's difficult to measure the exact size of an animal on camera. Certain angles can be deceiving, making animals look three times their actual size. Mick paced out two metres directly in front of the water, then paced out another five metres towards the baited tree from the bank. I knocked some pegs in the ground as Mick walked. The markers would help us better identify the size of the croc on camera when it approached the bait. Then we got back in the boat and travelled downstream, even closer towards where Hendri was taken, to set the second bait. It was an eerie feeling looking around and seeing no signs of wildlife – not a monkey, not a mozzie, barely a bird in the sky. It didn't seem right, we were in the heart of the Congo and the environment was dead. We pulled up and set more motion-sensor cameras in the area then headed home.
On the way, we approached the bridge hanging over the river, with green banks on either side and open sky stretching out behind. A group of locals greeted us at the wash bay, an inlet of shallow water on the riverbanks where they washed their clothes. It was a peaceful scene until the water moved. I jumped out of the boat and into the muddy water, grabbing hold of a lively three-foot salty. The locals looked at me like I was mad as I pulled it out of the water.
'If this little guy is hanging around here, then his mum and dad won't be far away, either,' I explained.
Chris and Ben relayed my words in their basic French to the village chief and the locals around them, but they didn't seem too worried. Crocs were part of daily life. They weren't concerned about the smaller crocs, they just wanted us to catch the big guy. I strapped another couple of cameras on the bamboo around the wash bay with the hope of building a picture of croc activity on the village doorstep.
'We know what makes crocs tick – the first step is finding proof the big croc is still around and then we'll go from there,' Mick said to camera.
We finished setting up our gear and operations in the chief's compound then went to meet some of the locals who had experienced close calls with crocs. So far the local victims had all been fishermen, and there was one lucky escapee who had come face to face with a big croc a few months earlier while he was out fishing with his brother. Chris, Ben and I went to meet the victim and his brother in their mud grass hut.
'My brother fell and the crocodile bit his leg. It swallowed his leg up to here.' The brother pointed to his knee.
'Was the croc trying to pull you in to the water?' Chris asked the victim in French. Ben translated his response for me.
'Yes, the crocodile bit down on my leg and tried to pull me in. I held on to a rock and my brother grabbed a big stick and put it in the crocodile's mouth.'
'He was only this far from the water then the croc let go.' The brother gestured an inch in the air with his fingers pinched closely together.
We also wanted to know more about the croc because they would have got a good view of him out of the water.
'How big was it?' Chris asked.
'As big as the room!' The guy became very animated and threw his arms in the air. The room was about 16 feet wide.
'The size matches the croc that took Hendri,' Ben concluded.
We wrapped up work for the day and set up our tents in a cluster in the middle of the village. After just one night of camping out and eating around the fire I really got a sense of their lives. It is based solely around the basics: shelter, fire, food and water.
The next morning I had to take Ben and Chris out again with the camera crew in our shadow.
'If any of you end up in the water, get down as low as you can to the river floor and swim silently without making a noise or splash until you get to the bank,' I warned the guys.
Our boat had a motor but it was essentially a rubber dinghy, which a croc could puncture in a heartbeat, and our local driver was apprehensive.
'Chris and Ben, when we get up there, just make sure you hold the bamboo poles in your hands for protection,' I instructed.
We got on the island and before we could prepare the cameras we could see evidence in the mud: claw marks, a smaller print and a bigger print.
'It looks like about a seven- foot and a 12- foot croc,' Mick said. 'There are some smooth lines here, so I imagine the 12-footer has wiped out the little one.'
'Oi! Stay a couple of metres back from that water . . . more! Step back more, stand there!' I called to Chris and Ben. They were both extremely capable fellas, but you can't learn about crocs overnight so I was constantly on the lookout for them.
We switched out the sensor cameras as the rain started pelting down around us.
'We'll check this footage on the laptop. But the crocs are obviously hungry,' said Mick. 'He would definitely be in this area, it's just a matter of time,' he added.
'This section of the river always creeps me out,' Chris said.
'Yeah, it has a spooky feel to it, and this rain is adding some intensity. The good thing is that rains like this bring crocs to the surface because they enjoy the showering sensation, so hopefully we'll get the big fella,' I reasoned.
'All right, lads, come on, let's get out of here. We don't want to be hanging around here too long with this heavy rain,' Mick said.
We tracked back to the village in the boat. It was quiet and I could feel everyone thinking. We got off at the wash bay and I walked up to the bridge and looked down at the river.
It was peaceful. An old man paddled quietly along the bank, hunting for some food. I could hear the faint noise of his canoe's slipstream and watched the pretty little whirlpools coming off the backs of his handcrafted paddles. Every couple of minutes a small patch of bubbles rose to the surface in scattered spots across the river. The man looked around. He knew the crocs were there but he didn't seem worried. He looked up at me and smiled, and I smiled back. In the distance I could hear the village kids playing and laughing and I watched as the farmers on the other side of the river dug into the ground with their homemade garden tools – a rock attached to a hard stick, crafted into an axe-like shape for digging. The village was a harmonious community – the only problem was its unwelcome members lurking in the water.
The old man enjoyed the river, pulling up fish and paddling along when he was out of luck. He secured another catch and fist-pumped into the air. He was so proud of himself. He hauled it up, and as the fish exited the water so did a dark black shadow. Out launched a crocodile, taking the man's fish and snapping his line. The man didn't jump back or react, he just sat down and accepted the steal. He peered over the boat for a moment then put his thumbs up to me to show he was okay.
We got in and downloaded all of the footage onto my laptop. The first lot of action was two crocs, just like Mick had guessed, coming over to investigate the bait. The next slide was a big hippo approaching the island. It hopped up onto the bank, wandered around and sniffed the bait then turned sideways to the water, tilted its head back and opened its jaws right up.
'That looks like a territorial move,' said Chris. 'It wouldn't surprise me if it was facing off with the croc in the water, staring it down.'
Chris was right. The hippo moved out, and a couple of slides later one hell of a croc crept out from the water.
'Wow, that's the sort of croc we're looking for.' Mick was impressed. We sized it up to our markers on the ground and left it up to Mick to determine its measurements. 'There's no question that animal is a metre wide. That's really big, I reckon about 15 to 16 feet long. That's a good-sized crocodile.'
I clicked to the next slide.
'Oh shit, that's crazy!' Ben couldn't believe it. The very next shot, less than 10 minutes after the croc disappeared, was us pulling up in our boat.
'That's the scary thing, if we go back to that area there could be a croc loitering. Or if that hippo's hiding in the bushes, he'll come at us and those things can run 30 miles an hour, they're even worse than crocs,' Chris said. 'Out of all the footage from the three locations, the island has the most activity so let's focus on that.'
The rest of the team agreed and we hit the sack early, ready for the morning.
I spoke to camera the next day on the way back to the island, as it was important for me to explain to the general public why I wanted to catch this croc.
'My work is about more than just helping the village; it's about saving animals, relocating and protecting them. Not destroying them or their environment. If these attacks keep continuing, the croc will be shot, which is not what I want. If we can take him out and move him to a nearby park in South Africa, where the owners have said they would love to have him, then to me that's a job well done.'
Mick and I hooked in getting the trap sorted, and the boys came and lent a hand a couple of hours later. I created a box with fence panels and bamboo. It was a simple enclosure that mimicked the same design as the traps I made in Australia – three walls and a gate tied together. I put bait on the end of a piece of rope, which would then trigger the gate to drop down when the rope was pulled. I baited the rope with more goat meat and dug a narrow channel so that the crocs could get in and the hippos stayed out.
I overheard Chris speaking to camera, sceptical about the trap: 'I mean, it seems like it'll work. We'll wait and see, I guess.'
I yelled back to him, 'Oh, it'll work, mate!'
But then big rains came in and so we had to steam back to camp faster than usual.
We were heading along the river when I heard Mick call out, 'Stop!' I turned and saw our cameraman falling overboard, along with his camera. I was driving and it was hard to see in the heavy rain, but I spun the boat around and made him out in the water. I could see his camera bobbing along, slowly sinking, and him swimming behind trying to get a hold of it.
'Mate, forget the fucking camera get in the boat!' I yelled. 'Leave it, get in the boat now!'
He was egg-beating in the water to keep his head up, basically inviting a crocodile to eat him.
'You'll be gone in a minute!' Mick shouted out and flung his hand over the edge of the boat. The cameraman hesitated and looked back at his device, then paddled over and Mick pulled him in. He was in shock, we all were.
'That's a $100,000 camera,' he said to me.
'Oh well, we've got another one at camp. You're lucky to be alive – who cares about the camera.'
No one spoke for the rest of the ride back to camp.
'Crocodile or no crocodile? Crocodile or no crocodile?' I asked the boys as we made our way back to the trap the next morning.
'Crocodile,' Ben said.
Everyone seemed optimistic, but I was uncertain.
To minimise our sounds and presence in the river we went upstream on the bank by foot until we got to a vista point in a tree. I climbed halfway up.
'I'll see if the trap has been activated. Chuck us the binoculars, Chris; it looks like the door has gone down.' I was optimistic for a moment. 'We're going to have to go and check it out.'
We took our boat from the bridge to the island and me and a few others walked cautiously towards the trap door.
'Wow, wow, just go steady, mate! Bait's gone and no croc,' Mick yelled out to me from the boat.
'Yeah, I can see that's a problem.' I climbed over and checked out the tracks. There was a large claw print and a biggish slide mark.
'It's a decent-sized croc, so how the hell did it get out?' Ben asked.
'The door is down and the bait is gone and . . . where's your knife, mate?' Mick asked me. I'd used my knife in part of the trigger mechanism I'd set in the trap, but now it was nowhere to be seen.
'Shit, the croc nicked my best bowie knife!' I exclaimed. 'Very cheeky. It looks like he's managed to wedge himself out through the bamboo panel on the bottom here.'
The panel closest to the ground was bent upwards. The trap clearly needed fortification but options were pretty limited in the Congo.
We spent the next few days sourcing new bits and pieces for it, but the best we could do was use sheets of chicken wire and hessian. We patched up the trap and reset it with new bait. Every day for the next two weeks was like groundhog day – head out in the morning, check the traps and swap out the cameras, then go back to the village for the afternoon. There were no crocs to be seen, they'd become uninterested. When it's wet, crocs often move out onto high ground trying to find food, so they're not in the main streams. I was frustrated because I knew if we had come in dry season we would have caught the croc in the first couple of days, but the heavy rain and water was making it almost impossible.
We brought a heap of soccer balls with us, so we spent our downtime in the afternoon playing with the kids and sharing stories with the locals around the campfire at night. Nights in the Congo were beautiful. No commercial lights, no interference – everything was pitch black with millions of stars that made the sky glow.
We had three days left in the Congo and had no croc to show for our month of hard work. Mick and I were spending our time scoping out the banks of the river right next to the village, working out the best place to build a caged wash bay and swimming area for the locals. It wouldn't last forever, but it was better than nothing.
I was walking along the bank with Mick a little way behind, when I heard rustling in the grass. I pushed through to a small clearing and saw a croc tangled up in rope, death-rolling around and causing a commotion.
'Oi, Mick!' I shouted. 'Come here quick and give me a hand to catch this croc while it's trapped.'
'Roger!' Mick called back.
The croc had obviously swallowed the bait from our trap with the rope still attached to it, and then got caught up in a tree on the bank. Just as Mick pushed through the clearing, the croc broke the rope around the tree and took off back into the water.
'Damn! He's gone. He was big but I don't think it was our fella!' I said.
'Hey, Matty, check this out,' Mick said from behind me. He leant down and picked something up. 'It's your knife, mate. The croc didn't fancy eating it.'
I couldn't believe it. We were kilometres from the island and a crocodile had returned my knife to me! That was the last thing I expected.
'Clever bloody crocs around here,' Mick remarked, and I had a good laugh. I was stoked.
Our last day in the Congo was full of mixed emotions. I felt like I'd failed Chris and Ben and let down the locals. I knew if the weather had been dry it would have been a different story, but I couldn't help but beat myself up.
All of us went on the boat to check the trap on the last day, and Pete, one of Hendri's friends, along with a local scientist, came with us. We were planning to take down the trap but, unexpectedly, we were greeted with action: two crocs throwing their weight around.
'There's a croc, oh shit! There's two! Game on, guys, here we go!' I launched out of the boat and jumped straight onto the trap's rails. There was a whole lot of yelling and commotion to follow.
'Mick, give me two ropes!' The crocs were thrashing around big time.
Ben passed me some rope.
'No, not that shit! Small rope, small rope!'
'Is there any other rope?' Mick asked the boys in the boat.
Before Mick could say anything else I yelled over everyone, 'Yes, there's other rope! Come on, let's get it!'
This was no time for messing around. I was worried the trap wouldn't hold the crocs for much longer.
'I want to secure the bigger croc first, then I'll go in and get the other one.' I climbed in on the inside of the cage and positioned myself to get a head rope over the croc.
'Watch yourself, man,' I heard Ben say as I jumped off the railing into the muddy, slushy water of the trap, pulling the head rope tight.
I handed the rope through to the guys. 'Secure this croc, give the rope some slack, and now pull!' I told them.
'I got him, I got him!' Chris and Ben held tightly on the other side of the railing. One down, one to go.
But the next croc was not giving in. I was moving around, making sure the tail of the first croc didn't whip me and the mouth of the second croc didn't nip me. Tension was high. I got into position and looped the head rope into place on my first go. The croc was secured.
'Ben and Chris, do you boys want to jump in with me?' I asked.
I figured it would help them face their fear. They looked anxious but they didn't hesitate and got in. I guided them through what to do. They worked the ropes and helped me put hessian over the crocs' eyes and tape up their mouths.
'In comparison to the head you guys saw on the croc that took Hendri, what are these guys like?' I asked.
'Yeah, I mean, they're definitely huge animals but it's not the one. The head isn't as massive,' Chris said.
I had a feeling it wasn't – these were a bit over 12 foot and from all accounts and the footage we saw on the first day, the big fella we were after was at least 16 foot.
'Okay, well, we should let these ones go then,' I said.
'But I'm concerned about this one,' Ben said, gesturing to the larger one. 'There's all those kids playing in the wash bay up stream. Can't we relocate it? This one, at least?' He didn't want me to let them go.
Mick cut through with reason, like he always does. 'Look, we came here with a particular goal, to catch a killer crocodile – we haven't seen a killer croc in person, we haven't caught a killer croc, so we're not just going to relocate one of these crocs and say close enough is good enough.'
Mick was right. The reality was that these crocs hadn't eaten people and there were hundreds more like these guys, so it wouldn't make a difference even if we did relocate them to a wildlife park. I called over the scientist who had come with us. I explained that I was going to mark the crocodile and asked him to record the markings for future tracking and reference.
Metal clip-on tags, like cattle tags, can be punched through a croc's tail scute (bony plate) but most of the time they get ripped out, so I tend not to use them. My preference is cutting the crocodile's scutes. If scutes are cut off at the base with a very sharp blade, then the marks are permanent and the scute heals without trouble. The scute measuring and identification system is a bit confusing but once you do it a few times it's simple. The point between the double row and single row of tail scutes is considered to be zero. Some people never cut these three scutes, so your starting point is always present. Counting down the tail from this point, the single scutes represent hundreds. The double scutes represent tens on the croc's left side and single numbers on the right. So you can create a unique number for any crocodile. (The last few single scutes should not be cut off, because crocs often lose their tail tips in territorial battles.) Using this numbering system, I cut off three scutes (100, 30 and 7) giving the croc the ID number of 137. The scientist wrote this ID number down along with a description of the croc. He was stoked; it was a good system for him to learn, especially because he worked a lot with crocodiles in the Congo.
'Okay, fellas, that's it, I guess,' I told everyone. 'Let's pack up this trap and use the gear to reinforce the wash bay barrier some more.'
It was the first time in my life I hadn't caught the croc I was after. I was so unsatisfied.
We said our goodbyes to the locals the very next morning and made our way home to Australia. I got straight back into egg collecting and thought about that bloody croc every night. It drove me nuts. I tried to get Nat Geo to fund Mick and me to go back in the dry season, but the ship had sailed. Chris and Ben returned to the village six months later and helped build and open the big new well in the town. I was glad to hear that we had made some sort of a difference in the community.
In November 2012, I got a call from our fixer telling me the croc had been caught and killed. I was devastated. Apparently it was spotted in the river while the military were around. They stood up on the bridge and opened fire on the water, then two days later the croc rose to the surface. It measured 16.1 feet and the locals feasted on it for weeks. Its fate made me feel sick. I know it was a man-eater, but it didn't deserve a gruesome death. It should have gone to a park, keeping both people and the crocodile safe. Still, I want to go back to the Congo again one day and see if life has improved in the village since the big fella got caught.
#
From cocaine hippos in Bogotá to king cobras in Borneo, I had a pocket full of cash and was ready to make my mark on the world with Outback Wrangler. Pulling together season one of the TV show was pretty nerve-racking. It was my first time working on a big production and there were so many unknowns. The plan was to head to Bogotá, Colombia and Borneo to film two international shows and do two within Australia.
A fair swag of an entire floor of our production company's office in Sydney was taken over by people solely responsible for the show's preproduction. We had researchers, accountants, lawyers and assistants for assistants. I was a little frustrated because I'd recently requested a Cineflex high-definition camera system, which was fundamental to filming, but my request was shut down due to budget cuts. I can't throw stones because I've overemployed people in my own businesses before, but my hopes and dreams were tied up in the success of the first season and I wasn't going to let it be a flop. Although without the company there wouldn't have been a show, and what was great about the big team were the film and TV uni students. They did a lot of the groundwork for the main producers, tracking down the details for all of the contacts on the ground.
In particular, they connected me with a guy named Carlos, a wildlife ranger based in Bogotá. He had good English and knew his stuff, and I spoke to him every couple of days in the lead-up to the trip. Carlos and the other local rangers needed my help with resolving a major human–animal conflict.
Colombian drug lord Pablo Escobar built a private zoo back in the early 1980s on his ranch Hacienda Nápoles, about halfway between Medellín and Bogotá. Most of the exotic animals housed there were transferred to other zoos around the world, but not the hippos. Nobody knew exactly how many there were but the local authorities estimated around 50 to 60. We were going to use confiscated drug money, which the government held in a community beneficiary, to fund an operation to catch, rehome and de-sex about 30 of the rogue hippos. The hippos were territorial and aggressive and had been left to roam around the surrounding communities as they pleased, so there was a high risk to the safety of the people living in the villages. There had been a couple of attempts by other people to fix the problem but no real progress had been made. We wanted to move the hippos into a large enclosure so that they were still protected, but away from people.
Unfortunately, we ran into a curveball two weeks before filming. The GFC hit hard and the US dollar plummeted. The only option was to cut a show. Bogotá was the most expensive and logistically challenging, so it was a no-brainer for the production company: the episode was scrapped. I was pissed off, but I knew there was nothing we could do about it, and I didn't have a spare half a mil lying around.
So all of our international efforts turned to Borneo, where there were saltwater crocodiles and lots of other animals in conflict with humans, due to industry and development. Similar to the Northern Territory, the crocs in Borneo had been protected since the 1980s, which had caused human and animal conflict to increase, and the local rangers were looking for fair and peaceful solutions. My main focus was to train the wildlife rangers in catching and relocating saltwater crocodiles.
I was also going to provide support for the work with the local orangutans, elephants and snakes, all of which were being impacted by palm oil plantations. Malaysia, along with Indonesia, produces more than 80 per cent of the world's supply of palm oil. People call it the golden crop because it has lifted poor farmers out of poverty, but at a devastating cost to the environment. Over 90 per cent of orangutan habitat has been destroyed in the last 20 years, in what the United Nations calls a 'conservation emergency'. The more I learnt about it the sooner I wanted to get over to Borneo to film the show and spread the word about the horrific demise of the environment and its animals.
I was excited to check it all out, but I wasn't excited about what seemed like a cast of thousands that were heading over with me. I had just been in Australia filming two of the other Outback Wrangler episodes alongside the large team in Sydney, and the ground team in Borneo was just as big. There were risk-assessment people, medics, assistants, a producer, a researcher, a sound guy, a cameraman and a second cameraman. I knew all we needed was a soundy, producer and my two cameramen, that's it, but I went along with it nonetheless.
We landed in Kota Kinabalu – or KK, as it's commonly called – the state capital of Sabah, Malaysia. A convoy of 1980s Toyota Camry taxis with patchy, faded paint jobs pulled up to the rank and we jumped in. My car was filled with fluffy pink dice, eccentric bobbing figurines and other trinkets blu-tacked to the dashboard. The cab driver was friendly, a reflection of everyone else I met in Malaysia. We went straight to our hotel, passing through streets clogged with traffic, and bustling markets and people buzzing around everywhere. The city was alive. But as we pulled up to our accommodation I became enraged: it was a five-star hotel! We were in Malaysia to shoot a remote grassroots doco on a limited budget, and the production company had booked what I was sure was the most expensive hotel in town.
'Jesus Christ,' I said, as we got out of the cab.
'I know, it's just incredible, isn't it?!' the producer replied eagerly.
'Incredibly expensive,' I said back, sarcastically.
I dumped my bags in the hotel room and met up with Simon, our British fixer, on the ground, and we all headed to Lok Kawi Wildlife Park, a rescue park run by the Sabah Wildlife Rangers.
The Sabah Wildlife rangers' team leaders, JB and Dr Zen, were a lot savvier than I'd anticipated but their team was still very raw. JB told me about the poaching and loss of habitat, which had led to the pygmy elephant becoming gravely endangered; another sad example of humans and animals in conflict.
They asked me: 'Mr Matt, do you want to meet our latest rescue animal? She's just a baby. Her mum was shot by palm oil farmers.'
I was honoured and happily accepted the offer. One of the most memorable moments in my life was the time I spent with this cute cuddly girl. She was only 12 months old and weighed in at about 300 kilos. She was friendly and playful and just wanted affection. Her eyes were so innocent and kind and she deserved to be with her mother, but her mother had been killed.
Elephants know no better; they are so used to walking through the jungle looking for a feed, and now palm oil plants are the only option. And when they eat the hearts of the plants, the farmers just aim and shoot. The local wildlife team was working on creating rainforest corridors from one jungle to the next, laid out in between the palm oil plantations so the elephants can access the environments they need to survive. It's far from ideal but it's something, because the palm oil industry isn't slowing down, no matter how much backlash it receives. There is a lot of money tied up with people getting richer, and that greed is overcoming the love and need to protect our most valuable commodity – the environment.
I left the park with a heavy heart and travelled to a nearby croc enclosure I'd heard about just for a stickybeak, as the Malaysian saltwater crocodile or buaya air masin is identical to our crocs in Oz. Regrettably, the park stank like shit and the crocs were living in crappy conditions, so I made sure not to send any crocs I caught back to that facility.
After getting our feet on the ground in KK, it was time to pack up our gear and hit the road. On the way back to our stupidly lavish hotel I asked the cab driver to stop at the local market. I'd read a sign at the front of our hotel saying DURIAN STRICTLY BANNED BEYOND THIS POINT! I thought the durian was some sort of weapon, but when I asked JB at the wildlife park he said, 'Very, very stinky fruit like farts. Very popular but very bad.' I was intrigued and wanted to see if JB was right.
For those of you who haven't tried one: don't. The smell was so pungent that I gagged before taking a bite, but I blocked my nose and hooked into it – the mushy, slimy fruit squirmed around my mouth. I tried to swallow it but it was too much, I spat it out and the local ladies lost it laughing behind me. My orange-juice chaser didn't even disguise the taste.
I was happy to never see or smell the fruit ever again. After I got home to Australia, I remember reading food writer Richard Sterling's take on the fruit: 'Its odour is best described as . . . turpentine and onions, garnished with a gym sock.' I couldn't have said it better myself.
JB and the wildlife rescue team convoyed a couple of large trucks to Sepilok, a town about eight hours' drive from KK, while the crew and I flew there and met them at the other end. We landed at Sandakan Airport and drove the short distance to Sepilok, where we packed the truck with steel and equipment.
JB took me to meet his wife, who ran Sepilok Orangutan Rehabilitation Centre where around 80 orphaned orangutans live freely in the 43-square-kilometre reserve. Orangutan mothers are killed so poachers can take the babies to sell them illegally. Orangutans are also killed if they are found on the palm-oil plantations. I sat around the table with JB and his wife and talked about the orangutan program and the work we were about to do with the local animals. As we chatted, the orangutans swung around in the jungle behind us. It was beautiful to watch.
I held one fragile little female who hugged me like I was her dad.
'What's her story?' I asked. I wasn't expecting to hear what I did.
'We pulled her out of a brothel last week. She was used illegally as a play toy for entertainment and sex. This is very common,' JB said.
I was outraged. How could someone be so sick?
'At the end of our trip I'd like to come through here and do some filming with you all, so I can take your work and stories back to Australia to help raise money for you, if that's okay?' I asked.
JB's wife was stoked, but I felt it was the least I could do.
We were on the road again, headed for our main destination, Kinabatangan. All the shanties along the road caught my attention. I saw mothers feeding their newborns on the dirt ground outside their homes made of tin and salvaged wood planks, while kids were playing on dumpsites nearby. We'd just come from an opulent hotel and now we were amid poverty – Malaysia was a place of both hope and despair. We came up onto a ridgeline and looked out. Standing tall in the near distance was a bare hill. On one side of it were palm-oil plantations and on the other side was cleared rainforest. There was a solitary tree left on the top of the hill, and sitting beneath it was one lonesome orangutan. The scene spoke volumes about the environmental crisis sweeping not just Malaysia but the world. Everyone was saddened by the sight.
'I need a goat, pig or cow – anything, really,' I said to JB later that day. 'Just some sort of meat that I can use to bait all of our traps. I'd like to get an animal that's good for us to eat too.'
The rangers were excited because most of their meals were rice and veggies and, if they were lucky, sometimes fish, but meat was a luxury.
'Oh, very good, very good, of course, Mr Matt,' JB replied. He took us to a local market along the way where I bought a pig.
'That's 1200 ringgit, sir,' the young man said to me.
'Holy hell, I can see why you mob don't eat meat around here!' I was so used to catching as many wild pigs as I wanted back home for nothing, I converted the currency in my head and it was about 600 Australian dollars, which made a dent in my wallet, that's for sure.
I cut up the pig there and then, bagging it up and giving the ribs to the rangers, their grins were quite literally from ear to ear.
We left our cars next to a jetty and were picked up by boat and taken to our resort, Proboscis Lodge Bukit Melapi, located right on the Kinabatangan River. The river is the second longest in Malaysia, measuring about 560 kilometres long. It flows through Borneo's forests, rising from the mountains of southwest Sabah and running into the Sulu Sea. The river is home to ten species of primate as well as crocodiles, elephants, snakes, leopards and sun bears. I'm told there are also over 200 bird species. The forests along the river have been subject to widespread deforestation and poaching. JB and his team established the Kinabatangan Wildlife Sanctuary in 2005 as a response to the culling of forests and animals in the area.
The film crew and I checked in to the Balinese-style resort, then I grabbed a pillow and sheet from my room and headed straight back out the door.
'Where are you going, Matty?' Ash, my good mate and cameraman, asked, when he saw me heading off.
'I'm not staying here with this circus,' I told him. 'I want to earn the trust and respect of the rangers, so I'll go camp up the river where they are.'
'Fair enough, mate,' Ash replied.
I couldn't stay in a flash hotel and only be with the rangers when the cameras were rolling – what a joke. I taxied along the river to the rangers' accommodation. The river was long and wide and very muddy. All of the logging that was happening upstream meant there was nothing protecting the soil, and the big rains had caused erosion.
The wildlife team were surprised to see me. I found myself a spare room to put my gear in and offered to help the lads fix a leak in the toilet. I was almost done when the hose burst, squirting water all over me. Everyone paused in nervous silence to see my reaction. I burst out laughing and I could see the relief spread across their faces; they were trying to work out what sort of bloke I was.
The local village, located about 10 minutes away, was the closest street market to us, so JB and I went for a gander to get some extra grub to go with our rib dinner. I'd been to the amazing street markets in Bangkok before, but these markets were a lot less regulated. I came across Larb Leuat Neua – raw beef with uncooked blood, Mok Huak – developing tadpoles, and Takatan – grasshoppers. And as for refrigeration, well, there wasn't any. The flies fancied the fish and raw meat section, which I tactfully directed JB away from.
'JB, how about we just get some rice, veggies and lots of chilli for tonight,' I suggested. 'I love chilli.'
Thankfully, he agreed. 'Okay, Mr Matt.'
I figured I'd drown all my feeds with chilli to flush my system and avoid any food poisoning. My plan worked out well: even though I had bad gas the whole trip, I was the only person out of the entire crew who didn't fall victim to any food-related illnesses, even when the rest of the guys stayed at the fancy hotel.
A cold shower freshened me up after the market and I was outside talking through the next day's plan with JB when I started sneezing profusely and my eyes began to water.
'It's dinnertime now, the girls are cooking,' JB added with a big smile.
I was sitting downwind from the open communal kitchen and my senses were overtaken by the strong and powerful smells of chilli and other spices. The aroma was livening up the air and I became hungry all of a sudden. When the food came we all ate and chatted, and the warmth of the local people soaked through me. I felt at home.
One of the main things I wanted to do for the rangers while I was in Borneo was build them crocodile traps similar to the ones we used in the Territory. Our traps are light, collapsible and easy to transport. The ranger team were doing all of their work with these ginormous heavy steel traps, which took half a day to set up with trucks, machinery and a lot of manpower. The traps weren't effective or efficient, but I wanted to check them out first-up.
The rangers and I hit the road at first light. We met up with Ash and the soundy, and drove for one-and-a-half hours to the local palm-oil plantation to set up four crocodile traps along the river. One of the nearby landowners came over to JB and me and pleaded with us to find a large lizard that was eating his wife's kittens.
'Please catch the lizard. It's seven feet long and has eaten three of our kittens and we only have a couple of them left.' His wife and kids came out of the house and pleaded with me too, tears rolling down their cheeks. I was a croc-catcher not a kitten-saver, but I promised them I would do what I could.
As we walked away I said to JB, 'I will track down that lizard for them.' He brushed it off and laughed. 'Good luck! We have tried everything, but it always gets away.'
We had a backhoe loader and a large crane helping us get the traps into position. Back home it takes 15 minutes to set up a trap, but this took just over an hour for each one. It was a huge operation and there was no helicopter to help out, either. It was mentally draining more than physically exhausting because it dragged on and on. Finally we successfully set the traps and booted home.
Our little film crew were in the lead car and the rangers were following in file. I turned a bend and slammed on the brakes. There, lying across the road, was the biggest wild snake I've ever seen, a Burmese reticulated python. I was ecstatic. I had heard so much about the snakes of Borneo and was dying to work with one.
The Borneo rainforest is one of the oldest on Earth – it's been around for over 140 million years, so the animals have had plenty of time to develop. During the last ice age, land bridges linked Borneo to the mainland so species were able to migrate over to Borneo, leaving the island with an astounding array of animals. The snakes are particularly diverse. There are said to be about 150 species on the island and a lot of them are man-eaters.
I jumped out of the car to get a closer look at the python. It was about 16 feet long, with the girth of an AFL footy. It was a striking army-camo colour with a rainbow reflection across its back. It must have just had a feed, I thought to myself, because it wasn't moving. I stepped in to grab it by the tail and it bucked around with its mouth open.
'Wow! It's a bit spirited,' I called out.
Ash and the soundy were getting in position to film me. But after a few more attempts and a few more retaliations, I wasn't having any luck.
'"Not going to move" my arse. This thing is aggressive!' I called out again. I grabbed the soundy's boom stick and managed to pin the snake's head to the ground and picked it up.
'You need a hand?' JB and the boys had arrived and were out of their car, sounding very concerned for my safety. The snake had managed to wrap itself around my body like fairy lights around a Christmas tree, squeezing my chest and legs. I was holding its head, with its row of razor-sharp needle-like teeth moving about, wanting to lock on. Its head was the size of my hand. JB and the female vet nurse jumped in and helped me to unravel the snake while I stood there with a shit-heap grin on my face.
'Now that was cool,' I said.
'You're crazy, Mr Matt!' JB said, laughing.
'Hold on to the tail there and help me get this guy into a bag. I can move him somewhere else and away from the road tomorrow,' I said.
I had already seen monitor lizards and snakes as road kill, but they didn't last long around here. The locals were quick to collect any kill for their tucker.
Day three was trap-building day. Luckily, I had brought some gear from home in my duffel bag – battery-operated drills, grinders, a tool kit and other bits and bobs. I was welding away in the morning sun. I didn't have a mask, but I was making do and getting the job done nonetheless. I wanted to build four collapsible traps to leave with the team.
'Matt, Matt, Matt!' JB's call interrupted me as he came running into the workshop area. 'King cobra, big deadly king cobra near our camp, can you help?!'
Two snakes in less than 24 hours? I couldn't believe my luck.
'Awesome. Ask one of the lads to run up and get Ash and the crew to film.'
Pumping with adrenaline, I followed JB to the car and we made our way to the nearby picnic park, a popular place for tourists and a less-than-ideal place for a venomous snake. We waited for the crew to catch up, then ventured into the little patch of jungle.
I have handled a lot of snakes in my time, and to me a snake is a snake, so I didn't think much different of a king cobra. I knew its toxin was deadly, but I still thought I'd manage just fine.
Boy, was I wrong. The snake was 11 feet long and it took me all of five seconds to work out it had one hell of a striking distance, as I moved straight in on it and it launched at me from five feet away.
'Ah, shit!' I exclaimed. 'This is going to make my life a little trickier.'
I was nervous and excited at the same time. It was a spur-of-the-moment catch, so there was no evacuation plan or antivenom available. If I was bitten, it would be an instant death.
After running around the jungle for a while, quite literally playing chasey with the snake, I was tiring out, and thinking, Matt, what the hell have you got yourself into here?
The wildlife team were standing back, taking it all in. They usually used baited traps and carefully planned a capture, so they were a little shell-shocked to see this foreign white man diving in around a king cobra.
After about half an hour of toing and froing, I realised the situation wasn't getting any better and was still thinking through the best way of getting this snake into a bag. I was in the middle of a face-off with a highly aggressive snake, trying to compose my thoughts, sweat pouring down my brow and panting like no one's business.
All of a sudden, I heard someone behind me. 'Stop, stop, stop! Stop this, Matt, and stop filming, Ash! We're not doing this as part of the show.' It was our producer, sounding like a nagging old teacher.
I paused for a second, focusing in on the snake and, without losing eye contact with it, yelled back: 'Listen here! YOU need to be quiet and go back to the car! It is not your place to advise me of what I can and can't do on my own fucking show and now is NOT the time!'
'But we don't have antivenom! This isn't in my risk assessment policy. I'll get sued if anything happens,' she shouted at me.
'Well, why don't you take your shitty risk assessment all the way to the car and back to Australia, because if you hang around here I WILL get killed. Now piss off!'
It was dead silent. No one said a word, and I saw her storm off to the car in my periphery. Thank God for that, I thought. I was furious – like hell was I going to stop filming. I pride myself on the content of Outback Wrangler being authentic and unscripted – that's what makes the show so unique, and my battle with the snake was as raw and real as it gets.
'JB, you got any tongs in your car?' I shouted. 'This bamboo stick isn't cutting it.'
'One moment!' I heard someone run off and return a couple of minutes later. I was expecting proper snake tongs at least a couple of metres long, but JB passed over what looked like a standard set of barbecue tongs.
'Mate, you hold onto those. I want you to grab the snake by its tail and I'll get him with my hands behind its neck. One, two, three, go!'
We launched in. The timing was perfect but shit, did the snake go ballistic, wrapping itself around my hand, trying to work its fang into my thumb. The trick with snakes is to not hesitate and be as precise as possible, handling them with the right amount of pressure. You need to hold the head firm, otherwise those two big fangs will connect with your skin.
The other wildlife boys brought over a bag and I dropped it in, making sure I kept my hands well clear, and tied the bag up as quickly as possible.
Phew, what a relief. I'd got the job done.
Well, almost . . .
'I am wondering, Mr Matt, only if you don't mind, can we take this snake to the clearing and let it out of the bag again, and you can take us through step by step what you did so we can learn this method without big tongs and traps?' JB asked.
I was completely wrecked and all I wanted to do was clock off for the day but he was so hopeful, and I wanted to help him and his team out.
'Yeah, I'd love to, mate,' I said.
Off we went and out came the snake again. I talked them through how to minimise stress for the animal, how to tell when a snake is about to strike, and how to hold onto the tail without it swinging around. The more we worked with the snake the quieter it got, and the rangers learnt that once a snake realises that you don't want to hurt it, it calms down completely.
By the end of the afternoon, I could hold the snake just under the belly without even having to grip its head. The rangers were in awe. I took the two snakes back to my room in separate bags and kept them next to my bed ready to release the next day, up the river, away from human activity.
I spent every spare minute I had in between filming making harpoons and traps. I'd burnt out the rangers in the first few days – I had them cutting steel, welding and grinding in the blistering heat. On the fourth day, the team had the day off, so I hopped in the boat and went to the hotel to see if any of the TV crew was free to lend a hand. Walking through the lush gardens of the resort I could hear bongo drums playing. I walked up the stairs to the entertaining area, expecting to see a traditional ceremony taking place. My jaw dropped.
'Oh, get off it,' I said under my breath. There, in the main compound next to the pool, was the assistant producer, playing bongo drums and surrounded by most of the crew, who were throwing back cocktails at 10 o'clock in the morning. The drum playing stopped as I approached, and everyone turned to me as if they had all seen a ghost. Dead silence.
'Oh, hey, Matt,' our researcher said, in a guilt-drenched voice.
'Where are the others?' I barked.
'Err . . . um, I think Ash is editing footage in his room and . . . the others might be sleeping?' the researcher said.
'You're kidding me! Jesus Christ, you're all useless. I'm working my arse off out here trying to make this show work and you guys are back here chewing up money, sinking piss, doing sweet FA! You're a pack of lazy leeches. I'm calling the production company and sending you home. There are too many hangers-on here and I'm sick of it!'
I feel shitty when I yell, and I keep my cool most of the time but I really was at wit's end. I stormed over to Ash's room where I found him downloading footage, and together we called the production company and got the go-ahead to send a bit of the deadweight home. A few of them packed their bags and left the very next morning. They hated my guts and I thought theirs stank pretty bad too.
Dr Zen appeared in the workshop early the next working.
'Sorry to interrupt, Mr Matt, but our spotters have confirmed a croc in one of our large traps in the palm-oil plantation.'
Dr Zen had already been sent the footage caught on the motion-sensor cameras, so we had a quick look to see what we were up against. For three days, the only action was from a big water monitor, the same lizard responsible for nicking the kittens. It would swim into the trap, set it off by eating the bait, and swim away through one of the gaps. These traps weren't like the cage traps we used back home, they had gaps big enough for the little sucker to get out. We all laughed as we watched this cheeky mongrel have a field day until we eventually saw our croc move in and get caught.
'Bingo!' I cried. 'That's our guy! Let's go, fellas!'
JB and the rangers usually relocated the croc in the same trap that the animal got caught in. Logistically, it's a nightmare, and doing it like that isn't good for the croc. It normally ends up with skin lacerations from being bumped around. So I was there to show the rangers how to safely remove it from the trap, tie it up and move it out in the open.
The team were keen to learn, and listened intently to what I had to say. 'First thing's first is getting the rope over the croc's snout while it's in the cage. Then get him up and away from the water. Pressure on the rope is a must. If he tries to run, hold the pressure and we stay together. No one gets in its way, okay?'
'Gotcha, Matt,' JB said, and a resounding chorus of okay followed from the rest of the 12 rangers in the team. We were good to go.
Busting out of the trap, the 13-foot croc was as feisty as anything, putting up a colossal fight.
'Wait, wait, don't pull, let him roll! Let him tire himself out,' I instructed, guiding the team through the fundamentals of croc-handling.
The rangers were sporadically yelling out things in English, which added to the intensity of the situation. The croc growled, snorted and death-rolled up the bank, so I went in and jumped on it, securing it with the usuals – hessian, gaffer tape and rope.
'I can almost guarantee this is a male, but let's see if we can find the penis, fellas.' Everyone looked at me, mystified.
Generally speaking, males have thicker tail bases and are stronger in build. And this one looked a little too big to be a female, as girls never get bigger than 12 or so feet. But, unless you know a croc's age, it's virtually impossible to determine their sex without inspecting their bits.
Crocs have internal sex organs, which are housed inside a small slit below its belly, towards the tail. You have to insert your finger into the slit and feel for the hard internal penis to work out if it's a male. It's a lot more straightforward when the crocs are mating because the male's penis extends out of the slit, making it visible.
I picked up the tail of the croc, found the slit, inserted my finger and confirmed it was a male. JB did the same so he could understand firsthand what I was talking about. Showing this to the guys was one of the most valuable, if not the most valuable, lessons I taught them, because knowing the sex of crocodiles is critical when you're trying to manage population numbers.
Locals with motorbike helmets on, barefoot teenagers and even a couple of police officers came scattering down the banks to be involved. I felt like a giant amongst the five-foot-tall locals who were all gathering around, staking out a space alongside the croc's body to be a part of the action.
We had close to 50 fascinated onlookers now, and when it came time to move the croc, everyone wanted to lend a hand. Usually you need three to four blokes to move a 13-foot croc, but we'd ended up with at least 20 local volunteers carting out the 400-kilo animal. I kept hold of the croc's head, so if he shook it didn't knock any of them out.
We loaded the croc on the back of the truck and I sent my film crew on ahead, instructing them to grab the snakes from my room. They were to meet the rangers and me at the boat ramp near the tourist information centre where we'd launch off and take the animals to their final spot for release.
The day's comedy show kept coming. The rangers and I arrived, unloaded the croc and intended to carry it down the path to the boat ramp. What I didn't realise was that the path cut straight through the front doors of the information centre and out the other side. I couldn't believe it. This relocation was meant to be incognito and there we were carrying a croc bigger than two men put together right past picnic benches where people were eating lunch and taking holiday photos. This is like a Fawlty Towers episode, I thought to myself.
We pushed open the doors and were greeted by a small group of female trekkers who fell to the floor like they were taking part in a 'stop, drop and roll' fire drill. And fair enough, six blokes casually carrying a croc through the information centre was probably the last thing they expected to see. We just smiled and kept walking. (It made for a good story, though. The incident ended up on the front page of every newspaper around Malaysia with the headline LOCAL HERO JB AND HIS WILDLIFE TEAM RELOCATE BIG CROC FROM POPULAR SWIMMING HOLE with a picture of us trudging through the tourist centre.)
When you're filming, everything takes twice as long, with interviews and retakes. By the time we met up with the crew again, loaded the animals onto the boat and scoffed down some lunch, it was 2 p.m. and we'd only just set off along the river.
'We are going to move this croc up to the research station, where university students come to study Borneo wildlife,' JB told us.
'How long will that take?' Ash asked.
'About two hours,' JB and the vet nurse replied in unison.
The boat was comfortable enough for the journey, with a large centre console and a fair bit of room for the rangers and film crew. As we poked along the river, we could see the trees moving around us and the long-nosed proboscis monkeys dancing through the branches. Their honking calls echoed around us, and every now and then an orangutan showed itself in a clearing.
We stopped off after an hour and released the cranky cobra and reticulated python safely into an expansive area of jungle where they could roam around without being harmed or causing harm. Then up the Kinabatangan River we continued. We had to be constantly on the lookout for the logs that were all through the river from the deforestation. It was okay going while it was light because we had visibility, but we'd been on the river for at least six or seven hours when the sun began to set. It certainly wasn't a two-hour trip as promised. Every 30 minutes or so JB would say 'Not far now, not far now,' which lost its meaning after the third or fourth time.
The moment it got dark the driver dropped our speed significantly, and we putted along the river at a painful snail's pace. The croc got restless a couple of times, swiping me with his head and knocking our soundy to the ground with his tail. After what seemed like an eternity, we reached the research centre, where we discovered the real reason for our visit.
It wasn't just to relocate the crocodile. A group of university students studying zoology, crocodiles in particular, were staying at the centre. None of the students had ever caught, let alone seen, a crocodile, and they were studying them! I ended up having to do a lesson at 9 o'clock at night on crocodiles, before I showed them how to handle and release the animal correctly. It was now 10 p.m. and we'd been going since 6.30 a.m. I was wet with muddied water, my muscles ached and I could feel the skin coming off my prune feet. I was tired beyond belief.
Thick dew and fog set in as we headed back to the tourist centre, making it colder and even more unpleasant. The thought of the seven-hour journey ahead was tormenting me. I ended up on the deck covering myself with a stinking wet rag that I'd used to cover the croc's eyes. It's amazing – even when you're wet – that if your body warmth is there you can actually stay relatively snug, so I was trying to stay as warm as possible.
I dozed off for a while and woke up hoping to see the tourist centre, but . . . nothing. I looked over the side of the boat and then turned to Ash, who looked like he'd just woken up too.
'Mate, are we going downstream or upstream?'
'Only one way home, mate, and that's downstream,' Ash replied confidently.
I frowned. 'Are you sure?'
'Yes, I'm sure.'
I got my head lamp out and checked overboard again. 'I don't think we are, man.'
I stared closely at the flow of the river. The water was bubbling through the reeds and up against the bank in the wrong direction.
'I bet my left arm we're going back upstream!' I called out.
'Nah, mate, don't be silly. There is no way we could have turned around,' Ash said to me.
No one else agreed with me either, so the driver kept going.
'I guarantee you we are!' I insisted.
'No, no, we are going downstream,' JB chimed in.
I was bursting with frustration that no one was listening to me and I wasn't having any more of it. 'Just stop! Just stop now! Completely stop the boat and pull over to the bank and let's have a look,' I said.
This time, the driver listened. We stopped, and immediately started drifting backwards.
'Oh my god, we're going upstream! How long have we been going this way for?' I yelled out again.
Everyone looked at one another, puzzled. No one had any idea.
It turned out that we had somehow rotated around in the black of the night and travelled even further past the research centre. I couldn't believe the carelessness of the crew. I sat there for a moment while What the hell were you thinking? Why wasn't anyone paying attention? What an absolute waste of time! came and went from the tip of my tongue, restraining myself and my loose lips. No one said a word, everyone just sat there completely deflated. We turned around and drove home, finally arriving back to base at 6.30 a.m., just as the sun was rising. At that point even I needed some creature comforts, and traded in the staff quarters for a hotel room to sleep off the drowsiness.
It was the longest day of my life.
The first thing that comes to mind when you think of an elephant is an animal big in size, but Borneo elephants, also known as pygmy elephants, are neither gigantic nor aggressive. They stand between 7 to 10 feet in size and weigh between 3000 to 5000 kilos. They're very cute and I felt privileged to have had the opportunity to work so closely with them in the wild.
A couple of days after the boat incident, a report came through from JB's spotters that there were a few herds of elephants in the palm-oil plantation not far from where we caught the croc. If we didn't move them, they would be shot by workers there. Before the days of extensive plantations, pygmy elephants roamed around in herds of 40 or so. Nowadays you're lucky to see a herd of 10 because so many are killed by plantation workers.
The sad part about relocating them out of harm's way is that even though you're doing some good you're also splitting up the small families that are left. Six or seven is a standard number for a herd, but the rangers can only relocate two to three at a time, so by the time the rangers return to move the remaining elephants, they have already moved on to another part of the plantation and can't be found again. But still, the rangers would rather move two or three at a time than leave them in the plantation to get killed.
We drove out in a convoy of three trucks with big cages, four smaller vehicles, a front-end loader, a 30-tonne excavator and associated equipment, along with the full team of 16 rangers, some scientists and PhD students. It felt like a cast of thousands but, from what I understood, we would need as many hands as possible to move the elephants once we had them tied up because they are such solid animals.
'All right, when we get there I only want Matt and me on foot with the camera crew. Everyone else is to stay in the trucks until we need all hands on deck,' Dr Zen said across the radio to everyone. 'The elephants are usually pretty calm but can charge sometimes if startled, so we need to reduce our risk.'
He turned to me. 'Mr Matt, if the animal comes at you just run and get out of the way.' I heard him out but didn't agree with the running away part. Panicking and running usually draws more attention and reason for an animal to run you down.
We spotted the herd and snuck through the plantation with the film crew tracking closely behind us. Dr Zen singled out a cow and her calf along with a bull, then called in the rest of the team over the radio to move in behind. The elephants appeared to be a family unit, so he focused on keeping those three together for the relocation. He wanted to get the cow first because he knew that the calf would stay near her, and then he planned on moving quickly to the calf, and then the bull.
Dr Zen stopped, pulled out his little case of drugs and .50-calibre dart, and loaded the syringe with the immobilising drug. Once we got close enough to the elephants he released the dart, which was propelled from the gun with the help of compressed gas.
The dart hit the cow, administering the drugs. We thought all was going well but the bull must have heard us moving in behind them, because it turned back to face us and started to charge.
Holy shit, I thought.
The commotion caused the other herds to start stampeding around us. Everyone scattered and skidded behind the palm trees like panicked mice. There were some close calls with the camera crew when a couple of bulls started chasing the guys up the arse, but the elephants continued on, flapping their ears and trumpeting through the tall timber.
I'd been in situations similar to this many times before with a loose brumby during a muster, so, while everyone ran around like headless chooks, I just waited for the elephants to charge right at me and stepped behind a big tree at the very last minute. Then I sat back and watched the circus unfold for a bit. It was hysterical.
Dr Zen quickly loaded half of the next dose in another dart ready for the calf, as he didn't want her to run off and be without her mum. But the same bull we were trying to relocate turned back towards us and started charging again, and Dr Zen had no other choice than to let off the half-dose on the bull. That slowed him down for a moment while Dr Zen reloaded another half-dose, and then we lunged towards him and put another dart in him.
We had to be so careful with the second dart, because if you overdose an elephant it falls over and can't get back up on its feet. And if they're on their side for too long, the pressure on their lungs gets too much and they can drown in their own fluids.
They tried to move away but the bull had now been darted with its two half-doses, and the cow was really starting to slow as the drugs kicked in. Luckily for us, the calf just wanted to be with her mum, so she moved quietly alongside the other two. All three animals began to lag behind the rest of the herd and we could follow them easily, with their large tracks marking the ground. We pursued them, shadowed by the team of rangers, until the bull began to stumble.
'Okay, go!' Dr Zen called out, and the rangers swarmed the bull, attaching metal cuffs and chains to his legs and tying him to a tree to ensure he stayed upright. The moment the elephant was secured they administered the antidote, bringing him back to his senses.
'Oh no, look at this,' Dr Zen said from behind me.
There was a huge cyst, the size of a soccer ball, on the bull's arse. Dr Zen got out his scalpel and made a tiny incision right on the cyst's surface. The bull didn't even flinch. Out oozed buckets, and I mean buckets, of pus, followed by a bit of blood. It made me feel so sick to watch and I dry-retched at the smell of the open wound.
'There will be a bullet in here, Mr Matt.'
Dr Zen was right. Once the fluid was released, out came a little bronze bullet from a .22 shotgun, or something similar.
'Poor fella. Lucky he's still a little groggy,' I said.
All three elephants were now chained up, calm and ready to be moved into their pens for transportation. Timing and accuracy were critical, and JB, Dr Zen and their team had nailed it.
It took us a full day of mucking around and a hell of a lot of people to tie up and move the animals. Then torrential rain started pouring down. We couldn't move them any further towards the trucks in that weather, so we had to leave them tied up comfortably with a few of the wildlife carers camped out in the cars nearby to look after them.
There was massive rain that night, and over the next few days. We couldn't even make our way back out to the plantation, the rain was that heavy. I didn't want to do nothing, so I moved my workshop gear to the hotel and finished welding up all of my traps to leave with the rangers.
I walked out of my room on the morning that the rain stopped, and couldn't believe my eyes. There had been over a metre of rainfall and the river had risen to meet my doorstep, which was about four or five steps from the pathway that led down to the riverbank. It was nuts. Boats were floating past me like I was in a ground-floor apartment next to a water canal in Venice. A piece of driftwood bobbed by, followed closely by a 12-foot crocodile.
'G'day, mate,' I greeted it. 'Coming in for a visit?'
It had obviously smelt the pig bait I had lying around and was hoping for a feed. I waited for the croc to piss off along the river and made my way down the steps, finding myself touching the bottom submerged in water up to my waist.
JB came around with a little dinghy and picked me up from my doorstep. He couldn't have come at a better time.
'The elephants are okay, but the calf had to swim around for a bit because she couldn't touch the ground. There's a lot of water where they are too, so the relocation's going to be difficult,' JB said.
The truck drivers navigated the muddy roads into the plantation, backing themselves in as close to the animals as they could manage. We had a 25-tonne excavator that came in and lifted the pens off the backs of the trucks. It was a muddy shitfight but we managed to walk the elephants into the open pens with the aid of prongs, which JB gently prodded in behind their ears. After another drawn-out day we got them into the caged pens and the crane on the back of the truck, lifted them up and lowered them securely on the tray. I could see why relocations like this only happened once a week in good weather – it was one hell of a job.
I could see so many improvements that could be made and thought of lots of different ideas, but I kept my mouth shut. There's nothing worse than an outsider coming in and making suggestions that are good in theory but practically have zero likelihood of being achieved. It's not because they weren't good, but I knew I had a limited knowledge of how they could actually be put into practice in an environment with challenges and issues I wasn't overly familiar with.
It was a four-hour drive to the elephants' new home. The area of rainforest backed onto another plantation, but the thing was that this particular plantation was surrounded by an electric fence, so there was no way the elephants could become displaced. Essentially, they would be trapped in their own habitat with a river forming one boundary and an electric fence forming another. We turned off the electricity and flattened one area of fencing to get the trucks in and released the beautiful creatures. It was such a good feeling seeing them amble off happily together. We put the fence back up, turned the electricity on and made our way home. I wish we'd had even more machinery and manpower, so we could have relocated the herds together and kept families intact but at least we saved some lives. It was a good start.
The next story I am about to tell you involves the most dangerous experience of my life. How I didn't die on this particular day, I still don't know.
It was time for me to show JB and the team how we collect crocodile eggs back home. I explained to them that using helicopters and other machinery makes life a lot easier, and I decided to hire a chopper to demonstrate exactly what we do.
Four thousand dollars later, in flies a long-range-helicopter with a bloke I later named Captain Courageous (can you sense the sarcasm?). He flew in with a co-pilot, which was unnecessary, as well as a rope specialist.
I walked up to the co-pilot and shook his hand, then turned to the pilot to shake his.
'G'day, mate. I'm Matt, how ya going?'
He stood up straight, folded his arms and said in short, sharp syllables: 'You must call me Captain.'
'Righto, Captain,' I said back, turning to Ash with a half-smirk on my face, thinking, Is this bloke for real?
To top it off, he was wearing epaulettes (the shoulder badges worn by fixed-wing pilots), and his badges had four stripes each, the very top ranking available. That in itself was a joke to me. If you ever see a chopper pilot wearing epaulettes, well, they shouldn't be flying a chopper. Back in Australia if you're a chopper pilot you're just seen as a glorified ringer, a cowboy in the sky, but it was obviously different in Malaysia. I thought about it some more and realised it was probably quite a notable job for a lot of the locals, and perhaps that's how protocol was in their country so I respected his wishes.
The rope specialist was a great lad and showed me the ropes, quite literally. We obviously didn't have the double-hook sling set-up we used back in the Territory, so for some harebrained reason it was decided that abseiling down from the chopper was the best way into the nest. Why on earth we didn't just hook ourselves up while on the ground and sling in that way has got me fucked. But I do remember someone from the helicopter company suggesting we abseil from the chopper and me thinking, Hell yeah, that'd fun. I can't do it in Australia so if it's legal here and they're going to sign it off, why not give it a crack!
'Have you ever abseiled before, mate?' the rope specialist asked me.
I was playing around with the abseil catch, trying to suss out how the rope fed through it. 'Yeah, yeah, sure have.'
It was a flat-out lie. I'd never abseiled before, let alone seen anyone do it.
'And you've done it out of a chopper before, yeah?'
'Yeah, mate, of course, heaps of times.'
The lie was expanding by the second. I had no idea what I was doing, but in life, I never say no to anything because I don't like opportunities passing me by. And I definitely didn't want this adventure to slip through my fingers. I just figured I'd say yes, then work it out quick-smart as we went along.
The rope guy briefed us on abseiling from the chopper, which was a little more complicated than I'd expected. We all went through the ins and outs of how JB and I were going to make our way down the rope and on and off the crocodile nests. I put a helmet on and had GoPro cameras strapped all over me to get footage for the show. I was ready to rock and roll.
'Who am I?' the pilot asked me as we were getting into the chopper.
I looked at him, confused.
'What is my name?' he asked me again.
I caught on to what he was saying. He was trying to reaffirm that I knew to call him Captain. This guy had to have tickets on himself.
'You're Captain Courageous,' I said back, joking around.
'No, just Captain,' he snapped back at me. 'You must call me Captain at all times. I am Captain, you are passenger.'
'Jesus Christ, here we go,' I said to Ash, who chuckled and hopped into the chopper. We'd agreed that he'd spend the day switching between the chopper and the support boat on the river to film, while JB and I taxied around in the chopper with the two pilots.
'Have fun with Captain, my man!' Ash called out sarcastically.
Before we headed out to the crocs, we decided to practise a bit. JB and I had our ropes hooked onto our harnesses at one end and underneath the chopper at the other end. I gave it a go first. I shimmied myself out of the helicopter door and onto the edge of the skid. While I was sitting there, leaning in towards the chopper, I made sure my rope wasn't tangled and then gently pushed myself off, swinging under the machine. I was relying on equipment and a system that was completely thrown together. If I had fallen from this height of 100 feet, I would have been in a pretty bad way. There were no testing or safety precautions taken, either; I'd just jimmied up a carabiner and was hanging on a 10-mm-thick rope that was linked to a pulley system I didn't know how to use. In hindsight, it's pretty ridiculous but I was so excited to give abseiling from the chopper a go that I didn't really think about it.
I punched off and pretty much zip-lined down this 100-foot rope, almost in freefall because I forgot how to pull on the mechanism.
I was ripping down as fast as anything, thinking, Oh shit, gotta stop, gotta stop. I jerked the rope just as I got to the end, and bounced back up.
'Right, I've got that part worked out now,' I said to myself, heart racing.
After the fifth test JB and I had it sorted and were feeling confident enough to get in on a croc nest so I could show him how to approach, open and clear a nest safely and efficiently.
Apart from the 'Higher, Captain', 'Lower, Captain', 'Please go this way, please go that way, Captain', which I had to say on repeat, I was really enjoying myself, and even though what I was doing was risky, I didn't feel like my life was ever really in danger.
But boy, was that about to change. JB and I had collected a couple of the nests and our slinging and abseiling in and out of the chopper and nests was going perfectly. JB explained to the pilot that there was a particular nest about a kilometre away that he wanted us to check out. Instead of climbing back into the helicopter for the journey, I stayed on the end of the line. This is no different to what we do in the Territory, but the pilot usually flies only 10 or so feet above the ground or trees to ferry us safely to the next location.
Without warning, I felt myself being lifted higher and higher into the air.
'Stop, Captain, stop! What are you doing?! Too high, Captain, too high!' I called out over my handheld radio.
I went from 100 feet to 200 feet and that figure was doubling and doubling as he flew higher and higher. I started shouting every single swearword under the sun, screaming and screaming at the stop of my lungs for him to fly lower but nothing was happening. My messages weren't getting relayed. He began doing a full flight circuit and was flying up in circles until he got to over 1000 feet. I was completely nauseous, anticipating the thin rope snapping at any minute.
This is it, I thought, picturing myself plummeting to my death.
I tried one more time at the top of my lungs 'STOP, YOU FUCKING IDIOT!' but it was lost in the wind and my voice was breaking I had worn it out completely from all the yelling.
At that point I stopped trying to communicate with Captain altogether and hung on for dear life, just hoping that by some miracle I would make it back to ground safely.
I eventually felt the chopper descending, bringing me in to land. My feet touched the ground near a nest, and I unhooked and sprinted into a clearing nearby, waving my hands wildly out of rage. The pilot came in to land next to me.
I have never felt rage like it in my life. I know this sounds terrible but to be honest I wanted to grab the pilot by the head and skull-drag that idiot out of the captain's seat and punch the shit out of him. Fuck, I was angry. But I held back, absolutely served it to him verbally and then waited until Ash and the rangers arrived by car to collect me. They'd been tracking our progress on the boat. Ash told me he could hear everything I was saying over his radio on the boat, but the pilot had obviously been tuned to another frequency. I have no idea what made him fly so high. No one in their right mind would do something like that, so I was happy to never see Captain Courageous and his stupid face ever again.
That night, Ash showed me a picture of me in the air, a tiny speck of a thing. You couldn't even make it out that I was a person, that's how high up I was. It took me a bit to wind down before getting to sleep, and the memory of that day still haunts me.
It was our last day with the rangers before heading home. We packed up, said our goodbyes and made our way back to the airport, taking a small detour past the orangutan sanctuary on the way.
'JB, let's just stop over in the swimming hole one more time on the way out and make sure there aren't any crocs in the traps,' I said.
As we came to the bridge my eyes lit up.
'The lizard, the bloody lizard, that's it!' I called. 'Stop the car, Ash! I'm getting out, come with me!'
I took off without shoes or a microphone, skidding down the banks and into the shallows. The lizard looked even bigger in real life; it was at least seven feet long and fat as anything. It scurried into the water when I arrived, and I traced its movement by watching the weeds waving around on the surface as it waded beneath. The bank was littered with fallen palm fronds, which were stabbing my bare feet with every step I took. It was agonising, but I wanted to get this kitten-eating lizard so badly. I took a big step forward, ready to leap on the lizard while it was in the water, then hesitated.
'Ah shit, I wonder if there's another big croc in here,' I called out.
Ash and our soundy were standing on the bank, trying to film. The soundy was stretching the arm of his boom mic as close to me as he could get it.
'Nah, Matty,' Ash called back. 'Go for it, mate, you'll be sweet!'
I listened to Ash and went in hard, landing right on top of the lizard and boy, did it kick up a fight. Its feet were as big as mine and its talons were otherworldly, larger than any claws I'd ever seen on a wedge-tailed eagle.
I pulled the lizard in as close to me as possible, and its back legs scrabbled up mine, ripping through my jeans on both legs and cutting my skin. It hurt, but I held on tight, and ran up the bank with the lizard and straight to JB. The smile on his face was priceless, and he just started laughing. 'Only you, Mr Matt, only you.'
I gave it to him and he passed it onto another ranger, who put it into a cage in the back of their little truck to release on their next trip to the rainforest.
There was no better way to finish off the trip than at the orangutan sanctuary with JB and his wife. They let me help with the release of two orphaned orangutans back into the wild. JB took one and the other, a two-year-old girl, jumped on me. She held onto me in a piggyback, just chilling as we walked through two-and-a-half kilometres of lush jungle to the release platform. It was a beautiful experience.
One of the handlers at the sanctuary came with us, and warned me about a dominant male orangutan that had been known to hang around where we were going to release the two orphans.
'He's a bully, and has a habit of pooing and throwing it at people,' the handler explained. 'Just keep an eye out for him, and don't make eye contact with him or any of the bigger animals the deeper we get into the jungle.'
I've seen footage of orangutans pulling people's limbs off before, so I didn't like the sound of coming face to face with this cranky male.
I walked my girl up the steps of the release platform where we would let her go.
Slap!
Something hit the side of my face. I looked around but couldn't see anything.
Slap! Something hit me again, and I looked down to see two banana peels at my feet.
'You okay, Matt?' JB asked.
'Yep, all good.'
'It's the big guy we were telling you about,' JB continued. 'Just ignore him, it will be okay.'
We carried on with the release and the two orphans swung out of our arms with ease. I've released lots of rehabilitated animals before and it's always an emotional experience. I'd only spent the short walk with my girl but I bonded with her instantly. She took a tiny piece of my heart with her, that's for sure.
'Go well, girl,' I said as she finally disappeared into the distance.
JB and the handler stepped down the ladder first and I followed them, making my way to the ground. Unexpectedly, the shadow of a big orangutan appeared from above me, grabbing my arm with force. I froze.
'Matt, squat down. Kneel into a ball and don't look up,' I heard the handler say to me.
I sat on the ground for a few minutes, with JB and the handler squatting beside me. The orangutan prodded me a few times, putting his weight on my shoulder. After leaning on me for a good minute, he eventually let go, paused momentarily, then swung away.
'Okay, Matt, you need to stand up slowly, don't look back, and just walk away with us at a normal pace.'
I followed the handler's instructions and off we went, without any further problems. My heart was still pounding, thinking about what could have been.
'Well, that seems like a very fitting way to finish off a trip of a lifetime, full of every single possible wildlife encounter that most people can only dream of!' I said to JB and the team when we got back to the group and I told them the story.
I gave my hat, boots, jacket, belts and knives to JB as a thank-you present. Leaving him with my things was the least I could do, when he, Dr Zen and the team of rangers gave me experiences and memories I'll never forget.
#
CROCODILES INVADE FOOTY FIELD, TOURISTS WADE THROUGH CROCODILE-INFESTED WATERS, MONSTER CROC MAKES A MEAL OUT OF A KANGAROO and CROCODILE ATTACKS MAN are just some of the news headlines to come out of Daly River.
You get the picture. Daly River, located midway between Darwin and Katherine in the Northern Territory, has crocs – big ones, and lots of them. It's pretty fitting, then, that Tripod, my Instagram-famous 17-foot pet crocodile, called the Daly River home. It is where he was caught back in 1986 for causing havoc and trying to eat the local traditional owners, who were camped out along the river. Nick Robinson, a famous NT croc catcher, pulled a then-15-foot Tripod out of the main waterway with the help of a young Indigenous fella by the name of James Perry.
The story has it that when Nick asked James to help him bait the trap, the young fella took it upon himself to get the job done while Nick wasn't there. James got a horse leg, tied it over his shoulder and swam across the croc-infested Daly River to the other side of the bank, where he baited the trap. James emerged from the river completely unscathed and lived to tell the tale. The croc was caught that very same night James baited the trap, so when the croc was finally pulled out to be relocated, everyone named him 'James' and the story became a bit of a Territory legend. The croc had his front right leg torn off in a battle with another croc before he was caught on the Daly, and it was actually me who renamed him 'Tripod' when he ended up at my place years later. It's unfortunate for Tripod, but it means he is easier to work with because his mobility is pretty limited, and his remaining front leg doesn't do much for him either. Essentially, Tripod is part prehistoric monster, part immobilised slug. But there's no denying he'd still eat you in a heartbeat if you gave him the chance – slowness does not mean passiveness.
Tripod was moved to the NT Crocodile Park, where he was kept in the tourism section, wowing guests with his impressive size and presence. I started running my Outback tours in 2005 and, as part of the experience, I took guests through the park for a private feeding show with some of my favourite crocs, including Tripod. The tourism side of the park shut down in 2007 due to biosecurity restrictions. The facility was then converted into a fully functioning farm while Crocosaurus Cove, a crocodile tourism park, opened up in Darwin so tourists could still experience crocodiles up close. Many of the pens in the old facility were knocked down and opened up to create more spacious breeding facilities, and part and parcel of this was having to integrate the crocs who were once kept separately. Just like they do in the wild, the males immediately asserted their dominance over one another, and some pretty impressive fights broke out.
It was the middle of the breeding season, and a small number of crocs had hatched from uncollected eggs in the facility's new open areas, which comprised about 20 acres of expansive swampland. Jimmy and I got tasked with catching and relocating the little ones into smaller enclosures. Given that crocs are primarily nocturnal, we went in the dead of night, when they are most active out of the water so we could see them more clearly. We drove out in a ute on the lookout for crocs around two-foot in length. We intended to catch them, tape them up and put them in the back of the tray.
We reached the first large pond and the ute's headlights revealed well over 100 crocs, which if you can imagine, is a hell of a lot of crocs moving around in the one area. It's an amazing sight, even for someone like me who sees that sort of thing a lot. Some were partially submerged in the water, but most were stretching out along the banks.
We idled in the ute for a moment and then I spotted a little one about four metres from the water's edge. I hopped out of the truck with my net and started to creep up on it. I became more cautious the closer I got to the water, well aware of the big aggressive males lurking in there. I tip-toed up and scooped the first one into my net, no worries. But as I walked back to the ute with it, it started calling out to its mum.
A croc's call sounds kind of like a cross between a duck's quack and a frog's croak. It's distinctive, and it's loud. This croc would not stop calling and it was getting louder.
'Shhhh,' I shushed it under my breath, and increased the speed of my powerwalk back to the ute, where Jimmy had already caught another little one.
'Their calls are going to attract some unwanted attention,' I said to him.
'You're telling me! I just had a gnarly female fly metres into the air to have a crack at me. She literally flew, mate, got completely airborne. I ain't ever seen nothing like it!' Jimmy was pretty wound up, and it did feel like we were asking for trouble.
There was another little one I saw near the bank that I wanted to collect before we called it a night. So I crept back down to the water and jumped on her from behind. She immediately started calling her parents for help and then, sure enough, out of the water came two big, angry females, making a beeline for me. Luckily, they emerged from the same direction, so I had a clear path to run the other way. They both chased me right up to the ute and I had to high-jump into the tray. Jimmy was already sitting in there, panting and sweating.
'I'm done with this, Matty,' he said. 'We've been out for two hours, caught three little crocs and nearly been chewed twice each. It's not worth it.'
'Roger,' I agreed. 'But I want to go give Tripod some love before we head home. I've got a chook for him.'
He rolled his eyes. 'You're obsessed with that animal, Matty. The way you talk about him, it's like he's a fluffy puppy.'
It was true, there was something about Tripod. I felt like he knew who I was and vice versa. He was my mate.
We drove over to his pen but it was empty. We noticed that his fence had been knocked down during the recent renovations, so we drove around the entire park looking for him for an hour but couldn't find him.
'He's gone walkabout in the open area,' I said to Jimmy.
But he wanted out. 'Mate, I don't think we're going to find him tonight. It's 1 a.m., let's come back another time.'
So we left.
Even though I was fed up with catching the small crocs I kept at it until the end of breeding season, using it as a way to scour the park, searching for Tripod. The whole park holds more than 1000 breeding crocodiles, so trying to locate one particular croc in a cast of a thousands was a challenge. I had no luck for 12 weeks. By that point, I was sure he was gone, and had been beaten in a battle.
The scenery at night-time and the movements of crocs was unbelievable, so I took my camera out on one of the last nights of breeding season. I headed out to a part of the farm where a large pond is canopied by trees. It's called Jurassic Park, and is home to about 30 enormous males. It was a full moon with a clear sky – the lighting was insane.
I brought my mate Jai along with me. I worded him up.
'There's a lot of big fellas in the one waterhole, so this could get interesting,' I told him. 'Just have your wits about you.'
'When isn't it interesting with you, Matt?' Jai replied.
The moment I hopped out of the ute a small croc showed interest and chased me around a little woodpile. Jai had to run out in her periphery and entice her over his way with a stick for her to chew on.
'Far out, there's always one cranky one,' I remarked.
I shone my torch across the pond and, standing in front of me, about 10 metres away, was Tripod.
'Hey, my man!' I called out. God, it made me happy to see him.
He started slugging his way towards me and, at the same time, a beautiful spectacle unfolded right in front of my eyes. At least 20 smaller crocodiles, from different spots around the pond's edge, speared through the air and into the water at the exact same time. It was like synchronised diving. I kicked myself for missing the shot, but I had a pig leg in my hand and a croc coming at me.
'That was the sickest thing I've ever seen!' Jai called out.
'I'm with you there,' I agreed. Then I remembered Tripod, who was still making his way towards me. 'Come on, hey! Come onnnn. Come on. Hey, come onnnn!' I continued, echoing out my usual call while I slapped a pig leg on the ground, causing vibrations and bringing him closer.
Tripod would have recognised the sound of my ute's engine and associated it with a feed. He stopped a few feet away from me and waited patiently. The algae in the water had given him a slimy green coat, which shone fluorescent in the moonlight. He looked supernatural. I walked behind him, giving him a good rub and taking some pictures of his striking colour. He gulped down his pig leg and then slowly returned to the water, heading back out into its depths.
Without warning, he launched with full gusto at a 14-foot croc to his right, engulfing it in his jaws and thrashing it around like it was a rag doll. The other croc tried to wriggle away, but Tripod crunched and snapped down on him some more before flinging him through the air and back onto the bank, before continuing into the water. The mangled croc lay there on the shore, lifeless.
I was shocked. I've never forgotten that moment because I witnessed this usually docile, slow and lazy crocodile turn into a killing machine in an instant. No matter how laidback Tripod seems to me I am never ever complacent with him, because even though he knows me he'd still eat me if ever given the chance.
I knew that same fate could very easily happen to Tripod given his disability and I didn't want him to end up that way, so I asked the owner of the croc farm if I could get a permit from Parks and Wildlife and keep him as a pet at my place.
'You can have him, if you find him and work out how to move him,' the owner said to me.
I got moving on the offer before he changed his mind and thought it was fitting to call Nick Robinson and get him involved in the relocation, given the history. Nick, Willow and I returned the next morning with a tilt tray, crane, equipment and a strategy. Enticed with a chook, Tripod slunk out of the waterhole and straight towards me. He stopped at my feet and even opened his jaws, making it so easy to slip a rope over his snout. I couldn't believe it. It was plain sailing and I had him just like that. Willow jumped in behind me and we began to pull him towards the back of the tilt tray.
And that was when Tripod decided to flip out. He flung around, kicking us both on our arses. We immediately scrambled up but we couldn't regain control of him. He took off and we had no choice but to start skiing on the end of the rope behind him, through the mud towards the water, where a heap of other crocs would have happily met our arrival. Nick came running in to help. He grabbed the rope and dug his heels into the mud, throwing more deadweight on the end of the rope. We finally held Tripod in one position, albeit on an obscure angle.
'Okay, you fellas, hold tight while I get some drugs in him,' Nick instructed. He darted around to Tripod's tail and jabbed him with the dose. Finally, after 20 minutes of awkwardly wedging ourselves into the ground to hold him still, he fell asleep. We secured him safely and used the crane to lift him onto the mattress on the back of the tilt-tray truck.
It was impressive watching the crane lift a 17-foot, one-tonne animal into the air and onto a queen-size mattress. We got it done with ease, and Tripod stayed calm (and asleep) the entire time. I had a 30-tonne digger at home, so I got a few of my mates to hook in and cut out a big hole in less than 48 hours. We filled it with water, added a high, sturdy fence, and dropped Tripod into his new home at the back of my property.
The most common question I get asked by people, by media in particular, is 'How old is Tripod?' The truth is, I don't really know. It's always been a bit of a guesstimate, but I usually say he's between 70 and 80 years old, mainly because of his large size and the fact that crocodiles don't ever stop growing over their lifetimes so, in theory, the bigger the croc is the older it should be. However, this notion is also a little flawed because it doesn't take into account genetics, diet and whether the croc has lived in captivity or not – all factors that would affect the growth rate of a croc. Additionally, determining the age of crocodiles is tricky because they live for such a long time and, quite simply, reliable age records are hard to come by.
I feel confident saying that Tripod is between 70 and 80 years old because the average age of a saltwater crocodile is 70 years, and some have been said to live up to 110, so 70 to 80 seems like a realistic, even conservative, age range to me. Another thing that not a lot of people know about crocs is they show almost zero signs of aging. In other words, a 10-year-old crocodile matches a 100-year-old one in terms of their physical capabilities. This biological immortality is referred to as negligible senescence, and very few animals have it. It's pretty cool to think that crocs never age or deteriorate, and that their only real chance of dying is via disease, misfortune or a predator. It's no wonder they've been around for 200 million years.
Tripod has been happily homed at my place for nearly 10 years now, and in that time has become a social media sensation. Videos and pictures of Tripod reach millions of people around the world who are in awe of his dinosaur-like presence.
I've never had a close call with Tripod in my daily interactions with him, but I do remember one incident that nearly cost me a limb. I give Tripod's pen a good clean-out a couple of times a year. On this particular occasion I'd pumped him in some fresh water, cleared out loose debris and was whipper-snippering the long grass in his pen. I climbed a tree on my way over to a patch of grass on the other side and cut a couple of its branches down in the process to clean things up. As I swung myself around the tree with my left hand, ready to launch in the air and jump off around the other side, Tripod's massive jaws slammed down at my feet.
'Holy shit!'
I shifted my foot up the next flimsy tree branch, with the whipper snipper still turned on in my right hand. Tripod snapped again, going for the whipper snipper. I had no one around to deter him and this tree, which was more like a scrub bush, was no place to retreat for safety. I fidgeted and fumbled with the snipper, eventually finding the 'off' button at the same time as Tripod manoeuvred himself up at me again.
In the same motion, I had to jump from my position in the tree to the ground and use the whipper snipper as my defence stick, like I do on crocodile nests, prodding and poking him back into the water. He had a couple more cracks, but once he realised there was nothing different to see he retreated.
I can't believe I was so silly not to think that the sound of the motor would draw him out of the water. Whether it's a boat engine, a car idling or a lawnmower running, crocs are always drawn to those mechanical sounds. I only half-tidied his pen that time and am pretty wary with any sounds around him these days.
I love the big fella and I will be very interested to see how long he hangs around for. He's pretty unlikely to catch a disease, he's not going to get eaten by anything and he's fed more than enough food, so maybe Tripod will stand the test of time and continue to grow and outlive all of us!
#
Anyone who knows me knows that I am the king of grand ideas and love nothing more than showing people a good time. A lot of my business decisions have favoured making people happy over making lots of money, which any good businessman will know is the first way to go arse-up pretty quick. Instead of crunching numbers and looking at the commercial viability of something, I tend to imagine the coolest, most unique and adventurous business idea and work back from there, which, again, is good for your soul but not your back pocket. To add to this combination, I also have a habit of biting off more than I can chew and my expectations of what I can achieve are often far greater than what eventuates. And there is no better example of where these personality traits of mine come into play than my first large-scale tourism endeavour, Outback Floatplane Adventures.
I love life in the Territory, whether it's flying my chopper, catching crocs, fishing or exploring the outback. I truly believe that life doesn't get much better and I've always had a desire to share it with people. Over the years I've dreamt up a range of weird and wonderful ideas, but I kept coming back to an adventure with crocs, choppers, airboats, fishing and four-wheel driving. But I didn't know where I would do it, I didn't have access to any land, I didn't have a plane to reach remote locations, and I had no cash to invest in it, so my dream stayed a dream for a long time.
In 2011, I got a gig flying some high-end lads on a corporate trip around the Kimberley, where we visited what David Attenborough refers to as one of the modern natural wonders of the world, the Horizontal Falls. I met a bloke called Troy Thomas, who'd started a small tourism operation out there off the back of a fishing charter company. He grew it over the years to eventually comprise an all-inclusive trip with a seaplane, jetboats, choppers and a pontoon structure with overnight accommodation. The tour actually went on to be named the number-one adventure tour in Australia. The lads and I booked one of Troy's overnight tours while we were in the area, and Troy and I hit it off straight away. I told him about my idea for a tourism operation in the Territory, and we agreed then and there that we'd do something. Again, going into business with someone I'd shared beers with over a couple of days was another brash decision on my behalf, but it felt right so I went with it.
We became business partners and Outback Floatplane Adventures was born. With it came a testing but triumphant journey that was full of hiccups, headaches, hard work and an absolute labour of love to bring a new high-quality tourism experience to the Northern Territory.
Troy and I agreed that the tour would involve a floatplane flight from Darwin, which would land at a remote water location and dock on a pontoon set-up. From there, we would take guests for a river cruise, followed by a rainforest tour in an airboat, a scenic flight in a helicopter, and finishing with a little off-road Polaris quad-bike adventure. It took us 10 months from the time Troy and I met until our first day of operation. I already owned a chopper, Troy was sending his 12-metre cruise vessel, Cyclone Creek, from Broome, Troy's dad, Reece, a veteran pilot, helped us secure a floatplane and the Polaris and pontoons were easy enough to organise. The one thing that Troy and I both didn't know much about at the time was airboats, so I left the sourcing and purchasing with Troy, who then passed the job onto a middleman by the name of Brendan Smith, an 'airboat specialist'. Well, what a mistake that was.
Troy looped me in on a quick three-way phone call with Brendan to go over the plan for obtaining the airboats, and the pricing.
'Righto, lads,' Brendan said. 'Diamondback Airboats in Orlando, Florida, is the best in the business, so we'll get everything done in the States. Instead of using their petrol V8 engines, let's put a Duramax Diesel fuel-injector engine in. It will have way more grunt.'
I followed the conversation along with the odd 'okay' and 'Sounds good' but didn't have much to add. This guy was the specialist, so I figured I'd leave it to him.
'And pricing?' Troy asked.
'You'll be looking at 150,000 Australian dollars and a 14-seat airboat delivered to your doorstep in Darwin. I've got it all covered so, if you're happy with the price, leave it with me and I'll get it to you in three months.'
'Roger, sounds good,' I said, and we both agreed to go ahead, paying our deposit directly to Brendan.
Every now and then, a group email came through from Diamondback Airboats with picture updates of the boat. A couple of times the team in Orlando raised their concerns about the use of a different engine, but Brendan always wrote back to them, insisting it would be fine. After construction of the boat was underway, Brendan told us he needed to fly to Orlando to ensure that the airboat met all of our survey requirements with the correct flotation, safety rails and equipment, so we paid for his flights, accommodation and associated costs.
The requests continued to roll in after his return. Brendan asked for an additional $5000 for more engine parts, $15,000 for freighting, and other dribs and drabs for cash reimbursements. The airboat was starting to cost us in excess of $170,000, and I smelt something fishy. Troy knew Brendan, but I was yet to meet him in person and I didn't have a good feeling about him. When the time came for the final product to be delivered, we were told that the airboat's arrival into Brisbane had been delayed by two weeks, which meant it would be another four weeks until it made it to Darwin. I called Brendan to see what was going on.
'Can you shed some light on what's happening, Brendan?'
'Oh, hey, Matt, yeah it's just one of those things, mate. When you're transporting something from country to country, timings can change. Don't stress, I'm going to base myself here in Brisbane until the boat arrives, at no charge to you guys, to make sure I see it to Darwin without a glitch.'
'Oh, so you're in Brisbane already?' I asked.
'Yeah, mate, yeah, I am,' he replied.
Things just seemed off with him. I hung up the phone and did some sniffing around over the next week or so. Ten days later I saw a picture on Facebook of Brendan at the Sanctuary Cove Boat Show on the Gold Coast – and he was there with our airboat promoting it!
I called Troy, overwhelmed with rage. 'You've got to be shitting me!' I shouted. 'That broker is a bloody shark! Our boat is here, and he's got it on the Gold Coast. He's taken it down to the boat show and is telling people it's his boat and that he can build airboats like this in Australia!'
'Ya kidding!' Troy replied. 'What a con artist. I'll get onto him now and get him AND the boat up to Darwin to have some words.'
I was flat out organising all of the other moving parts of our new business, and the last thing I needed was this headache.
Troy called me to say he'd spoken with Brendan. 'He apologised for taking it and not keeping us in the loop. Apparently he didn't think it would be an issue.'
'I don't know what planet he lives on to think that's okay!' I snapped back.
'Yeah,' Troy agreed. 'But there's one more thing. The flea doesn't even have transport organised to Darwin like he promised.'
I couldn't even respond to this, so he continued. 'And that's not the worst of it: he's just left our airboat in the convention centre car park in Brisbane, so I've just had to organise its collection and transport.'
I was absolutely dumbfounded that someone could do this sort of thing.
'I didn't reveal to Brendan how livid we are because I don't want to scare him off coming to Darwin,' Troy said.
'Good! I at least want the opportunity to have it out with him!' I replied.
The airboat arrived in Darwin two weeks later. Even though it had cost us an arm and a leg we were excited to finally see it in the flesh, and we took it straight out to Manton Dam to test it. We pulled the cover off.
'Far out, Troy,' I remarked. 'Look at the state of this thing. This Brendan fella has burnt us in the arse something fierce.' There was grease, sand, scuff marks and even chewing gum all over the floor. He must have had people walking on it at the show.
The flooring was made of non-greasy sea deck, which usually looks great but not when someone has trodden all over it with shit on their feet. Gee, I was wild. In a way, it was my own fault for not having my finger on the pulse, but setting everything else up with the business had me preoccupied. I had trusted that Troy had the right contacts, but he was just as busy as I was and didn't have time to oversee things.
I always give people the benefit of the doubt and presume they are as honest and fair, but sadly I continued to be proven wrong more times than I did right. My mate Macca, who lives locally in Darwin and owns an airboat himself, came out to test drive the airboat with us and give us his thoughts. We finally got the airboat in the water and, five minutes in, it stopped with error messages showing up.
'Yeah, guys, this engine is shithouse,' Macca told us. 'The boat is perfect, but this engine isn't made for this sort of thing. It's no good.'
Again, I just had nothing to say and Macca continued with his thoughts.
'Sure, a Duramax engine is the duck's nuts if you're putting it into a Dodge Ram but NOT an airboat. This thing is triple the weight of a normal airboat engine, and the boat's going to lag because of it.'
At this point, nothing he said surprised me. I was sitting there in silence, imagining all the different ways I could give Brendan the karma he deserved.
Troy tried to be a little more positive. 'Sure, it's got a lot of torque, but the immediate power isn't quite there.'
'Mate, it's the power you need the most. That's what gets an airboat up on land and gives it the lift it needs. Imagine if this had 14 passengers on it,' Macca added.
'Yeah, it would be impossible to manoeuvre,' I agreed.
The key to an airboat's power is instant velocity and this dumb-arse Brendan had not only asked to make the width of the boat thinner, which took away the water displacement on the hull, but then fitted a heavy engine without the pick-up power. And this engine had cost double the price of a normal Chevy V8 engine, which is absolutely perfect for airboat use. It was all a mess.
I called Diamondback in Florida, which I should have done a lot earlier, and started asking questions. What I heard from them just added to my disappointment.
Diamondback (which, for the record, is an awesome airboat company) had never dealt with Brendan before this ordeal. And Brendan had tried to take the airboat out of the workshop while he was in Orlando before he finished paying for it, plus he actually ripped Diamondback off, putting a ten-thousand-dollar payment on a bogus credit card. Unfortunately, after trying my damnedest to help Diamondback recover this money we had no luck and were unable to ever get the payment.
Everything was completely embarrassing. We had really been taken for a run and the worst part was that the conning didn't stop there. I clarified with Diamondback which foam they put in the airboat and it was nothing close to what we needed to meet our survey requirements. Brendan had instructed Diamondback to put polyester foam underneath the hull, which is like what you would have on a normal foam cooler box. This sort of foam isn't fire- or fuel-resistant, so if we were to get fuel on it, the foam would disintegrate and turn into a napalm gel, transforming the boat into a floating bomb and if that didn't cause the boat to blow up it would certainly sink. We had this new one hundred and seventy-thousand-dollar boat that wasn't even water-worthy.
So there I was the next day, in my shed with my beautiful new airboat, pulling out the seats, flooring and sea deck, and cutting through everything with a hot knife. I was crawling into crevices only a cat could reach, sucking in fumes and coughing my guts up. It ended up taking me two whole days to pull out the foam, replace it and refabricate the boat while I uttered every single swearword known to mankind. I was putting my life savings into this business and some weasel was trying to ruin me. It was the worst feeling. I was heartbroken. And the foam was the final straw.
I called Brendan, trying to remain as calm as possible, not letting on how upset I was. 'Hey mate, just checking to see if you're still planning on coming up next week and if you need me to book you accommodation?' I asked when he answered the phone.
'Hey, Matt, yeah that would be great. Looking forward to seeing the boat in the water!' he said, all excited.
'Yep, and I'm looking forward to seeing you too, mate,' I said, through gritted teeth.
I put the airboat to the back of my mind and focused on the rest of the set-up. I figured I'd wait for Brendan to reach Darwin before making any more major decisions, because I planned on getting him to foot the bill for a new engine at the very least. We only had eight weeks left until the tour launched and I had far more important things to work out, like which location we were going to run out of.
Because I collected crocodile eggs during the wet season, I knew every waterhole, nook and cranny around the Territory. I'd been flying over the landscape for the past 15 years and had earmarked two places that I thought would be able to handle our tour and that were under 30 minutes flying time from Darwin. One was called Bull Coin and the other was Sweets Lagoon. I locked in a time to fly out there with Reece and Troy for a reconnaissance-type mission. Reece flew us out in a little Cessna 182, which we attached floats to so we could land on water.
We tested the first spot at Bull Coin. Landing on a big waterhole, we all hopped out and immediately felt pretty underwhelmed by our surroundings. Don't get me wrong, it's beautiful country but it was too open and just wasn't right for what we were after. We returned to Darwin and switched from the Cessna to my chopper so I could take Reece and Troy for a fly to the second spot, Sweets Lagoon. Willow came along for the ride too.
I'd shown the guys the area before and Reece, who had been flying for many years, didn't think we could land a floatplane on the water strip because he felt it was too small, but I was sure we could and took everyone for a closer look. I flew down through the trees and along the water strip to give Reece a better perspective. I pointed out where the floatplane would land and where the pontoon set-up would go.
'You're right, mate, this won't be a problem at all. We'll just have to clear out the thick grass, floating mat and logs so the plane floats don't hit the crap on landing,' Reece said.
We were stoked with the area. The main lagoon was a prehistoric paradise abundant with wildlife, and we knew it would blow the public away. We wanted to operate there but had no idea who owned the land or how to access it from the main road.
After flying around some more, I spotted a few dirt roads, a mango farm and a little shack in an overgrown area. I landed at the shack. It was totally rundown and clearly unoccupied. There was tall gammon grass growing right to the top of the roof and old scrap metal lying around the place. It looked like a junkyard.
I pushed the front door open with my foot and walked into Noah's Ark. There was a pig wallow in one corner with a couple of feral boars moving about, a couple of bandicoots scurried around, and a possum ran along the ceiling beam.
'Ahh!' Willow jumped aside to make way for a wallaby that came bounding out of the kitchen. It hopped past us and out the door. We all laughed at where we were and what was going on. The front door of the shack opened up to a large concrete breezeway, which was screened at either end, and there was a bathroom, kitchen and toilet on one side and three rooms with air-conditioning units on the other (whether they worked or not was another story!). There was worn-out furniture around the place and most of the flyscreens were torn apart. The floor was covered in bat shit and other faeces from the menagerie of animals living in there. I did a quick poke around the kitchen, which had all the basics. I opened a gas oven and out slithered a decent-sized brown snake. 'Jesus! It's been yonks since anyone's been in here!' I tailed the animal, walked outside and let it go in the bushes.
We walked along a narrow road nearby overgrown with pandan plants, shrubs and small trees. It would need a lot of slashing and grading for us to be able to get from the shack down the two-kilometre dirt road to the lagoon. We finished off the rest of our recce and flew home. There was no question, we all had our hearts set on Sweets Lagoon and the shack.
I discovered that the lagoon was on crown land, so we could operate there but I still needed to work out who owned what around the water and the shack: was it private property or Indigenous land? The next day I flew back there and started conversations with Paul Venturin from Finniss River Station. His fence line ran into the lagoon on one side, so I got permission to access the lagoon from his land to begin with, just to get the equipment in. He gave me the details of David, a Vietnamese man who owned the land directly connected to the lagoon as well as the shack. I planned on reaching out to him at some point too.
A week later, we trucked Cyclone Creek to Sweets Lagoon along with another truck to cart two front-end loaders equipped with forklifts and buckets. The seven-kilometre dirt road from Cox Peninsula Road to the station boundary was full of divots, culverts and holes, and recent rain meant that it was wet and muddy. The truck got stuck a couple of times and the lads had to drive the loaders off the second truck to pull the main truck out.
Eventually we reached the boundary and cut the fence. Taking the boat down through the paddock was easy going until we reached a patch of large cathedral termite mounds, which we had to zig zag through until we finally got to the only spot that would allow us to get close enough to the river to slide the boat in. We flattened a pad with the loaders and created it on a bit of an angle to help edge the boat into the water. There was a large five-foot-round solid melaleuca paperback tree on the other side of the bank. I figured I could hook a block and tackle onto the tree and create a big pulley system with the front-end loaders and the boat. That way, when the loaders went back, the boat would slide forward into the water.
I thought the job would be a headache, but it ended up working really well.
'This is the easiest thing we've had to do out of everything!' I said to Troy and the crew. The boat was in and it was time for the trickier part – getting it through a clogged-up channel about three metres wide. It was just after the wet season, so there had been constant rain with trees falling down all over the shop. This particular part of the river was like a beaver dam – it was totally clogged up with logs, floating mat and debris. We had to clear it all out to fit the boat through. The work needed to be done anyway, because a beaver dam locks a river up and stops the flow of water, which is bad for the environment. Healthy waterways need to flow so that the fish and aqua life can continually move around.
Macca had his airboat out with us. My mate Steve and I jumped on the front with Mac and he steered the way through the debris, breaking everything up like a food processor. Every so often Mac had to stop the airboat so I could cut up some of the bigger logs lying in the water. Towards the end of the channel, we came across an entire trunk laying right in our path. Stepping out carefully, I started cutting from the opposite end to where the trunk was still half-wedged into the ground. I cut slowly and carefully until I was suddenly taken by surprise: I felt the trunk give way underneath me.
Plunging into the water with the chainsaw blaring I thought I was cactus. The chainsaw fell on top of my right arm as I hit the water, nipping me with a painful sting. I resurfaced holding the chainsaw away from me with my good arm. Blood was pouring out, turning the water red. Macca leant down and grabbed the chainsaw from me, passed it to Steve and pulled me up.
'Heck, Matty, give me a look at ya!' Mac said, worry spread across his face.
I was dreading seeing the damage. 'I don't know, Mac, but it bloody hurts. Let's make sure I'm all connected still,' I said, jokingly. I was in pain but trying to make light of the situation.
Much to my amazement, after cleaning it up I was relieved to see a large surface-cut – nothing too serious. I got lucky and was even luckier that someone was there to help me back onto the boat, because bleeding out in croc-infested waters is a death sentence.
There was a really dense bit of mat right at the end of the channel before the lagoon opened up and we just couldn't break it with the airboat, we had to go back and forth over it, throwing anchors and big poles trying to separate it. After two hours, we had broken the mat up as much as possible and lifted the motors on Cyclone Creek, so it could glide over without damage. We finally broke onto the open lagoon, and everyone boarded the big boat and opened the beers to celebrate. We put some tunes on and took her up the river for a cruise before dropping the anchor and calling it a day.
Early the next morning I flew the chopper back to the lagoon to do some reconnaissance for the pontoon placement. I landed on the helipad on top of Cyclone Creek and walked down the stairs to see a letter sitting on the kitchen bench with a rock on it.
Please call to discuss plans. I'm at the mango farm all day. David.
I booted straight over in the chopper and met up with David, the owner of the land next to the lagoon, and the shack. I started explaining what I wanted to do, but he didn't seem keen on the idea.
'No tourism here, sorry. It's too dangerous, too many bad people!' he said.
'What do you mean?' I asked.
'People come and steal off me,' he told me. 'The pig hunters, they come and they steal diesel, shoot the pigs and cut my fences.'
He was a lovely guy who'd had a bad run and I felt sorry for him. He told me that someone had even tried to steal his forklift recently but ran out of petrol at the front gate.
'Look,' I said. 'I would love to rent your shack off you and do it up. We would use it and the road down to the lagoon. I'll put up proper security, help grade your roads and make the place look brand new again and I promise there won't be any more trouble on the property.'
David was hesitant, for obvious reasons. He didn't know me, I didn't know him, but Paul from Finniss River Station had put in a good word and so David agreed to give me a go. We worked out a land-use agreement for the first 12 months. The shack had a little generator and a bore, so we had power and water, which is all that matters in the bush.
At this point I had secured the lagoon, the shack and access to the shared road. I was on a roll, and it was time for Brendan's arrival into Darwin.
My mate Macca is a big, burly bloke with an intimidating presence, so I asked him to pick Brendan up from the airport and bring him straight to the shack. Mac knew about Brendan's dodgy business and was furious about the whole thing.
'Don't worry, Matty,' he assured me. 'I'll sort this Brendan guy out and make sure we get your airboat going.' And then off he went.
Troy and I were pressure-cleaning the shack when Macca's car pulled up. Out hopped Brendan with a beer-carton box wrapped around his hips. He shuffled towards the shack door.
'Is there a shower in there?' he asked, on the way through.
'Huh?' I replied.
But Brendan didn't stop to say anything else, he just kept heading for the bathroom.
'What happened, Mac?' Troy asked.
'I just gave him the shake-up he deserved and told him to sort his mess out before things turn ugly. I think the pressure was too much for the poor guy and he soiled himself. He came back to the car with his pants off, smelling like shit, so I had to sort him out with the box,' Mac said.
'He what?' Troy asked.
'His pants. He pooed them,' Macca said, in a matter-of-fact way.
I'd never seen anything like it, but I didn't ask Mac any more questions. I figured he'd laid down the law and he'd got the message through. I chucked a pair of old shorts at the door for Brendan because I didn't particularly want him wandering around naked. He appeared a little while later, speechless, sitting down at the trestle table outside. I couldn't stand to look at him.
'Troy, I actually can't even look at the bloke. I thought I could face him, but the sight of him just makes me sick. Macca's told him he needs to cover our new airboat engine, so that's it from me. I'm not going to waste my breath,' I said.
'I'll sort him out, then,' Troy said, speaking like Brendan wasn't even around. He turned to Brendan. 'Come on, ya flea. I'll take you back to Darwin. You're not welcome here.'
Brendan stood up and walked away with his head down, not saying a word. That was the last I ever saw of the guy. We engaged lawyers to get our money back, but it turned out he had nothing. His wife had taken him to the cleaners and he'd ripped off a heap of other people, so we were just one of the many. We would have spent more in legal fees trying to fight him, but I truly believe that every dog has their day. We had to foot the bill for the new engine from America, which cost another twenty-five thousand US dollars plus the money for shipping, air freight and installation. After the mess-around from Brendan, we worked out that the airboat cost us a quarter of a million Australian dollars. A big blow to a new business, but we kept pushing on.
We finally had the new engine, and got the airboat onto the lagoon. The machine was perfect, but Troy's and my driving was far from it. We weren't at all savvy to start with. On Troy's first expedition he sideswiped a tree, denting the cage, and on my first adventure I took a corner a little too hard and ended up high and dry.
At least Troy made it home after his incident, whereas I ended up in the middle of the rainforest, stranded. In the process of banking myself I cleaned up a tree, jerking the boat, creating an almighty bang as the props went ballistic and the engine screamed until it came to a standstill. I had no mobile, no sat phone and no radio. I knew Steve was on Cyclone Creek with the chopper because I could hear music playing in the distance. I was in prime croc territory, so I grabbed a stick, legged it to an elevated point and started yelling. I yelled and yelled and yelled for a good 10 minutes and eventually the music stopped. I yelled again and then again. And all of a sudden I heard the chopper start.
'Yes, he's heard me!'
I watched Steve fly around, looking in all the wrong places. He was four days into having his chopper licence and the poor guy was on a search and rescue mission AND I was about to signal him instructions for a very tricky landing to pick me up.
He eventually spotted me and I had him come in and land in this extremely tight area, light on his skids so I could jump in the machine and get out of there. He was a little slow, but he did it with precision and I was very impressed.
Then it was time to tow the airboat out, fix it up AGAIN and get it back on the water, which took another couple of days. Believe it or not, things with the airboat improved after that and it ran glitch-free for most of our first tour season.
We bought one of our pontoons second-hand and Troy arranged for another two to be built. Once we got them to the shack, I had to airlift them in an R66 helicopter. I started with the biggest of the three, using a ratchet strap to sling the pontoon under the chopper. It was one I used to tie things down on the back of my trailer. It wasn't the best choice, but it was doing the job fine until a gust of wind came through. The strap blew underneath the main hook, twisted and released itself and began floating down. All I could do was sit there and watch as a ten-thousand-dollar pontoon flew through the air.
'Oh noooo, oh noooo. Please land on the floating mat, please land on the floating mat,' I said as it flipped and floated down further and further below. One section landed on the mat and the other part hit hard into the water, shattering and bending on impact. I landed to assess the situation. It was totally unsalvageable.
I used the airboat to tow it back to the landing. No one said a word. They all would have known exactly what I was feeling and how annoyed I would have been at myself. I secured the other pontoons with double the amount of straps and got them out safely.
I had to fix the pontoons to the bottom of the lagoon. Steve was on the lookout for crocs while I swam down to fix the mooring lines to the cleats. It was challenging enough fiddling with the pontoons and moorings, let alone the fact I was in the heart of croc country. It was bloody scary swimming below the surface because I knew there were 16-foot man-eaters in the area, and I was fully aware of the love that crocs have for engines, after many incidents with my pet Tripod and his interest in metallic grinding sounds!
After an hour of being on edge, I eventually got the pontoons fixed with all limbs intact. We were a week out from launching by this point and Steve, Mac and I finished the day off on one of the airboat rainforest tracks. We were waist-deep in crystal-clear water, cutting the last of the floating branches. I wasn't too concerned about getting chomped on because I could see our surrounds so clearly. I got caught up in the job at hand and dropped my guard. In the corner of my eye I saw this shadow underneath the water coming towards us.
'Holy hell, that looks like a crocodile. Shit, it is! Croc, croc, croc, get out!' I yelled. Ditching everything in our hands we scooted up onto the airboat and out popped this croc, right where we were. She was only eight foot and a skinny little animal.
'Well, aren't you a sweet thing?' Mac said.
'She's sweet when we're out of the water, not so much when we're in it,' I said.
'All right, sweet thing, see ya next time!'
We named the croc Sweet Thing and she became one of the resident crocs at Sweets Lagoon, always popping her head up in the same place to say g'day to guests on tour.
Then it was time to add the final piece of the Outback Floatplanes puzzle, the Polaris. It was too heavy to swing underneath the chopper, so we pulled out the seats on Macca's airboat and putted the new 800-kilogram buggy across the lagoon at a snail's pace.
'Mac, please, take it slow. I can't afford to have this buggy end up at the bottom of the lagoon. We've had enough disasters for now.'
'Yeah mate, yeah mate, it will be fine,' Mac said, without a care in the world. 'I'll take it steady, don't worry.'
Macca's 'steady' is like a bull at a gate at the best of times, but he did us proud and got the buggy safely to the other side of the lagoon. I couldn't believe it. After 10 months of going harder than I ever had before, we had everything ready. I had no cash left to my name, was completely sleep-deprived and had lost a ton of weight from all the stress, but had a sense of pride like never before. I had a dream, took a risk (actually, I took a number of risks), and I made it happen. A lot of people told me I was nuts and I know a lot of people thought that Troy and I couldn't do it, but we did.
Our first tour ran in 2012 with seven guests who absolutely loved it, and things grew from there. We ran tours on top of tours, and the bookings continued to roll in with five-star reviews. In the first six months we had become the number-one tour in Darwin on TripAdvisor and were running three tours a day, seven days a week. It was the best thing I had ever achieved, and it was great for the Territory.
In early 2018, I sold my share of Outback Floatplane Adventures to Troy so I could focus on other business opportunities and filming for Outback Wrangler. It was very hard to part with the business because of everything I put into it, but it still runs today and I'm proud of the small legacy I left behind.
#
The Osborne Islands are found on the eastern side of the Admiralty Gulf in Kimberley country, WA. They are both beautiful and forbidding with large cliffy islands surrounded by rocks, smaller isles and impressive waterfalls. This remote part of the world is best known for its surface pearl farms owned and operated by Australian South Sea pearling company Paspaley. A mothership, barges, Mallard seaplanes, farm workers and crew support the operations.
But Osborne's striking turquoise waters aren't just filled with white gems, they're also home to a high population of sharks and saltwater crocodiles. Divers navigate the ocean's perils with strategies in place to keep them out of the jaws of these hungry predators. But, back in 2014, one particular croc started causing headaches. No matter where the divers went, this curious croc was there, and one day the big fella got too close for comfort. A bloke was sitting on the edge of one of the barges and got a rude shock when the croc launched up and missed him by an inch. They decided that something had to be done.
Paspaley didn't feel right about getting a permit to kill the croc, so they called me up to see if I'd be interested in relocating it. I'd never faced a challenge like it – catching a saltwater crocodile in open ocean was no easy feat. I had no idea at the time how the hell I'd manage the job, but I wasn't going to pass up this unique opportunity. And besides, the job had all the right ingredients for my favourite thing in life: adventure.
'I'm all over it!' I said down the phone to the Paspaley operations manager. 'Leave it with me and I'll get back to you with a plan. Just one thing, though: do you think I'll be able to film it for the show?'
'I don't see why not,' the Paspaley bloke replied.
With a shit-heaped grin on my face, I hung up the phone and got planning.
Paspaley was going to supply me with all the equipment and a dedicated Mallard seaplane to relocate the crocodile to the mainland. I also spoke to the Indigenous traditional landowners to let them know what was happening, contacted the Parks and Wildlife Commission of the Northern Territory, and asked Willow and Jono to give me a hand with the logistics of it all.
I came across my first hurdle during my meeting with NT Parks and Wildlife.
'Where are you going to put the croc once you catch it, Matty?' the head ranger, Tommy Nichols, asked me. 'The croc is in WA waters, not the Territory, so I don't like your chances of getting it back to a park here.'
'Shit,' I replied. 'I didn't think of that.'
I got on the blower to WA Parks and Wildlife, who confirmed what Tommy had said. So, my next port of call was the Malcolm Douglas Crocodile Park. Malcolm, who was famous for his 1968 Australian doco series Across the Top, opened a popular croc park near Broome's Cable Beach in 1983. After his sudden death in 2010 all of the crocs moved to the current park with better facilities, which his wife, Valerie, continues to run.
The new park is located about 15 minutes outside of Broome, and was going to be the closest and most practical option for the Osborne Islands croc. Valerie didn't want a bar of me or the croc, but unfortunately for her she was obliged to take it because WA Parks and Wildlife requested we keep it in WA and send it to her park, which was the only suitable place with capacity at the time.
'I don't want to be a part of your bloody filming spectacle. I've got no time for you or this,' she told me when I called her to make arrangements.
'Sorry we're not seeing eye to eye here, Valerie, but I look forward to seeing you in a couple of months when we drop the croc off. It will be great to meet you,' I said, trying the 'kill them with kindness' technique.
'Whatever, you idiot,' she replied, and hung up abruptly.
It was a shame she was so hostile. I had the croc's best interests at heart but it seemed like she didn't give a shit about it. Nevertheless, she signed the relocation agreement, so I had that box ticked.
It was the height of the wet season in 2016 and Jono and I were in the thick of collecting croc eggs, so Willow got the guernsey to do the reconnaissance. I sent him over in the Mallard seaplane with about 500 kilos of frozen pig legs stored in eskies to start baiting traps on floats. He was also going to set motion-sensor cameras along the main creek and around the outskirts of the biggest waterhole.
Willow lost a few cameras to big tides, but he managed to get footage of a couple of crocs, one of which looked like the guy we were after. More than anything, his visit gave us good intel on the crazy tides we would be dealing with, and we decided it was necessary for all three of us to go over.
We constructed four traps and got them on the Paspaley boats. We figured there would be floats there that Paspaley used for the pearling lines, so we could just attach them to the traps. It took a couple of weeks for the traps to get there, then everything was in place.
We had caught a couple of crocs at a cattle station during egg-collecting season, so Jono stayed back to organise their relocation and I booted over to the Osbornes with Willow and our cameraman, Ash. After four hours scouting the area, the reality of the job hit. The tides were so extreme – they came in and out aggressively – and looking out at the vast open ocean in front of us was overwhelming.
'Right, this is going to be one hell of a mission and we won't be able to get the job done without Jono. We need him here and we need him here now,' I said to Willow.
'Mate, the workers on the boat said there aren't any more planes coming until the next shift change in a week's time,' he replied.
I groaned. 'Ya kidding? Can we charter a light plane here? We need to get the croc now because of the way these tides are.'
Willow kept breaking the bad news to me. 'Nope, there are no airstrips. That's why they've got the Mallards, because they land on water.'
I sat there for a minute, thinking, and an idea came to me.
'Matty, you've got that sparkle in your eye like you're up to no good. What's on your mind?' Willow knows me well.
'Well, Jono has an A licence for skydiving, so let's charter a plane and get him to jump in. It'll be sick, just like Mad Max.'
Willow laughed, and shook his head. 'More like Mad Matt I reckon, mate.'
I don't think Willow believed me, but that arvo I called Jono, booked a plane and we made it happen.
'Check on the winds, mate, and let me know the best time to land. I'll send you the GPS spot on this little stretch of beach that I reckon will be ideal for landing,' I said to Jono over the satellite phone. I was pumped, it was all coming together. Just how I liked it.
We had our plan in place. Willow and I headed to the beach right after lunch and got the flares ready at the coordinates we'd given Jono. I had a handheld radio, so I could communicate with the pilot as they got closer. We heard the plane fly above us, and then, at 5000 feet, out popped Jono. It was game on.
He had a GoPro on him to capture the footage of the epic Kimberley scenery on the way down. He's the first and probably the last person to ever jump out of a plane over that part of the country.
Everything was going to plan but just as Jono got airborne the wind changed direction. Fuck. We had planned everything out depending on the way the wind was blowing, so when it changed unexpectedly, panic set in. I was worried, so I can't imagine what Jono was thinking when I let the flares off and he saw the way the wind was blowing.
'Shit, old son, there's not many options here,' Willow said. 'If he can't work out how to land on that narrow patch of beach it's the water or the cliffs, none of which are favourable.'
I was thinking the same thing. We were in a bit of a situation. The wind was meant to be running down along the skinny stretch of beach, so Jono could land into the wind and use the beach as a landing runway. Now that the wind was blowing perpendicular to the length of the beach it meant that he had to nail the landing on about five metres' worth of sand, otherwise he'd end up in the ocean, meeting our croc prematurely.
Jono came in closer and closer then turned his parachute in the air, coming into the wind the right way and just missing the top of the cliff. My palms were clammy with nerves. He skipped down on the rocks and over the wall to the beach. He was completely on point and came in sideways to the beach like a pro.
The whole time I was thinking, This is nuts, we might be plucking him out of the trees here or scooping him out of the ocean. I was second-guessing my wild idea, but it turned out okay, thank god, and he landed right next to our red flaring smoke.
'Woohoooo yewwww, Jonsy!' Willow and I ran over to him, giving high fives all around.
'Yeah, the boys!' Jono yelled back, smiling from ear to ear.
'Now let's got get this croc, hey?!' I said. We were high on life and ready to hook in to get the real job done.
Running Jonsy over everything quickly, we rebaited the traps in the areas Willow had scouted and got our first sighting on camera that night. There was the croc, coming and going from a waterhole at the top of a rocky creek that ran down into the ocean. We wanted to lure him to the waterhole rather than in the ocean's open water, so we set three traps around the big waterhole in the little channels, nooks and crannies. We got back on the Paspaley pearling mothership Roslynne for dinner, and the chefs cooked up fresh barramundi, which Willow and I had caught the day before. The fishing around the islands was second to none: there were one-metre-long-plus barra cruising the waters by the truckload, so anytime we had a lull during the day we were out fishing.
The next morning, we took the harpoons along to check the traps, but I didn't like our chances of using them. It would be tough to harpoon anything in that setting because of the rocks, mangroves and barnacles. Even though we had set the traps, the waterhole itself worked as an environmental trap for us. Because saltwater crocs can live in saltwater but need fresh water to survive, they would proactively make their way up to this big open freshwater waterhole. And on a high tide, the water from the sea would rise right up to meet it, making it even easier for crocs to access. We walked over there with some of the Paspaley workers who hung around to check out the action. They told us that sometimes crocs would get stuck there high and dry when the tide went back out because, without water to lubricate the sharp rocky creek down to the ocean, it was far too rough for a croc to move along. It was our plan to keep the croc up in the waterhole with attractive bait until it got landlocked.
The other helpful part was that the waterhole would be dirty at night-time when the crazy tide hit, but then it would completely clean out in the calm of the morning, making it clear and extremely easy to see. That's how I managed our first sighting in the flesh.
'And there he is, lads!' I could see the outline of a large croc at the bottom because of the clear water but it was only partially in view, so I wasn't exactly sure if it was the right one. I took off my boots and slid into the water.
'Matty, what the hell? Are you serious?' Willow's unnerved voice called out.
'What?' I shouted back.
'You can't get in the water with the bloody croc, you're asking to get eaten! He's got the upper hand, big time.'
Jono stood there silently as Willow and I argued.
'Of course I can,' I objected. 'It's so clear, I can see everything.'
'Don't, please don't do it.'
I didn't hear the rest of Willow's pleading as I submerged myself in the chilly water to get a better look at the croc. The water was shallow in parts and deep in others, with little rocky ledges around the edge. The croc had wedged himself under one of the ledges, which is why I couldn't see his full body from out of the water. I needed to get closer to him to see if he was the croc we were after. I returned to the surface and, without even having to ask, there was Jono handing me my mask and snorkel.
'I got you, mate,' he reassured me.
Jono was standing above me at the edge of the waterhole while Willow had climbed the big formation of rocks at the back of the waterhole to get on high ground. He was pacing around like a worried parent.
'I'm going to swim right down and get a nose rope on him underwater,' I said to Jono. 'I reckon I can do it, I just need you ready to pull hard once the rope's on. He's sitting on the deepest point at the bottom, wedged into a little concave in the rock. It will be fine.'
I didn't feel nervous. Actually, I felt really focused and calm, and very connected to the croc's energy. I could tell the animal was on the run, not on the hunt.
'He's trying to hide, fellas. As long as I don't swim right up to his head I'll be okay.'
'How deep is it?' Willow called out.
'About seven feet down, I reckon! This is where all my years of free-diving on the Great Barrier will come in handy!' I yelled back.
'Go for it, Matty. I'd be in the water if it was my TV show but I'm just here to support ya, mate,' Jono said, laughing.
I laughed along with him because I knew he'd be dying to get in the water with a wild croc too, but only one of us could do it without causing too much commotion.
'Let's harpoon him and get him that way!' Willow tried convincing me otherwise.
'I'm not 100 per cent sure it's the right croc, Willow, and there's no chance with the harpoon with all these rocks and the way he's positioned.' There was no talking me out of this one. 'Willow, you keep an eye out up top and make sure there's nothing else in the water with the big guy. That's the last thing I need,' I said, and then down I went.
I chased the croc around in circles along the bottom of the waterhole but he wasn't moving out from under the ridge so I couldn't manage a clear sighting. I'd lose him momentarily every time he disturbed the ground at the bottom, creating a cloud of sand and turning the water murky. Then I'd hear a faint 'He's coming back at you!' from Willow above me. The snorkel and mask meant I could stay underwater to follow him, but I'd lose him again each time he tucked into a deep ledge. It had become a game of croc chasey. I was just glad I was the one doing the chasing. Finally, I got the croc up onto a shallow ledge situated right on a corner, which I could use as a barrier between the croc and me. I peered around the corner and saw that his tail was within reach. I gently floated to the surface to see Jono standing above.
'This is epic, Matty,' he whispered. 'You need to take the GoPro with you and get footage of this, it's nuts.'
Ash came over and I handed him my Leatherman knife. 'Gaffer-tape the GoPro to the knife so I can use it as a stand for the camera.'
I swam down and stuck the knife with the GoPro into the sand, angling the camera towards the croc. I then went back up, got the head rope and a stick, and lowered myself into the water. I poked the croc on the tail, provoking him so that he would turn towards me and, sure enough, he spun around to face me, knocking the camera flat on the ground. I can't really remember what I thought, coming face to face with a wild croc underwater, but I don't recall feeling nervous. I just did what I knew I had to do, which was getting the head rope around him as soon as possible.
He closed his jaws slowly, which was not what I wanted because I needed them open to be able to get the noose around his snout. I got the stick and jabbed him gently on the side of his jaw, which made him turn slightly to the left, and then I looped the rope around his snout. We were on.
I got out of the water as quickly as I could, tugging on the head rope in the same direction as Jono, who was pulling on the end of the rope from above water. Willow was scampering down the rocks to get on the end of the rope and help us hold the croc without all of us getting pulled into the water.
The croc was going nuts. It was thrashing around, snapping about and putting up an almighty fight. Ash was running around us, trying to get shots of the croc under water as well as shots of us pulling the croc out without slipping on the rocks or getting in our way. It was mayhem.
We found a flat surface where the water overflowed down to the creek bed. It was the perfect spot to pull the animal out. The workers called in the Paspaley operations manager to identify the croc. After a bit of manoeuvring we finally hoisted the croc up and onto dry land. And by this time the ops manager had arrived.
'Do you want the bad news or the bad news? This isn't the right croc,' the ops manager said to us. 'This one is missing part of its tail and the one we're after has its tail fully intact.'
Our excitement turned to disappointment. Jono eventually piped up. 'Oh well, that was one hell of an adventure and good practice for when we get the real guy! Let's let this fella go, then.'
It was bittersweet releasing him after the rigmarole we went through, but he wasn't a problem, so he deserved to be left alone. We walked him back along the creek bed like he was a dog on a leash, pulling him along and keeping tension on the rope. It had enough water to ease him all the way down to the ocean, where we let him go. He swam off happily.
'I think we need a beer,' I said to Willow.
'Roger that!' he replied.
We were done for the day.
A croc-catching mission usually takes a lot of waiting around, reassessing and changing of plans until you finally manage to get the croc you're after. This trip typified the waiting-around part, but we didn't mind. Usually in the Territory things drag on in the stinking heat, but this time we got to hang out in a Kimberley paradise with incredible swimming holes, crisp sea breezes, unbelievably insane fishing and four-course meals cooked fresh every day by the talented chefs on the Roslynne.
Each morning we'd check and rebait the cage traps near the waterhole then we'd go fishing, eat and have a few beers. In the early arvo we'd scout the area again, recheck the traps and hope there was a croc waiting there. We did this for a few days, but the croc was nowhere to be seen. For such an apparently inquisitive croc it didn't make sense that he'd gone to ground. We felt like we were encroaching on the workers' space too much, so we made the call to fly home.
We woke up on our last morning and shot straight down to the bottom of the creek that connected up to the waterhole. The tide was as low as we'd ever seen it. Jono, Willow and I had to walk at least a kilometre through the muddied rocks and barnacles before we reached any water. The tides had shifted the traps all over the joint. One had capsized and tipped over, another was wedged up in a tree, and one was on its side on another ledge in the waterhole. We took a closer look.
'Matty, look!' Willow yelled. 'It's a croc!'
Willow was right. A decent-sized croc was inside the third trap.
'It's got a full tail on it too!' Willow said.
'You beauty! I think we've got him.' I said, heart racing with excitement.
Jono radioed back to the Roslynne and let them know we had the croc. A couple of Paspaley employees turned up 30 minutes later.
'That's the one!' one of the workers said when he saw croc.
They called a crew who were back in Darwin and arranged for the next available Mallard to fly the crocodile box over for the trip to Broome. The croc was about 13 feet long, skinny, and a full-blooded little thing. We secured him in all the correct places and carried him down the creek and, with the help of the workers, onto the tender.
'The Mallard is coming in to land now. They've taken the seats out of it and put the croc box in ready to transport out of here,' one of the workers confirmed with us.
We collected our gear from the mothership and waited on the tender, ready to go. The pearly white Grumman Mallard seaplane approached in the distance. Watching the amphibious aircraft land on water is truly spectacular, and surprisingly quiet. Only a faint hissing sound can be heard as the spray of water splashes against the turbine. It is an impressive piece of machinery and the plane's white exterior really pops against the blue waters.
I administered the tranquilising drug to the croc and we got him into the box on the plane smoothly. Lots of 'thank you's were coming through from the Paspaley workers, who were relieved to see the animal moved on. An hour and a half later we were all in Broome, safely handing the croc over to Valerie. She was still less than impressed with the whole idea. But she had the space and, as much as we didn't get on, I knew she was an animal-lover and would give that Kimberley croc a good life.
#
'Someone is going to get eaten soon, it's only a matter of time.'
Harold's soft voice echoed across the delayed phone line.
'Is there one croc or are there a few problems out there?' I asked him.
'Well,' Harold said, 'most of the crocs are big enough to eat any of the kids who play near the water, but there's one monster that has us all worried.'
'How big?' I asked.
'Big,' he replied with certainty.
Nobody knew the land and the crocs around Peppimenarti quite like Harold. A local traditional owner and Aboriginal Elder, Harold was born in Peppi and returned to his hometown as a young adult to raise his family. He had lived there ever since.
'Are we talking 15 foot big or 18 foot big?' I asked, trying to extrapolate more information from him.
'I reckon more like 20. He's been around since I was a kid,' he said calmly.
Nine times out of 10 I call bullshit when someone says there's a big croc, but I believed Harold. He then told me story after story about the close encounters community members had experienced with this colossal croc. And the more I heard, the more I knew that it was a job for me.
I planned to fly to Peppi later that week to meet Harold and scope out Tom Turner Creek, where the croc allegedly hung out. The creek ran directly alongside the small Aboriginal community of Peppimenarti, located just over 300 km southwest of Darwin. About 200 people call the town home, most of whom are local Ngan'gikurunggurr people.
Coincidentally, on the flight down to Peppi my manager, Nick Fordham, called.
'I've been chatting with 60 Minutes and they want to fly Peter Stefanovic up to Darwin to do a segment on you and your work,' Nick told me. 'They're hoping to film soon, so we need an interesting story.'
'I think I have one,' I said. 'I'll get back to you tomorrow.'
Station owners and locals often call me to relocate crocs. I try to do as many jobs as I can and we film some of the more exciting relocations for Outback Wrangler. Every now and then a current-affairs show, a blogger or a print news outlet wants to do a special feature story, but it's often hard to give the media what they want because I can't just create a story or a problem croc out of thin air. So the timing this time around was a perfect, but only if Harold was up to it.
Harold's warm smile greeted me as I landed the chopper.
'Let me introduce you to our people,' he said. Harold showed me around proudly, stopping me along the way to meet his family and friends. The main language spoken in Peppimenarti is Tyemirri, so Harold translated as everyone told me stories about their encounters with the infamous local croc.
A couple of women carrying newborn babies came out to see me. They started calling things out to me, sounding distressed.
Harold translated: 'They are telling you that the family dogs go down to the creek to drink, and not many come back.'
'I'm sorry to hear that,' I said.
'We've lost three dogs this month,' Harold added. 'The women are scared for their young kids, and the young fellas who fish down there have been stalked a bit too.'
Harold and I walked around to the back of his place with another young lad to check out the surrounds. The creek was Harold's backyard – about three metres from his doorstep was the bank leading down to the water, and in between the two stretched an unfenced patch of grass with a kids' plastic playground, basketball ring and sprinkler, which I imagined Harold's grandkids played around to cool off. It was a recipe for disaster.
'Harold, can you show me how big you think this croc's head is?'
He pointed to me. 'The size of your torso.'
The young fella nodded in agreement. 'Yeah, when I saw him out of the water last year his head was at least a metre.'
The reports were all the same: this was a bloody big croc.
Harold confirmed that he was happy for us to film everything, so I went home, locked in 60 Minutes and returned to Peppi a week later with Willow. I'd just got a new land cruiser, so we took it for a burn and navigated the unsealed roads out to Peppi. We kept an eye out for feral pigs on the way to use as bait. There weren't many around, but we did spot a dingo on the edge of the scrub near the road.
'Shit, take a look at that thing,' Willow said, scrunching up his face.
The right side of the dingo's face had opened up and a cyst the size of a softball was protruding from it. The wound was bloody and infected, and the flesh was hanging from the dingo's face. We waited in the car and watched as it staggered across the road in front of us. It was half-dead, the poor thing.
'That is so messed up,' Willow said. 'Is it diseased or what, Matty?'
I shook my head. 'I don't know, but I think we need to put the poor bastard out of his misery. He's in a hell of a lot of pain and there is no saving him.'
'Righto.' Without any more talk Willow got out of the car, pulled out his 44-Desert Eagle and shot the dingo in one hit. He walked over, picked it up by its legs and dragged it off into the bushes, hidden away to lay to rest.
'That dingo looked like it was out of an alien movie,' I said to Willow as we drove off. 'I've never seen something so mangled. Good thing it's at peace now.'
We managed to find two pigs for bait just before we reached Peppi, arriving mid-afternoon. We drove straight to the creek bed. I was hoping we'd find any trace of the croc, so I could show Willow what we were up against.
'Flipping hell, Matty, check this out.'
There, clear as day, was a thumping crocodile slide mark stamped into the mud. There was a crocodile footprint too. It was at least six times the size of Willow's hand and the markings were noticeably bigger than the tracks left by my 17-foot pet croc, Tripod. There was no denying we had a monster on our hands. We put the bait near the slide marks and set up a couple more cameras in the trees. I caught myself looking over my shoulder every now and then, which I don't often do, but an animal of that size could swallow me whole. I decided not to put a trap out because I wanted to get footage of what we were dealing with first, so we set up the usual cameras to get some intel.
The location was not where we would normally catch a croc; it was in a township, not in the middle of Woop Woop, so we even had access to wi-fi. This meant that our sensor cameras could send live notifications and pictures to Willow's phone every time there was movement.
His phone started going nuts that afternoon when we were back at Harold's place, but it was a false alarm. We watched as a couple of smaller 12-foot crocs entered the area to suss the bait out. While we waited, Harold took us to the community's workshop and salvage yard to get started on building the trap.
'Are you kidding?!' I exclaimed, looking around the workshop. 'You've got everything here that we need to make the trap. This makes things a hell of a lot easier.'
The workshop was about 200 metres from the creek and was stocked with a mountain of steel, old cattle pens, thick mesh and fencing: too much to choose from, really. Every now and then, Willow's phone pinged with a notification about movement down at the creek, but nothing worth taking notice of.
'Still no sign of the big boy, Harold,' I said.
'Oh, the Baru is in there, just give him time and he'll show himself,' he said.
'Why don't we work here over the next week welding up the traps and in that time hopefully we'll get some good footage,' suggested Willow, and I agreed.
Aboriginal people call the saltwater crocodile Baru, which means 'crocodile's ancestor'. Many local people in Peppi hold crocodiles as their totem and are responsible for looking after them, so harming one in any way would not be acceptable. Even if the croc ate a person, the local people still wouldn't want the animal shot, and that's why they were keen for me to safely relocate it.
We based ourselves at Elizabeth Downs Station, which was only an hour's drive to get to Peppi each day. The welding was more work than we had anticipated but we built a bloody good trap door, and I already had a couple of traps in Darwin that I planned on bringing back to use as well.
Each morning Willow and I rebaited near the creek, then got to work. The sensor cameras continued to reveal that there were several crocs in the area – and big ones at that – but none big enough to wow me and make me think it was the one we were after.
Once we finished welding the main trap, we headed back to Darwin and I let Nick know that we were ready for the 60 Minutes crew to arrive. Then I drove back to Peppi ahead of the shoot to set up the big panel trap, using old pens from cattle yards and other large sturdy pieces of steel, which can be welded together to create a trap that's more like an enclosure, with a door that releases when the bait is pulled on. This time Jono came along to help me erect the trap, tying up the trigger rope with the bait attached on one end and the trap door on the other. This meant there would be no mucking around once the camera crew arrived.
'There are some big marks over here, Matt, check it out.' Jono tiptoed around the trap's steel panels and pointed to the bank on the opposite side of the river. There was a similar-sized slide mark to the one Willow and I had seen a few weeks earlier along the bank.
I nodded. 'I reckon we've seen just about every croc in the area on camera except the big one, but he's leaving a good trail for us.'
Setting up the trap was a mission, but we got there in the end and then booted back to Darwin.
The cars were packed, the traps were stacked and Willow, Jono and I were up early the next day to greet the small 60 Minutes crew from Sydney. After a briefing and a lot of questions, the crew hit the road for the four-hour drive. I flew out with Pete Stefanovic, Jono and Benny the cameraman to have a yarn and get a feel for the story.
'As you've heard, there's been a ton of reports of crocs getting too close for comfort, so it's our job to go in and move them out,' I told them. 'Unfortunately, we can't relocate these ones back into the wild elsewhere because they're deemed as "problem animals". If we catch them and I can't get them into the local farm, I'll take them back to my joint and build a pen to keep them safe there.'
We made a quick stop at my bush shack and visited the lagoon that backs onto it. I got Pete out on the lagoon in the airboat with me and some of my crocs that I work with regularly. I wanted to give him some hands-on croc experience before the real deal, but I don't think the poor bloke knew what he was getting into.
'All right, Pete, jump off here and come say hello to Bone Cruncher,' I said, parking the airboat along the bank.
'What, me get down there WITH you?' he asked, uncertainly.
'Yep, that's the one.'
Hesitantly, he hopped onto the ground and stood close beside me as I fended off Bone Cruncher, a 14-foot resident female croc.
'So, Pete, the biggest thing when a crocodile is coming at you is to always make sure you've got an exit. You've got a tree behind you now, so just be aware of that, and always check your surroundings while you're moving about.'
Pete shuffled from foot to foot, looking around. 'Matt, I just am not sure exactly what to say right now. There's nothing between us and this apex predator but a stick.'
'Correct,' I confirmed. 'But in this particular situation we've got the advantage. We're up high on a slope so we've just got to make sure we stay out of the kill zone and we've got nothing to worry about.'
Pete didn't seem overly convinced but we finished feeding Bone Cruncher and I was glad he got a taster of what was in store out at Peppi. I also made sure we stopped by Tripod's pen on the way back to the chopper.
'Pete, please meet Tripod. Tripod, this is Pete. Tripod, please don't eat Pete.'
Pete let out a nervous laugh.
'I can't quite believe what I'm seeing right now to be honest,' he said to me as we got back into the air.
'Mate, you ain't seen nothing yet,' I said.
Just outside of Peppi, we spotted a good mob of wild pigs from the chopper, so I landed us and shut the machine down. We needed a few pigs as bait – one for the big panel trap and three more for the smaller cage traps I planned on slinging into other areas of the creek. We were upwind, so the pigs couldn't smell or hear us, and we continued to creep up as they moved about on a patch of grass at the edge of a water channel. Slowly, we crept forward, hearing the pigs' wallowing getting louder and louder. All of a sudden, a big boar noticed us and started storming straight at me, so I let my first shot off.
The pigs began scattering everywhere. I let a few more rounds go and in the flurry of bullets I downed four more pigs. We dragged them back to the chopper and tied them up, ready to sling out. I couldn't manage to lift them all, so I took the legs off one of them. I dropped the legs off, along with Jono, Pete and Benny, then returned for the rest of the hogs.
Once everyone had arrived at the creek, I introduced Pete to Harold. After some brief hellos Pete got Harold's take on it all.
'How scared are you?' Pete asked him.
'Very scared, very concerned,' Harold replied.
'Do you think it could take someone?' Pete continued.
'Yes, that possibility is real. It's scary,' he said. 'There are a few crocs here we would like moved out, but one of them is really no good.'
I explained to Pete that we would also be setting up a couple of smaller cage traps to capture the crocs that were lurking around on local sacred sites used for men's business. Aside from local Aboriginal men, no one else is allowed to visit these areas.
'This will be the first time a white man has ever visited these areas,' I explained to camera. This illustrated more than anything how desperate the locals were to have these problem crocs caught and moved out. 'Harold has asked me to place two traps in this particular spot in an attempt to move out a couple of the smaller crocs who have been hassling the local men.'
I baited and set two of the traps into the area, but we were forbidden to film any of it.
Then there were two more cage traps to bait and put down the other end of the creek, near a swimming hole. Willow baited the first trap while I chucked a pig leg up the back of the second trap and invited Pete to lend a hand.
'Now the pig leg is in there, we have to crawl into the back of the cage and secure it,' I said to him.
'Who's doing that?' Pete asked.
'YOU,' I told him.
A smile edged onto his face. He got down on his hands and knees.
'Don't shut that gate on me,' he said.
'No, no, I won't,' I reassured him, with a twinkle in my eye.
So Pete crawled into the trap and of course I let the gate down immediately, locking him inside.
'Yep, the trap works,' I joked.
There was leftover bait at the back from weeks before, so it stunk like all buggery in there. Pete dry-retched but handled it like a trooper, laughing it off. He was proving to be pretty easy-going and adaptable.
Once the remaining cage traps were baited, I slung them into place and explained the difference between the two types of traps.
'The large panel traps are for your 15-foot-plus crocs and these cage traps are for your smaller ones. Now let's go set up the big one.'
Willow showed Pete some of the markings on our way over to the main trap. 'See the marks here? That's his hands and that's his claws. These are fresh tracks left just a couple of days ago, I'd say.'
'Yeah, right, that is a big animal. It looks like a dinosaur print,' Pete observed. Then he fixed his attention on something else. 'What's that? Is that a gun in your pocket?' he asked, pointing to the shiny gold Desert Eagle sticking out of Willow's pants. I'm sure he was half taking the mickey out of him.
Willow defended his case. 'Yeah, mate, you need one of these out here. I've never had to use it before but it's just in case a big fella gets a hold of your leg.'
'The man with the golden gun,' Pete proclaimed, and Jono and I had a laugh.
Willow shrugged. 'I've copped a lot of slack about it, but I bet the lads won't tease me if I ever had to use it.'
'Fair enough, mate,' Pete said politely.
I took Pete through the ins and outs of catching a crocodile, and the mechanics of the large panel trap, showing him how to tie a pig to the trigger rope. His face lit up as I demonstrated to him how to navigate everything. I imagine it was all pretty fascinating for a city lad.
The area the trap was erected in was a perfect little dam off to the side of the main creek; the entrance was exactly the right size to fit the big trap door.
We walked around the upper rows of the panels, making sure everything was tied up tight enough. Then we climbed onto the outside of the trap to make our way back onto dry land. Pete lost his footing a bit moving about.
'DO NOT fall back, whatever you do,' I warned him. 'There's a good chance the croc is lying right there.'
Pete understood, and held on tighter. I lifted the trap door up with the help of everyone and set the trigger rope. We had the trap set and ready for action by the afternoon.
'The waiting game begins, guys,' I told them. 'I'll be surprised if we don't catch a croc in one of the traps straight-up, given the time of year. In the dry season, crocs go dormant and don't eat as much, but now that it's the build-up to the wet season they've started mating and are very hungry, very horny and more aggressive.'
I flew back to Elizabeth Downs Station with Pete and Benny, while the others drove. We all crashed around the outside table with a beer in hand.
Ping, ping, ping!
Willow's phone started going nuts. He passed it around for us to have a closer look, and we saw footage of big splashes and pig guts flying about the place: the trap was alive with activity. This went on for a while and then the trap door finally dropped down at 9 p.m., securing something inside.
'Let's hope our guy is in there,' I said. 'It'll be an early start tomorrow – you mob will get trucking at 5.30 in the cars to get to the trap at first light. And we won't be far behind in the chopper.'
The next morning the anticipation was high, but we were met with disappointment. There was nothing in the trap. I scouted the area and could see where the croc had got out by the tracks that went from the water's edge, up over the patch of dry land and though a gap in one of the cattle pens. Willow's footage showed a decent-sized croc but, judging by the size of the gap in the pen, it couldn't have been bigger than 15 feet. Harold was keen for us to catch and move anything larger than 14 feet, so I reinforced the panels with mesh to keep anything and everything inside the trap's walls. We put more pig meat on the trigger and were done by 10 o'clock that morning. There was nothing else we could do, so we drove back to Lizzy Downs.
We had just sat down to lunch when Willow's phone ran riot again. Less than two hours after we'd left, the croc had gone in and set the trap off. We scoffed down our food and jumped in the car for another hour-long trip.
We convoyed all the way back to Peppi and pulled the croc out. The animal was about 15 foot and painfully malnourished. Given that he was a hungry croc and would keep on causing headaches, we had to move him along.
'This one can go straight to the farm, Willow,' I instructed.
'Roger.'
We pulled him out and Willow arranged for one of the workers from the croc farm to bring a truck and drive him to Darwin.
The next part was the same – reset, rebait, repeat. Then we returned to the station.
We were sitting around the campfire watching the sun set over the Lizzy Downs floodplains and spinning yarns about the day when Willow came outside and let us know we had another croc.
'Another early start, team,' I told them. 'The trap door is down, and we've got a big animal in there.'
It's quite unique seeing two big crocs going into the same trap within 24 hours of each other. Even after the trap door dropped down, the croc continued to eat an entire pig and didn't at all seem fazed that he was locked up. The croc was completely motivated by food, another hungry animal in kill mode.
We went to bed on a high, eager to wake. The crew hit the road and the usual suspects went with me in the chopper at first light. Pete and I spoke to camera while Benny filmed us and got some sweeping scenery shots. I flew over the trap to see a croc resting on the water's surface.
'He's got a big head on him but still doesn't match the mammoth description I've been given,' I said to camera.
We landed and walked down to the trap nice and quietly and waited for the rest of the crew to rock up with the trailer, mattresses, harpoons and sedative drugs. Then we got into it. I poked around, and the croc roared about. There wasn't much room to move inside the pen, so Benny was sitting up on the rail filming. The croc hurled out and hit the rail halfway up, right at Benny's feet. The cameraman lost his balance and wobbled back and forth, like a seesaw. I was already in full flight towards him, ready to harpoon the croc or lure it towards me, before Benny fell right on top of it.
But, by a stroke of absolute luck, Benny's balance threw him ever so slightly backwards and he fell outside of the trap, landing amongst the spikey pandan plants and loose logs. He had prickles, twigs and mud all over him, but to his credit he got right back up on the railing and continued filming Pete and me, who were walking on the outskirts of the dam within the trap. I secured the harpoon line in the croc, to bring its head above the water. I pulled it in as it put up a fight, swimming about and resisting me. Without warning Pete stumbled backwards, looking at me with confusion.
'Shit, Pete, move! Jump up on that railing!'
Pete shot up immediately. The dam's edge must have been underhung in the exact spot where Pete was standing. The croc was pushing its nose up from underneath Pete, trying to break free. Pete was one lucky bastard because that croc could have very easily broken that sandy barrier and snapped his feet off! The croc resurfaced in the shallow water, hurling itself onto the air, trying to snap on anything and everything.
'Whoa, he's a nice strong animal, lads!' I called out.
It took a fair while to get the croc out of the waterhole, but we got there eventually and Jono put a head rope on him. Willow opened up the panel traps and we had everything ready to get the beast out.
By this time the whole community had gathered around to see what was going on. The day before we'd left pretty quietly, but now the word had got around, we had the local coppers, nurses, teachers, store owners and others young and old watching us.
I didn't realise at the time, but 60 Minutes had worked with my manager, Nick, to secretly fly my mum up from Adelaide to surprise me for the segment. Mum had raised me in the country and taught me a lot about living off the land and working with animals, but in the 20 years of me living up north she had never seen me catch a croc in person. So, right in the middle of it all, mid-harpooning and knee-deep in water I turned around to see my old girl standing there, looking at me with complete pride. I beamed back at her. That small moment meant a lot to me and to her.
As the croc catch unfolded, the main camera was on me while a second camera was on Mum, who was getting asked to commentate. Each time I looked over at her she smiled: she was loving every minute of it.
At one point I heard her say, 'He should know what he's doing by now, so if he screws up it's his own bloody fault.'
'Thanks, Mum!' I called back to her, laughing.
The croc kept retreating to the bottom of the trap and there was only really Willow, Jono, Pete and me there to muscle the croc out. I needed to pull him up enough to get the needle with the drugs in him.
'Go and grab the car, unhitch the trailer and we'll try and winch him out,' I said to Willow.
There was a steep embankment above where we were, so the car could have rolled in a heartbeat, but Willow navigated it well, and reached us safely. I usually don't like using a winch because it's too slow for my liking, but we didn't have another option in this case. I managed to attach the winch line to the croc, no worries.
Each time Willow activated the winch to pull, we'd tug on the line, lifting the croc's jaw so it didn't get caught on anything. I was trying to get the croc out as quickly as possible because it was getting hot and I wanted to avoid a build-up of lactic acid in the croc's body, which can be life-threatening. As Willow reversed the car up the hill, the wheels began to spin on the spot, getting bogged in the loose sand. He slammed his foot down and the car hopped backwards and into a pile of star pickets, puncturing a tyre.
Ya kidding me, I thought but I held my tongue. There was no time for a blue.
I had to get all the lads on the end of the rope still attached to the croc as well as some young strong Aboriginal blokes to keep holding it taut, while I helped Willow replace the tyre and reverse up the hill with the croc in tow. One thing's for sure, catching crocs is never smooth sailing.
We finally hauled him out onto the embankment, got a needle into his tail and put hessian over his eyes to calm him down. I let him sit for half an hour in the shade and then it was time to get him on the trailer.
Ten local Aboriginal ladies appeared out of nowhere and came down close to the car and began wailing for their totem. I didn't understand what they were saying, but they were crying and sounded like they were in pain. I had aimed on creating next to no commotion, but I wasn't doing a very good job at it. Harold consoled the ladies and assured me it was okay to carry on.
'They are sad to see him leave but they know it is the right thing, and they are blessing him and wishing him a good journey in the future,' Harold told me.
We hooked the car back to the trailer at the top of the embankment. The croc was up on the flat, ready to go. Jono got the mattress prepped and wet down the croc because it was a pretty warm day. We kept his head visible as well, so we could monitor his breathing. When a croc is anaesthetised, you have to watch the animal's nostrils, ensuring they shut every minute or so at least.
At this point, I pulled Harold aside to give him my thoughts. 'I still don't know if this is the croc you're after.'
Yes, the croc was a big croc at 16-and-a-half feet, but it wasn't the 20-foot giant they'd been talking about.
'I'm not sure either, Matty,' he replied.
'Well, do you want me to set the trap again tomorrow? We got two big crocs in two days, so it's just a matter of time, really. I guarantee you the next one will be a thumper.'
But because of all the upset and commotion that the croc-catching had caused within the community, Harold called the search off for good.
We pulled the trap down and packed it up while the croc continued to rest under the tree. I slung everything else back into the workshop and onto our equipment trailer, and sent Willow and Jono, along with the producer and the second cameraman, back in the vehicle to take the croc back to Lizzy Downs. It was just me, Mum and Pete left.
'Come on,' I said to them. 'Let's go for a cheeky fish on the flight back to the station.'
We flew over a waterhole called Wilson's, famous for an 18-foot croc named Wilson that was pulled out of the area 10 years earlier. Wilson's is a shaded tropical oasis full of life, including massive barramundi. Mum had never caught a barra before, so I was hoping for a good session.
'Come on, old girl, flick it in there!' I said.
She was blessed with beginner's luck, and on her first cast she hooked onto a big one. Pulling it in with all her might, she was struggling with the fish as it darted around.
'Keep pulling, Mum!' I shouted. 'Do you want a hand?'
'Not a chance!' she bit back, jokingly. 'You're not taking any of the glory!'
The fish finally tired out after 10 minutes of reeling and repositioning, and she pulled it up.
'No way! Mum, it's a stonker! Hang on, stay here.'
I bolted to the chopper and found the measuring tape.
'Okay, bets are on. Pete, what do you reckon?'
'Ninety centimetres,' he said.
'Mum?'
She shrugged. 'Hmmm, no idea. Very big is all I've got.'
'Righto, I'm going for a metre!' I laid out the tape and, I kid you not, the old girl bagged a 104-centimetre long barra on her first ever cast during her first ever barra expedition.
'I don't believe it! At least we have dinner sorted now!' I said.
We caught a couple more big fish, and even Pete caught one around the 90-cm mark. We were stoked with our efforts and booted back to Lizzy Downs after our perfect little fishing outing.
Pete interviewed Mum and me that night as the sun set behind the cathedral termite mounds. It was nice hearing her reflect on the relationship we have and the adventures and misadventures that we've shared along the way.
'Well, the fruit definitely doesn't fall far from the tree, does it?' he remarked after the interview.
There was no denying me and my old girl have the same love of wildlife and zest for adventure. There aren't too many mums out there who would encourage their son to catch one of the deadliest animals in the world, but she thinks it's all a bit of fun and doesn't see what the fuss is all about. She'd be out there doing the same as me if she were younger!
We took the second croc to my neighbour's place near Dundee and spent a few days working around the clock to build a pen at the back of his joint to keep the croc out of harm's way. The cros still lives there today. I had a ball with Pete and the crew, and the 60 Minutes segment turned out mint.
Every now and then I would still think about Peppi and wonder if that monster was a myth or not. And then, about a month after the 60 Minutes episode went to air, I received a call from a coroner in Katherine.
'Is this Matt Wright?'
'Roger.'
'G'day, mate,' she said. 'I'm from the Katherine Police and we're conducting an investigation. I'm hoping to ask you a few questions, if that's okay?'
'Sure thing,' I replied.
'Have you been catching crocodiles around Peppimenarti recently?'
'Yes, about five weeks ago I removed two.'
'How many did you see in the area?'
'There's a few around there. How come?'
'Unfortunately, we've just lost a local down there. He's been taken by a crocodile.'
'You're kidding me.' My heart sank. I knew straight away that we definitely hadn't caught the croc we were after.
'Wish I was.'
'Thank you for letting me know. I'll have a chat to the locals and see if they want me to come back again,' I said.
Sadly, Harold and Captain, another senior traditional owner, had passed away from health-related issues soon after that attack. When I found out, I realised that's why I hadn't heard from anyone in the community.
I would still like to go back to Peppi one day – that's if the locals wanted me to – and see if I can catch the cunning croc, because I would hate for him to take any more innocent lives.
#
When preproduction for season three of Outback Wrangler came around we began planning and working through our stories, and one of the biggest things we wanted to incorporate was more educational information about crocodiles. I work with crocodiles every day and know a lot about the animal. But I haven't learnt everything I know from books – it has come from my interactions with the animals over the past 20 years.
When I talk about crocodiles on the show, a lot of what I say is observational commentary as opposed to hard-hitting facts. The knowledge and insight I have is pretty unique because I live and breathe the animal in the wild, and it works on Wrangler, but my theoretical and scientific understanding of crocodiles could do with some work.
In my younger croc-catching years I knew how to read and catch a croc better than anyone. I never felt the need to learn more, because that's all I cared about back then. I was a croc catcher, not a croc expert. Of course, I've picked up a hell of a lot of information over the years working alongside scientists in the field, but as I do more TV shows and media interviews the more I am asked specialist questions.
During an interview a couple of years ago, when I was promoting season two of Outback Wrangler, I was talking about a crocodile's tail and the scute-marking system we use to identify crocs.
The presenter interrupted me mid-discussion. 'Sorry, Matt, and what exactly is a scute?'
'A scute, I guess, is similar to a hard bit of cartilage covered in skin,' I said.
My answer didn't quite cut the mustard and the presenter prompted me some more.
'And a more scientific explanation?'
I just sat there without a clue in the world and made a joke about how I catch crocs, not study them, and the audience laughed along with me. I went home that night and googled 'scutes' and it came up with 'Scutes are the keratinous sheathes that cover the osteoderms'. I remember thinking, Shit. I do need to learn more about the inner workings of these animals, especially if people are referring to me as a 'croc expert'.
So educating myself in the lead-up to Wrangler season three was a big focus, not just for the show but for the publicity tour that would follow. That way, I could be in a position to go into greater scientific detail if I was asked. I lived on the net and read a few books for a good couple of months, trawling information, and called Webby every now and then to validate things.
I've always spent a lot of time around Darwin-based reptile scientist Professor Grahame 'Webby' Webb. Given that we're both locals and work closely with crocodiles in our respective fields, we often find ourselves at presentations, talks and sometimes out in the field together, talking all things crocodile. Grahame began researching crocs in the late 1960s, and in my opinion, he is the world's leading authority on croc research and management.
And just before we began filming for Wrangler, I was invited to co-host an event with Webby at Charles Darwin University, where I got the opportunity to listen to his insights more than ever before, taking notes and picking his brain when I found something interesting. No matter how much you know about a topic, there is always something new to learn, and I loved every minute of getting to know crocodiles even better than before. And even after studying for a few months and continuing to research regularly to this day, there is so much out there for me still to learn and so much that all of us are yet to discover about this prehistoric predator.
I wanted to incorporate more science into what we do in future shows, particularly with tracking devices to learn more about crocodiles' territorial behaviours and whether they do have a homing instinct, as is often thought.
From my experience and recent study of the animal, here are ten things that I find interesting, which are less commonly known:
* The length of a croc can be calculated by multiplying the length of their head by seven.
* Crocs' osteoderms (located on their back), are like big solar panels. A croc can cook to death if it's in the sun for too long.
* Crocs swallow stones to grind down food in their tummy and cough up big fur balls from eating wild pigs.
* Crocs are good climbers and can get up extremely steep surfaces.
* Crocs can't die of biological causes. (See here for explanation.)
* Their eyes, ears and nostrils are all located on the top of their head.
* They have a highly acidic tummy, so they can quickly digest bones, horns and hooves.
* The saying 'crocodile tears' is true! Crocs produce tears, but it's because of swallowing air not crying.
* Crocs open their jaws to cool themselves because they don't have sweat glands.
* The muscles that open their jaws have little strength, hence why gaffer tape does the trick for me!
By the time filming arrived for Outback Wrangler season three, I was geared up with info and ready to weave new facts and information into our adventures. If you watch the shows, you'll see that info incorporated mostly through onscreen pop-ups, and the voiceover. I tried to tie things into the action where possible, but the reality is that the moment you have a croc in front of you, you're focused intently on catching the animal and not getting chewed, rather than turning to the camera and going on about how a croc can swim up to 15 km per hour. That's not really a priority in a high-risk situation. Nevertheless, I managed to throw in bits and pieces where I could. It was more informative than previous seasons, but there is still room for improvement when season four comes around, that's for sure.
We had eight episodes in season three of Outback Wrangler, all shot around the Top End. Two of the most memorable and demanding episodes were at Adelaide River Station and Moroak Station. We shot these episodes in tandem, which wasn't evident in the final cut of the show. The episodes aired as two stand-alone events, when in fact they were very much intertwined. This was not at all planned, especially given the large geographical distance between the two locations – Adelaide River is 130 kilometres south of Darwin, while Moroak is 600 kilometres southeast of Darwin, next to the Roper River. But in an unexpected series of events, I ended up having to catch crocs in the two locations at pretty much the same time, which was a mission to say the least.
My mate Mick Jacobi manages Adelaide River Station and had been on my case for years to get out there and catch, according to him, 'the biggest croc in the Territory'. I finally agreed to do it because it was taking more and more cattle from his property.
And then a friend of a friend reached out to me about going to Moroak to relocate a problem crocodile after a couple of incidents. The owner of Moroak Station is a lad by the name of Simon Hoar, and he and his wife and their four boys under the age of six moved there not long ago. The homestead is so close to the river that you can cast a fishing line and catch a barra from the front porch.
It's a pretty setting, but then crocs started popping up on the front lawn and humbugging the family while they were fishing. They put up fencing to keep the crocs away, but there was one big fella that kept coming back. It got to the point where, the moment the family ventured beyond the fence or out on their boat for the day, the croc would have a crack at them.
Jono and I, along with my dog Naish, made the long six-hour drive to Moroak. We offloaded our traps and gear and then planned to hit the sack as we had an early start the next morning. We just wanted to get a couple of pigs, set the cameras and boot straight back to Darwin, because the crew weren't arriving for another couple of weeks and all we needed for now was some insight into where the crocs were hanging out.
The next morning we drove around for an hour trying to find a pig. I'm used to spotting one from the air while flying a chopper, and it was the first time in a long time I'd had to look for one from the ground.
'This is hard yakka, Jonsy. I don't know how we're going to do this, especially because I'm not familiar with the country around here.'
'Yeah, I'm with you,' Jono agreed.
We came across an old cattle yard and finally heard and saw some movement in the bushes.
'There, Matty!' Jono called.
Naish must have heard him, and leapt from the tray, towards the noise. But it wasn't pigs, it was a couple of wallabies. They jumped out across the open plains.
'Oh noooo,' I said, stalling the car and flinging the door open. 'Naish!' I screamed, but she was gone, chasing the wallabies through the bushes. She had no chance of catching them but was obsessed with them, and had been ever since she was a pup.
'That's no good, Jonsy. We're going to have Buckley's chance of finding her again, especially without a chopper.'
We did a few bog laps, but we couldn't see her, as I expected.
'There's not much we can do right now, so let's drive around until we get a pig, set them along the banks with the cameras, and when the boys are back from mustering I'll ask Simon to borrow a chopper,' I said.
'Okay, sounds good,' Jono replied. 'Sorry about yelling out, mate. I didn't think.'
'All good. What will be will be, I guess,' I said, praying that I hadn't lost Naish.
It was 2 p.m. We'd found some pigs, set the cameras and I had my hands on a chopper. Two hours, and a lot of money wasted in fuel later, I still couldn't see Naish anywhere. We covered kilometres and kilometres from the air, scoping anywhere that looked like a possibility. Feeling completely defeated and heartbroken, I started flying back.
'We can't sleep at home tonight, Jonsy,' I said. 'I want to take the swags in the back of the car and sleep out in the old cattle yard where we lost her, in case she makes her way back.'
'Roger. I was going to suggest the same thing.'
About five kilometres from the cattle yard, I saw a small clearing with bushes scattered around just to the side of the dirt road we'd driven down earlier that day, and there, lying next to one of the bushes, was a white speck.
'No way, I think that's her.'
I dropped the chopper and hooked in to land. And there was Naish, curled up in the shade of a bush, totally exhausted. She couldn't get up, but she was wagging her tail and smiling. I ran towards her and smothered her in kisses and hugs. God, I was relieved. And tucked underneath a shrub next to her was a male dingo. He was the most placid fella, he didn't even get up to come at me, he just sat there. I'd never seen anything like it.
Jono came in behind me. 'How funny is that! Naish has found herself a fling out here!' he joked.
'I can't believe he's just sitting there, with us here. Mate, can you get the sat phone and ask the guys to come fetch her in the ute?'
'Roger,' said Jono, and went off to make the call.
'Come on, girl, come sit in the chopper with me and have some water before the ute arrives. Say goodbye to your mate.'
I carried Naish to the chopper, sitting down with her in the back seat. I turned around and saw that the dingo had followed me to the chopper. It jumped in.
'I actually can't believe it, it's like he's domesticated!' I said.
The dingo sat there with us until the guys came in the ute to pick up Naish and we parted ways with the dingo. I felt bad leaving him behind, but he was happy and healthy, and I was sure he would be okay.
Two weeks later Jono, Ash and I flew back out to Moroak to get things ready for the rest of the camera crew, who would be landing in Darwin two days later. On the way, we stopped by Adelaide River Station and set up a large panel trap with Mick's help. I figured we would get a croc trap going there first and test the waters before heading to Moroak to get things sorted properly there. Things were organised quick-smart with Mick and we took to the skies early arvo en route to our final destination at Moroak.
We flew around where we dropped the pigs a few weeks earlier to have a look at what was going on. We had with us four big cage traps, which we placed down first and then got stuck into setting up the smaller panel traps in two key locations we'd chosen based on croc activity from our camera footage.
Lots of little crocs and two decent-sized animals cruised along the waterway while we were setting the traps.
'This place is full of them!' Jono commented.
'Yeah, it is,' I said. 'We're not going to have trouble getting one in there, but it's going to be a headache if we keep catching the wrong one!'
My mate Munro happened to be mustering in the area, so he flew in to lend a hand reinforcing the trap. The boys double-tied the panels together while I set up the trigger rope. I turned around to check on their progress and there was Munro, all five foot seven of him, on the outside of the trap, up to his neck in the water.
'Christ, Munro, get in here now!'
He was bloody lucky his leg didn't get chewed; we'd seen so many big animals swimming past only metres from where he was.
Munro scampered up the trap and jumped inside it. 'Yeah crap, sorry mate, I had a bit of a brain fart there.'
'YA RECKON?!' Jono said.
The day was done and it was time for us to set up camp. We could have stayed at the homestead but it was dry season, and we'd found a spot near a crystal-clear shallow creek where we set up an epic little site with chairs, tents and swags. It was a camping paradise.
The next morning, after campfire baked beans and crusty bread for breakfast, I flew with Jono over the trap. We saw a heap of dirty water and, sure enough, a big crocodile.
'Oh shit,' I said. 'We've caught one within two days of setting it.'
It wasn't what I expected or wanted. We were trying to save money, so we didn't have the camera crew. They were all still in Darwin because we thought it would take a few days at least to set the traps and lure the crocs in.
'Hmmm, bit of a problem,' I said to Jono.
I flew higher to 2000 feet to get service to message Willow to get the camera crew and boot straight out as soon as possible because we had an animal in the trap already. As I got bars on my phone a message came through from Mick over at Mount Bundy Station: Croc in the trap.
'Oh, ya kidding. Jono, we've got one at Mount Bundy as well!' I said. 'The camera crew arrive in Darwin today, so I'm thinking we leave this guy here and fly to Adelaide River, meet the crew there first and film that one.'
'He's only just got in there and he's in the shade, so he'll be fine,' Jono agreed.
Willow was now across the plan, had picked up the remaining crew members and was driving southbound to Adelaide River Station.
'I still can't believe we already caught the croc, I didn't even bait the trap properly!' I said.
I'd put a bit of pig up on the rail of the trap to entice the croc in so we could get some footage of the size we were dealing with before the camera crew arrived, but the croc must have swam in and knocked the gate down in the process.
We flew over the croc in the trap to suss it out on our way to the Adelaide River homestead, and confirmed Mick's size misjudgement.
'Is it the big one we're after?' he asked, the moment I walked in the kitchen door.
'Well, it's big, mate, but I hate to break it to you, I don't think it's the 20-foot monster you've been talking about.'
'Yeah, righto,' he replied. 'Well it's probably not the one we're after then.'
Mick was married to the idea of this supposed 'monster'. I'd been wrong in the past so a small part of me thought there could be a bigger croc out there, but I doubted it.
'Mate, you're a small fella, so I think your point of reference is a bit off!' I joked. 'You think everything is big, like a Chihuahua thinking a kelpie is a Great Dane!'
'Hey, hey, come on now. I'm telling you it is, I've seen it fair and square,' he argued.
'Righto, well, let's get this one out tomorrow morning and see what else shows itself after that,' I said.
The crew arrived that night, along with Willow's dad, Robert, who brought a trailer, and Tommy Nichols from NT Parks and Wildlife, who brought the drugs for the croc. I was mentally drained but pleased I'd managed to align all of the moving parts and people.
At dawn, I flew the chopper by myself to check on the croc. I wanted to ensure everything was kosher before filming. I landed and I had a look over the trap. The water was muddy, but there wasn't much in there for the croc to hide in.
No croc. He'd managed to squeeze out through the gaps in the bars.
'Shit.' I pulled the motion sensors off the trap so I could see exactly what had happened and flew back to the homestead. Everyone was up and buzzing, ready for the first day. All I could think was, What a waste of time, we could be at Moroak pulling the other croc out and here we are for nothing.
'Do you want the good news or the bad news?' I said as I walked through the kitchen door.
'What?' everyone asked.
'Bad news is the croc is gone. Good news is there wasn't a big struggle from the looks of things, so we haven't scared him off and he's likely to come back in the next day or two. But we have to reinforce the trap, so he doesn't get out. And I can tell you one thing for sure, Mick, he's certainly no beast because he's managed to slip through the unmeshed rails. So, I'm guessing he's around the 15-foot mark, still a big croc, but not a next-level mammoth.'
Everyone looked at me like, 'Are you for real?!' but I rousted everyone up and we got going.
'Come on, you lot, the croc isn't going to catch itself, and we all like a challenge, so let's get moving!' I said lightheartedly.
We went back out to the trap and strapped some big melaleuca branches to the rails to make the gaps smaller.
'Well, that didn't go to plan, guys, sorry about that,' I said.
Jono, Ash, Dan and I then flew over to Moroak to check on the other croc, while Tommy and Robert went home to Darwin, defeated. The rest of the crew were going to drive from Adelaide River to Moroak, and wouldn't be getting in until dark.
Holy hell, did the croc in the Moroak trap go ballistic when we arrived. He'd found a little gap at the bottom of the embankment where the ground was uneven and started wedging his nose under it, lifting the trap up and bending the mesh outwards, trying to escape. His tail thrashed around like a whip, and he was death-rolling every now and then, causing the trap to wobble about.
I walked to one side of the trap and pulled a patch of mesh off. I poked a stick through the gap between the rails, enticing the croc towards me. He came straight over and shoved his snout through the gap.
'Rope, quick Jono!' I looped the rope over his head and pushed him back inside, making sure my hands didn't get chomped in the process. I managed to pull him back inside with a head rope attached.
I thought I had him but then, without warning, the croc darted for the gap he'd made under the trap and pulled me straight into the one-metre-deep water in the trap alongside him. Everyone stopped filming, dropped their gear and jumped up on the horizontal rails of the trap to pull me out, yelling all sorts. I thought I was dead meat; I honestly thought that was it for me in that moment. But, luckily for me, the croc had clawed so hard under the gap trying to escape that he had wedged himself in there and couldn't back out to eat me. His tail thrashed me around for a bit, and I felt like I was getting dumped by waves at the beach, but I eventually made it over the rails.
'Matty, you good?' Willow asked.
'Yeah guys, shit, that was close. Don't worry about me, let's pull him back out, otherwise we'll lose him as well and that'll be two crocs in two days and I'm not having that.'
It was all hands on deck to secure the animal and get him out of the trap.
'He's decent, just under 16 foot, I reckon. Call Simon and get him to bring his buggy down to pull the croc up the steep bank,' I said to Willow.
Simon arrived and was happy. 'That's definitely one of the crocs we're after,' he confirmed. 'I've seen him out of the water before, he's one of the culprits for sure.'
Willow got him on the trailer and a couple of the station workers drove him to the croc park in Darwin. The footage from my near-death moment wasn't captured and therefore not included in the episode because the cameras ended up in the mud. But the editors pieced together the rest of the day's adventure perfectly around it, making everything flow nicely, so at least our story wasn't ruined.
We started with that particular croc at 6 a.m. and it was 10 a.m. by the time we got him out and onto the trailer. I flew up high to get service again and touch base with Mick. The first text message that beeped through was Come ASAP he's back in.
Usually I'm hanging out for that kind of news, but I was so tired from all the travel that I couldn't think of anything worse than going back to Mount Bundy, but we didn't have a choice. This time I left the chopper at Moroak because Mick said he had a spare one not being used for the next couple of days. Jono, Willow, the camera crew and I drove together for the six-hour journey. We covered some serious ground and arrived at the trap in the late arvo, exhausted.
Tommy Nichols had left me some tranquilising needles before he had to return to Darwin, so I had them at hand for when we got the animal out. Thank god the croc was an easy one to move and get on the back of Mick's trailer. The croc was around 16 feet, a very similar size to the one at Moroak.
Mick was still in denial. 'There has to be another one, Matty, it can't be that croc. I don't think we need to move him out because he's not our guy.'
'Mate, he's already on the trailer and I can guarantee you it's the croc,' I assured him. 'That is one tiny waterhole so there are definitely no other big males in there. You are looking at the big monster you've been talking about.'
'You really think?' he asked, finally getting close to admitting defeat.
'Yes, mate, but the good thing is he won't be eating your cattle anymore.'
I felt bad because he was a little deflated that it wasn't the croc he thought it would be, especially because he'd been following it and tracking it for three years, telling me story after story about this monster. All his expectations were crushed in a heartbeat.
He took the croc to the Darwin croc park and the rest of us decided to suck it up and drive back to Moroak at 6 p.m. so we were ready for action first thing the next day. We didn't get in until 1 a.m. We fell into our swags, completely rooted.
The next morning, I called everyone around the fire. 'Okay, guys, we've caught one at Moroak, one at Adelaide River and we still have four cage traps and one panel trap here to rebait. Let's see if we can get a couple of the other bigger problem crocs. Once we're done here we'll go back to Adelaide River to finish off the story with Mick but no more to and froing, we'll go broke,' I said, alluding to the fact that, by this point, the show's travel expenses had spun out of control.
We gave ourselves another week at Moroak and popped to the neighbouring property, Flying Fox Station, in our downtime to help my mate Sully catch a couple of problem crocs on his property. It was reasonably easy work. The camera guys didn't even bother with filming any of it and came and had some fun with us. We got a lot done there and got back to our camp set-up at night for beers and a feed.
On the second last day at Moroak we caught a little croc next to the homestead. Even though it was small it was a good one to catch, given its proximity to the kids' play area. And it was a great opportunity to educate the kids about crocs. At only 10 or so feet long, he wasn't a major threat but we made sure we moved him a fair way up the river. There was a chance he might return, but it was a short-term deterrence and was better than doing nothing.
'All right, lads, you get trucking back to Darwin, I think we're done here. Jono and I will stay one more night to pack up, sling the traps out and stop in at Adelaide River to see if there's anything else Mick needs.'
About an hour after the guys took off, Jono and I flew over the panel traps to start hooking them on and slinging them out. We couldn't believe our eyes as we flew over one of the larger traps.
'No way, that can't be. There's the other big fella!' I said to Jono.
We'd been at Moroak for an entire week and it was so typical that on the last day we caught the one we'd been after. It was such a major part of the story because we'd been out there on the hunt for him for ages, but the obvious problem was that everyone had left.
'Just message Ash and he'll get it when he's at the next town. They don't really have much of a choice, they have to come back,' Jono said.
I flew up higher for reception and flicked the text over to Ash. I got a response from him once they'd reached Mataranka. Ok, back we come. See you in the morning.
Jono and I slept at the homestead because we'd already packed up the camp area. We were up at 6 a.m. and used the homestead wi-fi to confirm that the crew were on the road, heading for us. Then we flew straight over to the trap to get everything ready. There was a big mob of water, so I couldn't really see him.
'This guy is a bit camera shy,' I said to Jono. I poked around the trap until I noticed that an area of mesh had been warped. And then, just beneath the water's surface, was the bunged-up railing . . . and a big hole.
'No way, it can't be, this is just mean. What is the universe doing to us, Jonsy?! The bloody croc isn't here anymore!' I said.
Jono half-laughed and half-looked like he was going to cry with frustration.
And when the crew arrived, their reaction was much the same as Jono's.
'I'm so sorry, guys. Just hop right back in the utes, drive to Darwin, book in at the casino, my shout, and have cocktails by the pool. We all need a break,' I said, as the boys wandered back to the cars.
The post-production team did an awesome job at editing all of the behind-the-scenes headaches out of the two episodes. You wouldn't even know we had those problems or that I nearly got eaten. My planned contribution of factual information was also very limited in those two shows in the end, probably because I was so mentally exhausted from the running around. Nevertheless, we got the job done at both stations and I plan on going back to Moroak to get the one that got away. But, from all reports from Simon, there haven't been any more issues since we left, which is great to hear.
#
10.11.2017
My dearest Kaia,
Everyone likes to dream . . . me more than most. Kaia, having met you has exceeded any dreams I've ever had of meeting the best possible girl – I never imagined that I would be so lucky.
When it comes to explaining why this is so . . . to be honest, I don't even really know where to start because there are so many things that I love about you.
From the very onset, your zest for life, your passion for living and your desire to always learn captured my heart. The appreciation for some of the smallest things won over my admiration. You taught me that being rich on life experiences are what counts, not what's in the back pocket.
Since you have come into my life, I've learnt to see the positive in things, with your influence and optimistic outlook on life. Your selflessness and the time that you give so willingly to help others blows me away. Your kindness and generosity are just the start of your exceptional qualities that I have fallen in love with.
Every day, week and year, there is more and more that I see in you. The life that we live is one that is full of adventure, fun, excitement, love and travel, and the only thing that I would ever wish for is more time. This lifetime will never be enough, for you and me.
Kiki, there are so many generic internet vows and promises that I could rip off and say to you, like I will love you forever, protect you forever, provide for you forever, but you already know all that shit.
What I do promise is that I will always write you letters, cook your dinner (every now and then), hang up all your pictures around the house and even hook that magic fridge up or at least get Jono to do that for you. I promise to keep doing all the small things that count and all the big things that keep the magic alive, so you know just how much you mean to me.
I love you, Kiki, now and forever.
My wedding vows say a lot about what my wife Kaia means to me, but how we met and the adventures we've had along the way are pretty good telling as well.
The day I met Kiki was an interesting day. I was flying from Darwin to Perth to talk some things through with Troy, my business partner, who I was a bit disgruntled with at the time. Suited and booted, I was ready for a day of long meetings and negotiations in the office. But when I landed in Perth I learnt that Troy was in fact over on Rottnest Island, so my mate flew me over in his chopper and dropped me off on the beach.
I was taken aback by the beauty of the place. It was magnificent – with the whitest, finest sand and crystal-clear blue water. Ten or so boats, including Troy's, were rafted up in a little cove with music playing and people partying. I hopped onto Troy's boat still in my boots, jeans and collared business shirt. People were probably thinking, What the? Who is this random?
Troy was with his mates, and on the boat next to him was a group of incredibly beautiful women. I had been living up in the Northern Territory working on cattle stations, spending most of my days out in the bush. I wasn't really up with the times and didn't get exposed to women much. I was a complete fish out of water, with my scruffy hair and cowboy boots, amongst the bronzed sheilas in bikinis and manicured lads all wearing boardies that were so short they looked more like boxers.
'Let's forget about business today and have a few drinks together. We'll sort through things tomorrow,' Troy said to me.
'That suits me fine, I don't think I've stopped for a break in about four months.'
I sat back for a bit and watched everyone having fun. It felt good taking a moment out to do nothing. After a couple of beers, I thought I better try to blend in a little, so I got my shorts out of my backpack. They were marked with white paint and had a big tear in them, but they were better than my RMs and business shirt.
Everyone was having a great time. I got chatting to a young lad on the boat next to Troy's and while we were talking, one bird in particular caught my eye because she was beaming. She was smiling and laughing, and I mean properly smiling where her whole face lit up. It made me want to speak with her. All of a sudden, she came up to me.
'Hi, I'm Kaia,' she said, with so much warmth.
I didn't know what to say, I was shocked that she'd taken an interest in me. Usually, women like her think someone like me is a grubby bogan and tell me to get stuffed, but she approached me!
No one knew who the hell I was, and they all seemed like a pretty tight group of people. I was just this random gatecrashing a party and here I was, talking to the most beautiful girl of them all – actually, the most beautiful girl I'd ever seen.
'Where have you come from?' she asked me.
'Just up in Darwin.'
'Oh cool, have you been to the rock before?'
'Yeah, a few times, but I don't drive through the red centre much these days,' I said.
'Huh, red centre? No, not Uluru! The rock as in Rottnest – that's what we call it!' she said to me, laughing.
'Ah shit, I stuffed that one up,' I said, a bit embarrassed.
She didn't seem fazed by how out of place I was and wanted to talk. It was so refreshing. It was such a new thing for me to sit down and have a normal conversation with a woman and get along really well. I couldn't remember the last time a conversation with a bird flowed so smoothly – she even made me laugh, especially when her beer foamed up and spurted all over her face. We chatted the night away.
'What's got into you?' Troy said, making fun of me. 'Time to head back to my joint in Mandurah.'
'Nah, mate, I think I'll hang on the boat here and head back to Perth.'
Troy shook his head and laughed. 'Righto, see ya in the morning then!'
We wave-jumped all the way to Perth on the boat, trying to catch air each time we hit a wave, laughing our heads off. When Kaia leaned in and kissed me, I felt like a kid in a candy store.
I've had my fair share of nightmare relationships, and I was in and out of a pretty tumultuous one at the time. I wasn't happy, and I knew I had to end it for good, but I also wasn't looking for anyone else. I was over relationships and wanted to focus on work and my career. But out of nowhere this beautiful person popped up. There were a couple of other girls that day showing interest in me and a lot of the blokes trying to have a crack at Kaia, so we weren't a very liked couple. But we didn't give a shit about anyone else, we were in our own little world.
The next morning I asked Kaia for her number and she tapped it into my phone.
'Put yours in mine,' she instructed, and handed her mobile over.
I typed in the digits and gave it back. 'You okay?' I asked.
She looked really embarrassed. 'Um, sorry, this is really embarrassing, but what's your name again?'
I laughed. I never remember names but I remembered 'Kaia' of all names, and she couldn't remember something as simple as 'Matt'.
'I'm so sorry, I'm not usually like that, I just had a mental blank,' she said.
I could tell she was mortified, so I didn't make a thing of it. 'All good, Kaia.'
Troy picked me up, and he and I spent a couple of days in the office getting on top of things, but I couldn't get Kaia off my mind.
I had one night left in WA, so I called her and asked her to dinner. 'I'm in Mandurah at the moment,' I told her. 'How far is that from you?'
She laughed. 'You're over 70 km away from me and it's already 5 p.m. Maybe next time we'll catch up.'
'Yeah, righto, no worries. We'll keep in touch then,' I said, disappointed.
Troy had to stay in Mandurah and needed his car so I didn't have a way to get to her.
'You can catch the train,' Troy said to me after I hung up.
I was stoked to hear that I'd be able to get there. 'Oh yeah, I might just do that.'
Troy thought the whole situation was hilarious. 'Jesus, you're actually going to catch the train? I was only joking, but okay.'
I called Kaia back to tell her I was planning on taking the train and asked her what stop to get off at and if she'd pick me up in an hour's time. She also thought it was hilarious, but said she'd meet me. I was rapt.
I jumped off the train at Canning Highway in South Perth and her car was there waiting for me. I was happy to see that she actually showed up and didn't bail out, because she didn't really know me after all. We had pizzas at a joint near her place, which I noticed was very small, and we sat in a private booth at the back of the restaurant, away from anyone who might spot her. I was sure she was trying to hide me to start with. Nevertheless, we had another cracking night together and I went back to Darwin with a full heart.
We kept in contact over the next couple of months. Nothing full-on, though, because I was working through the aftermath of ending my last relationship and I wanted time to breathe. We met right at the start of egg-collecting season in 2014, which was actually perfect because I was out bush a lot and had time for my own thoughts.
One day I was in a buffalo wallow, trying to get some epic wildlife images and my phone fell out of my pocket and into the mud. At the same moment, a female croc came in on me and had a crack. I had to fend her off and, in the process, ended up completely covered in mud, with no crate, no stick and after rummaging around in the mud the phone wasn't to be found anywhere. I was in no rush to get back to town to get a new phone; I relished in the idea of being uncontactable.
A little while later, after I got back to town, I got my shit together, bought a phone and called Kaia.
'Hiya! How are you going?!' she cheered down the phone.
'Sorry I haven't been in touch,' I said. 'I lost my phone.'
She laughed lightheartedly, as if to say, Yeah right, I'm not stupid.
'No, I really did lose it,' I went on. 'You see, it fell out of my pocket in a buffalo wallow, while I was out collecting croc eggs. I stopped to take a photo, then a female croc came in to humbug me while I was on the nest, so I couldn't get it back.'
She laughed even harder. 'Wow, you're a really good liar!' she said.
Even though it was true, I realised how ridiculous the excuse sounded. I also knew that trying to convince her it was true was not going to work either. The thing is, she didn't seem all that fazed that we hadn't spoken in so long, so we just caught up and chatted like no time had passed. From then on, we spoke on the phone and texted every day and sent each other video messages too. We've got all the videos saved to a hard drive now, and Kaia dies with humiliation if I try to rewatch the ones of her.
During that time I made lots of trips to Perth, Margaret River and back to Rottnest, where I met up with Kaia. We hung out and got to know each other. On some of those trips I was so exhausted from working up north that I would sleep for two days straight before I could even function. But Kaia's energy was so calming I felt recharged after being around her.
On a trip to Melbourne together, she told me she had landed a job in London and was planning on moving over in a few months' time. I knew that if she left I wouldn't see her again, so a couple of weeks later when we were chatting on the phone I just came out and asked, 'Hey, why don't you move in with me in Darwin instead of going to London?' We'd only been seeing each other for a few months, so it was a big call and I had no idea what she was going to say.
'Yeah, why not? If it doesn't work out it'll be a good story!' she replied.
I knew that it would work, though, because I knew I'd finally met the love of my life.
One of the best things about my wife is her love of adventure. She's not afraid to roll up her sleeves, hook in and get grubby, and she puts up with all of my antics. There are too many stories to share where I've got her to push her boundaries. She happily stands guard on a croc nest while I collect eggs, catches pigs when I need tucker for Tripod, hops on the back of crocs when I'm after an extra set of hands, keeps me company on long musters and puts up with my pet animals. There really isn't too much that bothers her.
When I first flew Kaia up to Darwin I wanted to put her through her paces and suss out if she was a princess. We'd been fishing for barra and flying around the waterfalls having some fun. Then the rain started to pour down and the swamps turned muddy.
'Okay, I'm going to land the chopper in a sec and I need you to jump on a pig. I've got to give my croc a feed today.'
She turned and looked at me with a half-smile on her face, like she wasn't sure if what I'd said was for real.
'Okay, go!' I yelled, once I'd landed the chopper.
Without hesitating she undid her seatbelt and jumped out, falling face first into the mud. She completely and totally ate shit. She stumbled up onto her feet, leaving her thongs behind, and sprinted towards the pack of pigs. Grabbing hold of a small one, she turned back to me, losing it laughing. How she caught it in the mud I don't know. The rain was thrashing down around her.
'What do I do?! What do you want me to do?!' she was screaming at me.
I swiped my hand across my neck in an 'abort mission' movement, signalling her to let the pig go. She got the gist and let it run off. After fishing her thongs from the mud, she jumped back in the chopper with me.
'I'm confused, don't you need a pig?' she asked.
'Nah, I saw you were wearing thongs and knew you'd eat shit in the mud. To be honest, I just wanted to have a laugh.'
'Oh my god, that's so mean!' She was not impressed, but I was.
When we bought our first house together in 2016 in a rural area 20 minutes from Darwin, Kaia set about decorating it into a beautiful home. She spent a week sugar-soaping, vacuuming, mopping and rearranging all our new furniture. We had a barney about the colour of our new couch and dining room chairs because Kaia wanted cream and I told her that it was a dumb idea in the bush, and that they'd be filthy in no time. She didn't listen and went on creating a display home in the Aussie outback, which I thought was a little silly, but I let her do her thing knowing it was only a matter of time before the cream turned crappy.
I guess I didn't help the cause, turning our house into a zoo within the first couple of weeks we were living there.
'Hun, come look here! I've got three dingo pups!' I called out to Kaia from downstairs one morning. I was so excited. 'We've got Albie, Ernie and Laya. Their mum must have been shot. We'll raise them here for a bit until they're old enough to let out near the shack where I found them.' Kaia didn't seemed fussed at the time, but now I know she actually wasn't keen on the idea at all! She just didn't want to say anything because we were still pretty early on in the relationship and she could tell I had my heart set on the dingoes.
I was busy working at the campsite and was coming and going from our new home, so Kaia had to keep an eye on the puppies as well as our two hounds, Naish and Apollo.
Two days later, I was driving back to our place and I saw a dead wallaby on the side of the Stuart Highway. I pulled over to inspect its pouch and discovered a tiny joey. I stopped off and got some wallaroo powder from the vet to feed the wallaby and took her home to Kaia in a little pouch bag.
'Here ya go, hun. You don't mind taking her around with you, do ya?'
'I don't think I've got much of a choice, do I?' she replied, but she was smiling.
'She'll be right,' I said, and Kaia looked at me funny.
'What?' I asked.
'That should be her name,' Kaia said. 'Shelby Wright as in your favourite saying, she'll be right.'
With a name like that we had to keep her – and we had ourselves another new pet. I was stoked, but I could tell Kaia wasn't as wrapped with the idea. That same night I got a call from one of my employees asking me if I could take on a big cage with a heap of snakes in it, as the owner was moving interstate and had to get rid of them quickly. He explained that there were a couple of grass snakes, olive pythons and two epic albino pythons, which is what got me really excited.
'Yeah, righto, why not.'
I broke the news to Kaia. We agreed that I was only allowed to have them if the cage went in my office. But she wasn't so agreeable when I left on a tourism trip the next day and she had to look after seven snakes, two dogs, three baby dingoes, a wallaby and the six resident green tree frogs that had taken over our bathroom. She was defrosting mice for the snakes to eat, toileting the wallaby every three hours throughout the night (this is where you get up and prompt the animal to wee by rubbing their private parts with a wet wipe), and running the dingoes around while they clawed and nipped at her. Not to mention that Kaia had her own work as a communications manager to uphold. I'm sure at that point she was thinking a nice corporate banker in London might have been a much better option as a partner.
I got home from the trip and was cleaning the helicopter when I heard a scratching noise at the bottom of an old empty water tank on our property. I climbed up the ladder and looked in to see a beaten-up possum trapped at the bottom. I scaled down the wonky interior ladder and cupped it in my hat. It was malnourished and needed our help.
'Man, Kaia is going to kill me,' I said to myself. I walked through the front door and Kaia looked straight at my hat.
'What the fuck is that?' She was not impressed.
'Look, I know you've got a lot on your hands with work and the new animals, but this little girl was going to die if I didn't get her out of the water tank. I didn't have any choice. I had to bring her home.'
Kaia put her head in her hands and laughed. I don't think she knew what else to do.
That night, she went out for a drink with her girlfriends and left me to look after my menagerie of animals. I had a hell of a time. I let the two olive pythons out of the cage in my office and into our bedroom so they could roam around for a bit, and I did the same thing with the two albinos in the spare room. I had the possum slung across my chest in a pouch-like bag and the wallaby in another, feeding them together. Then I decided to let the dingo pups inside, just for a quick play and a bit of interaction with Naish and Apollo. All five dogs ran riot. Ernie was jumping over Apollo on the couch, Laya got into the cushions, Albie knocked over the bin and was tearing its contents apart, Naish wanted to rip into the possum while poor Shelby Wright was wriggling around in the bag like no one's business. All I could think was, Kaia is going to die when she sees the couch. It had red and brown paw prints everywhere.
I fed the animals quickly, cleaned where I could, hung up the possum and Shelby in their pouches high on the hatch on our window louvres and lay back on the couch ready for Kaia to come home.
She arrived with some news. 'I made some calls tonight and found a home for Shelby, the possum and three dingoes,' she said, sitting on the couch with me. 'And even better news – everyone's coming to get them tomorrow morning!'
She was stoked, and I didn't blame her. The rest of the animals required full-time care and attention, so they were better off going to someone else who could offer that. And as long as I could keep my snakes, I was happy.
We went to bed early that night but at 2 a.m. I woke up to a piercing scream: 'MATT!!!'
I jumped up and turned the light on, and there was Kaia starfished on the bed with the big olive python wrapped around her left leg. She leapt up and started hopping around the place naked, trying to unravel it. If you were a fly on the wall you would have been in fits of laughter. I scurried around looking for the second snake and found it in the undies drawer.
'What THE HELL were you thinking, Matt?!' Kaia yelled, once the snakes were safely locked back in my office. 'Never ever do that again, that was mortifying!'
It was so fricken funny, but I did kind of feel bad because it would have come as a big fright for her.
Time and time again Kaia has proved her resilience, and tolerance for me and my shenanigans. She's well and truly gone through the ropes with me. She's my girl for good reason, and I wouldn't want to go through life with anyone else. Getting engaged and then married is my proudest achievement so far, and now I can't wait to have a family of little rascals together . . . but I hope they're more like her than me.
#
'You must get an unlimited supply of beer', 'You're a lucky bastard' and 'You're living a bloke's dream' is what most people say about my partnership with Great Northern Brewing Co. Sure, I enjoy drinking and promoting the beer, but what I like most is that the brand is built around things I value – friendship, adventure and fun. Back in 2016, Great Northern was an up-and-coming beer. It was popular in northern Australia, and was building its presence elsewhere. Lots of people around Darwin drank it, including me, and people used to say to me, 'You drink enough of that stuff, Matty, you should do something with them one day.'
Then 'one day' happened and my manager, Nick, called me.
'Mate, I've been chatting with CUB [Carlton & United Breweries] and they're keen to get you on as the brand ambassador for Great Northern. I think it's a good fit, what do you reckon?' he asked.
'Sounds good to me. Actually, it sounds great,' I replied.
Great Northern came up with True North, a miniseries of ads shared as short films featuring Willow, Jono and me. Exploits and escapades are second nature to us, so the miniseries was essentially us doing what we do best – running amok and enjoying beers together. The funny part was that each ad came with a detailed script and storyline, but when it came time to filming, we didn't follow any of it. We were in our natural element so our banter, misadventures and one-liners happened organically, without the need to stage anything. The final product ended up being a miniseries that was mostly unscripted and shot documentary-style. In that way, the ads feel real, and I think people can tell there is an authenticity to them. That's why the ads and the campaign have been so successful.
Our first gig working with Great Northern took us to Cairns to go marlin fishing, hiking and diving. Each adventure formed one of three ads for True North, part one. I'd never been marlin fishing, before but I'd heard and seen a lot about it and was itching to get on the water.
I flew Naish, my dog and best mate of 12 years, over for the adventure too. She passed away unexpectedly from kidney failure at the beginning of 2018, so the trip ended up being my last big getaway with her and holds very special memories. Naish followed me around for that whole day of fishing, sitting in my shadow, licking my face and standing at the front of the boat, smiling into the wind. I remember looking at her and thinking how happy she was.
The weather itself was rough as guts. It was coming into the wet so the weather wasn't very forgiving, and the fish weren't biting which was a bit of a problem, given that the script was pretty much: Boys go fishing, catch a big marlin, cook up dinner and enjoy beers as the sun sets. None of these things were happening or even looked like they were going to happen. Instead, Willow got revoltingly seasick and we only caught three small fish. But it didn't end up mattering because a sub-story effortlessly developed. As the day went on the lack of fish evolved into a light-hearted competition between the three of us.
'Whoever catches the first fish in these conditions really is the best fisherman, because it's near on impossible,' Willow declared.
So it was game on, and the jokes and set-ups between us all kept coming, and Naish turned into the fourth star of the show with her big presence and personality. It ended up being a cracking day on the water and an epic first ad for the True North series.
The other good thing with Great Northern is that everyone I've worked with really believes in the brand, loves the outdoors lifestyle, thrives on adventure and thinks outside of the box. I haven't come across anyone who works for Great Northern, or CUB for that matter, who is caught in a rigid corporate mindset. They are all open to new ideas and new ways of doing things.
The Great Northern marketing team, along with our producer, drafted another great script for the second ad of the first series, which involved a hike through a rainforest to a waterfall. We saw pictures and videos of the waterfall before we left; it was spectacular, and we loved the sound of the planned activities and storyline, which was Guys hike through the rainforest, up to a waterfall, have a swim, then finish with beers and camping out.
Dan Walkington, who is the producer and main editor of Outback Wrangler, was the contracted producer for the True North series. Dan is a legend of a bloke and is responsible for editing all of my TV content, so he knows what he's doing. However, on this particular shoot, he was a little misinformed on the logistical side of things. Us lads were from Darwin and Dan is from Sydney so none of us knew the lay of the land around Cairns and Dan got his information about the waterfall's whereabouts over the phone, not on the ground. He called ahead to speak to the local tour guides and accommodation owners we were engaging and got the down-low, but in hindsight he probably should have done a recce himself. But Dan seemed confident, so we packed our swags, camping gear and a big esky full of beers and headed off down the path towards the waterfall.
'So, Dan the man, how long's this walk?' I asked.
'Not far, Matty,' he said. 'The guy I spoke to said it's close by, so I'm figuring it will be about 15 or so minutes until we get there.'
We wound through the rainforest along a narrow path. Everyone was taken aback by the beauty, commenting on the trees and wildlife. Then the path began to steepen, and by 'steepen' I mean it became almost vertical.
'Have we gone the right way, Dan? I don't imagine we'd be walking up that,' Willow asked, pointing to the mini-Mount Everest in front of us.
'Sorry, lads, all the signs say it's this way. I must have misjudged the distance a bit,' Dan said.
I had thought we were walking a couple of ks to a relaxing swimming hole when in fact we were only halfway through a five-kilometre trek up a mountain with a 30-kilogram esky, backpacks and camping gear. Our camera man, Jez, stopped filming momentarily as things turned a little tense. None of us were angry, we were just completely exhausted as we walked and walked, passing the esky from person to person every hundred metres. The path flattened out briefly as it connected with a bitumen road with a sign that said we were one kilometre away from the main waterfall.
'Gee, still another kilometre. I don't know if my legs have it in them,' I said.
Dan was up ahead and waved over a four-wheel drive coming down the incline. He exchanged words with the driver then came jogging towards us with a big smile.
'I offered the bloke fifty bucks to drive us up the rest of the hill! Come on, jump in!' He was so chuffed with his little advance.
'Dan, you do know, though, this lift is an hour-and-a-half and 10 blisters too late?' I joked.
'I know, Matty, I'm sorry about that,' he replied.
'All good, these things happen, mate. It's probably a good thing we're burning off the beer we're about to drink!' I said, lightheartedly.
We made it to the waterfall, which was beautiful and made the hike worthwhile. Once we were up there it was bliss. It was a Monday afternoon and out of the peak holiday season, so no one else was there. Always the most adventurous of us all, Jono scaled up the rocks the moment we arrived, jumping off into the pool with a double backflip. I followed him but chickened out mid-move and mid-air, and hit the water with an awkward bellyflop. And Willow, well, Willow didn't even fancy going in. We were the three musketeers with different levels of musketeering when it came to waterfalls. Willow kept the beers cold and the fire going and started cooking the steaks. We had an awesome feed around the campfire before calling it a night.
The next morning, we jumped into two hired four-wheel drives. Dan had reviewed the previous day's hiking footage and decided we all looked drained, so he wanted us to recreate our arrival at the waterfall, making out in the advert that we'd driven there.
'Re-enacting like this never goes well,' I grumbled to Jono as we climbed into the car.
Dan overheard. 'Yeah, I agree. I'm not a fan either, but we don't have much of a choice for this one, unfortunately.'
So we hit the road and soon found a river crossing with a bit of shallow water and a sandy bank.
'Okay, Matty, can you please go through that bit of water there and make out like you're four-wheel driving?' Dan asked.
'Sure can,' I said. I was a bit gung-ho behind the wheel, and accelerated too hard, nosediving the car straight into the water. The wheels lost traction with the ground and we began to float down the river.
'Oh, big problem,' I remarked.
Willow and Jono thought it was hilarious because I had insisted on taking the wheel and then had completely messed things up while the cameras were rolling. I was really stuffing up Dan's retake.
We eventually got traction again and inched out onto the bank. Then we jumped into the other car so we could try a second take while the crew sorted out the bogged vehicle. We found a shallower spot in the river to cross and Willow climbed onto the back of the car with a GoPro, to capture everything from another vantage point. But I hit a little ditch as we were crossing, which threw Willow off balance and he fell off the back and straight into the creek, saturating himself. He jumped back in and we eventually got across and booted up to the waterfall.
Dan was stoked, the misadventures had woven perfectly into the story, and so we called it a day. We finished the trip off the next morning with a diving voyage out on the ocean, which was fun but very cold. Jono's suggestion of rounding off the day with impromptu beer-battered fish on the barbie with a Great Northern made the perfect end to our Cairns trip.
At the end of 2017, Jono, Willow and I went to Port Douglas for round two of True North filming. Kaia works in communications, and at the time, she and I were flat out in the middle of writing this very book, so I felt a little bad leaving her in the office and heading off to Port Douglas.
'Honey, next week I'll be away again for a week working,' I said to her. 'You have enough recorded content to continue writing, don't you?'
'Yeah, I've got a heap,' she replied. 'What do you have on?'
'Ah . . . just something for Great Northern,' I said.
'Oh yeah, a corporate talk?' she asked.
'Ah, not quite,' I said. 'We've got another True North series to film.'
She looked at me with a big grin. 'Ohhhh, that sort of work.'
'Haha, yep, I'm afraid so! This time it's horseriding, fishing and whitewater rafting, so should be sick!' I said, bouncing around with excitement.
'Yes, I'm sure it will be, hun. Well, you enjoy that "work" of yours and I'll see you tonight to hear more about your plans,' she said, kissing me.
'They say if you love what you do, you never have to work a day in your life!' I said cheekily.
The A-team was back together with the Great Northern marketing crew, our cameraman Jez, soundy Row, and Dan. By this point, Naish's health was deteriorating, so I brought my younger dog Apollo along for his first proper adventure. This time around, we all got to choose our activity ahead of the shoot and work with Dan to flesh out a storyline to go with it. Willow chose fishing, Jono chose horseriding and I chose whitewater rafting.
After an early first night we woke up at dawn and drove out to the Daintree River, ready for Willow's fishing adventure. We launched the boat at the ramp and while Willow was giving us an overview of the day's plans, I got distracted by some movements in the mangroves. I sprinted to the water's edge and jumped on a three-foot croc.
'You're a goose!' Jono called out to me.
I was thrilled with the little fella and brought it back over to show the crew. Everyone had a hold of it and we talked crocs for a bit.
'Come on now, Matty, we're filming a fishing adventure not a croc-catching mission. We're not in Darwin anymore, put the croc back and let's get going!' Willow said. He wasn't pleased with me hijacking his activity, but I couldn't help myself. And it was worthwhile because the croc catch actually did end up making the final cut.
Once we were out on the river we discovered fishing was hard going because there was very little movement in the water, and we gave Willow a bit of slack for it. We get pretty spoilt with fishing up in the Territory, so an afternoon consisting of only a few bites and a couple of trevally felt very quiet. After stitching Willow up by putting a dead fish on the end of his line, blowing a boat engine and trying to paddle back to shore with an esky lid (we ended up getting towed by the film crew's boat) we arrived at our camping spot. The rest of the support crew met us there and fixed our boat, and then we took it for another quick adventure.
At sunset, we came across some pythons wrapped around a tree branch overhanging a narrow channel and stopped to do a bit more filming. However, this slice of the story didn't make it into the ads, thanks to a minor mishap between Jono and a tree branch.
Willow was steering the boat and we had Jez and Row on board, filming. We turned around in a tight area with lots of overhanging trees.
'Shit, watch out, watch out, duck!' I yelled to Jez and Row, who had their backs to us as they filmed at the front of the boat. They ducked and missed but Jono, who was standing just ahead of Jez and Row, turned right around to see what was going on and headbutted a low-lying branch.
'Ahhhh!' Jono yelled, and we stopped the boats.
He was holding his head, laughing at how silly he'd been. Blood was pissing out everywhere. The guys kept filming for a bit, but then Dan called it quits.
'Let's wrap it up, fellas. I think this misadventure is too much of a miss than a hit. No one wants to see blood.'
We got back to camp and Dan pulled out his make-up kit, plugging up Jono's face with thick foundation. Jono was less than impressed by the whole ordeal, but we all thought it was pretty funny. If you watch the '3 Men in a Tinnie' ad, you'll see that at the start Jono looks nice and fresh, but once we get to the camping spot he has a noticeable cut on his head, which features in the rest of the ads we filmed on this shoot.
Before we went on Jono's horseriding adventure we spent a day getting things sorted for whitewater rafting. A mate of mine, Jim, owns a cane farm property up near Cairns, so I organised to go out to his place to source some bamboo and gear to build our rafts with. I hadn't been to Jim's Cairns place before, and he wasn't around to help us with directions, so it took a bit of guesswork to find the joint. We located the shed and helped ourselves to the saws hanging up. There was some fish netting and a few plastic drums lying around too, which we loaded into our trailer, then got hacking into the bamboo.
Dan was stoked with the whole set-up. 'This is ideal, Matty. There's everything we need here for the rafts!'
Mucking around and doing our thing, we sawed our way through a little patch of bamboo. Apollo was running rampant, chasing birds around the paddock, having a great time.
Then a ute pulled up.
'G'day, guys, what are you doing?' the old fella driving it asked.
'Hey, mate, we're friends of Jim's. We're just taking some timber off his block for some rafts we're making,' I told him.
'Yeah, okay. Well, I'm not sure who Jim is but this is actually my block you're on,' he told us.
'Oh, really?! I'm so sorry, man, I had no idea!' I said to him, shocked.
I felt terrible. There we were, on some random person's property, with three of his saws in our hands, standing in front of a pile of his cane we'd just cut down.
'Don't stress, Matty, all good. Love your show, by the way,' he said.
Phew, I thought. Boy, were we lucky that he knew who we were and was okay about the entire situation, because we could have been seriously stung for trespassing.
'My young fellas are fans. If you don't mind, can I bring them up here for a picture?' he asked.
'God, yeah, it's the least we can do, mate!' I said, feeling totally relieved and equally embarrassed.
The guy came back up with his kids, their cousins, their cousins' friends, his neighbours and his neighbours' neighbours. We said hello and got a few photos. They were lovely people and the bloke's two young fellas were keen to help.
'You can leave all the material here if you like and come back when you're ready to film,' they said. We took up the offer and said we'd come by again.
Day three was horseriding, which was a nostalgic choice by Jono, who reminisced about the rides he and I went on as kids. Apollo was great around the horses and followed closely on the ground, running up terrain and paddling through water. Willow decided he wanted to go on a motorbike, which worked well for all of us because he could cart the esky with him.
Jono and I bounced around the place and began to feel confident in the saddle.
'I'm really starting to get the hang of this,' Jono said to me as we picked up into a canter down one of the fire breaks.
'Yeah, I'm not feeling too rusty myself,' I said back.
We got into the rhythm and made our way through the rainforests and water channels. But Jono had spoken a little too soon. He fell off his horse twice along the way, into the water, eating humble pie.
After a beautiful day we arrived at our camping spot to see Willow with beers on ice and fresh mud crabs for dinner.
'You beauty! You've delivered the goods today,' I said.
We didn't have any injuries, breakdowns or things go terribly wrong. It was actually one of the most picturesque and peaceful days I've had, and it was finished off perfectly. We needed some rest, especially ahead of what I had planned for the next day.
I called the two young lads from the cane farm and let them know we were coming over to collect our rafting gear. We rocked up, and there they were with these big grins on their faces. They're standing there with these three perfectly made rafts ready for us to take.
'Wow, lads, these are insane!' Jono said.
Each raft was solidly constructed with plastic water drums, bamboo sticks, rope and netting pulling everything together. I didn't want to tell them that part of the storyline was us constructing the rafts, so I asked if they minded if we took some material to make a couple more as backups and invited them along for the fun.
'Well, there's three rafts here,' Willow said, 'and you and Jono are making one more each, so I may as well use one of these sick rafts to put it to the test.'
'Ah, cheating are we, Willow? You don't back your own raft-building abilities?' Jono joked.
'Nah, nah, not at all, I'm just interested to see how it goes.'
'Okay then, mate, you take one of them and I bet I'll still pip ya!' I said, stirring up a bit of a challenge between us.
We arrived at the rapids and they didn't look like a simple walk in the park.
'Wow, Matty, you've really thrown us in the deep end here,' Willow remarked. 'You've always got to take it to that next level, don't you? That turbulence is definitely no joke.' He was right. The rapids were raging, and had some good power behind them.
'All for one and one for all, hey!' I said, as we walked to the top of the rapids. The young fellas were mucking about with their own rafts, taking it all in, as happy as can be.
The race was on to see who could reach the full Great Northern esky first, located about a kilometre downstream. I thought I was going to win, Jono was also pretty confident about his rafting abilities and I'd successfully made Willow doubt himself, as he'd revised his own prediction by saying he would be happy just to make it down in one piece. We bounced into the water, but as we got going I turned around to see Apollo jump off the rocks, following me in.
'Oh no!' I started paddling backwards as I watched him get caught in a whirlpool, going around and around in circles. I kept paddling back until I reached him, and then pulled him out and into the arms of one of the young fellas. I looked down the rapid and saw Willow fly in front of Jono, taking the lead. Jono moved closely behind, riding his raft like a rodeo bull. Once the guys noticed what had happened they jumped out and legged it back up to the top so we could do a rerun but, sadly for me, Willow took the win again fair and square, followed by Jono, and I came in deadset last. But the adventures weren't over yet.
'I'm in big trouble, guys!' Willow called as he got out of the water, rummaging around the rocks in the shallow water.
'Why, what happened?' I asked.
'My wedding ring got caught and came off me finger.'
'UH OH! BIG trouble, my man, you've got no hope finding it in these rapids,' I told him.
'I got my phone number engraved on the inside in case I ever lost it. I just didn't plan on losing it somewhere like this.' He looked around, worried.
'Wait, did you just say your ring is engraved with your number?!' Jono and I said at the same time, laughing.
'That's the weirdest thing I've ever heard,' Jono continued.
'Yeah, how does a phone number even fit on a ring?' I added.
'Shut up you two,' Willow said. He sighed. 'I'll wait and tell the missus when I get home.'
'Good thinking, mate,' Jono said.
Our trip ended the next day and we headed home, talking about all the good times from filming.
'I hope we can do a True South next time and head down the coast somewhere to do another series. They're the best fun,' Willow said.
'Ditto. They're bloody good, mate,' I agreed.
At the start of 2018, Willow was around at my joint having a barbie and, strangely enough, we were talking about our trips to Cairns, reminiscing about the fun, when Willow's phone rang.
'Get out of here, there's no way,' he exclaimed, after answering it.
'What is it?' I asked once he'd hung up.
He turned to me. 'A bloke found my ring!'
'You're joking?!' I said.
'For real, he's posting it up to me tomorrow. He reckons he found it in a side pool, stuck in a crevice. Then he saw my number on the inside. It wasn't a bad idea doing that after all, was it?!' Willow said.
'I just can't believe your luck, mate, that's incredible!'
#
It was a nice cool night in the middle of the dry season, and Jono and I were camping out in my caravan on La Belle Station, located approximately 180 km southwest of Darwin. I had bought the 1970s caravan off Gumtree a week earlier for 400 bucks. It resembled an empty tin can and was falling apart in most places. It was short on a toilet but had a working stove and provided good enough shelter for Jono and me. We needed a base during a week-long croc-catching mission on the cattle station, and with the cool nights I thought a caravan would provide better shelter than mozzie dome tents.
I'm usually a pretty heavy sleeper, but I woke up in the middle of the night with a full bladder. I was so comfy wrapped up in my sleeping bag with the crisp air blowing through the open windows chilling my face, that I prolonged my pee for a good hour until my bladder felt like it was going to burst. Reluctantly, I got up in my jocks, still half-asleep, and stumbled out the door, walking straight into something.
'Shit!' I blurted out.
Our tarpaulin awning had partly fallen down and had swiped me across the face, bringing me to my senses. I tied the blue tarp back to the pole and closed the botchy door behind me, then turned to look out over the lagoon. We'd parked the caravan about 100 metres from the water in a really pretty spot, with the big silver moon splashing its reflection down over the water's surface. The night owls hooted in the trees and I could hear other birds chirping too: the moon was giving off such a strong glow, they must have thought it was daytime. The frogs were also making one hell of a racket, and I was surprised the chorus of animals hadn't woken me earlier.
La Belle shares an eastern boundary with Litchfield National Park and, in some places, the property is just as picturesque as the Park. Most of the land is made up of marine floodplains full of wildlife and covered in native wetland grasses, with lots of crocodiles and heaps of big ones at that. The property is divided into 34 paddocks and can hold about 30,000 head of cattle. Cattle and crocs don't mix well, so the station owner had asked Jono and me to do a big relocation of about 30 crocs. It was perfect timing because the local crocodile park needed a top-up of breeding stock. The station owner also told us to keep an eye out for a massive croc that had been giving him grief, intimidating the station workers while they were fishing.
I'd heard the mustering pilots talking about this croc over the years; it had become a bit of a legend. People said it was around 18 feet, and it had a reputation for being able to carry an entire cow in its jaws across the floodplains. You usually see big crocs with a leg or head but never a full cow because of the sheer weight. So this croc had to be one hell of a monster animal. Jono and I knew he was hanging around because we'd seen him on our motion-sensor cameras, but we hadn't sighted him in the flesh yet.
The air outside felt like it was getting cooler by the second and I didn't want to stand still for long. I walked over to a bunch of trees nearby, where our semitrailer was parked, and relieved my bladder. Down that way were four crocs tied to the bases of tree trunks with a pen around the outside of them, on long ropes so they could still move around a bit.
We had slung these crocs out of a turkey nest, which is kind of like an above-ground dam with built up sides. I used a chopper to get the job done the day before, and we were keeping the crocs in the cool until a truck came by to relocate them to the farm. The four medium-sized females in this pen were carefully taped up with hessian over their eyes to keep them calm. I could hear the thud of one croc's tail slapping the ground, and another one sliding around the pen. The smell of the rotten leftover bait in the semitrailer wafted towards me, so I quickly finished my business and went to head back to bed. I turned and kicked something hard.
'Ugh!' I put my foot down and then stuck it out again, this time more gently, to try to feel what was in my way. I squatted to get a closer look.
'Strike me dead!' I said, as the blood starting pumping through my body. Lying right there, stretched out on the ground in front of me, was an enormous thumping croc tail. There was no denying it – it was the monster croc. He must have come up to investigate the rotting pig bait and got caught out high and dry. I stood still for a moment, then suddenly the croc swiped its tail, clipping me and sending me arse-up into the dirt. It reared up into the air, seized one of the trailer tyres in its jaws and shook the shit out of it, nearly ripping it off. I scrambled to my feet and moved around to the other side of the trailer.
'Jonoooo!' I screamed, at the top of my lungs. 'Come here now!'
The croc moved a couple of metres back towards the water but stopped in its tracks, uncharacteristically.
Jono scampered out of the caravan in his jocks. Thank god he had heard me.
'Get the rope and harpoon!' I called out to him. He ran, scooped up the gear and bolted in my direction, eyes wide open. I ripped the rope out of his hands and sprinted for the croc with Jono close behind me. I wanted to get in front of it to put a head rope on his snout before he reached the water but I wasn't fast enough. The croc had started moving again, heading into the lagoon.
'Quick, get one into him, Jonsy!'
Jono speared the harpoon into the back of the croc's neck, hitting it bang-on. With the croc attached, the harpoon line started reeling further and further out with the croc attached, while Jono ran behind, keeping a strong hold of the reel at the other end. I went like the clappers, slipping and sliding in the mud, down to the bank where our tinnie was parked. 'Jump in, mate, and get another one ready!'
We needed reinforcement with a croc this big, one harpoon wouldn't cut it. Jono was already all over it, prepping the second harpoon in the boat.
The technique of harpooning is often misunderstood. When people hear the word 'harpoon' they automatically think negative thoughts because of the horrific practice used by Japanese whalers. Those harpooning techniques kill whales slowly and painfully, and the thought of it makes me sick with anger. The hunters often use a toggling harpoon, which involves the top half of a harpoon point detaching into the body of the whale; twisting horizontally into the flesh, lodging the head of the harpoon through its skin and blubber and underneath its muscle. Whales have a non-valvular structure of blood vessels, so when even the smallest of harpoons pierces their skin a deadly drain of blood gushes out and the animal bleeds out. It is nothing short of grotesque.
Crocodile harpooning, on the other hand, is a different kettle of fish. Firstly, the wellbeing of the animal is our primary concern and we harpoon to relocate, not to kill. Our technique follows the Australian government's Code of practice for the humane treatment of wild and farmed Australian crocodiles. It is a quick, efficient method for capturing crocs, which reduces the period of struggle time and therefore stress on the animal. All harpoons are designed and built to ensure that its barbs penetrate no deeper than below the crocodile's skin so that it doesn't damage muscle and other under lying body tissues. The barbs are never longer than 2.5– 3 cm to prevent unnecessary penetration and we always try to place the harpoon in the neck area of a crocodile, which is the safest and most pain-free for the animal. It is also the most practical place because it helps bring the croc to the surface. If we were to hit the croc on the tail, the animal would have the upper hand: it would whip its tail around and be able to keep itself at the bottom of the water, so we would be attempting to catch it all day long.
We had no one to call for back-up and were in a tiny tinnie in the middle of a croc-infested lagoon, wearing nothing but our undies.
'We're not the most prepared are we, mate,' Jono said, grinning.
'We're pretty good at getting ourselves into situations like this and we're usually just as good at getting ourselves out of them, so let's hope the same applies tonight,' I replied.
'What do you mean "tonight"?! Mate, it's one-thirty in the MORNING and we're half-naked trying to catch the biggest croc in Australia!'
We both lost it laughing. So far, things were going well. The croc seemed pretty worn out; he still had the first harpoon attached to the rope in him and he wasn't putting up a fight. I continued to steer the boat, and Jono reeled in some of the loose line as we caught up to the croc in the water.
'All right, mate,' I said. 'Any minute now it's going to come up alongside the boat for a breath of fresh air. You need to get another harpoon in the same spot.'
'I can't see anything under that water,' Jono commented.
'Yeah, I know, that's why you've got to wait for that head to emerge. When it comes up for air, get another one of those harpoons into him.'
'Yep, yep, righty-o.' Jono took the headlamp and secured it to his head.
A big spray of water burst through the lagoon's surface as the croc rose, releasing a burst of air from his nostrils. Jono propelled another harpoon at him and clipped him in the back of the neck, right beside the other harpoon.
'Perfect shot, well done!' I shouted.
Away he darted, with a burst of power.
'This is ideal, we've got two in him,' I said, confidently.
'If we get him, I'll be very surprised if he's not the biggest croc ever caught in Australia,' Jono replied. 'It's the biggest I've ever seen in the wild, that's for sure.'
I agreed with Jono. This beast was more prehistoric dinosaur than modern crocodile.
'Now it's time for the head rope, then we'll work out how to get him out of here.'
By this time it was 2 a.m. We were tired and the insects were thicker than ever. With every breath I got another mouthful of mozzies, and their relentless biting was driving me mad.
'I can even feel these suckers biting at my balls,' Jono said.
So could I. They were biting through the material, having a crack at every inch of skin. I couldn't think straight.
'Pass me the lines and make yourself some clothes,' I said to Jono, nodding towards the hessian, gaffer tape and a small knife in the corner of the boat. He looked confused, and didn't move. Then the croc jerked the lines, so I turned the motor on to move along with him some more until he stopped again for a rest.
'Here, like this, I'll show you,' I said, switching the motor off and turning to Jono.
I cut some sheets of hessian and wrapped them around my calf, gaffer-taping them to hold them in place. I did my calves and quads on both legs, the full length of my arms and then mummified myself with the hessian around my torso, taping it tight. Jono looked impressed and followed suit.
Every time there was a bit of reprieve with the croc and we became stagnant in the water, I would turn the motor off and hold the lines while Jono taped up another part of his body. We looked like idiots, but it stopped us from getting cold and bitten.
We booted up the river again. I thought I could see something faint in the near distance so I cut the engine and let the croc pull us along. Sure enough there was a big log coming up in the water that I didn't see early enough. It was too late: we hit the log and Jono flipped off sideways into the pitch-black water. Luckily, he still had a hold of the rope. I grabbed the two reels wedged in the footwell to make sure they didn't fall overboard. Jono had one hand on the rope and was pulling himself in towards the boat, kicking frantically at the same time.
'Try to keep your splashing to a minimum!' I called out.
He stopped moving about and started doing breaststroke with his head above water. He swam up to the back end of the tinnie, panting crazily, with complete shock in his eyes.
'That spooked me out, that was close, mate,' he gasped, when he was safely back on board.
I knew it was too close, and I also knew we weren't at all equipped to catch the croc, but it was now or never with this guy.
'Jono,' I said, 'I promise we'll go back if we can just get a head rope on the croc first.'
Jono wasn't happy. 'Okay then.'
I turned to restart the motor and up came the croc behind me, attracted by the noise. He grabbed the boat's engine in his jaws and tore the whole thing out, leaving a gaping hole.
'We just lost our engine, Jono,' I said, with a touch of panic.
'Shit, now what?'
We both moved to the front of the tinnie to balance out the sinking end of the boat. But water started flowing into the back. We were right in the middle of the lagoon, with about 50 metres on either side of us until dry land.
'Grab those empty water bottles and foam balls,' I instructed Jono. 'We'll attach floats to the ends of the lines so we can locate the croc in the morning.'
We tied the croc off quick-smart, and hoped the floats would hold up until daybreak. There we were, Jono and me, sitting up the front of the tinnie trying to keep the back up and out of the water, paddling desperately with a couple of bottles. Each time I looked out across the water, I saw little crocodile eyes glistening like fairy lights; the water was covered in them. The big croc had left us alone, but there were hundreds of other crocs that wouldn't have minded having a crack. More and more water was pouring into the boat and then, about 10 metres from the bank, our tinnie did a Titanic and plummeted into the water. We launched ourselves forward and swam the last couple of metres to the bank.
'We made it!' Jono was relieved.
I looked around me. 'Well, not quite, mate. We're on the wrong side of the lagoon. Our camp is set up across the other side.'
'Ya fucking kidding me?'
'I wish I was.'
We made our way back up around the lagoon, moving around a few estuaries as we walked. The jungle was thick and totally dark in places, so going barefoot wasn't great. Sharp sticks stabbed into the arches of my feet and I walked with my hands in front of me, trying to feel my way. I just prayed and prayed that we didn't find ourselves coming head to head with a big bull or buffalo. After another hour and a half we staggered into the camp clearing. We had cuts and scratches, busted feet and busted brains. Without even rinsing off, we ripped off our hessian outfits and stumbled into our beds at 4.30 a.m.
My watch alarm went off at 7 a.m. and we jumped in the chopper to get an aerial view of the lagoon.
'Look, Jonsy,' I said, pointing. 'The two floats are still there! Let's hope the crocodile is too.'
We flew straight to the homestead and asked for a new boat and some manpower.
The station owner immediately agreed. 'We're not worried about the boat, just don't get yourself chewed on,' he said.
'Nah, we'll be right, hey, Jono?'
'Yep,' he replied bluntly. Jono was probably thinking, Don't give Matty your bloody boat, you'll never see it again! He was done with it all, I could tell.
'Will four fellas do?' the station owner asked.
'That will be perfect,' I said.
We got everything sorted: the guys would take the new boat down and bring a couple of utes in case we needed them.
There was no messing about. When the boat arrived, Jono and I put it in the water and headed for the floats. Two big marble-like brown eyes popped up to say hello as soon as we reached them.
I was stoked. 'We've still got him!'
Then out came his set of snapping jaws. Each tooth stretched two-and-a-half inches out of his mouth, his head was almost two feet wide and about four feet long.
'No wonder he tore the back of the boat off,' I said to Jono. 'It would've been like a little toy for a croc this size.'
As I was talking, the croc pushed out of the water even more and Jono dropped the head rope around its top jaw, pulling it tight. He got him first go, once again.
'You're on fire, mate!' I yelled, delighted.
We slowly towed the croc through the water and close to the bank. We had to get him onto land before I could tape him up. He was too big to do it in the water, and if I tried I knew I would be dead meat.
'Righto, mate, jump out onto the bank and tie the croc to that tree,' I said to Jono. 'I'll run up to the caravan and get the boys to bring the utes closer. If we need to, we can probably use the chopper to pull him up a bit.'
The croc was still underwater when I arrived back with the boys. Jono was keeping tension on the rope, concentrating and staying calm.
'He's just at the end of the rope, about six feet under, but when he emerges he's going to come out with a lot of force so we have to be careful that when we pull him, his bottom jaw doesn't get stuck in the mud, okay?' I said to the fellas.
'Okay,' everyone answered.
We had the rope wrapped around the tree and tied to the car, set up like a pulley to manoeuvre the croc out of the water. All eyes were on the monster as it appeared unwillingly. With each acceleration and pull, he moved about a foot or so. I could hear the station workers gasp as the croc showed its impressive size. But we were so focused on the animal that the guys in the ute didn't realise how much pressure they were putting on the rope around the tree.
'Argh!' I yelled, as the rope ripped backwards through my hands, burning them raw.
I turned around to hear an almighty crack and watched as the tree snapped in half. The rope went to pieces and the car catapulted forward. Jono fell over and I just stood there, dead still, in shock. In that split second the monster took the opportunity to belly down the bank like a snake into the water. It swam away frantically, with its tail whipping to and fro. I scrambled around trying to get a hold of what was left of the rope, but it was too late and the rope was too short.
That was it. The croc got away and there was no getting him back in the one day we had left. He escaped before we could properly measure him, so I never got to know how big he really was. I have gone back to try to catch him on six different occasions but have had no luck. I still to this day get texts and pictures from mustering pilots of the elusive croc carrying half-chewed cows in his mouth or of his kill that he's stashed in the jungle somewhere and left to rot. La Belle Station has changed hands in recent years, but the same croc is giving the same headaches to the new owners and it's still one of my missions to return and move him out. He was once a myth, but now we know he's the real deal and he's become a bit of a legend in the area.
#
As a young lad, I seized every opportunity to have fun, and loved nothing more than being free. Toeing the line and being responsible wasn't something I ever thought or cared about. I just lived every day as it came, doing whatever I felt like, and went to bed each night planning the next day's mischief. School was a bore and I didn't like my teachers much. The more I got in trouble and punished, the better I became at not getting caught. Punishment is meant to deter bad behaviour, but that never worked for me.
Growing up, everyone would always say Get good marks, perform in class and you'll succeed. My question was always, Yeah, but succeed in what? As a troublesome teenager I had no idea what the future would hold. All I knew was that life was for living and every minute I was asleep was a minute lost. 'You can sleep when you're dead' is one of my favourite sayings. Wasting time has never been on my agenda. I have a tendency to swing too far over into the land of work and forget to stop and smell the roses, so where is the balance I asked myself. My goals as a young bloke were to:
* work hard
* be financially stable
* buy a house
* meet a life partner
* have kids to pass on everything I know.
That all sounded reasonably easy to achieve and like the sort of life that I would be happy with. But as I grew into a young adult, being happy with those typical 'great Australian dream' goals wasn't enough. Becoming financially stable didn't happen overnight, and meeting the love of my life wasn't working for me (I felt like girlfriends just slowed me down). The only thing that was working were my jobs, so I threw all my efforts into that, and improving myself as a person, and figured the rest would fall into line as time went on.
I wasn't always a good kid as a young fella. Actually, I was totally out of control and did quite a few things that let my mum down. And it was seeing that disappointment in my single, hardworking mum that made me change. I wanted to prove to myself, and to Mum, that I could become a better person and a son that a mum could be proud of. I promised her that I would finish school. I didn't graduate with flying colours, that's for sure, but I graduated nonetheless.
When I was 17, I said my goodbyes and hit the road, leaving South Australia to head north with no plan or idea of what lay ahead. I had quite a few jobs along the way and a lot, and I mean a lot, of ups and downs. But I kept pushing through. What I learnt very quickly is that without any trade qualification or tertiary education, I had to invest in myself, upskilling in any way I could, whether that was doing a short course or getting a specialised machinery licence. I also said yes to everything and anything, no matter how crap the work was, because I knew that as long as I was working and trying my hand at new things I was adding new skills, insights and knowledge to my resumé.
At the heart of who I was as a kid and a young adult, and who I am today, is valuing and respecting people. I always make the time to develop and maintain friendships and business relationships because that is what life, love and work is built on. I've also seen that, in life, it is often more a case of who you know, not what you know, so if you maintain integrity and build a solid reputation as an honest, loyal, hardworking person then you will go places.
I was never 'too good for' or 'above' any job, and never said no to an opportunity for work. Even when I had no idea what I was meant to do, I just made sure I worked it out quick-smart. I think that coming from a working-class family and battling at times was a blessing in disguise. I knew that anything is better than having nothing, so I did everything in my power to create a life for me and have something. And the one thing I did have was good groundings and a solid work ethic, instilled in me by my mum, as well as a passion to achieve; to actually be someone and do something that made a difference in the world.
I see a lot of self-entitled kids these days, and some of them don't get very far because they scrunch their nose up about working at a supermarket checkout, or don't want to have to work hard to prove themselves in a low-paying job before progressing to the next step. I didn't have that luxury, so I was a toilet cleaner, bed-maker, driller, camp host, barman – the list goes on.
I finally felt like I'd achieved something when I saved up enough money to get my chopper licence, which was no walk in the park for someone who isn't book-smart. I was fine practically but the theoretical stuff . . . I don't know how many times I sat and resat some of those units. But the feeling of having a piece of paper with my name on it that enabled me the freedom to fly and do it as a job was one of the best feelings.
When I was 23 I worked as a ringer on Moroak Cattle Station, fencing, mustering and cleaning. As any ringer knows, to succeed in that job, you've got to be a jack-of-all-trades.
'Hey, Matty, can you shoe a horse?' my boss asked me one day.
'Yeah, no problem,' I said. 'Mum showed me a few times on the farm when I was younger, so I'll work it out again pretty easy.'
'Righto then, there's thirty horses out the back, bring them in tomorrow morning and start chucking a few shoes on. We've got to muster in a few days.'
Holy shit, I thought to myself, it's going to take me more than a few days to sort this one out. So that night I went around the camp asking a few lads about putting shoes on. Everyone was in the same boat as me: they had absolutely no idea. It soon became obvious as to why the boss asked me, because there was no one else. Anyway, I had basic knowledge and called my mum that night, asking her 101 questions. Morning time broke and I was dreading the job, but wasn't going to quit now. I ran the horses in, ready for action with all the gear but no clue.
The first fella was pretty good and quiet – happy for me to lift his feet, cut the shit out and rasp down his long-overgrown nail, bend the shoe into place and nail it on.
Yes! I thought. That wasn't too hard. It turned out I had declared victory too soon.
I drafted the next one in, a mare on heat, and she didn't look very happy. Same deal: I secured the halter and started with a nice touch around the feet, trying to lift them one at a time. She had her ears back, trying to nip at me every so often. A little more time passed, and I thought she had calmed down and was at ease. I got the front shoes on with a few jumps and a couple of nips, but all in all she was going well. Time for the back feet. I got her hoof cradled in my crutch and started cleaning the foot out. The next thing I knew I was flying through the air before crumpling into the cattle yards with a massive pain shooting through my mid-section.
'Shit, that hurt,' I said to myself.
I heard a voice from the back of the yard. 'Jesus, are you okay?'
I stood up and brushed myself off. 'How ya going? I'm Matt,' I said to this fella, who I'd never seen before.
'I'm Nolan,' the bloke said, in an American accent. 'I've just got in from Texas. A friend of mine has lined me up with a little work here on the cattle station.'
'You know anything about shoeing horses?' I asked.
'Yeah, that's my main job back home,' he replied.
That's music to my ears, I thought, wanting to do the Toyota jump then and there.
'Any chance you could give me a hand to get through this mob?' I asked. 'We gotta go mustering in a few days and if every horse is the same as this cranky bitch I'm gonna come out of here looking like I've just done 12 rounds with Muhammad Ali.'
'Yes, sir, it would be my pleasure to give you a hand,' he said.
Over the next three days we hooked in and got the rest of the horses done, and over that time I learnt so many new things about what to do and what not to do when you shoe a horse: all the tricks of the trade.
'You've really got the hang of this now, Matt. You'll be teaching everyone else soon,' Nolan said, when we were done.
I was so grateful to him and stoked to have another skill under my belt. I knew it would come in handy again in the future, especially as a ringer. It was sink or swim at the time and a lot of young lads were getting the flick, but I stepped up and got the job done, much to my boss's surprise. But saying yes to that job kept me in a job, and my boss was none the wiser.
After a few more months on the ground, my boss decided to put me in a chopper to start mustering. I was thrilled, but I also knew in the back of my mind, if I ballsed this one up I was gone, and there were 100 pilots sitting on the fence waiting to nick my job. Another old boss of mine once said to me, 'Matt, if you don't fuck up you're not really trying'.
I didn't mind the saying, except I've often felt like I'm cursed. I'm absolutely one of those people that bad stuff always happens to and especially in my twenties it felt like everything I touched would go wrong; it must have been all the karma coming back to bite me in the arse from when I was a kid. But as clichéd as it is, you've got to make mistakes in order to learn from them. There are so many people now that have the ability to achieve great things. But they're too worried about taking a risk or making a mistake that they just coast through life not ever stepping outside of their comfort zone.
After four years of working as a mustering pilot, I started to head home every so often to see my mum and old school friends. Everyone in South Australia was interested in what I'd been up to in northern Australia: flying, drilling and exploring the outback. The more I came home the more I would hear mates say, 'Gee, you're living the dream' or 'You're so lucky with the life you live, someone's really been looking over you'.
It started to get to me. In a way, I was lucky that I was able to do what I was passionate about, but what a lot of these people didn't see were the hard yards and sacrifices I had made to get where I was. They just saw me as this footloose and fancy-free cowboy, flying the skies and drinking beers at the pub. What they didn't know about was the months and years on end of working around the clock to make a dollar, or to achieve the next goal.
I used to write in a diary every day, and not long ago I came across all of the diaries I'd kept as a young lad, trying to make something of myself. At the start of each week I had written each one of my expected expenses, along with my weekly income and my savings. Then beneath it I would write my savings goal for the next week: sometimes it was only $50, but that was a lot to me back then.
When I started flying my first wage was $17,500 a year, which never bothered me because I was doing what I loved and was gaining valuable experience along the way. I learnt from everyone that I could, kept my mouth closed and my head down. Life went on, and I gained more insight and, most of all, good relationships. These became my most prized possessions. When you're a young pilot, you love hearing stories of local legend chopper pilots who are the best in the business. Their names were always spoken of as if they were gods – the way they flew, how they could muster some of the toughest cattle and, most of all, never have an accident. As my flying career excelled and I felt myself getting better, I knew I wanted to become one of those pilots, a leader in the field.
I worked for North Australian Helicopters until I was 27. My chief pilot there was John Logan and I remember him saying once, 'Matty, if you make it to 5000 hours without an accident you're good pilot. If you get to 10,000 hours then you're a great pilot. Take it steady and don't kill yourself over a crappy cow that doesn't want to go to the yards. Pick your battles and fly smart.' So, I set my goals to never screw up in a chopper, paying attention to detail at all times and never getting too confident. I've had one flying mishap in Canada, in my early days, but I never damaged the chopper or myself, which is a pretty solid run for nearly 20 years of flying.
Starting the filming side of things was one of my biggest challenges. I was still working on cattle stations and everyone there enjoyed having a laugh with me when I pulled out my handy cam. I loved the idea of putting together little fun videos for my family and friends and didn't mind looking like an absolute idiot. I wasn't any good in front of the camera and the blokes on the station would wet themselves laughing calling me 'Hollywood' and making me feel like an absolute twat. But I knew that I could do it. I loved telling stories, and figured it was just a matter of time and that, with practice, it wouldn't take long before I became more natural on camera. I had jackaroos, cooks, girlfriends and pretty much any ring-in I could find to film for me while I was doing my job. I'd save all my footage and then on my days off when everyone went into town to sink piss I'd book a flight to Cairns, hire a cheap hotel room and book an editing suite for the day to start cutting my footage together. At this stage, I was just sending the videos back home to show everyone what I was getting up to up north – mustering, catching wild bulls, fishing, catching crocs and collecting croc eggs. It was so foreign to everyone down south, so they loved what I was putting together. The hardest thing was having to watch myself back and cringing when I saw myself on screen, with my corny one-liners and daggy jokes, but the more I watched it the more I could see how to improve, and I slowly got better.
Imagination, drive and thinking outside the box is key to me and my success. I've never earnt a great deal of money but have always been happy with the path I'm on. When I hit my 30s I still had no real commitments or responsibilities. I continued working hard on investing in myself and always looking for the next challenge. I got to a point where I'd exceeded most of my expectations of what I dreamt of as a kid. I had a chopper licence, had flown around the world working in Canada and owned (well, the bank owned) two small properties. But I still felt like there was more.
I had to work out whether I wanted to pursue my flying career and or follow the TV. I decided to move back to Darwin and see where the TV side of things would take me, giving up a career path I had worked so hard to get. Moving into a small donga out the back of my mate's place, I started chopper tours taking people fishing and on adventures around the Northern Territory. I cross hired a mate's chopper to do the work and used it to film more stuff in an attempt to get traction for a TV show scissor reel.
After lots of empty promises, a stint with the Discovery Channel, a broken contract and a lot of 'no's, 'come back later's, 'maybe next year's, 'you don't have what it takes's and 'you'll never get a show's, I finally got a gig with Nat Geo and Outback Wrangler was born. It was no easy feat. I had to sell one of my properties and spent every cent I had on chopper hire and filming. There were a lot of times where I felt like giving up, especially because everything I was investing in and pushing for came out of my back pocket and it was already a very shallow pocket. So at the end of filming I had nothing left in the bank but I had achieved what I set out to do. And following the first episode of Wrangler, more doors started to open for me.
More people wanted to come on my tourism trips and I started partnering with different brands and doing TV talk shows. But the more I did, the more the tall poppy syndrome I experienced, and the great (or not so great) Aussie comments continued to roll in again and again: 'You're a lucky bastard', 'You've been kissed on the dick' and 'Who did you screw to get your show?'.
It made my blood boil, because I came from nothing, still had virtually nothing and was trying to have a crack so I could finally be something. I wasn't in Hollywood playing the lead role in Gladiator, it was just me doing a little outback show on a doco channel. All the same, I was chuffed to get the gig, especially since I'd been working hard on it for six years and, just as I was about to give up, it all paid off.
With the success of Outback Wrangler, even though it wasn't a large personal financial gain, it was a great stepping stone and fantastic profile-building, and with that came a lot more credibility to me and my brand. I thought that with Wrangler out there and having a little bit of a profile, tourism was the next best thing to start a stable cash-flow business to tide me over while I pursued filming. I got a great break there. My mate Mick Burns gave me his chopper and said I could pay him back whenever I had a chance. I'd worked with Mick for some time, so he knew I would commit to paying it off.
I was shitting myself because it was a $500,000 machine and the most expensive thing I'd ever bought. I dreaded the idea of crashing it or something happening to it because I didn't want to do anything to compromise our friendship. Every single cent I earnt I put back to Mick. It was $20,000 and $30,000 that I gave him some months and others I'd be left with a couple grand, but I was happy to live in a shed because I knew I was getting ahead.
It took three years to pay off. I had $20,000 left to pay and ironically that same week my last payment was due, I landed on a reef, the tide came in too quickly and the chopper sank. I actually laughed and laughed at the time, thinking, Only me, only me would this happen to. But at that point it didn't matter, because insurance covered it and I was able to pay Mick the last $20,000 and buy myself another chopper outright.
Outback Wrangler slowed for a bit, so I focused on building my tourism business and with that came Outback Floatplane Adventures. This brought a new level of challenges, non-stop work and success of a different kind. Seeing the joy on people's faces and the wildlife not only flourishing but enjoying interacting with humans, thanks to all the work I did out there, was so satisfying. The business grew practically overnight and everyone loved it.
But again people would say 'How lucky are you to do this as a job', 'You call fishing work?' and 'You've got the best job in the world.' Yes, I love my job and consider myself a lucky bloke, but what everyone doesn't see is the behind-the-scenes midnight fuel runs, working on airboats into the early hours of the morning and the logistical nightmares and big expenses involved with managing a lot of staff and machinery. A series of staff stuff-ups cost me a quarter of a million dollars in one month with an airboat engine blowing up, a helicopter over-speeding, and my brand-new land cruiser rolling and crashing. It was hard selling out the company in 2018, but it has left me with a huge sense of success, leaving behind a fantastic tourism product in the Territory.
Today my focus is around Matt Wright, Explore the Wild, which is my new tourism company offering a number of new and exciting tour products to people in the Territory. And the rest of my time goes to Outback Wrangler, which is going from strength to strength both here and overseas. It's paying its own way now and getting better and better with each season. It's been a long journey with its fair share of setbacks and wins, but following my dreams, passions and wild ideas to make a buck has finally got me to where I want to be. Now, at nearly 40, I have a sense of contentment I've never had before, and I can honestly say that my two proudest achievements area making my mum proud and marrying my amazing wife, Kaia. Nothing compares . . .
Willow heading back to camp on the sling after a day's work collecting croc eggs on La Belle Station. Craig and Mick are on foot.
Age 25, egg-collecting on the Victoria River back when Jimmy and I were the 'two rascals'.
Flying the chopper over the nests at Arafura Swamp in Arnhem Land, NT.
One of the bigger crocs I caught on the Vic River back in 2004, when I was working on Moroak Station as a ringer.
The Outback Wrangler core team: Jono (r), Willow (c) and me. Here we are catching and relocating a handful of small female crocs on Elizabeth Downs Station.
An 18-foot monster crocodile caught during filming for season three of Outback Wrangler.
17-foot croc Axel getting moved from Wyndham Crocodile Farm during the 'wild west' mass relocation. Today, he lives at Crocosaurus Cove in Darwin.
Using an excavator to pull out another big crocodile, just one of the methods we used to get the crocs out of their pens at Wyndham Crocodile Farm.
Carrying the Devil to the back of the truck to join Axel.
The back of the truck, filled with hay and covered by shade sails, providing a comfortable spot for the Wyndham crocs.
Crossing the river in the Congo on the so called 'ferry' made up of canoes strapped together with a steel sheet laid across the top.
Chatting to the crew in the Congo, making jokes about the precarious river crossing we were about to make.
Mick and me in the Congo, admiring the incredibly straight wooden harpoon poles, handcrafted by the locals with only a small knife and some sandpaper.
Practising erecting the panel trap just outside Kabeya Maji to ensure we had everything we needed before we set the trap for real.
Our main camp set-up in Kabeya Maji, where we slept and ate around the fire most nights, sharing stories with one another.
Talking to camera while sitting on the back of the bigger of the two crocs that I caught while I was in the Congo.
The set-up of Outback Floatplane Adventures in the very early days of operation. We didn't even have a helipad so I had to land the chopper on the roof of Cyclone Creek.
Macca and me doggedly transporting the Polaris, ready for the first ever tour for Outback Floatplane Adventures.
Hanging out with my 17-foot pet croc Tripod in his pen at the back of the shack.
My three-legged scaly mate in all his glory, turning it on for the cameras.
Welding up the traps with one of the wildlife rangers in the workshop shed in Borneo.
Baiting and securing a cage trap in Borneo, showing two of the rangers how to activate the trapdoor.
The Burmese reticulated python I found lying across the road in Borneo.
Relocating the python and a crocodile upriver during the infamous boat trip that lasted until the next morning.
'Captain Courageous' flying me insanely high in the sky while I screamed and swore into oblivion.
JB and me connecting with the bull elephant we caught and relocated in the jungle.
Releasing the beautiful orangutans at the wildlife sanctuary.
Kaia and me just after we said our vows at our wedding in Dunsborough, WA.
Suited and booted with my closest mates on my wedding day – groomsmen from left to right: Mick, Clayton, Jono, Dhani and Jai.
Kaia and me on the Injidup cliffs, getting photos with the same chopper I landed on Rottnest Island in when we first met.
# Also by Matt Wright
The Outback Wrangler: True Tales of Crocs,
Choppers and Shockers
# MICHAEL JOSEPH
UK | USA | Canada | Ireland | Australia
India | New Zealand | South Africa | China
Penguin Books is part of the Penguin Random House group of companies
whose addresses can be found at global.penguinrandomhouse.com.
First published by Penguin Random House Australia Pty Ltd, 2018
Text copyright © Matt Wright 2018
The moral right of the author has been asserted.
All rights reserved. Without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form or by any means (electronic, mechanical, photocopying, recording or otherwise), without the prior written permission of both the copyright owner and the above publisher of this book.
Some of the names of people in this book have been changed to protect their privacy.
Every effort has been made to acknowledge and contact the copyright holders for permission to reproduce material contained in this book. Any copyright holders who have been inadvertently omitted from acknowledgements and credits should contact the publisher and omissions will be rectified in subsequent editions.
All photographs courtesy of Matt Wright
Cover design by Alex Ross © Penguin Random House Australia Pty Ltd
penguin.com.au
ISBN: 9781760141929
#
# Contents
1. Cover
2. About the Book
3. Title Page
4. Contents
5. Introduction
6. 1 Two rascals
7. 2 Training the crew
8. 3 The wild west
9. 4 Under investigation
10. 5 Operation Congo
11. 6 Unparalleled wildlife
12. 7 My mate Tripod
13. 8 Hiccups and headaches
14. 9 Island adventure
15. 10 Colossal croc
16. 11 Outback Wrangler, season three
17. 12 My wife
18. 13 A bloke's dream
19. 14 Caught off guard
20. 15 Success
21. Picture Section
22. Also by Matt Wright
23. Imprint
24. Read more at Penguin Books Australia
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 4,472
|
\section{Introduction}\label{s:intro}
It is well known that the $n$-fold symmetric product of a Riemann surface, $n\geq 2$,
is an $n$-dimensional complex manifold. One has a precise
description of all proper holomorphic maps between $n$-fold products of bounded
planar domains\,---\,provided by the Remmert--Stein Theorem
\cite{remmertStein1960propMap} (also see
\cite[pp.\!~71--78]{narasimhan1971SCV})\,---\,and, more recently, of finite proper
holomorphic maps between products of Riemann surfaces; see
\cite{janardhanan2014properMap}, for instance. It is therefore natural to investigate
the structure of such maps between symmetric products of Riemann surfaces. To this
end, we are motivated by the following result of Edigarian and Zwonek
\cite{edigarian2005geometry} (the notation used will be explained below):
\begin{result}[paraphrasing {\cite[Theorem~1]{edigarian2005geometry}}]\label{r:e-z}
Let $\mathbb{G}^n$ denote the $n$-dimensional symmetrized polydisk and
let $f : \mathbb{G}^n\to \mathbb{G}^n$ be a proper holomorphic map. Then,
there exists a finite Blaschke product $B$ such that
\[
f\big(\pi^{(n)}(z_1,\dots, z_n)\big)\,=\,\pi^{(n)}\big(B(z_1),\dots, B(z_n)\big) \; \;
\forall (z_1,\dots, z_n)\in \mathbb{D}^n.
\]
\end{result}
In this result, and in what follows, we denote the open unit disk with centre
$0\in \mathbb{C}$ by $\mathbb{D}$. Throughout this paper $\sigma_j$, $j = 1,\dots, n$, will denote the
elementary symmetric polynomial of degree $j$ in $n$ indeterminates (when there is
no ambiguity, we shall\,---\,for simplicity of notation\,---\,suppress the parameter
$n$). The map $\pi^{(n)} : \mathbb{C}^n\to \mathbb{C}^n$ is defined as:
\[
\pi^{(n)}(z_1,\dots, z_n)\,:=\,\big(\sigma_1(z_1,\dots, z_n), \sigma_2(z_1,\dots, z_n),
\dots, \sigma_n(z_1,\dots, z_n)\big), \; \; (z_1,\dots, z_n)\in \mathbb{C}^n.
\]
The {\em symmetrized polydisk}, $\mathbb{G}^n$, is defined as
$\mathbb{G}^n := \pi^{(n)}(\mathbb{D}^n)$. It is easy to see that $\mathbb{G}^n$ is a domain
in $\mathbb{C}^n$, whence $\mathbb{G}^n$ is a holomorphic embedding of the $n$-fold symmetric
product of $\mathbb{D}$ into $\mathbb{C}^n$.
Given a Riemann surface $X$, we shall denote its $n$-fold symmetric product by
${\rm Sym}^n(X)$. The complex structure on $X$ induces a complex structure on
${\rm Sym}^n(X)$, which is described in brief in Section~\ref{s:symm_prelim} below.
In this paper, we shall extend Result~\ref{r:e-z}\,---\,see
Corollary~\ref{c:two_symmprods} below\,---\,to proper holomorphic maps between
the $n$-fold symmetric products of certain non-compact Riemann surfaces. At this
juncture, the reader might ask whether there is an analogous generalization of
Result~\ref{r:e-z} to $n$-fold symmetric products of {\em compact} Riemann
surfaces. Before we answer this question, we state the following result and
note that Corollary~\ref{c:two_symmprods} is its non-compact analogue. Indeed,
the following result (the notation therein is explained below) was among our
motivations for the investigation in this paper.
\begin{fact}[an {\bf adaptation} of the results in \cite{cilibertoSernesi:symmetric93}
by Ciliberto--Sernesi]\label{f:compact_isom}
Let $X$ and $Y$ be compact Riemann surfaces with
${\rm genus}(X) = {\rm genus}(Y) = g$, where $g > 2$. Let
$F : {\rm Sym}^n(X)\to {\rm Sym}^n(Y)$ be a surjective holomorphic map, where
$n = 1, 2, 3,\dots, 2g-3$, $n\neq g-1$. Then:
\begin{itemize}
\item[(1)] $X$ is biholomorphic to $Y$;
\item[(2)] The map $F$ is a biholomorphism; and
\item[(3)] There exists a biholomorphic map $\phi : X\to Y$ such that
\[
F(\langle x_1,\dots, x_n\rangle)\,=\,\langle\phi(x_1),\dots \phi_n(x_n)\rangle
\; \; \forall \langle x_1,\dots, x_n\rangle\in {\rm Sym}^n(X).
\]
\end{itemize}
\end{fact}
\noindent{In the above, and in what follows,
we denote by $\langle x_1,\dots, x_n\rangle$ the orbit of
$(x_1,\dots, x_n)\in X^n$ under the $S_n-$action on $X^n$ that permutes
the entries of $(x_1,\dots, x_n)$. The map
\[
X^n\ni (x_1,\dots, x_n)\,\longmapsto\,\langle x_1,\dots, x_n\rangle
\; \;\forall (x_1,\dots, x_n)\in X^n
\]
will be denoted by $\pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}^X$. When there is no ambiguity, we shall drop
the superscript.
\begin{remark}
The paper \cite{cilibertoSernesi:symmetric93} does not contain a
statement of Fact~\ref{f:compact_isom} in the specific form given above.
Therefore, we provide a justification. We first consider the
case $1\leq n\leq g-2$. A special case of a theorem by Martens
\cite{martens:torelli63} gives us (1) and (3)
of Fact~\ref{f:compact_isom}, assuming that $F$ is a biholomorphism.
In \cite[Section~2]{cilibertoSernesi:symmetric93}, Ciliberto and Sernesi give a
different proof of Martens's theorem. It is straightforward to check that the
proof in \cite{cilibertoSernesi:symmetric93} yields (1)--(3) above\,---\,taking
$1\leq n\leq g-2$\,---\,even when $F$ is just a surjective holomorphic map.
Now consider the case $g\leq n\leq 2g-3$. This time, the first two
paragraphs following the heading ``Proof of Theorem~(1.3)''
in \cite{cilibertoSernesi:symmetric93} give us
(1) and (3)
of Fact~\ref{f:compact_isom}, assuming again that $F$ is a biholomorphism.
Again, it is straightforward to check that the requirement that $F$ be a
biholomorphism is not essential. The argument in those paragraphs gives
us (1)--(3) above\,---\,taking
$g\leq n\leq 2g-3$\,---\,even when $F$ is just a surjective
holomorphic map. \hfill $\blacktriangleleft$
\end{remark}
The restrictions on the pair $(g, n)$ in Fact~\ref{f:compact_isom} are
essential. It is classically known that there exist nonisomorphic compact
Riemann surfaces of genus 2 having isomorphic Jacobians, hence isomorphic
$2$-fold symmetric products.
Next, consider a non-hyperelliptic compact
Riemann surface $X$ of genus 3. Given any $\langle x_1, x_2\rangle \in {\rm Sym}^2(X)$,
there is a unique
point $\langle y_1, y_2\rangle \in {\rm Sym}^2(X)$ such that the divisor
$(x_1 + x_2 + y_1 + y_2)$ represents the
holomorphic cotangent bundle. The automorphism of ${\rm Sym}^2(X)$ given by
$\langle x_1,x_2\rangle\mapsto \langle y_1, y_2\rangle$ is {\em not} given
by any automorphism of $X$ (here, $g = 3$, $n = 2$, whence $n = g-1$).
Furthermore, we expect
any generalization of Fact~\ref{f:compact_isom} to be somewhat intricate
because, among other things:
\begin{itemize}
\item Any generalization wherein ${\rm genus}(X) \neq {\rm genus}(Y)$
will place restrictions on the pair $({\rm genus}(X), {\rm genus}(Y))$ owing to
the Riemann--Hurwitz formula.
\item The geometry of ${\rm Sym}^n(X)$ varies considerably depending on
whether $1\leq n\leq {\rm genus}(X)-1$ or $n\geq {\rm genus}(X)$.
\end{itemize}
In short, any generalization of Fact~\ref{f:compact_isom} would
rely on techniques very different from those involved in
proving Corollary~\ref{c:two_symmprods}. Thus, we shall address
the problem of the structure of surjective holomorphic maps in the
compact case in forthcoming work.
We now focus on $n$-fold symmetric products of {\em non-compact}
Riemann surfaces. We should
mention here that Chakrabarti and Gorai have extended Result~\ref{r:e-z}
to $n$-fold symmetric products of bounded planar domains in
\cite{debraj2015function}
Their result as well as Result~\ref{r:e-z} rely on an interesting
adaptation\,---\,introduced in \cite{edigarian2005geometry}\,---\,of an
argument by Remmert--Stein. The latter argument relies on two
essential analytical ingredients:
\begin{itemize}
\item[$(i)$] The ability to extract subsequences\,---\,given an auxiliary
sequence constructed from the given proper map\,---\,that converge
locally uniformly; and
\item[$(ii)$] A vanishing-of-derivatives argument that stems from the
mean-value inequality.
\end{itemize}
These ingredients continue to be relevant when planar domains are replaced
by Riemann domains and, indeed, parts of our proofs emulate the argument
in \cite{edigarian2005geometry}.
However, our proofs of the theorems below
do {\bf not} reduce to a mere application of Result~\ref{r:e-z} to appropriate
coordinate patches. An explanation of this is presented in the paragraph that
follows \eqref{e:const} below. Equally significantly, we need to identify a class
of Riemann surfaces $X$ for which some form of the ingredient $(i)$ above is
available for ${\rm Sym}^n(X)$, $n\geq 2$. This is the objective of our first
theorem\,---\,which might also be of independent interest.
\begin{theorem}\label{t:symm_taut}
Let $X$ be a connected bordered Riemann surface with
$\mathcal{C}^2$-smooth boundary. Then ${\rm Sym}^n(X)$ is Kobayashi
complete, and hence taut, for each $n\in \mathbb{Z}_+$.
\end{theorem}
We must clarify that in this paper the term {\em connected bordered
Riemann surface with $\mathcal{C}^2$-smooth boundary} refers to a non-compact
Riemann surface $X$ obtained by excising from a compact Riemann $S$
a finite number of closed, pairwise disjoint disks $D_1,\dots, D_m$ such that
$\partial{D_j}$, $j = 1,\dots, m$, are $\mathcal{C}^2$-smooth. The complex
structure on $X$ is the one it inherits from $S$: i.e., a holomorphic chart
of $X$ is of the form
$(\varphi, U\!\setminus\!(D_1\sqcup\dots \sqcup D_m))$,
where $(\psi, U)$ is a holomorphic chart of $S$ and $\varphi$
is the restriction of $\psi$ to $U\!\setminus\!(D_1\sqcup\dots \sqcup D_m)$.
The ingredients $(i)$ and $(ii)$ above allow us to analyse
proper holomorphic maps between a product manifold of dimension
$n$ and an $n$-fold symmetric product, where the factors of the
product manifold need not necessarily be the same. This is formalised
by our next theorem. A similar result
is proved in \cite{debraj2015function} where the factors of the
products involved are bounded planar domains.
Corollary~\ref{c:two_symmprods} is obtained as an easy
consequence of the following:
\begin{theorem}\label{t:bord}
Let $X=X_1\times \cdots \times X_n$ be
a complex manifold where each $X_j$ is a connected non-compact Riemann surface
obtained by excising a non-empty indiscrete set from a compact Riemann surface $R_j$.
Let $Y$ be a connected
bordered Riemann surface with $\mathcal{C}^2$-smooth boundary. Let
$F: X \to {\rm Sym}^n(Y)$ be a
proper holomorphic map. Then, there exist proper holomorphic maps
$F_j : X_j \to Y$, $j = 1,\dots, n$, such that
\[
F(x_1,\dots, x_n)\,=\,\pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}\circ \big(F_1(x_1),\dots, F_n(x_n)\big)
\; \; \; \forall (x_1,\dots, x_n)\in X.
\]
\end{theorem}
The complex structure on each of the factors $X_1,\dots, X_n$ has a
description analogous to the one given above for bordered Riemann surfaces.
Finally, we can state the corollary alluded to above. Observe that it is the
analogue, in a non-compact setting, of Fact~\ref{f:compact_isom}. It also
subsumes Result~\ref{r:e-z}: recall that the proper holomorphic self-maps of
$\mathbb{D}$ are precisely the finite Blaschke products.
\begin{corollary}\label{c:two_symmprods}
Let $X$ be a connected non-compact Riemann surface obtained by
excising a non-empty indiscrete set
from a compact Riemann surface $R$, and let $Y$ be a connected
bordered Riemann surface with $\mathcal{C}^2$-smooth boundary. Let
$F: {\rm Sym}^n(X) \to {\rm Sym}^n(Y)$ be a
proper holomorphic map. Then, there exists a proper holomorphic map
$\phi : X \to Y$ such that
\[
F\circ \pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}^X(x_1,\dots, x_n)\,=\,\pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}^Y\big(\phi(x_1),\dots, \phi(x_n)\big) \; \; \;
\forall (x_1,\dots, x_n)\in X^n.
\]
\end{corollary}
This corollary follows immediately from Theorem~\ref{t:bord} since
$F\circ \pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}^X : X^n\to {\rm Sym}^n(Y)$ is proper.
We conclude this section with an amusing observation that follows from
Corollary~\ref{c:two_symmprods}. We first make an explanatory
remark. It is well known that if $M_1$ and $M_2$ are two
non-compact complex manifolds of the same dimension and
$F: M_1\to M_2$ is a proper holomorphic map, then there exists a
positive integer $\mu$ such that, for any generic point $p\in M_2$,
$F^{-1}\{p\}$ has cardinality $\mu$. We call this number the {\em multiplicity}
of $F$, which we denote by ${\rm mult}(F)$.
\begin{corollary}
Let $X$ and $Y$\,---\,a pair of connected non-compact Riemann
surfaces\,---\,be exactly as in Corollary~\ref{c:two_symmprods}.
If $F: {\rm Sym}^n(X) \to {\rm Sym}^n(Y)$ is a proper holomorphic map,
then ${\rm mult}(F)$ is of the form $d^n$, where $d$ is some
positive integer.
\end{corollary}
\section{Preliminaries about the symmetric products}\label{s:symm_prelim}
In this section we shall give a brief description, given a
Riemann surface $X$, of the complex structure on ${\rm Sym}^n(X)$, $n\geq 2$,
that makes it a complex manifold. We shall use the notation introduced in
Section~\ref{s:intro}. Given this notation:
\begin{itemize}
\item Recall that $\langle x_1,\dots, x_n \rangle := \pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}(x_1,\dots,x_n)$,
\item Given subsets $V_j\subseteq X$ that are open, let us write:
\[
\langle V_1,\dots, V_n \rangle := \left\{\langle x_1,\dots x_n\rangle : x_j \in V_j,
\; j = 1, \dots, n\right\}.
\]
\end{itemize}
Since ${\rm Sym}^n(X)$ is endowed with the quotient topology relative to $\pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}$,
$\langle V_1,\dots, V_n\rangle$ is, by definition, open in ${\rm Sym}^n(X)$.
\smallskip
${\rm Sym}^n(X)$ is endowed with a complex structure as follows.
Given a point $p\in {\rm Sym}^n(X)$, $p = \langle p_1,\dots p_n \rangle$,
choose a holomorphic chart $(U_j, \varphi_j)$ of $X$ at $p_j$,
$j = 1,\dots, n$, such that
\[
U_j\cap U_k = \emptyset \ \ \text{if $p_j\neq p_k$} \qquad \text{and}
\qquad U_j = U_k \ \ \text{if $p_j = p_k$}.
\]
The above choice of local charts ensures that the map
$\varPsi_p: \langle U_1,\dots, U_n\rangle\to \mathbb{C}^n$ given by
\[
\varPsi_p: \langle x_1,\dots, x_n\rangle\,\longmapsto\,\big(\elsym{1}(\varphi_1(x_1),\dots,
\varphi_n(x_n)),\dots, \elsym{n}(\varphi_1(x_1),\dots,\varphi_n(x_n))\big),
\]
(where $\sigma_1,\dots, \sigma_n$ are the elementary symmetric polynomials
that were introduced in Section~\ref{s:intro}) is a homeomorphism. This
follows from the Fundamental Theorem of Algebra.
The collection of such charts
$(\langle U_1,\dots, U_n\rangle, \varPsi_p)$ produces a holomorphic
atlas on ${\rm Sym}^n(X)$. We shall call such a chart a {\em model coordinate
chart at $p\in {\rm Sym}^n(X)$}.
Finally, let $Z$ be a compact Riemann surface, $X\varsubsetneq Z$ be an embedded
open complex submanifold of $Z$, and let $\mathfrak{A}(Z)$ denote the complex structure
on $Z$. Then, since\,---\,for any point
$p\in X$\,---\,there is a chart $(U, \varphi)\in \mathfrak{A}(Z)$ such that $U\subset X$,
the above discussion shows that
${\rm Sym}^n(X)$ is an embedded complex submanifold of ${\rm Sym}^n(Z)$. We refer the
reader to \cite{whitney1972varieties} for details.
\smallskip
\section{Hyperbolicity and its consequences}
The proof of Theorem~\ref{t:bord} will require several results about
holomorphic mappings into Kobayashi hyperbolic spaces. We summarize the
relevant results
in this section. An encyclopedic reference for the results in this is section
is \cite{kobayashi98hyp}.
In the theory of holomorphic functions of one variable, the behaviour of
holomorphic functions near an isolated singularity is well-studied.
Among the important results in this area are the famous theorems of Picard.
A consequence of Picard's big theorem is that any
meromorphic mapping on $\mathbb{D} \setminus \{0\}$ that misses three
points automatically extends to a meromorphic function defined on
the whole of $\mathbb{D}$. One of the proofs of Picard's theorem
relies on the fact that the sphere with three points
removed is a hyperbolic Riemann surface. This perspective allows one
to generalize the aforementioned extension theorem to holomorphic
mappings into Kobayashi hyperbolic spaces. To this end, we need a definition.
\begin{definition}
Let $Z$ be a complex manifold and let $Y$ be a relatively compact complex
submanifold of $Z$. We call a point $p \in \overline{Y}$ a
\textit{hyperbolic point} if every $Z$-open neighbourhood $U$ of $p$ contains a
smaller neighbourhood $V$ of $p,\ \overline{V} \subset U$, such that
\begin{equation}\label{e:hyp_imb}
K_{Y}(\overline{V} \cap Y, Y\setminus U) := \inf\{K_Y(x,y) : x \in
\overline{V} \cap Y, y \in Y \setminus U\} > 0,
\end{equation}
where $K_{Y}$ denotes the Kobayashi pseudo-distance on $Y$. We say that $Y$ is
\emph{hyperbolically embedded} in $Z$ if every point of $\overline{Y}$ is a
hyperbolic point.
\end{definition}
The following result is an example of an extension result in higher dimensions
of the type alluded to above.
\begin{result}[Kiernan {\cite[Theorem~3]{kiernan72extension}}]
\label{r:kwack}
Let $X$ be a complex space and let $\mathscr{E} \subset X$ be a closed complex subspace.
Let $Y$ be a complex manifold that is hyperbolically embedded in a complex
manifold $Z$. Then every holomorphic map $f:X\setminus\mathscr{E} \to Y$ extends
to a meromorphic map $\widetilde{f}: X\to Z$.
\end{result}
This result will play a role in the final stages of proving
Theorem~\ref{t:bord}. To this end, we would also need\,---\,naturally, given the
statement of Theorem~\ref{t:bord}\,---\,conditions under which
a meromorphic map between complex spaces is actually
holomorphic. One situation where this happens is when the complex spaces
are manifolds and the target space is Kobayashi hyperbolic.
\begin{result}[Kodama \cite{kodama79bimero}]\label{r:merhyp}
Let $f : X\to Y$ be a meromorphic map, where $X$ is a complex manifold and
$Y$ is a Kobayashi hyperbolic manifold. Then $f$ is holomorphic.
\end{result}
The following lemma enables us\,---\,as we shall see in Section~\ref{s:bord}\,---\,to
use the preceding results in our specific set-up.
\begin{lemma}\label{l:embed}
Let $Y$, a non-compact Riemann surface, be as in
Theorem~\ref{t:bord} and let $S$ be the compact connected Riemann
surface from which $Y$ is obtained by excising a finite number
of closed disks. Then
${\rm Sym}^n(Y)$ is hyperbolically embedded in ${\rm Sym}^n(S)$.
\end{lemma}
\begin{proof}
Let $\mathcal{W} \subset S$ be another
connected bordered Riemann surface with $\mathcal{C}^2$-smooth boundary
such that
$\overline{Y}\subset \mathcal{W}$. By Theorem~\ref{t:symm_taut}, ${\rm Sym}^n(Y)$
and ${\rm Sym}^n(\mathcal{W})$ are both Kobayashi complete. In particular,
$K_Y$ and $K_{\mathcal{W}}$ are {\bf distances}. It follows from the
discussion at the end of Section~\ref{s:symm_prelim} that
${\rm Sym}^n(Y)$ and ${\rm Sym}^n(\mathcal{W})$ are embedded submanifolds
of ${\rm Sym}^n(S)$. Observe that it suffices to show
that each $p\in \partial{{\rm Sym}^n(Y)}$ is holomorphically embedded in $S$.
Fix a point $p\in \partial{{\rm Sym}^n(Y)}$. Given any $S$-open neighbourhood $U$ of
$p$, we choose a neighbourhood $V$ of $p$ such that $\overline{V}\subset U$ and
$\overline{V}\subset {\rm Sym}^n(\mathcal{W})$. For any $x\in \overline{V}\cap Y$
and $y\in Y\setminus U$, we have
$K_Y(x, y)\,\geq\,K_{\mathcal{W}}(x,y) > 0$.
We know that $\overline{V}\cap \overline{Y}$ and $\overline{Y\setminus U}$ are
compact in $\mathcal{W}$. Thus, the inequality in \eqref{e:hyp_imb} follows from the
last inequality.
\end{proof}
As the proof of the above lemma shows, Theorem~\ref{t:symm_taut} is
an essential ingredient in the proof of Theorem~\ref{t:bord}. In the remainder
of this section, we shall present some prerequisites for proving
Theorem~\ref{t:symm_taut}. We begin with a couple of definitions.
\begin{definition}
Let $Z$ be a complex manifold and $Y\subset Z$ be a connected open subset of $Z$.
Let $p\in \partial{Y}$. We say that {\em $p$ admits a weak peak function for $Y$}
if there exists a continuous function $f_p: \overline{Y}\to \mathbb{C}$ such that
$f|_Y$ is holomorphic,
\[
f_p(p) = 1 \qquad \text{and} \qquad |f_p(y)| < 1 \; \; \forall y\in Y.
\]
We say that {\em $p$ admits a local weak peak function for $Y$} if $p$
admits a weak peak function for $Y\cap U_p$, where $U_p$ is some open neighbourhood
(in $Z$) of $p$.
\end{definition}
In what follows, given a complex manifold $X$, $C_X$ will denote the Carath{\'e}odory
pseudo-distance on $X$. The term {\em Carath{\'e}odory hyperbolic} has a
meaning analogous to that of the term Kobayashi hyperbolic. Furthermore, we say
that $X$ is {\em strongly $C_X$-complete} if $X$ is Carath{\'e}odory hyperbolic
and each closed ball in $X$, with respect to the distance $C_X$, is compact.
We shall also need the following:
\begin{result}\label{R:s-Car_complt}
Let $Z$ be a Stein manifold and $Y\subset Z$ a relatively compact connected open
subset of $Z$. If each point of $\partial{Y}$ admits a weak peak function for $Y$,
then $Y$ is strongly $C_Y$-complete.
\end{result}
The above result has been established with $Z = \mathbb{C}^n$ and $Y$ a bounded domain
in $\mathbb{C}^n$ in \cite[Theorem~4.1.7]{kobayashi98hyp}. Its proof applies
{\em mutatis mutandis} for $Y$ and $Z$ as in Result~\ref{R:s-Car_complt} (that the class of
bounded holomorphic functions on $Y$ separates points is routine to show
with our assumptions on the pair $(Y, Z)$).
\smallskip
\section{The proof of Theorem~\ref{t:symm_taut}}
Before we provide a proof, some remarks on notation are in order. For simplicity of notation,
in this section ({\bf unlike} in subsequent sections), the symbol $D_j$, $j\in \mathbb{N}$, will denote
a closed topological disk. Given non-empty open subsets $A$ and $B$ of the Riemann
surface $S$ (explained below), we shall denote
the relation $\overline{A}\subset B$ (especially when there is a sequence of such relations)
as $A\Subset B$, where the closure is taken in $S$.
\begin{proof}[The proof of Theorem~\ref{t:symm_taut}]
We begin with the following
\smallskip
\noindent{{\bf Claim.} {\em Each $y\in \partial{X}$ admits a weak peak function for $X$.}}
\vspace{0.5mm}
\noindent{Each of the individual ingredients in this construction is classical, so we shall be brief.
Fix $y\in \partial{X}$. Let $S$ be the compact Riemann surface such that
\[
X = S\setminus \big(D_1\sqcup\dots \sqcup D_m\big),
\]
where each $D_j$ is a {\bf closed} topological disk with $\mathcal{C}^2$-smooth boundary. We may
assume without loss of generality that $y\in \partial{D_1}$. Let us write
$X^* = S\setminus (\Delta_1\sqcup\dots \sqcup \Delta_m)$, where each $\Delta_j$,
$j = 1,\dots, m$, is a closed topological disk such that
\[
\Delta_j \varsubsetneq (D_j)^\circ, \; \; j = 2,\dots, m.
\]
Before we describe $\Delta_1$, let us choose a holomorphic chart $(U, \psi)$
centered at $y$ such that:
\begin{itemize}
\item $\psi : (U, y) \longrightarrow (\mathbb{D}, 0)$,
\item $\psi^{-1}((0, 1]\,)\subset S\setminus \overline{X}$, and
\item $U$ is {\bf so small} that
$\psi^{-1}(\mathbb{D}\cap \overline{D(1; 1)}\,)\cap \overline{X} = \{y\}$.
\end{itemize}
The last requirement is possible because $\partial{X}$ is of class $\mathcal{C}^2$.
It is easy to construct a {\em local} peak function $\phi$ at $y$ for $X$ that is, in
fact, holomorphic on $U$ and such that
\begin{align}
|\phi(x)| &< 1 \; \; \forall x\in U\!\setminus\!\psi^{-1}(\mathbb{D}\cap \overline{D(1; 1)}\,),
\label{E:small} \\
|\phi(x)| &> 1 \; \; \forall x\in \psi^{-1}(\mathbb{D}\cap D(1; 1)), \notag \\
|\phi(x)| &= 1 \; \; \forall x\in \psi^{-1}(\mathbb{D}\cap \partial{D(1; 1)}) \ \;\text{with}
\; \ \phi^{-1}\{1\} = \{y\} \notag
\end{align}
(the interested reader is referred to the proof of Proposition~\ref{p:INDEP}
for further details). Let $\Delta_1\varsubsetneq (D_1)^\circ$ and be such that
$\Delta_1\cap U\neq \emptyset$ and
$\partial{\Delta_1}$ intersects $\partial{U}$ at exactly two points. In fact, we can
choose $\Delta_1$ such that, in addition to these properties,
$\partial{\Delta_1}$ also intersects $\psi^{-1}(\{\zeta\in \mathbb{C}: |\zeta| = 1-\varepsilon\})$ in exactly
two points for some positive $\varepsilon\ll 1$.
Pick two $S$-open neighbourhoods, $V_1$ and $V_2$, of $y$ such that
\[
V_1\Subset V_2\Subset U \quad\text{and}
\quad \psi^{-1}((1-\varepsilon)\mathbb{D}\cap \overline{D(1; 1)}\,)\cap X^*\subset V_1.
\]
Let $\chi_1, \chi_2 \longrightarrow [0,1]$ be two functions in $\mathcal{C}^\infty(X^*)$ with
\begin{align*}
\left.\chi_1\right|_{V_1\cap X^*}\equiv 1 \quad\text{and} \quad
\left.\chi_1\right|_{X^*\setminus V_2}\equiv 0, \\
\left.\chi_2\right|_{V_2\cap X^*}\equiv 1 \quad\text{and} \quad
\left.\chi_2\right|_{X^*\setminus U}\equiv 0.
\end{align*}
Finally, consider the function:
\[
G(x):=\begin{cases}
(1-\phi(x))\chi_2(x), &\text{if $x\in (X^*\cap U)$}, \\
0, &\text{if $x\in (X^*\setminus U)$}.
\end{cases}
\]}
In what follows, it will be understood that any expression of the form $\Psi/G$
is {\bf $\boldsymbol{0}$ by definition} outside ${\sf supp}(\Psi)$. Define the $(0,1)$-form
$\omega\in \Gamma\big(T^{\boldsymbol{*}\,(0,1)}S|_{X^*}\big)$ as
\[
\omega = \frac{\overline{\partial}\chi_1}{G}
\]
By construction, $\omega$ is of class $\mathcal{C}^\infty$, vanishes on
$(X^*\cap \overline{V}_1)\cup (X^*\!\setminus V_2)$, and
\begin{equation}\label{E:closed}
x\in X\cap (V_2\setminus \overline{V}_1)\,\Longrightarrow\,\omega(x)
= \left.\overline{\partial}\left(\frac{\chi_1}{1-\phi}\right)\right|_x.
\end{equation}
By the Behnke--Stein theorem \cite{behnkeStein:eaFRF48}, $X^*$ is Stein. Thus,
it admits a solution to the $\overline{\partial}$-problem
\begin{equation}\label{E:d-bar}
\overline{\partial} u\,=\,\omega \; \ \text{on $X^*$.}
\end{equation}
Furthermore, it is a classical fact that there exists a solution, say $u^{\raisebox{-1pt}{$\scriptstyle\infty$}}$, to
\eqref{E:d-bar} of class $\mathcal{C}^\infty(X^*)$.
Write $u_y := \left.u^{\raisebox{-1pt}{$\scriptstyle\infty$}}\right|_{\overline{X}}$.
As $X\Subset X^*$, $u_y$ is bounded. Thus\,---\,by subtracting a large
positive constant if necessary\,---\,we may assume that ${\sf Re}(u_y) < 0$ on
$\overline{X}$. Observe that, by \eqref{E:closed}, \eqref{E:d-bar} and the
construction of $G$,
\[
\big(-\!(\chi_1/G) + u_y\big)^{-1}\in \mathscr{O}(X).
\]
By \eqref{E:small} and by our adjustment of ${\sf Re}(u_y)$, we have
\[
{\sf Re}\big( (-(\chi_1/G) + u_y )^{-1} \big)(x) < 0 \; \; \forall x\in X.
\]
From this, it is easy to check that $f_y(x) := e^{(\,1/(-(\chi_1/G) + u_y))(x)}$,
$x\in \overline{X}$, is a weak peak function at $y$ for $X$. Hence our claim.
\smallskip
In this paragraph, we assume that $n\geq 2$.
Let us pick a point $\langle y_1,\dots, y_n\rangle\in \partial{\rm Sym}^n(X)$.
It is routine to see that $\pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}$ is a proper map. Thus, we may assume
without loss of generality that $y_1\in \partial{X}$. Our Claim above gives
us a weak peak function for $X$ at $y_1$: call it $f$. Set
\[
{\sf h}(z)\,:=\,\frac{1+z}{1-z},
\]
which maps $\mathbb{D}$ biholomorphically to the open right half-plane $\boldsymbol{{\sf H}}_+$ and
maps $(0, 1)\longmapsto (1, +\infty)$. Let $(\boldsymbol{\cdot})^{1/n}$ denote
the holomorphic branch on $\boldsymbol{{\sf H}}_+$ of the $n$-th root such that
\begin{equation}\label{E:branch}
z^{1/n} \in \big\{w\in \mathbb{C} : {\sf Re}(w) > 0, \ |{\sf Im}(w)| < \arctan(\pi/2n){\sf Re}(w)\big\}
\; \; \forall z\in \boldsymbol{{\sf H}}_+\,.
\end{equation}
Furthermore, note that
\begin{itemize}
\item[$(*)$] $(\boldsymbol{\cdot})^{1/n}$ extends to $\partial{\boldsymbol{{\sf H}}_+}$ as a
continuous function
such that $\lim_{\overline{\boldsymbol{{\sf H}}}_+\ni z\,\to\,\infty}\,z^{1/n} = \infty$.
\end{itemize}
If $n = 1$ then set $F := f$. If $n\geq 2$, then define
\[
F(\langle x_1,\dots, x_n\rangle)\,:=\,{\sf h}^{-1}\!\left(
\prod\nolimits_{1\leq j\leq n}\Big(\frac{1 + f(x_j)}{1 - f(x_j)}\Big)^{1/n}\right) \; \;
\forall \langle x_1,\dots, x_n\rangle\in \overline{{\rm Sym}^n(X)}.
\]
By \eqref{E:branch} we see that $F\in \mathcal{C}\big(\overline{{\rm Sym}^n(X)}\big)\cap
\mathscr{O}({\rm Sym}^n(X))$. By $(*)$ and the properties of $f$ it follows that $F$ is
a weak peak function for ${\rm Sym}^n(X)$ at
$\langle y_1,\dots, y_n\rangle\in \partial{\rm Sym}^n(X)$.
We have just shown that, whether $n = 1$ or $n\geq 2$, each point in
$\partial{\rm Sym}^n(X)$ admits a weak peak function for ${\rm Sym}^n(X)$. Recall that
$X^*$ is Stein. It follows from
Result~\ref{R:s-Car_complt}, by taking $Z = {\rm Sym}^n(X^*)$, that
${\rm Sym}^n(X)$ is strongly Carath{\'e}odory complete.
In particular, ${\rm Sym}^n(X)$ is Kobayashi complete.
By a result of Kiernan \cite{kiernan:rbtthm70}, it follows that
${\rm Sym}^n(X)$ is taut.
\end{proof}
\smallskip
\section{Technical propositions}\label{s:technical}
In proving Theorem~\ref{t:bord}, we will need to understand the behaviour of
holomorphic maps $f : Z\to\overline{{\rm Sym}^n(Y)}$, $n\geq 2$\,---\,where $Z$ is
connected and $Y$ is as in Theorem~\ref{t:bord}\,---\,in the event that
${\sf range}(f)\not\subset {\rm Sym}^n(Y)$.
To this end, we shall use the notation introduced in Sections~\ref{s:intro}
and~\ref{s:symm_prelim}.
Thus, given a Riemann surface $Y$ and $(y_1,\dots, y_n)\in Y^n$, $\pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}$
is as introduced in Section~\ref{s:intro}, and
\[
\langle y_1,\dots, y_n\rangle\,:=\,\pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}(y_1,\dots, y_n).
\]
For a point $y\in Y$, a presentation of $y$ having the form of the left-hand side
of the above equation will be called the {\em quotient representation of $y$}.
We require one further observation. For a Riemann surface $Y$, let
$D_1,\dots, D_n$ be non-empty subsets of
$Y$ such that $D_1\times\dots\times D_n$
is not necessarily closed under the $S_n-$action on $Y^n$, $n\geq2$. In
any circumstance, we shall use $\pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}(D_1\times\dots\times D_n)$
to denote the image of the set $D_1\times\dots\times D_n$ under the
map $\pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}} : Y^n\to {\rm Sym}^n(Y)$.
We begin with the following simple lemma:
\begin{lemma}\label{l:hol}
Let $X$ be a Riemann surface, $n\geq 2$, and let $D_1,\dots,D_n \subset X$
be open subsets. Write $\mathcal{D} := \bigcup_{j=1}^n D_j$.
Define $H :=
\pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}(D_1\times\dots\times D_n)$. Suppose $\phi : \mathcal{D}\to \mathbb{C}$ is a
bounded holomorphic map and $\mathscr{S}$ a symmetric polynomial
in $n$ indeterminates. Then, the relation
$\Gamma \subset H\times\mathbb{C}$ defined by
\begin{multline*}
\Gamma\,:=\,\big\{\big(\langle v_1,\dots, v_n\rangle, w\big)\in H\times\mathbb{C} :
w = \mathscr{S}(\phi(x_1),\dots , \phi(x_n)) \; \text{and} \\
(x_1,\dots, x_n)\in (\pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}^{-1}\{\langle v_1,\dots, v_n\rangle\}
\cap D_1\times\dots\times D_n)\big\}.
\end{multline*}
is the graph of a holomorphic function defined on $H$.
\end{lemma}
\begin{proof}
Let $\pi_1$ (resp., $\pi_2$) denote the projection onto the first (resp., second)
factor of $H\times \mathbb{C}$.
Consider any point $\langle v_1,\dots, v_n\rangle\in H$. That
$\pi_1^{-1}\{\langle v_1,\dots, v_n\rangle\}\cap \Gamma$ is a singleton follows
clearly from the fact that $\mathscr{S}$ is a symmetric polynomial. It is thus
the graph of a function $\Phi$. Consider the mapping
$\Phi^\prime: D_1\times\dots\times D_n \to \mathbb{C}$ given by
\[
(x_1,\dots,x_n) \mapsto \mathscr{S}(\phi(x_1),\dots,\phi(x_n)).
\]
Let us write $\Delta := D_1\times\dots\times D_n$. By construction,
$\Phi^\prime = \Phi \circ \big(\!\left.\pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}\right|_{\Delta}\big)$. Since
$\pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}$ admits holomorphic branches of local inverses around any of its
regular values, the above construction shows that
$\Phi$ is holomorphic outside the set of critical values of
$\pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}$ in $H$. Since $\phi$, and thus $\Phi^\prime$, are bounded,
Riemann's removable singularities theorem implies that $\Phi$
is holomorphic on $H$.
\end{proof}
The key result needed is the following, which
generalizes Lemma~5 of \cite{edigarian2005geometry}.
\begin{proposition}\label{p:INDEP}
Let $Y$ be a connected bordered Riemann surface with $\mathcal{C}^2$-smooth
boundary and let $S$ be the compact Riemann
surface from which $Y$ is obtained by excising a finite number of
closed disks. Let $Z$ be a connected complex manifold and
let $f: Z \to {\rm Sym}^n(Y)$, $n\geq 2$, be a holomorphic
map such that $f(Z) \subset \overline{{\rm Sym}^n(Y)}$ (where the closure is in
${\rm Sym}^n(S)$). Suppose there exists a $z_0\in Z$ such that
$f(z_0)$ is of the form $\langle y_1, \boldsymbol{\ast} \rangle$, $y_1\in \partial Y$.
Then
\[
f(z)\;\;\text{is of the form $\langle y_1, \boldsymbol{\ast} \rangle$ for
all}\;\;z \in Z.
\]
Moreover, if $y_1$ appears
$k$ times, $1\leq k\leq n$, in the quotient representation of $f(z_0)$ then the same is
true for $f(z)$ for all $z\in Z$.
\end{proposition}
\begin{proof}
Let $y_2,\dots, y_l$ be the other \emph{distinct} points that
appear in the quotient representation of $f(z_0)$. Let $D_1,\dots,D_l$
be small coordinate disks in $S$ centered at
$y_1,\dots,y_l$, respectively, whose closures are pairwise disjoint.
Let $(D_j, \psi_j)$, $j = 1,\dots, l$ , denote the coordinate charts. By ``coordinate
disks centered at $y_j$'', we mean that $\psi_j(D_j) = \mathbb{D}$ and
$\psi_j(y_j) = 0$, $j = 1,\dots, l$. Furthermore,
as $Y$ has $\mathcal{C}^2$-smooth boundary, we can (by shrinking $D_1$ and scaling
$\psi_1$ if necessary) ensure that
\begin{itemize}
\item $\psi_1(\partial Y\cap D_1)\cap \{\zeta\in \mathbb{C} : |{\sf Re}(\zeta) - 1|^2
+ |{\sf Im}(\zeta)|^2 = 1\}\,=\,\{\psi_1(y_1)\}\,=\,\{0\}$; and
\item $\psi_1(Y\cap D_1)\subset \{\zeta\in \mathbb{D} : |{\sf Re}(\zeta) - 1|^2
+ |{\sf Im}(\zeta)|^2 > 1\}$.
\end{itemize}
Let us define $\phi\in \mathscr{O}(D_1)$ by
\[
\phi(y)\,:=\,\exp\left\{\frac{\psi_1(y)}{2 - \psi_1(y)}\right\} \; \; \; \forall y\in D_1.
\]
Using the fact that the M{\"o}bius transformation $\zeta\mapsto \zeta/(2 - \zeta)$ maps
the circle $\{\zeta\in \mathbb{C} : |{\sf Re}(\zeta) - 1|^2 + |{\sf Im}(\zeta)|^2 = 1\}$ onto $\{\zeta\in \mathbb{C} : {\sf Re}(\zeta) = 0\}$,
it is routine to verify that
\begin{equation}\label{E:peak}
\phi(y_1) =1 \quad\text{and} \quad |\phi(y)| < 1 \; \; \;
\forall y \in D_1 \cap (\overline{Y}\!\setminus\!\{y_1\}),
\end{equation}
and that $\phi$ is a bounded function.
Write $\mathcal{D} := \sqcup_{j=1}^l D_j$ and define a function $\widetilde{\phi}\in \mathscr{O}(\mathcal{D})$
as follows
\[
\widetilde{\phi}(y)\,:=\,\begin{cases}
\phi(y), & \text{if $y\in D_1$}, \\
0, & \text{otherwise}.
\end{cases}
\]
Let $H := \pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}(D_1^k \times D_2^{k_2} \times \dots \times D_l^{k_l})$,
where $k_j$ is the number of times $y_j$ appears in the quotient
representation of $f(z_0)$, $j = 2,\dots, l$. Now consider the following relation
$\Gamma \subset H\times\mathbb{C}$ defined by
\begin{multline}
\Gamma\,:=\,\big\{\big(\langle v_1,\dots, v_n\rangle, w\big)\in H\times\mathbb{C} :
w = \widetilde{\phi}(x_1) + \dots + \widetilde{\phi}(x_n) \\
\text{and} \; (x_1,\dots, x_n)\in (\pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}^{-1}\{\langle v_1,\dots, v_n\rangle\}
\cap D_1^k \times D_2^{k_2} \times \dots \times D_l^{k_l})\big\}.
\end{multline}
It follows from Lemma~\ref{l:hol} that $\Gamma$ is the graph
of a function, say $\Phi$, that is holomorphic on $H$.
Now as $f(z_0) \in H$ and $H$ is an open neighborhood of $f(z_0)$,
we can find a small connected open set $U\varsubsetneq Z$ around $z_0$
such that $f(U) \subset H$. Consider the holomorphic map
$\Phi\circ \big(\left.f\right|_U\big)$. As $f(U)\subset \overline{{\rm Sym}^n(Y)}$,
we have, by construction:
\[
\Phi\circ \big(\left.f\right|_U\big)(z_0)\,=\,k\,=\,\sup\nolimits_{z\in U}
\big|\Phi\circ \big(\left.f\right|_U\big)(z)\big|.
\]
By the maximum modulus theorem, $\Phi\circ \big(\left.f\right|_U\big)\equiv k$.
By the definition of the function $\Phi$, we deduce that the conclusion of our
proposition holds true on the open set $U$.
Let $E$ be the set of points of $Z$ for which the conclusion of the proposition
holds true. By hypothesis, $E$ is non-empty. The above argument shows
that $E$ is an open set. Let $z \in Z\setminus E$. If $y_1$ does not
appear in the quotient representation $f(z)$ at all, then, by continuity, there
exists a neighbourhood $U_z$ of $z$
such that the same is true for every point in $U_z$. On the other hand, if
$y_1$ does appear in the quotient representation of $f(z)$ but not
$k$ times, then the argument given prior to this paragraph
shows that we can find a neighbourhood $U_z$ of $z$ such that the same is true
for every point in $U_z$. In either case, therefore, $U_z \subset (Z \setminus E)$.
This shows that $E$ is closed. Therefore $E = Z$.
\end{proof}
\smallskip
\section{The proof of Theorem~\ref{t:bord}}\label{s:bord}
In proving Theorem~\ref{t:bord} we will find it convenient to use a certain expression,
which we now define.
\begin{definition}\label{d:deps_only}
Let $M_1,\dots, M_n$ and $N$ be complex manifolds, and $\mathfrak{V}$ a proper (possibly empty)
analytic subvariety of $M_1\times\dots\times M_n$. Let
$f : (M_1\times\dots\times M_n)\setminus \mathfrak{V}\to N$ be a holomorphic map. We say
that $f$ {\em depends only on the $j$-th coordinate on
$(M_1\times\dots\times M_n)\setminus \mathfrak{V}$}, $1\leq j\leq n$,
if for each $x\in M_j$ lying outside
some proper (possibly empty) analytic subvariety of $M_j$,
\begin{multline}
f(x_1,\dots,x_{j-1}, x, x_{j},\dots, x_{n-1}) = f(y_1,\dots,y_{j-1}, x, y_{j},\dots, y_{n-1}) \\
\text{for all $(x_1,\dots,x_{n-1})\neq (y_1,\dots,y_{n-1})\in \prod_{i\neq j}M_i$}
\end{multline}
such that $(x_1,\dots,x_{j-1}, x, x_{j},\dots, x_{n-1}),
(y_1,\dots,y_{j-1}, x, y_{j},\dots, y_{n-1})\notin \mathfrak{V}$.
\end{definition}
Before we give the proof of Theorem~\ref{t:bord}, we ought to point out to the reader a convention
that will be used below. Given a product space, $\pi_j$ will denote the projection onto the $j$-th coordinate.
If several product spaces occur in a discussion, we {\bf shall not add} additional labels
to $\pi_j$ to indicate the domain of this projection unless there is scope for ambiguity.
\begin{proof}[The proof of Theorem~\ref{t:bord}]
Let $S$ be a compact Riemann surface such that $Y$ is obtained from
$S$ by excising a finite number of closed disks such that $\partial Y$ is
$\mathcal{C}^2$-smooth. Theorem~\ref{t:bord} is a tautology when $n = 1$,
so it will be understood here that $n\geq 2$.
Let $R_j$, $j = 1,\dots,n$, be as in the
statement of Theorem~\ref{t:bord}.
Let $p=(p_1,\dots,p_n)$ be a point in $R_1 \times
\cdots \times R_n$ such that, for each $1 \leq j \leq n$, $p_j$ is a limit point of $R_j\setminus X_j$.
Also by hypothesis, we can choose $p_j$ to belong to $\partial{X}_j$.
Let $(U_j,\psi_j)$ be holomorphic coordinate charts of $R_j$ chosen in such a way that:
\begin{itemize}
\item $p_j \in U_j$; and
\item Each $U_j$ is biholomorphic to a disk.
\end{itemize}
Let $W_j := U_j \cap X_j$ and $V_j := \psi_j(W_j)$.
For $(z_1,\dots,z_n) \in V_1 \times \dots \times V_n$, let
\begin{equation}\label{e:g}
g(z_1,\dots,z_n) := F(\psi_1^{-1}(z_1),\dots,\psi_n^{-1}(z_n)).
\end{equation}
Fix a point $q\in {\partial X_n}\cap U_n$. Consider a sequence $\{w_\nu\} \subset V_n$ such that
$w_\nu \to \psi_n(q)$.
Let
\[
\phi_\nu :V_1 \times \dots \times V_{n-1} \to {\rm Sym}^n(Y) :=
g(z_1,\dots,z_{n-1},w_\nu).
\]
We claim that we can extract a subsequence $\{w_{\nu_m}\}\subset \{w_\nu\}$ such that
$\{\phi_{\nu_m}\}$ converges
uniformly on compacts to a holomorphic mapping $h :V_1 \times \dots \times
V_{n-1} \to {\rm Sym}^n(S)$. To this end, fix another
connected bordered Riemann surface, $Y^*\subset S$, with $\mathcal{C}^2$-smooth boundary
such that $Y\Subset Y^*$.
By Theorem~\ref{t:symm_taut}, both ${\rm Sym}^n(Y)$ and
${\rm Sym}^n(Y^*)$ are taut. Owing to the tautness of ${\rm Sym}^n(Y)$, and
as $F$ is proper, we can extract a subsequence $\{w_{\nu_m}\}\subset \{w_\nu\}$ such that
$\phi_{\nu_m}$ is compactly divergent\,---\,i.e., given compacts
$K_1\subset V_1 \times \dots \times V_{n-1}$ and $K_2\subset {\rm Sym}^n(Y)$, there
exists an integer $M(K_1, K_2)$ such that
\begin{equation}\label{e:comp_div}
\phi_{\nu_m}(K_1)\cap K_2\,=\,\emptyset \; \; \; \forall m\geq M(K_1, K_2).
\end{equation}
We now view each $\phi_{\nu_m}$ as a map into ${\rm Sym}^n(Y^*)$. This time,
owing the tautness of ${\rm Sym}^n(Y^*)$, there exists a holomorphic map
$h :V_1 \times \dots \times V_{n-1} \to {\rm Sym}^n(S)$ such that\,---passing to
a subsequence of $\{\phi_{\nu_m}\}$ and {\bf relabelling} if
necessary\,---\,$\{\phi_{\nu_m}\}$ converges uniformly on compacts to $h$.
This establishes our claim. From this and \eqref{e:comp_div} it follows
that $h(V_1 \times \dots \times V_{n-1}) \subset \partial{\rm Sym}^n(Y)$. It follows from
Proposition~\ref{p:INDEP} that there exists a point $\xi\in \partial Y$
such that
\begin{equation}\label{e:const}
h(z)\;\;\text{is of the form $\langle \xi, \boldsymbol{\ast} \rangle$ for
all}\;\;z \in V_1 \times \dots \times V_{n-1}.
\end{equation}
It is a classical fact\,---\,see \cite[Chapter~5]{jost2005riemann}, for instance\,---\,that
there exists a bounded, non-constant function $\chi$ that is holomorphic on some open
connected set $\mathcal{W}$ that contains $\overline{Y}$. Let $\Psi : {\rm Sym}^n(\mathcal{W}) \to \mathbb{C}^n$ be
defined by
\[
\langle z_1,\dots,z_n \rangle \longmapsto \big(\elsym{1}(\chi(z_1),\dots,\chi(z_n)),
\elsym{2}(\chi(z_1),\dots,\chi(z_n)),\dots, \elsym{n}(\chi(z_1),\dots,\chi(z_n))\big).
\]
A remark on the purpose of the map $\Psi$ is in order. If we could, by shrinking each
$U_j$ if necessary, find a single model coordinate chart $(\Omega, \varPsi)$ on ${\rm Sym}^n(Y)$
(refer to Section~\ref{s:symm_prelim} for some remarks on the term ``model coordinate chart'')
so that
\begin{itemize}
\item[$i)$] $F(W_1\times\dots\times W_n)\subset \Omega$, and
\item[$ii)$] $\overline{W}_j\cup \partial X_j$ is indiscrete for each $j = 1,\dots, n$,
\end{itemize}
then the principal part of our proof would reduce to an application of
\cite[Theorem~1.2]{debraj2015function} by Chakrabarti--Gorai. However, it is far
from clear that one can find coordinate charts that satisfy {\bf both} $(i)$ and
$(ii)$. The role of the map $\Psi$ is to compensate for this difficulty.
\smallskip
\noindent{{\bf Step 1.} {\em Finding local candidates for $F_1,\dots,F_n$}}
\vspace{0.5mm}
\noindent{The argument at this stage of our proof closely follows that of
Edigarian--Zwonek \cite{edigarian2005geometry} and Chakrabarti--Gorai
\cite{debraj2015function}. But since we must modify the map $g$ (see
\eqref{e:g} above) in order to use the latter
argument\,---\,which has consequences on what follows\,---\,we shall present
parts of this argument in some detail. We begin by defining $G := \Psi\circ g$
(the need for this map is hinted at by our preceding remarks).
By the definition of the map $\Psi$, its holomorphic derivative
is non-singular on an open dense subset of
${\rm Sym}^n(\mathcal{W})$. Therefore\,---\,since $F$ is a proper holomorphic map\,---\,the complex
Jacobian of $G$ does not vanish identically on
$V_1 \times \dots \times V_n$. We expand this latter
determinant along the last column to conclude that there exists $\mu \in
\{1,\dots,n\}$ such that
\begin{equation}\label{e:mu}
\det \left[ \partl{G_i}{z_j}\right]_{i=1,\dots,n,\,i \neq \mu,\,j = 1,\dots,n-1}
\not \equiv 0 \;\; \text{on $V_1 \times \dots \times V_n$}.
\end{equation}}
Let us write $\theta := \Psi \circ h$, $\theta^{(m)} := \Psi \circ \phi_{\nu_m}$, $m = 1, 2, 3,\dots$,
and $\mathcal{V} := V_1 \times \dots \times V_{n-1}$.
Owing to \eqref{e:const}, there exists a $C \in \mathbb{C}$ such that
\begin{equation}\label{e:polybdy}
C^n - C^{n-1}\theta_1+ \dots + (-1)^{n-1}C\theta_{n-1} + (-1)^n \theta_n\,\equiv\,0 \; \;\text{on
$\mathcal{V}$},
\end{equation}
where $\theta = (\theta_1,\dots,\theta_n)$. Differentiating with respect to $z_j$,
$j = 1,\dots,n-1$, we get
\begin{equation}\label{e:sys1}
-C^{n-1} \partl{\theta_1}{z_j} + \dots + (-1)^{n-1} C \partl{\theta_{n-1}}{z_j}
+ (-1)^n \partl{\theta_n}{z_j}\,\equiv\,0 \;\; \text{on $\mathcal{V}$}.
\end{equation}
Rearranging \eqref{e:sys1}, we get the following system of $(n-1)$ equations:
\begin{equation}\label{e:sys2}
\sum_{k =1,\dots,n,\,k \neq \mu} (-1)^kC^{n-k}\partl{\theta_k}{z_j}\,=\,(-1)^{\mu+1}C^{n-\mu}
\partl{\theta_\mu}{z_j} \;\; \text{on $\mathcal{V}$}, \;\; j = 1,\dots,n-1.
\end{equation}
Given an $(n-1)\times n$ matrix $B$ and
$l \in\{1,\dots,n\}\setminus\{\mu\}$,
denote by $\Delta_l(B)$ the determinant of the
$(n-1) \times (n-1)$ matrix obtained by:
\begin{itemize}
\item deleting the $\mu$-th column of $B$; and
\item replacing the $l$-th column by the $\mu$-th column of $B$.
\end{itemize}
Denote by $\Delta_\mu(B)$ the
determinant of the $(n-1)\times (n-1)$ matrix obtained by deleting the
$\mu$-th column of $B$. Note that each of the functions $\Delta_j$ is a
polynomial in the entries of the matrix $B$.
We now introduce the $(n-1)\times n$ matrices
\[
D_{n-1}\theta(z^\prime)\,:=\,\left[
\partl{\theta_k}{z_j}(z^\prime)\right]_{1\leq j\leq n-1,\,1\leq k\leq n} \; \; \text{and}
\; \;
{}^m\!D_{n-1}\theta(z^\prime)\,:=\,\left[
\partl{\theta^{(m)}_k}{z_j}(z^\prime)\right]_{1\leq j\leq n-1,\,1\leq k\leq n},
\]
where $z^\prime := (z_1,\dots,z_{n-1})$. We also set:
\[
\mathfrak{A}\,:=\,\big\{z^\prime\in \mathcal{V} : \Delta_\mu\big(D_{n-1}\theta(z^\prime)\big) = 0\big\}.
\]
Depending on $\mathfrak{A}$, we need to consider two cases.
\noindent{{\bf Case 1.} $\mathfrak{A}\varsubsetneq \mathcal{V}$.}
\vspace{0.5mm}
\noindent{By applying Cramer's rule to the system described by \eqref{e:sys2},
we get:
\[
(-1)^l C^{n-l}\,=\,(-1)^\mu C^{n-\mu}\frac{\Delta_l(D_{n-1}\theta)}{\Delta_\mu
(D_{n-1}\theta)} \;\; \text{on $(\mathcal{V}\!\setminus\!\mathfrak{A})$ and} \;\; l\in
\{1,\dots,n\}\!\setminus\!\{\mu\}.
\]
If $\mu \neq 1$, we shall argue by taking $l = \mu-1$ in the above. If $\mu = 1$,
we shall take $l = 2$. We
shall first consider the case $\mu\neq 1$. In this case, the above equation gives
\begin{equation}\label{e:relation}
-C \Delta_\mu(D_{n-1}\theta) = \Delta_{\mu-1}(D_{n-1}\theta) \;\; \text{on $\mathcal{V}$}.
\end{equation}}
\noindent{{\bf Case 2.} $\mathfrak{A} = \mathcal{V}$.}
\vspace{0.5mm}
\noindent{As in Case~1, we assume $\mu\neq 1$. Since the system \eqref{e:sys2}\,---\,treating $C$
as the indeterminate\,---\,admits a solution, $\Delta_{\mu}(D_{n-1})\equiv 0$ forces on us the
conclusion \eqref{e:relation} for trivial reasons.}
So, in each of the above cases, we get the identity \eqref{e:relation}.
Differentiating this identity with respect to $z_j$ and eliminating $C$, we get
the relations
\[
\Delta_\mu(D_{n-1}\theta) \partl{\Delta_{\mu-1}(D_{n-1}\theta)}{z_j} - \Delta_{\mu-1}(D_{n-1}\theta)
\partl{\Delta_\mu(D_{n-1}\theta)}{z_j}\,\equiv\,0 \;\; \text{on $\mathcal{V}$}, \;\; j = 1,\dots,n-1.
\]
The left-hand sides of the above relations are constituted of polynomial expressions involving
\[
\lim_{m\to \infty}g_s\left(z_1,\dots,z_{n-1}, w_{\nu_m}\right), \; \; s = 1,\dots,n,
\]
their compositions with the function $\chi$,
and their partial derivatives (with respect to $z_1,\dots,z_{n-1}$) up to order two. Hence, by
Weierstrass's theorem on the derivatives of holomorphic functions, we have
\begin{align}
\lim_{m\to \infty}&\Delta_\mu({}^m\!D_{n-1}\theta)(z^\prime)
\partl{\Delta_{\mu-1}({}^m\!D_{n-1}\theta)}{z_j}(z^\prime)
- \Delta_{\mu-1}({}^m\!D_{n-1}\theta)(z^\prime)
\partl{\Delta_\mu({}^m\!D_{n-1}\theta)}{z_j}(z^\prime) \notag \\
=&\,\Delta_\mu(D_{n-1}\theta)(z^\prime) \partl{\Delta_{\mu-1}(D_{n-1}\theta)}{z_j}(z^\prime)
- \Delta_{\mu-1}(D_{n-1}\theta)(z^\prime)
\partl{\Delta_\mu(D_{n-1}\theta)}{z_j}(z^\prime) \notag \\
=&\,0 \; \; \; \forall z^\prime\in \mathcal{V}, \; \; j = 1,\dots,n-1. \label{e:lim_0}
\end{align}
Consider the functions $\tau_j : V_1 \times \dots \times V_{n} \to \mathbb{C}$ defined as follows:
\begin{align*}
\tau_j(z^\prime, z_n)\,:=\,\Delta_\mu & (D_{n-1}G)(z^\prime, z_n)
\partl{\Delta_{\mu-1}(D_{n-1} G)}{z_j}(z^\prime, z_n)\\
& - \Delta_{\mu-1}(D_{n-1} G)(z^\prime, z_n)
\partl{\Delta_\mu(D_{n-1}G)}{z_j}(z^\prime, z_n),
\end{align*}
for each $j = 1,\dots,n-1$. Here, $D_{n-1}G(\boldsymbol{\cdot}, z_n)$ is an $(n-1)\times n$ matrix
that is defined in the same way as $D_{n-1}\theta$.
Observe that \eqref{e:lim_0} holds for any subsequence
$\{w_{\nu_m}\}\subset \{w_\nu\}$ with the properties discussed right after \eqref{e:g},
where $V_n\ni w_\nu\to q$. Finally, as $q\in \partial X\cap U_n$ was picked arbitrarily,
\eqref{e:lim_0} implies that
\[
\tau_j(z^\prime, \zeta)\longrightarrow 0 \; \; \text{as $\zeta\to \psi(U_n)\cap \partial V_n$
for each $z^\prime\in \mathcal{V}$},
\]
and for each $j = 1,\dots,n-1$.
Thus we can extend each $\tau_j$ to a continuous function $\widetilde{\tau}_j$
defined on $V_1 \times \dots \times V_ {n-1} \times \psi(U_n)$ by setting
$\widetilde{\tau}_j(z^\prime, z_n) = 0$ whenever $z_n \in \psi(U_n) \setminus V_n$. By Rado's
theorem\,---\,see \cite[Chapter~4]{narasimhan1971SCV}\,---\,$\widetilde{\tau}_j$ is holomorphic
on $V_1 \times \dots \times V_{n-1} \times \psi(U_n)$. Let us now fix
$z^\prime\in \mathcal{V}$ and $j\,:\,1\leq j\leq n-1$. By construction,
$\psi(U_n) \setminus V_n$ has at least one limit point in $\psi(U_n)$. Thus, by the identity
theorem, $\widetilde{\tau}_j(z^\prime, \boldsymbol{\cdot})$ is identically $0$. As this holds true
for every $z^\prime$ and $j$, it follows that each $\tau_j$ is identically $0$.
Set
\[
\gamma_n := -\frac{\Delta_{\mu-1}(D_{n-1} G)}{\Delta_\mu(D_{n-1}
G)}.
\]
The function $\gamma_n$ is well-defined on the set
$(V_1 \times \dots \times V_n)\setminus \mathcal{A}$, where
\[
\mathcal{A}\,:=\,\left\{z \in V_1 \times \dots \times V_n: \Delta_\mu\big(D_
{n-1}G(z)\big) = 0\right\}.
\]
By \eqref{e:mu}, $\mathcal{A}$ is a proper analytic subvariety of
$V_1 \times \dots \times V_n$. Observe that
$\left.\tau_j\right|_{(V_1 \times \dots \times V_n)\setminus \mathcal{A}}$ is
the numerator of $\partl{\gamma_n}{z_j}$, whence
\[
\partl{\gamma_n}{z_j}\,\equiv\,0 \;\; \text{on $(V_1 \times \dots \times V_n)\setminus
\mathcal{A}$},
\]
for $j=1, \dots, n-1$. Since $\mathcal{A}$ is a proper analytic subvariety, this implies
that $\gamma_n$ depends only on
$z_n$\,---\,in the sense of Definition~\ref{d:deps_only}\,---\,on each
set of the form $\mathscr{M}\setminus \mathcal{A}$, where
$\mathscr{M}$ is a connected component of $V_1 \times \dots \times V_n$.
Appealing to \eqref{e:relation}, and arguing in the same manner as above, we get
\[
\gamma_n(z^\prime, \zeta)\longrightarrow C \; \; \text{as $\zeta\to \psi(U_n)\cap \partial V_n$
for each $z^\prime\in \mathcal{V}$},
\]
where, we now recall, $C$ satisfies the equation \eqref{e:polybdy}.
Again, by an argument involving Rado's
theorem\,---\,see \cite{edigarian2005geometry} or \cite{debraj2015function}\,---\,that
is analogous to the one above, it follows that
\begin{multline} \label{eq:gamma}
\gamma_n^n(z) - \gamma_n^{n-1}(z)G_1(z) + \dots +
(-1)^{n-1}\gamma_n(z)G_{n-1}(z) \\
+ (-1)^n G_n(z)\,\equiv\,0 \; \; \forall z\in
(V_1 \times \dots \times V_n)\setminus \mathcal{A},
\end{multline}
where we write $G = (G_1,\dots, G_n)$. This
shows that $\gamma_n((V_1 \times \dots \times V_n) \setminus \mathcal{A})
\subset \chi(Y)$ which, by the choice of $\chi$, is bounded.
By Riemann's removable singularities theorem,
$\gamma_n$ extends to be holomorphic on $V_1\times\dots \times V_n$.
A completely analogous argument can be given\,---\,which results in a slightly
different expression for $\gamma_n$\,---\,when $\mu = 1$ (in which case, we take
$l = 2$, $l$ as introduced at the beginning of Step~1).
Repeating this argument with some $i$ replacing $n$ above
yields us maps $\gamma_i: V_1\times\dots \times V_n\to \mathbb{C}$ that satisfy equations
analogous to \eqref{eq:gamma}.
What we have at this stage is summarized by
the following commutative diagram:
\begin{figure}[ht]
\begin{tikzcd}
X_1\times\dots\times X_n \arrow[r, "F"]
& {\rm Sym}^n(Y) \arrow[rr, "\Psi"]
&
& \mathbb{C}^n \\
& Y^n \arrow[rr, "{(\chi\circ \pi_1,\dots,\,\chi\circ\pi_n)}"]
\arrow[u, "\pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}"]
&
& \mathbb{C}^n \arrow[u, "{\pi^{(n)}}"] \\
& & \\
V_1\times\dots\times V_n \arrow[uuu, "{(\psi^{-1}_1\circ\pi_1,\dots,\,\psi^{-1}_n\circ\pi_n)}"]
\arrow[uurrr, "{(\gamma_1,\dots,\gamma_n)}"]
\end{tikzcd}
\end{figure}
\noindent{where we use $\pi_j$, $j = 1,\dots,n$, to denote the projection onto the $j$-th coordinate
(where the product domain in question is understood from the context). Let us write:
\begin{align*}
\mathscr{C}\,&:=\,\text{the set of critical points of $\pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}} : Y^n \to {\rm Sym}^n(Y)$}, \\
\mathscr{C^*}\,&:=\,\text{the set of critical points of $\pi^{(n)} : \mathbb{C}^n \to \mathbb{C}^n$}.
\end{align*}
Here $\pi^{(n)}$ is as introduced in Section~\ref{s:intro}.
We now find connected open sets
$W^*_j\subset W_j$, $j = 1,\dots, n$, that are so small that:
\begin{itemize}
\item[$a)$] $F(W^*_1\times\dots\times W^*_n)\cap \pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}(\mathscr{C})\cap
\Psi^{-1}\big(\pi^{(n)}(\mathscr{C}^*)\big) = \emptyset$;
\vspace{0.5mm}
\item[$b)$] $\pi^{(n)}$ is invertible on $\Psi\big(F(W^*_1\times\dots\times W^*_n)\big)$; and
\vspace{0.5mm}
\item[$c)$] The map $(\chi\circ\pi_1,\dots, \chi\circ\pi_n)$ is invertible on each image
of $\Psi\big(F(W^*_1\times\dots\times W^*_n)\big)$ under a branch of a local inverse of
$\pi^{(n)}$ that intersects the image of $(\chi\circ\pi_1,\dots, \chi\circ\pi_n)$.
\end{itemize}
Let $\big(\pi^{(n)}\big)^{-1}_s$, $s = 1,\dots, n!$, denote the branches introduced in $(c)$.
The definition of the map $\Psi$ ensures that, in fact, the images of
$\Psi\big(F(W^*_1\times\dots\times W^*_n)\big)$ under each $\big(\pi^{(n)}\big)^{-1}_s$
are contained in $(\chi\circ\pi_1,\dots, \chi\circ\pi_n)(Y)$. From this
and a routine diagram-chase\,---\,since, by construction, the arrow representing
$\pi^{(n)}$ can be reversed on $\Psi\big(F(W^*_1\times\dots\times W^*_n)\big)$\,---\,we
see that there exists a number $s^0$, $1\leq s^0\leq n!$ such that
\[
(\gamma_1,\dots, \gamma_n)\circ \big(\psi_1(W^*_1)
\times\dots\times \psi_n(W^*_n)\big)\,=\,\big(\pi^{(n)}\big)^{-1}_{s^0}\big(
\Psi(F(W^*_1\times\dots\times W^*_n))\big).
\]
Thus, by $(c)$, there is a local holomorphic inverse\,---\,call it $\mathscr{I}
\equiv (\mathscr{I}_1,\dots, \mathscr{I}_n)$\,---\,of
$(\chi\circ\pi_1,\dots, \chi\circ\pi_n)$ such that the maps
\[
{\sf f}_j\,:=\,\mathscr{I}_j\circ (\gamma_1,\dots, \gamma_n)\circ (\psi_1\circ\pi_1,
\dots, \psi_n\circ\pi_n)
\]
are well-defined on $W^*_1\times\dots\times W^*_n$ and holomorphic, $j = 1,\dots,n$.
From the above commutative diagram, we see that
\[
\left.F\right|_{W^*_1\times\dots\times W^*_n} = \pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}\circ({\sf f}_1,\dots, {\sf f}_n).
\]}
Since, by construction, $\psi_1(W^*_1)\times\dots\times \psi_n(W^*_n)$ lies in a connected
component of $V_1\times\dots\times V_n$, $\gamma_j$ depends only on $z_j$
on $\psi_1(W^*_1)\times\dots\times \psi_n(W^*_n)$ for each
$j = 1,\dots, n$. Then, owing to the structure of the map $(\chi\circ\pi_1,\dots, \chi\circ\pi_n)$,
of which $\mathscr{I}$ is a local inverse, it follows that
\begin{multline}\label{e:indep}
\text{for each $j$, $j = 1,\dots,n$, the map ${\sf f}_j : W^*_1\times\dots\times W^*_n
\to Y_j$ depends} \\
\text{only on the $j$-th coordinate on $W^*_1\times\dots\times W^*_n$.}
\end{multline}
\noindent{{\bf Step 2.} {\em Establishing the (global) existence of $F_1,\dots,F_n$}}
\vspace{0.5mm}
\noindent{We abbreviate $X_1\times\dots\times X_n$ to $X$. Let
$\mathscr{E} := F^{-1}\big(\pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}(\mathscr{C})\big)$ ($\mathscr{C}$ is as introduced
above), which
is a proper analytic subset of $X$. If $x \in X\setminus \mathscr{E}$, then
we can find a connected product neighbourhood $\Omega_x$ of $x$ such that the map
$F|_{\Omega_x}$ lifts to $Y^n$ (i.e., it admits a holomorphic map
$f : \Omega_x\to Y^n$ such that $\pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}\circ f = F|_{\Omega_x}$).}
Fix an $x_0 \in X\setminus \mathscr{E}$. Consider any path
$\Gamma: [0,1] \to X\setminus \mathscr{E}$ such that $\Gamma(0)$ is in $W^*_1 \times
\dots \times W^*_n$ and $\Gamma(1) = x_0$. Here,
$W^*_j\subset X_j$, $j = 1,\dots, n$, are the domains introduced towards the end of the
argument in Step~1.
We can cover $\Gamma([0,1])$ by finitely many product neighbourhoods\,---\,call
them $\Omega^0, \Omega^1, \dots \Omega^s$\,---\,on which the map $F$ lifts to $Y^n$.
Moreover, it is easy to see that we can find
$\Omega^0, \Omega^1, \dots \Omega^s$ and lifts $(f_1^i,\dots, f_n^i) : \Omega^i \to Y^n$
of $F|_{\Omega^i}$ to $Y^n$ for each $i$ such that:
\begin{itemize}
\item $\Omega^0 = W^*_1 \times\dots \times W^*_n$;
\vspace{0.5mm}
\item $(f_1^0,\dots, f_n^0) : \Omega^0 \to Y^n$ is the map $({\sf f}_1,\dots, {\sf f}_n)$
provided by Step~1;
\vspace{0.5mm}
\item $\Omega^i\cap \Omega^{i-1} \neq \emptyset$ for $i = 1,\dots,s$;
\vspace{0.5mm}
\item For each $i = 1,\dots,s$,
$f^{i}_j\big|_{K^i} \equiv f^{i-1}_j\big|_{K^i}$ for each $j = 1,\dots,n$,
where $K^i$ is some connected component of $\Omega^{i}\cap \Omega^{i-1}$.
\end{itemize}
Then, owing to \eqref{e:indep}, it follows from the identity theorem and
induction that each $f^s_j$ depends only on the $j$-th coordinate on $\Omega^s$.
In short, given any $x_0 \in X \setminus\mathscr{E}$, we can find a
product neighborhood $N = N_1 \times \dots \times N_n\ni x_0$ and maps
$f_j : N \to Y$ that depend only on the $j$-th coordinate on $N$
such that $F|_N = \pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}\circ (f_1,\dots,f_n)$.
\vspace{1mm}
\noindent{{\bf Claim.} {\em This $(f_1,\dots, f_n)$ does not depend on the choice of path
$\Gamma$ joining $x_0$ to $W^*_1 \times\dots \times W^*_n$ or the choice of $\Omega^i$,
$i = 1,\dots,s$, covering $\Gamma([0,1])$.}}
\vspace{0.5mm}
\noindent{To see this, suppose $(\varphi_1,\dots, \varphi_n)$ is a lift of $F$ to $Y^n$ on a
neighbourhood of $x_0$ obtained by carrying out the above procedure along some different path
or via a different cover of $\Gamma([0,1])$. Then, there exists a permutation
$\rho$ of $\{1,\dots,n\}$ such that
\[
(\varphi_1,\dots, \varphi_n)\,\equiv\,(f_{\rho(1)},\dots, f_{\rho(n)}) \; \; \text{on a
neighbourhood of $x_0$}.
\]
Now, $\varphi_j$ depends {\bf only} on the $j$-coordinate. The above equation implies
that $\varphi_j$ depends only on the $\rho(j)$-th coordinate, $j = 1,\dots,n$. This
is impossible unless $\rho$ is the identity permutation. Hence the claim.}
\vspace{1mm}
Since the $x_0\in X \setminus\mathscr{E}$ mentioned above is completely arbitrary, it follows
from the above Claim that we have holomorphic maps $\widetilde{F}_j :
X\setminus \mathscr{E} \to Y$, $j = 1,\dots n$, such that
$\widetilde{F}_j$ depends only on the $j$-th coordinate on $X \setminus\mathscr{E}$ (in
the sense of Definition~\ref{d:deps_only})
and such that
\begin{equation}\label{e:prelim_rel}
F|_{X\setminus \mathscr{E}} = \pi_{\!{\raisebox{-1.5pt}{$\scriptstyle{Sym}$}}}\circ(\widetilde{F}_1,\dots, \widetilde{F}_n).
\end{equation}
By Lemma~\ref{l:embed}, $Y$ is hyperbolically embedded in $S$ ($S$ is as
introduced at the beginning of this proof). Then, by
Results~\ref{r:kwack}~and~\ref{r:merhyp}, each $\widetilde{F}_j$ extends
to a holomorphic map on $X_j$, $j = 1,\dots, n$. By continuity, we can now
view these extended maps as holomorphic maps $F_j : X_j\to Y$. In view of
\eqref{e:prelim_rel}, we have our result. The properness of each of the maps
$F_j$ is straightforward.
\end{proof}
\section*{Acknowledgements}
A part of the research in this paper was conducted during a visit by Gautam Bharali
and Indranil Biswas to the International Centre for Theoretical Sciences (ICTS) to
participate in the meeting {\em Complex Geometry} in March 2017
(programme code: {\bf ICTS/Prog-compgeo/2017/03}). They thank the ICTS for
its support. Gautam Bharali is supported by a Swarnajayanti Fellowship (grant
no.~DST/SJF/MSA-02/2013-14). Indranil Biswas is supported by a J.C.~Bose Fellowship.
Divakaran Divakaran is supported by a project grant from the Indian Institute of
Science Education and Research Bhopal (grant no.~IISERB/INS/MATH/2016091).
Jaikrishnan Janardhanan is supported by a DST-INSPIRE fellowship from the Department
of Science and Technology, India.
\bibliographystyle{amsalpha}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,232
|
# Table of Contents
1. Title Page
2. Table of Contents
3. Copyright
4. Dedication
5. Acknowledgments
6. Epigraph
7. Working for Human Happiness
8. WANT
9. The Good Father
10. Flood of Humanity
11. DOING
12. City Missionary
13. Draining the City, Saving the Children
14. Journey to Dowagiac
15. A Voice Among the Newsboys
16. Happy Circle
17. Photos
18. Almost a Miracle
19. REDOING
20. Invisible Children
21. Neglect of the Poor
22. The Trials of Charley Miller
23. The Death and Life of Charles Loring Brace
24. Legacy
25. Notes
26. Bibliography
27. Index
28. About the Author
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 3,202
|
The Newark Branch is a branch of the Erie Railroad in New Jersey, United States, running between Jersey City and Paterson with stops in the Broadway Section in North Newark. Inaugurated in the 1870s, the line was last used for passenger service on September 30, 1966 and was later used for freight service.
History
The Paterson and Newark Railroad, a subsidiary of the Erie Railroad, was founded in 1864 and by 1869 had developed a right-of way (ROW) along the western banks of the Passaic between the two cities for which it was named. The line was conceived as a connection between Newark and Paterson, where a transfer was possible to Erie's Main Line southbound service to the Hudson Waterfront and ferries across the Hudson River to New York or northbound to New York State and the Midwest. Service began by 1870 but was hindered by unresolved issues with landowners opposed to the seizure of their riverfront property.
Originally a crossing of the Lower Passaic River was planned so trains from Newark could travel east using the New Jersey Railroad bridge, ROW, and terminal at Exchange Place in Jersey City. In 1871, construction began on a new alignment from Newark to Jersey City. The company was re-organized in 1872 and renamed the Paterson, Newark, and New York Railroad when a crossing was developed at the site of NX Bridge. Eventually trackage from the river crossing converged with the New York and Greenwoood Lake Railway, which crossed the Passaic to the north over the WR Draw. From that junction in the Kearny Meadows, the two lines continued east over the Hackensack River on the DB Draw to the Long Dock Tunnel through Bergen Hill, terminating at Erie's Pavonia Terminal.
Service
Passenger service on the line became known as the Newark Branch. From Pavonia Terminal, and later Hoboken Terminal, service ran west to Harrison and Kearny. After crossing the Passaic into Newark, it ran west of and parallel to the river to Belleville, Nutley, Clifton and Paterson with some continuing service to Glen Rock, Ridgewood, Ho-Ho-Kus, and Waldwick Like the Bergen County Line, the Newark Branch was a branch of the Main Line, both with service extending north to Waldwick laying over at nearby Waldwick Yard. Commuter operations on the Newark Branch were discontinued in October 1966.
Status and re-use study
By the 1960s, only one of the two tracks was in a suitable condition, The line became part of Conrail and later Norfolk Southern Railway. (NS). In 1977 the line was severed when the NX Bridge over the Passaic River was taken out of service and left in the open position. By 2002 the line east of the bridge was out of service. That was due to the loss of the last shipper on that portion of the branch, SparTech Poly-Com.
A portion of the line along the west bank of the Passaic River, known as the Newark Industrial Track, is still used to serve one customer in Clifton, Van Ness Plastics.
In 2020, the North Jersey Transportation Planning Authority, in conjunction with the Passaic County Planning Board, produced the Paterson–Newark Transit Market Study report, which examined the potential of restoring passenger service on the line.
See also
Timeline of Jersey City area railroads
List of bridges, tunnels, and cuts in Hudson County, New Jersey
Notes
Defunct New Jersey railroads
Erie Railroad
Railway companies established in 1864
Transportation in Hudson County, New Jersey
Transportation in Essex County, New Jersey
Transportation in Passaic County, New Jersey
Transportation in Newark, New Jersey
Erie Railroad lines
External link
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 135
|
Published 04/19/2019 10:25:48 pm at 04/19/2019 10:25:48 pm in White Linen Blackout Curtains.
white linen blackout curtains white linen curtains 108 white linen curtains linen drapes white linen curtains white linen blackout curtains .
amazoncom best dreamcity room darkening thermal insulated solid hversailtex classical grommet top thermal insulated heavy weight textured tiny plaid linen look innovated, cotton blackout curtains linen blackout curtain fabric cotton cotton blackout curtains cotton blackout curtains linen cotton curtain stone white west elm throughout white linen cotton blackout curtains white , white linen blackout curtains navy and bedroom living room gray white linen blackout curtains drapes curtain modern grommet chair rectangular white linen blackout curtains , linen curtains large size of blackout drapes natural linen linen curtains full size of white curtains dazzling white curtains white linen blackout curtains , white linen blackout curtains gallery incredible along with white linen blackout curtains simple and modern brown linen blackout energy saving curtains throughout sweet white white linen blackout curtains , smart sheer curtains smart linen curtain new sheer white curtains smart ,belgian flax linen curtain white west elm belgian flax linen curtain white,, riviera stripe drape charcoal pottery barn , white linen curtains blackout curtains inch long curtain white white linen curtains white linen curtains white linen curtains flax linen d pottery barn white white linen curtains , white linen curtains linen cotton grommet curtain white white linen white linen curtains linen curtains pure white linen curtains inches wide white linen curtains blackout .
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,331
|
A motorcycle and automobile accident resulting in the temporary closure of Westbound Bee Ridge between School Avenue and U.S. 41 is being investigated by the The Sarasota Police Department.
north of the Laurel Road interchange on Interstate 75 a traffic accident has backed up traffic in the northbound lanes for about three miles. Emergency personnel were on the scene at 7:30 p.m. , but the traffic jam remained. Details of the injuries and damages resulting from the accident are not yet known.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,161
|
{"url":"https:\/\/www.albert.io\/learn\/ap-calculus-ab-bc\/question\/mean-value-theorem-to-find-the-maximum-dollarfbdollar-given-dollarfadollar","text":"Limited access\n\nYou are given a function $f(x)$ that is continuous on the closed interval $[2,10]$ and differentiable on the open interval $(2,10)$.\n\nIf $f(2)=5$ and $1\\le f'(x)\\le 3$, what is the largest possible value for $f(10)$?\n\nA\n\n$13$\n\nB\n\n$19$\n\nC\n\n$24$\n\nD\n\n$29$\n\nSelect an assignment template","date":"2017-04-23 23:29:44","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\": 0.3724522888660431, \"perplexity\": 61.72784899190272}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-17\/segments\/1492917118851.8\/warc\/CC-MAIN-20170423031158-00156-ip-10-145-167-34.ec2.internal.warc.gz\"}"}
| null | null |
Teacher Charged in Hit-and-Run That Severely Injured Bicyclist in Silver Lake
Posted 10:20 AM, December 17, 2019, by Tracy Bloom, Updated at 01:27PM, December 17, 2019
A 52-year-old woman pleaded not guilty on Tuesday to charges tied to a hit-and-run crash that left a male bicyclist severely injured in Silver Lake , the Los Angeles County District Attorney's Office said.
Molly Jane Hoene has been charged with two felony counts of hit-and-run in connection with the incident, prosecutors said in a news release.
Hoene entered the not guilty plea during her arraignment Tuesday, according to a public information officer for L.A. Superior Court.
She is due back in court Jan. 23.
Hoene, a teacher with the Los Angeles Unified School District, is accused of being behind the wheel of a vehicle that slammed head on into a bicyclist in the area of Berkeley Avenue and Berkeley Circle on Oct. 25, investigators said.
A man was struck and seriously injured by a Mini Cooper in a hit-and-run in Silver Lake on Oct. 25, 2019. (Credit: Los Angeles Police Department)
She fled the scene without rendering aid to the seriously injured 52-year-old man, according to the DA's office. Police called his injuries severe, and it's not clear if he has recovered.
The incident was caught on a surveillance camera, and a tip led detectives to a blue Mini Cooper believed to be involved in the crash.
The car was found at an auto body shop in Glendale five days after the collision. It was impounded and searched for forensic evidence, Los Angeles Police Department officials said.
Hoene was apprehended on Nov. 26 at a gated community in Palm Desert where she had been staying with relatives, according to LAPD.
The defendant has since been released on $50,000 bond, according to L.A. County inmate records. A booking photo has not been released.
Hoene face a maximum prison sentence of four years if convicted.
KTLA's Dianne Sanchez contributed to this story.
Teacher Arrested in Silver Lake Hit-and-Run Crash That Severely Injured Bicyclist
Video: Driver Sought After Mini Cooper Hits Bicyclist Head-On in Silver Lake
Topics: Los Angeles County District Attorney's Office, Los Angeles Police Department, Molly Hoene, silver lake
Hit-and-Run Driver Arrested in Lake View Terrace Crash That Killed Horses and Injured Riders: LAPD
Driver Sentenced for 2016 Hit-and-Run Death of Motorcyclist in Pasadena
Ex-East L.A. High School Teacher Sentenced to 2 Years for Sexual Assault of Teenage Student
Felon Who Shot and Dumped Dog in La Mirada Charged With 6 Counts, Including Animal Cruelty: DA
Man Accused of Intentionally Running Over Woman in Long Beach Is Charged With Murder
2 Horses Killed, 2 Riders Seriously Injured in Lake View Terrace Hit-and-Run
Man Serving Time for Murder Is Convicted of Raping, Killing Silver Lake Woman in 1980
Man Indicted on Murder Charge in Killing of Malibu Creek State Park Camper, Other Shootings
Former Fugitive Convicted of Murder in 2008 Shooting of Handyman in Silver Lake
Man Who Murdered Handyman in 2008 Sentenced to 50 Years to Life
Woman Who Fatally Hit Pedestrian While Texting, Driving in Westlake Pleads No Contest
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 1,741
|
A hook and painters' can holder will free up your hands.
Fits both the right and left-hand side of the ladder; suitable therefore for right-handed and left-handed people.
Supplied as a set. Can holder for 1 litre cans and hook for cans with handles.
Additional information: Intended for: 1) Extra wide one-piece ladder, 8000 ladder, Extra wide double and triple combination ladder, Extra wide double and triple extension ladder, Modular ladders.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,701
|
{"url":"https:\/\/www.kunxi.org\/2007\/04\/meet-mr-dtrace-part-4\/","text":"# Meet Mr. DTrace - Part 4\n\ndtrace\n\nEventually, we got the building environment working. Thanks to Ben and help from\n\n# gnusol, the missing piece is sunwbtool\n\n## Using OpenSolaris binutils\n\nMake sure you are using the OpenSolaris toolchain instead of GNU\u2019s, otherwise the ld may crash with objects compiled with cc. Be cautious to you environment variable $PATH, and make sure \/usr\/ccs\/bin and $SUNWspro\/bin precede the generic \/usr\/bin.\n\nAnd copy this configuration to your $HOME\/.mozconfig, copied from Alex .$topsrcdir\/browser\/config\/mozconfig\n\nac_add_options --enable-svg\n\n## Building, building, building\u2026\n\nRun the following commands:\n\n.\/configure\ngmake\n\nwait it to fail\n\ncd config; gmake\ncd ..\/js; gmake\n\nIf it works, we can apply the patch to add static probes:\n\ncd js\/src; patch -p2\n\nIf you are in luck, you could get libmozjs.so. Now we need to preload it using run-firefox.sh:\n\nlibmozjs=\/export\/home\/bookstack\/work\/firefox-1.5.0.7\/trunk\/js\/src\/libmozjs.so\n\/usr\/lib\/firefox\/firefox-bin\nUnfortunately, there is a core dump when loading the libmozjs.so whenever the dtrace support is added. I doubt that the Firefox in NexentaOS is built by GCC, and it is not binary compatible with the SunStudio, I might have to build the whole Firefox. Ooooh, that is painful, I even does not do that in my Gentoo Linux\u2026","date":"2021-04-13 01:25:36","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\": 0.5885982513427734, \"perplexity\": 9215.65213608806}, \"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-2021-17\/segments\/1618038071212.27\/warc\/CC-MAIN-20210413000853-20210413030853-00535.warc.gz\"}"}
| null | null |
/*jshint strict:false */
/*
Native TreeWalker is buggy in IE and Opera:
* IE9/10 sometimes throw errors when calling TreeWalker#nextNode or
TreeWalker#previousNode. No way to feature detect this.
* Some versions of Opera have a bug in TreeWalker#previousNode which makes
it skip to the wrong node.
Rather than risk further bugs, it's easiest just to implement our own
(subset) of the spec in all browsers.
*/
var typeToBitArray = {
// ELEMENT_NODE
1: 1,
// ATTRIBUTE_NODE
2: 2,
// TEXT_NODE
3: 4,
// COMMENT_NODE
8: 128,
// DOCUMENT_NODE
9: 256,
// DOCUMENT_FRAGMENT_NODE
11: 1024
};
function TreeWalker ( root, nodeType, filter ) {
this.root = this.currentNode = root;
this.nodeType = nodeType;
this.filter = filter;
}
// There is a javascript TreeWalker already that I don't want to write over
window.STreeWalker = TreeWalker
TreeWalker.prototype.nextNode = function () {
var current = this.currentNode,
root = this.root,
nodeType = this.nodeType,
filter = this.filter,
node;
while ( true ) {
node = current.firstChild;
while ( !node && current ) {
if ( current === root ) {
break;
}
node = current.nextSibling;
if ( !node ) { current = current.parentNode; }
}
if ( !node ) {
return null;
}
if ( ( typeToBitArray[ node.nodeType ] & nodeType ) &&
filter( node ) ) {
this.currentNode = node;
return node;
}
current = node;
}
};
// Nate: The best way to think of this traversal is that starting from the current node it goes up one level
// and then down one level to the right if possible, then from there down and to the left as far as possible.
// r
// a b
// d e f g
//
// If the currentNode is d, nextNONode will return e. Calling it again returns 'a' since there are no more branches
// of a. Calling again gives f, then g, b, and finally r. I'm not certain if this is post order so I refrained from
// using a completely analogous name to previousPONode.
// NATE: We now assume the breakoutFunction can take root as second argument
TreeWalker.prototype.nextNONode = function (breakoutFunction) {
var current = this.currentNode,
root = this.root,
nodeType = this.nodeType,
filter = this.filter,
node;
while ( true ) {
if ( current === root ) {
return null;
}
node = current.nextSibling;
//modified to let us break on an element satisfying the breakoutFunction
if ( node ) {
if(breakoutFunction && breakoutFunction(node, root)){
this.currentNode = node;
return node;
}
else{
while ( current = node.firstChild ) {
node = current;
}
}
} else {
node = current.parentNode;
}
if ( !node ) {
return null;
}
if ( ( typeToBitArray[ node.nodeType ] & nodeType ) &&
filter( node ) ) {
this.currentNode = node;
return node;
}
current = node;
}
};
// NOTE: haven't included a fliter/ breakut function yet
TreeWalker.prototype.nextSNode = function (){
var current = this.currentNode,
root = this.root,
nodeType = this.nodeType,
filter = this.filter,
node, sib, parent;
while ( true ) {
if(sib = current.nextSibling){
this.currentNode = sib
return sib
} else {
parent = current.parentNode
sib = parent.nextSibling
if (!sib){
this.currentNode = null
return null
} else {
this.currentNode = sib
return sib.firstChild ? sib.firstChild : sib
}
}
}
}
// NATE: the breakoutFunction takes (node, root)
TreeWalker.prototype.previousNode = function (breakoutFunction) {
var current = this.currentNode,
root = this.root,
nodeType = this.nodeType,
filter = this.filter,
node;
while ( true ) {
if ( current === root ) {
return null;
}
node = current.previousSibling;
//modified to let us break on an element satisfying the breakoutFunction
if ( node ) {
if(breakoutFunction && breakoutFunction(node, root)){
this.currentNode = node;
return node;
}
else{
while ( current = node.lastChild ) {
node = current;
}
}
} else {
node = current.parentNode;
}
if ( !node ) {
return null;
}
if ( ( typeToBitArray[ node.nodeType ] & nodeType ) &&
filter( node ) ) {
this.currentNode = node;
return node;
}
current = node;
}
};
// Previous node in post-order.
// Nate: Analogous to nextNONode, this function goes up one level, then down one level to the left,
// and then down and to the right as far as possible.
TreeWalker.prototype.previousPONode = function () {
var current = this.currentNode,
root = this.root,
nodeType = this.nodeType,
filter = this.filter,
node;
while ( true ) {
node = current.lastChild;
while ( !node && current ) {
if ( current === root ) {
break;
}
node = current.previousSibling;
if ( !node ) { current = current.parentNode; }
}
if ( !node ) {
return null;
}
if ( ( typeToBitArray[ node.nodeType ] & nodeType ) &&
filter( node ) ) {
this.currentNode = node;
return node;
}
current = node;
}
};
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,645
|
true: true.c
gcc -Wall --static -o true true.c
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,168
|
When thinking about ways to improve company culture, many employers mistakenly focus their efforts on creating a "fun" atmosphere. However, company culture isn't made up of superficial offerings – it's made up of people. Oftentimes, the key to building a great company culture starts by hiring for POTENTIAL.
During the hiring process, keeping an open mind will help you find the new hire you didn't know you needed. To do this, consider candidates with applicable transferable skills.
While a candidate may lack a skill on your "must-have" list, a similar experience or a proven ability to learn quickly can be a valid substitute. Additionally, looking outside of your normal candidate pool can bring an entirely new set of skills and ideas to your team.
When deciding whether a candidate has the potential to grow into a great employee, be sure to ask situational questions during the interview.
Understanding how an employee thinks in the face of a challenge is an excellent way to find out how they will handle themselves under pressure.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,472
|
{"url":"https:\/\/www.projecteuclid.org\/euclid.kjm\/1250271316","text":"## Journal of Mathematics of Kyoto University\n\n### On parabolic geometry of type PGL(d,C)\/P\n\nIndranil Biswas\n\n#### Abstract\n\nLet $P$ be the maximal parabolic subgroup of $\\text{PGL}(d, {\\mathbb C})$ defined by invertible matrices $(a_{ij})_{i,j=1}^d$ with $a_{dj}\\,=\\, 0$ for all $j\\, \\in\\, [1\\, ,d-1]$. Take a holomorphic parabolic geometry $(M\\, ,E_P\\, ,\\omega)$ of type $\\text{PGL}(d,{\\mathbb C})\/P$. Assume that $M$ is a complex projective manifold. We prove the following: If there is a nonconstant holomorphic map $f\\, :\\, {\\mathbb C} {\\mathbb P}^1\\,\\longrightarrow \\, M$, then $M$ is biholomorphic to the projective space ${\\mathbb C}{\\mathbb P}^{d-1}$.\n\n#### Article information\n\nSource\nJ. Math. Kyoto Univ., Volume 48, Number 4 (2008), 747-755.\n\nDates\nFirst available in Project Euclid: 14 August 2009\n\nPermanent link to this document\nhttps:\/\/projecteuclid.org\/euclid.kjm\/1250271316\n\nDigital Object Identifier\ndoi:10.1215\/kjm\/1250271316\n\nMathematical Reviews number (MathSciNet)\nMR2513584\n\nZentralblatt MATH identifier\n1175.53039\n\n#### Citation\n\nBiswas, Indranil. On parabolic geometry of type PGL(d,C)\/P. J. Math. Kyoto Univ. 48 (2008), no. 4, 747--755. doi:10.1215\/kjm\/1250271316. https:\/\/projecteuclid.org\/euclid.kjm\/1250271316","date":"2019-10-23 17:26:29","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\": 0.8929795026779175, \"perplexity\": 759.9791431853236}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570987834649.58\/warc\/CC-MAIN-20191023150047-20191023173547-00192.warc.gz\"}"}
| null | null |
{"url":"https:\/\/python-bloggers.com\/2019\/05\/network-theory-and-game-of-thrones-a-perfect-combination\/","text":"[This article was first published on Python Programming - Data Science Blog | AI, ML, big data analytics , and kindly contributed to python-bloggers]. (You can report issue about the content on this page here)\n\nGame of Thrones is arguably one of the biggest pop culture phenomena to hit the public consciousness in the last decade. Since the hype for the final season\u2019s arrival has gone down a bit, especially mine after episode three , I thought I could use this time to finally explore a side of Data Science that has always intrigued me \u2013 Network Theory, and combine it with a topic I am very invested in \u2013 Game of Thrones. Just to be clear I won\u2019t be making any claims or predictions about the plot of the show \u2013 No Spoilers. I just want to use Game of Thrones as a hopefully relatable context for discussing the analysis techniques.\n\nAt a high level, Network Theory is the study of relationships between objects, more specifically it is a subfield of Graph Theory with extra attributes attached to the nodes and edges. If you\u2019re confused by these terms, don\u2019t worry I\u2019ll explain everything in a bit. For the rest, you might be familiar with graph theory and have not-so-fond memories associated with it, but bear with me for a while. I first learned basic graph theory in my university\u2019s algorithms course and, I\u2019ll be honest, I found absolutely nothing of interest in the entire topic. Sure, I could find the shortest path between two cities or find the best way to lay down routers in a computer network, but these topics never seemed fun to me. That is until I started exploring data science and learned about network analysis. That really opened my eyes to what the graph theory concepts were capable of. I encourage you to check out this video about exploring opposing factions and their effects on each other using graphs.\n\nI will go about my exploration using two separate but related lines. The first will be a theoretical overview of each topic, and the second will be some implementation to see the concepts in action.\n\n## What is Network Theory?\n\nLet\u2019s get some basic definitions out of the way so everyone is on the same page and we can get to the fun bits.\n\nNetwork Theory is the study of objects and their relationships with each other. This setup is often represented as a Graph which is just a collection of nodes and edges. Nodes are the individual objects in a network and Edges are the links representing the relationships between nodes.\n\nLet\u2019s see an example of this:\n\n## Undirected, Directed, and Weighted Graphs\n\nI want to clarify something right in the beginning, a graph is one model of a given situation or system, that does not mean that the graph is the only model. What a node or edge represents is your choice (or informed by the data). In this case I can determine that each node is a person and an edge between them means that they are friends with each other. This is called an Undirected Graph.\n\nNow consider the same situation but I tell you that it is possible for a person to be friends with someone who\u2019s not friends with them, a one-sided friendship. This changes the situation and we need a different (more informative) model to represent it. Enter Directed Graphs:\n\nThis graph tells a completely different story from the first one. Now I give you even more information, I tell you how much a person values their friendship with the other person. This means there is a notion of edges being heavier or more important than other edges. This is represented using a Weighted graph which can be either directed or undirected.\n\n## Getting our feet wet\n\nI will be using Python to implement the algorithms since I\u2019m more comfortable with it and while learning a new topic I don\u2019t want to be bogged down by language specifics.\n\nStarting with the data, it is available on Andrew Beveridge\u2019s GitHub. He (in his own words) does maths for fun, which I feel is a very appropriate mindset for me too. The data contains records of character interactions for all seasons (upto 7). I am only using the latest season\u2019s data but I encourage you to explore the temporal information in the dataset. You could even create animations to show how the data has changed over time.\n\nLet\u2019s discuss the packages I will be using briefly. Numpy, pandas, matplotlib (and seaborn) are the usual companions in any Data Science project. I also use the defaultdict data structure here.\n\nThe interesting stuff is in the last three lines. I will be using networkx for the general purpose graph handling, nxviz for to do the heavy lifting visualization tasks and the community package is used for one particular algorithm.\n\nimport numpy as np\nimport pandas as pd\nimport matplotlib.pyplot as plt\n%matplotlib inline\nimport seaborn as sns\n\nfrom collections import defaultdict\n\nimport networkx as nx\nimport nxviz as nxv\nimport community\n\n\nA brief note about loading a graph dataset before we continue our exploration further. You might be familiar with common data formats like CSV or JSON which can be loaded as a dataframe object using the pandas library. Graphs are a bit different in this regard since they\u2019re usually saved as 2 separate files (one each for nodes and edges) which need to be loaded and combined into a graph object manually. The object is provided by the networkx library but the exact mechanics of constructing the graph will be different for each dataset. I\u2019ve added my code here for you to see.\n\ndef make_graph(nodes_df, edges_df):\ng = nx.Graph()\n\nfor i,row in nodes.iterrows():\nkeys = row.index.tolist()\nvalues = row.values\n# The dict contains all attributes\n\nfor i,row in edges.iterrows():\nkeys = row.index.tolist()\nvalues = row.values\n**dict(zip(keys,values)))\n\nreturn g\n\ng = make_graph(nodes, edges)\n\n\nNow let\u2019s get to know the dataset. The network we have is for interactions between the characters on the show. These interactions are set up using the fan-made script of the show, and details can be seen in the original source. I\u2019ve modified the data a bit by adding categorical variables to the nodes (gender, allegiance and culture). The nodes represent characters and have some attributes like their gender and house. The edges just have a \u2018weight\u2019 which is the number of interactions between the characters. You should always look at the original source of the data to understand how it was collected (remember this when you see interactions between characters who\u2019ve never met in the show ).\n\nWe previously saw a very simple graph visualization (the one with the emoji faces). It\u2019s called a node-link diagram. It\u2019s very easy to read and get an overview of the data, but it gets very unwieldy for slightly bigger networks. Let\u2019s see how our data turns out:\n\nnx.draw(g, with_labels=True)\n\n\nAs expected, the nodes are shown as a jumbled mess with too much overlap and absolutely no thought given to the arrangement. The locations of the individual nodes can be tweaked using the pos argument of the nx.draw function, but it is too much work to manually figure out the best arrangement.\n\nCan we do better? Yes we can. Let\u2019s make a Circos Plot using a very simple library called nxviz. If you\u2019ve ever used seaborn for making plots, this is very similar to work with.\n\nc = nxv.CircosPlot(g, node_color='Gender', node_grouping='Gender',\nedge_width=(edges['Weight'] \/ edges['Weight'].quantile(0.97)).tolist(),\nnode_labels=True, node_label_layout='rotation',\ngroup_label_position=\"middle\",\ngroup_label_offset=12,\nfigsize=(8,8))\nc.draw()\n\n\nThis plot has slightly less overall information (it shows fewer edges), but it is much easier to read. We can see patterns in the connectivity of different houses and the relative sizes of different houses.\n\n## Which gender has more interactions?\n\nLet\u2019s try to explore this data further. I want to see how males and females interact with each other. For this we can create a transformed version of our original graph. Which kind of graph would be most suitable to represent this information? We know we have two genders, and we want to model relationships between them. This means we have two nodes. We have four possible kinds of interactions:\n\n\u2022 female -> female\n\u2022 male -> male\n\u2022 female -> male\n\u2022 male -> female\n\nwhich means we will have 4 edges, but note that these are directed edges and not undirected ones like in our original graph. So, I have created a weighted, directed graph with two nodes, one for each gender in the data.\n\nThe weights will be the the sum of all interactions in the data for that pair. This means the weight of the female -> female edge will be the sum of weights for all edges in the original graph where the people on both ends were female.\n\nI\u2019ll introduce another useful graph concept here, the adjacency list. For a graph with $n$ nodes this is an $n \\times n$ matrix where the entry at index $(u,v)$ contains the weight for the edge between nodes $u$ and $v$. For an unweighted graph this is a binary matrix, just containing 1 and 0 and is symmetric for undirected graphs (think about why this is).\n\nI\u2019ll plot this matrix as a heatmap and normalize the values because there are overall more males in the show\u2019s cast.\n\ncounter = defaultdict(int)\nfor frm, to in g.edges:\n\nsg = nx.DiGraph()\nfor (frm,to),w in counter.items():\nsg.edges[('male', 'female')]\n\nm = nx.to_numpy_matrix(sg, nodelist=sg.nodes)\nsns.heatmap(m\/m.sum(), annot=True,\nxticklabels=list(sg.nodes), yticklabels=list(sg.nodes));\nplt.gca().set_aspect('equal')\n\n\nWe can see that 45% of the interactions in the show happen between males and 33% happen between males and females. All in all males are part of about 80% of all interactions in the show.\n\nThis might be a good time to remind you to keep in mind how an \u2018interaction\u2019 has been defined for this dataset.\n\nNevertheless, this result can be explained by the fact that the show is based in a Medieval setting of Kings, Lords and Knights \u2013 all of whom used to me males. So, the results we have obtained make sense in the context of the show.\n\n## Which character is most important?\n\nNow I want to find out who the most important characters are in the show. The condition for there being an edge between two nodes is that the two characters must have interacted, and the weight of the edge represents the number of interactions. It is reasonable to say that more important characters will have more interactions with various characters overall.\n\nIntuitively, the more the weights of edges connected to a given node, the higher the node\u2019s importance. So, to represent importance I can sum the weights of all edges a node has. This is called the Degree Centrality of the node, formally defined as the number of edges which are incident upon a node (for an undirected graph this is same as the outgoing nodes). There are other measures of Centrality defined in graph theory, each is useful to identify different characteristics of a network.\n\nLet\u2019s look at the trend of degree centrality measure in our data.\n\ndeg_cen = nx.degree_centrality(g)\nnodes['deg_cen'] = nodes['Id'].apply(lambda x: deg_cen[x])\ng = make_graph(nodes, edges)\n\nfig, ax = plt.subplots(figsize=(20,8))\nsns.barplot(data=nodes.nlargest(50, 'deg_cen'),\nx='Id', y='deg_cen', hue='Gender',\nax=ax);\nax.set_xticklabels(ax.get_xticklabels(), rotation=90);\n\n\nFans of the show should be able to immediately identify that the main characters are indeed in the lead here \u2013 Jon, Dany, Tyrion, Cersei and Sansa make up for most of the screen time of the show.\n\nThis is not the only way to visualize degree centrality. There is another type of plot, called Arc Plot which could be more informative. Let\u2019s see:\n\na = nxv.ArcPlot(g, node_color='Gender', node_grouping='Gender',\nnode_size=[10*g.nodes[n]['deg_cen'] for n in g.nodes],\nedge_width=(edges['Weight'] \/ edges['Weight'].quantile(.95)).tolist(),\nfigsize=(10,10))\na.draw()\n\n\nHere the size of the node is based on the degree centrality and the edge thickness is based on its weight. The nodes are also colored based on gender.\n\n## Which characters are in the same faction?\n\nIf you\u2019ve seen or even heard about the show, you know that it features some serious political rivalries and factions. Is there a way to visualize these alliances? Can we draw some lines on a graph to indicate these alliances? Drawing these lines is known as Graph Partitioning.\n\nFinding Graph Partitions is a very common task in graph theory. One very simple algorithm to achieve this is the min-cut method. A cut is a partition of the nodes of a graph into 2 disjoint sets. The weight of this cut (for weighted graphs only) is the sum of weights of all edges which cross the cut. This means that there will always be some cut of a graph which has the minimum weight.\n\nSo, the problem of finding the two factions in Game of Thrones now becomes a problem of finding the min-cut of the graph. We can extend this into a k-cut problem to find an arbitrary number of partitions.\n\nWhile I will not be using the min-cut method to find partitions, I find the graph cut definition to be the most intuitive explanation for the process. The reason I use a different method is that the min-cut method does not generate good partitions for the Game of Thrones dataset.\n\nThe implementation I use comes from the community package in Python.\n\ncmt = community.best_partition(g, weight='Weight')\nnodes['cmt'] = [v for c,v in cmt.items()]\ng = make_graph(nodes, edges)\n\nc = nxv.CircosPlot(g, node_color='cmt', node_grouping='cmt',\nnode_labels=True, node_label_layout='rotation',\nedge_width=(edges['Weight'] \/ edges['Weight'].quantile(0.98)).tolist(),\nfigsize=(8,8))\nc.draw()\n\n\nWe have identified 4 communities. There are some interesting things we have extracted. All the Martells\/Sands are in the same community (so is the Mountain ). The Starks are in the same community, except for Jon who is with Dany and Tyrion\u2019s group. These groupings make sense if we think about the plot of the show. Jon has been with Dany\u2019s group more than with the Starks.\n\nSince we have split the graph in 4 partitions, maybe looking at the node-link diagrams for each partition separately isn\u2019t such a bad idea anymore.\n\npos = nx.spring_layout(g) # compute graph layout\nfor cmt_num in np.unique([v for c,v in cmt.items()]):\npartition = community.best_partition(g)\n\nnodelist = []\n# one community against the others\nfor node, c in partition.items():\nif c == cmt_num:\nnodelist.append(node)\nelse:\npartition[node] = -1 # put all the other communities in one communitiy\n\nsg = g.subgraph(nodelist)\nedge_widths = [sg.edges[e]['Weight'] for e in sg.edges]\nedge_widths = [w\/np.quantile(edge_widths, 0.9) for w in edge_widths]\n\nfig, ax = plt.subplots(figsize=(15,10))\nnx.draw(sg, pos, with_labels=True, width=edge_widths)\nax.set_title(f'Community {cmt_num}');\n\n\nFor these smaller sub-graphs these diagrams are much easier to read. The edges are shaded based on their weight and for the first 3 communities we can see a sort of central cluster connected with thick edges \u2013 basically supporting the rest of the nodes in the partition. The last community is less well-knit in this regard.\n\nLike I said in the beginning, I just wanted to learn more about network analysis and share my exploration with you, and I think I succeeded in doing that. Do not consider the analysis or insights from this blog to be any kind of predictions. If you have some knowledge about the show, then you might have noticed that there were a few results that don\u2019t exactly match with the show\u2019s storyline. A more thorough analysis is required to make any form of claims based on the data.\n\nOne of my inspirations to learn network analysis was a SciPy 2018 talk by Eric Ma and Mridul Seth, titles Network Analysis Made Simple: Network Fundamentals. It goes over basics of the data structures and libraries along with a few slightly advanced topics in the end which I did not cover here.\n\nI hope this blog has helped you learn some fundamental concepts of Network Theory. I am hoping you would use network theory in a project soon! .","date":"2020-09-23 13:09:29","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\": 0.4257960915565491, \"perplexity\": 697.3135761003671}, \"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-2020-40\/segments\/1600400210996.32\/warc\/CC-MAIN-20200923113029-20200923143029-00397.warc.gz\"}"}
| null | null |
Q: REST не возвращает страницу Метод REST.
@RequestMapping(value = "/create_HTML_Page_with_new_coordinates/{x}/{y}", method = RequestMethod.GET,
produces = {
MediaType.APPLICATION_XML_VALUE,
MediaType.APPLICATION_JSON_VALUE})
public @ResponseBody
ResponseEntity<String> getLinktoHTMLPageFromJBossTempDirectory(@PathVariable("x") int x, @PathVariable("y") int y) {
String link;
ResponsePageHtml responsePageHtml = new ResponsePageHtml(x, y);
responsePageHtml.writeHTMLtoTempJBOSS();
link = responsePageHtml.getFullFilePATH();
return new ResponseEntity<String>(link, HttpStatus.OK);
}
тут суть такая. у меня есть класс ResponsePageHtml который создает страницу html которая содержит небольшой яндекс апи по отображению карты по переданным координатам.
я создаю страницу, она записывается в директорию bin Jboss и потом я возвращаю ссылку на нее.
Но после выполнения метода вместо того чтобы отобразилась страница отображается такая надпись.
This page contains the following errors:
error on line 1 at column 1: Document is empty
Below is a rendering of the page up to the first error.
страница html создается и при открытии нормально отображает карту.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,320
|
This website is designed for general educational and information purposes only and does not render medical advice, unless otherwise indicated. The information provided through this website should not be used for diagnosing or treating a health problem or illness. It is not a substitute for professional health care, and is designed to support - not replace - the relationship you have with your health care provider. If you have or suspect you may have a health problem, you should consult your health care provider. If you think you may have a medical emergency, call your doctor or emergency medical services immediately. If you are seeking substance abuse rehabilitation, including substance detoxification, please contact the Passages Ventura Intake Counselors at (888) 926-4859 .
Passages Ventura disclaims any liability whatsoever for any documentation, information, programs, software, or other material that is or may become a part of the website. Passages Ventura does not warrant or guarantee that the information will not be offensive to any user and/or may not meet your needs and requirements. Our website may include links to third party Web service providers. These links are provided solely as a convenience to you and not as an endorsement of the content on such sites. We are not responsible for the content of linked third-party sites.
Modern information and communication technologies play a fundamental role in the activities of an organization like Passages Ventura. We are based in the United States of America. Our principal activity is the delivery of substance abuse rehabilitation services.
If you think you may have a medical emergency, call your doctor or emergency medical services immediately. If you are seeking substance abuse rehabilitation, including substance detoxification, please contact the Passages Ventura Intake Counselors at (888) 926-4859.
Passages Ventura disclaims any liability whatsoever for any documentation, information, programs, software, or other material that is or may become a part of the website. Passages Ventura does not warrant or guarantee that the information will not be offensive to any user and/or may not meet your needs and requirements.
We do not collect information about our visitors from sources outside of Passages Ventura, such as public records or bodies, or private organizations.
Except as requested by you, we do not disclose your personal information to other organizations. As the Internet is not secure or private, unauthorized people may be able to intercept, read and possibly modify e-mail you send or are sent by Passages Ventura.
As your employer may claim ownership of, or the right to access, the e-mail account issued to you by your e-mail, you should avoid using an employer issued e-mail account to communicate with Passages Ventura.Passages Ventura may forward e-mails as appropriate for treatment, and other related reasons. As such, Passages Ventura staff members, other than the recipient, may have access to e-mails that you send. Such access will only be to such persons who have a right to access your e-mail to provide services to you. Otherwise, Passages Ventura will not, otherwise, forward e-mails to independent third parties without your prior written consent, except as authorized or required by law.
Passages Ventura reserves the right to save your e-mail and include your e-mail or information contained within your e-mail in your medical record.
E-mails may not be the only form of communication that Passages Ventura will use to communicate with you. Additionally, Passages Ventura may decide that it is not in your best interest to continue to communicate with you by e-mail. In such case, Passages Ventura will notify that it no longer intends to communicate with you by e-mail.
Passages Ventura is committed to complying with all federal, state, and local laws, as well as applicable regulations, standards, and guidelines established by government agencies and accepted accrediting organizations.
Passages Ventura's goal is to provide high quality information through the content on its website. Every effort has been made to provide accurate and comprehensive information that will be useful to its readers.
Passages Ventura uses original, stock, and purchased photography on the website. The individuals portrayed in the images are models and are not intended to imply endorsement or any association with any addiction or condition.
The information provided on this website was authored by, or has been reviewed and approved by, health care and substance abuse rehabilitation professionals at Passages Ventura.
We welcome your feedback about the content provided on the Passages Ventura website. However, we must make the final decisions about the information that is posted on our site to ensure that it meets the needs of all of our users.
Passages Ventura owns the information on this website. The entire contents and design of this site are protected under U.S. and international copyright laws. You may look at our website online, download information to your computer to read later, or print copies of pages for yourself, your family, or your friends. However, you must not remove any copyright notices from our materials. We reserve all other rights, which means that you must not sell our information, rewrite or modify it, redistribute it, use it on your own website, or use it for any other commercial purpose without our prior written permission.
Passages Ventura will use reasonable means to protect the privacy of your information sent by e-mail. However, because of the risks outlined below, Passages Ventura cannot guarantee that e-mail communications will be confidential. Additionally, Passages Ventura will not be liable in the event that you or anyone else inappropriately uses your e-mail. Passages Ventura will not be liable for improper disclosure of your information that is not caused by Passages Ventura's intentional misconduct.
At the discretion of Passages Ventura, its staff, physicians and agents and upon your agreement to the terms outlined within this consent form, you may use e-mail to communicate with Passages Ventura. These e-mails may contain your personal health information. If you decide to use e-mail to communicate with Passages Ventura, you should be aware of the following risks and/or your responsibilities.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,065
|
Enric Badia Romero (även känd som Enrique Badía Romero), född 1930 i Barcelona, är en katalansk (spansk) serieskapare. Han har bland annat tecknat Modesty Blaise och sin egen skapelse Axa.
Hans första serier antogs och publicerades i spanska tidningar 1947, och fram till 1958 var han en flitig bidragsgivare till flera tidningar i hemstaden Barcelona. 1959 började Romero söka sig utomlands och etablerade kontakter med amerikanska, franska, tyska, italienska och brittiska förlag, bland andra Fleetway Publications.
Romero tecknade flera serier för Fleetway, såsom Kathy & Wendy, Isometrics och Cassius Clay, innan han 1970 fick arbetet som tecknare av Modesty Blaise 1970. Den serien tecknade han fram till 1978. 1976 tecknade han några episoder av Rahan för tidningen Pif Gadget. 1978 skapade han Axa tillsammans med Donne Avenell. Han återvände till Modesty Blaise 1990.
Referenser
Noter
Externa länkar
Badía Romero (officiell webbplats)
Spanska serieskapare
Personer från Barcelona
Födda 1930
Levande personer
Män
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,872
|
Q: How to set user.is_staff to True by default in Django Admin? I need to set the is_staff value to True when creating a user in Admin interface.
How can I do that?
Thanks
A: You can define a custom ModelAdmin and add your custom logic there:
class UserAdmin(admin.ModelAdmin):
def save_model(self, request, obj, form, change):
if request.user.is_superuser:
obj.is_staff = True
obj.save()
admin.site.register(User, UserAdmin)
You can read more about it here.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,029
|
Chuadanga is een district (zila) in de divisie Khulna van Bangladesh. Het district telt ongeveer 1,1 miljoen inwoners en heeft een oppervlakte van 1158 km². De hoofdstad is de stad Chuadanga.
Bestuurlijk
Chuadanga is onderverdeeld in 4 upazila (subdistricten), 31 unions, 514 dorpen en 4 gemeenten.
Subdistricten: Chuadanga Sadar, Alamdanga, Damurhuda en Jibannagar
Externe links
District Chuadanga
Banglapedia (webarchive)
District van Bangladesh
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 388
|
Q: Celery or urllib3 is redirecting me to https I have a problem with making requests trough urllib3. So, I'm connecting through proxy and running script trough celery.
urllib3 setup:
self.http = urllib3.ProxyManager('http://127.0.0.1:24000')
Urllib3 request:
page = self.http.request('get', self.start_url, headers=self.headers)
And after that I see in celery logs something like this:
[2019-11-19 16:13:54,038: INFO/ForkPoolWorker-2] Redirecting http://www.olx.pl/nieruchomosci/mieszkania/wynajem/wroclaw/ -> https://www.olx.pl/nieruchomosci/mieszkania/wynajem/wroclaw/
How can I disable this redirect?
A: It's not urllib3 or celery, it's the remote server.
$ curl -D- http://www.olx.pl/nieruchomosci/mieszkania/wynajem/wroclaw/
HTTP/1.1 301 Moved Permanently
Content-Length: 0
Location: https://www.olx.pl/nieruchomosci/mieszkania/wynajem/wroclaw/
Expires: Tue, 19 Nov 2019 16:33:49 GMT
Cache-Control: max-age=0, no-cache, no-store
Pragma: no-cache
Date: Tue, 19 Nov 2019 16:33:49 GMT
Connection: keep-alive
Server: OLXcdn
X-T: True
As you can see there the server is redirecting you to HTTPS, so you can't disable this on the client side.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,392
|
It's not yet dark
Cast and Contact
"A crowd-pleasing message of the triumph of will and art over physical adversity"
Wendy Ide - Screen International
Soon after premiering his short film The Sound of People at the 2008 Sundance Film Festival, promising young Irish director Simon Fitzmaurice was tragically diagnosed with motor neurone disease (ALS). At just 34 years old, he was given four years to live.
Fitzmaurice and his wife were expecting their third child, and a career in storytelling lay at his feet. Reeling from the shock, Fitzmaurice drew strength from his deepest desires—instead of being stuck in that painful moment, he realized his greatest defiance of ALS would be to direct his first feature film. Seven years later, despite total physical incapacitation, Fitzmaurice completed My Name is Emily (2015), directing it only with the use of his eyes.
This emotional journey of self-realization and personal triumph over life-crushing adversity is nothing short of inspiring. All of it is captured with intimate home movies, photographs, and an affectionate voice-over by compatriot Colin Farrell, transporting us into Fitzmaurice's creative world where every physical and psychological challenge is met with positivity and the desire to fulfill a dream.
"a powerful call to live in the moment...
an illuminating experience."
David Ehrlich - IndieWire
"..incredibly powerful... Fenton has a strong, poetic eye"
Brian Tallerico - Rogerebert.com
About It's Not Yet Dark
About the film makers
See the film
Some press articles that the film has generated
© 2021 Copyright Newgrange Pictures & Kennedy Films. Web Design by Inner Circle.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 9,927
|
\section{Introduction}
Interference alignment studies \cite{Jafar_FnT} have spurred much interest in the degrees of freedom (DoF) of wireless communication networks. While much progress has been made under the assumption of perfect channel knowledge, the degrees of freedom under channel uncertainty at the transmitters have remained mostly a mystery. A prime example is the, heretofore unresolved, conjecture by Lapidoth, Shamai and Wigger from the Allerton conference in 2005 \cite{Lapidoth_Shamai_Wigger_BC}, also featured at the ``\emph{Open Problems Session}" at the Inaugural Information Theory and its Applications (ITA) workshop in 2006 \cite{Lapidoth_Shamai_Wigger_ITA}, which claims that the DoF collapse under finite precision channel state information at the transmitter (CSIT). Specifically, Lapidoth et al. conjecture that the DoF of a 2 user multiple input single output (MISO) broadcast channel (BC) with 2 antennas at the transmitter and 1 antenna at each of the receivers, must collapse to unity (same as single user) if the probability distribution of the channel realizations, from the transmitter's perspective, is sufficiently well behaved that the differential entropy rate is bounded away from $-\infty$. The condition excludes not only settings where some or all channel coefficients are perfectly known, but also scenarios where some channel coefficients are functions of others, even if their values remain unknown. The best DoF outer bound under such channel uncertainty, also obtained by Lapidoth et al., is $\frac{4}{3}$. Deepening the mystery is the body of evidence on both sides of the conjecture. On the one hand, supporting evidence in favor of the collapse of DoF is available if the channel is essentially degraded, i.e., the users' channel vector directions are statistically indistinguishable from the transmitters' perspective \cite{Huang_Jafar_Shamai_Vishwanath, Varanasi_noCSIT}. On the other hand, the idea of blind interference alignment introduced by Jafar in \cite{Jafar_corr} shows that the 2 user MISO BC achieves $\frac{4}{3}$ DoF (which is also an outer bound, thus optimal), even without knowledge of channel realizations at the transmitter, provided that one user experiences time-selective fading and the other user experiences frequency-selective fading. Since the time-selective channel is assumed constant across frequency and the frequency-selective channel is assumed constant across time, it makes some channel coefficients functions of others (they are equal if they belong to the same coherence time/bandwidth interval), so that the model does not contradict the conjecture of Lapidoth et al. Thus, quite remarkably, this conjecture of Lapidoth, Shamai and Wigger, which predates interference alignment in wireless networks, has remained unresolved for nearly a decade.
Following in the footsteps of Lapidoth et al., subsequent works have made similar, sometimes even stronger conjectures, as well as partial attempts at proofs. For instance, the collapse of DoF of the MISO BC was also conjectured by Weingarten, Shamai and Kramer in \cite{Weingarten_Shamai_Kramer} under the finite state compound setting. However, this conjecture turned out to be too strong and was shown to be false by Gou, Jafar and Wang in \cite{Gou_Jafar_Wang}, and by Maddah-Ali in \cite{Maddah_Compound}, who showed that, once again, $\frac{4}{3}$ DoF are achievable (and optimal) for almost all realizations of the finite state compound MISO BC, regardless of how large (but finite) the number of states might be. Since the differential entropy of the channel process is not defined (approaches $-\infty$) for the finite state compound setting, this result also does not contradict the conjecture of Lapidoth et al. A related refinement of the conjecture, informally noted on several occasions (including by Shlomo Shamai at the ITA 2006 presentation) and mentioned most recently (although in the context of i.i.d. fading channels) by Tandon, Jafar, Shamai and Poor in \cite{Tandon_Jafar_Shamai_Poor} --- is that the DoF should collapse even in the ``PN" setting, where perfect (P) CSIT is available for one of the two users, while no (N) CSIT is available for the other user. A valiant attempt at proving this conjecture is made in \cite{Hao_Rasouli_Clerckx}, but it turns out to be unsuccessful because it relies critically on an incorrect use of the extremal inequality of \cite{Liu_Viswanath} under channel uncertainty.\footnote{Similar problems arise in \cite{Rassouli_Hao_Clerckx, Hao_Clerckx}. A simple counter example is the MISO BC with finitely many channel states, where the same arguments as used in these works would imply a collapse of DoF to 1, whereas this setting is known to have $\frac{4}{3}$ DoF as shown in \cite{Gou_Jafar_Wang, Maddah_Compound}.} Thus the ``PN" conjecture has also thus far remained unresolved.
That these conjectures remain unresolved, is emblematic of a broader lack of understanding of the DoF of wireless networks under non-degenerate forms of channel uncertainty. For instance, by extension, under non-degenerate channel uncertainty we also do not know the DoF of the vector broadcast channel with more than 2 users, or the DoF of interference networks, $X$ networks, cellular, multi hop, or two-way relay networks, with or without multiple antennas, or any of a variety of settings with partial uncertainty, such as mixed \cite{Gou_Jafar, Sheng_Kobayashi_Gesbert_Yi} or alternating \cite{Tandon_Jafar_Shamai_Poor} channel uncertainty. Thus, the resolution of these conjectures is likely to have a broad impact on our understanding of the ``robustness" of the DoF of wireless networks. This is the motivation for our work in this paper.
\subsection{Overview of Contribution}
The main contribution of this work is to prove the conjecture of Lapidoth, Shamai and Wigger, thereby closing the ITA 2006 open problem, as well as the ``PN" conjecture of Tandon et al., for all non-degenerate forms of finite precision CSIT, which includes all settings where density functions of the unknown channel realizations exist and are bounded. For all such settings, we show that the DoF collapse to unity as conjectured. Remarkably, this is the first result to show the total collapse of DoF under channel uncertainty without making assumptions of degradedness, or the (essentially) statistical equivalence of users.
Our approach is based on bounding the expected number of codewords that are resolvable at their desired receiver whose images align (within bounded noise distortion) at the undesired receiver under finite precision CSIT. We show that this quantity is $\approx O((\log(P))^{n})$ where $n$ is the length of codewords, and $P$ is the power constraint which defines the DoF limit as $P\rightarrow \infty$. This is negligible relative to the total number of resolvable codewords, which is $\approx O(P^{nd/2})$ when the desired information is sent at rate $\frac{d}{2}\log(P)$, i.e., with DoF $d>0$ (normalization by $\frac{1}{2}\log(P)$ is because we deal with real channels). The difference between the entropy contributed by any set of codewords at their desired receiver (desired DoF) and the entropy contributed by the same set of codewords at the undesired receiver (DoF consumed by interference) tends to zero in the DoF sense. Under non-degenerate channel uncertainty, it is not possible to utilize the DoF at the desired receiver without sacrificing the same number of DoF at the undesired receiver due to interference. Therefore, the DoF are bounded above by unity, the same as with a single user.
We also generalize this result in two directions. First, we extend it to include CSIT that improves as $P\rightarrow\infty$, e.g., through quantized feedback at rate $\frac{\alpha}{2}\log(P)$, so that the probability density function of unknown channel coefficients concentrates around the correct realizations. This refinement of CSIT is captured by the growth in the peak value of the probability density function. We show that if the peak of the probability density function of unknown channel coefficients grows no faster than $O(P^\frac{\alpha}{2})$, representing e.g., improving channel quantization from feedback at rate $\frac{\alpha}{2}\log(P)$, then the total DoF are bounded above by $1+\alpha$. Furthermore, with quantized feedback of rate $\frac{\alpha}{2}\log(P)$ this DoF bound is achievable.
Finally, we go beyond 2 users and generalize the result to the $K$ user MISO broadcast channel where the transmitter has $K$ antennas and there are $K$ users with a single antenna each. Here also, we prove that the DoF collapse to unity under non-degenerate channel uncertainty. Since the outer bound for this MISO BC is also an outer bound for MISO BC's with fewer than $K$ antennas at the transmitter or fewer than $K$ users, for $K$ user interference networks, for $M\times N$ X channels where $K\geq\max(M,N)$, our result establishes the collapse of DoF to unity for all such networks under non-degenerate channel uncertainty. Remarkably, the best known outer bounds for $K$ user interference and $M\times N$ user $X$ networks under non-degenerate channel uncertainty (except for essentially degraded settings) prior to this work were $\frac{K}{2}$ and $\frac{MN}{M+N-1}$ (same as with perfect CSIT). Thus, at least for DoF, this work represents a pessimistic leap of the same magnitude as the ones made in the optimistic direction in \cite{Cadambe_Jafar_int, Cadambe_Jafar_X}.
\subsection{Notation} We use the Landau $O(\cdot)$, $o(\cdot)$, and $\Theta(\cdot)$ notations as follows. For functions $f(x), g(x)$ from $\mathbb{R}$ to $\mathbb{R}$, $f(x)=O(g(x))$ denotes that $\limsup_{x\rightarrow\infty}\frac{|f(x)|}{|g(x)|}<\infty$. $f(x)=o(g(x))$ denotes that $\limsup_{x\rightarrow\infty}\frac{|f(x)|}{|g(x)|}=0$. $f(x)=\Theta(g(x))$ denotes that there exists a positive finite constant, $M$, such that $\frac{1}{M} g(x)\leq f(x)\leq Mg(x)$, $\forall x$. We use $\mathbb{P}(\cdot)$ to denote the probability function $\mbox{Prob}(\cdot)$. We define $\lfloor x\rfloor$ as the largest integer that is smaller than or equal to $x$ when $x>0$, the smallest integer that is larger than or equal to $x$ when $x<0$, and $x$ itself when $x$ is an integer. The index set $\{1,2,\cdots, n\}$ is represented compactly as $[1:n]$ or simply $[n]$ when it would cause no confusion. Arbitrary subsets of $[n]$ may be denoted as $[s]\subset[n]$. The difference of sets $[n]/[s]$ represents the set of elements that are in $[n]$ but not in $[s]$. $X^{[s]}$ represents $\{X(t): t\in[s]\}$. For example, $X^{[n]}=\{X(1), X(2),\cdots, X(n)\}$. With some abuse of notation we use $\{X\}$ to denote the set of values that can be taken by the random variable $X$. The cardinality of a set $A$ is denoted as $|A|$.
\section{The 2 User MISO BC with Perfect CSIT for One User}
To prove the collapse of DoF in the strongest sense possible, let us first enhance the 2 user MISO BC by allowing perfect CSIT for user 1. Consider the vector broadcast channel with 2 users where the transmitter is equipped with $2$ antennas, each user is equipped with $1$ receive antenna, and there are $2$ independent messages $W_1,W_2$ that originate at the transmitter and are desired by users $1$ and $2$, respectively. The transmission takes place over $n$ channel uses. The channel state information at the transmitter (CSIT) is denoted as $\mathcal{T}$, and includes perfect channel state information for the channel vector of user but not for the channel vector of user 2. In the terminology of Tandon et al. \cite{Tandon_Jafar_Shamai_Poor}, this is the PN setting, although not restricted to any statistical equivalence assumptions.
The best outer bound for the DoF of the PN setting based on known results so far is $\frac{3}{2}$, which is obtained from the finite state compound model by Weingarten, Shamai and Kramer in \cite{Weingarten_Shamai_Kramer} and is applicable to finite precision CSIT as well. While Weingarten et al. conjectured that their outer bound was loose even in the finite state compound setting, predicting a collapse of DoF, this conjecture was shown to be false by Gou, Jafar and Wang in \cite{Gou_Jafar_Wang}, who showed that $\frac{3}{2}$ DoF are achievable under the finite state compound model, through the DoF tuple $(d_1, d_2)=(1,0.5)$. The key to achievability is to split user 1's 1 DoF into two parts that carry 0.5 DoF each. These parts align at user 2, consuming half the available signal space of user 2, while remaining resolvable at user 1. User 2's signal, carrying 0.5 DoF, is then sent in the null space of user 1's channel, and is resolvable from the 0.5 dimensional interference-free space at user 2. Note that zero forcing at user 1 is possible because perfect CSIT for user 1 is assumed to be available.
The $\frac{3}{2}$ DoF outer bound is also applicable in the blind interference alignment setting (BIA) introduced by Jafar in \cite{Jafar_corr}, where user 1 experiences time or frequency selective fading but user 2 experiences a relatively flat fading channel. Here also the outer bound is shown to be achievable through the tuple $(d_1,d_2)=(1, 0.5)$. The key is to send two symbols for user 1, one from each antenna, repeated over two channel realizations where the channel of user 1 changes but the channel of user 2 remains the same. Thus, user 1 sees two linear combinations of the two symbols from which both symbols can be resolved, whereas user 2 only sees the same linear combination over both channel uses. Thus the interference occupies only 0.5 DoF at user 2. The remaining 0.5 DoF at user 2 is utilized by sending his desired signal, carrying 0.5 DoF, into the null space of user 1.
The finite state compound setting and the blind interference alignment setting reveal some of the challenges of proving the collapse of DoF for the PN setting. Any attempt at proving a collapse of DoF must carefully exclude such scenarios from the channel model. With this cautionary note, we are now ready to introduce the channel model for our problem.
\subsection{General Channel Model}
The channel is described as follows:
\begin{eqnarray}
\left[\begin{array}{l}
\tilde{Y}_1(t)\\\tilde{Y}_2(t)
\end{array}\right]
&=&\underbrace{\left[\begin{array}{ll}
\tilde{G}_{11}(t)&\tilde{G}_{12}(t)\\
\tilde{G}_{21}(t)&\tilde{G}_{22}(t)
\end{array}\right]}_{{\bf \tilde{G}}(t)}
\left[\begin{array}{l}
\tilde{X}_1(t)\\
\tilde{X}_2(t)
\end{array}\right]+\left[\begin{array}{l}
\tilde{Z}_1(t)\\\tilde{Z}_2(t)
\end{array}\right]\label{eq:channelmodel}
\end{eqnarray}
where all symbols are real. At time $t\in\mathbb{N}$, $\tilde{Y}_k(t)$ is the symbol received by user $k$, $\tilde{Z}_k(t)\sim\mathcal{N}(0,1)$ is the real additive white Gaussian noise (AWGN), $\tilde{X}_k(t)$ is the symbol sent from transmit antenna $k$, and $\tilde{G}_{kj}(t)$ is the channel fading coefficient between the $j^{th}$ transmit antenna and user $k$. The channel coefficients are not restricted to i.i.d. realizations, i.e., they may be correlated across space and time but are assumed to be drawn from a continuous distribution such that their joint density exists. The transmitter is subject to the power constraint:
\begin{eqnarray}
\frac{1}{n}\sum_{t=1}^n[(\tilde{X}_1(t))^2+(\tilde{X}_2(t))^2]&\leq&\tilde{P}=O(P), \label{eq:power}
\end{eqnarray}
To avoid degenerate situations we will assume that the range of values of each of the elements $\tilde{G}_{ij}$ is bounded away from zero and infinity, as is the determinant of the overall channel matrix --- i.e., $|\tilde{G}_{ij}(t)|, \mbox{det}(\tilde{\bf G}(t))$ are all $\Theta(1)$. Stated explicitly, there exists positive finite constant $M$, such that
\begin{eqnarray}
\frac{1}{M}\leq |\tilde{G}_{ij}(t)|, \mbox{det}(\tilde{\bf G}(t))\leq M
\end{eqnarray}
Note that this is not a major restriction because by choosing the bounding constants large enough, the omitted neighborhoods can be reduced to a probability measure less than $\epsilon$ for arbitrarily small $\epsilon$, and thus has only a vanishing impact on the DoF.
\subsection{Canonical Form}
Without loss of generality, for the purpose of deriving a DoF outer bound the channel model is reduced to the following form, which is preferable due to the consolidation of channel parameters (See Appendix \ref{app:z} for justification). This is the canonical form that we will use throughout the paper.
\begin{figure}[h]
\center
\includegraphics[width=9cm]{canonicalbc}
\caption{Canonical form of the 2 user MISO BC with perfect CSIT for user 1. }\label{fig:canonicalbc}
\end{figure}
The canonical form of the channel model, shown in Fig. \ref{fig:canonicalbc} has the same outputs $Y_1(t), Y_2(t)\in\mathbb{R}$, but the inputs are $X_1(t), X_2(t)\in\mathbb{R}$, so that:
\begin{eqnarray}
\left[\begin{array}{l}
Y_1(t)\\Y_2(t)
\end{array}\right]
&=&\left[\begin{array}{cc}
1&0\\
{G}(t)&1
\end{array}\right]
\left[\begin{array}{l}
{X}_1(t)\\
{X}_2(t)
\end{array}\right]+\left[\begin{array}{l}
Z_1(t)\\Z_2(t)
\end{array}\right]\label{eq:canonical}
\end{eqnarray}
The channel coefficient $G(t)$ is also bounded away from zero and infinity, i.e., there exists finite positive $M$, such that $|G(t)|\in\left(\frac{1}{M}, M\right)$.
The new power constraint is expressed as
\begin{eqnarray}
\frac{1}{n}\sum_{t=1}^n[({X}_1(t))^2+({X}_2(t))^2]&\leq&{P}, \label{eq:powercanonical}
\end{eqnarray}
where $P=\Theta(\tilde{P})$. Further, for notational convenience let us define the set of admissible inputs.
\begin{eqnarray}
\mathcal{X}^{[n]}&\triangleq&\{(X_1^{[n]},X_2^{[n]}): \frac{1}{n}\sum_{t=1}^n[({X}_1(t))^2+({X}_2(t))^2]\leq{P}\}
\end{eqnarray}
\subsection{Messages, Rates, Capacity, DoF}
The messages $W_1, W_2$ are jointly encoded at the transmitter for transmission over $n$ channel uses at rates $R_1, R_2$, respectively, into a $2^{nR_1+ nR_2}\times n$ codebook matrix over the input alphabet. The codebook is denoted by $C(n,[R_1,R_2], P)$. For given power constraint parameter $P$, the rate vector $[R_1, R_2]$ is said to be achievable if there exists a sequence of codebooks $\mathcal{C}(n,[R_1,R_2], P)$, indexed by $n$, such that the probability that all messages are correctly decoded by their desired receivers approaches 1 as $n$ approaches infinity. The closure of achievable rate vectors is the capacity region $\mathcal{C}(P)$. The DoF tuple $(d_1, d_2)$ is said to be achievable if there exist $(R_1(P), R_2(P))\in\mathcal{C}(P)$ such that
\begin{eqnarray*}
d_1&=&\lim_{P\rightarrow\infty} \frac{R_1(P)}{\frac{1}{2}\log(P)}\\
d_2&=&\lim_{P\rightarrow\infty} \frac{R_2(P)}{\frac{1}{2}\log(P)}\\
\end{eqnarray*}
The closure of all achievable DoF tuples $(d_1, d_2)$ is called the DoF region, $\mathcal{D}$. The sum-DoF value is defined as
\begin{eqnarray*}
\mathcal{D}_\Sigma&=&\max_{(d_1,d_2)\in\mathcal{D}}(d_1+d_2)
\end{eqnarray*}
\subsection{Non-degenerate Channel Uncertainty}
Beyond the assumptions that are already stated, the non-degenerate channel uncertainty model is defined by the additional assumption that the joint probability density function of the channel coefficients, $G^{[n]}$, conditioned on the available CSIT, $\mathcal{T}$, which we denote as $f_{G^{[n]}|\mathcal{T}}(g^{[n]})$, exists and is bounded. Specifically, the conditions that we require are the following.
\subsubsection{Peak of Density Function is Bounded for Fixed $P$}
For a given $P$, there exists a finite constant $f_{\max}$, $1\leq f_{\max}<\infty$, such that the probability that a subset of channel coefficients takes values in any measurable set is no more than the $\left(f_{\max}\right)^{|[s]|}$ times the Lebesgue measure of that set, where $|[s]|$ is the dimension of the space containing the set, i.e., the set is drawn from $\mathbb{R}^{|[s]|}$.
\begin{eqnarray}
\mathbb{P}(G^{[s]}\in\mathcal{G}^{[s]}_o)&\leq&f_{\max}^{|[s]|}(P)\int_{\mathcal{G}^{[s]}_0}dG^{[s]}, ~~\forall [s]\subset[n], \forall \mathcal{G}^{[s]}_o\subset\mathcal{G}^{[s]}\label{eq:ourcondition}
\end{eqnarray}
The condition implies that a zero measure space cannot carry a non-zero probability.
So it precludes scenarios where e.g., the channel is perfectly known or when one channel coefficient is a function of the rest. In all such cases, a zero measure space carries a non-zero probability, thus precluding the existence of a bounded constant $f_{\max}(P)$ as defined above. This restriction essentially accomplishes the same goal as the restriction by Lapidoth et al. \cite{Lapidoth_Shamai_Wigger_BC} that the differential entropy should be greater than $-\infty$.
Note that if all joint and conditional density functions are bounded then (\ref{eq:ourcondition}) is immediately satisfied, and $f_{\max}$ may be chosen to be the peak value of all joint and conditional density functions.
\begin{eqnarray}
f_{\max}=\max\left(1,\max_{{[s]: [s]\subset[n]}}\sup_{\mathcal{G}^{[s]}} \left(f_{G^{[s]}|\mathcal{T},G^{[n]/[s]}}(g^{[s]})\right)^{\frac{1}{|[s]|}}\right)
\end{eqnarray}
Further, while we do not restrict channel realizations to be independent, suppose, just as an illustrative example, that we consider such a setting, which is of some interest. Then the joint/conditional density is the $|[s]|$-fold product of the marginal density functions,
\begin{eqnarray}
f_{G^{[s]}|\mathcal{T}, G^{[n]/[s]}}(g^{[s]})&=&\prod_{t\in[s]} f_{G(t)|\mathcal{T}}(g(t))
\end{eqnarray}
and one can simply choose:
\begin{eqnarray}
f_{\max}&=&\max\left(1,\max_{t\in[1:n]}\sup_{g(t)\in\mathcal{G}(t)}f_{G(t)|\mathcal{T}}(g(t))\right)
\end{eqnarray}
Thus, a simple way to interpret our condition is that we require that the densities exist and are bounded.
\subsubsection{Peak of Density Function is Allowed to Scale with $P$}
\noindent As a function of $P$, we allow $f_{\max}(P)$ to scale as $O((\sqrt{P})^\alpha)$ for some $\alpha\in[0,1]$.
\begin{eqnarray}
f_{\max}(P)&=&O(P^{\frac{\alpha}{2}})
\end{eqnarray}
The case studied by Lapidoth et al. in \cite{Lapidoth_Shamai_Wigger_BC}, where the density does not depend on $P$, is represented here by setting $\alpha=0$. The positive values of $\alpha$ allow us to address settings where the CSIT improves with $P$, e.g., due to quantized channel feedback of rate $\frac{\alpha}{2}\log(P)$, so that the weight of the distribution is increasingly concentrated around the true channel realizations. Note that the maximum value of $\alpha$ is unity, because a feedback rate of $ \frac{1}{2}\log(P)$, implying 1 real DoF worth of feedback, is sufficient to approach perfect CSIT performance over channels that take only real values.
Since the receivers have full channel state information, $\mathcal{T}$ is globally known. For compact notation, we will suppress the conditioning, writing $f_{G^{[n]}}(g^{[n]})$ directly instead.
\subsection{$K$ User Extension}
Extending beyond the 2 user case, the canonical channel model in the $K$ user setting is described as follows.
\begin{figure}[h]
\center
\includegraphics[width=12cm]{canonicalbc3}
\caption{Canonical form of the 3 user MISO BC with Graded Channel Uncertainty. }\label{fig:canonicalbc3}
\end{figure}\begin{eqnarray}
Y_1(t)&=&X_1(t)+Z_1(t)\\
Y_2(t)&=&G_{21}(t)X_1(t)+X_2(t)+Z_2(t)\\
&\vdots&\nonumber\\
Y_K(t)&=&G_{K1}(t)X_1(t)+G_{K2}(t)X_2(t)+\cdots+G_{K(K-1)}(t)X_{K-1}(t)+X_K(t)+Z_K(t)
\end{eqnarray}
where the inputs, $X_k(t)\in\mathbb{R}$, are subject to the power constraint
\begin{eqnarray}
\frac{1}{n}\sum_{t=1}^n\left((X_1(t))^2+(X_2(t))^2+\cdots+(X_K(t))^2\right)&\leq& P
\end{eqnarray}
The $G_{ij}(t)$ terms are known to the transmitter only up to finite precision and are assumed to be bounded away from 0 and infinity. Further, the density of the $k^{th}$ users' unknown channel coefficients, $k>1$, is bounded by $f_{\max,k}(P)=O\left(P^{\frac{\alpha_k}{2}}\right)$.
\section{Results}
We state the main result in its most general form, for $K$ users. The $2$ user case, corresponds to setting $\alpha_2\triangleq\alpha$.
\begin{theorem}\label{theorem:main}
For the $K$ user MISO BC with non-degenerate channel uncertainty, the sum-GDoF are bounded above as
\begin{eqnarray}
\mathcal{D}_\Sigma&\leq&1+\alpha_2+\alpha_3+\cdots+\alpha_K
\end{eqnarray}
\end{theorem}
\subsubsection*{Settling the Conjecture by Lapidoth et al. in \cite{Lapidoth_Shamai_Wigger_BC}}
The $2$ user setting studied by Lapidoth et al., where the joint pdf is fixed, i.e., it does not depend on $P$, is captured here when $\alpha=0$ (equivalently, $\alpha_2=0)$. When $\alpha=0$, the sum-GDoF are bounded above by unity, thus settling the conjecture of Lapidoth et al. for all non-degenerate channel uncertainty models.
\subsubsection*{Settling the ``PN" Conjecture}
Since we allow perfect CSIT for one user, and one may assume (as a special case of our result) that the channels are i.i.d., the collapse of DoF for $\alpha=0$, also proves the conjecture of Tandon et al. for the 2 user setting.
\subsubsection*{Interference and $X$ networks}
Consider any one-hop wireless network where all receivers are equipped with a single antenna each. This includes all interference and $X$ networks. Allowing the transmitters to cooperate produces a MISO BC setting. Since cooperation cannot hurt, the outer bound for the MISO BC under non-degenerate channel uncertainty applies to interference and $X$ networks as well. In all cases, the DoF collapse to unity.
\subsubsection*{Limited rate feedback ($\alpha>0$)}
Consider the 2 user setting, with $\alpha=\alpha_2>0$. This case is interesting because it has direct implications to the achievable DoF under limited rate quantized channel state feedback for the channel vector of user 2. If the feedback link has $\alpha$ DoF, i.e., the feedback rate scales as $\frac{\alpha}{2}\log(P)$ bits per channel use, then this corresponds to $\sim P^{\frac{\alpha}{2}}$ channel quantization levels, so that the size of a quantization interval scales as $\frac{1}{(\sqrt{P})^{\alpha}}$ and the channel density restricted to a quantization interval, i.e., $f_{G^{[n]}}(g^{[n]}|\mathcal{T})$ scales as $P^{\frac{\alpha}{2}}$. Theorem \ref{theorem:main} tells us that in this case the GDoF are bounded above as $\mathcal{D}_\Sigma\leq 1+\alpha$. It is also easy to see that under such quantized feedback, the DoF tuple $(d_1, d_2)=(1, \alpha)$ is achievable, simply by best-effort zero-forcing at the transmitter and treating residual interference as noise at the receiver 2. Thus, $\mathcal{D}_\Sigma=1+\alpha$ is the optimal sum-DoF value if the quantized channel state feedback is limited to rate $\frac{\alpha}{2}\log(P)$. This generalizes the result of Caire, Jindal and Shamai from \cite{Caire_Jindal_Shamai} who showed\footnote{While the result in \cite{Caire_Jindal_Shamai} is for the complex setting, the statement here is specialized to the real setting.} that in order to achieve the same DoF as with perfect CSIT, i.e., $\mathcal{D}_\Sigma=2$, the quantized feedback rate should scale as $\frac{1}{2}\log(P)$, i.e., carry one full degree of freedom $(\alpha=1)$. The bound for the $K$ user extension is similarly tight as well.
\allowdisplaybreaks
\section{Aligned Image Sets under Channel Uncertainty}
We begin this section with a disclaimer, that this section is included mainly to share an intuitive understanding of the proof. The proof itself will be presented in Section \ref{sec:proof2}. To facilitate an intuitive discussion, no attempt is made to be mathematically precise or rigorous here. The main idea we want to illustrate intuitively is a geometrical notion of aligned images of codewords---loosely related to Korner and Marton's work on the images of a set in \cite{Korner_Marton_images} but under a much more specialized setting---which is the key to our proof. As the proof in Section \ref{sec:proof2} will show, the problem boils down to the difference of two terms when only information to user 1 is being transmitted,
\begin{eqnarray}
\mathcal{D}_\Sigma &\leq& 1 + \limsup_{P\rightarrow\infty}\limsup_{n\rightarrow\infty}\frac{1}{\frac{n}{2}\log(P)}\left(h(Y_1^{[n]}|G^{[n]})-h(Y_2^{[n]}|G^{[n]})\right)\label{eq:start}
\end{eqnarray}
The first term, $h(Y_1^{[n]}|G^{[n]})$, we wish to maximize because it represents the rate of desired information being sent to user 1. The second, $h(Y_2^{[n]}|G^{[n]})=h(G^{[n]}X_1^{[n]}+X_2^{[n]}+Z_2^{[n]}|G^{[n]})$ we wish to minimize, because it represents the interference seen by user 2, due to the information being sent to user 1. If $G^{[n]}$ was perfectly available to the transmitter, then $X_2^{[n]}$ could be chosen to cancel $G^{[n]}X_1^{[n]}$ thus eliminating interference entirely at user 2. With only statistical knowledge of $G^{[n]}$, zero forcing is not possible. Indeed, the purpose of $X_2^{[n]}$ is mainly to align interference into as small a space as possible. However, instead of consolidating interference in the sense of vector space dimensions, as is typically the case in DoF studies involving interference alignment, here the goal is for $X_2^{[n]}$ to minimize the size of the image, as seen by user 2, of the codewords that carry information for user 1. This is the new perspective that is the key to the proof.
\subsection{Toy Setting to Introduce Aligned Image Sets}
For illustrative purposes, let us start with a rather extreme over-simplification, by considering the case with $n=1$, ignoring noise, and using the log of the cardinality of the codewords as a surrogate for the entropy. With this simplification, the quantity that we are interested in is the difference:
\begin{eqnarray}
\log|\{X_1\}|-\log|\{GX_1+X_2\}|
\end{eqnarray}
averaged over $G$. By $|\{A\}|$ is meant the cardinality of the set of values taken by the variable $A$.
The codebook is the set of ($X_1, X_2$) values. Note that $|\{X_1\}|$, the number of distinct values of $X_1$, is the number of distinct ``codewords" as seen by user 1, who (once noise is ignored) only sees $Y_1=X_1$, so that his ``rate" is $\log|\{X_1\}|$. Given the set of $X_1$ values, we would like to associate each $X_1$ value with a corresponding $X_2$ value, such that the number of distinct values of $Y_2=GX_1+X_2$ is minimized. In other words, we wish to minimize the image of the set of codewords as seen by user 2, by choosing $X_2$ to be a suitable function of $X_1$.
\begin{figure}[h]
\center
\includegraphics[width=7cm]{images}
\caption{Two codewords, $\nu$ and $\gamma$, and their equivalence classes, $S_\nu$ and $S_\gamma$, containing all codewords that have the same image at user 2 as $\nu$ and $\gamma$, respectively. The partitioning into equivalence classes depends on the channel realization. The figure shows the distinct equivalence classes for two channel realizations, $G$ and $G'$. }\label{fig:eqclass}
\end{figure}
Consider two codewords $(X_1, X_2)=(x_1, x_2)$ and $(X_1,X_2)=(x_1',x_2')$. If $x_1\neq x_1'$ then these codewords are distinct from user 1's perspective, and thus capable of carrying information to user 1 via the transmitter's choice to transmit one or the other. Suppose the channel is $G$. Then for these two codewords to ``\emph{align}" where they cause interference, they must have the same image as seen by user 2. This gives us the condition for aligned images that is central to this work.
\begin{eqnarray}
Gx_1+x_2&=&Gx_1'+x_2'\\
\Rightarrow G&=&-\left(\frac{x_2'-x_2}{x_1'-x_1}\right)
\end{eqnarray}
In other words, $G$ must be the negative of the slope of the line connecting the codeword $(x_1,x_2)$ to the codeword $(x_1',x_2')$ in the $X_1,X_2$ plane. For a given channel realization $G$, all codewords that align with $(x_1,x_2)$ (i.e., whose images align with the image of $(x_1,x_2)$) as seen by user 2, must lie on the same line that passes through $(x_1, x_2)$ and has slope $-G$. Conversely, all codewords that lie on this line have images that align with the image of $(x_1,x_2)$ at user 2. For any codeword that does not lie on this line, there is a parallel line with the same slope, $-G$, that represents the set of codewords whose images align with the image of that codeword. Thus, these lines of the same slope, $-G$, partition the set of codewords into equivalence classes, such that codewords that lie on the same line have the same image at user 2.
Also note that a different channel realization, $G'$, gives rise to a different equivalent class partition, corresponding to lines with slope $-G'$. This is illustrated in Fig. \ref{fig:eqclass}. Since the $X_2$ values are functions of $X_1$ values, in the figure we label the codewords only on the $X_1$ axis. The codeword $\nu$ belongs to the equivalence class $S_\nu(G)$ under the channel realization $G$ and to the equivalence class $S_\nu(G')$ under the channel realization $G'$. Also, note that two codewords that belong to the same equivalence class under one channel realization, cannot belong to the same equivalence class under any other channel realization. For instance, codewords $\lambda$ and $\nu$ belong to the same equivalence class $S_\nu(G)$ under channel realization $G$, but they belong to different equivalence classes, $S_\nu(G')$ and $S_\gamma(G')$, under a different channel realization $G'$.
Recall that our goal is to minimize the number of distinct images seen by user 2, (averaged) across all channel realizations. It is not preferable to have too few codewords aligned into the same image, because it would create too many distinct images across all codewords. However, remarkably, it is also not preferable to have too many codewords aligned into the same image for any given channel realization, because for every \emph{other} channel realization, each of these codewords must have a different image. Thus,
\begin{eqnarray}
\mbox{No. of distinct images}&\geq&\mbox{No. of codewords per image}
\end{eqnarray}
A rough calculation (assuming uniformity for this intuitive argument so averages can be ignored) will shed light on the best case scenario.
\begin{eqnarray}
\mbox{No. of codewords} &=& \mbox{No. of distinct images }\times\mbox{ No. of codewords per image}\\
&\leq& \mbox{No. of distinct images }\times\mbox{ No. of distinct images}\\
\Rightarrow \mbox{No. of distinct images} &\geq& \sqrt{\mbox{No. of codewords}}
\end{eqnarray}
Mathematically,
\begin{eqnarray}
|\{GX_1+X_2\}|&\geq&\sqrt{|\{X_1\}|}
\end{eqnarray}
and therefore we have the bound
\begin{eqnarray}
\log(|\{X_1\}|) - \log(|\{GX_1+X_2\}|)&\leq&\log(|\{X_1\}|)-\frac{1}{2}\log(|\{X_1\}|)=\frac{1}{2}\log(|\{X_1\}|)
\end{eqnarray}
Now, realizing that $\log(|\{X_1\}|)$ can at most be user 1's single interference free capacity, i.e., that it corresponds to at most rate $\frac{1}{2}\log(P)$, and substituting back into (\ref{eq:start}) the intuitive argument leads us to $\mathcal{D}_\Sigma \leq \frac{3}{2}$. This is indeed the DoF outer bound for the finite state compound channel setting, and is, quite surprisingly achievable, as shown by Gou, Jafar, Wang in \cite{Gou_Jafar_Wang} and Maddah-Ali in \cite{Maddah_Compound} when the number of possible channel realizations, $|\mathcal{G}|$, is finite and fixed, even as the cardinality of codebooks approaches infinity as $P\rightarrow\infty$.
\subsection{Sketch of Proof}
Unfortunately, the utility of the over-simplified noise free setting with $n=1$ starts wearing thin as we go beyond this point. While the picture of aligned image sets illustrated here will continue to be meaningful, the remaining nuances of the proof are not easily conveyed in this toy setting. Staying with the intuitive character of this section, let us conclude with an outline which will be useful to navigate the structure of the proof that appears in the subsequent section.
From the perspective of DoF studies, the presence of noise essentially imposes a resolution threshold, e.g., $\delta$, such that the codewords with images that differ by less than $\delta$, are unresolvable. As the first step of the proof, this effect is captured by discretizing the input and output alphabet and eliminating noise, as is done in a variety of deterministic channel models that have been used for DoF studies \cite{Avestimehr_Diggavi_Tse, Bresler_Tse}, so that instead of differential entropies we now need to deal only with entropies $H(\bar{Y}_1^{[n]}|G^{[n]})$ and $H(\bar{Y}_2^{[n]}|G^{[n]})$. Here $\bar{X}_1, \bar{X}_2$ represent the discretized inputs, $\bar{Y}_1, \bar{Y}_2$ the discretized outputs, and $\bar{Y}_1=\bar{X}_1$. Next step is to note that we are only interested in the maximum value of the difference $H(\bar{Y}_1^{[n]}|G^{[n]})-H(\bar{Y}_2^{[n]}|G^{[n]})$. It then follows that without loss of generality, $\bar{X}_2^{[n]}$ can be made a function of $\bar{X}_1^{[n]}$, and therefore $\bar{Y}_2^{[n]}$ becomes a function of $\bar{Y}_1^{[n]}, G^{[n]}$. This implies that $H(\bar{Y}_1^{[n]}|G^{[n]})=H(\bar{Y}_1^{[n]}, \bar{Y}_2^{[n]}|G^n)$ $=H(\bar{Y}_2^{[n]}|G^{[n]})+H(\bar{Y}_1^{[n]}|\bar{Y}_2^{[n]},G^{[n]})$. Thus, the difference of entropies is equal to $H(\bar{Y}_1^{[n]}|\bar{Y}_2^{[n]},G^{[n]})=H(\bar{X}_1^{[n]}|\bar{Y}_2^{[n]},G^{[n]})$. Now, conditioned on $\bar{Y}_2^{[n]}, G^{[n]}$, the set of feasible values of $\bar{X}_1^{[n]}$ is precisely an aligned image set $S(G^{[n]})$, i.e., all these $\bar{X}_1^{[n]}$ produce the same value of $\bar{Y}_2^{[n]}$ for the given channel realization $G^{[n]}$. Since entropy is maximized by a uniform distribution, $H(\bar{X}_1^{[n]}|\bar{Y}_2^{[n]},G^{[n]})\leq \mbox{E}_{G^n}\left[\log\left(|S(G^{[n]})|\right)\right]\leq\log\left(\mbox{E}|S(G^{[n]})|\right)$, where the last step followed from Jensen's inequality. Thus, the difference of entropies is bounded by the log of the expected cardinality of the aligned image sets. The most critical step of the proof then is to bound the expected cardinality of aligned image sets. This is done by bounding the probability that two given $\bar{X}_1^{[n]}$ are in the same aligned image set, i.e., the probability of the set of channels for which the two produce the same image $\bar{Y}_2^{[n]}$. Recall that for two codewords to belong to the same aligned set in the absence of noise, the channel realization over each channel use must be the slope of the vector connecting the corresponding codeword vectors. The blurring of $\delta$ around the two codewords also blurs the slope of the line connecting them, but by no more than $\pm \delta/\Delta$, where $\Delta$ is the distance (difference in magnitudes) between the two codeword symbols over that channel use. Thus, the probability that the given two codewords that are resolvable at user 1 cast the same image at user 2 is bounded above by $\approx f_{\max}\frac{2\delta}{\Delta}$. The power constraint of $P$ implies that there are at most $\approx \sqrt{P}/{\delta}$ resolvable codeword symbols per channel use. Summing over all possible resolvable codeword symbols, gives us $\approx f_{\max}\sum_{\Delta\in[0:\sqrt{P}/\delta]}\frac{2\delta}{\Delta}= f_{\max}\log(P)+o(\log(P))$, per channel use, so that the average cardinality of an aligned image set, $E|S(G^{[n]}|$, turns out to be bounded above by $\approx(f_{\max}\log(P))^n$, and $\log(E|S(G^{[n]}|)$ is bounded above by $\approx n\log(f_{\max})+n\log(\log(P))$. Since $f_{\max}=O(P^{\frac{\alpha}{2}})$, normalizing by $\frac{n}{2}\log(P)$ and sending first $n$ and then $P$ to infinity sends this term to $\alpha$. Thus, combining with (\ref{eq:start}) produces the sum-DoF outer bound value $1+\alpha$, giving us the result of Theorem 1. Note that in the DoF limit, $\delta=\Theta(1)$, and it will be useful to think of it as $1$ for simplicity, so that the inputs and outputs are restricted to integer values. With this sketch as the preamble, we now proceed to the actual proof.
\section{Proof of Theorem \ref{theorem:main} for $K=2$ Users }\label{sec:proof2}
For ease of exposition, the proof is divided into several key steps. The first step is the discretization of the channel to capture the effect of noise, leading to a deterministic channel model, whose DoF will be an outer bound to the DoF of the canonical channel model, which in turn is an outer bound on the DoF of the general channel model.
\begin{enumerate}
\item {\bf Deterministic Channel Model}\\
The deterministic channel model has inputs $\bar{X}_1(t), \bar{X}_2(t)\in\mathbb{Z}$ and outputs $\bar{Y}_1(t),\bar{Y}_2(t)\in\mathbb{Z}$, defined as
\begin{eqnarray}
\bar{Y}_1(t)&=&\bar{X}_1(t)\\
\bar{Y}_2(t)&=&\lfloor G(t)\bar{X}_1(t)\rfloor+\bar{X}_2(t)
\end{eqnarray}
with the power constraint
\begin{eqnarray}
\bar{X}_1(t), \bar{X}_2(t)\in\{0,1,\cdots,\lceil\sqrt{P}\rceil\}, ~\forall t\in\mathbb{N}
\end{eqnarray}
and the set of inputs that satisfy the power constraints defined as
\begin{eqnarray}
\bar{\mathcal{X}}^{[n]}&=&\{(\bar{X}_1^{[n]},\bar{X}_2^{[n]})\in\mathbb{Z}^{[n]}\times\mathbb{Z}^{[n]} : \bar{X}_1(t), \bar{X}_2(t)\in\{0,1,\cdots,\lceil \sqrt{P} \rceil\}, ~\forall t\in[1:n]\}
\end{eqnarray}
The assumptions on the unknown channel coefficients sequence $G^{[n]}$ are the same as before.
\begin{lemma}\label{lemma:detn}
The DoF of the canonical channel model are bounded above by the DoF of the deterministic channel model.
\end{lemma}
The proof of Lemma \ref{lemma:detn} appears in Appendix \ref{app:detn} and follows along the lines of similar proofs by Bresler and Tse in \cite{Bresler_Tse}.
\item{\bf Difference of Entropies Representing Desired Signal and Interference Dimensions}\\
Starting from Fano's inequality, we proceed as follows.
\begin{eqnarray}
nR_1&\leq&I(W_1; \bar{Y}_1^{[n]}|W_2, G^{[n]})+o(n)\label{eq:startn}\\
&=&H(\bar{Y}_1^{[n]}|W_2, G^{[n]})+o(n)\\
nR_2&\leq&I(W_2;\bar{Y}_2^{[n]}|G^{[n]})+o(n)\\
&=&H(\lfloor G^{[n]} \bar{X}_1^{[n]}\rfloor +\bar{X}_2^{[n]}|G^{[n]})-H(\bar{Y}_2^{[n]}|W_2, G^{[n]})+o(n)\\
&\leq & \frac{n}{2}\log(P)-H(\bar{Y}_2^{[n]}|W_2, G^{[n]})+n~o(\log(P))+o(n)\label{eq:onedof}\\
\Rightarrow n(R_1+R_2)&\leq&\frac{n}{2}\log(P)+[H(\bar{Y}_1^{[n]}|W_2,G^{[n]})-H(\bar{Y}_2^{[n]}|W_2,G^{[n]})]+n~o(\log(P))+o(n)\\
\Rightarrow \mathcal{D}_\Sigma&\leq&1+\limsup_{P\rightarrow\infty}\limsup_{n\rightarrow\infty}\frac{[H(\bar{Y}_1^{[n]}|W_2,G^{[n]})-H(\bar{Y}_2^{[n]}|W_2,G^{[n]})]}{\frac{n}{2}\log(P)}\\
&\leq&1+\limsup_{P\rightarrow\infty}\limsup_{n\rightarrow\infty}\max_{w_2\in[1:2^{nR_2}]}\frac{[H(\bar{Y}_1^{[n]}|W_2=w_2,G^{[n]})-H(\bar{Y}_2^{[n]}|W_2=w_2,G^{[n]})]}{\frac{n}{2}\log(P)}\nonumber\\
&=&1+\bar{\mathcal{D}}_\Delta\label{eq:Deltan}
\end{eqnarray}
so that what remains is to bound the difference of entropy terms:
\begin{eqnarray}
\bar{\mathcal{D}}_\Delta&\triangleq&\limsup_{P\rightarrow\infty}\limsup_{n\rightarrow\infty}\max_{\substack{\mathbb{P}(\bar{X}_1^{[n]},\bar{X}_2^{[n]})\\ (\bar{X}_1^{[n]},\bar{X}_2^{[n]})\in\bar{\mathcal{X}}^{[n]}}}\frac{H(\bar{X}_1^{[n]}|G^{[n]})-H(\lfloor G^{[n]}\bar{X}_1^{[n]}\rfloor+\bar{X}_2^{[n]}|G^{[n]})}{\frac{n}{2}\log(P)} \label{eq:detkey}
\end{eqnarray}
Note that in (\ref{eq:onedof}) we bounded $H(\lfloor G^{[n]} \bar{X}_1^{[n]}\rfloor +\bar{X}_2^{[n]}|G^{[n]})$ as follows.
\begin{eqnarray}
H(\lfloor G^{[n]} \bar{X}_1^{[n]}\rfloor +\bar{X}_2^{[n]}|G^{[n]})&\leq&\sum_{t=1}^nH(\lfloor G(t) \bar{X}_1(t)\rfloor +\bar{X}_2(t)|G(t))\\
&\leq&\sum_{t=1}^n\mbox{E}\log\left((|G(t)|+1)(\lceil\sqrt{P}\rceil+1)\right)\label{eq:card}\\
&=&\frac{n}{2}\log(P)+n~o(\log(P))
\end{eqnarray}
where (\ref{eq:card}) follows from the observation that for a given $G(t)$ value, $\lfloor G^{[n]} \bar{X}_1^{[n]}\rfloor +\bar{X}_2^{[n]}$ can take at most $(1+G(t))(1+\lceil\sqrt{P}\rceil)$ values (all integers) and the entropy of a variable that can take finitely many values is at most the log of the number of values.
\item{\bf Functional Dependence $\bar{X}_2^{[n]}(\bar{X}_1^{[n]})$}\\
Next we show that one can assume that $\bar{X}_2^{[n]}$ is a function of $\bar{X}_1^{[n]}$. Given the sets of codeword vectors $\{\bar{X}_1^{[n]}\}$, $\{\bar{X}_2^{[n]}\}$, define $\mathcal{L}$ as the mapping from $\bar{X}_1^{[n]}$ to $\bar{X}_2^{[n]}$, i.e.,
\begin{eqnarray}
\bar{X}_2^{[n]}&=&\mathcal{L}(\bar{X}_1^{[n]})
\end{eqnarray}
In general, because the mapping may be random, $\mathcal{L}$ is a random variable. Because conditioning cannot increase entropy,
\begin{eqnarray}
H\left(\lfloor G^{[n]}\bar{X}_1^{[n]}\rfloor+\mathcal{L}(\bar{X}_1^{[n]})\right|G^{[n]})&\geq&H\left(\lfloor G^{[n]}\bar{X}_1^{[n]}\rfloor+\mathcal{L}(\bar{X}_1^{[n]})\right|G^{[n]}, \mathcal{L})\\
&\geq&\min_{L\in\{\mathcal{L}\}}H\left(\lfloor G^{[n]}\bar{X}_1^{[n]}\rfloor+\mathcal{L}(\bar{X}_1^{[n]})\right|G^{[n]}, \mathcal{L}=L)
\end{eqnarray}
Let $L_o\in\mathcal{L}$ be the mapping that minimizes the entropy term. Then, choosing
\begin{eqnarray}
\bar{X}_2^{[n]}(\bar{X}_1^{[n]})&=&L_o(\bar{X}_1^{[n]})
\end{eqnarray}
we have
\begin{eqnarray}
\bar{\mathcal{D}}_\Delta&\leq&\hat{\mathcal{D}}_\Delta\triangleq\limsup_{P\rightarrow\infty}\limsup_{n\rightarrow\infty}\max_{\substack{\mathbb{P}(\bar{X}_1^{[n]}), \bar{X}_2^{[n]}(\bar{X}_1^{[n]})\\ (\bar{X}_1^{[n]},\bar{X}_2^{[n]})\in\bar{\mathcal{X}}^{[n]}}}\frac{H(\bar{X}_1^{[n]}|G^{[n]})-H(\lfloor G^{[n]}\bar{X}_1^{[n]}\rfloor+\bar{X}_2^{[n]}(\bar{X}_1^{[n]})|G^{[n]})}{\frac{n}{2}\log(P)}\nonumber\\
&&\label{eq:Deltahatn}
\end{eqnarray}
because the choice of the mapping function does not affect the positive entropy term, and it minimizes the negative entropy term. Henceforth, because $\bar{X}_2^{[n]}$ is a function of $\bar{X}_1^{[n]}$, we will refer to codewords only through $\bar{X}_1^{[n]}$ values.
\item {\bf Definition of Aligned Image Sets}\\
The aligned image set containing the codeword $\bar{\nu}^{[n]}\in\{\bar{X}_1^{[n]}\}$ for channel realization $G^{[n]}$ is defined as the set of all codewords that cast the same image as $\bar{\nu}^{[n]}$ at user 2.
\begin{eqnarray}
S_{\bar{\nu}^{[n]}}(G^{[n]})&\triangleq&\{\bar{x}_1^{[n]}\in\{\bar{X}_1^{[n]}\}: \lfloor G^{[n]}\bar{x}_1^{[n]}\rfloor+\bar{X}_2^{[n]}(\bar{x}_1^{[n]})=\lfloor G^{[n]}\bar{\nu}^{[n]}\rfloor+\bar{X}_2^{[n]}(\bar{\nu}^{[n]})\}
\end{eqnarray}
Since we will need the average (over $G^{[n]}$) of the cardinality of an aligned image set, E$|S_{\bar{\nu^{[n]}}}(G^{[n]})|$, it is worthwhile to point out that the cardinality $|S_{\bar{\nu^{[n]}}}(G^{[n]})|$ as a function of $G^{[n]}$, is a bounded simple function, and therefore measurable. It is bounded because its values are restricted to natural numbers not greater than $(1+\lceil \sqrt{P}\rceil)^{2n}$. To see that it is a simple function, note that $|S_{\bar{\nu}^{[n]}}(G^{[n]})|$ is continuous on irrational numbers. Therefore, the set $\{G^{[n]}: |S_{\bar{\nu}^{[n]}}(G^{[n]})|=m\}, \forall m\in \{1,\cdots,(1+\lceil \sqrt{P}\rceil)^{2n}\}$ is the union of two measurable sets. The first is an intersection of an open set and the set of irrational numbers, so it is measurable. The second is a subset of rational numbers which generally has zero measure. So, $\{G^{[n]}: |S_{\bar{\nu}^{[n]}}(G^{[n]})|=m\}$ is a measurable set.
\item{\bf Bounding Difference of Entropies, $\hat{\mathcal{D}}_\Delta$, in Terms of Size of Aligned Image Sets}\\
\begin{eqnarray}
H(\bar{X}_1^{[n]}|G^{[n]})&=& H(\bar{X}_1^{[n]}, S_{\bar{X}_1^{[n]}}(G^{[n]})|G^{[n]})\\
&=&H(S_{\bar{X}_1^{[n]}}(G^{[n]})|G^{[n]})+H(\bar{X}_1^{[n]}| S_{\bar{X}_1^{[n]}}(G^{[n]}),G^{[n]})\\
&=&H(\lfloor G^{[n]}\bar{X}_1^{[n]}\rfloor+\bar{X}_2^{[n]}(\bar{X}_1^{[n]})|G^{[n]})+H(\bar{X}_1^{[n]}| S_{\bar{X}_1^{[n]}}(G^{[n]}),G^{[n]})\\
&\leq&H(\lfloor G^{[n]}\bar{X}_1^{[n]}\rfloor+\bar{X}_2^{[n]}(\bar{X}_1^{[n]})|G^{[n]})+\mbox{E}\left[\log(|S_{\bar{X}_1^{[n]}}(G^{[n]})|)\right]\label{eq:uniform}\\
&\leq&H(\lfloor G^{[n]}\bar{X}_1^{[n]}\rfloor+\bar{X}_2^{[n]}(\bar{X}_1^{[n]})|G^{[n]})+\log\left(\mbox{E}\left[|S_{\bar{X}_1^{[n]}}(G^{[n]})|\right]\right)\label{eq:jensensn}
\end{eqnarray}
where (\ref{eq:uniform}) follows because uniform distribution maximizes entropy, and (\ref{eq:jensensn}) follows from Jensen's inequality. Rearranging terms, we note that
\begin{eqnarray}
\hat{\mathcal{D}}_\Delta&\leq&\limsup_{P\rightarrow\infty}\limsup_{n\rightarrow\infty}\max_{\substack{P(\bar{X}_1^{[n]}), \bar{X}_2^{[n]}(\bar{X}_1^{[n]})\\ (\bar{X}_1^{[n]},\bar{X}_2^{[n]})\in\bar{\mathcal{X}}^{[n]}}}\frac{\log\left(\mbox{E}\left[|S_{\bar{X}_1^{[n]}}(G^{[n]})|\right]\right)}{\frac{n}{2}\log(P)}\label{eq:DS}
\end{eqnarray}
\item{\bf Bounding the Probability of Image Alignment}\\
Given two codewords $\bar{x}_1^{[n]}$ and $\bar{\nu}^{[n]}$, let us bound the probability that their images align at user 2. Note that for $\bar{x}_1^{[n]}\in S_{\bar{\nu}^{[n]}}(G^{[n]})$ we must have
\begin{eqnarray}
\lfloor G^{[n]}\bar{x}_1^{[n]}\rfloor-\lfloor G^{[n]}\bar{\nu}^{[n]}\rfloor&=&\bar{X}_2^{[n]}(\bar{\nu}^{[n]})-\bar{X}_2^{[n]}(\bar{x}_1^{[n]})\\
\Rightarrow G^{[n]}(\bar{x}_1^{[n]}-\bar{\nu}^{[n]})&\in&\bar{X}_2^{[n]}(\bar{x}_1^{[n]})-\bar{X}_2^{[n]}(\bar{\nu}^{[n]})+\Delta_{(-1,1)}^{[n]}
\end{eqnarray}
where $\Delta_{(-1,1)}(t)\in(-1,1), \forall t\in[1:n]$. Thus, for all $t\in[1:n]$ such that $\bar{x}_1(t)\neq\bar{\nu}(t)$, the value of $G(t)$ must lie within an interval of length no more than $\frac{2}{|\bar{x}_1(t)-\bar{\nu}(t)|}$. Since the maximum value of the joint probability density function of $\{G(t): \mbox{ such that } \bar{x}_1(t)\neq\bar{\nu}(t), t\in[1:n]\}$ is bounded by $f_{\max}^{\sum_{t=1}^n 1(\bar{x}_1(t)\neq\bar{\nu}(t))}\leq f_{\max}^{[n]}$, we can bound the probability that the images of two codewords align as follows.
\begin{eqnarray}
\mathbb{P}(\bar{x}_1^{[n]}\in S_{\bar{\nu}^{[n]}}(G^{[n]}))&\leq&f_{\max}^{n}\prod_{t:\bar{x}_1(t)\neq\bar{\nu}(t)}\frac{2}{\left|\bar{x}_1(t)-\bar{\nu}(t)\right|}
\end{eqnarray}
\item{\bf Bounding the Average Size of Aligned Image Sets}\\
\begin{eqnarray}
\mbox{E}\left[\left|S_{\bar{\nu}^{[n]}}(G^{[n]})\right|\right]&=&\sum_{\bar{x}_1^{[n]}\in\{\bar{X}_1^{[n]}\}}\mathbb{P}\left(\bar{x}_1^{[n]}\in S_{\bar{\nu}^{[n]}}(G^{[n]}) \right)\\
&=&1+\sum_{\substack{\bar{x}_1^{[n]}\in\{\bar{X}_1^{[n]}\}\\ \bar{x}_1^{[n]}\neq \bar{\nu}^{[n]}}}\mathbb{P}\left(\bar{x}_1^{[n]}\in S_{\bar{\nu}^{[n]}}(G^{[n]}) \right)\\
&\leq&1+(2f_{\max})^{n}\sum_{\substack{\bar{x}_1^{[n]}\in\{\bar{X}_1^{[n]}\}\\ \bar{x}_1^{[n]}\neq \bar{\nu}^{[n]}}}\prod_{t:\bar{x}_1(t)\neq\bar{\nu}(t)}\frac{1}{\left|\bar{x}_1(t)-\bar{\nu}(t)\right|}\\
&\leq&1+(2f_{\max})^{n}\prod_{t=1}^n\left(1+\sum_{\Delta_{\bar{x}}=1}^{\lceil\sqrt{P}\rceil}\frac{2}{\Delta_{\bar{x}}}\right)\\
&\leq&1+(2f_{\max})^{n}\prod_{t=1}^n\left(\log(\sqrt{P})+o(\log(P))\right)
\end{eqnarray}
Since this is true for all $\bar{\nu}^{[n]}\in\{\bar{X}_1^{[n]}\}$
\begin{eqnarray}
\mbox{E}\left[\left|S_{\bar{X}_1^{[n]}}(G^{[n]})\right|\right]&\leq&1+(2f_{\max})^{n}\left(\log(\sqrt{P})+o(\log(P))\right)^{n}\label{eq:avesizen}
\end{eqnarray}
\item{\bf Combining the Bounds to Complete the Proof}\\
Combining (\ref{eq:DS}) and (\ref{eq:avesizen}) we have
\begin{eqnarray}
\hat{\mathcal{D}}_\Delta&\leq&\limsup_{P\rightarrow\infty}\limsup_{n\rightarrow\infty}\frac{\log\left(1+(2f_{\max})^{n}\left(\log(\sqrt{P})+o(\log(P))\right)^{n} \right)}{\frac{n}{2}\log(P)}\\
&=&\limsup_{P\rightarrow\infty}\frac{\log(f_{\max})}{\frac{1}{2}\log(P)}+\limsup_{P\rightarrow\infty}\frac{\log(\log(P))}{\frac{1}{2}\log(P)}\\
&\leq&\alpha \label{eq:alphan}
\end{eqnarray}
where (\ref{eq:alphan}) follows because $f_{\max}=O(P^{\frac{\alpha}{2}})$.
Finally combining (\ref{eq:alphan}) with (\ref{eq:Deltan}) and (\ref{eq:Deltahatn}) we have the desired outer bound
\begin{eqnarray}
\mathcal{D}_\Sigma&\leq&1+\alpha
\end{eqnarray}
\end{enumerate}
\hfill$\Box$
\section{Proof of Theorem \ref{theorem:main} for $K$ Users}\label{sec:proofk}
The generalization of the proof to the $K$ user setting is, for the most part, straightforward based on the 2 user case studied earlier. To avoid repetition our presentation will only briefly summarize the aspects that follow directly and use detailed exposition for only those aspects that require special attention. We divide the proof into a similar set of steps for ease of reference with the 2 user case.
\begin{enumerate}
\item {\bf Deterministic Channel Model}\\
As in the 2 user case, the deterministic channel model is described as:
\begin{eqnarray}
\bar{Y}_k&=&\sum_{i=1}^{k-1}\lfloor G_{ki}(t)\bar{X}_i(t)\rfloor +\bar{X}_k(t)
\end{eqnarray}
where the integer inputs satisfy the following per-symbol power constraint
\begin{eqnarray}
\bar{X}_k(t)&\in&\{0,1,\cdots, \lceil\sqrt{P}\rceil\}, ~~\forall k\in[1:K]
\end{eqnarray}
As before, let us define $\bar{\mathcal{X}}^{[n]}$ as the set of codewords that satisfy the power constraint. We have the following bound.
\begin{lemma}\label{lemma:kdet}
The DoF of the canonical model are bounded above by the DoF of the deterministic model.
\end{lemma}
We omit the proof of Lemma \ref{lemma:kdet} since it is a straightforward extension of the 2 user proof which was already presented in much detail.
\item {\bf Difference of Entropy Terms}\\
For the $k^{th}$ user we bound the rate as
\begin{eqnarray}
nR_k&\leq&I(W_k; \bar{Y}_k^{[n]}|G^n,W_{k+1}, W_{k+2}, \cdots, W_K)+o(n)\\
&\leq& H(\bar{Y}_k^{[n]}|G^n,W_{k+1}, \cdots, W_K)-H(\bar{Y}_k^{[n]}|G^n,W_k, W_{k+1}, \cdots, W_K)+o(n)
\end{eqnarray}
where $G^n$ includes all channel realizations. Adding the rate bounds we obtain
\begin{eqnarray}
n\sum_{k=1}^KR_k&\leq&\frac{n}{2}\log(P)+\sum_{k=2}^K\left(H(\bar{Y}_{k-1}^{[n]}|G^n,W_{k},\cdots, W_K)-H(\bar{Y}_k^{[n]}|G^n,W_k, \cdots, W_K)\right)\nonumber\\
&&+n~o(\log(P))+o(n)
\end{eqnarray}
\begin{eqnarray}
&\leq&\frac{n}{2}\log(P)++n~o(\log(P))+o(n)\nonumber\\
&&+\sum_{k=2}^K\left[\max_{{w}_i\in\{W_i\},i\in[k:K]}\left(H(\bar{Y}_{k-1}^{[n]}|G^n,W_i={w}_i, \forall i\in[k:K])-H(\bar{Y}_k^{[n]}|G^n,W_i={w}_i, \forall i\in[k:K])\right)\right]\nonumber\\
\end{eqnarray}
In DoF terms,
\begin{eqnarray}
\bar{\mathcal{D}}_\Sigma &\leq&1 + \sum_{k\in[2:K]}\bar{\mathcal{D}}_{\Delta,k}\label{eq:final1}
\end{eqnarray}
So we need to bound each of the following difference of entropy terms, $\forall k\in[2:K]$
\begin{eqnarray}
\bar{\mathcal{D}}_{\Delta,k}&\triangleq&\limsup_{P\rightarrow\infty}\limsup_{n\rightarrow\infty}\max_{\substack{\mathbb{P}(\bar{X}_1^{[n]},\cdots, \bar{X}_K^{[n]})\\ (\bar{X}_1^{[n]},\cdots,\bar{X}_K^{[n]})\in\bar{\mathcal{X}}^{[n]}}}\frac{H(\bar{Y}_{k-1}^{[n]}|G^{[n]})-H( \bar{Y}_k^{[n]}|G^{[n]})}{\frac{n}{2}\log(P)} \label{eq:detkey}
\end{eqnarray}
We will bound these terms one at a time. The remainder of the proof will show that $\bar{\mathcal{D}}_{\Delta,k}\leq \alpha_k$.
\item{\bf Functional Dependence $\bar{Y}_k^{[n]}(\bar{Y}_{k-1}^{[n]}, G^{[n]})$}\\
For a given channel realization for user $k-1$, $G_{k-1}^{[n]}$, there are multiple vectors $(\bar{X}_1^{[n]}, \bar{X}_2^{[n]},\cdots,\bar{X}_{k}^{[n]})$ that cast the same image in $\bar{Y}_{k-1}^{[n]}$. Thus, given the channel for user $k-1$, the mapping from $\bar{Y}_{k-1}^{[n]}$ to one of these vectors $(\bar{X}_1^{[n]}, \bar{X}_2^{[n]},\cdots,\bar{X}_{k}^{[n]})$ is random. Let us denote it by $\mathcal{L}$, i.e.,
\begin{eqnarray}
(\bar{X}_1^{[n]}, \bar{X}_2^{[n]},\cdots,\bar{X}_{k}^{[n]})&=&\mathcal{L}(\bar{Y}_{k-1}^{[n]}, G_{k-1}^{[n]})
\end{eqnarray}
Now note that
\begin{eqnarray}
H(\bar{Y}_k^{[n]}|G^n)&\geq&H(\bar{Y}_k^{[n]}|G^n,\mathcal{L})\\
&\geq&\min_{{L}\in\{\mathcal{L}\}}H(\bar{Y}_k^{[n]}|G^n,\mathcal{L}={L})
\end{eqnarray}
Let a minimizing mapping be ${L}_o$. Fix this as the deterministic mapping,
\begin{eqnarray}
(\bar{X}_1^{[n]}, \bar{X}_2^{[n]},\cdots,\bar{X}_{k}^{[n]})&=&L_o(\bar{Y}_{k-1}^{[n]}, G_{k-1}^{[n]})
\end{eqnarray}
This implicitly allows the transmitter to have full knowledge of the channel vector of user $k-1$. We note that the choice of mapping does not affect the positive entropy term $H(\bar{Y}_{k-1}^{[n]}|G^{[n]})$ but it minimizes $H(\bar{Y}_k^{[n]}|G^{[n]})$, so that we can bound $\bar{\mathcal{D}}_{\Delta,k}$ as follows.
\begin{eqnarray}
\bar{\mathcal{D}}_{\Delta,k}&\leq&\hat{\mathcal{D}}_{\Delta,k}\triangleq\limsup_{P\rightarrow\infty}\limsup_{n\rightarrow\infty}\max_{\substack{\mathbb{P}(\bar{Y}_{k-1}^{[n]}|G^{[n]}), \bar{Y}_k^{[n]}(\bar{Y}_{k-1}^{[n]}, G^{[n]})\\ (\bar{X}_1^{[n]},\cdots,\bar{X}_K^{[n]})\in\bar{\mathcal{X}}^{[n]}}}\frac{H(\bar{Y}_{k-1}^{[n]}|G^{[n]})-H(\bar{Y}_{k}^{[n]}|G^{[n]})}{\frac{n}{2}\log(P)}\nonumber\\
&&\label{eq:final2}
\end{eqnarray}
Henceforth, note that $\bar{Y}_{k}^{[n]}$ is a function of $\bar{Y}_{k-1}^{[n]}, G^{[n]}$.
\item {\bf Define Aligned Image Sets}\\
For channel realization $G^{[n]}$, define the aligned image set $S_{\bar{Y}_{k-1}^{[n]}}(G^{[n]})$ as the set of all $\bar{Y}_{k-1}^{[n]}$ that have the same image in $\bar{Y}_{k}^{[n]}$, i.e.,
\begin{eqnarray}
S_{\bar{\nu}^{[n]}}(G^{[n]})&\triangleq&\{\bar{y}_{k-1}^{[n]}\in\{\bar{Y}_{k-1}^{[n]}\}: \bar{Y}_k^{[n]}(\nu^{[n]}, G^{[n]})=\bar{Y}_k^{[n]}(\bar{y}_{k-1}^{[n]}, G^{[n]})\}
\end{eqnarray}
\item{\bf Bounding Difference of Entropies in Terms of Size of Aligned Image Sets}
\begin{eqnarray}
H(\bar{Y}_{k-1}^{[n]}|G^{[n]})&=&H(\bar{Y}_{k-1}^{[n]}, \bar{Y}_{k}^{[n]}|G^{[n]})\\
&=&H(\bar{Y}_{k}^{[n]}|G^{[n]})+H(\bar{Y}_{k-1}^{[n]}|{\bar{Y}_{k}^{[n]}},G^{[n]})\\
&=&H(\bar{Y}_{k}^{[n]}|G^{[n]})+H(S_{\bar{Y}_{k-1}^{[n]}}(G^{[n]})|G^{[n]})\\
&\leq&H(\bar{Y}_{k}^{[n]}|G^{[n]})+\mbox{E}\left[\log\left|S_{\bar{Y}_{k-1}^{[n]}}(G^{[n]})\right|\right]\\
&\leq&H(\bar{Y}_{k}^{[n]}|G^{[n]})+\log\mbox{E}\left[\left|S_{\bar{Y}_{k-1}^{[n]}}(G^{[n]})\right|\right]\label{eq:jensensk}
\end{eqnarray}
where we used Jensen's inequality in (\ref{eq:jensensk}). Rearranging terms, we note that
\begin{eqnarray}
\hat{\mathcal{D}}_{\Delta,k}&\leq&\limsup_{P\rightarrow\infty}\limsup_{n\rightarrow\infty}\max_{\substack{\mathbb{P}(\bar{Y}_{k-1}^{[n]}, G^{[n]}), \bar{Y}_k^{[n]}(\bar{Y}_{k-1}^{[n]},G^{[n]})\\ (\bar{X}_1^{[n]},\cdots,\bar{X}_K^{[n]})\in\bar{\mathcal{X}}^{[n]}}}\frac{\log\left(\mbox{E}\left[\left|S_{\bar{Y}_{k-1}^{[n]}}(G^{[n]})\right|\right]\right)}{\frac{n}{2}\log(P)}\label{eq:DSK}
\end{eqnarray}
\item{\bf Bounding the Probability of Image Alignment}\\
Given the channel, $G^{[n]}_{k-1}$, of user $k-1$ and two realizations of $\bar{Y}_{k-1}^{[n]}$, say $\bar{y}^{[n]}$ and $\bar{y'}^{[n]}$, which map to $\bar{X}_j^{[n]}(\bar{y}^{[n]}, G_{k-1}^{[n]})=\bar{x}_j^{[n]}$ and $\bar{X}_j^{[n]}(\bar{y'}^{[n]},G_{k-1}^{[n]})=\bar{x'}_j^{[n]}$, $\forall j\in[1:k]$, let us bound the probability that they produce the same image $\bar{Y}_k^{[n]}$. For notational compactness let us define $G_{kk}(t)=1, \forall k\in[1:K], \forall t\in[1:n]$. Note that for $\bar{y'}^{[n]}\in S_{\bar{y}^{[n]}}(G^{[n]})$ we must have, $\forall t\in[1:n]$
\begin{eqnarray}
\sum_{j\in[1:k]}\lfloor G_{kj}(t)\bar{x'}_j(t)\rfloor &=&\sum_{j\in[1:k]}\lfloor G_{kj}(t)\bar{x}_j(t)\rfloor \\
\Rightarrow \lfloor G_{kj^*(t)}(t)\bar{x'}_{j^*(t)}(t)\rfloor-\lfloor G_{kj^*(t)}(t)\bar{x}_{j^*(t)}(t)\rfloor &=&\sum_{\substack{j\in[k:k],j\neq j^*(t)}}\left(\lfloor G_{kj}(t)\bar{x}_j(t)\rfloor -\lfloor G_{kj}(t)\bar{x'}_j(t)\rfloor\right)\\
\Rightarrow G_{kj^*(t)}(t)\left(\bar{x'}_{j^*(t)}(t)-\bar{x}_{j^*(t)}(t)\right) &\in&\sum_{j\in[1:k],j\neq j^*(t)}\left(\lfloor G_{kj}(t)\bar{x}_j(t)\rfloor -\lfloor G_{kj}(t)\bar{x'}_j(t)\rfloor\right)+\Delta_{(-1,1)}\nonumber\\
\end{eqnarray}
where $\Delta_{(-1,1)}\in(-1,1)$, and we define
\begin{eqnarray}
j^*(t)&\triangleq& \arg \max_{j\in[1:k-1]}|\bar{x'}_{j}(t)-\bar{x}_{j}(t)|
\end{eqnarray}
Thus, for all $t\in[1:n]$ such that $\bar{x'}_{j^*(t)}(t)\neq\bar{x}_{j^*(t)}(t)$, the value of $G_{kj^*(t)}(t)$ must lie within an interval of length no more than $\frac{2}{\left| \bar{x'}_{j^*}(t)-\bar{x}_{j^*}(t)\right|}$. Therefore, the probability that the images due to $\bar{y}^{[n]}$ and $\bar{y'}^{[n]}$ align at user $k$, is bounded as follows.
\begin{eqnarray}
\mathbb{P}\left(\bar{y'}^{[n]}\in S_{\bar{y}^{[n]}}(G^{[n]})\right)&\leq&f_{\max,k}^n\prod_{t:\bar{x'}_{j^*(t)}(t)\neq\bar{x}_{j^*(t)}(t)}\frac{2}{\left| \bar{x'}_{j^*(t)}(t)-\bar{x}_{j^*(t)}(t)\right|}
\end{eqnarray}
It will be useful to express the bound in terms of $\bar{y'}(t), \bar{y}(t)$. To this end, let us proceed as follows.
\begin{eqnarray}
\bar{y'}(t)-\bar{y}(t)&=&\sum_{j=1}^{k-1}\left(\lfloor G_{k-1,j}(t)\bar{x'}_j(t)\rfloor-\lfloor G_{k-1,j}(t)\bar{x}_j(t)\rfloor\right)\\
&\in&\sum_{j=1}^{k-1}\left(\lfloor G_{k-1,j}(t)\left(\bar{x'}_j(t)- \bar{x}_j(t)\right)\rfloor\right) + (-K,K)\\
|\bar{y'}(t)-\bar{y}(t)|&\leq&|\bar{x'}_{j^*(t)}(t)- \bar{x}_{j^*(t)}(t)|\sum_{j=1}^{k-1}| G_{k-1,j}(t)| + K\\
\Rightarrow \frac{1}{|\bar{x'}_{j^*(t)}(t)- \bar{x}_{j^*(t)}(t)|}&\leq&\frac{\sum_{j=1}^{k-1}| G_{k-1,j}(t)|}{|\bar{y'}(t)-\bar{y}(t)|-K}
\end{eqnarray}
whenever $|\bar{y'}(t)-\bar{y}(t)|>K$. Therefore,
\begin{eqnarray}
\mathbb{P}\left(\bar{y'}^{[n]}\in S_{\bar{y}^{[n]}}(G^{[n]})\right)&\leq&\bar{g}^n(f_{\max,k})^n\prod_{t:|\bar{y'}(t)-\bar{y}(t)|>K}\frac{1}{|\bar{y'}(t)-\bar{y}(t)|-K}
\end{eqnarray}
where
\begin{eqnarray}
\bar{g}^n&\triangleq&\max\left(1,\prod_{t:\bar{x'}_{j^*(t)}(t)\neq\bar{x}_{j^*(t)}(t)}2\sum_{j=1}^{k-1}| G_{k-1,j}(t)| \right)
\end{eqnarray}
\item{\bf Bounding the Average Size of Aligned Image Sets}
\begin{eqnarray}
\mbox{E}\left[S_{\bar{y}}^{[n]}(G^n)\right]&=&\sum_{\bar{y'}^{[n]}\in\{{\bar{Y}_{k-1}^{[n]}}\}}\mathbb{P}\left(\bar{y'}^{[n]}\in S_{\bar{y}^{[n]}}(G^n)\right)\\
&\leq&\bar{g}^n(f_{\max,k})^n\prod_{t=1}^n\left(\sum_{\bar{y'}(t): |\bar{y'}(t)-\bar{y}(t)|\leq K}1+\sum_{\bar{y'}(t): K< |\bar{y'}(t)-\bar{y}(t)|\leq Q_y(t)}\frac{1}{|\bar{y'}(t)-\bar{y}(t)|-K}\right)\nonumber\\
&&\\
&\leq&\bar{g}^n(f_{\max,k})^n\left(\log(\sqrt{P})+o(\log(P))\right)^n\label{eq:aveSK}
\end{eqnarray}
where $Q_y(t)\leq\lceil\sqrt{P}\rceil\sum_{j\in[1:k-1]}(|G_{k-1,j}(t)|+K)$.
\item{\bf Combining the Bounds to Complete the Proof}\\
Combining (\ref{eq:aveSK}) and (\ref{eq:DSK}) we have
\begin{eqnarray}
\hat{\mathcal{D}}_{\Delta,k}&\leq&\limsup_{P\rightarrow\infty}\limsup_{n\rightarrow\infty}\frac{\log\left((\bar{g}f_{\max,k})^n(\log(\sqrt{P})+o(\log(P)))^n\right)}{\frac{n}{2}\log(P)}\\
&\leq&\alpha_{k}\label{eq:finalfinal}
\end{eqnarray}
Finally, combining (\ref{eq:final1}), (\ref{eq:final2}) and (\ref{eq:finalfinal}) we have the result,
\begin{eqnarray}
\bar{\mathcal{D}}_\Sigma &\leq &1+\alpha_2+\cdots+\alpha_K
\end{eqnarray}
\end{enumerate}
\section{Discussion}
Since CSIT is almost never available with infinite precision, the collapse of DoF under finite precision channel uncertainty is a sobering result that stands in stark contrast against the tremendous DoF gains shown to be possible with perfect channel knowledge \cite{Cadambe_Jafar_int, Cadambe_Jafar_X}. However, as evident from the conjecture of Lapidoth, Shamai and Wigger, the pessimistic outcome is not unexpected. In terms of practical implications, just like the extremely positive DoF results, the extremely negative DoF results should be taken with a grain of salt. The collapse of DoF under finite precision CSIT is very much due to the asymptotic nature of the DoF metric, and may not be directly representative of finite SNR scenarios which are of primary concern in practice. From a technical perspective, the new outer bound technique offers hope for new insights through the studies of more general forms of CSIT, such as finite precision versions of delayed \cite{Maddah_Tse}, mixed \cite{Sheng_Kobayashi_Gesbert_Yi,Gou_Jafar}, topological \cite{Jafar_TIM}, blind \cite{Jafar_corr} and alternating \cite{Tandon_Jafar_Shamai_Poor} CSIT.
\bigskip
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,498
|
\section{Introduction}
Understanding how the system of quarks and gluons produced in
ultrarelativistic heavy ion collisions evolves towards thermal
equilibrium is one of the central questions in our theoretical
understanding of nuclear collisions. On the one hand, there exists a
strong experimental evidence for equilibration of the quark-gluon
system produced in a heavy ion collision at RHIC. The evidence is
based on the success of hydrodynamic models of the collisions
\cite{bj,EKR,hydro1,hydro2,HN}, indicating a collective behavior of
the quark-gluon system, and on the discovery of jet quenching
\cite{dAtaphen,dAtaphob,dAtastar,brahms,aaphenix,aaphobos,aastar,Bj,EL,BDMPSfull,EL2,Zak,SW},
which demonstrated the presence of strong final state
interactions. However, hydrodynamic models of the evolution of the
quark gluon system only work well, especially for the elliptic flow
$v_2$ \cite{Ollie}, if equilibration occurs in fact at a {\sl very}
early time \cite{hydro1,hydro2}, $t \stackeven{<}{\sim} 0.5 \,{\rm fm}/c$, a time
whose smallness is difficult to reconcile with current dynamical
pictures of equilibration \cite{BMSS,Mueller_eq,SS,BV,DG,MG,Wongc}.
On the other hand, a complete theoretical understanding of
thermalization is still lacking. The success of saturation/Color Glass
approach \cite{glr,mq,bm,mv,k1,jkmw,dip,bk,jimwlk,GM1} in describing
particle multiplicities \cite{KN} in heavy ion collisions and particle
spectra in deuteron--gold collisions
\cite{brahms-1,brahms-2,phenix,phobos,star,klm,kkt1,aaksw,bkw,jmnv,km,dmc,kt,braun}
(see also \cite{kst,ag}) appears to indicate that saturation/Color
Glass formalism is valid for the initial stages of heavy ion collisions
at RHIC. Our understanding of the very early pre-equilibration stages
of the collision and their description in terms of classical gluon
fields has significantly advanced in the recent years
\cite{KMW,KR,GM,KV,K2,KNV,Lappi,MS}. Based on this saturation initial conditions,
Baier, Mueller, Schiff and Son proposed the so-called ``bottom-up''
thermalization scenario \cite{BMSS} in which multiple $2 \rightarrow
2$, $2 \rightarrow 3$ and $3 \rightarrow 2$ rescattering processes,
the importance of which was originally underlined in \cite{Wong},
would drive the system to thermal equilibration over the time scales
of the order of $\tau_0 \sim 1/\alpha_s^{13/5} Q_s$. While the estimates in
\cite{BMSS} were mostly parametric, the numerical value of this thermalization
time appears to be much larger than $0.5$~fm needed by hydrodynamic
simulations \cite{hydro1,hydro2}.
More recently it was argued by Arnold, Lenaghan and Moore that the
``bottom-up'' thermalization scenario could be susceptible to plasma
instabilities \cite{ALM,AL}, which were advocated previously in
\cite{Uli,Mrow,RM,SR,MMR}. Such instabilities might help facilitate the
equilibration process making the thermalization time shorter than
predicted by the ``bottom-up'' scenario. However, in \cite{AL} Arnold
and Lenaghan proved a lower bound on the thermalization time, which
happened to be surprisingly close to the ``bottom-up'' estimate. In
\cite{ALMY} it has been suggested that, while complete thermalization
may not happen until later times, an isotropization of the produced
particle distribution in momentum space may happen much faster,
leading to generation of longitudinal pressure needed for hydrodynamic
description to work.
Here we take a different approach to the problem of
thermalization. Thermalization could be thought of as a transition
between the initial conditions, which are characterized by the energy
density scaling like $\epsilon \sim 1/\tau$, and the Bjorken
hydrodynamics, which, in case of the ideal gas equation of state has
$\epsilon \sim 1/\tau^{4/3}$ \cite{bj}. (Of course at realistic
temperatures achieved in heavy ion collisions the power of $4/3$ may
become somewhat smaller: however, it is always greater than $1$ for
hydrodynamic expansion.) Therefore it appears that corrections to the
saturation/Color Glass initial conditions
\cite{KMW,KR,GM,KV,K2,KNV} would contribute towards modifying
the $\epsilon \sim 1/\tau$ scaling to some higher power. Thus one
should be interested in Feynman diagrams which would bring in
$\tau$-dependent corrections to $\epsilon \sim 1/\tau$ scaling of the
(classical) gluon fields in the initial stages of the
collisions. Unfortunately, after examining a number of diagrams, we
noticed that while many of them introduce $\tau$-dependent corrections
to the initial conditions, such corrections are subleading and small
at large $\tau$ and do not modify $\epsilon \sim 1/\tau$ scaling at
late times. After reaching this conclusion we have constructed a
general argument proving that $\epsilon \sim 1/\tau$ scaling always
dominates at late times, both for classical fields and quantum
corrections, which we are presenting here.
The paper is structured in the following way. We begin in Section 2 by
calculating the energy density of a lowest-order non-trivial classical
gluon field from \cite{KR}. As expected the energy density of the
classical field scales as $\epsilon \sim 1/\tau$. We then continue in
Section 3 by considering the most general case of boost-invariant
gluon production, which is, indeed, not limited to classical
fields. We argue that $\epsilon \sim 1/\tau$ scaling persists to all
orders in the coupling constant $\alpha_s$, as shown in \eq{ed}. The
argument is based on a simple observation (see \eq{I1}) that
$\tau$-dependent corrections to the classical gluon field may only
come in through powers of gluon virtuality $k^2$ in momentum space
with each power of $k^2$ giving rise to a power of $1/\tau$. In order
for the on-mass shell amplitude (at $k^2 =0$) to be non-singular only
positive powers of $k^2$ are allowed: hence, the corrections come in
only as inverse extra powers of $\tau$ and are negligible at late
times. In Section 3 we generalize our results to rapidity-dependent
distributions. The $\epsilon \sim 1/\tau$ scaling does not get
modified by rapidity-dependent corrections either (see
\eq{ede}). Rapidity-dependent corrections come in as, for example, powers of $k_+$,
which is one of light cone components of the gluon's
momentum. However, as could be seen from, say, \eq{J7}, powers of
$k_+$ do not modify the $\tau$-dependence of energy density. In
Section 4 we argue that $\epsilon \sim 1/\tau$ scaling persists even
when massless quarks are included in the problem. Therefore it appears
that perturbative thermalization can not happen in heavy ion
collisions. We conclude in Section 5 by arguing that if perturbative
thermalization is impossible, than the non-perturbative QCD effects
must be responsible for the formation of quark-gluon plasma (QGP) at
RHIC \cite{DK}. We list the non-perturbative effects which we believe may be
responsible for thermalization.
\section{Energy-Momentum Tensor of Classical Gluon Field}
We start by calculating the energy-momentum tensor of the lowest order
gluon field produced in an ultrarelativistic heavy ion collision. This
field has been found analytically in \cite{KMW,KR} and the
corresponding Feynman diagrams are depicted here in
\fig{twonuc}. The cross in \fig{twonuc} denotes the space-time point in which we
measure the gluon field.
\begin{figure}[b]
\begin{center}
\epsfxsize=15cm
\leavevmode
\hbox{\epsffile{twonuc.eps}}
\end{center}
\caption{Lowest order ($\sim g^3$) gluon field produced in nuclear collisions.}
\label{twonuc}
\end{figure}
The gluon field in $\partial_\mu A^\mu = 0$ covariant gauge given by
diagrams in \fig{twonuc} can be written as \cite{KR}
\begin{equation}\label{lofi}
A_{\mu}^{(3) \, a} (x) = - i \int \frac{ d^4 k}{(2 \pi)^4}\, \frac{e^{- i k
\cdot x} }{k^2 + i \epsilon k_0 } \, J_\mu^{(3) \, a} (k),
\end{equation}
with
\begin{equation}\label{loj}
J_\mu^{(3) \, a} (k) \, = \, \sum_{i,j = 1}^{A_1,A_2} \, \int d^2 q \,
\frac{g^3}{ (2 \pi)^2} \, f^{abc}\, (T_i^b) \, (\tilde{T}_j^c)\, e^{i
[k_+ x_{i-} + k_- y_{j+} -
\underline{k} \cdot \underline{y}_j - \underline{q} \cdot
(\underline{x}_i - \underline{y}_j)]} \, \frac{C_\mu (k,
\underline{q})}{\underline{q}^2 (\underline{k} - \underline{q})^2}
\end{equation}
where $C_\mu (k, \underline{q})$ is the Lipatov vertex \cite{bfkl}
\begin{equation}\label{lipa}
C_\mu (k, \underline{q}) \, = \, \left( \frac{
\underline{q}^2 }{k_- + i \epsilon} - k_+ \, , \, - \frac{
(\underline{k} - \underline{q})^2 }{k_+ + i \epsilon} + k_- \, , \,
2\underline{q} - \underline{k}
\right).
\end{equation}
The field of \eq{lofi} is given for a collision of a quark $i$ in one
of the nuclei having transverse coordinate ${\underline x}_i$ and light cone
coordinate $x_{i-}$ with a quark $j$ in the other nucleus having
transverse coordinate ${\underline y}_j$ and the light cone coordinate
$y_{j+}$. The matrices $(T_i^b)$ and $(\tilde{T}_j^c)$ act in the
color spaces of the quarks $i$ and $j$ correspondingly. Indeed we sum
over all quark pairs in \eq{loj}, with valence quarks $i$ being in any
of the $A_1$ nucleons in the first nucleus and valence quarks $j$
being in any one of the $A_2$ nucleons in the second nucleus.
The energy-momentum tensor of a gluon field is given by
\begin{equation}\label{tmng}
T^{\mu\nu} \, = \, - F^{a \, \mu\rho} \, F^{a \, \nu}_{\ \ \ \rho} +
\frac{1}{4} \, g^{\mu\nu} \, (F^a_{\rho\sigma})^2.
\end{equation}
We need to calculate $T^{\mu\nu}$ averaged in the wave functions of
both nuclei \cite{k1,KR}
\begin{equation}\label{tmnga}
\left< T^{\mu\nu} \right> \, = \, \left< - F^{a \, \mu\rho} \,
F^{a \, \nu}_{\ \ \ \rho} +
\frac{1}{4} \, g^{\mu\nu} \, (F^a_{\rho\sigma})^2 \right>,
\end{equation}
where the averaging implies integrating over all possible positions of
quarks in the nucleons and nucleons in the nuclei, and taking traces
(divided by $N_c$) in the color spaces of the quarks \cite{k1,KR}. The
averaging can be represented as
\begin{equation}\label{ave}
\left< \ldots \right> \, = \, \prod_{i,j=1}^{A_1, A_2} \frac{d^2 x_i}{S_{1\perp}}
\frac{d^2 y_j}{S_{2\perp}} \frac{d x_{i-}}{a_-} \frac{d y_{j+}}{a_+} \,
\frac{1}{N_c^2} \, \mbox{Tr}_i [ \mbox{Tr}_j [ \ldots ]]
\end{equation}
where $S_{1\perp}$ and $S_{2\perp}$ are the cross sectional areas of
the two nuclei, which we for simplicity assume to be cylindrical with
the cylinder axis pointing in the beam ($z$-) direction. $a_-$ and
$a_+$ are Lorentz-contracted nucleon sizes in the $-$ and $+$
directions correspondingly, which are very small, making averaging
over $x_{i-}$ and $y_{j+}$ equivalent to just putting $x_{i-} = 0$ and
$y_{j+} =0$ \cite{KR}.
We are interested in calculating $T_{\mu\nu}$ for the gluon field
produced in a central nuclear collisions in the forward light
cone. Since the gluon field of \eq{lofi} is $o(g^3)$, we may only use
it to compute $o(g^6)$ contribution to the energy-momentum tensor, for
which we will need only the Abelian part of the field strength tensor
$F^a_{\mu\nu}$. To compute higher orders in $T_{\mu\nu}$ one would
also need higher orders in $A_\mu^a$. Using the field (\ref{lofi}) in
the Abelian part of \eq{tmnga}, performing the averaging defined in
\eq{ave} and remembering that the multiplicity distribution given by
the diagrams of \fig{twonuc} for gluons with transverse momentum ${\underline
k}$, rapidity $y$, located at impact parameter ${\underline b}$, is
\begin{equation}\label{lomult}
\frac{d N}{d^2 k \, dy \, d^2 b} \, = \, \frac{8 \, \alpha_s^3 \, C_F}{\pi} \,
\frac{A_1 \, A_2}{S_{1\perp} \, S_{2\perp}} \, \frac{1}{{\underline k}^4} \,
\ln \frac{k_T}{\Lambda}
\end{equation}
we obtain after lengthy algebra (and dropping the averaging sign around
$T_{\mu\nu}$)
\begin{eqnarray}\label{tmncl}
&&T_{++} \, = \, \left( \frac{x_+}{\tau} \right)^2 \, \pi \, \int d^2 k \, \frac{d N}{d^2 k \, d \eta \, d^2 b} \ k_T^2 \, \left[ J_1 (k_T \tau) \right]^2 \nonumber ~\\
&&T_{--} \, = \, \left( \frac{x_-}{\tau} \right)^2 \, \pi \, \int d^2 k \, \frac{d N}{d^2 k \, d \eta \, d^2 b} \ k_T^2 \, \left[ J_1 (k_T \tau) \right]^2 \nonumber ~\\
&&T_{+-} \, = \, \frac{x_+ \, x_-}{\tau^2} \, \pi \, \int d^2 k \, \frac{d N}{d^2 k \, d \eta \, d^2 b} \ k_T^2 \, \left[ J_0 (k_T \tau) \right]^2 \nonumber ~\\
&&T_{ij} \, = \, \delta_{ij} \, \frac{\pi}{2} \, \int d^2 k \, \frac{d N}{d^2 k \, d \eta \, d^2 b} \ k_T^2 \, \left[ J_0 (k_T \tau) \right]^2, \ \ \ T_{+i} \, = \, T_{-i} \, = \, 0.
\end{eqnarray}
Here we used $x_\pm = (t \pm z)/\sqrt{2}$, $\tau = \sqrt{t^2 - z^2} =
\sqrt{2 x_+ x_-}$, and $k_T = |{\underline k}|$. We also took advantage
of the fact that the multiplicity distribution (\ref{lomult}) is
rapidity independent and, since $T_{\mu\nu}$ should depend only on
space-time coordinates, replaced momentum space rapidity $y$ with the
space-time rapidity $\eta = (1/2) \ln (x_+ / x_-)$. (At this point
such substitution makes no difference: in Section \ref{arg2} we will
show how this substitution is formally justified in the
rapidity-dependent case.) In arriving at \eq{tmncl} we have used the
integral defined in \eq{I1} and the one given by \eq{J5} in Appendix B
with $\Delta = 0$ along with
\begin{equation}
\int \frac{d^2 q}{{\underline q}^2 ({\underline k} - {\underline q})^2} \, = \, \frac{4 \pi}{{\underline k}^2}
\, \ln \frac{k_T}{\Lambda}
\end{equation}
where $\Lambda$ is some infrared cutoff. \eq{tmncl} is derived in the
leading logarithmic approximation in $\ln k_T/\Lambda$.
Eq. (\ref{tmncl}) gives us the energy-momentum tensor in the forward
light cone of the lowest order gluon field from \fig{twonuc} produced
in a central collision of two identical nuclei. While it is written in
a non-specific form with regards to the order of the coupling constant
$g$, we have proven Eq. (\ref{tmncl}) only at the order $o(g^6)$.
\section{Energy Density in the Boost-Invariant Approximation}
\subsection{Region of Applicability}
Let us first consider the case of high energy heavy ion collisions,
where the total rapidity interval is large enough to allow for eikonal
approximation, but not large enough for quantum BFKL-type corrections
\cite{bfkl} to become important. This is the quasi-classical regime of
McLerran-Venugopalan model \cite{mv,k1,jkmw}. To achieve it one needs
the Bjorken $x$ variable to be small enough such that \cite{ks}
\begin{equation}\label{xmv}
x \, < \, \frac{1}{2 m_N R} \, \sim \, A^{-1/3},
\end{equation}
which is the condition ensuring that coherent eikonal interactions are
possible in the nuclear wave functions. For corresponding rapidities,
$Y = \ln 1/x$, the condition of \eq{xmv} means that
\begin{equation}\label{xmv2}
Y \, > \, \ln A^{1/3}.
\end{equation}
Remembering that McLerran-Venugopalan model corresponds to resummation
of multiple\\ rescatterings parameter $\alpha_s^2 A^{1/3} \sim 1$ we rewrite
\eq{xmv2} as
\begin{equation}\label{xmv3}
Y \, > \, \ln \frac{1}{\alpha_s^2} \, \sim \, \ln \frac{1}{\alpha_s}.
\end{equation}
On the other hand, the boost invariant approximation is broken down by
quantum evolution corrections, which, in the dominant leading
logarithmic approximation, bring in powers of $\alpha_s Y$
\cite{bfkl,dip,bk,jimwlk}. Indeed these corrections are negligible when $\alpha_s Y \stackeven{<}{\sim}
1$, such that
\begin{equation}\label{xmv4}
Y \, < \, \frac{1}{\alpha_s}.
\end{equation}
Eqs. (\ref{xmv3}) and (\ref{xmv4}) define the rapidity interval for
nuclear collisions in which the boost invariant approximation, which
we will consider in this Section, is valid.
\subsection{Most General Boost Invariant form of $T^{\mu\nu}$}
Similar to the Bjorken approach \cite{bj} we will consider a central
collision of two very large nuclei, such that the gluon production is
translationally invariant in the transverse direction. Defining two
four-vectors in terms of light-cone coordinates
\begin{equation}\label{umu}
u_\mu = \left(\frac{x_+}{\tau}, \frac{x_-}{\tau}, {\underline 0} \right)
\end{equation}
and
\begin{equation}\label{vmu}
v_\mu = \left(\frac{x_+}{\tau}, - \frac{x_-}{\tau}, {\underline 0} \right)
\end{equation}
we can write the most general energy-momentum tensor for the system as
\begin{equation}\label{tmn1}
T_{\mu\nu} \, = \, A(\tau) \, u_\mu u_\nu + B(\tau) \, (u_\mu v_\nu + u_\nu
v_\mu) + C(\tau) \, v_\mu v_\nu + D(\tau) \, g_{\mu\nu},
\end{equation}
where in our convention $g_{\mu\nu} = \mbox{diag}
\{ 1,-1,-1,-1 \}$. Here the parameters $A,B,C,D$ in \eq{tmn1} are
functions of $\tau$ only, since the large transverse extent and
azimuthal cylindrical symmetry for central collisions of the nuclei
allow us to neglect the transverse coordinate dependence, and the
assumption of boost invariance makes functions $A,B,C,D$ independent
of space-time rapidity $\eta = (1/2) \ln (x_+/x_-)$.
From \eq{tmn1} we see that
\begin{equation}\label{tpp}
T_{++} \, = \, [A(\tau) + B(\tau) + C(\tau)] \left( \frac{x_+}{\tau}
\right)^2
\end{equation}
and
\begin{equation}\label{tmm}
T_{--} \, = \, [A(\tau) - B(\tau) + C(\tau)] \left( \frac{x_-}{\tau}
\right)^2.
\end{equation}
Due to $+ \leftrightarrow -$ symmetry of the collision of two
identical nuclei it should be possible to obtain $T_{--}$ from
$T_{++}$ after changing all $+$ indices to $-$ indices of all the
relevant four-vectors in it. This condition, when applied to
Eqs. (\ref{tpp}) and (\ref{tmm}), demands that $B(\tau) = 0$.
Rewriting the remaining non-zero functions $A, C, D$ as
\begin{equation}\label{relab}
A(\tau) = \epsilon (\tau) + p (\tau), \ \ \ C(\tau) = p_3 (\tau) - p
(\tau), \ \ \ \mbox{and} \ \ \ D(\tau) = - p (\tau)
\end{equation}
we obtain for the non-zero components of the energy momentum tensor
\begin{eqnarray}\label{tmngen}
&&T_{++} \, = \, [\epsilon (\tau) + p_3 (\tau)] \, \left( \frac{x_+}{\tau}
\right)^2, \nonumber ~\\
&&T_{--} \, = \, [\epsilon (\tau) + p_3 (\tau)] \, \left(
\frac{x_-}{\tau} \right)^2, \nonumber ~\\
&&T_{+-} \, = \, [\epsilon (\tau) - p_3 (\tau)] \,
\frac{x_+ x_-}{\tau^2} = [\epsilon (\tau) - p_3 (\tau)] \, \frac{1}{2}, \nonumber ~\\
&&T_{ij} \, = \, \delta_{ij} \, p (\tau),
\end{eqnarray}
where the indices $i,j = 1,2$ denote the transverse components of the
tensor. \eq{tmngen} gives us the most general boost-invariant
energy-momentum tensor for a collision of two very large nuclei with
the total rapidity interval satisfying conditions (\ref{xmv3}) and
(\ref{xmv4}) allowing for a boost-invariant description of the gluon
production.
At $z=0$ in the center-of-mass frame the energy-momentum tensor from
\eq{tmngen} can be written as
\begin{eqnarray}\label{tmngen0}
T^{\mu\nu} &=&
\left( \matrix{ \epsilon (\tau) & 0 & 0 & 0 \cr
0 & p (\tau) & 0 & 0 \cr
0 & 0 & p (\tau) & 0 \cr
0 & 0 & 0 & p_3 (\tau) \cr} \right)\, .
\end{eqnarray}
Now we can see the physical meaning of the parameterization introduced
in \eq{relab}: $\epsilon$ is the energy density and $p_3$ is the
longitudinal pressure along the beam axis ($z$-direction), which in
principle does not have to be equal to the transverse pressure
$p$. Indeed, for the case of boost-invariant Bjorken hydrodynamics
\cite{bj}, the two pressures are identical, $p_3 (\tau) = p (\tau)$. However,
as one can show, for classical gluon fields generated in heavy ion
collisions \cite{KMW,KR,GM,KV,K2}, the longitudinal pressure component
is zero at sufficiently late times, $p_3 (\tau) = 0$, while $\epsilon
(\tau)= 2 \, p (\tau) \neq 0$ \cite{KNV}.
Applying the conservation of energy-momentum tensor condition
\begin{equation}\label{cons}
\partial_\mu T^{\mu\nu} \, = \, 0
\end{equation}
to the tensor in \eq{tmngen} we obtain
\begin{equation}\label{hydroeq}
\frac{d \epsilon}{d \tau} \, = \, - \frac{\epsilon + p_3}{\tau}
\end{equation}
similar to Bjorken hydrodynamics \cite{bj}.
\eq{hydroeq} shows that if energy density scales with proper time as
$\epsilon \sim 1/\tau$ then the longitudinal pressure is zero, $p_3
=0$. This is indeed the case for classical gluon production in the
initial stages of the heavy ion collisions considered in Section 2
(see also \cite{KNV}). To see this let us use the energy momentum
tensor from \eq{tmncl} in \eq{tmngen} to write
\begin{eqnarray}\label{bes}
&&\epsilon \, = \, \frac{\pi}{2} \, \int d^2 k \, \frac{d N}{d^2 k \, d \eta \, d^2 b} \ k_T^2 \, \left\{ \left[ J_1 (k_T \tau) \right]^2 + \left[ J_0 (k_T \tau) \right]^2 \right\} \nonumber ~\\
&&p_3 \, = \, \frac{\pi}{2} \, \int d^2 k \, \frac{d N}{d^2 k \, d \eta \, d^2 b} \ k_T^2 \, \left\{ \left[ J_1 (k_T \tau) \right]^2 - \left[ J_0 (k_T \tau) \right]^2 \right\} \nonumber ~\\
&&p \, = \, \frac{\pi}{2} \, \int d^2 k \, \frac{d N}{d^2 k \, d \eta \, d^2 b} \ k_T^2 \, \left[ J_0 (k_T \tau) \right]^2.
\end{eqnarray}
Using the large-argument asymptotics of the Bessel functions we write
\begin{equation}\label{loed}
\epsilon \bigg|_{\tau \gg 1/\langle k_T \rangle} \, \approx \, \frac{1}{\tau} \, \int d^2 k \, \frac{d N}{d^2 k \, d \eta \, d^2 b} \ k_T \, = \, \frac{1}{\tau} \, \frac{d E_T}{d \eta \, d^2 b},
\end{equation}
which precisely agrees with the Bjorken energy density estimate
\cite{bj}. Here we assumed that the gluon spectrum is characterized by some
typical transverse momentum $\langle k_T \rangle$, such that large
time asymptotics is defined by $\tau \gg 1/\langle k_T \rangle$. (Strictly speaking
such ``typical'' momentum for lowest order gluon field of \eq{lofi} is
the infrared cutoff $\Lambda$, but it would become the saturation
scale $Q_s \gg
\Lambda$ once multiple rescatterings are included \cite{KV,K2}.)
Similarly, using the large-argument asymptotics of the Bessel
functions one can show that
\begin{equation}\label{p3}
p_3 \bigg|_{\tau \gg 1/\langle k_T \rangle} \, \approx \, 0
\end{equation}
in agreement with \eq{loed} and \eq{hydroeq}. Thus the large time
asymptotics of the energy-momentum tensor of the lowest order
classical gluon field is given by $T_{\mu\nu} = \mbox{diag}\{\epsilon,
p, p, 0\}$ with $\epsilon = 2\, p$ and $\epsilon$ given by
\eq{loed}\footnote{It is interesting to point out that in approaching
the asymptotics of Eqs. (\ref{loed}) and (\ref{p3}) both $\epsilon$
and $p_3$ oscillate, such that $\epsilon /3$ becomes temporarily
comparable to $p_3$ at proper time $\tau \sim 1/Q_s$. While the
mathematical origin of these oscillations is clearly due to the Bessel
functions in \eq{bes}, their physical interpretation (if it exists) is
presently unclear.}. Similar results were obtained in numerical
simulations of the full classical gluon field including all orders in
multiple rescatterings \cite{KNV}.
The onset of thermalization or isotropization of the system
\cite{ALMY} should come with generation of the non-zero longitudinal
pressure $p_3$ comparable to the transverse pressure $p$. In order for
that to happen \eq{hydroeq} necessarily requires the energy density to
start scaling with $\tau$ as $\epsilon \sim 1/\tau^{1 + \Delta}$,
where $\Delta$ is some positive number. In the case of ideal Bjorken
hydrodynamics $\Delta = 1/3$.
Thus the process of thermalization in heavy ion collisions can be
viewed as a transition from the $\epsilon \sim 1/\tau$ scaling,
characteristic of free-streaming classical fields (\ref{loed}), to
$\epsilon \sim 1/\tau^{1 + \Delta}$ scaling. Below we are going to
study whether such transition can result from Feynman diagram
resummation.
\subsection{Can Boost-Invariant Bjorken Hydro Result from Feynman Diagrams?}
\label{arg1}
Let us explore what kinds of energy-momentum tensor may result from
Feynman diagram resummation. We will concentrate only on gluon fields,
and later will generalize our conclusions to include quark fields as
well. We will assume that the initial gluon field is given by the
classical field of McLerran-Venugopalan model
\cite{mv,k1,KMW,KR,GM,KV,K2,KNV}, though our results would not depend much
on this assumption\footnote{There is a common misconception in the
community that in McLerran-Venugopalan model one assumes that $y =
\eta$: while this assumption was made in the original works on the
subject \cite{KMW,mv}, it is actually not necessary, with all the
results of McLerran-Venugopalan model easily derivable without making
any assumptions on correlations between $\eta$ and $y$ (see
\cite{KR}).}. In the saturation scenario, the gluon fields at the
early times with $\tau \sim 1/Q_s$ are strong
\begin{equation}
A^a_\mu \, \sim \, \frac{Q_s}{g}.
\end{equation}
In calculating corresponding field strength tensor $F^a_{\mu\nu}$, one
would require both the Abelian and the non-Abelian parts of
it. However, as classical fields and their energy density (\ref{loed})
decrease with proper time, for $\tau \stackeven{>}{\sim} 1/Q_s$ the Abelian part of
$F^a_{\mu\nu}$ would dominate. This is also true for quantum
corrections to classical fields. Therefore, in the following
discussion we will first restrict ourselves to calculating the Abelian
part of $T_{\mu\nu}$ only, and will later show that inclusion of
non-Abelian parts of $T_{\mu\nu}$ would not change our argument.
The most general gluon field generated through any-order Feynman
diagrams in $\partial_\mu A^\mu = 0$ covariant gauge can be written as
\begin{equation}\label{ffi}
A_{\mu}^{a} (x) = - i \int \frac{ d^4 k}{(2 \pi)^4}\, \frac{e^{- i k
\cdot x} }{k^2 + i \epsilon k_0 } \, J_\mu^a (k),
\end{equation}
where we are using the retarded regularization for the outgoing gluon
propagator $-i/k^2$ to ensure causality. The function $J_\mu^a (k)$
denotes the rest of the diagram (the truncated part). In general
$J_\mu^a (k)$ is an abbreviated notation for $J_\mu^a (k; q_1,
\lambda_1; q_2, \lambda_2; \ldots)$, which depends on momenta $q_i$ of
extra gluons (or quarks) in the final state and on their polarizations
(helicities) $\lambda_i$. In case of the classical gluon field there
are no extra particles in the final state and momenta $q_i$'s do not
enter the expression (\ref{ffi}). (In fact, the possible particles in
the final state for classical fields can be removed by using retarded
regularization of gluon propagators \cite{Ian04}.) Quantum corrections
to the classical gluon field would inevitably bring in extra final
state particles. The resulting ``quantum'' field $A_{\mu}^{a} (x)$ in
\eq{ffi} would also depend on momenta $q_i$: again we suppress this
dependence in the notation. Indeed, once quantum corrections are
included there is no dominant gluon field anymore: in that sense the
gluon field in \eq{ffi} is not really a field, but more like a
scattering amplitude (located on one side of the cut), with one ($k$)
of the many outgoing particle lines ($q_i$'s) being off-mass shell
with its propagator ending at a space-time point $x_\mu$.
Substituting the field from \eq{ffi} into the expression for the
energy-momentum tensor (\ref{tmnga})
\begin{eqnarray*}
\left< T^{\mu\nu} \right> \, = \, \left< - F^{a \, \mu\rho} \,
F^{a \, \nu}_{\ \ \ \rho} +
\frac{1}{4} \, g^{\mu\nu} \, (F^a_{\rho\sigma})^2 \right>,
\end{eqnarray*}
and keeping only the Abelian parts of $F^a_{\mu\nu}$'s we obtain
\begin{eqnarray*}
T_{\mu\nu} \, = \, \int \frac{ d^4 k \, d^4 k'}{(2 \pi)^8}\,
\frac{e^{-i k \cdot x - i k' \cdot x}}{(k^2 + i \epsilon k_0) \, (k'^2
+ i \epsilon k'_0)} \, \bigg< - [k_\mu J^{a \, \rho} (k) - k^\rho
J^a_\mu (k)] \, [k'_\nu J^a_\rho (k') - k'_\rho J^a_\nu (k')]
\end{eqnarray*}
\begin{equation}\label{tmnab}
+ \frac{1}{4} \, g_{\mu\nu} \, [k^\rho J^{a \, \sigma} (k) - k^\sigma
J^{a \, \rho} (k)] \, [k'_\rho J^a_\sigma (k') - k'_\sigma J^a_\rho
(k')]\bigg>,
\end{equation}
where the brackets $\langle \ldots \rangle$ are defined by \eq{ave}
and now also include integration over all momenta $q_i$'s. The gluon
field in \eq{ffi} is generated by the color sources in two colliding
nuclei, which are modeled by valence quarks, just like in
McLerran-Venugopalan model \cite{mv}. Of course, the resulting gluon
field from \eq{ffi} is not necessarily classical, it includes extra
quark and gluon emissions as well as loops, just like any production
diagram with incoming valence quarks of the nuclei providing the
initial condition for the scattering process.
Since performing the transverse averaging over a very large nucleus in
the brackets on the right hand side of \eq{tmnab} puts ${\underline k} = -
{\underline k}'$, which results from transverse translational invariance of
gluon production, we can rewrite it as
\begin{eqnarray*}
T_{\mu\nu} \, = \, \int \frac{ d^4 k \, d^4 k'}{(2 \pi)^8}\,
\frac{e^{-i k \cdot x - i k' \cdot x}}{(k^2 + i \epsilon k_0) \, (k'^2
+ i \epsilon k'_0)} \, \bigg<\bigg< - [k_\mu J^{a \, \rho} (k) - k^\rho
J^a_\mu (k)] \, [k'_\nu J^a_\rho (k') - k'_\rho J^a_\nu (k')]
\end{eqnarray*}
\begin{equation}\label{tmnab1}
+ \frac{1}{4} \, g_{\mu\nu} \, [k^\rho J^{a \, \sigma} (k) - k^\sigma
J^{a \, \rho} (k)] \, [k'_\rho J^a_\sigma (k') - k'_\sigma J^a_\rho
(k')]\bigg>\bigg> \, \frac{(2 \pi)^2}{S_\perp} \, \delta ({\underline k} + {\underline
k}'),
\end{equation}
where the double brackets $\langle\langle \ldots \rangle\rangle$
denote now the color averaging, integration over $q_i$'s, summation
over nucleons in the nuclei and averaging over longitudinal and
remaining transverse coordinates. $S_\perp$ is the cross sectional
area of the nuclei which we assume to be identical.
Let us define the following correlation function
\begin{equation}\label{ddef}
D_{\mu\nu} \, \equiv \, \left<\left< J_\mu (k) \, J_\nu (k')
\right>\right>\bigg|_{{\underline k} = - {\underline k}'}.
\end{equation}
Using the covariant gauge condition $\partial_\mu A^\mu = 0$, which
translates into $k_\mu \, J^\mu (k) \, = \, k'_\mu \, J^\mu (k') \, =
\, 0$, along with the $k_+ \leftrightarrow k_-$
(and $k'_+ \leftrightarrow k'_-$) symmetry of the collision, we can
derive the following relations between different components of
$D_{\mu\nu}$:
\begin{eqnarray}\label{drel}
D_{+i} \, = \, \frac{k_j}{2 \, k_-} \, D_{ji} \hspace*{.5cm} && \hspace*{.5cm} D_{-i} \, = \, \frac{k_j}{2 \, k_+} \, D_{ji} \nonumber \\
D_{i+} \, = \, \frac{k'_j}{2 \, k'_-} \, D_{ij} \hspace*{.5cm} && \hspace*{.5cm} D_{i-} \, = \, \frac{k'_j}{2 \, k'_+} \, D_{ij} \nonumber \\
\frac{D_{++}}{k_+ \, k'_+} \, &=& \, \frac{D_{--}}{k_- \, k'_-},
\end{eqnarray}
where the Latin indices $i, j = 1,2$ indicate the transverse
components of the correlators and two lowercase repeated Latin indices
indicate contraction over that index.
We are interested in calculating the energy density $\epsilon$ given
by (see
\eq{tmngen})
\begin{equation}\label{edens}
\epsilon \, = \, \frac{1}{2} \, \left( \frac{\tau}{x_+} \right)^2 \, T_{++} + T_{+-}.
\end{equation}
Using \eq{tmnga} we write
\begin{equation}\label{tpp1}
T_{++} \, = \, - \left< F^{a \, \rho}_+ \, F^a_{+\rho} \right> \, = \,
\left< F^a_{+i} \, F^a_{+i} \right>
\end{equation}
and
\begin{equation}\label{tpm1}
T_{+-} \, = \, \frac{1}{2} \, \left< F^a_{+-} \, F^a_{+-} \right> +
\frac{1}{4} \, \left< F^a_{ij} \, F^a_{ij} \right>.
\end{equation}
Using \eq{ffi} together with relations from \eq{drel} in
Eqs. (\ref{tpp1}) and (\ref{tpm1}) we obtain
\begin{eqnarray*}
T_{++} \, = \, \int \frac{ d^4 k \, d^4 k'}{(2 \pi)^8}\,
\frac{e^{-i k \cdot x - i k' \cdot x}}{(k^2 + i \epsilon k_0) \, (k'^2
+ i \epsilon k'_0)} \, \left[ k_+ \, k'_+ \, D_{ii} - {\underline k}^2 \,
D_{++} - \frac{1}{2} \, \left( \frac{k'_+}{k_-} + \frac{k_+}{k'_-}
\right) \, k_i \, k_j \, D_{ij} \right]
\end{eqnarray*}
\begin{equation}\label{tpp2}
\times \, \frac{(2 \pi)^2}{S_\perp} \, \delta ({\underline k} + {\underline k}')
\end{equation}
and
\begin{eqnarray*}
T_{+-} \, = \, \int \frac{ d^4 k \, d^4 k'}{(2 \pi)^8}\,
\frac{e^{-i k \cdot x - i k' \cdot x}}{(k^2 + i \epsilon k_0) \, (k'^2
+ i \epsilon k'_0)} \, \left[ - \frac{1}{2} \, {\underline k}^2 \, D_{ii} + 2
k_- \, k'_- \, D_{++} + k_i \, k_j \, D_{ij} \right]
\end{eqnarray*}
\begin{equation}\label{tpm2}
\times \, \frac{(2 \pi)^2}{S_\perp} \, \delta ({\underline k} + {\underline k}').
\end{equation}
Substituting Eqs. (\ref{tpp2}) and (\ref{tpm2}) into \eq{edens} yields
\begin{eqnarray*}
\epsilon \, = \, \int \frac{ d^4 k \, d^4 k'}{(2 \pi)^8}\,
\frac{e^{-i k \cdot x - i k' \cdot x}}{(k^2 + i \epsilon k_0) \, (k'^2
+ i \epsilon k'_0)} \, \left\{ \left[ \frac{1}{2} \, \left(
\frac{\tau}{x_+} \right)^2 \, k_+ \, k'_+ - \frac{1}{2} \, {\underline k}^2
\right] \, D_{ii} \, + \right.
\end{eqnarray*}
\begin{equation}\label{edens11}
+ \left. \left[ 2 \, k_- \, k'_- - \frac{1}{2} \, \left(
\frac{\tau}{x_+} \right)^2 \, {\underline k}^2 \right] \, D_{++} + \left[
- \frac{1}{4} \, \left( \frac{\tau}{x_+} \right)^2 \,
\left( \frac{k'_+}{k_-} + \frac{k_+}{k'_-} \right) \, + \, 1 \right]
\, k_i \, k_j \, D_{ij} \right\} \, \frac{(2 \pi)^2}{S_\perp} \, \delta ({\underline k} + {\underline k}').
\end{equation}
Since the tensor structure of the correlators $D_{\mu\nu}$ from
\eq{ddef} is symmetric under $k \leftrightarrow k'$, and using the
last relation in \eq{drel}, without any loss of generality one can
write
\begin{equation}\label{f2def}
D_{++} \, = \, k_+ \, k'_+ \, f_2 (k^2, k'^2, k \cdot k', k_T),
\end{equation}
where $f_2 (k^2, k'^2, k \cdot k', k_T)$ is some unknown
boost-invariant function, which, due to rapidity independence of the
problem, depends only on $k^2$, $k'^2$, $k \cdot k'$ and on the
magnitude of the transverse momentum $k_T$. In general, dependence of
$f_2$ on $k \cdot k'$ might lead to rapidity dependence: however, as
we will see below the resulting leading energy density is still boost
invariant. $f_2 (k^2, k'^2, k \cdot k', k_T)$ is symmetric under the
interchange $k^2 \leftrightarrow k'^2$. Similarly
\begin{equation}\label{f1def}
D_{ii} \, = \, f_1 (k^2, k'^2, k \cdot k', k_T)
\end{equation}
and
\begin{equation}\label{f3def}
k_i \, k_j \, D_{ij} \, = \, f_3 (k^2, k'^2, k \cdot k', k_T)
\end{equation}
with $f_1$ and $f_3$ also some symmetric functions under $k^2
\leftrightarrow k'^2$. Using these redefinitions we can rewrite
\eq{edens11} as
\begin{eqnarray*}
\epsilon \, = \, \int \frac{ d^4 k \, d^4 k'}{(2 \pi)^8}\,
\frac{e^{-i k \cdot x - i k' \cdot x}}{(k^2 + i \epsilon k_0) \, (k'^2
+ i \epsilon k'_0)} \, \left\{ \left[ \frac{1}{2} \, \left(
\frac{\tau}{x_+} \right)^2 \, k_+ \, k'_+ - \frac{1}{2} \, {\underline k}^2
\right] \, f_1 (k^2, k'^2, k \cdot k', k_T) \, + \right.
\end{eqnarray*}
\begin{eqnarray*}
+ \left[ 2 \, k_- \, k'_- - \frac{1}{2} \, \left(
\frac{\tau}{x_+} \right)^2 \, {\underline k}^2 \right] \, k_+ \, k'_+ \,
f_2 (k^2, k'^2, k \cdot k', k_T) +
\end{eqnarray*}
\begin{equation}\label{edens2}
+ \left. \left[ - \frac{1}{4} \, \left( \frac{\tau}{x_+} \right)^2 \,
\left( \frac{k'_+}{k_-} + \frac{k_+}{k'_-} \right) \, + \, 1 \right]
\, f_3 (k^2, k'^2, k \cdot k', k_T) \right\} \, \frac{(2 \pi)^2}{S_\perp}
\, \delta ({\underline k} + {\underline k}').
\end{equation}
For reasons which will become apparent in a moment, we are interested
in determining the following combination of $f$'s
\begin{eqnarray*}
f_1 (k^2=0, k'^2=0, k \cdot k' =0, k_T) - k_T^2 f_2 (k^2=0, k'^2=0, k \cdot k' =0, k_T)
\end{eqnarray*}
\begin{equation}
- \frac{2}{k_T^2} \, f_3 (k^2=0, k'^2=0, k \cdot k' =0, k_T).
\end{equation}
To calculate it we compare \eq{tpp2} with
\begin{eqnarray*}
T_{++} \, = \, \int \frac{ d^4 k \, d^4 k'}{(2 \pi)^8}\,
\frac{e^{-i k \cdot x - i k' \cdot x}}{(k^2 + i \epsilon k_0) \, (k'^2
+ i \epsilon k'_0)} \, \bigg<\bigg< - [k_+ J^{a \, \rho} (k) - k^\rho
J^a_+ (k)]
\end{eqnarray*}
\begin{equation}\label{f2}
\times \, [k'_+ J^a_\rho (k') - k'_\rho J^a_+ (k')] \bigg>\bigg>
\, \frac{(2 \pi)^2}{S_\perp} \, \delta ({\underline k} + {\underline k}'),
\end{equation}
which follows from \eq{tmnab}. Equating the integrands of
Eqs. (\ref{tpp2}) and (\ref{f2}) we derive
\begin{eqnarray*}
k_+ \, k'_+ \, f_1 (k^2, k'^2, k \cdot k', k_T)
- {\underline k}^2 \, k_+ \, k'_+ \, f_2 (k^2, k'^2, k \cdot k', k_T) -
\end{eqnarray*}
\begin{equation}\label{f3}
- \frac{1}{2} \, \left( \frac{k'_+}{k_-} + \frac{k_+}{k'_-}
\right) \, f_3 (k^2, k'^2, k \cdot k', k_T)
\, = \, \bigg<\bigg< - [k_+ J^{a \, \rho} (k) - k^\rho J^a_+ (k)] \,
[k'_+ J^a_\rho (k') - k'_\rho J^a_+ (k')] \bigg>\bigg>.
\end{equation}
Putting $k = - k'$ and $k^2 = k'^2 =0$ in \eq{f3} and employing the
fact that in covariant gauge $k^\rho J^a_\rho (k) =0$ we obtain
\begin{eqnarray*}
f_1 (k^2=0, k'^2=0, k \cdot k' =0, k_T) - k_T^2 f_2 (k^2=0, k'^2=0, k \cdot k' =0, k_T) -
\end{eqnarray*}
\begin{equation}\label{f5}
- \frac{2}{k_T^2} \, f_3 (k^2=0, k'^2=0, k \cdot k' =0, k_T) \,
= \, - \bigg\langle\bigg\langle J^{a \, \rho} (k)
\, J^a_\rho (-k) \bigg\rangle\bigg\rangle \bigg|_{k^2 = 0}.
\end{equation}
Finally, since in order to construct the amplitude out of the field
given by \eq{ffi} one needs to truncate the field and put the outgoing
gluon's momentum on the mass shell, $k^2 = 0$, we see that $J^{a \,
\rho} (k)$ at $k^2 = 0$ is nothing but a production amplitude for a
real gluon carrying momentum $k$ (without convolution with the
polarization vector). The corresponding multiplicity distribution of
the produced gluons is given by
\begin{equation}
\frac{dN}{d^2 k \, dy} \, = \, \frac{1}{2 (2 \pi)^3} \,
\bigg\langle\bigg\langle J^{a \, \rho} (k)
\, J^a_\rho (-k) \bigg\rangle\bigg\rangle \bigg|_{k^2 = 0}.
\end{equation}
Therefore,
\begin{eqnarray*}
f_1 (k^2=0, k'^2=0, k \cdot k' =0, k_T) - k_T^2 f_2 (k^2=0, k'^2=0, k \cdot k' =0, k_T)
\end{eqnarray*}
\begin{equation}\label{f6}
- \frac{2}{k_T^2} \, f_3 (k^2=0, k'^2=0, k \cdot k' =0, k_T) \, = \,
- 2 (2 \pi)^3 \, \frac{dN}{d^2 k \, dy}
\end{equation}
and, for a cylindrical nucleus,
\begin{eqnarray*}
\frac{1}{S_\perp} \, \bigg[ f_1 (k^2=0, k'^2=0, k \cdot k' =0, k_T)
- k_T^2 f_2 (k^2=0, k'^2=0, k \cdot k' =0, k_T)
\end{eqnarray*}
\begin{equation}\label{f7}
- \frac{2}{k_T^2} \, f_3 (k^2=0, k'^2=0, k \cdot k' =0, k_T) \bigg] \,
= \, - 2 (2 \pi)^3 \, \frac{dN}{d^2 k \, dy \, d^2 b}.
\end{equation}
Now let us get back to \eq{edens2}. Rewriting for each of the $f$'s
\begin{eqnarray*}
f_i (k^2 , k'^2 , k \cdot k', k_T) \, = \, f_i (k^2 =0, k'^2 =0, k
\cdot k' =0, k_T) +
\end{eqnarray*}
\begin{equation}\label{iter1}
+ [f_i (k^2 , k'^2 , k \cdot k', k_T) - f_i (k^2 =0, k'^2 =0, k \cdot k' =0,
k_T)]
\end{equation}
and keeping only the $f_i (k^2 =0, k'^2 =0, k \cdot k' =0, k_T)$ in
\eq{edens2} we can perform the longitudinal momenta ($k_+, k_-, k'_+,
k'_-$) integrations with the help of \eq{J7} from Appendix B obtaining
\begin{eqnarray*}
\epsilon \, \approx \, - \frac{1}{8 \, S_\perp} \, \int \frac{d^2 k}{(2 \, \pi)^2} \,
k_T^2 \, \left\{ \left[ J_1 (k_T \tau) \right]^2 + \left[ J_0 (k_T \tau) \right]^2
\right\}
\end{eqnarray*}
\begin{eqnarray*}
\times \, \bigg[ f_1 (k^2=0, k'^2=0, k \cdot k' =0, k_T) - k_T^2 f_2 (k^2=0, k'^2=0,
k \cdot k' =0, k_T)
\end{eqnarray*}
\begin{equation}\label{edens33}
- \frac{2}{k_T^2} \, f_3 (k^2=0, k'^2=0, k \cdot k' =0, k_T) \bigg],
\end{equation}
which, after employing \eq{f7} becomes
\begin{equation}\label{edens3}
\epsilon \, \approx \, \frac{\pi}{2} \, \int d^2 k \,
\frac{d N}{d^2 k \, d \eta \, d^2 b}
\ k_T^2 \, \left\{ \left[ J_1 (k_T \tau) \right]^2 + \left[ J_0 (k_T \tau) \right]^2
\right\}.
\end{equation}
(Again we have used the rapidity-independence of the gluon spectrum
$\frac{d N}{d^2 k \, d \eta \, d^2 b}$ to replace $y$ with $\eta$.)
One might worry that the functions $f_i (k^2 , k'^2 , k \cdot k',
k_T)$ may not have a finite $k^2,k'^2, k\cdot k^\prime \rightarrow 0$
limit, which would be dangerous for the decomposition of
\eq{iter1} \cite{AMY2}. However, let us remind the reader that the quantity $J_\mu^a
(k)$ defined in \eq{ffi} has the meaning of (truncated) gluon
production amplitude for the off-shell gluon with virtuality $k^2$. In
the $k^2 \rightarrow 0$ limit $J_\mu^a (k)$ becomes the gluon
production amplitude for an on-shell gluon, and is indeed
finite. Therefore, the correlation functions $D_{\mu\nu}$ from
\eq{ddef}, which in the $k^2,k'^2, k\cdot k^\prime \rightarrow 0$
limit have the meaning of the gluon production amplitude squared (but
without the Lorentz index contraction), are also finite in this
limit. This implies that the functions $f_i (k^2 , k'^2 , k \cdot k',
k_T)$, defined in terms of various components of $D_{\mu\nu}$ in
Eqs. (\ref{f2def}), (\ref{f1def}) and (\ref{f3def}), are finite in the
$k^2,k'^2, k\cdot k^\prime \rightarrow 0$ limit.
For the proper time $\tau$ much larger than $1/\langle k_T \rangle$,
with $k_T$ the typical transverse momentum in the distribution
$\frac{d N}{d^2 k \, d
\eta \, d^2 b}$, \eq{edens3} becomes
\begin{equation}\label{ed}
\epsilon \bigg|_{\tau \gg 1/\langle k_T \rangle} \, \approx \, \frac{1}{\tau} \, \int d^2 k \,
\frac{d N}{d^2 k \, d \eta \, d^2 b} \ k_T \, = \, \frac{1}{\tau} \,
\frac{d E_T}{d \eta \, d^2 b},
\end{equation}
i.e., it falls off as $1/\tau$.
Therefore, we have shown that the energy density $\epsilon$ of a gluon
field produced in a heavy ion collision always has a non-zero term
scaling as $\sim 1/\tau$. However, to demonstrate that this term
dominates at late times, we still need to prove that it does not get
canceled by the terms we left out in writing down the decomposition of
\eq{iter1} and keeping the first terms only. Thus we have to analyze
the contribution arising from substituting the terms from the square
brackets of \eq{iter1} into \eq{edens2}. We need to show that such
contributions fall off faster than $1/\tau$, and, therefore, can be
neglected at late times. Here we will demonstrate that this is true
for one of the terms --- the $f_1$-term in \eq{edens2}. The proof for
the other two terms on the right hand side of \eq{edens2} would be
analogous.
Substituting the square brackets from \eq{iter1} into \eq{edens2} we
obtain the following contribution to the energy density, which we have
to prove to be small:
\begin{eqnarray*}
\frac{1}{2} \, \int \frac{ d^4 k \, d k'_+ \, d k'_-}{(2 \pi)^6 \, S_\perp}\,
\frac{e^{-i k \cdot x - i k' \cdot x}}{(k^2 + i \epsilon k_0) \, (k'^2
+ i \epsilon k'_0)} \, \left[ \left( \frac{\tau}{x_+} \right)^2 \,
k_+ \, k'_+ - {\underline k}^2 \right]
\end{eqnarray*}
\begin{equation}\label{contr1}
\times \,[f_1 (k^2, k'^2, k \cdot k', k_T) - f_1 (k^2 =0, k'^2 =0, k \cdot k' =0, k_T)].
\end{equation}
For a wide range of amplitudes one can write
\begin{eqnarray*}
f_1 (k^2, k'^2, k \cdot k', k_T) - f_1 (k^2 =0, k'^2 =0, k \cdot k'
=0, k_T) \, = \, (k^2 \, k'^2 )^{\Delta_1} \, g^{(1)} (k^2, k'^2, k
\cdot k', k_T) +
\end{eqnarray*}
\begin{equation}\label{iter2}
+ [(k+k')^2]^{\Delta_2} \, g^{(2)} (k^2, k'^2, k
\cdot k', k_T),
\end{equation}
where $g^{(1)} (k^2 =0, k'^2 =0, k \cdot k' =0, k_T) \neq 0$, $g^{(2)}
(k^2 =0, k'^2 =0, k \cdot k' =0, k_T) \neq 0$, and $\Delta_1, \Delta_2 > 0$. In
arriving at \eq{iter2} we have also used the fact that $f_1 (k^2,
k'^2, k \cdot k', k_T) \, = \, f_1 (k'^2, k^2, k \cdot k', k_T)$, which follows from the $k
\leftrightarrow k'$ symmetry in the definition of $f_1 (k^2, k'^2, k \cdot k',
k_T)$ given by \eq{f1def} along with \eq{ddef}. In \eq{iter2} we put a
power of $(k+k')^2$ instead of a power of $k \cdot k'$ in front of
$g^{(2)}$. Similarly to \eq{iter1} we write
\begin{eqnarray*}
g^{(i)} (k^2 , k'^2 , k \cdot k', k_T) \, = \, g^{(i)} (k^2 =0, k'^2
=0, k \cdot k'=0, k_T) +
\end{eqnarray*}
\begin{equation}\label{iter3}
+ [g^{(i)} (k^2 , k'^2 , k \cdot k', k_T) - g^{(i)} (k^2 =0,
k'^2 =0, k \cdot k'=0, k_T)]
\end{equation}
for $i=1,2$. Substituting the first term on the right hand side of
\eq{iter3} for $g^{(1)}$ into the first term on the right hand side of
\eq{iter2}, and then plugging the resulting contribution into \eq{contr1} yields
\begin{equation}\label{iter4}
\frac{1}{2} \, \int \frac{ d^4 k \, d k'_+ \, d k'_-}{(2 \pi)^6 \,
S_\perp}\, \frac{e^{-i k \cdot x - i k' \cdot x}}{(k^2 + i \epsilon
k_0) \, (k'^2 + i \epsilon k'_0)} \, \left\{ \left( \frac{\tau}{x_+}
\right)^2 \, k_+ \, k'_+ - {\underline k}^2 \right\} \, (k^2 \, k'^2
)^{\Delta_1} \, g^{(1)} (0, 0, 0, k_T).
\end{equation}
Performing the $k_+, k_-, k'_+, k'_-$ integrations in \eq{iter4} with
the help of \eq{I1} in Appendix A and \eq{J5} with $\lambda =1$ in
Appendix B we obtain
\begin{equation}\label{iter5}
\frac{e^{2 \pi i \Delta_1}}{8 \, S_\perp \, \Gamma (1-\Delta_1)^2} \,
\int \frac{ d^2 k}{(2 \pi)^2} \, g^{(1)} (0, 0, 0, k_T) \, k_T^2 \,
\left( \frac{2 \, k_T}{\tau} \right)^{2 \, \Delta_1} \, \left\{ \left[ J_{-\Delta_1 -1} (k_T \tau)
\right]^2 + \left[ J_{-\Delta_1} (k_T \tau) \right]^2
\right\},
\end{equation}
which, for $\tau \gg 1/\langle k_T \rangle$, scales as
\begin{equation}\label{iter6}
\sim \frac{1}{\tau^{1 + 2 \Delta_1}},
\end{equation}
and is, therefore, negligibly small at late proper times compared to
the leading contribution to energy density given by \eq{ed}. (Here we
assume that the typical transverse momentum $\langle k_T \rangle$ is
the same for $\frac{d N}{d^2 k \, d \eta \, d^2 b}$ in \eq{edens3} and
for $g^{(1)} (0, 0, k_T)$ in \eq{iter5}: both functions result from
expanding the same amplitude in powers of $k^2 \, k'^2$, and no new
scale can arise from such an expansion, which justifies our
assumption.) The particular way of regularizing the $k^2$ branch cut
used in \eq{I1} and \eq{J1} is not essential for arriving at
\eq{iter6}, since other regularizations would yield the same result.
The second term on the right hand side of \eq{iter2} gives a similarly
small contribution. To see this we substitute the first term on the
right hand side of \eq{iter3} for $g^{(2)}$ into the second term on
the right hand side of \eq{iter2}, and then substitute the result into
\eq{contr1} obtaining
\begin{eqnarray*}
\int \frac{ d^4 k \, d k'_+ \, d k'_-}{(2 \pi)^6 \, 2 \,
S_\perp}\, \frac{e^{-i k \cdot x - i k' \cdot x}}{(k^2 + i \epsilon
k_0) \, (k'^2 + i \epsilon k'_0)} \, \left\{ \left( \frac{\tau}{x_+}
\right)^2 \, k_+ \, k'_+ - {\underline k}^2 \right\} [(k+k')^2]^{\Delta_2}
\, g^{(2)} (0, 0, 0, k_T) \, =
\end{eqnarray*}
\begin{equation}\label{iter7}
= \, [- \partial_\mu \partial^\mu]^{\Delta_2} \frac{1}{8 \, S_\perp} \,
\int \frac{ d^2 k}{(2 \pi)^2} \, g^{(2)} (0, 0, 0, k_T) \, k_T^2 \,
\, \left\{ \left[ J_{1} (k_T \tau) \right]^2 + \left[ J_{0} (k_T \tau)
\right]^2 \right\}.
\end{equation}
For $\tau \gg 1/\langle k_T \rangle$ the integral on the right of
\eq{iter7} scales as $\sim 1/\tau$: applying the derivatives we see that
the whole expression in \eq{iter7} scales as
\begin{equation}\label{iter8}
\sim \frac{1}{\tau^{1 + 2 \Delta_2}},
\end{equation}
and is also negligibly small at late proper times compared to the
leading contribution to energy density given by \eq{ed}.
For the second term on the right hand side of \eq{iter3}, $g^{(i)}
(k^2, k'^2, k \cdot k', k_T) - g^{(i)} (0,0,0,k_T)$ with $i=1$(or
$i=2$), one can repeat the procedure outlined above for $f_1 (k^2,
k'^2, k \cdot k', k_T) - f_1 (0,0,0,k_T)$, using the redefinition just
like in \eq{iter2} and showing that the leading term in the resulting
decomposition, similar to \eq{iter3}, falls off faster with $\tau$
than \eq{iter6} (or \eq{iter8}). Iterating the procedure would
generate a series of corrections falling off at progressively higher
powers of $1/\tau$, all of which could be neglected at $\tau \gg
1/\langle k_T \rangle$.
Of course the assumption of \eq{iter2}, while quite general, does not
include all the possibilities. One might imagine other ways for $f_1
(k^2, k'^2, k \cdot k', k_T) - f_1 (0,0,0, k_T)$ to approach zero as
$k^2, k'^2, k \cdot k' \rightarrow 0$: it might scale as $1 / (\ln k^2
\, \ln k'^2)$, or, less likely, as $e^{- k_T^2 / k^2 - k_T^2 / k'^2
}$. In any case, \eq{I1} suggests that each power of $k^2$ (or each
power of $k'^2$ or of $k \cdot k'$) gives a power of $1/\tau$ for
energy density $\epsilon$ in coordinate space: $k^2 \rightarrow
1/\tau$. (Indeed the powers of $k_T$ do not modify the
$\tau$-dependence of $T_{\mu\nu}$ at all.) Therefore, one may argue
that after the momentum integration is done in \eq{edens3}, the $f_1
(k^2, k'^2, k \cdot k', k_T) - f_1 (0,0,0, k_T)$ term, when
substituted into \eq{edens2}, yields approximately the following
contribution to $\epsilon$
\begin{equation}\label{rem}
\frac{1}{\tau} \, \left[ f_1 \left(\frac{1}{\tau}, \frac{1}{\tau}, \frac{1}{\tau},
\langle k_T \rangle\right) - f_1 (0,0,0,\langle k_T \rangle) \right],
\end{equation}
which falls off faster than $1/\tau$ and can thus be neglected
compared to \eq{ed}. This conclusion is natural, since the term in
\eq{rem}, or, equivalently, the second term on the right hand side
of \eq{iter1}, does not contribute to the production cross section, as
follows from \eq{f6}, which is another way of saying that it is not
important at late times.
The proofs that the contributions to energy density $\epsilon$
generated by substituting $f_2 (k^2, k'^2, k \cdot k', k_T)$ $- f_2
(0,0,0,k_T)$ and $f_3 (k^2, k'^2, k \cdot k', k_T)$ $-f_3 (0,0,0,k_T)$
instead of $f_2$ and $f_3$ into \eq{edens2} are also subleading at
large $\tau$ can be constructed by analogy to the above.
Finally, we have to comment on our use of the Abelian part of
$T_{\mu\nu}$ only in \eq{tmnab} and throughout this Section. Including
the non-Abelian parts of the field strength tensor $F_{\mu\nu}$ would
generate higher powers of $A_\mu^a$ in the definition (\ref{tmnga}) of
$T_{\mu\nu}$. Using \eq{ffi} those extra powers can be rewritten as
extra integrals over $k''$ and $k'''$ in the extra terms which would
be added to \eq{tmnab}. Due to \eq{I1}, each of these extra integrals
would (at least) generate a Bessel function $J_{-\Delta} (k_T \,
\tau)$, which at large $\tau$ scales as $(1/\sqrt{\tau}) \, \cos (k_T
\, \tau + \frac{\pi}{2} \Delta - \frac{\pi}{4})$. Even without the cosine, one
can immediately see that the cubic in $A_\mu^a$ term in $T_{\mu\nu}$
would fall off at least like $1/\tau^{3/2}$ at large $\tau$. The
quadric terms would fall off at least like $1/\tau^{2}$. Both of these
terms would be negligibly small compared to the leading quadratic term
scaling as $1/\tau$ shown in \eq{ed}.
\eq{ed} has a straightforward physical interpretation. Every Feynman
diagram has a final state in which the particles are propagating as
non-interacting plane waves until the infinite late times. Indeed the
energy density of such a ``free-streaming'' state scales as $\sim
1/\tau$, and this is exactly what \eq{ed} represents.
Therefore, in this Section we have proven that in the
rapidity-independent case, defined by Eqs. (\ref{xmv3}) and
(\ref{xmv4}) for the total rapidity interval in the collision of two
very large nuclei, at any order in the perturbative expansion in the
strong coupling $g$, the resulting gluon field's energy density has a
non-vanishing term which is dominant at late times giving $\epsilon
\sim 1/\tau$ (\ref{ed}). Hence it appears that, in this boost-invariant case,
thermalization leading to Bjorken hydrodynamic description of the
evolution of produced gluonic system, can not result from resummation
of perturbative QCD diagrams.
\section{Generalization to the Rapidity-Dependent Case}
For rapidity intervals $Y \, \stackeven{>}{\sim} \, \frac{1}{\alpha_s}$ in heavy ion
collisions the quantum evolution corrections \cite{bfkl,dip,bk,jimwlk}
would become important making the produced particle distribution
rapidity dependent. Below we are first going to argue that
rapidity-dependent hydrodynamic description may only change the
$\epsilon \sim 1/\tau^{4/3}$ scaling of the ideal Bjorken energy
density to a higher power, $\epsilon \sim 1/\tau^{4/3 + \Delta}$. We
will then demonstrate that the rapidity-dependent quantum corrections,
such as the ones introduced by the BFKL evolution \cite{bfkl}, would
{\sl not} modify the $\epsilon \sim 1/\tau$ scaling derived in the
previous Section.
\subsection{Rapidity-Dependent Hydrodynamics}
In the rapidity dependent case the most general form of the
energy-momentum tensor is given by the equation similar to \eq{tmn1}
\begin{equation}\label{tmnr1}
T_{\mu\nu} \, = \, A(\tau, \eta) \, u_\mu u_\nu + B(\tau, \eta) \,
(u_\mu v_\nu + u_\nu v_\mu) + C(\tau, \eta) \, v_\mu v_\nu + D(\tau, \eta)
\, g_{\mu\nu},
\end{equation}
with $u_\mu$ and $v_\mu$ still given by Eqs. (\ref{umu}) and
(\ref{vmu}) and where now all the coefficients $A,B,C,D$ are also
functions of the space-time rapidity $\eta$. Due to this
$\eta$-dependence the $+ \leftrightarrow -$ symmetry argument no
longer applies in general. However, it still holds at mid-rapidity
($\eta = 0$) for a collision of two identical nuclei leading to
\begin{equation}\label{Bcond}
B (\tau, \eta = 0) \, = \, 0.
\end{equation}
Applying the conservation of energy-momentum tensor condition
(\ref{cons}) to the tensor in \eq{tmnr1} yields
\begin{eqnarray}\label{eom1}
\tau \, \partial_\tau B - 2 \, \partial_\eta D + 2 \, \partial_\eta C + 2 \, B &=& 0
\nonumber \\
2 \, \tau \, \partial_\tau A + \partial_\eta B + 2 \, \tau \,
\partial_\tau D + 2 \, A + 2 \, C &=& 0.
\end{eqnarray}
The energy-momentum tensor in \eq{tmnr1} would describe a hydrodynamic
system if it could be reduced to the standard hydrodynamic form
\begin{equation}\label{hydro}
T_{\mu\nu} \, = \, (\epsilon + p) \, w_\mu \, w_\nu - p \, g_{\mu\nu},
\end{equation}
where $w_\mu$ is the four-vector of the fluid velocity, $w_\mu \,
w^\mu = 1$. For the $1 + 1$-dimensional expansion of the system
created in a collisions of two very large nuclei considered here the
fluid velocity has zero transverse component, ${\underline w} = 0$, such that
$w_\mu = (w_+, w_-, {\underline 0})$. Matching \eq{tmnr1} onto \eq{hydro} we
obtain
\begin{equation}\label{aep}
A \, = \, \epsilon + p + C
\end{equation}
and
\begin{equation}\label{dp}
D \, = \, - p.
\end{equation}
For the hydrodynamic energy momentum tensor (\ref{hydro}) the
following relation holds
\begin{equation}
T_{++} \, T_{--} \, = \, (T_{+-} + p)^2,
\end{equation}
leading to a constraint
\begin{equation}\label{Cond}
C \, = \, \frac{B^2}{4 \, A}.
\end{equation}
Combining \eq{eom1} and \eq{Cond} with the equation of state relating
$\epsilon$ and $p$, would give us a complete set of rapidity-dependent
hydrodynamic equations. However the resulting system of equations is
nonlinear and is hard to solve analytically. Instead we are going to
construct a perturbative solution for small rapidity-dependent
corrections to Bjorken hydrodynamics \cite{bj}. We begin by noting
that, since $B = 0$ in the boost-invariant case considered in the
previous Section, we can assume that non-zero $B$ reflects the
deviation from the ideal Bjorken hydrodynamics, and could be assumed
small if we are interested in small corrections to the
latter. Assuming that $B \ll A$ and keeping only linear in $B$
corrections allows us to neglect $C$, since, due to \eq{Cond}, $C \sim
B^2$. Than, using Eqs. (\ref{aep}) and (\ref{dp}) in \eq{eom1} yields
\begin{eqnarray}\label{eom2}
\tau \, \partial_\tau B + 2 \, \partial_\eta p + 2 \, B &=& 0
\nonumber \\
2 \, ( \tau \, \partial_\tau \epsilon + \epsilon + p) + \partial_\eta
B \, &=& 0.
\end{eqnarray}
We are interested in the solution for the ideal gas equation of state:
therefore we put $\epsilon = 3 \, p$. The most general solution of
\eq{eom2} satisfying the condition of \eq{Bcond} and mapping back onto
Bjorken hydrodynamic behavior for small $B$ is
\begin{equation}\label{esol}
\epsilon \, = \, \epsilon_0 \, \cos (\sqrt{\Delta} \, \eta) \,
\frac{1}{\tau^{\frac{1}{3} \, (5 - \sqrt{1 - 3 \Delta})}}
\end{equation}
with
\begin{equation}\label{bsol}
B \, = \, - \frac{2}{3} \, \epsilon_0 \, \frac{\sqrt{1 - 3 \Delta} -
1}{\sqrt{\Delta}} \, \sin (\sqrt{\Delta} \, \eta) \,
\frac{1}{\tau^{\frac{1}{3} \, (5 - \sqrt{1 - 3 \Delta})}},
\end{equation}
where $\Delta$ and $\epsilon_0$ are arbitrary constants. The
corresponding flow velocity components are given by
\begin{equation}\label{wpm}
w_\pm \, \approx \, \frac{x_\pm}{\tau} \, \left( 1 \pm \frac{3 \, B}{8 \,
\epsilon} \right).
\end{equation}
Looking at the solution given by \eq{esol} one may wonder why the
energy density is not positive definite. Indeed for $\Delta < 0$ the
energy density $\epsilon$ from \eq{esol} becomes positive definite,
since $\cos (\sqrt{\Delta} \, \eta)$ would be replaced by $\cosh
(\sqrt{|\Delta|} \, \eta)$. However, the resulting rapidity
distribution of energy density would increase as one moves further
away from mid-rapidity, which is unphysical. Therefore one has to have
$\Delta > 0$. Resolution of the positivity problem for $\epsilon$
comes from the necessity to satisfy the $B \ll A$ assumption which we
have made at the beginning of this calculation. It translates into $B
\ll \epsilon$ condition, which is satisfied by Eqs. (\ref{esol}) and
(\ref{bsol}) only if $\sqrt{\Delta} \, \eta \ll 1$. Since in this
Section we are interested in large rapidity intervals, $\eta \sim Y
\stackeven{>}{\sim} 1/\alpha_s$, the $\sqrt{\Delta} \, \eta \ll 1$ requires that
$\Delta \, \stackeven{<}{\sim} \, \alpha_s \ll 1$. Hence, for large rapidities,
Eqs. (\ref{esol}) and (\ref{bsol}) are valid only at the lowest order
in $\Delta$
\begin{equation}\label{esole}
\epsilon \, \approx \, \frac{\epsilon_0}{\tau^{\frac{4}{3} + \frac{\Delta}{2}}} \,
\left(1 - \frac{1}{2} \, \Delta \, \eta^2 \right),
\end{equation}
\begin{equation}\label{bsole}
B \, \approx \, \frac{\epsilon_0}{\tau^{\frac{4}{3} +
\frac{\Delta}{2}}} \, \Delta \, \eta,
\end{equation}
where we did not expand $\tau^{-\Delta/2}$ since, at late times, $\Delta
\, \ln \tau$ does not have to be small for $B \ll \epsilon$ condition
to hold. For small $\sqrt{\Delta} \, \eta$ the energy density in
\eq{esole} is indeed positive.
\eq{esole} has an important feature which we would like to emphasize:
since $\Delta > 0$, it shows that the energy density of the
boost-non-invariant ideal hydrodynamics falls off with $\tau$ {\sl
faster} than the energy density of the boost-invariant ideal Bjorken
hydrodynamics \cite{bj}. Here we have proven it only for a small
rapidity-dependent perturbation of the Bjorken solution. However one
should expect our conclusion to hold in a general case of a
rapidity-dependent hydrodynamics. In the case of a rapidity-dependent
hydrodynamics, the longitudinal pressure is higher than in the
boost-invariant Bjorken case, generating the longitudinal acceleration
of the flow (see e.g. \eq{wpm}). The central-rapidity high-density
system starts expanding faster than in Bjorken case, leading to a
faster depletion of the energy density with $\tau$. In other words,
once the longitudinal homogeneity of pure Bjorken hydrodynamics is
broken by some rapidity-dependent phenomena, the system starts doing
more work in the longitudinal direction than it was doing in pure
Bjorken hydrodynamics case, and this leads to a faster decrease of
energy density with proper time.\footnote{The author would like to
thank Ulrich Heinz for explaining to him this argument.}
\subsection{Rapidity-Dependent Energy Density}
\label{arg2}
Here we are going to generalize the argument of Sect. \ref{arg1} to
the case of rapidity-dependent gluon fields. It is impossible to
define co-moving energy density for a general energy-momentum tensor
like the one given in \eq{tmnr1}, since, in the general not
necessarily hydrodynamic case, one can not define the co-moving frame,
and in the case of hydrodynamics (\ref{hydro}) one needs to know the
flow velocity to define the co-moving frame, which is impossible to do
without solving the hydrodynamics equations (\ref{eom1}). Therefore we
will restrict our analysis to the case of mid-rapidity, $\eta = 0$,
where, for a collision of two identical nuclei, the co-moving frame is
just the center of mass frame of the two nuclei. There \eq{edens}
would apply, such that
\begin{equation}\label{edense}
\epsilon (\tau , \eta=0) \, = \, \frac{1}{2} \, \left( \frac{\tau}{x_+} \right)^2 \,
T_{++} (\tau , \eta=0) + T_{+-} (\tau , \eta=0) \, = \, T_{++} (\tau ,
\eta=0) + T_{+-} (\tau , \eta=0).
\end{equation}
Repeating the steps from Section \ref{arg1} we write
\begin{eqnarray*}
\epsilon (\tau , \eta=0) \, = \, \int \frac{ d^4 k \, d^4 k'}{(2 \pi)^8}\,
\frac{e^{-i k \cdot x - i k' \cdot x}}{(k^2 + i \epsilon k_0) \, (k'^2
+ i \epsilon k'_0)}\bigg|_{\eta = 0} \, \left\{ \left[ \frac{1}{2} \, \left(
\frac{\tau}{x_+} \right)^2 \, k_+ \, k'_+ - \frac{1}{2} \, {\underline k}^2
\right] \, \right.
\end{eqnarray*}
\begin{eqnarray*}
\times \, f_1 (k^2, k_+, k'^2, k'_+, k \cdot k', k_T) \, + \left[ 2 \, k_- \, k'_- - \frac{1}{2} \, \left(
\frac{\tau}{x_+} \right)^2 \, {\underline k}^2 \right] \, k_+ \, k'_+ \, f_2 (k^2, k_+, k'^2, k'_+, k \cdot k', k_T)
\end{eqnarray*}
\begin{equation}\label{edense1}
+ \left. \left[ - \frac{1}{4} \, \left( \frac{\tau}{x_+} \right)^2 \,
\left( \frac{k'_+}{k_-} + \frac{k_+}{k'_-} \right) \, + \, 1 \right]
\, f_3 (k^2, k_+, k'^2, k'_+, k \cdot k', k_T) \right\} \,
\frac{(2 \pi)^2}{S_\perp} \, \delta ({\underline k} + {\underline k}').
\end{equation}
where now, in the rapidity dependent case, $f_i$'s are functions of
$k_\pm$ and $k'_\pm$ as well. However, since we can always rewrite
$k_- = (k^2 + k_T^2)/2 k_+$ and $k'_- = (k'^2 + k_T^2)/2 k'_+$, we put
only $k_+$ and $k'_+$ in the arguments of the functions $f_i$.
Rapidity-dependent quantum evolution corrections come in as logarithms
of Bjorken $x$ variable \cite{bfkl}. If $p_+$ is a large longitudinal
momentum carried by a nucleon in the nucleus moving in the
$+$-direction, than $x = k_+ / p_+$. The rapidity-dependent
corrections would then bring in powers of $\alpha_s \, \ln 1/x \, = \, \alpha_s
\, \ln p_+ / k_+$. Resummation of such corrections for the gluon production
amplitudes generates powers of $1/x$, or, equivalently, $p_+ /
k_+$. Therefore, to verify whether such corrections modify the
$\tau$-dependence of $\epsilon$, we can consider the following general
form for the functions $f_i$'s
\begin{equation}\label{frap}
f_i (k^2, k_+, k'^2, k'_+, k \cdot k', k_T) \, = \, \left( \frac{p_+}{k_+}
\, \frac{p_+}{k'_+} \right)^{\lambda} \,
\tilde{f}_i (k^2, k'^2, k \cdot k', k_T),
\end{equation}
where we again used the fact that $f$'s are symmetric under $k
\leftrightarrow k'$ interchange. For simplicity we assume the
power $\lambda$ to be the same for $f_1$, $f_2$ and $f_3$: this
assumption is not crucial and can be easily relaxed. The logarithmic
corrections to $f_i$'s, i.e., terms with $\ln p_+ / k_+$ and $\ln p_+
/ k'_+$, can be obtained from $f_i$'s in \eq{frap} by differentiating
it with respect to $\lambda$. Indeed that would give only logarithms
like $\ln (p_+^2/k_+ k'_+)$, but not $\ln k_+/k'_+$: while we are
quite confident that the latter terms never appear in perturbation
theory, our approach can be easily generalized to include both types
of logarithms by putting different powers for $p_+/k_+$ and $p_+/k'_+$
factors in \eq{frap}. (A careful reader may worry that the
$p_+$-dependence was never explicitly shown in the argument of $f_i$'s
and was explicitly assumed there: in fact, due to boost-invariance,
the $p_+$-dependence enters in $f_i$'s only through the ratios of $p_+
/ k_+$ and $p_+ / k'_+$. In the rapidity-independent case of Section
\ref{arg1}, $f_i$'s were independent of $p_+$, which corresponds to the
eikonal limit.)
Substituting $f_i$'s from \eq{frap} into \eq{edense1} we obtain
\begin{eqnarray*}
\epsilon (\tau , \eta=0) \, = \, \int \frac{ d^4 k \, d^4 k'}{(2 \pi)^8}\,
\frac{e^{-i k \cdot x - i k' \cdot x}}{(k^2 + i \epsilon k_0) \, (k'^2
+ i \epsilon k'_0)}\bigg|_{\eta = 0} \, \left\{ \left[ \frac{1}{2} \, \left(
\frac{\tau}{x_+} \right)^2 \, k_+ \, k'_+ - \frac{1}{2} \, {\underline k}^2
\right] \, \right.
\end{eqnarray*}
\begin{eqnarray*}
\times \,
\tilde{f}_1 (k^2, k'^2, k \cdot k', k_T) \, + \left[ 2 \, k_- \, k'_- - \frac{1}{2} \, \left(
\frac{\tau}{x_+} \right)^2 \, {\underline k}^2 \right] \, k_+ \, k'_+ \, \
\tilde{f}_2 (k^2, k'^2, k \cdot k', k_T) +
\end{eqnarray*}
\begin{equation}\label{edense2}
+ \left. \left[ - \frac{1}{4} \, \left( \frac{\tau}{x_+} \right)^2 \,
\left( \frac{k'_+}{k_-} + \frac{k_+}{k'_-} \right) \, + \, 1 \right]
\, \tilde{f}_3 (k^2, k'^2, k \cdot k', k_T) \right\} \, \left( \frac{p_+}{k_+}
\, \frac{p_+}{k'_+} \right)^{\lambda} \, \frac{(2 \pi)^2}{S_\perp} \, \delta ({\underline k} + {\underline k}').
\end{equation}
To perform the longitudinal momentum integrals we will use the
integral in Eqs. (\ref{J1}) and (\ref{J6}) of Appendix B. There we can
see that, similar to the rapidity-independent case, each positive
extra power of $k^2$ (or $k'^2$ or $k \cdot k'$) gives a power of
$1/\tau$. Therefore, we again are interested in contribution of $f_i
(k^2 =0, k_+, k'^2 =0, k'_+, k \cdot k'=0, k_T)$ as in the terms
giving the leading-$\tau$ behavior. Similar to \eq{f7} we can write
\begin{equation}\label{fe7}
\frac{1}{S_\perp} \, \left( \frac{p_+}{k_+}
\, \frac{p_+}{- k_+} \right)^{\lambda} \, \left[ \tilde{f}_1 (0, 0, 0, k_T)
- k_T^2 \tilde{f}_2 (0, 0, 0, k_T) -
\frac{2}{k_T^2} \, \tilde{f}_3 (0, 0, 0, k_T) \right] \,
= \, - 2 (2 \pi)^3 \,
\frac{dN}{d^2 k \, dy \, d^2 b}.
\end{equation}
which shows that this combination of $f_i$'s is not zero. Using
\eq{fe7} in \eq{edense2}, where we put $k^2 = k'^2 =0$ in the
arguments of all $f_i$'s, and performing the longitudinal integrations
using the formulas from Appendix B yields for the leading term in
energy density
\begin{equation}\label{edense3}
\epsilon (\tau , \eta=0) \, \approx \, \frac{\pi}{2} \, \int d^2 k \,
\frac{d N}{d^2 k \, d \eta \, d^2 b}\bigg|_{\eta = 0} \ k_T^2 \,
\left\{ \left[ J_{-1-\lambda} (k_T \tau) \right]^2 + \left[ J_{-\lambda}
(k_T \tau) \right]^2 \right\}.
\end{equation}
In arriving at \eq{edense3} we have noticed that, according to
\eq{J6}, each power of $k_+$ or $k'_+$ gives a power of $k_T \, e^\eta /\sqrt{2}$
after the integration. For on-mass shell gluons in \eq{fe7} one has
$k_+ = k_T \, e^y /\sqrt{2}$. Therefore, the powers of $k_T \, e^\eta
/\sqrt{2}$ were absorbed in $\frac{d N}{d^2 k \, d \eta \, d^2 b}$ by
just replacing $y \rightarrow \eta$.
The large-$\tau$ asymptotics of \eq{edense3} is the same as in
\eq{ed}:
\begin{equation}\label{ede}
\epsilon (\tau , \eta=0) \bigg|_{\tau \gg 1/\langle k_T \rangle} \, \approx \,
\frac{1}{\tau} \, \int d^2 k \, \frac{d N}{d^2 k \, d \eta \, d^2 b}\bigg|_{\eta = 0}
\ k_T \, = \, \frac{1}{\tau} \, \frac{d E_T}{d \eta \, d^2
b}\bigg|_{\eta = 0}.
\end{equation}
Therefore, we have proven that even in the rapidity-dependent case the
mid-rapidity energy density given by the Feynman diagrams falls off as
$1/\tau$ at large $\tau$. This conclusion could be easily derived by
just analyzing Eqs. (\ref{J1}) and (\ref{J6}): one can see there that
each extra power of $k_+$ does not bring in any new powers of $\tau$,
and only modifies the order of the Bessel function, which can not
change the $\tau$-dependence, as follows from \eq{edense3}.
\eq{ede} shows that the scaling of energy density of the
rapidity-dependent solution of the hydrodynamics equations given by
\eq{esole}, $\epsilon \sim 1/\tau^{\frac{4}{3} + \frac{\Delta}{2}}$,
can not come from the leading contribution of Feynman
diagrams. Indeed, the subleading contributions may still lead to
energy density falling off with $\tau$ faster than $1/\tau$: however,
due to \eq{I1}, such contributions must come in with extra positive
powers of $k^2$ in momentum space. They would go to zero in the
on-mass shell $k^2 \rightarrow 0$ limit and, thus, would not
contribute to the production cross section. Such corrections are
probably irrelevant for all physical observables.
\section{Including Quarks}
\label{arg3}
To generalize our conclusion to massless quarks we will restrict our
discussion to rapidity-independent case only: generalization to
the rapidity-dependent case can be easily done following the procedure
outlined in Section \ref{arg2}. We start with the energy-momentum
tensor for a single massless quark flavor:
\begin{equation}\label{tmnq}
T_{\mu\nu}^{quark} \, = \, \frac{i}{2} \, \overline{\psi} \,
(\gamma_\mu \, D_\nu + \gamma_\nu \, D_\mu) \, \psi.
\end{equation}
The corresponding energy density is given by \eq{edens}, which we
again want to rewrite as a double integral, just like \eq{edens2}, by
Fourier transforming the quark field
\begin{equation}\label{psi1}
\psi (x) \, = \, i \, \int \frac{d^4 k}{(2 \pi)^4} \,
\frac{k \cdot \gamma}{k^2 + i \epsilon k_0} \, e^{-i k \cdot x} \, \xi (k)
\end{equation}
and
\begin{equation}\label{psi2}
\overline{\psi} (x) \, = \, i \, \int \frac{d^4 k'}{(2 \pi)^4} \,
\tilde{\xi} (k') \, \frac{k' \cdot \gamma}{k'^2 + i \epsilon k'_0} \, e^{-i k' \cdot x}
\end{equation}
with $\xi (k)$ and $\tilde{\xi} (k')$ some spinors. In the following,
similar to Section \ref{arg1}, we will keep only the Abelian part of
$T_{\mu\nu}^{quark}$. The non-Abelian corrections are suppressed at
late times and can be neglected, since, just like in Section
\ref{arg1}, they fall off faster than the Abelian term by at least a
factor of $1/\sqrt{\tau}$. Replacing the covariant derivatives $D_\mu$
in \eq{tmnq} by a regular derivative $\partial_\mu$ and substituting
Eqs. (\ref{psi1}) and (\ref{psi2}) in it we obtain
\begin{equation}\label{tmnq1}
T_{\mu\nu}^{quark} \, = \, - \frac{1}{2} \, \int \frac{ d^4 k \, d^4
k'}{(2 \pi)^8} \, \frac{e^{-i k \cdot x - i k' \cdot x}}{(k^2 + i
\epsilon k_0) \, (k'^2 + i \epsilon k'_0)} \, \left< \tilde{\xi} (k') \,
k' \cdot \gamma \, (\gamma_\mu \, k_\nu + \gamma_\nu \, k_\mu) \, k
\cdot \gamma \, \xi (k) \right>.
\end{equation}
Rewriting
\begin{equation}\label{xhh}
\left<\left< \tilde{\xi} (k') \, k' \cdot \gamma \, \gamma_\mu \, k
\cdot \gamma \, \xi (k) \right>\right> = - \left[ (k_\mu - k'_\mu) \, h_1 (k^2,
k'^2, k \cdot k', k_T) + (k_\mu + k'_\mu) \, h_2 (k^2, k'^2, k \cdot k', k_T) \right]
\end{equation}
in \eq{tmnq1} and using \eq{edens} we write
\begin{eqnarray*}
\epsilon^{quark} (\tau) \, = \,
\int \frac{ d^4 k \, d^4 k'}{(2 \pi)^8}\,
\frac{e^{-i k \cdot x - i k' \cdot x}}{(k^2 + i \epsilon k_0) \, (k'^2
+ i \epsilon k'_0)} \,
\end{eqnarray*}
\begin{eqnarray*}
\times \, \Bigg\{ \Bigg[ \frac{1}{2} \, \left(
\frac{\tau}{x_+}\right)^2 \, (k_+ - k'_+) \, k_+ + k_+ \, k_-
- \frac{1}{2} \, (k'_+ \, k_- + k_+ \, k'_-) \Bigg] \, h_1 (k^2,
k'^2, k \cdot k', k_T) \, +
\end{eqnarray*}
\begin{equation}\label{qedens1}
+ \, \Bigg[ \frac{1}{2} \, \left(
\frac{\tau}{x_+}\right)^2 \, (k_+ + k'_+) \, k_+ + k_+ \, k_- + \frac{1}{2} \,
(k'_+ \, k_- + k_+ \, k'_-) \Bigg]
\, h_2 (k^2, k'^2, k \cdot k', k_T) \, \Bigg\} \, \frac{(2 \pi)^2}{S_\perp} \, \delta ({\underline k}
+ {\underline k}').
\end{equation}
Similar to Section \ref{arg1}, by putting $k = - k'$ and $k^2 = k'^2
=0$ in \eq{xhh} we derive
\begin{equation}\label{h1}
\frac{1}{S_\perp} \, h_1 (k^2 =0, k'^2 =0, k \cdot k' =0, k_T) \, = \, \frac{1}{S_\perp}
\left<\left< \tilde{\xi} (-k) \,
k \cdot \gamma \, \xi (k) \right>\right>\bigg|_{k^2 =0} \, = \, 2 (2
\pi)^3 \,
\frac{dN^q}{d^2k \, dy \, d^2 b},
\end{equation}
where $\frac{dN^q}{d^2k \, dy \, d^2 b}$ is the multiplicity of the
produced quarks. Arguing, just like we did for gluons, that the
leading-$\tau$ behavior for the quark energy density is given by $h_1
(k^2 =0, k'^2 =0, k \cdot k' =0, k_T)$ and $h_2 (k^2 =0, k'^2 =0, k
\cdot k' =0, k_T)$ in \eq{qedens1} we can integrate over the longitudinal
momenta in \eq{qedens1} using \eq{J7} in Appendix B obtaining
\begin{eqnarray*}
\epsilon^{quark} (\tau ) \, \approx \, \frac{\pi}{2} \, \int d^2 k \,
\frac{dN^q}{d^2k \, d\eta \, d^2 b} \, k_T^2 \, \left\{ - J_0 (k_T \tau) \,
J_2 (k_T \tau) + 2 [J_1 (k_T \tau)]^2 + [J_0 (k_T \tau)]^2 \right\} +
\end{eqnarray*}
\begin{equation}\label{qedens2}
+ \, \frac{1}{8 \, S_\perp} \, \int \frac{d^2 k}{(2 \, \pi)^2} \, h_2
(0, 0, 0, k_T) \, k_T^2 \,
\left\{ - J_0 (k_T \tau) \, J_2
(k_T \tau) - 2 [J_1 (k_T \tau)]^2 + [J_0 (k_T \tau)]^2 \right\},
\end{equation}
where we also substituted space-time rapidity $\eta$ instead of $y$ in
the quark multiplicity distribution, which makes no difference in the
boost invariant case we consider. We can assume that $h_2 (k^2 =0,
k'^2 =0, k \cdot k' =0, k_T)$ is a steeply falling function of $k_T$ for $k_T \gg
\langle k_T \rangle \sim Q_s$. The assumption is justified since $h_2 (k^2 =0,
k'^2 =0, k \cdot k' =0, k_T)$ comes from the same amplitude that gave $h_1 (k^2 =0,
k'^2 =0, k \cdot k' =0, k_T)$, which is equal to the quark spectrum, as shown in
\eq{h1}, which in turn is always a steeply falling function of
$k_T$ scaling at least like $\sim 1/k_T^4$ for $k_T$ above some scale
$\langle k_T \rangle \sim Q_s$. By the same argument $h_2 (k^2 =0,
k'^2 =0, k \cdot k' =0, k_T)$ should be regular (or at most
logarithmically divergent) at $k_T = 0$. Using these assumptions we
can argue that the integral in the second term on the right hand side
of \eq{qedens2} is dominated by $k_T \sim Q_s$, which allows us to
rewrite it as
\begin{equation}\label{t2}
\frac{1}{16 \, \pi \, S_\perp} \, h_2 (0, 0, 0, Q_s) \,
\int_0^{Q_s} dk_T \, k_T^3 \, \left\{ - J_0 (k_T \tau) \, J_2
(k_T \tau) - 2 [J_1 (k_T \tau)]^2 + [J_0 (k_T \tau)]^2 \right\}.
\end{equation}
Performing the integration in \eq{t2} one would obtain a linear
combination of hypergeometric functions, which can be shown to fall
off at least as $\sim 1/\tau^2$ at large $\tau$. Therefore the second
term on the right hand side of \eq{qedens2} falls off with $\tau$ at
least as $\sim 1/\tau^2$, and can be neglected if the first term on
the right hand side of \eq{qedens2} falls off with $\tau$ slower than
$\sim 1/\tau^2$.
This can be easily verified. At large $\tau$ the first term on the
right hand side of \eq{qedens2} gives
\begin{equation}\label{qed}
\epsilon^{quark} (\tau )\bigg|_{\tau \gg 1/\langle k_T \rangle} \, \approx \,
\frac{2}{\tau} \, \int d^2 k \, \frac{dN^q}{d^2k \, d\eta \, d^2 b} \, k_T \, = \,
\frac{1}{\tau} \, \frac{dE_T^{quarks}}{d\eta \, d^2 b},
\end{equation}
since the factor of $2$ accounts for the anti-quark contribution. We
can see that indeed the term in \eq{t2} is negligibly small compared
to \eq{qed}, which gives us the dominant contribution to the energy
density due to quarks at late times $\tau$. (Of course the typical
transverse momentum $\langle k_T \rangle$ in \eq{qed} does not have to be exactly
equal to the similar typical momentum for gluons in \eq{ed}: however,
the difference between the two is usually given by the ratio of the
Casimir operators, which is just a constant ($4/9$) and does not
change our argument above.)
We have shown that inclusion of massless quarks does not change the
conclusion of the previous Sections that the leading diagrammatic
contribution to energy density scales as $1/\tau$ at large $\tau$ for
any order in the coupling $g$. Therefore, it appears that inclusion of
quarks does not affect the onset of thermalization.
\section{Conclusions}
We have shown above in Eqs. (\ref{ed}) and (\ref{qed}) that gluon and
quark fields generated by Feynman diagrams in high energy heavy ion
collisions lead to energy densities scaling as $\epsilon \sim 1/\tau$
at $\tau~\gg~1/\langle k_T \rangle$. In the saturation/Color Glass picture of heavy
ion collisions, the typical momentum $\langle k_T \rangle$ is proportional to the
saturation scale $Q_s$. Therefore, the $1/\tau$ scaling of energy
density sets in at relatively {\sl early} times $\tau \sim 1/Q_s$
(even though throughout the paper we called these proper times {\sl
late} times). The remaining evolution of the system in the
perturbative scenario considered above, characterized by $\epsilon
\sim 1/\tau$ scaling, is reminiscent of the so-called {\sl free streaming},
where the system simply falls apart without particles interacting.
\begin{figure}[b]
\begin{center}
\epsfxsize=15cm
\leavevmode
\hbox{\epsffile{frst.eps}}
\end{center}
\caption{(A) Gluon field produced in a collision with all the interactions
throughout its proper time evolution. (B) The dominant contribution to
the gluon field comes from early-time interactions. }
\label{frst}
\end{figure}
To explain how this happens, let us provide a diagrammatic
interpretation of our conclusion of $\epsilon \sim 1/\tau$
scaling. Let us imagine a general gluon field produced in a heavy ion
collision as shown in \fig{frst}A. There the gluon field is first
produced in the nuclear collision at $\tau = 0$ denoted by the
$\otimes$ sign. The cross at the other end denotes the later point in
$\tau$ where we measure the energy density of the gluon field. In the
evolution of the system the gluon interacts with other gluon fields
produced in the collision in all possible ways, as shown in
\fig{frst}A. However, the proper times of these interactions are not fixed:
they are integrated over the whole range of $\tau$. Interactions may
also happen at different impact parameters, which are also integrated
over. The $\epsilon \sim 1/\tau$ scaling conclusion from
Eqs. (\ref{ed}) and (\ref{qed}) appears to indicate that the dominant
diagrams are given by \fig{frst}B, where all the interactions happen
at early times, after which the system simply falls apart. In other
words, the integrations over proper times of the interactions in
\fig{frst}A are dominated by early times of \fig{frst}B. $\langle k_T \rangle$, or
$Q_s$, being the only scale in the problem, sets the typical time
scale for the end of interaction period and the onset of free
streaming, $\tau_0 \sim 1/Q_s$. Such behavior has been previously
observed in the numerical simulations of the classical gluon fields
\cite{KV,KNV}.
Another way to physically understand our conclusion of $\epsilon \sim
1/\tau$ scaling is as follows. At any order in the coupling constant
$\alpha_s$ the gluon (or quark) field has a diagram (or several diagrams)
which is (are) non-zero if the gluon is put on mass-shell. This is just a statement
that gluon (or quark) multiplicity distribution can be expanded in a
perturbation series in $\alpha_s$. As we have shown above in Sections
\ref{arg1}, \ref{arg2} and \ref{arg3}, such diagrams always give energy
density scaling as $1/\tau$. Each diagram is dominated by the on-shell
particles free streaming away, which always leads to $\epsilon \sim
1/\tau$.
Therefore we have shown that the onset of thermalization and the
subsequent Bjorken or rapidity-dependent hydrodynamic expansion of the
system of quarks and gluons produced in heavy ion collisions can not
result from summation of Feynman diagrams. Nevertheless, there
exists a solid phenomenological evidence for the strong final state
interactions
\cite{dAtaphen,dAtaphob,dAtastar,brahms,aaphenix,aaphobos,aastar,Bj,EL,BDMPSfull,EL2,Zak,SW}
and hydrodynamic behavior
\cite{EKR,hydro1,hydro2,HN} of the system produced in heavy ion collisions
at RHIC, indicating a formation of strongly interacting quark-gluon
plasma (QGP). To reconcile it with the above argument that Feynman
diagrams lead to a free streaming behavior for both quarks and gluons,
one must conclude that non-perturbative QCD effects are instrumental
in QGP formation at RHIC. These could be the the non-perturbative
effects associated with infrared modes having momenta of the order of
$\Lambda_{QCD}$ which can not be represented by Feynman
diagrams. Therefore, the above argument does not apply to such
modes. Alternatively, the non-perturbative effects might be of the
nature similar to the ultra-soft modes in finite temperature
non-Abelian field theories, which have momenta of the order of $g^2 \,
T$ with $T$ the temperature of the system. It is well-known that
resummation of ultra-soft modes is a non-perturbative problem in
finite temperature QCD \cite{Linde}. It is also known that ultra-soft
modes are very important for many physical observables for equilibrium
QCD matter at finite temperature \cite{Bodeker,ASY,BI}. If they are
important for equilibrium QCD matter, it would be natural to suggest
that the ultra-soft modes could also play a major role in
non-equilibrium phenomena such as the onset of
thermalization. However, a more careful analysis of the issue is
needed in order to draw any conclusions. Such analysis is beyond the
scope of this paper.
Distinguishing which one of the two types of non-perturbative effects
plays a more important role in the process of thermalization would
also be important for our understanding of LHC heavy ion data. The
non-perturbative effects characterized by the scale $\Lambda_{QCD}$
are likely to be of little importance at LHC where the saturation
scale $Q_s$ is predicted to be much larger than $\Lambda_{QCD}$
shifting most partons away from the infrared region. At the same time,
the non-perturbative ultra-soft modes carrying momenta $g^2 \, T$ may
remain important even at high LHC energies if the relevant temperature
scales with the saturation scale, $T \sim Q_s$, increasing at high
energy.
\section*{Acknowledgments}
The author would like to thank Ian Balitsky, Eric Braaten, Ulrich
Heinz, Larry McLerran, Al Mueller and Dam Son for many informative
discussions. The author is also grateful to Ulrich Heinz for
proofreading the manuscript.
This work is supported in part by the U.S. Department of Energy under
Grant No. DE-FG02-05ER41377.
\renewcommand{\theequation}{A\arabic{equation}}
\setcounter{equation}{0}
\section*{Appendix A}
Here we are going to prove the following formula
\begin{equation}\label{I1}
I \, \equiv \, \int_{-\infty}^{\infty} d k_+ \, dk_- \, e^{- i k_+ x_-
- i k_- x_+} \, (k^2 + i \epsilon k_0)^{\Delta - 1} \, = \, - \frac{2
\pi^2}{\Gamma (1-\Delta)} \, \left( \frac{2 \, k_T}{\tau} \right)^\Delta
\, e^{i \, \pi \, \Delta} \, J_{-\Delta} (k_T \tau)
\end{equation}
with $k^2 = 2 \, k_+ \, k_- - {\underline k}^2$ and for $x_+ > 0$, $x_- > 0$,
and $\Delta >0$. Let us first rewrite the integral (\ref{I1}) as
\begin{equation}\label{I2}
I \, = \, 2^{\Delta - 1} \, \int_{-\infty}^{\infty} \frac{d k_+ \,
dk_- \, e^{- i k_+ x_- - i k_- x_+}}{(k_+ + i \epsilon )^{1-\Delta}} \,
\left( k_- - \frac{{\underline k}^2}{2 (k_+ + i \epsilon)} + i \epsilon
\right)^{\Delta - 1}.
\end{equation}
Defining $\tilde{k}_- = k_- - {\underline k}^2 / 2 k_+$ we write
\begin{equation}\label{I3}
I \, = \, 2^{\Delta - 1} \, \int_{-\infty}^{\infty} \frac{d k_+}{(k_+
+ i \epsilon )^{1-\Delta}} \, e^{- i k_+ x_- - i \frac{{\underline k}^2}{2
k_+ + i \epsilon} \, x_+} \, \int_{-\infty}^{\infty} d\tilde{k}_- \,
e^{- i \tilde{k}_- \, x_+} \, ( \tilde{k}_- + i \epsilon )^{\Delta - 1}.
\end{equation}
The $\tilde{k}_-$ integral can be easily performed by distorting the
integration contour around the branch cut. We obtain
\begin{equation}\label{I4}
I \, = \, - 2^{\Delta - 1} \, \frac{2 \pi i \, e^{i \frac{\pi}{2}
\Delta}}{\Gamma (1-\Delta)} \, x_+^{-\Delta} \, \int_{-\infty}^{\infty} \frac{d
k_+}{(k_+ + i \epsilon )^{1-\Delta}} \, e^{- i k_+ x_- - i \frac{{\underline
k}^2}{2 k_+ + i \epsilon} \, x_+}.
\end{equation}
Expanding the second term in the power of the exponent in \eq{I4} in a
Taylor series we write
\begin{equation}\label{I5}
I \, = \, - 2^{\Delta - 1} \, \frac{2 \pi i \, e^{i \frac{\pi}{2}
\Delta}}{\Gamma (1-\Delta)} \, x_+^{-\Delta} \, \sum_{n=0}^\infty \, \frac{1}{n!} \,
\left(\frac{- i {\underline k}^2 \, x_+}{2}\right)^n \int_{-\infty}^{\infty}
\frac{d k_+}{(k_+ + i \epsilon )^{n + 1-\Delta}} \, e^{- i k_+ x_-}.
\end{equation}
Performing the $k_+$ integration just like we did the $\tilde{k}_-$
integral above yields
\begin{equation}\label{I6}
I \, = \, - 2^{\Delta - 1} \, \frac{(2 \pi)^2 \, e^{i \pi
\Delta}}{\Gamma (1-\Delta)} \, (x_+ \, x_-)^{-\Delta} \, \sum_{n=0}^\infty \,
\frac{1}{n! \, \Gamma (n+1-\Delta)} \,
\left(\frac{- {\underline k}^2 \, x_+ \, x_-}{2}\right)^n.
\end{equation}
Remembering that $2 x_+ x_- = \tau^2$ and performing the summation
over $n$ we obtain \eq{I1} as desired.
\renewcommand{\theequation}{B\arabic{equation}}
\setcounter{equation}{0} \section*{Appendix B}
Our goal in this appendix is to perform the following integration
\begin{equation}\label{J1}
J \, \equiv \, \int_{-\infty}^{\infty} d k_+ \, dk_- \, e^{- i k_+ x_-
- i k_- x_+} \, (k^2 + i \epsilon k_0)^{\Delta - 1} \, (k_+ + i
\epsilon)^\lambda.
\end{equation}
Repeating the steps from Appendix A which led to \eq{I4} we write
\begin{equation}\label{J2}
J \, = \, - 2^{\Delta - 1} \, \frac{2 \pi i \, e^{i \frac{\pi}{2}
\Delta}}{\Gamma (1-\Delta)} \, x_+^{-\Delta} \, \int_{-\infty}^{\infty} \frac{d
k_+}{(k_+ + i \epsilon )^{1-\Delta - \lambda}} \, e^{- i k_+ x_- - i \frac{{\underline
k}^2}{2 k_+ + i \epsilon} \, x_+}.
\end{equation}
Expanding the second term in the exponent yields
\begin{equation}\label{J3}
J \, = \, - 2^{\Delta - 1} \, \frac{2 \pi i \, e^{i \frac{\pi}{2}
\Delta}}{\Gamma (1-\Delta)} \, x_+^{-\Delta} \, \sum_{n=0}^\infty \, \frac{1}{n!} \,
\left(\frac{- i {\underline k}^2 \, x_+}{2}\right)^n \int_{-\infty}^{\infty}
\frac{d k_+}{(k_+ + i \epsilon )^{n + 1-\Delta-\lambda}} \, e^{- i k_+ x_-}.
\end{equation}
Performing the $k_+$-integration we obtain
\begin{equation}\label{J4}
J \, = \, - 2^{\Delta - 1} \, \frac{(2 \pi)^2 \, e^{i \pi
\Delta + i \frac{\pi}{2} \lambda}}{\Gamma (1-\Delta)} \,
(x_+ \, x_-)^{-\Delta} \, x_-^{-\lambda} \, \sum_{n=0}^\infty \,
\frac{1}{n! \, \Gamma (n+1-\Delta-\lambda)} \,
\left(\frac{- {\underline k}^2 \, x_+ \, x_-}{2}\right)^n,
\end{equation}
which, after summing over $n$ gives
\begin{equation}\label{J5}
J \, = \, - \frac{2 \pi^2}{\Gamma (1-\Delta)} \,
\left( \frac{2 \, k_T}{\tau} \right)^\Delta
\, \left( \frac{k_T \, \tau}{2 \, x_-} \right)^\lambda
\, e^{i \, \pi \, \Delta + i \frac{\pi}{2} \lambda} \,
J_{-\Delta - \lambda} (k_T \tau).
\end{equation}
Noting that $x_\pm = \tau e^{\pm \eta} /\sqrt{2}$ with $\eta$ the
space-time rapidity we rewrite \eq{J5} as
\begin{equation}\label{J6}
J \, = \, - \frac{2 \pi^2}{\Gamma (1-\Delta)} \,
\left( \frac{2 \, k_T}{\tau} \right)^\Delta
\, \left( \frac{k_T}{\sqrt{2}} \right)^\lambda \, e^{\lambda \, \eta}
\, e^{i \, \pi \, \Delta + i \frac{\pi}{2} \lambda} \, J_{-\Delta - \lambda} (k_T \tau).
\end{equation}
As one can see from \eq{J6}, extra powers of $k_+$ in \eq{J1} as
opposed to \eq{I1} do not bring in any extra inverse powers of $\tau$:
they only modify the order of the Bessel function.
Finally, let us list here another useful integral, which can be easily
obtained by direct integration
\begin{equation}\label{J7}
\int_{-\infty}^{\infty} d k_+ \, dk_- \, \frac{e^{- i
k_+ x_- - i k_- x_+}}{k^2 + i \epsilon k_0} \, k_+^n \, k_-^m \, = \,
- 2 \, \pi^2 \, \left( \frac{i \, k_T \, \tau}{2 \, x_-} \right)^n \,
\left( \frac{- i \, k_T \, \tau}{2 \, x_+} \right)^m \, J_{m-n} (k_T \tau),
\end{equation}
where $n$ and $m$ are integers.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,335
|
using Orangemile.Cy.Param;
using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Text;
using System.Text.RegularExpressions;
using System.Threading.Tasks;
namespace Orangemile.Cy.Holder
{
public class TdsUtils
{
public static string normalize(string str)
{
if (str == null)
{
return "_";
}
str = str.Trim().ToLower();
List<char> chars = new List<char>(str.Length);
foreach ( char c in str )
{
if ( c == ' ')
{
chars.Add('_');
}
else if (Char.IsLetterOrDigit(c))
{
chars.Add(c);
}
}
return new string(chars.ToArray());
}
public static string toCsv(Tds tds, string rowdelim, string coldelim)
{
StringBuilder sb = new StringBuilder();
int rows = tds.getRowCount();
int columns = tds.getColumnCount();
object [,] data = tds.getData();
for (int r = 0; r < rows; r++)
{
if (r != 0)
{
sb.Append(rowdelim);
}
for (int c = 0; c < columns; c++)
{
object value = data[r, c];
if (c != 0)
{
sb.Append(coldelim);
}
sb.Append("" + value);
}
}
return sb.ToString();
}
public static Tds readFile(string tdsname, string fileName, Params ctrlParams)
{
string fieldSeperatorStr = (string)ctrlParams.getOneValue(new string[] { "sep", "fieldSep", "field_sep", "delimiter", "delim", "field_seperator", "seperator" });
char delimiter;
if (fieldSeperatorStr == null)
{
// derive from file extension
string ext = Path.GetExtension(fileName);
if (ext == ".csv" || ext == ".txt" )
{
delimiter = ',';
}
else if (ext == ".psv")
{
delimiter = '|';
}
else if (ext == ".tsv")
{
delimiter = '\t';
}
else
{
delimiter = ',';
}
} else {
if (fieldSeperatorStr.Trim().Length != 1)
{
throw new Exception("Invalid Field Seperator - only 1 character is supported");
}
delimiter = fieldSeperatorStr.Trim()[0];
}
int headerSkip = ctrlParams.getOneValueAsInt(new string[] {"skip", "header_skip", "header_rows", "topSkip", "top"}, 0);
int footerSkip = ctrlParams.getOneValueAsInt(new string[] { "footer_skip", "footer", "footer_rows", "bottomSkip", "bottom" }, 0);
List<object> headers = ctrlParams.getValues(new string[] { "header", "headers", "columns", "column_names", "columnNames", "names"});
string quoteStr = (string)ctrlParams.getOneValue(new string[] { "quote" });
char quote = '"';
bool hasQuotes = false;
if (quoteStr != null)
{
if (quoteStr.Trim().Length != 1)
{
throw new Exception("Invalid quote - only 1 character is supported");
}
hasQuotes = true;
quote = quoteStr.Trim()[0];
}
object[,] data;
Dictionary<string, string> _internPool = new Dictionary<string, string>();
using (StreamReader sr = new StreamReader(fileName))
{
int count = 0;
string line;
List<object[]> rows = new List<object []>();
while ( (line = sr.ReadLine()) != null)
{
if (line.Trim().Length == 0)
{
continue;
}
if (count < headerSkip)
{
count++;
continue;
}
List<string> row = Split(line, delimiter, hasQuotes, quote, _internPool);
rows.Add(row.ToArray());
count++;
}
int rowsCount = rows.Count;
if (rowsCount == 0 || rowsCount <= (1 + footerSkip))
{
throw new Exception("No rows read from file");
}
int headerLength = rows[0].Length;
data = new object[rowsCount - footerSkip, headerLength];
for (int i = 0; i < (rowsCount - footerSkip); i++)
{
object [] row = rows[i];
for (int k = 0; k < headerLength; k++)
{
if (row.Length > k)
{
data[i, k] = row[k];
}
}
}
}
return new InMemoryTds(tdsname, data);
}
public static List<string> Split(string line, char delimiter, bool hasQuotes, char quote, Dictionary<string, string> internPool)
{
List<string> fields = new List<string>();
char[] chars = line.ToCharArray();
bool fieldStart = true;
bool inquote = false;
int bufferPos = 0;
char[] buffer = new char[chars.Length];
for (int i = 0; i < chars.Length; i++)
{
if (fieldStart)
{
if (hasQuotes && chars[i] == quote)
{
fieldStart = false;
inquote = true;
continue;
}
else if (chars[i] == delimiter)
{
fields.Add(null);
fieldStart = true;
continue;
}
buffer[bufferPos++] = chars[i];
fieldStart = false;
}
else if (hasQuotes && chars[i] == quote)
{
// check next value for delimiter
if (chars.Length > (i + 1))
{
if (chars[i + 1] == delimiter)
{
// next char is delimiter
inquote = false;
continue;
}
// next char is not delimiter, so add, and continue
buffer[bufferPos++] = chars[i];
}
else
{
// reached end of line
break;
}
}
else if (chars[i] == delimiter)
{
if (inquote)
{
buffer[bufferPos++] = chars[i];
continue;
}
// found delimiter - and not in quotes
fields.Add(intern(new string(buffer, 0, bufferPos), internPool));
bufferPos = 0;
fieldStart = true;
continue;
}
else
{
buffer[bufferPos++] = chars[i];
}
}
// add remainder
if (bufferPos != 0)
{
fields.Add(intern(new string(buffer, 0, bufferPos), internPool));
}
return fields;
}
public static string intern(string value, Dictionary<string, string> internPool)
{
string result;
if (!internPool.TryGetValue(value, out result))
{
internPool[value] = value;
result = value;
}
return result;
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 39
|
Catalepsia patológica é uma doença rara em que os membros se tornam rígidos, mas não há contrações, embora os músculos se apresentem mais ou menos rijos. A pessoa fica o tempo todo consciente e quem passa por ela pode ficar horas nesta situação.
Causas
A catalepsia patológica ocorre em determinadas doenças nervosas, debilidade mental, histeria, intoxicação e alcoolismo.
Pode ser um sintoma de certas perturbações do sistema nervoso ou síndromes como o mal de Parkinson, síndrome neuroléptica maligna e epilepsia. É também um sintoma característico de abstinência de anfetaminas como cocaína. Além disso, pode ser causada no tratamento da esquizofrenia por antipsicóticos como o haloperidol ou do anestésico cetamina.
Catalepsia também é um termo usado pelo hipnotizador para descrever um braço ou perna "morto" (sem capacidade aparente de movimento) ou para o transe completo.
História
No passado já existiram casos de pessoas que foram enterradas vivas e na verdade estavam passando pela catalepsia patológica.
Muitos especialistas, contudo, afirmam que isso não seria possível nos dias de hoje pois já existem equipamentos tecnológicos que, quando corretamente utilizados, não falham ao definir os sinais vitais e permitem atestar o óbito com precisão. E como o tradicional exame da causa de morte envolve abrir o corpo para analisar os órgãos, se um paciente não chegar morto, certamente ele sairá morto do exame.
Caso Janina Kolkiewicz
Em novembro de 2014, uma polonesa de 91 anos, Janina Kolkiewicz, despertou dentro de um saco no necrotério de Ostrów Lubelski (leste da Polónia), 11 horas depois de uma médica ter certificado seu falecimento. A promotoria local iniciou uma investigação sobre este caso, segundo informaram os meios de comunicação locais, que relatam o terror vivido pelos funcionários do necrotério quando viram que algo se movimentava dentro do saco onde tinham sido depositado o corpo de Janina.
A médica Wieslawa Cyz, profissional que certificou o falecimento da mulher, assegurou que estava totalmente convencida de que a idosa havia morrido.
"Não havia pulso, não havia sinais de respiração e nem ritmo cardíaco." disse a médica ao jornal local "Dziennik Wschodni".
"Seus olhos estavam muito abertos, mas não eram sensíveis à luz." acrescentou Cyz, que confessou estar envergonhada por ter expedido o certificado de falecimento de uma paciente que ainda estava viva.
"Não há explicação." lamentou a doutora.
A sobrinha da idosa e também sua cuidadora, Bogumila Kolkiewicz, explicou à rede de televisão "TVP" que felizmente sua tia não tem nenhuma lembrança do ocorrido, porque sofre de demência senil e não tem consciência do que aconteceu.
A mulher de 91 anos está bem, de acordo com a sobrinha, que diz que ela "retornou da morte com frio e muito apetite".
"Quando me encontrei com Janina ela me pediu sopa, uma xícara de chá e dois crepes, portanto parece que tinha fome." acrescentou.
Na literatura
Um grande número de obras literárias possui personagens com esse transtorno, dentre eles um dos mais famosos é a obra de Alexandre Dumas, O Conde de Monte Cristo, onde Abbé Faria tem crises de catalepsia, de tempos em tempos, antes de finalmente morrer de uma dessas crises. É um tema recorrente nas obras de Edgar Allan Poe. No Brasil, o poeta ultrarromântico Álvares de Azevedo, em sua obra Noite na Taverna, menciona um caso de catalepsia patológica: o personagem Solfieri relata ter mantido relações sexuais com uma mulher cataléptica, após confundi-la com um cadáver.
Doenças neurológicas
Psicopatologias
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,254
|
Q: Почему TryParse работает не так как в C# У меня есть такой код
[bool] Read ([TextReader]$i)
{
[string]$text = $i.ReadLine();
[string[]]$numbers = $text.Split('.');
if ($numbers.Length -eq 4)
{
for ([int]$j = 0; $j -lt $numbers.Length;++ $j)
{
if (-not [byte]::TryParse($numbers[$j],[ref]$null))
{
return $false;
}
$this.data[$j] = [byte]::Parse($numbers[$j]);
}
return $true;
}
return $false;
}
Он работает, но почему не работает так
[bool] Read ([TextReader]$i)
{
[string]$text = $i.ReadLine();
[string[]]$numbers = $text.Split('.');
if ($numbers.Length -eq 4)
{
for ([int]$j = 0; $j -lt $numbers.Length;++ $j)
{
if (-not [byte]::TryParse($numbers[$j],[ref]$this.data[$j]))
{
return $false;
}
}
return $true;
}
return $false;
}
В C# TryParse меняет значение элемента массива, а в Powershell нет...
Что я упускаю?
A: Спасибо PetSerAl за ответ.
Изменил код так:
[bool] Read ([TextReader]$i)
{
[string]$text = $i.ReadLine();
[string[]]$numbers = $text.Split('.');
if ($numbers.Length -eq 4)
{
for ([int]$j = 0; $j -lt $numbers.Length;++ $j)
{
[byte]$temp = 0;
if (-not [byte]::TryParse($numbers[$j],[ref]$temp))
{
return $false;
}
$this.data[$j] = $temp;
}
return $true;
}
return $false;
}
Действительно, [ref] работает с переменными, но не работает с элементами массива. Буду Знать.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,591
|
{"url":"https:\/\/socratic.org\/questions\/if-cos-a-4-5-how-do-you-find-tan-2a#617823","text":"# If cos A = -4\/5 how do you find tan 2A?\n\n##### 2 Answers\nMay 21, 2018\n\n\u00ad\u00b140\/7 * 3\/5 = \u00b1 24\/7\n\n#### Explanation:\n\n$\\cos x = - \\frac{4}{5}$\n\n$\\tan 2 x = \\frac{\\sin 2 x}{\\cos 2 x} = \\frac{2 \\sin x \\cos x}{{\\cos}^{2} x - {\\underbrace{{\\sin}^{2} x}}_{1 - {\\cos}^{2} x}} = \\frac{- 2 \\cdot \\sin x \\cdot \\frac{4}{5}}{2 \\cdot \\frac{16}{25} - 1}$\n\n\u00b1 sin x = sqrt {1 - 16\/25} = sqrt{{25-16}\/25} = 3\/5\n\n$\\tan 2 x = - \\frac{8}{5} \\cdot \\sin x \\cdot \\frac{25}{32 - 25} = - \\frac{40}{7} \\sin x$\n\nMay 22, 2018\n\n24\/7\n\n#### Explanation:\n\n$\\cos A = - \\frac{4}{5}$.\nA could be in Quadrant 2 or Quadrant 3.\n${\\sin}^{2} A = 1 - {\\cos}^{2} A = 1 - \\frac{16}{25} = \\frac{9}{25}$\n$\\sin A = \\pm \\frac{3}{5}$\n$\\sin 2 A = 2 \\sin A . \\cos A = 2 \\left(\\pm \\frac{3}{5}\\right) \\left(- \\frac{4}{5}\\right) = \\pm \\frac{24}{25}$\n${\\cos}^{2} A = 1 - {\\sin}^{2} 2 A = 1 - \\left(\\frac{576}{625}\\right) = \\frac{49}{625}$\n$\\cos 2 A = \\pm \\frac{7}{25}$\n$\\tan 2 A = \\frac{\\sin 2 A}{\\cos 2 A} = \\left(\\pm \\frac{24}{25}\\right) \\left(\\pm \\frac{25}{7}\\right) = \\pm \\frac{24}{7}$\nIf A lies in Quadrant 2, then, 2A lies in Quadrant 3, and tan 2A is positive\nIf A lies in Q. 3, then, 2A lies in Q. 1, then, tan 2A is positive.\nFinally\n$\\tan 2 A = \\frac{24}{7}$","date":"2021-10-26 20:43:09","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 13, \"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\": 0.614814281463623, \"perplexity\": 7663.966944533275}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323587926.9\/warc\/CC-MAIN-20211026200738-20211026230738-00260.warc.gz\"}"}
| null | null |
Basic ES5 shim for older browsers.
Contains shims for the following objects:
* Function
* String
* Object
* Array
If any of the shimmed methods exist in the environment, they are not replaced.
Function
---------
Shim to Function.bind
String
------
Shim to String.trim
Object
-------
Shim to Object.keys and Object.create. For Object.create, only the first param is considered, that is, the prototype of the object to be created. Propeties descriptors are not parsed.
Array
------
Shim to
* forEach
* indexOf
* map
* filter
* some
* every
* lastIndexOf
* reduce
* reduceRight
* Array.isArray
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,190
|
module Slippers
class TemplateGroup
def initialize(params={})
@templates = params[:templates]
@super_group = params[:super_group]
@missing_handler = params[:missing_template_handler] || Slippers::Engine::MISSING_HANDLER
@default_string = params[:default_string] || Slippers::Engine::DEFAULT_STRING
end
attr_reader :missing_handler, :default_string
def find(subtemplate)
return nil unless @templates
return create_template(subtemplate.to_sym) if @templates.include?(subtemplate.to_sym)
find_in_super_group(subtemplate)
end
def has_registered?(class_name)
return false unless @templates
return true if @templates.include?(class_name)
return false unless @super_group
@super_group.has_registered?(class_name)
end
def render(item)
return '' unless @templates
return @templates[item.class].render(item) if has_registered?(item.class)
return '' unless @super_group
@super_group.render(item)
end
private
def find_in_super_group(subtemplate)
return nil unless @super_group
@super_group.find(subtemplate)
end
def create_template(subtemplate)
template = @templates[subtemplate]
return template unless template.is_a?(String)
Slippers::Engine.new(template)
end
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,122
|
\section{Introduction}
The Science4Cast 2021 competition \cite{Science4Cast, bigdata2021} addresses the challenge of predicting future connections in the ever-growing semantic network of scientific concepts used in the fields of Machine Learning and Artificial Intelligence in order to foresee which research topics will emerge in the coming years. In the competition we are given two instances of this dataset: one corresponding to the state of the network in 2014, and one to 2017. The ultimate goal would be to predict the state of the network in 2020, but the task at hand in the challenge is a simpler one: given a random set composed of $10^6$ links that do not exist in 2017 we must order them according to which we think have the best chances of being connected by 2020. For the 2014 to 2017 predictions we are given a similar random set for which we know the solution in 2017, allowing us to train a prediction model.
The challenge in this competition is closely related to the active field of link prediction in complex networks \cite{Liben:2007, Lu:2011, wang2015link}. Many well-known network-based link prediction methods such as Common Neighbours \cite{Liben:2007}, Resource-Allocation \cite{Zhou:2009} and Adamic-Adar \cite{adamic2003friends} follow one simple principle: two nodes are more likely to connect if they have many common neighbours. This is equivalent to counting paths of length 2, and typically implies similarity between nodes, favouring the creation of triangles in the network ($A\sim B$, $B\sim C$ $\Rightarrow$ $A\sim C$). In the context of predicting protein-protein interactions it has been shown that counting paths of length 3 is in fact a better predictor than paths of length 2 \cite{kovacs2019network}, the insight being that proteins interact not because they are similar between themselves, but because they are similar to each other's neighbours. This principle favours the creation of squares in the network. A more recent result \cite{qlp} shows that a quantum algorithm for link prediction based on continuous-time quantum walks can identify predictions based on both even and odd-length paths, which sometimes improves the precision over the aforementioned methods. The method encodes the prediction scores in the amplitudes of a quantum superposition, allowing the best links to be sampled from the distribution, avoiding the need to explicitly calculate all pair-wise scores in the network. When ran on a quantum computer, this method can potentially provide a speedup in link prediction.
\subsection{Technical details}
In both the 2014 and 2017 versions of the dataset we are given a list of links $(i,j,t_{ij})$ where $i$ and $j$ are two nodes that formed a link at a certain time $t_{ij}$. In our prediction models we mostly deal with the adjacency matrix $A$ of the network, which we build by adding a weight $w_{ij}$ to each entry $A_{ij}$ and $A_{ji}$ for each $(i, j)$ present in the original list. For the unweighted case we simply have $w_{ij} = 1$. We note that the dataset often has repeated links for the same pair of nodes $(i,j)$, which makes sense given that two concepts can be linked more than once at different times in different research articles. As such, even the unweighted case where all $w_{ij} = 1$ leads to an adjacency matrix where the entries $A_{ij}$ can be greater than 1, which is by itself a weighted adjacency matrix.
We also deal with the degree of the nodes which we denote by $k_i$ for a node $i$, and which can be computed directly from the adjacency matrix,
\begin{equation}
k_i=\sum_jA_{ij}.
\end{equation}
Finally we deal with common neighbours between nodes. Let $\Gamma(i)$ be the set of nodes neighbouring $i$. The number of common neighbours between $i$ and $j$ can be computed from the second power of the adjacency matrix,
\begin{equation}
|\Gamma(i)\cap\Gamma(j)| = (A^2)_{ij}.
\end{equation}
\section{Preliminary data analysis}
\subsection{Prediction set degree distribution}
As mentioned, we are given a list of randomly selected pairs of nodes that we must order from best to worst prediction. In both the training data and competition data this list contains $10^6$ pairs, which is only a very small subset of the full list of possible predictions, with a size on the order of $10^9$ for a sparse network with $N\sim 64,000$. In order to gain some insight into the types of models that will work best for this task it is useful to see how the nodes in each potential link that we must score are connected to the network. As such, for each pair $(v_1, v_2)$ we calculated the degree values $(k_1, k_2)$ and identified three categories of potential links: links between two nodes with zero degree, links between one node with zero degree and one with degree greater than zero, and links between two nodes with degree greater than zero. We show in Table \ref{tab:predictiondegrees} the number of pairs that fall in each category for both the 2014 and 2017 networks.
\begin{table}
\centering
\caption{Number of pairs of nodes that fall in each of the three degree categories.}
\label{tab:predictiondegrees}
\begin{tabular}{ccc}
\hline\hline
& Training (2014) & Competition (2017) \\\hline
$(k = 0, k = 0)$ & 265966 & 32571 \\
$(k = 0, k > 0)$ & 500060 & 297545\\
$(k > 0, k > 0)$ & 233974 & 669884\\\hline\hline
\end{tabular}
\end{table}
From Table \ref{tab:predictiondegrees} we note that there is a large quantity of links to be scored which include nodes with $k=0$, especially in the training set. This indicates that models solely based on paths between nodes, which are common in network-based link prediction, should not perform well in this task since they can only score links belonging to the same connected component of the network. Furthermore, we note that the distribution of pairs between each category is quite different between the training set and the competition set, which may introduce some difficulties in training a model suitable for the competition set.
\subsection{Network degree distribution}
We now look at the degree distribution of the whole network. Real-world complex networks are often called \textit{scale-free}, meaning that their degree distribution follows a power law \cite{barabasi2016network, RevModPhys.80.1275, Newman2010},
\begin{equation}
P(k)\propto k^{-\gamma},
\end{equation}
with $\gamma>2$, and where $P(k)$ is the density function describing the probability of finding a node with degree $k$ in that network. Values in the $2<\gamma<3$ range are common, and it is a region of interest since the second moment of the distribution diverges \cite{barabasi2016network, RevModPhys.80.1275, Newman2010}. Networks with $\gamma\leq 2$ are not well defined given that in such scenario the average degree diverges.
A scale-free distribution can be explained from a preferential attachment process, where new nodes that are added the network tend to connect to other nodes with a probability proportional to their degree,
\begin{equation}
\Pi(k_i)\propto k_i+B,
\end{equation}
with $\Pi(k_i)$ being the probability that a new node connects to a node with degree $k_i$, and $B$ is just a constant. As $B$ increases, the degree of the nodes becomes less relevant. $B$ can be related to $\gamma$ as $\gamma=4+B$ \cite{barabasi2016network,RevModPhys.80.1275,Newman2010}. Lower values of $\gamma$ thus mean that new nodes added to the network are more likely to be connected to nodes with higher degrees. Although this model does not consider new links that are created between nodes that are already part of the network, the same principle can be easily generalized to such a scenario.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{degree_distribution.pdf}
\caption{Degree distribution for both the 2014 and 2017 datasets.}
\label{fig:degreedistribution}
\end{figure}
In Figure \ref{fig:degreedistribution} we plot $P(k)$ for both the 2014 and 2017 networks, and fit a power law to the tail of the distribution. We find that in both cases we get a good fit for $\gamma=2.1$, which is close to the lower bound typically found in complex networks. This indicates that this dataset is likely to grow via a preferential attachment mechanism, and that will be the first hypothesis we will test. Intuitively, given the context of this dataset, this makes sense, as it is reasonable to consider that the popularity of each concept (measured by the degree of the respective node) will play an important role in how new connections are made with that concept.
\section{Network-based methods}
In this section we overview the two main methods we used to predict new connections in this competition. For the results described in this section we disregarded the time-stamp information provided in the data and simply built an adjacency matrix for the 2014 and 2017 networks given by the full list of links in the respective data files.
\subsection{Preferential attachment}
In the previous section we concluded that the growth of this dataset is likely to follow a preferential attachment mechanism. Preferential attachment scores in link prediction are often quantified as
\begin{equation}
s_{ij}^\text{PA}=k_i*k_j.
\end{equation}
However, this assumes the scoring of links between nodes that are already connected to the network, that is $k_{i,j}>0$, which as we already saw is not the case for all the links we must score in the competition. A possible description for the more general case of scoring both incoming nodes and already connected nodes can be described as
\begin{equation}
s_{ij}^\text{PA}=k_i + k_j + \epsilon\sqrt{k_i*k_j}.
\end{equation}
with $\epsilon$ a free parameter. In our tests with the training set we found that $\epsilon=0$ always performed better, and thus we defined our preferential attachment model (PA model) as
\begin{equation}
s_{ij}^\text{PA}=k_i + k_j. \label{eq:pa}
\end{equation}
Using this simple model we ran our first test on the competition data. Evaluating Eq. \ref{eq:pa} for each pair $(i,j)$ in the set of $10^6$ unconnected pairs we submitted the ordered list to the competition and obtained an AUC value of 0.89715. Immediately we note that PA outperforms the baseline Machine Learning model provided in the competition, an indication that our initial hypothesis was correct.
\subsection{Path-based}
While the preferential attachment model we derived performed well, it uses no information about the distance between $i$ and $j$, which is a popular feature used in link prediction methods, as mentioned in the introduction. As such we decided to test a selection of different path-based methods, including L3 \cite{kovacs2019network}, Common Neighbours \cite{Liben:2007}, Resource Allocation \cite{Zhou:2009} and Adamic-Adar \cite{adamic2003friends}. Ultimately, we found that all of them performed very similarly, indicating that different path structures are not very well defined in the network. Due to time and resource limitations we were unable to test a classical simulation of the quantum walk based method QLP \cite{qlp}, which includes contributions from higher order paths. Nevertheless, given the similar performance between the tested methods, it is unlikely that QLP would have performed better. Although there were only slight differences between the tested methods, we found Adamic-Adar (AA Method) to be the best performing one, with the scores defined in the unweighted case as
\begin{equation}
s_{ij}^\text{AA}=\sum_{u\in\Gamma(i)\cap\Gamma(j)}\frac{1}{\log k_u}
\label{eq:aa}
\end{equation}
where $\Gamma(i)\cap\Gamma(j)$ defines the set of common neighbours between nodes $i$ and $j$. Adamic-Adar is essentially a normalized count of nodes $u$ in this set. The penalization given by $1/\log(k_u)$ increases the importance of common neighbours with lower degree, the intuition being that those neighbours are more unique to $i$ and $j$ compared to other neighbours that may have more connections to other nodes. While Eq. \ref{eq:aa} is written for the unweighted case, in our code we use an adjacency matrix implementation which automatically incorporates the weights in the matrix.
Evaluating Eq. \ref{eq:aa} for each pair $(i,j)$ in the set of $10^6$ unconnected pairs and submitting the ordered list to the competition we obtained an AUC value of 0.87091, which is not as good as the PA method, but is still close to the baseline Machine Learning model.
\subsection{Combining methods}
So far we have described two different models to score links in the dataset: the popularity based PA method, and the similarity based AA method. In order to see if we can improve our overall results we wish to join the results from both methods. To do so we chose a simple linear combination of the normalized score arrays:
\begin{equation}
\textbf{s}^\text{PA+AA}=\epsilon*\textbf{s}^\text{AA}+(1-\epsilon)*\textbf{s}^\text{PA}
\end{equation}
with $\epsilon$ a free parameter, and where $\textbf{s}^\text{AA}$ and $\textbf{s}^\text{PA}$ are the full arrays of scores for the set of $10^6$ unconnected pairs. Running a grid search for $\epsilon$ between 0 and 1 in the training set we found the best result for $\epsilon=0.92$. Using this value we submitted a combination of the previous AA and PA scores to the competition to obtain an AUC of 0.91385, a small improvement over the standalone PA method. This result indicates that the neighbourhood of the nodes does have some information that impacts the predictions which is not captured by their degree.
\section{Time-weighted adjacency matrix}
In the analysis made so far we purposefully neglected the information that we have about the time $t_{ij}$ at which these links were created. Indeed, the fact that this network isn't a static one but is continuously adding new link is a fact that can't be ignored. Links which are older in time might, for example, represent really well established links between core concepts of the fields, whether newer links might suggest which are the "hot topics" of the research at the moment. In order to include this information in our link prediction methods, we decided to weight the links in the dataset as a function of the time at which they were created. More precisely we considered:
\begin{equation}
(i, j, t_{ij}) \rightarrow (i, j, f_{\theta}(\tilde{t}_{ij})),
\end{equation}
where $\tilde{t}_{ij}$ is the time $t_{ij}$ at which the link $(i,j)$ was created normalized to a value between 0 and 1, and $f(\cdot)$ is a generic function parameterized by some parameters $\theta$ that we will optimize in order to fit our problem. The purpose here is that by building the adjacency matrix for the network with the weighted links we effectively alter their importance for each method.
\begin{figure}
\centering
\includegraphics[width=0.8\columnwidth]{time_function.pdf}
\caption{Function used to define the time-weights in the network, where $t$ corresponds to the time-stamp in each link normalized to a value between 0 and 1, and f(t) corresponds to the link weight attributed to that time-stamp.}
\label{fig:time_function}
\end{figure}
To optimize $f_{\theta}$ one could resort to various techniques that make use of really general function approximators (e.g. Neural Networks). However, given the size of our problem and the computational constraints we started by restricting our optimization with some theoretical assumptions. As we hinted at earlier, our assumptions were exactly that the most important links are both those that are older and newer in time, representing rooted concepts and "hot topics", respectively. These assumptions can be easily quantified by a convex polynomial, and thus we parameterized our time-weighting function as:
\begin{equation}
f_{\theta}(t) = \theta_0 + (\theta_2\cdot(t-\theta_1))^{\theta_3}
\end{equation}
as illustrated in Figure \ref{fig:time_function} and optimized over the training set using both a general purpose optimizer from scipy and a greedy grid-search while maximizing the AUC for the PA method, given that this is both the more important and more efficient method. In the end the best set of parameters obtained from the training set was $(\theta_0, \theta_1, \theta_2, \theta_3) = (0.0, 0.45, 3, 6)$. We then repeated all the scoring methods described in the previous section, substituting the original adjacency matrix with our time-weighted one, and got an improved result of $\text{AUC}=0.90364$ for the PA method alone, and $0.91385$ for a combination of AA and PA with $a=0.95$. Through some more manual tests on the competition set we ultimately found the best result combining PA and AA of $\text{AUC}=0.91853$ for $(\theta_0, \theta_1, \theta_2, \theta_3) = (0.5, 0.5, 3, 6)$.
It is interesting to see that our hypothesis regarding older and newer links did indeed improve the results. A different optimization routine using a small Neural Network also produced functions showing a similar shape to those obtained from the polynomial model, albeit with reduced performance. Overall though, the improvement we have obtained in this section is quite small, and thus we can not claim this is a strong organizational principle in the growth of the network.
\section{Conclusions}
\begin{table}
\centering
\caption{Summary of results and learned parameters. The ranking does not necessarily reflect our preferred Bacalhau dishes.}
\label{tab:summary}
\begin{tabular}{c|c|cccc|c|c}
\hline\hline
Method & $\epsilon$ & $\theta_0$ & $\theta_1$ & $\theta_2$ & $\theta_3$ & AUC & Bacalhau Code\\\hline
PA & - & - & - & - & - & 0.89715 & à Brás \\
AA & - & - & - & - & - & 0.87091 & à G. de Sá\\
PA + AA & 0.92 & - & - & - & - & 0.91385 & com Todos\\
PA & - & 0.0 & 0.45 & 3.0 & 6.0 & 0.90364 & com Natas\\
PA + AA & 0.95 & 0.5 & 0.5 & 3.0 & 6.0 & 0.91853 & à Lagareiro \\\hline\hline
\end{tabular}
\end{table}
We present in Table \ref{tab:summary} a summary of the results obtained in this competition. Despite it being a simple model composed of handcrafted features based on known organizational principles of complex networks, our scoring method resulted to be competitive with respect to state of the art Machine Learning techniques in the Science4Cast 2021 competition, achieving 4th place in the leaderboard with less than $0.01$ difference in the AUC compared to the top positions.
It is worth mentioning that we did test other more complex models using more features such as predictions from the L3 method, Resource Allocation, Principal Component Analysis, and other node popularity measures such as the Eigenvector Centrality and Average Neighbour Degree. Ultimately, we found that neither of these methods improved our results, even when running a more complex optimization routine combining all of them. Regarding the time-weighting, as we mentioned earlier, we also tried a more general optimization routine using a small Neural Network representation of $f$, but found similar function shapes to those produced by the polynomial model.
Ultimately, the simplicity of our model is also its strongest feature. Even considering just the unweighted preferential attachment model from Eq.\ref{eq:pa}, we were able to reach an AUC close to 0.9 without training any parameters and negligible computational cost. This method can be easily applied to predict new links in the full dataset of 64000 nodes, or even larger datasets that follow the same growth principles. Adding the Adamic-Adar scores also does not require too many resources, consisting essentially of one sparse matrix product and one free parameter to train afterwards for the linear combination of the two score vectors. Optimizing the time-weights is what requires the most resources during training, but impose no added complexity to the models afterwards.
As for a general comment and future directions for the underlying challenge of this competition, we would like to briefly discuss the dataset itself. The dataset provided consists of a semantic network where links represent connections made between concepts in scientific papers. This network is in fact a projection of a larger bipartite network with one group of nodes representing papers and another representing concepts, and links existing solely between papers and concepts. This larger bipartite network is considered a primary network, as it is a more accurate description of the underlying system being modeled. When projecting down to a concept only network, it is likely that some of the organizational principles behind the growth of this system become lost. This type of analysis is described in more detailed in \cite{vasques2018degree}, where, for example, it is shown how the degree distribution of the projected network (as we plotted in Fig. \ref{fig:degreedistribution} for this network) relates to the degree distributions of both node types in the original network. As such, we believe that studying the underlying bipartite system will allow for the creation of better and more precise models to predict future connections. At the same time, the prediction task also becomes harder, as the goal will now be to predict the appearance of new nodes (papers) and the links that come with it (concepts studied).
\section*{Code Availability}
The code used in this work is available at:\\ \url{https://github.com/Buffoni/quantum-link-prediction}
\section*{Acknowledgements}
The authors thank the support from Funda\c{c}\~{a}o para a Ci\^{e}ncia e a Tecnologia (FCT, Portugal), namely through project UIDB/50008/2020, as well as from projects TheBlinQC and QuantHEP supported by the EU H2020 QuantERA ERA-NET Cofund in Quantum Technologies and by FCT (QuantERA/0001/2017 and QuantERA/0001/2019, respectively), and from the EU H2020 Quantum Flagship project QMiCS (820505). Furthermore, JPM acknowledges the support of FCT through scholarship SFRH/BD/144151/2019 and BC thanks the support from FCT through project CEECINST/00117/2018/CP1495.
\bibliographystyle{IEEEtran}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 2,884
|
Le sedicenni è un film del 1965 diretto da Luigi Petrini.
Trama
Collegamenti esterni
Film sentimentali
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,910
|
\section{Introduction}
In the era of big data, interpretable methods for compressing a high-dimensional dataset into a lower dimensional set which shares the same essential characteristics are imperative. Since the work of \citet{hotelling1933analysis}, principal component analysis (PCA) has been one of the most popular approaches for completing this task. Formally, given centered data $\bm{A} \in \mathbb{R}^{n \times p}$ and its normalized empirical covariance matrix $\bm{\Sigma}:=\frac{\bm{A} \bm{A}^\top }{{n-1}} \in \mathbb{R}^{p \times p}$, PCA selects one or more leading eigenvectors of $\bm{\Sigma}$ and subsequently projects $\bm{A}$ onto these eigenvectors. This can be achieved in $O(p^3)$ time by taking a singular value decomposition {\color{black}$\bm{\Sigma}=\bm{U}\bm{\Lambda}\bm{U}^\top$}.
A common criticism\footnote{\color{black}A second criticism of PCA is that, as set up here, it uses the sample correlation or covariance matrix. This is a drawback, because sample covariance matrices are poorly conditioned estimators which over-disperses the sample eigenvalues, particularly in high-dimensional settings. In practice, this can be rectified by, e.g., using a shrinkage estimator \citep[see, e.g.,][]{ledoit2004well}. We do not do so here for simplicity, but we recommend doing so if using the techniques developed in this paper in practice.} of PCA is that the columns of {\color{black}$\bm{U}$} are not interpretable, since each eigenvector is a linear combination of all $p$ original features. This causes difficulties because:
\begin{itemize}\itemsep0em
\item {\color{black}In medical applications such as cancer detection, PCs generated during exploratory data analysis need to supply interpretable modes of variation \citep{hsu2014sparse}.}
\item In scientific applications such as protein folding, each original co-ordinate axis has a physical interpretation, and the reduced set of co-ordinate axes should too.
\item In finance applications such as investing capital across index funds, each non-zero entry in each eigenvector used to reduce the feature space incurs a transaction cost.
\item If $p \gg n$, PCA suffers from a curse of dimensionality and becomes physically meaningless \citep{amini2008high}.
\end{itemize}
One common method for obtaining interpretable principal components is to stipulate that they are sparse, i.e., maximize variance while containing at most $k$ non-zero entries. This approach leads to the following non-convex mixed-integer quadratically constrained problem \citep[see][]{d2005direct}:
\begin{align}\label{OriginalSPCA}
\max_{\bm{x} \in \mathbb{R}^p} \ \bm{x}^\top \bm{\Sigma} \bm{x} \quad
\text{s.t.} \quad \bm{x}^\top \bm{x}= 1,\ \vert \vert \bm{x} \vert \vert_0 \leq k,
\end{align}
where the constraint $\vert \vert \bm{x} \vert \vert_0 \leq k$ forces variance to be explained in a compelling fashion.
\subsection{Background and Literature Review}
Owing to sparse PCA's fundamental importance in a variety of applications including best subset selection \citep{d2008optimal}, natural language processing \citep{zhang2012sparse}, compressed sensing \citep{candes2007dantzig}, and clustering \citep{luss2010clustering}, three distinct classes of methods for addressing Problem \eqref{OriginalSPCA} have arisen. Namely, (a) heuristic methods which obtain high-quality sparse PCs in an efficient fashion but do not supply guarantees on the quality of the solution, (b) convex relaxations which obtain certifiably near-optimal solutions by solving a convex relaxation and rounding, and (c) exact methods which obtain certifiably optimal solutions, albeit in exponential time
\paragraph{Heuristic Approaches: }
The importance of identifying a small number of interpretable principal components has been well-documented in the literature since the work of \citet{hotelling1933analysis} \citep[see also][]{jeffers1967two}, giving rise to many distinct heuristic approaches for obtaining high-quality solutions to Problem \eqref{OriginalSPCA}. Two interesting such approaches are to rotate dense principal components to promote sparsity \citep{kaiser1958varimax, richman1986rotation, jolliffe1995rotation}, or apply an $\ell_1$ penalty term as a convex surrogate to the cardinality constraint \citep{jolliffe2003modified, zou2006sparse}. Unfortunately, the former approach does not provide performance guarantees, while the latter approach {still results in} a non-convex optimization problem.
More recently, motivated by the need to rapidly obtain high-quality sparse principal components at scale, a wide variety of first-order heuristic methods have emerged. The first such \textit{modern} heuristic was developed by \citet{journee2010generalized}, and involves combining the power method with thresholding and re-normalization steps. By pursuing similar ideas, several related methods have since been developed \citep[see][]{witten2009penalized, hein2010inverse, richtarik2012alternating, luss2013conditional, yuan2013truncated}. Unfortunately, while these methods are often very effective in practice, they sometimes badly fail to recover an optimal sparse principal component, and a practitioner using a heuristic method typically has no way of knowing when this has occurred. Indeed, \citet{berk2017} recently compared $7$ heuristic methods, including most of those reviewed here, on $14$ instances of sparse PCA, and found that none of the heuristic methods successfully recovered an optimal solution in all $14$ cases {(i.e., no heuristic was right all the time).}
\paragraph{Convex Relaxations: }
Motivated by the shortcomings of heuristic approaches on high-dimensional datasets, and the successful application of semi-definite optimization in obtaining high-quality approximation bounds in other applications \citep[see][]{goemans1995improved, wolkowicz2012handbook}, a variety of convex relaxations have been proposed for sparse PCA. The first such convex relaxation was proposed by \citet{d2005direct}, who reformulated sparse PCA as the rank-constrained mixed-integer semidefinite optimization problem (MISDO):
\begin{equation}\label{sdospca1}
\begin{aligned}
\max_{\bm{X} \succeq \bm{0}} \quad \langle \bm{\Sigma}, \bm{X} \rangle\ \text{s.t.} \ \mathrm{tr}(\bm{X})=1, \ \Vert\bm{X}\Vert_0 \leq k^2,\ \mathrm{Rank}(\bm{X})=1,
\end{aligned}
\end{equation}
where $\bm{X}$ models the outer product $\bm{x}\bm{x}^\top$. {\color{black}Note that, for a rank-one matrix $\bm{X}$, the constraint $\Vert \bm{X}\Vert_0 \leq k^2$ in \eqref{sdospca1} is equivalent to the constraint $\Vert\bm{x}\Vert_0 \leq k$ in \eqref{OriginalSPCA}, since a vector $\bm{x}$ is $k$-sparse if its outer product $\bm{x}\bm{x}^\top$ is $k^2$-sparse.}
{After performing this reformulation,} \citet{d2005direct} relaxed both the cardinality and rank constraints and instead solved
\begin{equation}\label{sdospca1.relax}
\begin{aligned}
\max_{\bm{X} \succeq \bm{0}} \quad \langle \bm{\Sigma}, \bm{X} \rangle\ \text{s.t.} \ \mathrm{tr}(\bm{X})=1, \ \Vert\bm{X}\Vert_1 \leq k,
\end{aligned}
\end{equation}
which supplies a valid upper bound on Problem \eqref{OriginalSPCA}'s objective.
The semidefinite approach has since been refined in a number of follow-up works. Among others, \citet{d2008optimal}, building upon the work of \citet{ben2002tractable}, proposed a different semidefinite relaxation which supplies a sufficient condition for optimality via the primal-dual KKT conditions, and \citet{d2014approximation} analyzed the quality of the semidefinite relaxation in order to obtain high-quality approximation bounds. A common theme in these approaches is that they require solving large-scale semidefinite optimization problems. This presents difficulties for practitioners because state-of-the-art implementations of interior point methods such as \verb|Mosek| require $O(p^6)$ memory to solve Problem \eqref{sdospca1.relax}, and therefore currently cannot solve instances of Problem \eqref{sdospca1.relax} with $p \geq 300$ \citep[see][for a recent comparison]{bertsimas2019polyhedral}. {\color{black}Techniques other than interior point methods, e.g., ADMM or augmented Lagrangian methods as reviewed in \cite{majumdar2019survey} could also be used to solve Problem \eqref{sdospca1.relax}, although they tend to require more runtime than IPMs to obtain a solution of a similar accuracy and be numerically unstable for problem sizes where IPMs run out of memory \citep{majumdar2019survey}.}
{A number of works have also studied the statistical estimation properties of Problem \eqref{sdospca1.relax}, by assuming an underlying probabilistic model. Among others, \citet{amini2008high} have demonstrated the asymptotic consistency of Problem \eqref{sdospca1.relax} under a spiked covariance model once the number of samples used to generate the covariance matrix exceeds a certain threshold; see \cite{vu2012minimax, berthet2013optimal, wang2016statistical} for further results in this direction, \cite{miolane2018phase} for a recent survey.}
{ In an complementary direction, \citet{dey2018convex} has recently questioned the modeling paradigm of lifting $\bm{x}$ to a higher dimensional space by instead considering the following (tighter) relaxation of sparse PCA in the original problem space
\begin{align}\label{prob:l1relax_small}
\max_{\bm{x} \in \mathbb{R}^p}\quad \bm{x}^\top \bm{\Sigma}\bm{x}\quad \text{s.t.}\quad \Vert\bm{x}\Vert_2=1, {\color{black}\Vert \bm{x}\Vert_1 \leq \sqrt{k}}.
\end{align}
Interestingly, Problem \eqref{prob:l1relax_small}'s relaxation provides a $\left(1+\sqrt{\tfrac{k}{k+1}}\right)^2$-factor bound approximation of Problem \eqref{OriginalSPCA}'s objective, while Problem \eqref{sdospca1.relax}'s upper bound may be exponentially larger in the worst case \citep{amini2008high}. This additional tightness, however, comes at a price: Problem \eqref{prob:l1relax_small} is NP-hard to solve—indeed, providing a constant-factor guarantee on sparse PCA is NP-hard \citep{magdon2017np}—and thus \eqref{prob:l1relax_small} is best formulated as a MIO, while Problem \eqref{sdospca1.relax} can be solved in polynomial time.
}
More recently, by building on the work of \citet{kim2001second}, \citet[]{bertsimas2019polyhedral} introduced a second-order cone relaxation of \eqref{sdospca1} which scales to $p=1000s$, and matches the semidefinite bound after imposing a small number of cuts. Moreover, it typically supplies bound gaps of less than $5\%$. However, it does not supply an \textit{exact} certificate of optimality, which is often desirable, for instance in medical applications.
A fundamental drawback of existing convex relaxation techniques is that they are not coupled with rounding schemes for obtaining high-quality feasible solutions. This is problematic, because optimizers are typically interested in obtaining high-quality solutions, rather than certificates. In this paper, we take a step in this direction, by deriving new convex relaxations that naturally give rise to greedy and random rounding schemes. The fundamental point of difference between our relaxations and existing relaxations is that we derive our relaxations by rewriting sparse PCA as a MISDO and dropping an integrality constraint, rather than using more ad-hoc techniques.
\paragraph{Exact Methods:} Motivated by the successful application of mixed-integer optimization for solving statistical learning problems such as best subset selection \citep{bertsimas2020sparse} and sparse classification \citep{bertsimas2017sparse}, several exact methods for solving sparse PCA to certifiable optimality have been proposed. The first branch-and-bound algorithm for solving Problem \eqref{OriginalSPCA} was proposed by \citet{moghaddam2006spectral}, by applying norm equivalence relations to obtain valid bounds. However, \citet{moghaddam2006spectral} did not couple their approach with high-quality initial solutions and tractable bounds to prune partial solutions. Consequently, they could not scale their approach beyond $p=40$.
A more sophisticated branch-and-bound scheme was recently proposed by \citet{berk2017}, which couples tighter Gershgorin Circle Theorem bounds \citep[][Chapter 6]{horn1990matrix} with a fast heuristic due to \cite{yuan2013truncated} to solve problems up to $p=250$. However, their method cannot scale beyond $p=100$s, because the bounds obtained are too weak to avoid enumerating a sizeable portion of the tree.
Recently, the authors developed a framework for reformulating convex mixed-integer optimization problems with logical constraints \citep[see][]{bertsimas2019unified}, and demonstrated that this framework allows a number of problems of practical relevance to be solved to certifiably optimality via a cutting-plane method. In this paper, we build upon this work by reformulating Problem \eqref{OriginalSPCA} as a \textit{convex} mixed-integer semidefinite optimization problem, and leverage this reformulation to design a cutting-plane method which solves sparse PCA to certifiable optimality. A key feature of our approach is that we need not solve any semidefinite subproblems. Rather, we use {concepts} from SDO to design a semidefinite-free approach which uses simple linear algebra techniques.
{ Concurrently to our initial submission, \citet{li2020exact} also attempted to reformulate sparse PCA as an MISDO, and proposed valid inequalities for strengthening their formulation and local search algorithms for obtaining high-quality solutions at scale. Our work differs in the following two ways. First, we propose strengthening the MISDO formulation using the Gershgorin circle theorem and demonstrate that this allows our MISDO formulation to scale to problems with $p=100$s of features, while they do not, to our knowledge, solve any MISDOs to certifiable optimality where $p>13$. Second, we develop tractable second-order cone relaxations and greedy rounding schemes which allow practitioners to obtain certifiably near optimal sparse principal components even in the presence of $p=1,000$s of features. More remarkable than the differences between the works however is the similarities: more than $15$ years after \citet{d2005direct}'s landmark paper first appeared, both works proposed reformulating sparse PCA as an MISDO less than a week apart. In our view, this demonstrates that the ideas contained in both works transcend sparse PCA, and can perhaps be applied to other problems in the optimization literature which have not yet been formulated as MISDOs.}
\subsection{Contributions and Structure}
The main contributions of the paper are twofold. First, we reformulate sparse PCA exactly as a mixed-integer semidefinite optimization problem; a reformulation which is, to the best of our knowledge, novel. Second, we leverage this MISDO formulation to design efficient algorithms for solving non-convex mixed-integer quadratic optimization problems, such as sparse PCA, to certifiable optimality or {\color{black}within $1-2\%$ of optimality in practice} at a larger scale than existing state-of-the-art methods.
The structure and detailed contributions of the paper are as follows:
\begin{itemize}\itemsep0em
\item In Section \ref{sec:exact.misdo}, we reformulate Problem \eqref{OriginalSPCA} as a mixed-integer SDO. { We propose a cutting-plane method which solves it to certifiable optimality in Section \ref{sec:exact.oa}. Our algorithm decomposes the problem into a purely binary master problem and a semidefinite separation problem. Interestingly, we show in Section \ref{sec:exact.subpb} that the separation problems can be solved efficiently via a leading eigenvalue computation and does not require any SDO solver. Finally, {\color{black}the} Gershgorin Circle theorem has been empirically successful for deriving upper-bounds on the objective value of \eqref{OriginalSPCA} \citep{berk2017}. We theoretically analyze the quality of such bounds in Section \ref{sec:exact.circle} and show in Section \ref{sec:exact.oval} that tighter bounds derived from Brauer's ovals of Cassini theorem can also be imposed via mixed-integer second-order cone constraints.}
\item In Section \ref{sec:relaxandround}, we analyze the semidefinite reformulation's convex relaxation, and introduce a greedy rounding scheme (Section \ref{ssec:relax.bool}) which supplies high-quality solutions to Problem \eqref{OriginalSPCA} in polynomial time, {\color{black} together with a sub-optimality gap (see numerical experiments in Section \ref{sec:numres})}. To further improve the quality of rounded solution and the optimality gap, we introduce strengthening inequalities (Section \ref{ssec:validineq}). While solving the strengthened formulation exactly would result in an intractable MISDO problem, solving its relaxation and rounding the solution appears as an efficient strategy to return high-quality solutions with a {\color{black} numerical certificate} of near-optimality.
\item In Section \ref{sec:numres}, we apply the cutting-plane and random rounding methods to derive optimal and near optimal sparse principal components for problems in the UCI dataset. We also compare our method's performance against the method of \citet{berk2017}, and find that our exact cutting-plane method performs comparably, while our relax+round approach successfully scales to problems an order of magnitude larger {\color{black}and often returns solutions which outperform the exact method at sizes which the exact method cannot currently scale to}. A key feature of our numerical success is that we sidestep the computational difficulties in solving SDOs at scale by proposing semidefinite-free methods for solving the convex relaxations, i.e., solving second-order cone relaxations.
\end{itemize}
\paragraph{Notation: }
We let nonbold face characters such as $b$ denote scalars, lowercase bold faced characters such as $\bm{x}$ denote vectors, uppercase bold faced characters such as $\bm{X}$ denote matrices, and calligraphic uppercase characters such as $\mathcal{Z}$ denote sets. We let $[p]$ denote the set of running indices $\{1, ..., p\}$. We let $\mathbf{e}$ denote a vector of all $1$'s, $\bm{0}$ denote a vector of all $0$'s, and $\mathbb{I}$ denote the identity matrix, with dimension implied by the context.
We also use an assortment of matrix operators. We let $\langle \cdot,\cdot \rangle$ denote the Euclidean inner product between two matrices, $\Vert \cdot \Vert_F$ denote the Frobenius norm of a matrix, $\Vert \cdot \Vert_\sigma$ denote the spectral norm of a matrix, $\Vert \cdot \Vert_*$ denote the nuclear norm of a matrix, $\bm{X}^\dag$ denote the Moore-Penrose psuedoinverse of a matrix $\bm{X}$ and $S_+^p$ denote the $p \times p$ positive semidefinite cone; see \citet{horn1990matrix} for a general theory of matrix operators.
\section{An Exact Mixed-Integer Semidefinite {\color{black} Optimization Algorithm}}\label{sec:reformulation}
In Section \ref{sec:exact.misdo}, we reformulate Problem \eqref{OriginalSPCA} as a convex mixed-integer semidefinite optimization problem. From this formulation, we propose an outer-approximation scheme (Section \ref{sec:exact.oa}) which, as we show in Section \ref{sec:exact.subpb}, does not require solving any semidefinite problems. We improve convergence of the algorithm by deriving quality upper-bounds on Problem's \eqref{OriginalSPCA} objective value in Section \ref{sec:exact.circle} and \ref{sec:exact.oval}
{\color{black}
\subsection{A Mixed-Integer Semidefinite Reformulation} \label{sec:exact.misdo}
}
Starting from the rank-constrained SDO formulation \eqref{sdospca1}, we introduce binary variables $z_i$ to model whether $X_{i,j}$ is non-zero, via the logical constraint $X_{i,j}=0$ if $z_i=0$; note that we need not require that $X_{i,j}=0$ if $z_j=0$, since $\bm{X}$ is a symmetric matrix. By enforcing the logical constraint via $-M_{i,j}z_i \leq X_{i,j}\leq M_{i,j}z_i$ for sufficiently large $M_{i,j}>0$, Problem \eqref{sdospca1} becomes
\begin{align*}
\max_{\bm{z} \in \{0, 1\}^p: \bm{e}^\top \bm{z} \leq k} \ \max_{\bm{X} \in S^p_+} \quad & \langle \bm{\Sigma}, \bm{X} \rangle\\
\text{s.t.} \quad & \mathrm{tr}(\bm{X})=1,\ -M_{i,j}z_i \leq X_{i,j} \leq M_{i,j}z_i \ \forall i, j \in [p],\ \mathrm{Rank}(\bm{X})=1.
\end{align*}
To obtain a MISDO reformulation, we omit the rank constraint. In general, omitting a rank constraint generates a relaxation and induces some loss of optimality. Remarkably, this omission
is without loss of optimality in this case. Indeed, the objective is convex and therefore some rank-one extreme matrices $\bm{X}$ is optimal. We formalize this observation in the following theorem; note that a similar result—although in the context of computing Restricted Isometry constants and with a different proof—exists \citep[][]{gally2016computing}:
\begin{theorem}\label{thm:misdpreformthm}
Problem \eqref{OriginalSPCA} attains the same optimal objective value as the problem:
\begin{equation}\label{misdpprimal}
\begin{aligned}
\max_{\bm{z} \in \{0, 1\}^p: \bm{e}^\top \bm{z} \leq k} \ \max_{\bm{X} \in S^p_+} \quad & \langle \bm{\Sigma}, \bm{X} \rangle&\\ \text{s.t.} \quad & \mathrm{tr}(\bm{X})=1 \quad & \\
& \vert X_{i,j}\vert \leq M_{i,j}z_i \quad & \forall i, j \in [p],\\
& \color{black}\sum_{j=1}^p \vert X_{i,j}\vert \leq \sqrt{k} z_i \quad & \forall i \in [p],
\end{aligned}
\end{equation}
where $M_{i,i}=1$, {\color{black}and} $M_{i,j}=\frac{1}{2}$ if $j \neq i$.
\end{theorem}
\begin{remark}
Observe that if {\color{black} $k \leq \sqrt{n}$ and} we set $M_{i,j}=1 \ \forall i,j \in [p]$ in Problem \eqref{misdpprimal} {\color{black}and omit the valid inequality $\sum_{j=1}^p \vert X_{i,j}\vert \leq \sqrt{k} z_i$} then the optimal value of the continuous relaxation is trivially $\lambda_{\max}(\bm{\Sigma})$. Indeed, letting $\bm{x}$ be a leading eigenvector of the unconstrained problem (where $\Vert \bm{x}\Vert_2=1$), we can set $z_i=\vert x_i\vert \geq \vert x_i \vert \vert x_j\vert${\color{black}, where the inequality holds since $\Vert \bm{x}\Vert_2=1$,} and $X_{i,j}=x_ix_j$, meaning {\color{black}(a) $\sum_i z_i=\Vert \bm{x}\Vert_1 \leq k \leq \sqrt{n}$ {\color{black}by norm equivalence} and (b) $\vert X_{i,j}\vert \leq z_i$} and thus $(\bm{X}, \bm{z})$ solves this continuous relaxation. Therefore, setting $M_{i,j}=\frac{1}{2}$ if $j\neq i$ {\color{black}and/or imposing the valid inequality $\sum_{j=1}^p \vert X_{i,j}\vert \leq \sqrt{k} z_i$} is necessary for obtaining non-trivial relaxations {\color{black}whenever $k$ is small}.
\end{remark}
{\color{black}
}
\begin{proof}
It suffices to demonstrate that for any feasible solution to \eqref{OriginalSPCA} we can construct a feasible solution to \eqref{misdpprimal} with an equal or greater payoff, and vice versa.
\begin{itemize}\itemsep0em
\item Let $\bm{x} \in \mathbb{R}^{p}$ be a feasible solution to \eqref{OriginalSPCA}. Then, {\color{black}since $\Vert \bm{x}\Vert_1 \leq \sqrt{k}$}, $(\bm{X}:=\bm{x}\bm{x}^\top, \bm{z})$ is a feasible solution to \eqref{misdpprimal} with equal cost, where $z_i=1$ if $\vert x_i\vert>0$, $z_i=0$ otherwise.
\item Let $(\bm{X}, \bm{z})$ be a feasible solution to Problem \eqref{misdpprimal}, and let $\bm{X}=\sum_{i=1}^p \sigma_i \bm{x}_i\bm{x}_i^\top$ be a Cholesky decomposition of $\bm{X}$, where $\bm{e}^\top \bm{\sigma}=1, \bm{\sigma} \geq \bm{0}$, {\color{black}and $\Vert \bm{x}_i\Vert_2=1 \ \forall i \in [p]$}. Observe that $\Vert\bm{x}_i\Vert_0 \leq k\ \forall i \in [p], $ since we can perform the Cholesky decomposition on the submatrix of $\bm{X}$ induced by $\bm{z}$, and ``pad'' out the remaining entries of each $\bm{x}_i$ with $0$s to obtain the decomposition of $\bm{X}$. Therefore, let us set $\hat{\bm{x}}:=\arg\max_{i}[\bm{x}_i^\top \bm{\Sigma}\bm{x}_i]$. Then, $\hat{\bm{x}}$ is a feasible solution to \eqref{OriginalSPCA} with an equal or greater payoff.
\end{itemize}
Finally, we let $M_{i,i}=1$, $M_{i,j}=\frac{1}{2}$ if $i \neq j$, as the $2 \times 2$ minors imply $X_{i,j}^2 \leq X_{i,i}X_{j,j}\leq \frac{1}{4}$ whenever $i \neq j$ \citep[c.f.][Lemma 1]{gally2016computing}.
\end{proof}
Theorem \ref{thm:misdpreformthm} reformulates Problem \eqref{OriginalSPCA} as a mixed-integer SDO. Therefore, we can solve Problem \eqref{misdpprimal} using general branch-and-cut techniques for semidefinite optimization problems \citep[see][]{gally2018framework, kobayashi2019branch}. However, this approach is not scalable, as it comprises solving a large number of semidefinite subproblems and the community does not know how to efficiently warm-start interior point methods (IPMs) for SDOs.
Alternatively, we propose a saddle-point reformulation of Problem \eqref{misdpprimal} which avoids the computational difficulty of solving a large number of SDOs by exploiting problem structure, as we will show in Section \ref{sec:exact.subpb}. The following result reformulates Problem \eqref{misdpprimal} as a max-min saddle-point problem amenable to outer-approximation:
\begin{theorem}\label{thm:saddlepointtheorem}
Problem \eqref{misdpprimal} attains the same optimal value as the following problem:
\begin{align}\label{prob:saddlepointproblem}
\max_{\bm{z} \in \{0, 1\}^p: \ \bm{e}^\top \bm{z} \leq k} \quad f(\bm{z})\\ \label{eqn:separation}
\quad \text{ where } \quad f(\bm{z}):= \min_{\lambda \in \mathbb{R}, \bm{\alpha} \in \mathbb{R}^{p \times p} {\color{black},\bm{\beta} \in \mathbb{R}^p}} \quad & \lambda +\sum_{i=1}^p {z}_i {\color{black}\left(\sum_{j=1}^p M_{i,j}\max(0, \vert \alpha_{i,j}\vert-\beta_i)+{\color{black}\sqrt{k}\beta_i}\right)}\\
\text{s.t.} \quad & \lambda\mathbb{I}+\bm{\alpha} \succeq \bm{\Sigma}.\nonumber
\end{align}
\end{theorem}
\begin{remark}
The above theorem demonstrates that $f(\bm{z})$ is concave in $\bm{z}$, by rewriting it as the infimum of functions which are linear in $\bm{z}$ \citep[][]{boyd2004convex}.
\end{remark}
\begin{proof}
Let us {\color{black}introduce auxiliary variables $U_{i,j}$ to model the absolute value of $X_{i,j}$ and rewrite the inner optimization problem of \eqref{misdpprimal} as
\begin{equation}\label{prob:sparsepcainnerprimal}
\begin{aligned}\color{black}
f(\bm{z}):= \quad & \max_{\bm{X} \succeq \bm{0}, \bm{U}} \quad & \langle \bm{\Sigma}, \bm{X}\rangle\\
\text{s.t.} \quad & \mathrm{tr}(\bm{X})=1, \quad & & [\lambda]\\\
& \color{black} U_{i,j} \leq M_{i,j}z_i \ &\forall i, j \in [p], \quad & [\sigma_{i,j}]\\
& \color{black} \vert X_{i,j}\vert \leq U_{i,j} \ &\forall i, j \in [p],\quad & [\alpha_{i,j}]\\
& \color{black} \sum_{j=1}^p U_{i,j} \leq \sqrt{k} z_i &\forall i \in [p], \quad & [\beta_{i}]\\
\end{aligned}
\end{equation}
where we associate dual constraint multipliers with primal constraints in square brackets.}
For $\bm{z}$ such that $\bm{e}^\top \bm{z} \geq 1$, the maximization problem induced by $f(\bm{z})$ satisfies Slater's condition \citep[see, e.g.,][Chapter 5.2.3]{boyd2004convex}, strong duality applies and leads to
{\color{black}
\begin{align*}
f(\bm{z}) = \min_{\substack{\lambda \\ \bm{\sigma},\bm{\alpha},\bm{\beta} \geq \bm{0}}} \quad & \lambda+\sum_{i,j}\sigma_{i,j}M_{i,j}z_i+\sum_{i=1}^p \beta_i \sqrt{k}z_i\\
\text{s.t.} \quad & \lambda \mathbb{I}+\bm{\alpha} \succeq \bm{\Sigma}, \vert \alpha_{i,j}\vert \leq \sigma_{i,j}+\beta_i.
\end{align*}}
{\color{black}
We eliminate $\bm{\sigma}$ from the dual problem above by optimizing over $\sigma_{i,j}$ and setting $\sigma^\star_{i,j}=\max(0, \vert \alpha_{i,j}\vert-\beta_i)$.}
Note that for $\bm{z}=\bm{0}$, the primal subproblem is infeasible and the dual subproblem has objective $-\infty$, but this can safely be ignored since $\bm{z}=\bm{0}$ is certainly suboptimal.
\end{proof}
\subsection{A Cutting-Plane Method} \label{sec:exact.oa}
Theorem \ref{thm:saddlepointtheorem} shows that evaluating $f(\bm{\hat{z}})$ yields the globally valid overestimator: $$f(\bm{z}) \leq f(\hat{\bm{z}})+\bm{g}_{\hat{\bm{z}}}^\top(\bm{z}-\hat{\bm{z}}),$$ where $\bm{g}_{\hat{\bm{z}}}$ is a supergradient of $f$ at $\bm{\hat{z}}$, at no additional cost. In particular, we have
$$g_{\hat{\bm{z}},i}={\color{black}\left(\sum_{j=1}^p M_{i,j}\max\left(0, \vert \alpha_{i,j}^\star(\bm{\hat{z}})\vert-\beta_i(\hat{\bm{z}})\right)+{\color{black}\sqrt{k}\beta_i(\hat{\bm{z}})}\right)},$$
where {\color{black}$\bm{\alpha}^\star(\hat{\bm{z}})$, $\bm{\beta}^\star(\hat{\bm{z}})$ constitutes an optimal choice of $(\bm{\alpha}, \bm{\beta})$ for a fixed $\bm{\hat{\bm{z}}}$}. This observation leads to an efficient strategy for maximizing $f(\bm{z})$: iteratively maximizing and refining a piecewise linear upper estimator of $f(\bm{z})$. This strategy is called outer-approximation (OA), and was originally proposed by \citet{duran1986outer}. OA works by iteratively constructing estimators of the following form at each {\color{black}iteration} $t$:
\begin{align}
f^t(\bm{z})=\min_{1 \leq i \leq t} \left\{f(\bm{z}_i)+\bm{g}_{\bm{z}_i}^\top (\bm{z}-\bm{z}_i)\right\}.
\end{align}
After constructing each overestimator, we maximize $f^t(\bm{z})$ over $\{0, 1\}^p$ to obtain $\bm{z}_t$, and evaluate $f(\cdot)$ and its supergradient at $\bm{z}_t$. This procedure yields a non-increasing sequence of overestimators $\{f^t(\bm{z}_t)\}_{t=1}^T$ which converge to the optimal value of $f(\bm{z})$ within a finite number of iterations {\color{black}$T \leq {p \choose 1}+\ldots+{p \choose k}$}, since $\{0, 1\}^p$ is a finite set and OA never visits a point twice. Additionally, we can avoid solving a different MILO at each OA iteration by integrating the entire algorithm within a single branch-and-bound tree, as proposed by \cite{quesada1992lp}, using \verb|lazy constraint callbacks|. Lazy constraint callbacks are now standard components of modern MILO solvers such as \verb|Gurobi| or \verb|CPLEX| and substantially speed-up OA. We formalize this procedure in Algorithm \ref{alg:cuttingPlaneMethod}; note that $\partial f(\bm{z}_{t+1})$ denotes the set of supergradients of $f$ at $\bm{z}_{t+1}$.
\begin{algorithm*}
\caption{An outer-approximation method for Problem \eqref{OriginalSPCA}}
\label{alg:cuttingPlaneMethod}
\begin{algorithmic}\normalsize
\REQUIRE Initial solution $\bm{z}_1$
\STATE $t \leftarrow 1 $
\REPEAT
\STATE Compute $\bm{z}_{t+1}, \theta_{t+1}$ solution of
{\vspace{-2mm}
\begin{align*}
\max_{\bm{z} \in\{0, 1\}^p: \bm{e}^\top \bm{z} \leq k, \theta} \: \theta \quad \mbox{ s.t. } \theta \leq f(\bm{z}_i) + \bm{g}_{\bm{z}_i}^\top (\bm{z}-\bm{z}_i) \ \forall i \in [t],
\end{align*}}\vspace{-5mm}
\STATE Compute $f(\bm{z}_{t+1})$ and $\bm{g}_{\bm{z}_{t+1}} \in \partial f(\bm{z}_{t+1})$ by solving \eqref{eqn:separation}
\STATE $t \leftarrow t+1 $
\UNTIL{$ f(\bm{z}_t)-\theta_t \leq \varepsilon$}
\RETURN $\bm{z}_t$
\end{algorithmic}
\end{algorithm*}
\subsection{A { Semidefinite-free} Subproblem Strategy} \label{sec:exact.subpb}
Our derivation and analysis of Algorithm \ref{alg:cuttingPlaneMethod} indicates that we can solve Problem \eqref{OriginalSPCA} to certifiable optimality by solving a (potentially large) number of semidefinite subproblems \eqref{eqn:separation}, which might be prohibitive in practice.
Therefore, we now derive a computationally efficient subproblem strategy which crucially does not require solving \textit{any} semidefinite programs. Formally, we have the following result:
\begin{theorem}\label{compefficientsubproblem}\color{black}
For any $\bm{z} \in \{0, 1\}^p: \bm{e}^\top \bm{z} \leq k$, optimal dual variables in \eqref{eqn:separation} are
\begin{align}\color{black}
\lambda=\lambda_{\max}\left(\bm{\Sigma}_{1,1}\right),\
\hat{\bm{\alpha}}=\begin{pmatrix} \hat{\bm{\alpha}}_{1,1} & \hat{\bm{\alpha}}_{1,2}\\ \hat{\bm{\alpha}}_{1,2}^\top & \hat{\bm{\alpha}}_{2,2}\end{pmatrix}=\begin{pmatrix} \bm{0} & \bm{0}\\ \bm{0} & \bm{\Sigma}_{2,2}-\lambda \mathbb{I}+\bm{\Sigma}_{1,2}^\top \left(\lambda \mathbb{I}-\bm{\Sigma}_{1,1}\right)^\dag \bm{\Sigma}_{1,2}\end{pmatrix},\\
{\color{black}\beta_{i}=(1-z_i)\begin{pmatrix}\vert\hat{\alpha}_{i,1}\vert, \vert\hat{\alpha}_{i,2}\vert, \ldots, \vert\hat{\alpha}_{i,i}\vert, \vert\hat{\alpha}_{i,i}\vert, \ldots, \vert\hat{\alpha}_{i,p}\vert\end{pmatrix}_{[\ceil{\sqrt{k}\ }]} \ \forall i \in [p],}
\end{align}
where $\lambda_{\max}(\cdot)$ denotes the leading eigenvalue of a matrix, $\hat{\bm{\alpha}}=\begin{pmatrix} \hat{\bm{\alpha}}_{1,1} & \hat{\bm{\alpha}}_{1,2}\\ \hat{\bm{\alpha}_{1,2}^\top} & \hat{\bm{\alpha}}_{2,2}\end{pmatrix}$ is a {\color{black}permutation} such that $\hat{\bm{\alpha}}_{1,1}$ (resp. $\hat{\bm{\alpha}}_{2,2}$) denotes the entries of $\hat{\bm{\alpha}}$ where $z_i=z_j=1$ ($z_i=z_j=0$); $\bm{\Sigma}$ is similar{\color{black}, and $(\bm{x})_{[k]}$ denotes the $k$th largest element of $\bm{x}$.}
\end{theorem}
\begin{remark}
By Theorem \ref{compefficientsubproblem}, Problem \eqref{eqn:separation} can be solved by computing the leading eigenvalue of $\bm{\Sigma}_{1,1}$ and solving a linear system. This justifies our claim that we need not solve any SDOs in our algorithmic strategy.
\end{remark}
\begin{proof}
We appeal to strong duality and complementary slackness. Observe that, for any $\bm{z} \in \{0, 1\}^p$, $f(\bm{z})$ is the optimal value of a maximization problem over a closed convex compact set. Therefore, there exists some optimal primal solution $\bm{X}^\star$ without loss of generality. Moreover, since the primal has non-empty relative interior {with respect to the non-affine constraints, it satisfies the Slater constraint qualification and} strong duality holds \citep[see, e.g.,][Chapter 5.2.3]{boyd2004convex}. Therefore, by complementary slackness \citep[see, e.g.,][Chapter 5.5.2]{boyd2004convex}, there must exist some dual-optimal solution $(\lambda, \hat{\bm{\alpha}}, \bm{\beta})$ which obeys complementarity with $\bm{X}^\star$. Moreover, $\vert X_{i,j}\vert \leq M_{i,j}$ is implied by $\mathrm{tr}(\bm{X})=1, \bm{X} \succeq \bm{0}$, while $\sum_{j=1}^p \vert X_{i,j}\vert \leq z_i \sqrt{k}$ is implied by $\vert X_{i,j}\vert
\leq M_{i,j}z_i$ and $\bm{e}^\top \bm{z} \leq k$. Therefore, by complementary slackness, we can take the constraints $\vert X_{i,j}\vert \leq M_{i,j}z_i$, {\color{black}$\sum_{j=1}^p \vert X_{i,j}\vert \leq z_i \sqrt{k}$} to be inactive when $z_i=1$ without loss of generality, which implies that $\hat{\alpha}_{i,j}^\star, {\color{black} \beta_i^\star}=0$ if $z_i=1$ in some dual-optimal solution. Moreover, we also have $\hat{\alpha}_{i,j}^\star=0$ if $z_j=1$, since $\hat{\bm{\alpha}}$ obeys the dual feasibility constraint $\lambda \mathbb{I}+\hat{\bm{\alpha}}\succeq \bm{\Sigma}$, and therefore is itself symmetric.
Next, observe that, by strong duality, $\lambda=\lambda_{\max}(\bm{\Sigma}_{1,1})$ in this dual-optimal solution, since $\bm{\alpha}$ only takes non-zero values if $z_i=z_j=0$ and does not contribute to the objective{\color{black}, and $\bm{\beta}$ is similar}.
Next, observe that, by strong duality and complementary slackness, any dual feasible $(\lambda, \hat{\bm{\alpha}}, {\color{black} \bm{\beta}})$ satisfying the above conditions is dual-optimal. Therefore, we need to find an $\hat{\bm{\alpha}}_{2,2}$ such that
\begin{align*}
\begin{pmatrix}\lambda\mathbb{I} -\bm{\Sigma}_{1,1} & -\bm{\Sigma}_{1,2}\\ -\bm{\Sigma}_{2,1} & \lambda\mathbb{I}+\hat{\bm{\alpha}}_{2,2}-\bm{\Sigma}_{2,2}
\end{pmatrix}\succeq \bm{0}.
\end{align*}
By the generalized Schur complement lemma \citep[see][Equation 2.41]{boyd1994linear}, this is PSD if and only if
\begin{enumerate}\itemsep0em
\item $\lambda\mathbb{I} -\bm{\Sigma}_{1,1} \succeq \bm{0}$,
\item $\left(\mathbb{I}-(\lambda\mathbb{I} -\bm{\Sigma}_{1,1})(\lambda\mathbb{I} -\bm{\Sigma}_{1,1})^\dag\right) \bm{\Sigma}_{1,2}=\bm{0}$, and
\item $\lambda \mathbb{I}+\hat{\bm{\alpha}}_{2,2}-\bm{\Sigma}_{2,2}\succeq \bm{\Sigma}_{1,2}^\top \left(\lambda \mathbb{I}-\bm{\Sigma}_{1,1}\right)^\dag \bm{\Sigma}_{1,2}$.
\end{enumerate}
The first two conditions hold {because, as argued above, $\lambda$ is optimal and therefore feasible, and the conditions are independent of $\hat{\bm{\alpha}}_{2,2}$}. Therefore, it suffices to pick $\hat{\bm{\alpha}}_{2,2}$ in order that the third condition holds. We achieve this by setting $\hat{\bm{\alpha}}_{2,2}$ so the PSD constraint in condition (3) holds with equality.
Finally, let us {\color{black} optimize for} $\bm{\beta}$ to obtain stronger cuts (when $z_i=0$ we can pick any feasible $\beta_i$, but optimizing to set $\partial f(\bm{z})_i$ to be as small as possible gives stronger cuts). {\color{black} This is equivalent to solving the univariate minimization problem for each $\beta_i$:
\begin{align*}
\min_{\beta_i} \left(\sum_{j=1}^p M_{i,j}\max(0, \vert \alpha_{i,j}\vert-\beta_i)+{\color{black}\sqrt{k}\beta_i}\right).
\end{align*}
Moreover, it is a standard result from max-$k$ optimization \citep[see, e.g.,][]{zakeri2014optimization, todd2018max} that
this is achieved by setting $\beta_i$ to be the $\lceil \sqrt{k}\ \rceil$ largest element of $\{{\alpha}_{i,j}\}_{j \in [p]} \cup \{\alpha_{i,i}\}$ in absolute magnitude, where we include $\alpha_{i,i}$ twice since $M_{i,i}=1$ while $M_{i,j}=1/2$ if $j \neq i$. }
\end{proof}
\subsection{Strengthening the Master Problem via the Gershgorin Circle Theorem} \label{sec:exact.circle}
To accelerate Algorithm \ref{alg:cuttingPlaneMethod},
we strengthen the master problem by imposing bounds from the circle theorem. Formally, we have the following result, which can be deduced from \citep[Theorem 6.1.1]{horn1990matrix}:
\begin{theorem}\label{thm:circletheorem}
For any vector $\bm{z} \in \{0, 1\}^p$ we have the following upper bound on $f(\bm{z})$
\begin{align}
f(\bm{z}) \leq \max_{j \in [p]: z_j=1}\sum_{i \in [p]}z_i \vert \Sigma_{i,j}\vert.
\end{align}
\end{theorem}
Observe that this bound cannot be used to \textit{directly} strengthen Algorithm \ref{alg:cuttingPlaneMethod}'s master problem, since the bound is not convex in $\bm{z}$. Nonetheless, it can be successfully applied if we (a) impose a big-M assumption on Problem \eqref{OriginalSPCA}'s optimal objective and (b) introduce $p$ additional binary variables $\bm{s} \in \{0, 1\}^p: \bm{e}^\top \bm{s}=1$ {which model whether the $i$th Gershgorin disc is active; recall that each eigenvalue is contained in the union of the discs}. Formally, we impose the following valid inequalities in the master problem:
\begin{align}\label{eqn:gershgorincircle}
\exists \bm{s} \in \{0, 1\}^p: \ & \theta \leq \sum_{i \in [p]} z_i \vert \Sigma_{i,j}\vert+M(1-s_j) \ \forall j \in [p], \bm{e}^\top \bm{s}=1, \bm{s} \leq \bm{z},
\end{align}
{ where $\theta$ is the epigraph variable maximized in the master problem stated in Algorithm \ref{alg:cuttingPlaneMethod}{, and $M$ is an upper bound on the sum of the $k$ largest absolute entries in any column of $\bm{\Sigma}$.} Note that we set $\bm{s}\leq \bm{z}$ since if $z_i=0$ the $i$th column of $\bm{\Sigma}$ does not feature in the relevant submatrix of $\bm{\Sigma}$.}
In the above inequalities, a valid $M$ is given by any bound on the optimal objective. Since Theorem {\color{black}\ref{thm:circletheorem}} supplies one such bound for any given $\bm{z}$, we can compute \begin{align}
M:=\max_{j \in [p]}\max_{\bm{z} \in \{0, 1\}^p: \bm{e}^\top \bm{z} \leq k} \ \sum_{i \in [p]}z_i \vert \Sigma_{i,j}\vert,
\end{align} {which can be done in $O(p^2)$ time.
{\color{black}To further improve Algorithm \ref{alg:cuttingPlaneMethod}, we also make use of the Gershgorin circle theorem before generating each outer-approximation cut. Namely, at a given node in a branch-and-bound tree, there are indices $i$ where $z_i$ has been fixed to $1$, indices $i$ where $z_i$ has been fixed to $0$, and indices $i$ where $z_i$ has not yet been fixed. Accordingly, we compute the worst-case Gershgorin bound—by taking the worst-case bound over each index $j$ such that $z_j$ has not yet been fixed to $0$, i.e., $$\max_{j: z_j\neq 0}\left\{\max_{\bm{s} \in \{0, 1\}^p: \bm{e}^\top \bm{s} \leq k}\left\{\sum_{i \in [p]}s_i \vert \Sigma_{i,j}\vert \ \text{s.t.} \ s_i=0\ \text{if}\ z_i=0, s_i=1 \ \text{if}\ z_i=1\right\}\right\}.$$ If this bound is larger than our incumbent solution then we generate an outer-approximation cut, otherwise the entire subtree rooted at this node does not contain an optimal solution and we use instruct the solver to avoid exploring this node via a \verb|callback|.}
{
Our numerical results in Section \ref{sec:numres} echo the empirical findings of \citet{berk2017} and indicate that Algorithm \ref{alg:cuttingPlaneMethod} performs substantially better when the Gershgorin bound is supplied in the master problem. Therefore, it is interesting to theoretically investigate the tightness, or at least the quality, of Gershgorin's bound.
We supply some results in this direction in the following proposition:
\begin{proposition}\label{prop:gershgorinthmapprox}
Suppose that $\bm{\Sigma}$ is a scaled diagonally dominant matrix as defined by \cite{boman2005factor}, i.e., there exists some vector $\bm{d}>0$ such that $$d_i\Sigma_{i,i} \geq \sum_{j \in [p]: j \neq i}d_j\vert \Sigma_{i,j}\vert \ \forall i \in [p].$$ Then, letting $\rho:=\max_{i,j \in [p]} \{\frac{d_i}{d_j}\}$, the Gershgorin circle theorem provides a $(1+\rho)$-factor approximation, i.e., \begin{align}
f(\bm{z}) \leq \max_{j \in [p]}\left\{\sum_{i \in [p]} z_i \vert \Sigma_{i,j}\vert \right\}\leq (1+\rho) f(\bm{z}) \quad \forall \bm{z} \in \{0, 1\}^p.
\end{align}
\end{proposition}
{\color{black}
\begin{remark}
Observe that, for a fixed $\bm{z}$, the ratio $\rho:=\max_{i,j \in [p]} \{\frac{d_i}{d_j}\}$ need only be computed over indices $i,j$ such that $z_i,z_j=1$. Moreover, for a partially specified $\bm{z}$—which might arise at an intermediate node in a branch-and-bound tree generated by Algorithm \ref{alg:cuttingPlaneMethod}—the ratio $\rho$ need only be computed over indices $i$ where $z_i$ is unspecified or set to $1$. This suggests that the quality of the Gershgorin bound improves upon branching.
\end{remark}
}
\begin{remark}
In particular, if $\bm{\Sigma} \in S^n_+$ is a diagonal matrix, then {\color{black}Equation \eqref{eqn:gershgorincircle}'s} bound is tight - which follows from the fact that the spectrum of $\bm{\Sigma}$ and the discs coincide if and only if $\bm{\Sigma}$ is diagonal \citep[see, e.g,][Chapter 6]{horn1990matrix}. Alternatively, if $\bm{\Sigma}$ is a diagonally dominant matrix then $\rho=1$ and the Gershgorin circle theorem provides a $2-$factor approximation.
\end{remark}
\begin{proof}
Scaled diagonally dominant matrices have scaled diagonally dominant principal minors—this is trivially true because $$d_i\Sigma_{i,i} \geq \sum_{j \in [p]: j \neq i}d_j\vert \Sigma_{i,j}\vert \ \forall i \in [p]\implies d_i\Sigma_{i,i} \geq \sum_{j \in [p]: j \neq i}d_j z_j\vert \Sigma_{i,j}\vert \ \forall i \in [p]: z_i=1$$for the same vector $\bm{d}>\bm{0}$ and therefore the following chain of inequalities holds
\begin{align*}
f(\bm{z}) \leq & \max_{j \in [p]}\{\sum_{i \in [p]} z_i \vert \Sigma_{i,j}\vert \}=\max_{j \in [p]}\{z_j\Sigma_{j,j}+ \sum_{i \in [p]: j \neq i}z_i\vert \Sigma_{i,j}\vert \}\\
& \leq \max_{j \in [p]}\{z_j\Sigma_{j,j}+\sum_{i \in [p]: j \neq i}\rho \frac{d_i}{d_j}z_i\vert \Sigma_{i,j}\vert\}\leq (1+\rho)\max_{j \in [p]}\{z_j\Sigma_{j,j}\}\leq (1+\rho) f(\bm{z}) \quad \forall \bm{z} \in \{0, 1\}^p, \end{align*}
where the second inequality follows because $\rho\geq \frac{d_i}{d_j}$, the third inequality follows from the scaled diagonal dominance of the principal submatrices of $\bm{\Sigma}$, and the fourth inequality holds because the leading eigenvalue of a PSD matrix is at least as large as each diagonal
\end{proof}
}
To make clear the extent our numerical success depends upon Theorem \ref{thm:circletheorem}, our results in Section \ref{sec:numres} present implementations of Algorithm \ref{alg:cuttingPlaneMethod} both with and without the bound.
{
\subsection{Beyond Gershgorin: Further Strengthening via Brauer's Ovals of Cassini} \label{sec:exact.oval}
Given the relevance of Gershgorin's bound, we propose, in this section, a stronger —yet more expensive to implement— upper bound, based on an generalization of the Gershgorin Circle theorem, namely Brauer's ovals of Cassini.
First, we derive a new upper-bound on $f(\bm{z})$ that is at least as strong as the one presented in Theorem \ref{thm:circletheorem} and often strictly stronger \citep[][Chapter 6]{horn1990matrix}
\begin{theorem}\label{thm:cassini1}
For any vector $\bm{z} \in \{0, 1\}^p$, we have the following upper bound on $f(\bm{z})$:
\begin{align} \label{eqn:bound.ovals}
f(\bm{z}) \leq \max_{i,j \in [p]: i>j, z_i=z_j=1} \left\{\frac{\Sigma_{i,i}+\Sigma_{j,j}}{2}+\frac{\sqrt{(\Sigma_{i,i}-\Sigma_{j,j})^2+4R_i(\bm{z}) R_j(\bm{z})}}{2}\right\},
\end{align}
where $R_i(\bm{z}):=\sum_{j \in [p]: j \neq i} z_j \vert \Sigma_{i,j}\vert$ is the absolute sum of off-diagonal entries in the $i$th column of the submatrix of $\bm{\Sigma}$ induced by $\bm{z}$.
\end{theorem}
\begin{proof}
Let us first recall that, per \citet{brauer1946limits}'s original result, all eigenvalues of a matrix $\bm{\Sigma} \in S^p_+$ are contained in the union of the following $p(p-1)/2$ ovals of Cassini:
\begin{align*}
\bigcup_{i \in [p], j \in [p]: i < j} \left\{\lambda \in \mathbb{R}_+: \vert \lambda-\Sigma_{i,i}\vert \vert \lambda-\Sigma_{j,j}\vert \leq R_i R_j \right\},
\end{align*}
where $R_i:=\sum_{j \in [p]: j \neq i} \vert \Sigma_{i,j}\vert$ is the absolute sum of off-diagonal entries in the $i$th column of $\bm{\Sigma}$. Next, let us observe that, if $\lambda$ is a dominant eigenvalue of a PSD matrix $\bm{\Sigma}$ then $\lambda \geq \Sigma_{i,i} \ \forall i $ and, in the $(i,j)$th oval, the bound reduces to
\begin{align}\label{eqn:dominanteigenvaluebound}
\lambda^2 -\lambda(\Sigma_{i,i}+\Sigma_{j,j})+\Sigma_{i,i}\Sigma_{j,j}-R_i R_j \leq 0,
\end{align}
which, by the quadratic formula, implies an upper bound is $\frac{\Sigma_{i,i}+\Sigma_{j,j}}{2}+\frac{\sqrt{(\Sigma_{i,i}-\Sigma_{j,j})^2+4R_i R_j}}{2}$. The result follows because if $z_i=0$ the $i$th row of $\bm{\Sigma}$ cannot be used to bound $f(\bm{z})$.
\end{proof}
Theorem \ref{thm:cassini1}'s inequality can be enforced numerically as mixed-integer second order cone constraints. Indeed, the square root term in \eqref{eqn:bound.ovals} can be modeled using second-order cone, and the bilinear terms only involve binary variables and can be linearized.
Completing the square in Equation \eqref{eqn:dominanteigenvaluebound}, \eqref{eqn:bound.ovals} is equivalent to the following system of $p(p-1)/2$ mixed-integer second-order cone inequalities:
\begin{align*}
\left(\theta-\frac{1}{2}(\Sigma_{i,i}+\Sigma_{j,j})\right)^2 \leq \sum_{s,t \in [p]: s\neq i, t \neq j} W_{s,t}\vert \Sigma_{i,s}\Sigma_{j,t}\vert -\frac{3}{4}\Sigma_{i,i}\Sigma_{j,j}+M(1-s_{i,j}) \ \forall i,j \in [p]: i <j,\\
\sum_{i,j \in [p]: i<j} s_{i,j}=1, s_{i,j} \leq \min(z_i, z_j) \ i,j \in [p]: i<j, \ s_{i,j} \in \{0, 1\} \ i,j \in [p]: i<j.
\end{align*}
where $W_{i,j}=z_i z_j$ is a product of binary variables which can be modeled using, e.g., the {\color{black}McCormick} inequalities $\max(0, z_i+z_j-1) \leq W_{i,j} \leq \min(z_i, z_j)$, and $M$ is an upper bound on the right-hand-side of the inequality for any $i,j: i \neq j$, which can be computed in $O(p^3)$ time in much the same manner as a big-$M$ constant was computed in the previous section. Note that we do not make use of these inequalities directly in our numerical experiments, due to their high computational cost. However, an interesting extension would be to introduce the binary
variables dynamically, via branch-and-cut-and-price \citep{barnhart1998branch}.
Since the bound derived from the ovals of Cassini (Theorem \ref{thm:cassini1}) is at least as strong as the Gershgorin circle's one (Theorem \ref{thm:circletheorem}), it satisfies the same approximation guarantee (Proposition \ref{prop:gershgorinthmapprox}). In particular, it is tight when $\bm{\Sigma}$ is diagonal and provides a $2-$factor approximation for diagonally dominant matrices.
Actually, we now prove a stronger result and demonstrate that Theorem \ref{thm:cassini1} provides a $2-$factor bound on $f(\bm{z})$ for doubly diagonally dominant matrices—a broader class of matrices than diagonally dominant matrices \citep[see][for a general theory]{li1997doubly}:
\begin{proposition}
Let $\bm{\Sigma}\in S^p_+$ be a doubly diagonally dominant matrix, i.e.,
\begin{align*}
\Sigma_{i,i}\Sigma_{j,j}\geq R_i R_j \ \forall i,j \in [p]: i >j,
\end{align*}
where $R_i:=\sum_{j \in [p]: j \neq i} \vert \Sigma_{i,j} \vert$ is the sum of the off-diagonal entries in the $i$th column of $\bm{\Sigma}$. Then, we have that
\begin{align}
f(\bm{z}) \leq \max_{i,j \in [p]: i>j, z_i=z_j=1} \left\{\frac{\Sigma_{i,i}+\Sigma_{j,j}}{2}+\frac{\sqrt{(\Sigma_{i,i}-\Sigma_{j,j})^2+4R_i(\bm{z}) R_j(\bm{z})}\}}{2}\right\} \leq 2f(\bm{z}).
\end{align}
\end{proposition}
\begin{proof}
Observe that if $\Sigma_{i,i}\Sigma_{j,j}\geq R_i R_j$ then $$\sqrt{(\Sigma_{i,i}-\Sigma_{j,j})^2+4 R_i R_j}\leq \sqrt{(\Sigma_{i,i}-\Sigma_{j,j})^2+4 \Sigma_{i,i}\Sigma_{j,j}}=\Sigma_{i,i}+\Sigma_{j,j}.$$ The result then follows in essentially the same fashion as Proposition \ref{prop:gershgorinthmapprox}.
\end{proof}
}
\section{Convex Relaxations and Rounding Methods}\label{sec:relaxandround}
For large-scale instances, high-quality solutions can be obtained by solving a convex relaxation of Problem \eqref{misdpprimal} and rounding the optimal solution. In Section \ref{ssec:relax.bool}, we propose relaxing $\bm{z} \in \{0, 1\}^p$ in \eqref{misdpprimal} to $\bm{z} \in [0, 1]^p$ and applying a greedy rounding scheme. We further tighten this relaxation using second-order cones constraints in Section \ref{ssec:validineq}.
\subsection{A Boolean Relaxation and a Greedy Rounding Method} \label{ssec:relax.bool}
We first consider a Boolean relaxation of \eqref{misdpprimal}, which we obtain\footnote{\color{black}Note that we omit the {\color{black}$\sum_{j=1}^p \vert X_{i,j}\vert \leq \sqrt{k} z_i$} constraints when we develop our convex relaxations, since they are essentially dominated by the $\Vert \bm{X}\Vert_1 \leq k$ constraint we introduce in the next section; we introduced these inequalities to improve our semidefinite-free subproblem strategy for the exact method.} by relaxing $\bm{z} \in \{0, 1\}^p$ to $\bm{z} \in [0, 1]^p$. This gives $\displaystyle \max_{\bm{z} \in [0, 1]^p: \bm{e}^\top \bm{z} \leq k} \: f(\bm{z})$, {\color{black}i.e.},
\begin{equation}
\begin{aligned}\label{prob:lprelax2}
\max_{\substack{\bm{z} \in [0, 1]^p}: \bm{e}^\top \bm{z} \leq k} \: \max_{\bm{X} \succeq \bm{0}} \quad & \langle \bm{\Sigma}, \bm{X} \rangle \ \text{s.t.} \ \mathrm{tr}(\bm{X})=1, \vert X_{i,j}\vert \leq M_{i,j}z_i\ \forall i,j \in [p].
\end{aligned}
\end{equation}
A useful strategy for obtaining a high-quality feasible solution is to solve \eqref{prob:lprelax2} and set $z_i=1$ for $k$ indices corresponding to the largest $\bm{z}_j$'s in \eqref{prob:lprelax2} {\color{black}as proposed in the randomized case for general integer optimization problems by \cite{raghavan1987randomized}}. We formalize this in Algorithm \ref{alg:greedymethod}. {\color{black}We remark that rounding strategies for sparse PCA have previously been proposed \citep[see][]{fountoulakis2017randomized, dey2017sparse, chowdhury2020approximation}, however, the idea of rounding $\bm{z}$ and then optimizing for $\bm{X}$ appears to be new.}
\begin{algorithm*}
\caption{A greedy rounding method for Problem \eqref{OriginalSPCA}}
\label{alg:greedymethod}
\begin{algorithmic}\normalsize
\REQUIRE Covariance matrix $\bm{\Sigma}$, sparsity parameter $k$
\STATE Compute $\bm{z}^\star$ solution of \eqref{prob:lprelax2} or \eqref{spca:sdpplussocp}
\STATE Construct $\bm{z} \in \{0, 1\}^p: \bm{e}^\top \bm{z}=k$ such that $z_i\geq z_j$ if $z^\star_i \geq z^\star_j$.
\STATE Compute $\bm{X}$ solution of \vspace{-2mm}
\begin{align*}
\max_{\bm{X} \in S^p_+} \ \langle \bm{\Sigma}, \bm{X} \rangle \ \text{s.t.} \ \mathrm{tr}(\bm{X})=1, X_{i,j}=0 \ \text{if} \ z_i z_j=0 \ \forall i,j \in [p].
\end{align*} \vspace{-5mm}
\RETURN $\bm{z}, \bm{X}$.
\end{algorithmic}
\end{algorithm*}
{
\begin{remark}\label{remark:scalability}
Our numerical results in Section \ref{sec:numres} reveal that explicitly imposing a PSD constraint on $\bm{X}$ in the relaxation \eqref{prob:lprelax2}—or the ones derived later in the following section—prevents our approximation algorithm from scaling to larger problem sizes than the exact Algorithm \ref{alg:cuttingPlaneMethod} can already solve. Therefore, to improve scalability, the semidefinite cone can be safely approximated via its second-order cone relaxation, $X_{i,j}^2 \leq X_{i,i}X_{j,j}\ \forall i,j \in [p]$, plus a small number of cuts of the form $\langle \bm{X}, \bm{x}_t\bm{x}_t^\top\rangle \geq 0$ as presented in \citet{bertsimas2019polyhedral}.
\end{remark}
}
{
\begin{remark}
Rather than relaxing and greedily rounding $\bm{z}$, one could consider a higher dimensional relax-and-round scheme where we let $\bm{Z}$ model the outer product $\bm{z}\bm{z}^\top$ via $\bm{Z}\succeq \bm{z}\bm{z}^\top$, $\max(0, z_i+z_j-1)\leq Z_{i,j} \leq \min(z_i, z_j) \ \forall i,j \in [p]$, $Z_{i,i}=z_i$, and require that $\sum_{i,j \in [p]}Z_{i,j} \leq k^2$. Indeed, a natural ``round'' component of such a relax-and-round scheme is precisely Goemans-Williamson rounding \citep[][]{goemans1995improved,bertsimas1998semidefinite}, which performs at least as well as greedy rounding in both theory and practice. Unfortunately, some preliminary numerical experiments indicated that Goemans-Williamson rounding is not actually much better than greedy rounding in practice, and is considerably more expensive to implement. Therefore, we defer the details of the Goemans-Williamson scheme to Appendix \ref{sec:relax.goemans}, and do not consider it any further in this paper.
\end{remark}
}
\subsection{Valid Inequalities for Strengthening Convex Relaxations}\label{ssec:validineq}
We now propose valid inequalities which allow us to improve the quality of the convex relaxations discussed previously.
Note that as convex relaxations and random rounding methods are two sides of the same coin \citep{barak2014rounding}, applying these valid inequalities also improves the quality of the randomly rounded solutions.
\begin{theorem}
Let $\mathcal{P}_{strong}$ denote the optimal objective value of the following problem:
\begin{equation}
\label{spca:sdpplussocp}
\begin{aligned}
\max_{\bm{z} \in [0, 1]^p: \bm{e}^\top \bm{z} \leq k} \max_{\bm{X} \in S^p_+} \ \langle \bm{\Sigma}, \bm{X} \rangle \ \text{s.t.} \quad & \mathrm{tr}(\bm{X})=1,
\vert X_{i,j}\vert \leq M_{i,j}z_i\ \forall i,j \in [p] ,\\
&\sum_{j \in [p]} X_{i,j}^2 \leq X_{i,i}z_i, \Vert \bm{X}\Vert_1 \leq k.
\end{aligned}
\end{equation}
Then, \eqref{spca:sdpplussocp} is a stronger relaxation than \eqref{prob:lprelax2}, i.e., the following inequalities hold:
\begin{align*}
\max_{\bm{z} \in [0, 1]^p: \bm{e}^\top \bm{z} \leq k} f(\bm{z})\geq \mathcal{P}_{strong} \geq \max_{\bm{z} \in \{0, 1\}^p: \bm{e}^\top \bm{z} \leq k} f(\bm{z}).
\end{align*}
Moreover, suppose an optimal solution to \eqref{spca:sdpplussocp} is of rank one. Then, the relaxation is tight: $$\mathcal{P}_{strong}= \max_{\bm{z} \in \{0, 1\}^p: \bm{e}^\top \bm{z} \leq k} f(\bm{z}).$$
\end{theorem}
\begin{proof}
The first inequality $\max_{\bm{z} \in [0, 1]^p: \bm{e}^\top \bm{z} \leq k} f(\bm{z})\geq \mathcal{P}_{strong}$ is trivial. The second inequality holds because $\mathcal{P}_{strong}$ is indeed a valid relaxation of Problem \eqref{OriginalSPCA}. Indeed, $\| \bm{X} \|_1 \leq k$ follows from the cardinality and big-M constraints. The semidefinite constraint $\bm{X} \succeq 0$ impose second-order cone constraints on the $2\times 2$ minors of $\bm{X}$, $X_{i,j}^2 \leq z_i X_{i,i} X_{j,j}$, which can be aggregated into $\sum_{j \in [p]} X_{i,j}^2 \leq X_{i,i} z_i $ \citep[see][for derivations]{bertsimas2019polyhedral}.
Finally, suppose that an optimal solution to Problem \eqref{spca:sdpplussocp} is of rank one, i.e., the optimal matrix $\bm{X}$ can be decomposed as $\bm{X}=\bm{x}\bm{x}^\top$. Then, the SOCP inequalities imply that $\sum_{j \in [p]} x_i^2 x_j^2 \leq x_i^2 z_i$. However, $\sum_{j \in [p]}x_j^2=\mathrm{tr}(\bm{X})=1$, which implies that $x_i^2 \leq x_i^2 z_i$, i.e., $z_i=1$ for any index $i$ such that $\vert x_i\vert>0$. Since $\bm{e}^\top \bm{z} \leq k$, this implies that $\Vert \bm{x}\Vert_0 \leq k$, i.e., $\bm{X}$ also solves Problem \eqref{sdospca1}.
\end{proof}
{ As our numerical experiments will demonstrate and despite the simplicity of our rounding mechanism in Algorithm \ref{alg:greedymethod}, the relaxation \eqref{spca:sdpplussocp} provides high-quality solutions to the original sparse PCA problem \eqref{OriginalSPCA}, without introducing any additional variables.} {\color{black}We remark that other inequalities, including the second-order cone inequalities proposed in \citet[Lemma 2 (ii)]{li2020exact}, could further improve the convex relaxation; we leave integrating these inequalities within our framework as future work.}
\section{Numerical Results}\label{sec:numres}
We now assess the numerical behavior of the algorithms proposed in Section \ref{sec:reformulation} and \ref{sec:relaxandround}. To bridge the gap between theory and practice, we present a \verb|Julia| code which implements the described convex relaxation and greedy rounding procedure on GitHub\footnote{\href{github.com/ryancorywright/ScalableSPCA.jl}{https://github.com/ryancorywright/ScalableSPCA.jl}}. The code requires a conic solver such as \verb|Mosek| and several open source Julia packages to be installed.
\subsection{Performance of Exact Methods}
In this section, we apply Algorithm \ref{alg:cuttingPlaneMethod} to medium and large-scale sparse principal component analysis problems, with and without Gershgorin circle theorem bounds in the master problem. All experiments were implemented in \verb|Julia| $1.3$, using \verb|Gurobi| $9.1$ and \verb|JuMP.jl| $0.21.6$, and performed on a standard Macbook Pro laptop, with a $2.9$GHz $6$-Core Intel i9 CPU, using $16$ GB DDR4 RAM. We compare our approach to the branch-and-bound algorithm\footnote{\color{black}The solve times for their method, as reported here, differ from those reported in \citet{berk2017} due to a small typo in their implementation (line $110$ of their branchAndBound.jl code should read ``if $y[i]==-1$ $||$ $y[i]==1$'', not ``if $y[i]==-1$'' in order to correctly compute the Gershgorin circle theorem bound); correcting this is necessary to ensure that we obtain correct results from their method.} developed by \cite{berk2017} on the UCI \verb|pitprops|, \verb|wine|, \verb|miniboone|, \verb|communities|, \verb|arrythmia| and \verb|micromass| datasets, both in terms of runtime and the number of nodes expanded; we refer to \cite{berk2017, bertsimas2019polyhedral} for descriptions of these datasets. Note that we normalized all datasets before running the method (i.e., we compute the leading sparse principal components of correlation matrices). Additionally, we warm-start all methods with the solution from
the method of \cite{yuan2013truncated}, to maintain a fair comparison.
Table \ref{tab:comparison} reports the time for Algorithm \ref{alg:cuttingPlaneMethod} (with and without Gershgorin circle theorem bounds in the master problem) and the method of \cite{berk2017} to identify the leading $k$-sparse principal component for {\color{black}$k \in \{5, 10, 20\}$}, along with the number of nodes expanded, and the number of outer approximation cuts generated. We impose a relative optimality tolerance of $10^{-3}$ for all approaches {\color{black}, i.e., terminate each method when $(UB-LB)/UB\leq 10^{-3}$ where $UB$ denotes the current objective bound and $LB$ denotes the current incumbent objective value}. Note that $p$ denotes the dimensionality of the correlation matrix, and $k \leq p$ denotes the target sparsity.
{
\begin{table}[h]
\centering\footnotesize
\caption{{\color{black}Runtime in seconds per approach. We impose a time limit of $600$s. If a solver fails to converge, we report the relative bound gap at termination in brackets
}}
\begin{tabular}{@{}l l l r r r r r r r r r@{}} \toprule
Dataset & $p$ & $k$ & \multicolumn{3}{c@{\hspace{0mm}}}{Alg. \ref{alg:cuttingPlaneMethod}} & \multicolumn{3}{c@{\hspace{0mm}}}{Alg. \ref{alg:cuttingPlaneMethod}+ Circle Theorem} & \multicolumn{2}{c@{\hspace{0mm}}}{Method of B.+B.} \\
\cmidrule(l){4-6} \cmidrule(l){7-9} \cmidrule(l){10-11} & & & Time(s) & Nodes & Cuts & Time(s) & Nodes & Cuts & Time(s) & Nodes \\\midrule
Pitprops & $13$ & $5$ & \color{black} $0.30$ & \color{black} $1,608$ & \color{black} $1,176$ & \color{black} $\textbf{0.06}$ & \color{black} $38$ & \color{black} $8$ & $1.49$ & $22$ \\
& & $10$ & \color{black} $0.14$ & \color{black} $414$ & \color{black} $387$ & \color{black} $\textbf{0.02}$ & \color{black} $18$ & \color{black} $21$ & $\textbf{0.02}$ & $14$ \\\midrule
Wine & $13$ & $5$ & \color{black} $0.57$ & \color{black} $2,313$ & \color{black} $1,646$ & \color{black} $\textbf{0.02}$ & \color{black} $46$ & \color{black} $11$ & $0.04$ & $34$ \\
& & $10$ & \color{black} $0.17$ & \color{black} $376$ & \color{black} $311$ & $\color{black} 0.03$ & \color{black} $54$ & \color{black} $58$ & $\textbf{0.02}$ & $12$ \\\midrule
Miniboone & $50$ & $5$ & \color{black} $\textbf{0.01}$ & \color{black} $0$ & \color{black} $11$ & \color{black} $\textbf{0.01}$ & \color{black} $0$ & \color{black} $3$ & $0.04$ & $2$ \\
& & $10$ & \color{black} $\textbf{0.01}$ & \color{black} $0$ & \color{black} $16$ & \color{black} $0.02$ & \color{black} $0$ & \color{black} $3$ & $0.04$ & $2$ \\
& & \color{black} $20$ & \color{black} $0.03$ & \color{black} $0$ & \color{black} $26$ & \color{black} $\textbf{0.01}$ & \color{black} $0$ & \color{black} $3$ & \color{black} $1.30$ & \color{black} $5,480$ \\\midrule
Communities & $101$ & $5$ & \color{black} ($2.87\%$) & \color{black} $28,462$ & \color{black} $25,483$ & \color{black} $\textbf{0.20}$ & \color{black} $201$ & \color{black} $3$ & $0.57$ & $101$ \\
& & $10$ & \color{black}($13.3\%$) & \color{black} $37,479$ & \color{black} $36,251$ & \color{black} $\textbf{0.34}$ & \color{black} $406$ & \color{black} $39$ & $0.94$ & $1,298$ \\
& & \color{black} $20$ & \color{black} ($39.6\%)$ & \color{black} $24,566$ & \color{black} $24,632$ & \color{black} ($12.1\%)$ & \color{black} $42,120$ & \color{black} $37,383$ & \color{black} $(\textbf{9.97\%})$ & \color{black} $669,500$ \\\midrule
Arrhythmia & $274$ & $5$ & \color{black} ($18.1\%$) & \color{black} $22,771$ & \color{black} $20,722$ & \color{black}$6.07$ & \color{black} $135$ & \color{black} $1,233$ & $\textbf{4.17}$ & $1,469$ \\
& & $10$ & \color{black} ($32.6\%)$ & \color{black} $19,500$ & \color{black} $19,314$ & \color{black} ($2.92\%$) & \color{black} $15,510$ & \color{black} $6,977$ & $\textbf{(0.83\%)}$ & $471,680$ \\
& & \color{black} $20$ & \color{black} $(74.4\%)$ & \color{black} $33,773$ & \color{black} $12,374$& \color{black} ($24.3\%$) & \color{black} $33,123$ & \color{black} $19,662$ & \color{black} $(\textbf{18.45\%})$ & \color{black} $311,400$ \\\midrule
Micromass & $1300$ & $5$ & \color{black} $(1.29\%)$ & \color{black} $3,859$ & \color{black} $3,099$ & \color{black} $163.60$ & \color{black} $2,738$ & \color{black} $6$ & $\textbf{24.31}$ & $1,096$ \\
& & $10$ & \color{black} $(10.6\%)$ & \color{black} $3,366$ & \color{black} $3,369$ & \color{black} $\textbf{241.86}$ & \color{black} $3,233$ & \color{black} $121$ & $362.4$ & $36,690$ \\
& & \color{black} \color{black} $20$ & \color{black} ($35.9\%)$ & \color{black} $2,797$ & \color{black} $2,839$ & \color{black} ($35.9\%$) & \color{black} $2,676$ & \color{black} $2,115$ & \color{black} $(\textbf{10.34}\%)$ & \color{black} $31,990$ \\
\bottomrule
\end{tabular}
\label{tab:comparison}
\end{table}
}
Our main findings from these experiments are as follows:
\begin{itemize}\setlength\itemsep{0em}
\item For smaller problems, the strength of Algorithm \ref{alg:cuttingPlaneMethod}'s cuts allows it to outperform state-of-the-art methods such as the method of \cite{berk2017}. Moreover, for larger problem sizes, the adaptive branching strategy {\color{black}performs comparably to} Algorithm \ref{alg:cuttingPlaneMethod}. This suggests that {\color{black}the relative merits of both approaches are roughly even, and which method is preferable may depend on the problem data.}
\item Generating outer-approximation cuts and valid upper bounds from the Gershgorin circle theorem are both powerful ideas, but the greatest aggregate power appears to arise from intersecting these bounds, rather than using one bound alone.
\end{itemize}
\begin{itemize}
\item {\color{black}Once both $k$ and $p$ are sufficiently large (e.g. $p>300$ and $k>10$), no approach is able to solve the problem to provable optimality within $600$s. This motivates our study of convex relaxations and randomized rounding methods in the next section.}
\end{itemize}
\vspace{-5mm}
\subsection{Convex Relaxations and Randomized Rounding Methods}
In this section, we apply Algorithm \ref{alg:greedymethod} to obtain high quality convex relaxations and feasible solutions for the datasets studied in the previous subsection, and compare the relaxation to a difference convex relaxation developed by \citet{d2008optimal}, in terms of the quality of the upper bound and the resulting greedily rounded solutions. All experiments were implemented using the same specifications as the previous section. { Note that \citet{d2008optimal}'s upper bound\footnote{ Strictly speaking, \citet{d2008optimal} does not actually write down this formulation in their work. Indeed, their bound involves dual variables which cannot be used directly to generate feasible solutions via greedy rounding. However, the fact that this bound and \citep[Problem (8)]{d2008optimal} are dual to each other follows directly from strong semidefinite duality, and therefore we refer to this formulation as being due to \cite{d2008optimal} (it essentially is).} which we compare against is:
\begin{equation}
\begin{aligned}\label{prob:lprelax3}
\max_{\substack{\bm{z} \in [0, 1]^p}: \bm{e}^\top \bm{z} \leq k} \: \max_{\bm{X} \succeq \bm{0}, \bm{P}_i \succeq \bm{0}\ \forall i \in [p]} \quad & \sum_{i \in [p]}\ \langle \bm{a}_i\bm{a}_i^\top, \bm{P}_i\rangle \ \text{s.t.} \ \mathrm{tr}(\bm{X})=1,\ \mathrm{tr}(\bm{P}_i)=z_i, \ \bm{X}\succeq \bm{P}_i \ \forall i \in [p],
\end{aligned}
\end{equation}
where $\bm{\Sigma}=\sum_{i=1}^p \bm{a}_i\bm{a}_i^\top$ is a Cholesky decomposition of $\bm{\Sigma}$, and we obtain feasible solutions from this relaxation by greedily rounding an optimal $\bm{z}$ in the bound \textit{\`{a} la} Algorithm \ref{alg:greedymethod}. {\color{black} To allow for a fair comparison,} we also consider augmenting this formulation with the inequalities derived in Section \ref{ssec:validineq} to obtain the following stronger yet more expensive to solve relaxation:
\begin{equation}
\label{spca:sdpplussocp2}
\begin{aligned}
\max_{\substack{\bm{z} \in [0, 1]^p}: \bm{e}^\top \bm{z} \leq k} \: \max_{\substack{\bm{X} \succeq \bm{0},\\ \bm{P}_i \succeq \bm{0}\ \forall i \in [p]}} \quad & \sum_{i \in [p]}\ \langle \bm{a}_i\bm{a}_i^\top, \bm{P}_i\rangle \ \text{s.t.} \ \mathrm{tr}(\bm{X})=1,\ \mathrm{tr}(\bm{P}_i)=z_i, \ \bm{X}\succeq \bm{P}_i \ \forall i \in [p],\\
&\sum_{j \in [p]} X_{i,j}^2 \leq X_{i,i}z_i, \Vert \bm{X}\Vert_1 \leq k.
\end{aligned}
\end{equation}
}
{\color{black} We first apply these relaxations on datasets where Algorithm \ref{alg:cuttingPlaneMethod} terminates, hence the optimal solution is known and can be compared against.} We report the quality of both methods with and without the additional inequalities discussed in Section \ref{ssec:validineq}, in Tables \ref{tab:comparison_convrelaxations}-\ref{tab:comparison_convrelaxations2} respectively\footnote{For the instances of \eqref{prob:lprelax3} or \eqref{spca:sdpplussocp2} where $p>13$ we used SCS version $2.1.1$ (with default parameters) instead of Mosek, since Mosek required more memory than was available in our computing environment, and SCS takes an augmented Lagrangian approach which is less numerically stable but requires significantly less memory. That is, \eqref{prob:lprelax3}'s formulation is too expensive to solve via IPMs on a laptop when $p=50$.}.
\begin{table}[h]
\centering\footnotesize
\caption{{\color{black}Quality of relaxation gap (upper bound vs. optimal solution-denoted R. gap), objective gap (rounded solution vs. optimal solution-denoted O. gap) and runtime in seconds per method.}}
\begin{tabular}{@{}l l l r r r r r r@{}} \toprule
Dataset & $p$ & $k$ & \multicolumn{3}{c@{\hspace{0mm}}}{Alg. \ref{alg:greedymethod} with \eqref{prob:lprelax2}} & \multicolumn{3}{c@{\hspace{0mm}}}{Alg. \ref{alg:greedymethod} with \eqref{prob:lprelax3}} \\
\cmidrule(l){4-6} \cmidrule(l){7-9} & & & R. gap $(\%)$& O. gap $(\%)$ & Time(s) & R. gap $(\%)$ & O. gap $(\%)$ & Time(s) \\\midrule
Pitprops & $13$ & $5$ &$23.8\%$ & $0.00\%$ &$0.02$ &$23.8\%$ & $16.1\%$ &$0.46$\\
& & $10$ & $1.10\%$ &$0.30\%$ &$0.03$ &$1.10\%$ & $1.33\%$ &$0.46$\\\midrule
Wine & $13$ & $5$ &$36.8\%$ & $0.00\%$ &$0.02$ &$36.8\%$ & $40.4\%$ &$0.433$\\
& & $10$ &$2.43\%$ & $0.26\%$ &$0.03$ &$2.43\%$ & $15.0\%$ &$0.463$\\\midrule
Miniboone & $50$ & $5$ & $781.3\%$ & $235.6\%$ & $7.37$ & $781.2\%$ & $34.7\%$ & $1,191.0$\\
& & $10$ & $340.6\%$ & $117.6\%$ & $7.50$ & $340.6\%$ & $44.9\%$ & $1,102.6$\\
& & \color{black} $20$ & \color{black} $120.3\%$ & \color{black} $38.08\%$ & \color{black} $6.25$ & \color{black} $120.3\%$ & \color{black} $31.9\%$ & \color{black} $1,140.2$ \\
\bottomrule
\end{tabular}
\label{tab:comparison_convrelaxations}
\end{table}
\begin{table}[h]
\centering\footnotesize
\caption{\color{black} Quality of relaxation gap (upper bound vs. optimal solution-denoted R. gap), objective gap (rounded solution vs. optimal solution-denoted O. gap) and runtime in seconds per method, with additional inequalities from Section \ref{ssec:validineq}.}
\begin{tabular}{@{}l l l r r r r r r@{}} \toprule
Dataset & $p$ & $k$ & \multicolumn{3}{c@{\hspace{0mm}}}{Alg. \ref{alg:greedymethod} with \eqref{spca:sdpplussocp}} & \multicolumn{3}{c@{\hspace{0mm}}}{Alg. \ref{alg:greedymethod} with \eqref{spca:sdpplussocp2}} \\
\cmidrule(l){4-6} \cmidrule(l){7-9} & & & R. gap $(\%)$& O. gap $(\%)$ & Time(s) & R. gap $(\%)$ & O. gap $(\%)$ & Time(s) \\\midrule
Pitprops & $13$ & $5$ & $0.71\%$ &$0.00\%$ & $0.17$ &$1.53\%$ & $0.00\%$ &$0.55$ \\
& & $10$ & $0.12\%$ &$0.00\%$ & $0.27$ &$1.10\%$ & $0.00\%$ &$3.27$\\\midrule
Wine & $13$ & $5$ &$1.56\%$ & $0.00\%$ &$0.24$ &$2.98\%$ & $15.03\%$ &$0.95$\\
& & $10$ &$0.40\%$ & $0.00\%$ &$0.22$ &$2.04\%$ & $0.00\%$ &$1.15$\\\midrule
Miniboone & $50$ & $5$ & $0.00\%$ & $0.00\%$ & $163.3$ & $0.00\%$ & $0.01\%$ & $500.7$\\
& & $10$ & $0.00\%$ & $0.00\%$ & $148.5$ & $0.00\%$ & $0.02\%$ & $489.9$\\%\midrule
& & \color{black} $20$ & \color{black} $0.00\%$ & \color{black} $0.00\%$ & \color{black} $194.5$ & \color{black} $0.00\%$ & \color{black} $0.00\%$ & \color{black} $776.3$\\
\bottomrule
\end{tabular}
\label{tab:comparison_convrelaxations2}
\end{table}
Observe that applying Algorithm \ref{alg:greedymethod} without the additional inequalities (Table \ref{tab:comparison_convrelaxations}) yields rather poor relaxations and randomly rounded solutions. However, by intersecting our relaxations with the additional inequalities from Section \ref{ssec:validineq} (Table \ref{tab:comparison_convrelaxations2}), we obtain extremely high quality relaxations. Indeed, with the additional inequalities, Algorithm \ref{alg:greedymethod} using formulation \eqref{spca:sdpplussocp} identifies the optimal solution in all instances (0\% O. gap), and always supplies a bound gap of less than $2\%$. Moreover, in terms of obtaining high-quality solutions, the new inequalites allow Problem \eqref{spca:sdpplussocp} to perform
as well or better as Problem \eqref{prob:lprelax3}, despite optimizing over one semidefinite matrix, rather than $p+1$ semidefinite matrices. This suggests that Problem \eqref{spca:sdpplussocp} should be considered as a viable, more scalable and more accurate alternative to existing SDO relaxations such as Problem \eqref{prob:lprelax3}. For this reason, we shall only consider using Problem \eqref{spca:sdpplussocp}'s formulation for the rest of the paper.
We remark however that the key drawback of applying these methods is that, as implemented in this section, they do not scale to sizes beyond which Algorithm \ref{alg:cuttingPlaneMethod} successfully solves. This is a drawback because Algorithm \ref{alg:cuttingPlaneMethod} supplies an exact certificate of optimality, while these methods do not. In the following set of experiments, we therefore investigate numerical techniques to improve the scalability of Algorithm \ref{alg:greedymethod}.
\subsection{Scalable Dual Bounds and {\color{black}Randomized} Rounding Methods}
To improve the scalability of Algorithm \ref{alg:greedymethod}, we relax the PSD constraint on $\bm{X}$ in \eqref{prob:lprelax2} and \eqref{spca:sdpplussocp}. With these enhancements, we demonstrate that Algorithm \ref{alg:greedymethod} can be successfully scaled to generate high-quality bounds for $1000s \times 1000s$ matrices.
{ As discussed in Remark \ref{remark:scalability}, we can replace the PSD constraint $\bm{X} \succeq \bm{0}$ by requiring that the $p(p-1)/2$ two by two minors of $\bm{X}$ are non-negative: $X_{i,j}^2 \leq X_{i,i} X_{j,j}$. Second, we consider adding $20$ linear inequalities of the form $\langle \bm{X}, \bm{x}_t\bm{x}_t^\top\rangle \geq 0$, for some vector $\bm{x}_t$ \citep[see][for a discussion]{bertsimas2019polyhedral}.} Table \ref{tab:comparison_convrelaxations3} reports the performance of Algorithm \ref{alg:greedymethod} (with the relaxation \eqref{spca:sdpplussocp}) with these two approximations of the positive semidefinite cone, ``Minors'' and ``Minors + 20 inequalities'' respectively. {\color{black}Note that we report the entire duality gap (i.e. do not break the gap down into its relaxation and objective gap components) since, as reflected in Table \ref{tab:comparison}, some of these instances are currently too large to solve to optimality.}
\begin{table}[h]
\centering\footnotesize
\caption{Quality of bound gap (rounded solution vs. upper bound) and runtime in seconds of Algorithm \ref{alg:greedymethod} with \eqref{spca:sdpplussocp}, outer-approximation of the PSD cone.}
\begin{tabular}{@{}l l l r r r r@{}} \toprule
Dataset & $p$ & $k$ & \multicolumn{2}{c@{\hspace{0mm}}}{Minors} & \multicolumn{2}{c@{\hspace{0mm}}}{Minors + 20 inequalities} \\
\cmidrule(l){4-5} \cmidrule(l){6-7} & & & Gap $(\%)$ & Time(s) & Gap $(\%)$ & Time(s) \\\midrule
Pitprops & $13$ & $5$ & $1.51\%$ & $0.02$ &$0.72\%$ &$0.36$ \\
& & $10$ & $5.29\%$ & $0.02$ &$1.12\%$ &$0.36$ \\\midrule
Wine & $13$ & $5$ & $2.22\%$ & $0.02$ &$1.59\%$ &$0.38$ \\
& & $10$ & $3.81\%$ & $0.02$ &$1.50\%$ &$0.37$ \\\midrule
Miniboone & $50$ & $5$ & $0.00\%$ & $0.11$ &$0.00\%$ &$0.11$ \\
& & $10$ & $0.00\%$ & $0.12$ &$0.00\%$ &$0.12$ \\
& & \color{black} $20$ & \color{black} $0.00\%$ & \color{black} $0.39$ & \color{black} $0.00\%$ & \color{black} $0.39$ \\ \midrule
Communities & $101$ & $5$ & $0.07\%$ & $0.67$ &$0.07\%$ &$14.8$ \\
& & $10$ & $0.66\%$ & $0.68$ &$0.66\%$ &$14.4$ \\
& & \color{black} $20$ & \color{black} $3.32\%$ & \color{black} $1.84$ & \color{black} $2.23\%$ & \color{black} $33.5$ \\\midrule
Arrhythmia & $274$ & $5$ & $3.37\%$ & $27.2$ &$1.39\%$ &$203.6$ \\
& & $10$ & $3.01\%$ & $25.6$ &$1.33\%$ &$184.0$ \\
& & \color{black} $20$ & \color{black} $8.87\%$ & \color{black} $21.8$ & \color{black} $4.48\%$ & \color{black} $426.8$ \\\midrule
Micromass & $1300$ & $5$ & $0.04\%$ & $239.4$ &$0.01\%$ &$4,639$ \\
& & $10$ & $0.63\%$ & $232.6$ &$0.32\%$ &$6,392$ \\
& & \color{black} $20$ & \color{black} $13.1\%$ & \color{black} $983.5$ & \color{black} $5.88\%$ & \color{black} $16,350$ \\
\bottomrule
\end{tabular}
\label{tab:comparison_convrelaxations3}
\end{table}
Observe that if we impose constraints on the $2\times 2$ minors only then we obtain a solution {\color{black} certifiably} within {\color{black}$13\%$} of optimality in seconds (resp. minutes) for $p=100$s (resp. $p=1000$s). Moreover, adding $20$ linear inequalities, we obtain a solution within $6\%$ of optimality in minutes (resp. hours) for $p=100$s (resp. $p=1000$s). {\color{black}Moreover, the bound gaps compare favorably to Algorithm \ref{alg:cuttingPlaneMethod} and the method of \cite{berk2017} for instances which these methods could not solve to certifiable optimality. For instance, for the Arrhythmia dataset when $k=20$ we obtain a bound gap of less than $9\%$ in $20$s, while the method of \cite{berk2017} obtains a bound gap of $18.45\%$ in $600$s. This illustrates the value of the proposed relax+round method on datasets which are currently too large to be optimized over exactly.}
To conclude this section, we explore Algorithm \ref{alg:greedymethod}'s ability to scale to even higher dimensional datasets in a high performance setting, by running the method on one Intel Xeon E5--2690 v4 2.6GHz CPU core using 600 GB RAM. Table \ref{tab:comparison_convrelaxations4} reports the methods scalability and performance on the Wilshire $5000$, and \verb|Arcene| UCI datasets. For the \verb|Gisette| dataset, we report on the methods performance when we include the first $3,000$ and $4,000$ rows/columns (as well as all $5,000$ rows/columns). Similarly, for the \verb|Arcene| dataset we report on the method's performance when we include the first $6,000$, $7,000$ or $8,000$ rows/columns. We do not report results for the \verb|Arcene| dataset for $p>8,000$, as computing this requires more memory than was available (i.e. $>600$ GB RAM).
We do not report the method's performance when we impose linear inequalities for the PSD cone, as solving the relaxation without them is already rather time consuming. Moreover, we do not impose the $2 \times 2$ minor constraints to save memory, do not impose $\vert X_{i,j}\vert \leq M_{i,j}z_i$ {\color{black}when $p \geq 4000$} to save even more memory, and report the overall bound gap, as improving upon the randomly rounded solution is challenging in a high-dimensional setting.
\begin{table}[h]
\centering\footnotesize
\caption{Quality of bound gap (rounded solution vs. upper bound) and runtime in seconds.}
\begin{tabular}{@{}l l l r r @{}} \toprule
Dataset & $p$ & $k$ & \multicolumn{2}{c@{\hspace{0mm}}}{Algorithm \ref{alg:greedymethod} (SOC relax)+Inequalities}\\
\cmidrule(l){4-5} & & & Bound gap $(\%)$ & Time(s) \\\midrule
Wilshire $5000$ & $2130$ & $5$ & $0.38\%$ & $1,036$\\
& & $10$ & $0.24\%$ & $1,014$\\
& & \color{black} $20$ & \color{black} $0.36\%$ & \color{black} $1,059$\\
\midrule
Gisette & $3000$ & $5$ & $1.67\%$ & $2,249$\\
& & $10$ & $35.81\%$ & $2,562$\\
& & \color{black} $20$ & \color{black} $10.61\%$ & \color{black} $3,424$\\
\midrule
Gisette & $4000$ & $5$ & $1.55\%$ & $1,402$\\
& & $10$ & $54.4\%$ & $1,203$\\
& & \color{black} $20$ & \color{black} $11.84\%$ & \color{black} $1,435$\\
\midrule
Gisette & $5000$ & $5$ & $1.89\%$ & $2,169$\\
& & $10$ & $2.22\%$ & $2,455$\\
& & \color{black} $20$ & \color{black} $7.16\%$ & \color{black} $2,190$\\\midrule
Arcene & $6000$ & $5$ & $0.01\%$ & $3,333$\\
& & $10$ & $0.06\%$ & $3,616$\\
& & \color{black} $20$ & \color{black} $0.14\%$ & \color{black} $3,198$\\\midrule
Arcene & $7000$ & $5$ & $0.03\%$ & $4,160$\\
& & $10$ & $0.05\%$ & $4,594$ \\
& & \color{black} $20$ & \color{black} $0.25\%$ & \color{black} $4,730$\\
\midrule
Arcene & $8000$ & $5$ & $0.02\%$ & $6,895$\\
& & $10$ & $0.17\%$ & $8,479$\\
& & \color{black} $20$ & \color{black} $0.21\%$ & \color{black} $6,335$\\
\bottomrule
\end{tabular}
\label{tab:comparison_convrelaxations4}
\end{table}
These results suggest that if we solve the SOC relaxation using a first-order method rather than an interior point method, our approach could successfully generate certifiably near-optimal PCs when $p=10,000$s, particularly if combined with a feature screening technique \citep[see][]{d2008optimal, atamturk2020feature}.
{\color{black}
\subsection{Performance of Exact and Approximate Methods on Synthetic Data}
We now compare the exact and approximate methods against existing state-of-the-art methods in a spiked covariance matrix setting.
We use the experimental setup laid out in \citet[Section 7.1]{d2008optimal}. We recover the leading principal component of a test matrix\footnote{\color{black}This statement of the test matrix is different to \citet[Section 7.1]{d2008optimal}, who write $\bm{\Sigma}=\bm{U}^\top \bm{U}+\sigma \bm{v}\bm{v}^\top$, rather than $\bm{\Sigma}=\frac{1}{n}\bm{U}^\top \bm{U}+\frac{\sigma}{\Vert \bm{v}\Vert_2^2} \bm{v}\bm{v}^\top$. However, it agrees with their source code.} $\bm{\Sigma} \in S^{p}_+$, where $p=150$, $\bm{\Sigma}=\frac{1}{n}\bm{U}^\top \bm{U}+\frac{\sigma}{\Vert \bm{v}\Vert_2^2} \bm{v}\bm{v}^\top$, $\bm{U} \in [0, 1]^{150 \times 150}$ is a noisy matrix with i.i.d. standard uniform entries, $\bm{v} \in \mathbb{R}^{150}$ is a vector of signals such that
\begin{align}
v_i=\begin{cases}
1, & \text{if} \ i \leq 50,\\
\frac{1}{i-50}, & \text{if} \ 51 \leq i \leq 100,\\
0, & \text{otherwise,}
\end{cases}
\end{align}
and $\sigma=2$ is the signal-to-noise ratio. The methods which we compare are:
\begin{itemize}\itemsep0em
\item \textbf{Exact}: Algorithm \ref{alg:cuttingPlaneMethod} with Gershgorin inequalities and a time limit of $600$s.
\item \textbf{Approximate:} Algorithm \ref{alg:greedymethod} with Problem \eqref{spca:sdpplussocp}, the SOC outer-approximation of the PSD cone, no PSD cuts, and the additional SOC inequalities.
\item \textbf{Greedy:} as proposed by \cite{moghaddam2006spectral} and laid out in \citep[Algorithm 1]{d2008optimal}, start with a solution $\bm{z}$ of cardinality $1$ and iteratively augment this solution vector with the index which gives the maximum variance contribution. Note that \cite{d2008optimal} found this method outperformed the $3$ other methods (approximate greedy, thresholding and sorting) they considered in their work.
\item \textbf{Truncated Power Method:} as proposed by \citet{yuan2013truncated}, alternate between applying the power method to the solution vector and truncating the vector to ensure that it is $k$-sparse. Note that \cite{berk2017} found that this approach performed better than $5$ other state-of-the-art methods across the real-world datasets studied in the previous section of this paper and often matched the performance of the method of \cite{berk2017}—indeed, it functions as a warm-start for the later method.
\item \textbf{Sorting:} sort the entries of $\bm{\Sigma}_{i,i}$ by magnitude and set $z_i=1$ for the $k$ largest entries of $\bm{\Sigma}$, as studied in \cite{d2008optimal}. This naive method serves as a benchmark for the value of optimization in the more sophisticated methods considered here.
\end{itemize}
Figures \ref{fig:sensitivitytok} depicts the ROC curve (true positive rate vs. false positive rate for recovering the support of $\bm{v}$) over $20$ synthetic random instances, as we vary $k$ for each instance. We observe that among all methods, the sorting method is the least accurate, with a substantially larger false detection rate for a given true positive rate than the remaining methods (AUC$=0.7028$). The truncated power method and our exact method\footnote{\color{black}Note that the exact method would dominate the remaining methods if given an unlimited runtime budget. Its poor performance reflects its inability to find the true optimal solution within $600$ seconds.} then offer a substantial improvement over sorting, with respective AUCs of $0.7482$ and $0.7483$. The greedy method then offers a modest improvement over them (AUC$=0.7561$) and the approximate relax+round method is the most accurate (AUC$=0.7593$).
In addition to support recovery, Figure \ref{fig:sensitivitytok2} reports average runtime (left panel) and average optimality gap (right panel) over the same instances. Observe that among all methods, only the exact and the approximate relax+round methods provide optimality gaps, i.e., {\color{black} numerical certificates} of near optimality. On this metric, relax+round supplies average bound gaps of $1\%$ or less on all instances, while the exact method typically supplies bound gaps of $30\%$ or more. This comparison illustrates the tightness of the valid inequalities from Section \ref{ssec:validineq} that we included in the relaxation. Moreover, the relax+round method converges in less than one minute on all instances. All told, the relax+round method is the best performing method overall, although if $k$ is set to be sufficiently close to $0$ or $p$ all methods behave comparably.
In particular, the relax+round method should be preferred over the exact method, even though the exact method performs better at smaller problem sizes.
\begin{figure}[h]\centering
\includegraphics[scale=0.6]{roccurve_synthetic.pdf}
\caption{\color{black}ROC curve over $20$ synthetic instances where $p=150$, $k_{\text{true}}=100$ is unspecified.}
\label{fig:sensitivitytok}
\end{figure}
\begin{figure}[h]
\begin{subfigure}[t]{.45\linewidth}
\includegraphics[scale=0.4]{runtimes_synthetic.pdf}
\end{subfigure}
\begin{subfigure}[t]{.45\linewidth}
\centering
\includegraphics[scale=0.4]{boundgap_synthetic.pdf}
\end{subfigure}
\caption{\color{black}Average time to compute solution, and optimality gap over $20$ synthetic instances where $p=150$, $k_{\text{true}}=100$ is unspecified.}
\label{fig:sensitivitytok2}
\end{figure}
}
{\color{black}
\subsection{Summary and Guidelines From Experiments}
In summary, our main findings from our numerical experiments are as follows:
\begin{itemize}
\item For small or medium scale problems where $p \leq 100$ or $k \leq 10$, exact methods such as Algorithm \ref{alg:cuttingPlaneMethod} or the method of \cite{berk2017} reliably obtain certifiably optimal or near-optimal solutions in a short amount of time, and should therefore be preferred over other methods. However, for larger-scale sparse PCA problems, exact methods currently do not scale as well as approximate or heuristic methods. \color{black}
\item For larger-scale sparse PCA problems, our proposed combination of solving a second-order cone relaxation and rounding greedily reliably supplies certifiably near-optimal solutions in practice (if not in theory) in a relatively small amount of time. Moreover, it outperforms other state-of-the-art heuristics including the greedy method of \cite{moghaddam2006spectral, d2008optimal} and the Truncated Power Method of \cite{yuan2013truncated}. Accordingly, it should be considered as a reliable and more accurate alternative for problems where $p=1000$s.\color{black}
\item In practice, for even larger-scale problem sizes, we recommend using a combination of these methods: a computationally cheaper method (with $k$ set in the $1000$s) as a feature screening method, to be followed by the approximate relax+round method (with $k$ set in the $100$s) and/or the exact method, if time permits.
\end{itemize}
}
\section{Three Extensions and their Mixed-Integer Conic Formulations}
We conclude by discussing {\color{black}three} extensions of sparse PCA where our methodology applies.
\subsection{Non-Negative Sparse PCA}
One potential extension to this paper would be to develop a certifiably optimal algorithm for non-negative sparse PCA \citep[see][for a discussion]{zass2007nonnegative}, i.e., develop a tractable reformulation of
\begin{align*}
\max_{\bm{x} \in \mathbb{R}^p} \quad & \langle \bm{x}\bm{x}^\top, \bm{\Sigma} \rangle \ \text{s.t.}\ \bm{x}^\top \bm{x}=1, \bm{x} \geq \bm{0}, \Vert \bm{x}\Vert_0 \leq k.
\end{align*}
Unfortunately, we cannot develop a MISDO reformulation of non-negative sparse PCA \textit{mutatis mutandis} Theorem \ref{thm:misdpreformthm}. Indeed, while we can still set $\bm{X}=\bm{x}\bm{x}^\top$ and relax the rank-one constraint, if we do so then, by the non-negativity of $\bm{x}$, lifting $\bm{x}$ yields:
\begin{equation}\label{misdpprimal_cp}
\begin{aligned}
\max_{\bm{z} \in \{0, 1\}^p: \bm{e}^\top \bm{z} \leq k} \ \max_{\bm{X} \in \mathcal{C}_n} \quad & \langle \bm{\Sigma}, \bm{X} \rangle\\ \text{s.t.} \quad & \mathrm{tr}(\bm{X})=1,\ X_{i,j}=0 \ \text{if} \ z_i=0, \ X_{i,j}=0 \ \text{if} \ z_j=0 \ \forall i, j \in [p].
\end{aligned}
\end{equation}
where $\mathcal{C}_n:=\{\bm{X}:\ \exists \ \bm{U} \geq \bm{0}, \bm{X}=\bm{U}^\top \bm{U}\}$ denotes the completely positive cone, which is NP-hard to separate over and cannot currently be optimized over tractably \citep{dong2013separating}. Nonetheless, we can develop relatively tractable mixed-integer conic upper and lower bounds for non-negative sparse PCA. Indeed, we can obtain a fairly tight upper bound by replacing the completely positive cone with the larger doubly non-negative cone $\mathcal{D}_n:=\{\bm{X} \in S^p_+: \bm{X} \geq \bm{0}\}$, which is a high-quality outer-approximation of $\mathcal{C}_n$, indeed exact when $k \leq 4$ \citep{burer2009difference}.
Unfortunately, this relaxation is strictly different in general, since the extreme rays of the doubly non-negative cone are not necessarily rank-one when $k \geq 5$ \citep{burer2009difference}. Nonetheless, to obtain feasible solutions which supply lower bounds, we could inner approximate the completely positive cone with the cone of non-negative scaled diagonally dominant matrices \citep[see][]{ahmadi2019dsos,bostanabad2018inner}.
\subsection{Sparse PCA on Rectangular Matrices}
A second extension would be to extend our methodology to the non-square case:
\begin{align}
\max_{\bm{x} \in \mathbb{R}^m, \bm{y} \in \mathbb{R}^n} \quad & \bm{x}^\top \bm{A}\bm{y} \ \text{s.t.} \ \Vert \bm{x}\Vert_2=1, \Vert \bm{y}\Vert_2=1, \Vert \bm{x}\Vert_0 \leq k, \Vert \bm{y}\Vert_0 \leq k.
\end{align}
Observe that computing the spectral norm of a matrix $\bm{A}$ is equivalent to:
\begin{align}
\max_{\bm{X} \in \mathbb{R}^{n \times m}} \quad \langle \bm{A}, \bm{X}\rangle \ \text{s.t.} \ \begin{pmatrix} \bm{U} & \bm{X}\\ \bm{X}^\top & \bm{V}\end{pmatrix} \succeq \bm{0}, \mathrm{tr}(\bm{U})+\mathrm{tr}(\bm{V})=2,
\end{align}
where, in an optimal solution, $\bm{U}$ stands for $\bm{x}\bm{x}^\top$, $\bm{V}$ stands for $\bm{y}\bm{y}^\top$ and $\bm{X}$ stands for $\bm{x}\bm{y}^\top$—this can be seen by taking the dual of \citep[Equation 2.4]{recht2010guaranteed}.
Therefore, by using the same argument as in the positive semidefinite case, we can rewrite sparse PCA on rectangular matrices as the following MISDO:
\begin{equation}
\begin{aligned}
\max_{\bm{w} \in \{0, 1\}^m, \bm{z} \in \{0, 1\}^n}\max_{\bm{X} \in \mathbb{R}^{n \times m}} \quad & \langle \bm{A}, \bm{X}\rangle \\ \text{s.t.} & \ \begin{pmatrix} \bm{U} & \bm{X}\\ \bm{X}^\top & \bm{V}\end{pmatrix} \succeq \bm{0}, \mathrm{tr}(\bm{U})+\mathrm{tr}(\bm{V})=2,\\ & U_{i,j}=0 \ \text{if} \ w_i=0\ \forall i,j \in [m], \\
& V_{i,j}=0 \ \text{if} \ z_i=0\ \forall i,j \in [n], \bm{e}^\top \bm{w} \leq k, \bm{e}^\top \bm{z} \leq k.
\end{aligned}
\end{equation}
\subsection{Sparse PCA with Multiple Principal Components}
A third extension where our methodology is applicable is the problem of obtaining multiple principal components simultaneously, rather than deflating $\bm{\Sigma}$ after obtaining each principal component. As there are multiple definitions of this problem, we now discuss the extent to which our framework encompasses each case.
\paragraph{Common Support:} Perhaps the simplest extension of sparse PCA to a multi-component setting arises when all $r$ principal components have common support. By retaining the vector of binary variables $\bm{z}$ and employing the
Ky-Fan theorem \citep[c.f.][Theorem 2.3.8]{wolkowicz2012handbook} to cope with multiple principal components, we obtain the following formulation in much the same manner as previously:
\begin{align}
\max_{\bm{z} \in \{0, 1\}^p: \bm{e}^\top \bm{z} \leq k}\ \max_{\bm{X} \in S^p_+} \quad & \langle \bm{X}, \bm{\Sigma}\rangle\ \text{s.t.} \ \bm{0} \preceq \bm{X} \preceq \mathbb{I}, \ \mathrm{tr}(\bm{X})=r,\ X_{i,j}=0 \ \text{if} \ z_{i}=0\ \forall i \in [p].
\end{align}
Notably, the logical constraint $X_{i,j}=0$ if $z_i=0$, which formed the basis of our subproblem strategy, still successfully models the sparsity constraint. This suggests that (a) one can derive an equivalent subproblem strategy under common support, and (b) a cutting-plane method for common support should scale equally well as with a single component.
\paragraph{Disjoint Support:}
In a sparse PCA problem with disjoint support \citep{vu2012minimax}
, simultaneously computing the first $r$ principal components is equivalent to solving:
\begin{equation}
\begin{aligned}
\max_{\substack{\bm{z} \in \{0, 1\}^{p \times r}: \bm{e}^\top \bm{z}_t \leq k\ \forall t \in [r], \\\bm{z}\bm{e} \leq \bm{e}}} \max_{\bm{W} \in \mathbb{R}^{p \times r}} \quad & \langle \bm{W}\bm{W}^\top, \bm{\Sigma}\rangle\\
& \bm{W}^\top \bm{W}=\mathbb{I}_{r},\ W_{i,j}=0 \ \text{if} \ z_{i,t}=0\ \forall i \in [p], t \in [r],
\end{aligned}
\end{equation}
where $z_{i,t}$ is a binary variable denoting whether feature $i$ is a member of the $t$th principal component. By applying the technique used to derive Theorem \ref{thm:misdpreformthm} \textit{mutatis mutandis}, and invoking the Ky-Fan theorem \citep[c.f.][Theorem 2.3.8]{wolkowicz2012handbook} to cope with the rank-$r$ constraint, we obtain:
\begin{equation}
\begin{aligned}
\max_{\bm{z} \in \{0, 1\}^p: \bm{e}^\top \bm{z} \leq k} \max_{\bm{X} \in S^p} \quad & \langle \bm{X}, \bm{\Sigma}\rangle\\
& \bm{0} \preceq \bm{X} \preceq \mathbb{I}, \ \mathrm{tr}(\bm{X})=r,\ X_{i,j}=0 \ \text{if} \ Y_{i,j}=0\ \forall i \in [p],
\end{aligned}
\end{equation}
where $Y_{i,j}=\sum_{t=1}^r z_{i,t}z_{j,t}$ is a binary matrix denoting whether features $i$ and $j$ are members of the same principal component; this problem can be addressed by a cutting-plane method in much the same manner as when $r=1$.
{\color{black}
\section*{Acknowledgments}
We are grateful to the three anonymous referees and the associate editor for many valuable comments which improved the paper.
}
{\footnotesize
\bibliographystyle{abbrvnat}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 291
|
{"url":"http:\/\/mathoverflow.net\/revisions\/23492\/list","text":"$0^0$ is undefined.\nEDIT: People write things like $\\sum_{k=0}^\\infty x^k$ all day, but somehow $x=k=0$ is still scary when written as $0^0$.\n$0^0$ is undefined.","date":"2013-05-20 01:45:03","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\": 0.8825010061264038, \"perplexity\": 568.1316413090891}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2013-20\/segments\/1368698203920\/warc\/CC-MAIN-20130516095643-00073-ip-10-60-113-184.ec2.internal.warc.gz\"}"}
| null | null |
{"url":"https:\/\/www.deepdyve.com\/lp\/ou_press\/lending-standards-over-the-credit-cycle-QjfsLoFyi2","text":"# Lending Standards over the Credit Cycle\n\nLending Standards over the Credit Cycle Abstract We analyze how firms\u2019 segmentation into credit classes affects the lending standards applied by banks to small and medium enterprises over the cycle. We exploit an institutional feature of the Italian credit market that generates a discontinuity in the allocation of comparable firms into the performing and substandard classes of credit risk. In the boom period, segmentation results in a positive interest rate spread between substandard and performing firms. In the bust period, the increase in banks\u2019 cost of wholesale funds implies that substandard firms are excluded from credit. These firms then report lower values of production and capital investments. Received January 22, 2016; editorial decision December 18, 2017 by Editor Robin Greenwood. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online. A growing empirical literature shows that segmentation between investment-grade and speculative-grade firms can have important implications for their access to capital markets (e.g., Kisgen and Strahan 2010; Lemmon and Roberts 2010; Chernenko and Sunderam 2012). Segmentation implies that firms of different credit quality have access to different pools of investor capital, and that the price and quantity available of this capital vary over time. An unanswered question is whether the effects of such asset class segmentation extend into bank lending policies, and lead to substantially different access to credit for otherwise similar small- and medium-sized enterprises (SME). This question is relevant not only because SME account for up to 70% of jobs in most Organisation for Economic Co-operation and Development (OECD) countries, but also because they nearly exclusively rely on bank financing (OECD 1997). In this paper, then, we study whether segmentation influences the bank lending standards applied to SME, and, relatedly, how the consequences of firm segmentation vary over the credit cycle. The empirical identification of the link between SME segmentation and bank lending standards is a challenging one. The reason is that the adjustment of lending standards can conform to different mechanisms. In neoclassical theories of financial intermediation, banks tighten credit by raising the credit spread and quantity drops along the credit demand. Alternatively, for given price, lenders can tighten standards by rationing risky firms\u2019 quantity of credit\u2014as in models with informational frictions. Consequently, to distinguish between these mechanisms requires detailed contract-level information on price and quantity of bank credit.1 To address this challenge, our analysis relies on a unique loan-level data set collected by the Italian central bank. This data set allows us to observe the total quantity of credit granted and the per-loan interest rate charged by financial intermediaries to SME. Our sample is composed of 144,000 firm-year observations in the manufacturing sector and 253,000 funding contracts covering the period between 2004 and 2011. Like other OECD economies, Italy was experiencing a credit cycle that reached its peak between 2006 and 2007 (Drehmann, Borio, and Tsatsaronis 2012) and then culminated with the Great Recession. To study the consequences of segmentation for firms\u2019 real decisions, we also use a comprehensive data set containing information on firms\u2019 balance sheet statements. Our full data sets then give us an untapped opportunity to study how firm segmentation shapes the relationship between banks and SME over the cycle. An additional empirical challenge to the analysis is how to isolate changes in banks\u2019 lending supply from changes in firms\u2019 desire to borrow. To do so, we exploit the institutional features of the Italian credit market for SME. First, for historical reasons, the credit risk assessment of SME performed by Italian banks uses a common credit rating (the Score) that banks purchase from an external agency (Centrale dei Bilanci, or CEBI). Unlike U.S. corporate credit ratings, the Score is unsolicited, available for all SME, and computed based only on firms\u2019 past balance sheets. Second, within this rating methodology, firms are allocated into two main rating classes\u2014performing and substandard\u2014based on the value of a continuous variable. Importantly, the bank has access to information on both the risk class and the continuous value of the firm\u2019s rating when making its decisions, but, when reporting its loan portfolio to financial markets, it classifies firms based only on their rating classes. The structure of the rating, and its construction, give us the opportunity to follow an intuitive empirical strategy to identify the effects of segmentation on financial contracts. Specifically, we exploit the sharp discontinuity in the allocation of firms into the performing and substandard classes of credit risk. As our measure of lending standards, then, we take the differences in the credit conditions between a firm marginally classified as performing and one that is marginally classified as substandard based on the value of the rating\u2019s continuous variable. These threshold differences inform us about how banks\u2019 supply of credit is affected by segmentation, while holding constant the demand for credit. The classification between substandard and performing risks is important for bank lending choices because it affects the banks\u2019 cost of financing. The national banking regulator adopts a conservative definition of nonperforming loans (NPL), which also includes loans of substandard credit quality (World Bank 2002; Bank of Italy 2013; Barisitz 2013; Jassaud and Kang 2015; Bholat et al. 2016).2 Banks then allocate loans in the category of nonperforming loans based, among others, on the risk status provided by their credit scoring (e.g., Intesa 2015). This has implications for bank capital and investor assessment of bank balance sheets. Indeed, NPL absorb valuable bank capital (Jassaud and Kang 2015), and their volume is often referenced as the major indicator of banks\u2019 asset quality by rating agencies (Moody\u2019s 2015; Fitch 2016). We empirically confirm the importance of banks\u2019 choice of exposure toward performing and substandard credit quality by relating the cost of funding borne by Italian banks to the composition of their loan portfolio. Our main findings on the impact of segmentation on lending conditions follow. In the boom period, the substandard and performing firms at the threshold are treated differently mainly in terms of the interest rates applied to new loans. Indeed, we find an interest rate spread of about 4% (or 20 basis points), and a positive but not statistically significant difference in the amount of granted credit. As a consequence of the financial crisis that hit the Italian banking sector, in the bust period banks tightened their lending standards mainly by acting on the quantity margin: specifically, the performing firms obtain 39% more financing than comparable substandard firms (at a similar interest rate). For the final years in our sample (2010\u20132011), our estimates point to a reduction in the differences of bank lending at the threshold, and an increase in the interest rate spread. All these results are consistent with those arising from a model of financial contracting in the presence of informational frictions and market segmentation. To quantify the importance of segmentation for bank lending, we compare the estimates of our threshold analysis to those arising from a na\u00efve specification that analyzes differences in the lending conditions between all performing and substandard firms. We find that, in the bust period, segmentation can account for a significantly larger part of the observed na\u00efve differential in the amount of credit than in the boom period. Another key insight arising from our discontinuity strategy relates to the patterns of the interest rate spread. While the na\u00efve interest rate differences are increasing throughout the cycle, we show that, during the crisis, the threshold spread is close to zero\u2014reflecting the implementation of lending standards\u2019 adjustment primarily via a restriction of substandard firms\u2019 access to credit. We then trace the implications of lending standards for firms\u2019 real activity. The production choices of the firms at the threshold significantly diverge during the crisis, to the point that the marginally performing firms report up to 50% larger values of production than the marginally substandard ones. After decomposing production values into firms\u2019 investment in inputs, we find that an increase in the interest rate spread induces firms to adjust their expenditures in variable inputs (i.e., intermediates and employment). Instead, in the bust period, when banks act on the quantity margin to adjust lending standards, firms respond by cutting capital investments, which typically have a long-run nature. The richness of our contract-level data allows us to study the economic mechanism driving the sensitivity of bank lending to segmentation. Specifically, we test for the relative importance of bank capitalization and bank investor composition in explaining the relationship between segmentation and lending policies. In line with, among others, Ivashina and Scharfstein (2009) and Iyer, Puri, and Ryan (2016), we show that the degree of exposure to funding from short-term investors is quantitatively more important than bank capitalization to explain our threshold differences. Finally, we compare the lending conditions applied to two comparable firms, one of which is downgraded to the substandard class as a result of a small change in the value of its continuous rating (which is observed only by the bank, not by its investors). This analysis shows that the negative impact of a downgrade on credit allocations becomes progressively larger and statistically significant in crisis and recovery. We confirm the internal validity of our results by presenting the following robustness checks to our empirical design. First, we find no systematic evidence of manipulation of the rating, which confirms the fact that it is very difficult for firms to manipulate the Score. Second, we show that, close to the threshold, firms feature comparable economic characteristics, and are thus \u201cas if\u201d randomly sampled. Third, we confirm the relevance of the threshold that assigns firms to the performing and substandard classes. In particular, we run our threshold analysis at all the other six thresholds associated with the categorical value of the rating, and find that most of the estimates are not statistically significant. This suggests that our results capture a form of market segmentation, not a simple rating effect. In addition to the literature on the consequences of market segmentation for financial contracts, our paper also contributes to the macrofinance literature studying the dynamics of credit over the cycle.3 Specifically, Greenwood and Hanson (2013) show that the deterioration of credit quality during booms forecasts low excess returns to bondholders. Similarly, in their historical account of credit cycles, Lopez-Salido, Stein, and Zakraj\u0161ek (2017) find that elevated credit sentiment is associated with a more aggressive pricing of risk and a subsequent contraction in economic activity. Consistent with these studies, we provide evidence of how the 2004\u20132011 cycle affected the transmission of market segmentation into bank lending policies. Our paper is also related to the body of work on empirical banking (e.g., Jim\u00e9nez et al. 2012, 2014; Chodorow-Reich 2014). We extend this literature by showing that, to understand the dynamics of bank lending standards, one needs to jointly analyze the price and quantity of lending.4 1. Documenting Segmentation in the Credit Market The goal of this section is to establish the presence of segmentation in the Italian credit market for SME. We will first present the institutional features of this market that generate segmentation, and then document the relationship between segmentation and the banks\u2019 cost of wholesale funds. 1.1 The Score rating system Evidence from the 2006 Bank of Italy survey of Italian banks indicates that 90% of the banks using a firm\u2019s rating find it important when deciding on whether to process a loan application, 76% of them use the rating to set the amount of lending, and 62% use it to formulate an interest-rate offer. For historical reasons, Italian banks use a common credit rating produced by Centrale dei Bilanci (CEBI) when making decisions about lending to SME. CEBI is a credit agency founded in 1983 as a joint initiative of the Italian Central Bank and the Italian Banking Association to record and process firms\u2019 financial statements. According to Standard & Poor\u2019s (2004), \u201cBanks are the main users of the outputs of CEBI,\u201d referring to the Score rating produced by CEBI as the major tool used to assess SME credit risk. In 2004, the share of credit granted to SME by banks subscribing to the Score rating system was 73%. The following features of the Score are of particular interest to our research design: The Score is unsolicited by firms and is computed based on firms\u2019 past balance sheets. Although its exact algorithm is a business secret of CEBI, information provided to the regulator by the agency that produces the Score shows that the construction of the rating is based on multiple discriminant analyses of past firm balance sheet information (Altman 1968).5 These features make the manipulation of the rating very unlikely. The system generates two continuous variables that determine the assignment to discrete rating categories. Based on predetermined thresholds, the first continuous variable is used to allocate the firms among the first five rating categories (1\u20135), the second to allocate the firms among categories 6 to 9. The Score therefore ranges from 1, for firms that are the least likely to default, to 9, for those most likely to default.6 We obtained from CEBI direct access to the information on the values of the continuous and discrete variables for the manufacturing firms rated by the agency. We also have access to the exact thresholds that determine the allocation of firms into the different rating categories. This means that we can reconstruct the exact firm allocation mechanism implemented within the Score rating system. Figure 1 illustrates some of the key empirical features of the Score. The left panel of Figure 1 plots the Score variable of firms in year $$t$$ against the share of delinquent firms in year $$t+1$$. To construct this figure, we combine information from Italian chambers of commerce and the credit register of the Italian central bank for the period 2004\u20132011. We define a firm as delinquent if it entered a formal bankruptcy process, or if its loan was flagged as late\/defaulted in the credit register. Finally, we decompose the informativeness of the rating variable across three periods: boom (2004\u20132007), bust (2008\u20132009), and recovery (2010\u20132011). The panel suggests a monotonic relationship between the rating variable and future credit events. Indeed, the share of delinquent firms with a Score of up to 4 in a given year hovers around 4%. This share rises to about 10% for firms with a Score of 7. At the same time, the decomposition of default rates across subperiods indicates that the informativeness of the rating variable is relatively stable over the cycle. More specifically, the increase in delinquency rates between the boom period and the bust period for a Score of 7 is less than one percentage point. Figure 1. View largeDownload slide Characteristics of the Score assignment variable The left panel plots the Score variable against the share of defaults within the next year in the boom (dashed), crisis (solid), and recovery (dotted). The right panel plots the average loan rate by Score category for the first quarter of 2005. Figure 1. View largeDownload slide Characteristics of the Score assignment variable The left panel plots the Score variable against the share of defaults within the next year in the boom (dashed), crisis (solid), and recovery (dotted). The right panel plots the average loan rate by Score category for the first quarter of 2005. The right panel of Figure 1 plots the rating variable against the interest rate on loans for the first quarter of 2005. A strong positive relationship exists between the rating variable and interest rates on loans. The best (lowest) Score, in terms of creditworthiness, is on average associated with a loan interest rate of 4%, and the worst (highest) category pays an average loan interest rate of around 5%.7 Figure 1 therefore suggests that the Score rating provides a reliable estimate of the expected likelihood of a firm\u2019s delinquency, which is then taken into account by the banks for their lending decisions. 1.2 Segmentation of SME in the Italian credit market Within the Score rating methodology, the distinction between the performing and substandard classes of credit risk stands out as particularly relevant for banks and their stakeholders. The performing class consists of the firms with a Score category between 1 and 6, and the substandard class comprises firms with a Score between 7 and 9.8 The importance of this classification stems from its implications for bank disclosure and reporting of their loan portfolio. National regulators decide on the loan categories that enter the class of NPL: this is relevant for our purposes because the Bank of Italy adopts a conservative definition of NPL, which includes loans of substandard credit quality (World Bank 2002; Bank of Italy 2013; Barisitz 2013; Jassaud and Kang 2015; Bholat et al. 2016).9 NPL absorb valuable bank capital: the capital charge for NPL amounts on average to 12% of banks\u2019 risk-weighted assets, and are estimated to tie up more than 6% of bank capital (Jassaud and Kang 2015).10 Moreover, a bank\u2019s exposure to NPL is often referenced as the major indicator of asset quality by the bank\u2019s rating agencies.11 Banks then allocate loans in the category of nonperforming loans based, among others, on the risk status provided by their credit scoring (e.g., Intesa 2015). Moreover, in their annual reports, they clearly distinguish between their exposure to the firms classified as substandard and performing by the rating (e.g., Unicredit 2008). As a consequence, investors monitor the volume of substandard lending to assess a bank\u2019s risk profile. The presence of such segmentation gives rise to clear, testable implications. First, one expects outside investors to charge a higher cost of funding to those banks that carry a higher volume of substandard loans in their loan book. Second, one should find that the continuous variables should not contain any useful information to explain the bank cost of funding on wholesale funding markets. 1.3 Segmentation and bank cost of financing We now provide evidence consistent with the presence of segmentation in the Italian credit market. We use three confidential data sets from the Bank of Italy. The first provides us with information on the amount and interest rate at which Italian banks raise financing from repo markets, households, and firms at a monthly frequency between 2004 and 2011. The second data set contains yearly bank balance sheets between 2006 and 2011, and provides us with information about a bank\u2019s size, capitalization, and liquidity. Finally, we use information from the credit register to determine the composition of each bank\u2019s SME portfolio based on the categorical and continuous variables of the rating system. To estimate the relationship between a bank\u2019s cost of financing and its lending portfolio, we use the following ordinary least squares (OLS) specification: \\begin{align} r_{b,t} &= \\alpha_{0} +\\alpha_{1} \\mbox{Substandard to Total Credit}_{b,t-1} +\\alpha_{2} \\mbox{Continuous Score 1}_{b,t-1} \\nonumber \\\\ &\\quad +\\alpha_{3} \\mbox{Continuous Score 2}_{b,t-1} + X_{b,t-1}\\Psi + I_{b,t}\\Phi + \\pi_{t} +\\epsilon_{b,t}. \\end{align} (1) In Equation (1), $$b$$ denotes a bank in our data set, and $$t$$ is taken at the monthly level. The dependent variable, $$r_{b,t}$$, is the (volume) weighted average interest rate paid by banks across all investors. Substandard to Total Credit$$_{b,t-1}$$ is the share of a bank\u2019s volume of lending to SME in the substandard rating class relative to total lending. Continuous Variable 1$$_{b,t-1}$$ and Continuous Variable 2$$_{b,t-1}$$ characterize the SME portfolio of the bank in terms of the average continuous ratings. $$X_{b,t-1}$$ denotes a vector of bank characteristics; $$I_{b,t}$$ denotes issuance characteristics such as amounts, maturity, and investor composition; and $$\\pi_{t}$$ are month-year fixed effects. All explanatory variables, except for issuance characteristics, are measured before the issuance. Standard errors are clustered at the bank level. In Columns (1) and (2) of Table 1, we show that external investors monitor banks by pricing lending portfolios based on banks\u2019 exposure to the substandard and performing classes. The estimate in Column (1) implies that a 25% higher share of substandard lending in the bank portfolio is associated with an increase in the bank\u2019s interest rate of approximately 28%, or 31 basis points. Column (2) extends the baseline specification in Equation (1) by including the continuous values produced by the rating system. The coefficient on the share of substandard loans remains significant and economically identical to the first specification. Instead, the coefficients on the values of the continuous variables are neither statistically nor economically significant. Our evidence is therefore consistent with the presence of market-driven segmentation in the Italian credit market for SME. Investors observe the distribution of loans into rating classes, and set a higher interest-rate premium to compensate for a larger exposure to substandard loans. Table 1 Banks\u2019 cost of financing and rating segmentation Pre-2008 Post-2008 (1) (2) (3) (4) Substandard to Total Credit 1.26*** 1.24* \u20130.37 1.34** (0.46) (0.66) (0.29) (0.68) Continuous Variable 1 \u20130.2 (0.15) Continuous Variable 2 0.09 (0.31) Bank & Issuance Characteristics Yes Yes Yes Yes Bank Fixed Effects No No Yes Yes Time Fixed Effects Yes Yes Yes Yes $$R^2$$ 0.76 0.76 0.85 0.54 N 4,788 4,728 2,233 2,212 Pre-2008 Post-2008 (1) (2) (3) (4) Substandard to Total Credit 1.26*** 1.24* \u20130.37 1.34** (0.46) (0.66) (0.29) (0.68) Continuous Variable 1 \u20130.2 (0.15) Continuous Variable 2 0.09 (0.31) Bank & Issuance Characteristics Yes Yes Yes Yes Bank Fixed Effects No No Yes Yes Time Fixed Effects Yes Yes Yes Yes $$R^2$$ 0.76 0.76 0.85 0.54 N 4,788 4,728 2,233 2,212 The table reports the estimates of the specification in Equation (1), using as a dependent variable the interest rate at which Italian banks raise financing. In Columns (1) and (2), the dependent variable is the (volume) weighted average interest rate at which banks raised financing across different types of investors (repo markets, households, firms) between 2004 and 2011. In Columns (3) and (4), we reestimate our pricing equation for the periods before and after 2008, respectively. Accordingly, the dependent variable is the interest rate at which banks raised financing on repurchase markets before 2008 in Column (3) and after 2008 in Column (4). Substandard to Total Credit is the share of a bank\u2019s volume of lending to SME in the substandard rating class relative to total lending. Continous Variable 1 denotes the mean of the continuous variable of firms in rating categories 1 to 5. Continous Variable 2 denotes the mean of the continuous variable of firms in rating categories 6 to 9. The specification includes a vector of bank and issuance characteristics. Issuance characteristics include amounts raised, maturity, and investor composition. Bank characteristics include size (in terms of total assets), the value of the tier 1 capitalization ratio, and the bank\u2019s liquidity ratio. The specification includes monthly fixed effects, with standard errors clustered at the bank level. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Table 1 Banks\u2019 cost of financing and rating segmentation Pre-2008 Post-2008 (1) (2) (3) (4) Substandard to Total Credit 1.26*** 1.24* \u20130.37 1.34** (0.46) (0.66) (0.29) (0.68) Continuous Variable 1 \u20130.2 (0.15) Continuous Variable 2 0.09 (0.31) Bank & Issuance Characteristics Yes Yes Yes Yes Bank Fixed Effects No No Yes Yes Time Fixed Effects Yes Yes Yes Yes $$R^2$$ 0.76 0.76 0.85 0.54 N 4,788 4,728 2,233 2,212 Pre-2008 Post-2008 (1) (2) (3) (4) Substandard to Total Credit 1.26*** 1.24* \u20130.37 1.34** (0.46) (0.66) (0.29) (0.68) Continuous Variable 1 \u20130.2 (0.15) Continuous Variable 2 0.09 (0.31) Bank & Issuance Characteristics Yes Yes Yes Yes Bank Fixed Effects No No Yes Yes Time Fixed Effects Yes Yes Yes Yes $$R^2$$ 0.76 0.76 0.85 0.54 N 4,788 4,728 2,233 2,212 The table reports the estimates of the specification in Equation (1), using as a dependent variable the interest rate at which Italian banks raise financing. In Columns (1) and (2), the dependent variable is the (volume) weighted average interest rate at which banks raised financing across different types of investors (repo markets, households, firms) between 2004 and 2011. In Columns (3) and (4), we reestimate our pricing equation for the periods before and after 2008, respectively. Accordingly, the dependent variable is the interest rate at which banks raised financing on repurchase markets before 2008 in Column (3) and after 2008 in Column (4). Substandard to Total Credit is the share of a bank\u2019s volume of lending to SME in the substandard rating class relative to total lending. Continous Variable 1 denotes the mean of the continuous variable of firms in rating categories 1 to 5. Continous Variable 2 denotes the mean of the continuous variable of firms in rating categories 6 to 9. The specification includes a vector of bank and issuance characteristics. Issuance characteristics include amounts raised, maturity, and investor composition. Bank characteristics include size (in terms of total assets), the value of the tier 1 capitalization ratio, and the bank\u2019s liquidity ratio. The specification includes monthly fixed effects, with standard errors clustered at the bank level. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. In Columns (3) and (4), we focus on the cost of financing on the repurchase market, the primary source of funds for the securitized banking system. This market is of particular interest, because Gorton and Metrick (2012) describe the crisis as a \u201crun on repo\u201d that was triggered by concerns about bank solvency. We therefore reestimate our pricing equation in Equation (1) separately for the period before and after 2008, and augment our specification with bank fixed effects. In the boom phase of the credit cycle, the correlation between interest rates on the repurchase market and the composition of banks\u2019 lending portfolio is low and statistically nonsignificant. In the bust period, the correlation is positive and economically significant, implying an increase in the interest rate premium required by investors from banks that are relatively more exposed to substandard credit risk. 2. Theoretical Framework To motivate our empirical analysis, we propose a model of credit with market segmentation and moral hazard. Specifically, we extend the basic framework in Tirole (2006, chap. 3), to accommodate the institutional features of the Italian credit market for SME. As in Tirole (2006), entrepreneurs need capital to fund a project. The bank-firm relationship is frustrated by moral hazard: by putting effort, the firm succeeds with positive probability. By shirking, the firm defaults with certainty, but the entrepreneur then gains private benefits. Finally, we allow firms to use the bank\u2019s monitoring technology. Differently from the standard setup, firms are segmented into two rating classes, performing and substandard. Moreover, we assume that the conditions at which banks receive funding depend on the phase of the credit cycle. We show that a bank\u2019s ability to tame the firm\u2019s moral hazard problem can be impaired when funding conditions on the wholesale market heat up (i.e., in the crisis period). This can push the bank to reduce lending at the expense of the substandard firms.12 The model features three categories of agents: the bank, its investors, and two firms. The two firms are allocated by the rating system used by the bank into the performing and substandard classes of credit risk. We consider the case in which the two firms fall exactly at the threshold between the two classes. The bank knows this, and understands that they are economically identical. The cost of funds to the bank is set by external investors who, consistent with our empirical evidence, observe only the firms\u2019 rating class. The existence of market segmentation has then two main implications for bank lending.13 First, the bank\u2019s cost of funding to a firm will reflect the composition of demand in the credit class. Second, the cost of financing paid by the bank will vary over the cycle according to the conditions on the wholesale funding market. In the boom period, the low cost at which the bank raises financing in the wholesale market implies that both firms can obtain access to unmonitored credit. More specifically, both firms receive the same amount of lending, but the substandard firm pays a higher interest rate\u2014mirroring the higher risk in their class. In the bust period, worse conditions on the market for wholesale funds erode firms\u2019 net worth, and imply that lending is not viable for the bank. Then, firms have two options: the first is to use the bank\u2019s monitoring technology, which comes at a cost, but also alleviates the moral hazard problem. Alternatively, they are (partially) rationed from credit. Assume that monitoring works with the performing firm, so that the bank can break even on this firm\u2019s project. Instead, the monitoring technology does not work for the substandard firm: that is, the rise in the cost of wholesale funding for the bank, combined with the cost of monitoring, implies that the net present value of the substandard firm\u2019s project remains negative. Then, the substandard firm is credit rationed at equilibrium. To sum up: in the bust period, quantity differences arise in the credit contracts offered to the two firms at the threshold. As we will further discuss later, these results will guide the interpretation of the credit differences arising in our empirical analysis. 3. Data Preview and Economic Environment To test the link between segmentation of firms and bank lending standards, we use a confidential data sets from the Bank of Italy that contain information on bank balance sheets and the financial contracts signed between banks and SME. We instead obtain firm balance sheets and rating information from CEBI. Our final sample is composed of about 144,000 firm-year observations in the manufacturing sector and 253,000 funding contracts signed between the first quarter of 2004 and the last quarter of 2011. Further details on the data set and its organization can be found in Online Appendix A. This section first documents the presence of substantial heterogeneity across rating classes. This heterogeneity suggests that a na\u00efve comparison between the credit conditions of firms in different rating classes is likely to yield misleading conclusions on the pattern of lending standards, because the resulting credit differences could simply reflect differences in firms\u2019 demand for credit. Then, we show the patterns of firms\u2019 financial contracts over time, which document how the phases of the credit cycle that Italy experienced between 2004 and 2011 affect financial allocations. Finally, we present key developments in the Italian banking environment that occurred during our sample period, illustrating the significant effects of the financial crisis on the wholesale funding and capitalization of Italian banks. 3.1 Firm financing environment We begin by presenting the sources of cross-sectional heterogeneity in our data set and the time-series variation in firm financial contracts. 3.1.1 Cross-sectional descriptive statistics Table 2 provides the cross-sectional characteristics of the full sample in Column (1). Columns (2) and (3) show corresponding results for the group of performing and substandard firms, and Columns (4) and (5) show the same for categories 6 and 7. Finally, Column (6) reports the mean difference between the values of the variables in categories 6 and 7. Table 2 Descriptive statistics All Performing Substandard Score 6 Score 7 6\u20137 Comparison (1) (2) (3) (4) (5) (6) Panel A: Loan information Term Loans: Interest Rate 4.57 4.32 5.3 4.79 5.29 \u20130.48*** (1.62) (1.56) (1.6) (1.58) (1.59) Term Loans: Amount 816 885 617 451 569 \u2013118 (9850) (5156) (17300) (1623) (17700) Term Loans: Maturity 0.66 0.66 0.65 0.77 0.72 0.05*** (0.47) (0.47) (0.48) (0.44) (247) N 253,502 188,026 65,475 49,265 60,326 109,591 Panel B: Aggregate financing information All Bank Financing Granted 8,503 9,237 6,167 7,542 6,392 1,150*** (37,200) (40,600) (23,100) (24,600) (21,100) Share of Term Loans Granted 0.35 0.35 0.36 0.33 0.35 \u20130.02*** (0.25) (0.25) (0.25) (0.21) (0.25) Share of Write-downs 0.01 0.01 0.03 0.00 0.01 \u20130.01*** (0.09) (0.04) (0.17) (0.05) (0.09) N 543,855 414,041 129,754 63,722 104,253 167,975 Panel C: Balance sheet information Employment 92 95 76 73 72 1 (294) (295) (290) (170) (207) Investment to Assets 0.05 0.05 0.04 0.04 0.039 0.001** (0.06) (0.06) (0.06) (0.06) (0.06) Return to Assets 0.05 0.07 0.00 0.05 0.03 0.02*** (0.10) (0.08) (0.13) (0.07) (0.07) Leverage 0.67 0.61 0.86 0.79 0.85 \u20130.06*** (0.19) (0.18) (0.10) (0.10) (0.09) N 143,953 108,353 35,600 16,432 27,350 43,782 All Performing Substandard Score 6 Score 7 6\u20137 Comparison (1) (2) (3) (4) (5) (6) Panel A: Loan information Term Loans: Interest Rate 4.57 4.32 5.3 4.79 5.29 \u20130.48*** (1.62) (1.56) (1.6) (1.58) (1.59) Term Loans: Amount 816 885 617 451 569 \u2013118 (9850) (5156) (17300) (1623) (17700) Term Loans: Maturity 0.66 0.66 0.65 0.77 0.72 0.05*** (0.47) (0.47) (0.48) (0.44) (247) N 253,502 188,026 65,475 49,265 60,326 109,591 Panel B: Aggregate financing information All Bank Financing Granted 8,503 9,237 6,167 7,542 6,392 1,150*** (37,200) (40,600) (23,100) (24,600) (21,100) Share of Term Loans Granted 0.35 0.35 0.36 0.33 0.35 \u20130.02*** (0.25) (0.25) (0.25) (0.21) (0.25) Share of Write-downs 0.01 0.01 0.03 0.00 0.01 \u20130.01*** (0.09) (0.04) (0.17) (0.05) (0.09) N 543,855 414,041 129,754 63,722 104,253 167,975 Panel C: Balance sheet information Employment 92 95 76 73 72 1 (294) (295) (290) (170) (207) Investment to Assets 0.05 0.05 0.04 0.04 0.039 0.001** (0.06) (0.06) (0.06) (0.06) (0.06) Return to Assets 0.05 0.07 0.00 0.05 0.03 0.02*** (0.10) (0.08) (0.13) (0.07) (0.07) Leverage 0.67 0.61 0.86 0.79 0.85 \u20130.06*** (0.19) (0.18) (0.10) (0.10) (0.09) N 143,953 108,353 35,600 16,432 27,350 43,782 All panels use data for the period 2004.Q1\u20132011.Q4, and monetary values expressed in KE (1,000 euro). Standard deviations are reported in brackets. The last column reports the difference in means of each variable between categories 6 and 7. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Panel A uses pooled loan-level data with observations at the loan-quarter level. Interest Rate is the gross annual interest rate inclusive of participation fees, loan origination fees, and monthly service charges. Amount is the granted amount of the issued term loan. Maturity is a binary variable indicating whether the maturity of the newly issued loans is up to one year, or longer. Panel B uses credit register data with observations at the firm-quarter level. All Bank Financing Granted is the firms\u2019 total amount of bank financing granted summing across all categories (loans, credit lines, backed loans). Share of Term Loans Granted is the firms\u2019 total amount of term loans granted, divided by the total amount of bank financing granted for all categories. Share of Write-downs is a binary variable indicating whether the firms\u2019 total amount of bank financing granted for all categories has experienced write-downs by banks. Panel C uses balance sheet and cash-flow statements at the firm-year level. Employment is defined as the firms\u2019 average employment over the year. Investment to Assets is the firms\u2019 investment in material fixed assets over total fixed assets. Returns to Assets is defined as the firms\u2019 earnings before interest and taxes over total assets. Leverage is the firms\u2019 ratio of debt (both short and long term) over total assets. In all panels, $$N$$ corresponds to the pooled number of observations in our sample. Table 2 Descriptive statistics All Performing Substandard Score 6 Score 7 6\u20137 Comparison (1) (2) (3) (4) (5) (6) Panel A: Loan information Term Loans: Interest Rate 4.57 4.32 5.3 4.79 5.29 \u20130.48*** (1.62) (1.56) (1.6) (1.58) (1.59) Term Loans: Amount 816 885 617 451 569 \u2013118 (9850) (5156) (17300) (1623) (17700) Term Loans: Maturity 0.66 0.66 0.65 0.77 0.72 0.05*** (0.47) (0.47) (0.48) (0.44) (247) N 253,502 188,026 65,475 49,265 60,326 109,591 Panel B: Aggregate financing information All Bank Financing Granted 8,503 9,237 6,167 7,542 6,392 1,150*** (37,200) (40,600) (23,100) (24,600) (21,100) Share of Term Loans Granted 0.35 0.35 0.36 0.33 0.35 \u20130.02*** (0.25) (0.25) (0.25) (0.21) (0.25) Share of Write-downs 0.01 0.01 0.03 0.00 0.01 \u20130.01*** (0.09) (0.04) (0.17) (0.05) (0.09) N 543,855 414,041 129,754 63,722 104,253 167,975 Panel C: Balance sheet information Employment 92 95 76 73 72 1 (294) (295) (290) (170) (207) Investment to Assets 0.05 0.05 0.04 0.04 0.039 0.001** (0.06) (0.06) (0.06) (0.06) (0.06) Return to Assets 0.05 0.07 0.00 0.05 0.03 0.02*** (0.10) (0.08) (0.13) (0.07) (0.07) Leverage 0.67 0.61 0.86 0.79 0.85 \u20130.06*** (0.19) (0.18) (0.10) (0.10) (0.09) N 143,953 108,353 35,600 16,432 27,350 43,782 All Performing Substandard Score 6 Score 7 6\u20137 Comparison (1) (2) (3) (4) (5) (6) Panel A: Loan information Term Loans: Interest Rate 4.57 4.32 5.3 4.79 5.29 \u20130.48*** (1.62) (1.56) (1.6) (1.58) (1.59) Term Loans: Amount 816 885 617 451 569 \u2013118 (9850) (5156) (17300) (1623) (17700) Term Loans: Maturity 0.66 0.66 0.65 0.77 0.72 0.05*** (0.47) (0.47) (0.48) (0.44) (247) N 253,502 188,026 65,475 49,265 60,326 109,591 Panel B: Aggregate financing information All Bank Financing Granted 8,503 9,237 6,167 7,542 6,392 1,150*** (37,200) (40,600) (23,100) (24,600) (21,100) Share of Term Loans Granted 0.35 0.35 0.36 0.33 0.35 \u20130.02*** (0.25) (0.25) (0.25) (0.21) (0.25) Share of Write-downs 0.01 0.01 0.03 0.00 0.01 \u20130.01*** (0.09) (0.04) (0.17) (0.05) (0.09) N 543,855 414,041 129,754 63,722 104,253 167,975 Panel C: Balance sheet information Employment 92 95 76 73 72 1 (294) (295) (290) (170) (207) Investment to Assets 0.05 0.05 0.04 0.04 0.039 0.001** (0.06) (0.06) (0.06) (0.06) (0.06) Return to Assets 0.05 0.07 0.00 0.05 0.03 0.02*** (0.10) (0.08) (0.13) (0.07) (0.07) Leverage 0.67 0.61 0.86 0.79 0.85 \u20130.06*** (0.19) (0.18) (0.10) (0.10) (0.09) N 143,953 108,353 35,600 16,432 27,350 43,782 All panels use data for the period 2004.Q1\u20132011.Q4, and monetary values expressed in KE (1,000 euro). Standard deviations are reported in brackets. The last column reports the difference in means of each variable between categories 6 and 7. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Panel A uses pooled loan-level data with observations at the loan-quarter level. Interest Rate is the gross annual interest rate inclusive of participation fees, loan origination fees, and monthly service charges. Amount is the granted amount of the issued term loan. Maturity is a binary variable indicating whether the maturity of the newly issued loans is up to one year, or longer. Panel B uses credit register data with observations at the firm-quarter level. All Bank Financing Granted is the firms\u2019 total amount of bank financing granted summing across all categories (loans, credit lines, backed loans). Share of Term Loans Granted is the firms\u2019 total amount of term loans granted, divided by the total amount of bank financing granted for all categories. Share of Write-downs is a binary variable indicating whether the firms\u2019 total amount of bank financing granted for all categories has experienced write-downs by banks. Panel C uses balance sheet and cash-flow statements at the firm-year level. Employment is defined as the firms\u2019 average employment over the year. Investment to Assets is the firms\u2019 investment in material fixed assets over total fixed assets. Returns to Assets is defined as the firms\u2019 earnings before interest and taxes over total assets. Leverage is the firms\u2019 ratio of debt (both short and long term) over total assets. In all panels, $$N$$ corresponds to the pooled number of observations in our sample. The table shows that there is significant heterogeneity among firms with different risk profiles, not only with respect to financial characteristics, but also in terms of balance sheet characteristics. More specifically, panel A of Table 2 shows that in the full sample, the average nominal interest rate charged for a loan is 4.57%. However, the interest rates applied to performing and substandard firms are 4.32% and 5.3%, respectively. Although the average loan in the sample is approximately 816,000 euro, it is about 617,000 euro for a firm in the substandard class. Moreover, the loans in our sample are mostly short term, as these account for around two-thirds of the total value of granted loans. Panel B reports the aggregate financing characteristics of the firms in our sample. On average, total bank lending amounts to 8.5 million euro (ME) per firm, 35% of which is in the form of loans. While firms in the performing class receive bank financing that adds up to about 9.2ME, firms in the substandard class receive an average of 6ME. Panel C provides an overview of the main balance sheet characteristics of Italian manufacturing firms based on unique firm-year observations. Firms in our sample are relatively small. On average, they employ 92 workers, with firms in the performing class being relatively larger than those in the substandard class. While the investment-to-asset ratio is stable across classes, the values of leverage and return to assets are not. The leverage ratio increases from 0.61 for firms in the performing class to 0.86 for those in the substandard class. Moreover, return on assets decreases from 0.07 to 0 for firms in these two classes. Finally, Column (6) of panel C shows that the heterogeneity in firm characteristics extends to rating categories 6 and 7. The cost and availability of bank financing suggests significantly tighter conditions for firms in category 7 as opposed to category 6. For instance, interest rates for firms in category 6 are 50 points lower than those of firms in category 7. At the same time, these firms are significantly different in terms of characteristics related to the demand for credit, such as the value of investment and profitability. Taken together, the descriptive statistics show the importance of obtaining a measure of lending standards that is not biased by demand heterogeneity. 3.1.2 Time-series descriptive statistics In Figure 2, we document the variation in financial contracts across time. Figure 2. View largeDownload slide Descriptive statistics across time In the left panel, we plot the per-firm aggregate value of bank financing for different rating categories across time. In the right panel, we plot the nominal average interest rates applied to firms in different rating categories across time. Figure 2. View largeDownload slide Descriptive statistics across time In the left panel, we plot the per-firm aggregate value of bank financing for different rating categories across time. In the right panel, we plot the nominal average interest rates applied to firms in different rating categories across time. The left panel illustrates that, like other OECD economies (Drehmann, Borio, and Tsatsaronis 2012), between 2004 and 2011 Italy was experiencing a credit cycle that reached its peak in 2007. The right panel focuses on firms\u2019 nominal average interest rates, showing that nominal rates mirrored the pattern of the indicators for the monetary policy of the European Central Bank. More specifically, the left panel shows that the time series of the amount of bank financing to Italian SME features a humped shape. From the first quarter of 2004 to the fourth quarter of 2007, bank financing increased by 18%, on average. It then decreased by 11% through the end of the sample period. Although this pattern is qualitatively similar across risk classes, the variation in bank financing is larger for substandard firms: between 2004 and 2008 bank financing to performing firms increased by 13%, while it rose by 29% for substandard firms. This evidence is consistent with the historical account of credit booms by Greenwood and Hanson (2013), who show that the quality of credit deteriorates as aggregate credit increases. Finally, the right panel of Figure 2 shows that nominal interest rates increased from 4.3% in 2004 to 6.11% in late 2008. Similar to the patterns in the left panel, the levels of the interest rate spreads are consistent with the risk categories in our rating system. 3.2 Banking environment In Figure 3, we illustrate the key developments in the Italian banking environment that occurred during our sample period. We use bank balance sheet data between 2006 and 2011 from Bank of Italy. Figure 3. View largeDownload slide Bank capital and credit risk In the top panel, we plot the ratio of the volume of repo financing over total assets for the five largest banks in our data set. In the middle panel, we plot the tier 1 capital ratio for the five largest banks in our data set across time. In the bottom panel, we use data from the European Central Bank statistical data warehouse to plot the credit risk capital allocations over total capital requirements (black line), the fraction of capital allocations computed using the standardized approach (gray line), and the fraction computed using the internal rating-based (IRB) approach (dashed black line). Figure 3. View largeDownload slide Bank capital and credit risk In the top panel, we plot the ratio of the volume of repo financing over total assets for the five largest banks in our data set. In the middle panel, we plot the tier 1 capital ratio for the five largest banks in our data set across time. In the bottom panel, we use data from the European Central Bank statistical data warehouse to plot the credit risk capital allocations over total capital requirements (black line), the fraction of capital allocations computed using the standardized approach (gray line), and the fraction computed using the internal rating-based (IRB) approach (dashed black line). The top panel of Figure 3 plots the share of repo financing of banks relative to their total assets for the five largest banks in our sample. In the expansionary phase of the cycle, the dependence of banks on repo financing grew from 5% in 2005 to nearly 12% at the beginning of 2008. During the financial crisis, this source of financing plummeted to 2.5% and remained at low levels until the end of our sample period. The middle panel of Figure 3 illustrates the capitalization of Italian banks: we compute the tier 1 capital ratio for the five largest banks in our sample by dividing banks\u2019 tier 1 capital by their total assets. The figure shows that the average value of banks\u2019 capital ratio at the beginning of the financial crisis period was approximately 4.5%. In 2008 the ratio fell to around 3.6%, before rising above 5% toward the end of the sample period. The patterns in these two panels are shared by the banking systems of other European countries during the same time interval. The bottom panel of Figure 3 provides evidence on the implementation of the Basel II agreements. Credit risk capital allocations account for more than 100% of total capital requirements through 2008 and 2010, implying that credit risk management was critical for Italian banks during our sample period. Moreover, the transition from Basel I to Basel II is unlikely to drive the evolution of lending standards in our sample. Indeed, the total fraction of capital allocations calculated using internal rating systems oscillates around 20%. 4. The Empirical Model 4.1 Identification strategy Empirically identifying how segmentation influences bank lending standards is challenging for two reasons. First, it requires a setup where the econometrician observes the exact information held by the bank about the firm credit risk profile. Then, to isolate demand from supply considerations, the econometrician would like to compare firms that are identical from the perspective of the loan officer, but classified into different classes of credit risk. To address these challenges, we exploit the institutional features of the Italian credit market for SME introduced in Section 1. The structure of the rating, and its construction, give us the opportunity to follow an intuitive empirical strategy to identify the effects of segmentation on financial contracts. Specifically, we exploit the sharp discontinuity in the allocation of firms into the performing and substandard classes of credit risk. As our measure of lending standards, then, we take the differences in the credit conditions between a firm marginally classified as performing and one that is marginally classified as substandard based on the value of the rating\u2019s continuous variable. These threshold differences inform us about how banks\u2019 supply of credit is affected by segmentation, while holding constant the demand for credit. The support of the continuous variable for categories 6 and 7 ranges between \u20130.6 and 1.5, and the threshold is 0.15. Below this threshold, a firm\u2019s Score is 7 and thus the firm falls into the substandard class. Above the threshold, a firm\u2019s Score is 6 and it is in the performing class. Throughout the analysis, we normalize the threshold to 0 and only use the support of the continuous variable that spans between categories 6 and 7. Thus, if $$s_{i}$$ is the value of firm $$i$$\u2019s continuous variable, the allocation of this firm into a rating class takes place according to the following sharp mechanism: \\begin{eqnarray} {\\textit{Score}}_{i,t} = \\left\\{ \\begin{array}{@{}llll} 6\\ \\text{(Performing)} & \\quad \\text{If $0 \\leq s_{i,t} <1.35$} \\\\\\ \\\\ 7\\ \\text{(Substandard)} & \\quad \\text{If $-0.75 \\leq s_{i,t}<0$} \\end{array} \\right. . \\end{eqnarray} (2) 4.2 Main specification Let $$\\bar{s}$$ denote the normalized threshold that allocates firms into rating categories 6 and 7. Our main specification follows: \\begin{align} y_{i,t} &= \\beta_{0} +\\beta_{1}(\\mbox{Performing}_{i,t}\\times\\mbox{Boom}_{t})+\\beta_{2} (\\mbox{Performing}_{i,t}\\times\\mbox{Crisis}_{t})\\nonumber\\\\ & \\quad +\\beta_{3} (\\mbox{Performing}_{i,t}\\times\\mbox{Recovery}_{t})+f_{t}(s_{i,t}-\\bar{s})\\nonumber\\\\ & \\quad +\\mbox{Performing}_{i,t}\\times g_{t}(s_{i,t}-\\bar{s}) + \\pi_{t} + u_{i,t}. \\end{align} (3) The dependent variable capturing the supply of bank financing is the (log) total value of bank lending granted to firm $$i$$ in quarter $$t$$. This measure accounts for the possibility that firms obtain credit from multiple banks. The variable capturing the cost of bank financing is the (log) value of the interest rate applied to a new loan granted to firm $$i$$ in quarter $$t$$. The indicator Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 between the first quarter of 2004 until the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 between the first quarter of 2008 until the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 between the first quarter of 2010 until the last quarter of 2011, and 0 otherwise. Functions $$f_{t}(\\cdot)$$ and $$g_t(\\cdot)$$ correspond to flexible sixth-order polynomials whose goal is to fit the smoothed curves on either side of the cutoff as closely to the data as possible. Function $$f_{t}(\\cdot)$$ is estimated from $$0$$ to the left, whereas the Performing$$_{i,t}\\times g_{t}(\\cdot)$$ term is estimated from $$0$$ to the right. The subscript $$t$$ for $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ indicates that the polynomials are separately estimated for each time period through interactions with $$\\pi_{t}$$, the quarter-year fixed effects. $$u_{i,t}$$ is a mean-zero error term clustered at the firm level.14 As a robustness check, we will estimate a version of the specification that also includes the past value of the rating, and its interaction with each time period. The interpretation of Equation (1) is the following. First, note that, at the cutoff, the $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ polynomials are evaluated at 0 and drop out of the calculation. This allows us to interpret the parameters $$(\\beta_{1}, \\beta_{2}, \\beta_{3})$$ as capturing the magnitude of the discontinuity in credit conditions at the threshold $$\\bar s$$. The null hypothesis of our framework is that if a bank uses all its information on the borrowing firm, there should be no discontinuity in lending contracts at the threshold. In other words, under our null hypothesis, segmentation should not matter for lending decisions. Second, the estimated discontinuity parameters $$(\\beta_{1}, \\beta_{2}, \\beta_{3})$$ have an intuitive interpretation. The estimate of $$\\beta_{1}$$ measures differences in credit allocations between marginally performing firms and substandard firms in the period between 2004 and 2007. The estimate of $$\\beta_{2}$$ measures differences in credit allocations in the period between 2008 and 2009. Finally, the estimate of $$\\beta_{3}$$ captures the difference between marginally performing firms and substandard firms in the period between 2010 and 2011. In the main specification, we restrict our attention to the sample of firms that remain in the same rating category for at least two consecutive years. This condition limits two potential concerns. The first is that the bank reports to investors a firm as performing on the basis of its rating in $$t-1$$, even though it is already downgraded in $$t$$. The second is related to the possibility that large variations in the value of the continuous rating that then lead to downgrades might themselves be correlated to the firms\u2019 demand for credit. We then separately study the source of variation coming from a firm downgrade for financial contracting, and provide evidence based on downgrades caused by small changes in the value of the continuous rating. We extend our main specification in two directions. First, we study whether, via its impact on lending standards, segmentation is relevant for firms\u2019 real choices. Specifically, we estimate Equation (3) using as dependent variables firms\u2019 expenditures in production inputs and the value of production. The balance sheet information we use for this analysis is reported in end-of-the-year statements; thus, it reflects a firm\u2019s lending conditions throughout the year. Second, we look at the differences between the lending conditions at the threshold within each phase of the credit cycle. To this end, we estimate Equation (4) separately for each quarter-year cross-section of firms at the threshold in our sample period: $$y_{i,.}=\\beta_{0}+\\beta_{1} \\mbox{Performing}_{i,.}+f(s_{i,.} -\\bar{s})+\\mbox{Performing}_{i,.}\\times g(s_{i,.}-\\bar{s}) + u_{i,.}.$$ (4) In Equation (4), the dot indicates that we fix the time period. This exercise is meant to understand whether there are distinct credit dynamics within each of the subperiods of the credit cycle. 4.3 Mechanism for the transmission of market segmentation In this section, we first exploit the heterogeneity of the banks in our data set to study how banks\u2019 financial structure affects the sensitivity of lending to market segmentation. Then, we analyze the implications of segmentation for marginally downgraded firms over the cycle. 4.3.1 Banks\u2019 financial structure We consider two channels through which financial structure can affect banks\u2019 sensitivity to market segmentation: capital requirements and investor composition. Intuitively, low levels of regulatory capital can help explain a bank\u2019s greater sensitivity to market segmentation. Similarly, investor composition can account for the sensitivity of banks to market segmentation: certain investor categories are more responsive than others to bank solvency risk, and update their assessment of bank loan quality over the cycle (Ivashina and Scharfstein 2009; Iyer, Puri, and Ryan 2016). To explore the relative merits of these two channels in determining bank sensitivity to segmentation, we compute the following measures of bank heterogeneity. To study the role played by capital requirements, we compute, for the pre-crisis period, each bank\u2019s tier 1 capital ratio. To study heterogeneity in investor composition, we focus on the importance of repo markets for a bank funding structure. As we show in Table 1, during the crisis, investors in repo markets updated their interest rate conditions based on banks\u2019 exposure to substandard firms. We therefore measure each banks\u2019 pre-crisis share of financing from repo markets. We augment our main specification with interactions between the Performing$$_{it}$$ indicator and these bank-specific characteristics: \\begin{align} y_{i,b,t} & = \\beta_{0} +\\beta_{1}(\\mbox{Performing}_{i,t}\\times\\mbox{Boom}_{t})+\\beta_{2}(\\mbox{Performing}_{i,t}\\times\\mbox{Crisis}_{t})\\nonumber\\\\ & \\quad +\\beta_{3}(\\mbox{Performing}_{i,t}\\times\\mbox{Recovery}_{t})\\nonumber\\\\ & \\quad +\\gamma_{1}(\\mbox{Performing}_{i,t}\\times\\mbox{Tier1}_{b}\\times\\mbox{Boom}_{t})\\nonumber\\\\ & \\quad +\\gamma_{2} (\\mbox{Performing}_{i,t}\\times\\mbox{Tier1}_{b}\\times\\mbox{Crisis}_{t})\\nonumber\\\\ & \\quad +\\gamma_{3} (\\mbox{Performing}_{i,t}\\times\\mbox{Tier1}_{b}\\times \\mbox{Recovery}_{t})\\nonumber\\\\ & \\quad +\\delta_{1} (\\mbox{Performing}_{i,t}\\times\\mbox{Repo}_{b}\\times\\mbox{Boom}_{t})\\nonumber\\\\ & \\quad +\\delta_{2} (\\mbox{Performing}_{i,t}\\times\\mbox{Repo}_{b}\\times\\mbox{Crisis}_{t})\\nonumber\\\\ & \\quad +\\delta_{3} (\\mbox{Performing}_{i,t}\\times\\mbox{Repo}_{b}\\times\\mbox{Recovery}_{t})\\nonumber\\\\ & \\quad +f_{t}(s_{i,t}-\\bar{s})+\\mbox{Performing}_{i,t}\\times g_{t}(s_{i,t}-\\bar{s}) + X_{i,b,t}\\Psi + \\pi_{t} + u_{i,t}. \\end{align} (5) In Equation (5), Tier1$$_b$$ is defined as a bank $$b$$\u2019s core equity capital divided by its total assets, and Repo$$_{b}$$ is defined as the share of the bank\u2019s total financing from repo markets. $$X_{i,b,t}$$ is a vector that includes the levels and interactions of all the variables in the set of triple interactions. Standard errors are clustered at the firm-bank level. As an additional robustness check, we augment Equation (5) by including firm-year fixed effects. 4.3.2 Analysis of downgraded firms Finally, we study how market segmentation affects the lending policies set on firms that are marginally downgraded over the cycle. More specifically, we ask what is the implication of a downgrade to substandard quality for credit conditions over the cycle, and whether the bank exploits its superior information on the company\u2019s downgrade. We compare two firms that fall in the performing class until year $$t-1$$, but differ in their rating class in year $$t$$.15 The specification follows: \\begin{align} y_{i,b,t} & = \\beta_{0} + \\beta_{1} (\\mbox{Down}_{i,t}\\times\\mbox{Boom}_{t})+\\beta_{2} (\\mbox{Down}_{i,t}\\times\\mbox{Crisis}_{t})\\nonumber\\\\ &\\quad +\\beta_{3} (\\mbox{Down}_{i,t}\\times\\mbox{Recovery}_{t})\\nonumber\\\\ &\\quad +f_{t}(s_{i,t}-s_{i,t-1})+\\mbox{Down}_{i,t}\\times g_{t}(s_{i,t}-s_{i,t-1})\\nonumber\\\\ &\\quad +m_{t}(s_{i,t-1})+\\mbox{Down}_{i,t}\\times n_{t}(s_{i,t-1}) + \\pi_{t} + u_{i,t}. \\end{align} (6) Down$$_{i,t}$$ is a binary variable equal to 1 if the firm is downgraded from category 6 to category 7 in year $$t$$, and is 0 otherwise. In Equation (5), the polynomials in $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ are a function of the change in the continuous variable between $$t-1$$ and $$t$$. By evaluating these polynomials close to 0, our analysis considers those firms that were downgraded as a consequence of a similar and small change in the value of the continuous rating. To make sure that we implement a local identification of downgraded and non-downgraded firms around the threshold, we augment the specification by including also polynomials for the continuous assignment variable in $$t-1$$, $$m_{t}(\\cdot)$$ and $$n_{t}(\\cdot)$$. Consequently, we compare firms that not only experienced a similar small change in the continuous variable, but were also close to the threshold.16 5. Results In this section, we present the results on the differences in credit conditions\u2014specifically, differences in the interest rates and in the total amount of bank financing\u2014for firms at the threshold between the performing and the substandard classes. We then decompose the changes in lending standards within each phase by estimating differences in credit allocations separately for each quarter. Finally, we explore whether differences in credit conditions give rise to differences in firms\u2019 production and input choices. 5.1 Results on credit allocations Table 3 reports the estimates related to credit allocations. The dependent variable in Columns (1) to (3) is the log amount of total bank financing granted to the firm, while in Columns (4) to (6) the dependent variable is the log interest rate on new bank loans. In Columns (1) and (4) we report the estimates of the main specification in Equation (3), while Columns (2) and (5) augment this specification by interacting past ratings with quarter-year fixed effects. Finally, Columns (3) and (6) report the results of a na\u00efve specification comparing lending conditions to all performing and substandard firms. Table 3 Credit effects Quantity Price Dependent variable (1) (2) (3) (4) (5) (6) Boom $$\\times$$ Performing 0.11 0.11 0.35*** \u20130.04** \u20130.04** \u20130.15*** (0.09) (0.09) (0.02) (0.02) (0.02) (0.01) Crisis $$\\times$$ Performing 0.33*** 0.33*** 0.32*** 0.03 0.03 \u20130.15*** (0.11) (0.11) (0.02) (0.03) (0.03) (0.01) Recovery $$\\times$$ Performing 0.20* 0.20* 0.30*** \u20130.08** \u20130.08** \u20130.27*** (0.12) (0.11) (0.02) (0.04) (0.04) (0.01) Lagged Rating \u20130.01*** 0.00*** 0.00*** 0.00*** (0.00) (0.00) (0.00) (0.00) Polynomial Yes Yes No Yes Yes No Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes $$R^2$$ 0.06 0.06 0.01 0.38 0.38 0.43 N 157,775 157,775 518,047 105,865 105,865 246,240 Quantity Price Dependent variable (1) (2) (3) (4) (5) (6) Boom $$\\times$$ Performing 0.11 0.11 0.35*** \u20130.04** \u20130.04** \u20130.15*** (0.09) (0.09) (0.02) (0.02) (0.02) (0.01) Crisis $$\\times$$ Performing 0.33*** 0.33*** 0.32*** 0.03 0.03 \u20130.15*** (0.11) (0.11) (0.02) (0.03) (0.03) (0.01) Recovery $$\\times$$ Performing 0.20* 0.20* 0.30*** \u20130.08** \u20130.08** \u20130.27*** (0.12) (0.11) (0.02) (0.04) (0.04) (0.01) Lagged Rating \u20130.01*** 0.00*** 0.00*** 0.00*** (0.00) (0.00) (0.00) (0.00) Polynomial Yes Yes No Yes Yes No Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes $$R^2$$ 0.06 0.06 0.01 0.38 0.38 0.43 N 157,775 157,775 518,047 105,865 105,865 246,240 In Columns (1), (2), (4), and (5) the table reports OLS estimates of the threshold specification in Equation (3). Instead, Columns (3) and (6) estimate a simple mean difference specification using data for all firms in the rating system. The dependent variable in the first three columns is the (log) total value of bank lending granted to firm $$i$$ in quarter $$t$$. The dependent variable in the last three columns is the (log) value of the interest rate applied to a new loan granted to firm $$i$$ in quarter $$t$$. The indicator Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 from the first quarter of 2004 to the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 from the first quarter of 2008 to the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 from the first quarter of 2010 to the last quarter of 2011, and 0 otherwise. Functions $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ correspond to flexible sixth-order polynomials estimated separately for each quarter-year. Function $$f_{t}(\\cdot)$$ is estimated from $$0$$ to the left, whereas the Performing$$_{i,t}\\times g_{t}(\\cdot)$$ term is estimated from $$0$$ to the right. Finally, Lagged Rating controls for the past value of the rating. In Columns (2) and (5) the variable is interacted with quarter-year fixed effects. Standard errors, clustered at the firm level, are reported in brackets. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Table 3 Credit effects Quantity Price Dependent variable (1) (2) (3) (4) (5) (6) Boom $$\\times$$ Performing 0.11 0.11 0.35*** \u20130.04** \u20130.04** \u20130.15*** (0.09) (0.09) (0.02) (0.02) (0.02) (0.01) Crisis $$\\times$$ Performing 0.33*** 0.33*** 0.32*** 0.03 0.03 \u20130.15*** (0.11) (0.11) (0.02) (0.03) (0.03) (0.01) Recovery $$\\times$$ Performing 0.20* 0.20* 0.30*** \u20130.08** \u20130.08** \u20130.27*** (0.12) (0.11) (0.02) (0.04) (0.04) (0.01) Lagged Rating \u20130.01*** 0.00*** 0.00*** 0.00*** (0.00) (0.00) (0.00) (0.00) Polynomial Yes Yes No Yes Yes No Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes $$R^2$$ 0.06 0.06 0.01 0.38 0.38 0.43 N 157,775 157,775 518,047 105,865 105,865 246,240 Quantity Price Dependent variable (1) (2) (3) (4) (5) (6) Boom $$\\times$$ Performing 0.11 0.11 0.35*** \u20130.04** \u20130.04** \u20130.15*** (0.09) (0.09) (0.02) (0.02) (0.02) (0.01) Crisis $$\\times$$ Performing 0.33*** 0.33*** 0.32*** 0.03 0.03 \u20130.15*** (0.11) (0.11) (0.02) (0.03) (0.03) (0.01) Recovery $$\\times$$ Performing 0.20* 0.20* 0.30*** \u20130.08** \u20130.08** \u20130.27*** (0.12) (0.11) (0.02) (0.04) (0.04) (0.01) Lagged Rating \u20130.01*** 0.00*** 0.00*** 0.00*** (0.00) (0.00) (0.00) (0.00) Polynomial Yes Yes No Yes Yes No Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes $$R^2$$ 0.06 0.06 0.01 0.38 0.38 0.43 N 157,775 157,775 518,047 105,865 105,865 246,240 In Columns (1), (2), (4), and (5) the table reports OLS estimates of the threshold specification in Equation (3). Instead, Columns (3) and (6) estimate a simple mean difference specification using data for all firms in the rating system. The dependent variable in the first three columns is the (log) total value of bank lending granted to firm $$i$$ in quarter $$t$$. The dependent variable in the last three columns is the (log) value of the interest rate applied to a new loan granted to firm $$i$$ in quarter $$t$$. The indicator Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 from the first quarter of 2004 to the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 from the first quarter of 2008 to the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 from the first quarter of 2010 to the last quarter of 2011, and 0 otherwise. Functions $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ correspond to flexible sixth-order polynomials estimated separately for each quarter-year. Function $$f_{t}(\\cdot)$$ is estimated from $$0$$ to the left, whereas the Performing$$_{i,t}\\times g_{t}(\\cdot)$$ term is estimated from $$0$$ to the right. Finally, Lagged Rating controls for the past value of the rating. In Columns (2) and (5) the variable is interacted with quarter-year fixed effects. Standard errors, clustered at the firm level, are reported in brackets. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. The estimates related to the period between 2004 and 2007 in Columns (1) and (4) suggest that segmentation mainly results in a positive interest rate spread between substandard and performing firms at the threshold. Firms in the substandard class are charged up to 4%,17 or 20 basis points, higher interest rates on new bank loans than similar firms in the performing class. The difference in the total amount of lending granted to these firms, instead, is positive (11%) but not statistically significant. The size of this coefficient reflects the large within-period dynamics occurring in the boom period, as we discuss later. Through 2008 and 2009, the financial crisis that hit the Italian banking sector led to an exacerbation of the consequences of segmentation for lending policies. Importantly, we find that tighter lending standards essentially translate into differences in the quantity of lending for the firms at the threshold. Indeed, marginally performing firms obtain 39% more bank financing than similar firms across the threshold. Instead, interest rate differences remain stable and close to zero (in economic and statistical terms). These results are consistent with the prediction of our theoretical framework. A rise in the interest rates paid by banks to outside investors, together with the increase in the opportunity cost of lending to substandard firms, translate into an equilibrium in which banks monitor the performing firms and (partly) exclude substandard firms from lending. Between 2010 and 2011, our estimates are in line with an incomplete recovery of bank lending. During this period, segmentation means a reduction in the differences in the quantity of credit from 39% to 22%. However, this reduction is accompanied by an increase in the interest rate spread to approximately 8%, or 40 basis points. To better understand the dynamics of credit within each phase, we report, in Figure 4, the quarterly estimates obtained with the specification in Equation (4). Figure 4. View largeDownload slide Discontinuity estimates for quantity and price The figure plots the estimates and 90% confidence intervals of the threshold specification in Equation (4), run on each distinct quarter in our sample period (2004.Q1\u20132011.Q4). The dependent variable in the top panel is the (log) total value of bank lending granted to firm $$i$$ in quarter $$t$$ (top panel). The dependent variable in the bottom panel is the (log) value of the interest rate applied to a new loan granted to firm $$i$$ in quarter $$t$$ (bottom panel). The plotted discontinuity estimates refer to Performing$$_{i,t}$$, an indicator variable that takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. Figure 4. View largeDownload slide Discontinuity estimates for quantity and price The figure plots the estimates and 90% confidence intervals of the threshold specification in Equation (4), run on each distinct quarter in our sample period (2004.Q1\u20132011.Q4). The dependent variable in the top panel is the (log) total value of bank lending granted to firm $$i$$ in quarter $$t$$ (top panel). The dependent variable in the bottom panel is the (log) value of the interest rate applied to a new loan granted to firm $$i$$ in quarter $$t$$ (bottom panel). The plotted discontinuity estimates refer to Performing$$_{i,t}$$, an indicator variable that takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. The top panel of the figure shows that early in the boom period (2004 and 2005), differences in the total amount of lending to firms at the threshold are positive and large, but progressively vanish in 2006 and 2007. These patterns then are likely to explain the economically large differences in total lending for the boom period in Column (1) of Table 3. Similarly to total lending, the interest rate spreads between firms at the threshold (bottom panel) narrow throughout the boom phase and disappear at the peak of the cycle. Differences in credit allocations are relatively stable within the crisis period, but vary again during the recovery period. Specifically, the estimates for 2010 and 2011 imply a gradual decrease in the differences in the amount of bank lending. To quantify the impact of segmentation on bank lending, we contrast the results obtained with our threshold analysis to those arising from a na\u00efve specification that compares lending conditions to all performing and substandard firms (Columns (3) and (6)). First, segmentation is relatively more important to explain the na\u00efve specification\u2019s differences in total lending in the bust than in the boom period. In the boom period, the na\u00efve estimates imply a 42% differential in the amount of bank credit. During that period, the threshold differences amount to only 12%, suggesting that segmentation alone cannot explain the large estimate in the na\u00efve specification. In 2008\u20132009, the overall differential between the quantity of lending across rating classes remains stable, while Column (1) indicates a 39% differential for the firms at the threshold. In the bust period, then, segmentation can account for a larger part of the observed differential in the amount of credit than in the boom period. Second, the analysis of the interest rate spreads arising from a na\u00efve comparison would lead to misleading conclusions, not only quantitatively but also qualitatively. Indeed, the results of the na\u00efve regression suggest that the interest rate differences are persistently large in economic terms, and increasing throughout the cycle. Instead, we show that, within our discontinuity design, the interest rate spread narrows over the boom phase, and disappears during the crisis. This reflects the fact that, in the bust period, bank lending standards\u2019 adjustments are implemented primarily by changing the quantity of credit. 5.2 Implications for firms\u2019 real activity Table 4 reports the results of our baseline regression in Equation (3) using as dependent variables the log of firms\u2019 sales and expenditures in investment, employment, and intermediates. The balance sheet reports contain only partial information about employment choices; thus, to fill this data gap, we obtain employment figures from firms\u2019 mandatory contributions to the Italian pension system, and merge this information based on the firms\u2019 fiscal identifier. Table 4 Real effects Production Investment Intermediates Employment Dependent variable (1) (2) (3) (4) (5) (6) (7) (8) Boom $$\\times$$ Performing 0.16* 0.16* 0.14 0.14 0.11 0.11 0.14* 0.14* (0.09) (0.09) (0.15) (0.15) (0.09) (0.09) (0.08) (0.08) Crisis $$\\times$$ Performing 0.44*** 0.44*** 0.55** 0.55** 0.43*** 0.43*** 0.39*** 0.38*** (0.12) (0.12) (0.23) (0.23) (0.13) (0.13) (0.16) (0.11) Recovery $$\\times$$ Performing 0.31** 0.31** 0.12 0.12 0.28* 0.28* 0.28** 0.28** (0.14) (0.14) (0.23) (0.23) (0.15) (0.15) (0.15) (0.14) Lagged Rating \u20130.01*** \u20130.00* \u20130.01*** \u20130.01*** (0.00) (0.00) (0.00) (0.00) Polynomial Yes Yes Yes Yes Yes Yes Yes Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes Yes Yes $$R^2$$ 0.09 0.09 0.07 0.07 0.09 0.09 0.02 0.02 N 41,157 41,157 33,889 33,889 40,585 40,585 39,041 39,041 Production Investment Intermediates Employment Dependent variable (1) (2) (3) (4) (5) (6) (7) (8) Boom $$\\times$$ Performing 0.16* 0.16* 0.14 0.14 0.11 0.11 0.14* 0.14* (0.09) (0.09) (0.15) (0.15) (0.09) (0.09) (0.08) (0.08) Crisis $$\\times$$ Performing 0.44*** 0.44*** 0.55** 0.55** 0.43*** 0.43*** 0.39*** 0.38*** (0.12) (0.12) (0.23) (0.23) (0.13) (0.13) (0.16) (0.11) Recovery $$\\times$$ Performing 0.31** 0.31** 0.12 0.12 0.28* 0.28* 0.28** 0.28** (0.14) (0.14) (0.23) (0.23) (0.15) (0.15) (0.15) (0.14) Lagged Rating \u20130.01*** \u20130.00* \u20130.01*** \u20130.01*** (0.00) (0.00) (0.00) (0.00) Polynomial Yes Yes Yes Yes Yes Yes Yes Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes Yes Yes $$R^2$$ 0.09 0.09 0.07 0.07 0.09 0.09 0.02 0.02 N 41,157 41,157 33,889 33,889 40,585 40,585 39,041 39,041 The table reports the estimates of the threshold specification in Model (3) using as dependent variables the (log) sales, investment, employment, and intermediates of firm $$i$$ in year $$t$$. The indicator Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 from the first quarter of 2004 to the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 from the first quarter of 2008 to the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 from the first quarter of 2010 to the last quarter of 2011, and 0 otherwise. Functions $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ correspond to flexible sixth-order polynomials estimated separately for each year. Function $$f_{t}(\\cdot)$$ is estimated from $$0$$ to the left, whereas the Performing$$_{i,t}\\times g_{t}(\\cdot)$$ term is estimated from $$0$$ to the right. Finally, Lagged Rating controls for the past value of the rating. In Columns (2), (4), (6), and (8) the variable is interacted with quarter-year fixed effects. Standard errors, clustered at the firm level, are reported in brackets. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Table 4 Real effects Production Investment Intermediates Employment Dependent variable (1) (2) (3) (4) (5) (6) (7) (8) Boom $$\\times$$ Performing 0.16* 0.16* 0.14 0.14 0.11 0.11 0.14* 0.14* (0.09) (0.09) (0.15) (0.15) (0.09) (0.09) (0.08) (0.08) Crisis $$\\times$$ Performing 0.44*** 0.44*** 0.55** 0.55** 0.43*** 0.43*** 0.39*** 0.38*** (0.12) (0.12) (0.23) (0.23) (0.13) (0.13) (0.16) (0.11) Recovery $$\\times$$ Performing 0.31** 0.31** 0.12 0.12 0.28* 0.28* 0.28** 0.28** (0.14) (0.14) (0.23) (0.23) (0.15) (0.15) (0.15) (0.14) Lagged Rating \u20130.01*** \u20130.00* \u20130.01*** \u20130.01*** (0.00) (0.00) (0.00) (0.00) Polynomial Yes Yes Yes Yes Yes Yes Yes Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes Yes Yes $$R^2$$ 0.09 0.09 0.07 0.07 0.09 0.09 0.02 0.02 N 41,157 41,157 33,889 33,889 40,585 40,585 39,041 39,041 Production Investment Intermediates Employment Dependent variable (1) (2) (3) (4) (5) (6) (7) (8) Boom $$\\times$$ Performing 0.16* 0.16* 0.14 0.14 0.11 0.11 0.14* 0.14* (0.09) (0.09) (0.15) (0.15) (0.09) (0.09) (0.08) (0.08) Crisis $$\\times$$ Performing 0.44*** 0.44*** 0.55** 0.55** 0.43*** 0.43*** 0.39*** 0.38*** (0.12) (0.12) (0.23) (0.23) (0.13) (0.13) (0.16) (0.11) Recovery $$\\times$$ Performing 0.31** 0.31** 0.12 0.12 0.28* 0.28* 0.28** 0.28** (0.14) (0.14) (0.23) (0.23) (0.15) (0.15) (0.15) (0.14) Lagged Rating \u20130.01*** \u20130.00* \u20130.01*** \u20130.01*** (0.00) (0.00) (0.00) (0.00) Polynomial Yes Yes Yes Yes Yes Yes Yes Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes Yes Yes $$R^2$$ 0.09 0.09 0.07 0.07 0.09 0.09 0.02 0.02 N 41,157 41,157 33,889 33,889 40,585 40,585 39,041 39,041 The table reports the estimates of the threshold specification in Model (3) using as dependent variables the (log) sales, investment, employment, and intermediates of firm $$i$$ in year $$t$$. The indicator Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 from the first quarter of 2004 to the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 from the first quarter of 2008 to the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 from the first quarter of 2010 to the last quarter of 2011, and 0 otherwise. Functions $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ correspond to flexible sixth-order polynomials estimated separately for each year. Function $$f_{t}(\\cdot)$$ is estimated from $$0$$ to the left, whereas the Performing$$_{i,t}\\times g_{t}(\\cdot)$$ term is estimated from $$0$$ to the right. Finally, Lagged Rating controls for the past value of the rating. In Columns (2), (4), (6), and (8) the variable is interacted with quarter-year fixed effects. Standard errors, clustered at the firm level, are reported in brackets. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Columns (1) and (2) yield three main findings. First, between 2004 and 2007, marginally performing firms on average produce 17% more than marginally substandard firms. A yearly decomposition of these estimates, which is reported in the Online Appendix, suggests that, consistent with the economically large differences in credit allocations arising in 2004 and 2005 (Figure 4), production differences are mainly concentrated in the early years of the boom and vanish in 2007. Our second finding is that production choices of firms at the threshold diverge significantly during the period in which access to credit is limited for the marginally substandard firms: in 2008 and 2009, the marginally performing firms report about 50% larger values of production than the marginally substandard ones. Finally, consistent with the partial recovery of lending taking place between 2010 and 2011, we find that, in this period, production differences gradually decrease but remain larger than the pre-crisis ones. To further the analysis of the implications of shifts in lending standards for firm real activity, we report the differences in input choices made by the firms at the threshold. We estimate our discontinuity design using as dependent variables the value of firms\u2019 investment in capital, expenditures in intermediates, and employment. The main finding is that the divergence in production outcomes during the crisis is driven mainly by investment choices. During the most acute phase of the financial crisis, on average, performing firms invest about 70% more than substandard firms. In recovery, instead, lower values of production are essentially driven by reduced expenditures in intermediate and labor inputs. 6. The Economic Mechanism In this section, we investigate the economic mechanism driving the transmission of segmentation onto bank lending standards. 6.1 Bank heterogeneity Table 5 investigates the possible channels through which bank heterogeneity can explain how segmentation affects credit supply. In Columns (1) and (2), we jointly test for the relative importance of bank capitalization and investor composition in determining the sensitivity of bank lending to segmentation. Recall that, to proxy bank capitalization, we measure banks\u2019 tier 1 capital ratio. Instead, as a measure of investor composition, we take the banks\u2019 dependence on fundings from repo markets. Both measures are taken as a pre-2008 average at the bank level. Table 5 Bank heterogeneity Quantity Price Dependent variable (1) (2) (3) (4) Boom $$\\times$$ Performing 0.08 \u20130.05* (0.07) (0.03) Crisis $$\\times$$ Performing 0.26** \u20130.02 (0.09) (0.04) Recovery $$\\times$$ Performing 0.08 \u20130.12* (0.11) (0.07) Boom $$\\times$$ Performing $$\\times$$ Tier1 0.16 \u20130.07 0.09 0.11 (0.29) (0.23) (0.16) (0.14) Crisis $$\\times$$ Performing $$\\times$$ Tier1 \u20130.90** \u20130.83*** \u20130.00 0.29 (0.43) (0.33) (0.27) (0.27) Recovery$$\\times$$Performing$$\\times$$Tier1 \u20130.71 \u20130.96** 0.15 0.33 (0.55) (0.41) (.39) (.41) Boom $$\\times$$ Performing $$\\times$$ Repo 0.02 \u20130.02 \u20130.21** 0.06 (0.15) (0.11) (0.09) (0.08) Crisis $$\\times$$ Performing $$\\times$$ Repo 0.39* 0.31* 0.25* 0.16 (0.23) (0.19) (0.13) (0.16) Recovery $$\\times$$ Performing $$\\times$$ Repo 0.58* 0.36 \u20130.03 0.04 (0.35) (0.28) (0.25) (0.30) Polynomial Yes Yes No Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Firm $$\\times$$ Year Fixed Effects No Yes No Yes $$R^2$$ 0.02 0.55 0.37 0.77 N 787,634 787,634 89,140 89,140 Quantity Price Dependent variable (1) (2) (3) (4) Boom $$\\times$$ Performing 0.08 \u20130.05* (0.07) (0.03) Crisis $$\\times$$ Performing 0.26** \u20130.02 (0.09) (0.04) Recovery $$\\times$$ Performing 0.08 \u20130.12* (0.11) (0.07) Boom $$\\times$$ Performing $$\\times$$ Tier1 0.16 \u20130.07 0.09 0.11 (0.29) (0.23) (0.16) (0.14) Crisis $$\\times$$ Performing $$\\times$$ Tier1 \u20130.90** \u20130.83*** \u20130.00 0.29 (0.43) (0.33) (0.27) (0.27) Recovery$$\\times$$Performing$$\\times$$Tier1 \u20130.71 \u20130.96** 0.15 0.33 (0.55) (0.41) (.39) (.41) Boom $$\\times$$ Performing $$\\times$$ Repo 0.02 \u20130.02 \u20130.21** 0.06 (0.15) (0.11) (0.09) (0.08) Crisis $$\\times$$ Performing $$\\times$$ Repo 0.39* 0.31* 0.25* 0.16 (0.23) (0.19) (0.13) (0.16) Recovery $$\\times$$ Performing $$\\times$$ Repo 0.58* 0.36 \u20130.03 0.04 (0.35) (0.28) (0.25) (0.30) Polynomial Yes Yes No Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Firm $$\\times$$ Year Fixed Effects No Yes No Yes $$R^2$$ 0.02 0.55 0.37 0.77 N 787,634 787,634 89,140 89,140 The table reports OLS estimates of the threshold specification in Equation (5). The dependent variable in the first two columns is the (log) total value of bank lending granted by bank $$b$$ to firm $$i$$ in quarter $$t$$. The dependent variable in the last two columns is the (log) value of the interest rate applied to a new loan granted by bank $$b$$ to firm $$i$$ in quarter $$t$$. The indicator Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 from the first quarter of 2004 to the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 from the first quarter of 2008 to the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 from the first quarter of 2010 to the last quarter of 2011, and 0 otherwise. Tier1$$_b$$ is defined as a bank $$b$$\u2019s core equity capital divided by its total assets, and Repo$$_{b}$$ is defined as the share of the bank\u2019s total financing from repo markets. All of the bank specific variables are measured pre-crisis. Functions $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ correspond to flexible sixth-order polynomials estimated separately for each quarter-year. Function $$f_{t}(\\cdot)$$ is estimated from $$0$$ to the left, whereas the Performing$$_{i,t}\\times g_{t}(\\cdot)$$ term is estimated from $$0$$ to the right. Finally, Lagged Rating controls for the past value of the rating and is interacted with quarter-year fixed effects. Standard errors, clustered at the firm-bank level, are reported in brackets. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Table 5 Bank heterogeneity Quantity Price Dependent variable (1) (2) (3) (4) Boom $$\\times$$ Performing 0.08 \u20130.05* (0.07) (0.03) Crisis $$\\times$$ Performing 0.26** \u20130.02 (0.09) (0.04) Recovery $$\\times$$ Performing 0.08 \u20130.12* (0.11) (0.07) Boom $$\\times$$ Performing $$\\times$$ Tier1 0.16 \u20130.07 0.09 0.11 (0.29) (0.23) (0.16) (0.14) Crisis $$\\times$$ Performing $$\\times$$ Tier1 \u20130.90** \u20130.83*** \u20130.00 0.29 (0.43) (0.33) (0.27) (0.27) Recovery$$\\times$$Performing$$\\times$$Tier1 \u20130.71 \u20130.96** 0.15 0.33 (0.55) (0.41) (.39) (.41) Boom $$\\times$$ Performing $$\\times$$ Repo 0.02 \u20130.02 \u20130.21** 0.06 (0.15) (0.11) (0.09) (0.08) Crisis $$\\times$$ Performing $$\\times$$ Repo 0.39* 0.31* 0.25* 0.16 (0.23) (0.19) (0.13) (0.16) Recovery $$\\times$$ Performing $$\\times$$ Repo 0.58* 0.36 \u20130.03 0.04 (0.35) (0.28) (0.25) (0.30) Polynomial Yes Yes No Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Firm $$\\times$$ Year Fixed Effects No Yes No Yes $$R^2$$ 0.02 0.55 0.37 0.77 N 787,634 787,634 89,140 89,140 Quantity Price Dependent variable (1) (2) (3) (4) Boom $$\\times$$ Performing 0.08 \u20130.05* (0.07) (0.03) Crisis $$\\times$$ Performing 0.26** \u20130.02 (0.09) (0.04) Recovery $$\\times$$ Performing 0.08 \u20130.12* (0.11) (0.07) Boom $$\\times$$ Performing $$\\times$$ Tier1 0.16 \u20130.07 0.09 0.11 (0.29) (0.23) (0.16) (0.14) Crisis $$\\times$$ Performing $$\\times$$ Tier1 \u20130.90** \u20130.83*** \u20130.00 0.29 (0.43) (0.33) (0.27) (0.27) Recovery$$\\times$$Performing$$\\times$$Tier1 \u20130.71 \u20130.96** 0.15 0.33 (0.55) (0.41) (.39) (.41) Boom $$\\times$$ Performing $$\\times$$ Repo 0.02 \u20130.02 \u20130.21** 0.06 (0.15) (0.11) (0.09) (0.08) Crisis $$\\times$$ Performing $$\\times$$ Repo 0.39* 0.31* 0.25* 0.16 (0.23) (0.19) (0.13) (0.16) Recovery $$\\times$$ Performing $$\\times$$ Repo 0.58* 0.36 \u20130.03 0.04 (0.35) (0.28) (0.25) (0.30) Polynomial Yes Yes No Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Firm $$\\times$$ Year Fixed Effects No Yes No Yes $$R^2$$ 0.02 0.55 0.37 0.77 N 787,634 787,634 89,140 89,140 The table reports OLS estimates of the threshold specification in Equation (5). The dependent variable in the first two columns is the (log) total value of bank lending granted by bank $$b$$ to firm $$i$$ in quarter $$t$$. The dependent variable in the last two columns is the (log) value of the interest rate applied to a new loan granted by bank $$b$$ to firm $$i$$ in quarter $$t$$. The indicator Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 from the first quarter of 2004 to the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 from the first quarter of 2008 to the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 from the first quarter of 2010 to the last quarter of 2011, and 0 otherwise. Tier1$$_b$$ is defined as a bank $$b$$\u2019s core equity capital divided by its total assets, and Repo$$_{b}$$ is defined as the share of the bank\u2019s total financing from repo markets. All of the bank specific variables are measured pre-crisis. Functions $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ correspond to flexible sixth-order polynomials estimated separately for each quarter-year. Function $$f_{t}(\\cdot)$$ is estimated from $$0$$ to the left, whereas the Performing$$_{i,t}\\times g_{t}(\\cdot)$$ term is estimated from $$0$$ to the right. Finally, Lagged Rating controls for the past value of the rating and is interacted with quarter-year fixed effects. Standard errors, clustered at the firm-bank level, are reported in brackets. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. We begin by interpreting our results in Column (1). First, notice that the baseline differences remain qualitatively very similar to those obtained with the main specification. Second, in the pre-crisis period, bank heterogeneity does not seem to affect how banks set the amount of total lending. This is intuitive: in the boom period, banks expect favorable financing conditions on wholesale markets. This means that they can lend \u201cas if\u201d unconstrained by segmentation, and make full use of their information on the firms\u2019 risk profile. These patterns change dramatically during the crisis. The negative sign on the interaction (Crisis$$\\times$$Performing$$\\times$$Tier1) indicates that highly capitalized banks are less likely to offer different amounts of credit to borrowers at the threshold. Similarly, those banks that are less dependent on short-term investors are also less likely to cut on lending as a consequence of market segmentation. Interestingly, the sensitivity of bank lending to these factors remains high even in the recovery period. Column (2) augments the discontinuity design by including firm-year fixed effects. This means that we exploit heterogeneity in the amount of lending to the same firm and in the same year from different banks. The estimates remain similar despite the increase in the number of estimated parameters. Columns (3) and (4) repeat the analysis by looking at the differences in interest rates. Our estimates suggest that bank heterogeneity is not particularly helpful to explain banks\u2019 price differences at the threshold. For instance, there is no evidence of significant differences in the spreads set by highly and lowly capitalized banks. Moreover, although, in principle, investor composition could affect the interest rate spreads, the evidence arising from the estimated parameters in Table 5 is rather mixed and, thus, inconclusive. To analyze the quantitative importance of bank capitalization and investor composition, we relate the results in the table to the drop in capitalization and repo financing that happened between 2007 and 2009. During that period, Italian banks\u2019 tier 1 capitalization fell by almost one percentage point. If we take the implied cumulative effect of segmentation and multiply it by the drop in capitalization, we obtain a differential tightening at the threshold of only $$0.6\\%$$ (or $$\\left(\\exp\\left\\{-0.90\\right\\}-1\\right)\\times0.01$$). Instead, the share of repo financing by banks went from 10% in 2007, to approximately 2% at the end of 2009. This suggests that the investor composition channel can account for a differential quantity tightening of approximately $$3.8\\%$$ (or $$\\left(\\exp\\left\\{0.39\\right\\}-1\\right)\\times0.08$$). This represents 10% of the observed threshold difference during the crisis, and indicates that investor composition is quantitatively an important channel to explain the consequences of segmentation on lending policies.18 6.2 Evidence from downgrades In Table 6, we report estimates of lending conditions to downgraded firms. Table 6 Downgrades from performing to substandard Quantity Price Production Dependent variable (1) (2) (3) (4) (5) (6) Boom$$\\times$$Down 0.10*** 0.04 0.03*** \u20130.02 0.08*** 0.06 (0.03) (0.14) (0.00) (0.03) (0.03) (0.12) Crisis$$\\times$$Down 0.03 \u20130.12 0.01 \u20130.06 \u20130.00 \u20130.14 (0.04) (0.21) (0.01) (0.06) (0.04) (0.19) Recovery$$\\times$$Down \u20130.12*** \u20130.33* 0.03** 0.12** \u20130.01 \u20130.31* (0.04) (0.18) (0.01) (0.06) (0.04) (0.16) Polynomial No Yes No Yes No Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes $$R^2$$ 0.01 0.04 0.39 0.40 0.02 0.07 N 88,830 88,830 70,848 70,848 22,978 22,978 Quantity Price Production Dependent variable (1) (2) (3) (4) (5) (6) Boom$$\\times$$Down 0.10*** 0.04 0.03*** \u20130.02 0.08*** 0.06 (0.03) (0.14) (0.00) (0.03) (0.03) (0.12) Crisis$$\\times$$Down 0.03 \u20130.12 0.01 \u20130.06 \u20130.00 \u20130.14 (0.04) (0.21) (0.01) (0.06) (0.04) (0.19) Recovery$$\\times$$Down \u20130.12*** \u20130.33* 0.03** 0.12** \u20130.01 \u20130.31* (0.04) (0.18) (0.01) (0.06) (0.04) (0.16) Polynomial No Yes No Yes No Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes $$R^2$$ 0.01 0.04 0.39 0.40 0.02 0.07 N 88,830 88,830 70,848 70,848 22,978 22,978 The table reports OLS estimates of the threshold specification in Equation (6) using the sample of firms downgraded from Score 6 to 7. The dependent variable in the first two columns is the (log) total value of bank lending granted to firm $$i$$ in quarter $$t$$. The dependent variable in the third and fourth columns is the (log) value of the interest rate applied to a new loan granted to firm $$i$$ in quarter $$t$$. The dependent variable in the fifth and sixth columns is the (log) value of sales of firm $$i$$ in year $$t$$. The indicator Down$$_{i,t}$$ is a binary variable equal to 1 if the firm is downgraded from category 6 to category 7 in year $$t$$, and is 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 from the first quarter of 2004 to the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 from the first quarter of 2008 to the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 from the first quarter of 2010 to the last quarter of 2011, and 0 otherwise. Polynomial functions $$f_{t}(\\cdot)$$, $$g_{t}(\\cdot)$$, $$m_{t}(\\cdot)$$, $$n_{t}(\\cdot)$$ correspond to flexible sixth-order polynomials estimated separately for each quarter-year. The polynomials in $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ are a function of the change in the continuous variable between $$t-1$$ and $$t$$. The polynomials in $$m_{t}(\\cdot)$$ and $$n_{t}(\\cdot)$$ are a function of the continuous variable in $$t-1$$. Standard errors, clustered at the firm level, are reported in brackets. In addition, we include polynomials for the level of the continuous variable in $$t-1$$. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Table 6 Downgrades from performing to substandard Quantity Price Production Dependent variable (1) (2) (3) (4) (5) (6) Boom$$\\times$$Down 0.10*** 0.04 0.03*** \u20130.02 0.08*** 0.06 (0.03) (0.14) (0.00) (0.03) (0.03) (0.12) Crisis$$\\times$$Down 0.03 \u20130.12 0.01 \u20130.06 \u20130.00 \u20130.14 (0.04) (0.21) (0.01) (0.06) (0.04) (0.19) Recovery$$\\times$$Down \u20130.12*** \u20130.33* 0.03** 0.12** \u20130.01 \u20130.31* (0.04) (0.18) (0.01) (0.06) (0.04) (0.16) Polynomial No Yes No Yes No Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes $$R^2$$ 0.01 0.04 0.39 0.40 0.02 0.07 N 88,830 88,830 70,848 70,848 22,978 22,978 Quantity Price Production Dependent variable (1) (2) (3) (4) (5) (6) Boom$$\\times$$Down 0.10*** 0.04 0.03*** \u20130.02 0.08*** 0.06 (0.03) (0.14) (0.00) (0.03) (0.03) (0.12) Crisis$$\\times$$Down 0.03 \u20130.12 0.01 \u20130.06 \u20130.00 \u20130.14 (0.04) (0.21) (0.01) (0.06) (0.04) (0.19) Recovery$$\\times$$Down \u20130.12*** \u20130.33* 0.03** 0.12** \u20130.01 \u20130.31* (0.04) (0.18) (0.01) (0.06) (0.04) (0.16) Polynomial No Yes No Yes No Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes $$R^2$$ 0.01 0.04 0.39 0.40 0.02 0.07 N 88,830 88,830 70,848 70,848 22,978 22,978 The table reports OLS estimates of the threshold specification in Equation (6) using the sample of firms downgraded from Score 6 to 7. The dependent variable in the first two columns is the (log) total value of bank lending granted to firm $$i$$ in quarter $$t$$. The dependent variable in the third and fourth columns is the (log) value of the interest rate applied to a new loan granted to firm $$i$$ in quarter $$t$$. The dependent variable in the fifth and sixth columns is the (log) value of sales of firm $$i$$ in year $$t$$. The indicator Down$$_{i,t}$$ is a binary variable equal to 1 if the firm is downgraded from category 6 to category 7 in year $$t$$, and is 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 from the first quarter of 2004 to the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 from the first quarter of 2008 to the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 from the first quarter of 2010 to the last quarter of 2011, and 0 otherwise. Polynomial functions $$f_{t}(\\cdot)$$, $$g_{t}(\\cdot)$$, $$m_{t}(\\cdot)$$, $$n_{t}(\\cdot)$$ correspond to flexible sixth-order polynomials estimated separately for each quarter-year. The polynomials in $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ are a function of the change in the continuous variable between $$t-1$$ and $$t$$. The polynomials in $$m_{t}(\\cdot)$$ and $$n_{t}(\\cdot)$$ are a function of the continuous variable in $$t-1$$. Standard errors, clustered at the firm level, are reported in brackets. In addition, we include polynomials for the level of the continuous variable in $$t-1$$. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. In Columns (1), (3), and (5), we estimate a na\u00efve version of the specification in Equation (6), without the polynomial terms in the continuous variables ($$f_t(\\cdot),g_t(\\cdot),m_t(\\cdot),n_t(\\cdot)$$). The estimate obtained with this specification relative to the boom phase (Column (1)) suggests that downgraded firms obtain 10% more bank financing than non-downgraded firms. This puzzling result is most likely caused by some unobserved heterogeneity across these groups. Indeed, if banks were to use the information on the change in the rating\u2019s value to shape their response to downgrades, we would at most expect the absence of negative effects or the existence of small effects. In Columns (2), (4), and (6), we estimate the full specification, and thus compare firms that not only experienced a similar small change in the continuous variable, but were also close to the threshold. Consistent with our intuition earlier, we find that downgraded firms do not obtain higher volumes of credit than non-downgraded firms in the boom period. In crisis and recovery, the negative impact of a downgrade on credit allocations becomes progressively larger and statistically significant. During the recovery period, downgraded firms obtain 39% less bank financing than non-downgraded firms. The estimates in Column (4) also show that the restricted access to credit during the recovery period is accompanied by a higher cost of funds for downgraded firms. Finally, Column (6) shows that differences in the amount of production between marginally downgraded and non-downgraded firms are small and not statistically significant during the boom period. Intuitively, consistent with the credit patterns, these production differences are reversed during the subsequent phases of the cycle. 7. Empirical Tests In this section, we test the three identifying assumptions underlying our empirical setting. First, we show that firms do not seem to manipulate their ratings to self-select into more favorable categories. Second, we show that firms at the threshold are balanced in terms of their economic characteristics. Finally, we present placebo tests to provide further evidence on the relevance of the threshold between the substandard and performing classes of credit risk. Given that the Score is computed on a yearly basis, we perform these tests on the yearly cross-section of firms, unless otherwise stated. 7.1 Manipulation of the Score and self-selection Given the importance of the Score in bank credit decisions, a natural question to ask is whether firms are able to manipulate their credit rating and self-select into a better category. Manipulation of the rating is very unlikely, not only because the Score is unsolicited by firms and is computed based on firms\u2019 past balance sheets, but also because its exact algorithm is a business secret. Nevertheless, manipulation can be detected empirically: it would result in a systematic discontinuity of firms\u2019 distribution at the threshold, due either to the absence of observations near the threshold or to the presence of clusters of observations on the side of the threshold assigning a firm to the safer category. In Table 7, we test for the presence of a discontinuity in firm density at that threshold. Table 7 Self-selection into rating categories 6 and 7 Year 2004 2005 2006 2007 2008 2009 2010 2011 McCrary Density Estimate 0.10 0.13 0.02 0.08 0.3*** \u20130.00 0.08 0.17 (0.06) (0.07) (0.07) (0.06) (0.07) (0.08) (0.10) (0.10) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 Year 2004 2005 2006 2007 2008 2009 2010 2011 McCrary Density Estimate 0.10 0.13 0.02 0.08 0.3*** \u20130.00 0.08 0.17 (0.06) (0.07) (0.07) (0.06) (0.07) (0.08) (0.10) (0.10) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 The table reports, at a yearly level, the McCrary density estimates of the continuous variable\u2019s distribution. For each year, we run a kernel local linear regression of the log of the density on both sides of the threshold separating substandard firms in category 7 from performing firms in category 6. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Table 7 Self-selection into rating categories 6 and 7 Year 2004 2005 2006 2007 2008 2009 2010 2011 McCrary Density Estimate 0.10 0.13 0.02 0.08 0.3*** \u20130.00 0.08 0.17 (0.06) (0.07) (0.07) (0.06) (0.07) (0.08) (0.10) (0.10) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 Year 2004 2005 2006 2007 2008 2009 2010 2011 McCrary Density Estimate 0.10 0.13 0.02 0.08 0.3*** \u20130.00 0.08 0.17 (0.06) (0.07) (0.07) (0.06) (0.07) (0.08) (0.10) (0.10) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 The table reports, at a yearly level, the McCrary density estimates of the continuous variable\u2019s distribution. For each year, we run a kernel local linear regression of the log of the density on both sides of the threshold separating substandard firms in category 7 from performing firms in category 6. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Following McCrary (2008), for each year we run a kernel local linear regression of the log of the density on both sides of the threshold separating substandard firms in category 7 from performing firms in category 6. Table 7 shows that, with the exception of 2008, there is no evidence of significant discontinuities in the distribution of firms at the threshold. The discontinuity in 2008 is most likely coincidental for two reasons. First, if firms had discovered the exact formula of the Score and how to manipulate their assignment, a discontinuity should emerge systematically in every year following 2008. Second, had strategic manipulation occurred, it would mean that firms had anticipated by at least one year the financial crisis and the associated benefits of being classified as marginally performing entities.19 7.1.1 Policy experiment We also exploit a policy experiment to address the potential concern that the discontinuity arising in the McCrary tests for 2008 reflects firms\u2019 strategic manipulation of the Score. In November 2008, Law 185 (decreto legislativo n. 185) granted firms the possibility to revaluate fixed assets. Crucially, differently from previous laws with the same goal, Law 185 does not require the firm to pay taxes on the higher values of the assets in its balance sheet. We exploit this policy experiment in the following way: we run our main specification in Equation (3) using as dependent variable the (log) value of revalued assets. If the Score was manipulated, then we should observe that those firms that marginally fall in the performing class during the crisis were also those that revaluated assets disproportionally more than the marginally substandard firms. Table 8 shows that there is no significant difference in the outcome variable across the three phases of the credit cycle. This evidence further confirms that manipulation of the assignment variable is highly unlikely. Table 8 Manipulation: Revaluations Log revaluations Dependent variable (1) (2) Boom $$\\times$$ Performing \u20130.04 \u20130.05 (.05) (0.06) Crisis $$\\times$$ Performing \u20130.01 \u20130.03 (0.06) (0.07) Recovery $$\\times$$ Performing 0.00 \u20130.02 (0.05) (0.06) Polynomial Yes Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Control Variables No Yes $$R^2$$ 0.12 0.21 N 77,079 57,243 Log revaluations Dependent variable (1) (2) Boom $$\\times$$ Performing \u20130.04 \u20130.05 (.05) (0.06) Crisis $$\\times$$ Performing \u20130.01 \u20130.03 (0.06) (0.07) Recovery $$\\times$$ Performing 0.00 \u20130.02 (0.05) (0.06) Polynomial Yes Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Control Variables No Yes $$R^2$$ 0.12 0.21 N 77,079 57,243 The table reports OLS estimates of the threshold specification in Model 3. The dependent variable is the (log) value of revalued assets of firm $$i$$ in year $$t$$. The indicator Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 from the first quarter of 2004 to the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 from the first quarter of 2008 to the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 from the first quarter of 2010 to the last quarter of 2011, and 0 otherwise. Functions $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ correspond to flexible sixth-order polynomials estimated separately for each quarter-year. Function $$f_{t}(\\cdot)$$ is estimated from $$0$$ to the left, whereas the Performing$$_{i,t}\\times g_{t}(\\cdot)$$ term is estimated from $$0$$ to the right. Standard errors, clustered at the firm level, are reported in brackets. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Table 8 Manipulation: Revaluations Log revaluations Dependent variable (1) (2) Boom $$\\times$$ Performing \u20130.04 \u20130.05 (.05) (0.06) Crisis $$\\times$$ Performing \u20130.01 \u20130.03 (0.06) (0.07) Recovery $$\\times$$ Performing 0.00 \u20130.02 (0.05) (0.06) Polynomial Yes Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Control Variables No Yes $$R^2$$ 0.12 0.21 N 77,079 57,243 Log revaluations Dependent variable (1) (2) Boom $$\\times$$ Performing \u20130.04 \u20130.05 (.05) (0.06) Crisis $$\\times$$ Performing \u20130.01 \u20130.03 (0.06) (0.07) Recovery $$\\times$$ Performing 0.00 \u20130.02 (0.05) (0.06) Polynomial Yes Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Control Variables No Yes $$R^2$$ 0.12 0.21 N 77,079 57,243 The table reports OLS estimates of the threshold specification in Model 3. The dependent variable is the (log) value of revalued assets of firm $$i$$ in year $$t$$. The indicator Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 from the first quarter of 2004 to the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 from the first quarter of 2008 to the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 from the first quarter of 2010 to the last quarter of 2011, and 0 otherwise. Functions $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ correspond to flexible sixth-order polynomials estimated separately for each quarter-year. Function $$f_{t}(\\cdot)$$ is estimated from $$0$$ to the left, whereas the Performing$$_{i,t}\\times g_{t}(\\cdot)$$ term is estimated from $$0$$ to the right. Standard errors, clustered at the firm level, are reported in brackets. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. 7.2 Balancing tests In Table 9, we analyze whether firms close to the threshold are as if randomly sampled, a critical identification assumption within regression discontinuity models. If firms are nonrandomly sorted into specific rating classes, we would expect firm characteristics to differ systematically across the threshold. Following the regression discontinuity literature, the firm characteristics we test are those logically unaffected by the threshold but plausibly related to firm financing. Table 9 Model diagnostics: Balancing checks Year 2004 2005 2006 2007 2008 2009 2010 2011 Panel A: Presample characteristics Leverage 0.00 0.01 \u20130.04 \u20130.03 0.05 \u20130.01 \u20130.04 0.01 (.03) (.04) (.03) (.03) (.04) (.04) (.05) (.06) N 3,967 3,636 3,595 3,678 2,888 2,705 2,168 2,024 Return to Assets 0.00 0.00 0.00 \u20130.01 \u20130.02 0.00 0.00 0.00 (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.02) (0.02) N 5,306 4,844 4,750 4,836 3,776 3,504 2,721 2,508 Investment to Assets 0.02 0.02 0.01 0.02 0.02 \u20130.02 \u20130.03 \u20130.02 (0.01) (0.02) (0.01) (0.02) (0.02) (0.02) (0.03) (0.02) N 4,501 4,136 4,083 4,174 3,353 3,100 2,414 2,237 Panel B: Bank balancing characteristics Credit Event 0.01 0.00 0.01 0.00 \u20130.01 0.00 \u20130.03 (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.03) N 5,736 5,944 6,358 5,411 5,276 4,235 4,045 Asked 0.02 0.00 \u20130.02 \u20130.07 \u20130.03 0.04 0.03 \u20130.07 (0.04) (0.05) (0.04) (0.05) (0.04) (0.04) (0.05) (0.05) N 5,687 5,677 5,889 6,306 5,370 5,264 4,217 4,030 Bank Size \u20130.12 \u20130.05 \u20130.02 0.23** 0.1 0.09 0.04 0.23 (0.14) (0.14) (0.11) (0.12) (0.14) (0.17) (0.19) (0.18) N 5,652 5,641 5,855 6,287 5,356 5,108 4,105 3,937 Panel C: Time-invariant characteristics Activity: Food Industry 0.03 \u20130.04 0.03 \u20130.01 0.05 0.04 0.06 \u20130.06 (0.04) (0.05) (0.04) (0.04) (0.04) (0.04) (0.06) (0.06) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 Location: Top 5 Cities 0.06 0.03 0.05 \u20130.06 0.02 \u20130.01 0.07 0.05 (0.06) (0.06) (0.06) (0.06) (0.06) (0.06) (0.08) (0.07) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 Year 2004 2005 2006 2007 2008 2009 2010 2011 Panel A: Presample characteristics Leverage 0.00 0.01 \u20130.04 \u20130.03 0.05 \u20130.01 \u20130.04 0.01 (.03) (.04) (.03) (.03) (.04) (.04) (.05) (.06) N 3,967 3,636 3,595 3,678 2,888 2,705 2,168 2,024 Return to Assets 0.00 0.00 0.00 \u20130.01 \u20130.02 0.00 0.00 0.00 (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.02) (0.02) N 5,306 4,844 4,750 4,836 3,776 3,504 2,721 2,508 Investment to Assets 0.02 0.02 0.01 0.02 0.02 \u20130.02 \u20130.03 \u20130.02 (0.01) (0.02) (0.01) (0.02) (0.02) (0.02) (0.03) (0.02) N 4,501 4,136 4,083 4,174 3,353 3,100 2,414 2,237 Panel B: Bank balancing characteristics Credit Event 0.01 0.00 0.01 0.00 \u20130.01 0.00 \u20130.03 (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.03) N 5,736 5,944 6,358 5,411 5,276 4,235 4,045 Asked 0.02 0.00 \u20130.02 \u20130.07 \u20130.03 0.04 0.03 \u20130.07 (0.04) (0.05) (0.04) (0.05) (0.04) (0.04) (0.05) (0.05) N 5,687 5,677 5,889 6,306 5,370 5,264 4,217 4,030 Bank Size \u20130.12 \u20130.05 \u20130.02 0.23** 0.1 0.09 0.04 0.23 (0.14) (0.14) (0.11) (0.12) (0.14) (0.17) (0.19) (0.18) N 5,652 5,641 5,855 6,287 5,356 5,108 4,105 3,937 Panel C: Time-invariant characteristics Activity: Food Industry 0.03 \u20130.04 0.03 \u20130.01 0.05 0.04 0.06 \u20130.06 (0.04) (0.05) (0.04) (0.04) (0.04) (0.04) (0.06) (0.06) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 Location: Top 5 Cities 0.06 0.03 0.05 \u20130.06 0.02 \u20130.01 0.07 0.05 (0.06) (0.06) (0.06) (0.06) (0.06) (0.06) (0.08) (0.07) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 The table estimates differences in presample firm characteristics at the threshold. In all rows, the dependent variable is measured in 2003. The estimates refer to the indicator variable Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. Credit Event is a binary variable equal to 1 if any of a given firm\u2019s banks classified the firm\u2019s credit as nonperforming. Asked is a binary variable equal to 1 if any non-current bank requested information on the firm during the year. Bank Size corresponds to the value of a bank\u2019s total assets. Food Industry is a binary variable indicating whether the firms\u2019 SIC code belongs to the food industry. Top 5 Cities is a binary variable indicating whether the firms\u2019 headquarters zip code is in one of the largest five cities. See Tables 2 for the definition of the other variables. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Table 9 Model diagnostics: Balancing checks Year 2004 2005 2006 2007 2008 2009 2010 2011 Panel A: Presample characteristics Leverage 0.00 0.01 \u20130.04 \u20130.03 0.05 \u20130.01 \u20130.04 0.01 (.03) (.04) (.03) (.03) (.04) (.04) (.05) (.06) N 3,967 3,636 3,595 3,678 2,888 2,705 2,168 2,024 Return to Assets 0.00 0.00 0.00 \u20130.01 \u20130.02 0.00 0.00 0.00 (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.02) (0.02) N 5,306 4,844 4,750 4,836 3,776 3,504 2,721 2,508 Investment to Assets 0.02 0.02 0.01 0.02 0.02 \u20130.02 \u20130.03 \u20130.02 (0.01) (0.02) (0.01) (0.02) (0.02) (0.02) (0.03) (0.02) N 4,501 4,136 4,083 4,174 3,353 3,100 2,414 2,237 Panel B: Bank balancing characteristics Credit Event 0.01 0.00 0.01 0.00 \u20130.01 0.00 \u20130.03 (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.03) N 5,736 5,944 6,358 5,411 5,276 4,235 4,045 Asked 0.02 0.00 \u20130.02 \u20130.07 \u20130.03 0.04 0.03 \u20130.07 (0.04) (0.05) (0.04) (0.05) (0.04) (0.04) (0.05) (0.05) N 5,687 5,677 5,889 6,306 5,370 5,264 4,217 4,030 Bank Size \u20130.12 \u20130.05 \u20130.02 0.23** 0.1 0.09 0.04 0.23 (0.14) (0.14) (0.11) (0.12) (0.14) (0.17) (0.19) (0.18) N 5,652 5,641 5,855 6,287 5,356 5,108 4,105 3,937 Panel C: Time-invariant characteristics Activity: Food Industry 0.03 \u20130.04 0.03 \u20130.01 0.05 0.04 0.06 \u20130.06 (0.04) (0.05) (0.04) (0.04) (0.04) (0.04) (0.06) (0.06) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 Location: Top 5 Cities 0.06 0.03 0.05 \u20130.06 0.02 \u20130.01 0.07 0.05 (0.06) (0.06) (0.06) (0.06) (0.06) (0.06) (0.08) (0.07) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 Year 2004 2005 2006 2007 2008 2009 2010 2011 Panel A: Presample characteristics Leverage 0.00 0.01 \u20130.04 \u20130.03 0.05 \u20130.01 \u20130.04 0.01 (.03) (.04) (.03) (.03) (.04) (.04) (.05) (.06) N 3,967 3,636 3,595 3,678 2,888 2,705 2,168 2,024 Return to Assets 0.00 0.00 0.00 \u20130.01 \u20130.02 0.00 0.00 0.00 (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.02) (0.02) N 5,306 4,844 4,750 4,836 3,776 3,504 2,721 2,508 Investment to Assets 0.02 0.02 0.01 0.02 0.02 \u20130.02 \u20130.03 \u20130.02 (0.01) (0.02) (0.01) (0.02) (0.02) (0.02) (0.03) (0.02) N 4,501 4,136 4,083 4,174 3,353 3,100 2,414 2,237 Panel B: Bank balancing characteristics Credit Event 0.01 0.00 0.01 0.00 \u20130.01 0.00 \u20130.03 (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.03) N 5,736 5,944 6,358 5,411 5,276 4,235 4,045 Asked 0.02 0.00 \u20130.02 \u20130.07 \u20130.03 0.04 0.03 \u20130.07 (0.04) (0.05) (0.04) (0.05) (0.04) (0.04) (0.05) (0.05) N 5,687 5,677 5,889 6,306 5,370 5,264 4,217 4,030 Bank Size \u20130.12 \u20130.05 \u20130.02 0.23** 0.1 0.09 0.04 0.23 (0.14) (0.14) (0.11) (0.12) (0.14) (0.17) (0.19) (0.18) N 5,652 5,641 5,855 6,287 5,356 5,108 4,105 3,937 Panel C: Time-invariant characteristics Activity: Food Industry 0.03 \u20130.04 0.03 \u20130.01 0.05 0.04 0.06 \u20130.06 (0.04) (0.05) (0.04) (0.04) (0.04) (0.04) (0.06) (0.06) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 Location: Top 5 Cities 0.06 0.03 0.05 \u20130.06 0.02 \u20130.01 0.07 0.05 (0.06) (0.06) (0.06) (0.06) (0.06) (0.06) (0.08) (0.07) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 The table estimates differences in presample firm characteristics at the threshold. In all rows, the dependent variable is measured in 2003. The estimates refer to the indicator variable Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. Credit Event is a binary variable equal to 1 if any of a given firm\u2019s banks classified the firm\u2019s credit as nonperforming. Asked is a binary variable equal to 1 if any non-current bank requested information on the firm during the year. Bank Size corresponds to the value of a bank\u2019s total assets. Food Industry is a binary variable indicating whether the firms\u2019 SIC code belongs to the food industry. Top 5 Cities is a binary variable indicating whether the firms\u2019 headquarters zip code is in one of the largest five cities. See Tables 2 for the definition of the other variables. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. In panel A of Table 9, the dependent variables are a broad set of firm financing, investment, and profitability measures taken in 2003. In the first row, we show that firms at the threshold do not differ in terms of leverage choices in the pre-sample period. Moreover, we find no significant difference in firms\u2019 return on assets or investments. Panel B tests for differences in bank-firm relationships at the threshold. The first row in the table focuses on the banks\u2019 probability of reporting a delinquent loan. If there were a discontinuity in the probability of a firm\u2019s credit event at the threshold, then our results could be explained by the fact that banks correctly price this difference. However, we find no statistically or economically significant differences at the threshold. In the second row, the variable $$Asked$$ is a binary indicator equal to 1 if a bank requests information on a new loan applicant. The estimates suggest that firms at the threshold do not display a different propensity to apply for loans to new banks. The last row of the panel tests for the presence of assortative matching between banks and firms at the threshold (Paravisini et al. 2014). For each firm, we compute its bank\u2019s average size.20 Again, we find no evidence of a systematic difference at the threshold. Panel C focuses on differences in time-invariant firm characteristics. In the first row, the dependent variable is the firm\u2019s activity sector proxied by its SIC code. The yearly estimates indicate no statistically or economically significant evidence of firms clustering into sectors such as food industries. Next, we look at time-invariant characteristics related to firms\u2019 geographic locations. This is a particularly interesting dimension to study within this setting because Italian geography is correlated with heterogeneity in economic development, crime rates, and political accountability (Brollo et al. 2013) and could thus be associated with opportunistic manipulation. The variable capturing location in the largest cities or the most entrepreneurial areas does not display a statistically significant discontinuity.21 7.3 Empirical relevance of the threshold We now provide further evidence on the relevance of the threshold between performing and substandard firms. First, we confirm the local interpretation of our estimates by providing nonparametric plots of the outcome variable as a function of the continuous assignment variable. Second, we implement placebo tests in which we randomly re-label the value of the threshold. Finally, we investigate whether banks use alternative ratings\u2019 cutoffs to formulate lending standards. 7.3.1 Nonparametric plots In the left panel of Figure 5, we focus on data from the second quarter of 2009, when our results at the threshold feature quantity differences and no interest rate differences. We divide the domain of $$s$$ into mutually exclusive bins of size $$0.03$$.22 For each bin, we compute the average and the 90% confidence interval of the outcome variable, and plot these values at the bin\u2019s midpoint. The fitted gray line shows how close the sixth-order polynomial approximates the variation in bank financing conditions at the threshold. Figure 5. View largeDownload slide Second quarter of 2009 The figure focuses on the second quarter of 2009. We divide the domain of $$s_i$$ into mutually exclusive bins with a size of $$0.03$$. For each bin, we compute the average and the 90% confidence interval of the outcome variable, and plot these values at the bin\u2019s midpoint. The fitted gray line shows how closely the sixth-order polynomial approximates the variation in bank financing conditions at the threshold. Figure 5. View largeDownload slide Second quarter of 2009 The figure focuses on the second quarter of 2009. We divide the domain of $$s_i$$ into mutually exclusive bins with a size of $$0.03$$. For each bin, we compute the average and the 90% confidence interval of the outcome variable, and plot these values at the bin\u2019s midpoint. The fitted gray line shows how closely the sixth-order polynomial approximates the variation in bank financing conditions at the threshold. The top panel of Figure 5 shows that a clear discontinuity arises in the total amount of bank financing close to the threshold. The magnitude of this discontinuity can be quantified by comparing the mean value of the variable of interest in the two bins next to the threshold. Immediately to the left of the threshold, the average value of (log) granted credit is approximately 14.6, whereas immediately to the right this value is 15, implying that the estimated value of $$\\beta$$ captures the variation arising directly at the threshold. The bottom panel of Figure 5 repeats this exercise for the interest rates on new bank loans. It shows that when there is no discontinuity in the value of the conditional regression function at the threshold, the polynomial fit does not display any significant discontinuity. Figure 6 repeats this analysis by focusing on the second quarter of 2011, when our results at the threshold feature significant interest rate differences and no quantity differences.23 Figure 6. View largeDownload slide Second quarter of 2011 The figure focuses on the second quarter of 2011. We divide the domain of $$s_i$$ into mutually exclusive bins with a size of $$0.03$$. For each bin, we compute the average and the standard deviation of the outcome variable, and plot these values at the bin\u2019s midpoint. The fitted gray line shows how closely the sixth-order polynomial approximates the variation in bank financing conditions at the threshold. Figure 6. View largeDownload slide Second quarter of 2011 The figure focuses on the second quarter of 2011. We divide the domain of $$s_i$$ into mutually exclusive bins with a size of $$0.03$$. For each bin, we compute the average and the standard deviation of the outcome variable, and plot these values at the bin\u2019s midpoint. The fitted gray line shows how closely the sixth-order polynomial approximates the variation in bank financing conditions at the threshold. 7.3.2 Placebo tests Finding a significant discontinuity in lending conditions at the threshold, as shown in Figure 4, might not necessarily establish a causal relationship between the threshold and the design of financial contracts. For example, analogous results might arise when comparing financing conditions borne by firms whose Score lies further away from the true threshold. We thus implement the following falsification tests: we draw approximately 100 randomly distributed placebo thresholds along the support of Score categories 6 and 7, and rerun our specification on the cross-section of firms at the threshold in all the quarters in our sample. We plot in Figure 7 the distribution of the placebo estimates for the second quarters of 2009 and 2011. Figure 7. View largeDownload slide Placebo estimates: Second quarters of 2009 and 2011 The figure plots the empirical distribution of discontinuity estimates based on approximately 100 randomly drawn placebo thresholds. The vertical dotted line represents the estimate obtained from the true threshold. The top panel figures focus on the second quarter of 2009, while the bottom panel focuses on the second quarter of 2011. Figure 7. View largeDownload slide Placebo estimates: Second quarters of 2009 and 2011 The figure plots the empirical distribution of discontinuity estimates based on approximately 100 randomly drawn placebo thresholds. The vertical dotted line represents the estimate obtained from the true threshold. The top panel figures focus on the second quarter of 2009, while the bottom panel focuses on the second quarter of 2011. Figure 7 illustrates that the contractual differences identified by the true threshold estimates (vertical dotted line) are not due to a coincidental discontinuity. If this were the case, then we should observe similar estimates arising when considering randomly placed thresholds. In the top-left panel, we find that the 100 placebo estimates for the differences in the quantity of bank financing are approximately normally distributed around 0. Similarly, the bottom-right panel shows that in the second quarter of 2011 the interest rate differences of 20% that we find in the main analysis are well outside the normal variation arising from randomly placed thresholds.24 This evidence demonstrates the relevance of the categorical value of the Score for Italian banks\u2019 lending decisions. If banks were not using the categorical rating when making their credit choices, then the threshold should not yield financial outcomes that are significantly and systematically different from those obtained using a randomly set threshold along the support of the continuous variable. Our evidence rejects this claim on the basis of the distribution of placebo estimates within and across the sample period. 7.3.3 Other rating thresholds Finally, as in Agarwal et al. (Forthcoming), we investigate whether banks use alternative ratings\u2019 cutoffs to formulate lending standards. We estimate our specification on the cross-section of firms at all the other six thresholds associated with the categorical value of the rating system.25 In Table 10, the reported dummy variable is equal to 1 for firms in the better\u2014that is, lower-value\u2013rating category, and 0 otherwise. Table 10 Yearly discontinuity estimates: Other thresholds Year 2004 2005 2006 2007 2008 2009 2010 2011 Threshold between categories 1 and 2 Quantity \u20130.3 \u20130.15 0.07 0.17 \u20130.28 \u20130.19 \u20130.3 \u20130.32 (0.24) (0.26) (0.26) (0.31) (0.27) (0.25) (0.23) (0.21) N 2,555 2,693 2,648 2,684 2,886 2,975 2,677 2,773 Price 0.04 0.13 0.08 0.03 \u20130.12 \u20130.23 \u20130.04 \u20130.22 ( 0.11) (0.12) (0.11) (0.08) (0.08) (0.2) (0.18) (0.22) N 583 716 782 815 715 712 832 775 Threshold between categories 2 and 3 Quantity \u20130.12 \u20130.19 \u20130.45 \u20130.3 \u20130.25 \u20130.2 \u20130.45 \u20130.51 ( 0.39) (0.4) (0.39) (0.35) (0.41) (0.34) (0.36) (0.35) N 2,311 2,508 2,480 2,383 2,265 2,243 2,243 2,375 Price 0.00 0.16 \u20130.1 0.01 \u20130.02 \u20130.1 \u20130.23 0.7*** ( 0.13) (0.12) (0.11) (0.08) (0.14) (0.27) (0.22) (0.22) N 1,099 1,427 1,595 1,702 1,475 1,260 1,406 1,825 Threshold between categories 3 and 4 Quantity \u20130.24 \u20130.03 \u20130.14 0.29 0.11 \u20130.29 \u20130.15 0.29 ( 0.31) (0.3) (0.35) (0.29) (0.33) (0.32) (0.29) (0.3) N 6,087 6,361 6,371 6,526 6,040 5,968 5,840 6,128 Price \u20130.03 0.03 0.09 \u20130.03 \u20130.08 \u20130.01 \u20130.12 \u20130.03 ( 0.08) (0.09) (0.08) (0.04) (0.06) (0.13) (0.15) (0.12) N 7,197 9,359 10,255 10,547 9,033 8,625 11,153 13,158 Threshold between categories 4 and 5 Quantity \u20130.33 0.22 \u20130.44* \u20130.18 \u20130.2 \u20130.06 \u20130.26 \u20130.41* ( 0.24) (0.24) (0.24) (0.21) (0.24) (0.24) (0.24) (0.23) N 7,019 7,359 7,437 7,616 6,960 6,878 6,711 7,058 Price 0.00 \u20130.05 0.03 \u20130.01 0.00 \u20130.02 \u20130.23*** 0.07 ( 0.05) (0.06) (0.04) (0.03) (0.03) (0.1) (0.08) (0.07) N 11,072 14,972 16,561 17,056 14,662 13,505 17,687 19,743 Threshold between categories 7 and 8 Quantity \u20130.25 \u20130.28 \u20130.29 \u20130.06 \u20130.36 \u20130.63 1.44* 1.01 ( 0.48) (0.51) (0.55) (0.55) (0.63) (0.66) (0.73) (0.88) N 4,160 4,136 4,256 4,602 3,752 3,472 2,875 2,688 Price .00 \u20130.2 0.1 \u20130.22** \u20130.08 0.35* \u20130.56 \u20130.12 ( 0.19) (0.17) (0.11) (0.09) (0.1) (0.2) (0.56) (0.27) N 6,058 8,394 10,412 13,192 8,280 6,047 5,883 5,791 Threshold between categories 8 and 9 Quantity \u20130.9 0.18 0.51 \u20131.31 \u20131.26 \u20130.42 \u20130.97 \u20131.68 ( 1.4) (1.16) (1.12) (1.36) (1.09) (1.24) (0.95) (1.2) N 596 649 598 646 595 668 517 616 Price \u20131.29 \u20130.01 0.21 0.09 \u20130.02 0.07 0.4 \u20130.31 ( 54.98) (0.53) (0.26) (0.27) (0.13) (0.5) (0.47) (0.4) N 380 494 655 761 518 701 471 489 Year 2004 2005 2006 2007 2008 2009 2010 2011 Threshold between categories 1 and 2 Quantity \u20130.3 \u20130.15 0.07 0.17 \u20130.28 \u20130.19 \u20130.3 \u20130.32 (0.24) (0.26) (0.26) (0.31) (0.27) (0.25) (0.23) (0.21) N 2,555 2,693 2,648 2,684 2,886 2,975 2,677 2,773 Price 0.04 0.13 0.08 0.03 \u20130.12 \u20130.23 \u20130.04 \u20130.22 ( 0.11) (0.12) (0.11) (0.08) (0.08) (0.2) (0.18) (0.22) N 583 716 782 815 715 712 832 775 Threshold between categories 2 and 3 Quantity \u20130.12 \u20130.19 \u20130.45 \u20130.3 \u20130.25 \u20130.2 \u20130.45 \u20130.51 ( 0.39) (0.4) (0.39) (0.35) (0.41) (0.34) (0.36) (0.35) N 2,311 2,508 2,480 2,383 2,265 2,243 2,243 2,375 Price 0.00 0.16 \u20130.1 0.01 \u20130.02 \u20130.1 \u20130.23 0.7*** ( 0.13) (0.12) (0.11) (0.08) (0.14) (0.27) (0.22) (0.22) N 1,099 1,427 1,595 1,702 1,475 1,260 1,406 1,825 Threshold between categories 3 and 4 Quantity \u20130.24 \u20130.03 \u20130.14 0.29 0.11 \u20130.29 \u20130.15 0.29 ( 0.31) (0.3) (0.35) (0.29) (0.33) (0.32) (0.29) (0.3) N 6,087 6,361 6,371 6,526 6,040 5,968 5,840 6,128 Price \u20130.03 0.03 0.09 \u20130.03 \u20130.08 \u20130.01 \u20130.12 \u20130.03 ( 0.08) (0.09) (0.08) (0.04) (0.06) (0.13) (0.15) (0.12) N 7,197 9,359 10,255 10,547 9,033 8,625 11,153 13,158 Threshold between categories 4 and 5 Quantity \u20130.33 0.22 \u20130.44* \u20130.18 \u20130.2 \u20130.06 \u20130.26 \u20130.41* ( 0.24) (0.24) (0.24) (0.21) (0.24) (0.24) (0.24) (0.23) N 7,019 7,359 7,437 7,616 6,960 6,878 6,711 7,058 Price 0.00 \u20130.05 0.03 \u20130.01 0.00 \u20130.02 \u20130.23*** 0.07 ( 0.05) (0.06) (0.04) (0.03) (0.03) (0.1) (0.08) (0.07) N 11,072 14,972 16,561 17,056 14,662 13,505 17,687 19,743 Threshold between categories 7 and 8 Quantity \u20130.25 \u20130.28 \u20130.29 \u20130.06 \u20130.36 \u20130.63 1.44* 1.01 ( 0.48) (0.51) (0.55) (0.55) (0.63) (0.66) (0.73) (0.88) N 4,160 4,136 4,256 4,602 3,752 3,472 2,875 2,688 Price .00 \u20130.2 0.1 \u20130.22** \u20130.08 0.35* \u20130.56 \u20130.12 ( 0.19) (0.17) (0.11) (0.09) (0.1) (0.2) (0.56) (0.27) N 6,058 8,394 10,412 13,192 8,280 6,047 5,883 5,791 Threshold between categories 8 and 9 Quantity \u20130.9 0.18 0.51 \u20131.31 \u20131.26 \u20130.42 \u20130.97 \u20131.68 ( 1.4) (1.16) (1.12) (1.36) (1.09) (1.24) (0.95) (1.2) N 596 649 598 646 595 668 517 616 Price \u20131.29 \u20130.01 0.21 0.09 \u20130.02 0.07 0.4 \u20130.31 ( 54.98) (0.53) (0.26) (0.27) (0.13) (0.5) (0.47) (0.4) N 380 494 655 761 518 701 471 489 The table reports estimates from our baseline specification at all the seven thresholds associated with the categorical value of the rating system. We report standard errors in brackets. The dependent variable is either All Bank Financing Granted or Interest Rate for each year between 2004.Q1\u20132011.Q4. We estimate the discontinuity $$\\left( s_{i}\\geq 0 \\right)$$ using a flexible sixth-order polynomial on either side of each normalized threshold between each contiguous Score category, allowing for a discontinuity at 0. The reported estimates refer to $$S_{i}$$, a binary variable that takes a value of 1 if the continuous variable $$s_i \\geq 0$$, that is, if the firm is allocated to the lower credit risk category as opposed to the higher credit risk category. See Table 2 for other variable definitions. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Table 10 Yearly discontinuity estimates: Other thresholds Year 2004 2005 2006 2007 2008 2009 2010 2011 Threshold between categories 1 and 2 Quantity \u20130.3 \u20130.15 0.07 0.17 \u20130.28 \u20130.19 \u20130.3 \u20130.32 (0.24) (0.26) (0.26) (0.31) (0.27) (0.25) (0.23) (0.21) N 2,555 2,693 2,648 2,684 2,886 2,975 2,677 2,773 Price 0.04 0.13 0.08 0.03 \u20130.12 \u20130.23 \u20130.04 \u20130.22 ( 0.11) (0.12) (0.11) (0.08) (0.08) (0.2) (0.18) (0.22) N 583 716 782 815 715 712 832 775 Threshold between categories 2 and 3 Quantity \u20130.12 \u20130.19 \u20130.45 \u20130.3 \u20130.25 \u20130.2 \u20130.45 \u20130.51 ( 0.39) (0.4) (0.39) (0.35) (0.41) (0.34) (0.36) (0.35) N 2,311 2,508 2,480 2,383 2,265 2,243 2,243 2,375 Price 0.00 0.16 \u20130.1 0.01 \u20130.02 \u20130.1 \u20130.23 0.7*** ( 0.13) (0.12) (0.11) (0.08) (0.14) (0.27) (0.22) (0.22) N 1,099 1,427 1,595 1,702 1,475 1,260 1,406 1,825 Threshold between categories 3 and 4 Quantity \u20130.24 \u20130.03 \u20130.14 0.29 0.11 \u20130.29 \u20130.15 0.29 ( 0.31) (0.3) (0.35) (0.29) (0.33) (0.32) (0.29) (0.3) N 6,087 6,361 6,371 6,526 6,040 5,968 5,840 6,128 Price \u20130.03 0.03 0.09 \u20130.03 \u20130.08 \u20130.01 \u20130.12 \u20130.03 ( 0.08) (0.09) (0.08) (0.04) (0.06) (0.13) (0.15) (0.12) N 7,197 9,359 10,255 10,547 9,033 8,625 11,153 13,158 Threshold between categories 4 and 5 Quantity \u20130.33 0.22 \u20130.44* \u20130.18 \u20130.2 \u20130.06 \u20130.26 \u20130.41* ( 0.24) (0.24) (0.24) (0.21) (0.24) (0.24) (0.24) (0.23) N 7,019 7,359 7,437 7,616 6,960 6,878 6,711 7,058 Price 0.00 \u20130.05 0.03 \u20130.01 0.00 \u20130.02 \u20130.23*** 0.07 ( 0.05) (0.06) (0.04) (0.03) (0.03) (0.1) (0.08) (0.07) N 11,072 14,972 16,561 17,056 14,662 13,505 17,687 19,743 Threshold between categories 7 and 8 Quantity \u20130.25 \u20130.28 \u20130.29 \u20130.06 \u20130.36 \u20130.63 1.44* 1.01 ( 0.48) (0.51) (0.55) (0.55) (0.63) (0.66) (0.73) (0.88) N 4,160 4,136 4,256 4,602 3,752 3,472 2,875 2,688 Price .00 \u20130.2 0.1 \u20130.22** \u20130.08 0.35* \u20130.56 \u20130.12 ( 0.19) (0.17) (0.11) (0.09) (0.1) (0.2) (0.56) (0.27) N 6,058 8,394 10,412 13,192 8,280 6,047 5,883 5,791 Threshold between categories 8 and 9 Quantity \u20130.9 0.18 0.51 \u20131.31 \u20131.26 \u20130.42 \u20130.97 \u20131.68 ( 1.4) (1.16) (1.12) (1.36) (1.09) (1.24) (0.95) (1.2) N 596 649 598 646 595 668 517 616 Price \u20131.29 \u20130.01 0.21 0.09 \u20130.02 0.07 0.4 \u20130.31 ( 54.98) (0.53) (0.26) (0.27) (0.13) (0.5) (0.47) (0.4) N 380 494 655 761 518 701 471 489 Year 2004 2005 2006 2007 2008 2009 2010 2011 Threshold between categories 1 and 2 Quantity \u20130.3 \u20130.15 0.07 0.17 \u20130.28 \u20130.19 \u20130.3 \u20130.32 (0.24) (0.26) (0.26) (0.31) (0.27) (0.25) (0.23) (0.21) N 2,555 2,693 2,648 2,684 2,886 2,975 2,677 2,773 Price 0.04 0.13 0.08 0.03 \u20130.12 \u20130.23 \u20130.04 \u20130.22 ( 0.11) (0.12) (0.11) (0.08) (0.08) (0.2) (0.18) (0.22) N 583 716 782 815 715 712 832 775 Threshold between categories 2 and 3 Quantity \u20130.12 \u20130.19 \u20130.45 \u20130.3 \u20130.25 \u20130.2 \u20130.45 \u20130.51 ( 0.39) (0.4) (0.39) (0.35) (0.41) (0.34) (0.36) (0.35) N 2,311 2,508 2,480 2,383 2,265 2,243 2,243 2,375 Price 0.00 0.16 \u20130.1 0.01 \u20130.02 \u20130.1 \u20130.23 0.7*** ( 0.13) (0.12) (0.11) (0.08) (0.14) (0.27) (0.22) (0.22) N 1,099 1,427 1,595 1,702 1,475 1,260 1,406 1,825 Threshold between categories 3 and 4 Quantity \u20130.24 \u20130.03 \u20130.14 0.29 0.11 \u20130.29 \u20130.15 0.29 ( 0.31) (0.3) (0.35) (0.29) (0.33) (0.32) (0.29) (0.3) N 6,087 6,361 6,371 6,526 6,040 5,968 5,840 6,128 Price \u20130.03 0.03 0.09 \u20130.03 \u20130.08 \u20130.01 \u20130.12 \u20130.03 ( 0.08) (0.09) (0.08) (0.04) (0.06) (0.13) (0.15) (0.12) N 7,197 9,359 10,255 10,547 9,033 8,625 11,153 13,158 Threshold between categories 4 and 5 Quantity \u20130.33 0.22 \u20130.44* \u20130.18 \u20130.2 \u20130.06 \u20130.26 \u20130.41* ( 0.24) (0.24) (0.24) (0.21) (0.24) (0.24) (0.24) (0.23) N 7,019 7,359 7,437 7,616 6,960 6,878 6,711 7,058 Price 0.00 \u20130.05 0.03 \u20130.01 0.00 \u20130.02 \u20130.23*** 0.07 ( 0.05) (0.06) (0.04) (0.03) (0.03) (0.1) (0.08) (0.07) N 11,072 14,972 16,561 17,056 14,662 13,505 17,687 19,743 Threshold between categories 7 and 8 Quantity \u20130.25 \u20130.28 \u20130.29 \u20130.06 \u20130.36 \u20130.63 1.44* 1.01 ( 0.48) (0.51) (0.55) (0.55) (0.63) (0.66) (0.73) (0.88) N 4,160 4,136 4,256 4,602 3,752 3,472 2,875 2,688 Price .00 \u20130.2 0.1 \u20130.22** \u20130.08 0.35* \u20130.56 \u20130.12 ( 0.19) (0.17) (0.11) (0.09) (0.1) (0.2) (0.56) (0.27) N 6,058 8,394 10,412 13,192 8,280 6,047 5,883 5,791 Threshold between categories 8 and 9 Quantity \u20130.9 0.18 0.51 \u20131.31 \u20131.26 \u20130.42 \u20130.97 \u20131.68 ( 1.4) (1.16) (1.12) (1.36) (1.09) (1.24) (0.95) (1.2) N 596 649 598 646 595 668 517 616 Price \u20131.29 \u20130.01 0.21 0.09 \u20130.02 0.07 0.4 \u20130.31 ( 54.98) (0.53) (0.26) (0.27) (0.13) (0.5) (0.47) (0.4) N 380 494 655 761 518 701 471 489 The table reports estimates from our baseline specification at all the seven thresholds associated with the categorical value of the rating system. We report standard errors in brackets. The dependent variable is either All Bank Financing Granted or Interest Rate for each year between 2004.Q1\u20132011.Q4. We estimate the discontinuity $$\\left( s_{i}\\geq 0 \\right)$$ using a flexible sixth-order polynomial on either side of each normalized threshold between each contiguous Score category, allowing for a discontinuity at 0. The reported estimates refer to $$S_{i}$$, a binary variable that takes a value of 1 if the continuous variable $$s_i \\geq 0$$, that is, if the firm is allocated to the lower credit risk category as opposed to the higher credit risk category. See Table 2 for other variable definitions. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Table 10 shows that most of our estimates on the other thresholds of the Score are not statistically significant. This confirms that our results capture a form of market segmentation, not a simple rating effect, as the only rating values that matter are those moving firms between the performing and substandard classes of credit. 8. Conclusions In this paper, we ask whether the effects of firm segmentation into performing and substandard rating classes can affect the lending policies of banks across the credit cycle. We take advantage of the institutional features of the Italian credit market for SME in order to obtain a quasi-random assignment of firms into these classes of credit risk. The resulting patterns of lending differences give us a new, contract-level measure for the bank lending standards. In this setting, bank lending standards are driven by market segmentation and reflect banks\u2019 sensitivity to the markets for banks\u2019 capital. While our analysis focuses on the single credit cycle that interested the Italian economy between 2004 and 2011, there are two considerations that support both the external validity and the interest of our results. First, the aggregate financing patterns of the Italian economy during this period were similar to those of other OECD economies. Second, the credit cycle in our data culminates with the great recession. This renders the analysis particularly interesting, as it allows us to provide implications for the qualitative and quantitative features of lending standards before and during those years, and the consequences for real allocations. Finally, we discuss the implications of our analysis for the allocative efficiency of banks\u2019 credit policies. By construction, firms in our empirical design are ex ante identical and should, absent the threshold, receive the same credit conditions. This means that, whenever we observe differences in the credit terms at the threshold, there is an inefficiency caused by segmentation in the relative allocation of credit. We thank the editor (Robin Greenwood) and two anonymous referees for insightful comments. The paper also benefited from comments by Klaus Adam, Allen Berger, Steve Bond, Elena Carletti, Antonio Ciccone, Decio Coviello, Matteo Crosignani, Andrew Ellul, Carlo Favero, Nicola Gennaioli, Simon Gilchrist, Martin Hellwig, Victoria Ivashina, Rajkamal Iyer, Nobuhiro Kiyotaki, Augustin Landier, Rocco Macchiavello, Tommaso Nannicini, Steven Ongena, Marco Pagano, Nicola Pavanini, Nicola Persico, Jos\u00e9-Luis Peydr\u00f3, Andrea Polo, Andrea Pozzi, Manju Puri, Antoinette Schoar, Amit Seru, Enrico Sette, Andrei Shleifer, Jeremy Stein, Javier Su\u00e1rez, Adi Sunderam, Michele Tertilt, David Thesmar, Franco Varetto, Egon Zakraj\u0161ek, and participants in the Banque de France (ACPR), Bank of Italy, Bank of Spain, Bocconi, CSEF, Danmarks Nationalbank, EIEF, Goethe University (Frankfurt), HEC Montreal, IFN (Stockholm), Italian Treasury Department, University of Mannheim, Max Planck Institute (Bonn), Tilburg University, Universit\u00e0 Tor Vergata (Rome) seminars and in the NBER Summer Institute (Capital Markets and the Economy), Swiss Conference on Financial Intermediation, Annual Bank Research Conference FDIC\/JFSR, European Winter Finance Summit, ESSFM, Csef-Igier Symposium on Economics and Institutions, First Young Scholars Finance Consortium (Texas A&M), Petralia Workshop and 4Nations Cup conferences for helpful comments. The views expressed are those of the authors and do not necessarily reflect those of the Bank of Italy. Emanuele Tarantino thanks the EIEF for its hospitality. Supplementary data can be found on The Review of Financial Studies web site. Footnotes 1 A possibility would be to look at the U.S. syndicated loan market, which allows us to use a long time series of data within a well-known environment. However, borrowers in this market tend to be significantly larger than a typical SME (Sufi 2007; Ivashina 2009). 2 Specifically, the definition of NPL includes bad loans, past due, and loans to insolvent firms other than substandard credits. The latter are defined as exposures to counterparties facing temporary difficulties defined on the basis of objective factors. 3 This literature finds that the flow of credit (e.g., Covas and Den Haan 2011; Jermann and Quadrini 2012; Becker and Ivashina 2014) and the value of credit spreads (Gilchrist, Yankov, and Zakraj\u0161ek 2009) are both highly procyclical. 4 Our results also inform the (growing) theoretical literature on lending standards over the cycle (e.g., Dell\u2019Ariccia and Marquez 2006; Martin 2008; Kovbasyuk and Spagnolo 2017; Gete 2017). 5 While the formula in the original Altman\u2019s model is publicly known, the agency uses its own version. Specifically, to our knowledge, CEBI\u2019s version of the model uses approximately fifteen factors taken from firms\u2019 balance sheets; however, the exact composition and weights in the formula are a business secret. That is, they are not shared with the regulator or the banks. 6 The continuous variables are difficult to interpret because their value is industry specific. Moreover, differently from the discrete value of the rating, by construction, they do not provide the bank with a direct estimate of the firm default probability (Altman 2004). 7 Descriptive statistics on firms\u2019 distribution in the rating categories can be found in Online Appendix B (Figure B1). 8 To understand the consequences for firms of this classification in terms of S&P\u2019s ratings, note that a Score of 6 corresponds to class B, and a Score of 7 to class CCC (Altman 2004). 9 Specifically, the definition of NPL includes bad loans, past due, and loans to insolvent firms other than substandard credits. The latter are defined as exposures to counterparties facing temporary difficulties defined on the basis of objective factors. 10 Additionally, NPL weigh in the banks\u2019 balance sheets for two main reasons. The first is that there are very limited fiscal and accounting incentives for banks to write off and sell NPL. The second is related to the lengthy Italian bankruptcy system (Rodano, Serrano-Velarde, and Tarantino 2016), and the small number of asset management companies willing to buy these assets. 11 For example, in their banks\u2019 rating guidelines, (Moody\u2019s 2015, 33) reports that \u201c[asset] risks are captured, to a considerable degree, by a single financial ratio, problem loans\/gross loans (which we term the problem loan ratio),\u201d and Fitch (2016) specifies that the \u201ccore metric\u201d to measure asset quality is the problem loan ratio. 12 In this section, we present the model\u2019s main insights. The full derivation can be found in Online Appendix D. While this theoretical framework relies on \u201cex post monitoring,\u201d the intuition extends to models of \u201cex ante screening.\u201d A previous version of the paper explored this mechanism and showed the robustness of the conclusions. In the boom period, when screening is costly and bank liquidity is aplenty, the bank pools the firms at the threshold with the other firms in the same asset class. This means that all borrowers receive lending at a return that reflects the average degree of risk in a class (thus leading to price differences at the threshold). In the bust period, the exacerbation of the adverse selection problem, combined with a shortage of the banking sector\u2019s liquidity, implies that the bank engages in screening at equilibrium. Screening then leads to differences in the quantity of credit offered to the firms at the threshold that penalize those borrowers falling in the substandard class. 13 In the absence of segmentation, the two firms would always obtain the same contract with the bank at equilibrium. 14 We estimate alternative specifications in which we scale the supply of bank financing by assets or express interest rates in terms of basis point differences, and we obtain the same results. To simplify the analysis, we restrict $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ to be of the same polynomial order. However, our results are not sensitive to this choice. Finally, we also use local-linear functions to estimate differences in credit conditions at the threshold. Our results remain robust to these additional checks. 15 Clearly, one limitation of this analysis is that the reason for the downgrade might itself be correlated to the demand for credit of the firm. 16 We thank the anonymous referee for very helpful suggestions on this point. 17 To obtain the exact percentage changes, we compute $$\\left[\\left(\\exp\\left\\{\\hat{\\beta}\\right\\}-1\\right)\\times100 \\right]$$, where $$\\hat\\beta$$ is the per-period coefficient. 18 We also explored the sensitivity of bank lending to other sources of bank heterogeneity. For instance, consistent with the previous results, we find that the banks that were highly exposed to the interbank market significantly cut lending to the substandard firms at the threshold in 2008 and 2009. Similarly, during that period, intermediaries putting more weight on soft information when setting credit policies were less likely to cut their lending to substandard borrowers. One needs to be careful when interpreting this last result, as bank organizational structure is likely to be correlated with differences in size and investor composition. 19 Figure C1 in Online Appendix C provides the year-by-year plots associated with these tests. We also plot the distribution of firms that enter rating categories 6 or 7 in any given year. If firms were able to determine the value of their own continuous variable, then we should observe a disproportionate number of new firms clustering just above the threshold, in category 6. Confirming the lack of manipulation, Figure C2 of Online Appendix C shows that a significant mass of firms enters the sample with a value of the continuous variable that lies just below the threshold, in category 7. Finally, we also jointly test for manipulation across the entire cycle and find no evidence of bunching. 20 This evidence is important since small banks are typically seen as more efficient in generating private information about borrowers. Thus, one possibility would be that differences in lending are due to borrowers self-selecting into different bank relations. 21 Table C3 of Online Appendix C shows the results of additional balancing tests. 22 Our results remain the same when plotting bins of different size, like $$0.02$$ or $$0.01$$. 23 Note that, around the threshold, the relationship between credit outcomes and the continuous value of the rating is not necessarily monotonic. Two comments are in order here. First, deriving the identification of the estimates from the units closest to the threshold is precisely the focus of the applied literature on discontinuity designs. Second, on average, the relationship between the value of the rating and the interest rates of the loans is monotonic. To address potential concerns on the sensitivity of our results with respect to bandwidth choices, we reestimate our specification using lower polynomial orders, and local linear methods. Our results are robust to these changes, and can be found in Table C5 of Online Appendix C. 24 In Online Appendix C, Table C4 reports the descriptive statistics about the mean, median, and statistical significance of these placebo tests across all quarters. The estimated values are about zero and are not significant in most of the quarters. Finally, Figure C3 illustrates that a randomly drawn placebo threshold is also unlikely to yield an economically sensible pattern of estimates across time. 25 Due to the construction of the CEBI rating, the threshold between categories 5 and 6 cannot be used (see Section 1). References Agarwal, S., Chomsisengphet, S. Mahoney, N. and Stroebel. J. Forthcoming. Do banks pass through credit expansions? The marginal profitability of consumer lending during the Great Recession. Quarterly Journal of Economics . Altman, E. I. 1968 . Financial ratios, discriminant analysis and the prediction of corporate bankruptcy. Journal of Finance 23 ( 4 ): 589 \u2013 609 . Google Scholar CrossRef Search ADS Altman, E. I. Managing credit risk: the challenge for the new millennium. Presentation updated through 2004, New York University Stern School of Business, http:\/\/people.stern.nyu.edu\/ealtman\/2-%20CopManagingCreditRisk.pdf. CrossRef Search ADS Ashcraft, A. B. 2005 . Are banks really special? New evidence from the FDIC-induced failure of healthy banks. American Economic Review 95 ( 5 ): 1712 \u2013 30 . Google Scholar CrossRef Search ADS Bank of Italy . 2013 . The recent asset quality review on non-performing loans conducted by the Bank of Italy: Main features and results. July 29 . https:\/\/www.bancaditalia.it\/media\/approfondimenti\/2013\/analisi-prestiti-deteriorati\/Asset_quality_review.pdf?language_id=1. Barisitz, S. 2013 . Nonperforming loans in Western Europe: A selective comparison of countries and national definitions. Focus on European Economic Integration, Oesterreichische Nationalbank ( Austrian Central Bank ), no. 1 , pp. 28 \u2013 47 . Becker, B., and Ivashina V. 2014 . Cyclicality of credit supply: Firm level evidence. Journal of Monetary Economics 62 ( C ): 76 \u2013 93 . Google Scholar CrossRef Search ADS Bholat, D., Lastra, R. Markose, S. Miglionico, A. and Sen. K. 2016 . Non-performing loans: Regulatory and accounting treatments of assets. Bank of England Working Papers, no. 594 . Brollo, F., Nannicini, T. Perotti, R. and Tabellini. G. 2013 . The political resource curse. American Economic Review 103 ( 5 ): 1759 \u2013 96 . Google Scholar CrossRef Search ADS Calonico, S., Cattaneo, M. D. and Titiunik. R. 2014 . Robust nonparametric confidence intervals for regression discontinuity designs. Econometrica 82 ( 6 ): 2295 \u2013 2326 . Google Scholar CrossRef Search ADS Chernenko, S., and Sunderam. A. 2012 . The real consequences of market segmentation. Review of Financial Studies 25 ( 7 ): 2041 \u2013 69 . Google Scholar CrossRef Search ADS Chodorow-Reich, G. 2014 . The employment effects of credit market disruptions: Firm-level evidence from the 2008\u20132009 financial crisis. Quarterly Journal of Economics 129 ( 1 ): 1 \u2013 59 . Google Scholar CrossRef Search ADS Covas, F., and Den Haan. W. J. 2011 . The cyclical behavior of debt and equity finance. American Economic Review 101 ( 2 ): 877 \u2013 99 . Google Scholar CrossRef Search ADS Dell\u2019Ariccia, G., and Marquez. R. 2006 . Lending booms and lending standards. Journal of Finance 61 ( 5 ): 2511 \u2013 46 . Google Scholar CrossRef Search ADS Drehmann, M., Borio, C. and Tsatsaronis. K. 2012 . Characterising the financial cycle: Don\u2019t lose sight of the medium term! BIS Working Papers, no. 380 , Bank for International Settlements . Fitch . 2016 . Global bank rating criteria. July 15 . https:\/\/www.fitchratings.com\/site\/re\/884135. Gete, P. 2017 . Banking crises, lending standards and misallocation. http:\/\/dx.doi.org\/10.2139\/ssrn.2905308. Gilchrist, S., Yankov, V. and Zakraj\u0161ek. E. 2009 . Credit market shocks and economic fluctuations: Evidence from corporate bond and stock markets. Journal of Monetary Economics 56 ( 4 ): 471 \u2013 93 . Google Scholar CrossRef Search ADS Gorton, G. B., and Metrick. A. 2012 . Securitized banking and the run on repo. Journal of Financial Economics 104 ( 3 ): 425 \u2013 51 . Google Scholar CrossRef Search ADS Greenwood, R., and Hanson. S. G. 2013 . Issuer quality and corporate bond returns. Review of Financial Studies 26 ( 6 ): 1483 \u2013 1525 . Google Scholar CrossRef Search ADS Imbens, G., and Kalyanaraman. K. 2014 . Optimal bandwidth choice for the regression discontinuity estimator. Review of Economic Studies 79 ( 3 ): 933 \u2013 59 . Google Scholar CrossRef Search ADS Intesa . 2015 . Consolidated financial statements (Part E). http:\/\/www.group.intesasanpaolo.com\/scriptIsir0\/si09\/contentData\/view\/20150408_RischiCredito_uk.pdf?id=CNT-05-000000025BE48&ct=application\/pdf. Ivashina, V. 2009 . Asymmetric information effects on loan spreads. Journal of Financial Economics 92 ( 2 ): 300 \u2013 319 . Google Scholar CrossRef Search ADS Ivashina, V., and Scharfstein. D. 2009 . Bank lending during the financial crisis of 2008. Journal of Financial Economics 97 ( 3 ): 319 \u2014 38 . Google Scholar CrossRef Search ADS Iyer, R., Puri, M. and Ryan. N. 2016 . A tale of two runs: Depositor responses to bank solvency risk. Journal of Finance 71 ( 6 ): 2687 \u2013 2726 . Google Scholar CrossRef Search ADS Jassaud, N., and Kang. K. H. 2015 . A strategy for developing a market for nonperforming loans in Italy. IMF Working Paper, no. 15\/24 , International Monetary Fund. Google Scholar CrossRef Search ADS Jermann, U., and Quadrini. V. 2012 . Macroeconomic effects of financial shocks. American Economic Review 102 ( 1 ): 238 \u2013 71 . Google Scholar CrossRef Search ADS PubMed Jim\u00e9nez, G., Ongena, S. Peydr\u00f3, J.-L. and Saurina. J. 2012 . Credit supply and monetary policy: Identifying the bank balance-sheet channel with loan applications. American Economic Review 102 ( 5 ): 2301 \u2013 26 . Google Scholar CrossRef Search ADS Jim\u00e9nez, G., Ongena, S. Peydr\u00f3, J.-L. and Saurina. J. 2014 . Hazardous times for monetary policy: What do 23 million loans say about the impact of monetary policy on credit risk-taking? Econometrica 82 ( 2 ): 463 \u2013 505 . Google Scholar CrossRef Search ADS Kisgen, D. J., and Strahan. P. E. 2010 . Do regulations based on credit ratings affect a firm\u2019s cost of capital. Review of Financial Studies 23 ( 12 ): 4324 \u2013 47 . Google Scholar CrossRef Search ADS Khwaja, A. I., and Mian. A. 2008 . Tracing the impact of bank liquidity shocks: Evidence from an emerging market. American Economic Review 98 ( 4 ): 1413 \u2013 42 . Google Scholar CrossRef Search ADS Kovbasyuk, S., and Spagnolo. G. 2017 . Memory and markets. EIEF Working Papers, no. 1606, Einaudi Institute for Economics and Finance. Lemmon, M., and Roberts. M. R. 2010 . The response of corporate financing and investment to changes in the supply of credit. Journal of Financial and Quantitative Analysis 45 ( 3 ): 555 \u2013 87 . Google Scholar CrossRef Search ADS Lopez-Salido, D., Stein, J. C. and Zakraj\u0161ek. E. 2017 . Credit-market sentiment and the business cycle. Quarterly Journal of Economics 132 ( 3 ): 1373 \u2013 1426 . Google Scholar CrossRef Search ADS McCrary, J. 2008 . Manipulation of the running variable in the regression discontinuity design: a density test. Journal of Econometrics 142 ( 2 ): 698 \u2013 714 . Google Scholar CrossRef Search ADS Moody\u2019s. 2015 . Rating methodology: Banks. March 16 . www.moodys.com\/methodologies. Martin, A. 2008 . Endogenous credit cycles. Economics Working Papers, no. 916, Department of Economics and Business, Universitat Pompeu Fabra. Google Scholar CrossRef Search ADS OECD . 1997 . Small Businesses, Job Creation and Growth: Facts, Obstacles and Best Practices. Paravisini, D., Rappaport, V. Schnabl, P. and Wolfenzon. D. 2014 . Dissecting the effect of credit supply on trade: Evidence from matched credit-export data. Review of Economic Studies 82 ( 1 ): 333 \u2013 359 . Google Scholar CrossRef Search ADS Rajan, R. G. 2005 . Has financial development made the world riskier? Proceedings of the Economic Policy Symposium, Jackson Hole, Federal Reserve Bank of Kansas City , August , 313 \u2013 69 . Rodano, G., Serrano-Velarde, N. and Tarantino. E. 2016 . Bankruptcy law and bank financing. Journal of Financial Economics 120 ( 2 ): 363 \u2013 82 . Google Scholar CrossRef Search ADS Stein, J. C. 2002 . Information production and capital allocation: Decentralized versus hierarchical firms. Journal of Finance 57 ( 5 ): 1891 \u2013 1921 . Google Scholar CrossRef Search ADS Standard & Poor\u2019s. 2004 . Credit risk tracker Italy. Standard & Poor\u2019s Risk Solutions. Sufi, A. 2007 . Information asymmetry and financing arrangements: Evidence from syndicated loans. Journal of Finance 62 ( 2 ): 629 \u2013 68 . Google Scholar CrossRef Search ADS Tirole, J. 2006 . The theory of corporate finance . Princeton, NJ : Princeton University Press. Unicredit Bank. 2008 . Unicredit S.p.A. 2008 Annual Report. World Bank . 2002 . Bank loan classification and provisioning practices in selected developed and emerging countries (a survey of current practices in countries represented on the Basel Core Principles Liaison Group). \u00a9 The Author(s) 2018. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https:\/\/academic.oup.com\/journals\/pages\/about_us\/legal\/notices) http:\/\/www.deepdyve.com\/assets\/images\/DeepDyve-Logo-lg.png The Review of Financial Studies Oxford University Press\n\n# Lending Standards over the Credit Cycle\n\n, Volume Advance Article (8) \u2013 Apr 24, 2018\n40 pages\n\n\/lp\/ou_press\/lending-standards-over-the-credit-cycle-QjfsLoFyi2\nPublisher\nOxford University Press\nISSN\n0893-9454\neISSN\n1465-7368\nD.O.I.\n10.1093\/rfs\/hhy023\nPublisher site\nSee Article on Publisher Site\n\n### Abstract\n\nAbstract We analyze how firms\u2019 segmentation into credit classes affects the lending standards applied by banks to small and medium enterprises over the cycle. We exploit an institutional feature of the Italian credit market that generates a discontinuity in the allocation of comparable firms into the performing and substandard classes of credit risk. In the boom period, segmentation results in a positive interest rate spread between substandard and performing firms. In the bust period, the increase in banks\u2019 cost of wholesale funds implies that substandard firms are excluded from credit. These firms then report lower values of production and capital investments. Received January 22, 2016; editorial decision December 18, 2017 by Editor Robin Greenwood. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online. A growing empirical literature shows that segmentation between investment-grade and speculative-grade firms can have important implications for their access to capital markets (e.g., Kisgen and Strahan 2010; Lemmon and Roberts 2010; Chernenko and Sunderam 2012). Segmentation implies that firms of different credit quality have access to different pools of investor capital, and that the price and quantity available of this capital vary over time. An unanswered question is whether the effects of such asset class segmentation extend into bank lending policies, and lead to substantially different access to credit for otherwise similar small- and medium-sized enterprises (SME). This question is relevant not only because SME account for up to 70% of jobs in most Organisation for Economic Co-operation and Development (OECD) countries, but also because they nearly exclusively rely on bank financing (OECD 1997). In this paper, then, we study whether segmentation influences the bank lending standards applied to SME, and, relatedly, how the consequences of firm segmentation vary over the credit cycle. The empirical identification of the link between SME segmentation and bank lending standards is a challenging one. The reason is that the adjustment of lending standards can conform to different mechanisms. In neoclassical theories of financial intermediation, banks tighten credit by raising the credit spread and quantity drops along the credit demand. Alternatively, for given price, lenders can tighten standards by rationing risky firms\u2019 quantity of credit\u2014as in models with informational frictions. Consequently, to distinguish between these mechanisms requires detailed contract-level information on price and quantity of bank credit.1 To address this challenge, our analysis relies on a unique loan-level data set collected by the Italian central bank. This data set allows us to observe the total quantity of credit granted and the per-loan interest rate charged by financial intermediaries to SME. Our sample is composed of 144,000 firm-year observations in the manufacturing sector and 253,000 funding contracts covering the period between 2004 and 2011. Like other OECD economies, Italy was experiencing a credit cycle that reached its peak between 2006 and 2007 (Drehmann, Borio, and Tsatsaronis 2012) and then culminated with the Great Recession. To study the consequences of segmentation for firms\u2019 real decisions, we also use a comprehensive data set containing information on firms\u2019 balance sheet statements. Our full data sets then give us an untapped opportunity to study how firm segmentation shapes the relationship between banks and SME over the cycle. An additional empirical challenge to the analysis is how to isolate changes in banks\u2019 lending supply from changes in firms\u2019 desire to borrow. To do so, we exploit the institutional features of the Italian credit market for SME. First, for historical reasons, the credit risk assessment of SME performed by Italian banks uses a common credit rating (the Score) that banks purchase from an external agency (Centrale dei Bilanci, or CEBI). Unlike U.S. corporate credit ratings, the Score is unsolicited, available for all SME, and computed based only on firms\u2019 past balance sheets. Second, within this rating methodology, firms are allocated into two main rating classes\u2014performing and substandard\u2014based on the value of a continuous variable. Importantly, the bank has access to information on both the risk class and the continuous value of the firm\u2019s rating when making its decisions, but, when reporting its loan portfolio to financial markets, it classifies firms based only on their rating classes. The structure of the rating, and its construction, give us the opportunity to follow an intuitive empirical strategy to identify the effects of segmentation on financial contracts. Specifically, we exploit the sharp discontinuity in the allocation of firms into the performing and substandard classes of credit risk. As our measure of lending standards, then, we take the differences in the credit conditions between a firm marginally classified as performing and one that is marginally classified as substandard based on the value of the rating\u2019s continuous variable. These threshold differences inform us about how banks\u2019 supply of credit is affected by segmentation, while holding constant the demand for credit. The classification between substandard and performing risks is important for bank lending choices because it affects the banks\u2019 cost of financing. The national banking regulator adopts a conservative definition of nonperforming loans (NPL), which also includes loans of substandard credit quality (World Bank 2002; Bank of Italy 2013; Barisitz 2013; Jassaud and Kang 2015; Bholat et al. 2016).2 Banks then allocate loans in the category of nonperforming loans based, among others, on the risk status provided by their credit scoring (e.g., Intesa 2015). This has implications for bank capital and investor assessment of bank balance sheets. Indeed, NPL absorb valuable bank capital (Jassaud and Kang 2015), and their volume is often referenced as the major indicator of banks\u2019 asset quality by rating agencies (Moody\u2019s 2015; Fitch 2016). We empirically confirm the importance of banks\u2019 choice of exposure toward performing and substandard credit quality by relating the cost of funding borne by Italian banks to the composition of their loan portfolio. Our main findings on the impact of segmentation on lending conditions follow. In the boom period, the substandard and performing firms at the threshold are treated differently mainly in terms of the interest rates applied to new loans. Indeed, we find an interest rate spread of about 4% (or 20 basis points), and a positive but not statistically significant difference in the amount of granted credit. As a consequence of the financial crisis that hit the Italian banking sector, in the bust period banks tightened their lending standards mainly by acting on the quantity margin: specifically, the performing firms obtain 39% more financing than comparable substandard firms (at a similar interest rate). For the final years in our sample (2010\u20132011), our estimates point to a reduction in the differences of bank lending at the threshold, and an increase in the interest rate spread. All these results are consistent with those arising from a model of financial contracting in the presence of informational frictions and market segmentation. To quantify the importance of segmentation for bank lending, we compare the estimates of our threshold analysis to those arising from a na\u00efve specification that analyzes differences in the lending conditions between all performing and substandard firms. We find that, in the bust period, segmentation can account for a significantly larger part of the observed na\u00efve differential in the amount of credit than in the boom period. Another key insight arising from our discontinuity strategy relates to the patterns of the interest rate spread. While the na\u00efve interest rate differences are increasing throughout the cycle, we show that, during the crisis, the threshold spread is close to zero\u2014reflecting the implementation of lending standards\u2019 adjustment primarily via a restriction of substandard firms\u2019 access to credit. We then trace the implications of lending standards for firms\u2019 real activity. The production choices of the firms at the threshold significantly diverge during the crisis, to the point that the marginally performing firms report up to 50% larger values of production than the marginally substandard ones. After decomposing production values into firms\u2019 investment in inputs, we find that an increase in the interest rate spread induces firms to adjust their expenditures in variable inputs (i.e., intermediates and employment). Instead, in the bust period, when banks act on the quantity margin to adjust lending standards, firms respond by cutting capital investments, which typically have a long-run nature. The richness of our contract-level data allows us to study the economic mechanism driving the sensitivity of bank lending to segmentation. Specifically, we test for the relative importance of bank capitalization and bank investor composition in explaining the relationship between segmentation and lending policies. In line with, among others, Ivashina and Scharfstein (2009) and Iyer, Puri, and Ryan (2016), we show that the degree of exposure to funding from short-term investors is quantitatively more important than bank capitalization to explain our threshold differences. Finally, we compare the lending conditions applied to two comparable firms, one of which is downgraded to the substandard class as a result of a small change in the value of its continuous rating (which is observed only by the bank, not by its investors). This analysis shows that the negative impact of a downgrade on credit allocations becomes progressively larger and statistically significant in crisis and recovery. We confirm the internal validity of our results by presenting the following robustness checks to our empirical design. First, we find no systematic evidence of manipulation of the rating, which confirms the fact that it is very difficult for firms to manipulate the Score. Second, we show that, close to the threshold, firms feature comparable economic characteristics, and are thus \u201cas if\u201d randomly sampled. Third, we confirm the relevance of the threshold that assigns firms to the performing and substandard classes. In particular, we run our threshold analysis at all the other six thresholds associated with the categorical value of the rating, and find that most of the estimates are not statistically significant. This suggests that our results capture a form of market segmentation, not a simple rating effect. In addition to the literature on the consequences of market segmentation for financial contracts, our paper also contributes to the macrofinance literature studying the dynamics of credit over the cycle.3 Specifically, Greenwood and Hanson (2013) show that the deterioration of credit quality during booms forecasts low excess returns to bondholders. Similarly, in their historical account of credit cycles, Lopez-Salido, Stein, and Zakraj\u0161ek (2017) find that elevated credit sentiment is associated with a more aggressive pricing of risk and a subsequent contraction in economic activity. Consistent with these studies, we provide evidence of how the 2004\u20132011 cycle affected the transmission of market segmentation into bank lending policies. Our paper is also related to the body of work on empirical banking (e.g., Jim\u00e9nez et al. 2012, 2014; Chodorow-Reich 2014). We extend this literature by showing that, to understand the dynamics of bank lending standards, one needs to jointly analyze the price and quantity of lending.4 1. Documenting Segmentation in the Credit Market The goal of this section is to establish the presence of segmentation in the Italian credit market for SME. We will first present the institutional features of this market that generate segmentation, and then document the relationship between segmentation and the banks\u2019 cost of wholesale funds. 1.1 The Score rating system Evidence from the 2006 Bank of Italy survey of Italian banks indicates that 90% of the banks using a firm\u2019s rating find it important when deciding on whether to process a loan application, 76% of them use the rating to set the amount of lending, and 62% use it to formulate an interest-rate offer. For historical reasons, Italian banks use a common credit rating produced by Centrale dei Bilanci (CEBI) when making decisions about lending to SME. CEBI is a credit agency founded in 1983 as a joint initiative of the Italian Central Bank and the Italian Banking Association to record and process firms\u2019 financial statements. According to Standard & Poor\u2019s (2004), \u201cBanks are the main users of the outputs of CEBI,\u201d referring to the Score rating produced by CEBI as the major tool used to assess SME credit risk. In 2004, the share of credit granted to SME by banks subscribing to the Score rating system was 73%. The following features of the Score are of particular interest to our research design: The Score is unsolicited by firms and is computed based on firms\u2019 past balance sheets. Although its exact algorithm is a business secret of CEBI, information provided to the regulator by the agency that produces the Score shows that the construction of the rating is based on multiple discriminant analyses of past firm balance sheet information (Altman 1968).5 These features make the manipulation of the rating very unlikely. The system generates two continuous variables that determine the assignment to discrete rating categories. Based on predetermined thresholds, the first continuous variable is used to allocate the firms among the first five rating categories (1\u20135), the second to allocate the firms among categories 6 to 9. The Score therefore ranges from 1, for firms that are the least likely to default, to 9, for those most likely to default.6 We obtained from CEBI direct access to the information on the values of the continuous and discrete variables for the manufacturing firms rated by the agency. We also have access to the exact thresholds that determine the allocation of firms into the different rating categories. This means that we can reconstruct the exact firm allocation mechanism implemented within the Score rating system. Figure 1 illustrates some of the key empirical features of the Score. The left panel of Figure 1 plots the Score variable of firms in year $$t$$ against the share of delinquent firms in year $$t+1$$. To construct this figure, we combine information from Italian chambers of commerce and the credit register of the Italian central bank for the period 2004\u20132011. We define a firm as delinquent if it entered a formal bankruptcy process, or if its loan was flagged as late\/defaulted in the credit register. Finally, we decompose the informativeness of the rating variable across three periods: boom (2004\u20132007), bust (2008\u20132009), and recovery (2010\u20132011). The panel suggests a monotonic relationship between the rating variable and future credit events. Indeed, the share of delinquent firms with a Score of up to 4 in a given year hovers around 4%. This share rises to about 10% for firms with a Score of 7. At the same time, the decomposition of default rates across subperiods indicates that the informativeness of the rating variable is relatively stable over the cycle. More specifically, the increase in delinquency rates between the boom period and the bust period for a Score of 7 is less than one percentage point. Figure 1. View largeDownload slide Characteristics of the Score assignment variable The left panel plots the Score variable against the share of defaults within the next year in the boom (dashed), crisis (solid), and recovery (dotted). The right panel plots the average loan rate by Score category for the first quarter of 2005. Figure 1. View largeDownload slide Characteristics of the Score assignment variable The left panel plots the Score variable against the share of defaults within the next year in the boom (dashed), crisis (solid), and recovery (dotted). The right panel plots the average loan rate by Score category for the first quarter of 2005. The right panel of Figure 1 plots the rating variable against the interest rate on loans for the first quarter of 2005. A strong positive relationship exists between the rating variable and interest rates on loans. The best (lowest) Score, in terms of creditworthiness, is on average associated with a loan interest rate of 4%, and the worst (highest) category pays an average loan interest rate of around 5%.7 Figure 1 therefore suggests that the Score rating provides a reliable estimate of the expected likelihood of a firm\u2019s delinquency, which is then taken into account by the banks for their lending decisions. 1.2 Segmentation of SME in the Italian credit market Within the Score rating methodology, the distinction between the performing and substandard classes of credit risk stands out as particularly relevant for banks and their stakeholders. The performing class consists of the firms with a Score category between 1 and 6, and the substandard class comprises firms with a Score between 7 and 9.8 The importance of this classification stems from its implications for bank disclosure and reporting of their loan portfolio. National regulators decide on the loan categories that enter the class of NPL: this is relevant for our purposes because the Bank of Italy adopts a conservative definition of NPL, which includes loans of substandard credit quality (World Bank 2002; Bank of Italy 2013; Barisitz 2013; Jassaud and Kang 2015; Bholat et al. 2016).9 NPL absorb valuable bank capital: the capital charge for NPL amounts on average to 12% of banks\u2019 risk-weighted assets, and are estimated to tie up more than 6% of bank capital (Jassaud and Kang 2015).10 Moreover, a bank\u2019s exposure to NPL is often referenced as the major indicator of asset quality by the bank\u2019s rating agencies.11 Banks then allocate loans in the category of nonperforming loans based, among others, on the risk status provided by their credit scoring (e.g., Intesa 2015). Moreover, in their annual reports, they clearly distinguish between their exposure to the firms classified as substandard and performing by the rating (e.g., Unicredit 2008). As a consequence, investors monitor the volume of substandard lending to assess a bank\u2019s risk profile. The presence of such segmentation gives rise to clear, testable implications. First, one expects outside investors to charge a higher cost of funding to those banks that carry a higher volume of substandard loans in their loan book. Second, one should find that the continuous variables should not contain any useful information to explain the bank cost of funding on wholesale funding markets. 1.3 Segmentation and bank cost of financing We now provide evidence consistent with the presence of segmentation in the Italian credit market. We use three confidential data sets from the Bank of Italy. The first provides us with information on the amount and interest rate at which Italian banks raise financing from repo markets, households, and firms at a monthly frequency between 2004 and 2011. The second data set contains yearly bank balance sheets between 2006 and 2011, and provides us with information about a bank\u2019s size, capitalization, and liquidity. Finally, we use information from the credit register to determine the composition of each bank\u2019s SME portfolio based on the categorical and continuous variables of the rating system. To estimate the relationship between a bank\u2019s cost of financing and its lending portfolio, we use the following ordinary least squares (OLS) specification: \\begin{align} r_{b,t} &= \\alpha_{0} +\\alpha_{1} \\mbox{Substandard to Total Credit}_{b,t-1} +\\alpha_{2} \\mbox{Continuous Score 1}_{b,t-1} \\nonumber \\\\ &\\quad +\\alpha_{3} \\mbox{Continuous Score 2}_{b,t-1} + X_{b,t-1}\\Psi + I_{b,t}\\Phi + \\pi_{t} +\\epsilon_{b,t}. \\end{align} (1) In Equation (1), $$b$$ denotes a bank in our data set, and $$t$$ is taken at the monthly level. The dependent variable, $$r_{b,t}$$, is the (volume) weighted average interest rate paid by banks across all investors. Substandard to Total Credit$$_{b,t-1}$$ is the share of a bank\u2019s volume of lending to SME in the substandard rating class relative to total lending. Continuous Variable 1$$_{b,t-1}$$ and Continuous Variable 2$$_{b,t-1}$$ characterize the SME portfolio of the bank in terms of the average continuous ratings. $$X_{b,t-1}$$ denotes a vector of bank characteristics; $$I_{b,t}$$ denotes issuance characteristics such as amounts, maturity, and investor composition; and $$\\pi_{t}$$ are month-year fixed effects. All explanatory variables, except for issuance characteristics, are measured before the issuance. Standard errors are clustered at the bank level. In Columns (1) and (2) of Table 1, we show that external investors monitor banks by pricing lending portfolios based on banks\u2019 exposure to the substandard and performing classes. The estimate in Column (1) implies that a 25% higher share of substandard lending in the bank portfolio is associated with an increase in the bank\u2019s interest rate of approximately 28%, or 31 basis points. Column (2) extends the baseline specification in Equation (1) by including the continuous values produced by the rating system. The coefficient on the share of substandard loans remains significant and economically identical to the first specification. Instead, the coefficients on the values of the continuous variables are neither statistically nor economically significant. Our evidence is therefore consistent with the presence of market-driven segmentation in the Italian credit market for SME. Investors observe the distribution of loans into rating classes, and set a higher interest-rate premium to compensate for a larger exposure to substandard loans. Table 1 Banks\u2019 cost of financing and rating segmentation Pre-2008 Post-2008 (1) (2) (3) (4) Substandard to Total Credit 1.26*** 1.24* \u20130.37 1.34** (0.46) (0.66) (0.29) (0.68) Continuous Variable 1 \u20130.2 (0.15) Continuous Variable 2 0.09 (0.31) Bank & Issuance Characteristics Yes Yes Yes Yes Bank Fixed Effects No No Yes Yes Time Fixed Effects Yes Yes Yes Yes $$R^2$$ 0.76 0.76 0.85 0.54 N 4,788 4,728 2,233 2,212 Pre-2008 Post-2008 (1) (2) (3) (4) Substandard to Total Credit 1.26*** 1.24* \u20130.37 1.34** (0.46) (0.66) (0.29) (0.68) Continuous Variable 1 \u20130.2 (0.15) Continuous Variable 2 0.09 (0.31) Bank & Issuance Characteristics Yes Yes Yes Yes Bank Fixed Effects No No Yes Yes Time Fixed Effects Yes Yes Yes Yes $$R^2$$ 0.76 0.76 0.85 0.54 N 4,788 4,728 2,233 2,212 The table reports the estimates of the specification in Equation (1), using as a dependent variable the interest rate at which Italian banks raise financing. In Columns (1) and (2), the dependent variable is the (volume) weighted average interest rate at which banks raised financing across different types of investors (repo markets, households, firms) between 2004 and 2011. In Columns (3) and (4), we reestimate our pricing equation for the periods before and after 2008, respectively. Accordingly, the dependent variable is the interest rate at which banks raised financing on repurchase markets before 2008 in Column (3) and after 2008 in Column (4). Substandard to Total Credit is the share of a bank\u2019s volume of lending to SME in the substandard rating class relative to total lending. Continous Variable 1 denotes the mean of the continuous variable of firms in rating categories 1 to 5. Continous Variable 2 denotes the mean of the continuous variable of firms in rating categories 6 to 9. The specification includes a vector of bank and issuance characteristics. Issuance characteristics include amounts raised, maturity, and investor composition. Bank characteristics include size (in terms of total assets), the value of the tier 1 capitalization ratio, and the bank\u2019s liquidity ratio. The specification includes monthly fixed effects, with standard errors clustered at the bank level. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Table 1 Banks\u2019 cost of financing and rating segmentation Pre-2008 Post-2008 (1) (2) (3) (4) Substandard to Total Credit 1.26*** 1.24* \u20130.37 1.34** (0.46) (0.66) (0.29) (0.68) Continuous Variable 1 \u20130.2 (0.15) Continuous Variable 2 0.09 (0.31) Bank & Issuance Characteristics Yes Yes Yes Yes Bank Fixed Effects No No Yes Yes Time Fixed Effects Yes Yes Yes Yes $$R^2$$ 0.76 0.76 0.85 0.54 N 4,788 4,728 2,233 2,212 Pre-2008 Post-2008 (1) (2) (3) (4) Substandard to Total Credit 1.26*** 1.24* \u20130.37 1.34** (0.46) (0.66) (0.29) (0.68) Continuous Variable 1 \u20130.2 (0.15) Continuous Variable 2 0.09 (0.31) Bank & Issuance Characteristics Yes Yes Yes Yes Bank Fixed Effects No No Yes Yes Time Fixed Effects Yes Yes Yes Yes $$R^2$$ 0.76 0.76 0.85 0.54 N 4,788 4,728 2,233 2,212 The table reports the estimates of the specification in Equation (1), using as a dependent variable the interest rate at which Italian banks raise financing. In Columns (1) and (2), the dependent variable is the (volume) weighted average interest rate at which banks raised financing across different types of investors (repo markets, households, firms) between 2004 and 2011. In Columns (3) and (4), we reestimate our pricing equation for the periods before and after 2008, respectively. Accordingly, the dependent variable is the interest rate at which banks raised financing on repurchase markets before 2008 in Column (3) and after 2008 in Column (4). Substandard to Total Credit is the share of a bank\u2019s volume of lending to SME in the substandard rating class relative to total lending. Continous Variable 1 denotes the mean of the continuous variable of firms in rating categories 1 to 5. Continous Variable 2 denotes the mean of the continuous variable of firms in rating categories 6 to 9. The specification includes a vector of bank and issuance characteristics. Issuance characteristics include amounts raised, maturity, and investor composition. Bank characteristics include size (in terms of total assets), the value of the tier 1 capitalization ratio, and the bank\u2019s liquidity ratio. The specification includes monthly fixed effects, with standard errors clustered at the bank level. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. In Columns (3) and (4), we focus on the cost of financing on the repurchase market, the primary source of funds for the securitized banking system. This market is of particular interest, because Gorton and Metrick (2012) describe the crisis as a \u201crun on repo\u201d that was triggered by concerns about bank solvency. We therefore reestimate our pricing equation in Equation (1) separately for the period before and after 2008, and augment our specification with bank fixed effects. In the boom phase of the credit cycle, the correlation between interest rates on the repurchase market and the composition of banks\u2019 lending portfolio is low and statistically nonsignificant. In the bust period, the correlation is positive and economically significant, implying an increase in the interest rate premium required by investors from banks that are relatively more exposed to substandard credit risk. 2. Theoretical Framework To motivate our empirical analysis, we propose a model of credit with market segmentation and moral hazard. Specifically, we extend the basic framework in Tirole (2006, chap. 3), to accommodate the institutional features of the Italian credit market for SME. As in Tirole (2006), entrepreneurs need capital to fund a project. The bank-firm relationship is frustrated by moral hazard: by putting effort, the firm succeeds with positive probability. By shirking, the firm defaults with certainty, but the entrepreneur then gains private benefits. Finally, we allow firms to use the bank\u2019s monitoring technology. Differently from the standard setup, firms are segmented into two rating classes, performing and substandard. Moreover, we assume that the conditions at which banks receive funding depend on the phase of the credit cycle. We show that a bank\u2019s ability to tame the firm\u2019s moral hazard problem can be impaired when funding conditions on the wholesale market heat up (i.e., in the crisis period). This can push the bank to reduce lending at the expense of the substandard firms.12 The model features three categories of agents: the bank, its investors, and two firms. The two firms are allocated by the rating system used by the bank into the performing and substandard classes of credit risk. We consider the case in which the two firms fall exactly at the threshold between the two classes. The bank knows this, and understands that they are economically identical. The cost of funds to the bank is set by external investors who, consistent with our empirical evidence, observe only the firms\u2019 rating class. The existence of market segmentation has then two main implications for bank lending.13 First, the bank\u2019s cost of funding to a firm will reflect the composition of demand in the credit class. Second, the cost of financing paid by the bank will vary over the cycle according to the conditions on the wholesale funding market. In the boom period, the low cost at which the bank raises financing in the wholesale market implies that both firms can obtain access to unmonitored credit. More specifically, both firms receive the same amount of lending, but the substandard firm pays a higher interest rate\u2014mirroring the higher risk in their class. In the bust period, worse conditions on the market for wholesale funds erode firms\u2019 net worth, and imply that lending is not viable for the bank. Then, firms have two options: the first is to use the bank\u2019s monitoring technology, which comes at a cost, but also alleviates the moral hazard problem. Alternatively, they are (partially) rationed from credit. Assume that monitoring works with the performing firm, so that the bank can break even on this firm\u2019s project. Instead, the monitoring technology does not work for the substandard firm: that is, the rise in the cost of wholesale funding for the bank, combined with the cost of monitoring, implies that the net present value of the substandard firm\u2019s project remains negative. Then, the substandard firm is credit rationed at equilibrium. To sum up: in the bust period, quantity differences arise in the credit contracts offered to the two firms at the threshold. As we will further discuss later, these results will guide the interpretation of the credit differences arising in our empirical analysis. 3. Data Preview and Economic Environment To test the link between segmentation of firms and bank lending standards, we use a confidential data sets from the Bank of Italy that contain information on bank balance sheets and the financial contracts signed between banks and SME. We instead obtain firm balance sheets and rating information from CEBI. Our final sample is composed of about 144,000 firm-year observations in the manufacturing sector and 253,000 funding contracts signed between the first quarter of 2004 and the last quarter of 2011. Further details on the data set and its organization can be found in Online Appendix A. This section first documents the presence of substantial heterogeneity across rating classes. This heterogeneity suggests that a na\u00efve comparison between the credit conditions of firms in different rating classes is likely to yield misleading conclusions on the pattern of lending standards, because the resulting credit differences could simply reflect differences in firms\u2019 demand for credit. Then, we show the patterns of firms\u2019 financial contracts over time, which document how the phases of the credit cycle that Italy experienced between 2004 and 2011 affect financial allocations. Finally, we present key developments in the Italian banking environment that occurred during our sample period, illustrating the significant effects of the financial crisis on the wholesale funding and capitalization of Italian banks. 3.1 Firm financing environment We begin by presenting the sources of cross-sectional heterogeneity in our data set and the time-series variation in firm financial contracts. 3.1.1 Cross-sectional descriptive statistics Table 2 provides the cross-sectional characteristics of the full sample in Column (1). Columns (2) and (3) show corresponding results for the group of performing and substandard firms, and Columns (4) and (5) show the same for categories 6 and 7. Finally, Column (6) reports the mean difference between the values of the variables in categories 6 and 7. Table 2 Descriptive statistics All Performing Substandard Score 6 Score 7 6\u20137 Comparison (1) (2) (3) (4) (5) (6) Panel A: Loan information Term Loans: Interest Rate 4.57 4.32 5.3 4.79 5.29 \u20130.48*** (1.62) (1.56) (1.6) (1.58) (1.59) Term Loans: Amount 816 885 617 451 569 \u2013118 (9850) (5156) (17300) (1623) (17700) Term Loans: Maturity 0.66 0.66 0.65 0.77 0.72 0.05*** (0.47) (0.47) (0.48) (0.44) (247) N 253,502 188,026 65,475 49,265 60,326 109,591 Panel B: Aggregate financing information All Bank Financing Granted 8,503 9,237 6,167 7,542 6,392 1,150*** (37,200) (40,600) (23,100) (24,600) (21,100) Share of Term Loans Granted 0.35 0.35 0.36 0.33 0.35 \u20130.02*** (0.25) (0.25) (0.25) (0.21) (0.25) Share of Write-downs 0.01 0.01 0.03 0.00 0.01 \u20130.01*** (0.09) (0.04) (0.17) (0.05) (0.09) N 543,855 414,041 129,754 63,722 104,253 167,975 Panel C: Balance sheet information Employment 92 95 76 73 72 1 (294) (295) (290) (170) (207) Investment to Assets 0.05 0.05 0.04 0.04 0.039 0.001** (0.06) (0.06) (0.06) (0.06) (0.06) Return to Assets 0.05 0.07 0.00 0.05 0.03 0.02*** (0.10) (0.08) (0.13) (0.07) (0.07) Leverage 0.67 0.61 0.86 0.79 0.85 \u20130.06*** (0.19) (0.18) (0.10) (0.10) (0.09) N 143,953 108,353 35,600 16,432 27,350 43,782 All Performing Substandard Score 6 Score 7 6\u20137 Comparison (1) (2) (3) (4) (5) (6) Panel A: Loan information Term Loans: Interest Rate 4.57 4.32 5.3 4.79 5.29 \u20130.48*** (1.62) (1.56) (1.6) (1.58) (1.59) Term Loans: Amount 816 885 617 451 569 \u2013118 (9850) (5156) (17300) (1623) (17700) Term Loans: Maturity 0.66 0.66 0.65 0.77 0.72 0.05*** (0.47) (0.47) (0.48) (0.44) (247) N 253,502 188,026 65,475 49,265 60,326 109,591 Panel B: Aggregate financing information All Bank Financing Granted 8,503 9,237 6,167 7,542 6,392 1,150*** (37,200) (40,600) (23,100) (24,600) (21,100) Share of Term Loans Granted 0.35 0.35 0.36 0.33 0.35 \u20130.02*** (0.25) (0.25) (0.25) (0.21) (0.25) Share of Write-downs 0.01 0.01 0.03 0.00 0.01 \u20130.01*** (0.09) (0.04) (0.17) (0.05) (0.09) N 543,855 414,041 129,754 63,722 104,253 167,975 Panel C: Balance sheet information Employment 92 95 76 73 72 1 (294) (295) (290) (170) (207) Investment to Assets 0.05 0.05 0.04 0.04 0.039 0.001** (0.06) (0.06) (0.06) (0.06) (0.06) Return to Assets 0.05 0.07 0.00 0.05 0.03 0.02*** (0.10) (0.08) (0.13) (0.07) (0.07) Leverage 0.67 0.61 0.86 0.79 0.85 \u20130.06*** (0.19) (0.18) (0.10) (0.10) (0.09) N 143,953 108,353 35,600 16,432 27,350 43,782 All panels use data for the period 2004.Q1\u20132011.Q4, and monetary values expressed in KE (1,000 euro). Standard deviations are reported in brackets. The last column reports the difference in means of each variable between categories 6 and 7. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Panel A uses pooled loan-level data with observations at the loan-quarter level. Interest Rate is the gross annual interest rate inclusive of participation fees, loan origination fees, and monthly service charges. Amount is the granted amount of the issued term loan. Maturity is a binary variable indicating whether the maturity of the newly issued loans is up to one year, or longer. Panel B uses credit register data with observations at the firm-quarter level. All Bank Financing Granted is the firms\u2019 total amount of bank financing granted summing across all categories (loans, credit lines, backed loans). Share of Term Loans Granted is the firms\u2019 total amount of term loans granted, divided by the total amount of bank financing granted for all categories. Share of Write-downs is a binary variable indicating whether the firms\u2019 total amount of bank financing granted for all categories has experienced write-downs by banks. Panel C uses balance sheet and cash-flow statements at the firm-year level. Employment is defined as the firms\u2019 average employment over the year. Investment to Assets is the firms\u2019 investment in material fixed assets over total fixed assets. Returns to Assets is defined as the firms\u2019 earnings before interest and taxes over total assets. Leverage is the firms\u2019 ratio of debt (both short and long term) over total assets. In all panels, $$N$$ corresponds to the pooled number of observations in our sample. Table 2 Descriptive statistics All Performing Substandard Score 6 Score 7 6\u20137 Comparison (1) (2) (3) (4) (5) (6) Panel A: Loan information Term Loans: Interest Rate 4.57 4.32 5.3 4.79 5.29 \u20130.48*** (1.62) (1.56) (1.6) (1.58) (1.59) Term Loans: Amount 816 885 617 451 569 \u2013118 (9850) (5156) (17300) (1623) (17700) Term Loans: Maturity 0.66 0.66 0.65 0.77 0.72 0.05*** (0.47) (0.47) (0.48) (0.44) (247) N 253,502 188,026 65,475 49,265 60,326 109,591 Panel B: Aggregate financing information All Bank Financing Granted 8,503 9,237 6,167 7,542 6,392 1,150*** (37,200) (40,600) (23,100) (24,600) (21,100) Share of Term Loans Granted 0.35 0.35 0.36 0.33 0.35 \u20130.02*** (0.25) (0.25) (0.25) (0.21) (0.25) Share of Write-downs 0.01 0.01 0.03 0.00 0.01 \u20130.01*** (0.09) (0.04) (0.17) (0.05) (0.09) N 543,855 414,041 129,754 63,722 104,253 167,975 Panel C: Balance sheet information Employment 92 95 76 73 72 1 (294) (295) (290) (170) (207) Investment to Assets 0.05 0.05 0.04 0.04 0.039 0.001** (0.06) (0.06) (0.06) (0.06) (0.06) Return to Assets 0.05 0.07 0.00 0.05 0.03 0.02*** (0.10) (0.08) (0.13) (0.07) (0.07) Leverage 0.67 0.61 0.86 0.79 0.85 \u20130.06*** (0.19) (0.18) (0.10) (0.10) (0.09) N 143,953 108,353 35,600 16,432 27,350 43,782 All Performing Substandard Score 6 Score 7 6\u20137 Comparison (1) (2) (3) (4) (5) (6) Panel A: Loan information Term Loans: Interest Rate 4.57 4.32 5.3 4.79 5.29 \u20130.48*** (1.62) (1.56) (1.6) (1.58) (1.59) Term Loans: Amount 816 885 617 451 569 \u2013118 (9850) (5156) (17300) (1623) (17700) Term Loans: Maturity 0.66 0.66 0.65 0.77 0.72 0.05*** (0.47) (0.47) (0.48) (0.44) (247) N 253,502 188,026 65,475 49,265 60,326 109,591 Panel B: Aggregate financing information All Bank Financing Granted 8,503 9,237 6,167 7,542 6,392 1,150*** (37,200) (40,600) (23,100) (24,600) (21,100) Share of Term Loans Granted 0.35 0.35 0.36 0.33 0.35 \u20130.02*** (0.25) (0.25) (0.25) (0.21) (0.25) Share of Write-downs 0.01 0.01 0.03 0.00 0.01 \u20130.01*** (0.09) (0.04) (0.17) (0.05) (0.09) N 543,855 414,041 129,754 63,722 104,253 167,975 Panel C: Balance sheet information Employment 92 95 76 73 72 1 (294) (295) (290) (170) (207) Investment to Assets 0.05 0.05 0.04 0.04 0.039 0.001** (0.06) (0.06) (0.06) (0.06) (0.06) Return to Assets 0.05 0.07 0.00 0.05 0.03 0.02*** (0.10) (0.08) (0.13) (0.07) (0.07) Leverage 0.67 0.61 0.86 0.79 0.85 \u20130.06*** (0.19) (0.18) (0.10) (0.10) (0.09) N 143,953 108,353 35,600 16,432 27,350 43,782 All panels use data for the period 2004.Q1\u20132011.Q4, and monetary values expressed in KE (1,000 euro). Standard deviations are reported in brackets. The last column reports the difference in means of each variable between categories 6 and 7. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Panel A uses pooled loan-level data with observations at the loan-quarter level. Interest Rate is the gross annual interest rate inclusive of participation fees, loan origination fees, and monthly service charges. Amount is the granted amount of the issued term loan. Maturity is a binary variable indicating whether the maturity of the newly issued loans is up to one year, or longer. Panel B uses credit register data with observations at the firm-quarter level. All Bank Financing Granted is the firms\u2019 total amount of bank financing granted summing across all categories (loans, credit lines, backed loans). Share of Term Loans Granted is the firms\u2019 total amount of term loans granted, divided by the total amount of bank financing granted for all categories. Share of Write-downs is a binary variable indicating whether the firms\u2019 total amount of bank financing granted for all categories has experienced write-downs by banks. Panel C uses balance sheet and cash-flow statements at the firm-year level. Employment is defined as the firms\u2019 average employment over the year. Investment to Assets is the firms\u2019 investment in material fixed assets over total fixed assets. Returns to Assets is defined as the firms\u2019 earnings before interest and taxes over total assets. Leverage is the firms\u2019 ratio of debt (both short and long term) over total assets. In all panels, $$N$$ corresponds to the pooled number of observations in our sample. The table shows that there is significant heterogeneity among firms with different risk profiles, not only with respect to financial characteristics, but also in terms of balance sheet characteristics. More specifically, panel A of Table 2 shows that in the full sample, the average nominal interest rate charged for a loan is 4.57%. However, the interest rates applied to performing and substandard firms are 4.32% and 5.3%, respectively. Although the average loan in the sample is approximately 816,000 euro, it is about 617,000 euro for a firm in the substandard class. Moreover, the loans in our sample are mostly short term, as these account for around two-thirds of the total value of granted loans. Panel B reports the aggregate financing characteristics of the firms in our sample. On average, total bank lending amounts to 8.5 million euro (ME) per firm, 35% of which is in the form of loans. While firms in the performing class receive bank financing that adds up to about 9.2ME, firms in the substandard class receive an average of 6ME. Panel C provides an overview of the main balance sheet characteristics of Italian manufacturing firms based on unique firm-year observations. Firms in our sample are relatively small. On average, they employ 92 workers, with firms in the performing class being relatively larger than those in the substandard class. While the investment-to-asset ratio is stable across classes, the values of leverage and return to assets are not. The leverage ratio increases from 0.61 for firms in the performing class to 0.86 for those in the substandard class. Moreover, return on assets decreases from 0.07 to 0 for firms in these two classes. Finally, Column (6) of panel C shows that the heterogeneity in firm characteristics extends to rating categories 6 and 7. The cost and availability of bank financing suggests significantly tighter conditions for firms in category 7 as opposed to category 6. For instance, interest rates for firms in category 6 are 50 points lower than those of firms in category 7. At the same time, these firms are significantly different in terms of characteristics related to the demand for credit, such as the value of investment and profitability. Taken together, the descriptive statistics show the importance of obtaining a measure of lending standards that is not biased by demand heterogeneity. 3.1.2 Time-series descriptive statistics In Figure 2, we document the variation in financial contracts across time. Figure 2. View largeDownload slide Descriptive statistics across time In the left panel, we plot the per-firm aggregate value of bank financing for different rating categories across time. In the right panel, we plot the nominal average interest rates applied to firms in different rating categories across time. Figure 2. View largeDownload slide Descriptive statistics across time In the left panel, we plot the per-firm aggregate value of bank financing for different rating categories across time. In the right panel, we plot the nominal average interest rates applied to firms in different rating categories across time. The left panel illustrates that, like other OECD economies (Drehmann, Borio, and Tsatsaronis 2012), between 2004 and 2011 Italy was experiencing a credit cycle that reached its peak in 2007. The right panel focuses on firms\u2019 nominal average interest rates, showing that nominal rates mirrored the pattern of the indicators for the monetary policy of the European Central Bank. More specifically, the left panel shows that the time series of the amount of bank financing to Italian SME features a humped shape. From the first quarter of 2004 to the fourth quarter of 2007, bank financing increased by 18%, on average. It then decreased by 11% through the end of the sample period. Although this pattern is qualitatively similar across risk classes, the variation in bank financing is larger for substandard firms: between 2004 and 2008 bank financing to performing firms increased by 13%, while it rose by 29% for substandard firms. This evidence is consistent with the historical account of credit booms by Greenwood and Hanson (2013), who show that the quality of credit deteriorates as aggregate credit increases. Finally, the right panel of Figure 2 shows that nominal interest rates increased from 4.3% in 2004 to 6.11% in late 2008. Similar to the patterns in the left panel, the levels of the interest rate spreads are consistent with the risk categories in our rating system. 3.2 Banking environment In Figure 3, we illustrate the key developments in the Italian banking environment that occurred during our sample period. We use bank balance sheet data between 2006 and 2011 from Bank of Italy. Figure 3. View largeDownload slide Bank capital and credit risk In the top panel, we plot the ratio of the volume of repo financing over total assets for the five largest banks in our data set. In the middle panel, we plot the tier 1 capital ratio for the five largest banks in our data set across time. In the bottom panel, we use data from the European Central Bank statistical data warehouse to plot the credit risk capital allocations over total capital requirements (black line), the fraction of capital allocations computed using the standardized approach (gray line), and the fraction computed using the internal rating-based (IRB) approach (dashed black line). Figure 3. View largeDownload slide Bank capital and credit risk In the top panel, we plot the ratio of the volume of repo financing over total assets for the five largest banks in our data set. In the middle panel, we plot the tier 1 capital ratio for the five largest banks in our data set across time. In the bottom panel, we use data from the European Central Bank statistical data warehouse to plot the credit risk capital allocations over total capital requirements (black line), the fraction of capital allocations computed using the standardized approach (gray line), and the fraction computed using the internal rating-based (IRB) approach (dashed black line). The top panel of Figure 3 plots the share of repo financing of banks relative to their total assets for the five largest banks in our sample. In the expansionary phase of the cycle, the dependence of banks on repo financing grew from 5% in 2005 to nearly 12% at the beginning of 2008. During the financial crisis, this source of financing plummeted to 2.5% and remained at low levels until the end of our sample period. The middle panel of Figure 3 illustrates the capitalization of Italian banks: we compute the tier 1 capital ratio for the five largest banks in our sample by dividing banks\u2019 tier 1 capital by their total assets. The figure shows that the average value of banks\u2019 capital ratio at the beginning of the financial crisis period was approximately 4.5%. In 2008 the ratio fell to around 3.6%, before rising above 5% toward the end of the sample period. The patterns in these two panels are shared by the banking systems of other European countries during the same time interval. The bottom panel of Figure 3 provides evidence on the implementation of the Basel II agreements. Credit risk capital allocations account for more than 100% of total capital requirements through 2008 and 2010, implying that credit risk management was critical for Italian banks during our sample period. Moreover, the transition from Basel I to Basel II is unlikely to drive the evolution of lending standards in our sample. Indeed, the total fraction of capital allocations calculated using internal rating systems oscillates around 20%. 4. The Empirical Model 4.1 Identification strategy Empirically identifying how segmentation influences bank lending standards is challenging for two reasons. First, it requires a setup where the econometrician observes the exact information held by the bank about the firm credit risk profile. Then, to isolate demand from supply considerations, the econometrician would like to compare firms that are identical from the perspective of the loan officer, but classified into different classes of credit risk. To address these challenges, we exploit the institutional features of the Italian credit market for SME introduced in Section 1. The structure of the rating, and its construction, give us the opportunity to follow an intuitive empirical strategy to identify the effects of segmentation on financial contracts. Specifically, we exploit the sharp discontinuity in the allocation of firms into the performing and substandard classes of credit risk. As our measure of lending standards, then, we take the differences in the credit conditions between a firm marginally classified as performing and one that is marginally classified as substandard based on the value of the rating\u2019s continuous variable. These threshold differences inform us about how banks\u2019 supply of credit is affected by segmentation, while holding constant the demand for credit. The support of the continuous variable for categories 6 and 7 ranges between \u20130.6 and 1.5, and the threshold is 0.15. Below this threshold, a firm\u2019s Score is 7 and thus the firm falls into the substandard class. Above the threshold, a firm\u2019s Score is 6 and it is in the performing class. Throughout the analysis, we normalize the threshold to 0 and only use the support of the continuous variable that spans between categories 6 and 7. Thus, if $$s_{i}$$ is the value of firm $$i$$\u2019s continuous variable, the allocation of this firm into a rating class takes place according to the following sharp mechanism: \\begin{eqnarray} {\\textit{Score}}_{i,t} = \\left\\{ \\begin{array}{@{}llll} 6\\ \\text{(Performing)} & \\quad \\text{If $0 \\leq s_{i,t} <1.35$} \\\\\\ \\\\ 7\\ \\text{(Substandard)} & \\quad \\text{If $-0.75 \\leq s_{i,t}<0$} \\end{array} \\right. . \\end{eqnarray} (2) 4.2 Main specification Let $$\\bar{s}$$ denote the normalized threshold that allocates firms into rating categories 6 and 7. Our main specification follows: \\begin{align} y_{i,t} &= \\beta_{0} +\\beta_{1}(\\mbox{Performing}_{i,t}\\times\\mbox{Boom}_{t})+\\beta_{2} (\\mbox{Performing}_{i,t}\\times\\mbox{Crisis}_{t})\\nonumber\\\\ & \\quad +\\beta_{3} (\\mbox{Performing}_{i,t}\\times\\mbox{Recovery}_{t})+f_{t}(s_{i,t}-\\bar{s})\\nonumber\\\\ & \\quad +\\mbox{Performing}_{i,t}\\times g_{t}(s_{i,t}-\\bar{s}) + \\pi_{t} + u_{i,t}. \\end{align} (3) The dependent variable capturing the supply of bank financing is the (log) total value of bank lending granted to firm $$i$$ in quarter $$t$$. This measure accounts for the possibility that firms obtain credit from multiple banks. The variable capturing the cost of bank financing is the (log) value of the interest rate applied to a new loan granted to firm $$i$$ in quarter $$t$$. The indicator Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 between the first quarter of 2004 until the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 between the first quarter of 2008 until the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 between the first quarter of 2010 until the last quarter of 2011, and 0 otherwise. Functions $$f_{t}(\\cdot)$$ and $$g_t(\\cdot)$$ correspond to flexible sixth-order polynomials whose goal is to fit the smoothed curves on either side of the cutoff as closely to the data as possible. Function $$f_{t}(\\cdot)$$ is estimated from $$0$$ to the left, whereas the Performing$$_{i,t}\\times g_{t}(\\cdot)$$ term is estimated from $$0$$ to the right. The subscript $$t$$ for $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ indicates that the polynomials are separately estimated for each time period through interactions with $$\\pi_{t}$$, the quarter-year fixed effects. $$u_{i,t}$$ is a mean-zero error term clustered at the firm level.14 As a robustness check, we will estimate a version of the specification that also includes the past value of the rating, and its interaction with each time period. The interpretation of Equation (1) is the following. First, note that, at the cutoff, the $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ polynomials are evaluated at 0 and drop out of the calculation. This allows us to interpret the parameters $$(\\beta_{1}, \\beta_{2}, \\beta_{3})$$ as capturing the magnitude of the discontinuity in credit conditions at the threshold $$\\bar s$$. The null hypothesis of our framework is that if a bank uses all its information on the borrowing firm, there should be no discontinuity in lending contracts at the threshold. In other words, under our null hypothesis, segmentation should not matter for lending decisions. Second, the estimated discontinuity parameters $$(\\beta_{1}, \\beta_{2}, \\beta_{3})$$ have an intuitive interpretation. The estimate of $$\\beta_{1}$$ measures differences in credit allocations between marginally performing firms and substandard firms in the period between 2004 and 2007. The estimate of $$\\beta_{2}$$ measures differences in credit allocations in the period between 2008 and 2009. Finally, the estimate of $$\\beta_{3}$$ captures the difference between marginally performing firms and substandard firms in the period between 2010 and 2011. In the main specification, we restrict our attention to the sample of firms that remain in the same rating category for at least two consecutive years. This condition limits two potential concerns. The first is that the bank reports to investors a firm as performing on the basis of its rating in $$t-1$$, even though it is already downgraded in $$t$$. The second is related to the possibility that large variations in the value of the continuous rating that then lead to downgrades might themselves be correlated to the firms\u2019 demand for credit. We then separately study the source of variation coming from a firm downgrade for financial contracting, and provide evidence based on downgrades caused by small changes in the value of the continuous rating. We extend our main specification in two directions. First, we study whether, via its impact on lending standards, segmentation is relevant for firms\u2019 real choices. Specifically, we estimate Equation (3) using as dependent variables firms\u2019 expenditures in production inputs and the value of production. The balance sheet information we use for this analysis is reported in end-of-the-year statements; thus, it reflects a firm\u2019s lending conditions throughout the year. Second, we look at the differences between the lending conditions at the threshold within each phase of the credit cycle. To this end, we estimate Equation (4) separately for each quarter-year cross-section of firms at the threshold in our sample period: $$y_{i,.}=\\beta_{0}+\\beta_{1} \\mbox{Performing}_{i,.}+f(s_{i,.} -\\bar{s})+\\mbox{Performing}_{i,.}\\times g(s_{i,.}-\\bar{s}) + u_{i,.}.$$ (4) In Equation (4), the dot indicates that we fix the time period. This exercise is meant to understand whether there are distinct credit dynamics within each of the subperiods of the credit cycle. 4.3 Mechanism for the transmission of market segmentation In this section, we first exploit the heterogeneity of the banks in our data set to study how banks\u2019 financial structure affects the sensitivity of lending to market segmentation. Then, we analyze the implications of segmentation for marginally downgraded firms over the cycle. 4.3.1 Banks\u2019 financial structure We consider two channels through which financial structure can affect banks\u2019 sensitivity to market segmentation: capital requirements and investor composition. Intuitively, low levels of regulatory capital can help explain a bank\u2019s greater sensitivity to market segmentation. Similarly, investor composition can account for the sensitivity of banks to market segmentation: certain investor categories are more responsive than others to bank solvency risk, and update their assessment of bank loan quality over the cycle (Ivashina and Scharfstein 2009; Iyer, Puri, and Ryan 2016). To explore the relative merits of these two channels in determining bank sensitivity to segmentation, we compute the following measures of bank heterogeneity. To study the role played by capital requirements, we compute, for the pre-crisis period, each bank\u2019s tier 1 capital ratio. To study heterogeneity in investor composition, we focus on the importance of repo markets for a bank funding structure. As we show in Table 1, during the crisis, investors in repo markets updated their interest rate conditions based on banks\u2019 exposure to substandard firms. We therefore measure each banks\u2019 pre-crisis share of financing from repo markets. We augment our main specification with interactions between the Performing$$_{it}$$ indicator and these bank-specific characteristics: \\begin{align} y_{i,b,t} & = \\beta_{0} +\\beta_{1}(\\mbox{Performing}_{i,t}\\times\\mbox{Boom}_{t})+\\beta_{2}(\\mbox{Performing}_{i,t}\\times\\mbox{Crisis}_{t})\\nonumber\\\\ & \\quad +\\beta_{3}(\\mbox{Performing}_{i,t}\\times\\mbox{Recovery}_{t})\\nonumber\\\\ & \\quad +\\gamma_{1}(\\mbox{Performing}_{i,t}\\times\\mbox{Tier1}_{b}\\times\\mbox{Boom}_{t})\\nonumber\\\\ & \\quad +\\gamma_{2} (\\mbox{Performing}_{i,t}\\times\\mbox{Tier1}_{b}\\times\\mbox{Crisis}_{t})\\nonumber\\\\ & \\quad +\\gamma_{3} (\\mbox{Performing}_{i,t}\\times\\mbox{Tier1}_{b}\\times \\mbox{Recovery}_{t})\\nonumber\\\\ & \\quad +\\delta_{1} (\\mbox{Performing}_{i,t}\\times\\mbox{Repo}_{b}\\times\\mbox{Boom}_{t})\\nonumber\\\\ & \\quad +\\delta_{2} (\\mbox{Performing}_{i,t}\\times\\mbox{Repo}_{b}\\times\\mbox{Crisis}_{t})\\nonumber\\\\ & \\quad +\\delta_{3} (\\mbox{Performing}_{i,t}\\times\\mbox{Repo}_{b}\\times\\mbox{Recovery}_{t})\\nonumber\\\\ & \\quad +f_{t}(s_{i,t}-\\bar{s})+\\mbox{Performing}_{i,t}\\times g_{t}(s_{i,t}-\\bar{s}) + X_{i,b,t}\\Psi + \\pi_{t} + u_{i,t}. \\end{align} (5) In Equation (5), Tier1$$_b$$ is defined as a bank $$b$$\u2019s core equity capital divided by its total assets, and Repo$$_{b}$$ is defined as the share of the bank\u2019s total financing from repo markets. $$X_{i,b,t}$$ is a vector that includes the levels and interactions of all the variables in the set of triple interactions. Standard errors are clustered at the firm-bank level. As an additional robustness check, we augment Equation (5) by including firm-year fixed effects. 4.3.2 Analysis of downgraded firms Finally, we study how market segmentation affects the lending policies set on firms that are marginally downgraded over the cycle. More specifically, we ask what is the implication of a downgrade to substandard quality for credit conditions over the cycle, and whether the bank exploits its superior information on the company\u2019s downgrade. We compare two firms that fall in the performing class until year $$t-1$$, but differ in their rating class in year $$t$$.15 The specification follows: \\begin{align} y_{i,b,t} & = \\beta_{0} + \\beta_{1} (\\mbox{Down}_{i,t}\\times\\mbox{Boom}_{t})+\\beta_{2} (\\mbox{Down}_{i,t}\\times\\mbox{Crisis}_{t})\\nonumber\\\\ &\\quad +\\beta_{3} (\\mbox{Down}_{i,t}\\times\\mbox{Recovery}_{t})\\nonumber\\\\ &\\quad +f_{t}(s_{i,t}-s_{i,t-1})+\\mbox{Down}_{i,t}\\times g_{t}(s_{i,t}-s_{i,t-1})\\nonumber\\\\ &\\quad +m_{t}(s_{i,t-1})+\\mbox{Down}_{i,t}\\times n_{t}(s_{i,t-1}) + \\pi_{t} + u_{i,t}. \\end{align} (6) Down$$_{i,t}$$ is a binary variable equal to 1 if the firm is downgraded from category 6 to category 7 in year $$t$$, and is 0 otherwise. In Equation (5), the polynomials in $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ are a function of the change in the continuous variable between $$t-1$$ and $$t$$. By evaluating these polynomials close to 0, our analysis considers those firms that were downgraded as a consequence of a similar and small change in the value of the continuous rating. To make sure that we implement a local identification of downgraded and non-downgraded firms around the threshold, we augment the specification by including also polynomials for the continuous assignment variable in $$t-1$$, $$m_{t}(\\cdot)$$ and $$n_{t}(\\cdot)$$. Consequently, we compare firms that not only experienced a similar small change in the continuous variable, but were also close to the threshold.16 5. Results In this section, we present the results on the differences in credit conditions\u2014specifically, differences in the interest rates and in the total amount of bank financing\u2014for firms at the threshold between the performing and the substandard classes. We then decompose the changes in lending standards within each phase by estimating differences in credit allocations separately for each quarter. Finally, we explore whether differences in credit conditions give rise to differences in firms\u2019 production and input choices. 5.1 Results on credit allocations Table 3 reports the estimates related to credit allocations. The dependent variable in Columns (1) to (3) is the log amount of total bank financing granted to the firm, while in Columns (4) to (6) the dependent variable is the log interest rate on new bank loans. In Columns (1) and (4) we report the estimates of the main specification in Equation (3), while Columns (2) and (5) augment this specification by interacting past ratings with quarter-year fixed effects. Finally, Columns (3) and (6) report the results of a na\u00efve specification comparing lending conditions to all performing and substandard firms. Table 3 Credit effects Quantity Price Dependent variable (1) (2) (3) (4) (5) (6) Boom $$\\times$$ Performing 0.11 0.11 0.35*** \u20130.04** \u20130.04** \u20130.15*** (0.09) (0.09) (0.02) (0.02) (0.02) (0.01) Crisis $$\\times$$ Performing 0.33*** 0.33*** 0.32*** 0.03 0.03 \u20130.15*** (0.11) (0.11) (0.02) (0.03) (0.03) (0.01) Recovery $$\\times$$ Performing 0.20* 0.20* 0.30*** \u20130.08** \u20130.08** \u20130.27*** (0.12) (0.11) (0.02) (0.04) (0.04) (0.01) Lagged Rating \u20130.01*** 0.00*** 0.00*** 0.00*** (0.00) (0.00) (0.00) (0.00) Polynomial Yes Yes No Yes Yes No Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes $$R^2$$ 0.06 0.06 0.01 0.38 0.38 0.43 N 157,775 157,775 518,047 105,865 105,865 246,240 Quantity Price Dependent variable (1) (2) (3) (4) (5) (6) Boom $$\\times$$ Performing 0.11 0.11 0.35*** \u20130.04** \u20130.04** \u20130.15*** (0.09) (0.09) (0.02) (0.02) (0.02) (0.01) Crisis $$\\times$$ Performing 0.33*** 0.33*** 0.32*** 0.03 0.03 \u20130.15*** (0.11) (0.11) (0.02) (0.03) (0.03) (0.01) Recovery $$\\times$$ Performing 0.20* 0.20* 0.30*** \u20130.08** \u20130.08** \u20130.27*** (0.12) (0.11) (0.02) (0.04) (0.04) (0.01) Lagged Rating \u20130.01*** 0.00*** 0.00*** 0.00*** (0.00) (0.00) (0.00) (0.00) Polynomial Yes Yes No Yes Yes No Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes $$R^2$$ 0.06 0.06 0.01 0.38 0.38 0.43 N 157,775 157,775 518,047 105,865 105,865 246,240 In Columns (1), (2), (4), and (5) the table reports OLS estimates of the threshold specification in Equation (3). Instead, Columns (3) and (6) estimate a simple mean difference specification using data for all firms in the rating system. The dependent variable in the first three columns is the (log) total value of bank lending granted to firm $$i$$ in quarter $$t$$. The dependent variable in the last three columns is the (log) value of the interest rate applied to a new loan granted to firm $$i$$ in quarter $$t$$. The indicator Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 from the first quarter of 2004 to the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 from the first quarter of 2008 to the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 from the first quarter of 2010 to the last quarter of 2011, and 0 otherwise. Functions $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ correspond to flexible sixth-order polynomials estimated separately for each quarter-year. Function $$f_{t}(\\cdot)$$ is estimated from $$0$$ to the left, whereas the Performing$$_{i,t}\\times g_{t}(\\cdot)$$ term is estimated from $$0$$ to the right. Finally, Lagged Rating controls for the past value of the rating. In Columns (2) and (5) the variable is interacted with quarter-year fixed effects. Standard errors, clustered at the firm level, are reported in brackets. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Table 3 Credit effects Quantity Price Dependent variable (1) (2) (3) (4) (5) (6) Boom $$\\times$$ Performing 0.11 0.11 0.35*** \u20130.04** \u20130.04** \u20130.15*** (0.09) (0.09) (0.02) (0.02) (0.02) (0.01) Crisis $$\\times$$ Performing 0.33*** 0.33*** 0.32*** 0.03 0.03 \u20130.15*** (0.11) (0.11) (0.02) (0.03) (0.03) (0.01) Recovery $$\\times$$ Performing 0.20* 0.20* 0.30*** \u20130.08** \u20130.08** \u20130.27*** (0.12) (0.11) (0.02) (0.04) (0.04) (0.01) Lagged Rating \u20130.01*** 0.00*** 0.00*** 0.00*** (0.00) (0.00) (0.00) (0.00) Polynomial Yes Yes No Yes Yes No Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes $$R^2$$ 0.06 0.06 0.01 0.38 0.38 0.43 N 157,775 157,775 518,047 105,865 105,865 246,240 Quantity Price Dependent variable (1) (2) (3) (4) (5) (6) Boom $$\\times$$ Performing 0.11 0.11 0.35*** \u20130.04** \u20130.04** \u20130.15*** (0.09) (0.09) (0.02) (0.02) (0.02) (0.01) Crisis $$\\times$$ Performing 0.33*** 0.33*** 0.32*** 0.03 0.03 \u20130.15*** (0.11) (0.11) (0.02) (0.03) (0.03) (0.01) Recovery $$\\times$$ Performing 0.20* 0.20* 0.30*** \u20130.08** \u20130.08** \u20130.27*** (0.12) (0.11) (0.02) (0.04) (0.04) (0.01) Lagged Rating \u20130.01*** 0.00*** 0.00*** 0.00*** (0.00) (0.00) (0.00) (0.00) Polynomial Yes Yes No Yes Yes No Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes $$R^2$$ 0.06 0.06 0.01 0.38 0.38 0.43 N 157,775 157,775 518,047 105,865 105,865 246,240 In Columns (1), (2), (4), and (5) the table reports OLS estimates of the threshold specification in Equation (3). Instead, Columns (3) and (6) estimate a simple mean difference specification using data for all firms in the rating system. The dependent variable in the first three columns is the (log) total value of bank lending granted to firm $$i$$ in quarter $$t$$. The dependent variable in the last three columns is the (log) value of the interest rate applied to a new loan granted to firm $$i$$ in quarter $$t$$. The indicator Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 from the first quarter of 2004 to the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 from the first quarter of 2008 to the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 from the first quarter of 2010 to the last quarter of 2011, and 0 otherwise. Functions $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ correspond to flexible sixth-order polynomials estimated separately for each quarter-year. Function $$f_{t}(\\cdot)$$ is estimated from $$0$$ to the left, whereas the Performing$$_{i,t}\\times g_{t}(\\cdot)$$ term is estimated from $$0$$ to the right. Finally, Lagged Rating controls for the past value of the rating. In Columns (2) and (5) the variable is interacted with quarter-year fixed effects. Standard errors, clustered at the firm level, are reported in brackets. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. The estimates related to the period between 2004 and 2007 in Columns (1) and (4) suggest that segmentation mainly results in a positive interest rate spread between substandard and performing firms at the threshold. Firms in the substandard class are charged up to 4%,17 or 20 basis points, higher interest rates on new bank loans than similar firms in the performing class. The difference in the total amount of lending granted to these firms, instead, is positive (11%) but not statistically significant. The size of this coefficient reflects the large within-period dynamics occurring in the boom period, as we discuss later. Through 2008 and 2009, the financial crisis that hit the Italian banking sector led to an exacerbation of the consequences of segmentation for lending policies. Importantly, we find that tighter lending standards essentially translate into differences in the quantity of lending for the firms at the threshold. Indeed, marginally performing firms obtain 39% more bank financing than similar firms across the threshold. Instead, interest rate differences remain stable and close to zero (in economic and statistical terms). These results are consistent with the prediction of our theoretical framework. A rise in the interest rates paid by banks to outside investors, together with the increase in the opportunity cost of lending to substandard firms, translate into an equilibrium in which banks monitor the performing firms and (partly) exclude substandard firms from lending. Between 2010 and 2011, our estimates are in line with an incomplete recovery of bank lending. During this period, segmentation means a reduction in the differences in the quantity of credit from 39% to 22%. However, this reduction is accompanied by an increase in the interest rate spread to approximately 8%, or 40 basis points. To better understand the dynamics of credit within each phase, we report, in Figure 4, the quarterly estimates obtained with the specification in Equation (4). Figure 4. View largeDownload slide Discontinuity estimates for quantity and price The figure plots the estimates and 90% confidence intervals of the threshold specification in Equation (4), run on each distinct quarter in our sample period (2004.Q1\u20132011.Q4). The dependent variable in the top panel is the (log) total value of bank lending granted to firm $$i$$ in quarter $$t$$ (top panel). The dependent variable in the bottom panel is the (log) value of the interest rate applied to a new loan granted to firm $$i$$ in quarter $$t$$ (bottom panel). The plotted discontinuity estimates refer to Performing$$_{i,t}$$, an indicator variable that takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. Figure 4. View largeDownload slide Discontinuity estimates for quantity and price The figure plots the estimates and 90% confidence intervals of the threshold specification in Equation (4), run on each distinct quarter in our sample period (2004.Q1\u20132011.Q4). The dependent variable in the top panel is the (log) total value of bank lending granted to firm $$i$$ in quarter $$t$$ (top panel). The dependent variable in the bottom panel is the (log) value of the interest rate applied to a new loan granted to firm $$i$$ in quarter $$t$$ (bottom panel). The plotted discontinuity estimates refer to Performing$$_{i,t}$$, an indicator variable that takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. The top panel of the figure shows that early in the boom period (2004 and 2005), differences in the total amount of lending to firms at the threshold are positive and large, but progressively vanish in 2006 and 2007. These patterns then are likely to explain the economically large differences in total lending for the boom period in Column (1) of Table 3. Similarly to total lending, the interest rate spreads between firms at the threshold (bottom panel) narrow throughout the boom phase and disappear at the peak of the cycle. Differences in credit allocations are relatively stable within the crisis period, but vary again during the recovery period. Specifically, the estimates for 2010 and 2011 imply a gradual decrease in the differences in the amount of bank lending. To quantify the impact of segmentation on bank lending, we contrast the results obtained with our threshold analysis to those arising from a na\u00efve specification that compares lending conditions to all performing and substandard firms (Columns (3) and (6)). First, segmentation is relatively more important to explain the na\u00efve specification\u2019s differences in total lending in the bust than in the boom period. In the boom period, the na\u00efve estimates imply a 42% differential in the amount of bank credit. During that period, the threshold differences amount to only 12%, suggesting that segmentation alone cannot explain the large estimate in the na\u00efve specification. In 2008\u20132009, the overall differential between the quantity of lending across rating classes remains stable, while Column (1) indicates a 39% differential for the firms at the threshold. In the bust period, then, segmentation can account for a larger part of the observed differential in the amount of credit than in the boom period. Second, the analysis of the interest rate spreads arising from a na\u00efve comparison would lead to misleading conclusions, not only quantitatively but also qualitatively. Indeed, the results of the na\u00efve regression suggest that the interest rate differences are persistently large in economic terms, and increasing throughout the cycle. Instead, we show that, within our discontinuity design, the interest rate spread narrows over the boom phase, and disappears during the crisis. This reflects the fact that, in the bust period, bank lending standards\u2019 adjustments are implemented primarily by changing the quantity of credit. 5.2 Implications for firms\u2019 real activity Table 4 reports the results of our baseline regression in Equation (3) using as dependent variables the log of firms\u2019 sales and expenditures in investment, employment, and intermediates. The balance sheet reports contain only partial information about employment choices; thus, to fill this data gap, we obtain employment figures from firms\u2019 mandatory contributions to the Italian pension system, and merge this information based on the firms\u2019 fiscal identifier. Table 4 Real effects Production Investment Intermediates Employment Dependent variable (1) (2) (3) (4) (5) (6) (7) (8) Boom $$\\times$$ Performing 0.16* 0.16* 0.14 0.14 0.11 0.11 0.14* 0.14* (0.09) (0.09) (0.15) (0.15) (0.09) (0.09) (0.08) (0.08) Crisis $$\\times$$ Performing 0.44*** 0.44*** 0.55** 0.55** 0.43*** 0.43*** 0.39*** 0.38*** (0.12) (0.12) (0.23) (0.23) (0.13) (0.13) (0.16) (0.11) Recovery $$\\times$$ Performing 0.31** 0.31** 0.12 0.12 0.28* 0.28* 0.28** 0.28** (0.14) (0.14) (0.23) (0.23) (0.15) (0.15) (0.15) (0.14) Lagged Rating \u20130.01*** \u20130.00* \u20130.01*** \u20130.01*** (0.00) (0.00) (0.00) (0.00) Polynomial Yes Yes Yes Yes Yes Yes Yes Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes Yes Yes $$R^2$$ 0.09 0.09 0.07 0.07 0.09 0.09 0.02 0.02 N 41,157 41,157 33,889 33,889 40,585 40,585 39,041 39,041 Production Investment Intermediates Employment Dependent variable (1) (2) (3) (4) (5) (6) (7) (8) Boom $$\\times$$ Performing 0.16* 0.16* 0.14 0.14 0.11 0.11 0.14* 0.14* (0.09) (0.09) (0.15) (0.15) (0.09) (0.09) (0.08) (0.08) Crisis $$\\times$$ Performing 0.44*** 0.44*** 0.55** 0.55** 0.43*** 0.43*** 0.39*** 0.38*** (0.12) (0.12) (0.23) (0.23) (0.13) (0.13) (0.16) (0.11) Recovery $$\\times$$ Performing 0.31** 0.31** 0.12 0.12 0.28* 0.28* 0.28** 0.28** (0.14) (0.14) (0.23) (0.23) (0.15) (0.15) (0.15) (0.14) Lagged Rating \u20130.01*** \u20130.00* \u20130.01*** \u20130.01*** (0.00) (0.00) (0.00) (0.00) Polynomial Yes Yes Yes Yes Yes Yes Yes Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes Yes Yes $$R^2$$ 0.09 0.09 0.07 0.07 0.09 0.09 0.02 0.02 N 41,157 41,157 33,889 33,889 40,585 40,585 39,041 39,041 The table reports the estimates of the threshold specification in Model (3) using as dependent variables the (log) sales, investment, employment, and intermediates of firm $$i$$ in year $$t$$. The indicator Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 from the first quarter of 2004 to the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 from the first quarter of 2008 to the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 from the first quarter of 2010 to the last quarter of 2011, and 0 otherwise. Functions $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ correspond to flexible sixth-order polynomials estimated separately for each year. Function $$f_{t}(\\cdot)$$ is estimated from $$0$$ to the left, whereas the Performing$$_{i,t}\\times g_{t}(\\cdot)$$ term is estimated from $$0$$ to the right. Finally, Lagged Rating controls for the past value of the rating. In Columns (2), (4), (6), and (8) the variable is interacted with quarter-year fixed effects. Standard errors, clustered at the firm level, are reported in brackets. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Table 4 Real effects Production Investment Intermediates Employment Dependent variable (1) (2) (3) (4) (5) (6) (7) (8) Boom $$\\times$$ Performing 0.16* 0.16* 0.14 0.14 0.11 0.11 0.14* 0.14* (0.09) (0.09) (0.15) (0.15) (0.09) (0.09) (0.08) (0.08) Crisis $$\\times$$ Performing 0.44*** 0.44*** 0.55** 0.55** 0.43*** 0.43*** 0.39*** 0.38*** (0.12) (0.12) (0.23) (0.23) (0.13) (0.13) (0.16) (0.11) Recovery $$\\times$$ Performing 0.31** 0.31** 0.12 0.12 0.28* 0.28* 0.28** 0.28** (0.14) (0.14) (0.23) (0.23) (0.15) (0.15) (0.15) (0.14) Lagged Rating \u20130.01*** \u20130.00* \u20130.01*** \u20130.01*** (0.00) (0.00) (0.00) (0.00) Polynomial Yes Yes Yes Yes Yes Yes Yes Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes Yes Yes $$R^2$$ 0.09 0.09 0.07 0.07 0.09 0.09 0.02 0.02 N 41,157 41,157 33,889 33,889 40,585 40,585 39,041 39,041 Production Investment Intermediates Employment Dependent variable (1) (2) (3) (4) (5) (6) (7) (8) Boom $$\\times$$ Performing 0.16* 0.16* 0.14 0.14 0.11 0.11 0.14* 0.14* (0.09) (0.09) (0.15) (0.15) (0.09) (0.09) (0.08) (0.08) Crisis $$\\times$$ Performing 0.44*** 0.44*** 0.55** 0.55** 0.43*** 0.43*** 0.39*** 0.38*** (0.12) (0.12) (0.23) (0.23) (0.13) (0.13) (0.16) (0.11) Recovery $$\\times$$ Performing 0.31** 0.31** 0.12 0.12 0.28* 0.28* 0.28** 0.28** (0.14) (0.14) (0.23) (0.23) (0.15) (0.15) (0.15) (0.14) Lagged Rating \u20130.01*** \u20130.00* \u20130.01*** \u20130.01*** (0.00) (0.00) (0.00) (0.00) Polynomial Yes Yes Yes Yes Yes Yes Yes Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes Yes Yes $$R^2$$ 0.09 0.09 0.07 0.07 0.09 0.09 0.02 0.02 N 41,157 41,157 33,889 33,889 40,585 40,585 39,041 39,041 The table reports the estimates of the threshold specification in Model (3) using as dependent variables the (log) sales, investment, employment, and intermediates of firm $$i$$ in year $$t$$. The indicator Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 from the first quarter of 2004 to the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 from the first quarter of 2008 to the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 from the first quarter of 2010 to the last quarter of 2011, and 0 otherwise. Functions $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ correspond to flexible sixth-order polynomials estimated separately for each year. Function $$f_{t}(\\cdot)$$ is estimated from $$0$$ to the left, whereas the Performing$$_{i,t}\\times g_{t}(\\cdot)$$ term is estimated from $$0$$ to the right. Finally, Lagged Rating controls for the past value of the rating. In Columns (2), (4), (6), and (8) the variable is interacted with quarter-year fixed effects. Standard errors, clustered at the firm level, are reported in brackets. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Columns (1) and (2) yield three main findings. First, between 2004 and 2007, marginally performing firms on average produce 17% more than marginally substandard firms. A yearly decomposition of these estimates, which is reported in the Online Appendix, suggests that, consistent with the economically large differences in credit allocations arising in 2004 and 2005 (Figure 4), production differences are mainly concentrated in the early years of the boom and vanish in 2007. Our second finding is that production choices of firms at the threshold diverge significantly during the period in which access to credit is limited for the marginally substandard firms: in 2008 and 2009, the marginally performing firms report about 50% larger values of production than the marginally substandard ones. Finally, consistent with the partial recovery of lending taking place between 2010 and 2011, we find that, in this period, production differences gradually decrease but remain larger than the pre-crisis ones. To further the analysis of the implications of shifts in lending standards for firm real activity, we report the differences in input choices made by the firms at the threshold. We estimate our discontinuity design using as dependent variables the value of firms\u2019 investment in capital, expenditures in intermediates, and employment. The main finding is that the divergence in production outcomes during the crisis is driven mainly by investment choices. During the most acute phase of the financial crisis, on average, performing firms invest about 70% more than substandard firms. In recovery, instead, lower values of production are essentially driven by reduced expenditures in intermediate and labor inputs. 6. The Economic Mechanism In this section, we investigate the economic mechanism driving the transmission of segmentation onto bank lending standards. 6.1 Bank heterogeneity Table 5 investigates the possible channels through which bank heterogeneity can explain how segmentation affects credit supply. In Columns (1) and (2), we jointly test for the relative importance of bank capitalization and investor composition in determining the sensitivity of bank lending to segmentation. Recall that, to proxy bank capitalization, we measure banks\u2019 tier 1 capital ratio. Instead, as a measure of investor composition, we take the banks\u2019 dependence on fundings from repo markets. Both measures are taken as a pre-2008 average at the bank level. Table 5 Bank heterogeneity Quantity Price Dependent variable (1) (2) (3) (4) Boom $$\\times$$ Performing 0.08 \u20130.05* (0.07) (0.03) Crisis $$\\times$$ Performing 0.26** \u20130.02 (0.09) (0.04) Recovery $$\\times$$ Performing 0.08 \u20130.12* (0.11) (0.07) Boom $$\\times$$ Performing $$\\times$$ Tier1 0.16 \u20130.07 0.09 0.11 (0.29) (0.23) (0.16) (0.14) Crisis $$\\times$$ Performing $$\\times$$ Tier1 \u20130.90** \u20130.83*** \u20130.00 0.29 (0.43) (0.33) (0.27) (0.27) Recovery$$\\times$$Performing$$\\times$$Tier1 \u20130.71 \u20130.96** 0.15 0.33 (0.55) (0.41) (.39) (.41) Boom $$\\times$$ Performing $$\\times$$ Repo 0.02 \u20130.02 \u20130.21** 0.06 (0.15) (0.11) (0.09) (0.08) Crisis $$\\times$$ Performing $$\\times$$ Repo 0.39* 0.31* 0.25* 0.16 (0.23) (0.19) (0.13) (0.16) Recovery $$\\times$$ Performing $$\\times$$ Repo 0.58* 0.36 \u20130.03 0.04 (0.35) (0.28) (0.25) (0.30) Polynomial Yes Yes No Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Firm $$\\times$$ Year Fixed Effects No Yes No Yes $$R^2$$ 0.02 0.55 0.37 0.77 N 787,634 787,634 89,140 89,140 Quantity Price Dependent variable (1) (2) (3) (4) Boom $$\\times$$ Performing 0.08 \u20130.05* (0.07) (0.03) Crisis $$\\times$$ Performing 0.26** \u20130.02 (0.09) (0.04) Recovery $$\\times$$ Performing 0.08 \u20130.12* (0.11) (0.07) Boom $$\\times$$ Performing $$\\times$$ Tier1 0.16 \u20130.07 0.09 0.11 (0.29) (0.23) (0.16) (0.14) Crisis $$\\times$$ Performing $$\\times$$ Tier1 \u20130.90** \u20130.83*** \u20130.00 0.29 (0.43) (0.33) (0.27) (0.27) Recovery$$\\times$$Performing$$\\times$$Tier1 \u20130.71 \u20130.96** 0.15 0.33 (0.55) (0.41) (.39) (.41) Boom $$\\times$$ Performing $$\\times$$ Repo 0.02 \u20130.02 \u20130.21** 0.06 (0.15) (0.11) (0.09) (0.08) Crisis $$\\times$$ Performing $$\\times$$ Repo 0.39* 0.31* 0.25* 0.16 (0.23) (0.19) (0.13) (0.16) Recovery $$\\times$$ Performing $$\\times$$ Repo 0.58* 0.36 \u20130.03 0.04 (0.35) (0.28) (0.25) (0.30) Polynomial Yes Yes No Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Firm $$\\times$$ Year Fixed Effects No Yes No Yes $$R^2$$ 0.02 0.55 0.37 0.77 N 787,634 787,634 89,140 89,140 The table reports OLS estimates of the threshold specification in Equation (5). The dependent variable in the first two columns is the (log) total value of bank lending granted by bank $$b$$ to firm $$i$$ in quarter $$t$$. The dependent variable in the last two columns is the (log) value of the interest rate applied to a new loan granted by bank $$b$$ to firm $$i$$ in quarter $$t$$. The indicator Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 from the first quarter of 2004 to the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 from the first quarter of 2008 to the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 from the first quarter of 2010 to the last quarter of 2011, and 0 otherwise. Tier1$$_b$$ is defined as a bank $$b$$\u2019s core equity capital divided by its total assets, and Repo$$_{b}$$ is defined as the share of the bank\u2019s total financing from repo markets. All of the bank specific variables are measured pre-crisis. Functions $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ correspond to flexible sixth-order polynomials estimated separately for each quarter-year. Function $$f_{t}(\\cdot)$$ is estimated from $$0$$ to the left, whereas the Performing$$_{i,t}\\times g_{t}(\\cdot)$$ term is estimated from $$0$$ to the right. Finally, Lagged Rating controls for the past value of the rating and is interacted with quarter-year fixed effects. Standard errors, clustered at the firm-bank level, are reported in brackets. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Table 5 Bank heterogeneity Quantity Price Dependent variable (1) (2) (3) (4) Boom $$\\times$$ Performing 0.08 \u20130.05* (0.07) (0.03) Crisis $$\\times$$ Performing 0.26** \u20130.02 (0.09) (0.04) Recovery $$\\times$$ Performing 0.08 \u20130.12* (0.11) (0.07) Boom $$\\times$$ Performing $$\\times$$ Tier1 0.16 \u20130.07 0.09 0.11 (0.29) (0.23) (0.16) (0.14) Crisis $$\\times$$ Performing $$\\times$$ Tier1 \u20130.90** \u20130.83*** \u20130.00 0.29 (0.43) (0.33) (0.27) (0.27) Recovery$$\\times$$Performing$$\\times$$Tier1 \u20130.71 \u20130.96** 0.15 0.33 (0.55) (0.41) (.39) (.41) Boom $$\\times$$ Performing $$\\times$$ Repo 0.02 \u20130.02 \u20130.21** 0.06 (0.15) (0.11) (0.09) (0.08) Crisis $$\\times$$ Performing $$\\times$$ Repo 0.39* 0.31* 0.25* 0.16 (0.23) (0.19) (0.13) (0.16) Recovery $$\\times$$ Performing $$\\times$$ Repo 0.58* 0.36 \u20130.03 0.04 (0.35) (0.28) (0.25) (0.30) Polynomial Yes Yes No Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Firm $$\\times$$ Year Fixed Effects No Yes No Yes $$R^2$$ 0.02 0.55 0.37 0.77 N 787,634 787,634 89,140 89,140 Quantity Price Dependent variable (1) (2) (3) (4) Boom $$\\times$$ Performing 0.08 \u20130.05* (0.07) (0.03) Crisis $$\\times$$ Performing 0.26** \u20130.02 (0.09) (0.04) Recovery $$\\times$$ Performing 0.08 \u20130.12* (0.11) (0.07) Boom $$\\times$$ Performing $$\\times$$ Tier1 0.16 \u20130.07 0.09 0.11 (0.29) (0.23) (0.16) (0.14) Crisis $$\\times$$ Performing $$\\times$$ Tier1 \u20130.90** \u20130.83*** \u20130.00 0.29 (0.43) (0.33) (0.27) (0.27) Recovery$$\\times$$Performing$$\\times$$Tier1 \u20130.71 \u20130.96** 0.15 0.33 (0.55) (0.41) (.39) (.41) Boom $$\\times$$ Performing $$\\times$$ Repo 0.02 \u20130.02 \u20130.21** 0.06 (0.15) (0.11) (0.09) (0.08) Crisis $$\\times$$ Performing $$\\times$$ Repo 0.39* 0.31* 0.25* 0.16 (0.23) (0.19) (0.13) (0.16) Recovery $$\\times$$ Performing $$\\times$$ Repo 0.58* 0.36 \u20130.03 0.04 (0.35) (0.28) (0.25) (0.30) Polynomial Yes Yes No Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Firm $$\\times$$ Year Fixed Effects No Yes No Yes $$R^2$$ 0.02 0.55 0.37 0.77 N 787,634 787,634 89,140 89,140 The table reports OLS estimates of the threshold specification in Equation (5). The dependent variable in the first two columns is the (log) total value of bank lending granted by bank $$b$$ to firm $$i$$ in quarter $$t$$. The dependent variable in the last two columns is the (log) value of the interest rate applied to a new loan granted by bank $$b$$ to firm $$i$$ in quarter $$t$$. The indicator Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 from the first quarter of 2004 to the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 from the first quarter of 2008 to the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 from the first quarter of 2010 to the last quarter of 2011, and 0 otherwise. Tier1$$_b$$ is defined as a bank $$b$$\u2019s core equity capital divided by its total assets, and Repo$$_{b}$$ is defined as the share of the bank\u2019s total financing from repo markets. All of the bank specific variables are measured pre-crisis. Functions $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ correspond to flexible sixth-order polynomials estimated separately for each quarter-year. Function $$f_{t}(\\cdot)$$ is estimated from $$0$$ to the left, whereas the Performing$$_{i,t}\\times g_{t}(\\cdot)$$ term is estimated from $$0$$ to the right. Finally, Lagged Rating controls for the past value of the rating and is interacted with quarter-year fixed effects. Standard errors, clustered at the firm-bank level, are reported in brackets. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. We begin by interpreting our results in Column (1). First, notice that the baseline differences remain qualitatively very similar to those obtained with the main specification. Second, in the pre-crisis period, bank heterogeneity does not seem to affect how banks set the amount of total lending. This is intuitive: in the boom period, banks expect favorable financing conditions on wholesale markets. This means that they can lend \u201cas if\u201d unconstrained by segmentation, and make full use of their information on the firms\u2019 risk profile. These patterns change dramatically during the crisis. The negative sign on the interaction (Crisis$$\\times$$Performing$$\\times$$Tier1) indicates that highly capitalized banks are less likely to offer different amounts of credit to borrowers at the threshold. Similarly, those banks that are less dependent on short-term investors are also less likely to cut on lending as a consequence of market segmentation. Interestingly, the sensitivity of bank lending to these factors remains high even in the recovery period. Column (2) augments the discontinuity design by including firm-year fixed effects. This means that we exploit heterogeneity in the amount of lending to the same firm and in the same year from different banks. The estimates remain similar despite the increase in the number of estimated parameters. Columns (3) and (4) repeat the analysis by looking at the differences in interest rates. Our estimates suggest that bank heterogeneity is not particularly helpful to explain banks\u2019 price differences at the threshold. For instance, there is no evidence of significant differences in the spreads set by highly and lowly capitalized banks. Moreover, although, in principle, investor composition could affect the interest rate spreads, the evidence arising from the estimated parameters in Table 5 is rather mixed and, thus, inconclusive. To analyze the quantitative importance of bank capitalization and investor composition, we relate the results in the table to the drop in capitalization and repo financing that happened between 2007 and 2009. During that period, Italian banks\u2019 tier 1 capitalization fell by almost one percentage point. If we take the implied cumulative effect of segmentation and multiply it by the drop in capitalization, we obtain a differential tightening at the threshold of only $$0.6\\%$$ (or $$\\left(\\exp\\left\\{-0.90\\right\\}-1\\right)\\times0.01$$). Instead, the share of repo financing by banks went from 10% in 2007, to approximately 2% at the end of 2009. This suggests that the investor composition channel can account for a differential quantity tightening of approximately $$3.8\\%$$ (or $$\\left(\\exp\\left\\{0.39\\right\\}-1\\right)\\times0.08$$). This represents 10% of the observed threshold difference during the crisis, and indicates that investor composition is quantitatively an important channel to explain the consequences of segmentation on lending policies.18 6.2 Evidence from downgrades In Table 6, we report estimates of lending conditions to downgraded firms. Table 6 Downgrades from performing to substandard Quantity Price Production Dependent variable (1) (2) (3) (4) (5) (6) Boom$$\\times$$Down 0.10*** 0.04 0.03*** \u20130.02 0.08*** 0.06 (0.03) (0.14) (0.00) (0.03) (0.03) (0.12) Crisis$$\\times$$Down 0.03 \u20130.12 0.01 \u20130.06 \u20130.00 \u20130.14 (0.04) (0.21) (0.01) (0.06) (0.04) (0.19) Recovery$$\\times$$Down \u20130.12*** \u20130.33* 0.03** 0.12** \u20130.01 \u20130.31* (0.04) (0.18) (0.01) (0.06) (0.04) (0.16) Polynomial No Yes No Yes No Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes $$R^2$$ 0.01 0.04 0.39 0.40 0.02 0.07 N 88,830 88,830 70,848 70,848 22,978 22,978 Quantity Price Production Dependent variable (1) (2) (3) (4) (5) (6) Boom$$\\times$$Down 0.10*** 0.04 0.03*** \u20130.02 0.08*** 0.06 (0.03) (0.14) (0.00) (0.03) (0.03) (0.12) Crisis$$\\times$$Down 0.03 \u20130.12 0.01 \u20130.06 \u20130.00 \u20130.14 (0.04) (0.21) (0.01) (0.06) (0.04) (0.19) Recovery$$\\times$$Down \u20130.12*** \u20130.33* 0.03** 0.12** \u20130.01 \u20130.31* (0.04) (0.18) (0.01) (0.06) (0.04) (0.16) Polynomial No Yes No Yes No Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes $$R^2$$ 0.01 0.04 0.39 0.40 0.02 0.07 N 88,830 88,830 70,848 70,848 22,978 22,978 The table reports OLS estimates of the threshold specification in Equation (6) using the sample of firms downgraded from Score 6 to 7. The dependent variable in the first two columns is the (log) total value of bank lending granted to firm $$i$$ in quarter $$t$$. The dependent variable in the third and fourth columns is the (log) value of the interest rate applied to a new loan granted to firm $$i$$ in quarter $$t$$. The dependent variable in the fifth and sixth columns is the (log) value of sales of firm $$i$$ in year $$t$$. The indicator Down$$_{i,t}$$ is a binary variable equal to 1 if the firm is downgraded from category 6 to category 7 in year $$t$$, and is 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 from the first quarter of 2004 to the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 from the first quarter of 2008 to the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 from the first quarter of 2010 to the last quarter of 2011, and 0 otherwise. Polynomial functions $$f_{t}(\\cdot)$$, $$g_{t}(\\cdot)$$, $$m_{t}(\\cdot)$$, $$n_{t}(\\cdot)$$ correspond to flexible sixth-order polynomials estimated separately for each quarter-year. The polynomials in $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ are a function of the change in the continuous variable between $$t-1$$ and $$t$$. The polynomials in $$m_{t}(\\cdot)$$ and $$n_{t}(\\cdot)$$ are a function of the continuous variable in $$t-1$$. Standard errors, clustered at the firm level, are reported in brackets. In addition, we include polynomials for the level of the continuous variable in $$t-1$$. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Table 6 Downgrades from performing to substandard Quantity Price Production Dependent variable (1) (2) (3) (4) (5) (6) Boom$$\\times$$Down 0.10*** 0.04 0.03*** \u20130.02 0.08*** 0.06 (0.03) (0.14) (0.00) (0.03) (0.03) (0.12) Crisis$$\\times$$Down 0.03 \u20130.12 0.01 \u20130.06 \u20130.00 \u20130.14 (0.04) (0.21) (0.01) (0.06) (0.04) (0.19) Recovery$$\\times$$Down \u20130.12*** \u20130.33* 0.03** 0.12** \u20130.01 \u20130.31* (0.04) (0.18) (0.01) (0.06) (0.04) (0.16) Polynomial No Yes No Yes No Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes $$R^2$$ 0.01 0.04 0.39 0.40 0.02 0.07 N 88,830 88,830 70,848 70,848 22,978 22,978 Quantity Price Production Dependent variable (1) (2) (3) (4) (5) (6) Boom$$\\times$$Down 0.10*** 0.04 0.03*** \u20130.02 0.08*** 0.06 (0.03) (0.14) (0.00) (0.03) (0.03) (0.12) Crisis$$\\times$$Down 0.03 \u20130.12 0.01 \u20130.06 \u20130.00 \u20130.14 (0.04) (0.21) (0.01) (0.06) (0.04) (0.19) Recovery$$\\times$$Down \u20130.12*** \u20130.33* 0.03** 0.12** \u20130.01 \u20130.31* (0.04) (0.18) (0.01) (0.06) (0.04) (0.16) Polynomial No Yes No Yes No Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Yes Yes Yes Yes $$R^2$$ 0.01 0.04 0.39 0.40 0.02 0.07 N 88,830 88,830 70,848 70,848 22,978 22,978 The table reports OLS estimates of the threshold specification in Equation (6) using the sample of firms downgraded from Score 6 to 7. The dependent variable in the first two columns is the (log) total value of bank lending granted to firm $$i$$ in quarter $$t$$. The dependent variable in the third and fourth columns is the (log) value of the interest rate applied to a new loan granted to firm $$i$$ in quarter $$t$$. The dependent variable in the fifth and sixth columns is the (log) value of sales of firm $$i$$ in year $$t$$. The indicator Down$$_{i,t}$$ is a binary variable equal to 1 if the firm is downgraded from category 6 to category 7 in year $$t$$, and is 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 from the first quarter of 2004 to the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 from the first quarter of 2008 to the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 from the first quarter of 2010 to the last quarter of 2011, and 0 otherwise. Polynomial functions $$f_{t}(\\cdot)$$, $$g_{t}(\\cdot)$$, $$m_{t}(\\cdot)$$, $$n_{t}(\\cdot)$$ correspond to flexible sixth-order polynomials estimated separately for each quarter-year. The polynomials in $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ are a function of the change in the continuous variable between $$t-1$$ and $$t$$. The polynomials in $$m_{t}(\\cdot)$$ and $$n_{t}(\\cdot)$$ are a function of the continuous variable in $$t-1$$. Standard errors, clustered at the firm level, are reported in brackets. In addition, we include polynomials for the level of the continuous variable in $$t-1$$. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. In Columns (1), (3), and (5), we estimate a na\u00efve version of the specification in Equation (6), without the polynomial terms in the continuous variables ($$f_t(\\cdot),g_t(\\cdot),m_t(\\cdot),n_t(\\cdot)$$). The estimate obtained with this specification relative to the boom phase (Column (1)) suggests that downgraded firms obtain 10% more bank financing than non-downgraded firms. This puzzling result is most likely caused by some unobserved heterogeneity across these groups. Indeed, if banks were to use the information on the change in the rating\u2019s value to shape their response to downgrades, we would at most expect the absence of negative effects or the existence of small effects. In Columns (2), (4), and (6), we estimate the full specification, and thus compare firms that not only experienced a similar small change in the continuous variable, but were also close to the threshold. Consistent with our intuition earlier, we find that downgraded firms do not obtain higher volumes of credit than non-downgraded firms in the boom period. In crisis and recovery, the negative impact of a downgrade on credit allocations becomes progressively larger and statistically significant. During the recovery period, downgraded firms obtain 39% less bank financing than non-downgraded firms. The estimates in Column (4) also show that the restricted access to credit during the recovery period is accompanied by a higher cost of funds for downgraded firms. Finally, Column (6) shows that differences in the amount of production between marginally downgraded and non-downgraded firms are small and not statistically significant during the boom period. Intuitively, consistent with the credit patterns, these production differences are reversed during the subsequent phases of the cycle. 7. Empirical Tests In this section, we test the three identifying assumptions underlying our empirical setting. First, we show that firms do not seem to manipulate their ratings to self-select into more favorable categories. Second, we show that firms at the threshold are balanced in terms of their economic characteristics. Finally, we present placebo tests to provide further evidence on the relevance of the threshold between the substandard and performing classes of credit risk. Given that the Score is computed on a yearly basis, we perform these tests on the yearly cross-section of firms, unless otherwise stated. 7.1 Manipulation of the Score and self-selection Given the importance of the Score in bank credit decisions, a natural question to ask is whether firms are able to manipulate their credit rating and self-select into a better category. Manipulation of the rating is very unlikely, not only because the Score is unsolicited by firms and is computed based on firms\u2019 past balance sheets, but also because its exact algorithm is a business secret. Nevertheless, manipulation can be detected empirically: it would result in a systematic discontinuity of firms\u2019 distribution at the threshold, due either to the absence of observations near the threshold or to the presence of clusters of observations on the side of the threshold assigning a firm to the safer category. In Table 7, we test for the presence of a discontinuity in firm density at that threshold. Table 7 Self-selection into rating categories 6 and 7 Year 2004 2005 2006 2007 2008 2009 2010 2011 McCrary Density Estimate 0.10 0.13 0.02 0.08 0.3*** \u20130.00 0.08 0.17 (0.06) (0.07) (0.07) (0.06) (0.07) (0.08) (0.10) (0.10) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 Year 2004 2005 2006 2007 2008 2009 2010 2011 McCrary Density Estimate 0.10 0.13 0.02 0.08 0.3*** \u20130.00 0.08 0.17 (0.06) (0.07) (0.07) (0.06) (0.07) (0.08) (0.10) (0.10) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 The table reports, at a yearly level, the McCrary density estimates of the continuous variable\u2019s distribution. For each year, we run a kernel local linear regression of the log of the density on both sides of the threshold separating substandard firms in category 7 from performing firms in category 6. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Table 7 Self-selection into rating categories 6 and 7 Year 2004 2005 2006 2007 2008 2009 2010 2011 McCrary Density Estimate 0.10 0.13 0.02 0.08 0.3*** \u20130.00 0.08 0.17 (0.06) (0.07) (0.07) (0.06) (0.07) (0.08) (0.10) (0.10) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 Year 2004 2005 2006 2007 2008 2009 2010 2011 McCrary Density Estimate 0.10 0.13 0.02 0.08 0.3*** \u20130.00 0.08 0.17 (0.06) (0.07) (0.07) (0.06) (0.07) (0.08) (0.10) (0.10) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 The table reports, at a yearly level, the McCrary density estimates of the continuous variable\u2019s distribution. For each year, we run a kernel local linear regression of the log of the density on both sides of the threshold separating substandard firms in category 7 from performing firms in category 6. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Following McCrary (2008), for each year we run a kernel local linear regression of the log of the density on both sides of the threshold separating substandard firms in category 7 from performing firms in category 6. Table 7 shows that, with the exception of 2008, there is no evidence of significant discontinuities in the distribution of firms at the threshold. The discontinuity in 2008 is most likely coincidental for two reasons. First, if firms had discovered the exact formula of the Score and how to manipulate their assignment, a discontinuity should emerge systematically in every year following 2008. Second, had strategic manipulation occurred, it would mean that firms had anticipated by at least one year the financial crisis and the associated benefits of being classified as marginally performing entities.19 7.1.1 Policy experiment We also exploit a policy experiment to address the potential concern that the discontinuity arising in the McCrary tests for 2008 reflects firms\u2019 strategic manipulation of the Score. In November 2008, Law 185 (decreto legislativo n. 185) granted firms the possibility to revaluate fixed assets. Crucially, differently from previous laws with the same goal, Law 185 does not require the firm to pay taxes on the higher values of the assets in its balance sheet. We exploit this policy experiment in the following way: we run our main specification in Equation (3) using as dependent variable the (log) value of revalued assets. If the Score was manipulated, then we should observe that those firms that marginally fall in the performing class during the crisis were also those that revaluated assets disproportionally more than the marginally substandard firms. Table 8 shows that there is no significant difference in the outcome variable across the three phases of the credit cycle. This evidence further confirms that manipulation of the assignment variable is highly unlikely. Table 8 Manipulation: Revaluations Log revaluations Dependent variable (1) (2) Boom $$\\times$$ Performing \u20130.04 \u20130.05 (.05) (0.06) Crisis $$\\times$$ Performing \u20130.01 \u20130.03 (0.06) (0.07) Recovery $$\\times$$ Performing 0.00 \u20130.02 (0.05) (0.06) Polynomial Yes Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Control Variables No Yes $$R^2$$ 0.12 0.21 N 77,079 57,243 Log revaluations Dependent variable (1) (2) Boom $$\\times$$ Performing \u20130.04 \u20130.05 (.05) (0.06) Crisis $$\\times$$ Performing \u20130.01 \u20130.03 (0.06) (0.07) Recovery $$\\times$$ Performing 0.00 \u20130.02 (0.05) (0.06) Polynomial Yes Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Control Variables No Yes $$R^2$$ 0.12 0.21 N 77,079 57,243 The table reports OLS estimates of the threshold specification in Model 3. The dependent variable is the (log) value of revalued assets of firm $$i$$ in year $$t$$. The indicator Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 from the first quarter of 2004 to the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 from the first quarter of 2008 to the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 from the first quarter of 2010 to the last quarter of 2011, and 0 otherwise. Functions $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ correspond to flexible sixth-order polynomials estimated separately for each quarter-year. Function $$f_{t}(\\cdot)$$ is estimated from $$0$$ to the left, whereas the Performing$$_{i,t}\\times g_{t}(\\cdot)$$ term is estimated from $$0$$ to the right. Standard errors, clustered at the firm level, are reported in brackets. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Table 8 Manipulation: Revaluations Log revaluations Dependent variable (1) (2) Boom $$\\times$$ Performing \u20130.04 \u20130.05 (.05) (0.06) Crisis $$\\times$$ Performing \u20130.01 \u20130.03 (0.06) (0.07) Recovery $$\\times$$ Performing 0.00 \u20130.02 (0.05) (0.06) Polynomial Yes Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Control Variables No Yes $$R^2$$ 0.12 0.21 N 77,079 57,243 Log revaluations Dependent variable (1) (2) Boom $$\\times$$ Performing \u20130.04 \u20130.05 (.05) (0.06) Crisis $$\\times$$ Performing \u20130.01 \u20130.03 (0.06) (0.07) Recovery $$\\times$$ Performing 0.00 \u20130.02 (0.05) (0.06) Polynomial Yes Yes Quarter $$\\times$$ Year Fixed Effects Yes Yes Control Variables No Yes $$R^2$$ 0.12 0.21 N 77,079 57,243 The table reports OLS estimates of the threshold specification in Model 3. The dependent variable is the (log) value of revalued assets of firm $$i$$ in year $$t$$. The indicator Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. It is interacted with three indicator variables associated with the phases of the credit cycle. Boom$$_{t}$$ takes a value of 1 from the first quarter of 2004 to the last quarter of 2007, and 0 otherwise. Crisis$$_{t}$$ takes a value of 1 from the first quarter of 2008 to the last quarter of 2009, and 0 otherwise. Recovery$$_{t}$$ takes a value of 1 from the first quarter of 2010 to the last quarter of 2011, and 0 otherwise. Functions $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ correspond to flexible sixth-order polynomials estimated separately for each quarter-year. Function $$f_{t}(\\cdot)$$ is estimated from $$0$$ to the left, whereas the Performing$$_{i,t}\\times g_{t}(\\cdot)$$ term is estimated from $$0$$ to the right. Standard errors, clustered at the firm level, are reported in brackets. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. 7.2 Balancing tests In Table 9, we analyze whether firms close to the threshold are as if randomly sampled, a critical identification assumption within regression discontinuity models. If firms are nonrandomly sorted into specific rating classes, we would expect firm characteristics to differ systematically across the threshold. Following the regression discontinuity literature, the firm characteristics we test are those logically unaffected by the threshold but plausibly related to firm financing. Table 9 Model diagnostics: Balancing checks Year 2004 2005 2006 2007 2008 2009 2010 2011 Panel A: Presample characteristics Leverage 0.00 0.01 \u20130.04 \u20130.03 0.05 \u20130.01 \u20130.04 0.01 (.03) (.04) (.03) (.03) (.04) (.04) (.05) (.06) N 3,967 3,636 3,595 3,678 2,888 2,705 2,168 2,024 Return to Assets 0.00 0.00 0.00 \u20130.01 \u20130.02 0.00 0.00 0.00 (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.02) (0.02) N 5,306 4,844 4,750 4,836 3,776 3,504 2,721 2,508 Investment to Assets 0.02 0.02 0.01 0.02 0.02 \u20130.02 \u20130.03 \u20130.02 (0.01) (0.02) (0.01) (0.02) (0.02) (0.02) (0.03) (0.02) N 4,501 4,136 4,083 4,174 3,353 3,100 2,414 2,237 Panel B: Bank balancing characteristics Credit Event 0.01 0.00 0.01 0.00 \u20130.01 0.00 \u20130.03 (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.03) N 5,736 5,944 6,358 5,411 5,276 4,235 4,045 Asked 0.02 0.00 \u20130.02 \u20130.07 \u20130.03 0.04 0.03 \u20130.07 (0.04) (0.05) (0.04) (0.05) (0.04) (0.04) (0.05) (0.05) N 5,687 5,677 5,889 6,306 5,370 5,264 4,217 4,030 Bank Size \u20130.12 \u20130.05 \u20130.02 0.23** 0.1 0.09 0.04 0.23 (0.14) (0.14) (0.11) (0.12) (0.14) (0.17) (0.19) (0.18) N 5,652 5,641 5,855 6,287 5,356 5,108 4,105 3,937 Panel C: Time-invariant characteristics Activity: Food Industry 0.03 \u20130.04 0.03 \u20130.01 0.05 0.04 0.06 \u20130.06 (0.04) (0.05) (0.04) (0.04) (0.04) (0.04) (0.06) (0.06) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 Location: Top 5 Cities 0.06 0.03 0.05 \u20130.06 0.02 \u20130.01 0.07 0.05 (0.06) (0.06) (0.06) (0.06) (0.06) (0.06) (0.08) (0.07) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 Year 2004 2005 2006 2007 2008 2009 2010 2011 Panel A: Presample characteristics Leverage 0.00 0.01 \u20130.04 \u20130.03 0.05 \u20130.01 \u20130.04 0.01 (.03) (.04) (.03) (.03) (.04) (.04) (.05) (.06) N 3,967 3,636 3,595 3,678 2,888 2,705 2,168 2,024 Return to Assets 0.00 0.00 0.00 \u20130.01 \u20130.02 0.00 0.00 0.00 (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.02) (0.02) N 5,306 4,844 4,750 4,836 3,776 3,504 2,721 2,508 Investment to Assets 0.02 0.02 0.01 0.02 0.02 \u20130.02 \u20130.03 \u20130.02 (0.01) (0.02) (0.01) (0.02) (0.02) (0.02) (0.03) (0.02) N 4,501 4,136 4,083 4,174 3,353 3,100 2,414 2,237 Panel B: Bank balancing characteristics Credit Event 0.01 0.00 0.01 0.00 \u20130.01 0.00 \u20130.03 (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.03) N 5,736 5,944 6,358 5,411 5,276 4,235 4,045 Asked 0.02 0.00 \u20130.02 \u20130.07 \u20130.03 0.04 0.03 \u20130.07 (0.04) (0.05) (0.04) (0.05) (0.04) (0.04) (0.05) (0.05) N 5,687 5,677 5,889 6,306 5,370 5,264 4,217 4,030 Bank Size \u20130.12 \u20130.05 \u20130.02 0.23** 0.1 0.09 0.04 0.23 (0.14) (0.14) (0.11) (0.12) (0.14) (0.17) (0.19) (0.18) N 5,652 5,641 5,855 6,287 5,356 5,108 4,105 3,937 Panel C: Time-invariant characteristics Activity: Food Industry 0.03 \u20130.04 0.03 \u20130.01 0.05 0.04 0.06 \u20130.06 (0.04) (0.05) (0.04) (0.04) (0.04) (0.04) (0.06) (0.06) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 Location: Top 5 Cities 0.06 0.03 0.05 \u20130.06 0.02 \u20130.01 0.07 0.05 (0.06) (0.06) (0.06) (0.06) (0.06) (0.06) (0.08) (0.07) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 The table estimates differences in presample firm characteristics at the threshold. In all rows, the dependent variable is measured in 2003. The estimates refer to the indicator variable Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. Credit Event is a binary variable equal to 1 if any of a given firm\u2019s banks classified the firm\u2019s credit as nonperforming. Asked is a binary variable equal to 1 if any non-current bank requested information on the firm during the year. Bank Size corresponds to the value of a bank\u2019s total assets. Food Industry is a binary variable indicating whether the firms\u2019 SIC code belongs to the food industry. Top 5 Cities is a binary variable indicating whether the firms\u2019 headquarters zip code is in one of the largest five cities. See Tables 2 for the definition of the other variables. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Table 9 Model diagnostics: Balancing checks Year 2004 2005 2006 2007 2008 2009 2010 2011 Panel A: Presample characteristics Leverage 0.00 0.01 \u20130.04 \u20130.03 0.05 \u20130.01 \u20130.04 0.01 (.03) (.04) (.03) (.03) (.04) (.04) (.05) (.06) N 3,967 3,636 3,595 3,678 2,888 2,705 2,168 2,024 Return to Assets 0.00 0.00 0.00 \u20130.01 \u20130.02 0.00 0.00 0.00 (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.02) (0.02) N 5,306 4,844 4,750 4,836 3,776 3,504 2,721 2,508 Investment to Assets 0.02 0.02 0.01 0.02 0.02 \u20130.02 \u20130.03 \u20130.02 (0.01) (0.02) (0.01) (0.02) (0.02) (0.02) (0.03) (0.02) N 4,501 4,136 4,083 4,174 3,353 3,100 2,414 2,237 Panel B: Bank balancing characteristics Credit Event 0.01 0.00 0.01 0.00 \u20130.01 0.00 \u20130.03 (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.03) N 5,736 5,944 6,358 5,411 5,276 4,235 4,045 Asked 0.02 0.00 \u20130.02 \u20130.07 \u20130.03 0.04 0.03 \u20130.07 (0.04) (0.05) (0.04) (0.05) (0.04) (0.04) (0.05) (0.05) N 5,687 5,677 5,889 6,306 5,370 5,264 4,217 4,030 Bank Size \u20130.12 \u20130.05 \u20130.02 0.23** 0.1 0.09 0.04 0.23 (0.14) (0.14) (0.11) (0.12) (0.14) (0.17) (0.19) (0.18) N 5,652 5,641 5,855 6,287 5,356 5,108 4,105 3,937 Panel C: Time-invariant characteristics Activity: Food Industry 0.03 \u20130.04 0.03 \u20130.01 0.05 0.04 0.06 \u20130.06 (0.04) (0.05) (0.04) (0.04) (0.04) (0.04) (0.06) (0.06) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 Location: Top 5 Cities 0.06 0.03 0.05 \u20130.06 0.02 \u20130.01 0.07 0.05 (0.06) (0.06) (0.06) (0.06) (0.06) (0.06) (0.08) (0.07) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 Year 2004 2005 2006 2007 2008 2009 2010 2011 Panel A: Presample characteristics Leverage 0.00 0.01 \u20130.04 \u20130.03 0.05 \u20130.01 \u20130.04 0.01 (.03) (.04) (.03) (.03) (.04) (.04) (.05) (.06) N 3,967 3,636 3,595 3,678 2,888 2,705 2,168 2,024 Return to Assets 0.00 0.00 0.00 \u20130.01 \u20130.02 0.00 0.00 0.00 (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.02) (0.02) N 5,306 4,844 4,750 4,836 3,776 3,504 2,721 2,508 Investment to Assets 0.02 0.02 0.01 0.02 0.02 \u20130.02 \u20130.03 \u20130.02 (0.01) (0.02) (0.01) (0.02) (0.02) (0.02) (0.03) (0.02) N 4,501 4,136 4,083 4,174 3,353 3,100 2,414 2,237 Panel B: Bank balancing characteristics Credit Event 0.01 0.00 0.01 0.00 \u20130.01 0.00 \u20130.03 (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) (0.03) N 5,736 5,944 6,358 5,411 5,276 4,235 4,045 Asked 0.02 0.00 \u20130.02 \u20130.07 \u20130.03 0.04 0.03 \u20130.07 (0.04) (0.05) (0.04) (0.05) (0.04) (0.04) (0.05) (0.05) N 5,687 5,677 5,889 6,306 5,370 5,264 4,217 4,030 Bank Size \u20130.12 \u20130.05 \u20130.02 0.23** 0.1 0.09 0.04 0.23 (0.14) (0.14) (0.11) (0.12) (0.14) (0.17) (0.19) (0.18) N 5,652 5,641 5,855 6,287 5,356 5,108 4,105 3,937 Panel C: Time-invariant characteristics Activity: Food Industry 0.03 \u20130.04 0.03 \u20130.01 0.05 0.04 0.06 \u20130.06 (0.04) (0.05) (0.04) (0.04) (0.04) (0.04) (0.06) (0.06) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 Location: Top 5 Cities 0.06 0.03 0.05 \u20130.06 0.02 \u20130.01 0.07 0.05 (0.06) (0.06) (0.06) (0.06) (0.06) (0.06) (0.08) (0.07) N 5,951 5,876 6,098 6,514 5,551 5,360 4,307 4,110 The table estimates differences in presample firm characteristics at the threshold. In all rows, the dependent variable is measured in 2003. The estimates refer to the indicator variable Performing$$_{i,t}$$ takes a value of 1 if a firm is in the performing class (i.e., $$s_{i,t} \\geq 0$$ implying a Score of 6), and 0 otherwise. Credit Event is a binary variable equal to 1 if any of a given firm\u2019s banks classified the firm\u2019s credit as nonperforming. Asked is a binary variable equal to 1 if any non-current bank requested information on the firm during the year. Bank Size corresponds to the value of a bank\u2019s total assets. Food Industry is a binary variable indicating whether the firms\u2019 SIC code belongs to the food industry. Top 5 Cities is a binary variable indicating whether the firms\u2019 headquarters zip code is in one of the largest five cities. See Tables 2 for the definition of the other variables. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. In panel A of Table 9, the dependent variables are a broad set of firm financing, investment, and profitability measures taken in 2003. In the first row, we show that firms at the threshold do not differ in terms of leverage choices in the pre-sample period. Moreover, we find no significant difference in firms\u2019 return on assets or investments. Panel B tests for differences in bank-firm relationships at the threshold. The first row in the table focuses on the banks\u2019 probability of reporting a delinquent loan. If there were a discontinuity in the probability of a firm\u2019s credit event at the threshold, then our results could be explained by the fact that banks correctly price this difference. However, we find no statistically or economically significant differences at the threshold. In the second row, the variable $$Asked$$ is a binary indicator equal to 1 if a bank requests information on a new loan applicant. The estimates suggest that firms at the threshold do not display a different propensity to apply for loans to new banks. The last row of the panel tests for the presence of assortative matching between banks and firms at the threshold (Paravisini et al. 2014). For each firm, we compute its bank\u2019s average size.20 Again, we find no evidence of a systematic difference at the threshold. Panel C focuses on differences in time-invariant firm characteristics. In the first row, the dependent variable is the firm\u2019s activity sector proxied by its SIC code. The yearly estimates indicate no statistically or economically significant evidence of firms clustering into sectors such as food industries. Next, we look at time-invariant characteristics related to firms\u2019 geographic locations. This is a particularly interesting dimension to study within this setting because Italian geography is correlated with heterogeneity in economic development, crime rates, and political accountability (Brollo et al. 2013) and could thus be associated with opportunistic manipulation. The variable capturing location in the largest cities or the most entrepreneurial areas does not display a statistically significant discontinuity.21 7.3 Empirical relevance of the threshold We now provide further evidence on the relevance of the threshold between performing and substandard firms. First, we confirm the local interpretation of our estimates by providing nonparametric plots of the outcome variable as a function of the continuous assignment variable. Second, we implement placebo tests in which we randomly re-label the value of the threshold. Finally, we investigate whether banks use alternative ratings\u2019 cutoffs to formulate lending standards. 7.3.1 Nonparametric plots In the left panel of Figure 5, we focus on data from the second quarter of 2009, when our results at the threshold feature quantity differences and no interest rate differences. We divide the domain of $$s$$ into mutually exclusive bins of size $$0.03$$.22 For each bin, we compute the average and the 90% confidence interval of the outcome variable, and plot these values at the bin\u2019s midpoint. The fitted gray line shows how close the sixth-order polynomial approximates the variation in bank financing conditions at the threshold. Figure 5. View largeDownload slide Second quarter of 2009 The figure focuses on the second quarter of 2009. We divide the domain of $$s_i$$ into mutually exclusive bins with a size of $$0.03$$. For each bin, we compute the average and the 90% confidence interval of the outcome variable, and plot these values at the bin\u2019s midpoint. The fitted gray line shows how closely the sixth-order polynomial approximates the variation in bank financing conditions at the threshold. Figure 5. View largeDownload slide Second quarter of 2009 The figure focuses on the second quarter of 2009. We divide the domain of $$s_i$$ into mutually exclusive bins with a size of $$0.03$$. For each bin, we compute the average and the 90% confidence interval of the outcome variable, and plot these values at the bin\u2019s midpoint. The fitted gray line shows how closely the sixth-order polynomial approximates the variation in bank financing conditions at the threshold. The top panel of Figure 5 shows that a clear discontinuity arises in the total amount of bank financing close to the threshold. The magnitude of this discontinuity can be quantified by comparing the mean value of the variable of interest in the two bins next to the threshold. Immediately to the left of the threshold, the average value of (log) granted credit is approximately 14.6, whereas immediately to the right this value is 15, implying that the estimated value of $$\\beta$$ captures the variation arising directly at the threshold. The bottom panel of Figure 5 repeats this exercise for the interest rates on new bank loans. It shows that when there is no discontinuity in the value of the conditional regression function at the threshold, the polynomial fit does not display any significant discontinuity. Figure 6 repeats this analysis by focusing on the second quarter of 2011, when our results at the threshold feature significant interest rate differences and no quantity differences.23 Figure 6. View largeDownload slide Second quarter of 2011 The figure focuses on the second quarter of 2011. We divide the domain of $$s_i$$ into mutually exclusive bins with a size of $$0.03$$. For each bin, we compute the average and the standard deviation of the outcome variable, and plot these values at the bin\u2019s midpoint. The fitted gray line shows how closely the sixth-order polynomial approximates the variation in bank financing conditions at the threshold. Figure 6. View largeDownload slide Second quarter of 2011 The figure focuses on the second quarter of 2011. We divide the domain of $$s_i$$ into mutually exclusive bins with a size of $$0.03$$. For each bin, we compute the average and the standard deviation of the outcome variable, and plot these values at the bin\u2019s midpoint. The fitted gray line shows how closely the sixth-order polynomial approximates the variation in bank financing conditions at the threshold. 7.3.2 Placebo tests Finding a significant discontinuity in lending conditions at the threshold, as shown in Figure 4, might not necessarily establish a causal relationship between the threshold and the design of financial contracts. For example, analogous results might arise when comparing financing conditions borne by firms whose Score lies further away from the true threshold. We thus implement the following falsification tests: we draw approximately 100 randomly distributed placebo thresholds along the support of Score categories 6 and 7, and rerun our specification on the cross-section of firms at the threshold in all the quarters in our sample. We plot in Figure 7 the distribution of the placebo estimates for the second quarters of 2009 and 2011. Figure 7. View largeDownload slide Placebo estimates: Second quarters of 2009 and 2011 The figure plots the empirical distribution of discontinuity estimates based on approximately 100 randomly drawn placebo thresholds. The vertical dotted line represents the estimate obtained from the true threshold. The top panel figures focus on the second quarter of 2009, while the bottom panel focuses on the second quarter of 2011. Figure 7. View largeDownload slide Placebo estimates: Second quarters of 2009 and 2011 The figure plots the empirical distribution of discontinuity estimates based on approximately 100 randomly drawn placebo thresholds. The vertical dotted line represents the estimate obtained from the true threshold. The top panel figures focus on the second quarter of 2009, while the bottom panel focuses on the second quarter of 2011. Figure 7 illustrates that the contractual differences identified by the true threshold estimates (vertical dotted line) are not due to a coincidental discontinuity. If this were the case, then we should observe similar estimates arising when considering randomly placed thresholds. In the top-left panel, we find that the 100 placebo estimates for the differences in the quantity of bank financing are approximately normally distributed around 0. Similarly, the bottom-right panel shows that in the second quarter of 2011 the interest rate differences of 20% that we find in the main analysis are well outside the normal variation arising from randomly placed thresholds.24 This evidence demonstrates the relevance of the categorical value of the Score for Italian banks\u2019 lending decisions. If banks were not using the categorical rating when making their credit choices, then the threshold should not yield financial outcomes that are significantly and systematically different from those obtained using a randomly set threshold along the support of the continuous variable. Our evidence rejects this claim on the basis of the distribution of placebo estimates within and across the sample period. 7.3.3 Other rating thresholds Finally, as in Agarwal et al. (Forthcoming), we investigate whether banks use alternative ratings\u2019 cutoffs to formulate lending standards. We estimate our specification on the cross-section of firms at all the other six thresholds associated with the categorical value of the rating system.25 In Table 10, the reported dummy variable is equal to 1 for firms in the better\u2014that is, lower-value\u2013rating category, and 0 otherwise. Table 10 Yearly discontinuity estimates: Other thresholds Year 2004 2005 2006 2007 2008 2009 2010 2011 Threshold between categories 1 and 2 Quantity \u20130.3 \u20130.15 0.07 0.17 \u20130.28 \u20130.19 \u20130.3 \u20130.32 (0.24) (0.26) (0.26) (0.31) (0.27) (0.25) (0.23) (0.21) N 2,555 2,693 2,648 2,684 2,886 2,975 2,677 2,773 Price 0.04 0.13 0.08 0.03 \u20130.12 \u20130.23 \u20130.04 \u20130.22 ( 0.11) (0.12) (0.11) (0.08) (0.08) (0.2) (0.18) (0.22) N 583 716 782 815 715 712 832 775 Threshold between categories 2 and 3 Quantity \u20130.12 \u20130.19 \u20130.45 \u20130.3 \u20130.25 \u20130.2 \u20130.45 \u20130.51 ( 0.39) (0.4) (0.39) (0.35) (0.41) (0.34) (0.36) (0.35) N 2,311 2,508 2,480 2,383 2,265 2,243 2,243 2,375 Price 0.00 0.16 \u20130.1 0.01 \u20130.02 \u20130.1 \u20130.23 0.7*** ( 0.13) (0.12) (0.11) (0.08) (0.14) (0.27) (0.22) (0.22) N 1,099 1,427 1,595 1,702 1,475 1,260 1,406 1,825 Threshold between categories 3 and 4 Quantity \u20130.24 \u20130.03 \u20130.14 0.29 0.11 \u20130.29 \u20130.15 0.29 ( 0.31) (0.3) (0.35) (0.29) (0.33) (0.32) (0.29) (0.3) N 6,087 6,361 6,371 6,526 6,040 5,968 5,840 6,128 Price \u20130.03 0.03 0.09 \u20130.03 \u20130.08 \u20130.01 \u20130.12 \u20130.03 ( 0.08) (0.09) (0.08) (0.04) (0.06) (0.13) (0.15) (0.12) N 7,197 9,359 10,255 10,547 9,033 8,625 11,153 13,158 Threshold between categories 4 and 5 Quantity \u20130.33 0.22 \u20130.44* \u20130.18 \u20130.2 \u20130.06 \u20130.26 \u20130.41* ( 0.24) (0.24) (0.24) (0.21) (0.24) (0.24) (0.24) (0.23) N 7,019 7,359 7,437 7,616 6,960 6,878 6,711 7,058 Price 0.00 \u20130.05 0.03 \u20130.01 0.00 \u20130.02 \u20130.23*** 0.07 ( 0.05) (0.06) (0.04) (0.03) (0.03) (0.1) (0.08) (0.07) N 11,072 14,972 16,561 17,056 14,662 13,505 17,687 19,743 Threshold between categories 7 and 8 Quantity \u20130.25 \u20130.28 \u20130.29 \u20130.06 \u20130.36 \u20130.63 1.44* 1.01 ( 0.48) (0.51) (0.55) (0.55) (0.63) (0.66) (0.73) (0.88) N 4,160 4,136 4,256 4,602 3,752 3,472 2,875 2,688 Price .00 \u20130.2 0.1 \u20130.22** \u20130.08 0.35* \u20130.56 \u20130.12 ( 0.19) (0.17) (0.11) (0.09) (0.1) (0.2) (0.56) (0.27) N 6,058 8,394 10,412 13,192 8,280 6,047 5,883 5,791 Threshold between categories 8 and 9 Quantity \u20130.9 0.18 0.51 \u20131.31 \u20131.26 \u20130.42 \u20130.97 \u20131.68 ( 1.4) (1.16) (1.12) (1.36) (1.09) (1.24) (0.95) (1.2) N 596 649 598 646 595 668 517 616 Price \u20131.29 \u20130.01 0.21 0.09 \u20130.02 0.07 0.4 \u20130.31 ( 54.98) (0.53) (0.26) (0.27) (0.13) (0.5) (0.47) (0.4) N 380 494 655 761 518 701 471 489 Year 2004 2005 2006 2007 2008 2009 2010 2011 Threshold between categories 1 and 2 Quantity \u20130.3 \u20130.15 0.07 0.17 \u20130.28 \u20130.19 \u20130.3 \u20130.32 (0.24) (0.26) (0.26) (0.31) (0.27) (0.25) (0.23) (0.21) N 2,555 2,693 2,648 2,684 2,886 2,975 2,677 2,773 Price 0.04 0.13 0.08 0.03 \u20130.12 \u20130.23 \u20130.04 \u20130.22 ( 0.11) (0.12) (0.11) (0.08) (0.08) (0.2) (0.18) (0.22) N 583 716 782 815 715 712 832 775 Threshold between categories 2 and 3 Quantity \u20130.12 \u20130.19 \u20130.45 \u20130.3 \u20130.25 \u20130.2 \u20130.45 \u20130.51 ( 0.39) (0.4) (0.39) (0.35) (0.41) (0.34) (0.36) (0.35) N 2,311 2,508 2,480 2,383 2,265 2,243 2,243 2,375 Price 0.00 0.16 \u20130.1 0.01 \u20130.02 \u20130.1 \u20130.23 0.7*** ( 0.13) (0.12) (0.11) (0.08) (0.14) (0.27) (0.22) (0.22) N 1,099 1,427 1,595 1,702 1,475 1,260 1,406 1,825 Threshold between categories 3 and 4 Quantity \u20130.24 \u20130.03 \u20130.14 0.29 0.11 \u20130.29 \u20130.15 0.29 ( 0.31) (0.3) (0.35) (0.29) (0.33) (0.32) (0.29) (0.3) N 6,087 6,361 6,371 6,526 6,040 5,968 5,840 6,128 Price \u20130.03 0.03 0.09 \u20130.03 \u20130.08 \u20130.01 \u20130.12 \u20130.03 ( 0.08) (0.09) (0.08) (0.04) (0.06) (0.13) (0.15) (0.12) N 7,197 9,359 10,255 10,547 9,033 8,625 11,153 13,158 Threshold between categories 4 and 5 Quantity \u20130.33 0.22 \u20130.44* \u20130.18 \u20130.2 \u20130.06 \u20130.26 \u20130.41* ( 0.24) (0.24) (0.24) (0.21) (0.24) (0.24) (0.24) (0.23) N 7,019 7,359 7,437 7,616 6,960 6,878 6,711 7,058 Price 0.00 \u20130.05 0.03 \u20130.01 0.00 \u20130.02 \u20130.23*** 0.07 ( 0.05) (0.06) (0.04) (0.03) (0.03) (0.1) (0.08) (0.07) N 11,072 14,972 16,561 17,056 14,662 13,505 17,687 19,743 Threshold between categories 7 and 8 Quantity \u20130.25 \u20130.28 \u20130.29 \u20130.06 \u20130.36 \u20130.63 1.44* 1.01 ( 0.48) (0.51) (0.55) (0.55) (0.63) (0.66) (0.73) (0.88) N 4,160 4,136 4,256 4,602 3,752 3,472 2,875 2,688 Price .00 \u20130.2 0.1 \u20130.22** \u20130.08 0.35* \u20130.56 \u20130.12 ( 0.19) (0.17) (0.11) (0.09) (0.1) (0.2) (0.56) (0.27) N 6,058 8,394 10,412 13,192 8,280 6,047 5,883 5,791 Threshold between categories 8 and 9 Quantity \u20130.9 0.18 0.51 \u20131.31 \u20131.26 \u20130.42 \u20130.97 \u20131.68 ( 1.4) (1.16) (1.12) (1.36) (1.09) (1.24) (0.95) (1.2) N 596 649 598 646 595 668 517 616 Price \u20131.29 \u20130.01 0.21 0.09 \u20130.02 0.07 0.4 \u20130.31 ( 54.98) (0.53) (0.26) (0.27) (0.13) (0.5) (0.47) (0.4) N 380 494 655 761 518 701 471 489 The table reports estimates from our baseline specification at all the seven thresholds associated with the categorical value of the rating system. We report standard errors in brackets. The dependent variable is either All Bank Financing Granted or Interest Rate for each year between 2004.Q1\u20132011.Q4. We estimate the discontinuity $$\\left( s_{i}\\geq 0 \\right)$$ using a flexible sixth-order polynomial on either side of each normalized threshold between each contiguous Score category, allowing for a discontinuity at 0. The reported estimates refer to $$S_{i}$$, a binary variable that takes a value of 1 if the continuous variable $$s_i \\geq 0$$, that is, if the firm is allocated to the lower credit risk category as opposed to the higher credit risk category. See Table 2 for other variable definitions. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Table 10 Yearly discontinuity estimates: Other thresholds Year 2004 2005 2006 2007 2008 2009 2010 2011 Threshold between categories 1 and 2 Quantity \u20130.3 \u20130.15 0.07 0.17 \u20130.28 \u20130.19 \u20130.3 \u20130.32 (0.24) (0.26) (0.26) (0.31) (0.27) (0.25) (0.23) (0.21) N 2,555 2,693 2,648 2,684 2,886 2,975 2,677 2,773 Price 0.04 0.13 0.08 0.03 \u20130.12 \u20130.23 \u20130.04 \u20130.22 ( 0.11) (0.12) (0.11) (0.08) (0.08) (0.2) (0.18) (0.22) N 583 716 782 815 715 712 832 775 Threshold between categories 2 and 3 Quantity \u20130.12 \u20130.19 \u20130.45 \u20130.3 \u20130.25 \u20130.2 \u20130.45 \u20130.51 ( 0.39) (0.4) (0.39) (0.35) (0.41) (0.34) (0.36) (0.35) N 2,311 2,508 2,480 2,383 2,265 2,243 2,243 2,375 Price 0.00 0.16 \u20130.1 0.01 \u20130.02 \u20130.1 \u20130.23 0.7*** ( 0.13) (0.12) (0.11) (0.08) (0.14) (0.27) (0.22) (0.22) N 1,099 1,427 1,595 1,702 1,475 1,260 1,406 1,825 Threshold between categories 3 and 4 Quantity \u20130.24 \u20130.03 \u20130.14 0.29 0.11 \u20130.29 \u20130.15 0.29 ( 0.31) (0.3) (0.35) (0.29) (0.33) (0.32) (0.29) (0.3) N 6,087 6,361 6,371 6,526 6,040 5,968 5,840 6,128 Price \u20130.03 0.03 0.09 \u20130.03 \u20130.08 \u20130.01 \u20130.12 \u20130.03 ( 0.08) (0.09) (0.08) (0.04) (0.06) (0.13) (0.15) (0.12) N 7,197 9,359 10,255 10,547 9,033 8,625 11,153 13,158 Threshold between categories 4 and 5 Quantity \u20130.33 0.22 \u20130.44* \u20130.18 \u20130.2 \u20130.06 \u20130.26 \u20130.41* ( 0.24) (0.24) (0.24) (0.21) (0.24) (0.24) (0.24) (0.23) N 7,019 7,359 7,437 7,616 6,960 6,878 6,711 7,058 Price 0.00 \u20130.05 0.03 \u20130.01 0.00 \u20130.02 \u20130.23*** 0.07 ( 0.05) (0.06) (0.04) (0.03) (0.03) (0.1) (0.08) (0.07) N 11,072 14,972 16,561 17,056 14,662 13,505 17,687 19,743 Threshold between categories 7 and 8 Quantity \u20130.25 \u20130.28 \u20130.29 \u20130.06 \u20130.36 \u20130.63 1.44* 1.01 ( 0.48) (0.51) (0.55) (0.55) (0.63) (0.66) (0.73) (0.88) N 4,160 4,136 4,256 4,602 3,752 3,472 2,875 2,688 Price .00 \u20130.2 0.1 \u20130.22** \u20130.08 0.35* \u20130.56 \u20130.12 ( 0.19) (0.17) (0.11) (0.09) (0.1) (0.2) (0.56) (0.27) N 6,058 8,394 10,412 13,192 8,280 6,047 5,883 5,791 Threshold between categories 8 and 9 Quantity \u20130.9 0.18 0.51 \u20131.31 \u20131.26 \u20130.42 \u20130.97 \u20131.68 ( 1.4) (1.16) (1.12) (1.36) (1.09) (1.24) (0.95) (1.2) N 596 649 598 646 595 668 517 616 Price \u20131.29 \u20130.01 0.21 0.09 \u20130.02 0.07 0.4 \u20130.31 ( 54.98) (0.53) (0.26) (0.27) (0.13) (0.5) (0.47) (0.4) N 380 494 655 761 518 701 471 489 Year 2004 2005 2006 2007 2008 2009 2010 2011 Threshold between categories 1 and 2 Quantity \u20130.3 \u20130.15 0.07 0.17 \u20130.28 \u20130.19 \u20130.3 \u20130.32 (0.24) (0.26) (0.26) (0.31) (0.27) (0.25) (0.23) (0.21) N 2,555 2,693 2,648 2,684 2,886 2,975 2,677 2,773 Price 0.04 0.13 0.08 0.03 \u20130.12 \u20130.23 \u20130.04 \u20130.22 ( 0.11) (0.12) (0.11) (0.08) (0.08) (0.2) (0.18) (0.22) N 583 716 782 815 715 712 832 775 Threshold between categories 2 and 3 Quantity \u20130.12 \u20130.19 \u20130.45 \u20130.3 \u20130.25 \u20130.2 \u20130.45 \u20130.51 ( 0.39) (0.4) (0.39) (0.35) (0.41) (0.34) (0.36) (0.35) N 2,311 2,508 2,480 2,383 2,265 2,243 2,243 2,375 Price 0.00 0.16 \u20130.1 0.01 \u20130.02 \u20130.1 \u20130.23 0.7*** ( 0.13) (0.12) (0.11) (0.08) (0.14) (0.27) (0.22) (0.22) N 1,099 1,427 1,595 1,702 1,475 1,260 1,406 1,825 Threshold between categories 3 and 4 Quantity \u20130.24 \u20130.03 \u20130.14 0.29 0.11 \u20130.29 \u20130.15 0.29 ( 0.31) (0.3) (0.35) (0.29) (0.33) (0.32) (0.29) (0.3) N 6,087 6,361 6,371 6,526 6,040 5,968 5,840 6,128 Price \u20130.03 0.03 0.09 \u20130.03 \u20130.08 \u20130.01 \u20130.12 \u20130.03 ( 0.08) (0.09) (0.08) (0.04) (0.06) (0.13) (0.15) (0.12) N 7,197 9,359 10,255 10,547 9,033 8,625 11,153 13,158 Threshold between categories 4 and 5 Quantity \u20130.33 0.22 \u20130.44* \u20130.18 \u20130.2 \u20130.06 \u20130.26 \u20130.41* ( 0.24) (0.24) (0.24) (0.21) (0.24) (0.24) (0.24) (0.23) N 7,019 7,359 7,437 7,616 6,960 6,878 6,711 7,058 Price 0.00 \u20130.05 0.03 \u20130.01 0.00 \u20130.02 \u20130.23*** 0.07 ( 0.05) (0.06) (0.04) (0.03) (0.03) (0.1) (0.08) (0.07) N 11,072 14,972 16,561 17,056 14,662 13,505 17,687 19,743 Threshold between categories 7 and 8 Quantity \u20130.25 \u20130.28 \u20130.29 \u20130.06 \u20130.36 \u20130.63 1.44* 1.01 ( 0.48) (0.51) (0.55) (0.55) (0.63) (0.66) (0.73) (0.88) N 4,160 4,136 4,256 4,602 3,752 3,472 2,875 2,688 Price .00 \u20130.2 0.1 \u20130.22** \u20130.08 0.35* \u20130.56 \u20130.12 ( 0.19) (0.17) (0.11) (0.09) (0.1) (0.2) (0.56) (0.27) N 6,058 8,394 10,412 13,192 8,280 6,047 5,883 5,791 Threshold between categories 8 and 9 Quantity \u20130.9 0.18 0.51 \u20131.31 \u20131.26 \u20130.42 \u20130.97 \u20131.68 ( 1.4) (1.16) (1.12) (1.36) (1.09) (1.24) (0.95) (1.2) N 596 649 598 646 595 668 517 616 Price \u20131.29 \u20130.01 0.21 0.09 \u20130.02 0.07 0.4 \u20130.31 ( 54.98) (0.53) (0.26) (0.27) (0.13) (0.5) (0.47) (0.4) N 380 494 655 761 518 701 471 489 The table reports estimates from our baseline specification at all the seven thresholds associated with the categorical value of the rating system. We report standard errors in brackets. The dependent variable is either All Bank Financing Granted or Interest Rate for each year between 2004.Q1\u20132011.Q4. We estimate the discontinuity $$\\left( s_{i}\\geq 0 \\right)$$ using a flexible sixth-order polynomial on either side of each normalized threshold between each contiguous Score category, allowing for a discontinuity at 0. The reported estimates refer to $$S_{i}$$, a binary variable that takes a value of 1 if the continuous variable $$s_i \\geq 0$$, that is, if the firm is allocated to the lower credit risk category as opposed to the higher credit risk category. See Table 2 for other variable definitions. One star denotes significance at the 10% level, two stars denote significance at the 5% level, and three stars denote significance at the 1% level. Table 10 shows that most of our estimates on the other thresholds of the Score are not statistically significant. This confirms that our results capture a form of market segmentation, not a simple rating effect, as the only rating values that matter are those moving firms between the performing and substandard classes of credit. 8. Conclusions In this paper, we ask whether the effects of firm segmentation into performing and substandard rating classes can affect the lending policies of banks across the credit cycle. We take advantage of the institutional features of the Italian credit market for SME in order to obtain a quasi-random assignment of firms into these classes of credit risk. The resulting patterns of lending differences give us a new, contract-level measure for the bank lending standards. In this setting, bank lending standards are driven by market segmentation and reflect banks\u2019 sensitivity to the markets for banks\u2019 capital. While our analysis focuses on the single credit cycle that interested the Italian economy between 2004 and 2011, there are two considerations that support both the external validity and the interest of our results. First, the aggregate financing patterns of the Italian economy during this period were similar to those of other OECD economies. Second, the credit cycle in our data culminates with the great recession. This renders the analysis particularly interesting, as it allows us to provide implications for the qualitative and quantitative features of lending standards before and during those years, and the consequences for real allocations. Finally, we discuss the implications of our analysis for the allocative efficiency of banks\u2019 credit policies. By construction, firms in our empirical design are ex ante identical and should, absent the threshold, receive the same credit conditions. This means that, whenever we observe differences in the credit terms at the threshold, there is an inefficiency caused by segmentation in the relative allocation of credit. We thank the editor (Robin Greenwood) and two anonymous referees for insightful comments. The paper also benefited from comments by Klaus Adam, Allen Berger, Steve Bond, Elena Carletti, Antonio Ciccone, Decio Coviello, Matteo Crosignani, Andrew Ellul, Carlo Favero, Nicola Gennaioli, Simon Gilchrist, Martin Hellwig, Victoria Ivashina, Rajkamal Iyer, Nobuhiro Kiyotaki, Augustin Landier, Rocco Macchiavello, Tommaso Nannicini, Steven Ongena, Marco Pagano, Nicola Pavanini, Nicola Persico, Jos\u00e9-Luis Peydr\u00f3, Andrea Polo, Andrea Pozzi, Manju Puri, Antoinette Schoar, Amit Seru, Enrico Sette, Andrei Shleifer, Jeremy Stein, Javier Su\u00e1rez, Adi Sunderam, Michele Tertilt, David Thesmar, Franco Varetto, Egon Zakraj\u0161ek, and participants in the Banque de France (ACPR), Bank of Italy, Bank of Spain, Bocconi, CSEF, Danmarks Nationalbank, EIEF, Goethe University (Frankfurt), HEC Montreal, IFN (Stockholm), Italian Treasury Department, University of Mannheim, Max Planck Institute (Bonn), Tilburg University, Universit\u00e0 Tor Vergata (Rome) seminars and in the NBER Summer Institute (Capital Markets and the Economy), Swiss Conference on Financial Intermediation, Annual Bank Research Conference FDIC\/JFSR, European Winter Finance Summit, ESSFM, Csef-Igier Symposium on Economics and Institutions, First Young Scholars Finance Consortium (Texas A&M), Petralia Workshop and 4Nations Cup conferences for helpful comments. The views expressed are those of the authors and do not necessarily reflect those of the Bank of Italy. Emanuele Tarantino thanks the EIEF for its hospitality. Supplementary data can be found on The Review of Financial Studies web site. Footnotes 1 A possibility would be to look at the U.S. syndicated loan market, which allows us to use a long time series of data within a well-known environment. However, borrowers in this market tend to be significantly larger than a typical SME (Sufi 2007; Ivashina 2009). 2 Specifically, the definition of NPL includes bad loans, past due, and loans to insolvent firms other than substandard credits. The latter are defined as exposures to counterparties facing temporary difficulties defined on the basis of objective factors. 3 This literature finds that the flow of credit (e.g., Covas and Den Haan 2011; Jermann and Quadrini 2012; Becker and Ivashina 2014) and the value of credit spreads (Gilchrist, Yankov, and Zakraj\u0161ek 2009) are both highly procyclical. 4 Our results also inform the (growing) theoretical literature on lending standards over the cycle (e.g., Dell\u2019Ariccia and Marquez 2006; Martin 2008; Kovbasyuk and Spagnolo 2017; Gete 2017). 5 While the formula in the original Altman\u2019s model is publicly known, the agency uses its own version. Specifically, to our knowledge, CEBI\u2019s version of the model uses approximately fifteen factors taken from firms\u2019 balance sheets; however, the exact composition and weights in the formula are a business secret. That is, they are not shared with the regulator or the banks. 6 The continuous variables are difficult to interpret because their value is industry specific. Moreover, differently from the discrete value of the rating, by construction, they do not provide the bank with a direct estimate of the firm default probability (Altman 2004). 7 Descriptive statistics on firms\u2019 distribution in the rating categories can be found in Online Appendix B (Figure B1). 8 To understand the consequences for firms of this classification in terms of S&P\u2019s ratings, note that a Score of 6 corresponds to class B, and a Score of 7 to class CCC (Altman 2004). 9 Specifically, the definition of NPL includes bad loans, past due, and loans to insolvent firms other than substandard credits. The latter are defined as exposures to counterparties facing temporary difficulties defined on the basis of objective factors. 10 Additionally, NPL weigh in the banks\u2019 balance sheets for two main reasons. The first is that there are very limited fiscal and accounting incentives for banks to write off and sell NPL. The second is related to the lengthy Italian bankruptcy system (Rodano, Serrano-Velarde, and Tarantino 2016), and the small number of asset management companies willing to buy these assets. 11 For example, in their banks\u2019 rating guidelines, (Moody\u2019s 2015, 33) reports that \u201c[asset] risks are captured, to a considerable degree, by a single financial ratio, problem loans\/gross loans (which we term the problem loan ratio),\u201d and Fitch (2016) specifies that the \u201ccore metric\u201d to measure asset quality is the problem loan ratio. 12 In this section, we present the model\u2019s main insights. The full derivation can be found in Online Appendix D. While this theoretical framework relies on \u201cex post monitoring,\u201d the intuition extends to models of \u201cex ante screening.\u201d A previous version of the paper explored this mechanism and showed the robustness of the conclusions. In the boom period, when screening is costly and bank liquidity is aplenty, the bank pools the firms at the threshold with the other firms in the same asset class. This means that all borrowers receive lending at a return that reflects the average degree of risk in a class (thus leading to price differences at the threshold). In the bust period, the exacerbation of the adverse selection problem, combined with a shortage of the banking sector\u2019s liquidity, implies that the bank engages in screening at equilibrium. Screening then leads to differences in the quantity of credit offered to the firms at the threshold that penalize those borrowers falling in the substandard class. 13 In the absence of segmentation, the two firms would always obtain the same contract with the bank at equilibrium. 14 We estimate alternative specifications in which we scale the supply of bank financing by assets or express interest rates in terms of basis point differences, and we obtain the same results. To simplify the analysis, we restrict $$f_{t}(\\cdot)$$ and $$g_{t}(\\cdot)$$ to be of the same polynomial order. However, our results are not sensitive to this choice. Finally, we also use local-linear functions to estimate differences in credit conditions at the threshold. Our results remain robust to these additional checks. 15 Clearly, one limitation of this analysis is that the reason for the downgrade might itself be correlated to the demand for credit of the firm. 16 We thank the anonymous referee for very helpful suggestions on this point. 17 To obtain the exact percentage changes, we compute $$\\left[\\left(\\exp\\left\\{\\hat{\\beta}\\right\\}-1\\right)\\times100 \\right]$$, where $$\\hat\\beta$$ is the per-period coefficient. 18 We also explored the sensitivity of bank lending to other sources of bank heterogeneity. For instance, consistent with the previous results, we find that the banks that were highly exposed to the interbank market significantly cut lending to the substandard firms at the threshold in 2008 and 2009. Similarly, during that period, intermediaries putting more weight on soft information when setting credit policies were less likely to cut their lending to substandard borrowers. One needs to be careful when interpreting this last result, as bank organizational structure is likely to be correlated with differences in size and investor composition. 19 Figure C1 in Online Appendix C provides the year-by-year plots associated with these tests. We also plot the distribution of firms that enter rating categories 6 or 7 in any given year. If firms were able to determine the value of their own continuous variable, then we should observe a disproportionate number of new firms clustering just above the threshold, in category 6. Confirming the lack of manipulation, Figure C2 of Online Appendix C shows that a significant mass of firms enters the sample with a value of the continuous variable that lies just below the threshold, in category 7. Finally, we also jointly test for manipulation across the entire cycle and find no evidence of bunching. 20 This evidence is important since small banks are typically seen as more efficient in generating private information about borrowers. Thus, one possibility would be that differences in lending are due to borrowers self-selecting into different bank relations. 21 Table C3 of Online Appendix C shows the results of additional balancing tests. 22 Our results remain the same when plotting bins of different size, like $$0.02$$ or $$0.01$$. 23 Note that, around the threshold, the relationship between credit outcomes and the continuous value of the rating is not necessarily monotonic. Two comments are in order here. First, deriving the identification of the estimates from the units closest to the threshold is precisely the focus of the applied literature on discontinuity designs. Second, on average, the relationship between the value of the rating and the interest rates of the loans is monotonic. To address potential concerns on the sensitivity of our results with respect to bandwidth choices, we reestimate our specification using lower polynomial orders, and local linear methods. Our results are robust to these changes, and can be found in Table C5 of Online Appendix C. 24 In Online Appendix C, Table C4 reports the descriptive statistics about the mean, median, and statistical significance of these placebo tests across all quarters. The estimated values are about zero and are not significant in most of the quarters. Finally, Figure C3 illustrates that a randomly drawn placebo threshold is also unlikely to yield an economically sensible pattern of estimates across time. 25 Due to the construction of the CEBI rating, the threshold between categories 5 and 6 cannot be used (see Section 1). References Agarwal, S., Chomsisengphet, S. Mahoney, N. and Stroebel. J. Forthcoming. Do banks pass through credit expansions? The marginal profitability of consumer lending during the Great Recession. Quarterly Journal of Economics . Altman, E. I. 1968 . Financial ratios, discriminant analysis and the prediction of corporate bankruptcy. Journal of Finance 23 ( 4 ): 589 \u2013 609 . Google Scholar CrossRef Search ADS Altman, E. I. Managing credit risk: the challenge for the new millennium. Presentation updated through 2004, New York University Stern School of Business, http:\/\/people.stern.nyu.edu\/ealtman\/2-%20CopManagingCreditRisk.pdf. CrossRef Search ADS Ashcraft, A. B. 2005 . Are banks really special? New evidence from the FDIC-induced failure of healthy banks. American Economic Review 95 ( 5 ): 1712 \u2013 30 . Google Scholar CrossRef Search ADS Bank of Italy . 2013 . The recent asset quality review on non-performing loans conducted by the Bank of Italy: Main features and results. July 29 . https:\/\/www.bancaditalia.it\/media\/approfondimenti\/2013\/analisi-prestiti-deteriorati\/Asset_quality_review.pdf?language_id=1. Barisitz, S. 2013 . Nonperforming loans in Western Europe: A selective comparison of countries and national definitions. Focus on European Economic Integration, Oesterreichische Nationalbank ( Austrian Central Bank ), no. 1 , pp. 28 \u2013 47 . Becker, B., and Ivashina V. 2014 . Cyclicality of credit supply: Firm level evidence. Journal of Monetary Economics 62 ( C ): 76 \u2013 93 . Google Scholar CrossRef Search ADS Bholat, D., Lastra, R. Markose, S. Miglionico, A. and Sen. K. 2016 . Non-performing loans: Regulatory and accounting treatments of assets. Bank of England Working Papers, no. 594 . Brollo, F., Nannicini, T. Perotti, R. and Tabellini. G. 2013 . The political resource curse. American Economic Review 103 ( 5 ): 1759 \u2013 96 . Google Scholar CrossRef Search ADS Calonico, S., Cattaneo, M. D. and Titiunik. R. 2014 . Robust nonparametric confidence intervals for regression discontinuity designs. Econometrica 82 ( 6 ): 2295 \u2013 2326 . Google Scholar CrossRef Search ADS Chernenko, S., and Sunderam. A. 2012 . The real consequences of market segmentation. Review of Financial Studies 25 ( 7 ): 2041 \u2013 69 . Google Scholar CrossRef Search ADS Chodorow-Reich, G. 2014 . The employment effects of credit market disruptions: Firm-level evidence from the 2008\u20132009 financial crisis. Quarterly Journal of Economics 129 ( 1 ): 1 \u2013 59 . Google Scholar CrossRef Search ADS Covas, F., and Den Haan. W. J. 2011 . The cyclical behavior of debt and equity finance. American Economic Review 101 ( 2 ): 877 \u2013 99 . Google Scholar CrossRef Search ADS Dell\u2019Ariccia, G., and Marquez. R. 2006 . Lending booms and lending standards. Journal of Finance 61 ( 5 ): 2511 \u2013 46 . Google Scholar CrossRef Search ADS Drehmann, M., Borio, C. and Tsatsaronis. K. 2012 . Characterising the financial cycle: Don\u2019t lose sight of the medium term! BIS Working Papers, no. 380 , Bank for International Settlements . Fitch . 2016 . Global bank rating criteria. July 15 . https:\/\/www.fitchratings.com\/site\/re\/884135. Gete, P. 2017 . Banking crises, lending standards and misallocation. http:\/\/dx.doi.org\/10.2139\/ssrn.2905308. Gilchrist, S., Yankov, V. and Zakraj\u0161ek. E. 2009 . Credit market shocks and economic fluctuations: Evidence from corporate bond and stock markets. Journal of Monetary Economics 56 ( 4 ): 471 \u2013 93 . Google Scholar CrossRef Search ADS Gorton, G. B., and Metrick. A. 2012 . Securitized banking and the run on repo. Journal of Financial Economics 104 ( 3 ): 425 \u2013 51 . Google Scholar CrossRef Search ADS Greenwood, R., and Hanson. S. G. 2013 . Issuer quality and corporate bond returns. Review of Financial Studies 26 ( 6 ): 1483 \u2013 1525 . Google Scholar CrossRef Search ADS Imbens, G., and Kalyanaraman. K. 2014 . Optimal bandwidth choice for the regression discontinuity estimator. Review of Economic Studies 79 ( 3 ): 933 \u2013 59 . Google Scholar CrossRef Search ADS Intesa . 2015 . Consolidated financial statements (Part E). http:\/\/www.group.intesasanpaolo.com\/scriptIsir0\/si09\/contentData\/view\/20150408_RischiCredito_uk.pdf?id=CNT-05-000000025BE48&ct=application\/pdf. Ivashina, V. 2009 . Asymmetric information effects on loan spreads. Journal of Financial Economics 92 ( 2 ): 300 \u2013 319 . Google Scholar CrossRef Search ADS Ivashina, V., and Scharfstein. D. 2009 . Bank lending during the financial crisis of 2008. Journal of Financial Economics 97 ( 3 ): 319 \u2014 38 . Google Scholar CrossRef Search ADS Iyer, R., Puri, M. and Ryan. N. 2016 . A tale of two runs: Depositor responses to bank solvency risk. Journal of Finance 71 ( 6 ): 2687 \u2013 2726 . Google Scholar CrossRef Search ADS Jassaud, N., and Kang. K. H. 2015 . A strategy for developing a market for nonperforming loans in Italy. IMF Working Paper, no. 15\/24 , International Monetary Fund. Google Scholar CrossRef Search ADS Jermann, U., and Quadrini. V. 2012 . Macroeconomic effects of financial shocks. American Economic Review 102 ( 1 ): 238 \u2013 71 . Google Scholar CrossRef Search ADS PubMed Jim\u00e9nez, G., Ongena, S. Peydr\u00f3, J.-L. and Saurina. J. 2012 . Credit supply and monetary policy: Identifying the bank balance-sheet channel with loan applications. American Economic Review 102 ( 5 ): 2301 \u2013 26 . Google Scholar CrossRef Search ADS Jim\u00e9nez, G., Ongena, S. Peydr\u00f3, J.-L. and Saurina. J. 2014 . Hazardous times for monetary policy: What do 23 million loans say about the impact of monetary policy on credit risk-taking? Econometrica 82 ( 2 ): 463 \u2013 505 . Google Scholar CrossRef Search ADS Kisgen, D. J., and Strahan. P. E. 2010 . Do regulations based on credit ratings affect a firm\u2019s cost of capital. Review of Financial Studies 23 ( 12 ): 4324 \u2013 47 . Google Scholar CrossRef Search ADS Khwaja, A. I., and Mian. A. 2008 . Tracing the impact of bank liquidity shocks: Evidence from an emerging market. American Economic Review 98 ( 4 ): 1413 \u2013 42 . Google Scholar CrossRef Search ADS Kovbasyuk, S., and Spagnolo. G. 2017 . Memory and markets. EIEF Working Papers, no. 1606, Einaudi Institute for Economics and Finance. Lemmon, M., and Roberts. M. R. 2010 . The response of corporate financing and investment to changes in the supply of credit. Journal of Financial and Quantitative Analysis 45 ( 3 ): 555 \u2013 87 . Google Scholar CrossRef Search ADS Lopez-Salido, D., Stein, J. C. and Zakraj\u0161ek. E. 2017 . Credit-market sentiment and the business cycle. Quarterly Journal of Economics 132 ( 3 ): 1373 \u2013 1426 . Google Scholar CrossRef Search ADS McCrary, J. 2008 . Manipulation of the running variable in the regression discontinuity design: a density test. Journal of Econometrics 142 ( 2 ): 698 \u2013 714 . Google Scholar CrossRef Search ADS Moody\u2019s. 2015 . Rating methodology: Banks. March 16 . www.moodys.com\/methodologies. Martin, A. 2008 . Endogenous credit cycles. Economics Working Papers, no. 916, Department of Economics and Business, Universitat Pompeu Fabra. Google Scholar CrossRef Search ADS OECD . 1997 . Small Businesses, Job Creation and Growth: Facts, Obstacles and Best Practices. Paravisini, D., Rappaport, V. Schnabl, P. and Wolfenzon. D. 2014 . Dissecting the effect of credit supply on trade: Evidence from matched credit-export data. Review of Economic Studies 82 ( 1 ): 333 \u2013 359 . Google Scholar CrossRef Search ADS Rajan, R. G. 2005 . Has financial development made the world riskier? Proceedings of the Economic Policy Symposium, Jackson Hole, Federal Reserve Bank of Kansas City , August , 313 \u2013 69 . Rodano, G., Serrano-Velarde, N. and Tarantino. E. 2016 . Bankruptcy law and bank financing. Journal of Financial Economics 120 ( 2 ): 363 \u2013 82 . Google Scholar CrossRef Search ADS Stein, J. C. 2002 . Information production and capital allocation: Decentralized versus hierarchical firms. Journal of Finance 57 ( 5 ): 1891 \u2013 1921 . Google Scholar CrossRef Search ADS Standard & Poor\u2019s. 2004 . Credit risk tracker Italy. Standard & Poor\u2019s Risk Solutions. Sufi, A. 2007 . Information asymmetry and financing arrangements: Evidence from syndicated loans. Journal of Finance 62 ( 2 ): 629 \u2013 68 . Google Scholar CrossRef Search ADS Tirole, J. 2006 . The theory of corporate finance . Princeton, NJ : Princeton University Press. Unicredit Bank. 2008 . Unicredit S.p.A. 2008 Annual Report. World Bank . 2002 . Bank loan classification and provisioning practices in selected developed and emerging countries (a survey of current practices in countries represented on the Basel Core Principles Liaison Group). \u00a9 The Author(s) 2018. Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https:\/\/academic.oup.com\/journals\/pages\/about_us\/legal\/notices)\n\n### Journal\n\nThe Review of Financial StudiesOxford University Press\n\nPublished: Apr 24, 2018\n\n## You\u2019re reading a free preview. Subscribe to read the entire article.\n\n### DeepDyve is your personal research library\n\nIt\u2019s your single place to instantly\nthat matters to you.\n\nover 18 million articles from more than\n15,000 peer-reviewed journals.\n\nAll for just $49\/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49\/month\n\\$360\/year\n\nSave searches from\nPubMed\n\nCreate lists to\n\nExport lists, citations","date":"2018-12-17 07:49:50","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\": 8, \"equation\": 2, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.33737945556640625, \"perplexity\": 1663.179691930869}, \"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-2018-51\/segments\/1544376828448.76\/warc\/CC-MAIN-20181217065106-20181217091106-00187.warc.gz\"}"}
| null | null |
Натуральне число 1865
1865 рік до нашої ери
1865 рік нашої ери
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 742
|
{"url":"https:\/\/socratic.org\/questions\/58f056cd11ef6b52b225b0b1","text":"# Question #5b0b1\n\nApr 14, 2017\n\n$105 m$\n\n#### Explanation:\n\nUsing the applicable kinematic expression\n\n$h = u t + \\frac{1}{2} g {t}^{2}$\nwhere $h$ is the height after time $t$. $u$ is initial velocity and $g$ is acceleration due to gravity.\n\nInserting giving numbers and remembering that gravity is acting against the direction of motion\n\n$h = 50 \\times 7 + \\frac{1}{2} \\left(- 10\\right) {\\left(7\\right)}^{2}$\n$\\implies h = 350 - 245$\n$\\implies h = 105 m$","date":"2020-06-06 18:45:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 9, \"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\": 0.8539968132972717, \"perplexity\": 1397.0236337043264}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590348517506.81\/warc\/CC-MAIN-20200606155701-20200606185701-00140.warc.gz\"}"}
| null | null |
\section{Introduction}
Interacting many-body spin ensembles exhibit a variety of
phenomena such as phase transitions \citep{Oja1997,Kessler2012}
spin waves \citep{deGennes1963,Shiomi2019} and emergent
thermodynamics \citep{Dorner2012,Bardarson2012}. Spin diffusion
\citep{Bloembergen1949,Blumberg1960} is one of the earliest
studied phenomena, where unitary quantum-mechanical evolution
results in an irreversible dissipation of a localized spin
polarization that is well described by the classical diffusion
model. Pure spin diffusion in homogeneous solids has been observed
in a few notable examples \citep{Zhang1998,Eberhardt2007}.
However, most systems of interest are inhomogeneous by nature. In
particular, magnetic (hyperfine) interaction with the central spin
of a localized electron [Fig.~\ref{Fig:Intro}(a)] causes shifts
(known as the Knight shifts\cite{Knight1949,Urbaszek2013}) in the
nuclear spin energy levels [Fig.~\ref{Fig:Intro}(b)]. The
resulting nuclear spin dynamics are complicated, as observed in a
wide range of solid state impurities
\citep{Leppelmeier1968,Wolfe1973,Paget1982,Lu2006,Hayashi2008,Wittmann2018,Ashok2018}
and semiconductor nanostructures
\citep{Tycko1995,Bayot1997,Hayashi2008,Nikolaenko2009,Reilly2010,Makhonin2010,Sallen2014}.
It is still an open question whether the inhomogeneous Knight
shifts accelerate \citep{Klauser2008,Reilly2010,Gong2011} or
suppress
\citep{Deng2005,Lai2006,Lu2006,Ramanathan2008,Gong2011,Sallen2014}
spin diffusion between the nuclei. Resolving this dilemma is both
of fundamental interest and practical importance for the recent
proposals to use nuclear spins as quantum memories and registers
\citep{Heshami2016,Denning2019,Chekhovich2020}, since spin
diffusion would set an ultimate limit to the longevity of any
useful quantum state.
\begin{figure}
\includegraphics[width=0.95\linewidth]{FigDiffSketch.pdf
\caption{\label{Fig:Intro} Schematic of a central spin model,
sketched for the one-dimensional case, along the growth axis $z$
of a GaAs/AlGaAs structure. (a) Wavefunction density
$|\psi_{\rm{e}}|^2$ of an electron ($e$, ball and arrow) localized
in GaAs. (b) Energy levels of the nuclei, that are depicted for
simplicity as spins 1/2, and can occupy states with $+1/2$ and
$-1/2$ spin projections (up and down arrows). Dashed lines show
the bulk nuclear spin energies dominated by the external magnetic
field $B_{\rm{z}}$. These bulk energies are generally different in
GaAs ($z\leq4$) and AlGaAs ($z\geq5$) due to the difference in
chemical shifts and homogeneous strain. The energies of the
individual nuclei are further shifted by the electron Knight field
(mainly in GaAs) and by the atomic-scale strain disorder in the
AlGaAs alloy. Magnetic dipole interaction between the nuclei can
result in spin exchange via a flip-flop process, sketched by the
curved arrows for nuclei at $z=-1$ and $z=0$ as an example. If
energy mismatch is larger than the nuclear spin level homogenous
broadening, for example for nuclei at $z=4$ and $z=5$, spin
exchange becomes prohibited, suppressing nuclear spin diffusion.}
\end{figure}
Figure~\ref{Fig:Intro} sketches the central spin model where an
electron can be trapped in a GaAs layer surrounded by the AlGaAs
barriers, and for simplicity spin-1/2 particles are used to
describe the energy levels of the nuclei subject to the strong
external magnetic field $B_{\rm{z}}$. Any two nuclear spins $i$
and $j$ are coupled by the dipole-dipole interaction
$\propto2\hat{I}_{{\rm{z}},i}\hat{I}_{{\rm{z}},j}-(\hat{I}_{{\rm{x}},i}\hat{I}_{{\rm{x}},j}+\hat{I}_{{\rm{y}},i}\hat{I}_{{\rm{y}},j})$,
where $\hat{I}_{{\rm{x}},i}$, $\hat{I}_{{\rm{y}},i}$ and
$\hat{I}_{{\rm{z}},i}$ are the Cartesian components of the spin
operator ${\bf{I}}_i$ of the $i$-th nucleus. The last two terms of
the dipole-dipole interaction describe a flip-flop spin exchange
process (dashed arrows at $z=-1$ and $0$ in
Fig.~\ref{Fig:Intro}(b)), responsible for the transfer of spin
polarization in space, known as spin diffusion. The electric
quadrupolar moments of the spin-3/2 nuclei make them sensitive to
electric field gradients (EFGs), which can be induced by the
GaAs/AlGaAs interfaces roughness ($z=4.5$) or atomic-scale strains
arising from random positioning of the aluminium atoms
\citep{MALINOWSKI2001,Knijn2010} in AlGaAs ($z\ge5$). The
resulting energy splitting mismatch between the adjacent nuclei
can impede nuclear spin diffusion.
When an electron is added, its spin $\bf{s}$ couples to the
nuclear spin ensemble via hyperfine interaction:
\begin{align}
\mathcal{\hat{H}}_{\rm{hf}}=\sum_{j}{A_j(\hat{s}_{\rm{x}}\hat{I}_{{\rm{x}},j}+\hat{s}_{\rm{y}}\hat{I}_{{\rm{y}},j}+\hat{s}_{\rm{z}}\hat{I}_{{\rm{z}},j})},\label{Eq:Hhf}
\end{align}
where the summation goes over all nuclei $j$, and the coupling
constants $A_j$ are proportional to the electron density
$|\psi_{\rm{e}}({\bf{r}}_j)|^2$ at the nuclear sites ${\bf{r}}_j$.
On the one hand, through the term
$\hat{s}_{\rm{z}}\hat{I}_{{\rm{z}}}$, the electron spin can
produce a further diffusion barrier
\citep{Deng2005,Lai2006,Lu2006,Ramanathan2008,Gong2011,Sallen2014},
at the points of strong Knight shift gradient ($z=3$ in
Fig.~\ref{Fig:Intro}(a)). On the other hand, the electron spin can
mediate spin flip-flops between two distant nuclei with similar
energy splitting (e.g. $z=-2$ and $z=2$), potentially opening a
new channel for spin diffusion, especially at low magnetic fields
\citep{Klauser2008,Reilly2010,Gong2011}.
Here, we examine electron-controlled nuclear spin diffusion in
high quality epitaxial GaAs/AlGaAs quantum dots (QDs), which
emerged recently as an excellent platform for quantum light
emitters \citep{Liu2019,Zhai2020,Tomm2021} as well as spin qubits
\citep{Chekhovich2020,Zaporski2022} and quantum memories
\citep{Denning2019}. Crucially, we design experiments where
nuclear spin dynamics are examined either in absence or in
presence of the electron central spin, but under otherwise
identical initial nuclear spin state. In this way, we distinguish
with high accuracy the effects specific to the electron spin, and
demonstrate that no observable Knight field barrier is formed.
Instead, the nuclear-nuclear interactions mediated by the electron
spin accelerate nuclear spin diffusion up to unexpectedly high
magnetic fields -- we attribute this to the previously neglected
impact of the electron spin flips. Our results answer a
long-standing question in spin physics, and provide practical
guidelines for the design and optimization of quantum dot
electron-nuclear spin qubits and quantum memories.
\begin{figure*}
\includegraphics[width=0.95\linewidth]{FigSamplePLNMR.pdf
\caption{\label{Fig:SampNMR} (a) Atomic force microscopy (AFM)
profile of the AlGaAs surface after nanohole etching. (b) Surface
level profiles taken along the horizontal and vertical lines
through the center of the nanohole in (a). (c) Schematic (not to
scale) of the sample structure. GaAs quantum dots (QDs) are formed
by infilling of the in-situ etched nanoholes in the bottom
Al$_{0.33}$Ga$_{0.67}$As barrier. The bottom (top)
Al$_{0.15}$Ga$_{0.85}$As layer is $n$ ($p$) type doped to form a
$p-i-n$ diode structure. External gate bias $V_{\rm{Gate}}$ is
applied for deterministic QD charging with electrons. (d)
Photoluminescence (PL) spectra of a negatively charged trion $X^-$
following $\sigma^+$ (triangles) and $\sigma^-$ optical pumping
which induces nuclear spin polarization that manifests in
hyperfine shifts of the Zeeman doublet spectral splitting $\Delta
E_{\rm{PL}}$. (e) Optically detected NMR of the $^{75}$As spin-3/2
nuclei measured in a single QD. Strain-induced quadrupolar shifts
of the nuclear spin-3/2 levels (left inset) give rise to an NMR
triplet with splitting $\nu_{\rm{Q}}\approx24$~kHz, observed in an
empty QD (0$e$, diamonds). Charging the QD with a single electron
(1$e$, circles) induces inhomogeneous Knight shifts observed as
NMR spectral broadening. The measurement is conducted using
``inverse NMR'' signal amplification technique
\citep{Chekhovich2012}, with spectral resolution shown by the
horizontal bars (smaller for 0$e$ and larger for 1$e$). The
measurement pump-probe cycle is shown in the top inset. The bias
$V_{\rm{Gate}}$ is tuned to 0$e$ charge state for optical pumping
of the nuclear spins and to 1$e$ state for their optical probing.
The radio frequency (RF) pulse is applied in the dark under either
0$e$ or 1$e$ bias.}
\end{figure*}
\section{Sample and experimental techniques}
The studied heterostructure is grown by in-situ etching of
nanoholes \cite{Heyn2009,Atkinson2012} in the AlGaAs surface
[Figs.~\ref{Fig:SampNMR}(a,b)], which are then infilled with GaAs
to form the quantum dots (QDs). The structure is processed into a
$p-i-n$ diode [Fig.~\ref{Fig:SampNMR}(c)] where an external bias
$V_{\rm{Gate}}$ is applied to charge QDs deterministically with
individual electrons (See details in Supplementary Section 1). In
this way, it is possible to study nuclear spin dynamics in an
empty (0$e$) or single-electron (1$e$) state. A static magnetic
field $B_{\rm{z}}$ is applied along the growth axis $z$ (Faraday
geometry) and the sample is kept at liquid helium temperature of
4.2~K. We use confocal microscopy configuration where QD
photoluminescence (PL) is excited and collected through an
aspheric lens with a focal distance of 1.45~mm and numerical
aperture of 0.58. The collected PL is dispersed in a two-stage
grating spectrometer, and recorded with a charge-coupled device
(CCD) camera.
The changes in the PL spectral splitting $\Delta E_{\rm{PL}}$ of a
negatively charged trion $X^-$ [see Fig.~\ref{Fig:SampNMR}(d)] is
the hyperfine shift $E_{\rm{hf}}$, which gives a measure of an
average nuclear spin polarization degree within the QD
\citep{Urbaszek2013}. The hyperfine shifts (also known as
Overhauser shifts) arise from the
$\hat{s}_{\rm{z}}\hat{I}_{{\rm{z}}}$ term of the hyperfine
interaction Hamiltonian (Eq.~\ref{Eq:Hhf}). Large nonequilibrium
nuclear spin polarization is generated on demand by exciting the
QD with a circularly polarized pump laser, which repeatedly
injects spin-polarized electrons into a QD, and causes nuclear
spin polarization build up via electron-nuclear spin flip-flops
described by the
$\hat{s}_{\rm{x}}\hat{I}_{{\rm{x}}}+\hat{s}_{\rm{y}}\hat{I}_{{\rm{y}}}$
part of Eq.~\ref{Eq:Hhf}. A small copper wire coil is placed near
the sample to produce radiofrequency (RF) oscillating magnetic
field perpendicular to the static magnetic field. Application of
the RF field allows for the energy spectrum of the nuclear spins
to be probed via nuclear magnetic resonance (NMR). Moreover, the
RF field can be used to depolarize the nuclear spins on-demand.
Further details can be found in the Supplementary Section 2,
including sample growth details, PL spectra, characterization of
QD charge state control, and additional results at an elevated
temperature of 15.2~K.
\section{Experimental results and discussion}
\subsection{Nuclear spin system of a GaAs quantum dot}
Figure~\ref{Fig:SampNMR}(e) shows NMR spectra of $^{75}$As in a
single GaAs QD, measured using ``inverse NMR'' technique with an
optical Pump-RF-Probe cycle shown in the top inset. For an empty
QD (open symbols), an NMR triplet is observed \citep{Ulhaq2016},
corresponding to the three magnetic-dipole transitions between the
four Zeeman-split states $I_{\rm{z}}=\{-3/2,-1/2,+1/2,+3/2\}$ of a
spin-3/2 nucleus (left inset). The central resolution-limited peak
originates from the $-1/2\leftrightarrow +1/2$ NMR transition that
is weakly affected by strain. The two satellite transition peaks
$\pm1/2\leftrightarrow \pm3/2$ are split from the central
transition peak by the strain-induced EFGs. The average splitting
$\nu_{\rm{Q}}\approx24$~kHz between the triplet components
corresponds to an average elastic strain of
$\approx2.6\times10^{-4}$ (Refs.
\citep{Chekhovich2018,Griffiths2019}). The satellite transitions
are inhomogeneously broadened, with non-zero NMR amplitudes
detected approximately in a range of $\nu_{\rm{Q}}\in[10,50]$~kHz,
indicating that elastic strain varies within the nanoscale volume
of the QD.
When a single electron occupies the QD, it induces inhomogeneous
Knight shifts that exceed the quadrupolar shifts, leading to a
broadened NMR peak [solid symbols in Fig.~\ref{Fig:SampNMR}(e)].
From the NMR peak width, the Knight shifts, characterizing the
typical coupling strength between the electron spin and an
individual nuclear spin, are estimated to be $A_j/h\approx50$~kHz,
where $h$ is the Planck's constant.
These NMR characterization results indicate a complex interplay of
dipolar, quadrupolar and hyperfine interactions governing the
nuclear spin dynamics, which we now investigate experimentally.
\subsection{Nuclear spin relaxation in a GaAs quantum dot}
In the nuclear spin relaxation (NSR) experiment [see timing
diagram in Fig.~\ref{Fig:Diff}(a)] any remnant nuclear spin
polarization is first erased by saturating the $^{75}$As,
$^{69}$Ga and $^{71}$Ga NMR resonances in the entire
heterostructure \citep{Barrett1995}. This is followed by a
variable-duration ($T_{\rm{Pump}}$) optical pumping
\citep{Paget1982,Tycko1995,Hayashi2008,Nikolaenko2009} with photon
energies below the AlGaAs barrier bandgap, which prepares nuclear
spin polarization localized around the QD nanoscale volume. After
the pump laser is turned off, the gate bias $V_{\rm{Gate}}$ is set
to a desired level for a dark time $T_{\rm{Dark}}$ -- this way
evolution under 0$e$ or 1$e$ QD charge state is studied for
nominally identical initial nuclear spin polarizations. Finally,
the remaining polarization within the QD volume is probed through
an optically detected hyperfine shift $E_{\rm{hf}}$.
\begin{figure*}
\includegraphics[width=0.95\linewidth]{FigT1N.pdf}
\caption{\label{Fig:Diff} (a) NSR measurement cycle starting with
a radio frequency erase pulse, followed by circularly polarized
($\sigma^+$) optical pumping and then optical probing of the QD
nuclear spin polarization after dark evolution delay
$T_{\rm{Dark}}$ (see details in Supplementary Section 2). The
sample gate bias $V_{\rm{Gate}}$ is varied allowing to choose
between 0$e$ (dashed line) and 1$e$ (dotted line) QD charge state
during $T_{\rm{Dark}}$. (b) Dark time dependence of the hyperfine
shift $E_{\rm{hf}}$, which probes the average polarization of
$\approx10^5$ QD nuclear spins, weighted by the QD electron
density $|\psi_{\rm{e}}|^2$. The nuclear spin decay is measured
(symbols) at $B_{\rm{z}}=0.39$~T for different pumping times
$T_{\rm{Pump}}$ while keeping the QD empty (0$e$, open symbols) or
charged with one electron (1$e$, solid symbols) during the dark
time. Lines show numerical solution of the spin diffusion
Eq.~\ref{Eq:DiffEq}. (c) Fitted QD nuclear spin half-life times
$T_{\rm{1,N}}$ (right scale) and the corresponding NSR rates
$\varGamma_{\rm{N}}=1/T_{\rm{1,N}}$ (left scale) at magnetic field
$B_{\rm{z}}=0.39$~T. (d) same as (c) for $B_{\rm{z}}=9.82$~T. All
results are for the same individual dot QD1. Error bars are 95\%
confidence intervals.}
\end{figure*}
Figure~\ref{Fig:Diff}(b) shows the average QD nuclear spin
polarization as a function of the pump-probe delay $T_{\rm{Dark}}$
during which the sample is kept in the dark. The decay is
non-exponential, thus we characterize the NSR timescale
$T_{\rm{1,N}}$ by the half-life time over which the QD hyperfine
shift $E_{\rm{hf}}$ decays to 1/2 of its initial value. The NSR
rate is then defined as $\varGamma_{\rm{N}}=1/T_{\rm{1,N}}$. When
the pumping time $T_{\rm{Pump}}$ is increased, $T_{\rm{1,N}}$
notably increases, as can be seen in Figs.~\ref{Fig:Diff}(c,d).
Such dependence of $T_{\rm{1,N}}$ on $T_{\rm{Pump}}$ is observed
both in an empty (0$e$) and charged (1$e$) QD states, and in a
wide range of magnetic fields.
\subsection{Nuclear spin diffusion}
In order to explain the results of Fig.~\ref{Fig:Diff}, we note
that nuclear spin dipole-dipole interactions conserve the nuclear
spin polarization for any magnetic field exceeding the dipolar
local field, typically $\lesssim1$~mT. Therefore, at high magnetic
field the decay of nuclear spin polarization can proceed via two
routes: either via spin-conserving diffusion to the surrounding
nuclei, or spin transfer to external degrees of freedom, including
quadrupolar coupling to lattice vibrations
\citep{McNeil1976,Lu2006} or a hyperfine interaction with a charge
spin \citep{Lu2006,Latta2011,Vladimirova2017,Gillard2021} that is
in turn coupled to the lattice or other charges. Spin diffusion
can only take place if the spatial profile of the initial nuclear
spin polarization is inhomogeneous. By contrast, direct
spin-lattice and hyperfine interactions have no explicit
dependence on the spin polarization spatial profile. Optical
pumping that is short compared to spin diffusion timescales
creates nuclear spin polarization localized to the QD volume and
the resulting short $T_{\rm{1,N}}$ is therefore a clear indicator
of spin diffusion as the dominant NSR mechanism
\citep{Paget1982,Tycko1995,Hayashi2008,Nikolaenko2009}.
Conversely, long pumping provides enough time for nuclear
polarization to diffuse from the QD into the surrounding AlGaAs
barriers, suppressing any subsequent spin diffusion out of the QD
and increasing $T_{\rm{1,N}}$, as observed in
Figs.~\ref{Fig:Diff}(c,d).
In order to complement our experimental investigation we model the
spatiotemporal evolution of the nuclear spin polarization degree
$P _{\rm{N}}(t,z)$ by solving numerically the one dimensional spin
diffusion equation
\begin{align}
\frac{\partial P_{\rm{N}}(t,z)}{\partial
t}=D(t)\frac{{\partial}^{2} P _{\rm{N}}(t,z)}{\partial z^2} +\nonumber\\
+w(t)|\psi_{\rm{e}}(z)|^2 (P _{\rm{N},0}-P
_{\rm{N}}(t,z)),\label{Eq:DiffEq}
\end{align}
where the last term describes optical nuclear spin pumping with a
rate proportional to electron density $|\psi_{\rm{e}}(z)|^2$ and
the time-dependent factor $w(t)$ equal to 0 or $w_0$ when optical
pumping is off or on, respectively. Correspondingly, the spin
diffusion coefficient $D(t)$ takes two discrete values
$D_{\rm{Dark}}$ or $D_{\rm{Pump}}$ when optical pumping is off or
on, respectively. $P_{\rm{N},0}$ is a steady state nuclear spin
polarization degree that optical pumping would generate in the
absence of spin diffusion. Eq.~\ref{Eq:DiffEq} is solved
numerically and the parameters such as $D^{(ne)}_{\rm{Dark}}$,
$w_0(B_{\rm{z}})$, $D_{\rm{Pump}}(B_{\rm{z}})$ are varied to
achieve the best fit to the entire experimental datasets of
$E_{\rm{hf}}(T_{\rm{Pump}},T_{\rm{Dark}})$ measured at
$B_{\rm{z}}=0.39, 9.82$~T for empty ($n=0$) and charged ($n=1$) QD
states. The best-fit calculated dynamics are shown by the lines in
Fig.~\ref{Fig:Diff}(b) and capture well the main features of the
experimentally measured nuclear spin decay, confirming the
validity of the spin diffusion picture. The one-dimensional
character of diffusion, occurring predominantly along the sample
growth $z$ direction, is justified by the large ratio of the QD
diameter $\approx70$~nm to QD height $<9~$nm, and is further
verified by modeling two-dimensional spin diffusion (see
Supplementary Section 4).
\subsection{Effect of central spin on nuclear spin diffusion}
Dividing the typical Knight shift of $\approx50$~kHz by half the
QD thickness (4.5~nm) we calculate the gradient and roughly
estimate the Knight shift difference of $\approx4.4$~kHz for the
two nearest-neighbor spins of the same isotope separated by
$a_0/\sqrt{2}$, where $a_0=0.565$~nm is the lattice constant. Such
difference significantly exceeds the energy that can be exchanged
with the dipole-dipole reservoir for a spin flip-flop to happen
\citep{Sallen2014} (the dipole-dipole energy is on the order of
$\approx h/T_{2,{\rm{N}}}$, where $T_{2,{\rm{N}}}\in[1,5]$~ms is
the nuclear spin-echo coherence time
\citep{Chekhovich2015,Chekhovich2020}). Therefore, one may naively
expect a Knight field gradient barrier to form and suppress spin
diffusion in an electron-charged QD (since the flip-flops would be
limited to the few nuclear spin pairs whose vector differences are
nearly orthogonal to the Knight field gradient). By contrast,
Figs.~\ref{Fig:Diff}(c,d) show that in experiment the NSR is
faster when the QD is occupied by a single electron (1$e$, solid
symbols) for all studied $T_{\rm{Pump}}$, demonstrating that no
significant Knight field barrier is formed. However, in order to
quantify the effect of the central spin on nuclear spin diffusion
we must distinguish it from other non-diffusion NSR mechanisms
introduced by the electron spin. To this end, we examine the
magnetic field dependence shown in Fig.~\ref{Fig:DiffBz}.
\begin{figure}
\includegraphics[width=0.95\linewidth]{FigT1NBz.pdf
\caption{\label{Fig:DiffBz} (a) Nuclear spin relaxation (NSR) rate
$\varGamma_{\rm{N}}$ as a function of $B_{\rm{z}}$ measured in
0$e$ (open symbols) and 1$e$ (solid symbols) states upon long
pumping $T_{\rm{Pump}}=990$~s. Top horizontal axis shows the
electron Zeeman splitting at zero nuclear spin polarization. (b)
Ratio $\varGamma_{\rm{N}}^{(1e)}/\varGamma_{\rm{N}}^{(0e)}$ of the
NSR rates in 1$e$ and 0$e$ charge states as a function of
$B_{\rm{z}}$ measured under long pumping $T_{\rm{Pump}}=990$~s
(squares) and short pumping $T_{\rm{Pump}}\in[0.08,0.48]$~s
(triangles). All results are for the same individual dot QD1,
except for the additional data from QD2 at $B_{\rm{z}}=0.5$~T in
(b). Error bars are 95\% confidence intervals.}
\end{figure}
First, we examine a case where long optical pumping is used to
suppress spin diffusion and highlight the non-diffusion
mechanisms. Fig.~\ref{Fig:DiffBz}(a) shows the experimental
dependence $\varGamma_{\rm{N}}(B_{\rm{z}})$ for long
$T_{\rm{Pump}}=990$~s. In an empty QD (0$e$) spin diffusion is
still the dominant NSR mechanisms -- indeed, the observed rates
$\varGamma_{\rm{N}}^{(0e)}\in[3\times10^{-3},6\times10^{-3}]$~s$^{-1}$
are considerably higher than
$\varGamma_{\rm{N}}\in[6\times10^{-5},1\times10^{-3}]$~s$^{-1}$,
found in bulk crystals \citep{McNeil1976} such as semi-insulating
GaAs \citep{Lu2006}, where spin diffusion is negligible. The
electron-induced rates under long pumping
$\varGamma_{\rm{N}}^{(1e)}\in[4\times10^{-3},2\times10^{-2}]$~s$^{-1}$
are nearly independent of $B_{\rm{z}}$, and exceed the 0$e$ rates
by no more than a factor of
$\varGamma_{\rm{N}}^{(1e)}/\varGamma_{\rm{N}}^{(0e)}<4$ [squares
in Fig.~\ref{Fig:DiffBz}(b)]. Such small effect of the electron is
explained by the small strain of the GaAs/AlGaAs structures, which
reduces the efficiency of the non-diffusion NSR mechanisms related
to phonon and electron cotunneling. This is in stark contrast to
the large magnetic-field-induced variation
$\varGamma_{\rm{N}}^{(1e)}\in[5\times10^{-4},1\times10^{1}]$~s$^{-1}$
in Stranski-Krastanov self-assembled InGaAs QDs
\citep{Gillard2021}, where phonon and cotunneling mechanisms
dominate, both enabled by the noncollinear hyperfine interaction
\citep{Latta2011,Gillard2021}, arising in turn from the large
strain-induced nuclear quadrupolar shifts. Overall, the absolute
QD NSR rates shown in Fig.~\ref{Fig:DiffBz}(a) are nearly
constant, with some residual irregular dependence on magnetic
field which we ascribe to uncontrollable parameters, such as
charge state of the nearby impurities, or the initial spatial
profile of the nuclear spin polarization defined by the optical
nuclear spin pumping rate. By contrast, for any given $B_{\rm{z}}$
and $T_{\rm{Pump}}$ the ratio
$\varGamma_{\rm{N}}^{(1e)}/\varGamma_{\rm{N}}^{(0e)}$ shown in
Fig.~\ref{Fig:DiffBz}(b) gives a reliable measure, which we use to
examine the electron spin's effect on spin diffusion.
In order to discriminate the diffusion-related effect of the QD
electron spin, in addition to the long-pumping measurements
[squares in Fig.~\ref{Fig:DiffBz}(b)], we choose for each magnetic
field a short pumping time, typically
$T_{\rm{Pump}}\in[0.08,0.48]$~s, that yields initial QD nuclear
spin polarization at $\approx1/2$ of the steady-state long-pumping
polarization. The resulting short-pumping ratio
$\varGamma_{\rm{N}}^{(1e)}/\varGamma_{\rm{N}}^{(0e)}$ is shown by
the triangles in Fig.~\ref{Fig:DiffBz}(b) -- its excess over the
long-pumping ratio
$\varGamma_{\rm{N}}^{(1e)}/\varGamma_{\rm{N}}^{(0e)}$ is ascribed
to spin diffusion alone, discriminating it from any non-diffusion
mechanisms introduced by the electron spin. The
electron-spin-induced acceleration of the nuclear spin diffusion
is seen to be particularly pronounced at low magnetic fields
$B_{\rm{z}}\lesssim0.5$~T, consistent with the influence of the
electron-mediated nuclear-nuclear spin interaction
\citep{Klauser2008,Reilly2010,Gong2011}. Such pairwise indirect
interaction of nuclei $j$ and $k$ is derived from second order
perturbation expansion of Eq.~\ref{Eq:Hhf}:
\begin{align}
\mathcal{H}_{\rm{hf},j,k}^{\rm{ind}}\propto \frac{A_j A_k}{\Delta
E_{\rm{e}}}\hat{s}_{\rm{z}}\hat{I}^{(+)}_{j}\hat{I}^{(-)}_{k},\label{Eq:HhfInd}
\end{align}
where $\hat{I}_{j}^{(\pm)}=\hat{I}_{{\rm{x}},j}\pm
i\hat{I}_{{\rm{y}},j}$ and $\Delta
E_{\rm{e}}=\mu_{\rm{B}}g_{\rm{e}}B_{\rm{z}}+E_{\rm{hf}}$ is the
electron spin splitting due to both the Zeeman effect and the
nuclear-spin-induced hyperfine shift $E_{\rm{hf}}$. In our
experiments both contributions are negative, so that any nuclear
spin polarization increases $|\Delta E_{\rm{e}}|$. The rate of the
indirect nuclear-nuclear spin flip-flops scales as $\propto\Delta
E_{\rm{e}}^{-2}$. Consequently, the resulting acceleration of
nuclear spin diffusion in gate-defined GaAs QDs was previously
found to be limited to the low fields $B<0.02 - 0.75~$T (Refs.
\citep{Reilly2010,Gong2011,Malinowski2017}). By contrast,
Fig.~\ref{Fig:DiffBz}(b) shows that such acceleration persists at
magnetic fields well above $B_{\rm{z}}\gtrsim2$~T, with short- and
long-pumping $\varGamma_{\rm{N}}^{(1e)}/\varGamma_{\rm{N}}^{(0e)}$
ratios converging only at the maximum field $B_{\rm{z}}=9.82$~T.
One contributing factor is the smaller electron $g$-factor
$g_{\rm{e}}\approx-0.1$ (see Supplementary Section 2) and an order
of magnitude smaller number of nuclei in the studied epitaxial
QDs, which result in a smaller $|\Delta E_{\rm{e}}|$ and larger
$A_j$, respectively, when compared to the gate-defined QDs with
$g_{\rm{e}}\approx-0.4$. While these factors lead to a stronger
hyperfine-mediated couplings in the studied QDs, they do not
explain the magnetic field dependence: At high field
$B_{\rm{z}}=9.82$~T the electron spin Zeeman splitting is $|\Delta
E_{\rm{e}}|\approx58~\mu$eV, whereas at low field
$B_{\rm{z}}=0.39$~T we take into account both the Zeeman splitting
$\approx-2.3~\mu$eV and the time-averaged hyperfine shift
$E_{\rm{hf}}\approx-2.5~\mu$eV (half of the initial
$E_{\rm{hf}}\approx-5~\mu$eV under the shortest used
$T_{\rm{Pump}}\approx8~$ms) to estimate $|\Delta
E_{\rm{e}}|\approx5~\mu$eV. This would correspond to a factor of
$(58/5)^2\approx130$ reduction in the hyperfine mediated rates,
while the measured short-pumping-limit NSR rate reduces only by a
factor of $\approx6$ from
$\varGamma_{\rm{N}}^{(1e)}\approx0.74~{\rm{s}}^{-1}$ at low field
[Fig.~\ref{Fig:Diff}(c)] to
$\varGamma_{\rm{N}}^{(1e)}\approx0.12~{\rm{s}}^{-1}$ at high field
[Fig.~\ref{Fig:Diff}(d)]. Prompted by these observations, we point
out that Eq.~\ref{Eq:HhfInd} treats electron spin as isolated,
while in a real system the electron is coupled to external
environments such as phonons and other charges. A fluctuating
electron spin can accelerate nuclear spin diffusion, provided
there is a frequency component in the time-dependent Knight field
that equals the energy mismatch of a pair of nuclei
\citep{Khutsishvili1966,Horvitz1970} -- this contribution has been
considered for deep impurities \citep{Wolfe1973}, but has been
previously ignored in the context of III-V semiconductor
nanostructures. For similar GaAs/AlGaAs QDs \citep{Zhai2020} the
electron spin lifetime was reported to be 48~$\mu$s, while for
InGaAs/GaAs QDs with a similar tunnel coupling it was found to
vary with magnetic field between $\approx$50~$\mu$s and a few
milliseconds \citep{Gillard2021}. These lifetimes correspond to
fluctuation frequencies in a $[1,10]$~kHz range, indeed matching
the typical differences in the nuclear spin energies as revealed
by NMR spectra of Fig.~\ref{Fig:SampNMR}(e). Thus we speculate
that the intrinsic electron spin flips, governed e.g. by the
phonon relaxation and cotunneling coupling to the electron Fermi
reservoir of the $n$-doped layer
\citep{Kroutvar2004,Lu2010,Gillard2021}, contribute to
acceleration of nuclear spin diffusion in the studied GaAs QDs,
especially at high magnetic fields.
\subsection{Comparison with previous results on nuclear spin diffusion}
In order to understand what controls the rate of spin diffusion we
first make a comparison with Stranski-Krastanov InGaAs/GaAs and
InP/GaInP self-assembled QDs, where quadrupolar shifts are so
large (MHz range \citep{Bulutay2012,Chekhovich2012}) that all
nuclear spins are essentially isolated from each other,
eliminating spin diffusion and resulting in very long nuclear spin
lifetimes $T_{\rm{1,N}}^{(0e)}>10^4~$s in empty (0$e$) QDs
\citep{Lai2006,Greilich2007,Klauser2008,Chekhovich2010,Latta2011,Latta2011,Gillard2021}.
Even in presence of the electron spin (1$e$) the nuclear spin
diffusion takes place only inside the QD
\citep{Klauser2008,Latta2011}, without diffusion into the
surrounding material.
In the lattice-matched GaAs QDs the strain-induced effects are
smaller but not negligible, characterized by quadrupolar shifts
$\nu_{\rm{Q}}$ ranging approximately between 10 and 50~kHz within
the QD, as revealed by NMR spectra in Fig.~\ref{Fig:SampNMR}(e).
Nuclei in $I_{\rm{z}}=\pm1/2$ and $|I_{\rm{z}}|>1/2$ states must
be considered separately. The central transition between the
$I_{\rm{z}}=-1/2$ and $+1/2$ spin states is affected only by the
second order quadrupolar shifts, which scale as
$\propto\nu_{\rm{Q}}^2/\nu_{\rm{L}}$ and are within a few kHz for
the studied range of nuclear spin Larmor frequencies
$\nu_{\rm{L}}\in[1,130]$~MHz. These second order quadrupolar
shifts are comparable to the homogeneous nuclear spin linewidth
$\propto1/T_{2,{\rm{N}}}$, and therefore spin diffusion in
GaAs/AlGaAs QDs is expected to be unimpeded for the nuclei in the
$I_{\rm{z}}=\pm1/2$ states. By contrast, the $I_{\rm{z}}=\pm3/2$
spin states experience first order quadrupolar shifts
$\nu_{\rm{Q}}$, which are tens of kHz, significantly exceeding the
homogeneous NMR linewidths in the studied GaAs QDs. The resulting
dynamics of the $I_{\rm{z}}=\pm3/2$ nuclei is therefore sensitive
to nanoscale inhomogeneity of the strain-induced $\nu_{\rm{Q}}$.
From the NSR experiments [Fig.~\ref{Fig:Diff}(b)] we observe that
nuclear spin polarization relaxes to zero, even in an empty QD
(0$e$). This can only happen if spin diffusion is unimpeded not
only for the $I_{\rm{z}}=\pm1/2$ states, but also for the
$I_{\rm{z}}=\pm3/2$ states that are subject to the larger first
order quadrupolar shifts. Our interpretation is that strain in the
studied GaAs/AlGaAs QDs is a smooth function of spatial
coordinates: for nearly each QD nucleus it is possible to find
some neighboring nuclei with a strain variation small enough to
form a chain that conducts spin diffusion out of the GaAs QD into
the AlGaAs barriers.
Similarly fast NSR was observed previously in neutral QDs formed
by monolayer fluctuations in GaAs/AlGaAs quantum wells
\citep{Nikolaenko2009}. However, the opposite scenario was
realized in QDs with nanoholes etched in pure GaAs
\citep{Ulhaq2016} where nuclear spin polarization in an empty QD
(0$e$) was preserved for over $T_{1,{\rm{N}}}>5000$~s, suggesting
that some of the nuclei were frozen in the $I_{\rm{z}}=\pm3/2$
states, akin to quadrupolar blockade of spin diffusion in
self-assembled QDs. This contrast is rather remarkable since the
average strain, characterized by the average
$\nu_{\rm{Q}}\approx20 - 30$~kHz, is very similar for QDs grown in
nanoholes etched in AlGaAs (studied here) and in GaAs (Ref.
\citep{Ulhaq2016}). This comparison suggests that nuclear spin
dynamics are sensitive to QD morphology down to the atomic scale,
and could be affected by factors such as QD shape, as well as
GaAs/AlGaAs interface roughness and intermixing
\citep{Jusserand1992,SaherHelmy1997,Braun1997}. One possible
contributing factor is the QD growth temperature, which was
610$^\circ$~C in the structures used here, considerably higher
than 520$^\circ$~C in the structures studied previously
\citep{Atkinson2012,Ulhaq2016}. Further work would be required to
elucidate the role of all the underlying growth parameters.
Conversely, NSR can be a sensitive probe of the QD internal
structure.
We now quantify the spin diffusion process and compare our results
to the earlier studies in GaAs-based structures. The best fit of
the experimental NSR dynamics [lines in Fig.~\ref{Fig:Diff}(b)]
yields $D^{(0e)}_{\rm{Dark}}=2.2^{+0.7}_{-0.5}$~nm$^2$~s$^{-1}$
for the diffusion coefficient in an empty QD and in the absence of
optical excitation, in agreement with $D = 1.0 \pm
0.15$~nm$^2$~s$^{-1}$ measured previously for spin diffusion
between two GaAs quantum wells across an Al$_{0.35}$Ga$_{0.65}$As
barrier \citep{MALINOWSKI2001}. This is approximately an order of
magnitude smaller than the first-principle estimate
\citep{Lowe1967,Redfield1969,Butkevich1988} of
$D^{(0e)}_{\rm{Dark}}\approx19$~nm$^2$~s$^{-1}$ for bulk GaAs (see
Supplementary Section 3) and the $D = 15.0 \pm 7$~nm$^2$~s$^{-1}$
value measured in pure AlAs \citep{Nguyen2014}. The reduced
diffusion in the AlGaAs alloy can be explained by the quadrupolar
disorder, arising from the random positioning of the aluminium
atoms \citep{MALINOWSKI2001}. Charging of the QD with a single
electron accelerates spin diffusion: we find
$D^{(1e)}_{\rm{Dark}}(9.82~{\rm{T}})=4.7^{+1.2}_{-1.0}$~nm$^2$~s$^{-1}$,
which increases to
$D^{(1e)}_{\rm{Dark}}(0.39~{\rm{T}})=7.7\pm1.9$~nm$^2$~s$^{-1}$ at
low magnetic fields where hyperfine-mediated nuclear-nuclear spin
exchange is enhanced in accordance with Eq.~\ref{Eq:HhfInd}. While
experimental data can be well described by the spin diffusion
Eq.~\ref{Eq:DiffEq}, it is worth noting the limited nature of the
model, which ignores the spatial variations of the nuclear-nuclear
couplings, the dependence of the electrons spin splitting $\Delta
E_{\rm{e}}$ on the instantaneous nuclear spin polarization, the
isotopic difference between $^{75}$As, $^{69}$Ga and $^{71}$Ga, as
well as neglecting any spin diffusion orthogonal to the sample
growth $z$ direction. As such, the diffusion coefficients $D$
should be treated as a coarse-grained description, aggregating the
numerous lattice-constant-scale parameters of the many-body spin
ensemble evolution.
\section{Discussion and Outlook}
The GaAs/AlGaAs QDs grown by nanohole infilling combine excellent
optical properties with low intrinsic strain, allowing for nuclear
spin qubit and quantum memory designs
\citep{Denning2019,Chekhovich2020,Zaporski2022}. The key
performance characteristic is the nuclear spin coherence time,
which can be extended up to $T_{2,{\rm{N}}}\approx10~$ms
(Ref.~\citep{Chekhovich2020}), but is ultimately limited by the
longitudinal relaxation time $T_{\rm{1,N}}$. Moreover, it is the
state longevity of the nuclei interfaced with the QD electron spin
that is relevant. Thus, one should consider the NSR time in the
regime of short pumping, found here to range from
$T_{\rm{1,N}}^{(1e)}\approx1~$s at low magnetic fields to
$T_{\rm{1,N}}^{(1e)}\approx10~$s at high fields. For nuclear spin
quantum computing with the typical 10~$\mu$s coherent control
gates \citep{Chekhovich2020}, a large number of operations
$\gtrsim 10^5$ would be possible without the disruptive effect of
spin diffusion.
In conclusion, we have addressed the long-standing dilemma of
whether the central spin of an electron accelerates or suppresses
diffusion in a nuclear spin lattice. We have used
variable-duration optical pumping
\citep{Paget1982,Tycko1995,Hayashi2008,Nikolaenko2009} to identify
nuclear spin diffusion as the dominant NSR mechanism. In contrast
to previous studies of nuclear spin diffusion
\cite{Paget1982,Lu2006,Tycko1995,Bayot1997,Makhonin2010,Sallen2014},
we use a charge tunable structure and probe nuclear spin dynamics
with and without the electron under otherwise identical conditions
-- importantly, our QD charge control is achieved without
reverting to optical pumping \citep{Makhonin2010,Sallen2014}, thus
eliminating the unwanted charge fluctuations. Combining these two
aspects, we conclude that in a technologically important class of
lattice matched GaAs/AlGaAs nanostructures the electron spin
accelerates the nuclear spin diffusion, with no signature of a
Knight-field-gradient barrier. We expect these findings to be
relevant for a range of lattice-matched QDs
\citep{Nikolaenko2009,Reilly2010,Gong2011,Malinowski2017} and
shallow impurities \citep{Lu2006}, whereas an efficient spin
diffusion barrier can arise from an electron with sub-nanometer
localization \citep{Wolfe1973}. Future work can examine reduction
of spin diffusion in low-strain nanostructures. The proximity of
the $n$-doped layer, acting as a sink for nuclear polarization, as
well as QD morphology can be optimized. Alternatively, pure AlAs
barriers can be used to grow GaAs QDs with well isolated Ga
nuclei, potentially offering long-lived spin memories and qubits.
\begin{acknowledgments}
Acknowledgements: P.M-H. and E.A.C. were supported by EPSRC
through a doctoral training grant and EP/V048333/1, respectively.
E.A.C. was supported by a Royal Society University Research
Fellowship. A.R. acknowledges support of the Austrian Science Fund
(FWF) via the Research Group FG5, I 4320, I 4380, I 3762, the
European Union's Horizon 2020 research and innovation program
under Grant Agreements No. 899814 (Qurope) and No. 871130
(Ascent+), the Linz Institute of Technology (LIT), and the LIT
Secure and Correct Systems Lab, supported by the State of Upper
Austria. E.A.C is grateful to Ren\'{e} Dost for advice on sample
processing. Author contributions: S.M., S.F.C.S and A.R.
developed, grew and processed the quantum dot samples. P.M-H. and
E.A.C. conducted the experiments. E.A.C. drafted the manuscript
with input from all authors. E.A.C. coordinated the project.
\end{acknowledgments}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,912
|
<?php
namespace Puszek\PuszekAdmin\AdminBundle\Document;
use Doctrine\ODM\MongoDB\Mapping\Annotations as MongoDB;
use JMS\Serializer\Annotation as Serializer;
/**
* @MongoDB\Document(collection="client")
*/
class Client
{
/**
* @MongoDB\Id
*/
protected $id;
/**
* @MongoDB\String
* @MongoDB\UniqueIndex
*/
protected $name;
/**
* @MongoDB\String
* @Serializer\SerializedName("privateKey")
*/
protected $privateKey;
/**
* @MongoDB\Date
*/
protected $createdAt;
/**
* Get id
*
* @return id $id
*/
public function getId()
{
return $this->id;
}
/**
* Set name
*
* @param string $name
* @return self
*/
public function setName($name)
{
$this->name = $name;
return $this;
}
/**
* Get name
*
* @return string $name
*/
public function getName()
{
return $this->name;
}
/**
* Set privateKey
*
* @param string $privateKey
* @return self
*/
public function setPrivateKey($privateKey)
{
$this->privateKey = $privateKey;
return $this;
}
/**
* Get privateKey
*
* @return string $privateKey
*/
public function getPrivateKey()
{
return $this->privateKey;
}
/**
* @return mixed
*/
public function getCreatedAt()
{
return $this->createdAt;
}
/**
* @param mixed $createdAt
*/
public function setCreatedAt($createdAt)
{
$this->createdAt = $createdAt;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,505
|
La coma de Holder és l'interval musical equivalent a de l'octava. La seva raó numèrica és igual a . La coma de Holder és una fracció del to que serveix per construir l'anomenat Sistema de Holder, una aproximació al sistema de Pitàgores per la via de la divisió de l'octava en parts iguals. L'elecció del nombre 53 com el divisor de la octava és un interessant troballa, perquè per si sol permet aconseguir les següents propietats del sistema de Holder:
* El to té 9 comes de Holder i el semitò diatònic té 4. Aquest semitò és «petit» com el pitagòric.
* El semitò cromàtic té 5 comes i per tant es diferencia del semitò diatònic en una coma de Holder. Aquí la coma de Holder es comporta com la coma pitagòrica i com la inversa de la coma sintònica del sistema Just, perquè així com en sistema de Holder i en el de Pitàgores el semitò cromàtic és més gran que el diatònic, en el sistema just passa al revés.
* Els tons són iguals (no hi ha to gran i to petit com en els sistemes Justs).
* La quinta justa té 31 comes i és sorprenentment propera a la quinta pura de Pitàgores, de relació 3:2.
Alguns inconvenients d'aquest sistema són que es basa en una divisió de l'octava, el que és totalment artificial, que les terceres majors són molt grans i dissonants, com el ditó pitagòric, i que els semitons són diferents (encara que això és una cosa que es anava buscant l'hora de triar la dimensió de la coma, precisament per aproximar-se al sistema de Pitàgores).
De la successió de tons i semitons de l'escala diatònica, és a dir: TTSTTTS, substituint els tons per 9 comes i els semitons per 4 comes, s'obté 9+9+4+9+9+9+4 = 53.
La coma de Holder té el seu origen en el sistema de Pitàgores quan l'espiral de quintes s'amplia fins a 53 d'elles, que equivalen aproximadament a 31/8 (veure coma de Mercator). Quan la coma de Mercator es reparteix entre les 53 quintes, s'obté un sistema de temperament igual que divideix l'octava en 53 parts, aquest és el sistema de Holder.
El nom d'aquesta coma ho va establir William Holder.
Intervals musicals
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,963
|
Q: Tutorial on how to model flooding using data from gauging stations along rivers in QGIS? Can someone please direct me to a tutorial where I can learn more about how to model flooding using data from gauging stations along rivers using QGIS?
A: If you use GRASS GIS as backend in QGIS (via QGIS-GRASS toolbox or Sextante plugin), you can use a range of hydrological tools. See http://grass.osgeo.org/wiki/Hydrological_Sciences for options.
A: The GRASS recommendation from @markusN is a good one. Another option, although it's not integrated into QGIS, is the Gerris Flow Solver. GFS is a tremendously powerful hydraulic and hydrological modeling tool. From the site:
Gerris is a Free Software program for the solution of the partial differential equations describing fluid flow. The source code is available free of charge under the Free Software GPL license.
Gerris was created by Stéphane Popinet and is supported by NIWA (National Institute of Water and Atmospheric research) and Institut Jean le Rond d'Alembert.
A brief summary of its main features:
*
*Solves the time-dependent incompressible variable-density Euler, Stokes or Navier-Stokes equations
*Solves the linear and non-linear shallow-water equations
*Adaptive mesh refinement: the resolution is adapted dynamically to the features of the flow
*Entirely automatic mesh generation in complex geometries
*Second-order in space and time
*Unlimited number of advected/diffused passive tracers
*Flexible specification of additional source terms
*Portable parallel support using the MPI library, dynamic load-balancing, parallel offline visualisation
*Volume of Fluid advection scheme for interfacial flows
*Accurate surface tension model
*Multiphase electrohydrodynamics
There is also a very detailed tutorial on the Karamea river, in New Zealand. If you follow that example you will learn a lot about the software, flood visualization, and hydraulic modeling.
A: Have a look at Crayfish for QGIS
http://plugins.qgis.org/plugins/crayfish/
"Crayfish is a plugin (extension) developed by Lutra Consulting for the free and open source GIS platform Quantum GIS (QGIS).
The Crayfish plugin aspires to be a complete set of pre and post-processing tools for hydraulic modellers using TUFLOW, ISIS 2D and other modelling packages."
http://www.tuflow.com/
Personally, I have no experience with it.
A: Sort of depends where you are (which country) on what software and methods are usually used or are standardised on - some councils, state governments and governments have required methods.
Also what sort of area are you looking at modelling ?
Urban, rural ???
eg for Australia have a read of the www.arr.org.au site.
They are in the process of releasing updated guidelines and some state governments are in the process of releasing their required methods.
drafts chapters of their new releases www.arr.org.au/downloads-and-software/chapters/
QGIS and SAGA are very useful
WBN have released a tuflow plugin for QGIS back in april (2014)
They also have a tutorial for using qgis to prepare models for tuflow
http://www.tuflow.com/GIS%20Platforms.aspx?QGIS_and_SAGA
http://wiki.tuflow.com/index.php?title=Main_Page
Drains for urban pipe and drain flow (required by some councils)
also for flows into pipe and drains for tuflow
www.watercom.com.au/download.html
Can download hecas for free
In the US be very careful about units as some states use metric and some US imperial for their data.
A: QGIS 3.x series has native Mesh Layer that uses MDAL library to read many formats used in flood modeling. I suggest doing online course on hydraulical modeling in QGIS or read book "QGIS for Hydrological Applications" by Hans van der Kwast and Kurt Menke
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,811
|
Jean "Jennie" Margaret Gheer (November 13, 1846 - June 20, 1910) was an American missionary and educator. In 1879, at the age of 33, she was sent by the Woman's Foreign Missionary Society of the Methodist Episcopal Church to Japan. She founded Eiwa Jo Gakko in Fukuoka in 1885, the origin of Fukuoka Jo Gakko and Fukuoka Jo Gakuin, an educational institution for girls and women that flourishes to this day.
Biography
Jennie Margaret Gheer was born in Bellwood, Pennsylvania. Her father was a furniture merchant. She graduated from the Normal School in Millersville and worked as a teacher in public schools in Antis, Tyrone, and Altoona in Blair County. She developed interest in foreign mission and participated in the New York branch of the Woman's Foreign Missionary Society of the Methodist Episcopal Church (WFMS). In October 1879, she was assigned to Nagasaki, Japan. After meeting Elizabeth Russell from the Cincinnati branch, the two women sailed from San Francisco for Nagasaki, via Yokohama, on October 25, 1879, reaching their destination on November 23, 1879. The WFMS had initially assigned them to Kolkata, India, but two weeks before their scheduled departure, they were suddenly reassigned to Japan. At the time, Gheer was 33 and Russell 43 years old, and they knew almost nothing about each other, nor about Japan.
The two women were cordially received at the port of Nagasaki by John Carrol Davidson and his wife, who had been sent there by the American Episcopal Methodist Church in 1873 when the edict prohibiting Christianity was abolished by the Meiji Government. Davidson had built a Methodist church in Dejima in 1876 and sent a letter to WFMS requesting two female missionaries to establish a girls' school. Shortly thereafter, Russell founded a mission school for girls in the foreign settlement in Higashi-Yamate, Nagasaki on December 1, 1879. Although there was only one student in 1879–80, the number increased to 18 in 1881 when the school was named Kwassui Jo Gakko, and 43 in 1882 when the school building was rebuilt. Gheer encouraged and supported Russell during these days. Gheer was talented at teaching the Old Testament, the New Testament, and music such as singing, playing the organ, and piano.
In 1884, the first Methodist church was built in the city of Fukuoka, and Russell or Gheer had to move there to establish a girls' school. Being missed, Gheer eventually left Nagasaki. On June 15, 1885, Gheer opened a girls' school, called Eiwa Jo Gakko, the origin of Fukuoka Jo Gakko, established in 1919 and Fukuoka Jo Gakuin Junior and Senior High School, established in 1947 and 1948, respectively. Gheer took leave from her job as the first principal in 1888 due to illness, returning to Japan in 1890. She stayed and worked in Japan for most of the following 20 years until 1910 when she returned to the States because of serious illness. She traveled all over Kyushu and Okinawa to train evangelists. She set up orphanages, kindergartens for the poor, and Sunday schools for illiteracy, health, and vocational training for women.
Gheer sailed to the States on May 17, 1910. Hearing the severity of the illness, her brother Thomas P Gheer managed to reserve a private train compartment from Seattle to her home. She arrived at her sister Anna's home in Bellwood on June 13, 1910, and died just a week later, on June 20, 1910, at the age of 63. Gheer had adopted a baby girl in 1880, named Elisaberta Forrsell. Her mother had died shortly after she was born. The father, a Finnish officer on a Russian ship, who had been seriously injured in a fall, also died within a year. The child, nicknamed Lisa, returned to the States with Gheer, and later entered the Curtis Institute of Music, Philadelphia.
See also
Fukuoka Jo Gakuin University
Kwassui Women's University
References
External links
Fukuoka Jo Gakuin
KWASSUI WOMEN'S UNIVERSITY
Female Christian missionaries
Foreign educators in Japan
1846 births
1910 deaths
Missionary educators
Woman's Foreign Missionary Society of the Methodist Episcopal Church
Methodists from Pennsylvania
University and college founders
Women founders
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,912
|
Compare Humalog prices, print discount coupons, find manufacturer promotions and details on available patient assistance programs.
Download our free Humalog discount card to use Lilly offers a coupon for Humalog; for the Humalog U-200 Kwikpen.
Merck Prescription Discount Program. This Humalog Coupon is accepted at Walmart, Walgreens, CVS, RiteAid and 59,000 other pharmacies nationwide.
Access Humalog Coupons Simple search with direct use of printable and online coupons. Find coupons by either brands or category search. Download Now.
Free pharmacy coupon for Humalog Mix 75/25 Insulin Vials. Get up to 75% discount on Humalog Mix 75/25 Insulin Vials prescription at CVS, Walgreens and other pharmacies nationwide.
Get Your Free Pharmacy Discount Card Step 1: this card or coupon will allow the user to save on all prescription benefits linked with the BIN, GRP and PCN codes.
Humalog KwikPen: View Coupon: Lilly Cares Patient Assistance Program This program provides brand name medications at no or low cost: Provided by: The Lilly Cares Foundation, Inc.
© Free location coupons Incorp coupon | Promo Codes & Deals 2018 Humalog kwikpen discount coupon.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,626
|
Félix Lemaréchal (Tours, 7 augustus 2003) is een Frans voetballer met Ivoriaanse roots die onder contract ligt bij AS Monaco.
Clubcarrière
Lemaréchal genoot zijn jeugdopleiding bij EB Saint-Cyr-sur-Loire, Tours FC, Girondins Bordeaux en AS Monaco. In het seizoen 2021/22 maakte hij zijn opwachting bij het tweede elftal van Monaco in de Championnat National 2. Hij maakte dat seizoen ook zijn officiële debuut in het eerste elftal van de club: op de tiende competitiespeeldag liet trainer Niko Kovač hem tegen Olympique Lyon in de 82e minuut invallen. In de zomer van 2021 had hij ook al in het eerste elftal gespeeld tijdens de vriendschappelijke wedstrijden tegen Antwerp FC, VfL Wolfsburg en Real Sociedad.
Clubstatistieken
Bijgewerkt op 4 januari 2022.
Interlandcarrière
Lemaréchal speelde in 2019 acht jeugdinterlands voor Frankrijk.
Zie ook
Lijst van spelers van AS Monaco
Frans voetballer
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,260
|
{"url":"https:\/\/tex.stackexchange.com\/questions\/319279\/how-to-exclude-just-one-chapter-number-in-the-table-of-contents","text":"I need to add an appendix to the end of my thesis. But I don't want LaTeX to enumerate that section as a chapter in the table of contents.\n\nDo you know how I can get it without numbering? I don't want that number 5 in the table of contents!\n\nBTW, I use book document style.\n\n\u2022 Are you using \\chapter{Appendix 1} to include the appendix? If so, use \\chapter*{Appendix 1} \\addcontentsline{toc}{chapter}{Appendix 1} instead... \u2013\u00a0Werner Jul 13 '16 at 18:38\n\nA \\chapter* is sufficient to remove numbering of the content in the ToC and in the chapter title. You'll have to manually insert a ToC-entry, as well as update the headers, if possible:\n\n\\documentclass{book}\n\n\\usepackage{lipsum}\n\n\\begin{document}\n\n\\tableofcontents\n\n\\chapter{A chapter}\n\\lipsum[1-10]\n\n\\appendix\n\\chapter*{Appendix 1}\n\n\\lipsum[1-10]\n\n\\end{document}\n\n\n\\backmatter should be sufficient to suppress the addition of numbers in the ToC:\n\n\\documentclass{book}\n\n\\begin{document}\n\n\\tableofcontents\n\n\\chapter{Foo}\n\n\\backmatter\n\\appendix\n\n\\chapter{Appendix Stuff}\n\n\\end{document}","date":"2021-03-04 18:05:36","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\": 0.9859582185745239, \"perplexity\": 1396.0316474499439}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178369512.68\/warc\/CC-MAIN-20210304174506-20210304204506-00613.warc.gz\"}"}
| null | null |
{"url":"https:\/\/moodle.org\/plugins\/view.php?plugin=auth_elcentra&moodle_version=10","text":"auth_elcentra\n77 sites\n6 fans\nMoodle 2.2, 2.3, 2.4, 2.5, 2.6\n\nOnce installed and configured, this plugin offers a hassle free method for Admins & end users. Accounts get created without any additional action from the admin.\n\n\u2022 VKontakte (New!)\n\n### Contributors\n\n\u2022 Mon, 30 Dec 2013, 9:21 AM\nThanks for your patience. I have installed and tested this on one of my production sites and it seems to work well. I think there are a few minor issues I would like to see tidied up before we get this approved. First, in the lang\/en\/auth_electra.php file the @package name is incorrect and the copyright is listed as Martin. Please correct those. Also, I like to see the boilerplate notice making the GPL license explicit. Ideally this is in all of the files but for starters it would be good if it were in the version.php file. I did receive a PHP warning when logging in for the first time with a Google account - Warning: Creating default object from empty value in \/moodle\/auth\/elcentra\/auth.php on line 230. Thanks for providing the Github source control URL. Please consider following the Moodle naming convention for repositories - namely, moodle-{type}_{name} so in this case moodle-auth_elcentra. Pointing to the README file in the repository is fine but you are also welcome to create a page in Moodle Docs. I am going to mark this as needing more work but the changes should be simple. If you have any questions just let me know. Peace - Anthony\n\u2022 Mon, 30 Dec 2013, 9:24 AM\np.s. - I second Aparup's recommendation to avoid $GET for state - any reason not to use the optional_param there? \u2022 Tue, 21 Jan 2014, 6:33 PM Thanks for your reply. We are now done with all your above requirements. Rather than moodle docs, we prefer using the README file since it would be a single place to update all our changes. If you insist us to create a moodle docs page, please send us the procedure to do so. We believe we have fixed all your reports and answered your queries. I am interested & eager to see this plugin go live. Please feel to let us know in case we have missed any. Thanks \u2022 Mon, 24 Feb 2014, 12:41 PM Hi, Thanks for addressing so many issues. ps: https:\/\/moodle.org\/plugins\/view.php?plugin=auth_googleoauth2 is very similar (twitter support seems to be the difference). It seems technically feasible that these are merged, so please have a look and consider. \u2022 Tue, 11 Mar 2014, 6:01 PM I would like to use ony facebook logi, how to remove google, twitter and linkedin? \u2022 Wed, 12 Mar 2014, 3:15 PM Just remove the image links of twitter, google, linkedin. That would stop the usage of these 3. You are only going to link it to facebook, so the users will have links to use only that \u2022 Wed, 12 Mar 2014, 3:16 PM @Banerjee We plan to proceed this in a different way with integration of lot more local sites and social networks. Hence the new plugin \u2022 Fri, 28 Mar 2014, 11:02 PM first of all nice plugin and it works flawlessly with me on version 2.6 how ever there are some issues for which guidance is needed .. 1. On fb login - as I am just using fb login its only capturing email and name from response code ... is there any way at least country can captured too.. with possible extension for picture in future ... 2. If author can only explain the login flow .. I guess many good things can be added to plugin .. as I am still confuse when it takes from fbreponse file to auth file ... and what happens after that but in general good work !!! \u2022 Wed, 9 Apr 2014, 5:00 PM I commented the line error_reporting(E_ALL); in twitter_request.php because was causing the error Internal server error 500 \u2022 Fri, 2 May 2014, 1:43 PM suhail shah - We arent taking country as fb blocks it for many users owing to their privacy settings. Will include them if more people ask for it. We will add image retrieval in the future versions. Thanks for your suggestions and please share more of them . \u2022 Fri, 2 May 2014, 1:45 PM Fortunato Borruto - Can you give the error you faced? You will be able to find it in your error_log \u2022 Sat, 5 Jul 2014, 5:09 PM L.s. Good plugin! I'm now using another authentication (linkedin) plugin. This plugin adds the Linkedin picture. Could this be added in your plugin? Best regards Bert \u2022 Thu, 10 Jul 2014, 3:02 AM Hi, I have installed this plugin and in status saying \"To be installed\". I'm getting this message both in 2.5 and 2.6 versions \u2022 Tue, 22 Jul 2014, 2:03 PM Hi, The plugin is great and works as per description. I wanted to know is there any way to stop self registration throught this plugin. (may be some code modifications) I will like to allow access to only those users who have been added by admin. \u2022 Sat, 23 May 2015, 1:46 PM I'm using your plugin just whith google authentication, but when updating Moodle to 2.9 version it crashed. The users got only a white screen when trying to login. Activating debug mode gave some tips. In the script moodle\/auth\/elcentra\/googleresponse.php line 26: \"require_once '..\/..\/lib\/pluginlib.php';\". (A kind of file not found error) Since https:\/\/tracker.moodle.org\/browse\/MDL-46122 , that file is not needed anymore. So deleting the line is OK. Next error is class \"plugin_manager\" is not defined (line 47): \"$pluginManager = plugin_manager::instance();\"\n\nA quick view to moodle\/lib\/classes\/pluginmanager.php shows that the new name for the class is \"core_plugin_manager\", so udpadtin the line 47 of moodle\/auth\/elcentra\/googleresponse.php whit this content: \"\\$pluginManager = core_plugin_manager::instance();\" will resolve the issue.\n\nCheers.","date":"2016-12-11 04:15:35","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\": 0.17069962620735168, \"perplexity\": 2388.6641780646896}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"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-2016-50\/segments\/1480698544097.11\/warc\/CC-MAIN-20161202170904-00320-ip-10-31-129-80.ec2.internal.warc.gz\"}"}
| null | null |
{"url":"https:\/\/www.physicsforums.com\/threads\/becoming-a-mathematician-how-important-is-iq.462166\/","text":"Becoming a mathematician - how important is IQ?\n\n\u2022 Math\nLevis2\nHello - im a 16 year old danish boy. I'm in what is equivalent in denmark to the 10th grade in the US, and i simply love math. It's funny though, since before i attended 10th grade, i dreaded math due to it being so boring - but i think that was due to the simple arithmetics we did in my previous school. Once i encountered a more pure math in 10th grade, i was sold!\n\nMy number 1 goal in this world - the thing that matters most to me - is becoming a mathematician. I want to take a phd in math, and teach at a university, and if im lucky, end up making a useful contribution. That's what matters most to me of all things atm.\n\nBut there's a problem - im not a child prodigy. I can't do topology or real analysis, and my iq is only 130 !!! Ever since i took that iq test, i have been so scared of not being able to make contributions to math, or even complete my degree in college. I'm afraid that it will get too complicated when i'm not that intelligent.\n\nFunny stuff is though, that i have taught myself basic calculus, and can set up differential equations on the saltconcentration in, lets say a lake, based on differences in in-and-out flows of water etc. My teacher says he's never seen anyone like me in 9 years of teaching in high schools, but i presume he hasn't met any real good mathematicians lol .. I have also invented a formula by myself for calculating the area of a triangle if one only knows its sides. It looks this this;\n\nA=1\/2*c*squareroot(a^2-(c-(b^2+c^2-a^2)\/2c)^2)\n\nWhere c has to be the biggest side in the triangle. The order of a and b doesnt matter :) All of this is easy stuff though ... nothing worthy a true mathematician :(\n\nNow my question is, can i take a phd in math and become a mathematician, even though i'm not that intelligent? And if i'm barely able to do my phd, will i then be a garbagety and lousy matehmatician ?\n\nit's a thought that takes up a lot of space in my head atm .. i'm so worried that i wont be able to take a degree or contribute to the art of mathematics :(\n\nHelp!\n\nHomework Helper\nBut there's a problem - im not a child prodigy. I can't do topology or real analysis, and my iq is only 130 !!!\n\nNobody would expect you to do topology and real analysis at 16 years old! But if you're aware of these subjects at such a young age, you have a very bright future ahead of you.\n\nEver since i took that iq test, i have been so scared of not being able to make contributions to math, or even complete my degree in college. I'm afraid that it will get too complicated when i'm not that intelligent.\n\nDon't let an IQ test hold you back from doing what you love. To be honest, a lot of IQ tests aren't very good anyways. And the ones that are decent measure how good you are at thinking logically. Mathematics is filled with logic and a lot of it comes with time and understanding of the subjects.\n\nFunny stuff is though, that i have taught myself basic calculus, and can set up differential equations on the saltconcentration in, lets say a lake, based on differences in in-and-out flows of water etc. My teacher says he's never seen anyone like me in 9 years of teaching in high schools\n\nIf you're doing stuff like this when you're 16, you'll be fine. I taught myself calculus from a book when I was 18, and a teacher I had didn't believe I'd do well in AP (college-level) calculus because I skipped a class, but I rose to the challenge and did just fine in the class. What's great is that you're only 16 and studying this stuff.\n\nNow my question is, can i take a phd in math and become a mathematician, even though i'm not that intelligent? And if i'm barely able to do my phd, will i then be a garbagety and lousy matehmatician ?\n\nIt sounds like you've got a lot of potential. Don't let an IQ test get in the way of what you want to do. A better question is how you do in your math classes?\n\nTylerH\nIQ isn't the end all test(You should note that 130 isn't bad. You're in the top 5-10% of the population.). You can do anything you can learn to do. Differential equations for a 16 yo is crazy. That's awesome.\n\nIMHO, love for the sciences, and inquisitiveness is the most important ingredient to a great mathematician.\n\nLevis2\nI really want you guys to be right :) pretty much all i care about (besides girls lol :P) is mathematics .. And no 130 isnt low, but seen from a a top mathematicians perspective, it is really low.\n\nI want to make a meaningful contribution, and become a good mathematician! I dont want to scrabe by, barely able to make it, and balancing on the edge of flunking out :(\n\nIts because when i read about mathematicians who have made contributions, they have always been child prodigies or near to .. So my future doent look very bright :)\n\nMy GPA in math classes is A (or max here in denmark - don't know what it translates to in the US) but the stuff they teach in class is trivial, and not worthy of a mathematician :)\n\nGold Member\nHigh IQ's are meaningless indicators of future success. That's all there is to be said.\n\nLevis2\nHigh IQ's are meaningless indicators of future success. That's all there is to be said.\n\nmeaningless in the financial world maybe - but i have always been told that iq almost exactly translates into mathematical ability? I hope you're right though :)\n\nIf that is true - and i hope it is - how come every great contemporary mathematician has an iq of >145-150? It's a strange coincidence then, nevertheless a good one :)\n\nHomework Helper\nMy GPA in math classes is A (or max here in denmark - don't know what it translates to in the US) but the stuff they teach in class is trivial, and not worthy of a mathematician :)\n\nWhat's trivial to you might be very difficult to many other people. It's good that you want to do really well, but don't worry so much. Go out and play some basketball. You'll be fine.\n\nGold Member\nmeaningless in the financial world maybe - but i have always been told that iq almost exactly translates into mathematical ability? I hope you're right though :)\n\nIf that is true - and i hope it is - how come every great contemporary mathematician has an iq of >145-150? It's a strange coincidence then, nevertheless a good one :)\n\nWell you aren't going to find many \"great\" people with IQs of 85 and 95. Having such an IQ means you are part of a small but not minuscule part of society. To be a \"great\" person in any field, yah you probably are going to be part of that end of the bell curve but that end of the bell curve represents tens of millions of people. There are plenty of threads around that talk about how meaningless IQ tests are when you begin to move away from the 2 standard deviations.\n\nHello - im a 16 year old danish boy. I'm in what is equivalent in denmark to the 10th grade in the US, and i simply love math. It's funny though, since before i attended 10th grade, i dreaded math due to it being so boring - but i think that was due to the simple arithmetics we did in my previous school. Once i encountered a more pure math in 10th grade, i was sold!\n\nMy number 1 goal in this world - the thing that matters most to me - is becoming a mathematician. I want to take a phd in math, and teach at a university, and if im lucky, end up making a useful contribution. That's what matters most to me of all things atm.\n\nBut there's a problem - im not a child prodigy. I can't do topology or real analysis, and my iq is only 130 !!! Ever since i took that iq test, i have been so scared of not being able to make contributions to math, or even complete my degree in college. I'm afraid that it will get too complicated when i'm not that intelligent.\n\nFunny stuff is though, that i have taught myself basic calculus, and can set up differential equations on the saltconcentration in, lets say a lake, based on differences in in-and-out flows of water etc. My teacher says he's never seen anyone like me in 9 years of teaching in high schools, but i presume he hasn't met any real good mathematicians lol .. I have also invented a formula by myself for calculating the area of a triangle if one only knows its sides. It looks this this;\n\nA=1\/2*c*squareroot(a^2-(c-(b^2+c^2-a^2)\/2c)^2)\n\nWhere c has to be the biggest side in the triangle. The order of a and b doesnt matter :) All of this is easy stuff though ... nothing worthy a true mathematician :(\n\nNow my question is, can i take a phd in math and become a mathematician, even though i'm not that intelligent? And if i'm barely able to do my phd, will i then be a garbagety and lousy matehmatician ?\n\nit's a thought that takes up a lot of space in my head atm .. i'm so worried that i wont be able to take a degree or contribute to the art of mathematics :(\n\nHelp!\n\nHey Levis and welcome to the forums\n\nThere are quite a number of people who weren't the so called \"child prodigies\" that saturates the history of our so called \"genius\" minds.\n\nOne example off the top of my head is the late George Polya. His last position if i remember correctly was at Stanford University. He actually did a law degree (and publicly noted how boring it was) before commencing study in mathematics.\n\nPersonally (and this is just my opinion), one of the more important traits, especially in a scientific field is to have a high level of curiosity, and also persistence. With genuine curiosity, you're bound to explore things deeply and find things that are unknown to the majority of people. The persistence part is just as crucial, because in most endeavors it separates the successful from the not so successful.\n\nAlso with regards to your IQ test, here is some trivia for you: A great mathematician (and also a qualified engineer) by the name of Henri Poincare failed an IQ test. If you're interested in more details go to a wiki site.\n\nI think you need to be more confident in yourself. You need to be confident enough in pursuing your goals to an end, but not to the point where you are blindly arrogant. Its kind of like how children just try things because they are born not knowing what is possible and what isn't. Its unfortunate that we grow up and develop an attitude of just accepting what other people say is possible and stop then and there in our tracks.\n\nSo summing I guess the words, be curious, be persistent, and be healthily confident are my advice for you.\n\nHomework Helper\nGold Member\nDearly Missed\nDo not EVER betray your great passions!\nWhat are thinking, quitting maths because you fear you won't be good enough??\n\nWhat are you planning to do, if NOT maths??\n\nV\u00e6r tro mot deg selv.\n\nStaff Emeritus\nHomework Helper\nHello - im a 16 year old danish boy. I'm in what is equivalent in denmark to the 10th grade in the US, and i simply love math. It's funny though, since before i attended 10th grade, i dreaded math due to it being so boring - but i think that was due to the simple arithmetics we did in my previous school. Once i encountered a more pure math in 10th grade, i was sold!\n\nMy number 1 goal in this world - the thing that matters most to me - is becoming a mathematician. I want to take a phd in math, and teach at a university, and if im lucky, end up making a useful contribution. That's what matters most to me of all things atm.\n\nBut there's a problem - im not a child prodigy. I can't do topology or real analysis, and my iq is only 130 !!! Ever since i took that iq test, i have been so scared of not being able to make contributions to math, or even complete my degree in college. I'm afraid that it will get too complicated when i'm not that intelligent.\n\nBut... 130 is above average... You should have no problem with an IQ of 130. Where did you take that IQ-test anyway? You didn't take it on the internet, did you?\n\nIn fact, a high IQ doesn't mean anything in mathematics. What matters is:\n- Work hard\n- Be creative\n- Be interested\nAs long as you are those three things, you shouldn't have any problem.\n\nFunny stuff is though, that i have taught myself basic calculus, and can set up differential equations on the saltconcentration in, lets say a lake, based on differences in in-and-out flows of water etc. My teacher says he's never seen anyone like me in 9 years of teaching in high schools, but i presume he hasn't met any real good mathematicians lol .. I have also invented a formula by myself for calculating the area of a triangle if one only knows its sides. It looks this this;\n\nA=1\/2*c*squareroot(a^2-(c-(b^2+c^2-a^2)\/2c)^2)\n\nWhere c has to be the biggest side in the triangle. The order of a and b doesnt matter :) All of this is easy stuff though ... nothing worthy a true mathematician :(\n\nHmm, you have basically rediscovered Heron's formula: http:\/\/en.wikipedia.org\/wiki\/Heron's_formula The formula you gave, was first proven by Qin Jiushao. That's nice work.\n\nBut really? Doing differential equations at your age?? That's crazy!!! So you do differential equations, and you still think you're a bad mathematician?? Don't fool yourself, if you can do all of this at your age, then you will have a bright mathematical future!!\n\nI don't believe that intelligence can be accurately measured by an IQ-test. So if you didn't do as good as you wanted on an IQ-test, then don't worry!! Intelligence can only be measured by your actions in life. And if you discorver Heron's formula and do differential equations at your age, then I've seen enough: you're obviously intelligent enough for mathematics...\n\nlitup\nI really want you guys to be right :) pretty much all i care about (besides girls lol :P) is mathematics .. And no 130 isnt low, but seen from a a top mathematicians perspective, it is really low.\n\nI want to make a meaningful contribution, and become a good mathematician! I dont want to scrabe by, barely able to make it, and balancing on the edge of flunking out :(\n\nIts because when i read about mathematicians who have made contributions, they have always been child prodigies or near to .. So my future doent look very bright :)\n\nMy GPA in math classes is A (or max here in denmark - don't know what it translates to in the US) but the stuff they teach in class is trivial, and not worthy of a mathematician :)\n\nDid you know Richard Feynman \"only\" had an IQ of 126, but he managed to win the Nobel prize in physics. What you are doing at 16 is awesome, you are plenty smart enough, besides, motivation is 99% of success and you seem to have that in spades!\n\nliquidFuzz\nFirst rule, what ever you want to become don't ever put yourself down. Don't make a list of reasons why you won't make it. If you ask me I'd say IQ is just a figure. If you read about the great once in history, not only mathematics - a personal favourite is Immanuel Kant, you'll often find that they blossomed late in life. Not at 16. ;-) What made them great was the perseverance that gave them the chance of finally prevail. This lead to rule numero duae. Follow your passion! Without it you'll never be more than a day trader, whatever you do. Furthermore, with passion it's easier to be creative, which is good while tinkering with plus, minus and what have you.\n\nAn everyday example, I had problems learning how to read and write, still do to an extent. If I had listen to all the things said about me I'd never taken my exam in Engineering Physics.\n\nLast edited:\nMsh1\nIf you want your question answered by a so called 'child prodigy' who also happens to be one of the greatest contemporary mathematicians and also a fields medalist. Read this:\n\nI am a senior university student in pure mathematics, who has the same ambitions as you. My experience is that even though having a sharp mind and a high IQ can sometimes help you solve problems faster and more easily (although some experts don't even believe in IQ), working HARD is FAR more important. But the important thing is that if you want to increase your odds of making it big in mathematics, you have to start NOW. There is usually two different approaches you can take. You can either try and learn a lot of new subjects and theories (generally not advised though better than nothing) or you can try and focus on competitions like Mathematics Olympiad. Although the subjects covered in Olympiad competitions are considered \"Elementary mathematics\" and greatly differ from what real research is like, I find them to be excellent for problem solving. So I recommend you compete (and do well) in as many competitions as you can.\n\nLast edited:\nYou should also note that most neuroscientists and psychologists agree that intelligence level can be helped by nature (read:genetics) but must be nurtured (read:hard work). It's something called brain plasticity. No matter how smart a test says you are, you can always get yourself to a higher level by spending your time working at it. There are very few people who are child prodigies, and a lot of highly successful people, even in mathematics, were not child prodigy super-geniuses. Find any person who's famous in mathematics, physics, or another difficult subject, I guarantee if you dig a little deeper you'll find that there was mountains and mountains of hard work to be done before they could get where they got.\n\nOf course, you love mathematics. You have to keep working at it, and you have to believe that the harder you work, the smarter you get. Even if it turns out to be a crock, you're still helping yourself a lot. I, and many other scientists who study the human brain, think it's not a hoax, and think that you really can 'get smarter' through hard work. As long as you don't overwork yourself to the point of hating the subject and burning out, do what you can, and always make sure you're doing it because you love to do it.\n\nIt took Einstein 10 years to come up with GR, and he was not a child prodigy. Don't go imposter syndrome on yourself, have some confidence in your ability and potential.\n\nOP, 130 is not \"above average\", but in the top 2% of the population (it's defined that way!). That means it is *very high*! Common understanding in psychology is that IQ 120 is sufficient to pursue any career in any field. This includes physics and mathematics.\n\nThe main factor in getting into the top levels is to do hard work, and a lot of it. Interestingly, this is precisely what is keeping many smart children from achieving any level of success: They stop once things get complicated, because they are not used to things being difficult, and it challenges their self-concept if they can't easily handle a problem:\nhttp:\/\/www.highlightsparents.com\/parenting_perspectives\/interview_with_dr_carol_dweckdeveloping_a_growth_mindset.html [Broken]\nhttp:\/\/nymag.com\/news\/features\/27840\/\n(actually, there were some better articles on Carol Dweck's research, but I did not find any non-original articles in a hurry). This is also why in science, just like in any other field, you will typically find smart-to-very smart people (IQ 120-140) in top positions, but actual genius level intelligence is very very rare (I've yet to meet one such professor in person, and I work in a branch of theoretical physics!).\n\nBecoming an expert is simple -- just constantly keep on improving your skills, never being satisfied. For 10 years or more. Note that this actually also applied to child prodigies like Mozart and the like. They just started earlier.\n\nLast edited by a moderator:\nHomework Helper\nThe main factor in getting into the top levels is to do hard work, and a lot of it. Interestingly, this is precisely what is keeping many smart children from achieving any level of success\n\nHard workers can take advantage of opportunities but if one is not competitive, there will be limited opportunities. Innate ability does count, as well as opportunity, as well as hard work. If one has the ability and opportunities are there, hard work becomes sufficient.\n\nIt sounds like Levis2 has the ability, and with hard work he may make use of whatever opportunities there are, that is up to him.\n\nGold Member\nAt some point in time, grades would be meaningless for you anyway.\nYou could care less what IQ you have.\n\nI once took something like 2-3 IQ tests, and I wasn't given any metric scale grade (even when I asked for it), I don't think genuine IQ tests which are conduceted by professional psychatrists give you any answer as for your grade.\n\nfasterthanjoao\nAnd no 130 isnt low, but seen from a a top mathematicians perspective, it is really low.\n\nThis isn't true. Mathematicians are just human beings too, though I suppose some might argue\n\nYou don't need to be a 'genius' to make a meaningful contribution. I am not a genius, and I work in mathematics research. I work with some people that I do consider to be 'genius' - but the majority of us are just regular, every-day, human beings.\n\nAll that, and I would just totally ignore the result of an IQ test anyway. There's no point in letting some (essentially arbitrary) number define anything about you. My thoughts are that they are a remnant of an antiquated way of thinking about intelligence and how people process information.\n\nLevis2\nThx a lot guys .. This really gives me hope you know! The reason all this insecurity sprouted in my mind, is due to all the reading ive done. I keep reading about \"great mathematicians\", and it always follows that their iq is >145 .. They are always very good in an early age. Einstein may not have have been a true child prodigy, but from what i've read he was a VERY good math student even in his 13's. When i was thirten, i was doing stuff like this; 2x+15=1x .. This may have been due to my lack of interest at that time - i wasnt really interested in math, so i didn't have the drive to study to become better. I will never know if i would have been able to grasp more advanced content in that age ..\n\nIt may all root in my perfective nature - i want to be the best! I want to be the best in everything .. Approx 1 year ago, i got a whim about airguns. I saved up, and bought an airgun worth 2200$, because i wasnt satisfied with my garbagety gamo springer airgun .. 1\/2 year ago, i had a whim about hunting, and i spent approx 3300$on weapons and equipment (yeah had to sell lots of stuff.. and i work a lot too) .. I always want to achieve and do my best! I'm a very stubborn person.\n\nTherefore i am worried that i will simply be a mediocre mathematician, not brilliant enough to achieve his dream; teach in a university, and hopefully discover something of significance.\n\nSo my problem is, that i keep comparing myself to the \"generel\" image of a mathematcian, which is an utter genius, an image that is far superior to me. It's really demoralising to read about the great mathematicians, like Terry Tao, who simply is superior to me in every way .. I keep thinking how unfair it is, that they have been gifted with those superior abilities, and i have to fight my way through math with mediocre abilities. It's of great annoyance to me !:)\n\nThe IQ-test was conducted on the internet, but i was told that it should be a reliable test. It's suppose to be the same guy who made mensas test that made this one, atleast for all i know. This is not good though, since it's likely that i would score less on a genuine test.\n\nBut math is truly a passion of mine, and i intend to pursue it. I just don't know how to convince myself, that i will be able to phd in math and achieve my goals :S\n\nSorry for the length of this reply lol .. :)\n\nStaff Emeritus\nHomework Helper\nThx a lot guys .. This really gives me hope you know! The reason all this insecurity sprouted in my mind, is due to all the reading ive done. I keep reading about \"great mathematicians\", and it always follows that their iq is >145 .. They are always very good in an early age. Einstein may not have have been a true child prodigy, but from what i've read he was a VERY good math student even in his 13's. When i was thirten, i was doing stuff like this; 2x+15=1x .. This may have been due to my lack of interest at that time - i wasnt really interested in math, so i didn't have the drive to study to become better. I will never know if i would have been able to grasp more advanced content in that age ..\n\nIt may all root in my perfective nature - i want to be the best! I want to be the best in everything .. Approx 1 year ago, i got a whim about airguns. I saved up, and bought an airgun worth 2200$, because i wasnt satisfied with my garbagety gamo springer airgun .. 1\/2 year ago, i had a whim about hunting, and i spent approx 3300$on weapons and equipment (yeah had to sell lots of stuff.. and i work a lot too) .. I always want to achieve and do my best! I'm a very stubborn person.\n\nStubborness is a very good quality in mathematics, you will need it. Say you can't find a solution to a math problem or say that you don't understand a theory that well, then it is stubborness that pulls you true\n\nTherefore i am worried that i will simply be a mediocre mathematician, not brilliant enough to achieve his dream; teach in a university, and hopefully discover something of significance.\n\nSo my problem is, that i keep comparing myself to the \"generel\" image of a mathematcian, which is an utter genius, an image that is far superior to me. It's really demoralising to read about the great mathematicians, like Terry Tao, who simply is superior to me in every way .. I keep thinking how unfair it is, that they have been gifted with those superior abilities, and i have to fight my way through math with mediocre abilities. It's of great annoyance to me !:)\n\nPlease, don't say that you have mediocre abilities, if you say it long enough you will start to believe it. The truth is, you don't know your math abilities. You will only know them if you start doing mathematics.\nBut if you're already doing ODE's, then I can say that you don't have mediocre abilities...\n\nAnd you don't need a high IQ to pursue math. Look at Henri Poincarre, one of the most brilliant mathematicians of our time!\n\nThe IQ-test was conducted on the internet, but i was told that it should be a reliable test. It's suppose to be the same guy who made mensas test that made this one, atleast for all i know. This is not good though, since it's likely that i would score less on a genuine test.\n\nEnough said. A test on the internet is NOT reliable, no matter what they want you to believe. I repeat: a test on the internet is NEVER reliable. The real IQ-test involves much more then just \"complete the sequence\"-questions. And another thing: an internet IQ-test probably won't factor in you age, a real IQ-test will.\nIf you want to know your IQ, then DON'T believe this test: this test will lie to you... Go to a psychologist, they are the only ones that know how to get your IQ.\n\nLevis2\nI hope that test is unreliable .. unless it gave me a higher score than i should have gotten :) I will never do an IQ test again.. if i should happen to score low, then i would properly underestimate my abilities even more, so i think it's best to just not do any iq-tests:)\n\nAlthough i must admit, that i can't solve differential equations, unless it's simple integration. I can only make models of the real world involving a function and its derivative, you know models about mixing to substances etc... Hell, i haven't even had any classes what so ever in functions and graphs .. If i hadn't looked it up myself, i would know what a function is :) I have not yet looked up exponential functions etc, so i cant really solve ODE's. It's weird we havent been taught anything about functions yet, which might be one of the, if not the, most important things in math.\n\nOne thing of GREAT annoyance occured to me today .. I was told by my teacher, that archimedes had found a way to calculate the area under a parabola using triangles. I figured, that if that old man could do it, so could i .. :) So i started working on it, and i decided in my mind that i should divide the area into an infinite amount of triangles, and calculate the limit as the number of triangles went to infinity. But to get a rational result, i had to find a proportion between the first and the second triangles .. I made a crappy drawing, and was not able to measure a useable proportion between the heights in the trinangles. So i gave up - i thought i might have got the idea and principles wrong, and i looked it up on the net (when i try figure something out, i must do it myself .. otherwise it's not me who's done it:) and i found out, that i had simply made an incorrect measurement, and my thoughts had been correct all along.. So i measured it in geogebra instead, and i found a proportion between the heights in the triangles to be 1\/8. Using this new info, i could calculate the area under a parabola with an infinite sum, where the number of triangles goes to infinity. so i made a formula :);\n\nI have attached a word document containing the formula, since i couldnt post my maple screenshot in any other way .. :) I would appreciate if anyone could tell if it's correct. '\n\nBut i must admit, that i am ahead of most students in my school mathematically, but compared to Terry Tao and other great mathematicians, my abilities are certainly mediocre :P One cannot decide whether someone is good at math, just because he can easily do the math in high school .. it's too easy to determine that :) Even though i don't understand why the vast majority of students are almost frightened by my \"line\" (you can choose different \"lines\" of study, which are specified for your area and interest\") which is Math and physics, with philosophy as a side-class. It's unbelieveable how bad the average student is in math in my age .. atleast here in Denmark :)\n\nAttachments\n\n\u2022 approximation.doc\n38.5 KB \u00b7 Views: 343\nSankaku\nIn every area of life you will ALWAYS find people better than you. Smarter, faster, stronger, tougher. However, you should be inspired by them, not frightened by them. My attitude has always been: \"They are human. I am human. If I want to, I can do what they do.\"\n\nIf you are going to be crushed by not being the best at everything, you are setting yourself up for serious psychological problems for the rest of your life. Let it go.\n\nBe good. Be really really good. But forget about being 'the best.' It doesn't exist - only in artificial testing systems like IQ or class grades. The real world doesn't work like that. As others have said, hard work counts for much more than raw talent.\n\nHomework Helper\n@Levis2: Well an IQ test, even an online one, should at least have comparative validity. Perhaps you would like others to take that same test to compare results.\n\nHomework Helper\nGold Member\nDearly Missed\nLevis2:\nHere's a far better way to gauge your competence as a mathematician than any IQ-test will be:\n\n1. When you read university textbooks on maths (if you haven't begun to read such now, DO SO!), hold your hand over the <i>proof section<\/i>.\n\n2. Be sure you understand the theorem or conjecture that is to be proven, and then make your best shot at it.\n\n3. Compare your effort with the proof in the text. try to gauge whose proof is the best, and, not the least, <i>why<\/i> that proof (either your own or the book's) is better.\n\nWhen asked how they developed their skills, many mathematicians cite this very technique as the most important one, according to their own self-evaluation.\n\nShackleford\nIQ tests were originally designed to determine the level of mental retardation.\n\nLevis2\nLevis2:\nHere's a far better way to gauge your competence as a mathematician than any IQ-test will be:\n\n1. When you read university textbooks on maths (if you haven't begun to read such now, DO SO!), hold your hand over the <i>proof section<\/i>.\n\n2. Be sure you understand the theorem or conjecture that is to be proven, and then make your best shot at it.\n\n3. Compare your effort with the proof in the text. try to gauge whose proof is the best, and, not the least, <i>why<\/i> that proof (either your own or the book's) is better.\n\nWhen asked how they developed their skills, many mathematicians cite this very technique as the most important one, according to their own self-evaluation.\n\nYeah ive done this, not with university books though. i do most of my reading on the internet .. University books are 3 years from my present educational level, but i plan on borrowing some from the library, if i am able to grasp the content :)\n\nAnd i know iq tests were originally designed for measuring mental retardation, but many people say they have been improved, since they are now widely used to detect brilliance in children.\n\nHomework Helper\nGold Member\nDearly Missed\nYeah ive done this, not with university books though. i do most of my reading on the internet .. University books are 3 years from my present educational level, but i plan on borrowing some from the library, if i am able to grasp the content :)\n\nAnd i know iq tests were originally designed for measuring mental retardation, but many people say they have been improved, since they are now widely used to detect brilliance in children.\nYou are already more than 3 years ahead than most people in mathematical maturity.\n\nYou should pick up introductory textbooks on calculus and linear algebra.\nyou will have no trouble whatsoever working with them.\n\nLykke til, fra en nordmann som \u00f8nsker deg alt vel!\n\nLevis2\nYou are already more than 3 years ahead than most people in mathematical maturity.\n\nYou should pick up introductory textbooks on calculus and linear algebra.\nyou will have no trouble whatsoever working with them.\n\nLykke til, fra en nordmann som \u00f8nsker deg alt vel!\n\n3 years is arguable, but ahead - yes idd :) The funniest thing is, that it was right around christmas i decided to study ahead. It was actually then, where i decided to become a mathematician of the highest quality, and i realised the high school math wasnt enough atm.\n\nMen som du nok har bem\u00e6rket, s\u00e5 er jeg MEGET interesseret i faget :) Jeg ser n\u00e6sten ikke tv mere, efter at jeg begyndte p\u00e5 mit \"forl\u00f8b\" her for 1 m\u00e5ned siden haha !:) Dog irriterer det mig at vi her i danmark kun har 6 universitet .. Konkurrencen er virkelig h\u00e5rd, s\u00e5 der skal fandme skrives et ret s\u00e5 banebrydende bidrag, hvis jeg skal undervise p\u00e5 en af dem senere, hvilket er min store dr\u00f8m !:P\n\nlawsofform\nHaving a non-genius IQ will not keep you from becoming a mathematician, but confusing correlation with causation might!\n\nSankaku\nHaving a non-genius IQ will not keep you from becoming a mathematician, but confusing correlation with causation might!\n\nHomework Helper\nGold Member\nDearly Missed\n3 years is arguable, but ahead - yes idd :) The funniest thing is, that it was right around christmas i decided to study ahead. It was actually then, where i decided to become a mathematician of the highest quality, and i realised the high school math wasnt enough atm.\n\nMen som du nok har bem\u00e6rket, s\u00e5 er jeg MEGET interesseret i faget :) Jeg ser n\u00e6sten ikke tv mere, efter at jeg begyndte p\u00e5 mit \"forl\u00f8b\" her for 1 m\u00e5ned siden haha !:) Dog irriterer det mig at vi her i danmark kun har 6 universitet .. Konkurrencen er virkelig h\u00e5rd, s\u00e5 der skal fandme skrives et ret s\u00e5 banebrydende bidrag, hvis jeg skal undervise p\u00e5 en af dem senere, hvilket er min store dr\u00f8m !:P\n\nHei, Levis2!\nHusk at du ligger minst 3 \u00e5r foran de fleste av konkurrentene dine, s\u00e5 du burde ha meget store sjanser.\n\nForslag fra meg:\nSnakk med l\u00e6reren din. Han vet hvor god du er. Sp\u00f8r ham om han kan skrive et brev, eller noe slikt, til n\u00e6rmeste universitet, at han har en unik elev (du!) som trenger konstruktiv veiledning i h\u00f8yere matematikk.\nPr\u00f8v \u00e5 f\u00e5 i stand et m\u00f8te med noen professorer p\u00e5 universitetet n\u00e6rmest der du bor, og fortell dem hva du holder p\u00e5 med p\u00e5 egen h\u00e5nd n\u00e5, og har f\u00e5tt til p\u00e5 egen h\u00e5nd. De vil bli gledelig overrasket over ditt faglige niv\u00e5, det kan jeg nok garantere!\n\nDisse vil v\u00e6re de beste til \u00e5 gi deg forslag om hva du burde lese, og eventuelt, om det er mulig for deg \u00e5 kunne f\u00e5 pr\u00f8vd deg selv formelt i noen fag, selvom du enn\u00e5 ikke har avlagt avsluttende skole-eksamener.\n\nI'll take the rest in English:\nYou show unique abilities, and the best thing you can do for yourself is, by the help of others, to get a constructive environment of learning suited to your level.\nPrecisely because you are quite rare, you must be willing to make much of that environment for yourself, but with thoughtful guidance of professionals who are interested in helping you onwards. Your teacher is one such professional, and as I've said, I'm sure some professor at your nearby university will be delighted to help you.\nYou must be the one to try and find him, though.\n\nLevis2\nHei, Levis2!\nHusk at du ligger minst 3 \u00e5r foran de fleste av konkurrentene dine, s\u00e5 du burde ha meget store sjanser.\n\nForslag fra meg:\nSnakk med l\u00e6reren din. Han vet hvor god du er. Sp\u00f8r ham om han kan skrive et brev, eller noe slikt, til n\u00e6rmeste universitet, at han har en unik elev (du!) som trenger konstruktiv veiledning i h\u00f8yere matematikk.\nPr\u00f8v \u00e5 f\u00e5 i stand et m\u00f8te med noen professorer p\u00e5 universitetet n\u00e6rmest der du bor, og fortell dem hva du holder p\u00e5 med p\u00e5 egen h\u00e5nd n\u00e5, og har f\u00e5tt til p\u00e5 egen h\u00e5nd. De vil bli gledelig overrasket over ditt faglige niv\u00e5, det kan jeg nok garantere!\n\nDisse vil v\u00e6re de beste til \u00e5 gi deg forslag om hva du burde lese, og eventuelt, om det er mulig for deg \u00e5 kunne f\u00e5 pr\u00f8vd deg selv formelt i noen fag, selvom du enn\u00e5 ikke har avlagt avsluttende skole-eksamener.\n\nI'll take the rest in English:\nYou show unique abilities, and the best thing you can do for yourself is, by the help of others, to get a constructive environment of learning suited to your level.\nPrecisely because you are quite rare, you must be willing to make much of that environment for yourself, but with thoughtful guidance of professionals who are interested in helping you onwards. Your teacher is one such professional, and as I've said, I'm sure some professor at your nearby university will be delighted to help you.\nYou must be the one to try and find him, though.\n\nNow if this is the case, then that's what ill do. It would be awesome to have some directions from a professor about what i should study. The only problem is, that i'm only advanced in the topics i have studied by myself - if i were to take a test for university \"classes\" you might say, then i would probably be able to do okay in calculus and logic, but would problably not be able to pass in, lets say, vectors, which i havent studied yet. But if i read all of the stuff, i might be able to though :) Could be cool to take weekend classes in higher math !!\n\nHomework Helper\nGold Member\nDearly Missed\nYou are confusing mathematical maturity and level of knowledge.\nThose are not the same, and it is with respect to maturity that you already are on a university level, from what i've seen.\nThe professor (AND your regular teacher at school) can easily help you fill in the relevant bits of knowledge, and you'll master them easily enough.\n\nHopefully, there ARE weekend classes in higher maths.\n\nIn a somewhat longer perspective, your school might be willing to give you, say, a few hours off to attend classes at the university, if they are told from a professional there that you are mature for that level of learning (based on what he has learned about you).\n\nAs I mentioned to you, your first step is to have a meeting with your teacher, and see if he is willing to help you contact the university in a constructive manner.\nHe is the person around you best able to gauge your abilities at the moment, and give a convincing argument to the university why they should be interested in helping you.\n\nLevis2\nYou are confusing mathematical maturity and level of knowledge.\nThose are not the same, and it is with respect to maturity that you already are on a university level, from what i've seen.\nThe professor (AND your regular teacher at school) can easily help you fill in the relevant bits of knowledge, and you'll master them easily enough.\n\nHopefully, there ARE weekend classes in higher maths.\n\nIn a somewhat longer perspective, your school might be willing to give you, say, a few hours off to attend classes at the university, if they are told from a professional there that you are mature for that level of learning (based on what he has learned about you).\n\nAs I mentioned to you, your first step is to have a meeting with your teacher, and see if he is willing to help you contact the university in a constructive manner.\nHe is the person around you best able to gauge your abilities at the moment, and give a convincing argument to the university why they should be interested in helping you.\n\nyeah i just realised that i was confusing those two terms aswell :) But how does the university\/teacher gauge my mathematical maturity, if they are not to gauge them on my knowledge? Is it a teachers instinct?\n\nAnd attending university (if i'm able to, university aint elementary school:) classes would be a dream coming true!!","date":"2022-08-08 00:50:54","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\": 0.55826336145401, \"perplexity\": 1404.8011679423378}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882570741.21\/warc\/CC-MAIN-20220808001418-20220808031418-00394.warc.gz\"}"}
| null | null |
package govmomi
import (
"github.com/vmware/govmomi/vim25/methods"
"github.com/vmware/govmomi/vim25/mo"
"github.com/vmware/govmomi/vim25/types"
)
type Folder struct {
types.ManagedObjectReference
c *Client
}
func NewFolder(c *Client, ref types.ManagedObjectReference) *Folder {
return &Folder{
ManagedObjectReference: ref,
c: c,
}
}
func (f Folder) Reference() types.ManagedObjectReference {
return f.ManagedObjectReference
}
func (f Folder) Children() ([]Reference, error) {
var mf mo.Folder
err := f.c.Properties(f.Reference(), []string{"childEntity"}, &mf)
if err != nil {
return nil, err
}
var rs []Reference
for _, e := range mf.ChildEntity {
if r := NewReference(f.c, e); r != nil {
rs = append(rs, r)
}
}
return rs, nil
}
func (f Folder) CreateVM(config types.VirtualMachineConfigSpec, pool *ResourcePool, host *HostSystem) (*Task, error) {
req := types.CreateVM_Task{
This: f.Reference(),
Config: config,
Pool: pool.Reference(),
}
if host != nil {
ref := host.Reference()
req.Host = &ref
}
res, err := methods.CreateVM_Task(f.c, &req)
if err != nil {
return nil, err
}
return NewTask(f.c, res.Returnval), nil
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,686
|
Adnan Abidi / Reuters A vendor arranges books by the roadside in the old quarters of Delhi, August 20, 2006.
Ananya Vajpeyi is a historian at the Centre for the Study of Developing Societies, New Delhi, and the author of Righteous Republic: The Political Foundations of Modern India.
The Sahitya Akademi controversy can best be understood as part of a larger battle: a culture war between India's literary establishment, dominated by left-leaning, secular writers, and India's right-wing government, led by Prime Minister Narendra Modi and his Hindu nationalist Bharatiya Janata Party. The BJP came to power in New Delhi in May 2014, securing a majority of seats in the Indian Parliament despite winning just 31 percent of the votes cast in the national election (an anomaly made possible by India's "first past the post" electoral system). Since then, the Indian intelligentsia and media have watched with alarm as government officials have intimidated minorities, sectarian violence has risen, cultural institutions have purged their liberal incumbents, and public debate has become ever more polarized.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 5,221
|
Harrington wins British Open, ends Norman's dream of major
SOUTHPORT, England " Turns out Padraig Harrington's wrist was strong enough to hit all the right shots in the British Open. Better yet, it was strong enough to lift the silver claret jug.
Harrington became the first European in more than a century to win golf's oldest championship two years is a row, smashing a pair of fairway metals into the par 5s Sunday that allowed him to pull away from misktake-prone Greg Norman and hold off a late charge by Ian Poulter for a four-shot victory.
Even in the relentless wind, Harrington managed to shoot 32 on the back nine to close with a 1-under 69.
And to think he gave himself only a 75 percent chance of teeing off on Thursday, and only a 50 percent chance of finishing.
No one bothered asking him the odds of winning.
.sd-donation .logo { width: 50%; margin: 1rem 0 1rem; }
.sd-donation h1 { font-size: 2rem; text-transform: none; color: #fff; } .sd-donation p { color: #fff; font-weight: 300; } .sd-donation hr { width: 20%; border-top: 4px solid #000; }
.sd-donation .btn { padding: .5rem 2rem; background-color: #fff !important; border-radius: 0; } .sd-donation .btn { color: #037BC1; }
.sd-donation .btn:hover { background-color: #005789 !important; } .sd-donation .btn:hover a { color: #fff !important; } .sd-donation .col-xl-5.p-0 { background-image: url('https://cdn.summitdaily.com/wp-content/uploads/sites/9/2020/03/sd-donate-cta-bg.jpg'); background-size: cover; min-height:330px; }
@media ( min-width: 768px ) { .sd-donation .logo { width: 35%; } }
@media ( min-width: 1440px ) { .sd-donation { text-align: left; } .sd-donation-mobile { display: none; }
.sd-donation hr { margin-left: 0; }
"I enjoyed the claret jug so much I didn't want to give it back," Harrington said.
The 36-year-old Irishman injured his right wrist eight days ago, and it was so sore when he arrived at Royal Birkdale that he stopped practice after nine holes on Tuesday and three swings on Wednesday.
But he was at full strength in gusts up to 40 mph off the Irish Sea, especially down the stretch. He ripped a 3-wood into the wind to about 40 feet on the par-5 15th and got down in two putts for birdie to build a two-shot lead. Then came another 3-wood that bounded up the green on the par-5 17th and settled 4 feet away for eagle.
A year ago, Harrington was an emotional wreck at Carnoustie after making double bogey on the final hole and beating Sergio Garcia in a playoff.
Backed by a four-shot lead, he was afforded a pleasurable walk along the dunes toward the 18th green, the only suspense his margin of victory.
He finished at 3-over 283, becoming the first European since James Braid in 1905-06 to win the Open in successive years.
It was his first victory since the British Open last year, and it could not have come at a better time. Harrington moved to the top of Europe's Ryder Cup standings, and the victory moved him to No. 3 in the world ranking behing Tiger Woods and Phil Mickelson.
"I'm quite enjoying this," Harrington said, cradling the claret jug. "I don't think I'll get down off the stage."
Norman played a familiar role as the tragic figure.
This had all the elements of a fairy tale like few others in golf. Norman, 53, married tennis great Chris Evert three weeks ago and was on the tail end of his honeymoon when he wound up with a two-shot lead going into the final round and a chance to become the oldest major champion. Instead, it ended like so many other majors when he was in his prime.
The Shark lost his two-shot lead after the third hole. He still had a one-shot lead going to the back nine, but bogeyed three of the next four holes and had to settle for a 77 and a tie for third with Henrik Stenson (71).
"I walk away from here disappointed, but with my head held high, because I hung in there," Norman said.
Poulter thought he could bring England its first British Open since Nick Faldo in 1992, playing bogey-free over his final 15 holes and making a 15-foot par on the 18th hole to finish off a 69. He went to the practice range in case of a playoff, but put his clubs away when he saw that Harrington made eagle on the 17th hole.
Norman tried to keep alive his hopes with a 35-foot par putt on the 14th, and a shot from a pot bunker that made him spin backward, turning to see the ball land 4 feet away for a birdie. Harrington, however, didn't back down.
"Padraig played brilliantly today, even though he tried to let it get away in the middle of the round," Norman said. "He came back and performed brilliantly, and he finished like a true champion."
Harrington walked off the 18th green with his children, Patrick and Cairan, and sat atop a pot bunker to pose with the jug.
The leaderboard featured a familiar name, missing an "s."
Chris Wood, a 20-year-old amateur from England, closed in on the lead until three straight bogeys on the back nine. He closed with a 72 and tied for fifth at 10-over 290 with Jim Furyk (71).
Arrow Insurance Mgmt Inc
Insurance Agency Service/Sales at Arrow Insurance Mgmt Inc in FRISCO
Local property/casualty insurance agency is accepting resumes to add the right person to our service/sales team. Insurance experience and license…
Natural Skincare Retailer
Customer Service at Natural Skincare Retailer in SILVERTHORNE
CSR Order Fulfillment Silverthorne online Natural skin care retailer. FT. Resume to patrick@onewellworld.com
CTL|Thompson
Project Geotechnical Engineer or Geologist at CTL|Thompson in BRECKENRIDGE
CTL|Thompson Inc. seeks a Project Geotechnical Engineer or Geologist for its Breckenridge, CO office. The successful applicant will work on…
Tubing Hill Attendant, Maintenance Operator at Town of Frisco in FRISCO
FREE COPPER PASS FT/PT Position Avail Adventure Park Tubing Hill Attendant $13.50 Copper Ski Pass Included for 16+ hrs. Many…
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,238
|
\section*{Supplementary information}
The supplementary information contains the explicit expressions for the Green's kernels of the Stokes equations, the calculation of the cortex flow in terms of the concentration fields, the description of the numerical procedure, the technique of shape reconstruction in the quasi-spherical limit and the physical values used to estimate non-dimensional parameters.
\section{Green's kernels}
The following kernels are to be used in eq. (2) of the main text:
\begin{align}
\label{kernels}
G_{ij}(\boldsymbol x,\boldsymbol x')&=\frac{1}{8\pi}\left[\frac{\delta_{ij}}{|\boldsymbol x-\boldsymbol x')|}+\frac{(\boldsymbol x-\boldsymbol x')_i(\boldsymbol x-\boldsymbol x')_j}{|\boldsymbol x-\boldsymbol x'|^3}\right],\notag\\
K_{ijk}(\boldsymbol x,\boldsymbol x')&=\frac{3}{4\pi}\frac{(\boldsymbol x-\boldsymbol x')_i(\boldsymbol x-\boldsymbol x')_j(\boldsymbol x-\boldsymbol x')_k}{|\boldsymbol x-\boldsymbol x'|^5}.
\end{align}
\section{Full solution}
The results in this section are presented in the most general form, without any assumptions about the relative values of $\eta_{in}$, $\eta_{out}$, $\eta_s$, and $\eta_b$.
\subsection{Spherical Harmonics}
All fields on the cell surface are expanded in spherical harmonics of the vector pointing from the center of the cell to a given point of its surface.
The spherical harmonics are defined as
\begin{equation}
\label{Y}
Y_{l,m}(\boldsymbol x)=\sqrt{\frac{2l+1}{4\pi}\frac{(l-m)!}{(l+m)!}}P_l^m\left(\frac{x_3}{x}\right)\left(\frac{x_1+ix_2}{|x_1+ix_2|}\right)^m,
\end{equation}
where $P_l^m$ are associated Legendre polynomials.
The following expansions are used
\begin{equation}
\label{sphericalca}
\begin{aligned}
&c^a(\boldsymbol x)=\sum\limits_{l=0}^\infty c^a_l(\boldsymbol x)\\
&c^a_l(\boldsymbol x)=\sum\limits_{m=-l}^lc^a_{l,m}Y_{l,m}(\boldsymbol x)\\
\end{aligned}
\end{equation}
\begin{equation}
\label{sphericalcm}
\begin{aligned}
&c^\mu(\boldsymbol x)=\sum\limits_{l=0}^\infty c^\mu_l(\boldsymbol x)\\
&c^\mu_l(\boldsymbol x)=\sum\limits_{m=-l}^lc^\mu_{l,m}Y_{l,m}(\boldsymbol x)\\
\end{aligned}
\end{equation}
\begin{equation}
\label{sphericalfn}
\begin{aligned}
&f^n(\boldsymbol x)=\sum\limits_{l=2}^\infty f^n_l(\boldsymbol x)\\
&f^n_l(\boldsymbol x)=\sum\limits_{m=-l}^lf^n_{l,m}Y_{l,m}(\boldsymbol x)\\
\end{aligned}
\end{equation}
\begin{equation}
\label{sphericalU}
\begin{aligned}
&U(\boldsymbol x)=\sum\limits_{l=1}^\infty U_l(\boldsymbol x)\\
&U_l(\boldsymbol x)=\sum\limits_{m=-l}^lU_{l,m}Y_{l,m}(\boldsymbol x)\\
\end{aligned}
\end{equation}
Note that $f^n_1=0$ is zero, as required by the condition of the total force acting on the cell being equal to zero, $f^n_0$ is irrelevant because it just shifts the osmotic pressure drop across the membrane, and $U_0$ is irrelevant because only gradients of $U$ enter equations.
We define the vector spherical harmonics as
\begin{equation}
\label{Y1}
\boldsymbol Y_{1,l,m} (\boldsymbol x) = [\boldsymbol \nabla^s -(l+1)\boldsymbol x] Y_{l,m}(\boldsymbol x)
\end{equation}
\begin{equation}
\label{Y2}
\boldsymbol Y_{2,l,m} (\boldsymbol x) = [\boldsymbol \nabla^s +l\boldsymbol x] Y_{l,m}(\boldsymbol x)
\end{equation}
\begin{equation}
\label{Y3}
\boldsymbol Y_{3,l,m} (\boldsymbol x) = \boldsymbol x \times \boldsymbol\nabla^s Y_{l,m}(\boldsymbol x)
\end{equation}
The following expansions are used
\begin{equation}
\label{vY}
\boldsymbol u^c=\sum\limits_{j=1}^3\sum\limits_{l=0}^\infty\sum\limits_{m=-l}^lu^c_{j,l,m}\boldsymbol Y_{j,l,m}(\boldsymbol x)
\end{equation}
\begin{equation}
\label{fY}
\boldsymbol f=\sum\limits_{j=1}^3\sum\limits_{l=0}^\infty\sum\limits_{m=-l}^lf_{j,l,m}\boldsymbol Y_{j,l,m}(\boldsymbol x)
\end{equation}
\subsection{Force calculation}
The force $\boldsymbol f$ can be represented as a sum of two contributions $f_{j,l,m}=f^v_{j,l,m}+f^e_{j,l,m},$ where
\begin{equation}
\label{f}
\begin{aligned}
&f^v_{1,l,m}=-\frac{(l+2)[2\eta_s(l^2+l+1)+\eta_b(l+1)(l+2)]}{2l+1}\frac{u^c_{1,l,m}}{R^2}-\frac{l(l-1)(l+2)(2\eta_s+\eta_b)}{2l+1}\frac{u^c_{2,l,m}}{R^2},\\
&f^v_{2,l,m}=-\frac{(l+1)(l+2)(l-1)(2\eta_s+\eta_b)}{2l+1}\frac{u^c_{1,l,m}}{R^2}-\frac{(l-1)[2\eta_s(l^2+l+1)+\eta_b(l-1)l]}{2l+1}\frac{u^c_{2,l,m}}{R^2},\\
&f^v_{3,l,m}=-\eta_s(l+2)(l-1)\frac{u^c_{3,l,m}}{R^2},\\
&f^e_{1,l,m}=\frac{l+2}{2l+1}\frac{\left[\chi c^\mu-\alpha c^a\right]_{l,m}}{R}+\frac{f^n_{l,m}}{2l+1},\\
&f^e_{2,l,m}=\frac{l-1}{2l+1}\frac{\left[\chi c^\mu-\alpha c^a\right]_{l,m}}{R}-\frac{f^n_{l,m}}{2l+1},\\
&f^e_{3,l,m}=0.
\end{aligned}
\end{equation}
The amplitudes $f^v_{j,l,m}$ contain the contribution of the surface viscosity terms in eq. (1) of the main text, while the amplitudes $f^e_{j,l,m}$ contain contributions of myosin contractility, cortex compressibility, and Lagrange multiplier $f^n\boldsymbol n$.
\subsection{Fluid dynamics}
The integrals in eq. (2) of the main text can be calculated analytically for a spherical cell.
The results are
\begin{equation}
\label{BIEsphere}
\frac{\eta_{in}+\eta_{out}}{2}u^c_{j,l,m}=Rg_{j,l}f_{j,l,m}+(\eta_{out}-\eta_{in})k_{j,l}u^c_{j,l,m},
\end{equation}
where the coefficients $g_{j,l}$ and $k_{j,l}$ are listed in table \ref{greentable}.
\begin{table}
\begin{center}
\begin{tabular}{c|ccc}
$j$ & 1 & 2 & 3 \\
\hline
$g_{j,l}$ & $\frac{l}{(2l+1)(2l+3)}$ & $\frac{l+1}{(2l-1)(2l+1)}$ & $\frac{1}{2l+1}$ \\
$k_{j,l}$ & $\frac{3}{2(2l+1)(2l+3)}$ & $-\frac{3}{2(2l-1)(2l+1)}$ & $-\frac{3}{2(2l+1)}$ \\
\end{tabular}
\end{center}
\caption{\label{greentable}Integrals of Green's kernels for a spherical cell.}
\end{table}
\subsection{Explicit solution}
We note that the Green's kernels (\ref{kernels}) are diagonal in the basis of vector spherical harmonics for spherical shape.
Furthermore, we see that the amplitudes of the surface viscosity force $f^v_{3,l,m}$ depend only on $u^c_{3,l,m}.$
This implies that the vector spherical harmonics with first index 3 are completely decoupled from the two other types.
Since $f^e_{3,l,m}=0$ for all $l$ and $m$, we conclude that $u^c_{3,l,m}=0$ for all $l$ and $m$ as well.
Adding the fixed shape condition $(\boldsymbol u^c-\boldsymbol v_s)\boldsymbol\cdot\boldsymbol n=0$ yields that $\boldsymbol u^c-\boldsymbol v_s$ can be written as a surface gradient of some surface potential $U$, as used in the main text.
Or, in spherical harmonics,
\begin{equation}
\label{grads}
u^c_{1,l,m}=\frac{U_{l,m}}{R}\frac{l}{2l+1},\,\,\, u^c_{2,l,m}=\frac{U_{l,m}}{R}\frac{l+1}{2l+1},\,\,\, u^c_{3,l,m}=0 \textrm{ for }l>1.
\end{equation}
Solving the equations (\ref{f}), (\ref{BIEsphere}), and (\ref{grads}) for $U$ and $f^n$ yields
\begin{equation}
\label{lambda}
U_{l,m}=\frac{R(\chi c^\mu-\alpha c^a)_{l,m}}{\lambda_l}.
\end{equation}
\begin{equation}
\label{lambdal}
\lambda_l=\begin{cases}
&3\eta_{in}R+2\eta_{out}R+2\eta_s+2\eta_b\textrm{, for }l=1,\\
&(2l+1)(\eta_{in}+\eta_{out})R+l(l+1)\eta_b+2(l^2+l-1)\eta_s\textrm{, for }l>1.
\end{cases}
\end{equation}
\begin{equation}
\label{vs}
v_s=-\frac{2}{3}\nabla U_1.
\end{equation}
\begin{equation}
\label{fnres}
f^n_{l,m}=-\left[(l+2)\eta_{out}R+(l-1)\eta_{in}R+2(l+2)(l-1)\eta_s\right]\frac{U_{l,m}}{R^2}\textrm{ for }l>1.
\end{equation}
Using the expressions (\ref{lambda}) and (\ref{vs}), the retrograde flow and the swimming velocity can be expressed as a function of the concentration fields.
The time evolution equations of the concentration fields are obtained by substituting $U$ into eqs. (3) and (4) of the main text.
The following equation can be used to reduce all calculations to scalar spherical harmonics
\begin{equation}
\label{Uc}
\boldsymbol\nabla^s\boldsymbol\cdot\left[c(\boldsymbol u^c-\boldsymbol v_s)\right]=\boldsymbol\nabla^s\boldsymbol\cdot(c\boldsymbol\nabla^s U)=\frac{\Delta^s(cU)+c\Delta^sU-U\Delta^sc}{2}.
\end{equation}
\section{Numerical procedure}
The numerical procedure consists in representing $c^a$, $c^\mu$ and $U$ by the amplitudes of the spherical harmonics for all values of $l<l_{max}$ and $|m|\le l$, where $l_{max}$ is the cut-off value.
We take $l_{max}=64$ for such calculations.
We observed that regardless of the initial conditions, the dynamics relaxed to an axisymmetric solution.
We therefore also performed simulations with the shape assumed axisymmetric from the beginning, which is achieved by setting all amplitudes for $m\ne 0$ to zero.
With this assumption, $l_{max}=1024$ was used, which proved to be necessary for strongly polarized cells.
The eqs. (3) and (4) of the main text were solved by an explicit Euler scheme by truncating the harmonic expansion of the advection terms to $l<l_{max}$.
The time step was chosen small enough to avoid the instability due to the stiffness of the diffusion equation (typically $10^{-4}$ in non-dimensional units).
In some cases, a small diffusion of actin (diffusion coefficient $10^{-3}$ in non-dimensional units) was added to enhance the stability of the actin advection equation.
The steady-state branches in Figs. 2 and 3 of the main text were obtained by solving eqs. (3) and (4) of the main text with $\dot c^a=\dot c^\mu=0$ using Newton's method.
\section{Model for the cell shape}
The calculation of the shape follows the method used in Ref. [28] of the main text.
Spherical shape of a cell in suspension can be physically achieved by a combination of high osmotic pressure $\Delta P$ inside the cell and the inextensibility of the membrane.
Further in this section, we allow the shape of the cell to deviate from a sphere, albeit weakly, taking the leading terms in the small-deformation expansion.
We parametrize the cell shape by a shape function $\rho$
\begin{equation}
\label{sphericalx}
|\boldsymbol x|=R_0\left[1+\rho(\boldsymbol x)\right],
\end{equation}
where $|\boldsymbol x|$ is the distance from the center of the cell to a given point on its boundary.
$\rho_0=0$ to the leading order in deformation because of the conservation of the membrane area.
$\rho_1=0$ because this term corresponds to a translation of the cell to the leading order.
We show below that $\rho$ scales as $\Delta P^{-1},$ which justifies an expansion in powers of $\rho$.
We consider the quasi-spherical limit, taking the leading terms in such expansions.
The function $\rho(\boldsymbol x)$ is expanded in spherical harmonics of $\boldsymbol x$ to be used below
\begin{equation}
\label{rho}
\begin{aligned}
&\rho(\boldsymbol x)=\sum\limits_{l=2}^\infty\rho_l(\boldsymbol x)\\
&\rho_l(\boldsymbol x)=\sum\limits_{m=-l}^l\rho_{l,m}Y_{l,m}(\boldsymbol x).
\end{aligned}
\end{equation}
Assuming the tension of the membrane $\zeta_0$ to be homogeneous (unaffected by the cortex flow) we can write for the tension force $\boldsymbol f^m$
\begin{equation}
\label{fm}
\boldsymbol f^m=-H\zeta_0\boldsymbol n=-\zeta_0H_0\boldsymbol n-(H-H_0)\boldsymbol n\zeta_0,
\end{equation}
where $H$ is the mean curvature of the membrane (sum of the principal curvatures) and $H_0=2/R$ is the value of $H$ for a perfectly spherical cell.
The term $-\zeta_0H_0\boldsymbol n$ in eq. (\ref{fm}) corresponds to an isotropic compression of the fluid inside the cell, which is balanced by the osmotic pressure.
This relates the tension of the membrane $\zeta_0$ to the pressure difference by the Laplace law:
\begin{equation}
\label{Laplace}
\zeta_0= \frac{R \Delta P}{2}.
\end{equation}
The term $(H-H_0)\boldsymbol n\zeta_0$ in eq. (\ref{fm}) corresponds to a position-dependent normal force, which we identify with the Lagrange multiplier $f^n\boldsymbol n$ used to maintain the shape of the cell.
This justifies that $H-H_0$ scales as $\zeta_0^{-1}$ or, equivalently, as $\Delta P^{-1}$ for fixed $f^n$.
Since $f^n$ is governed by the actomyosin dynamics, as follows from eq. (\ref{fnres}), we obtain that the shape of the cell can be indeed made as close to a sphere as necessary by choosing $\Delta P$ large enough.
This shows that all calculations made for perfectly spherical cells remain valid to the leading order in the limit of large $\Delta P$ even if the spherical-shape condition is relaxed.
The mean curvature can be related to the shape function by
\begin{equation}
\label{sphericalH}
H(\boldsymbol x)=\frac{2}{R}+\frac{1}{R}\sum\limits_{l=2}^\infty (l-1)(l+2)\rho_l(\boldsymbol x).
\end{equation}
This yields the final relation between the shape harmonics $\rho_{l,m}$ and the Lagrange multiplier $f^n$:
\begin{equation}
\label{rhores}
\rho_{l,m}=-\frac{2f^n_{l,m}}{(l-1)(l+2)\Delta P}.
\end{equation}
\section{Physical parameters}
We list in table.~\ref{t:valpar} the physical data we have considered to obtain rough estimates of the three non-dimensional parameters entering in the model.
\begin{table}
\scriptsize
\begin{tabular}{lll}
\hline\hline
name & symbol & typical value \\
\hline
cortical thickness & $h$ & $10^{-7}$ m \cite{clark2013monitoring,turlier2014furrow}\\
cortical viscosity & $\eta_s$ & $h\times (10^3-10^6)$ Pa m s \cite{turlier2014furrow,bergert2015force}\\
myosin contractility & $ \chi c_0^{\mu}$ &$h\times (10^2-10^3)$ Pa m \cite{bergert2015force,turlier2014furrow} \\
F-actin compressibility & $ \alpha c_0^{a}$ &$h\times 10^3$ Pa.m \cite{hawkins2011spontaneous}\\
myosin diffusion coefficient & $D^{\mu}$ &$10^{-13}-10^{-12}$ $\text{m}^{2}\text{s}^{-1}$ \cite{uehara2010determinants, hawkins2011spontaneous}\\
cell size & $R$ &$10^{-5}$ m \\
F-actin turnover & $\beta$ &$ 10^{-2}-10^{-1}$ ~$\text{s}^{-1}$ \cite{hawkins2011spontaneous, fritzsche2013analysis,turlier2014furrow} \\
\hline
characteristic length & $l_0=R$ &$10^{-5}$ m \\
characteristic time & $t_0=R^2/D^{\mu}$ & $10^2-10^3$ s \\
characteristic surface stress & $\sigma_0=D^{\mu}\eta_s/R^2$ & $1-10^4$ Pa m \\
\hline
contractility parameter & $\bar{\chi}=\chi c_0^{\mu}/\sigma_0$ & $10^{-2}-10^3$ \\
compressibility parameter & $\bar{\alpha}=\alpha c_0^{a}/\sigma_0$ & $10^{-1}-10^3$ \\
turnover parameter & $\bar{\beta}=\beta t_0$ & $1-10^2$\\
\hline\hline
\end{tabular}
\caption{\small Estimates of material coefficients and non dimensional parameters definitions.\label{t:valpar}}
\end{table}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,603
|
Communes
Allemans, commune française, située dans le département de la Dordogne ;
Allemans, ancienne commune française intégrée à celle de Penne-d'Agenais, située dans le département de Lot-et-Garonne ;
Allemans-du-Dropt, commune française, située dans le département de Lot-et-Garonne.
Voir aussi
Les Allemans, ancienne commune française, située dans le département de l'Ariège
Alamans ou Alémans, ensemble de tribus germaniques
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,918
|
Get an alert with the newest ads for "kitchenaid stand mixers" in Toronto (GTA).
Brand New, Two year warranty KitchenAid Commercial KSMC895 Four different colours to choose from 10 Speeds Watts 500 Capacity 8 Quart Volts 120 NSF-Certified Powerful enough to handle 8+ pounds of dough; quiet enough to sit on your counter Comes with bowl, flat beater, dough hook, and whip We ship all across Canada.
Excellent condition, lightly used, very clean. Pick up Oshawa, can arrange to meet at another location.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,732
|
{"url":"http:\/\/mathhelpforum.com\/trigonometry\/44668-find-vlue-4-cos-cos-b-cos-c-if.html","text":"# Thread: Find the vlue of 4(cos A + cos B + cos C) if...?\n\n1. ## Find the vlue of 4(cos A + cos B + cos C) if...?\n\nIn a triangle ABC, if 3R = 4r, value of 4(cos A + cos B + cos C) is? (R and r in the triangle have usual notations)\n\nAns: 7\n\nHow? I couldn't get 7.\n\n2. Originally Posted by fardeen_gen\nIn a triangle ABC, if 3R = 4r, value of 4(cos A + cos B + cos C) is? (R and r in the triangle have usual notations)\n\nAns: 7\n\nHow? I couldn't get 7.\nPlease explain what R and r are... Because your usual notations are not necessarily everybody's usual notations...\n\n3. Hello, moo!\n\nTraditionally, these are the accepted definitions . . .\n\n. . $R$ is the circumradius, the radius of the circumscribing circle.\n\n. . $r$ is the inradius, the radius of the inscribed circle.","date":"2016-08-24 09:06:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 2, \"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\": 0.9797248244285583, \"perplexity\": 3091.713392152017}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-36\/segments\/1471982291592.14\/warc\/CC-MAIN-20160823195811-00099-ip-10-153-172-175.ec2.internal.warc.gz\"}"}
| null | null |
{% extends 'taxbrain/input_base.html' %}
{% load staticfiles %}
{% load flatblocks %}
{% block style %}
{% load results %}
{{block.super}}
<link href="{% static 'css/vendor/bootstrap3-block-grid.min.css' %}" rel="stylesheet">
<link rel="stylesheet" href="https://cdn.datatables.net/1.10.18/css/dataTables.bootstrap.min.css">
<link rel="stylesheet" href="https://cdn.datatables.net/buttons/1.5.2/css/buttons.bootstrap.min.css">
<link rel="stylesheet" href="https://cdn.datatables.net/responsive/2.2.2/css/responsive.bootstrap.min.css">
<style>
.btn {
color: black;
}
.text-white {
color: white;
}
.table > thead > tr > th {
border: 0;
}
.single-line {
white-space: nowrap;
}
.param-text {
margin-left: 80px;
margin-right: 80px;
margin-top: 40px;
margin-bottom: 40px;
text-align: left;
}
.file-contents{
font-family: "Courier New", Courier, monospace;
font-weight: bold;
margin-left: 80px;
margin-right: 80px;
margin-top: 40px;
margin-bottom: 40px;
text-align: left;
}
.container {
width: 95%;
}
.output, .aggr-output {
display: none;
}
.output.selected-table-year.selected-table-tag, .aggr-output.selected-aggr-table-tag {
display: block;
}
.container > .row {
padding: 10px 0;
}
.output h1 {
text-transform: uppercase;
}
#table-select-container .nav-pills li {
padding-top: 40px;
}
#table-select-container .nav-pills li:first-child {
padding-top: 0;
}
.dt-btn-wrapper {
text-align: center;
}
</style>
{% endblock %}
{% block content %}
<div class="wrapper">
{% include 'taxbrain/header.html' %}
<div class="result-header">
<div class="result-header-control">
{{ view.result_header }}
</div>
<p class="meta">These results were generated by <a href="https://github.com/OpenSourcePolicyCenter/webapp-public/tree/v{{ object.webapp_vers }}"> PolicyBrain version {{ object.webapp_vers }}</a> on {{ object.creation_date | date:"D, M jS Y \a\t g:iA" }} UTC using <a href="https://github.com/open-source-economics/Tax-Calculator/tree/{{ object.upstream_vers }}"> Tax-Calculator version {{ object.upstream_vers }}.</a> </p>
{% if object.inputs.quick_calc %}
<p class="meta">This calculation used only a small sample of the available data and only calculated revenues for one year instead of ten. For the full results, <a href="/taxbrain/submit/{{ object.pk }}/">click here</a></p>
{% endif %}
{% if is_behavior %}
<p class="meta">The microsimulation upon which this dynamic simulation was based can be found <a href="{{microsim_url}}">here</a> </p>
{% endif %}
<div class="result-table">
<div class="result-table-controls">
{% if view.is_editable and not view.is_from_file %}
<a href="{{ object.get_absolute_edit_url }}" class="text-white btn btn-secondary">Edit Parameters</a>
{% endif %}
{% if view.has_link_to_dyn %}
<button onclick="buttonAction()" class="text-white btn btn-secondary">Link to Dynamic Simulations</button>
{% endif %}
<a href="{{ object.get_absolute_download_url }}" class="text-white btn btn-secondary">Download Files</a>
</div>
{% if view.is_from_file %}
<h2>The following reform file(s) were uploaded for this simulation: </h2>
{% else %}
<h2>The following parameters were used for this simulation: </h2>
{% endif %}
<div class="file-contents">
{%autoescape off %}
{{ view.inputs_to_display | nbsp | linebreaks | safe}}
{%endautoescape %}
</div>
</div>
</div>
<br>
<div id="table-drilldown-container">
<div>
<div class="container">
<div class="row">
{% if object.aggr_outputs %}
<div class="col-md-12" id="fiscal-totals-container">
<div class="panel panel-default">
<div class="panel-body">
{% include "taxbrain/select_tags.html" with tags=view.aggr_tags only %}
{% for x in object.aggr_outputs %}
<div class="aggr-output" {% for key, value in x.tags.items %} data-tag-{{ key }}="{{ value }}"{% endfor %}>
<h1 class="text-center">{{ x.title }}</h1>
{{ x.renderable | safe }}
</div>
{% endfor %}
</div>
</div>
</div>
{% endif %}
</div>
<div class="row">
{% if object.outputs %}
<div class="col-md-8 col-md-push-2" id="table-select-container">
<div class="panel panel-default">
<div class="panel-heading">
<h3 class="panel-title">Build Table</h3>
</div>
<div class="panel-body">
<div class="row">
<div class="col-md-6">
<h1 class="text-center">For Year: </h1>
</div>
<div class="col-md-6">
<h1 class="text-center">
<div class="dropdown dropdown-select dropdown-inline data-switch">
<a class="dropdown-toggle" id="selected-year" type="button" data-toggle="dropdown" aria-expanded="true">
{{ object.inputs.years.0 }}
<span class="caret"></span>
</a>
<ul class="dropdown-menu" id="year-selector" role="menu" aria-labelledby="years">
{% for year in object.inputs.years %}
<li role="presentation">
<a role="menuitem" tabindex="-1" href="#" data-year="{{ year }}">
{{ year }}
</a>
</li>
{% endfor %}
</ul>
</div>
</h1>
</div>
</div>
{% include "taxbrain/select_tags.html" with tags=view.tags only %}
</div>
</div>
</div>
{% endif %}
</div>
<div class="row">
{% if object.outputs %}
<div class="col-md-12" id="table-container">
<div class="panel panel-default">
<div class="panel-body" style="overflow: scroll">
{% for x in object.outputs %}
<div class="output" data-year="{{ x.year }}" {% for key, value in x.tags.items %} data-tag-{{ key }}="{{ value }}"{% endfor %}>
<h1 class="text-center">{{ x.title }}</h1>
{{ x.renderable | safe }}
</div>
{% endfor %}
</div>
</div>
</div>
{% endif %}
</div>
</div>
</div>
</div>
</div>
<div class="modal fade" id="block-link-modal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel">
<div class="modal-dialog" role="document">
<div class="modal-content">
<div class="modal-header">
<button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button>
<h4 class="modal-title" id="myModalLabel">Dynamic Simulation Link Unavailable</h4>
</div>
<div class="modal-body">
Oops. Your simulation already had behavioral parameters. You can't link this simulation to a dynamic simulation.
</div>
<div class="modal-footer">
<a type="button" class="btn btn-default" data-dismiss="modal">Got it.</a>
</div>
</div>
</div>
</div>
{% endblock %}
{% block bottom_scripts %}
{{ block.super }}
<script type="text/javascript" src="https://cdn.datatables.net/1.10.18/js/jquery.dataTables.min.js"></script>
<script type="text/javascript" src="https://cdn.datatables.net/1.10.18/js/dataTables.bootstrap.min.js"></script>
<script type="text/javascript" src="https://cdn.datatables.net/buttons/1.5.2/js/dataTables.buttons.min.js"></script>
<script type="text/javascript" src="https://cdn.datatables.net/buttons/1.5.2/js/buttons.bootstrap.min.js"></script>
<script type="text/javascript" src="https://cdn.datatables.net/buttons/1.5.2/js/buttons.colVis.min.js"></script>
<script type="text/javascript" src="https://cdn.datatables.net/buttons/1.5.2/js/buttons.flash.min.js"></script>
<script type="text/javascript" src="https://cdn.datatables.net/buttons/1.5.2/js/buttons.html5.min.js"></script>
<script type="text/javascript" src="https://cdn.datatables.net/buttons/1.5.2/js/buttons.print.min.js"></script>
<script type="text/javascript" src="https://cdn.datatables.net/responsive/2.2.2/js/dataTables.responsive.min.js"></script>
<script type="text/javascript" src="https://cdn.datatables.net/responsive/2.2.2/js/responsive.bootstrap.min.js"></script>
<script>
var buttonAction = function () {
if ({{view.has_link_to_dyn|yesno:"true,false,true"}} ) {
window.location.href = "/dynamic/{{ object.pk }}/";
} else {
$('#block-link-modal').modal('show');
}
};
var dataTableOptions = {
"paging": false,
"ordering": false,
"searching": false,
"bInfo": false,
'dom': 'Bt', /* Display the buttons before the table */
"buttons": [
'print',
{"text": "Copy table",
"extend": "copy"},
{"text": "Select columns",
"extend": "colvis"}
]
// "responsive": true
};
var initDataTable = function (el) {
var dt = el.dataTable(dataTableOptions);
dt.parent().find("div.dt-buttons.btn-group")
.wrap("<div class='dt-btn-wrapper'></div>");
};
var initVisibleTable = function (containerSelector) {
/* We only expect one table to be visible at once, i.e. if the right year is
selected, and the right tag is selected */
var table_to_show = $(containerSelector + " table:visible");
/* Make sure we don't initialize the DataTable twice */
if ( ! $.fn.DataTable.isDataTable(table_to_show) ) {
initDataTable(table_to_show);
}
};
var initVisibleYearTable = function () {
initVisibleTable('#table-container');
};
var initVisibleAggrTable = function () {
initVisibleTable('#fiscal-totals-container');
};
var activateYearTablesBySelectedYear = function () {
var containers_to_show = $('.output[data-year="' + $("#selected-year").text().trim() + '"]');
containers_to_show.addClass('selected-table-year');
$('.output').not(containers_to_show).removeClass('selected-table-year');
};
var activateTablesBySelectedTags = function (args) {
/* Get the current set of tags */
var tags_to_match = {};
$(args.selectorContainerSelector + ' .tag-selector:visible li.active').each(function (_index, el) {
$.each(el.attributes, function (_index, attr) {
if (attr.name.startsWith('data-tag-')) {
tags_to_match[attr.name] = attr.value;
}
})
});
/* Find the tables matching that set of tags */
var selector = '.' + args.outputClass;
$.each(tags_to_match, function (key, value) {
selector += '[' + key + '="' + value + '"]'
});
var containers_to_show = $(selector);
/* Label these tables as selected (subject to a year selection) and
unlabel the rest, then initialize visible tables */
containers_to_show.addClass(args.selectedTableTagClass);
$('.' + args.outputClass).not(containers_to_show).removeClass(args.selectedTableTagClass);
};
var activateYearTablesBySelectedTags = function () {
activateTablesBySelectedTags({
"selectorContainerSelector": "#table-select-container",
"outputClass": "output",
"selectedTableTagClass": "selected-table-tag"
});
};
var activateAggrTablesBySelectedTags = function () {
activateTablesBySelectedTags({
"selectorContainerSelector": "#fiscal-totals-container",
"outputClass": "aggr-output",
"selectedTableTagClass": "selected-aggr-table-tag"
});
};
$(document).ready(function () {
$('table').addClass('table').addClass('table-striped');
activateAggrTablesBySelectedTags();
initVisibleAggrTable();
activateYearTablesBySelectedTags();
activateYearTablesBySelectedYear();
initVisibleYearTable();
$("#year-selector li a").click(function (e) {
e.preventDefault();
activateYearTablesBySelectedYear();
initVisibleTable();
});
$(".tag-selector li a").click(function (e) {
/* Toggle the visible selection */
e.preventDefault();
var target_li = $(e.target).parent('li');
target_li.addClass('active');
var parent_container = target_li.parents('.tag-selector-container');
var unselected = target_li.parent('.nav').find('li').not(target_li);
unselected.removeClass('active');
/* Deactivate child tags for unselected options */
unselected.each(function (_index, el) {
var children_keys = $(el).attr('data-children-keys').trim().split(" ");
parent_container.find($(children_keys.map(function (key) {
return ".tag-" + key;
}).join(", ")))
.addClass('hidden');
});
/* Activate child tags */
var children_keys = target_li.attr('data-children-keys').trim().split(" ");
parent_container.find($(children_keys.map(function (key) {
return ".tag-" + key;
}).join(", ")))
.removeClass('hidden');
activateYearTablesBySelectedTags();
initVisibleYearTable();
activateAggrTablesBySelectedTags();
initVisibleAggrTable();
});
});
</script>
{% endblock %}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 9,591
|
Служба «Кини Мини» ( или ) — одна из первых британских частных военных компаний, основанная бывшими военнослужащими королевских вооружённых сил и спецслужб. Занималась различными видами охранной деятельности и, принимая участие в различных военных конфликтах второй половины XX века, просуществовала до начала 1990-х годов. Известно, что вплоть до 1987 года компания KMS координировала свою работу по подготовке кадров с ЦРУ.
Специалисты и бойцы «Кини Мини» неоднократно обвинялись в разнообразных нарушениях морально-этических норм и в совершении военных преступлений. Компания оправдывалась, что она занимается по большей части боевым тренингом и не может нести ответственность за противозаконные деяния обученного ею военного персонала. Тем не менее, процесс боевой подготовки военных специалистов компанией «Кини Мини» не прекратился даже после длинной череды совершённых ими зверств, что стало причиной публикации в международной прессе сомнений в искренности и обоснованности этих оправданий.
Общие сведения
Учреждение компании KMS состоялось в 1975 году. Её основателями стали экс-директор САС бригадир Майк Уингейт-Грей, полковник Джим Джонсон, майор Дэвид Уокер и майор разведывательной группы САС Эндрю Найтингейл. Помимо них в состав первых лиц KMS вошёл представитель известной страховой корпорации Blackwall Green Джон Саузерн. Регистрация нового юридического лица была проведена в оффшорной зоне на острове Джерси (Ла-Манш), где кроме неё была зарегистрирована ещё одна британская ЧВК — «Уотчгуард интернэшнл». В начале некоторое время для скрытия спонсоров KMS использовалось название Executives International.
Непосредственным руководством деятельностью компании занимались майор Дэвид Уолкер, бывший специалист Скотленд-Ярда по арабской преступности Рэй Такер и майор Эндрю Найтингейл.
Происхождение названия «Кини Мини» не имеет однозначного ответа. По одной из версий название компании пришло из арабского жаргона и обозначает «подполье», по второй — оно переводится с языка суахили как «змея в траве» и возникло во времена восстания Мау-Мау, по третьей — позаимствовано из сленга спецподразделений САС и означает «скрытные операции» (). Последняя версия появилась благодаря признанию бывшего директора САС генерала Питера де ля Бирилье.
В начале своей деятельности компания «Кини Мини» активно вербовала в свои ряды ветеранов боевых действий в Северной Ирландии и Омане. В целом считается, что методом, характерным для всех британских ЧВК, стал приём «на работу» кадровых офицеров, временно «уволенных» из спецслужб для проведения особо резонансных мероприятий. В случае успешного завершения заданий их немедленно принимали назад в их подразделения.
Головной офис компании «Кини Мини» располагался вместе с офисом её дочернего предприятия Saladin Security в Лондоне, недалеко от штаб-квартиры 22-го полка САС.
Компанию «Кини Мини» не следует путать с другой частной военной организацией, которая тоже использовала в качестве своего названия аббревиатуру KMS. Это предприятие было создано английскими военнослужащими задолго до «Кини Мини» в 60-х годах XX века для формального прикрытия британских операций на территории Йемена. В создании этой «ширмы» принимал участие всё тот же бригадир Майк Уингейт-Грей, представитель британской спецслужбы МИ-6 Фрэнк Стил (как специалист по Ближнему Востоку), полковник Дэвид «Динки» Сазерлэнд (как представитель контрразведывательной службы МИ-5) и майор САС Дэар Нюэлл. Помещение для офиса было предоставлено полковником Дэвидом Стирлингом из фондов его телевизионной компании TIE ().
История
Коммерческая деятельность компании «Кини Мини» началось с охраны британских дипломатов в Буэнос-Айресе. В 1976 году султан Омана нанял инструкторов KMS для обучения своих специальных сил, a министр нефтяной промышленности Саудовской Аравии и британский посол в Ливане завербовали себе телохранителей из рядов KMS. В 1979 году компания KMS получила контракты британского МИДа на охрану дипломатов в Уганде, Эль-Сальвадоре и Родезии.
В 1981 году Эндрю Найтингейл погиб в автомобильной аварии в Омане. В течение 1982 года сотрудники KMS продолжали охранять британских дипломатов в Уругвае, несмотря на угрозы со стороны Аргентины в ходе Фолкленской войны. В 1983 году компания KMS добилась нескольких государственных подрядов от правительства Шри-Ланки из-за вспыхнувшей на острове гражданской войны. С января 1984 года специалисты KMS начали тренировать полицейский спецназ Шри-Ланки. Подготовка шри-ланкийских спецподразделений велась по заказу британского правительства для борьбы с формированиями организации «Тигры освобождения Тамил-Илама». В это же время Дэвид Уолкер приступил к работе в Никарагуа. В этой стране «Кини Мини» работала во взаимодействии с ЦРУ. При участии eё специалистов была поставлена на поток боевая подготовка бойцов «контрас», осуществлено минирование портовой инфраструктуры в Манагуа и организовано несколько диверсионно-террористических актов. В дополнение к этому в декабре 1984 года Уолкер подал идею начать совместно с «контрас» отработку тактических приёмов уничтожения советских вертолётов Ми-24. Для этого в Чили были закуплены переносные зенитные ракетные комплексы «Блоупайп», однако их поставку осуществить не удалось.
В 1985 году компания KMS начинает работать с вертолётной авиатехникой, а на острове Шри-Ланка её персонал приступает к тренировке армейских коммандос и к управлению боевыми операциями. В этом же году специалисты KMS подорвали госпиталь в Манагуа, а на Шри-Ланке военный персонал KMS оказался причастен к пыткам и исчезновениям людей.
По заданию ЦРУ компании KMS и Saladin Security занимались тренингом афганских моджахедов, в частности, в 1986 году компания KMS пыталась обучать их навыкам разминирования. По мнению независимых журналистов из «Центра за целостность общества» компания KMS получала вознаграждение от ЦРУ не только за боевую подготовку афганских моджахедов, но и за военное обучение других вооружённых формирований исламских фундаменталистов. В том же году, в связи с обвинениями в военных преступлениях на территории Шри-Ланки, компанию покинул Дэвид Уолкер. В 1987 году выяснилось, что полицейский спецназ Шри-Ланки, обученный KMS, причастен к массовому убийству 85 гражданских лиц, а пилоты KMS обеспечивали поддержку с воздуха индийским войскам во время уничтожения гражданского населения.
С 1988 года мероприятия KMS по обучению войск на Шри-Ланке были свёрнуты, а в тренировочном процессе всё возрастающую роль начала играть компания Saladin Security.
В начале 1990-х годов компания KMS прекратила свою деятельность из-за многочисленных скандалов и постоянной критики в международной прессе.
Примечания
Источники
Военные компании Великобритании
Компании, основанные в 1975 году
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,157
|
{"url":"http:\/\/library.kiwix.org\/mathoverflow.net_eng_all_2018-08\/A\/tag\/soft-question\/1.html","text":"## Tag: soft-question\n\n277 Thinking and Explaining 2010-09-14T02:13:50.010\n\n240 Why do so many textbooks have so much technical detail and so little enlightenment? 2010-01-27T01:52:40.797\n\n235 Widely accepted mathematical results that were later shown to be wrong? 2010-08-13T10:12:59.930\n\n210 Mathematical games interesting to both you and a 5+-year-old child 2017-09-18T21:29:35.607\n\n207 A single paper everyone should read? 2009-10-23T18:42:05.273\n\n207 What are some examples of colorful language in serious mathematics papers? 2010-04-23T05:01:02.703\n\n203 Examples of unexpected mathematical images 2014-08-09T07:01:31.603\n\n191 What's a mathematician to do? 2010-10-26T16:53:25.853\n\n188 Refereeing a Paper 2010-08-24T22:45:04.400\n\n179 Awfully sophisticated proof for simple facts 2010-10-17T15:16:59.150\n\n177 Intuitive crutches for higher dimensional thinking 2010-05-26T09:49:53.130\n\n176 Fundamental Examples 2009-11-11T07:52:13.173\n\n170 Have any long-suspected irrational numbers turned out to be rational? 2010-07-22T16:06:17.797\n\n168 Your favorite surprising connections in Mathematics 2010-02-08T00:10:18.557\n\n151 Magic trick based on deep mathematics 2009-12-25T22:07:11.903\n\n150 Most interesting mathematics mistake? 2009-10-17T14:28:43.527\n\n147 Do good math jokes exist? 2009-10-18T21:47:45.010\n\n147 What are your favorite instructional counterexamples? 2010-03-02T05:57:45.173\n\n146 Best online mathematics videos? 2009-10-21T20:21:43.117\n\n141 Famous mathematical quotes 2009-11-29T20:18:38.147\n\n141 Interesting mathematical documentaries 2012-06-19T18:47:55.263\n\n138 Examples of great mathematical writing 2009-10-12T17:12:16.123\n\n138 Which math paper maximizes the ratio (importance)\/(length)? 2009-12-01T01:12:30.173\n\n137 Most harmful heuristic? 2009-10-24T20:54:56.270\n\n132 Too old for advanced mathematics? 2009-11-29T08:44:17.350\n\n126 How does one justify funding for mathematics research? 2014-07-11T13:55:15.687\n\n122 Generalizing a problem to make it easier 2010-09-26T08:09:31.003\n\n122 What non-categorical applications are there of homotopical algebra? 2014-06-06T06:50:41.107\n\n119 Jokes in the sense of Littlewood: examples? 2010-09-15T18:30:40.780\n\n115 When and how is it appropriate for an undergraduate to email a professor out of the blue? 2010-08-04T18:19:56.767\n\n115 Most memorable titles 2010-10-31T14:11:28.240\n\n115 How To Present Mathematics To Non-Mathematicians? 2010-11-24T09:29:31.743\n\n111 Most helpful math resources on the web 2009-10-23T18:54:36.030\n\n110 What are the most misleading alternate definitions in taught mathematics? 2009-12-02T15:59:43.357\n\n105 Why are flat morphisms \"flat?\" 2009-11-25T11:41:08.330\n\n104 How do you keep your research notes organized? 2009-10-22T02:38:40.237\n\n104 Are there examples of non-orientable manifolds in nature? 2010-11-12T15:04:51.417\n\n101 Counterexamples in Algebra? 2010-06-21T23:10:07.347\n\n100 Periods and commas in mathematical writing 2009-11-24T11:00:37.490\n\n97 eBook readers for mathematics 2010-07-04T12:59:45.767\n\n97 Mathematical habits of thought and action which would be of use to non-mathematicians 2011-09-07T03:45:14.740\n\n94 What recent discoveries have amateur mathematicians made? 2010-10-30T14:20:24.430\n\n94 Does Physics need non-analytic smooth functions? 2012-11-26T16:58:48.853\n\n93 Which popular games are the most mathematical? 2010-02-01T09:42:11.693\n\n93 What are some famous rejections of correct mathematics? 2010-02-03T00:50:28.910\n\n93 What are some very important papers published in non-top journals? 2015-11-06T16:38:14.340\n\n91 Which mathematicians have influenced you the most? 2009-11-14T13:28:17.443\n\n90 New grand projects in contemporary math 2012-12-30T21:08:08.743\n\n90 Mistakes in mathematics, false illusions about conjectures 2014-06-03T12:06:13.590\n\n90 What is the most useful non-existing object of your field? 2014-09-28T21:58:53.573\n\n89 When should a result be made into a paper? 2010-10-10T22:41:58.550\n\n86 How do you decide whether a question in abstract algebra is worth studying? 2010-09-24T06:17:01.583\n\n85 Work of plenary speakers at ICM 2014 2013-10-17T10:28:19.270\n\n84 How do I see LaTeX math on any web page and in email? 2010-04-22T02:19:12.500\n\n84 Analogues of P vs. NP in the history of mathematics 2014-03-13T20:12:02.987\n\n83 How Much Work Does it Take to be a Successful Mathematician? 2009-12-26T16:16:34.237\n\n81 Top specialized journals 2009-10-31T05:13:36.823\n\n80 Mathematicians who were late learners?-list 2009-10-31T20:34:53.330\n\n80 Is there a database for tracking the dependencies of mathematical theorems? 2016-09-08T17:53:43.413\n\n79 What is the definition of \"canonical\"? 2010-03-28T18:16:39.353\n\n78 Can a mathematical definition be wrong? 2010-07-11T04:18:27.257\n\n77 Which are the best mathematics journals, and what are the differences between them? 2009-09-30T22:14:11.017\n\n77 Quick proofs of hard theorems 2010-05-16T19:11:01.790\n\n77 Modern Mathematical Achievements Accessible to Undergraduates 2013-05-05T19:02:58.223\n\n76 Theorems with unexpected conclusions 2010-03-13T21:15:12.933\n\n76 What is the oldest open problem in mathematics? 2010-06-04T18:30:16.613\n\n76 Blackbox Theorems 2012-06-13T20:58:36.810\n\n75 Are there any good websites for hosting discussions of mathematical papers? 2011-01-03T20:14:46.187\n\n75 Why should one still teach Riemann integration? 2011-01-21T02:25:40.993\n\n74 Why is it a good idea to study a ring by studying its modules? 2009-11-12T19:55:23.767\n\n73 Favorite popular math book 2009-12-11T22:11:12.853\n\n73 Pseudonyms of famous mathematicians 2010-11-07T17:56:35.487\n\n73 How do you not forget old math? 2013-09-27T07:46:39.277\n\n72 How has \"what every mathematician should know\" changed? 2010-03-25T21:34:00.523\n\n71 What is sheaf cohomology intuitively? 2010-09-16T12:39:18.877\n\n71 More open problems 2010-12-04T20:12:03.513\n\n71 Theorems that are 'obvious' but hard to prove 2011-01-09T11:19:49.657\n\n71 What notions are used but not clearly defined in modern mathematics? 2011-02-25T20:43:24.550\n\n70 Have you solved problems in your sleep? 2014-04-27T00:19:23.330\n\n70 Parodies of abstruse mathematical writing 2015-03-14T00:19:01.867\n\n70 Coming out as transgender in the mathematical community 2017-11-26T00:01:09.333\n\n69 Tools for long-distance collaboration 2010-12-14T13:11:45.687\n\n68 What's a nice argument that shows the volume of the unit ball in $\\mathbb R^n$ approaches 0? 2009-12-08T22:08:04.490\n\n68 Why didn't Vladimir Arnold get the Fields Medal in 1974? 2010-06-05T09:07:41.550\n\n68 How to write popular mathematics well? 2011-12-21T19:04:01.580\n\n68 Proofs shown to be wrong after formalization with proof assistant 2018-01-21T01:29:30.550\n\n67 books well-motivated with explicit examples 2009-12-06T04:05:23.357\n\n67 Good papers\/books\/essays about the thought process behind mathematical research 2010-05-13T17:49:36.667\n\n67 Why are matrices ubiquitous but hypermatrices rare? 2010-12-02T13:21:14.013\n\n67 Examples of theorems with proofs that have dramatically improved over time 2012-05-03T10:07:24.083\n\n66 When should you, and should you not, share your mathematical ideas? 2010-10-27T03:39:47.607\n\n66 What could be some potentially useful mathematical databases? 2011-06-21T22:05:43.347\n\n66 Lost soul: loneliness in pursing math. Advice needed. 2012-10-19T02:00:19.053\n\n65 Which mathematical ideas have done most to change history? 2010-02-01T14:43:02.030\n\n65 Good ways to engage in mathematics outreach? 2010-12-28T04:37:08.930","date":"2019-02-19 06:39:55","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4565597474575043, \"perplexity\": 4253.385375403323}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-09\/segments\/1550247489425.55\/warc\/CC-MAIN-20190219061432-20190219083432-00608.warc.gz\"}"}
| null | null |
La battaglia di Liegi si svolse durante le giornate iniziali della prima guerra mondiale, quando l'esercito tedesco invase il neutrale Belgio. Si svolse dal 5 al 16 agosto del 1914, quando si arrese l'ultimo dei forti a protezione della città.
Le fortificazioni
La città belga di Liegi si trova alla confluenza dei fiumi Mosa ed Ourthe, con la foresta delle Ardenne a sud, e la città olandese di Maastricht a nord. La Mosa scorre in questo punto in una profonda valle, costituendo un ostacolo non da poco ad una forza di invasione.
Per rafforzare ulteriormente la posizione, nel XIX secolo venne realizzato un anello difensivo di dodici forti, in base ai metodi di fortificazione tedeschi dell'epoca, situati in un raggio di 6-10 chilometri dalla città. I forti si proteggevano reciprocamente con zone di fuoco incrociato, ed erano stati progettati in maniera tale che se uno di essi fosse caduto i due a lato avrebbero potuto comunque colpire una forza che intendesse avanzare attraverso la breccia.
Sei fortezze erano concepite come forti principali, avevano una pianta quadrangolare, ed erano circondate da un fossato con controscarpata e reticolati di filo spinato. La struttura, fuori terra solo in minima parte, era in cemento e sormontata da cupole in acciaio per l'osservazione e l'artiglieria. L'armamento era costituito da cannoni di vario calibro, da 57 a 210 mm, più le mitragliatrici; i pezzi erano montati entro cupole, di cui alcune mobili che venivano posizionate solo per il fuoco e quindi ritratte.
Alternati ai forti principali erano i forti secondari, a pianta triangolare, chiamati fortini; costruiti col medesimo criterio dei forti principali e destinati a difenderli dagli assalti di fanteria, erano armati con pezzi leggeri e mitragliatrici.
I forti erano collegati da gallerie ed ospitavano riserve di munizioni, viveri ed acqua per guarnigioni fino a ottanta uomini. Linee fortificate campali correvano su tutto il perimetro tra un forte e l'altro.
In totale i pezzi d'artiglieria tra leggeri e pesanti erano circa duecentocinquanta, sebbene ormai antiquati. Altri punti deboli erano la mancanza di pezzi d'artiglieria da campagna nello spazio fra una fortezza e l'altra e la scarsezza di truppe.
Gli schieramenti
Belgi
Il tenente generale Gérard Mathieu Leman era stato destinato al comando del sistema fortificato di Liegi con l'ordine da parte del sovrano di resistere fino alla fine; aveva a sua disposizione la 3ª Divisione che comprendeva: quattro brigate miste, ognuna delle quali di due reggimenti di fanteria, un gruppo di artiglieria da campagna, una compagnia mitraglieri ed un plotone di gendarmi. Le truppe da fortezza erano al comando del maggior generale Janssen, e consistevano in quattro reggimenti di fanteria della riserva, ognuno dei quali su tre battaglioni.
La guarnigione di Liegi era costituita da un reggimento di artiglieria, al comando del colonnello Marcin, che contava dodici batterie, una per fortezza, per un totale di 78 pezzi pesanti. Quattro altre batterie della riserva furono formate in seguito alla mobilitazione. La difesa tra fortezza e fortezza era affidata a sei batterie di pezzi da 80 mm, tre di mortai da 87 mm, due di pezzi da 120 mm e tre di pezzi da 150 mm. La difesa della cittadella poggiava su tre batterie da 80 mm.
Il colonnello Lemiere comandava un battaglione di genieri, un battaglione di pontieri, una compagnia di telegrafisti e due di minatori. Parte della difesa era pure la guardia civica, per un totale di 27 ufficiali e 463 uomini. Esclusa quest'ultima le truppe da fortezza allineavano 8490 fucili e 180 cannoni pesanti. Il totale dei difensori contava 31.990 fucili, 252 cannoni, 500 cavalieri e 30 mitragliatrici.
Tedeschi
In accordo col Piano Schlieffen tre armate tedesche, 1ª, 2ª e 3ª, dovevano attraversare il Belgio e quindi puntare verso sud per accerchiare le forze francesi, facendo perno sulla fortezza di Metz. L'ala destra era costituita dalla 1ª Armata del generale von Kluck; avrebbe iniziato la sua marcia ad Aquisgrana. La 2ª Armata, al comando del generale von Bülow, doveva attendere al confine della provincia di Liegi; il suo compito era quello di assicurare le vie di comunicazione. La 3ª Armata doveva entrare in Francia attraverso il Lussemburgo.
L'avanguardia tedesca, che doveva aprire la via per la 1ª e la 2ª Armata, fu denominata "Armata della Mosa", ed era comandata dal generale von Emmich. Comprendeva sei brigate di fanteria ciascuna composta da due reggimenti di fanti, un battaglione di esploratori, uno squadrone di cavalleria, un gruppo d'artiglieria con pezzi da 105 e 77 mm, una compagnia di pionieri. Inoltre l'armata schierava un corpo d'armata di cavalleria, due batterie di mortai 21 cm Mörser 10 da 210 mm, uno squadrone aereo, uno zeppelin da bombardamento. Il totale degli uomini era di 59.800 con 35.000 fucili, 100 mitragliatrici, 100 cannoni e 14.000 cavalleggeri.
L'assedio
Originariamente il Piano Schlieffen per l'invasione della Francia prevedeva l'invasione sia del Belgio che dell'Olanda; il pianeggiante territorio belga avrebbe consentito un comodo passaggio per la fanteria e l'artiglieria, al confronto del difficile e boscoso terreno della zona di confine più a sud, inoltre i tedeschi sapevano che la Francia non temeva un attacco proveniente dal neutrale Belgio, e per questo motivo non ne aveva fortificato la frontiera.
Tuttavia von Moltke apportò delle modifiche (in seguito rivelatesi fatali) al piano originale, riducendo le truppe d'invasione per rafforzare i restanti settori del fronte; ciò non consentì di invadere anche l'Olanda, e privò l'avanzata di importanti vie di comunicazione stradale e ferroviaria. La via per le pianure belghe era perciò obbligata a passare per la città fortificata di Liegi.
Le truppe al comando di von Emmich, cui era accanto anche Erich Ludendorff in qualità di osservatore, passarono il confine nel pomeriggio del 4 agosto 1914, poche ore dopo la dichiarazione di guerra. Avanzarono verso la Mosa, ma trovarono i ponti distrutti; riuscirono ad attraversarla il giorno successivo a Visé. La 3ª Divisione belga difendeva la città da trinceramenti allestiti in tutta fretta, tuttavia il 5 agosto riuscì a respingere la fanteria tedesca che tentava di infiltrarsi tra i forti. Anche un attacco contro Fort Barchon fu respinto con gravi perdite tedesche.
Si svolse quindi uno dei primi attacchi aerei della storia, con uno Zeppelin inviato a bombardare la città. Nel frattempo la cavalleria mosse da Visè verso sud per effettuare una manovra di accerchiamento. Con la prospettiva di un attacco immediato da più lati, Leman ordinò alla 3ª Divisione di ritirarsi dalla città e riunirsi al resto dell'esercito belga più ad ovest. Ludendorff prese quindi il comando della 14ª Brigata, che era riuscita ad infiltrarsi tra i forti, ed il giorno 7 riuscì a catturare la città.
Tuttavia l'anello dei forti continuava a rimanere intatto, e bloccava di fatto l'avanzata impedendo l'uso delle linee ferroviarie. Particolarmente importante era la linea che attraversava il Plateau d'Herve, da cui dovevano provenire i rifornimenti tedeschi, difesa dai forti Barchon, Fleron ed Evegnee, che furono quindi i primi ad essere attaccati. Fort Barchon si arrese il giorno 8, Fort Evegnee il giorno 10; Fort Fleron fu messo fuori uso quando il meccanismo di movimento della cupola fu distrutto da un'esplosione,
Nel complesso il sistema fortificato riuscì a far fronte ai costanti assalti e bombardamenti da parte delle truppe tedesche, e contribuì a rallentare l'avanzata. I tedeschi decisero quindi per sbloccare in fretta la situazione di far intervenire l'artiglieria d'assedio, tra cui il mortaio Krupp Morser L/14, o Gamma M, detto Grande Berta, da 420mm, ed alcuni cannoni austriaci Skoda da 305mm.
I pezzi arrivarono per ferrovia il giorno 11 ed iniziarono il bombardamento il giorno successivo. I forti vennero bombardati uno ad uno da più direzioni: i forti Pontisse, Embourg, e Chaudfontaine caddero il giorno successivo; Fort Fleron (come pure Fort Liers) cadde il giorno 14, dopo aver ricevuto 3000 colpi in dodici ore. Il giorno 15 si arresero i forti Boncelles, Latin e Loncin, dove lo stesso Leman venne fatto prigioniero. Gli ultimi due forti, Flemalle e Hollogne, si arresero il 16 e il 17.
Sviluppi successivi
L'Armata della Mosa fu sciolta ed integrata nelle armate 1ª e 2ª. Il generale von Emmich prese il comando del X Corpo d'armata. Il giorno 16 fu presa la cittadella di Huy, aprendo ai tedeschi la valle della Mosa sino in Olanda. Von Kluck e Von Bulow furono così in grado di attraversare il Belgio in direzione di Parigi.
La tenace resistenza di Liegi rallentò la tabella di marcia tedesca di alcuni giorni, guadagnando tempo prezioso per gli Alleati: il Corpo di Spedizione Britannico ebbe tempo di sbarcare a Boulogne e concentrarsi a Maubeuge per il giorno 14, la 5ª Armata francese mosse a nordovest verso la frontiera belga, mentre due corpi d'armata della 2ª Armata venivano richiamati dal Nord Africa.
I comandanti tedeschi tuttavia ne sminuirono l'importanza in tal senso, affermando che il loro esercito stava ancora schierandosi. Innegabilmente servì al morale delle forze alleate, e la Francia avrebbe in seguito conferito la Legion d'Onore alla città per il suo valore.
Bibliografia
Griess, Thomas E., The Great War, Avery Publishing, 1986.
Marshall, S.L.A., World War I, American Heritage, 1964.
Reynolds, F. J., The Story of the Great War, Vol. III, P.F. Collier & Son, New York, 1916.
Voci correlate
Fronte occidentale (Prima guerra mondiale)
Piano Schlieffen
Altri progetti
Collegamenti esterni
Guerra nel 1914
Liegi
Liegi
Liegi
Liegi
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,518
|
As a growing part of our committed service to our customers, and the community, Ott Funeral Home offers a variety of monuments for you to choose from. We can assist you in designing a monument that will best honor you or your loved one. Along with a wide selection of monuments, we also offer coping and rock to outline any size of grave plot.
For older existing monuments, we offer cleanings that will make markers look like new again. For cases where the original monument has an error, we can restore them to a corrected state. Plus, in today's challenging economy, we offer memorialization that will fit your needs and budget.
Browse our catalog below to see the products we offer.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,226
|
import paddle
from paddle.fluid import core
from paddle.fluid.wrapped_decorator import signature_safe_contextmanager
from .streams import Stream # noqa: F401
from .streams import Event # noqa: F401
__all__ = [
'Stream',
'Event',
'current_stream',
'synchronize',
'device_count',
'empty_cache',
'max_memory_allocated',
'max_memory_reserved',
'memory_allocated',
'memory_reserved',
'stream_guard',
'get_device_properties',
'get_device_name',
'get_device_capability',
]
def current_stream(device=None):
'''
Return the current CUDA stream by the device.
Parameters:
device(paddle.CUDAPlace()|int, optional): The device or the ID of the device which want to get stream from.
If device is None, the device is the current device. Default: None.
Returns:
CUDAStream: the stream to the device.
Examples:
.. code-block:: python
# required: gpu
import paddle
s1 = paddle.device.cuda.current_stream()
s2 = paddle.device.cuda.current_stream(0)
s3 = paddle.device.cuda.current_stream(paddle.CUDAPlace(0))
'''
device_id = -1
if device is not None:
if isinstance(device, int):
device_id = device
elif isinstance(device, core.CUDAPlace):
device_id = device.get_device_id()
else:
raise ValueError("device type must be int or paddle.CUDAPlace")
return core._get_current_stream(device_id)
def synchronize(device=None):
'''
Wait for the compute on the given CUDA device to finish.
Parameters:
device(paddle.CUDAPlace()|int, optional): The device or the ID of the device.
If device is None, the device is the current device. Default: None.
Examples:
.. code-block:: python
# required: gpu
import paddle
paddle.device.cuda.synchronize()
paddle.device.cuda.synchronize(0)
paddle.device.cuda.synchronize(paddle.CUDAPlace(0))
'''
device_id = -1
if device is not None:
if isinstance(device, int):
device_id = device
elif isinstance(device, core.CUDAPlace):
device_id = device.get_device_id()
else:
raise ValueError("device type must be int or paddle.CUDAPlace")
return core._device_synchronize(device_id)
def device_count():
'''
Return the number of GPUs available.
Returns:
int: the number of GPUs available.
Examples:
.. code-block:: python
import paddle
paddle.device.cuda.device_count()
'''
num_gpus = (
core.get_cuda_device_count()
if hasattr(core, 'get_cuda_device_count')
else 0
)
return num_gpus
def empty_cache():
'''
Releases idle cached memory held by the allocator so that those can be used in other GPU
application and visible in `nvidia-smi`. In most cases you don't need to use this function,
Paddle does not release the memory back to the OS when you remove Tensors on the GPU,
Because it keeps gpu memory in a pool so that next allocations can be done much faster.
Examples:
.. code-block:: python
import paddle
# required: gpu
paddle.set_device("gpu")
tensor = paddle.randn([512, 512, 512], "float")
del tensor
paddle.device.cuda.empty_cache()
'''
if core.is_compiled_with_cuda():
core.cuda_empty_cache()
def extract_cuda_device_id(device, op_name):
'''
Return the id of the given cuda device. It is just a utility that will not be exposed to users.
Args:
device(paddle.CUDAPlace or int or str): The device, the id of the device or
the string name of device like 'gpu:x'.
Default: None.
Return:
int: The id of the given device. If device is None, return the id of current device.
'''
if device is None:
return core.get_cuda_current_device_id()
if isinstance(device, int):
device_id = device
elif isinstance(device, core.CUDAPlace):
device_id = device.get_device_id()
elif isinstance(device, str):
if device.startswith('gpu:'):
device_id = int(device[4:])
else:
raise ValueError(
"The current string {} is not expected. Because {} only support string which is like 'gpu:x'. "
"Please input appropriate string again!".format(device, op_name)
)
else:
raise ValueError(
"The device type {} is not expected. Because {} only support int, str or paddle.CUDAPlace. "
"Please input appropriate device again!".format(device, op_name)
)
assert (
device_id >= 0
), f"The device id must be not less than 0, but got id = {device_id}."
assert (
device_id < device_count()
), f"The device id {device_id} exceeds gpu card number {device_count()}"
return device_id
def max_memory_allocated(device=None):
'''
Return the peak size of gpu memory that is allocated to tensor of the given device.
Note:
The size of GPU memory allocated to tensor is 256-byte aligned in Paddle, which may larger than the memory size that tensor actually need.
For instance, a float32 tensor with shape [1] in GPU will take up 256 bytes memory, even though storing a float32 data requires only 4 bytes.
Args:
device(paddle.CUDAPlace or int or str): The device, the id of the device or
the string name of device like 'gpu:x'. If device is None, the device is the current device.
Default: None.
Return:
int: The peak size of gpu memory that is allocated to tensor of the given device, in bytes.
Examples:
.. code-block:: python
# required: gpu
import paddle
max_memory_allocated_size = paddle.device.cuda.max_memory_allocated(paddle.CUDAPlace(0))
max_memory_allocated_size = paddle.device.cuda.max_memory_allocated(0)
max_memory_allocated_size = paddle.device.cuda.max_memory_allocated("gpu:0")
'''
name = "paddle.device.cuda.max_memory_allocated"
if not core.is_compiled_with_cuda():
raise ValueError(
f"The API {name} is not supported in CPU-only PaddlePaddle. Please reinstall PaddlePaddle with GPU support to call this API."
)
device_id = extract_cuda_device_id(device, op_name=name)
return core.device_memory_stat_peak_value("Allocated", device_id)
def max_memory_reserved(device=None):
'''
Return the peak size of GPU memory that is held by the allocator of the given device.
Args:
device(paddle.CUDAPlace or int or str): The device, the id of the device or
the string name of device like 'gpu:x'. If device is None, the device is the current device.
Default: None.
Return:
int: The peak size of GPU memory that is held by the allocator of the given device, in bytes.
Examples:
.. code-block:: python
# required: gpu
import paddle
max_memory_reserved_size = paddle.device.cuda.max_memory_reserved(paddle.CUDAPlace(0))
max_memory_reserved_size = paddle.device.cuda.max_memory_reserved(0)
max_memory_reserved_size = paddle.device.cuda.max_memory_reserved("gpu:0")
'''
name = "paddle.device.cuda.max_memory_reserved"
if not core.is_compiled_with_cuda():
raise ValueError(
f"The API {name} is not supported in CPU-only PaddlePaddle. Please reinstall PaddlePaddle with GPU support to call this API."
)
device_id = extract_cuda_device_id(device, op_name=name)
return core.device_memory_stat_peak_value("Reserved", device_id)
def memory_allocated(device=None):
'''
Return the current size of gpu memory that is allocated to tensor of the given device.
Note:
The size of GPU memory allocated to tensor is 256-byte aligned in Paddle, which may be larger than the memory size that tensor actually need.
For instance, a float32 tensor with shape [1] in GPU will take up 256 bytes memory, even though storing a float32 data requires only 4 bytes.
Args:
device(paddle.CUDAPlace or int or str): The device, the id of the device or
the string name of device like 'gpu:x'. If device is None, the device is the current device.
Default: None.
Return:
int: The current size of gpu memory that is allocated to tensor of the given device, in bytes.
Examples:
.. code-block:: python
# required: gpu
import paddle
memory_allocated_size = paddle.device.cuda.memory_allocated(paddle.CUDAPlace(0))
memory_allocated_size = paddle.device.cuda.memory_allocated(0)
memory_allocated_size = paddle.device.cuda.memory_allocated("gpu:0")
'''
name = "paddle.device.cuda.memory_allocated"
if not core.is_compiled_with_cuda():
raise ValueError(
f"The API {name} is not supported in CPU-only PaddlePaddle. Please reinstall PaddlePaddle with GPU support to call this API."
)
device_id = extract_cuda_device_id(device, op_name=name)
return core.device_memory_stat_current_value("Allocated", device_id)
def memory_reserved(device=None):
'''
Return the current size of GPU memory that is held by the allocator of the given device.
Args:
device(paddle.CUDAPlace or int or str): The device, the id of the device or
the string name of device like 'gpu:x'. If device is None, the device is the current device.
Default: None.
Return:
int: The current size of GPU memory that is held by the allocator of the given device, in bytes.
Examples:
.. code-block:: python
# required: gpu
import paddle
memory_reserved_size = paddle.device.cuda.memory_reserved(paddle.CUDAPlace(0))
memory_reserved_size = paddle.device.cuda.memory_reserved(0)
memory_reserved_size = paddle.device.cuda.memory_reserved("gpu:0")
'''
name = "paddle.device.cuda.memory_reserved"
if not core.is_compiled_with_cuda():
raise ValueError(
f"The API {name} is not supported in CPU-only PaddlePaddle. Please reinstall PaddlePaddle with GPU support to call this API."
)
device_id = extract_cuda_device_id(device, op_name=name)
return core.device_memory_stat_current_value("Reserved", device_id)
def _set_current_stream(stream):
'''
Set the current stream.
Parameters:
stream(paddle.device.cuda.Stream): The selected stream.
Returns:
CUDAStream: The previous stream.
'''
if not isinstance(stream, paddle.device.cuda.Stream):
raise TypeError("stream type should be paddle.device.cuda.Stream")
cur_stream = current_stream()
if id(stream) == id(cur_stream):
return stream
return core._set_current_stream(stream)
@signature_safe_contextmanager
def stream_guard(stream):
'''
**Notes**:
**This API only supports dygraph mode currently.**
A context manager that specifies the current stream context by the given stream.
Parameters:
stream(paddle.device.cuda.Stream): the selected stream. If stream is None, just yield.
Examples:
.. code-block:: python
# required: gpu
import paddle
s = paddle.device.cuda.Stream()
data1 = paddle.ones(shape=[20])
data2 = paddle.ones(shape=[20])
with paddle.device.cuda.stream_guard(s):
data3 = data1 + data2
'''
if stream is not None and not isinstance(stream, paddle.device.cuda.Stream):
raise TypeError("stream type should be paddle.device.cuda.Stream")
cur_stream = current_stream()
if stream is None or id(stream) == id(cur_stream):
yield
else:
pre_stream = _set_current_stream(stream)
try:
yield
finally:
stream = _set_current_stream(pre_stream)
def get_device_properties(device=None):
'''
Return the properties of given device.
Args:
device(paddle.CUDAPlace or int or str): The device, the id of the device or
the string name of device like 'gpu:x' which to get the properties of the
device from. If device is None, the device is the current device.
Default: None.
Returns:
_gpuDeviceProperties: The properties of the device which include ASCII string
identifying device, major compute capability, minor compute capability, global
memory available and the number of multiprocessors on the device.
Examples:
.. code-block:: python
# required: gpu
import paddle
paddle.device.cuda.get_device_properties()
# _gpuDeviceProperties(name='A100-SXM4-40GB', major=8, minor=0, total_memory=40536MB, multi_processor_count=108)
paddle.device.cuda.get_device_properties(0)
# _gpuDeviceProperties(name='A100-SXM4-40GB', major=8, minor=0, total_memory=40536MB, multi_processor_count=108)
paddle.device.cuda.get_device_properties('gpu:0')
# _gpuDeviceProperties(name='A100-SXM4-40GB', major=8, minor=0, total_memory=40536MB, multi_processor_count=108)
paddle.device.cuda.get_device_properties(paddle.CUDAPlace(0))
# _gpuDeviceProperties(name='A100-SXM4-40GB', major=8, minor=0, total_memory=40536MB, multi_processor_count=108)
'''
if not core.is_compiled_with_cuda():
raise ValueError(
"The API paddle.device.cuda.get_device_properties is not supported in "
"CPU-only PaddlePaddle. Please reinstall PaddlePaddle with GPU support "
"to call this API."
)
if device is not None:
if isinstance(device, int):
device_id = device
elif isinstance(device, core.CUDAPlace):
device_id = device.get_device_id()
elif isinstance(device, str):
if device.startswith('gpu:'):
device_id = int(device[4:])
else:
raise ValueError(
"The current string {} is not expected. Because paddle.device."
"cuda.get_device_properties only support string which is like 'gpu:x'. "
"Please input appropriate string again!".format(device)
)
else:
raise ValueError(
"The device type {} is not expected. Because paddle.device.cuda."
"get_device_properties only support int, str or paddle.CUDAPlace. "
"Please input appropriate device again!".format(device)
)
else:
device_id = -1
return core.get_device_properties(device_id)
def get_device_name(device=None):
'''
Return the name of the device which is got from CUDA function `cudaDeviceProp <https://docs.nvidia.com/cuda/cuda-runtime-api/group__CUDART__DEVICE.html#group__CUDART__DEVICE_1g1bf9d625a931d657e08db2b4391170f0>`_.
Parameters:
device(paddle.CUDAPlace|int, optional): The device or the ID of the device. If device is None (default), the device is the current device.
Returns:
str: The name of the device.
Examples:
.. code-block:: python
# required: gpu
import paddle
paddle.device.cuda.get_device_name()
paddle.device.cuda.get_device_name(0)
paddle.device.cuda.get_device_name(paddle.CUDAPlace(0))
'''
return get_device_properties(device).name
def get_device_capability(device=None):
'''
Return the major and minor revision numbers defining the device's compute capability which are got from CUDA function `cudaDeviceProp <https://docs.nvidia.com/cuda/cuda-runtime-api/group__CUDART__DEVICE.html#group__CUDART__DEVICE_1g1bf9d625a931d657e08db2b4391170f0>`_.
Parameters:
device(paddle.CUDAPlace|int, optional): The device or the ID of the device. If device is None (default), the device is the current device.
Returns:
tuple(int,int): the major and minor revision numbers defining the device's compute capability.
Examples:
.. code-block:: python
# required: gpu
import paddle
paddle.device.cuda.get_device_capability()
paddle.device.cuda.get_device_capability(0)
paddle.device.cuda.get_device_capability(paddle.CUDAPlace(0))
'''
prop = get_device_properties(device)
return prop.major, prop.minor
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 8,942
|
é a suspensão na atmosfera de produtos resultantes de uma combustão. Pode ser tóxica quando aspirada.
As partículas que constituem o fumo resultam da combustão incompleta de um qualquer material combustível. É assim um subproduto não desejado da combustão, produzido em fogueiras, brasas, motores de gasolina e gasóleo. Quando uma combustão é correta e completa, os únicos subprodutos são água, dióxido de carbono e compostos de diversos elementos.
A inalação do fumo é a causa principal de asfixia e morte nas vítimas dos incêndios. O fumo mata por intoxicação devido aos seus componentes tóxicos, como o monóxido de carbono e as pequenas partículas sólidas que bloqueiam os alvéolos pulmonares e asfixiam quem os inale. Pode conter várias partículas cancerígenas, e provocar câncer após um longo tempo. Por isso é recomendado não usar "estufas" ou "caldeiras" dentro das residências, pois podem permitir o escape destes gases nocivos.
Ver também
Tabaco, fumante, estomatite nicotínica - artigos relacionados com o fumo do tabaco em combustão
Intoxicação por monóxido de carbono
Escala de Ringelmann
Incêndios florestais
Reações químicas
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 3,574
|
{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/calculus\/calculus-early-transcendentals-8th-edition\/chapter-3-section-3-1-derivatives-of-polynomials-and-exponential-functions-3-1-exercises-page-181\/70","text":"## Calculus: Early Transcendentals 8th Edition\n\nThe equation of the parabola is $$y=f(x)=3x^2-2x+7$$\n$$y=f(x)=ax^2+bx+c$$ - The derivative of $f(x)$ would be $$f'(x)=2ax+b$$ 1) The slope of the tangent line of $f(x)$ at $A(x,y)$ would equal $f'(x)$ According to the question: - $f(x)$ has slope 4 at $x=1$. That means $f'(1)=4$. So, $$2a\\times1+b=4$$ $$2a+b=4\\hspace{1cm}(1)$$ - $f(x)$ has slope -8 at $x=-1$. That means $f'(-1)=-8$. So, $$2a\\times(-1)+b=-8$$ $$-2a+b=-8$$ $$2a-b=8\\hspace{1cm}(2)$$ Summing (1) and (2), we have $$4a=12$$ $$a=3$$ Therefore, $b=2a-8=2\\times3-8=-2$ So, $y=f(x)=3x^2-2x+c$ 2) The parabola passes through the point $(2,15)$. Therefore, $$3\\times2^2-2\\times2+c=15$$ $$8+c=15$$ $$c=7$$ Therefore, the equation of the parabola is $$y=f(x)=3x^2-2x+7$$","date":"2018-07-17 19:56:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7645890712738037, \"perplexity\": 71.49560589754861}, \"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\/1531676589892.87\/warc\/CC-MAIN-20180717183929-20180717203929-00321.warc.gz\"}"}
| null | null |
\section{Introduction}
A measure-valued solution to a partial differential equation (or a system of equations) is, roughly speaking, a map that gives for every point in the domain a probability distribution of values, and that satisfies the equation only in an average sense. If this probability distribution reduces to a point mass almost everywhere in the domain, then the measure-valued solution is simply a solution in the sense of distributions. The main advantage of measure-valued solutions is the fact that, in many situations, they can easily be obtained from weakly convergent sequences of approximate solutions, even when the convergence of the approximating sequence to a distributional solution may fail due to effects of oscillation and concentration.
Measure-valued solutions to hyperbolic conservation laws were introduced by DiPerna~\cite{diperna}. He showed for scalar conservation laws in one space dimension that measure-valued solutions exist and are, under the assumption of entropy admissibility, in fact concentrated at one point, i.e.\ they can be identified with a distributional (entropy) solution. In other words, in this case the formation of fast oscillations, which corresponds to a measure with positive variance, can be excluded.
In many other physically relevant systems, however, no such compactness arguments are available, and existence of admissible weak (i.e.\ distributional) solutions seems hopeless. In such cases, the existence of measure-valued solutions is the best one can hope for.
For the incompressible Euler equations, DiPerna and Majda~\cite{dipernamajda} showed the global existence of measure-valued solutions for any initial data with finite energy. The main point of their work was to introduce so-called \emph{generalised Young measures}, which take into account not only oscillations, but also concentrations.
Subsequently, measure-valued solutions were shown to exist for further models of fluid and gas dynamics, e.g.\ compressible Euler and Navier-Stokes equations~\cite{neustupa, kroner} or the Savage-Hutter avalanche model~\cite{gwiazda2005}.
Measure-valued solutions have been criticised for being a too weak notion of solution. Indeed, the by now fairly standard procedure of establishing measure-valued solutions by viscous approximation, thereby circumventing delicate problems of compactness, suggests that the solution thus obtained does not carry enough information to be of much use. In particular, in the absence of admissibility criteria, measure-valued solutions are obviously non-unique to a large extent, as they only contain information on certain moments of the measure.
It is therefore surprising that, in the case of the incompressible Euler equations, the so-called \emph{weak-strong uniqueness} property was proved, on the whole space, for admissible measure-valued solutions by Brenier, De Lellis, and Sz\'{e}kelyhidi~\cite{weak-strong}. This means that if there exists a sufficiently regular (classical) solution, then every admissible measure-valued solution with the same initial data will coincide with the classical solution. Admissibility means that the kinetic energy of the solution never exceeds the initial energy.
In fact, in~\cite{lions}, P.-L. Lions required any reasonable concept of (very) weak solution to satisfy global existence and weak-strong uniqueness. For the incompressible Euler equations, therefore, admissible measure-valued solutions qualify. It is important though to emphasize the necessity of admissibility: Without this assumption, various examples are known where weak-strong uniqueness fails even for distributional solutions of incompressible Euler~\cite{scheffer, shnirel1, euler1, eulerexistence}. Also, uniqueness need not hold for admissible solutions in the absence of a strong solution, see~\cite{euler2, euleryoung, daneri}.
We consider in this article two systems of equations in the realm of compressible fluid dynamics: The isentropic Euler equations,
\begin{equation}\label{eulerintro}
\begin{aligned}
\partial_th+\diverg(hu)&=0\\
\partial_t(hu)+\diverg(hu\otimes u)+\nabla(\kappa h^\gamma)&=hG,
\end{aligned}
\end{equation}
in any space dimension greater or equal one, and the Savage-Hutter equations
\begin{equation}\label{savhutintro}
\begin{aligned}
\partial_th+\diverg(hu)&=0\\
\partial_t(hu)+\diverg(hu\otimes u)+\nabla(ah^2)&=h\left(-dB(u)+f\right),
\end{aligned}
\end{equation}
which make sense (from a modelling viewpoint) in one or two space dimensions. Here, $G$ and $f$ are external force densities, and $B(u)$ is a maximal monotone set-valued map.
The latter system describes the evolution of the depth-averaged velocity and height of some material sliding over an inclined slope. The material is subject to the so-called Coulomb-Mohr friction law. For comprehensive studies, including derivation, numerical computations and experimental results on system~\eqref{savhutintro} and its various modifications, we refer to \cite{SaHu89, GrWiHu99, BoWe2004, BoMaPe2003, GrTaNo2003, GrCu2007, HuWaPu2005, PeBoMa2008, ZaPeTaNi2010}, among others.
We prove (cf. Theorems~\ref{Eweak-strong} and \ref{weak-strong} below):
\begin{theorem}\label{1}
Let (H,U) be a solution of~\eqref{eulerintro} or~\eqref{savhutintro} such that $H$ is Lipschitz continuous in $[0,T]\times\T^n$ and $U\in C^1([0,T]\times\T^n)$. Assume also $H\geq c>0$ for some constant $c$. Then every admissible measure-valued solution of~\eqref{eulerintro} or~\eqref{savhutintro}, respectively, with the same initial data as $(H,U)$ coincides with $(H,U)$.
\end{theorem}
Of course, the precise definitions of measure-valued solutions and admissibility will be given below. It should be mentioned that weak-strong uniqueness for admissible measure-valued solutions was proved in~\cite{weak-strong} for general hyperbolic systems of conservation laws, but this was done only for oscillation measures.
Moreover, the results in~\cite{weak-strong} are valid even for Lipschitz continuous strong solutions. Owing to commutator estimates analogous to the ones provided by Constantin, E and Titi for the incompressible Euler system in~\cite{constetiti}, the result of Theorem~\ref{1} can be obtained assuming only Lipschitz continuity of $U$, and Sobolev regularity of $H$. In fact, as in~\cite{weak-strong} it is sufficient to assume only that the symmetric part of $\nabla U$ be bounded. We omit details.
Weak-strong uniqueness for compressible Euler models appears important in the light of several recent examples of non-uniqueness of admissible weak solutions~\cite{euler2, chiodaroli, chiodarolikreml, chiodarolikremlfeireisl, chiodarolidelelliskreml, feireisl}. For the Savage-Hutter equations, such examples were very recently constructed in~\cite{feireislgwiazdaswierczewska}.
To prove the weak-strong uniqueness, we follow the general strategy of~\cite{feireisljinnovotny} (where weak-strong uniqueness was proved for the compressible Navier-Stokes equations), but we have to adjust these arguments to the measure-valued framework. For the Savage-Hutter system, an additional issue is to give a meaningful definition of measure-valued solutions that accounts for the multi-valued nature of the fricition term $B(u)$ in~\eqref{savhutintro}. Such a definition was proposed in~\cite{gwiazda2005} and we will use it here as well.
If the force $f$ is time-independent and $\norm{f}_\infty<d$, then a special class of solutions to~\eqref{savhutintro} is given by $u=0$ and $h$ independent of time and such that
\begin{equation*}
\left|\nabla h(x)-\frac{f(x)}{2a}\right|\leq\frac{d}{2a}\hspace{0.2cm}\text{for almost every $x$.}
\end{equation*}
Observe that our weak-strong uniqueness result allows to take $(H,U)$ as such a stationary solution, so that in particular every such solution enjoys uniqueness in the class of admissible measure-valued solutions.
For the Savage-Hutter model we also prove the following result:
\begin{theorem}\label{finiteintro}
There exists a finite time $0\leq T<\infty$, only depending on the parameters in~\eqref{savhutintro} and the initial data, such that every admissible measure-valued solution of~\eqref{savhutintro} starting from such initial data has zero momentum for almost every time $t>T$.
\end{theorem}
In particular, this implies that every admissible weak (distributional) solution becomes stationary after finite time. This highlights the importance of stationary solutions as well as the role played by the admissibility condition: Indeed, in~\cite{feireislgwiazdaswierczewska} non-admissible weak solutions were constructed whose momentum does not decrease to zero. The result is a rigorous justification of empirical and numerical observations of deposition of material after finite time,~\cite{SaHu91, Fe-Nietal2008, CoGuMo2012}. The finite-time runout of solutions is essentially used at the modelling stage as providing data for calibration of the system. This property was assumed in numerical simulations, however, to our knowledge, never proved.
Let us remark that for the one-dimensional Savage-Hutter model, we obtain a fairly complete picture: Existence of admissible global in time weak solutions is known~\cite{Gw2002}, they enjoy weak-strong uniqueness, and become stationary after finite time.
Similarly, for the compressible Euler system in the one-dimensional case there exist global in time admissible weak solutions having the weak-strong uniqueness property, see~\cite{DiPe83, LiPeSo96, LiPeTa94}.
Finally, let us point out some difficulties in extending our results to other domains than the torus. On the whole space, we can no longer require the denstity $H$ to be uniformly bounded away from zero and the initial energy
\begin{equation*}
\int_{\R^n}\frac{1}{2}h_0|u_0|^2+ah_0^2dx
\end{equation*}
(and similarly for~\eqref{eulerintro}) to be finite at the same time. On domains with physical boundaries, however, we do not even expect weak-strong uniqueness to hold, since a counterexample has been exhibited in~\cite{eulerboundary} in the case of the incompressible Euler equations.
\section{Notation}
We fix here some notation that will be used throughout the paper.
The $n$-dimensional torus will be denoted by $\T^n:=\R^n\slash \mathbb{Z}^n$.
Let $\Omega\subset\R^n$ be a measurable subset or $\Omega=\T^n$. The set of locally finite nonnegative measures on $\Omega$ will be denoted $\mathcal{M}^+(\Omega)$. If $X$ is a measurable subset of $\R^m$, then $\mathcal{P}(X)$ will be the set of probability measures on $X$.
Let $m\in\mathcal{M}^+(\Omega)$. The space $L^\infty_w(\Omega,m;\mathcal{P}(X))$ is then defined as the space of maps $\nu:\Omega\to\mathcal{P}(X)$, $x\mapsto\nu_x$, which are \term{weakly* measurable} with respect to $m$; that is, for every $\phi\in C_c(X)$ the map
\begin{equation*}
x\mapsto\int_{\Omega}\phi(\lambda)d\nu_x(\lambda)
\end{equation*}
is $m$-measurable. If $m$ is Lebesgue measure, we simply write $L^\infty_w(\Omega;\mathcal{P}(X))$. If $\Omega$ has the form $[0,T]\times\tilde{\Omega}$ for some measurable subset $\tilde{\Omega}\subset \R^n$, then $dx$ denotes $n$-dimensional Lebesgue measure and $dt$ one-dimensional Lebesgue measure. The Dirac mass centred at $x$ will be denoted as $\delta_x$, as usual.
The $m$-dimensional unit sphere is written $\mathbb{S}^m$. With $\bar{\Omega}$ we mean the topological closure of a subset of $\R^n$. We write $\R^+$ for the set of non-negative real numbers.
In the case $\Omega=[0,T]\times\tilde{\Omega}$, we will use measures of the form $m=m_t\otimes dt$; this means that, for every set of the form $\tau\times U$, where $\tau\subset[0,T]$ and $U\subset\tilde{\Omega}$ are measurable subsets,
\begin{equation*}
m(\tau\times U)=\int_\tau m_t(U)dt.
\end{equation*}
The differential operators $\nabla$ and $\diverg$ are applied only to the spatial variables. If $u$ and $v$ are vectors, then $u\otimes v$ denotes the matrix defined by $(u\otimes v)_{ij}=u_iv_j$. The divergence of a matrix field is understood to be taken row-wise.
Further notation will be introduced as we proceed.
\section{Generalised Young Measures}\label{young}
We recall briefly the notion of generalised Young measures, which were introduced by DiPerna and Majda~\cite{dipernamajda} and refined by Alibert and Bouchitt\'{e}~\cite{alibert}. Further details can be found e.g.\ in~\cite{k-r2, euleryoung}.
Young measures are used to represent weak limits of nonlinear functions of weakly convergent sequences. More precisely, suppose $\Omega\subset \R^n$ is a measurable set or $\Omega=\T^n$ ($n\geq1$), and $(u_k)_{n\in\mathbb{N}}$ is a sequence of maps bounded in $L^1(\Omega;\R^m)$ ($m\geq1$).
Then it was proved in~\cite{alibert} that there exists a subsequence (not relabeled) as well as a parametrised probability measure $\nu\in L_w^\infty(\Omega;\mathcal{P}(\R^m))$ (which is identical with the "classical" Young measure), a non-negative measure $m\in\mathcal{M}^+(\bar{\Omega})$, and a parametrized probability measure $\nu^\infty\in L_w^\infty(\Omega,m;\mathcal{P}(\mathbb{S}^{m-1}))$ such that
\begin{equation*}
f(x,u_n(x))dx\stackrel{*}{\rightharpoonup}\int_{\R^m}f(x,\lambda)d\nu_x(\lambda)dx+\int_{\mathbb{S}^{m-1}}f^\infty(x,\beta)d\nu^\infty_x(\beta)m
\end{equation*}
weakly* in the sense of measures. Here, $f:\Omega\times\R^m\to\R$ is any Carath\'{e}odory function (measurable in the first and continuous in the second argument) whose \term{recession function}
\begin{equation*}
f^\infty(x,\beta):=\lim_{{x'\rightarrow x\atop{\beta'\rightarrow
\beta\atop{s\rightarrow\infty}}}}\frac{f(x',s\beta')}{s}
\end{equation*}
is a well-defined and continuous function on $\bar{\Omega}\times\mathbb{S}^{m-1}$. Note that such an $f$ will have at most linear growth. If its growth is sublinear, then $f^\infty=0$.
Notice also that $\nu^\infty_{t,x}$ is only defined $m$-almost everywhere.
In order to properly define measure-valued solutions to compressible fluid equations within the framework of Alibert--Bouchitt\'{e}, we need a slight refinement which allows us to treat sequences whose components have different growth. Let $(u_k,w_k)_k$ be a sequence such that $(u_k)$ is bounded in $L^p(\Omega;\R^l)$ and $(w_k)$ is bounded in $L^q(\Omega;\R^m)$ ($1\leq p,q<\infty$). Define the ``nonhomogeneous unit sphere''
\begin{equation*}
\mathbb{S}^{l+m-1}_{p,q}:=\{(\beta_1,\beta_2)\in\R^{l+m}: |\beta_1|^{2p}+|\beta_2|^{2q}=1\}.
\end{equation*}
Then, there exists a a subsequence (not relabeled) and measures $\nu\in L_w^\infty(\Omega;\mathcal{P}(\R^{l+m}))$, $m\in\mathcal{M}^+(\bar{\Omega})$, $\nu^\infty\in L_w^\infty(\Omega,m;\mathcal{P}(\mathbb{S}^{l+m-1}_{p,q}))$ such that
\begin{equation*}
f(x,u_n(x),w_n(x))dx\stackrel{*}{\rightharpoonup}\int_{\R^{l+m}}f(x,\lambda_1,\lambda_2)d\nu_x(\lambda_1,\lambda_2)dx+\int_{\mathbb{S}^{l+m-1}_{p,q}}f^\infty(x,\beta_1,\beta_2)d\nu^\infty_x(\beta_1,\beta_2)m
\end{equation*}
in the sense of measures; this is valid for all integrands $f$ whose $p$-$q$-recession function exists and is continuous on $\bar{\Omega}\times\mathbb{S}^{l+m-1}_{p,q}$. The $p$-$q$-recession function is defined as
\begin{equation*}
f^\infty(x,\beta_1,\beta_2):=\lim_{{x'\rightarrow x\atop{(\beta_1',\beta_2')\rightarrow
(\beta_1,\beta_2)\atop{s\rightarrow\infty}}}}\frac{f(x',s^q\beta_1',s^p\beta_2')}{s^{pq}}.
\end{equation*}
The case $p=2$, $q=1$ was treated in Subsection 2.4.1 of~\cite{euleryoung} and the extension to general $p$ and $q$ is straightforward.
Let us quote another fact which is important for measure-valued solutions of time-dependent equations with bounded energy: If $\Omega=[0,T]\times\tilde{\Omega}$ for some measurable $\tilde{\Omega}\subset\R^n$ (or $\tilde{\Omega}=\T^n$), and if the sequence $(u_n,w_n)_n$ is bounded in $L^\infty([0,T];L^p(\tilde{\Omega})\times L^q(\tilde{\Omega}))$, then the corresponding concentration measure $m$ admits a disintegration of the form
\begin{equation*}
m=m_t(dx)\otimes dt,
\end{equation*}
where $t\mapsto m_t$ is bounded and measurable viewed as a map from $[0,T]$ into $\mathcal{M}^+(\overline{\tilde{\Omega}})$. The proof of this statement was given in~\cite{weak-strong}.
\section{Weak-strong uniqueness for measure-valued solutions of the compressible Euler equations}\label{Euler}
We consider the compressible Euler system
\begin{equation}\label{euler}
\begin{aligned}
\partial_th+\diverg(hu)&=0\\
\partial_t(hu)+\diverg(hu\otimes u)+\nabla(\kappa h^\gamma)&=hG.
\end{aligned}
\end{equation}
Here, $h:[0,T]\times\T^n\to\R$, $u:[0,T]\times\T^n\to\R^n$, and $G:[0,T]\times\T^n\to\R^n$, $\gamma>1$. We set the constant $\kappa>0$ equal to one in order to save some writing, remarking however that all computations remain unchanged for general $\kappa$.
\subsection{Measure-valued solutions}
We apply the abstract framework from the previous section, with $l=1$, $m=n$, $p=\gamma$, and $q=2$, in order to define the notion of \term{measure-valued solution} of~\eqref{euler}. Consider a generalised Young measure
\begin{equation*}
(\nu_{t,x},m,\nu^\infty_{t,x})\in L_w^\infty\left([0,T]\times\T^n;\mathcal{P}(\R^+\times\R^n)\right)\times\mathcal{M}^+([0,T]\times\T^n)\times L_w^\infty\left([0,T]\times\T^n,m;\mathcal{P}(\mathbb{S}^+)\right),
\end{equation*}
where we wrote
\begin{equation*}
\mathbb{S^+}:=\{(\beta_1,\beta')\in\mathbb{S}^{1+n}_{\gamma,2}: \beta_1\geq0\}.
\end{equation*}
We will use the variables $(\lambda_1,\lambda')\in\R^+\times\R^n$ and $(\beta_1,\beta')\in\mathbb{S}^+$ as dummy variables when integrating with respect to $\nu_{t,x}$ and $\nu_{t,x}^\infty$, respectively. One should think of $\lambda_1,\beta_1$ as representing $h$ and $\lambda', \beta'$ as representing $\sqrt{h}u$. We also use the common notation
\begin{equation*}
\langle F(\lambda_1,\lambda'),\nu_{t,x}\rangle:=\int_{\R^+\times\R^n}F(\lambda_1,\lambda')d\nu_{x,t}(\lambda_1,\lambda')
\end{equation*}
and analogously for $\nu^\infty$.
If we consider a function $f:[0,T]\times\T^n\times\R^+\times\R^n\to\R$ which has an appropriate $\gamma$-2-recession function as defined in Section~\ref{young}, we use the shorthand notation
\begin{equation*}
\bar{f}(dtdx):=\langle f(t,x,\cdot,\cdot),\nu_{t,x}\rangle dtdx+\langle f^\infty(t,x,\cdot,\cdot),\nu_{t,x}^\infty\rangle m(dtdx).
\end{equation*}
For instance, we have
\begin{equation*}
\begin{aligned}
\bar{h}&= \langle\lambda_1,\nu\rangle\\
\overline{h^\gamma}&=\langle\lambda_1^\gamma,\nu\rangle+\langle\beta_1^\gamma,\nu^\infty\rangle m\\
\overline{hu}&=\langle\sqrt{\lambda_1}\lambda',\nu\rangle\\
\overline{hu\otimes u}&=\langle\lambda'\otimes\lambda',\nu\rangle+\langle\beta'\otimes\beta',\nu^\infty\rangle m\\
\overline{h|u|^2}&=\langle|\lambda'|^2,\nu\rangle+\langle|\beta'|^2,\nu^\infty\rangle m\\
\overline{hG}&=\langle\lambda_1G,\nu\rangle=\bar{h}G.
\end{aligned}
\end{equation*}
We say that $(\nu,m,\nu^\infty)$ is a \term{measure-valued solution} of~\eqref{euler} with initial data $(h_0,u_0)$ if for every $\tau\in[0,T]$, $\psi\in C^1([0,T]\times\T^n;\R)$, $\phi\in C^1([0,T]\times\T^n;\R^n)$ it holds that
\begin{equation}\label{Emass_momentum}
\begin{aligned}
\int_0^\tau\int_{\T^n}\partial_t\psi \bar{h}+\nabla\psi\cdot\overline{hu}dxdt+\int_{\T^n}\psi(x,0)h_0-\psi(x,\tau)\bar{h}(x,\tau)dx&=0,\\
\int_0^\tau\int_{\T^n}\partial_t\phi\cdot\overline{hu}+\nabla\phi : \overline{hu\otimes u}+\diverg\phi\overline{h^\gamma}
-\phi\cdot\overline{hG}&dxdt\\
+\int_{\T^n}\phi(x,0)\cdot h_0u_0-\phi(x,\tau)\cdot\overline{hu}(x,\tau)dx&=0.
\end{aligned}
\end{equation}
It is part of the definition that all the integrals have to exist for any choice of test functions, in particular for the initial data we require $h_0\in L^1$, $h_0u_0\in L^1$.
Let us set
\begin{equation*}
E_{mvs}(t):=\int_{\T^n}\frac{1}{2}\overline{h|u|^2}(t,x)+\frac{1}{\gamma-1}\overline{h^\gamma}(t,x)dx
\end{equation*}
for almost every $t$, and
\begin{equation*}
E_0:=\int_{\T^n}\frac{1}{2}h_0|u_0|^2(x)+\frac{1}{\gamma-1}h_0^\gamma(x)dx.
\end{equation*}
We then say that a measure-valued solution is \term{admissible} if
\begin{equation}\label{EEmvsenergy}
\begin{aligned}
E_{mvs}(t)\leq E_0+\int_0^t\int_{\T^n}\overline{hG\cdot u}(s,x)dxds
\end{aligned}
\end{equation}
in the sense of distributions. An elementary computation yields the well-known fact that the energy is conserved for smooth solutions (i.e.~\eqref{EEmvsenergy} holds with equality), whereas the inequality becomes strict upon the formation of shocks.
\begin{remark}
The global existence of measure-valued solutions for \eqref{euler} was proved by Neustupa in \cite{neustupa}. However he used a different formulation of the Young measure, as the formalism of Alibert-Bouchitt\'{e} \cite{alibert} was not yet available. One can however rewrite the solutions of \cite{neustupa} in the form presented here. Neustupa's solutions can be seen to be admissible, as they can be obtained e.g.\ from an artificial viscosity approximation.
\end{remark}
\subsection{Weak-Strong Uniqueness}
\begin{theorem}\label{Eweak-strong}
Let $G\in L^{\infty}([0,T];L^2(\T^n))$ and suppose $H\in W^{1,\infty}([0,T]\times\T^n), U\in C^1([0,T]\times\T^n)$ is a solution of~\eqref{euler} with initial data $h_0\geq c>0$, $h_0\in L^\gamma(\T^n)$, $h_0|u_0|^2\in L^1(\T^n)$, and $H(x,t)\geq c>0$ for some constant $c$ and all $(t,x)\in[0,T]\times\T^n$. If $(\nu,m,\nu^\infty)$ is an admissible measure-valued solution with the same initial data, then
\begin{equation*}
\nu_{t,x}=\delta_{(H(t,x),\sqrt{H(t,x)}U(t,x))} \text{ for a.e. $t,x$, and $m=0$.}
\end{equation*}
\end{theorem}
\begin{proof}
Let us first define for a.e.\ $t\in[0,T]$ the \term{relative energy} between $(H,U)$ and the measure-valued solution as
\begin{equation*}
\begin{aligned}
E_{rel}(t)&=\int_{\T^n}\frac{1}{2}\overline{h|u-U|^2}+\overline{\frac{1}{\gamma-1}h^\gamma-\frac{\gamma}{\gamma-1}H^{\gamma-1}h+H^\gamma}dx\\
&=\int_{\T^n}\frac{1}{2}\langle|\lambda'-\sqrt{\lambda_1}U|^2,\nu_{t,x}\rangle dx+\frac{1}{2}\int_{\T^n}\langle|\beta'|^2,\nu_{t,x}^\infty\rangle dm_t(x)\\
&\hspace{1cm}+\int_{\T^n}\langle\frac{1}{\gamma-1}\lambda_1^\gamma-\frac{\gamma}{\gamma-1}H^{\gamma-1}\lambda_1+H^\gamma,\nu_{t,x}\rangle dx\\
&\hspace{1cm}+\int_{\T^n} \frac{1}{\gamma-1}\langle\beta_1^\gamma,\nu^\infty_{t,x}\rangle dm_t(x).
\end{aligned}
\end{equation*}
Here, the measure $m_t\in\mathcal{M}^+(\T^n)$ is obtained by the disintegration $m(dtdx)=m_t(dx)\otimes dt$, which is well-defined thanks to the admissibility (cf.\ Section~\ref{young}). Note that the strict convexity of $|\cdot|^\gamma$ implies that the relative energy is always non-negative. Then it is straightforward to observe that $E_{rel}(t)=0$ for a.e.\ $t$ implies Theorem~\ref{Eweak-strong}. Indeed, defining the projection operators $\pi^{\lambda_1}:(\lambda_1, \lambda')\mapsto \lambda_1$ and $\pi^{\lambda'}:(\lambda_1, \lambda')\mapsto \lambda'$, we observe that the strict convexity of $|\cdot|^\gamma$ implies that $\pi^{\lambda_1}\nu_{t,x}=\delta_{H(t,x)}$ for a.e. $t,x$ and hence $\nu_{t,x}=\delta_{(H(t,x))}\otimes \pi^{\lambda'}\nu_{t,x}$. Using the first term in the relative energy allows to conclude $\pi^{\lambda'}\nu_{t,x}=\delta_{(\sqrt{H(t,x)}U(t,x))}$.
Setting $\phi=U$ in the momentum equation (the second equation of~\eqref{Emass_momentum}), we obtain
\begin{equation}\label{Etest1}
\begin{aligned}
\int_{\T^n}\overline{hu}\cdot U(\tau)dx=&\int_{\T^n}h_0|u_0|^2dx+\int_0^\tau\int_{\T^n}\overline{hu}\cdot\partial_t U+\overline{hu\otimes u}:\nabla U dxdt\\
&+\int_0^\tau\int_{\T^n}\overline{h^\gamma}\diverg U+\overline{hG}\cdot U dxdt.
\end{aligned}
\end{equation}
Similarly, setting $\psi=\frac{1}{2}|U|^2$ and then $\psi=\gamma H^{\gamma-1}$ in~\eqref{Emass_momentum} yields
\begin{equation}\label{Etest2}
\frac{1}{2}\int_{\T^n}|U(\tau)|^2\bar{h}(\tau,x)dx=\int_0^\tau\int_{\T^n}U\cdot\partial_tU \bar{h}+\nabla U U\cdot\overline{hu}dxdt+\int_{\T^n}\frac{1}{2}|u_0|^2h_0dx
\end{equation}
and
\begin{equation}\label{Etest3}
\int_{\T^n}\gamma H^{\gamma-1}(\tau)\bar{h}(\tau)dx=\int_0^\tau\int_{\T^n}\gamma(\gamma-1)H^{\gamma-2}\partial_tH \bar{h}+\gamma(\gamma-1)H^{\gamma-2}\nabla H \cdot\overline{hu}dxdt+\int_{\T^n}\gamma h_0^\gamma dx,
\end{equation}
respectively.
Next, we can write the relative energy as
\begin{equation*}
\begin{aligned}
E_{rel}(\tau)&=\int_{\T^n}\frac{1}{2}\overline{h|u|^2}+\frac{1}{\gamma-1}\overline{h^\gamma}dx + \int_{\T^n}H^\gamma dx+\frac{1}{2}\int_{\T^n}|U|^2\bar{h}dx-\int_{\T^n}U\cdot\overline{hu}dx-\int_{\T^n}\frac{\gamma}{\gamma-1}H^{\gamma-1}\bar{h}dx\\
&=E_{mvs}(\tau)+ \int_{\T^n}H^\gamma dx+\frac{1}{2}\int_{\T^n}|U|^2\bar{h}dx-\int_{\T^n}U\cdot\overline{hu}dx-\int_{\T^n}\frac{\gamma}{\gamma-1}H^{\gamma-1}\bar{h}dx
\end{aligned}
\end{equation*}
(all integrands evaluated at time $\tau$). Next, using the balances~\eqref{Etest1},~\eqref{Etest2},~\eqref{Etest3} for the last three integrals, we obtain
\begin{equation*}
\begin{aligned}
E_{rel}(\tau)=E_{mvs}(\tau)&+\int_{\T^n}H^\gamma dx\\
&+\int_0^\tau\int_{\T^n}U\cdot\partial_tU \bar{h}+\nabla U U\cdot\overline{hu}dxdt+\int_{\T^n}\frac{1}{2}|u_0|^2h_0dx\\
&-\int_{\T^n}h_0|u_0|^2dx-\int_0^\tau\int_{\T^n}\overline{hu}\cdot\partial_t U+\overline{hu\otimes u}:\nabla U dxdt\\
&-\int_0^\tau\int_{\T^n}\overline{h^\gamma}\diverg U-\overline{hG}\cdot U dxdt\\
&-\int_0^\tau\int_{\T^n}(\gamma H^{\gamma-2}\partial_tH \bar{h}+\gamma H^{\gamma-2}\nabla H \cdot\overline{hu})dxdt-\int_{\T^n}\frac{\gamma}{\gamma-1} h_0^\gamma dx,
\end{aligned}
\end{equation*}
and using~\eqref{Emvsenergy} we have, for a.e.\ $\tau$,
\begin{equation}\label{Eintermediatestep}
\begin{aligned}
E_{rel}(\tau)\leq& -\int_{\T^n}h_0^\gamma dx +\int_0^\tau\int_{\T^n}\overline{hG\cdot u} +\int_{\T^n}H^\gamma dx\\
&+\int_0^\tau\int_{\T^n}U\cdot\partial_tU \bar{h}+\nabla U U\cdot\overline{hu}dxdt\\
&-\int_0^\tau\int_{\T^n}\overline{hu}\cdot\partial_t U+\overline{hu\otimes u}:\nabla U dxdt\\
&-\int_0^\tau\int_{\T^n}\overline{h^\gamma}\diverg U-\overline{hG}\cdot U dxdt\\
&-\int_0^\tau\int_{\T^n}(\gamma H^{\gamma-2}\partial_tH \bar{h}+\gamma H^{\gamma-2}\nabla H \cdot\overline{hu})dxdt.
\end{aligned}
\end{equation}
Next, we collect some terms and write
\begin{equation}\label{Eequality1}
\begin{aligned}
\int_{\T^n}&H^\gamma dx -\int_{\T^n}h_0^\gamma dx-\int_0^\tau\int_{\T^n}\gamma H^{\gamma-2}\partial_tH \bar{h}dxdt\\
&=\int_0^\tau\int_{\T^n}\frac{d}{dt}H^\gamma-\gamma H^{\gamma-2}\partial_tH \bar{h}dxdt\\
&=\int_0^\tau\int_{\T^n}\gamma H^{\gamma-1}\partial_tH-\gamma H^{\gamma-2}\partial_tH \bar{h}dxdt\\
&=\int_0^\tau\int_{\T^n}\gamma H^{\gamma-2}\partial_tH(H-\bar{h})dxdt,
\end{aligned}
\end{equation}
\begin{equation}\label{Eequality2}
\begin{aligned}
\int_0^\tau\int_{\T^n}&\overline{hG\cdot u}-\overline{hG}\cdot U dxdt\\
&=\int_0^\tau\int_{\T^n}\overline{hG\cdot (u-U)}dxdt,
\end{aligned}
\end{equation}
and
\begin{equation}\label{Eequality3}
\begin{aligned}
\int_0^\tau\int_{\T^n}&U\cdot\partial_tU \bar{h}+\nabla U U\cdot\overline{hu}-\overline{hu}\cdot\partial_t U-\overline{hu\otimes u}:\nabla U dxdt\\
&=\int_0^\tau\int_{\T^n}\partial_tU\cdot \overline{h(U-u)}+\nabla U :\overline{hu\otimes(U-u)}dxdt.
\end{aligned}
\end{equation}
Indeed, the last two equalities can be verified by writing the expressions in the "coarse-grained" overline notation explicitly in terms of the Young measure $(\nu,m,\nu^\infty)$.
Plugging equalities~\eqref{Eequality1},~\eqref{Eequality2},~\eqref{Eequality3} into~\eqref{Eintermediatestep}, we arrive at
\begin{equation}\label{Eintermediatestep2}
\begin{aligned}
E_{rel}(\tau)\leq&\int_0^\tau\int_{\T^n}\gamma H^{\gamma-2}\partial_tH(H-\bar{h})dxdt \\
&+\int_0^\tau\int_{\T^n}\overline{hG\cdot (u-U)}dxdt\\
&+\int_0^\tau\int_{\T^n}\partial_tU\cdot \overline{h(U-u)}+\nabla U :\overline{hu\otimes(U-u)}dxdt\\
&-\int_0^\tau\int_{\T^n}\overline{h^\gamma}\diverg U dxdt-\int_0^\tau\int_{\T^n}\gamma H^{\gamma-2}\nabla H \cdot\overline{hu}dxdt.
\end{aligned}
\end{equation}
For the last two integrals, we have by the divergence theorem
\begin{equation}\label{Eh^2estimate}
\begin{aligned}
-\int_0^\tau\int_{\T^n}&\overline{h^\gamma}\diverg U dxdt-\int_0^\tau\int_{\T^n}\gamma H^{\gamma-2}\nabla H \cdot\overline{hu}dxdt\\
&=\int_0^\tau\int_{\T^n}-\overline{h^\gamma}\diverg U +\gamma H^{\gamma-2}\nabla H \cdot(HU-\overline{hu})-\gamma H^{\gamma-2}\nabla H\cdot HUdxdt\\
&=\int_0^\tau\int_{\T^n}(H^\gamma-\overline{h^\gamma})\diverg U +\gamma H^{\gamma-2}\nabla H \cdot(HU-\overline{hu})dxdt.\\
\end{aligned}
\end{equation}
Inserting this back into~\eqref{Eintermediatestep2} and observing that, by the mass equation for $(H,U)$,
\begin{equation*}
\begin{aligned}
\gamma &H^{\gamma-2}\partial_tH(H-\bar{h})+\gamma H^{\gamma-2}\diverg UH(H-\bar{h}) +\gamma H^{\gamma-2}\nabla H \cdot HU
=\gamma H^{\gamma-2}U\cdot\nabla H\bar{h},
\end{aligned}
\end{equation*}
we get
\begin{equation}\label{Eintermediatestep3}
\begin{aligned}
E_{rel}(\tau)\leq&\int_0^\tau\int_{\T^n}\gamma H^{\gamma-2}\cdot\nabla H\overline{h(U-u)}dxdt \\
&+\int_0^\tau\int_{\T^n}\overline{hG\cdot (u-U)}dxdt\\
&+\int_0^\tau\int_{\T^n}\partial_tU\cdot \overline{h(U-u)}+\nabla U :\overline{hu\otimes(U-u)}dxdt\\
&-\int_0^\tau\int_{\T^n}\gamma H^{\gamma-1}\diverg U(H-\overline{h} )dxdt+\int_0^\tau\int_{\T^n}(H^\gamma-\overline{h^\gamma})\diverg U.
\end{aligned}
\end{equation}
The expression in the third line can be rewritten pointwise as
\begin{equation}\label{Epointwiseest}
\begin{aligned}
\partial_tU&\cdot \overline{h(U-u)}+\nabla U :\overline{hu\otimes(U-u)}\\
&=\partial_tU\cdot \overline{h(U-u)}+\nabla U :\overline{hU\otimes(U-u)}+\nabla U :\overline{h(u-U)\otimes(U-u)},
\end{aligned}
\end{equation}
and the integral of the last term as well as the last line in~\eqref{Eintermediatestep3} can both be estimated by
\begin{equation}\label{Eabsorb}
C\norm{U}_{C^1}\int_0^\tau E_{rel}(t)dt.
\end{equation}
For the remaining terms in~\eqref{Epointwiseest} we obtain, using the momentum equation for $(H,U)$,
\begin{equation}\label{Emomentum}
\begin{aligned}
\partial_tU&\cdot \overline{h(U-u)}+\nabla U :U\otimes\overline{h(U-u)}\\
&=\frac{1}{H}(\partial_t(HU)+\diverg(HU\otimes U))\cdot\overline{h(U-u)}\\
&=G\cdot\overline{h(U-u)} - \gamma H^{\gamma-2}\nabla H\cdot\overline{h(U-u)}.
\end{aligned}
\end{equation}
Putting together~\eqref{Eintermediatestep3},~\eqref{Eabsorb}, and~\eqref{Emomentum}, we obtain
\begin{equation*}
E_{rel}(\tau)\leq C\norm{U}_{C^1}\int_0^\tau E_{rel}(t)dt.
\end{equation*}
Finally, from Gronwall's inequality it follows that $E_{rel}(\tau)=0$ for a.e. $t$.
\end{proof}
\section{Savage-Hutter system}
We consider the two-dimensional Savage-Hutter model
\begin{equation}\label{savhut}
\begin{aligned}
\partial_th+\diverg(hu)&=0\\
\partial_t(hu)+\diverg(hu\otimes u)+\nabla(ah^2)&=h\left(-dB(u)+f\right).
\end{aligned}
\end{equation}
The one-dimensional case can be treated similarly. Here, $h:[0,T]\times\T^2\to\R$, $u:[0,T]\times\T^2\to\R^2$, $f:[0,T]\times\T^2\to\R^2$, and $a>0$ and $d>0$ are constant. By $B(u)$ we denote the subdifferential of $u\mapsto |u|$, so that $B(u)$ is multi-valued such that
\begin{equation*}
B(u)=\begin{cases}
\frac{u}{|u|} & \text{if $u\neq0$,}\\[0.2cm]
\overline{B_1(0)} & \text{if $u=0$.}
\end{cases}
\end{equation*}
Consequently, the equality sign in the second line of~\eqref{savhut} should really be an inclusion. We will stick however to the formulation~\eqref{savhut}, thereby slightly abusing notation.
\subsection{Stationary solutions}If in~\eqref{savhut} $f$ is independent of time, then a special class of solutions is given by $u\equiv0$ and any $h=h(x)>c$ such that
\begin{equation*}
\left|\nabla h-\frac{f}{2a}\right|\leq\frac{d}{2a}\hspace{0.3cm}\text{for a.e.\ $x$.}
\end{equation*}
\subsection{Measure-Valued Solutions}
We recall the notion of \term{measure-valued solution} of~\eqref{savhut} from \cite{gwiazda2005} in the notation used therein (in fact, there the problem was treated on the whole space, but it can easily be adapted to the torus). The author considers system~\eqref{savhut} with a right-hand side given by $h\tilde f(x, \sqrt{h} u)$, where
\begin{equation*}
\tilde f(x,\sqrt{h} u)=-d\frac{\sqrt h u}{\sqrt h |u|}+f
\end{equation*}
and for $u=0$ the mapping $\tilde f$ takes values in the closed unit ball.
To handle this multi-valued (monotone) term, let us first recall from~\cite{gwiazda2005} the following observation, see also \cite{GwZa2007, BuGwMaSw2009} for a similar approach.
\begin{lemma}\label{monotoniczna-ciaglosc}
Let $f:\R^n \rightarrow \R^n$ ($f:\R^n \rightarrow 2^{\R^n}$) be a monotone
function (monotone mapping).
Then $$(f+Id)^{-1}:\R^n\rightarrow \R^n $$
and
$$f\circ (f+Id)^{-1}:\R^n\rightarrow \R^n $$
are Lipschitz functions. Above we understand $f\circ(f+Id)^{-1}=Id-Id\circ(f+Id)^{-1}.$
Moreover, for any continuous function $g:\R^n \rightarrow \R^n$,
$$g\circ (f+Id)^{-1}:\R^n\rightarrow \R^n$$
is a continuous function.
\end{lemma}
Under the assumptions that
$(h_0,h_0u_0) \in L^1_{loc}(\R^2),$
$\int_{\R^2}\left\{\frac{1}{2} |u_0|^2h_0+a(h_0)^2 \right\}dx<\infty$,
there exists a triple of measures
\begin{equation*}
(\mu_{t,x},m,\mu^\infty_{t,x})\in L_w^\infty\left([0,T]\times\T^2;\mathcal{P}(\R^+\times\R^2)\right)\times\mathcal{M}^+([0,T]\times\T^2)\times L_w^\infty\left([0,T]\times\T^2,m;\mathcal{P}(\mathbb{S}^+)\right),
\end{equation*}
such that
\begin{equation*}
\begin{aligned}
\int_{[0,T)}\int_{\R^2}\overline{h}\partial_t \varphi_1
+\overline{m}\cdot \nabla_x
\varphi_1 dxdt&=\int_{\R^2}h_0\varphi_1(0)dx,\\
\int_{[0,T)} \left\{ \int_{\R^2}\overline{m}\partial_t \varphi_2 dx+
\langle \overline{e}+a \overline{p},\nabla_x\varphi_2 \rangle -
\int_{\R^2}\overline{f}\varphi_2 dx \right\} dt
&=\int_{\R^2}h_0u_0\varphi_2(0)dx
\end{aligned}
\label{MV-SH}
\end{equation*}
for all $\varphi_1, \varphi_2 \in C_c([0,T)\times\R^2)$, and
for almost all $t\in[0,T)$ it holds that
\begin{equation}\label{MV-NE}
\langle {\rm Tr}(\overline{e}(t))+a\overline{p}(t), 1\rangle
-\int_{\R^2}\left\{\frac{1}{2} |u_0|^2h_0+a(h_0)^2 \right\}dx
\leq\int_{[0,T)\times\R^2}\chi dxdt,
\end{equation}
where
\begin{equation*}
\begin{split}
\overline{h}(t,x)=&\int_{\R_+\times\R^2}
\lambda_1\,\,d\mu_{t,x}(\lambda),\\
\overline{p}(t,x)=&\int_{\R_+\times\R^2}
\lambda_1^2\,\,d\mu_{t,x}(\lambda)
+\Big(\int_{S^2_+}\beta_1^2\,\,d\nu_{t,x}^\infty(\beta) \Big) m,\\
\overline{m}(t,x)=&\int_{\R_+\times\R^2}
\sqrt{\lambda_1}\,(-\tilde{f}+Id)^{-1}
(x,(\lambda_2,\lambda_3))\,\,d\mu_{t,x}(\lambda),\\
\overline{e}(t,x)=&\int_{\R_+\times\R^2}
(-\tilde{f}+Id)^{-1}
(x,(\lambda_2,\lambda_3))\otimes(-\tilde{f}+Id)^{-1}
(x,(\lambda_2,\lambda_3))\,\,d\mu_{t,x}(\lambda)\\
&+\Big(\int_{S^2_+}(\beta_2,\beta_3)\otimes
(\beta_2,\beta_3)\,\,d\nu_{t,x}^\infty(\beta) \Big) m,\\
{\rm Tr}(\overline{e})(t,x)=&\int_{\R_+\times\R^2}
(-\tilde{f}+Id)^{-1}
(x,(\lambda_2,\lambda_3))\cdot(-\tilde{f}+Id)^{-1}
(x,(\lambda_2,\lambda_3))\,\,d\mu_{t,x}(\lambda)\\
&+\Big(\int_{S^2_+}\beta_2^2+\beta_3^2 \,\,d\nu_{t,x}^\infty(\beta) \Big) m,\\
\overline{f}(t,x)=&\int_{\R_+\times\R^2} \lambda_1
\tilde{f}\circ (-\tilde{f}+Id)^{-1}
(x,(\lambda_2,\lambda_3))d\mu_{t,x}(\lambda),\\
\chi(t,x)=&\int_{\R_+\times\R^2} \lambda_1 \tilde {f}\circ (-\tilde{f}+Id)^{-1}
(x,(\lambda_2,\lambda_3))\cdot (-\tilde{f}+Id)^{-1}
(x,(\lambda_2,\lambda_3))d\mu_{t,x}(\lambda)
\end{split}
\end{equation*}
for almost all $(t,x)\in[0,T]\times\T^2$.
Define for every $\mu$-measurable set $A\times B\subset\R^+\times\R^2$ the push-forward of $\mu$ through the map $(-\tilde f+Id)$ as
\begin{equation*}
\nu(A\times B):=(-\tilde f+Id)_\#\mu(A\times B)=\mu(A\times (-\tilde f+Id)^{-1}(B)).
\end{equation*}
Hence using again the variables $(\lambda_1,\lambda')\in\R^+\times\R^2$ (which correspond to $\lambda_1,\lambda_2, \lambda_3$) and $(\beta_1,\beta')\in\mathbb{S}^+$ (corresponding to $\beta_1, \beta_2, \beta_3$), when integrating with respect to $\nu_{t,x}$ and $\nu_{t,x}^\infty$, respectively, the problem can be translated to
\begin{equation*}
\begin{aligned}
\bar{h}&= \langle\lambda_1,\nu\rangle\\
\overline{h^2}&=\langle\lambda_1^2,\nu\rangle+\langle\beta_1^2,\nu^\infty\rangle m\\
\overline{hu}&=\langle\sqrt{\lambda_1}\lambda',\nu\rangle\\
\overline{hu\otimes u}&=\langle\lambda'\otimes\lambda',\nu\rangle+\langle\beta'\otimes\beta',\nu^\infty\rangle m\\
\overline{h|u|^2}&=\langle|\lambda'|^2,\nu\rangle+\langle|\beta'|^2,\nu^\infty\rangle m\\
\overline{h(-dB(u)+f)}&=
\langle \lambda_1
\tilde{f}\circ (-\tilde{f}+Id)^{-1}
(x,\lambda'),\mu\rangle.
\end{aligned}
\end{equation*}
We say that $(\mu,m,\mu^\infty)$ is a \term{measure-valued solution} of~\eqref{savhut} with initial data $(h_0,u_0)$ if
for every $\tau\in[0,T]$, $\psi\in C^1([0,T]\times\T^2;\R)$, $\phi\in C^1([0,T]\times\T^2;\R^2)$ it holds that
\begin{equation}\label{mass_momentum}
\begin{aligned}
\int_0^\tau\int_{\T^2}\partial_t\psi \bar{h}+\nabla\psi\cdot\overline{hu}dxdt+\int_{\T^2}\psi(x,0)h_0-\psi(x,\tau)\bar{h}(x,\tau)dx&=0,\\
\int_0^\tau\int_{\T^2}\partial_t\phi\cdot\overline{hu}+\nabla\phi : \overline{hu\otimes u}+a\diverg\phi\overline{h^2}
+\phi\cdot\overline{h(-dB(u)+f)}&dxdt\\
+\int_{\T^2}\phi(x,0)\cdot h_0u_0-\phi(x,\tau)\cdot\overline{hu}(x,\tau)&=0.
\end{aligned}
\end{equation}
For a.e.\ $t$, we set
\begin{equation*}
E_{mvs}(t):=\int_{\T^2}\frac{1}{2}\overline{h|u|^2}(t,x)+a\overline{h^2}(t,x)dx
\end{equation*}
and
\begin{equation*}
E_0:=\int_{\T^n}\frac{1}{2}h_0|u_0|^2(x)+h_0^2(x)dx.
\end{equation*}
We say that a measure-valued solution is \term{admissible}
if
\begin{equation}\label{Emvsenergy}
\begin{aligned}
E_{mvs}(t)\leq E_0-\int_0^t\int_{\T^n}d\overline{h(B(u)-f)\cdot u}(t,x)
\end{aligned}
\end{equation}
in the sense of distributions.
\subsection{Weak-Strong Uniqueness}
\begin{theorem}\label{weak-strong}
Let $f\in L^{\infty}([0,T];L^2(\T^2))$ and suppose $H\in W^{1,\infty}([0,T]\times\T^2), U\in C^1([0,T]\times\T^2)$ is a solution of~\eqref{savhut} with initial data $h_0\geq c>0$, $h_0\in L^2(\T^2)$, $h_0|u_0|^2\in L^1(\T^2)$ and $H(t,x)\geq c>0$ for some constant $c$ and every $(t,x)\in[0,T]\times\T^n$. If $(\nu,m,\nu^\infty)$ is an admissible measure-valued solution with the same initial data, then
\begin{equation*}
\nu_{t,x}=\delta_{(H(t,x),\sqrt{H(t,x)}U(t,x))} \text{ for a.e. $t,x$, and $m=0$.}
\end{equation*}
\end{theorem}
\begin{proof}
Let us first define for a.e.\ $t\in[0,T]$ the \term{relative energy} between $(H,U)$ and the measure-valued solution as
\begin{equation*}
\begin{aligned}
E_{rel}(t)&=\int_{\T^2}\frac{1}{2}\overline{h|u-U|^2}+a\overline{(h-H)^2}dx\\
&=\int_{\T^2}\frac{1}{2}\langle\left|\lambda'-\sqrt{\lambda_1}U\right|^2,\nu_{t,x}\rangle dx+\frac{1}{2}\int_{\T^2}\langle|\beta'|^2,\nu_{t,x}^\infty\rangle dm_t(x)\\
&\hspace{1cm}+a\int_{\T^2}\langle|\lambda_1-H|^2,\nu_{t,x}\rangle dx+a\int_{\T^2}\langle \beta_1^2,\nu^\infty_{t,x}\rangle dm_t(x).
\end{aligned}
\end{equation*}
Here, the measure $m_t\in\mathcal{M}^+(\T^2)$ is obtained by the disintegration $m(dtdx)=m_t(dx)\otimes dt$, which is well-defined thanks to the admissibility.
It is straightforward to observe that $E_{rel}(t)=0$ for a.e.\ $t$ implies Theorem~\ref{weak-strong}.
Following the computations of Section~\ref{Euler} we arrive at
\begin{equation*}
E_{rel}(\tau)+\int_0^\tau\int_{\T^2}\overline{h(dB(u)-dB(U))\cdot (u-U)}dxdt\leq C\norm{U}_{C^1}\int_0^\tau E_{rel}(t)dt.
\end{equation*}
Finally, since $B$ is monotone, the integral on the left hand side is non-negative, and from Gronwall's inequality it follows that $E_{rel}(\tau)=0$ for a.e. $t$.
\end{proof}
\section{Dissipation of Momentum in Finite Time}
\begin{theorem}\label{mvsfinitetime}
Let $(\nu,m,\nu^\infty)$ be an admissible measure-valued solution of the Savage-Hutter system~\eqref{savhut} with initial energy $E_0$ and
\begin{equation*}
\norm{f}_{L^\infty(\R^+\times\T^2)}<d.
\end{equation*}
Then there exists $0\leq T<\infty$ such that
\begin{equation*}
M(t):=\int_{\T^2}\overline{h|u|}(t,x)dx=0\hspace{0.3cm}\text{for almost every $t>T$.}
\end{equation*}
Moreover, there exists a constant $C$ depending only on $d-\norm{f}_\infty$ and $a$ such that
\begin{equation*}
T\leq CE_0^{1/4}.
\end{equation*}
\end{theorem}
\begin{proof}
For the momentum we have
\begin{equation*}
M(t)=\int_{\T^2}\overline{h|u|}(t,x)dx=\int_{\T^2}\langle\sqrt{\lambda_1}|\lambda'|,\nu_{t,x}\rangle dx.
\end{equation*}
Note in particular that the momentum does not concentrate, i.e.\ the 2-2-recession function of $(\lambda_1,\lambda')\mapsto \lambda_1|\lambda'|$ is zero.
For the following estimate, we use Jensen's inequality applied to the function $|\cdot|^{4/3}$ (recall that, according to our convention, the torus has measure 1), then Young's inequality,
\begin{equation*}
ab\leq\frac{a^p}{p}+\frac{b^q}{q},\hspace{0.3cm}\frac{1}{p}+\frac{1}{q}=1,
\end{equation*}
with the conjugate exponents $3$ and $3/2$, and finally the admissibility assumption:
\begin{equation*}
\begin{aligned}
M(t)^{4/3}&=\left(\int_{\T^2}\langle\sqrt{\lambda_1}|\lambda'|,\nu_{t,x}\rangle dx\right)^{4/3}\\
&\leq \int_{\T^2}\langle\sqrt{\lambda_1}^{4/3}|\lambda'|^{4/3},\nu_{t,x}\rangle dx\\
&\leq \frac{1}{3}\int_{\T^2}\langle\lambda_1^2,\nu_{t,x}\rangle dx+\frac{2}{3}\int_{\T^2}\langle|\lambda'|^{2},\nu_{t,x}\rangle dx\\
&\leq C(a) E_{mvs}(t)\leq C(a)\left(E_0-\int_0^t\int_{\T^2}(d-\norm{f}_\infty)\langle\sqrt{\lambda_1}|\lambda'|,\nu_{s,x}\rangle dxds\right)\\
&= C(a)\left(E_0-(d-\norm{f}_\infty)\int_0^tM(s)ds\right),
\end{aligned}
\end{equation*}
where
\begin{equation*}
C(a)=\max\left\{\frac{1}{3a},\frac{4}{3}\right\}.
\end{equation*}
Therefore, for almost every $t$, $M(t)$ is less than or equal to the solution of the integral equation
\begin{equation*}
\tilde{M}(t)^{4/3}=C(a)E_0-C(a)(d-\norm{f}_\infty)\int_0^t\tilde{M}(s)ds
\end{equation*}
or equivalently (after differentiating)
\begin{equation*}
\tilde{M}'(t)=-\frac{3}{4}C(a)(d-\norm{f}_\infty)\tilde{M}(t)^{2/3},\hspace{0.2cm}\tilde{M}(0)=(C(a)E_0)^{3/4}.
\end{equation*}
The solution of this ordinary differential equation is easily computed as
\begin{equation*}
M(t)=\begin{cases}\left[\frac{1}{3}(3(C(a)E_0)^{1/4}-\frac{3}{4}C(a)(d-\norm{f}_\infty)t)\right]^3 & \text{if $3(C(a)E_0)^{1/4}-\frac{3}{4}C(a)(d-\norm{f}_\infty)t\geq0$,}\\
0 & \text{otherwise.}
\end{cases}
\end{equation*}
In fact, $\left[\frac{1}{3}(3(C(a)E_0)^{1/4}-\frac{3}{4}C(a)(d-\norm{f}_\infty)t)\right]^3$ would also be a solution for all times, but we know a priori that $M(t)$ must be non-negative.
It follows that there is a time
\begin{equation*}
T\leq\frac{4}{d-\norm{f}_\infty}C(a)^{-3/4}E_0^{1/4}
\end{equation*}
after which $M(t)=0$ almost everywhere.
\end{proof}
\begin{corollary}
Let $(h,u)$ be an admissible weak solution of the Savage-Hutter equations with initial energy $E_0$ and $\norm{f}_\infty<d$. Then there exists a time $0\leq T<\infty$ such that for almost every $t>T$, $(h,u)$ is stationary, i.e.\ $u(t,x)=0$ for almost every $t>T$ and $x\in\T^2$, $\partial_th(t,x)=0$ for almost every $t>T$, $x\in\T^2$, and
\begin{equation*}
\left|\nabla h-\frac{f}{2a}\right|\leq\frac{d}{2a}.
\end{equation*}
Moreover, $T$ satisfies the estimate of Theorem~\ref{mvsfinitetime}.
\end{corollary}
\begin{proof}
As every admissible weak solution can be viewed as an admissible measure-valued solution via the identification $\nu=\delta_{(h,\sqrt{h}u)}$, $m=0$, from Theorem~\ref{mvsfinitetime} we obtain a time $T$ such that after this time, the momentum is zero:
\begin{equation*}
\int_{\T^2}h|u|dx=0\hspace{0.2cm}\text{for almost every $t>T$.}
\end{equation*}
Therefore, the Savage-Hutter equations reduce to $\partial_th=0$ and
\begin{equation*}
\nabla(ah^2)=-dhB(u)+hf.
\end{equation*}
The latter is clearly equivalent to $|\nabla h-f/2a|\leq d/2a$, given that $|B(u)|\leq1$.
\end{proof}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,369
|
Karolina Skog, Minister of Environment, Sweden, outlined an international alliance on chemicals and waste, launched at the HLPF, which aims to mobilize public interest on chemicals and waste beyond 2020.
Speakers underscored addressing chemicals a key enabler for achieving SDG 12.
Martin Kayser, BASF SE, described his company's development of a responsible care management system, including workshops to build capacity in safe transportation, storage and handling of chemicals.
17 July 2018: On Tuesday, 17 July 2018, on the sidelines of the High-level Political Forum on Sustainable Development (HLPF), the German Federal Ministry for the Environment, Nature Conservation and Nuclear Safety (BMU) organized a side event which provided a platform for participants to exchange perspectives on chemicals and waste, including challenges and opportunities for sound management, in the context of SDG 12 (responsible consumption and production).
During the event titled, 'Meeting the Challenge of Chemical Pollution beyond 2020: Working Together for Healthy People and a Healthy Planet,' Karolina Skog, Minister of Environment, Sweden, outlined an international alliance on chemicals and waste, launched at the HLPF, which aims to mobilize public interest on chemicals and waste beyond 2020. She underscored the need to reform the chemicals management system, including through, inter alia, trade and e-commerce regulations, and development of natural chemicals.
Asbestos kills more than 100,000 people each year, and up to 22 million people are at risk from lead-acid car battery recycling.
Speakers underscored addressing chemicals as a key enabler for achieving SDG 12, the benefits of addressing chemicals to realize the SDGs nationally and globally, and international cooperation to share experiences and mobilize support. They highlighted that asbestos kills more than 100,000 people each year, and that up to 22 million people are at risk from lead-acid car battery recycling.
Participants also stressed, inter alia, the need to: exchange information among countries; identify consumption patterns on chemical products in procurement during use and disposal; ensure the availability of information about safe or toxic chemicals for consumers; prevent the use of toxic chemicals in recycling products; and engage with the health industry.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 376
|
Biografia
Ramiro è conosciuto principalmente per alcuni film come Sólo un ángel nel ruolo di Eduardo Burone e Sin retorno nel ruolo di Carmona, ma anche per alcuni ruoli in televisione come quello di Diego Ayala nella soap Il segreto e El don de alba nel ruolo dell'uomo senza volto.
È noto in particolare per l'interpretazione del dottor Carlos nella serie TV spagnola Vis a vis.
Filmografia
Cinema
Héroes y demonios (1999)
Pasajero 10542 (2002)
Sólo un ángel (2005)
Nevar en Buenos Aires (2007)
Sin retorno (2010)
Bajo la rosa (2017)
Televisione
Los medicos (de hoy) (2000)
Franco Buenaventura, el profe (2002)
El precio del poder (2002)
Soy gitano (2003)
Resistiré (2003)
Culpable de este amor (2004)
Sálvame María (2005)
El comisario (2007)
Los hombres de Paco (2008)
Sin senos no hay paraíso (2008)
Hospital Central (2009)
Gavilanes (2010)
Fisica o chimica (Física o Química) (2010)
La fuga (2012)
El barco (2012)
Il segreto (El secreto de Puente Viejo) (2012)
El don de alba (2012-2013)
Vis a Vis (Vis a vis) (2015-2019)
Golpe al corazón (2017)
Vis a vis - L'Oasis (Vis a vis: El oasis) – serie TV, episodio 1x04-1x07 (2020)
El Internado: Las Cumbres (2021-in corso)
Collegamenti esterni
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,215
|
It's really hard to write a summary of a wedding that was so much more than just a wedding. For us it was an entire weekend, an entire month, of prepping and talking and travel and family! For my sister Amy and her now-husband John, family and time together were the most important element of their wedding day. If they were to be judged by the amount of family enjoying themselves on the dance floor that night, they would be rated a smashing success! That's not to say the entire day was without it's tangles ;) The weather took a crazy turn and their First Look happened in a car while it poured rain. The rings were almost missing, fingers were burned, flowers fell apart. Little things always happen. But the rain and wind stopped, the skies provided the most beautiful light and an even more amazing sunset, and everyone was together and happy for this couple we love so much. My sister is married and I am so excited to they trusted me to capture their day. Congrats guys!
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 2,614
|
Q: How Receiving WM_Notify in win32 python I want to get notification to my Form or Dialog handle that one of the control(Ex Button) got its state changed by performing some action(Ex Selecting an Item from combobox) in the same Form or Dialog
I have tried to implement on_notify, but for some reason Its not getting called on occurrence of any event change in the form or Dialog
I need to get on_notify method to be called on any kind of style or state changed in my Dialog or form and also find which control sent the notification to the Toplevel dialog
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 6,414
|
Dogwood is an unincorporated community in northwestern Douglas County, Missouri, United States. It is located on Missouri Route 14, approximately northwest of Ava and southwest of Seymour. The old store building sits at the intersection of Route 14 with route BB and the Dogwood cemetery is to the west at the intersection of routes 14 and Z.
History
In 1879, a Civil War veteran built a log store building and home near a spring surrounded by dogwoods along the route of the Springfield–Rockbridge portion of the old Salt Road along which salt, lumber and other materials had long been transported between northern Arkansas and Springfield. A post office was established at Dogwood in 1880 and remained in service until 1909. A school was established in 1888 and a log school building was built north of the store in 1891 and was replaced by a frame building in 1910. The school was north of the store.
In 1900, a missionary began holding church services in the school building. In 1905, a church building was erected and land was set aside for a cemetery. In 1932, due to widening and re-routing of Route 14, the church was moved to its current location north of the cemetery. In 1904, a second church was built north of the store and in 1931, a third church was built to the southeast on Route 14.
Geography
Dogwood is located on a ridge at approximate elevation of on the edge of the Springfield Plateau with the steep valleys of the Beaver Creek valley to the east and the Swan Creek valley to the southwest. The Dogwood fire tower and state hunting area on Tigris Peak is to the southeast.
References
Unincorporated communities in Douglas County, Missouri
1879 establishments in Missouri
Unincorporated communities in Missouri
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 80
|
Cheseaux-sur-Lausanne es una comuna suiza del cantón de Vaud, situada en el distrito de Lausana. Tiene una población estimada, a fines de 2020, de 4360 habitantes.
Limita al norte con las comunas de Boussens y Etagnières, al este con Morrens, al sureste con Lausana, al suroeste con Crissier y al oeste con Sullens.
La comuna formó parte hasta el 31 de diciembre de 2007 del círculo de Romanel.
Referencias
Enlaces externos
Sitio oficial de la comuna de Cheseaux-sur-Lausanne
Comunas de Vaud
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,097
|
By Stephen Baxter
_Published by Del Rey Books_
Manifold: Time
Manifold: Space
Manifold: Origin
A Del Rey® Book
Published by The Ballantine Publishing Group
Copyright © 2002 by Stephen Baxter
All rights reserved under International and Pan-American Copyright Conventions. Published in the United States by The Ballantine Publishing Group, a division of Random House, Inc., New York, and simultaneously in Canada by Random House of Canada by Random House of Canada Limited, Toronto.
Del Rey is a registered trademark and the Del Rey colophon is a trademark of Random
House, Inc.
www.delreydigital.com
Library of Congress Cataloging-in-Publication Data
Baxter, Stephen.
Manifold : origin / Stephen Baxter.— 1st ed.
p. cm.
I. Title.
PR 6052.A849 M34 2002
823′.914—dc21 2001052803
eISBN: 978-0-345-45547-5
v3.1_r1
_To my nephew, William Baxter_
# Contents
_Cover_
_Other Books by This Author_
_Title Page_
_Copyright_
_Dedication_
Part One: Wheel
Part Two: Red Moon
Part Three: Hominids
Part Four: World Engine
Part Five: Manifold
#
## _E mma Stoney_
Do you know me? Do you know where you are? Oh, Malenfant...
I know _you_. And you're just what you always were, an incorrigible space cadet. That's how we both finished up stranded here, isn't it? I remember how I loved to hear you talk when we were kids. When everybody else was snuggling at the drive-in, you used to lecture me on how space is a high frontier, a sky to be mined, a resource for humanity.
But is that all there is? Is the sky really nothing more than an empty stage for mankind to strut and squabble?
And what if we blew ourselves up before we ever got to the stars? Would the universe just evolve on, a huge piece of clockwork slowly running down, utterly devoid of life and mind?
How—desolating.
Surely it couldn't be like that. All those suns and worlds spinning through the void, the grand complexity of creation unwinding all the way out of the Big Bang itself... You always said you just couldn't believe that there was nobody out _there_ looking back at you down _here_.
But if so, where is everybody?
This is the Fermi Paradox—right, Malenfant? _If the aliens existed, they would be here_. I heard you lecture on that so often I could recite it in my sleep.
But I agree with you. It's powerful strange. I'm sure Fermi is telling us something very profound about the nature of the universe we live in. It is as if we are all embedded in a vast graph of possibilities, a graph with an axis marked _time_ , for our own future destiny, and an axis marked _space_ , for the possibilities of the universe.
Much of your life has been shaped by thinking about that cosmic graph. Your life and, as a consequence, mine.
Well, on every graph there is a unique point, the place where the axes cross. It's called the origin. Which is where we've finished up, isn't it, Malenfant? And now we _know_ why we were alone...
But, you know, one thing you never considered was the subtext. Alone or not alone—why do we _care_ so much?
I always knew why. We care because we are lonely.
I understood that because _I_ was lonely. I was lonely before you stranded me here, in this terrible place, this Red Moon. I lost you to the sky long ago. Now you found me here—but you're leaving me again, aren't you, Malenfant?
... Malenfant? Can you hear me? Do you know me? Do you know who you are?—Oh.
Watch the Earth, Malenfant. Watch the Earth...
## _M anekatopokanemahedo_
This is how it is, how it was, how it came to be.
It began in the afterglow of the Big Bang, that brief age when stars still burned.
Humans arose on an Earth. Emma, perhaps it was your Earth. Soon they were alone.
Humans spread over their world. They spread in waves across the universe, sprawling and brawling and breeding and dying and evolving. There were wars, there was love, there was life and death. Minds flowed together in great rivers of consciousness, or shattered in sparkling droplets. There was immortality to be had, of a sort, a continuity of identity through copying and confluence across billions upon billions of years.
Everywhere humans found life: crude replicators, of carbon or silicon or metal, churning meaninglessly in the dark.
Nowhere did they find mind—save what they brought with them or created—no _other_ against which human advancement could be tested.
They came to understand that they would forever be alone.
With time, the stars died like candles. But humans fed on bloated gravitational fat, and achieved a power undreamed of in earlier ages. It is impossible to understand what minds of that age were like, minds of times far downstream. They did not seek to acquire, to breed, or even to learn. They needed nothing. They had nothing in common with their ancestors of the afterglow.
Nothing but the will to survive. And even that was to be denied them by time.
The universe aged: indifferent, harsh, hostile and ultimately lethal.
There was despair and loneliness.
There was an age of war, an obliteration of trillion-year memories, a bonfire of identity. There was an age of suicide, as even the finest chose self-destruction against further purposeless time and struggle.
The great rivers of mind guttered and dried.
But some persisted: just a tributary, the stubborn, still unwilling to yield to the darkness, to accept the increasing confines of a universe growing inexorably old.
And, at last, they realized that something was wrong. _It wasn't supposed to have been like this_.
Burning the last of the universe's resources, the final downstreamers—lonely, dogged, all but insane—reached to the deepest past...
# _PART ONE_
# **Wheel**
#
## _R eid Malenfant_
"... Watch the Moon, Malenfant. Watch the Moon!"
So here was Reid Malenfant, his life down the toilet, chasing joky UFO reports around a desolate African sky. Emma's voice snapped him to full alertness, for just about the first time, he admitted to himself, since takeoff.
"What about the Moon?"
"Just look at it!"
Malenfant twisted his head this way and that, the helmet making his skull heavy, seeking the Moon. He was in the T-38's forward blister. Emma was in the bubble behind him, her head craned back. The jet trainer was little more than a brilliant shell around them, white as an angel's wing, suspended in a powder-blue sky. Where was the Moon—the west? He couldn't see a damn thing.
Frustrated, he threw the T-38 into a savage snap roll. A flat brown horizon twisted around the cockpit in less than a second.
"Jesus, Malenfant," Emma groaned.
He pulled out into a shallow climb toward the west, so that the low morning sun was behind him.
... And then he saw it: a Moon, nearly full, baleful and big— _too_ big, bigger than it had any right to be. Its colors were masked by the washed-out blue of the air of Earth, but still, it had _colors_ , yes, not the Moon's rightful palette of grays, but smatterings of a deep blue-black, a murky brown that even had tinges of green, for God's sake—but it was predominantly red, a strong scorched red like the dead heart of Australia seen from the flight deck of a Shuttle orbiter...
It was a Moon, but not _the_ Moon. A new Moon. A Red Moon.
He just stared, still pulling the T-38 through its climb. He sensed Emma, behind him, silent. What was there to say about this, the replacement of a Moon?
That was when he lost control.
## _F ire_
The people walk across the grass.
The sky is blue. The grass is sparse, yellow. The ground is red under the grass. Fire's toes are red with the dust. The people are slim black forms scattered on red-green.
They are called the Running-folk.
The people call to each other.
"Fire? Dig! Fire?"
"Dig, Dig, here! Loud, Loud?"
Loud's voice, from far away. "Fire, Fire! Dig! Loud!"
The sun is high. There are only people on the grass. The cats sleep when the sun is high. The hyenas sleep. The Nutcracker men and the Elf men sleep in their trees. Everybody sleeps except the Running-folk. Fire knows this without thinking.
As his legs walk, Fire holds his hands clamped together. Smoke curls up from between his thumbs. There is moss inside his hands. The fire is in the moss. He blows on the moss. More smoke comes. The fire hurts his palms and fingers. But his hands are hard.
His legs walk easily. Walking is for legs. Fire is not there in his legs. Fire is in his hands and his eyes. He makes his hands tend the fire, while his legs walk.
Fire is carrying the fire. That is his name. That is what he does.
It is darker. The people are quiet.
Fire looks up. A fat cloud hangs over him. The sun is behind the cloud. The edge of the cloud glows golden. His nose can smell rain. His bare skin prickles, cold. Immersed in this new moment, he has forgotten he is hungry.
The clouds part. There is a blue light, low in the sky. Fire looks at the blue light. It is not the sun. The blue light is new.
Fire fears anything new.
The fire wriggles in his hands.
He looks down, forgetting the blue light. There is no smoke. The moss has turned to ash. The fire is shrinking.
Fire crouches down. He shelters the moss under his belly. He feels its warmth on his bare skin. He hoots. "Fire, Fire! Fire, Fire!"
Stone is small-far. He turns. He shouts. He is angry. He begins to come back toward Fire.
Loud comes to Fire. Loud hoots. His voice is loud. Loud is his name. Loud kneels. He looks for bits of moss and dry grass. He pushes them into the bit of fire.
Dig comes to Fire. Her hand holds arrowhead roots. She squats beside Fire. Her taut dugs brush his arm. His member stiffens. He rocks. She grins. Her hands push a root into his mouth. He tastes her fingers, her salty sweat.
Loud hoots. His member is stiff too, sticking out under his belly. He crams bits of grass into Fire's hands.
Fire snaps his teeth. "Loud, Loud away!"
Loud hoots again. He grabs Dig's arm. She laughs. Her legs take her skipping away from both of them.
Others come to Fire. Here are women, Grass and Shoot and Cold and Wood. Here are their babies with no names. Here are children with no names. The children jabber. Their eyes are round and bright.
Here is Stone. Stone is dragging branches over the ground. Blue is helping Stone drag the branches. Sing is lying on the branches. Sing is white-haired. She is still. She is asleep.
Stone sees the dying fire. He sees Fire's stiff member. He roars. Stone's hands drop the branches.
Stone has forgotten Sing, on the branches. Sing tips to the ground. She groans.
Stone's axe clouts Fire on the back of the head. There is a hard sound. Stone shouts in Fire's face. "Fire, Fire! Hungry, feed!" His face is split by a scar. The scar is livid red.
"Fire, Fire," says Fire quietly. His arms drop and his head bows. He keeps hold of the fire.
Sing moans. Her eyes are closed. Her dugs are slack. The men pick her up by shoulders and legs and lift her back on the branches.
Stone and Blue grab the branches. Their legs walk them back the way they had come.
Fire tells his legs to stand him up. They can't. His hands are still clasped around the fire. Lights fill his head, more garish than that blue stripe in the sky. He nearly falls over backward.
Loud's hand grabs his armpit. Loud lifts him until his legs are straight.
Loud laughs. Loud walks away, fast, after Dig.
Fire's head hurts. Fire's hands hurt. Fire's member wants Dig.
He starts walking. He wants to stop thinking.
He thinks of the blue light.
## _E mma Stoney_
Emma had accompanied Malenfant, her husband, on a goodwill tour of schools and educational establishments in Johannesburg, South Africa. It had been a remarkably dismal project, a throwback to NASA PR malpractices of old, a trek through mostly prosperous, middle-class-and-up neighborhoods, with Malenfant running Barco shows from his two missions to the Space Station before rows of polite and largely uncaring teenagers.
In darkened classrooms Emma had watched the brilliance of the students' smiles, and the ruby-red winking of their earpiece phones like fireflies in the night. Between these children growing up in the fractured, complex, transformed world of 2015, and Reid Malenfant, struggling worker astronaut, all of fifty-five years old and still pursuing Apollo dreams from a boyhood long lost, there was a chasm as wide as the Rift Valley, she thought, and there always would be.
Still, for Emma, it had been a holiday in the African sun—the reason she had pried herself away from her work as financial controller of OnlineArt—and she and Malenfant had gotten along reasonably well, for them, even given Malenfant's usual Earthbound restless moodiness.
But that had been before the word had come through from the Johnson Space Center, headquarters of NASA's manned spaceflight program, that Malenfant had been washed out of his next mission, STS-194.
Well, that was the end of it. With a couple of phone calls Malenfant had cut short their stay in Joburg, and had begun to can the rest of the tour. He had been able to get out of all of it except for a reception at the US ambassador's residence in Nairobi, Kenya.
To her further dismay, Malenfant had leaned on Bill London—an old classmate from Annapolis, now a good buddy in the South African Navy—to let him fly them both up to Nairobi from out of a Joburg military airfield in a T-38, a sleek veteran supersonic jet trainer, a mode of transport favored by the astronauts since the 1960s.
It wasn't the first time Emma had been taken for a ride in one of those toy planes, and with Malenfant in this mood she knew she could expect to be thrown around the sky. And she shuddered at the thought of how Malenfant in this wounded state was going to behave when he got to Nairobi.
But she had gone along anyhow. Somehow she always did.
So that was how Emma Stoney, forty-five-year-old accountant, had found herself in a gear room getting dressed in a blue flight suit, oxygen mask, oversized boots, helmet, going through the procedures for using her parachute and survival kit and emergency oxygen, struggling to remember the purpose of the dozens of straps, lanyards and D-rings.
Malenfant was ready before she was, of course. He stomped out into the bright morning sunlight toward the waiting T-38. He carried his helmet and his flight plan, and his bald head gleamed in the sun, bronzed and smooth as a piece of machinery itself. But his every motion was redolent with anger and frustration.
Emma had to run to keep up with him, laden down with all her absurd right-stuff gear. By the time she reached the plane she was hot already. She had to be hoisted into her seat by two friendly South African female techs, like an old lady being lifted into the bath. Malenfant was in his cockpit, angrily going through a pre-takeoff checkout.
The T-38 was sleek and brilliant white. Its wings were stubby, and it had two bubble cockpits, one behind the other. The plane was disturbingly small; it seemed barely wide enough to squeeze in a whole person. Emma studied an array of controls and dials and softscreen readouts at whose purpose she could only guess. The venerable T-38 had been upgraded over the years—those shimmering softscreen readouts, for instance—but every surface was scuffed and worn with use, the metal polished smooth where pilots' gloved hands had rubbed against it, the leather of her seat extensively patched.
The last few minutes of the prep wore away quickly, as one of the ground crew took her through her final instructions: how she should close her canopy bubble, where to fasten a hook to a ring on a parachute, how to change the timing of her parachute opening. She watched the back of Malenfant's head, his jerky tension as he prepared his plane.
Malenfant taxied the jet to the end of the runway. Emma watched the stick move before her, slave to Malenfant's movements. Her oxygen mask smelled of hot rubber, and the roar of the jets was too loud for her to make out anything of Malenfant's conversation with the ground.
Do you _ever_ think of me, Malenfant?
There was a mighty shove at her back.
## _F ire_
Stone drops the branches. Sing rolls to the ground. Stone has forgotten her again.
The sun is low. They are close to a thick stand of trees. Fire can smell water.
Fire is tired. His stomach is empty. His hands are sore. "Hungry Fire hungry," he moans.
Sing, on the ground, looks up at him. She smiles. "Hungry Fire," she says. He thinks of her feeding him. But she is small and withered. She does not get up to feed him.
Stone walks over the branches he hauled across the savannah, the branches that transported Sing. He kicks them aside. He has forgotten he hauled them here. He bends. His hands seek out a piece of dung on the ground. His tongue tastes it. It is Nutcracker-man dung. The dung is old. The dung crumbles.
Fire is not fearful. There are no Nutcracker men near here.
Stone's feet kick aside more branches and twigs. He uncovers a round patch of black ground. Fire's nose smells ash. Stone hoots. "Hah! Fire Fire."
Fire crouches over the ash. The fire is warm in his hands.
Loud and Dig and others huddle near him. Their hands scrape dry stuff from the floor, dead leaves and dry moss and grass and bits of bark. Their hands pick up rocks, and rub the tinder against the rocks. Their fingers turn the tinder, making it fine and light.
Wood's legs walk to the forest. She comes back with a bundle of sticks, of wood. That is what she does. That is her name. She piles the sticks on the ground.
The hands of the others push the tinder into the middle of the pile of wood.
Working closely, the people jostle each other. They are hot from the walk. Their bare skin is slick with sweat. They grunt and yap, expressing tiredness, hunger, irritation. But they do not speak of the work. They are not thinking as their hands gather the fire materials. Their hands have done this all their lives. Their ancestors' hands have done this for hundreds of thousands of years.
Fire waits while they work.
He sees himself.
He is a child with no name. Another cups fire in his hands. He cannot see this other's face. The adults' huge hands make tinder. Fire is fascinated. They push him out of the way.
A woman picks him up. It is Sing. Her arms are strong. Her mouth smiles. She swings him in the air. The leaves are green and big.
... The leaves are small. The leaves are yellow. Sing is lying on the ground.
Fire's hands push into the tinder. He makes his hands put his precious bit of fire inside the tinder. His mouth blows on the fire. His hands want to come out of the prickling heat. He makes them stay in the tinder. Flame flickers. The wood smokes and pops, scorches and burns.
People laugh and hoot at the fire.
Fire pulls out his hands. His hands are sore.
## _E mma Stoney_
The plane shot almost vertically into the air, and its white nose plunged through a layer of fine, gauzy cloud. The ground imploded below her, the rectilinear patterns of the airfield shrinking into insignificance as the glittering carcass of Joburg itself shouldered over the horizon, agricultural land beyond showing as patches of grayish green and brown. On the eastern horizon the sun was unimaginably bright, sending shafts of light spearing through the cockpit glass, and to the west she spotted the Moon rising, almost full, its small gray face peering back at the sun's harsh glare.
Already the sky above was turning a deeper blue, shading to purple.
Emma felt her stomach lurch, but she knew it would pass. One of the many ironies of their relationship was that Emma was more resistant to motion sickness than her astronaut husband, who had spent around ten percent of the time on his two spaceflights throwing up.
Malenfant banked to the north, and the horizon settled down, sun to right, Moon to left. As they headed toward the interior of the continent, the land turned brown, parched, flat.
"What a shithole," Malenfant said, his voice a whisper over the jet's roar. "Africa. Cradle of mankind my ass."
"Malenfant—"
He hurled the T-38 forward with a powerful afterburner surge.
Within seconds they had reached 45,000 feet and had gone through a bone-shaking Mach 1. The vibrations damped away and the noise of the jets dwindled—for, of course, they were outstripping most of the sound they made—and the plane seemed to hang in shining stillness.
Emma, as she had before, felt a surge of exhilaration. It was at such paradoxical moments of stillness and speed that she felt closest to Malenfant.
But Malenfant was consumed by his gripes.
"Two years. I can't fucking believe it. Two years of training, two years of meetings and planning sessions, and paddling around in hydro labs and spinning around in centrifuges. All of it for nothing."
"Come on, Malenfant. It's not the end of the world. It's not as if Station work was ever such a prize anyhow. _Looking at stars, pissing in jars_. That's what you used to say—"
"Nobody was flying to fucking Mars. Station was all that was available, so I took it. Two flights, two lousy flights. I never even got to command a mission, for Christ's sake."
"You got washed out this time. That doesn't mean you won't fly again. A lot of crew are flying past your age." That was true, of course, partly because NASA was having such difficulty finding willing applicants from younger generations.
But Malenfant growled, "It's that asshole Bridges. He even called me into the JSC director's office to _explain_ the shafting. That fucking horse holder has always had it in for me. This will be the excuse he needs to send me to purgatory."
Emma knew who he meant. Joe Bridges was the director of flight operations—in effect, in NASA's Byzantine, smothering internal bureaucracy, in charge of astronaut selection for missions.
Malenfant was still muttering. "You know what Bridges offered me? ASP."
Emma riffled through her mental file of NASA acronyms. ASP: Astronaut Support Personnel, a nonflying astronaut assigned to support the crew of a mission.
"I'd have been point man on STS-194." Malenfant spat. "The Caped Crusader. Checking the soap dispensers in the orbiter john. Strapping some other asshole into _my_ seat on the flight deck."
"I gather you didn't take the job," Emma said dryly.
"I took it okay," he snapped. "I took it and shoved it sideways up that pencil-pusher's fat ass."
"Oh, Malenfant." She sighed.
She tried to imagine the meeting in that rather grand office, before a floor-to-ceiling office window with its view of the parklike JSC campus, complete with the giant Saturn V Moon rocket lying there on its side as if it had crash-landed beside the driveway. Even in these days of decline, there were too few seats for too many eager crewpersons, so—in what seemed to Emma his own very small world—Bridges wielded a great deal of power indeed.
She had never met this man, this Bridges. He might be an efficient bureaucrat, the kind of functionary the aviator types would sneer at, but who held together any major organization like NASA. Or perhaps this Bridges transcended his role; perhaps he was the type who had leveraged his position to accrete power beyond his rank. With the gifts at his disposal, she thought, he might have built up a network of debtors in the Astronaut Office and beyond, in all the places in NASA's sprawling empire ex-astronauts might reach.
Well, so what? Emma had encountered any number of such people in her own long, complex and moderately successful career in the financial departments of high-tech corporations. No organization was a rational place. Organizations were bear pits where people fought for their own projects, which might or might not have something to do with the organization's supposed mission. The wise person accepted that, and found a way to get what she wanted in spite of it all.
But to Malenfant—Malenfant the astronaut, an odd idealist about human behavior, always a loner, always impatient with the most minimal bureaucracy, barely engaged with the complexities of the world—to Malenfant, Joe Bridges, controlling the most important thing in his entire life (more important than me, she thought) could be nothing but a monster.
She stared out the window at the baked African plain. It was huge and ancient, she thought, a place that would endure all but unchanged long after the little white moth that buzzed over it today was corroded to dust, long after the participants in their tiny domestic drama were moldering bones.
Now she heard a whisper from the ground-to-air radio. It sounded like Bill London, good old bullshitter Bill from Annapolis, with some garbled report about UFOs over central Africa.
The plane veered to the right, and the rising sun wheeled around the cockpit, sparking from scuffs in the Plexiglas around her.
"Let's go UFO hunting," Malenfant snapped. "We got nothing better to do today, right?"
She wasn't about to argue; as so often in her relationship with Malenfant, she was, literally, powerless.
## _F ire_
Stone and Blue put branches into the fire. Leaves and twigs burn. Stone and Blue pull out the burning branches. Their legs carry them into the wood. Small animals squeal and run before the fire. Stone and Blue pursue, their eyes darting, their hands hurling rocks and bits of wood.
Fire's hands are very red and raw.
Dig comes to him. Water is in her mouth. The water spills on his hands. The water is cool. Dig has leaves. Her hands rub them on his burns.
Fire has no name. Sing is huge and smiling. Sing's hands rub his palms with leaves.
Fire has his name again. It is Dig who tends his burned hands, smiling.
"Blue light!" he shouts, suddenly.
Dig looks at him. Her eyes narrow. She tends his hands.
Fire's hand reaches out. It cups one conical breast. The breast is hot in his hand.
The fire is hot in his hand. A captured bat is hot in his hand.
His member does not rise. Dig tends his hands.
Blue and Stone return. Their hands carry rabbits. The rabbits are skinned. There is blood on the mouths of the men. The rabbits fall to the ground.
The children with no names fall on the rabbits. They jabber, snapping at each other. The children's small faces are bloody. The adults push the children aside, and growl and jostle over the rabbits. All the people work at the meat, stealing it from each other.
Grass and Cold throw some pieces of meat on the fire. The meat sizzles. Their hands pick out the meat. Their mouths chew the burned meat, swallowing some. Fire sees that their mouths want to swallow all the meat. But their fingers take meat from their mouths. They put the meat in the mouths of their babies with no names.
Sing groans. She is on the ground near the branches. Her nose can smell the food. Her hands can't reach it.
Fire is eating a twisted-off rabbit leg. His hands pluck meat off it, and put the meat in Sing's mouth.
Her head turns. Her mouth chews. Her eyes are closed. She chokes. Her mouth spits out meat.
Fire's hands pop the chewed meat in his mouth.
Sing is shivering.
Fire thinks of a bower.
There are branches here, on the ground. He has forgotten that they were used to transport Sing. He keeps thinking of the bower.
He makes his hands lay the branches on the ground. He thinks of twigs and grass and leaves. He gathers them, thinking of the bower. He makes his hands pile everything up on the branches.
He makes his arms pick up Sing.
It is sunny. He has no name. Sing is carrying Fire. Sing is large, Fire small.
It is dark. His name is Fire. Fire is carrying Sing. Fire is large, Sing shrunken.
He lays her on the crude bower. She sinks into the soft leaves and grass. The branches roll away. The grass scatters. Sing falls into the dirt, with a gasp.
Fire hoots and howls, kicking at the branches.
One of the branches is lodged against a rock. It did not roll away.
Fire makes his hands gather the branches again. He puts the branches down alongside the rock he found. His hands pile up more grass. At last he lowers Sing on the bower. The branches are trapped by the rocks. They do not roll away.
Sing sighs.
Every day he makes a bower for Sing. Every day he forgets how he did it before. Every day he has to invent a way to fix it, from scratch. Some days he doesn't manage it at all, and Sing has to sleep on the dirt, where insects bite her.
She sings. Her voice is soft and broken. Fire listens. He has forgotten the rocks and the branches.
She stops singing. She sleeps.
People are sleeping. People are huddled around the children. People are coupling. People are making water. People are making dung. People are chattering, for comfort, through rivalry.
Beyond the glow of the flames, the sky is dark. The land is gone. Something howls. It is far away.
Dig is sleeping near the fire.
Fire's legs walk to her. His hand touches her shoulder. She rolls on her back. She opens her eyes and looks at him.
His member is stiff.
"Hoo! Fire!"
It is Loud. He is on the ground. Fire's eyes had not seen him. Fire's eyes had seen only Dig.
Loud's hands throw red dirt into Fire's eyes. Fire blinks and sneezes and hoots.
Loud has crawled to Dig. His hands paw at her. His tongue is out, his member hard. Her hands are pushing him away. She is laughing.
Fire's hands grab Loud's shoulders. Loud falls off Dig and lands on his back. He pulls Fire to the ground and they roll. Fire feels hot gritty dirt cling to his back.
Stone roars. His scar shines in the fire light. His filth-grimed foot separates them with a shove. His axe clouts Loud on the head. Loud howls and scuttles away.
Stone's axe swings for Fire. Fire ducks and scrambles back.
Stone grunts. He moves to Dig. Stone's big hand reaches down to her, and flips her onto her belly.
Dig gasps. She pulls her legs beneath her. Fire hears the scrape of her skin on red dust.
Stone kneels. His hands push her legs apart. She cries out. He reaches forward. His hands cup her breasts. His member enters her. His hands clutch her shoulders, and his flabby hips thrust and thrust.
He gives a strangled cry. His back straightens. He shudders.
He pulls back and stands up. His member is bruised purple and moist. He turns. He kicks Fire in the thigh. Fire yells and doubles over.
Dig is on the ground, her hands tucked between her legs. She is curled up.
Loud is gone.
Fire's legs walk.
Fire stops.
Dig is far. The fire is far. He is in a mouth of darkness.
Eyes watch him.
He makes his legs walk him back to the fire.
Sing is lying on a bower. He has forgotten he made the bower. Her eyes watch him. Her arm lifts.
He kneels. His face rests on her chest. The bower rustles. Sing gasps.
Her hand runs over his belly. Her hand finds his member. It is painfully swollen. Her hand closes around it. He shudders.
She sings.
He sleeps.
## _E mma Stoney_
If this really was the close of Malenfant's career at NASA, Emma thought, it could be a good thing.
She wasn't the type of foolish ground-bound spouse who palpitated every moment Malenfant was in orbit (although she hadn't been able to calm her stomach during those searing moments of launch, as the Shuttle passed through one of NASA's "nonsurvivable windows" after another...). No, the sacrifices she had made went broader and deeper than that.
It had started as far back as the moment when, as a new arrival at the Naval Academy, he had broken his hometown girl's seventeen-year-old heart with a letter saying that he thought they should break off their relationship. Now he was at Annapolis, he had written, he wanted to devote himself "like a monk" to his studies. Well, that had lasted all of six months before he had started to pursue her again, with letters and calls, trying to win her back.
That letter had, in retrospect, set the course of their lives for three decades. But maybe that course was now coming to an end.
"You know," she said dreamily, "maybe if it is ending, it's fitting it should be like this. In the air, I mean. Do you remember that flight to San Francisco? You had just got accepted by the Astronaut Office..."
It had been Malenfant's third time trying to join the astronaut corps, after he had applied to the recruitment rounds of 1988—when he wasn't even granted an interview—and 1990. Finally in 1992, aged thirty-two, he had gotten an interview at the Johnson Space Center in Houston, and had gone back to his base in San Diego.
At last the Astronaut Office had called him. But he was sworn to secrecy until the official announcement, to be made the next day. Naturally he had kept the secret strictly, even from Emma.
So the next day they had boarded a plane for San Francisco, where they were going to spend a long weekend with friends of Emma's (Malenfant tended not to have the type of friends you could spend weekends with, not if you wanted to come home with your liver). Malenfant had given the pilot the NASA press release. Just after they got to cruise altitude, the pilot called Emma's name: _Would Emma Malenfant please identify herself? Would you please stand up?_
It had taken Emma a moment to realize she was being called, for she used her maiden name, Stoney, in business and her personal life, everywhere except the closed world of the Navy. Baffled—and wary of Malenfant's expressionless stillness—she had unbuckled her seat belt and stood up.
_I hope you like barbecue, Mrs. Malenfant_ , said the pilot, _because I have a press release here that says you are going to Houston, Texas. Commander ReidMalenfant, US Navy, has been selected to be a part of the 1992 NASA astronaut class_.
"... And everybody on the plane started whooping, just as if you were John Glenn himself, and the stewards brought us those dumb little plastic bottles of champagne. Do you remember, Malenfant?" She laughed. "But you couldn't drink because you were doubled over with airsickness."
Malenfant grunted sourly. "It starts in the air, so it finishes in the air. Is that what you think?"
"It does have a certain symmetry... Maybe this isn't the end, but the beginning of something new. Right? We could be at the start of a great new adventure together. Who knows?"
She could see how the set of his shoulders was unchanged.
She sighed. Give it time, Emma. "All right, Malenfant. What UFOs?"
"Tanzania. Some kind of sighting over the Olduvai Gorge, according to Bill."
"Olduvai? Where the human fossils come from?"
"I don't know. What does that matter? It sounds more authentic than most. The local air forces are scrambling spotter planes: Tanzania, Zambia, Kenya, Mozambique."
None of those names were too reassuring to Emma. "Malenfant, are you sure we should get caught up in that? We don't want some trigger-happy Tanzanian flyboy to mistake us for Eetie."
He barked laughter. "Come on, Emma. You're showing your prejudice. We trained half those guys and sold the planes to the other half. And they're only spotters. Bill is informing them we're coming. There's no threat. And, who knows? Maybe we'll get to be involved in first contact."
Under his veneer of cynicism she sensed an edge of genuine excitement. From out of the blue, here was another adventure for Reid Malenfant, hero astronaut. Another adventure that had nothing to do with her.
I was wrong, she thought. I'm never going to get him back, no matter what happens at NASA. But then I never had him anyhow.
Losing sympathy for him, she snapped, "You really told Joe Bridges to shove his job?"
"Sweetest moment of my life."
"Oh, Malenfant. Don't you know how it works yet? If you took your punishment, if you sweated out your time, you'd be back in rotation for the next assignment, or the one after that."
"Bullshit."
"It's the way of the world. I've had to go through it, in my own way. Everybody has. Everybody who wants to get on in the real world, with real people, anyhow. Everybody but you, the great hero."
"You sound like you're writing my appraisal," he said, a little ruefully. "Anyhow, ass-kissing wouldn't have helped. It was the Russians, that fucking Grand Medical Commission of theirs."
"The _Russians_ scrubbed you?"
"It was when I was in Star City."
Star City, the Russian military base thirty miles outside Moscow that served as the cosmonauts' training center.
"Malenfant, you got back from there a month ago. You never thought to tell me about it?"
Through two layers of Plexiglas, she could see him shrug. "I was appealing the decision. I didn't see the point of troubling you. Hell, Emma, I thought I would win. I _knew_ I would. I thought they couldn't scrub me."
Far off, to left and right, she saw contrails and glittering darts. Fighter planes, perhaps, converging on the strange anomaly sighted over Olduvai, whatever it was, if it existed at all.
She felt an odd frisson of anticipation.
"It took them a morning," Malenfant said. "They brought in a dozen Russian doctors to probe at my every damn orifice. A bunch of snowy-haired old farts with pubic hair growing out of their noses, with _no_ experience of space medicine. They ought to have no jurisdiction over the way we run our program."
"It's their program too," she said quietly. "What did they say?"
"One of them pulled me up over my shoulder." Malenfant suffered from a nerve palsy behind his right shoulder, the relic of an ancient football injury, a condition NASA had long ago signed off on. "Well, our guys gave them shit. But the fossil stood his ground.
"Then they took me into the Commission itself. I was sat on a stage with the guy who was going to be my judge, in front of an auditorium full of white-haired Russian doctors, and two NASA guys who were as mad as hell, like me. But the old asshole from the surgical group got up and said my shoulder was a 'disqualifying condition' that needed further tests, and our guys said I wasn't going to do that, and so the Russians said I was disqualified anyhow..."
Emma frowned, trying to puzzle it out. It sounded like a pretext to her; Malenfant had flown twice to the Station before after all, and the Russians must have known all about his shoulder, along with everything else about him. Why should it suddenly become a mission-threatening disability now?
Malenfant put the little jet through a gut-wrenching turn so tight she thought she heard the hull creak. "I knew we'd appeal," he said. "Those two NASA surgeons were livid, I'm telling you. They said they'd pass it all the way up the line, I should just get on with my training as if I was planning to fly, they'd clear me through. Hell, I believed them. But it didn't happen. When it got to Bridges—"
"Was your shoulder the only thing the Russians objected to?"
He hesitated.
"Malenfant?"
"No," he said reluctantly. "They smuggled shrinks' remarks into their final report to NASA. They should have presented them at the Commission... Hey, can you see something? Look, right on the horizon."
She peered into the north. The horizon was a band of dusty, mistladen air, gray between brown earth and blue sky, precisely curving. Was something there?—a spark of powder blue, a hint of a circle, like a lens flare?
But the day was bright, dazzling now the sun was climbing higher, and her eyes filled with water.
She sat back in her seat, and her various harnesses and buckles rustled and clinked around her, loud in the tiny cockpit. "What did it say, Malenfant? The Russian psych report."
He growled, " 'Peculiarities.' "
"What kind of peculiarities?"
"In my relations with the rest of the crew. They gave an example about how I was in the middle of some task and some Russkie came over nagging about how we were scheduled to do something else. Well, I nodded politely, and carried right on with what I was doing, until I was done..."
Now she started to understand. The Russians, who rightly believed they were still far ahead of the west in the psychology of the peculiarly cramped conditions of space travel, placed great collectivist emphasis on teamwork and sacrifice. They would not warm to a driven, somewhat obsessive loner-perfectionist like Malenfant.
"I should have socialized with the assholes," he said now. "I should have gone to the cosmonauts' cold-water apartments, and drunk their crummy vodka, and pressed the flesh with the guys on the gate."
She laughed gently. "Malenfant, you don't even socialize at NASA."
"My nature got me where I am now."
Yeah, washed out, she thought brutally. "But maybe it's not the nature you need for long-duration space missions. I guess not everybody forgives you the way I do."
"What is that supposed to mean?"
She ignored the question. "So the psych report is the real reason they grounded you. The shoulder was just an excuse."
"The Russians must have known the psych report would never stand up to scrutiny. If Joe Bridges had got his thumb out of his ass—"
"Oh, Malenfant, don't you see? They were giving you cover. If you're going to be grounded, do you want it to be because of your shoulder, or your personality? Think about it. They were trying to help you. They all were."
"That kind of help I can live without." Again he wrenched the plane through a savage snap roll.
Her helmet clattered against the Plexiglas, as varying acceleration tore at her stomach, and the brown African plain strobed around her. She was cocooned in the physical expression of his anger.
She glared at the back of Malenfant's helmeted head, which cast dazzling highlights from the African sun, with a mixture of fondness and exasperation. Well, that was Malenfant for you.
And because she was staring so hard at Malenfant she missed seeing the artifact until it was almost upon them.
Malenfant peeled away suddenly. Once again she glimpsed pale blue-white sky, dusty brown ground, shafts of glowering sunlight—and an arc, a fragment of a perfect circle, like a rainbow, but glowing a clear cerulean blue. Then it fell out of her vision.
"Malenfant—what was that?"
"Damned if I know." His voice was flat. Suddenly he was concentrating on his flying. The slaved controls in front of her jerked this way and that; she felt remote buffeting, some kind of turbulence perhaps, smoothed out by Malenfant's skillful handling.
He pulled the jet through another smooth curve, and sky and ground swam around her once more.
And he said, "Holy shit."
There was a circle in the sky.
It was facing them full on. It was a wheel of powder blue, like a hoop of the finest ribbon. It looked the size of a dinner plate held before her face—but of course it must be much larger and more remote than that.
Emma saw this beyond Malenfant's head and shoulders and the slim white fuselage. The jet's needle nose pointed straight at the center of the ring, so that the wheel framed her field of view with perfect symmetry, like some unlikely optical flare. Its very perfection and symmetry made it seem unreal. She had no idea of its scale—it would seem so close it must be hanging off the plane's nose, then something in her head would flip the other way and it would appear vast and distant, like a rainbow. She found it physically difficult to study it, as if it were an optical illusion, deliberately baffling; her eyes kept sliding away from it, evading it.
It's beyond my comprehension, she thought. Literally. Evolution has not prepared me for giant wheels suspended in the air.
## _F ire_
Water runs down his face.
He is lying on his back. The sky is flat and gray.
Rain falls. His ears hear it tapping on the ground. His eyes see the drops fall toward his face. They are fat and slow. Some of them fall on his face.
Water runs in his eyes. It stings. He sits up.
Fire is sitting on the ground. He is wet. His eyes hurt. His burned hands hurt.
He stands up. His legs walk him toward the trees.
People walk, run, stumble over muddy ground, adults and children. They move in silence, in isolation. Nobody is calling, nobody helping. They are cold and they hurt. They have each forgotten the other people, all save the mothers with their babies with no names. The mothers' arms carry the infants, sheltering them.
Fire reaches the trees.
The wind changes. His nose smells ash.
He remembers the fire. His legs run back.
The fire is out, drowned by the rain. The back of Fire's head hurts in anticipation of Stone's punishing axe.
Sing is calling. She is lying on a bower. The bower is falling apart, the leaves damp and shrivelled.
Loud is walking back to Sing.
Sing screams. Fire spins and crouches.
There is a Mouth. It is bright blue. The Mouth is skimming over the shining grass. The Mouth is approaching Fire, gaping wide.
Cats have mouths. A cat's mouth will take a person's head. This Mouth would take a whole person, standing straight. It is coming toward him, this Mouth with no body, this huge Mouth, widening.
It makes no noise. The rain hisses on the grass.
Fire screams. Fire's legs carry him off into the forest.
Still the Mouth comes. It towers into the sky.
Sing is at its base. Her arms push at the bower. Her legs can't stand up. She screams again.
Loud runs. His hands are throwing dirt at the Mouth.
The Mouth scoops him up.
There is a flash of light. Fire can see nothing but blue.
Loud screams.
## _E mma Stoney_
"Malenfant—you see it, too, right?"
He laughed. "It ain't no scratch in your contacts, Emma." He seemed to be testing the controls. Experimentally he veered away to the right. The ride got a lot more rocky.
The blue circle stayed right where it was, hanging in the African sky. No optical effect, then. This was _real_ , as real as this plane. But it hung in the air without any apparent means of support. And still she had no real sense of its scale.
But now she saw a contrail scraped across the air before the wheel, a tiny silver moth flying across its diameter. The moth was a plane, at least as big as their own.
"Damn thing must be a half-mile across," Malenfant growled. "A half-mile across, and hovering in the air eight miles high—"
"How appropriate."
"My God, it's the real thing," Malenfant said. "The UFO-nauts must be going crazy." She heard the grin in his voice. "Everything will be different now."
Now she made out more planes drawn up from the dusty ground below, passing before the artifact—if artifact it was. One of them looked like a fragile private jet, a Lear maybe, surely climbing well beyond its approved altitude.
Malenfant continued his turn. The artifact slid out of sight.
Dusty land wheeled beneath her. She was high above a gorge, cut deeply into a baked plain, perhaps thirty or forty miles long. Perhaps it was Olduvai itself, the miraculous gorge that cut through million-year strata of human history, the gorge that had yielded the relics of one ancient hominid form after another to the archaeologists' patient inspection.
How strange, she thought. Why here? If this wheel in the sky really is what it appears to be, an extraordinary alien artifact, if this is a first contact of a bewilderingly unexpected type (and what else could it be?) then _why here_ , high above the cradle of mankind itself? Why should this gouge into humanity's deepest past collide with this most unimaginable of futures?
The plane dropped abruptly. For a heartbeat Emma was weightless. Then the plane slammed into the bottom of an air pocket and she was shoved hard into her seat.
"Sorry," Malenfant muttered. "The turbulence is getting worse." The slaved controls worked before her. The plane soared and banked.
She suddenly wished she were on the ground, perhaps holed up in her well-equipped hotel room back in Joburg. The world must be going crazy over this. She would have every softscreen in the room turned to the coverage, filling her ears and eyes with a babble of instant commentary. Here, in this bubble of Plexiglas, she felt cut off.
But this is the real experience, she thought. I am here by the sheerest chance, at the moment when this vision appeared in the sky like the Virgin Mary over Lourdes, and yet I pine for my on-line womb. Well, I'm a woman of my time.
The artifact settled into place before Emma once more, vast, enigmatic, slowly approaching. Planes crisscrossed before it, puny. Emma spotted that small private jet, lumbering through the air so much more slowly than the military vehicles around it. She wondered if anybody had tried to make contact with the wheel yet—or if it had been fired on.
"Holy shit," said Malenfant. "Do you see that?"
"What?"
He lifted his arm and pointed; she could see the gesture through the Plexiglas blisters that encased them. "There. Near the bottom of the ring."
It looked like a very fine dark rain falling out of the ring, like a hail of iron filings.
Malenfant lifted small binoculars. "People," he said bluntly. He lowered the binoculars. "Tall, skinny, naked people."
She couldn't integrate the information. _People_ —thrust naked into the air eight miles high, to fall, presumably, all the way to the welcoming gorge of bones... Why? Where were they from?
"Can they be saved?"
Malenfant just laughed.
The plane buffeted again. As they approached the wheel the turbulence was growing stronger. It seemed to Emma that the air at the center of the ring was significantly disturbed; she made out concentric streaks of mist and dust there, almost like a sideways-on storm, neatly framed by the wheel's electric blue frame.
And now that lumbering business-type jet reached dead center of the artifact. It twisted once, twice, then crumpled like a paper cup in an angry fist. Glittering fragments began to hail into the ring.
It was over in seconds. There hadn't even been an explosion.
## _F ire_
Wind gusts. Lightning flashes.
There is no Loud.
People come spewing out of the Mouth. They fall to the grass.
The rain falls steadily on the grass, hissing.
## _E mma Stoney_
"Like it got sucked in," Malenfant said with grim fascination. "Maybe the wheel is a teleporter, drawing out our atmosphere." The plane juddered again, and she could see him wrestling with the stick. "Whatever it is it's making a mess of the air flow."
She could see the other planes, presumably military jets, pulling back to more cautious orbits. But the T-38 kept right on, battering its way into increasingly disturbed air. Malenfant's shoulders jerked as they hauled at the recalcitrant controls.
"Malenfant, what are you doing?"
"We can handle this. We can get a lot closer yet. Those African guys are half-trained sissies—"
The plane hit another pocket. They fell fifty or a hundred feet before slamming into a floor that felt hard as concrete.
Emma could taste blood in her mouth. "Malenfant!"
"Did you bring your Kodak? Come on, Emma. What's life for? This is history."
No, she thought. This is your washout. _That's_ why you are risking your life, and mine, so recklessly.
The artifact loomed larger in the roiling sky ahead of her, so large now that she couldn't see its full circle for the body of the plane. Those iron-filing people continued to rain from the base of the disk, some of them twisting as they fell.
"Makes you think," Malenfant said. "I spend my life struggling to get into space. And on the very day I get washed out of the program, _the very same day_ , space comes to me. Wherever the hell this thing comes from, whatever mother ship orbiting fucking Neptune, you can bet there's going to be a clamor to get out there. Those NASA assholes must be jumping up and down; it's their best day since Neil and Buzz. At last we've got someplace to go—but whoever they send, it isn't going to be _me_. Makes you laugh, doesn't it? If Mohammed can't get to the mountain..."
She closed her hand on the stick before her, letting it pull her passively to and fro. What if she grabbed the stick hard, yanked it to the left or the right? Could she take over the plane? And then what? "Malenfant, I'm scared."
"Of the UFO?"
"No. Of you."
"Just a little closer," he said, his voice a thin crackle over the intercom. "I won't let you come to any harm, Emma."
Suddenly she screamed. "... Watch the Moon, Malenfant. Watch the Moon!"
## _R eid Malenfant_
It was a Moon, but not _the_ Moon. A new Moon. A Red Moon.
It was a day of strange lights in the sky. But it was a sky that was forever barred to him.
The plane was flung sideways.
It was like a barrel roll. Suddenly his head was jammed into his shoulders and his vision tunnelled, worse than any eyeballs-back launch he had ever endured—and harder, much harder, than he would have wanted to put Emma through.
His systems went dead: softscreens, the clunky old dials, even the hiss of the comms, everything. He wrestled with the stick, but got no response; the plane was just falling through an angry sky, helpless as an autumn leaf.
The rate of roll increased, and the Gs just piled on. He knew he was already close to blacking out; perhaps Emma had succumbed already, and soon after that the damn plane was going to break up.
With difficulty he readied the ejection controls. "Emma! Remember the drill!" But she couldn't hear, of course.
... Just for a second, the panels flickered back to life. He felt the stick jerk, the controls bite.
It was a chance to regain control.
He didn't take it.
Then the moment was gone, and he was committed.
He felt exuberant, almost exhilarated, like the feeling when the solid boosters cut in during a Shuttle launch, like he was on a roller-coaster ride he couldn't get off.
But the plane plummeted on toward the sky wheel, rolling, creaking. The transient mood passed, and fear clamped down on his guts once more.
He bent his head, found the ejection handle, pulled it. The plane shuddered as Emma's canopy was blown away, then gave another kick as her seat hurled her clear.
And now his own canopy disappeared. The wind slammed at him, Earth and sky wheeling around, and all of it was suddenly, horribly real.
He felt a punch in the back. He was hurled upward like a toy and sent tumbling in the bright air, just like one of the strange iron-filing people, shocked by the sudden silence.
Pain bit savagely at his right arm. He saw that his flight-suit sleeve and a great swathe of skin had been sheared away, leaving bloody flesh. Must have snagged it on the rim of the cockpit on the way out.
Something was flopping in the air before him. It was his seat. He still had hold of the ejection handle, connected to the seat by a cable.
He knew he had to let go of the handle, or else it might foul his chute. Yet he couldn't. The seat was an island in this huge sky; without it he would be alone. It made no sense, but there it was.
At last, apparently without his volition, his hand loosened. The handle was jerked out of his grip, painfully hard.
Something huge grabbed his back, knocking all the air out of him again. Then he was dangling. He looked up and saw his chute open reassuringly above him, a distant roof of fully blossomed orange and white silk.
But the thin air buffeted him, and he was swaying alarmingly, a human pendulum, and at the bottom of each swing G forces hauled on his entrails. He was having trouble breathing; his chest labored. He pulled a green toggle to release his emergency oxygen.
The artifact hung above him, receding as he fell.
He had been flung west of it, he saw now, and it was closing up to a perfect oval, like a schoolroom demonstration of a planetary orbit. There was no sign of the other planes. Even the T-38 seemed to have vanished completely, save for a few drifting bits of light wreckage, a glimmer that must have been a shard of a Plexiglas canopy.
And he saw another chute. Half open. Hanging before the closing maw of the artifact like a speck of food before the mouth of some vast fish.
Emma, of course: she had ejected a half-second before Malenfant, so that she had found herself that much closer to the artifact than he had been.
And now she was being drawn in by the buffeting air currents.
He screamed, "Emma!" He twisted and wriggled, but there was nothing he could do.
Her chute fell into the portal. There was a flash of electric-blue light. And she was gone.
"Emma! _Emma!_ "
... Something fell past him, not ten yards away. It was a man: tall and lithe like a basketball player, stark naked. He was black, and under tight curls, his skull was as flat as a board. His mouth was working, gasping like a fish's. His gaze locked with Malenfant's, just for a heartbeat. Malenfant read astonishment beyond shock.
Then the man was gone, on his way to his own destiny in the ancient lands beneath.
A new barrage of turbulent air slammed into Malenfant. He rocked viciously. Nursing his damaged arm he fought the chute, fought to keep it stable—fought for his life, fought for the chance to live through this day, to find Emma.
As he spun, he glimpsed that new Red Moon, a baleful eye gazing down on his tiny struggles.
## _F ire_
The Mouth is gone.
The new people are nearby. The smallest is a child. They are all yelling. Their skin is bright, yellow-brown and blue. They are trying to stand up, but they stumble backwards.
Fire's legs walk forward. He walks over the soaked fireplace. The ashes are still hot. He yelps and his feet lift up, off the ashes.
Sing is nearby, on her branches, weeping.
Fire's eyes see Dig. They can't see Loud. Fire calls out. "Loud, Loud, Fire!" But Loud is gone.
Shrugging, the rain running down his back, he turns away. Fire will never think of his brother again.
A new person is coming toward him. This stranger has blue and brown skin on his body. Fire can't see his member. It is a woman. But he can't see breasts. It is a man.
The new person holds out empty hands. _"Please, can you help us? Do you know what happened to us? What place is this?"_
Fire hears: _"Help. What. Us. What."_ The voice is deep. It is a man.
Stone is standing beside Fire. "Nutcracker-man," he says softly.
"No," says Fire.
"Elf-man."
"No."
_"Please."_ The new person steps forward. _"I have a wife and child. Do you speak English? My wife is hurt. We need shelter. Is there a road near here, a phone we could use—"_
Stone's axe slams into the top of the new person's head. The head cracks open. Gray and red stuff splashes out.
The new person's eyes look at Fire. He shudders. He falls backwards.
Stone grunts. "Nutcracker-man." Stone slices off the new person's cheek and crams it into his mouth.
Fire hoots at the kill. Nutcracker-folk fight hard. This kill was easy.
Other people's legs bring them running from the trees to join Stone at his feast. They have forgotten the rain. They get wet again. But they are all drawn by the scent of the fresh meat.
The new person's skin yields easily to Stone's axe. It comes off in a sheet. Fire's fingers touch the sloughed skin. It is blue and brown, thick and dense. Fire is confused. It is skin. It is not skin.
The flesh under the strange skin is white. Stone's axe cuts into it easily. The axe butchers the body rapidly and expertly, an unthinking skill honed across a million years.
The other new people are screaming.
Fire had forgotten them. He straightens up. He has a chunk of flesh in his mouth. His teeth gnaw at it, while his hands pull on it.
The new people's legs are trying to run away. But the new people fall easily, as if they are weak or sick.
Grass and Cold catch the new people. They push them to Stone. One of the new people is bleeding from her head and staggering. Its arms are clutching the small one. When it screams its voice is high. It is a woman.
The other new person has no small one. It has blue skin all over its body. _"We don't mean you any harm. Please. My name is Emma Stoney."_ Its voice is high. It is a woman.
Shoot's hand grabs the hair of this one, pulls her head back.
The new woman's elbow rams into Shoot's belly. _"Get your hands off of me!"_ Shoot doubles over, gasping.
The men laugh at the women fighting.
The woman with the child speaks to Stone. _"Please. We're American citizens. My name is Sally Mayer. I—my husband... I know you can speak English. We heard you. Look, we can pay. American dollars."_ She holds out something green. Handfuls of leaves. Not leaves. Her arm is bleeding, he sees.
_I. You_. That is what Fire hears.
The woman has fallen silent. Her eyes are staring at the top of Stone's head. Her mouth is open.
The top of the woman's head is swollen.
Fire makes his hand run over his own brow. He feels thick eye ridges. He feels a sloping brow. He feels the small flat crown behind his brow. His fingers find a fly trapped in his greasy hair. He pulls it out. He pops it into his mouth.
Stone studies the new woman. Stone's fingers squeeze the woman's dug. It is large and soft, under its skin of green and brown. The woman yelps and backs away. The child, eyes wide, cringes from Stone's bloody hand.
Fire laughs. Stone will mount the woman. Stone will eat the woman.
_"No."_
The other new woman steps forward. Her hands pull the other woman behind her. _"We are like you. Look! We are people. We are not meat."_ She points to the child.
The child has no hair on his face. The child has wide round eyes. The child has a nose.
Nutcracker-folk have hair on their faces. Nutcracker-folk have no noses. Nutcracker-folk have nostrils flat against their faces.
Running-folk have no hair on their faces. They have round eyes. They have noses.
Stone's axe rises.
Fire takes a step forward. He is afraid of Stone and his axe. But he makes his hand grab Stone's arm.
"People," Fire says.
_"Yes."_ The new woman nods. _"Yes, that's right. We're people."_
Slowly, Stone's arm lowers.
The smell of meat is strong. One by one the people drift away from the new people, and cluster around the corpse.
Fire is left alone, watching the new people.
The fat new person is shaking, as if cold. Now she falls to the ground. The other puts the child down, and cradles the fat one's head on her lap.
The other's face lifts up to Fire. _"My name is Emma. Em-ma. Do you understand?"_
Fire carries the fire. That is his name. That is what he does.
Emma is her name. Emma is what she does. He doesn't know what _Em-ma_ is.
He says, "Em-ma."
_"Emma. Yes. Good. Please—will you help us? We need water. Do you have any water?"_
His eye spots something. Something moves on a branch on the ground nearby. He has forgotten that he used these branches to make a bower.
His hand whips out and grabs. His hand opens, revealing a caterpillar, fat and juicy. He did not have to think about catching it. It is just here. He pops it in his mouth.
_"Please."_
He looks down at the new people. Again he had forgotten they were there. "Em-ma." The caterpillar wriggles on his tongue. His hand pulls it out of his mouth. He remembers how he caught it, a sharp shard of recent memory.
He makes his hand hold out the caterpillar.
Emma's eyes stare at it. It is wet from his spit. Her hand reaches out and takes it.
The caterpillar is in her mouth. She chews. He hears it crunch. She swallows, hard. _"Good. Thank you."_
Fire's nose can smell meat more strongly now. Stone's axe has cracked the rib cage. Whatever is in the new person's belly may be good to eat.
The other new woman wakes up. Her eyes look at the corpse, at what the people are doing there. She screams. Emma's hand clamps over her mouth. The woman struggles.
The people crowd close around the corpse. Fire joins them.
He has forgotten the new people.
# _PART TWO_
# **Red Moon**
#
## _E mma Stoney:_
Her chest hurt. Every time she took a breath she was gasping and dragging, as if she had been running too far, or as if she was high on a mountainside.
That was the first thing Emma noticed.
The second thing was the dreaminess of moving here.
When she walked—even on the slippery grass, encumbered by her clumsy flight suit—she felt light, buoyant. But she kept tripping up. It was easy to walk slowly, but every time she tried to move at what seemed a normal pace she stumbled, as if about to take off. Eventually she evolved a kind of half-jog, somewhere between walking and running.
Also she was strong here. When she struggled to drag the woman—Sally?—out of the rain and into the comparative shelter of the trees, with the crying kid at her heels, she felt powerful, able to lift well above her usual limit.
The forest was dense, gloomy. The trees seemed to be conifers—impossibly tall, towering high above her, making a roof of green—but here and there she saw ferns, huge ancient broad-leafed plants. The forest canopy gave them some shelter, but still great fat droplets of water came shimmering down on them. When the droplets hit her flesh they clung—and they _stung_. She noticed how shrivelled and etiolated many of the trees' leaves looked. Acid rain?...
The forest seemed strangely quiet. No birdsong, she thought. Come to think of it she hadn't seen a bird in the entire time she'd been here.
The flat-head people—hominids, whatever—did not follow her into the forest, and as their hooting calls receded she felt vaguely reassured. But that was outweighed by a growing unease, for it was very dark here in the woods. The kid seemed to feel that, too, for he went very quiet, his eyes round.
But then, she thought resentfully, she was disoriented, spooked, utterly bewildered anyhow—she had just been through a plane wreck, for God's sake, and then hurled through time and space to wherever the hell—and being scared in a forest was scarcely much different from being scared on the open plain.
... _What_ forest? What plain? What is this place? _Where am I?_
Too much strangeness: Panic brushed her mind.
But the blood continued to pulse from that crude gash on Sally's arm, an injury she had evidently suffered on the way here, from wherever. And the kid sat down on the forest floor and cried right along with his mother, great bubbles of snot blowing out of his nose.
First things first, Emma.
The kid gazed up at her with huge empty eyes. He looked no older than three.
Emma got down on her knees. The kid shrank back from her, and she made an effort to smile. She searched the pockets of her flight suit, seeking a handkerchief, and finding everything but. At last she dug into a waist pocket of Sally's jacket—she was wearing what looked like designer safari gear, a khaki jacket and pants—and found a paper tissue.
"Blow," she commanded.
With his nose wiped, the boy seemed a bit calmer.
"What's your name?"
"Maxie." His tiny voice was scale-model Bostonian.
"Okay, Maxie. My name's Emma. I need you to be brave now. We have to help your mom. Okay?"
He nodded.
She dug through her suit pockets. She found a flat plastic box. It turned out to contain a rudimentary first aid kit: scissors, plasters, safety pins, dressings, bandages, medical tape, salves, and creams.
With the awkward little scissors she cut back Sally's sleeve, exposing the wound. It didn't look so bad: just a gash, fairly clean-edged, a couple of inches long. She wiped away the blood with a gauze pad. She could see no foreign objects in there, and the bleeding seemed mostly to have stopped. She used antiseptic salve to clean up, then pressed a fresh gauze pad over the wound. She wrapped the lower arm in a bandage, and taped it together.
... Was that right? How was she supposed to know? Think, damn it. She summoned up her scratchy medical knowledge, derived from what she had picked up secondhand from Malenfant's training—not that he'd ever told her much—and books and TV shows and movies... She pressed Sally's fingernail hard enough to turn it white. When she released it, the nail quickly regained its color. Good; that must mean the bandage wasn't too tight.
Now she propped the injured arm up in the air. With her free hand she packed up what was left of her first aid kit. She had already used one of only two bandages, half-emptied her only bottle of salve... If they were going to survive here, she would have to ration this stuff.
Or else, she thought grimly, learn to live like those nude hominids out there.
She turned to the kid. She wished she had some way to make this experience easier on him. But she couldn't think of a damn thing. "Maxie. I'm going to find something to keep the rain off. I need you to stay right here, with your mom. You understand? And if she wakes up you tell her I'll be right back."
He nodded, eyes fixed on her face.
She ruffled his hair, shaking out some of the water. Then she set off back toward the plain.
She paused at the fringe of the forest.
Most of the hominids were hunched over on themselves, as if catatonic with misery in the rain. One, apparently an old woman, lay flat out on the floor, her mouth open to the rain.
The rest seemed to be working together, loosely. They were upending branches and stacking them against each other, making a rough conical shape. Perhaps they were trying to build a shelter, like a teepee. But the whole project was chaotic, with branches sliding off the pile this way and that, and every so often one of them seemed to forget what she was doing and would simply wander off, letting whatever she was supporting collapse.
At last, to a great hoot of dismay from the workers, the whole erection just fell apart and the branches came clattering down.
The people scratched their flat scalps over the debris. Some of them made half-hearted attempts to lift the branches again, one or two drifted away, others came to see what was going on. At last they started to work together again, lifting the branches and ramming them into the ground.
It wasn't like watching adults work on a project, however unskilled. It was more like watching a bunch of eight-year-olds trying to build a bonfire for the very first time, figuring it out as they went along, with only the dimmest conception of the final goal.
But these hominids, these _people_ , weren't eight-year-olds. They were all adults, all naked, hairless, black. And they had the most beautiful bodies Emma had ever seen, frankly, this side of a movie screen anyhow. They were tall and lean—as tall as basketball players, probably—but much stronger-looking, with an all-around grace that reminded her of decathletes, or maybe Aussie Rules footballers (a baffling, sexy sport she'd tried to follow as a student, long ago).
With broad prominent noses and somewhat rounded chins, they had human-looking faces—human below the eye line, anyhow. Above the eyes was a powerful ridge of bone that gave each of them, even the smallest child, a glowering, hostile look. And above that came a flat forehead and a skull that looked oddly shrunken, as if the top of their heads had somehow been shaved clean off. Their hair was curly, but it was slicked down by the rain, showing the shape of their disturbingly small skulls too clearly.
The bodies of humans, the heads of apes. They spoke in hoots and fragmentary English words. And not one of them looked as if he or she had ever worn a stitch of clothing.
She had never heard of creatures like this. What _were_ these people? Some kind of chimp, or gorilla?—But with bodies like that? And what chimps used English?
What part of Africa had she landed in, exactly?
The rain came down harder still, reminding her she had a job to do.
She made her way out into the open, working across increasingly boggy ground, until she reached her parachute. She had been worried that the hominids might have taken it away, but it lay where it had fallen when she had come tumbling from out of the sky.
She took an armful of cloth and pulled it away from the ground. It came loose of the mud only with difficulty, and it was soaked through. She'd had vague plans of hauling the whole thing into the forest, but that was obviously impractical. She hunted through her pockets until she found a Swiss Army knife, kindly provided by the South African air force. She quickly discovered she had at her disposal a variety of screwdrivers, a can and bottle opener, a wood saw, scissors, a magnifying glass, even a nail file. At last she found a fat, sturdy blade. She decided she would cut loose a piece of cloth perhaps twenty feet square, which would suffice for a temporary shelter. Later, when the rain let up, she would come back and scavenge the rest of the silk.
She began to hack her way through the chute material. But it was slow work.
For the first time since that dreadful moment of midair disintegration, she had time to think.
It was all so fast, so blurred. She remembered Malenfant's final scream over the intercom, her sudden ejection—without warning, she had been thrust into the cold bright air, howling from the pain as the seat's rockets slammed into the small of her back—and then, even as her chute had begun to open, she saw the wheel opening like a mouth all around her—and she had realized that for better or worse she was going to fall _through_ it...
Blue light had bathed her face. There had been a single instant of pain, unbearable, agonizing.
And then, _this_.
She had found herself lying on scrubby grass, in a cloud of red dust, all the breath knocked out of her. _Lying on the ground_ , an instant after being forty thousand feet high. From the air to the ground: That was the first shock.
She was aware of the others, the strangers, the couple and the kid, who had appeared beside her, out of nowhere. And she glimpsed that blue portal, foreshortened, towering above her. But it had disappeared, just like that, stranding her here.
Yes, but where was _here_?
She had cut the chute section free. She sat back on her haunches, flexing arms that were not conditioned for manual work. She closed up the knife.
Then, on an impulse, she lifted up the knife and dropped it. It seemed to fall with swimming slowness.
Low gravity. As if she were on the Moon.
That was ridiculous. But if not the Moon, _where_?
Get a grip, Emma. Where you are surely matters a lot less than what you are going to do about it—specifically, how you plan to stay alive long enough for Malenfant to alert the authorities and come find you.
... _Malenfant_.
Had she been shying away from thinking about him? He certainly wasn't anywhere near here; he would be making enough noise if he was. Where, then? On the other side of the great blue portal?
But he'd been through the crash, too. Was he alive at all?
She shut her eyes, and found herself rocking gently, back and forth, on her haunches. She remembered how he had been in those last instants before the destruction of the plane, the reckless way he had hurled them both at the unknown.
Malenfant, Malenfant, what have you done?
A scream tore from the forest.
Emma bundled up her parachute cloth and ran back the way she had come.
On her bed of dead leaves, Sally was sitting up. With her good arm she held her kid to her chest. Maxie was crying again, but Sally's face was empty, her eyes dry.
Uneasy, Emma dumped the parachute cloth. In the seeping rain, she got to her knees and embraced them both. "It's all right."
The kid seemed to calm, sandwiched between the two women.
But Sally pushed her away. "How can you say that? Nothing's _right_." Her voice was eerily level.
Emma said carefully. "I don't think they mean us any harm... Not any more."
"Who?"
"The hominids."
_"I saw them,"_ Sally insisted.
"Who?"
"Ape-men. They were _here_. I just opened my eyes and there was this face over me. It was squat, hairy. Like a chimp."
Then not like the hominids out on the plain, Emma thought, wondering. Were there more than one kind of human-ape, running around this strange, dreamy forest?
"It was going through my pockets," Sally said. "I just opened my eyes and looked right in its face. I yelled. It stood up and ran away."
"It _stood up_? Chimps don't stand upright. Not habitually... Do they?"
"What do I know about chimps?"
"Look, the—creatures—out there on the plain don't sound like that description."
"They are ape-men."
"But they aren't squat and hairy." Emma said hesitantly. "We've been through a lot. You're entitled to a nightmare or two."
Doubt and hostility crossed Sally's face. "I know what I saw."
The kid was calm now; he was making piles of leaves and knocking them down again. Emma saw Sally take deep breaths.
At least Emma was married to an astronaut; at least she had had her head stuffed full of outré concepts, of other worlds and different gravities; at least she was used to the concept that there might be other places, other worlds, that Earth wasn't a flat, infinite, unchanging stage... To this woman and her kid, though, none of that applied; they had no grounding in weirdness, and all of this must seem unutterably bewildering.
And then there was the small matter of Sally's husband.
Emma was no psychologist. She did not kid herself that she understood Sally's reaction here. But she sensed this was the calm before the storm that must surely break.
She got to her feet. Be practical, Emma. She unwrapped her parachute silk and started draping it over the trees, above Sally. Soon the secondary forest-canopy raindrops pattered heavily on the canvas, and the light was made more diffuse, if a little gloomier.
As she worked she said hesitantly, "My name is Emma. Emma Stoney. And you—"
"I'm Sally Mayer. My husband is Greg." _Is?_ "I guess you've met Maxie. We're from Boston."
"Maxie sounds like a miniature JFK."
"Yes..." Sally sat on the ground, rubbing her injured arm. Emma supposed she was early thirties. Her brunette hair was cut short and neat, and she wasn't as overweight as she looked in her unflattering safari suit. "We were only having a joy ride. Over the Rift Valley. Greg works in software research. Formal methodologies. He had a poster paper to present at a conference in Joburg... Where are we, do you think?"
"I don't know any more than you do. I'm sorry."
Sally's smile was cold, as if Emma had said something foolish. "Well, it sure isn't _your_ fault. What do you think we ought to do?"
_Stay alive_. "Keep warm. Keep out of trouble."
"Do you think they know we are missing yet?"
_What "they"?_ "That wheel in the sky was pretty big news. Whatever happened to us probably made every news site on the planet."
Here came Maxie, kicking at leaves moodily, absorbed in his own agenda, like every kid who wasn't scared out of his wits. "I'm hungry."
Emma squeezed his shoulder. "Me, too." She started to rummage through the roomy pockets of her flight suit, seeing what else the South African air force had thought to provide.
She found a packet of dried foods, sealed in a foil tray. She laid out the colorful little envelopes on the ground. There was coffee and dried milk, dried meal, flour, suet, sugar, and high-calorie stuff like chocolate powder, even dehydrated ice cream.
Sally and Emma munched on trail mix, muesli, and dried fruits. Sally insisted Maxie eat a couple of crackers before he gobbled up the handful of hard candy he had spotted immediately.
Emma kept back one of the candies for herself, however. She sucked the cherry-flavored candy until the last sliver of it dissolved on her tongue. Anything to get rid of the lingering taste of that damn caterpillar.
_Caterpillar_ , for God's sake. Her resentful anger flared. She felt like throwing away the petty scraps of supplies, rampaging out to the hominids, demanding attention. Wherever the hell she was, she wasn't supposed to be here. She didn't want anything to do with this. She didn't want any responsibility for this damaged woman and her wretched kid—and she didn't want her head cluttered up with the memories of what had become of the woman's husband.
But nobody was asking what she wanted. And now the food was finished, and the others were staring at her, as if they expected her to supply them.
If not you, Emma, who else?
Emma took the foil box and went looking for water.
She found a stream a few minutes deeper into the forest. She clambered down into a shallow gully and scooped up muddy water. She sniffed at it doubtfully. It was from a stream of running water, so not stagnant. But it was covered with scummy algae, and plenty of green things grew in it. Was that good or bad?
She carried back as much water as she could to their improvised campsite, where Sally and Maxie were waiting. She set the water down and started going through her pockets again.
Soon she found what she wanted. It was a small tin, about the size of the tobacco tins her grandfather used to give her to save her coins and stamps. Inside a lot of gear was crammed tight; Maxie watched wonderingly as she pulled it all out. There were safety pins, wire, fish hooks and line, matches, a sewing kit, tablets, a wire saw, even a teeny-tiny button compass. And there was a little canister of dark crystals that turned out to be potassium permanganate.
Following the instructions on the can—to her shame she had to use her knife's lens to read them—she dropped crystals into the water until it turned a pale red.
Maxie turned up his nose, until his mother convinced him the funny red water was a kind of cola.
Habits from ancient camping trips came back to Emma now. For instance, you weren't supposed to _lose_ anything. So she carefully packed all her gear back into its tobacco tin, and put it in an inside pocket she was able to zip up. She took a bit of parachute cord and tied her Swiss Army knife around her neck, and tucked it inside her flight suit, and zipped that up, too.
And while she was fiddling with her toys, Sally began shuddering.
"Greg. My husband. Oh my God. _They killed him_. They just crushed his skull. The ape-men. Just like that. I saw them do it. It's true, isn't it?"
Emma put down her bits of kit with reluctance.
"Isn't it strange?" Sally murmured. "Greg isn't here. But I never thought to ask _why_ he isn't here. And all the time, in the back of my mind, I _knew_... Do you think there's something wrong with me?"
"No," Emma said, as soothingly as she could manage. "Of course not. It's very hard, a very hard thing to take—"
And then Sally just fell apart, as Emma had known, inevitably, she must. The three of them huddled together, in the rain, as Sally wept.
It was dark before Sally was cried out. Maxie was already asleep, his little warm form huddled between their two bodies.
The rain had stopped. Emma pulled down her rough canopy, and wrapped it around them.
Now Sally wanted to talk, whispering in the dark.
She talked of her holiday-of-a-lifetime in Africa, and how Maxie was doing at nursery school, another child, a daughter, at home, and her career and Greg's, and how they had been considering a third child or perhaps opting for a frozen-embryo deferred pregnancy, pending a time when they might be less busy.
And Emma told her about her life, her career, about Malenfant. She tried to find the gentlest, most undemanding stories she could think of.
Like the one about their engagement, at the end of Malenfant's junior year as a midshipman at the Naval Academy. He had received his class ring, and at the strange and formal Ring Dance she had worn his ring around her neck, while he carried her miniature version in his pocket. And then at the climax of the evening the couples took their turns to go to the center of the dance floor and climb up under a giant replica of the class ring. Filled with youth and love and hope, they dipped their rings in a bowl of water from the seven seas, and exchanged the rings, and made their vows to each other...
Oh, Malenfant, where are you now?
Eventually they slept: the three of them, brought together by chance, lost in this strange quasi-Africa, now huddled together on the floor of a nameless forest. But Emma came to full wakefulness every time she heard a leaf rustle or a twig snap, and every time a predator howled, in the huge lands beyond this sheltering forest.
Tomorrow we have to make a proper shelter, she thought. We can't sleep on the damn ground.
## _S hadow_
She woke early.
She turned on her back, stretching her long arms lazily. Her nest of woven branches was soft and warmed by her body heat, but where her skin was exposed to the cold, her hair prickled, standing upright. She found moist dew on her black fur, and she scooped it off with a finger and licked it.
Scattered through the trees she could see the nests of the Elf-folk, fat masses of woven branches with sleek bodies embedded, still slumbering.
She had no name. She had no need of names, nor capacity to invent them.
Call her Shadow.
The sky was growing light. She could see a stripe of dense pink smeared along one horizon. Above her head there was a lid of cloud. In a crack in the cloud an earth swam, bright, fat, blue.
Shadow stared at the earth. It hadn't been there last time she woke up.
Loose associations ran through her small skull: not thoughts, not memories, just shards, but rich and intense. And they were all blue. Blue like the sky after a storm. Blue like the waters of the river when it ran fat and high. Blue, blue, blue, clean and pure, compared to the rich dark green of night thoughts.
Blue like the light in the sky, yesterday.
Shadow's memories were blurred and unstructured, a corridor of green and red in which a few fragments shone, like bits of a shattered sculpture: her mother's face, the lightness of her own body as a child, the sharp, mysterious pain of her first bleeding. But nowhere in that dim green hall was there a flare of blue light like that. It was strange, and therefore it was frightening.
But memories were pallid. There was only the now, clear and bright: What came before and what would come after did not matter.
As the light gathered, the world began to emerge out of the dark green. Noise was growing with the light, the humming of insects and the whirring flight of bats.
Here, in this clump of trees high on an escarpment, she was at the summit of her world. The ground fell away to the sliding black mass of the river. The trees were scattered here, the ground bare and gray, but patches of green-black gathered on the lower slopes, gradually becoming darker and thicker, merging as they tumbled down the gullies and ravines that led to the river valley itself.
She knew every scrap of this terrain. She had no idea what lay beyond—no real conception that _anything_ lay beyond the ground she knew.
The others were stirring now. Her infant sister, Tumble, sat up on the belly of their mother, Termite. Termite stretched, and one shapely foot raised, silhouetted against the sky.
Shadow slid out of her nest. The pliant branches rustled back to their natural positions. This was a fig tree, with vines festooned everywhere. Shadow found a dense cluster of ripe fruit, and began to feed.
Soon there was a soft rain all around her, as discarded skins and seeds fell from the lips of the folk, toward the ground.
Above her there was a sharp, sudden crack. She flinched, looking up. It was Big Boss. His teeth bared, without so much as a stretch, he leapt out of his nest and went leaping wildly through the trees, swaying the branches and swinging on the vines.
Everywhere people abandoned their nests, scrambling to get out of the way of Big Boss. The last peace of the night was broken by grunts and screams.
But one man wasn't fast enough. It was Claw, Shadow's brother, hindered by his need to favor his useless hand, left withered by a childhood bout of polio.
Big Boss crashed directly into the nest of the younger male, smashing it immediately. Claw, screeching, fell crashing through the branches and down to the ground.
Big Boss scrambled after him, down to the ground. He strutted back and forth, waving his fists. He shook the vegetation and threw rocks and bits of dead wood. Then he sat, black hair bristling thick over his hunched shoulders.
One by one, Big Boss's acolytes approached him, weaker men he dominated with his fists and teeth and shows of anger. Big Boss welcomed them with embraces and brief moments of grooming.
Claw was one of the last, loping clumsily, his withered hand clutched to his belly. Shadow saw how his back was scratched and bleeding, a marker of his rude awakening. He bent and kissed Big Boss's thigh. But Claw's obeisance was rewarded only by a cuff on the side of his head, hard enough to send him sprawling.
The other men joined in, following their leader's example, kicking and punching at the howling Claw—but each of them retreated quickly after delivering his blow.
Big Boss spread his lips in a wide grin, showing his long canines.
Now Termite strode into the little clearing, calm and assured, her infant clinging to the thick black hair on her back. Claw ran to her and huddled at his mother's side, whimpering as if he was an infant himself.
One of the men pursued Claw, yelling. Like most of the men he was a head taller than Termite, and easily outweighed her. But Termite cuffed him casually, and he backed away.
Now Big Boss himself approached Termite. He slapped her, hard enough to make her stagger.
Termite stood her ground, watching Big Boss calmly.
With a last growl Big Boss turned away. He bent over and defecated explosively. Then he reached for leaves to wipe his backside, while his acolytes jostled to groom his long black fur.
Termite walked away, followed by Claw and her infant, seeking food.
The incident was over, power wielded and measured by all concerned.
Another day had begun in the forest of the Elf-folk.
Shadow, her long arms working easily, swung down to the ground to join her family.
The people lingered by the trees where they had slept. They sat with legs folded and groomed each other, picking carefully through the long black hairs, seeking dirt, ticks, and other insects.
Shadow sat her little sister on her lap. Tumble squirmed and wriggled—but with an edge of irritation, for she had picked up bloodsucking ticks some days before. Shadow found some of the tiny, purplish creatures in the child's scalp now. She plucked them away between delicate fingernails and popped them in her mouth, relishing the sharp tang of blood when they burst beneath her teeth.
All around her people walked, groomed, fed, locked into an intricate geometry of lust, loyalty, envy, power. The people were the most vivid thing in Shadow's world; everything else was a blur, barely more noticed than the steady swell of her own breathing.
At eleven years old, Shadow was three feet tall. She had long legs under narrow hips, long, graceful arms, a slim torso, a narrow neck and shoulders. She walked upright. But her legs were a little splayed, her gait clumsy, and her long, strong arms were capable of carrying her high in the trees. Her rib cage was high and conical, and her skull was small, her mouth with its red lips prominent. And over pink-black skin, her body was covered with long black fur.
Her eyes were clear, light brown, curious.
A few days before, Shadow had begun the bleeding, for the first time in her life. Several of the men and boys, smelling this, had begun to pursue her. Even now a cluster of the boys pressed close to her, dragging clumsy fingers through her hair, their eyes bright. But Shadow desired none of them, and when they got too persistent she approached her mother, who growled deeply.
Termite herself was surrounded by a group of attentive men and adolescent boys, some of them displaying spindly erections. Termite submitted to the gentle probing of their fingers. Though she was growing old now, and some of her fur was shot through with silver, Termite was the most popular woman in the group, as far as the men were concerned. On some patches of her head and shoulders her fur had been worn away by the constant grooming; her small skull was all but hairless, her black ears prominent.
That allure, of course, made her one of the most powerful women. Just as the weaker men would compete for the friendship of Big Boss, so the women were ambitious to be part of Termite's loose circle. Shadow—and Tumble, and even Claw—had special privileges, as Termite's children, arising from that power.
And it was real power, the only power, even if the women had to endure the blows and bites of the powerful men. Everybody knew her mother and her siblings, and that was where loyalty lay; for nobody knew her father. No man, not even Big Boss, would have achieved his status without the backing of a powerful mother and aunts.
At last it was time to move on. Little Boss—the brother of Big Boss, his closest lieutenant—led off, working his way down the hillside toward the river. He paused frequently, watching nervously to be sure that Big Boss followed.
The people gave up their grooming and wandered after them.
The Elf-folk entered thicker swathes of forest. The day grew hot, the air oppressive in the greenery. The people walked easily, save where the vines and brambles grew too dense, and then they would use their powerful arms to climb into the trees. They moved slowly, stopping to feed wherever the opportunity arose.
Even at its most dense the forest was sparse. Many of the trees' leaves were yellow, shriveled and sickly, and some of the trees themselves were dead, no more than gaunt stumps with broken-off branches at their roots. There was much space between the big trees, and the gaps in the forest canopy allowed the sunlight to reach the ground, where shoots and bushes grew thickly.
Shadow, like the others, kept away from the more open clearings. Though her long slim legs carried her easily over the clear ground, the denser green of the forest pulled at her, while the blue-white open sky and green-brown undergrowth repelled her.
They came to a knot of low shrubs.
Termite lowered Tumble to the ground. This was a bush Termite knew well, and her experienced eyes had spotted that some of the leaves had been rolled into tubes, held together by sticky threads. When Shadow opened up such a tube she was rewarded by a wriggling caterpillar, which she popped into her mouth.
The three of them rested on the ground, relishing the treat.
Little Tumble snuggled up to her mother, seeking her nipples. Gently Termite pushed the child away. At first Tumble whimpered, but soon her pleading turned to a tantrum, and the little ball of fur ran in circles and thumped the ground. Her mother held her close, subduing her struggles, until she was calm. Tumble took some of the caterpillars her mother unpacked for her. But later, Tumble made a pretence of having eaten her fill, and began to groom her mother with clumsy attentiveness. Termite submitted to this as she fed—and pretended not to notice as Tumble worked her way ever closer to her nipple, at last stealing a quick suck.
Shadow stretched out on the grass, legs comfortably crossed. She plucked caterpillar leaves from the bushes with one hand, holding the other crooked behind her head.
The sky was a washed-out blue, but clouds were tumbling across it. She had a dim sense of the future: Soon it would be dark, and it would rain, and she would get wet and cold. But she saw little further than that, little further than the bright sunny warmth of the sun and the softness of this patch of grass, and she relaxed, her thoughts warm and yellow.
She raised her free hand before her eyes. She stretched her fingers, making slats through which the sun peeked. She moved her hand back and forth, rapidly, making the sun flicker and dance.
Now, with a single graceful movement, she turned over and got to her knees. She gazed at the sharp shadow the sun cast on the leaf-strewn ground before her. She raised her hands, making the shadow do the same, and then she spread her fingers, making light shine through the hands of her shadow.
She got to her feet and began to whirl and dance, and the shadow, this other self, capered in response, its movements distorted and comical. Her dance was eerily beautiful.
The wind shifted, bringing a scent of smoke. Smoke and meat.
Big Boss stood tall and peered into the green. His nostrils flared.
He rooted around on the ground until he found a cobble the size of his fist. He hurled the cobble against a large rock embedded in the ground, smashing it. Then, with some care, he fingered the debris, searching for flakes of the right size and sharpness.
He stood tall, hands full of sharp flakes, a small trickle of blood oozing from one finger. He issued his summoning cry— _"Ai, ee!"_ —and, without looking back, he began to stalk off to the west, the way the smoke had come from. His brother Little Boss and another senior man, Hurler, scurried to follow him, keeping a submissive few paces back.
Claw had been crouching in the grass. He stood up now, and took a few steps after the men, uncertainly.
Little Boss slapped him so hard in the back that Claw was sent sprawling on his chest.
But Hurler helped him get back to his feet with a fast, savage yank. Hurler, a big man with powerful hands and a deadly accuracy with thrown rocks, was Termite's brother—Claw's uncle—and so favored him, more than the other men anyhow. The two of them trotted after Big and Little Boss.
As the men receded, Termite shrugged her slim shoulders and returned to her inspection of the shrubs.
## _E mma Stoney_
Emma clung to sleep as long as possible. When she could sleep no longer, she rolled onto her back, stiff and cold. There was sky above her, an ugly lid of cloud.
Still here, she thought. Shit. And there was an unwelcome ache in her lower bowels.
Nothing for it.
She went behind a couple of trees—close enough that she could still see her parachute canopy tent—and stripped to her underwear. She took a dump, her Swiss army knife dangling absurdly around her neck. The problem after that was finding a suitable wipe; the dried leaves she tried to use just crumbled in her hands.
Where am I? Answer came there none.
Maybe some kind of adrenaline rush had gotten her through yesterday. Today was going to be even worse, she thought. This morning she felt cold, stiff, dirty, lost, miserable—and with a fear that had sunk deep into her gut.
She got dressed and kicked leaves over the, umm, deposit she'd left. We have _got_ to build a latrine today.
Sally and Maxie, waking slowly, showed no desire to leave the forest. But Emma decided she ought to go say hello to the neighbors.
She stepped out of the forest.
It had stopped raining, but the sky was gray and solid and the grassy plain before her was bleak, uninviting. If she had not known otherwise she would have guessed it was uninhabited; the heapings of branches and stones seemed scarcely more than random.
And yet hominids— _people_ —sat and walked, jabbered and argued, from a distance, just as human as she was, every one of them as naked as a newborn. And they were talking English. The utter strangeness of that struck her anew.
I don't want to be here, facing this bizarreness, she thought. I want to be at home, with the net, and coffee and newspapers, and clean clothes, and a warm bathroom.
But it might not be long before she was begging at these hominids' metaphorical table. She had no doubt that those tall, powerful qua-people had a much better ability to survive in this wilderness than she did; she sensed that might become very important, unless they were rescued out of here in the next few days. So she forced herself forward.
Some of the women were tending to nursing infants. Older children were wrestling clumsily—and wordlessly, save for an occasional hoot or screech. The children seemed to her to have the least humanity; without the tall, striking, very human bodies of the adults, their low brows and flat skulls seemed more prominent, and they reminded her more of chimps.
Listening to the hominids yesterday, she had picked up a few of their functional names. The boy who had given her the caterpillar was called Fire. Right now Fire was tending the old woman on the ground, who was called Sing. He seemed to be feeding her, or giving her water. Evidence of kinship bonds, of care for the old and weak? It somewhat surprised Emma. But it was also reassuring, she thought, considering her own situation.
The largest man—Stone, the dominant type who had groped Sally—was sitting on the ground close to the smoking remains of the fire. He was picking through a pile of rocks. He was the leader, she figured—the leader of the men, anyhow.
She plucked up her courage and sat opposite him.
He glowered at her. His brown eyes, under a heavy lid of brow, were pits of hostility and suspicion. He actually raised his right fist at her, a mighty paw bearing a blunt rock.
But she sat still, her hands empty. Perhaps he remembered her. Or perhaps he was figuring out all over again that she was no threat. Anyhow, his hand lowered.
Seeming to forget her, he started working at the rocks again. He picked out a big lump of what looked like black glass; it must be obsidian, a volcanic glass. He turned it this way and that, inspecting it. His movements were very rapid, his gaze flickering over the rock surface.
His muscles were hard, his skin taut. His hair was tightly curled, but it was peppered with gray. His face would have passed in any city street—so long as he wore a hat, anyhow, to conceal that shrivelled skull. But an _Aladdin Sane_ zigzag crimson scar cut right across his face.
She thought he looked around fifty. Hard to tell in the circumstances.
He picked out another rock from his pile, a round pebble. He began to hammer at the obsidian, hard and confident. Shards flew everywhere, and for the first time Emma noticed that he had a patch of foliage over his lap, protecting his genitals from flying rock chips. He worked fast, confident, his eyes flickering—faster than a human would have, she thought, faster and more instinctively. It was less like watching the patient practicing of a human craft than a fast-reaction sport, like tennis or soccer, where the body takes over.
He may not have a wide repertoire of skills, she thought. Maybe this is the one type of tool he can make. But there was nothing limited in what she saw, nothing incomplete; it was as efficient a process as eating or breathing. The contrast with the way the people had struggled to build their heaped-up teepees couldn't have been more striking. How was it possible to be so smart about one thing, yet so dumb about another?
She felt her ideas adjust, her preconceptions dissolve. These people are not like me, she thought.
After a time, Stone abruptly stood up. He dropped his hammerstone, his lap cover, even the tool he had been making, and wandered away.
Emma stayed put.
Stone hunted around the grass, digging into the red dust beneath, picking out bits of rock or perhaps bone, discarding them where he found them. At last he seemed to have found what he wanted.
But then he was distracted by an argument between two of the younger men. He dropped the bone fragment and waded into what was fast becoming a wrestling match. Pretty soon all three of them were battling hard.
Others were gathering around, hooting and hollering. At last Stone floored one of the young men and drove off the other.
Breathing hard, sweating heavily enough to give him a pungent stink, he came back to the pile of rocks, where Emma waited patiently. When he got there he looked around for his bit of bone—but of course it had never made it this far. He bellowed, apparently frustrated, and got up again and resumed his search.
A human craftsman would have got all his tools together before he started, Emma supposed.
Stone came back with a fresh bit of bone. It was red, and bits of meat clung to it; Emma shuddered as she speculated where it might have come from. He used it to chip at the edge of his obsidian axe.
When he was done he dropped the improvised bone tool at his feet without another thought. He turned the axe over and over in his hands; it was a disc of shaped rock four inches across, just about right to fit into his powerful hand.
Then he hefted it and began to scrape at his neck with it.
My God, she thought. He's shaving.
He saw her looking. "Stone Stone!" he yelled. He turned away deliberately, suddenly as self-conscious as a teenager.
She got up and moved away.
## _S hadow_
The people were moving again, working deeper into the forest, seeking food. She spotted Termite and Tumble, walking hand-in-hand, and she followed them.
There had been a shower here. The vegetation was soaking, and droplets sprayed her as she pushed past bushes and low branches. But the droplets sparkled in the sun, and the wet leaves were a bright vivid green. The people's black hair was shot with flashes of rust brown, smelling rich and damp.
Termite came to an ants' nest, a mound punctured by small holes. She reached out and broke a long thin branch from a nearby bush. She removed the side branches and nibbled off the bark, leaving a long, straight stick half as long as her arm. She pushed one hand into the ants' nest and scooped out dirt.
Soon the ants began to swarm out of the nest. Termite plunged her stick into the nest, waited a few heartbeats, and then withdrew it. It was covered with squirming ants. She slid the tool through her free hand so that she was left with a palm filled with crushed and wriggling ants, which she scooped into her mouth, crunching quickly. There was a strong acid smell. Then she returned her stick to the mound and waited for a fresh helping.
Shadow and the other women and children joined in the feast with sticks of their own. Occasionally they had to slap at their feet and thighs as the ants swarmed to repel the invaders; these were big, strong ants that could bite savagely. But Shadow's stick was too spindly and it bent and finally snapped as she shoved it into the loose earth.
More people crowded around. The ants' nest became a mass of jostling and poking elbows and slaps and screeching.
Shadow quickly tired of the commotion. She straightened up, brushed dirt from her legs, and slipped farther into the forest.
She came to a tall palm. She thought she could see clusters of red fruit, high above the ground. Briskly she began to climb, her strong arms and gripping legs propelling her fast above the ground.
She found a cluster of fruit. She picked one, then another, stripping off the rich outer flesh, and letting the kernels fall with a whisper to the distant ground. This was one of the tallest trees in the forest. The sky seemed close here, the ground a distant place.
There were eyes, watching her.
She yelped and recoiled, gripping the palm's trunk with her arms.
She saw a face. But it was not like her own. The head was about the size of Shadow's, but there was a thick bony crest over the top of the skull, and immense cheekbones to which powerful muscles were fixed. The body, covered in pale brown fur, was squat, the belly distended. Two pink nipples protruded from the fur, and an infant clung there, peering back at Shadow with huge pale eyes. The infant might have been a twin of Tumble, but already that bony skull had started to evolve its strange, characteristic superstructure.
Mother and child were Nutcracker-folk.
## _E mma Stoney_
All the teepee shelters had fallen down.
One younger man was struggling, alone, to hoist branches upright. It was Fire, the teenager-type who had gifted her with the caterpillar. But nobody was helping him, so his branches had nothing to lean on, and they just fell over. Still he kept trying. At one point he even ran around his construction, trying to beat gravity, hoisting more branches before the others fell. Of course he failed. It was as if he knew what he wanted to build, but couldn't figure out how to achieve it.
Cautiously, Emma stepped forward.
Fire was startled. He stumbled backward. His branches fell with a crash.
She held her hands open and smiled. "Fire," she said. She pointed to herself. "Emma. Remember?"
At length he jabbered, "Fire Fire. Fire Emma."
"Emma, yes. Remember? You gave me the caterpillar." She pointed to her mouth.
His eyes widened. He ran away at startling speed, and came back with a scrap of what looked like potato. With impatient speed, he shoved it into her mouth. His fingers were strong, almost forcing her jaws open.
She chewed, feeling bruised, tasting the dirt on his fingers. The root was heavy and starchy. "Thank you."
He grinned and capered, like a huge child. She noticed that in his excitement he had sprouted an erection. She took care not to look at it; some complications could wait for another day.
"I'll help you," she said. She walked around his pile of branches. She picked up a light-looking sapling and hoisted it over her shoulder until it was upright. Though her strength still seemed boosted, she struggled to hold the sapling in place.
Mercifully Fire quickly got the idea. "Fire, Emma, Fire!" He ran around picking up more branches—some of them thick trunks, which he lifted as if they were made of polystyrene—and rammed them into place against hers.
The three or four branches propped each other up, a bit precariously, and the beginning of their makeshift teepee was in place. But, hooting with enthusiasm, Fire hurled more branches onto the tall conical frame. Soon the whole thing collapsed.
Fire shouted his disappointment. He did a kind of dance, kicking viciously at the branches. Then, with a kind of forgetful doggedness, he began to pick up the scattered branches once more.
Emma said, "I've a better idea." Raising her hands to make him wait, she jogged over to the muddy remnant of her parachute. She cut free a length of cord—taking care not to show her Swiss Army knife to any of the hominids—and hurried back.
Fire had, predictably, wandered away.
Emma squatted down on the ground to wait, as Fire dug more tubers from the ground, and spent some time throwing bits of stone, with startling accuracy, at a tree trunk, and went running after a girl—"Dig! Dig, Fire, Dig!" Then he happened to glance Emma's way, appeared to remember her and their project, and came running across as fast as a 100-meter record holder. Straightaway he began to pick up the branches again.
She motioned him to wait. "No. Look." She took one of the branches, and pulled another alongside, and then another. Soon he got the idea, and he helped her pile the branches close together. Now she wrapped her cord around them, maybe three feet below their upper extent, and tied a knot.
... Emma Stoney, frontier woman. What the hell are you doing? What if the knot slips or the cord breaks or your sad teepee just falls apart?
Well, then, she thought, I'll just think of something else, and try again. And again and again.
All the time the bigger issues were there in her mind, sliding under the surface like a shark: the questions of where she was, how she had got here, how long it was going to be before she got home again. How she felt about Malenfant, who had stranded her here. How come these ape-folk existed at all, and how come they spoke English... But this was _real_ , the red dust under her feet, the odd musk stink of the ape-boy before her, the hunger already gnawing at her belly. Right now there was nobody to take care of her, nobody but herself, and her first priority was survival. She sensed she had to find a way of working with these people. So far, in all this strange place, the only creature who had showed her any helpfulness or kindness at all was this lanky boy, and she was determined to build on that.
Find strength, Emma. You can fall apart later, when you're safely back in your apartment, and all this seems like a bad dream.
She labored to tie her knot tight and secure. When she was done, she backed away. "Up, up! Lift it up, Fire!"
With terrifying effortlessness he hoisted the three branches vertical. When he let go, they immediately crashed to the ground, of course, but she encouraged him to try again. This time she closed her hands around his, making him hold the branches in place, while she ran around pulling out the bases of the branches, making a pyramidal frame.
At last they finished up with a firmly secured frame, tied off at the top—and it was a frame that held as Fire, with exhilaration and unnerving vigour, hurled more branches over it.
Now all I have to do, Emma thought, is make sure he remembers this favor.
"... Emma! _Emma!_ "
Emma turned. Sally came running out of the forest, with Maxie bundled in her arms.
Creatures pursued her.
They looked like humans—no, not human, like chimps, with long, powerful arms, short legs, covered in fine black-brown hair—but _they walked upright_ , running, almost emulating a human gait. There were four, five, six of them.
Emma thought, dismayed, What now? What new horror is this?
One of the creatures, despite the relative clumsiness of his gait, was fast closing on Sally and the child.
Stone stepped forward. The old male stood stock still, reached back, and whipped his arm forward. His axe, spinning, flew like a Frisbee.
The axe sliced into the ape-thing's face. He, it, was knocked flat, dead immediately. The hominids hooted their triumph and ran to the fallen creature.
The other ape-things ran back to the forest's edge. They screeched their protest, but they weren't about to come out of the forest to launch a counterattack.
Sally kept running until she had reached Emma. They clutched each other.
"Now we know why our friends keep out of the forest," Emma said.
Fire was standing beside them. "Elf-folk," Fire said, pointing at the ape-things. "Elf-folk."
"That's what I saw yesterday," Sally murmured. "My God, Emma, they could have come on us while we slept. We're lucky to be alive—"
"They took the ice cream," Maxie said solemnly.
Sally patted his head. "It's true. They took all your food, Emma. I'm sorry. And the damn canopy."
Maxie said, "What are we going to eat now?"
It appeared the hominids had their own answer to this. From the spot where the apelike "Elf" had fallen came the unmistakable sounds of butchering.
## _S hadow_
For long moments Nutcracker-woman and Shadow gazed at each other, fearful, curious.
Then the Nutcracker-woman took a red fruit, stripped off the flesh, and popped the kernel into her mouth. She pressed up on her lower jaw with her free hand. Caught between her powerful molars, the shell neatly cracked in two. She extracted the nut's flesh and pushed it into her infant's greedy mouth.
Shadow's fear evaporated. She took a fruit herself and stripped it of flesh. But when she tried to copy the Nutcracker-woman's smooth destruction of the nut, she only hurt her jaw.
She spat out the shell and, cautiously, passed it to the Nutcracker-woman.
Just as hesitantly, the Nutcracker-woman took it. Her hand was just like Shadow's, the back coated with fine black hairs, the palm pink.
Shadow had grown used to meeting Nutcracker-folk.
The Elf-folk favored the fringes of the forest, for they could exploit the open land beyond, where meat could often be scavenged. The Nutcracker-folk preferred the dense green heart of the forest, where the vegetation grew richer. But as the forest shrank, the Elf-folk were forced to push deeper into the remaining pockets of green.
Sometimes there was conflict. The Nutcracker-folk were powerful and limber, more powerful than most Elf-folk, and they made formidable opponents.
All things considered, it was better to try to get along.
But now, as Shadow and the Nutcracker-woman amiably swapped fruit back and forth, there was a screech and crash at the base of the tree. The Nutcracker-woman peered down nervously, her child clinging to her shoulders.
It was the hunting party—or rather, what was left of them. She saw the two powerful brothers, Big Boss and Little Boss, and there was her own brother, Claw, trailing behind. They were empty-handed, and there was no blood around their mouths, or on their pelts. Big Boss seemed enraged. His hair bristled, making him a pillar of spiky blackness. As he stalked along he lashed out at the trees, at his brother—and especially at Claw, who was forced to flee, whimpering. But he needed to stay with the men, for he was in more danger from the predators of the forest than from their fists.
And there was no sign of Hurler, her uncle.
It was Hurler who had been killed by Stone's obsidian axe.
Images of him rattled through Shadow's memory. By tomorrow, though she would be aware of a loss, she would barely remember Hurler had existed.
The men abruptly stopped below Shadow's tree. They peered upward, silent, watchful.
The Nutcracker-woman had clamped her big hand over her baby's mouth, and it struggled helplessly. But now a nutshell slipped from the baby's paw, falling with a gentle clatter to the ground.
Big Boss grinned, his hair bristling. Little Boss and Claw spread out around the base of the tree.
Shadow slithered down the tree trunk. The men ignored her.
The three of them clambered into nearby trees. Soon there was an Elf-man in each of the trees to which the Nutcracker-woman could flee.
She began to call out, a piercing cry of fear. _"Oo-hah!"_ Nutcracker-people were fierce and strong, and would come rushing to the aid of their own.
But if any Nutcrackers were near, they did not respond.
Suddenly Big Boss made a leap, from his tree to the Nutcracker-woman's. The Nutcracker-woman screeched. She leapt to Claw's tree, her big belly wobbling.
But Claw, small as he was, was ready for her. As the Nutcracker-woman scrambled to get hold of a branch, Claw grabbed her infant from her.
He bit into its skull, and it died immediately.
The Nutcracker-woman screamed, and hurled herself toward Claw. But already, with his kill over his shoulder, Claw was scurrying down the tree trunk to the ground. Blood smeared around his mouth, he held up his limp prize, crying out with triumph.
But Big Boss and Little Boss converged on him. With a casual punch, Little Boss knocked Claw to the dirt, and Big Boss grabbed the infant. The two of them huddled over the carcass. With firm strong motions, they began to dismember it, twisting off the infant's limbs one by one as easily as plucking leaves from a branch. When Claw came close, trying to get a share of the meat, he was met by a punch or a kick. He retreated, screeching his anger.
In the tree above, the Nutcracker-woman could only watch, howling: _"Hah! Oo-hah!"_
Claw came up to the men time and again, pulling at their shoulders and beating their backs.
A powerful blow from Big Boss now sent Claw sprawling. Clutching his chest, he groaned and lay flat.
Shadow approached her brother. She held out a hand, fingers splayed, to groom him, calm him.
He turned on her.
There was blood on his mouth, and his hair bristled around him, and his eyes were crusted with tears. He punched her temple.
She found herself on the ground. The colors of the world swam, yellow leaching into the green.
Now Claw stood over her, breathing hard. He had an erection.
She reached for him.
He grabbed her hand and squeezed it, hard, so that her fingers were bent back. She cried out as bones bent and snapped.
Then he walked around her, legs splayed, erection sticking out of his fur. He grabbed at the trees and waved branches at her.
She understood the signs he was making. She knew what he wanted, in his frustration, in his rage. But he was her brother. The thought of him lying on her filled her head with blackness, her throat with bile.
She turned over and tried to stand. But when she put her injured hand on the ground, pain flared, and she fell forward.
He stamped hard on her back. She was driven flat into the undergrowth. She felt his hands on her ankles. He dragged her back toward him and pulled her legs apart. He was stronger than she was; sprawled facedown on the ground, she could not fight him.
His shadow fell over her, looming.
In another bloody heartbeat he was inside her. He screamed, in pain or pleasure. Shadow called for her mother, but she was far away.
## _E mma Stoney_
The days here lasted about thirty hours. Emma timed them with her wristwatch and a stick stuck in the ground to track shadows.
_Thirty hours_. No possibility of a mistake.
Not Earth, she thought reluctantly. But that thought was unreal. Absurd.
She knocked over her stick and took her watch off her wrist and stowed it in a pocket, so she wouldn't have to look at it.
After the Elf attack, the three of them stayed on the open plain.
But every morning it was strange, disorienting, to wake among the hominids. Whichever of them woke first would take one look at the strangers and hoot and holler in alarm. Soon they would all be awake, all of them yelling and brandishing their fists, and Emma and the others would have to cower away, waiting for the storm to pass. At last, somebody would recognize them—Fire, or Stone, or one of the younger women. "Em-ma. Sal-ly." After that the others would gradually calm down.
But Emma would have sworn that some of them _never_ regained their memories of the day before, that every day they woke up not recognizing Emma and the others. It seemed they came awake with the barest memory of the detail of their lives before, as if every day were like a new birth.
Emma wasn't sure if she pitied them for that, or envied them.
The days developed a certain routine. Emma and Sally worked to keep themselves and Maxie clean; they would rinse out their underwear—they had only one set each, the clothes they had arrived in—and scrub the worst of the dirt off the rest of their clothes and gear.
The women had precisely two tampons between them. When they were gone, they labored to improvise towels from bits of cloth.
As evening drew in Emma and little Maxie would help build the hominids' haphazard fire by throwing twigs and branches onto it. Paying dues, Emma thought; making sure we earn our place in the warmth.
In the dark the hominids gathered close to the fire, she supposed for safety and warmth. But they didn't form into anything resembling a circle, as humans would. There were little knots of them, men testing their strength against each other, women with their children, pairs coupling with noisy (and embarrassing) enthusiasm. But there was no storytelling, no singing, no dancing. They even ate separately, each hunched over her morsel, as if fearful of having it stolen.
The group did not have the physical grammar of a group bound by language, Emma thought. This was not a true hearth. Their bits of words, their proto-language, were surely a lot closer to the screeches of chimps, or even the songs of birds, than the vocalizations of humans. Though the Runners huddled together for security, they lived their lives as individuals, pursuing solitary projects, each locked forever inside her own head.
_They aren't human_ , Emma realized afresh, however much they might look like it. And this wasn't a community. It was more like a herd.
As night fell, Emma and the others would creep into the shelter she had made with Fire. A few of the hominids followed them, mothers with nursing infants. Maxie cried and complained at the pungent stink of their never-washed flesh. But Emma and Sally calmed him, and themselves, assuring each other that they were surely safer here than in the open, or in the forest.
One child, looking no more than five or six years old in human terms, fell ill. Her eyelids, cheeks, nose, and lips were encrusted with sores. The child was skinny, and was evidently in distress; her gestures were faint, her movements listless.
"I think it's yaws," Sally said. "I've seen it upriver, in Africa... It's related to syphilis. But it's transmitted by flies, who carry it from wound to wound. That's where the first signs show: little bumps in the corners of your eyes, or your nostrils, where the flies go to suck your moisture."
"What's the cure?"
"A shot of Extencilline. Safeguards you for life. But we don't have any."
Emma rummaged through her medical pocket. "What about Floxapen?"
"Maybe. But you're crazy to use it up on _them_. We're going to need it ourselves. We'll get ulcers. We need it."
Emma struggled to read the directions on the little bottle. She found a scrap of meat, embedded a pill in it, and fed it to the child. It was hard to hold her hand near that swollen, grotesque face.
The next morning, she did the same. She kept it up until the Floxapen was gone. It seemed to her the child was getting gradually better.
Maybe it helped the Runners accept them. She wasn't sure if they understood what she was doing, if they saw the cause-and-effect relationship between her treatment and any change in the girl's condition.
Sally didn't try to stop her. But Emma could see she was silently resentful at what she regarded as a waste of their scarce resources. It didn't help relations between them.
Five or six days after their arrival, she woke to find shards of deep blue sky showing through the loosely-stacked branches above her. She threw off her parachute-silk blanket and crawled out of the shelter's rough opening.
It was the first time the sky had been clear since she had got here. The sun was low, but it was strong, its warmth welcome on her face. The sky was a rich beautiful blue, and it was scattered with clouds, and it was _deep_. She saw low cumulus clouds, fat and gray and slow, and higher cirruslike clouds that scudded across the sky, and wispy traces even above that: layers of cloud that gave her an impression of tallness that she had rarely, if ever, seen on Earth.
She tried to orient herself. If the sun was _that_ way, at this hour, she was looking east. And when she looked to the west—oh, my Lord—there was a Moon: more than half-full, a big fat beautiful bright Moon.
... _Too_ big, too fat, too bright. It had to be at least twice the diameter of the pale gray Moon she was used to. And it was no mottled gray disk, like Luna. This was a vibrant dish of color. Much of it was covered with a shining steel-blue surface that glimmered in the light of the sun. Elsewhere she saw patches of brown and green. At either extreme of the disk—at the poles, perhaps—she saw strips of blinding white. And over the whole thing clouds swirled, flat white streaks and stripes and patches, gathered in one place into a deep whirlwind knot.
_Ocean_ : That was what that shining steel surface must be, just as the brown-green was land. That wasn't poor dead Luna: It was a planet, with seas and ice caps and continents and air.
And she quickly made out a characteristic continent shape on that brightly lit quadrant, almost bare of cloud, baked brown, familiar from schoolbook studies and CNN reports and Malenfant's schoolkid slide-shows. It was Africa, quite unmistakeably, the place she had come from.
That was no "Moon." That was _Earth_.
And if she was looking at Earth, up in the sky, her relentlessly logical mind told her, then she couldn't be _on_ Earth any more. "Stands to reason," she murmured.
It made sense, of course: the different air, the lightness of walking, these alien not-quite humans running around everywhere. She had known it the whole time, on some level, but she hadn't wanted to face it.
But, if not on Earth, _where was she_? How had she got here? How was she ever going to get home again? All the time she had been here, she realized, she had got not one whit closer to answering these most basic questions.
Now a shadow passed over them, and Emma felt immediately cold. A new cloud was driving overhead, flat, thick, dark.
Sally was standing beside her. "They talk English."
"What?"
"The flat-heads. They talk English. Just a handful of words, but it is _English_. Remember that. They surely didn't evolve it for themselves."
"Somebody must have taught it to them."
"Yes." She turned to Emma, her eyes hard. "Wherever we are, we aren't the first to get here. We aren't alone here, with these apes."
She's right, Emma realized. It wasn't much, but it was a hope to cling to, a shred of evidence that there was more to this bizarre experience than the plains and the forests and the hominids.
Emma peered into the sky, where Earth was starting to set.
Malenfant, where are you?
## _R eid Malenfant_
Malenfant parked at the Beachhouse car park. Close to the Kennedy Space Center, this was an ancient astronaut party house that NASA had converted into a conference center.
Malenfant, in his disreputable track suit, found the path behind the house. He came to a couple of wooden steps and trotted down to the beach itself. The beach, facing the Atlantic to the east, was empty, as far as he could see. This was a private reserve, a six-mile stretch of untouched coastline NASA held back for use by astronauts and their families and other agency personnel.
It wasn't yet dawn.
He stripped off his shoes and socks and felt the cool, moist sand between his toes. Tiny crabs scuttled across the sand at his feet, dimly visible. He wondered whether they had been disturbed by the new Moonlight, like so many of the world's animals. He stretched his hams, leaning forward on one leg, then the other. Too old to skip your stretching, Malenfant, no matter what else is on your mind.
The Red Moon was almost full—the first full Moon since its appearance, and Emma's departure. A month already. The light cast by the Red Moon was much brighter than the light of vanished silvery Luna, bright enough to wash out all but the brightest stars, bright enough to turn the sky a rich deep blue—but it was an eerie glow, neither day nor night. It was like being in a movie set, Malenfant thought, some corny old 1940s musical with a Moon painted on a canvas sky.
Malenfant hated it all: the light, the big bowl of mystery up there in the sky. To him the Red Moon was like a glowing symbol of his loss, of Emma.
Breathing deeply of the salty ocean air, he jogged through gentle dunes, brushing past thickets of palmetto. It wasn't as comfortable a jog as it used to be: The beach had been heavily eroded by the tide, and it was littered with swathes of sea-bottom mud, respectably large rocks, seaweed and other washed-up marine creatures—not to mention a large amount of oil smears and garbage, some of it probably emanating from the many Atlantic wrecks. But to Malenfant the solitude here was worth the effort of finding a path through the detritus.
It had been another sleepless night. He was consumed with his desire to reach the Red Moon.
Frustrated by the reception his proposals were receiving at NASA headquarters in Washington, he had decided to take his schemes, his blueprints and models and Barco shows, around the NASA centers, to Ames and Marshall and Kennedy and Johnson, trying to drum up grassroots support, and put pressure on the senior brass.
_We can do this. We've been to the Moon before—a Moon, anyhow—and this new mother is a lot more forgiving than old Luna. Now we have an atmosphere to exploit. No need to stand on your rockets all the way from orbit; you can glide to the ground... We can throw together a heavy-lift booster from Shuttle components in months_. That one the challenge for Marshall, where von Braun had built his Moon rockets. For Kennedy and Johnson, where the astronauts worked: _We have whole cadres of trained, experienced and willing pilots, specialists and mission controllers itching to take up the challenge of a new Moon. Hell, I'll go myself if you'll let me_... He had appealed to the scientists, too, the geologists and meteorologists and even the biologists who suddenly had a whole new world to study: _It will be a whole new challengein human spaceflight, a world with oceans and an atmosphere—an oxygen atmosphere, by God—just three days away. It's the kind of world we were hoping we might find when we sent our first fragile ships out on the ocean of space half a century ago. And who knows what we'll discover there..._
And then there were the groups he had come to think of as the xenoologists: the biologists and philosophers and astronomers and others who, long before the sudden irruption of the Red Moon, had considered the deeper mysteries of existence: Are we alone? Even if not, why does it _seem_ that we are alone? If we were to meet others—what would they be like?
_Come on, people. Our Moon disappeared, and was replaced by another. How the hell? Can this possibly be some natural phenomenon? If not_ , who's responsible? _Not us, that's for sure. The greatest mystery of this or any other age is hanging up there like some huge Chinese lantern. Shouldn't we go take a look?_
But, to his dismay and surprise, he had gotten no significant support from anybody—save the wacko UFO-hunting fringe types, who did him more harm than good. NASA, through the Jet Propulsion Laboratory, was working on a couple of unmanned orbital probes and a lander to go visit the Red Moon. But that was it. The notion of sending humans to Earth's new companion was definitively out of the question.
So he had been told, gently but firmly, by Joe Bridges.
"In these road shows of yours you underestimate the magnitude of the task, Malenfant. Whether you're doing that deliberately or not isn't for me to say. We know diddly about the structure of the Red Moon's atmosphere, which is somewhat essential data before you even begin to develop your gliding lander. And then what about the cost and schedule implications of putting together your 'Big Dumb Booster'—a brand new man-rated heavy-lift launcher, for God's sake? Our analysis predicts a schedule of years and a cost of maybe a hundred billion bucks. We just don't have that kind of money, Malenfant. And NASA can't go asking for it right now. Get your head out of your ass and take a look around. _The Tide_. The human race has other priorities..."
The first sunlight began to seep into the Atlantic horizon, smears of orange and pink banishing the Red Moon's unnatural light. Malenfant's calves were beginning to tingle, and he could feel his breathing deepening, his heart starting to pound.
Too long since I did this.
He had gotten hooked on running in the dawn light during the preparation for his first spaceflight. Emma had complained that he was spending even less time with her, but as long as he crept out of bed without waking her she had seemed to forgive him. But then there always had been a lot she had had to forgive him for. Is that why I want to reach her—just so I can say I'm sorry? Well, is that so bad? Or is it selfish—do I just want to get to her so I can project even more of my own shit onto her?...
_Emma!_
He pounded on, the moist sand cold under every footstep. As his blood pumped he felt the structure of his thoughts dissolve, his obsessive nighttime round of planning and worrying and agonizing over I-should-have-said and I-should-have-done, all of it washing away. The main reason to exercise, he thought: It stops your brain working, lets your body remind you you're still an animal.
It was the only respite he got from being himself.
He'd meant to run a couple of miles before doubling back. But when he reached his turnback point he spotted something on the beach, maybe a mile farther south: blocky, silhouetted, very large, returning crumpled orange highlights to the approaching sun. A beached whale? The tide had played hell with migration patterns. No, too angular for that. A wreck, then?
On impulse he continued on down the beach.
The washed-up object was the size of a small house, twenty-five or thirty feet high. It was heavily eroded, its walls sculpted by wind and water into pits and pillars. When Malenfant stood at its foot the sea breeze that washed over it was distinctly colder.
He ran his hand over its surface. Under stringy seaweed he found a gray, pitted surface, cold and slick under his palm. Ice, of course. The dawn light was still dim, but he could make out the cold clean blue-white shine of the harder ice beneath. He wondered how long the berg would sit here before it melted into the sand.
It was here because of the Tide.
The first few days had been the worst, when Earth's oceans, subject to a sudden discontinuous shock, had sloshed like water in a bathtub. Millions of square miles of coastal lowland had been scoured. In some places, pushed by currents or channelled by sea bottoms, the oceans had spawned waves several hundred feet high, walls of water that had crushed everything in their paths.
After that, with twenty times the mass of Luna, the Red Moon raised daily tides twenty times as high as before—roughly anyhow; the new Moon's spin complicated the complex gravitational dance of the worlds.
The coastlines of the world had been drastically reshaped. The English Channel was being widened as the soft white chalk of the lands that bordered it, including Dover's white cliffs, was worn away. Even rocky coastlines like Maine were being eroded. The lowest tides on the planet used to be in the Gulf of Mexico, the Mediterranean, and elsewhere: Now those tides of two feet or less had become forty feet, and around the shores of the Mediterranean many communities, with roots dating back to the dawn of civilization, had been smashed and worn away in a matter of weeks. Meanwhile the tides had forced their way into the mouths of many of the world's rivers, making powerful bores a hundred feet high, and vast floodplains filled and drained with each ebb and flow, drowning some of the planet's most fertile land in salt water.
People had fled inland, a secondary tide of misery, away from the devastated coasts. Already there had been too many deaths even to count, from flooding and tsunamis and quakes—and there were surely many more to come, as the displaced populations succumbed to disease, and flooded-out farmers failed to return a crop, and as the wars broke out over remaining stocks.
Meanwhile, as the polar seas flexed, titanic rafts of ice broke away from the shelves of Antarctica and the glaciers of Alaska and Greenland. The larger bergs broke up in the tempestuous seas, but many of them survived to the Equator, filling the oceans, already all but impassable, with an additional hazard. And so bergs like this one were now common sights at all latitudes on the seaboards of the Atlantic and Pacific. In some places they were actually being mined to make up for the disrupted local supplies of clean, fresh water. Always a silver lining, Malenfant thought sourly.
He stripped off his sweaty track suit and ran naked into the surf. Deeply mixed by the Tide with the waters of the deep ocean, the sea was icy cold and very salty, stinging when it splashed his eyes and the scar tissue on his healing arm. He took care not to go far out of his depth; he could feel a strong undercurrent as the sea drew back.
He swam a few strokes and then lay on his back, studying the sky, buoyant in the salty water.
The Red Moon was fat and swollen in the sky above him. Though it had (somehow) inserted itself into the same orbit as the old, vanished Moon, it was more than twice Luna's diameter, as large in area as five old Moons put together—and a lot more than five times as bright, because of its reflective cloud and water.
And this morning, the Red Moon was blue. The hemisphere facing him showed a vast, island-strewn ocean, blue-black and cloud-littered, with the shining white of ice caps at the northern and southern extremes. The Red Moon's north pole was tilted towards Earth by ten degrees or so, and Malenfant could see a huge high-pressure system sitting over the pole, a creamy swirl of cloud. But dark bands streaked around the equator, clouds of soot and smoke.
Malenfant, for all his personal animosity, admitted that the new Moon was hauntingly lovely. It even _looked_ like a world: obviously three-dimensional, with that shading of atmosphere at the sunlit limb, and sun casting a big fat highlight on its wrinkled ocean skin, as if it was some immense bowling ball. Poor Luna had been so dust-choked that its scattered light had made it look no more spherical than a painted dinner-plate.
Malenfant had, understandably obsessive, kept up with the evolving science of the Red Moon.
The new Moon turned on its axis relative to Earth—unlike departed, lamented Luna—with a "day" of about thirty hours, so that Earthbound watchers were treated to views of both sides. The other hemisphere was dominated by the worldlet's main landmass: a supercontinent, some called it, a roughly circular island-continent with a center red as baked clay, and fringed by gray-green smears that might be forests. The Red Moon was hemispherically asymmetric, then: like Mars and Luna, unlike Earth and Venus.
That great continent was pitted by huge, heavily eroded impact craters: To Malenfant they were an oddly pleasing reminder of true, vanished Luna. And the center of the supercontinent was marked by a single vast volcano that thrust much of the way out of the atmosphere. Its immense, shallow flanks, as seen in the telescope, were marked at successively higher altitudes by (apparent) rings of vegetation types, what appeared to be glaciers, and then by bare rock, giving it, to terrestrial observers, something of the look of a shooting target. (And so the commentators had called it Bullseye.)
The Red Moon's mightiest river rose on the flanks of the Bullseye. Perhaps that great magma upwelling had lifted and broken ancient aquifers. Or perhaps air uplifted by the great mountain was squeezed dry of its water by altitude. Anyhow the river snaked languidly across a thousand miles to the eastern coast, where it cut through a mountain chain there to reach the sea at a broad delta.
There were mountains on both east and west coasts of the supercontinent. They were presumably volcanoes. Those on the east coast appeared to be dormant; they were heavily eroded, and they seemed to cast a rain shadow over the desiccated interior of the continent. There was, however, a comparatively lush belt of vegetation between the mountains and the coast. The commentators had called it the Beltway. The greenery pushed its way into the interior of the continent in a narrow strip along the valley of that great river, which was a Nile for this small world.
But the mountains on the west coast were definitely not dormant. Presumably prompted by rock tides induced by Earth's gravity field, they had been observed to begin erupting a few days after the Red Moon's arrival in orbit around Earth.
They must have been spectacular eruptions. Thick, dense rock near the surface appeared to have blocked the magma flows, bottling up increasing pressure before yielding explosively like a champagne cork flying out of a bottle. On Earth, such stratovolcanoes—like Mount Fuji, Mount Rainier—could eject debris miles into the air. On the Moon the volcanoes had blown debris clear of the planet altogether. Meanwhile vast quantities of dust and gases had been pumped into the atmosphere, to spread in thick bands around much of the Moon's middle latitudes.
There was a great deal you could tell about the Red Moon, even from a quarter-million miles, with telescopes and spectrometers and radar, as the two hemispheres conveniently turned themselves up for inspection. For instance, those oceans really were water. The temperature range was right—as you'd expect since the Moon shared Earth's orbit around the sun—and examination of the visible and infrared spectra showed that the cloud caps were made of water vapor, just the right amount to have evaporated off the oceans.
The Red Moon's surface gravity was some two-thirds Earth's—a lot more than Luna's, and, crucially, enough for this miniature planet to have retained all the essential ingredients of an Earthlike atmosphere: oxygen, nitrogen, carbon, water vapor, carbon dioxide—unlike poor barren Luna. So the Red Moon had water oceans and a nitrogen-oxygen atmosphere.
Already the study of the Red Moon had revolutionized the young science of planetology. With a quarter of Earth's mass—but four times the mass of Mars, some twenty times the mass of Luna—the Red Moon was a planet in its own right, intermediate in size between the Solar System's small and large denizens, and so a good test bed for various theories of planetary formation and evolution.
It differed in key ways from Earth. Because it was so much smaller, it must have started its formation (wherever _that_ had occurred) with a much smaller supply of heat energy than Earth. And that inner heat had been rapidly dissipated through its surface.
Like a shrivelled orange, the Red Moon's rind was thick. Probably eons ago, the tectonic plates fused, and continents no longer slid over its face. There was no continental drift, no tectonic cycling, no oceanic ridges. Unlike Earth, the Moon's uncycled surface was very ancient; and that was why the interior of the continent bore those huge eroded craters, the scars left by immense impacts long ago.
And that was why the Bullseye was so vast. The huge shield mountain had probably formed over a fountain of magma erupting through a flaw in the crust layers. The crust beneath it must have been held in place over the flaw for hundreds of millions of years—so it more resembled Mars's Olympus Mons than, say, Earth's Hawaiian islands.
But there was more than geology up there. On the Red Moon, it appeared, there was life.
The air was Earthlike, containing around a sixth oxygen—a smaller proportion than Earth's atmosphere, but difficult to explain away by nonliving processes. It hadn't taken long to establish that the green-gray pigment that stained the fringes of the supercontinent and its wider river valleys, as well as the shallower sections of the world's ocean, was chlorophyll, the green of plants. There were other fingerprints of a living world: an excess of methane in the air, for example, put there perhaps by bacteria in bogs, or burning vegetation, or even the farts of Moon-calves. Though some scientists remained skeptical—and though nobody could say for sure if the Red Moon harbored anything like bogs or bacteria or cows—most people seemed to concur that there was indeed life on the Red Moon, life of some sort.
But was there intelligence?
Nobody had detected any structured radio signals. There had been no response to various efforts to signal to the Red Moon using radio and TV and laser, not to mention a few wacko methods, like the cutting of a huge right-angled triangle of ditches into the Saharan desert filled with burning oil.
But what were the mysterious lights that flickered over the night lands? Most observers claimed they were forest fires caused by lightning or drought. Perhaps, perhaps not. Could the streaming "wakes" sometimes visible on the great oceans be the wakes of ships, or were they simply peculiar meteorological features? And what about the geometrical traces—circles, rectangles, straight lines—that some observers claimed to have made out in clearings along the coasts and river valleys of the Red Moon's single huge continent? What were they but evidence of intelligence?
And if any of these signs were artificial, what kind of being might live up there to make them?
Malenfant was willing to admit that one manned expedition could do little to probe the mysteries of a world with fully half the surface area of Earth. But there were mysteries that no amount of remote viewing could unravel. The fact was, the most powerful telescope could not resolve an individual human being up there.
Malenfant was never going to find Emma by staring up from Earth.
But at this time of crisis, nobody wanted to see Malenfant's drawings of rocket boosters and gliding spaceships.
Of course there was the question of resources, of priorities. But Malenfant suspected that people were shying away from dealing with the most fundamental issue here: the existence of the Red Moon itself. It was just too big, too huge, impossible to rationalize or grasp or extrapolate. The Wheel was different. A blue circle in the air, a magic doorway? Yes, we can imagine ways we might do that, even if we can't think why we should. Peculiar-looking human beings falling out of the air? Yes, we know about the plasticity of the genome; we can even imagine time travel, the retrieval of our flat-browed ancestors. But _what kind of power hangs a new Moon in our sky?_
He didn't last long in the water; it was too cold. He took a few brisk strokes until the water was shallow enough for him to walk. He splashed out of the surf, shivering, briskly dried himself on his shirt, and began to pull on his pants.
There was somebody standing beside the beached berg fragment, just a slim shadow in the gray dawn light, watching him.
## _F ire_
Maxie is running around Fire's feet. _"Hide and seek. Hide and seek, Fire. Hide and seek."_
Fire stares at Maxie. To him the boy is a blur of movement and noise, unpredictable, incomprehensible, fascinating.
Maxie has leaves on his head. They flutter away as he runs. Sally puts them back on. _"No, Maxie,"_ she says. _"Be careful of the sun."_
_"Hide and seek, hide and seek."_ He stands still. His hands cover his eyes. _"Hands, Fire, eyes, Fire."_ His hands cover his eyes.
Fire puts his hands over his eyes. It is dark. The night is dark. He starts to feel sleepy.
Maxie calls, _"Eight nine ten ready! Fire Fire Fire!"_
Fire lowers his hands. It is not night. The sunlight is bright. The world is red and green and blue. He blinks.
Maxie has gone away.
Fire sees Sing on her bower of leaves. He walks toward her. He has forgotten Maxie.
Maxie is at his feet. _"Here I am, here I am!"_ Maxie stamps his foot. Red dust rises and sticks to Maxie's white flesh. _"You have to try, you silly. You have to play it right. Try again, try again. Eyes, Fire, hands, Fire."_ He covers his eyes.
As the sun climbs into the sky, the game goes on. Every time Maxie disappears Fire forgets about him. Every time he comes back Fire is surprised to see him.
Fire grows hungry. Fire thinks of himself in the forest, eating nuts and berries and leaves. Fire lopes toward the forest.
_"Come back, come back, you nasty!"_ Maxie falls to the dirt and howls.
Emma comes running to Fire. _"Fire, are you going to the forest? Can I come with you?"_
_Fire. Forest_. That is what Fire hears.
"Em-ma," he says.
Emma has blue hair. Fire frowns. He thinks of Emma with brown hair. Fire's hand touches Emma's hair. The blue hair is smooth like skin. It has bits of white vine stuck to it.
Emma says, _"It's just a hat, Fire. Just parachute silk."_ She puts the blue hair back on her head and pulls the vines under her chin. _"Can I come to the forest?"_
There is something on Emma's chest. It is bright red. Berries are bright red. Fire touches the berry. It is hard. It is stuck to a vine. The vine is around Emma's neck. His teeth bite the berry-thing. It is hard, like a nut. His teeth cannot break the shell.
Emma pulls it back from him. _"It's my knife, Fire. I showed you yesterday. And the day before. And the day before that. Look."_ Emma touches her knife. When she shows him again, there is a red part, and a part like a raindrop. There is a spot of light behind the raindrop, on Emma's hand. Emma is smiling. _"See, Fire? The lens? Remember this?"_
Fire sees the raindrop and the light. He hoots.
Emma steps away. _"Emma hungry. Emma forest. Fire forest. Emma Fire forest."_
Fire thinks of Emma and Fire in the forest, gathering berries, eating berries. He smiles. "Emma Fire forest. Berries trees nuts."
Emma smiles. _"Good. Let's go."_ She takes his hand.
The forest is a huge mouth. It is dark and green and cool.
He waits at the edge of the forest. His ears listen, his eyes see. The forest is still.
His legs walk into the wood. His feet explore the ground, finding soft bare earth. His arms and his torso and his head duck around branches. He is not thinking of how his body is moving.
His eyes learn to see the dark. His nose smells, his ears listen. He is not aware of time passing, of the sun climbing in the sky, of the dappled bits of light at his feet sliding over the forest floor detritus.
He sees a pitcher plant. It is a big purple sac, high above his head. His hands pull it down. There is water in the pitcher plant. There are insects in the water. His hand scoops out water and insects. He drinks the water. It tastes sweet. His teeth crunch the insects.
Emma is here. He has forgotten she was here. He gives her the pitcher plant.
Her hand lifts water and bugs to her mouth. She coughs. She spits out insects.
His eyes see a cloudberry plant. It has white flowers and pink fruit. His hands pull the fruit from the plant, avoiding the spiky brambles. His mouth chews the berries.
Here is Emma. Her hands explore the blue skin on her legs. Now she has a soft shining thing in her hands. Her hands open a mouth in the shining thing. She feeds the mouth with berries. He can see them in the stomach of the shining thing.
She holds up the shining thing. _"This is a bag, Fire. These berries are for Sally and Maxie. I can carry more in the bag than I can with my hands. You see?..."_
He thinks of Sally eating berries. He thinks of Maxie eating berries.
He thinks of Sing, on her bower. He thinks of Sing eating berries. His hands pluck berries. His mouth wants to eat the berries, but he thinks of Sing eating them. He keeps the berries in his hands.
His legs move him on. Soon he forgets about Sing, and his mouth eats the berries.
He finds a chestnut tree. It has leaves the size of his hands and sticky buds and nuts. Beneath the chestnut something white is growing. His hands and eyes explore it. It is a morel, a mushroom. His hands pull great chunks of it free, and lift them to his mouth.
Emma is here. Her hands are taking nuts from the chestnut. The nuts want to hurt Emma. He slaps her hands so they stop taking the nuts.
His ears hear a grunt, a soft rustle.
He stops thinking. He stops moving. His ears listen, his nose smells, his eyes flicker, searching.
His eyes see a dark form, squat. It has arms that move slowly. He sees eyes glinting in the green gloom. He sees ears that listen. He sees orange-brown hair, a fat heavy gut, a head with huge cheeks, a giant jaw.
It is a Nutcracker-man.
The Nutcracker-man grunts. He lifts pistachio nuts to his huge mouth. Fire can see his broad, worn teeth, glinting in the dappled light. The Nutcracker-man grinds the nuts between his giant teeth.
Fire's mouth fills with water, to tell him it wants the nuts.
Fire stands up suddenly. He rattles branches and throws twigs. "Nutcracker-man, Ho!"
The Nutcracker-man screeches, startled. His arms lift him into a tree and swing him away, crashing through foliage, bits of nut falling from his mouth.
Fire pushes through the brush. His hands cram the nuts into his grateful mouth.
Emma is here. Her hands are taking nuts and putting them into the mouth of the shining thing.
His nose can still smell the dung of the Nutcracker-man. He thinks of many Nutcracker-folk, out in the shadows of the forest.
His legs take him away from the place with the pistachio nuts, back toward the open daylight.
Emma follows him. But he has forgotten Emma. He remembers the nuts and the fungus and the Nutcracker-man.
## _R eid Malenfant_
He kept right on pulling on his pants. When he was done, his breath misting slightly, he walked up the slope of the eroded beach.
His silent observer was a woman: little more than a girl, really, slim, composed, dark. She was wearing a nondescript jumpsuit. She was very obviously Japanese.
"I know you," he said.
"We have not met." Her voice was deep, composed. "But, yes, I know you, too, Reid Malenfant."
"Just Malenfant," he said absently, trying to place her. Then he snapped his fingers. "You were on Station when—"
"—when the Moon changed. Yes. My name is Nemoto." She bowed. "I am pleased to meet you."
He bowed back. He felt awkward. He couldn't care less if she had glimpsed his wrinkly ass. But he wished, oddly, that he had his shoes on.
He looked up and down the beach. He saw no sign of transportation, not so much as a bicycle. "How did you get here?"
"I walked. I have a car, parked at the Beachhouse."
"As I have."
"Yes."
"Will you walk back that way with me?"
"Yes."
Side by side, in the gathering pink-gray light, they walked north along the beach.
Malenfant glanced sideways at Nemoto. Her face was broad, pale, her eyes black; her hair was elaborately shaved, showing the shape of her skull. She could have been no more than half Malenfant's age, perhaps twenty-five.
"The Red Moon is very bright," she said.
"Yes."
"It is a great spectacle. But it will be bad for the astronomers."
"You were an astronomer..."
"I am an astronomer."
"Yeah. Sorry."
Nemoto was a Japanese citizen trained as an astronaut at NASA. Her speciality had been space-based astronomy. She had been the brilliant kid who had made it all the way into space at the incredibly young age of twenty-four. He remembered Nemoto as being bright, excitable, even bubbly. Well, she wasn't bright and bubbly now. It was as if she had gone into eclipse.
"I have been looking for you," she said now. "I have missed you several times in your tour of the NASA centers. Malenfant, when you are not at your scheduled meetings, you are something of a recluse."
"Yeah," he said ruefully. "Nowadays more than I'd like to be."
"You miss your wife," she said bluntly.
"Yes. Yes, I miss my wife."
"I almost found you at your church."
"The chapel at Ellington Air Force Base?"
"I had not realized you are Catholic."
"I guess you should call me lapsed. I converted when I married Emma, back in '82. Emma, my wife. It was for the sake of her family. When I joined NASA we looked around for a chapel. Ellington was near Johnson, and a lot of my colleagues and their families went there, and we liked the priest..."
"Are you religious now?"
"No." He had tried, for the sake of the priest, Monica Chaum, as much as anybody else. But, unlike some who came back from space charged with religious zeal, Malenfant had lost it all when he made his first flight into orbit. Space was just too _immense_. Humans were like ants on a log, adrift in some vast river. How could any Earth-based ritual come close to the truth of the God who had made such a universe?
"So I gave up the chapel. It caused some problems with Emma's family. But she supported me. She always did."
"But now you have returned to the faith?"
"No. I do find the chapel kind of restful. But I get a lot more comfort from going out on a toot with Monica Chaum over at the Outpost. She has quite a capacity for a woman Catholic priest. I make no excuses; I've been through a lot." He eyed her. "As have you."
"Yes." Her face, never beautiful, was empty of expression. "As is well known."
Nemoto had been aboard the International Space Station, in low Earth orbit, when the Red Moon had made its dramatic entrance. Nemoto had been forced to watch from orbit as the first great tides battered at Japan.
"I returned to Earth as soon as I could. I and my colleague used our Japanese Hope shuttle. You may know that our landing facility was at Karitimati Island in the South Pacific—"
"Where? Oh, yeah, Christmas Island."
"There is little left of Karitimati. We were forced to come down here, at KSC."
He said carefully, "Where was your home?"
"I have no home now," was all she would reply.
He nodded. "Nor do I." It was true. He had an empty house in Clear Lake, but the hell with that. His home was with Emma—wherever she was.
Nemoto paused and looked into the sky. Although the first liquid glimmer of sun was resting on the horizon, the Red Moon still shone bright in the sky. "If you have abandoned your attempts to acquire faith, you do not believe that God is responsible for _that_?"
He grinned, rubbing his hand over his bare scalp, feeling a rime of salt there. "Not God, no. But I think _somebody_ is."
"And you would like to find out who."
"Wouldn't you?"
"Do you believe that the bodies which fell through the African portal were human?"
He frowned, taken aback by the question. "Nobody can make much of the mashed-up remains that they scraped out of the savannah."
"But they appear to be human, or a human variant. You _saw_ them, Malenfant. I've read your testimony. They share our DNA—much of it, though the recovered sequences show a large diversity from our own genome. There is speculation that they are more like one of our ancestors, a primitive hominid species."
"Yeah. So there are ape-men running all over our new Moon up there, right? I read the tabloids, too."
"Malenfant, what do _you_ believe?"
He said fiercely, "I believe that the Wheel was some kind of portal. I believe it linked Earth to its new moon. And I believe it transported those poor unevolved saps, here from there. What I don't know is what the hell it all means."
"And you believe your wife made the return journey. That she is still alive up there on the Red Moon, breathing its air, drinking its water, perhaps eating its vegetation."
"Where else could she be?... I'm sorry. It's what I want to believe, I guess. It's what I have to believe."
"Yes." She smiled. "Everybody knows this, Malenfant. Your longing to reach her is tangible. I can see it, now, in your eyes, the set of your body."
"You think I'm an asshole," he said brutally. "You think I should let go."
"No. I think you are fully human. This is to be admired."
He felt awkward again. He'd only just met this girl, yet somehow she'd already seen him naked every which way a person could be naked.
They reached the Beachhouse. They sat on its porch, facing the ocean. Malenfant sipped water from a plastic bottle. "So how come you've been pursuing me around NASA? What do you want, Nemoto?"
"I believe we can help each other. You want to set up a mission to reach the Red Moon. So do I. I believe we should. I believe we must. _I can get you there_."
Suddenly his heart was pumping. "How?"
Rapidly, with the aid of a pocket softscreen, she sketched out a cut-down mission profile, using a simplified version of Malenfant's Shuttle-based Big Dumb Booster design, topped by a Space Station evacuation lander, adapted for the Moon's conditions. "It will not be safe," she said. "But it will work. And it could be done, we believe, in a couple of months, at a cost of a few billion dollars."
It was fast and dirty, even by the standards of the proposals he had been touting himself. But it could work... "If we could get anybody to fund it."
"There are many refugee Japanese who would support this," Nemoto said gravely. "Of all the major nations it is perhaps the Japanese who have suffered most in this present disaster. Among the refugees, there is a strong desire at least to _know_ , to understand what has caused the deaths of so many. Thus there are significant resources to call on. But we would need to work with NASA, which has the necessary facilities for ground support."
"Which is where I come in." He drank his water. "Nemoto, maybe you're speaking to the wrong guy. I've already tried, remember. And I got nowhere. I come up against brick walls like Joe Bridges the whole time."
"We must learn to work with Mr. Bridges, not against him."
"How?"
She touched his hand. Her skin was cold. He was shocked by the sudden, unexpected contact. "By telling the truth, Malenfant. You care nothing for geology or planetology or the mystery of the Red Moon, or even the tide, do you? You want to find Emma." She withdrew her hand. "It is a motive that will awaken people's hearts."
"Ah. I get it. You want me to be a fundraiser. To blub on live TV."
"You will provide a focus for the project—a _human_ reason to pursue it. At a time when the waters are lapping over the grain fields, nobody cares about science. But they always care about family. We need a story, Malenfant. A hero."
"Even if that hero is a Quixote."
She looked puzzled. "Quixote's was a good story. And so will yours be."
She didn't seem in much doubt that he'd ultimately fall into line. And, looking into his heart, neither did he.
Irritated by her effortless command, he snapped, "So why are _you_ so keen to go exploring the new Moon, Nemoto? Just to figure out why Japan got trashed?... I'm sorry."
She shrugged. "There is more. I have read of your speeches on the Fermi Paradox."
"I wouldn't call them speeches. Bullshit for goodwill tours..."
"As a child, your eyes were raised to the stars. You wondered who was looking back. You wondered why you couldn't see them. Just as I did, half a world away."
He gestured at the Moon. "Is that what you think this is? We were listening for a whisper of radio signals from the stars. You couldn't get much less subtle a first contact than _this_."
"I think this huge event is more than that—even more significant. Malenfant, _people rained out of the sky_. They may or may not belong to a species we recognize, but they were people. It is clear to me that the meaning of the Red Moon is intimately bound up with us: what it is to be human—and why we are alone in the cosmos."
"Or _were_."
"Yes," she said. "And, consider this. This Red Moon simply appeared in our sky... It is not as if a fleet of huge starships towed it into position. We don't know how it got there. _And we don't know how long it will stay_ , conveniently poised next to the Earth. The Wheel disappeared just hours after it arrived. If we don't act now—"
"Yes, you're right. We must act urgently." The sun was a shimmering globe suspended on the edge of the ocean, and Malenfant began to feel its heat draw at the skin of his face. "We've a lot to talk about."
"Yes."
They walked up the path to their cars.
## _F ire_
The sun is above his head. The air is hot and still. The red ground shines brightly through brittle grass. People move to and fro on the red dust.
Fire thinks of Dig. He thinks of himself touching Dig's hair, her dugs, the small of her back. His member stiffens. His eyes and ears seek Dig. They don't find her.
He sees Sing.
Sing is lying flat her bower, in the sun. Her head does not rise. Her hand does not lift from where it is sprawled in the red dirt. Her legs are splayed. Flies nibble at her belly and eyes and mouth.
Fire squats. His hands flap at the flies, chasing them away. He shakes Sing's shoulder. "Sing Sing Fire Sing!"
She does not move. He puts his finger in her mouth. It is dry.
Fire picks up Sing's hand. It is limp, but her arm is stiff. He drops the hand. The arm falls back with a soft thump. Dust rises, falls back.
Emma is beside him.
_"Fire. Maxie is ill. Perhaps you can help. Umm, Maxie sore Maxie. Fire Maxie... Fire, is something wrong?"_
Her eyes look at Sing. Her hands press at Sing's neck. Emma's head drops over Sing's mouth, and her ear listens.
Fire thinks of Sing laughing. She is huge and looms over him. Her face blocks out the sun.
He looks at the slack eyes, the open mouth, the dried drool. This is not Sing.
His legs stand him up. He bends down and lifts the body over his shoulders. It is stiff. It is cold.
Emma stands. _"Fire? Are you all right?"_
Fire's legs jog downwind. They jog until his eyes see the people are far away. Then his arms dump the body on the ground. It sprawls. He hears bones snap. Gas escapes from its backside.
Bad meat.
He jogs away, back to the people.
He goes to Sing's bower. But the bower is empty. People are here, and then they are gone, leaving no memorials, no trace but their children, as transient as lions or deer or worms or clouds. Sing is gone from the world, as if she never existed. Soon he will forget her.
He scatters the branches with his foot.
Emma is watching him.
Sally is here, holding Maxie. Maxie is weeping. Emma says, _"Fire, I'm sorry. Can you help us? I don't know what to do..."_
Fire grins. He reaches for Maxie.
Maxie cringes. Sally pulls him back.
Emma says, _"No, Fire. He doesn't want to play. Fire Maxie ill sick sore."_
Fire frowns. He touches Maxie's forehead. It is hot and wet. He touches his belly. It is hard.
He thinks of a shrub with broad, coarse-textured leaves. He does not know why he thinks of the shrub. He doesn't even formulate the question. The knowledge is just there.
He lopes to the forest. His ears listen and his eyes peer into the dark greenery. There are no Nutcracker-folk. There are no Elf-folk.
He sees the shrub. He reaches out and plucks leaves.
His legs take him out of the forest.
Maxie stares at the leaves. Water runs down his face.
Fire pokes a leaf into his small, hot mouth. Maxie's mouth tries to spit it out. Fire pushes it back. Maxie's mouth chews the leaf. Fire holds his jaw so the mouth can't chew.
Maxie swallows the leaf, and wails.
Fire makes him swallow another. And another.
Somebody is shouting. "Meat! Meat!"
Fire's head snaps around. The voice is coming from upwind. Now his nose can smell blood.
Something big has died.
His legs jog that way.
He finds Stone and Blue and Dig and Grass and others. They are squatting in the dirt. They hold axes in their hands.
The meat is an antelope. It is lying on the ground.
Killing birds are tearing at the carcass.
The killing birds tower over the people. They have long gnarled legs, and stubby useless wings, and heads the size of Fire's thigh. The heads of the birds dig into the belly and joints of the antelope, pushing right inside the carcass.
The people wait, watching the birds.
A pack of hyenas circles, warily watching the birds and the people. And there are Elf-folk. They sit at the edge of the forest, picking at their black-brown hair. The bands of scavengers are set out in a broad circle around the carcass, well away from the birds, held in place by a geometry of hunger and wariness. The Running-folk are scavengers among the others—not the weakest, not the strongest, not especially feared. The people wait their turn with the others, waiting for the birds to finish, knowing their place.
One by one the birds strut away. Their heads jerk this way and that, dipping. Their eyes are yellow. They are looking for more antelopes to kill.
The hyenas are first to get to the corpse. Their faces lunge into its ripped-open rib cage. The hyenas start to fight with one another, forgetting the killing birds, forgetting the people.
Blue and Stone and Fire hurl bits of rock.
The dogs back away. Their muzzles are bloody red, their eyes glaring. Their mouths want the meat. But their bodies fear the stones and sticks of the people.
The people fall on the carcass.
Stone's axe, held between thumb and forefinger, slices through the antelope's thick hide. The axe rolls to bring more of its edge into play. It slices meat neatly from the bones. The birds have beaks to rip meat. The hyenas and cats have teeth. The people have axes. The people work without speaking, not truly cooperating.
Fire's hands cram bits of meat into his mouth, hot and raw. Fire thinks of the other people by the fire, the women and their infants and children with no name. He tells his mouth it must not eat all the meat. He holds great slabs of it in his hands, slippery and bloody.
Fire's ears hear a hollering. His head snaps around.
More Elf-folk are boiling out of the forest fringe, hooting, hungry. They have rocks and stones and axes in their hands. They run on their legs like people. But their legs are shorter than a person's, and they have big strong arms, longer and stronger than a person's.
Stone growls. His mouth bloody, he raises his axe at the Elf-folk.
The Elf-folk show their teeth. They hoot and screech.
A bat swoops from the sky. It is a hunter. Its wings are broad and flap slowly. The people scatter, fearing talons and beak.
The bat falls on the Elf-folk. It caws. It rises into the air. It has its talons dug into the scalp of an Elf-woman. She wriggles and cries, dugs swinging.
One Elf-man throws a rock at the bat. It misses. The others just watch. She is gone, in an instant, her life over.
Suddenly Stone charges forward at the Elf-folk. Blue follows. Dig follows.
The Elf-folk scamper away, into the safety of their forest.
Stone hoots his triumph.
The people return to the antelope. The hyenas have approached again, and bats have flown down, digging into the entrails of the antelope. The people hurl stones and shout. The people's hands take meat and bones from the carcass, until their hands are full. The people's mouths dig into the carcass and bite away final chunks of meat.
Other scavengers move in. Soon there will be nothing left of the antelope but scattered, crushed, chewed bones, over which insects will crawl.
The children fall on the meat. Their mouths snap and their hands punch and scratch as they fight over the meat.
Fire approaches Dig. He holds out meat. Her hands grab it. She throws it away. A child with no name falls on the discarded scrap.
Dig laughs. She turns her back on Fire.
Emma comes to Fire. She smiles, seeing the meat. His belly wants to keep all the meat, but he makes his hands give her some.
Emma takes it to the fire. There are rocks in the fire. Emma beats the meat flat and puts it on the hot rocks. She peels it off the rocks and carries it to Sally and Maxie.
Fire squats on the ground. His hands tear meat. His teeth crush it.
Emma stands before him. She is smiling. She pulls his hand.
His legs follow her.
She stops by a patch of dung. The dung is pale and watery and smelly. There is a leaf in the dung. There is a worm on the leaf, dead.
Emma says, _"I think you did it, Doctor Fire. You got the damn worm out of him."_
Fire does not remember the leaf, or Maxie. Emma's mouth is still moving, but he does not think about the noises she makes.
## _R eid Malenfant_
A flock of pigeons flew at the big Marine helicopter. Such was their closing speed that the birds seemed to explode out of the air all around them, a panicky blur of gray and white. The pilot lifted his craft immediately, and the pigeons fell away.
Nemoto's hands were over her mouth.
Malenfant grinned. "Just to make it interesting."
"I think the times are interesting enough, Malenfant."
"Yeah."
Now the chopper rolled, and the capital rotated beneath him. They flew over the Lincoln, Jefferson, and Washington monuments, set out like toys on a green carpet, and to the right the dome of the Capitol gleamed bright in the sunlight, showing no sign of the hasty restoration it had required after last month's food riots.
The helicopter leveled and began a gentle descent toward the White House, directly ahead. The old sandstone building looked as cute as it always had, depending on your taste. But now it was surrounded by a deep layer of defenses, even including a moat around the perimeter fence. And, save for a helipad, the lawn had been turned to a patchwork of green and brown, littered with small outbuildings. In a very visible (though hardly practical) piece of example-setting, the lawn had been given over to the raising of vegetables and chickens and even a small herd of pigs, and every morning the president could be seen by webcast feeding his flock. It was not a convincing portrait, Malenfant always thought, even if the prez was a farmer's son. But for human beings, it seemed, symbolism was everything.
The helicopter came down to a flawless landing on the pad. Nemoto climbed out gracefully, carrying a rolled-up softscreen. Malenfant followed more stiffly, feeling awkward to have been riding in a military machine in his civilian suit—but he was a civilian today, at the insistence of the NASA brass.
An aide greeted them and escorted them into the building itself. They had to pass through a metal-and-plastics detector in the doorway, and then spent a tough five minutes in a small security office just inside the building being frisked, photographed, scanned, and probed by heavily-armed Marine sergeants. Nemoto even had to give up her softscreen after downloading its contents into a military-issue copy.
Nemoto seemed to withdraw deeper into herself as they endured all this.
"Take it easy," Malenfant told her. "The goons are just doing their job. It's the times we live in."
"It is not that," Nemoto murmured. "It is this place, this moment. From orbit, I watched the oceans batter Japan. I felt I was in the palm of a monster immeasurably more powerful than me—a monster who would decide the fate of myself, and my family, and all I possessed and cared for, with an arbitrary carelessness I could do nothing to influence. And so, I feel, it is now. But I must endure."
"You really want to go on this trip, don't you?"
She glanced at him. "As you do."
"You always deflect my questions about yourself, Nemoto. You are a _koan_. An enigma."
She smiled at that fragment of Japanese.
At last they were done, and the aide, accompanied by a couple of the armed Marines, took them through corridors to the Oval Office, on the West Wing's first floor, which the vice president was using today. Her official residence, a rambling brick house on the corner of 34th Street and Massachusetts Avenue, was no longer considered sufficiently secure.
Nemoto said as they walked, "You say you know Vice President Della."
"Used to know her. She's had an interest in space all her career. As a senator she served on a couple of NASA oversight committees." Now the president had asked Della to take responsibility for Malenfant's project, in her capacity as chair of the Space Council.
Nemoto said, "If she is a friend of yours—"
"Hardly that. More an old sparring partner. Mutual, grudging respect. I haven't seen her for a long time—certainly not since she got _here_."
"Do you think she will support us?"
"She's from Iowa. She's a canny politician. She is—practical. But she has always seen a little further than most of the Beltway crowd. She believes space efforts have value. But she's a utilitarian. I've heard her argue for weather satellites, Earth resources programs. She even supports blue-sky stuff about asteroid mining and power stations in orbit. Moving the heavy industries off the planet might provide a future for this dirty old world... But robots can do all that. I don't think she sees much purpose in Man in Space. She never supported the Station, for instance."
"Then we must hope that she sees some utility in our venture to the Red Moon."
He grimaced. "Either that or we manage to twist her arm hard enough."
As they entered the Oval Office, Vice President Maura Della was working through documents on softscreens embedded in a walnut desk. The desk was positioned at one of the big office's narrow ends—the place really was oval-shaped, Malenfant observed, gawking like a tourist.
Della glanced up, stood, and came out from behind the desk to greet them. Dressed in a trim trouser suit, she was dark, slim, in her sixties. She shook them both briskly by the hand, waved them to green wing-back chairs before the desk, then settled back into her rocking chair.
The only other people in the room were an aide and an armed Marine at the door. Malenfant had been expecting Joe Bridges, and other NASA brass.
Without preamble Della said, "You're trying to get me over a barrel, aren't you, Malenfant?"
Malenfant was taken aback. This was, after all, the vice president. But he could see from the glint in Della's eye that if he wanted to win the play this was a time for straight talking. "Not you personally. But—yes, ma'am, that's the plan."
Della tapped her desk. Malenfant glimpsed his own image scrolling before her, accompanied by text and video clips and the subdued insect murmur of audio.
Maura Della always had been known for a straightforward political style. To Malenfant she looked a little lost in the cool grandeur of the Oval Office, even after three years in the job, out of place in the crispness of the powder-blue carpet and cream paintwork, and the many alcoves crammed with books, certificates, and ornaments, all precisely placed, like funerary offerings. This was clearly not a room you could feel you lived in.
There was a stone sitting on the polished desk surface, a sharp-edged fragment about the size of Malenfant's thumb, the color of lava pebbles. No, not stone, Malenfant realized, studying the fragment. _Bone_. A bit of skull, maybe.
Della said, "Your campaign has lasted two weeks already, in every media outlet known to man. Reid Malenfant the stricken hero, tilting at the new Moon to save his dead wife." She eyed him brutally.
"It has the virtue of being true, ma'am," Malenfant said frankly. "And she may not be dead. That's the whole point."
Nemoto leaned forward. "If I may—"
Della nodded.
"The response of the American public to Malenfant's campaign has been striking. The latest polls show—"
"Overwhelming support for what you're trying to do," Della murmured. She tapped her desk and shut down the images. "Of course they do. But let me tell you something about polls. The president's own approval ratings have been bouncing along the floor since the day the tides began to hit. You know why? Because people need somebody to blame.
"The appearance of a whole damn Moon in the sky is beyond comprehension. If as a consequence your house is smashed, your crops destroyed, family members injured or dead, you can't blame the Moon, you can't rage at the tide. In another age you might have blamed God. But now you blame whoever you think ought to be helping you climb out of your hole, which generally means all branches of the federal government, and specifically this office." She shook her head. "So polls don't drive me one way or the other. Because whatever I decide, your stunt isn't going to help _me_."
"Perhaps not," said Nemoto. "But it might help the people beyond this office. The people of the world. And that is what we are talking about, isn't it?"
Malenfant covered her hand. _Take it easy_.
Della glared. "Don't presume to tell me my job, young woman." Then she softened. "Even if you're right." She turned to a window. "God knows we need some good news... You know about the quakes."
"Yes, ma'am," Malenfant said grimly.
This was the latest manifestation of the Red Moon's baleful influence. Luna had raised tides in Earth's rock, just as in its water. Luna's rock tides had amounted to no more than a few inches.
But the Red Moon raised great waves several feet high.
Massive earthquakes had occurred in Turkey, Chile, and elsewhere, many of them battering communities already devastated by the effects of the tide. In fault zones like the San Andreas in California, the land above the faults was being eroded away much more rapidly than before, thus exposing the unstable rocks beneath, and exacerbating the tidal flexing of the rocks themselves.
Della said, "The geologists tell me that if the Red Moon stays in orbit around Earth, it is possible that the fault lines between Earth's tectonic plates—such as the great Ring of Fire that surrounds the Pacific—will ultimately settle down to constant seismic activity. _Constant_. I can't begin to imagine what that will mean for us, for humanity. No doubt devastating long-term impacts on the Earth's climate, all that volcanic dust and ash and heat being pumped into the air... When I look into the future now, the only rational reaction is dread and fear."
"People need to see that we are hitting back," Malenfant said. "That we are _doing_ something."
"Perhaps. That is the American way. The myth of action. But does our action hero have to be _you_ , Malenfant? And what happens when you crash up there, or die of starvation, or burn up on re-entry? How will _that_ play in the polls?"
"Then you find another hero," Malenfant said stonily. "And you try again."
"But even if you make it to the Moon, what will you find? You should know I've had several briefings in preparation for this meeting. One of them was with Dr. Julia Corneille, from the department of anthropology at the American Museum of Natural History. An old college friend, as it happens."
"Anthropology?"
"Actually Julia's speciality is paleoanthropology. Extinct homs, the lineage of human descent. You see the relevance."
"Homs?"
"Hominids." Della smiled. "Sorry. Field slang. You can tell I spent some time with Julia... She told me something of her life, her work in the field. Mostly out in the desert heartlands of Kenya."
"Looking for fossils," Malenfant said.
"Looking for fossils. People don't leave many fossils, Malenfant. And they don't just lie around. It took Julia years before she learned to pick them out, tiny specks against the soil. It's a tough place to work, harsh, terribly dry, a place where all the bushes have thorns on them... Fascinating story." She picked up the scrap of bone from her desk. "This was the first significant find Julia made. She told me she was engaged on another dig. She was walking one day along the bed of a dried-out river, when she happened to glance down... Well. It is a fragment of skull. A trace of a woman, of a species called _Homo erectus_. The _Erectus_ were an intermediate form of human. They arose perhaps two million years ago, and became extinct a quarter-million years ago. They had bodies close to modern humans, but smaller brains—perhaps twice the size of chimps'. But they were phenomenally successful. They migrated out of Africa and covered the Old World, reaching as far as Java."
Malenfant said dryly, "Fascinating, ma'am. And the significance—"
"The significance is that the homs who rained out of the sky, on the day you lost your wife, Malenfant, appear to have been _Homo erectus_. Or a very similar type."
There was a brief silence.
"But if _Erectus_ died out two hundred and fifty thousand years ago, what is he doing raining out of the sky?"
"That is what you must find out, Malenfant, if your mission is approved. Think of it. What if there _is_ a link between the homs of the Wheel and ancestral _Erectus_? Well, how can that be? What does it tell us of human evolution?" Della fingered her skull fragment longingly. "You know, we have spent billions seeking the aliens in the sky. But we were looking in the wrong place. The aliens aren't separated from us by distance, but by time. Here—" she said, holding out the bit of bone " _—here_ is the alien, right here, calling to us from the past. But we have to infer everything about our ancestors from isolated bits of bone—the ancient homs' appearance, gait, behavior, social structure, language, culture, tool-making ability—everything we know, or we think we know about them. We can't even tell how many species there were, let alone how they lived, how they _felt_. You, on the other hand, might be able to view them directly." She smiled. "Even _ask_ them. Think what it would mean."
Malenfant began to see the pattern of the meeting. In her odd mix of hard-nosed skepticism at his mission plans, and wide-eyed wonder at what he might find up there, Della was groping her way toward a decision. His best tactic was surely to play straight.
Nemoto had been listening coldly. She leaned forward. "Madam Vice President. You want this Dr. Corneille to have a seat on the mission."
Ah, Malenfant thought. Now we cut to the horse-trading.
Della sat back in her rocker, hands settling over her belly. "Well, they sent geologists to the Moon on Apollo."
" _One_ geologist," said Malenfant. "Only after years of infighting. And Jack Schmitt was trained for the job; he made sure he was, in fact. As far as I know there are no paleoanthropologists in the Astronaut Office."
"Would there be room for a passenger?"
Malenfant shook his head. "You've seen our schematics."
Della tapped her desk, and brought up computer-graphic images of booster rockets and spaceplanes. "You are proposing to build a booster from Space Shuttle components."
"Our Saturn V replacement, yes."
"And you will glide down into the Red Moon's atmosphere in a—what is it?"
"An X-38. It is a lifting body, the crew evacuation vehicle used on the Space Station. We will fit it out to keep us alive for the three-day trip. On the surface we will rendezvous with a package of small jets and boosters for the return journey, sent up separately. The whole mission design is based around a two-person crew. Madam Vice President, we just couldn't cram in anybody else."
"Not on the way out," Della said evenly. " _Two out, three back_. Isn't that your slogan, Malenfant?"
"That's the whole idea, ma'am. And those outbound two have to be astronauts. The best scientist in the world will be no use on the Red Moon dead."
"The same argument was used to keep scientists off Apollo," Della said.
"But it is still valid."
Nemoto said coldly, "The reality is that I must fly this mission because the Japanese funding depends on it. And Malenfant must fly the mission—"
"Because the American public longs for him to go," Della sighed. "You're right, of course. If this mission is approved, then it will be you two sorry jerks who fly it."
_If_. Malenfant allowed himself a flicker of hope.
Nemoto seemed to be growing agitated. "Madam Vice President, _we must do this_. If I may—" She leaned forward and unrolled her softscreen on Della's desktop.
Della watched her blankly. Malenfant had no idea where this was leading.
"There is evidence that similar events have touched human history before, evidence buried deep in our history and myths. Consider the story of Ezekiel, from the Old Testament: 'And when the living creatures went, the wheels went by them: and when the living creatures were lifted up from the Earth, the wheels were lifted up.' Or consider a tale from the ancient Persian Gulf, about an 'animal endowed with reason called Oannes, who used to converse with men but took no food... and he gave them an insight into letters and sciences and every kind of art—' "
Shit, Malenfant thought.
Della was keeping her face straight. "So is this your justification for a billion-dollar space mission? UFOs from the Bible?"
Nemoto said, "My point is that the irruption of the Red Moon is the greatest event in modern human history. It will surely shape our future— _as it has our past_. The emergence of the primitive hominids from Malenfant's portal tells us that. This one event is the pivot on which history turns."
"I feel I have enough on my plate without assuming responsibility for all human history."
Nemoto subsided, angry, baffled.
Della said bluntly, "However I do need to know why you are trying to kill yourselves."
Malenfant bridled. "The mission profile—"
"—is a death-trap. Come on, Malenfant; I've studied space missions before."
Malenfant sat up straight, Navy style. "We don't have time not to buy the risks on this one, ma'am."
"You're both obsessed enough to take those risks. That's clear enough. Nemoto I think I understand."
"You do?"
Della smiled at Nemoto. "Forgive me, dear. Malenfant, she may be an enigma to you, but that's because she's young. She lost her family, her home. She wants revenge."
Nemoto did not react to this.
"But what about you, Malenfant?"
"I lost my wife," he said angrily. "That's motive enough. With respect, ma'am."
She nodded. "But you are grounded. Let me put it bluntly, because others will ask the same question many times before you get to the launch pad. Are you going back to space to find your wife? Or are you using Emma as a lever to get back into space?"
Malenfant kept his face blank, his bearing upright. He wasn't about to lose his temper with the Vice President of the United States. "I guess Joe Bridges has been talking to you."
She drummed her fingers on her desk. "Actually he is pushing you, Malenfant. He wants you to fly your mission." She observed his surprise. "You didn't know that. You really don't know much about people, do you, Malenfant?"
"Ma'am, with respect, does it matter? If I fly to the Red Moon, whatever my motives, I'll still serve your purposes." He eyed her. "Whatever they are."
"Good answer." She turned again to her softscreen. "I'm going to sleep on this. Whether or not you bring back your wife, I do need you to bring us some good news, Malenfant. Oh, one more thing. Julia's ape-men falling from the sky... You should know there are a lot of people very angered at the interpretation that they might have anything to do with the origins of humankind."
Malenfant grunted. "The crowd that thinks Darwin was an asshole."
Della shrugged. "It's the times, Malenfant. Today only forty percent of American schools teach evolution. I'm already coming under a lot of pressure from the religious groups over your mission, both from Washington and beyond."
"Am I supposed to go to the Red Moon and convert the ape-men?"
She said sternly, "Watch your public pronouncements. You will go with God, or not at all." She fingered the bit of hominid skull on her desk. " 'O ye dry bones, hear the word of the Lord.' "
"Pardon?"
"Our old friend Ezekiel. Chapter thirty-seven, verse four. Good day."
## _E mma Stoney_
There were bees that swarmed at sunset. Some of them stung, but you could brush them away, if you were careful. But there were other species which didn't sting, but which gathered at the corner of the mouth, or the eyes, or at the edge of wet wounds, apparently feeding on the fluids of the body.
You couldn't relax, not for a minute.
Uncounted days after her arrival, Emma woke to find an empty shelter.
She threw off her parachute silk and crawled out of the shelter's rough opening. The sun was low, but it was strong, its warmth welcome on her face.
Sally's hair was a tangled mess, her safari suit torn, bloody and filthy. Maxie clung to her leg. Sally was pointing toward the sun. "They're leaving."
The Runners were walking away. They moved in their usual disorganized way, scattered over the plain in little groups. They seemed to be empty-handed. They had abandoned everything, in fact: their shelters, their tools. Just up and walked away, off to the east. Why?
"They left us," Maxie moaned.
A shadow passed over them, and Emma felt immediately cold. She glanced up at the deep sky. Cloud was driving over the sun.
A flake touched her cheek.
Something was falling out of the sky, drifting like very light snow. Maxie ran around, gurgling with delight. Emma held out her hand, letting a flake land there. It wasn't cold: in fact, it wasn't snow at all.
It was ash.
"We have to go, don't we?" Sally asked reluctantly.
"Yes, we have to go."
"But if we leave here, how will they find us?"
_They?_ What "they"? The question seemed almost comical to Emma.
But she knew Sally took it very seriously. They had spent long hours draping Emma's parachute silk over rocks and in the tops of trees, hoping its bright color might attract attention from the air, or even from orbit. And they had labored to pull pale-colored rocks into a vast rectangular sigil. None of it had done a damn bit of good.
There was, though, a certain logic to staying close to where they had emerged from the wheel-shaped portal. After all, who was to say the portal wouldn't reappear one day, as suddenly as it had disappeared, a magic door opening to take them home?
And beyond that, if they were to leave with the Runners—if they were to walk off in some unknown direction with these gangly, naked not-quite-humans—it would feel like giving up: a statement that they had thrown in their lot with the Runners, that they had accepted that _this_ was their life now, a life of crude shelters and berries from the forest and, if they were lucky, scraps of half-chewed, red-raw meat: _This_ was the way it would be for the rest of their lives.
But Emma didn't see what the hell else they could do.
They compromised. They spent a half-hour gathering the largest, brightest rocks they could carry, and arranging them into a great arrow that pointed away from the Runners' crude hearth, toward the east. Then they bundled up as much of their gear as they could carry in wads of parachute silk, and followed the Runners' tracks.
Emma made sure they stayed clear of a low heap of bones she saw scattered a little way away. She was glad it had never occurred to Sally to ask hard questions about what had become of her husband's body.
The days wore away.
Their track meandered around natural obstacles—a boggy marsh, a patch of dense forest, a treeless, arid expanse—but she could tell that their course remained roughly eastward, away from the looming volcanic cloud.
The Runners seemed to prefer grassy savannah with some scattered tree cover, and would divert to keep to such ground—and Emma admitted to herself that such parklike areas made her feel relatively comfortable, too, more than either dense forest or unbroken plains. Maybe it was no coincidence that humans made parks that reminded them, on some deep level, of countryside like this. I guess we all carry a little Africa around with us, she thought.
She was no expert on botany, African or otherwise. It did seem to her there were a lot of fernlike trees and relatively few flowering plants, as if the flora here was more primitive than on Earth. A walk in the Jurassic, then.
As for the fauna, she glimpsed herds of antelopelike creatures: Some of them were slim and agile, who would bolt as the Runners approached, but others were larger, clumsier, hairier, crossing the savannah in heavy-footed gangs. The animals kept their distance, and she was grateful for that. But again they didn't strike her as being characteristically _African_ : She saw no elephants, no zebra or giraffes. (But then, she told herself, there were barely any elephants left in Africa anyhow.)
It was clear there were predators everywhere. Once Emma heard the throaty, echoing roar of what had to be a lion. A couple of times she spotted cats slinking through brush at the fringe of forests: leopards, perhaps.
And once they came across a herd—no, a _flock_ —of huge, vicious-looking carnivorous birds.
The flightless creatures moved in a tight group with an odd nervousness, pecking at the ground with those savagely curved beaks, and scratching at their feathers and cheeks with claws like scimitars. Their behavior was very birdlike, but unnerving in creatures so huge.
The Runners took cover in a patch of forests for a full half-day, until the flock had passed.
The Runners called them "killing birds." A wide-eyed Maxie called the birds "dinosaurs."
And they did look like dinosaurs, Emma thought. Birds had evolved from dinosaurs, of course; here, maybe, following some ecological logic, birds had lost their flight, had forgotten how to sing, but they had rediscovered their power and their pomp, becoming lords of the landscape once more.
The Runners' gait wasn't quite human. Their rib cages seemed high and somewhat conical, more like a chimp's than a human's, and their hips were very narrow, so that each Runner was a delicately balanced, slim form with long striding legs.
Emma wondered what problems those narrow hips caused during childbirth. The heads of the Runners weren't that much smaller than her own. But there were no midwives here, and no epidurals either. Maybe the women helped each other.
Certainly each of them clearly knew her own children—unlike the men, who seemed to regard the children as small, irritating competitors.
The women even seemed to use sex to bond. Sometimes in the night, two women would lie together, touching and stroking, sharing gentle pleasures that would last much longer than the short, somewhat brutal physical encounters they had with the men.
By comparison, the men had no real community at all, just a brutish ladder of competition: They bickered and snapped among themselves, endlessly working out their pecking order. At that, Emma thought, this bunch of guys had a lot of common with every human mostly-male preserve she had ever come across, up to and including the NASA Astronaut Office.
Stone was the boss man; he used his fists and feet and teeth and hand-axes to keep the other men in their place, and to win access to the women. But he, and the other men, did not seek to injure or kill his own kind. It was all just a dominance game.
And Stone was not running a harem here. With all that fist-fighting he won himself more rolls in the hay than the other men, but the others got plenty, too; all they had to do was wait until Stone was asleep, or looking the other way, or was off hunting, or just otherwise engaged. Emma had no idea why this should be so. Maybe you just couldn't run a harem in a highly mobile group like this; maybe you needed a place to hold your female quasi-prisoners, a fortress to defend your "property" from other men.
It was what these people _lacked_ that struck Emma most strongly. They had no art, no music, no song. They didn't even have language; their verbless jabber conveyed basic emotions—anger, fear, demands—but little information. They only "talked" anyhow in social encounters, mating or grooming or fighting, never when they were working, making tools or hunting or even eating. She thought their "talk" had more in common with the purring and yowling of cats than information-rich human conversation.
Certainly the Runners never discussed where they were going. It was clear, though, from the way they studied animal tracks, and fingered shrubs, and sniffed the wind, that they had a deep understanding of this land on which they lived, and knew how to find their way across it.
... Yes, but how did that knowledge get there, if not through talking, learning? Maybe a facility for tracking was hard-wired into their heads at birth, she speculated, as the ability to pick up language seemed to be born with human infants.
Whatever, it was a peculiar example of how the Runners could be as smart as any human in one domain—say, tracking—and yet be dumber than the smallest child in another—such as playing Maxie's games of hide-and-seek and catch. It was as if their minds were chambered, some rooms fully stocked, some empty, all of the chambers walled off from each other.
When the Runners stopped for the night, they would scavenge for rocks and bits of wood and quickly make any tools they needed: hand axes, spears. But they carried nothing with them except chunks of food. In the morning, when it was time to move on, they would just drop their hand axes in the dirt and walk away, sometimes leaving the tools in the mounds of spill they had made during their creation.
Emma saw it made sense. It only took a quarter-hour or so to make a reasonable hand axe, and the Runners were smart at finding the raw materials they needed; they presumably wouldn't stop in a place that couldn't provide them in that way. To invest fifteen minutes in making a new axe was a lot better than spending all day carrying a lethally sharp blade in your bare hands.
All this shaped their lifestyle, in a way she found oddly pleasing. They had no possessions. If they wanted to move to some new place they just abandoned everything they had, like walking out of a house full of furniture leaving the doors unlocked. When they got to where they were going they would just make more of whatever they needed, and within half a day they were probably as well-equipped as they had been before the move. There must be a deep satisfaction in this way of life, never weighed down by possessions and souvenirs and memories. A clean self-sufficiency.
But Sally was dismissive. "Lions don't own anything either. Elephants don't. Chimps don't. Emma, these ape-men are animals, even if they are built like basketball players. The notion of possessing anything that doesn't go straight in their mouths has no more meaning to them than it would to my pet cat."
Emma shook her head, troubled. The truth, she suspected, was deeper than that.
Anyhow, people or animals, the Runners walked, and walked, and walked. They were black shadows that glided over bright red ground, hooting and calling to each other, nude walking machines.
Soon Emma's socks were a ragged bloody mess, and where her boots didn't fit quite right they chafed at her skin. A major part of each new day was the foot ritual, as Emma and Sally lanced blisters and stuffed their battered boots with leaves and grass. And if she rolled up her trousers, wet sores, pink on black, speckled her shins; Sally suffered similarly. They took turns carrying Maxie, but they were laden down with their parachute silk bundles, and a lot of the time he just had to walk as best he could, clinging to their hands, wailing protests.
During the long days of walking, Emma found herself inevitably spending more time than she liked with Sally.
Emma and Sally didn't much like each other. That was the blunt truth.
There was no reason why they should; they had after all been scooped at random from out of the sky, and just thrown together. At times, hungry or thirsty or frightened or bewildered, they would take it out on each other, bitching and arguing. But that would always pass. They were both smart enough to recognize how much they needed each other.
Still, Emma found herself looking down on Sally somewhat. Riding on her husband's high-flying career, Sally had gotten used to a grander style of life than Emma had ever enjoyed, or wanted. Emma had often berated herself for sacrificing her own aspirations to follow her husband's star, but it seemed to her that Sally had given up a lot more than she had ever been prepared to.
For the sake of good relations, she tried to keep such thoughts buried.
And Emma had to concede Sally's inner toughness. She had after all lost her husband, brutally slain before her eyes. Once she was through the shock of that dreadful arrival, Sally had shown herself to be a survivor, in this situation where a lot of people would surely have folded quickly.
Besides, she had achieved a lot of things Emma had never done. Not least raising kids. Maxie was as happy and healthy and sane as any kid his age Emma had ever encountered. And there turned out to be a girl, Sarah, twelve years old, left at home in Boston for the sake of her schooling while her parents enjoyed their extended African adventure.
Now, of course, this kid Sarah was left effectively orphaned. Sally told Emma that she knew that even if she didn't make it home her sister would take care of the girl, and that her husband's will and insurance cover would provide for the rest of her education and beyond. But it clearly broke her up to think that she couldn't tell Sarah what had become of her family.
It seemed odd to Emma to talk of wills and grieving relatives—as if they were corpses walking around up here on this unfamiliar Moon, too dumb to know they were dead—but she supposed the same thing must be happening in her family. _Her_ will would have handed over all her assets to Malenfant, who must be dealing with her mother and sister and family, and her employers would probably by now be recruiting to fill an Emma-shaped hole in their personnel roster.
But somehow she never imagined Malenfant grieving for her. She pictured him working flat-out on some scheme, hare-brained or otherwise, to figure out what had happened to her, to send her a message, even get her home.
Don't give up, Malenfant; I'm right here waiting for you. And it is, after all, your fault that I'm stuck here.
One day at around noon, with the sun high in the south, the group stopped at a water hole.
The three humans sat in the shade of a broad oaklike tree, while the Runners ate, drank, worked at tools, played, screwed, slept, all uncoordinated, all in their random way. Maxie was playing with one child, a bubbly little girl with a mess of pale brown hair and a cute, disturbingly chimplike face.
All around the Runners, a fine snow of volcano ash fell, peppering their dark skins white and gray.
The woman called Wood approached Emma and Sally shyly, her hand on her lower belly. Emma had noticed she had some kind of injury just above her pubis. She would cover it with her hand, and at night curl up around it, mewling softly.
Emma sat up. "Do you think she wants us to help?" Maybe the Runners had taken notice of her treatment of the child with yaws after all.
"Even if she does, ignore her. We aren't the Red Cross."
Emma stood and approached the woman cautiously. Wood backed away, startled. Emma made soothing noises. She got hold of the woman's arm, and, gently, pulled her hand away.
"Oh God," she said softly.
She had exposed a raised, black mound of infection, as large as her palm. At its center was a pit, deep enough for her to have put her fingertip inside, pink-rimmed. As Wood breathed the sides of the pit moved slightly.
Sally came to stand by her. "That's an open ulcer. She's had it."
Emma rummaged in their minuscule medical kit.
"Don't do it," Sally said. "We need that stuff."
"We're out of dressings," Emma murmured.
"That's because we already used them all up," Sally said tightly.
Emma found a tube of Savlon. She got her penknife and cut off a strip of chute fabric. The ulcer stank, like bad fish. She squeezed Savlon into the hole, and wrapped the strip of fabric around the woman's waist.
Wood walked away, picking at the fabric, amazed, somehow pleased with herself. Emma found she had used up almost all the Savlon.
Sally glowered. "Listen to me. While you play medicine woman with these flat-heads..." She made a visible effort to control her temper. "I don't know how long I can keep this up. My feet are a bloody mass. Every joint aches." She held up a wrist that protruded out of her grimy sleeve. "We must be covering fifteen, twenty miles a day. It was bad enough living off raw meat and insects while we stayed in one place. Now we're burning ourselves up."
Emma nodded. "I know. But I don't see we have any choice. It's obvious the Runners are fleeing something: the volcanism maybe. We have to assume they know, on some level anyhow, a lot more than we do."
Sally glared at the hominids. " _They killed my husband_. Every day I wake up wondering if today is the day they will kill and butcher me, and my kid. Yes, we have to stick with these flat-heads. But I don't have to be comfortable with it. I don't have to _like_ it."
A Runner hunting party came striding across the plain. They brought chunks of some animal: limbs covered in orange hair, a bulky torso. Emma saw a paw on one of those limbs: not a paw, a _hand_ , hairless, its skin pink and black, every bit as human as her own.
Nobody offered them a share of the meat, and she was grateful.
That night her sleep, out in the open, was disturbed by dreams of flashing teeth and the stink of raw red meat.
She thought she heard a soft padding, smelled a bloody breath. But when she opened her eyes she saw nothing but Fire's small blaze, and the bodies of the Runners, huddled together close to the fire's warmth.
She closed her eyes, cringing against the ground.
In the morning she was woken by a dreadful howl. She sat up, startled, her joints and muscles aching from the ground's hardness.
One of the women ran this way and that, pawing at the rust-red dirt. She even chased some of the children; when she caught them she inspected their faces, as if longing to recognize them.
Sally said, "It was the little brown-haired kid. You remember? Yesterday she played with Maxie."
"What about her?"
Sally pointed at the ground.
In the dust there were footprints, the marks of round feline paws, a few spots of blood. The scene of this silent crime was no more than yards from where Emma had slept.
After a time, in their disorganized way, the Runners prepared to resume their long march. The bereft mother walked with the others. But periodically she would run around among the people, searching, screaming, scrabbling at the ground. The others screeched back at her, or slapped and punched her.
This lasted three or four days. After that the woman's displays of loss became more infrequent and subdued. She seemed immersed in a mere vague unhappiness; she had lost something, but what it was, and what it had meant to her, were slipping out of her head.
Only Emma and Sally (and, for now, Maxie) remembered who the child had been. For the others, it was as if she had never existed, gone into the dark that had swallowed up every human life before history began.
## _R eid Malenfant_
As soon as Malenfant had landed the T-38 and gotten out of his flight suit, here was Frank Paulis, running across the tarmac in the harsh Pacific sunlight, round and fat, his bald head gleaming with sweat.
Paulis enclosed Malenfant's hand in two soft, moist palms. "I can't tell you what a pleasure it is to meet you at last. It's a great honor to have you here."
Malenfant extracted his hand warily. Paulis looked thirty-five, maybe a little older. His eyes shone with what Malenfant had come to recognize as hero worship.
That was why he was here at Vandenberg, after all: to scatter a little stardust on the overworked, underpaid legions of engineers and designers who were laboring to construct his Big Dumb Booster for him. But he hadn't expected it of a hard-headed entrepreneur type like Frank Paulis.
They clambered into an open-top car, Paulis and Malenfant side-by-side in the back. An aide, a trim young woman Paulis called Xenia, climbed into the driver's seat and cut in the SmartDrive. The car pulled smoothly away from the short airstrip.
They drove briskly along the empty roads here at the fringe of Vandenberg ASFB. To either side of the car there were low green shrubs speckled with bright yellow flowers. They were heading west, away from the sun and toward the ocean, and toward the launch facility.
Paulis immediately began to chatter about the work they were doing here, and his own involvement. "I want you to meet my engine man, an old buzzard called George Hench, from out of the Air and Space Force. Of course he still calls it just the Air Force. He started working on missile programs back in the 1950s..."
Malenfant sat back in the warm sunlight and listened to Paulis with half an ear. It was a skill he'd developed since the world's fascinated gaze had settled on him. Everybody seemed a lot more concerned to tell him what _they_ felt and believed, rather than listen to whatever he had to say. It was as if they all needed to pour a little bit of their souls into the cranium of the man who was going to the Red Moon on their behalf.
Whatever. So long as they did their work.
They rose along a slight incline and headed along a rise. Now Malenfant could see the ocean for the first time since landing. This was the Pacific coast of California, some hundred miles north of Los Angeles. The ocean was a heaving gray mass, its big waves growling. The ground was hilly, with crags and valleys along the waterline and low mountains in the background.
The area struck him as oddly beautiful. It wasn't Big Sur, but it was a lot prettier than Canaveral.
But the big Red Moon hung in the sky above the ocean, its parched desert face turned to the Earth, and its deep crimson color made the water look red as blood, unnatural.
The coastline here had not been spared by the Tide; shore communities like Surf had been comprehensively obliterated. But little harm had come to this Air and Space Force Base, a few miles inland. Canaveral, on the other hand, on Florida's Atlantic coast, had been severely damaged by the Tide. So Vandenberg had been the default choice to construct the launch facilities for Malenfant's unlikely steed.
The car slowed to a halt. They were in the foothills of the Casmalia Hills here. From this elevated vantage Malenfant could see a sweep of lowland speckled with concrete splashes linked by roadways: launch pads, many of them decommissioned.
Beyond that he made out blocky white structures. That was the Shuttle facility itself, the relic of grandiose 1970s Air Force dreams of pilots in space. The launch pad itself looked much like its siblings on the Atlantic coast: a gaunt service structure set over a vast flame pit, with gaping vents to deflect the smoke and flame of launch. The gantry was accompanied to either side by two large structures, boxy, white, open, both marked boldly with the USASF and NASA logos. The shelters were mounted on rails and could be moved in to enclose and protect the gantry itself.
It was nothing like Cape Canaveral. The place had the air of a construction site. There were trailers scattered over the desert, some sprouting antennae and telecommunications feeds. There weren't even any fuel tanks, just fleets of trailers, frost gleaming on their flanks. Engineers, most of them young, moved to and fro, their voices small in the desert's expanse, their hard hats gleaming like insect carapaces. There was an air of improvisation, of invention and urgency, about this pad being reborn after two decades under wraps.
"This has been a major launch center since 1958," Paulis said, sounding as proud as if he'd built the place himself. "Many of them polar launches. Good site for safety: If you go south of here, the next landmass you hit is Antarctica... Slick-six—sorry, SLC-6—is the southernmost launch facility here. It was originally built back in the 1960s to launch a spy-in-the-sky space station for the Air Force, which never flew. Then they modified it for the Air Force Shuttle program. But the Shuttle never flew from here either, and after _Challenger_ the facility was left dormant.'
"I guess it took a lot of un-mothballing," Malenfant said.
"You got that right."
And now, right at the heart of the gray industrial-looking equipment of the Shuttle facility, he made out a slim spire, brilliant white, nestling against its gantry as if for protection.
It looked something like the lower half of a Space Shuttle—two solid rocket boosters strapped to a fat, rust-brown external fuel tank—but there was no moth-shaped Shuttle orbiter clinging to the tank. Instead the tank was topped by a blunt-nosed payload cover almost as fat and wide as the tank itself. The stack vented vapor, and Malenfant could see ice glimmer on its unpainted flanks; evidently the engineers were running a fueling test.
Malenfant felt the hairs on the back of his neck stand up.
It was he who had produced the first back-of-the-envelope sketch of a Big Dumb Booster like this, sketches to show how Shuttle technology could be warped and mutated to manufacture a heavy-lift launcher, a remote descendant of the Saturn V, for this one-shot project. With Nemoto's backers in place he had led the way in fleshing out the design, based on ancient, never-funded studies from the 1970s and 1980s. He had overseen the computer-graphic simulations, the models. His fingerprints were all over the whole damn project.
But it was not until now, this oddly mundane moment here on this hillside, in a cheap car with jabbering Paulis and taciturn Xenia, that he had actually set eyes on his BDB: his Big Dumb Booster, the spaceship whose destiny would shape the rest of this life, one way or the other.
But it was Paulis who had got the thing built.
Even after Malenfant had been given presidential approval, such strict limits had been placed on budget and schedule that the NASA brass had soon realized they would need input from the private sector. They had turned to Boeing, their long-term partners in running the Shuttle, but Paulis had been quick to thrust himself forward. Frank J. Paulis had made his fortune from scratch; unusually for his generation he had made most of it from heavy engineering, specifically aerospace. He had made promises of impressive funding and the use of his design, manufacture, and test facilities around the country—in return for a senior management position on the BDB project.
NASA had predictably rebuffed him. Paulis had handed over his money and facilities anyhow.
But after a couple of months, when the first calamities had predictably hit the project and the schedule had begun to fall apart before it had properly started, NASA, under pressure from the White House, had turned to Paulis.
Paulis's first public act, in front of the cameras, had been to gather an immense heap of NASA documentation before the launch pad. "This isn't Canaveral, and this is not the Shuttle program," he'd told his bemused workers. "We can't afford to get tied up in a NASA paper trail. I invest the responsibility for quality in you, each and every one of you. I trust you to do your jobs. All I ask is that you do it right." And he set the documentation heap alight with a flame-thrower.
There were some, raised all their careers in NASA's necessarily safety-obsessed bureaucracy, who couldn't hack it; Paulis had had a twenty percent drop-out. But the rest had cheered him to the Pacific clouds.
After that, Paulis had proven himself something of a genius in raising public interest in the project. A goodly chunk of the booster when it lifted from its pad would be paid for by public subscriptions, raised every which way from Boy Scout lemonade stalls to major corporate sponsors; in fact when it finally took off the BDB's hide would be plastered with sponsors' logos. But Malenfant couldn't care less about that, as long as it _did_ ultimately take off, with him aboard.
Paulis, remarkably, was still talking, a good five minutes since Malenfant had last spoken.
"... The stack is over three hundred feet tall. You have a boat-tail of four Space Shuttle main engines here, attached to the bottom of a modified Shuttle external tank, so the lower stage is powered by liquid oxygen and hydrogen. You'll immediately see one benefit over the standard Shuttle design, which is in-line propulsion; we have a much more robust stack here. The upper stage is built on one Shuttle main engine. Our performance to low Earth orbit—"
Malenfant touched his shoulder. "Frank. I do know what we're building here."
"... Yes." Nervously, Paulis dug out a handkerchief and wiped sweat from his neck. "I apologize."
"Don't apologize."
"It's just that I'm a little over-awed."
"Don't be." Malenfant was still studying the somewhat squat lines of the booster stack. "Although I feel a little awe myself. I've come a long way from the first rocket I ever built."
At age seventeen, Malenfant was already building and flying model airplanes. With some high school friends he started out trying to make a liquid-fueled rocket, like the BDB, but failed spectacularly, and so they switched to solid fuels. They bought some gunpowder and packed it inside a cardboard tube, hoping it would burn rather than explode. "We propped it against a rock, stuck on some fins, and used a soda straw packed with powder for a fuse. We spent longer painting the damn thing than constructing it. I lit the fuse at a crouch and then ran for cover. The rocket went up fifty feet, whistling. Then it exploded with a bang—"
Paulis said, reverent, "And Emma was watching from her bedroom window, right? But she was just seven years old."
Malenfant was aware that the girl driver, Xenia, was watching him with a hooded, judgmental gaze.
Weeks back, in the course of his campaign to build support, he'd told the story of the toy rocket to one of his PR flacks, and she had added a few homey touches—of course Emma hadn't been watching; though she had been a neighbor at that time, at seven years old she had much more important things to do—and since then the damn anecdote had been copied around the planet.
His life story, suitably edited by the flacks, had become as well known as the Nativity story. His feelings of satisfaction at seeing the booster stack evaporated.
He really hadn't expected this kind of attention. But just as Nemoto had predicted, and just as Vice President Della's political instincts had warned her, Malenfant and his brave, lunatic stunt had raised public spirits at a time when many people were suffering grievously. In the end it wouldn't matter _what_ he did—people seemed to understand that there was no conceivable way he was going to "solve" the problem of the Red Moon—but as long as he pursued his mission with courage and panache, he would be applauded; it was as if everybody was escaping the suffering Earth with him.
But the catch was they all wanted a piece of him.
Paulis was still talking. "That thing in the sky changed everything. It didn't just deflect the tides. It deflected all our lives—mine included. When I woke up that first day, when I tuned my screens to the news and saw what it was doing to us, I felt—helpless. Swapping one jerkwater Moon for another is probably a trivial event, in a Galaxy of a hundred billion suns. Who the hell knows what else goes on out there? But I've never felt so small. I knew at that moment that my whole life could be shaped by events I can't control. Who knows what I might have become if not for _that_ , knocking the world off of its axis? Who knows what I might have achieved?"
"Life is contingent," the driver, Xenia, said unexpectedly. Her accent was vaguely east European. She reached back and covered Paulis's hand. "All we can do is try our best for each other."
"You're wise," Malenfant said.
She sat gravely, not responding.
"On our behalf, please go kick ass, sir," Frank Paulis said.
"I have less than twelve hours before I fly back out of here, Frank. Tell me who it is I have to meet."
The car pulled away from the viewpoint and headed toward the sprawling base. Malenfant took a last long breath of the crisp ocean air, bracing himself to be immersed in the company of people once more.
## _S hadow_
Shadow huddled under a tree, alone.
Claw came stalking past, panting, carrying yellow fruit in his good hand. She cowered away from him, seeking to hide in the deep brown dark of the tree's thick trunk. He hooted and slapped her. Then he stalked on, teeth bared.
Flies clustered around her hand. The webbing between her thumb and forefinger had been split open. Her inner thighs were scratched and sore. Her belly and breasts were bruised, and a sharp pain lingered deep inside her.
Claw had used her again.
Her hands reached for food—a sucked-out fruit skin dropped by somebody high in the tree above her, a caterpillar she spotted on a leaf. But her mouth chewed without relish, and her stomach did not want the food. Agony shot upward from her deepest belly to her throat. A thin, stinking bile spilled out of her mouth. She groaned and rolled over onto the ground, huddled over her wounded hand.
The light leaked out of the sky.
There was rustling and hooting as the people converged on the roosting site from wherever they had wandered during the day. The highranking women built their nests first, weaving branches together to make soft, springy beds, and settling down with their infants.
Somebody thumped Shadow's back, or kicked it. She didn't see who it was. She didn't care.
She stared at the dust. She did not eat. She did not drink. She did not climb the trees to build a nest. She only nursed the scarlet pain in her belly.
Just before the last sunlight faded, she heard screeching and crashing, far above her. Big Boss was making one last show of strength for the day, leaping from nest to nest, waking the women and throwing out the men.
The noises faded, like the light.
Something smelled bad.
She held up her hand in the blue-tinged dark. Something moved in the wound between thumb and forefinger, white and purposeful. She tucked the hand away from her face, deep under her belly.
She closed her eyes again.
Daylight.
She pushed at the ground. She sat up, and slumped back against the tree root.
The people were all around her, jostling, arguing, playing, eating. They didn't see her, here in her brown-green dark.
There was shit smeared on her fur. It was drying, but it smelled odd.
The man called Squat was trying to lead the people, to start the day. He was walking away from them, shaking a branch, stirring bright red dust that clung to his legs. He looked back at Big Boss, walked a little further, looked back again.
Big Boss followed, growling, his hair bristling all over his back. One by one the others followed, the adults feeding as they walked, the children playing with manic energy, as always.
Here was Little Boss. He squatted down on his haunches before Shadow. He was a big slab of hot, sweating muscle, bigger in height and weight than Big Boss himself. He picked up her damaged hand and turned it over. He poked at the edges of the wound, where pus oozed from broken flesh. He let go of the hand, so it fell into the dirt. He inspected her, wrinkling his nose.
He got up and walked a few paces away.
Then he turned. He ran back and, with all his momentum behind it, he kicked her, hard. She ducked her head out of the way, but the kick caught her shoulder and sent her sprawling.
Others came by: women, men, children. She received more slaps and kicks, and was confronted by teeth-baring displays of disgust. Shadow just lay in the dirt, where Little Boss's kick had thrown her.
But the beatings by the men were not severe today. They saved their energy for each other. Many of them jabbered and punched each other, in noisy, inconclusive bouts. The elaborate politics of the men was taking some new turn.
Then there were no more kicks or slaps. The people walked away, the rustle of their passing receding. Shadow was left alone. She dissolved, becoming only a mesh of crimson pain.
She knew herself only in relationship to other people: not through the place she lived, the skills she had. Ignored, it was as if she did not exist.
Now somebody crouched down before her. She smelled familiar warmth. She turned her head with difficulty; her neck was stiff. It was Termite, her mother. Beyond her Tumble, the infant, was playing with a lizard she had found, chasing it this way and that, picking it up by the tail and throwing it.
Termite, huge, strong, studied her daughter. Her face was twisted by uneasy disgust. But she probed at the scratches on Shadow's legs, dipped her fingers into the blood that had dried around Shadow's vagina, and tasted it. Then she inspected the ugly wound on Shadow's hand. Fly maggots were wriggling there.
Termite groomed carefully around the edge of the wound. She pulled out the maggots, squeezed out pus, and licked the edges of the wound. Then she gathered a handful of thick, dark-green leaves. She chewed these up, spitting them out into a green mass that stank powerfully, and scraped it over the wound.
It hurt sharply. Shadow squealed and pulled her hand back. But her mother was strong. Termite grabbed her hand and continued to tend the wound, despite Shadow's struggles.
Tumble kept her distance. She would approach her mother, stare at Shadow and wrinkle her small nose, and retreat; then she would forget whatever she had smelled, and approach once more. She hovered a few paces away, attraction and repulsion balanced.
Later, Termite put her powerful arms under Shadow's armpits, hauled her upright by main force, and dragged her into the shade of a fat, tall palm. She brought her food: figs, leaves, and shoots. Shadow tried to pull her face away. Termite grabbed her jaw and pinched the joints until Shadow opened her mouth. She forced the food between Shadow's lips, and pushed at her jaw until Shadow chewed and swallowed.
Shadow threw up.
Termite persisted.
By the time the roosting calls began to sound once more through the forest, Shadow was keeping down much of what she swallowed.
The people returned. The adults carried shaped cobbles, or bits of food. Some of the men had meat.
But there was much unrest. Squat and Little Boss were jabbering and throwing slaps at each other. Squat grabbed at a bloody animal leg Little Boss was carrying, trying to snatch it off him. Little Boss punched him hard in the nose, sending Squat flying back, and Little Boss took a defiant, bloody mouthful of his meat.
When the women started making their nests. Tumble climbed up her mother's legs and clung onto her shoulders and head.
Once again Termite tried to make Shadow stand, but Shadow fell back and sprawled in the dirt. So Termite leaned over and let Shadow fall across her shoulders. She stood straight with a grunt, and Shadow's arms and legs dangled at her back and belly.
With powerful gasps, Termite began to climb a palm, laden down by her infant and her nearly-grown daughter.
Shadow's head dangled at Termite's back. She saw Termite's legs and rump, a dark slope before her, powerful muscles working. With every jolt, Shadow felt her innards clench, and bright red pain flowed through her. Tumble's small hands delivered stinging slaps to her unprotected backside.
High in a palm, Termite let Shadow slide into the crook of a branch. Sweating and panting, Termite quickly pulled branches together to make a nest. Then she grabbed Shadow by the armpits and pulled her into the nest.
Termite settled herself, curling around her daughter's back. Whimpering, Tumble settled down in the nest at her mother's back, on the far side from Shadow.
The light slid away. The world was black and gray.
Shadow closed her eyes. She slept, entering a deep dreamless sleep, with her mother's warmth around her.
When she woke, in the first pink light of day, she found her thumb in her mouth, as if she were an infant. Memories flooded into her head. Her illness was like a tunnel of blood red, leading back to greener days beyond.
Her back was cold. Termite wasn't there.
She sat up. Termite and Tumble were in the nest, on its far side. Termite was assiduously grooming her infant's fur. Tumble was picking through a lump of feces, seeking undigested food.
Shadow inspected the wound in her hand. Green, chewed-up fiber clung to it. She licked away the green stuff. There was no sign of maggots or pus, and much of the damaged area was scabbed over, although the scabs cracked when she flexed her thumb.
She hooted and scrambled toward her mother.
Termite sat on the edge of the nest, her long arms wrapped around Tumble, watching Shadow with a hard, still face.
Shadow sat for long heartbeats in the center of the nest. She picked up bits of fur from the nest and teased it through her fingers. The scent of her mother was still there, mixed with the green smells of the tree. But there was a sourness, too.
The sourness was her own smell, Shadow's smell. Her mother, like her sister, could not bear to be with her, because of the smell. She ripped at her fur, screeching, and scattered handfuls of it around the disintegrating nest.
Termite watched impassively.
A stab of pain, lancing up from the depths of her gut, stopped Shadow dead.
She looked down at herself, her breasts and belly and legs. She felt a shiver of surprise that she was _here_ , inside this body that stank so strangely.
The pain stabbed again, hot and white. She doubled over, and vomit surged from her, sour and yellow.
It was a hard time for them all. With Big Boss weakening, the social order of the group was breaking down, and anger washed among the people like froth on a turbulent stream.
It went hard for Shadow. Pushed even from her mother's protective circle, suddenly she was the lowest woman in the group. They all hated her, not just for her low place, but because of what she had become, this stinking, bleeding monster. She could not defend herself, from their beatings and the theft of her food.
But still she clung to the group. Still she made her nest each night, high in the trees, away from the cats and other predators, as close to the others as she dared approach. Much as she feared their fists, she was drawn back, for there was nowhere else to go.
And she was still ill. Her bleeding had stopped. She was afflicted by stomach cramps and pain deep in her back. Her breasts and belly started to swell. She was violently sick each morning. Her days were a blur of pain and loneliness. When she saw her shadow, of a hunched-over creature with hair ragged and filthy, she did not recognize herself.
But then, one day, she felt something squirm in her belly, a kicking foot.
Her head filled with memories, of blood and shit and milk. She remembered a woman lying on her back, legs askew, other women working to pull a pink, slick mass from her body, their hands sticky with blood.
Her loneliness sharpened into fear.
Again she ran to her mother, reaching for her sparse fur, trying to groom, to get close.
Since the illness had started, Termite had never once struck her daughter, not as the others did. But now, as her broad nostrils widened with the stink of Shadow's body, her fists clenched.
Shadow cowered, whimpering.
Claw came running by, hair bristling, hooting inanely. He was grinning, but blood ran from a gouge in the side of his face. He was running from a fight. As he passed Shadow he aimed a kick at her that caught her in the small of her back.
Shadow dragged herself to the shade of a big palm. There she slumped down, and vomited copiously.
## _R eid Malenfant_
The next time he woke, Malenfant found the light that soaked through his parachute-canopy tent was a little less bright, the air perhaps a fraction cooler.
Night was coming, at last, to the desert.
He tried to sit up. His head banged as if his brain was rattling around in his skull. His mouth was a sandbox, and he felt a burning dryness right through his throat and nose. It felt like the worst hangover of all time.
But you're built for heat, Malenfant. You've got a body adapted to function away from the shelter of the trees, to walk upright in the heat of the day. That's why you sweat and the chimps don't. Haven't you learned anything from those paleo classes?...
He reached for his water flask and shook it. Still a quarter full, just as it had been before he slept. Deliberately he tucked it back under his blanket.
He got to his feet. He staggered, brushing his head against the hot, dusty canopy. The fabric rippled, and he heard sand hissing off it. He bent and found his broad stiff-brimmed hat, and jammed it on his bare scalp. Then, rubbing the stubble on his jaw, he stepped out of the makeshift tent.
Outside was like a dry sauna. He felt the moisture just suck straight out of his skin. The pain intensified around his temples and eyes, crumpling his forehead.
The world was elemental: nothing but sand, sky, and gnarled Joshua trees, over which their chutes were draped.
This was the Mojave desert. He and Nemoto had been dumped here as a survival training exercise. During the day the heat was flat and crushing; they could do nothing but lie in their tent of chutes. And at night they foraged for food.
Nemoto was crouched over a low fire. She was heating some kind of thin broth in a pan she'd made out of aluminum foil. She had a spare T-shirt wrapped around her head. _To survive you don't need equipment_ , the instructor had said. _All you need to pack is strength and ingenuity and determination. That, and a willingness to eat insects and lizards_.
Nemoto had proved ingenious at setting traps.
"I wonder—" His throat was so dry he had to start again. "I wonder what's in the soup this time."
Nemoto glanced up at him, and then looked back to her cooking. "Your speech is slurred. Drink some water, Malenfant."
He walked around their little campsite, stretching his legs. He could feel a tingling in his limbs, and the air felt thin. The horizon seemed blurred, perhaps by dust.
"I mean, why the hell are we here?" He lifted his arms and turned around. "Whatever we find on the Red Moon, it won't be like this."
"But on returning to Earth we might land in a desert area, and—"
He barked laughter, hurting his throat. "Let's face it, Nemoto. The chances of our returning healthy enough to play wild man in the desert are too remote to think about."
"Drink some water."
He stalked away, vainly seeking cooler air.
As the project had grown, as all such projects did, it had acquired its own logic, much of it loaned from NASA—to Malenfant's chagrin, and against his better judgment. While the ship was being prepared, the booster assembled and tested, nobody seemed to know what to do with the astronauts, except train them to death and send them on goodwill tours, just as NASA always had.
Some of the training Malenfant could swallow. He had, after all, flown in space twice before, and Nemoto, on her single trip to the Station, had logged an impressive number of days in orbit. So they endured hours in classrooms and in hastily mocked-up simulators going over every aspect of their unlikely craft's systems, and the procedures they would have to follow at their mission's major stages.
The major problem with that turned out to be the very volatility of the design. As teams of engineers struggled to cram in everything they thought they needed, key systems went through major redesigns daily—and all of it impacted in the crew's interface with their craft. In the end Malenfant had grown tired of the simulation programmers' laboring efforts. He had shut down the sims, had a dummy cabin mocked up from plywood, and had blown-up layouts of their instrument panels cut out of paper and pasted over the wood. It wasn't too interactive, but it familiarized them with systems and procedures—and it was easy to upgrade each morning with bits of tape and sticky paper, as news of each redesign came through.
But the spacecraft-specific training was the easy stuff. The rest was more problematic. How, after all, do you train to face a completely unknown world?
Malenfant and Nemoto had undergone a lot of altitude training, for it was clear that the Red Moon's air would be thinner than Earth's. Likewise they had been taken to tropical jungles, for it was planned to bring them down in a vegetated region close to the Moon's equator.
But beyond that, all was uncertain. Nobody knew if they would find water fresh enough to drink. Nobody knew if they would be able to eat the vegetation—always assuming the gray-green swathes visible through telescopes were vegetation at all. Nobody knew if there would be animals to hunt—or if there were animals that might hunt two human astronauts. It wasn't even clear if the air could be breathed unfiltered.
The ship would be packed with three days' ground supplies, including air filters and water and compressed food. If the makeshift explorers found they couldn't live off the land in that time, they were just going to have to climb back in their lander and depart (always supposing they could find the return-journey rocket pack that was supposed to follow them to the Moon).
And then there was the mystery of the hominids who had come tumbling through the Wheel in the sky.
Malenfant and Nemoto had sat through hours of lectures by Julia Corneille and others, trying to absorb the best understanding of the evolution of mankind, watching one species after another parade through dimly-realized computer animations— _Australopithecus, Homo habilis, Homo erectus_ , archaic _Homo sapiens, Homo heidelbergensis, Homo neandertalensis..._ It was a plethora of speculation as fragmentary, it seemed to Malenfant, as the bone scraps on which it was based. He had vaguely imagined that the newer evidence based on DNA variation might have cleared the picture, but it seemed only to have confused everybody further. Nobody knew where humanity was going, of course. It had startled Malenfant to find that if you dug deeper than pop science simplifications, nobody really knew where man had come from either.
The truth was that the sessions had been of little use. Malenfant had learned more than he wanted to know about archaeological techniques and dating methods and anatomical signifiers and all the rest. What he needed to know was how to handle a tribe of _Homo habilis_ , alive, fighting and breeding, should he crest a hillside on the Red Moon and discover them—or vice versa. But NASA's experts, curators of fragments all, simply weren't tuned to thinking that way. It was as if they could only see the bits of bone, and not the people that must once have lived to yield up these ancient treasures.
The only real consensus was that Malenfant and Nemoto should pack guns.
... He had lost his hat. He saw it on the ground.
There was a ringing in his ears. He ought to get his hat. He bent to reach it.
Next thing he knew, he was on his side. He lay there fuming.
The hat was too far away to reach, so he wriggled that way. Like a snake, he thought, cackling. When he had his hat he stuck it on the side of his head, so it shielded his face.
At least the paleo training had been relevant, he thought. Too much of the rest of his time had been filled up with pointless exercises like this. They had even threatened to put him back in a centrifuge. "I told them to stick the fucking centrifuge where the sun don't shine," he muttered.
The sand was hot and soft. Its pressure seemed to ease the pain in his head. Maybe he would sleep awhile.
There were hands under his hips and shoulders, pushing him onto his back. A face above him blocked out the sky. It, she, was saying something. Nemoto, of course.
He said, "Leave me alone."
She leaned closer. "Open your mouth." She lifted a flask and poured in water.
He made to spit it out, but that would be even more stupid. He swallowed it. "Stop that. We have to save it."
"You're dehydrated, Malenfant. You know the drill. You drink what you have until it's gone, and if you have not been found by then, you die of thirst. Simple logic. Either way it does no good to ration your water."
"Horse feathers," he said. But he let her pour more water into his mouth. It was the most delicious thing he had ever tasted.
## _E mma Stoney_
They continued to work their way east. A range of mountains, low and eroded almost to shapelessness, began to loom above the horizon. Though their outlines and colors were softened to blurs by the murky air, Emma thought she made out bands of vegetation, forest perhaps, on their lower slopes.
After another day's walking, the Runners paused by a shallow, slow-running stream.
Sally threw herself flat on the ground. She seemed to go to sleep at once. Maxie, as ever full of life at precisely the wrong time, ran off to play with the Runner children.
Emma sat on dusty grass and eased off her boots. Maybe her feet were toughening up; at least she didn't have to pour any blood out of her boots today. She limped to the stream to drink, wash her face, bathe her feet. She found a stand of root plants, a little like potatoes, small enough to dig out of the ground. It was a pleasure for once to be able to provide for herself.
Emma watched the Runners. The descending sun had turned the western sky a tall orange-pink—volcano sunset, she thought—and peering through the dusty air was like looking into a tank of shining water, through which exotic creatures swam.
The stream had washed down a rich supply of volcanic pebbles, and many of the adults were knapping tools. They squatted on their haunches in the stream, their lithe bodies folded up like penknives, tapping one stone against another. The axes they made were flattened slabs of stone, easy to grip, with clean sharp edges. Stone axes and wooden spears: the only tools the Runners ever made, over and over, tools they turned to every task from butchering carcasses to shaving—even though their hands were clearly just as capable of fine manipulation as Emma's.
There were a lot of oddities, if you watched carefully.
The toolmakers worked in silence and isolation, as if the others didn't exist. Emma never saw a Runner pick up a tool dropped by somebody else and use it, not once. A few children and young adults sat beside their elders, watching, trying to copy them. Mostly the adults ignored their apprentices; only very rarely did Emma see examples of coaching, such as when one woman picked a rock from out of a boy's hand and turned it around so it served to flake the anvil stone better.
All the tools turned out by the women, so far as Emma could tell, were functional. But some of the men's were different. Take Stone, for example, the bullying alpha-male. Sometimes he would sit and labor for hours at an axe, knocking off a chip here, a flake there. It was as if he pursued some impossible dream of symmetry or fineness, working at his axe far beyond the point where he could be adding any value.
Or, more strangely, he would sit with a pile of stones and work feverishly, turning out axe after axe. But some of these "axes" were mere flakes of rock the size of Emma's thumb—and some were great monsters that she could have held only in two hands, like a book opened for reading. These pathological designs seemed no use as tools; Stone would do no more than carry them around with him for a few hours, making sure everybody saw them, before dumping them, never used, their edges as sharp as the instant they were made.
Emma didn't know why Stone did this. Maybe it was a dim groping toward culture: hand axe as art form. After all, the hand axe was the only meaningful artifact they actually made, taking planning and vision and a significant skill; their other "tools," like their termite-digging sticks or even their spears, were little more than broken-off bits of wood or bone, based on serendipitous discoveries of raw materials, scarcely finished. The hand axe was the only way the Runners had to express themselves.
But if that was so, why didn't the women join in such "artistic" activities as well?
Or maybe the useless hand axes were about sex, not practicality or culture. After all to be able to make a decent axe showed a broad range of skills—planning, vision, manual skills, strength—essential for survival in this unforgiving wilderness. _Look at me, girls. I'm so fit and strong and full of food, I've got time to waste on these useless monsters and fingernail-sized scale models. Look at me!_ When everybody around you had a body as drop-dead beautiful as any athlete's she had ever seen, you needed something to stand out from the crowd.
Could that be true? The Runners had to enjoy something like full humanity, in planning and vision and concentration, when making the axes. But could they then abandon that humanity and revert to some lower level of instinct, as the axes became a symbol of sexual prowess, as unconscious as a bird's bright plumage?
It was all another reminder to her that no matter how human these beautiful creatures looked and sometimes behaved, they were not human. Their small heads contained shards of humanity, she thought, floating on a sea of animal drives and instincts: humans sometimes, not other times...
Or maybe she was just being anthropomorphic. Maybe she shouldn't be comparing the Runners to herself, seeing how human they were, or weren't; the Runners were simply Runners, and they fit into their world as well as she fit into hers.
Though it was a full hour since they had abandoned the trek for the day, Fire was still wandering around with his hands clasped together. He couldn't drop his hot burden until the others had gathered kindling and fuel for him, and as long as the sun was up and the air was warm they had no interest in doing that—in fact it didn't even seem to occur to them—and so Fire was stuck.
But he had more than that on his mind. He was vainly pursuing one of the girls, Dig: a real knock-out, Emma thought, with crisp auburn hair, full, high breasts and hips to die for. Poor Fire seemed to have no idea how to get through to her; he just followed her around, holding out his handfuls of ash, and plaintively calling her name. "Dig! Dig!"
Being the fire-carrier was obviously a key job, a cornerstone of this untidy little community. But as far as Emma could see his role didn't win Fire much respect from the other Runners, especially the men. Each night he would deliver his embers to the latest heap of kindling, and then would be pushed and slapped away. It was as if he was the runt of the litter. Certainly his handful of ashes didn't get him the girls the way the hand axes of the other boys and men did.
But this time, for once, Fire was getting closer to the object of his desire. She backed up against a tree, and he walked toward her, hands clasped, that ridiculous, tragic erection sticking out like a divining rod.
But a rock hit him hard in the side of the head.
The rock had been thrown by Stone.
Fire went down, toppling like a felled tree. He opened his hands to save himself before he hit the mud. His precious ashes scattered.
Runners ran forward. Dig and Blue got to their knees in the mud, and tried to scrape together the ashes and embers. But the embers were hissing, quickly extinguished in the mud.
Stone hadn't grasped the chain of events that led from his own hurled rock to the death of the fire, or else he just didn't want to know; either way he capered and howled, pressing the useless embers into the mud with his bare feet, and he aimed hefty kicks at Fire's ribs.
Fire curled up, arms wrapped over his head, whimpering in misery. Emma winced, but she knew better than to try to intervene.
After that, the daylight seemed to run out quickly. As the sun descended toward the horizon, the golden air turned to a dismal brown. The shadows of trees to the west lengthened, clutching at the cowering Runners like claws.
In the absence of a fire the Runners gathered more closely than usual, the women clutching their children, even the usually solitary men huddling close.
The first predators began to call.
Sally came to Emma. "You have to use your spyglass," she said. "Make a fire. And you have to do it now, before we run out of sun."
Emma sighed. "I'm frightened of showing them too much of what we've got."
"They aren't going to steal your glass and start using it all over the savannah," Sally said. "They don't _learn_."
"It's not that. Right now they seem to think we are like them. If they think we're too strange, they might reject us."
The shadow of a distant tree slid across Sally's face. "Sister, I don't think it's the time for philosophical dilemmas. In a couple of hours the hyenas are going to be chomping on our bones. And anyhow these guys have attention spans that make Maxie look like Michelangelo. By the morning, they'll have forgotten it all. Come on, Emma. Just do it."
"All right. Let's try to keep our tools out of their sight, though."
"Agreed."
They spent a few minutes gathering dry wood, and building a little teepee a couple of feet high. Then they scraped together dried leaves and tinder.
Emma crouched down on the ground, folded her magnifying glass out of her knife, and angled it until she caught the crimson light of the low sun. She moved it back and forth until she had focused a tight spot of light on a few bits of dry tinder. Then she waited, the cold of the ground seeping into her, her awkwardly angled arm growing stiff. She grumbled, "I don't know why the hell the South African air force didn't just give me a box of matches."
Some of the Runners came to watch what they were doing. They hooted excitedly, one woman even making rubbing-hands-warm motions. But when the tinder didn't catch light immediately, they became baffled and quickly lost interest.
Her spot of light disappeared. She looked up to see a small silhouetted figure, a grasping hand.
"Maxie's. Maxie's!"
Sally scooped him up. "Get away, Maxie, for heaven's sake." Maxie, denied the toy, began wailing.
Unnoticed, the tinder had started smoking.
Emma immediately dropped her glass. She cupped the thread of smoke with her hands and blew gently. The smoke trail billowed, nearly died.
She sat back and beckoned to Fire. "Hey. Come over here. Come on. This is your job."
Poor Fire sat squat on the ground, clutching his ribs, an immense lump forming on the side of his head.
"Umm, Fire smoke Fire. Fire Fire!"
At last he came forward, hobbling painfully. Shivering, he cupped his hands around the thread of smoke and blew, lips pursed.
It seemed to take him mere seconds to have a small flame going. With the precise motions of a surgeon, he began to feed the tiny red-yellow spot with bits of tinder.
When the smoke started to spread, the other Runners were drawn back. As the fire grew, they settled down around it, just as they did every night, and the men began to drag over heavy branches to make night logs.
Sally watched the Runners with cold contempt. "Not a word, not a gesture of congratulation or apology. Or surprise. Or relief. They've already forgotten how Fire lost his embers... The fire is just here, and they accept it. They really don't think like us, do they?"
Emma stretched stiff limbs. "Right now, I couldn't care less. Just so long as the fire keeps away the bad guys with the teeth."
As Emma fell into sleep, a rough hand grabbed her shoulder.
She froze. Her eyes snapped open. The sky, full of ash and smoke, retained a lingering purple-black glow, enough to show her a lithe, crouching silhouette. It, he, leaned over her. She was pushed onto her back. She could smell _Runner_ : a thick, pungent, meaty smell of flesh that had never once been washed.
In the back of her mind she had rehearsed for this, from the first day here. Don't resist, she told herself. Don't cry out. She had seen the Runners copulate, every day. It would be fast, brutal, and over.
For a moment her assailant was still, his breath hot. She stiffened, expecting hands to claw at her clothing. But that didn't come. Instead a head, heavy, topped by tight curls, descended to her breast. She felt shuddering, a low moan.
Gingerly she reached up. She explored a flat skull, those extraordinary brow ridges like motorcycle goggles. And she touched a swollen mass on one temple. Her assailant flinched away.
It was Fire.
He was weeping. She remembered how he used to go to the old woman, Sing, for comfort, before she died. She wrapped one arm around his back. His muscles were hard sheets, his skin slick with dirt and sweat.
He reached up and grabbed her fingers. With a sharpness that made her yelp, he pulled her hand down toward his crotch. She found an erection as stiff as a piece of wood. She tried to pull away, but he pushed her hand back.
Gently, hesitantly, she wrapped her fingers around his hot penis. His hand took her wrist and pushed it back and forth.
She rubbed him once, twice. He came quickly, in a rapid gush against her leg. He sighed, released her wrist, and lay more heavily against her.
Half-crushed, barely able to breathe, she waited until his breathing was regular. Then, gingerly, she pushed at his shoulder. To her intense relief, he rolled away.
In the morning, Fire scooped up his embers and ash, and the Runners dispersed for their walk. It was as if none of the previous evening's events had ever happened.
## _R eid Malenfant_
In the last hours he had to endure a visit from an Apollo astronaut: a walker on a now-vanished Moon, eighty-five years old, ramrod straight and tanned like a movie star. "You know, just before my flight we had a visit from Charles Lindbergh and his wife. He had figured that in the first second of my Saturn's flight, it would burn ten times more fuel than he had all the way to Paris. We laughed about that, I can tell you. Well, Lindbergh came to see me before I flew, and here I am come to see you before your flight. Passing on the torch, if you will..."
And so Malenfant, with a mixture of humility and embarrassment, shook the hand of a man who had shaken the hand of Lindbergh.
It was, at last, the night before launch.
At Vandenberg, he stood in the crisp Californian night air. The BDB's service structure was like an unfinished building, a steel cage containing catwalks and steps and elevators and enclosures. A dense tangle of pipes and ducts and tubing snaked through the metalwork. The slim booster itself was brilliantly lit, the sponsors' logos and NASA meatballs encrusting its hide shining brightly. Its main tanks were full of cryogenic propellants, and they spewed plumes of vapor into the air. No doubt in violation of a dozen safety rules, hard-hatted technicians, NASA and contractor grunts, scurried to and fro at the booster's base, and electric carts whirred by. It was a scene of industry, of competence, of achievement.
Malenfant stepped into an elevator and pushed the button for the service structure's crew level, three hundred feet high. He was escorted by a single tech, a Cape ape in clean room regalia of a white one-piece coverall, latex gloves and puffy plastic hat. Malenfant had met the guy before, and they nodded, grinning; he was a somewhat grizzled veteran, long laid off by NASA but rehired for this project.
They rose vertically in the clanking, swaying steel cage. Beyond the cage flashed steel beams, cables and work platforms, mostly unattended now. And beyond _that_ was the hide of the main tank itself: sleek, smooth, coated with ice where the cryogenic fuels had frozen the moisture out of the night air. It was such an immense cold mass that Malenfant felt the heat being drawn out of his own body, as if he were some speck of moisture that might end up glued to that glistening skin.
The elevator came to a stop. He stepped out, turned right, and walked over the access-arm catwalk. The walk was just a flimsy rail that spanned the rectilinear gulf between the tangled, rusted gantry, and the sleek hide of the booster. An ocean breeze picked up, laden with salt, and the catwalk creaked and swayed as if the gantry was mounted on springs. He grabbed a handrail for support. Through the chain-link fence he could see the lights of the base scattered in rectangles and straight lines over the darkened ground, and the more diffuse lights of the inland communities. The coast was black, of course, swept clean of habitation by the tide.
This was a noisy place. The Pacific wind moaned through the complex, and the huge propellant pipes groaned and cracked as rivers of the super-cold fluids surged through them. Fuel and wind: it was a noise of power, of gathering strength, and the hairs on the back of his neck prickled.
He reached the end of the walk. He stepped through the white room, the cramped enclosure where he would be inspected one last time before the launch, and he faced the streamlined fairing that would protect the Moon lander during launch. There was a hatchway cut into the fairing. A small wooden step led up to the hatch, a touch of home-workshop mundanity amid all this shining hardware.
From here he could see into the cabin of the lander itself: small, crammed with supplies, and with two canvas-frame couches side-by-side. The light was a subdued green. Instrument panels on the wall glowed with softscreen displays and telltale lights. It was like looking into a small cave, he thought, an undersea cave crusted with jewels.
Malenfant had been through it all before. Every space project, as it developed, became entangled and complex beyond the understanding of any single human. But from the astronaut's point of view that proliferating tangle reached a certain maximum, until, after some indefinable point—as the booster stack crept forward through its integration schedule, as launch day approached—the whole thing began to simplify, to focus.
In the end, he thought, every mission reduces to this: human beings climbing into the mouth of a monster, to be hurled away from the Earth. And all the technicians and managers and fundraisers and cheerleaders and paper-chasers in the world can do nothing but watch.
Emma's mother and her sister's family were staying in apartments on the ASFB. They had invited Malenfant to join them for Mass, celebrated by the base's Catholic chaplain.
Blanche Stoney, the mother, was an intimidating seventy-year-old. She offered Malenfant her hand without getting out of her chair. The sister, Joan, a little younger than Emma, had raised four kids alone, and had looked exhausted every time Malenfant had met her. But the kids were all now young teenagers and, it seemed to Malenfant, remarkably well behaved.
The priest said Mass for the family in a cramped living room.
Malenfant, upright in his civilian suit, tanned walnut brown by the desert sun, felt as out of place as a spanner in a sewing basket. But he endured the ceremony, and took his bread and wine with the others. He tried to find some meaning and comfort in the young priest's familiar words, and the play of light on the scraps of ornate cloth, the small chalices and the ruby-red wine.
The priest had asked Joan's two eldest boys to serve as altar boys. They did fine except during the communion service, when the younger boy held the ciborium upside down so that the hosts slid out and fell to the carpet, fluttering down one by one. In the background a softscreen showed live images of the preparation of the BDB. There were a lot of holds. Malenfant tried not to watch the _whole_ time.
When it was done, the priest packed up and went home with promises to call during the mission.
Joan brought Malenfant a beer. "I think we owe you this."
Blanche, the mother, snapped, "But you owed us your presence here tonight."
"I don't deny that, Blanche."
Malenfant spent some time trying to explain the technicalities of the mission to them—the countdown, the launch, the flight profile. Joan listened politely. At first the children seemed interested, but they drifted away.
In the end Malenfant was left alone with Blanche.
She skewered him with her gaze. "You wish you were anywhere but here, don't you?"
"Either that or I had another beer."
She laughed, clambered stiffly out of her chair, and, somewhat to his surprise, brought him a fresh can.
"I know you try," she said. "But you never really had much time for religion, did you? To you we're all just _ants on a log_ , aren't we? I heard you say that on some cast or other."
He winced at the overfamiliar words. "I think my wisdom has been spread a little thin recently."
She leaned forward. "Why are you going to the Red Moon? Is it really to find my daughter—or just vainglory? To prove you're not too old? I know what you flyboys are like. I know what really drives you. You have nobody here, do you? Nobody but Emma. So it's easy for you to leave."
"That's what the vice president thinks."
"Don't name-drop with me. What do _you_ say?"
"Blanche, I'm going up there for Emma. I really and truly am."
With sudden, savage intensity, she leaned forward and grabbed his hand. "Why?"
"Blanche, I don't—"
"You destroyed her. You started doing that from the moment you set your sights on her. I remember what you used to say. _You bake the cakes, I'll fly the planes_. From the moment she met you, she had to start making sacrifices. It was the whole logic of your relationship. And in the end, you fulfilled that logic. _You killed her_. And now you want to kill yourself to get away from the guilt. Look me in the eyes, damn it, and deny that's true."
For about the first time since it happened, he thought back to those final moments in the T-38, the clamor in that sun-drenched sky. He remembered the instant when he might have regained control, his sense of exhilaration as that huge disastrous Wheel approached.
He couldn't find words. Her rheumy eyes were like searchlights.
"I don't know, Blanche," he said honestly. "Maybe it's for me. Without her, I'm lonely. That's all."
She snorted contempt. "Every human being I know is lonely. I don't know why, but it's so. Children are consolation. You never let Emma have children, did you?"
"It was more complicated than that."
"Religion is comfort for the loneliness. But you rejected that, too, because we're just ants on a log."
"Blanche—I don't know what you want me to say. I'm sorry."
"No," she said more softly. Then she rested her hand on his head, and he bowed. "Don't say you're sorry. Just bring her back," she said.
"Yes."
"Where do you think she is now? What do you think she is going through?"
"I don't know," he said, honestly.
## _S hadow_
Relations among the men worsened. Every day there were increasingly savage and unpredictable fights, and many of the women and infants, not just Shadow, suffered punches and kicks and bites as a consequence.
One day it all came to a head.
Big Boss was sitting cross-legged on the ground with his back to a small clearing, working assiduously at a cluster of nut-palm fruit. Shadow was in the shade at the edge of the clearing, half-hidden as had become her custom.
Without warning Squat stalked into the clearing. All his hair stood on end, doubling his apparent bulk. He leaped up and grabbed branches, ripping them off the trees, shaking them and throwing them down before him. He picked up rocks and hurled them this way and that. His silence was eerie, but his lips were pursed tightly together, pulling his face into a harsh frown, his eyes fixed on Big Boss.
Big Boss ignored him. He kept on plucking at the fruit in his lap. Squat, and the other men, had made such displays before, and nothing had resulted.
But now Little Boss suddenly broke from the cover of the trees. Without warning or apparent provocation, he hurled himself on Big Boss.
Big Boss roared and faced his attacker, hair bristling. But Squat screeched and joined in. The three of them dissolved into a blur of flailing fists and thrashing limbs.
All around the clearing, other men ran to see what was happening. They circled the battlers, hooting and crying—but not one of them rushed to the aid of Big Boss.
Big Boss broke away. His eyes were round and white, and blood leaked over the side of his head, where one ear had been bitten so savagely it dangled by a thread of gristle. He ran toward the nearest tree, and tried to clamber into it. But he was limping, and Squat and Little Boss easily caught him. They pulled him back and hurled him to the ground, and punched and kicked and bit him. Squat began to jump on Big Boss's back, slamming his heels again and again into ribs and spine.
Now more of the men joined in, screaming and yelling. Though they concentrated their attentions on Big Boss, they squabbled and fought among themselves, vying for their places in the new order.
At last Little Boss climbed up on Big Boss's back. He stood straight and roared. His mouth was bloody. He grabbed one of Big Boss's arms, as if Big Boss was no more than a monkey he had caught in the forest. Little Boss twisted the arm this way and that, and Shadow heard bones snap, muscle tear.
The women and children huddled together beneath the trees, clutching each other or grooming tensely, shrinking from the aggression.
The men ran off into the forest, tense and excited, hair bristling. Big Boss lay where he had fallen, a bloody heap on the ground.
Slowly the women emerged from their sheltered places. Cautiously they fed and groomed each other and their children. None of them went near the fallen Big Boss—none save an overinquisitive child, who was hastily retrieved by his mother.
Only Shadow stayed in her pool of shade.
The day wore away. The shadows lengthened.
Big Boss raised his head, then let it fall flat again.
Then he got one arm under his body, and pushed himself upright. The other arm dangled. His flesh was ripped open, by teeth or chipped cobbles, so that flaps hung down from patches of gleaming gristle, and his skin was split by great gouges, crusted with dirt and half-dried blood. He had lost one ear completely, and one eye was a pit of blood from which a pale fluid leaked.
He opened his mouth. Spittle and blood looped between smashed teeth, and he moaned loudly.
The women and children ignored him.
Big Boss pulled his legs beneath him. He began to crawl toward the trees, one leg dragging, one arm dangling. Twice he fell flat. Twice he got himself up again, and continued to drag himself forward. Where he had been lying, the blood had soaked into the ground, leaving the dirt purple. And where he passed, he left a trail of sticky blood and spit and snot, like some huge snail.
When he got to the base of the tree, he twisted so he got his back against the bark of the trunk, and slumped back.
He was still for a long time. The sun, intermittently obscured by cloud, slid across the sky. Shadow thought Big Boss was dead.
But then he began to move again. Using the tree as a support, he pushed himself upright. He reached up with his less damaged arm to grab a low branch. He growled with pain. He got his chest over the branch, and fell forward, gasping. For a long time he was still once more, clinging to the branch. Then he carried on, hauling himself grimly from branch to branch, higher into the tree.
At last he reached a high point. Clinging to the tapering trunk with his legs, he pulled down branches with grim determination. Surrounded by clusters of yellow fruit, he slumped flat in this nest, the last he would ever make.
The women on the ground called, their panting hoots summoning each other and their children. The women climbed into the trees, infants clinging to their mothers' backs or chests. Shadow followed, keeping her distance. Soon she could see the women in their nests, clumpy shadows high in the trees, silhouetted against the deepening pink of the sky; here and there a limb stretched out, fingers working at a pelt or stroking a face.
Shadow glanced up at Big Boss's nest. One foot dangled in the air, toes clenching and unclenching. Until a new leader emerged, the ladder of rank was broken into chaos. The days to come would be stressful and trying for everyone.
As the last light seeped from the sky, the men returned. They swarmed around the bases of the trees. They were still squabbling, screeching, and fighting. Some of them clambered up into the trees and began to harass the women and children, smashing open their nests and chasing them across the branches; the women fought back grimly.
Now two of the men started climbing into Shadow's own tree, peering up at her, whispering and showing their white teeth. Shadow could smell the blood on their fur.
Forces worked in Shadow's mind: a fear of the dark unknown, a fear of further punishment at the hands of the people, a chill urge to cradle the thing in her womb. At last the forces reached a new equilibrium.
She slid out of her nest. As silently as she could, enduring the feeble kicking of the child in her womb, she clambered from the branches of her tree into the next, and then the next.
She slipped, alone, into the arboreal dark. Soon the sounds of the squabbling, roosting people were far behind her.
## _F ire_
Here is Fire. Here are his legs walking. Here he is, keeping his hands closed together, cupping the hot embers and the ash.
The sun is hot. The light is in his eyes. His eyes hurt him. His head hurts him.
He remembers why. He is lying on the ground. His eyes see bits of light, Stone's feet swinging at his head and belly and chest. Once again Stone had driven him away from Dig.
Fire wants not to be here. But it is Fire who holds the embers, not his hands. Fire must be here to make his hands hold the hot embers.
The sky grows dark. The air grows cold. Fire looks up. The sky is covered over by cloud.
Something falls before Fire. It is a flake. It is white and soft. There are many flakes, falling slowly, all around him.
A flake settles on his chest. Another on his shoulders. His skin cannot feel them. More flakes settle around him, on the floor. His feet make footprints in the thickening gray cover. He stops. He looks back at the prints. He laughs. He steps backwards into the prints he has made. He steps forward into the prints.
The ground is growing gray. The people are gray. The trees are gray. Some of the people are afraid. Their fingers wipe gray from their eyes and scalps. The children with no names whimper. Their faces hide in their mothers' bellies.
Fire is not afraid. The gray is ash. Fire sees himself in the morning light. He sees his hands sweeping through ash, gathering embers. Now everything is ash. His head tips back. Ash falls into his mouth. His tongue tastes it. Fire is happy in this ash world. His legs run, and his mouth gibbers and hoots.
But now his head is wet.
His legs stop running. He lifts his head. He sees big fat raindrops fall from the sky, slowly sliding toward his face. They hit his mouth and his cheeks and his nose and his eyes. His eyes sting.
The rain makes little pits in the ash. His toes explore the pits. The wet ash turns to gray mud.
The other people trudge around him. Their hair is flat. The mud sticks to their feet in great heavy cakes. The rain turns the ash on their bodies to gray streaks.
The people reach a bank of trees. They stand there, baffled.
Stone steps forward. His great nostrils flare. "River river river!" he cries. His legs march him into the trees. His arms push aside the foliage with great cracks and snaps.
Fire's legs carry him hurrying after Stone, into the forest.
The forest is green and dark and moist. Leaves and twigs clutch at Fire. His eyes look around fearfully, for Elf-folk, or worse. He sees nothing but people, like muddy shadows sliding through the bank of trees. He hears nothing but the crush of foliage by feet and hands, the soft breathing of the people.
Fire pushes out of the other side of the bank of trees.
The ground slopes down. There is rock here, purple-red, sticking out of the grass. Fire's feet carry him carefully over the slippery rocks.
He reaches water. The water is brown, and slides slowly past his feet. It is the river.
The people come down to the bank. Their hands splash water on their faces, washing away mud.
Fire does not touch the water. Fire's hands still hold the embers. Fire stands tall, and his eyes watch the river. To his left the river has scooped holes out from under the bank. A great lip of grass dangles toward the water. Fire sees that there is a gravel beach below the undercut, and deep dark openings behind it, caves.
"Fire Fire!" he cries. "Fire Fire!"
Fire walks toward the caves, cupping the embers. Grass and Wood, the women, follow him. They build a pile of the branches they have carried. They find the driest moss they can.
Inside the cave, Fire lowers his embers reverently into the moss. It smokes, but soon a flame is there, licking at the moss. Fire blows on it carefully.
When the fire is rising, Emma and Sally and Maxie come into the cave. Things cling to their backs, things of blue skin. Emma and Sally make the clinging things slide to the floor. They come to the fire and hold up their hands to its warmth. Sally rubs Maxie's wet hair.
Fire grins. Emma grins back.
The flames are bright. Fire has a shadow. It stretches into the back of the cave, across a bumpy, mottled floor of rock. Fire follows his shadow. It grows longer, leading deeper into the dark.
There are animals at the back of the cave. Fire's eyes open wide. Fire's legs prepare to run.
His nose cannot smell animals. His nose smells people. He makes his legs walk forward.
The animals are sprawled flat against the wall. He makes his hand touch an animal. The fur is ragged and loose. He grabs it and pulls. The skin of the animal comes away from the wall.
There is no animal. There is only the skin of the animal. It was stretched out over branches. He pushes. The whole frame falls over with a clatter.
Behind the fallen frame he sees spears. He picks up a spear. Its tip is a different color from the wood. His finger touches the tip. The tip is stone. It is an axe. No matter how hard he pulls, the stone wants to cling to its spear.
He drops the spear. He walks back along the cave, toward the light of the fire, the gray daylight.
People are gathered around the fire. Some children are sleeping. One woman sits in another's lap, gently cupping her breasts. A man and a woman are coupling noisily.
Emma and Sally and Maxie sit against a wall. Their eyes gaze at the fire, or out into the grayness beyond.
The people are not here, though their bodies are here. Emma and Sally and Maxie are here. They are always here.
Fire's body, warm and dry, wants to couple with Dig. His member stiffens quickly. He looks for Dig.
Dig is lying under Stone, on the floor. His hips thrust at her. Her eyes are closed.
Fire finds a rock on the floor. His fist closes around the rock and he raises it, over Stone's head.
Fire thinks of Stone's anger, his fists and feet.
He drops the rock.
He walks out of the cave, to the river.
The rain is less now. It makes little gray pits on the surface of the water that come and go, come and go. He watches the pits.
For a time he is not there. There is only his body, only the water at his toes, the rain on his head, the pits on the water.
He squats down. The water is a cloudy, muddy brown. A fine gray scum floats on its surface. His eyes cannot see fish. But the water pools here, quietly. And he sees bubbles, bursting on the water.
He slides his hands into the water. His hands like the water. It is cool and soothes his scarred palms. He waits, knees on the ground, hands in the water, the last rain pattering on the back of his neck.
He is not there.
A cold softness brushes his hands.
His hands grab and lift. A fish flies over his head, wriggling, silvery. His ears hear it land with a thump on the grass behind him.
He slides his hands back into the water.
He is not there.
## _R eid Malenfant_
So here was Malenfant, for better or worse in space once again, flying ass-backwards towards the Moon—a Moon, anyhow.
Nemoto and Malenfant sat upright, side by side, in a rounded bulge at the rear of the cramped, coffinlike, gear-crammed capsule. They were each encased in the heavy folds of their garish orange launch-and-entry suits, and a rubbery wet-raincoat stink filled the air.
Malenfant gazed into the tiny, scuffed, oil-smeared rectangle of glass before his face, trying to make out the greater universe into which he had been thrust. There was no sense of space, of openness; surrounded by the womblike ticking and purring of fans and pumps, immersed in the stench of rubber and metal, peering out through these tiny windows, it was like being stuck in a miniature submarine.
... But now Earth swam into view.
From the Station's low orbit Earth had always been immense to Malenfant, a vast glowing roof or floor to his world, ever present, dwarfing his petty craft. But now Earth was receding. First one precisely curved horizon slid into his window frame, and then the other, so that soon he could see the whole Earth, hanging like a Christmas-tree bauble in the velvet black, blue patches peeking out from beneath the white swirl of clouds, painted with the familiar continent-shapes. Malenfant could see Florida, Africa, Gibraltar, and even much of South America, his single glance spanning the Atlantic Ocean. The planet slowly shifted position, drifting from the top of his window to the bottom, so he had to crane forward to see it. Even from here he could see the damage done by the tide: smoke was smeared over a dozen coastal cities, and he saw the cold gleam of white-tops as angry waves continued to pound the land.
Malenfant had been somewhat relieved that the launch had gone through without significant hitches.
He had lain on his couch listening to the flexing of the tanks as they were laden with cryos, then the roar of propellants like a distant locomotive, the whine of the pumps, the waterfall shout of the pad's huge deluge system—and then the bursting roar of the engines. And he could think of nothing but the fact that this BDB booster stack on which he perched had never before flown in test, not even once—no time for that.
Anyhow they had gotten off the pad. The acceleration had been low at first. But as the engines far below had swivelled from side to side to adjust the direction of thrust, the two astronauts, stuck at the top of the stack, had been thrown back and forth, like ants clinging to the tip of a car antenna.
Then had come the violence of staging, as first the solid rocket boosters and then the big main engine cluster had cut out. Malenfant had been thrown forward against his harness, crashing his helmeted head against the curving bulkhead before him. After a heart-stopping moment of drift, the second stage had cut in, thrusting him back into his seat once more.
That second-stage burn had seemed to go on and on—six, seven, eight minutes, their craft growing lighter as fuel burned off, their velocity piling on. Not for Malenfant and Nemoto the old Apollo luxury of taking a couple of swings around the Earth to check out the systems; the BDB's last contribution had been to hurl them on a direct-ascent trajectory all the way out of Earth's gravity well without pausing.
Just ten minutes after leaving the pad at Vandenberg, the second stage finally cut out. Malenfant and Nemoto had listened to the clunk of the burnt-out stage disengaging itself from the lander, and the bull-snorts of nose-mounted attitude thrusters turning their little craft so it pointed nose-first to the Earth—ten minutes gone, and already Malenfant was bound irrevocably for the Moon.
Still the Earth shrank.
"There she goes," he murmured. "I feel as if I'm driving a car into a long, dark tunnel..."
It struck him that Nemoto hadn't said a single word since the pad rats had strapped them into their couches. Now, as they watched the Earth fall away, her small hand crept into his.
And then they broke. They began to work from panel to panel, throwing switches and checking dials, working through their postinsertion checklist, configuring the software that would run the craft's life support systems. Necessary work without which they would not survive, not even for an hour.
New Moon or old, Earth's satellite orbited just as far from the mother planet, and so it was going to take them three days to get there, just as it always had. But because they were flying backward, they weren't going to be able to see the Red Moon itself. Not until they got there.
For the first few hours the abandoned BDB second stage trailed after them, following its own independent path. It was scheduled to sail past the Moon and fly into interplanetary space. The stage was a lumpy cylinder, shining bright in the intense sunlight. Malenfant could clearly see the details of the attachment mechanisms at its upper face, and how its thin walls had crumpled during the launch. But it was venting unburnt fuel from three or four places. The small thrust of the fuel vents was making it tumble, like a garden sprinkler, and it was surrounded by a cloud of frozen fuel droplets that glimmered like stars.
The stage's subtly modified path was bringing it closer to the lander than Malenfant would have liked, at one point no more than a few hundred feet away. He stayed strapped into his seat, watching this potential hazard, and weighing out options. But after a couple more hours the stage began to drift away of its own accord.
When the lander was alone in the emptiness, Malenfant felt an odd pang of loneliness, and almost wished the booster stage would come swimming back, like some great metal whale.
After six hours in space, twelve since they had been woken before the launch, they unbuckled.
Malenfant felt a surge of validating freedom as he found himself floating up from his couch. His treacherous stomach gave a warning growl, however. Throwing up in this confined space would be even more of a catastrophe than on the Shuttle. He turned his back and popped a couple of tabs, trusting that the queasiness would pass.
Awkwardly, helping each other, they stripped out of their launch suits. Now they would wear lightweight jumpsuits and cloth booties, all the way to the Red Moon.
The X-38, hastily modified from a Space Station bail-out craft, was just thirty feet long, an ungainly shape the pilots likened to a potato with fins. Malenfant and Nemoto had been given couches in the rounded bulge at the craft's rear. The craft, designed for a couple of hours' flight down to Earth from low orbit, had been crammed with gear to keep them alive for ten, eleven, twelve days, the time it would take to reach the Red Moon, and come straight back again, if the natives didn't look friendly. Much of its interior was too cramped for the crew even to sit upright—but then, in its primary bail-out mode carrying injured or even unconscious crew back to Earth, reclining couches would have sufficed.
To the rear end of the lander was fixed a liquid-rocket pack. The engine and propellants were based on the simple, reliable systems of the old Apollo Lunar Module. This engine would be used to decelerate them into lunar orbit, and then, if they chose to commit, to slow them further, until the lander began its long glide down into the atmosphere, shedding its heat of descent in a long series of aerodynamic maneuvres, much like the Shuttle orbiter's entry to Earth's atmosphere.
During the last stages of the descent, a big blue and white parafoil, a steerable parachute a hundred and fifty feet wide, would blossom from the lander's rear compartment. That would be quite a ride. The parafoil, the largest steerable chute ever made, would be controlled by warping its wings, which was just the way the Wright brothers had steered their first-ever manned flying machine. That seemed somehow appropriate. Anyhow, thus they would steer their way to a final descent, landing gently on skids.
In theory.
In fact _they_ wouldn't be steering the craft anywhere. The whole descent was automated. This was something against which Malenfant had fought hard. To give up control of the rudders and flaps to some virus-ridden computer program went against every instinct he'd built up in thirty years of flying. But it was much easier and simpler for the engineers to devise a lander that could fly itself all the way down than to figure out how to give a pilot control. _Trust us, Malenfant. Trust the machine_.
The facilities were not glamorous, even compared to the Station and the Shuttle. To wash Malenfant had to strip to the buff and give himself a sponge-bath. It took longer to chase down floating droplets of water and soap than to bathe in the first place.
The toilet arrangements were even more basic. There was no lavatory compartment, as in the Shuttle and the Station, so they were thrown back to arrangements no more advanced than those used on Apollo, and earlier. There were receptacles for their urine, which wasn't so bad as long as you avoided spillage, but for anything more serious you had to strip to the buff again and try to dump your load into plastic bags you clamped over your ass with your hands.
In this cramped environment they had, of course, absolutely no privacy from each other. But it never became a problem. Nemoto was twenty-five years old, with a fine, lithe figure; but Malenfant never found her distracting—and vice versa applied, so far as he could tell. Their relationship was prickly, but they were easy together, even intimate, but like siblings.
It was as if he had known this odd, quiet girl for a long time. In some other life, perhaps.
After eighteen hours awake, they prepared for sleep.
Malenfant had always had trouble sleeping in orbit. Every time his thoughts softened he seemed to drift up from his couch, no matter how well he strapped himself down, and jerk himself to wakefulness, fearful of falling.
And on this trip it was even worse. He was acutely aware that he had travelled far from home this time—in particular, far beyond the invisible ceiling of Earth's magnetic field, which sheltered the world's inhabitants from the lethal radiation which permeated interplanetary space. When Malenfant closed his eyes he would see flashes and sparks—trails left in the fluid of his eyeballs by bits of flying cosmic debris that had come fizzing out of some supernova a hundred thousand years ago, perhaps—and he folded over on himself, imagining what that cold rain was doing to his vulnerable human body.
After a couple of hours he prescribed himself a sleeping pill.
On the couch next to his, Nemoto lay very still, and didn't react when he moved; he couldn't tell if she was asleep or awake.
When he woke up, the pure oxygen of the cabin's atmosphere had made his nose irritable and runny, and his skin was starting to flake off, bits of it floating around him in the gentle breezes.
The nearest thing to navigation in space Malenfant had performed before had been the not-inconsiderable task of sliding a Shuttle orbiter into its correct low-Earth orbit, and then nudging two giant spacecraft, the Space Station and the orbiter, into a hair's-width precise docking and capture.
Flying to the Red Moon was a whole different ball game.
The X-38 had left a planet whose surface was moving at around 1000 miles per hour. The craft was aiming to encounter a Moon moving at some 2300 miles per hour relative to the Earth, with an orbital plane that differed from the spacecraft's. Furthermore the X-38 had to aim, not at where the Moon was at time of launch, but where it would be three days later. For the sake of the air-to-ground public-consumption transmissions they were forced to endure, Malenfant sought metaphors for what they were trying to achieve. "It's like jumping from one moving train to another—and landing precisely in a top-price seat. No, more than that. Imagine jumping from a roller coaster car, and catching a bullet in your teeth as you fall..."
And the various computations had to be accurate to within one part in four _million_ , or the X-38 would slam too steeply into the Red Moon's atmosphere and burn up, or else go flying past the Moon and become lost, irretrievably, in interplanetary space. If they got the navigation wrong, they were both dead. It was as simple as that.
It didn't console Malenfant at all to consider that this feat of translunar navigation had been achieved by manned missions before—nine times, in fact, if you included Apollo 13—since here he was in an untried, utterly untested spacecraft, heading for an alien Moon, and everybody who had worked on those ancient missions was retired or dead.
So he labored at his astronomical sightings, in-situ position recordings which backed up tracking from the ground. He had a navigational telescope and sextant, and he used these to peer through the grimy windows of the lander to take sightings of the Earth, the sun, and the brighter stars. He kept checking the figures until he had "all balls," nothing but zeroes in his discrepancy analysis.
Oddly, it was this work, when he was forced to concentrate on what lay beyond the cabin's cozy walls, that gave him his deepest sense of the vastness he had entered. There was Earth, for example, the stage for (almost) all of human history, now reduced to a tiny blue marble in all that blackness. Sometimes it was simply impossible to believe that this wasn't just another sim, that the darkness beyond wasn't just blacked-out walls, a few feet away, close enough for him to touch if he reached out a hand.
But sometimes he got it, and the animal inside him quailed.
## _F ire_
It is morning. The rain has stopped. The sky is gray.
Fire's eyes watch a branch drift down the river.
Blue wades into the water, waist-deep. He catches the branch. It is heavy. He sets his shoulders and pushes until the branch is resting against the bank.
Another branch comes. Blue grabs it, and hauls and pushes it against the first.
More people come, men and women. Some of them remember the river. Some of them don't, and are startled to see it. They wade into the water. They catch branches and shove them against Blue's crude, growing raft.
Children play, running up and down the bank, jabbering.
A crocodile sits in the deeper water. Fire sees the ridges on his back, his yellow eyes. The crocodile's eyes watch the people. Its teeth want the children.
Fire walks back to the cave. The fire is still burning. People have brought more wood. The damp stuff makes billows of smoke that linger under the roof of the cave.
Maxie is standing before the fire. Maxie's hands hold a fish. The fish is small and silver. A stick is jammed into the fish's mouth. Maxie throws the fish on a rock at the center of the fire. The rock is hot. The fish's skin blisters. Its flesh spits and sizzles. There is a smell of fish and ash.
Sally helps Maxie get the fish out of the fire. _"Careful, Maxie. It's very hot."_
Stone is watching Sally, his eyes hard and unblinking. His member stiffens. His hand strokes it.
Maxie blows on the fish noisily. His white teeth bite into the belly of the fish.
Stone strides to Sally. She stumbles back, alarmed. Stone tucks his leg behind Sally's. She falls on her back. He falls on top of her. She yells. His hand rips at her brown skin. It tears open. Fire sees her pink breast, a shadow of hair below her belly.
Sally's fingers scramble on the floor of the cave. They find a rock. _"Keep off me, you fucking ape!"_ The rock slams into Stone's temple.
Stone grunts and slumps sideways.
Sally pulls herself out from under him. She scrambles away across the floor.
Stone's fingers touch his head. They come away bloody. He looks at
Sally.
His hand locks around her ankle. She screams. He hauls at her leg. She is thrown across the floor, screaming. She slams hard against a rock wall.
Fire's ears hear bone snap. Sally is silent.
Stone grabs her ankles. She lies there, limp, one arm bent above the elbow. He pries her legs apart. His strong fingers rip at brown skin.
Maxie is pressed against the wall. His mouth is wide open.
Emma has come into the cave. She runs to Stone. Her hand drags at his shoulder. _"Leave her alone!"_
Stone ignores her. Fire knows he cannot hear Emma. Stone is not in his ears and his head, but in his penis, his balls.
Fire thinks of Maxie, manipulating the fish in the fire. Maxie is smart. Maxie remembers. Maxie has hands to make good axes. Sally is Maxie's mother. Stone wants more babies like Maxie.
Stone is doing what is right for his people.
All this shimmers in Fire's head, like raindrop splashes on the water. But then it breaks up, like the splashes, and all he sees is an elemental logic: Stone with Sally, Fire with Dig.
Fire smiles.
Emma goes limp. She is sobbing. _"For God's sake."_
A rock flies past Fire's shoulder. It strikes Stone's arm. Stone roars. Blood spurts. He falls away from Sally. Sally lies limp. Fire sees he has not entered her.
Another rock flies in from the mouth of the cave. Stone drops flat. The rock flies over his head.
Fire faces the mouth of the cave. A person is standing there.
Not a person. Fire sees a short, stocky body draped with animal skins, a heavy, protruding face, a brow ridge as thick as a person's, straight black hair. One hand holds an axe. The other hand holds a spear.
It is not a person. It is a Ham. The Ham says, _"My home, Runner."_
Fire's hands ram into the Ham's belly.
The Ham falls back. Fire runs out of the cave.
People run this way and that, making for the river, screaming from fear or anger. Shadows flicker along the top of the undercut, flicker between caves. Spears stab, stone-tipped, so fast Fire can barely see them. Voices call. _"U-lu-lu-lu-lu!"_
A Ham drives a spear into the chest of the woman, Wood. She is knocked onto her back. The spear breaks and twists as she falls. Her body rips and spills. She cries out.
Fire is terrified, awed.
_"Help me. Fire, please."_
It is Emma. She has dragged Sally to her feet. Sally is lolling, unconscious. Sally's arm dangles, blood soaking into the brown skin over it.
Fire remembers the river. Fire remembers the raft. Fire's legs want to be on the raft, away from this blizzard of jabbing spears and shadows.
Fire's hands grab Sally by the waist. He hurls her over his shoulder. She cries out as her broken arm is jarred against his hip. He feels the cool flesh of her belly and breast against his shoulder. Emma has picked up Maxie. Her legs are running.
Stones hail around them, sticking into the ground. The people's legs run from the stones and the Hams' yells. _"U-lu-lu-lu-lu!"_
The people run splashing into the water. There is nowhere else to go. They scramble onto the raft. It is just a mass of floating branches, roughly pushed together. The raft is too small. The people fall off, or climb on each other's backs. As their legs and arms scrabble at the branches the raft drifts apart, in big floating chunks. The people call out and grab at each other's hands and ankles.
Fire runs onto the raft. His foot plunges through the soaked foliage and he falls forward. Sally falls off his shoulders and lands on a wriggling pile of children. The children push her away.
Emma is on the raft. Her hands slap at the children. _"Leave her alone!"_
Maxie sits by his mother, his hands clutching leaves and branches, wailing.
The raft is drifting away from the bank, into the deeper river. It twists, slowly. The people yell and sprawl, their hands clinging to the branches.
Stone comes running down the bank. His eyes are white. Hams pursue him. Stone hurls himself into the water. He goes under. His head comes up. He is coughing. Blue reaches out and grabs Stone. Stone clings to a branch, his body dangling in the water. Fire sees blood seep from Stone's shoulder.
The Hams run up and down the bank, yelling, hurling stones. _"U-lu-lu-lu-lu!"_ The stones fall harmlessly into the water.
The raft drifts toward the middle of the river, away from the bank with the undercut, the capering Hams.
Fire's shoulder stings. He looks around. Emma has slapped him. _"Help me."_
Emma's small axe cuts away Sally's brown, bloody skin. Underneath is more skin. It is pink, but it is mottled purple and black. Emma's hands run up and down the skin.
_"Good. The skin isn't broken. But I have no idea how to set a broken bone. Damn, damn."_ She produces a small gleaming thing. Water pours out over Sally's arm. No, not water: it stinks, like rotten fish. Her hands pull a chunk of branch from the raft. Fire can see water rippling underneath. Emma holds the branch against Sally's arm. _"Hold this,"_ she says. _"Fire hold. Hold it, damn it."_ Her hands wrap his around Sally's arm. His hands hold the branch against the arm. Emma takes a sheet of skin from around her neck. Her hands move over Sally's arm, very fast. When she pulls away her hands, the skin is wrapped around Sally's arm.
Fire stares and stares.
Emma lifts Sally's head and places it on her lap.
Maxie says, _"Is mommy going to be all right?"_
_"Yes. Yes, I hope so, Maxie."_
_"She needs a hospital."_
Emma laughs, but it is like a sob. _"Yes, Maxie. Yes, she needs a hospital."_
The raft is in the middle of the river, slowly turning. The banks to either side are far away, just lines of green and brown. The raft is small, and the river is large.
There is a scream.
Fire sees ridges. Yellow eyes. Teeth.
Stone roars. His arms lift his body. His bulk comes crashing down on the raft.
The whole raft shakes. People scream, clinging to each other. Branches splinter and separate. A child falls into the water, wailing.
Yellow eyes gleam. The crocodile's vast mouth opens.
The child's eyes are white. They stare at the people on the raft.
The mouth snaps shut.
The child is gone, forgotten.
The raft drifts down the river, slowly turning. The people cling to it in silence, locked inside their heads.
## _R eid Malenfant_
Ten minutes before lunar orbit insertion the cabin grew subtly darker. Gradually, as his eyes dark-adapted, Malenfant caught his first true view of the stars, a rich spangling carpet of them, glowing clear and steady.
They had fallen into the shadow of the Red Moon.
Malenfant and Nemoto were both strapped onto their couches. They had a checklist to work through, and settings on their various softscreen displays to confirm, just as if they were real pilots, like Borman and Anders, Armstrong and Collins. But the insertion sequence was completely automated, it either worked or it didn't, and there wasn't a damn thing Malenfant could do about it—nothing save slam his fist into the fat red abort button that would change the engine's firing sequence to send them straight home again. He would only do that in the event of a catastrophic control failure. Or, he mused, if somebody down there started shooting...
He glanced up at his window. There was a disc of darkness spreading across the stars, like an unwelcome tide.
It was, of course, the Red Moon. His heart thumped.
What were you thinking, Malenfant? Were you surprised to find that this huge object, this vast new Moon, is in fact real?
Well, maybe he was. Maybe he had spent too long in Shuttles and the Station, going around and around, boring a hole in the sky. He had become conditioned to believing that spaceflight wasn't about _going_ anywhere.
Passing behind the alien Moon, they abruptly lost the signal from Houston. For the first time since launch day, they were alone.
The cabin was warm—over eighty degrees—but his skin was cold where his clothes touched him.
## _E mma Stoney_
The river's broad body ran from west to east, so that the setting sun glimmered above its upstream sections, making the water shine like greasy tarmac. Thick black volcanic clouds streaked the glowing sky. And when she looked downstream, she saw the Earth, nearly full, hanging low over the horizon, directly above the dark water, as if the river were a great road leading her home.
The raft drifted over the brown, lazily swelling water, rotating slowly, heading roughly east. In fact it was scarcely a raft, Emma thought, just a jammed-together collection of branches, held together by no more than the tangle of the branches and twigs, and the powerful fingers of the Runners. Every so often a chunk of foliage would come loose and drift away, diminishing the raft further, and the Runners would huddle closer together, fearful. And the raft drifted: just that, with no oars or rudder or sail, completely out of any conscious control.
The Runners did not speak to each other, of course. Where humans would have been shouting, crying, yelling, debating what to do, comforting or blaming each other, the Runners just clung to the branches and to each other, silent, eyes wide and staring. Each Runner was locked in her own silent fear, almost as isolated as if she was physically alone. Emma was frightened, too, but at least she understood the fix they were in, and her head whirred busily seeking plans and options. All the Runners could do was wait passively while fate, and the river, took them where it would.
Emma, surrounded by naked, powerful, trembling bodies, had never been so forcefully struck by the Runners' limitations.
And meanwhile those Hams had looked for all the world to her like picture-book Neandertals. What was going on here?...
The river crowded through a section of swamp-forest. Here the trees were low, and the purple spikes of flowering water-hyacinths crowded close to the oily black water. They passed an inlet crowded with water-lilies, their white flowers cupped half-closed. Their leaves were oval, with serrated edges bright green on top and red-brown underneath. As Emma watched dully, a red-brown body of a bird unfolded from its well-concealed place at the base of one lily pad. Its neck and collar were white and gold, and it unfolded long legs and spindly toes, watching them suspiciously.
... Not a bird. A bat, apparently incubating its young on nests built on these floating weeds. She had never heard of bats behaving like that. As the Runner raft passed, the bat stepped with a surgical precision across the lily-pads, its leathery wings rustling. Then it scuttled back to its nest of weed, settling with an air of irritation.
Though the meal of the lost child seemed to have satisfied the huge creature that had first stalked them, Emma glimpsed ridges of skin and yellow eyes everywhere. The crocodiles watched as the raft eroded, inevitably approaching the point where it would dump all its hapless inhabitants in the water.
Sally turned her head. With a cough, she threw up. Pale yellow bile splashed over Emma's lap, stinking.
"Shit, oh shit." She got hold of Sally's leg, behind the knee, and strove to push her over on her side.
The raft rocked, its component branches rippling, and the Runners hooted and snapped.
Emma ignored them. At last she got Sally on her side. She pushed Sally's good arm under her head, with her broken arm on top of her torso, and one knee bent over so she wouldn't roll back. She tipped Sally's head back, hoping to ensure she wouldn't choke, and was rewarded with another gush of vomit that splashed over her hands.
And now she became aware of another problem: a fresh stink, a spreading patch of moisture over Sally's behind. Diarrhea, obviously.
Fire hooted and held his hands over his prominent nose.
There wasn't anything Emma could do about it, not for now. But it sure wasn't a good sign. Perhaps it was blood poisoning: one touch of a filthy Runner finger in a wound, one splash of river water, might have done the damage. Or it might be something worse, some disease such as hepatitis or cholera or typhoid, or even some virulent nasty native to this ugly little world; she didn't know enough about the symptoms of such things to be able to diagnose, one way or another.
And even if she did know what Sally was suffering from, what could she do about it? Her pocket-sized medical kit was gone, lost with the rest of her meager kit as they had fled from the huge skin-clad creatures called Hams. She began to go through the pockets of her ragged, filthy flight suit, hoping to find even a single antibiotic tablet that had gone astray.
Sally convulsed again, and her vomit turned more clear, just a thin, stringy fluid.
Maxie, squatting with the other children, watched all this in wide-eyed dismay. He had been silent since they had left the shore, and now he watched Emma wrestle with Sally as if she was a side of beef, no doubt storing up more problems in that tousled, bewildered head. Later, Emma; one patient at a time.
After an hour of random drifting, the raft began to approach the river's far shore. Shallow beaches strewn with purple-black pebbles slid by. More by chance than design, the Runners were completing the crossing of this huge, sluggish waterway.
Sand glimmered rust-red, a few feet beneath the surface, and it was snagging the raft's branches. The raft creaked and spun. It began to break up, its component branches drifting apart. The Runners cried out. One skinny woman fell into the water with a fearful hoot.
"Emma!" Maxie came stumbling to her, his little feet plunging into brown river water. He threw himself into her arms, and she clutched him close.
More of the Runners fell into the water, or leapt away from the raft toward the shore, splashing noisily and yelling with fear. They seemed to have a lot of difficulty swimming, and Emma wondered if their heavily-muscled bodies were denser than humans'. Wading clumsily, grabbing onto each other and their children, they began to flop out of the water and onto the beach, where they lay like sleek, muscular seals. They shook their heads to rid their tightly curled hair of water; droplets fell back to the river with eerie low-gravity slowness.
Emma felt cold water seeping into the legs of her track suit. Maxie cried out and squirmed higher up her body.
There was simply no way Emma was going to be able to get both Maxie and his mother across those few yards of deeper water.
Fire was one of the last to leave the raft. He actually stood upright on the raft, precariously, and its branches cracked and parted under his feet. Then, hooting, he leapt feet-first into the water. He staggered as his feet sank into the mud, but kept his balance. He looked down at the water lapping around his waist, as if amazed.
Emma called, "Fire! Help us, Fire. Fire Fire Emma Maxie!"
He looked around dully.
Emma held Maxie up above her head. The kid squealed and kicked; Emma wasn't going to be able to hold him like this for long. She cried, "Fire Fire!"
Fire reached out with a liquid motion. With one hand he grabbed Maxie under his armpit and lifted him away from Emma, as if the child were as light as balsa wood. Then he turned and began splashing his way to the shore, holding Maxie high.
Without allowing herself to think about it—without even looking out for crocs—Emma pushed away the last branches, the last of the raft, and let herself and Sally slide into the water. Sally lay facedown in the water, passive, but Emma managed to roll her onto her back. The makeshift sling was filthy, stained by blood and the muddy river water. Emma got the inert woman's head against her belly, and cupped her fingers under Sally's chin. Then, working with her feet and her one free arm, she began to swim backwards, towing Sally's floating form.
She was soon exhausted. Her soaked clothes were heavy and clinging, and her boots made her feet feel as if they were encased in concrete. It seemed an age before her kicking feet began to sink into a steeply rising river bottom. She stood up, gasping.
Sally was still floating, so Emma grabbed a handful of cloth at her shoulder and, still supporting her head, began to drag her out of the water. Nobody came to her assistance—nobody but Maxie, and he was more hindrance than help.
At last she got Sally out of the river, far enough that her feet were free of the lapping, muddy brown water, and she fell on her back with exhaustion.
On this side of the river, there was less evidence of the ash falls that had plagued the Runners for days. But beyond the narrow, pebble-strewn beach, the shore was heavily wooded. The Runners huddled together in suspicious silence, peering at the dense green banks above them.
Night was coming.
With barely a word exchanged, some of the Runners crept cautiously into the woods. Others walked down the beach, tentatively exploring, and Fire and a couple of the women began to drag branches from the edge of the forest, building a fire. Fire cast shy glances at Emma; evidently he remembered, in some dim way, how she had managed to start a fire even when he had lost his treasured handful of embers, probably a key moment in his tortured young life.
First things first, she thought.
She pulled Sally further up the beach. She turned Sally over once more to the recovery position, unzipped Sally's trousers and with some difficulty wrestled them off her, followed by her panties. The clothes were filthy, of course, from feces and river mud, and they clung to her flesh; but Emma was reluctant to use her knife—this was Sally's only set of clothing in the whole world, after all. When she had the pants off she used handfuls of leaves to clean Sally up as best she could, and covered her with her own T-shirt, briskly stripped off.
Then, leaving Maxie with his mother, she walked briskly down the beach. After fifty paces she came to a small stream, decanting from some source in the forest. It had cut itself a shallow, braided valley. Two of the children were playing here, splashing and wrestling. Emma walked a little way upstream of them and began to rinse out Sally's trousers and underwear in the shallow, sluggish water. When she was done she cleaned off her arms and hands, splashed cold water over her face, and took a deep drink. Then she dug her plastic bag out of her pocket—one of the few artifacts she had yet to lose—and dipped it to the stream to fill it with water.
More barely-remembered medical lore came back to her. Diarrhea and vomiting led to dehydration, which you ought to treat with sugar and salt, a teaspoon of each to a liter of water, if she remembered right. Fine, save that she had no sugar or salt, and no teaspoon for that matter...
She glanced up the beach.
Stone was squatting beside Sally. He had removed the T-shirt from her lower body, and was running his hands up her thigh. Maxie had cowered back to the edge of the woods, watching the huge man grope his mother.
Emma put down the water, straightened up, and began to walk back to Sally. She felt around her neck for her Swiss Army knife. She got to within a foot of Stone without him noticing she was there.
So where are you going to stick your blade, Emma? In his cheek, his rock-hard penis, his back? What makes you think this tiny little bee-sting blade will do more than goad him anyhow? He'll kill you, then do what he wants with Sally anyhow.
She pulled out the foldaway lens and lifted it up. She angled it so she caught the sun, and focused a bright spot on the back of Stone's broad neck. He howled, slapping his neck, and jumped up, whirling, his penis flopping. As calmly as she could she tilted back the lens so the spot of light shone in his eyes. He raised his hands, dazzled. She said, "Keep away from her, Stone, you asshole, or I will bring down the sun on you. Stone sun Stone sun! Understand?"
He growled, but still the light shone in his eyes. He stumbled away, his penis wilting.
Trembling, trying to give an impression of command, Emma walked back along the beach, picked up her bag of water, and hurried back to Sally.
Sally still lay on her side, her head resting on her good arm, eyes closed, mouth open. There was a bubble of saliva at her mouth. That bubble of saliva popped, abruptly.
"Oh shit," Emma said. She grabbed Sally and pushed her on her back. Sally sighed once, and then was still. Emma pinched Sally's cheeks until her lips parted. The skin was cool and waxy. She dug her fingers in Sally's mouth, and scooped out gobbets of vomit and flung it on the sand. Then she placed one hand under Sally's chin and tilted her head back. She could hear no breath, not a whisper.
She ran her hands over Sally's torso, seeking the end of the breastbone. Then she pulled her hands to the middle of her chest, placed the heel of her hand a little higher, and began to press down. "One-and-two-and..."
A child leapt out of the woods, a lithe hairy child, its face twisted into a snarl. Maxie scrambled away, screaming. Emma shrank back from Sally, gasping with terror.
... No, not a child. It was an ape, an adult—a female, in fact, with two small empty dugs, a skinny, naked body covered in spiky black-brown hair. She was maybe three feet tall. She had the face of a chimp, with lowering eyes gazing out of ridged sockets, and a protruding mouth with thick wrinkled lips covering angular teeth. Emma could have cupped her brain pan in one hand. But she walked and ran upright, human-style, like a clumsy mannequin—her feet were more human than not—and in one curved, bony hand, dangling below her knees, she clutched what looked like a shaped pebble.
She was a caricature, a shrunken, shrivelled, spellbound mix of ape and human, a dwarfish sprite: an Elf, just as the Runners called her kind. This ape-woman ran up to Emma and capered before her.
Emma picked up a handful of sand and hurled it in the Elf's face.
The Elf howled and staggered back, rubbing her eyes.
Fire came running out of the forest's shade. With a single, almost graceful swipe, he slammed a rock against the side of the Elf's head. She fell sprawled on the beach, unconscious or dying, half her face crushed.
Now there was screaming and yelling. All along the beach, Elf-folk were boiling out of the forest. They ran along the shore, rocks and sticks in their hands.
But the Runners fought back hard. Mothers grabbed their children and ran into the river, where the Elf-folk seemed reluctant to follow. Men and women threw rocks at the scampering Elf-folk, and swung at them with their fists and feet.
But there were many, many of the Elf-folk, and they fought with a mindless intensity that seemed to overwhelm even the Runners.
Emma, trying to ignore this hideous drama, threw herself back at Sally.
After fifteen compressions Emma pinched Sally's nose, clamped her mouth on Sally's, and breathed hard and deep. She tasted vomit and blood. She pulled her head away, let Sally's chest deflate, and tried again. After two breaths she searched again for a pulse, found none, and slammed the heel of her hand into Sally's chest once more.
The conflict went on, crude, animal-like.
It's not my battle, Emma told herself. These aren't people. If they are humans at all they are some kind of predecessor species. Really, they are just two breeds of animals fighting for space. But one breed was at least hollering simple words—"Stone!" "Stone, Blue, Blue!" "Away, away!"—and she couldn't help a deep sense of gratification every time one of those spindly Elf bodies went down, under Runner fists and feet.
Now Stone broke out of the squabbling pack. He had two Elves clinging to his back. One had its teeth sunk into his shoulder, and the other had torn off part of his scalp and a section of his right ear. Stone was howling, and blood poured over him from the glistening crimson wound in his head. More Elves swarmed over him, scratching, biting and beating. Stone went down, and rolled over into the water.
Emma heard an anguished scream. A woman burst out of the squabbling pack. It was Grass. Some of the Elves had closed in a pack around something that struggled, yelling, brown limbs flashing. It was a Runner child—perhaps Grass's child. Grass threw herself at the Elves' backs. They drove her off easily, but she came back for more, twice, three times, until at last a chipped rock was slammed against the side of her head, and she fell to her knees, grunting.
The Elf-folk slid into the forest with their prize, their screeching cries of triumph sounding like laughter.
... And still Emma could find no pulse. She sat back, arms hurting, lungs aching. She was aware of Maxie watching her, a little pillar of desolation, ominously silent. "Oh, Maxie, I'm sorry."
Stone was still in the water, on all fours, head lolling, his hair soaked, the water swirling crimson-brown under him.
Fire stood over him. He was holding a boulder, Emma saw, a slab of worn basaltic rock as big as his head.
Stone looked up, blood congealing over one eye. He raised a hand to Fire, reaching up for help.
Fire slammed the rock down on the crown of Stone's skull. There was a sound like a crunching apple.
Stone slumped. Thick red-black blood diffused in the water.
Fire stood staring at the body. Then he turned to Emma. His gait and eyes held a glittering hardness she had not seen before. She shrank back, scrambling over the ground, away from Sally's body.
Fire squatted down before her. His powerful, bloody fingers brushed her neck. She shuddered at his touch, feeling the burn scars on his palms. He pushed his hand inside her flight suit, and his hand closed around the Swiss Army knife. The lens was open. He snapped off the lens attachment as if breaking a matchstick.
Fire looked at the lens, and at Sally's body, and at Emma. Then he backed away from her, stinking of blood.
Maxie was a few feet away, backed up against a tree. His gaze was sliding over Runners, bloodstained sand, the river.
Emma stood, cautiously. Keeping her eyes on Fire, she reached out for Maxie. "Come on, Maxie. This is no place for us, not any more. It never was..."
"No!" Maxie pulled away from her, his face twisted.
She thought, Now I'm the woman who killed his mother. Nevertheless, I'm all he's got. She made a grab for him.
He ran along the beach.
"Maxie!"
Before she had taken a couple of strides after him he had joined the Runners, who were clustered together, fingering their wounds. She caught one last glimpse of his small face, hard resentful eyes peering back at her. He seemed to be pulling off his clothes.
Then he was lost.
There was a cry, a grisly, high-pitched cry, a child's cry, eloquent of unbearable pain. The woman, Grass, stood and peered mournfully into the forest.
Emma slid into the gloom of the forest, for she had no other place to go.
## _R eid Malenfant_
Events unfolded quickly now, faster than they had for the Apollo astronauts. The Red Moon's gravity, stronger than Luna's, was pulling hard at their falling spaceship, dragging it into a curve that would all but skim the atmosphere.
Nemoto murmured to herself, still working through her tasks as calmly as if they were in just another simulator in Houston.
Malenfant tried to focus on his checklist. But he kept looking up at the strange, shifting diorama beyond the window.
Suddenly he saw the dawn.
Light seeped into the edge of the great disc of blackness. At first it was a deep red, spreading smoothly out around the curve of this small world. Then the band of light began to thicken, growing orange-yellow, and finally shading into blue. The light was coalescing at its brightest point, as if gathering to give birth to the disc of sun itself. And now Malenfant saw shadows of low clouds in the atmosphere; they drew clear dark lines hundreds of miles long over deeper air layers. The surface began to pick up the first of the light—it was an ocean, dark and smooth and sleek, glowing a deep bloody red. And still the light continued to leak into the sky, diffusing higher and higher.
This was a sunrise, not on airless Luna, but on a world with an atmosphere actually deeper than Earth's—and an atmosphere left laden with dust by a chain of great stratovolcanoes. It was a startling, full-blooded dawn, somehow unexpected so far from home.
For the first time Malenfant's thoughts swivelled from Earth, his departure point, and turned with a rush to the world he was approaching. Suddenly he was eager to be down on the ground, to be sinking his fingers into the soil of a new world, and drinking in its air.
## _E mma Stoney_
The light seeped away, and the shadows turned a deeper green.
She moved as silently as she could. But still she was aware of every leaf she crushed, every twig that cracked. And each time she heard a rustle or snap, she expected an Elf to leap out at her.
She didn't know where she was going, what the hell she was doing. But she knew she had to get away from that beach.
The screaming began again, startling her. It was very close, very loud. She crouched down in the bush, staring, listening, too frightened to move.
And she glimpsed movement, through a screen of trees to her right. Smart, Emma. You walked right in on them.
They were the Elf-folk, of course. They had the Runner child spread-eagled against a bare patch of ground. His eyes were wide and staring. Elf teeth closed on the boy's upper thigh, and came away bloody, huge ape lips wrapped around a handful of meat.
The boy thrashed. Emma saw how his eyes turned white. And he screamed, and screamed, and screamed.
After that—as Emma watched, frozen in place by her fear of detection—the boy was steadily dismembered: the drinking of blood, the biting-off of genitals, the startlingly efficient twisting-off of an arm. And through all of this the boy was still alive, still screaming.
... There was a hand on her shoulder.
She gasped, swivelled, fell back in the bush with a soft crash. Someone was standing over her, a shadowy figure.
It was not an Elf, or a Runner. It was a woman. She was wearing a loose tunic of skin, bound around her waist with what looked like a rope plaited from greenery. There were tools stuck in the belt, tools of bone and wood. Her body looked shorter, stockier than a Runner's. Her face protruded. She had no chin. Her skull was large, larger than a Runner's, but she sported a thick ridge of bone over her eyes, and there were prominent crests of bones at her cheeks and over the crown of her head.
Not a human, then. This was one of the powerful, shadowy creatures the Runners had called a "Ham." Emma felt savage disappointment, renewed fear.
But the other beckoned, an unmistakeably human gesture.
Still Emma hesitated. Somewhere on this brutal world were the people who had taught the Runners to speak English. If she couldn't get back to Earth, then if her destiny lay anywhere, it was there—and not with this Ham.
But now she glanced back at the Elves. They had pulled open the boy's rib cage, and the child gave a final, exhausted moan as his heart was torn out.
You're kind of short on choices, Emma.
She followed the Ham.
The Ham glided away through the forest, pointing to the footsteps she made in the dead brush on the ground. When Emma stepped there, she made no sound.
## _R eid Malenfant_
Nemoto said laconically, "Three, two, one."
The booster pack fired, and Malenfant was pushed deep into his seat.
The light of their rockets illuminated the deserts and forests of the Red Moon. All over the little world, eyes were raised to the sky, curious and incurious.
# _PART THREE_
# **Hominids**
#
## _M anekatopokanemahedo_
Manekato lingered on the threshold of the room, held back by a mixture of respect and dread.
Her mother, Nekatopo, was dying.
Nekatopo, breathing evenly, gazed at the soft-glowing ceiling. A slim Worker waited beside the bed for her commands, as still as a polished rock.
Nekatopo's room was a hexagonal chamber whose form was the basis of the design of the House, indeed of the Farm itself. This room had been occupied by matriarchs throughout the deep history of the Lineage, and so it was Nekatopo's now—and would be Manekato's soon. But the room was stark. The ceiling was tall and the walls bare panels, glowing softly pink. The only piece of furniture was the bed on which Nekatopo lay, itself hexagonal.
Manekato remembered how her grandmother had decorated these same walls with exuberant fruits. But her daughter had stripped away all of that. "I honor my mother's memory," she had said. "But these walls are of Adjusted Space; they are not material. They do not tarnish or erode. They have a beauty beyond space and time, as our ancestors intended. Why deface them with transience?..."
But Manekato found the unreal simplicity as overwhelming, in its own way, as the happy clutter of her grandmother. When this room was hers, Manekato would find a middle way: her own way, as all the matriarchs had done—and she felt a sudden flush of shame, for her mother was not yet dead, and here she was calculating how she would use her room.
Now she saw that salty tears leaked over Nekatopo's cheeks, soaking the sparse hair, and trickled into her flat nose.
Manekato was troubled to her core. Her mother had never cried—not even on hearing the news of her imminent death—not even on the day when she had had to send away her only son, Babo, Mapping him to his marriage on a Farm on the other side of the world.
Manekato fled, hoping her mother had not noticed she had been here. She walked alone, along the path that led to the ocean. The Wind was gentle today, comparatively; she was barely aware of the way it ruffled the thick black hair on her back, and shivered over the trees that clung to the ground nearby.
To a human she would have looked something like a gorilla: Stocky, powerful, all of eight feet tall, she knuckle-walked elegantly. She pressed her knuckles into the crushed gravel of the path with gentleness, even reverence. Every speck of land on the Farm was precious to her, like an extension of her own heart. Even this humble path served its purpose with quiet dignity, and had borne the weight of her mother and her mother's mother, deep into the roots of time, as it bore her weight now.
Quiet dignity, she thought. That is what I must strive for, in the difficult days ahead.
The path ended at a shallow cliff top that overlooked the sea. The sea was gray and cloudy, laden with silt, and tall waves, generated by a storm raging far over the horizon, crashed with exorbitant violence on the heavily eroded shore. Manekato glimpsed the rectangular gridwork that covered the ocean floor—the boundary of the undersea Farms—a shining mesh that disappeared into the murk of the cloudy water.
The tides were shallow on this moonless Earth, so the beach was narrow and battered by waves. But still huge birds plummeted from the sky, their muscular wings folded, stabbing after the unwary fish and crabs who clung to life at this thin, inhospitable margin. Manekato swivelled her ears to hear the calls of the birds, deep-pitched and throaty to penetrate the unceasing roar of the Wind.
Manekato turned and looked back the way she had come, resting her weight easily on her knuckles. The Farm sprawled over a low hill—in fact it was the core of a volcano, Wind-eroded to a snub long before her Lineage had begun to work this land. The Farm was dominated by the low, streamlined House that sat at the crest of the hill, its prow facing the direction of the prevailing Wind like a beached ship. Around the House sprawled a glowing gridwork of light, in the hexagonal pattern that was the signature of the Poka Lineage. Each of the fields marked out by the grid bore a different crop, ranging from the most advanced self-recursive Worker designs—even from here she could see nubs of heads and stubby limbs pushing out of the ground—all the way back to the Lineage's first harvest, a fat-trunked, ground-hugging willow whose bark still provided some of the best tea available anywhere.
But the land itself was only a cross-section of the greater Farm. There were more cultivated layers stacked deep beneath her feet, fed by light piped from the surface, and mines for the water and hydrocarbons locked in the ground's deeper rocks, and even one mighty borehole that punched through the planet's crust and into the mantle, sipping at Earth's core heat. There were more ducts that pumped heat and carbon dioxide and other waste products back into the ground, of course, as the Poka Lineage contributed to the husbandry of the world.
Even above the ground the Farm's activities extended. Manekato could see engineered birds wheeling over the main House, snapping Wind-blown debris from the sky. The birds were restricted to the Farm's perimeter, and Manekato could see how they flocked in a great wedge-shaped slice of sky that projected up from the ground, so high that the uppermost birds were mere dots against the banded, rippling clouds that were the province of the Sky Farmers.
From the core of the Earth to the bellies of the clouds: That was the extent of the Poka Farm, every scrap of it worked and reworked, every speck of dirt, every molecule of air and water functional, every bacterium and insect and animal and bird with a well-designed role to play in the managed ecology.
There was not a patch of this world that was not similarly cultivated, cherished by its Lineage.
And the Farm would soon belong to Manekato, all of it—even though she was just eight years old: still a young adult, little more than a third of her life gone.
Even though she didn't want it.
Now Manekato heard a faint cry. She swivelled her parabolic ears towards the House, and picked out the voice of her mother, calling her name.
She hurried up the path, back sloping, powerful legs working, levering herself forward on her knuckles. As she passed, immature Workers called out to her, tinny voices piping from ill-formed mouths, already seeking to serve; and willow leaves swivelled frantically in her shadow as they strove to drink in all the light of the eight-hour day.
She returned to her mother's room, at the heart of the Farm. Unhappily she stepped forward, approaching the bed.
Her mother's bed looked like a simple hexagonal nest, woven of leafy branches. It was in fact a cluster of semisentient Workers, designed to mimic the nests of willow and birch branches that children learned to make for themselves from an early age. It had been manufactured to Manekato's design by Worker artisans, twelve generations removed from the crude self-recursive creatures budding in the fields outside.
The floor of the room was a pit filled with hard-compacted white dust. The dust was the ground-up bones of her ancestors. One day Nekatopo's bones would be added to the pit, and, not many years after that, Manekato's, too. Nobody knew how deep the dust pit extended. Manekato could feel the soft grittiness of the dust, but not a grain of it clung to her feet.
Nekatopo opened her eyes.
"... Mother?"
"Oh, Mane, Mane." It was a childish diminutive she had not used since Manekato was a baby. She reached up, her great arms withered and weak.
Manekato embraced her, feeling the tears soak into the hairs in her own chest.
"Oh, Mane, I'm so sorry. But you must go to the Market."
Manekato frowned. She knew that no woman had travelled to the Market since her grandmother's day. Manekato herself had never left the boundary of the Farm, and the prospect of travelling so far filled her with dread. "Why?"
Nekatopo struggled to sit up, and wiped her face with the back of her hand. "I don't even know how to tell you this. _We are going to lose the Farm_."
Manekato felt her mouth fall open. A change in the possession of a Farm occurred only when a Lineage became extinct, or when some member of a Lineage had committed a grave crime.
"I don't understand."
"I know you don't. Oh dear, dear Mane! It is the Astrologers. They have news for us which—well, it has gone around and around in my head, like the Astrologers' own wretched stars wheeling around the world. _The Farm is to be destroyed_. A great catastrophe is to befall the world—so say the Astrologers."
Manekato could not take in any of this. "Storms can be averted, waves tamed—"
"You must believe the Astrologers," Nekatopo whispered, insistent. "I'm sorry, Manekato. You must go to the Market and meet them."
Manekato pulled away from her frail mother, frightened, resentful. "Why? If all this is true, what use is talk?"
"Go to them," Nekatopo sighed, subsiding back into the arms of the semisentient branches.
Manekato walked to the door. Then—torn by shock, uncertainty, shame, doubt—she hesitated. "Nekatopo—if the Farm dies—what will become of me?"
Nekatopo lay on her bed, a dark brown bundle, breathing softly. She did not reply—but Manekato knew there was only one possible answer. If the Farm died, then the Lineage must die with it.
She burned with confusion, resentment.
But still she hesitated. It struck her that whatever the fate of the Farm, if she travelled to the Market, her mother might not be able to welcome her home again.
So, softly, she began to recite her true name. "Manekatopokanemahedo..."
Manekato's true name consisted of nearly fifty thousand syllables—one syllable more than her mother's name, two more than her grandmother's—one syllable added for each generation of the Lineage, back to the beginning, when members of a very different species, led by a matriarch called Ka, and her daughter called Poka, had first scratched at the unpromising slopes of the eroded hills here.
Manekato's people had farmed this scrap of land for fifty thousand generations, for more than a million years.
Nekatopo listened to this childlike performance, unmoving, but Manekato sensed her wistful pleasure.
## _J oshua_
Joshua crouched by a bubbling stream. His nostrils were filled by the musky smell of the hunters' skins, the soft green scent of grass.
The giant horse had become separated from its herd. It snorted, stamping a leg that seemed a little lame. Forgetting its peril in the foolish way of all horses, it nibbled at grass.
The Ham hunters crept forward. Most of them were men. There was no cover, here on the open plain, but they hunkered down in the long grass, and the drab brown skins they wore helped them blend into the background. They were patient. They worked toward the horse step by silent step, staying resolutely downwind of the animal. Lame or not, the heavy old stallion could still outrun any of them—or punish them with its hooves should they fail to trap it properly.
This small drama took place on a plain that stretched from the foot of a cliff. To the east, beyond a stretch of coarsely grassed dunes, the sea glimmered, a band of gray steel. And to the north a great river decanted into the sea via a broad, sluggish delta system. The plain was wet and scrubby, littered by pools. At the base of the cliff itself, a broad lake was fed by springs that sprouted from the cliff's rocks.
The coastal plain, with its caves and streams and pools and migrant herds, was the home of Joshua's people. They called themselves the People of the Gray Earth. Others called them Hams. They had lived here for two thousand generations.
To Joshua, the landscape was a blur, marked out by the position of the other hunters, as if they glowed brightly—and by the horse, the center of their attention.
A soft call came. Abel was waving his arm, indicating they should approach the horse a little closer. Abel was Joshua's older brother.
Joshua crouched lower and moved through the grass, toward the incurious horse.
But now his questing fingers found something new, lying hidden in the grass. It was a stick, long and straight. No, it was a _spear_ , with a stone tip fixed to the wood by some black, hard substance; he could see where twigs had been sheared away from it by a stone knife. He picked up the spear and hefted it, testing its weight. It was light and flimsy; it would surely break easily on a single thrust. Its shaft was oddly carved, into fine, baffling shapes.
A _bear_.
He dropped the spear, crying out, and stumbled back. Suddenly a bear had been looking at him, from out of the shaped wood in his hands.
A massive hand clamped over his mouth and he was pushed to the ground.
Abel loomed over him. His skins, of horse and antelope, were tightly bound about his body by lengths of rawhide thread. His eyes were dark pools under his bony brow. "Th' horse," he hissed.
"Bear," Joshua said, panting. "Saw bear."
Abel frowned and cast around, seeking the bear. Then he saw the broken spear. He picked it up, briefly fingering its dense carving, then hurled it from him with loathing. "Zealots," he said. "Or En'lish. Skinny-folk."
Yes, Joshua thought uneasily. Skinnies must have made the little spear. But nevertheless there had been, briefly, a bear glaring at him from out of the carved wood.
"Ho!" It was Saul, another of the Ham hunters. "Horse breakin'!"
Abel and Joshua struggled to their feet. The horse, startled, was coming straight toward the brothers, a mountain of meat and muscle, a giant as large as a carthorse.
Joshua grabbed a cobble, and Abel raised his thrusting spear. They grinned at each other in anticipation.
Joshua ran straight into the animal's mighty chest.
He was knocked flying, and he landed in the dirt in a tangle of loosened furs. Winded, he got straight up, and ran back toward the fray.
He saw that his brother had grabbed the horse around its neck. The horse was bucking, still running, and it carried Abel with it; but Abel was stabbing at the horse's throat with his spear. The spear was a short solid pillar of wood, stained deep with the blood of many kills. It was a weapon of strength and utility, without carving or decoration of any kind.
The slender spear of the Skinny-folk was meant to be thrown, so that an animal could be brought down from a distance, sparing such hard labor; the Hams had no such technology, and never would.
In a moment Abel's thrusts had reached some essential organ, and the animal crashed to the dust. The other men closed, yelling, hurling themselves on the animal to subdue it before it died. With a gleeful howl, ignoring the pain of his bruised chest and back, Joshua joined in. Before the animal was overpowered they all suffered bruises and cuts; one man broke a finger.
When the horse was dead, the butchery began.
Joshua found a flat cobble. He sat on the ground with one leg folded under him, tucked a flap of antelope skin over each hand, and began to work the cobble with fast, precise motions.
With fast blows of a pebble, he knocked away bits of stone, working around the cobble until he had left a series of thin ridges on a domed surface. After twenty or thirty strokes, with bits of stone littering the ground around him, he pulled a bone hammer from the cord around his waist. The hammer was a bit of antelope thigh bone, broken, discolored, heavily worn with use. With care, he struck one of the ridges. A thin, teardrop-shaped flake fell away. He picked it up and inspected it; it was fine and sharp, good enough for use without further work. He returned to his cobble and knocked out a series of flakes, with one confident blow after another, until the core had been returned to convexity. Then he began to prepare the core to make further flakes.
Joshua was good at working stone. It was a high art because each nodule of stone had its own unique properties; the toolmaker had to find a path through the stone to the tools he or she wanted. It was a question of seeing the tools in the raw stone. Men and women alike would watch his fast, precise movements, seeking to copy him. The women pushed their children toward him, making them watch. Nobody asked him about it, of course; people didn't _talk_ about toolmaking.
Making such tools was the thing Joshua did best, the thing for which he was most valued, the thing for which he valued himself. And yet it set him aside from the others.
He tucked his bone hammer back in his rawhide belt and took his flakes to the horse. He began work on a leg. With a series of swipes he cut down the skin on the inside of the limb, pulling it away from the muscle. Some of the horse's thick brown hair stuck to the edge of the tool. Then he moved to the belly, opening up the hide. He grasped the open skin and pulled it sideways. Where membranes clung to the skin, he swiped at them gently with his flake, holding the stone at its center between his fingers. The membranes parted easily. There was no blood, no mess.
When the horse was skinned, it was easily dismembered. Joshua cut away the meat of the neck. It fell open and was pulled away. He turned his axe over and over, seeking to use all its edge. When he was done he moved to the rib cage, and sliced down it with a crunch.
The people talked softly, steadily. They talked boastfully about their own and each other's prowess in the hunt of the horse, the people waiting for them at the hut—especially young Mary, whose breasts and hips were beginning to fill out, making her a center of intense interest among the men, and amusement for the women. Their attention was filled with each other; the horse, now it had turned to a mere mine of meat, had receded.
But even here, as the people worked together on the fallen horse, they sat a little away from Joshua. They were reluctant to look at him directly, and did not respond to what he said, as they responded to others.
Joshua was short, robust, heavily built. He was barrel-chested, and his arms and short, massive-boned legs were slightly bowed. His feet were broad, his toes fat and bony. His massive hands, with their long powerful thumbs, were scarred from stone chips. His skull, under a thatch of dark brown hair, was long and low with a pronounced bulge at the rear. His face was pulled forward into a great prow fronted by his massive, fleshy nose; his cheeks swept back as if streamlined, but his jaw, though chinless, was massive and thrust forward. Over each of his eyes a great ridge of bone thrust forward, masking his eyes. There was a pronounced dip above the brow ridges, before his shallow forehead led back into a tangle of hair.
He looked powerful, ferocious. But in his pale brown eyes there was uncertainty and confusion.
Joshua was twenty-five years old. Already he was one of the senior members of the group; only a handful of men and women were older than he was. And yet he still felt something of an outsider, as he had been all his life.
The problem was his toolmaking. He would always be valued for it. But others were suspicious of what lay at the center of that profound skill: his ability to see the tools in the stone.
It was uncomfortably like what the Zealots did, and the English. Skinny-folk spoke to the sky and the ground as if they were people. Their tools were carved and painted in ways that, sometimes, made even Joshua see people or animals that weren't really there.
Just as the knives and burins and scrapers he saw in the cobbles weren't really there either, not until he dug them out. The others sensed that his head was full of strangeness, and that was why there was a barrier around him, a barrier that never broke down.
Now the hunters had completed their butchery, and the meat lay scattered around them in neat crimson piles. Joshua dropped his stone flakes, and soon forgot them. The hunters picked up cobbles and smashed open the bones. They would bring the meat back to their hut at the base of the cliffs. But first they would enjoy the warm, greasy, delicious marrow, the privilege of successful hunters. There was a mood of contentment. They knew that they need not hunt again for several days, that the women and children would welcome their return with joy, and that the evening would be filled with good food, companionship, and sex.
And, while the men lolled contentedly, Abel began to talk of the Gray Earth.
The Gray Earth was the home of the people.
The Hams had fallen, baffled, to this strange place of red dirt and grass. They lived here, but it was not as the Gray Earth had been. On the Gray Earth, the animals ran past the people's caves like great rivers of meat. On the Gray Earth, there were no skinny Zealots or English or troublesome Elf-folk; on the Gray Earth there were only Hams, the people of the Gray Earth.
The men listened. The Gray Earth lay two thousand generations in the past, and now it made the people's only legend, relayed from one generation to the next, utterly unchanging and unembroidered; they were a people conservative even in their storytelling.
But Joshua looked up into the sky. The sun was fading now, and the earth shone brightly. This earth was not the Gray Earth, for it was not gray, but a bright, watery blue.
The Hams lived in an unchanging present. Joshua's sense of his life was of a series of days more or less like today, stretching ahead of and behind him like images in a hall of mirrors, reaching from his dimly-recalled days as a toddler begging scraps from his mother, all the way to no-longer-remote times when he would become as toothless and broken-down as old Jacob, back in the hut, again helpless and dependent on the kindness of others. The Hams knew of life and death and the cycle of their lives. But of the world beyond themselves they knew of no change.
... No change but one, Joshua reflected: in the past, they had lived on the Gray Earth, and now they did not.
Joshua looked at his companions as they rested, lolling against the ground, licking marrow from their fingertips, listening amiably to Abel's loose legends. He knew that not one of them would share his thoughts, of past and future and change, of knives buried in rocks.
Joshua kept silent, and peered up at the earth's cool loveliness.
The hut was in the overhang of the cliff, close to the lake. It was built of beech saplings stuck in the ground, bent over and tied at the top. Skins of horses and antelopes had been laid loosely over the frame, weighted down with rocks. More massive rocks had been dragged to the rim of the hut. The area around was scattered with debris, animal bones, abandoned tools, cobbles scooped from the hut floor, and handfuls of ashes.
As the hunters returned with their haul of meat, Joshua saw that smoke was already rising from rents in the roof. Only a few children were outside, playing with the scattered cobbles and bits of skin. Joshua saw bats pecking hopefully at the abandoned bones.
The children ran to the hunters, and playfully grabbed at their meat.
Inside the hut the air was smoky, but the fires in their shallow hearths gave off a yellow-red glow that sent long flickering shadows over the dome of skin above. Beside the hearths, many of the women and children were already eating. The women had been hunting, too. Impeded by their children and infants, women mostly did not tackle the huge game taken on by the men, but the steady flow of smaller game they returned, like beavers and rabbits and bats, provided more than half the group's provisions.
Joshua began to shuck off his skins, loosening or cutting rawhide ropes and letting the skins fall where they may. In the hot, stuffy air of the hut he began to scrape dirt and sweat from his skin with a bit of antelope jaw bone.
Soon everybody was naked. Men and women alike were muscular and stocky, as were all but the very youngest children, so that the hut was filled with brawny, glistening bodies, moving to and fro with slabs of meat and bits of stone and bone and skin, comparing fresh injuries and wounds. The Hams lived lives of constant exertion and physical stress, and injuries were common.
Nobody knew their fathers here. But people were tied by loyalty to their mothers and siblings, and couples were more or less monogamous while they stayed together. So the horse meat was distributed through the group, fairly evenly.
Joshua, with his own slab of meat, found a place on the fringe of the hearth built by Ruth, who coupled with Abel. The low fire was surrounded by heaps of dried seaweed, to be used as bedding. Abel sat with Ruth, and two small children settled down before them, noisily tearing at rabbit legs, blood running down their chins.
One of the younger men approached the pubescent girl Mary, but she huddled close to her mother.
Joshua ate his meat raw, tearing at it with his shovel-shaped teeth and cutting it with a flake knife; every so often he scraped his teeth with the knife. And as his powerful jaw ground at the meat, great muscles worked in his cheeks.
On the fringe of the firelight he sat alone, speaking to nobody.
He had had only brief relationships with some of the women. Abel, by comparison, had shared a hearth with this one woman, Ruth, for many seasons. Like the men and even some of the children, the women saw too much strangeness in Joshua.
In one corner of the hut sat old Jacob. He was sitting on a patch of cobbles, flat sides up, laid over a damp place on the floor. He watched the others, waiting without complaint.
Now Abel, his own hunger sated, sat beside the older man. He gossiped to him gently of the day, of who had said and done what to whom, and he tore at meat, cutting off strips with a small knife. But the old man had trouble chewing; he complained loudly about the pain of the pulpy stumps of his smashed teeth. So Abel chewed the meat himself, pulling at it until it was soft, and pushed it into Jacob's mouth as if feeding an infant. Jacob accepted it without comment or shame.
Jacob's body showed the traces of a long life's relentless work. A charge by an enraged horse had left him with smashed teeth, a shattered arm, a crushed left side and a sprained leg that stubbornly refused to heal. The suite of injuries had left him incapable of participating in the hunt, or even joining in the easier tasks of the hut, like building the fires or making tools.
Joshua recalled how a healthier Jacob had once helped Joshua tend Miriam, Joshua's mother, when she lay dying of an illness that had made her belly swell and caused her to cough blood. And now Abel tended Jacob. It was the way of things, accepted without question.
Jacob was the oldest individual in the group, at thirty-nine years old.
As the evening drew in the adults gathered in loose knots. Joshua joined a loose circle, saying little, cutting at a stick of fire-hardened wood to make a new thrusting spear. Ruth scraped at the skin of the horse to remove its fur, and dragged it through her teeth. Others settled into similar quiet chores.
Like the others Joshua listened intently to the talk, absorbing every detail of rumors, of promises made, romances broken, children praised or disciplined, injuries healed or acquired. His hands worked at the stick, but it was a simple, ancient task, so deeply ingrained by generations of practice that it was almost as unconscious as breathing. It was as if all that existed in the world was the circle of faces, orbiting the light of the fires. All they talked of was each other, never of the tools they made; those were things of doing, not talking.
As the last of the daylight seeped out of the bits of sky visible through the smoke vents, people drifted apart. Abel took Ruth's hand and led her to a dark corner of the hut, close to where toothless Jacob snored noisily.
Joshua lay down alone, close to the fire Ruth had built, on a rough pallet of seaweed. He stared into the fire, and he thought he saw creatures capering in the flames, Skinny people like the Zealots or the English. But though the dancing creatures amused him, they disturbed him, too, for there were only flames, no people or animals here.
It seemed to Joshua that he woke to hear a soft gasp, like surprise, from Jacob, and then silence. But Joshua ignored this, and fell deeper into sleep.
In the morning they found Jacob lying dead, slumped over on his damaged arm.
They would bury Jacob just outside the hut's main entrance.
Joshua swept away rubbish, picked-over animal bones and flakes of worked rock, and began to dig, using bare hands and stone scrapers, powerful muscles working.
When the grave was done it was about half Joshua's height in length, and so shallow that when he stood in it, its lip barely came up to his knees. Even so the diggers had disturbed other bones, yellow and brown from their immersion in the ground, the bones of people long forgotten.
Abel carried Jacob's corpse in his arms. The ruined body, toothless mouth gaping, was light, for it had been some time since Jacob could eat properly. Abel was weeping, for he had been fond of Jacob, who was now gone. Abel put the body on the ground. He tried to fold it up into a fetal position, knees tucked against the chest, head resting on a forearm, but the body was already too stiff. So Abel and others were forced to haul at the body until its joints cracked, and it folded as required. Then Abel bound up the wrists and ankles with rawhide thong.
Children watched wide-eyed.
Abel set the body into the grave among the yellowed bones of deeper, nameless ancestors. Then he used his broad feet to scuff dirt back into the hole.
Others joined in, with hands and feet, kicking at the piles of dirt around the grave. When the grave was roughly filled, Abel stamped on it to level it, and allowed the children to run over it.
People wept openly. Many of them had loved Jacob. But now Jacob was gone.
If the world of the Hams was unchanging, it was also a world of limits. If too many children were born, then they would starve, for the land afforded only so much food. No animal could be hunted save those small or old or weak enough to be brought down by the strength of a combination of hunters at close quarters. Every person went through life limited by their strength and their health and the richness of the land and the vagaries of the weather. Nobody, not even Joshua, could make a _new_ tool, of a type that had not been made before.
And here was the ultimate limit, the limit of death. Jacob was _gone_ , no more existent than in the days before he was born, beyond hope and pain and love. For now the people grieved, and they would speak of him as if he were alive. But soon those who remembered him would die in their turn, and even his name would fade from the world.
Absently Joshua looked up to the sky, his thick neck stiff, seeking the Blue Earth.
And that was when he saw it: a thing like a bat that sailed across the sky, black and white like a gull—and yet it was not a bat. Its wings were stiff, and it was huge and fat, and it drifted beneath a huge blue and white skin, suspended there by threads.
It sailed out of Joshua's sight, beyond the line of the cliffs. He watched, open-mouthed, noting where the extraordinary bat-creature fell.
## _S hadow_
Shadow didn't want to wake up. In her sleep she was warm and cushioned by the woven branches, dreaming arboreal dreams five million years old.
It was the baby that dispelled her dreams, with a bout of savage kicking that led to a stabbing stomach cramp.
Her green mood shattered in a hail of red. She rolled over, groaning, and her gullet flexed, as if she was about to vomit. But it was a dry retch; her stomach was empty.
She sat up, rubbing the base of her belly. Slowly the cramps eased. The sun was already above the horizon, the sky tinged subtly pink by the air's dust.
She inspected this tree to which she had fled in the dark. Elf-folk had been here. The branches were twisted and torn where they had been pulled together for nests, and much of the green fruit of the tree was missing.
She had not come far. She was still within the range of the people. The sun was already high, glimmering down through the canopy. The people woke with the dawn. They might be close already.
She grabbed a handful of fruit and pushed it into her mouth.
_The people_. As she did every time she woke, she remembered in grim red shards what had happened to her, Claw and Big Boss and Little Boss and the rejection by her mother. The fragmentary, terrified images broke up into a wash of green and red and blue. She hooted in alarm, as if some predator had come wheeling out of her own head to threaten her.
She abandoned her nest and scurried down the tree to the ground. She crashed through the undergrowth, twisting aside small branches and shrubs without a thought for the noise she was making. She saw no people, and did not hear them.
And she did not stop until she was in a place she did not know.
For the first time in her life, she was in a place without the guidance of her elders, who had known the position of every fruiting tree, every bubbling stream. _Everything was new_ : the trees, the rocks, the subtle crimson shades of the dust, even the way the sun lanced down through the canopy. She had no way to figure out a path through this new landscape, a way to survive. Her kind did not see patterns in the natural world; they learned the features of the environment around them—the dangers, the sources of food and water—by rote.
Panic struck her. She longed to run back the way she had come.
She thought of Claw.
One of the trees had a hole in its trunk, a little above her eye level. Suddenly she was thirsty. She probed at the hole with one finger. She was rewarded with cool dampness. She pulled out her finger and licked it. Hastily she gathered leaves, chewed them to a spongy mass, and stuck them in the hole. When she pulled out the leaf mass it was dripping wet, and she sucked the water gratefully.
Her stomach clenched abruptly. She squatted on her haunches and briskly, painfully, passed watery shit. She took some soft, crumbling wood from a rotting tree trunk, mashed it up to a wool, and used it to wipe her backside clean of the sour-smelling stuff.
She heard a distant hooting, an answering scream. It was the Elf-folk.
As soon as she was able, she got to her feet and walked on, feeding on whatever fruit and shoots she found, heading resolutely away from the noises of her people.
But soon, very soon, she ran out of forest. She stood on the fringe of the open savannah, clinging to the forest's green shade.
And a bat came drifting across the sky, a great black and white bat with blue wings.
She howled and lunged back into the green mouth of the forest.
## _E mma Stoney_
After getting away from Fire's Runner group, Emma had followed the beckoning Ham woman into the forest. It was an arduous trek, through increasingly dense foliage. But after perhaps a mile they came to a small clearing.
There were shelters here, made of skins stretched out over saplings driven into the ground. There was an overpowering stench, of people, of sweat, wood smoke, excrement, and burning fur. Even the walls of the huts stank, she found, a musty, disagreeable odor of a kind she associated with the clothing of old people who didn't wash or change enough.
But, stench or not, it was a kind of village.
A Ham village.
A village of Neandertals.
She approached cautiously, following the Ham who had found her.
The Hams barely seemed to notice her. They were utterly wrapped up in each other. Some of the children plucked at her clothing with their intimidating, strong fingers. But otherwise the Hams stepped around her, their eyes sliding away.
But however coolly the Hams greeted Emma, they did not expel her.
She dug out her own hearth and built a fire.
Nobody shared food with her that first night. But the next day she managed to catch a rabbit with a homemade snare, and she brought the meat back to the camp and cooked it, even sharing a little with the adults. They took the meat, sniffing the burned stuff gingerly, but ignored her.
So it went on.
There were many of them, she soon learned, perhaps eighty or ninety, in shelters that faded into the dense green forest background.
With their hulking bodies and broad bony faces the Hams seemed like extras in some dreadful old movie to Emma, wrapped up in their animal skins, knocking their crude tools out of the rock. Everything they did, from cracking open a bone to bouncing a child in the air, was suffused with strength—they seemed much more powerful even than the Runners—and Emma quailed before their brute power. But it was apparent that such strength was not always wisely applied, for she saw evidence of a large number of injuries, bone fractures, and crushing injuries, and scarred skin.
They were humans, of a sort, but humans who made a living about the hardest way she could imagine. Their favored hunting technique, for example, even for the largest prey, was to wrestle it to the ground. It was like living with a troupe of rodeo riders.
But they cared for their children, and for their ill and elderly.
_And they spoke English_ , just like Fire's people, the Runners. Who could have taught them? That central mystery nagged at her—and she sensed her own destiny lay in unravelling it.
The forest, like the savannah, was full of predators: cats and bears and dogs, not to mention snakes and insects, some of them giant-sized, that she didn't trust at all.
But the most dangerous creatures of all were the people.
There seemed to be many types of hominids wandering around this globe. She knew there were Hams and Runners and Elf-folk and Nutcracker-folk, and presumably others. The vegetarian Nutcrackers seemed content to chew on bamboo and nuts in the depths of the forest, following a sleepy, untroubled, almost mindless lifestyle that Emma sometimes envied. The Runners conversely generally stuck to the plains.
The forest-dwelling Elf-folk—three or four feet high, like upright, savage chimps—were, for Emma, the most dangerous factor in the landscape. Having glimpsed what that troupe of Elf-folk had done to the Runner child, to finish her life as a living food source in the hands of Elfmen remained her abiding nightmare.
But everybody pretty much left the Hams alone.
For one thing, with their clothing and comparatively elaborate tool kit and distorted English they were a lot smarter than the rest. And they were beefy besides, even the women and children, more than a match for any Elf.
For all the Hams jabbered their broken English, Emma knew she could never become part of this inward-looking, deeply conservative community. But she also knew she was a lot safer here than wandering around, alone in the forest.
And so she stayed, inhabiting a rough lean- to on the edge of the community, bit by bit building up her own survival skills and recovering her strength, and waiting for something to turn up.
The Hams' technology was more advanced than the Runners, but still, considering those big brain pans, remarkably limited. They had more advanced knapping techniques, manufacturing a range of flakes and points and burins in addition to the ubiquitous hand axes. They fitted stone tips to their thick thrusting-spears.
But that was about it. They had no piece of technology with more than two or three components. They didn't have innovations even Emma could think of, such as spear-throwers and bows.
Other gaps. If they weren't interested in something—a type of plant, for instance, which had no use for food or medicine or tools, nor carried any threat as a poison—they simply ignored it. If it didn't matter, it was as if it didn't even exist; as far as she could tell there were whole categories of such "useless" objects and phenomena which had no names.
There were no books here, of course—there was nothing like writing of any kind. And no art: no paintings on animal skins, no tattoos, not so much as a dab of crushed rock on a child's face.
Indeed, the Hams seemed to loathe symbology of any kind. The Hams tolerated the odd colors of Emma's skin and hair, her slimness of build, the way she spoke, even the garish blue of her clothes—but they could not bear the South African air force logo that adorned the breast of her flight suit, and she had to cut it out with a stone knife. (Loathe to throw away anything that had come from home, she had tucked the patch into a pocket on her sleeve.)
She came to suspect that what disturbed them wasn't the symbols themselves as much as the response of herself to them—and other _Skinny-folk_ , a class which seemed to include herself and the mysterious "Zealots" and "En'lish." The Hams would jabber about how Skinnies saw _people in the rock_ , as if the symbols themselves were somehow sentient.
As a result, the Hams' world was a startlingly drab place, lacking art and religion and story—save, of course, for their one great central myth of the Gray Earth, where they had come from. They didn't tell jokes. The children played only as baby chimps might, exercising their muscles and testing their animal reactions against each other.
And to them, death appeared to be a genuine termination, a singularity beyond which an individual, leaving no trace, had no meaning. To the Hams, today was everything, yesterday a minor issue—and if you weren't here tomorrow, you wouldn't matter.
In many ways, they were like the Runners, then. But, unlike the Runners, they talked and talked and talked. They seemed to have a wide vocabulary, much of it English, and they would hold long, complex conversations around their fires.
But it was only gossip. They never talked about how to make a better tool. Just about each other.
Emma thought she had gotten used to the Runners, who were a strange mixture of human and animal. These Hams were still not quite human as she was, nevertheless they had their own gaps in their heads, barriers between the rooms. As she watched them jabbering of who was screwing whom while their hands worked at one tool or another, apparently independently, she found it hard to imagine how it must be to _be_ a Ham.
Sometimes she envied them, however.
To her, a beautiful sunset was a comforting reminder of home, a symbol of renewal, of hope for a better day tomorrow. The Hams would watch such displays as intently as she did. But to them, she believed, a sunset was just a sunset, like the sound of some instrument lacking any overtones, a simple pure tone—but a tone with a beauty and purity which they experienced directly and without complication, as if it was the first sunset they had ever seen.
Day succeeded empty day.
At first, on arriving here, she dreamed of physical luxuries: running hot water; clean, well-prepared food; a soft bed. But as time wore on, it was as if her soul had been eroded down. She had simpler needs now: to sleep in the open on a bower of leaves no longer troubled her; to have her skin coated in slippery grime was barely noticeable.
But she longed for security, to be able to settle down to sleep without wondering if she would be alive to see the morning, to live without the brutality and death that permeated the forest.
And she longed for the sight of another human face. It didn't have to be Malenfant. Anybody.
One day her wish was granted.
They had been men, pushing their way through the forest, pursuing some project of their own. They wore clothing of animal skin, but it was carefully stitched—a long way beyond the crude wraps the Hams tied around their bodies—and they spoke English, with a strong, twisted accent.
Emma was electrified. She gazed on their thin, somewhat pinched faces with longing, as intently as one Ham might gaze at another. Were they the source of the Hams' and Runners' language? Her impulse was to call out to them, approach them.
But she saw that the Hams cowered from these _Zealots_ , as they called them, a label Emma found less than encouraging. So she, too, slipped back into the forest with her Hams.
Sometimes she raged inwardly. Or she worked through imaginary conversations with Malenfant—who had, after all, been flying the plane when she got stuck here, and so was the only person she could think of to blame.
But when the Hams saw her stalking around the forest lashing at branches and lianas, or, worse, muttering to herself, they became disturbed.
So she learned not to look inward.
She watched the Hams as they shambled about their various tasks, their brute bodies wrapped up in tied-on animal skins like Christmas parcels. One day at a time: That was how the Hams lived, with no significant thought for tomorrow—for they appeared simply to assume that tomorrow would be much like today, and like yesterday, and the day before that.
She did not abandon her shining thread of hope that someday she would get out of here—without that she would have feared for her sanity—but she tried to emulate the Hams in their focus on the now. One day at a time. It was almost comforting. She tried to accept the notion that the best prospect for _the rest of her life_ might be to dwell on the fringes of a group like this: physically safe, but excluded, utterly ignored, the only representative of a different, and uninteresting species.
The future stretched out in front of her, a long dark hall empty of hope.
Until she sighted the lander.
## _R eid Malenfant_
Malenfant took a tentative step away from the lander. Encumbered by his escape suit, breathing canned air, he peered out of a sealed-up helmet. His heavy black boots crunched on dead leaves and sparse grass, all of it overlaid on a ruddy, dusty soil. But he could barely hear the noise of his footsteps, and could not smell the grass or the leaves.
All around this little clearing, dense forest sprouted: a darkness through which green shadows flitted. He tipped back on his heels and peered up into a tall, washed-out sky. The Earth sailed there, fat and blue, the outline of a continent dimly visible.
So here was Reid Malenfant walking on the surface of a new world: a boyhood dream, realized at last. But he sure hadn't expected it to be like this.
Maybe he was unimaginative—it was something Emma had accused him of many times—maybe he had focused too much on the battle to assemble the mission in the first place, and the thrilling details of the three-day flight across space to get here. Maybe, somehow, he had been expecting this wandering Red Moon would be content to serve as no more than a passive stage for his designs. Now, for the first time, on some deep, gut level, he realized that this was a _whole world_ he was dealing with here—complex in its own right, with its own character and issues and dangers.
And his scheme to rescue Emma seemed as absurd and quixotic as many of his opponents at home had argued.
But what else could he have done but come here and try? Nemoto was walking around the clearing experimentally, slim despite the bulky orange escape suit and the parachute pack still strapped to her back. Her gait was something like a Moonwalk, halfway between a walk and a run. "Fascinating," she said. "Walking is a pendulumlike motion, an interchange between the body's gravitational potential energy and the forward kinetic energy. The body, seeking to minimize mechanical energy spent, aims for an optimal form of gait—walking or running—at any given speed. But the lower the gravity, the lower the speed at which walking breaks into running. It's all a question of scaling laws. The Froude number—"
"Give me a break, Nemoto."
She stopped, coming to stand beside him. And, before he could stop her, she unlocked her helmet and removed it.
She grinned at him. She looked green about the gills, but then she always did. And she hadn't dropped dead yet.
Malenfant lifted his own helmet over his head. He kept his hand on the green-apple pull that would activate his suit's emergency oxygen supply. His Snoopy-hat comms unit felt heavy, incongruous in this back-to-nature environment.
He took a deep breath.
The air was thin. But he'd anticipated as much, and the altitude training he'd gone through reduced the ache in his chest to a distant nuisance. (But Emma, he remembered, had had no altitude training; this thin air must have hurt her.) The air was moist, faintly cold, what he would describe as bracing. He could smell green, growing things—the autumn smell of dead leaves, a denser green scent that came from the forest.
And he could smell ash.
Nemoto was inspecting a small portable analyzer. "No unanticipated toxins," she said. "Thin but breathable." She stripped off her Snoopy hat, and started to shuck off her orange pressure suit. "In fact," she said, "the air here is healthier than in most locations on Earth."
After their three days in space cooped up in a volume no larger than the interior of a family car, Malenfant was no longer shy of Nemoto. But he felt oddly self-conscious getting naked, out here in the open, where who-knew-what eyes might be watching. But he began to unzip his suit anyhow. "I can smell ash."
"That is probably the Bullseye," Nemoto said. The big volcano had been observed to erupt more or less continuously since the Red Moon's arrival in Earth orbit, perhaps induced by the tides exerted by the Earth on its new Moon. "You should welcome the ash, Malenfant. This is a small world, with no tectonic activity. Weathering here is a one-way process, and without a restorative mechanism all the air would eventually get locked up in the rocks, with no way to recycle it."
"Like Mars."
"And yet not like Mars. We don't yet understand the geological and biological cycles on the Red Moon. Perhaps we never will. But the injection of gases into the air by the Bullseye surely serves to keep the atmosphere replenished. What else do you notice?"
He raised his head, sniffed, listened.
"Bird song," Nemoto said. "An absence rather than a presence."
"No birds? It ought to be easier for them to fly here, in the lower gravity."
"But the air is less dense. Wings would have less lift than on Earth. The bird would require more muscle power, respiration... We may see gliders, and flightless birds. But we cannot expect the diversity we see on Earth."
A pity, Malenfant thought.
Malenfant donned T-shirt, shorts, a thin sweater, and a bright blue coverall, and then pulled his boots back on. He was glad of the warmth of the clothes; the air here was damp and cold, though the sun's heat was sharp. Nemoto dressed the same way. They tucked their heavy Gore-Tex escape suits back into the lander, against the time when they would be needed during the return to Earth—an eventuality Malenfant was finding increasingly hard to visualize.
Malenfant settled his comms pack on his shoulder. This was a specialized piece of gear manufactured for them by technicians at the Johnson Space Center. On top of a small but powerful transceiver package sat a tiny, jewel-like camera. Antennae were built into their coveralls, and the signals were relayed by small comsats orbiting low around the Red Moon. The deal was that save for emergency the controllers would keep their mouths shut during the surface stay (which they insisted on calling an extravehicular activity, with, to Malenfant's mind, an absurd emphasis on the vehicle they had arrived in, as opposed to the place they had come to). But in return the ground had control of the cameras.
Soon the little camera on Malenfant's shoulder was swivelling back and forth with a minute whirring noise. "Good grief," he said. "I feel like Long John Silver."
Nemoto laughed, as she usually did when she detected one of his jokes. He wasn't sure whether she understood the reference or not.
With her own camera working, she walked across the flattened clearing. She began to load small sample bags with fast, random selections of the vegetation and the underlying crimson soil; these were contingency samples, to be lodged in the loader against the event that they had to leave here in a hurry. She found a shallow puddle, covered with a greenish scum, and she pushed the probe of her sensor pack into it. "Water," she said. "Though I wouldn't recommend you drink it."
Malenfant, his own camera peering here and there, turned to face the way the lander had come down, from the west. The route was somewhat easy to spot. The lander, suspended beneath its blue parafoil, had come bellying down out of the sky, crashing through the trees with abandon, and had left a clear trail of its glide-down in snapped trunks, crushed branches, and ripped-up bits of parafoil. The trail terminated in this small clearing, where shattered tree trunks clustered close around the lander's incongruous black and white carcass.
Malenfant stalked around the lander, inspecting the damage. The whole underside was scored, crushed, and gouged. Heat-resistant tiles had been plucked away and scattered through the forest, and all the aerosurfaces were scarred and crumpled.
The only good thing you could say about that landing was that it wasn't his fault.
After scouting out the Red Moon from orbit for a few days, the crew and the mission planners on the ground had settled on the largest settlement they had spotted as a suitable target for the landing. (Not that they could tell who or what had built that settlement...) It was close to the delta where the great continental river completed its long journey to the ocean. The plan had been to come down on a reasonably flat, open plain a few miles to the west of the Beltway, the thick belt of forest at the continent's eastern coast, close enough to that big settlement for Malenfant and Nemoto to complete their journey on foot. Later, the follow-up rocket pack would rendezvous with the lander on the ground.
That was the plan. The Red Moon hadn't proven quite so cooperative.
As soon as the lander had ducked into the thicker layers of this little world's surprisingly deep atmosphere, strong winds had gripped it. The mission planners had expected the unexpected; there had been no time or resources to model the Red Moon's meteorology in detail. But none of that had helped ease Malenfant's mind as he lay helpless in his bucket seat, buffeted like a toy in the hands of a careless child, watching their landing ellipse whip away beneath the lander's prow.
The lander's autonomous systems had looked actively for an alternative site suitable for a safe and controlled landing. But another gust stranded the lander over the Beltway itself. When it realized that it was running out of altitude—and that soon it would reach a line of cliffs, beyond which there was only ocean—the lander had taken a metaphorical deep breath and dumped itself in the forest.
"The trees appear to be predominantly spruce," Nemoto said. "The growths are tall, somewhat spindly. If we had come down in a forest more typical of Earth—"
"I know," Malenfant growled. "We'd have crumpled like a cardboard box. You know, that path we cut through the trees reminds me of Star City. Moscow. Yuri Gagarin's jet trainer came down into forest, and cut its way through the trees just like that. Ever since, they have cropped the trees to preserve the path. Gagarin's last walk from the sky."
"But our landing was not so terminal," Nemoto said dryly. "Not yet anyhow."
The sturdy little craft could never make another descent—but that didn't matter, for it didn't need to. The plan for the return to Earth was that Malenfant and Nemoto would fit a rocket pack to the lander's rear end, raise the assembly upright, and take off vertically. And since the lander's shell, sheltering its crew, hadn't crumpled or broken or otherwise lost its integrity, the return flight might still be possible. All Malenfant had to do to get home, then, was to find the rocket pack when it came floating down from the sky after its separate journey from Earth—completing its lunar surface rendezvous, as the mission planners had called it—fit it and launch.
Oh, and find Emma.
Malenfant turned away from the lander and walked tentatively toward the edge of the forest. The gravity was indeed eerie, and it was hard not to break into a run.
The trunks of the trees at the edge of the clearing were laden with parasites. Here a single snakelike liana wound around a trunk; here a rough-barked tree was covered by mosses and lichens; a third tree was a riot of ferns, orchids, and other plants. From a bole in one aged trunk, an eye peered out at him. It was steady, unblinking, like an owl's. He backed away, cautiously.
He found a tall, palmlike tree, with dead brown fronds piled at its base. He crouched down and rummaged in the litter until he had reached crimson dirt. It was dry and sandy, evidently poor in nutrients. When he touched it to his lips, it tasted sharply of blood, or iron. He spat out the grains. The dust seemed to drift slowly to the ground.
He picked out yellow fruit from the debris of fronds. With a sideways glance at his shoulder camera, he said, "Here's some fruit that seems to have fallen from the tree up there. You can see it is shaped like a bent cylinder. It is yellow, and its skin is smooth and soft to the touch—"
A small brown ball unrolled from the middle of the nest of fronds. Malenfant yelped, stumbling back. The ball sprouted four stubby legs and shot out into the clearing. Malenfant had glimpsed beady black eyes, a spiky hide, looking for all the world like a hedgehog.
Nemoto walked up to him, her camera tracking the small creature.
"The double-domes said there would be no small animals here," he grumbled. "Thin air, fast metabolism—"
"A pinch of observation is worth a mountain of hypothesis, Malenfant. Perhaps our small friend evolved greater lung surfaces through a novel strategy like folding, or even a fractal design. Perhaps she conserves energy by spending periods dormant, like some reptiles. We are here to learn, after all." She grabbed the fruit. "Your description of this banana was acute." She peeled it briskly, exposing soft white flesh, and bit into it. "But it is a banana. A little stringy, the taste thin, but definitely _Musa Sapientum_. And, of course, the thinness of the taste might be an artifact of the body fluid redistribution we have both suffered as a result of our spaceflight."
Malenfant took another banana, peeled it and bit into it savagely. "You're a real smart ass, Nemoto, you know that?"
"Malenfant, all the species here should be familiar, more or less. We have the hominid samples who fell through the portals to the Earth. Although their species is uncertain, their DNA sequencing was close to yours and mine..."
A shadow moved through the forest behind Nemoto: black on green, utterly silent, fluid.
"Holy shit," Malenfant said.
The shadow moved forward, resolved, stepped into the light.
It was a woman. And yet it was not.
She must have been six feet tall, as tall as Malenfant. Her eyes locked on Malenfant's, she bent, picked up the banana Nemoto had dropped, and popped it into her mouth, skin and all.
She was naked, hairless save for a dark triangle at her crotch and a tangle of tight curls on her head. She held nothing in her hands, wore no belt, carried no bag. She had the body of a nineteen-year-old tennis player, Malenfant thought, or a heptathlete: good muscles, high breasts. Perhaps her chest was a little enlarged, the ribs prominent, affording room for the larger lungs the theorists had anticipated, like an inhabitant of a 1950s dream of Mars. There was a liquid grace in her movements, a profound thoughtfulness in her stillness.
But over this wonderful body, and a small, childlike face, was the skull of a chimp. That was Malenfant's first impression anyhow: There were ridges of bone over the eyes, a forehead that sloped sharply back. Not a chimp, no, but not human either.
Her eyes were blue and human.
_"Homo erectus,"_ Nemoto was muttering nervously. "Or _H. ergaster_. Or some other species we never discovered. Or something unrelated to any hominid that ever evolved on Earth... And even if descended from some archaic stock, this is not a true _Erectus_ , of course, but a descendant of that lineage shaped by hundreds of thousands of years of evolution—just as a chimp is not like our common ancestor, but a fully evolved species in its own right."
"You talk too much, Nemoto."
"Yes... We have seen the reconstructions, inspected the bodies ejected from the Wheel. But to confront her alive, _moving_ , is eerie."
The hominid girl studied Malenfant with the direct, uncomplicated gaze of a child, without calculation or fear.
He stepped forward. He could _smell_ the girl: unwashed, not like an animal, an intense locker-room smell. He felt a deep charge, pulling him to her. At first he thought it was an erotic attraction—and that was present, too; the combination of that clear animal gaze and the beautiful, fully human body was undeniably compelling, even if he sensed those stringy arms could break his back if she chose. But what he felt was deeper than that. It was a kind of recognition, he thought.
"I know you," he said.
The girl stared back at him.
Nemoto fidgeted behind him. "Malenfant, we were given protocols for encounters like this."
He murmured, "I should offer her a candy and show her a picture card?" He returned his attention to the girl. _"I know you,"_ he repeated.
I know who you are. We evolved together. Once my grandmother and yours ran around the echoing plains of Africa, side by side.
This is a first contact, it struck him suddenly: a first contact between humanity and an alien intelligent species—for the intelligence in those eyes could not be denied, despite the absence of tools and clothing.
... Or rather, this is a contact renewed. How strange to think that buried deep in man's past was a _last_ contact, a last time we met one of these cousins of ours: perhaps a final encounter between one of my own ancestors and a girl like this in the plains of Asia, or a dying Neandertal on the fringe of the Atlantic, when we left them no place else to go.
The girl held her hands out, palms up. "Banana," she said, thickly, clearly.
Malenfant's jaw dropped. "Holy shit."
"English," Nemoto breathed. "She speaks English."
"En'lish," the girl said.
Now Malenfant's heart hammered. "That must mean Emma is here. She is near, and she survived."
Nemoto said cautiously, "We know very little, Malenfant; there is a whole world around us, a world of secrets."
There was a crackle behind Malenfant: a twig breaking, a footfall. He whirled.
There were more of the ape-people, eight or ten of them, male and female, all adults. They were as naked as the girl, though not all as handsome; some of them sported scars, gashes, and even burns, and some had hair streaked with gray. They were standing in a line, neatly fencing off Malenfant and Nemoto from the lander, and they were all gazing hard at the two of them.
"These do not seem quite so friendly," Nemoto murmured.
"Oh, really? You think now's a good time to start the sign language classes?"
"Malenfant, where are the guns?"
"... In the lander." _Shit_.
The silence stretched. The ape-people stood like statues.
"I am loath to abandon the lander," Nemoto hissed. "We have not even packed the contingency samples."
Malenfant suppressed a foolish laugh. "There go our science bonuses."
One of the ape-people stepped forward. Straggles of beard clung to his chin, though the longer strands seemed to have been cut, crudely. He opened his mouth and hissed. Malenfant thought his teeth were stained red.
Nemoto said, "Malenfant, I think—"
"Yeah. I think he's about to take a sample of _us_."
The big man raised his arm. Too late, Malenfant saw he was holding a stone in his fist. Malenfant ducked sideways. The stone missed his head, but it sliced through the layers of cloth over his shoulder, and nicked the flesh.
"Plan B," he gasped.
The two of them broke and ran for the forest. They pushed past the girl, who made a half-hearted effort to grab them. For a heartbeat Malenfant nursed a hope that he had made some connection, that she had on some level decided to let them go.
But then he was plunging into the green mouth of the forest after Nemoto, and there was no time for reflection.
The forest, away from the sunlight, was suffused by a clinging cloudy moistness that seemed to linger around every bush, and made every tree trunk slippery under Malenfant's palms. Soon they were both shivering.
And it was almost impossible to walk. Malenfant had done a little jungle survival training during his induction into the Shuttle program. But this forest was almost impassable, so deeply layered were the tangled roots, branches, leaves, and moss over the uneven ground. Malenfant was acutely aware that this was not a place for humans.
Still they blundered on, slipping, crashing, blundering, falling, making a noise that must have echoed off the flanks of the Bullseye itself.
He imagined the frantic activity in the back rooms of Mission Control in Houston, the buzzing calls to paleontologists and anthropologists and evolutionary psychologists. For once in his life he would have been glad to hear the tinny voices from the ground. But, though there was a hiss of static from the tiny speaker built into his shoulder pack, he could make out no voices.
Once he thought he confronted one of the ape-people. He caught a glimpse of someone—some _thing_ —in the dense green gloom ahead of him, upright like an ape-person, but smaller, chimp-sized, maybe hairy. It jabbered at him, reached up its long arms, and slipped out of sight into the forest canopy above.
After that, Malenfant found himself looking for possible threats upward as well as side to side.
At length, breathing hard in the thin air, shivering, they came to a halt, crouching close to the ground by a fat, fungus-laden tree trunk. Malenfant's face was slick with sweat and forest dew.
Nemoto's eyes were wide in the gloom, glancing this way and that, like a cornered animal.
"We haven't been too smart, have we?" he whispered.
"We were not expecting to come under immediate attack by a troupe of _Homo erectus_."
"Yeah, but it's taken us a bare half hour after opening the hatch to lose the lander, our supplies, and our weapons. I'm not even sure which way we're running."
"We will recover the lander."
"How do you know?"
"Because we must," Nemoto said simply.
A shadow slid across his field of view. It was subtle, difficult to distinguish from the swaying motion of a branch, the shifting coins of dappled sunlight that lay over the forest floor.
The camera on his shoulder swivelled to look into his face, and he forced a grin. "If you guys have any suggestions, now would be a good time..."
Eight, nine, ten shadows moved, all around them, shadows that coalesced into ape-people.
"The _Erectus_. They have been hunting us," Nemoto said. "Their intelligence is advanced enough for that, at least." She seemed calm, beyond fear.
The ape-people advanced. Some of them were grinning, and one of the men, perhaps excited by the prospect of a kill, sported an impressive erection.
Malenfant stood up slowly. The camera on his shoulder swivelled back and forth, whirring, somehow the most distracting object in his universe. He said, "I think—"
A vast, heavy creature came running out of the depths of the wood. It hurled itself at the largest ape-man. They rolled on the floor, wrestling.
The ape-men gathered around the combatants, hooting and hollering, their teeth showing between drawn-back lips—perhaps a rictus of fear—and they slapped ineffectually at the rolling figures.
Nemoto clutched Malenfant's arm, and they backed away.
Nemoto said, "I thought it was a bear."
"No," Malenfant said grimly.
No, not a bear: a _man_ —yet another sort of man, shorter than his naked opponent, but much more heavily muscled, and dressed in animal skins that were tied to his body with bits of red-black rope. Though the ape-man on the ground was a formidable opponent—surely more than a match for any human in hand-to-hand combat—the bear-man was stronger yet, and soon he had the ape-man pinned to the ground by sitting on his chest.
The bear-man snarled, "Enough?"
Once again the use of English, distorted but clear enough, startled Malenfant. Was it really credible that Emma could have taught the use of English to not one but _two_ species of other-men? But if not, what was going on?
The man on the ground snapped at the hand that slapped him, but it was clear that the fight had gone out of him. The bear-man sat back and let him up.
The ape-man rejoined his companions and, his defiance momentarily sparking, he growled at the bear-man. "Ham! Eat Ham good eat!"
The bear-man—the "Ham"—opened his huge mouth wide, exposing a row of flat brown teeth. He ran at the ape-people, making them scatter, and with a broad, bare foot he aimed a heavy kick at the naked rump of the last man.
Then the bear-man walked up to Malenfant and Nemoto. He was a good head shorter than Malenfant—no more than five-five, five-six—but he was broad as a barn door. Under the skins which wrapped him loosely, Malenfant could see muscles moving. His walk was somewhat ungainly, as if his legs were bowed, or his balance not quite perfect. His skull was long and flat, with a bulge at the back that showed beneath a sprawl of thick black hair. He had a vast cavernous nose, and brown eyes glinted beneath bony brows like two caves. Sweat had pooled in a hollow between the brow ridges and his low forehead.
"Neandertal," Nemoto muttered. "Or possibly _Homo heidelbergensis_. Most probably _Neandertalensis_ , of the so-called classic variant. Or rather a lineage evolved from Neandertal stock, in this unique place."
Malenfant could smell beer on the Neandertal's breath. "Holy shit," he said. _Beer?_
The Neandertal—or bear-man, or Ham—grinned at them. "Stupi' Runners," he said. "Scare easy." He stuck his tongue out and lunged forward. _"Boo!"_
Both Malenfant and Nemoto took a step back. The bear-man's voice was gravelly and thick, and his vowel sounds slurred one into the other. "But," Malenfant said, "he speaks better than I do after a couple of hours at the Outpost."
Now there was a crashing from the forest that resolved itself into clumsy, unconcealed footsteps. A new voice called, "What the devil is going on, Thomas?"
Malenfant frowned, trying to place the accent. English, of course—a British accent, maybe—but twisted in a way he didn't recognize.
The bear-man called, "Here, Baas. Runners. Chase off."
A man walked out of the shadows toward them—a human this time, a stocky man, white, age maybe fifty, with a grubby walrus moustache. He was dressed in a buckskin suit, and he had a kind of crossbow over his shoulder. What looked like a long-legged rabbit hung from his belt.
When he saw Malenfant and Nemoto, he stopped dead, mouth a perfect circle.
Malenfant spread his hands wide. "We're from America. NASA."
The man frowned. "From where?... _Have you come to rescue us?_ " Malenfant saw hope spark in his eyes, sudden, intense. He walked toward Malenfant, hand extended. "McCann. Hugh McCann. Oh, it has been so long in this place! Are you here to take us home?"
Malenfant felt a light touch on his shoulder, a soft crunch. When he looked, the camera he had worn there had gone, disappeared into the paw of the Neandertal.
## _E mma Stoney_
The spaceship had been quite unmistakeable as it drifted out of the sky, heading east, the Shuttle-orbiter black and white under a glowing blue-and-white canopy. Her eyes weren't what they used to be, but she'd swear she made out the round blue NASA meatball logo on its flank.
_Malenfant_. Who else?
She knew immediately she had to follow it. She couldn't stay with the Ham troupe any more. She couldn't rely on whoever had drifted down from the sky to come find her. Her destiny had been in her own hands since the moment she had fallen out of the sky of Earth into this strange place, and it was no different now. She had to get herself to that lander.
She gathered up her gear. She equipped herself with stone tools and spears from the Ham encampment—without guilt, for the Hams seemed to make most of their tools as they needed them and then abandoned them. With her hat of woven grasses and her poncho of animal skin, all draped over the remnants of her air force coverall, she must look like the wild woman of the woods, she thought.
She attempted to say good-bye to the Ham who had first found her, and to some of the others she had gotten to know. But she was met with only blankness or bafflement.
After all, since nobody ever went anywhere, nobody said _good-bye_ in a Ham community—except maybe at death.
She slipped into the forest.
## _S hadow_
Thanks to extended pulses of volcanism, this small world was steadily warming, and temperate forests were shrinking back in favor of more open grasslands. The range of Shadow's family group was only a little smaller than the remnant of forest to which they clung; with invisible, unconscious skill, Shadow's elders had always guided her away from the exposed fringes of the forest.
But now her people had turned on Shadow. And to escape them she would have to leave her forest home.
Emerging from the trees, she found herself at the foot of a shallow forest-covered slope, a foothill of taller mountains which reared up behind her. She faced a wide plain, a range of open, parklike savannah, grasslands punctuated by stands of trees. To the right of the plain a broad river ran, sluggish and brown. Away to the left a range of more rocky hills rose, their lower slopes coated with a thick carpet of forest. The hills marched away in a subtly curving ring; they were the rim mountains of a small crater.
She longed to slink back into the dark cool womb of the woods behind her.
She looked again at that smudge of green covering the crater wall. _Forest_ : the only other patch of it in her vision. She thought of food and water, nests high in the trees.
She took a step out into the open.
The sun's heat was like a warm hand on her scalp. She saw her shadow at her feet, shrunken by the height of the sun. The forest behind her tugged at her heart like the call of her mother. But she did not turn back.
She ran forward, alone, her footsteps singing in the grass.
She was soon hot, panting, dreadfully thirsty. Her thick fur trapped the heat of the sun. Her feet ached as they pounded the ground. Her arms dangled uselessly at her side; she longed to grasp, to climb. But there was nothing here to climb. She ran on, clumsy, determined, over ground that shone red through sparse yellow grass.
But as she ran she turned this way and that, fearing predators. A cat or a hyena would have little difficulty outrunning her, and still less in bringing her down. And she watched those remote woods. To her dismay they seemed to come no closer, no matter how hard she ran.
She came to a clear, shallow stream.
Unbearably thirsty, panting, she waded straight into the water. The stream was deliciously cool. The bed was of cobbles, laced with green growing things that streamed in the water. At its deepest the stream came up a little way beyond her knees.
She slid forward until she was on all fours. She rolled on her back, letting the water soak into her fur. She raised handfuls of water to her mouth. The water, leaking from her fingers, had a greenish tinge, and it was a little sour, but it was cold. She drank deeply, letting the water wash away the dust in her mouth and nose. She saw a thin trail of dust and blood seeping away from her.
A thin mucus clung to her wet hand. She saw that it contained tiny, almost transparent shrimps. She scraped the shrimps off her palm and popped them in her mouth. Their taste was sharp and creamy and delicious.
She stood up. With her gravid belly stroking the surface of the stream, she put her hands in the water, open like a scoop. She watched carefully as the water trickled through her fingers, and when the little crustaceans struck her palm she closed her hands around them.
Her thoughts dissolved, becoming pink and blue, like the sky, like the shrimp.
When she had had her fill of shrimp she clambered out of the stream, her fur dripping. She reclined on the bank. She folded her legs and inspected her feet. They were bruised and cut, and a big blister had swollen up on one toe. She washed her feet clean of the last of the grit between her toes, and then inspected the blister curiously; when she poked it with a fingernail the clear liquid in it moved around, accompanied by a sharp pain.
She heard a distant growl.
Startled, she tucked her feet underneath her, resting her knuckles on the ground. She peered around at the open plain.
The shadows, of rocks and isolated trees, had grown long. She had forgotten where she was: while she had played in the water, the day had worn away. She mewled and wrapped her long arms around her torso. She did not want to return to the running. But every instinct in her screamed that she must get off the ground before night fell.
She climbed out of the stream and began running toward the crater rim hills.
The light faded, terribly rapidly. Her shadow stretched out before her, and then dissolved into grayness.
Her face began to itch, as if some insect was working its way into her skin. She scratched her cheeks and brow. She looked for someone to groom her. But there was nobody here, and the itch wouldn't go away.
Still she ran, thirsty, dusty, exhausted.
And still those growls came, echoing across the savannah: the voices of predators calling to each other, marking out the territory they claimed.
It grew darker. The earth climbed in the sky. The land became drenched in a silvery blueness.
There was a growl, right in front of her. She glimpsed yellow eyes, like two miniature suns.
She screamed. She picked up handfuls of dirt and threw them at the yellow eyes. There was a howl.
She turned and ran, not caring where she went. But her gait was waddling and stiff, her feet broken and sore.
She could hear steady, purposeful footsteps behind her.
Memories clattered through her mind: of a bite that had crushed the skull of a child in a moment, of the remains of a predator's feast, bloody limbs and carcass, of the screams of a victim taken live to a nest, where cubs had fed long into the night. She screamed and ran and ran.
There was light ahead of her.
She ran toward the light, panting and hooting. She thought of daybreak in a safe treetop, her nest warm under her, her mother's massive body close by.
The light was yellow, and it flickered, and shadows moved before it. Afire.
She heard those scampering footsteps. There was a hot, panting breath on her neck.
A stone zinged through the air, past her head. It clattered against a rock, harmlessly. Now another stone flew. It caught her in the chest, knocking her flat on her back.
Behind her, the chasing cat yelped and yowled. When she sat up and turned, she saw its lithe silhouette sliding across the blue, glittering grass.
"Elf Elf away."
She yelled and scrabbled in the dirt.
She found herself looking up at a tall figure—a woman, perhaps twice as tall as she was, taller even than Big Boss had been, her torso long and ugly. She had small flat breasts. She was hairless, save for knots of hair on her head and between her legs. She had a small face and wide nose, and she carried a stick that she was pointing at Shadow.
She was a Runner.
Cautiously Shadow got to her feet. She jabbered at the woman, a series of intense pants, hoots, screeches, and cries. She expected the woman to respond. They would chatter together, sounds without words, their cries slowly matching in pitch and intensity as they greeted each other.
But the woman jabbed with the stick, coming close to piercing Shadow's skin. "Elf Elf away!"
Shadow feared the stick. But before her was the yellow fire. She could hear the fire pop and crackle, and she could smell food, the sharpness of leaves and burned meat. Many people were there—all tall and skinny and hairless like this stretched-out woman, but people nevertheless. Behind her there was only the darkness of the savannah, like a vast black mouth waiting to swallow her.
She took a pace toward the woman, hands outstretched. She tried to groom her, reaching for the hair on the woman's head.
The sharp stick jabbed in her shoulder. Again Shadow was thrown back into the dirt. She poked a finger in her latest wound; blood seeped slowly from it, soaking her fur. She whimpered in misery. The sharp noses of the cats would soon detect the blood.
Still the woman stood over her, arms akimbo, stick poised for another thrust.
Shadow tried to stand. A searing pain clamped around her stomach, making her stumble to the crimson dirt. She cried out, and beat her fists on her betraying belly. She looked up at the threatening, curious woman. She whimpered. She held out her feet, and flexed her toes. Helpless, she was reduced to the gestures of an infant.
The woman lowered the stick. She crouched down. Clear eyes looked into Shadow's. She reached out with her hand and stroked Shadow's fur. She touched the wounded shoulder, and the hand came away bloody; the woman wiped it in the dirt at her feet. Then she ran a curious hand over the bump in Shadow's belly.
Again Shadow reached for the woman's scalp and crotch to groom her. But the woman flinched back.
Shadow dropped her head, her energy exhausted. She lay in the dirt, on her back, her arms and legs splayed; Shadow was beaten.
The woman stared at her a while longer. Then she walked away, toward the fire.
Shadow curled over on her side.
Something hit her chest. She flinched back.
It was a piece of meat. It lay on the ground before her. She saw it had been cut from an animal—perhaps an antelope—by a sharp-edged stone. And people had bitten into it already; she saw where it had been ripped and torn by teeth. But still it was meat, a piece as big as her hand. She crammed it into her mouth, tearing at it with hands and teeth.
When she was done she lay down once more. The ground was hard and dusty, and she longed for the springy platform of a nest. But her arm made a pillow for her head.
Suspended between black night and the flickering firelight, she sank into redness.
## _R eid Malenfant_
On the walk through the forest with McCann, this oddball English guy, Malenfant got fixated on McCann's crossbow.
The crossbow, made purely of wood, was heavy. There was a long underslung trigger that neatly lifted a bowstring out of the notch. The trigger mechanism worked smoothly. The string itself was made of twisted vine, very fine, very strong. But there was no groove to direct the bolt. And the bolts themselves seemed crude to Malenfant: about as long as a pencil, but a lot thinner, and with a flight made from a single leaf, tucked into a slice in the wooden bolt, just one plane. It was hard to see how you could make an accurate shot with such a thing. But as they walked McCann did just that, over and over, apparently pleased to have an audience.
Nemoto's silent contempt for all this was obvious. Malenfant didn't care. His mind was tired of all the strangeness; to play with a gadget for a while was therapy.
It was getting dark by the time the Englishman led them to a fortress in the jungle. The two of them, bruised and bewildered, were led into the compound, taking in little. Surrounded by a tough-looking stockade, it turned out to be a place of straight lines and right angles, the huts lined up like ranks of soldiers, the line of the stockade walls as perfect as a geometrical demonstration.
"Shit," murmured Malenfant. "I can feel my anus clench just standing here."
Nemoto said, "They are very frightened people, Malenfant. That much is clear."
Malenfant glimpsed people moving to and fro in the gathering dark. No, not quite people. He shuddered.
McCann showed them hospitality, including food and generous draughts of some home-brew beer, thick and strong.
The hours passed in a blur.
He found himself in a sod hut, with Nemoto. His bed was a boxy frame containing a mattress of some vegetable fabric. It didn't look too clean.
They were both fried. They hadn't slept in around thirty-six hours. They had been through the landing, the assault by the _Erectus_ types, the march through the jungle. And, frankly, the beer hadn't helped. At least here, against all expectations, they had found what seemed like a haven. But still Malenfant inspected his lumpy bed suspiciously.
"I know what to do," Malenfant said. "Always turn your mattress. Then the body lice have to work their way back up to get to you." He lifted the corner of his mattress out of its wooden box.
"I would not do that," Nemoto said; but it was too late.
There was the sound of fingernails on wood, a smell like a poultry shed. Cockroaches poured out of the box, a steady stream of them, each the size of a mouse.
"Shit," Malenfant, said. "There are thousands in there." He stamped on one, briskly killing it.
"It's best to leave them," Nemoto said evenly. "They have glands on their backs. They only stink when disturbed."
Malenfant cautiously picked up a cockroach. Its antenna and palps hung limp, and it had a pale pink band over its head and thorax.
"Very ancient creatures, Malenfant," Nemoto said. "You find traces of them in carboniferous strata, three hundred million years deep."
"Doesn't mean I want to share my bed with one," Malenfant said. Carefully, as if handling a piece of jewelry, he set the cockroach on the floor. It scuttled out of sight under his bed frame.
Malenfant finally lowered his head to the pillow.
"Just think," Nemoto said from the darkness. "When you sleep with that pillow, you sleep with all the people who used it before."
Malenfant thought about that for a while. Then he dumped the pillow on the floor, rolled up his coverall, and stuck that under his head.
Later that night Malenfant was disturbed by a howl, like a lost child. Peering out, he spotted a small creature high in a palm tree, about the size of a squirrel.
"A hyrax," Nemoto murmured. "Close to the common ancestor of elephants, hippos, rhinos, tapirs, and horses."
"Another ancient critter, crying in the night. I feel like I've been lost in this jungle since God was a boy."
"I suspect we are very far from God. Try to get some sleep, Malenfant."
## _S hadow_
Pain stabbed savagely in her lower belly. It awoke her from a crimson dream of teeth and claws. She sat up screaming.
There was no cat. In the gray-pink light of dawn, she was sitting in the dirt. She was immediately startled to find herself on the ground, and not high in a tree.
Before her she could see skinny people walking around, pissing, children tumbling sleepily. Some of them turned to stare at her with their oddly flat faces.
But now more pain came, great waves of it that tore at her as if her whole body was clenched in some huge mouth.
Something gushed from between her legs. She looked down, parting her fur. She saw bloody water, seeping into the ground. She screamed again.
She scrabbled at the ground, seeking to find a tree, her mother, seeking to get away from this dreadful, wrenching agony. But the pain came with her. Her belly flexed and convulsed, like huge stones moving around inside her, and she fell back once more.
Now there was a face over hers: smooth and flat, shadowed against the pinkish sky. Strong hands pressed at her shoulders, pushing her back against the dirt. She lashed out, trying to scratch this creature who was attacking her. But she was feeble, and her blows were easily brushed aside. She could feel more hands on her ankles, prying her legs apart, and she thought of Claw, and screamed again. But the pressure, though gentle, was insistent, and kick as she might she could not free herself of these grasping, controlling hands.
Now the pain pulsed again, a red surge that overwhelmed her.
No more than half-conscious, she barely glimpsed what followed: the strong, skillful hands of the Runner women as they levered the baby from its birth canal, fingers clearing a plug of mucus from its mouth, the brisk slicing of the umbilical with a stone axe. All that Shadow perceived was the pain, the way it washed over her over and over, receding at last as the baby was taken from her—to be followed by a final agonizing pulse as the afterbirth emerged.
When it was done, Shadow struggled to prop herself up on her elbows. Her hair was matted with dust and blood. The ground between her legs was a mess of blood and mucus, drying in the gathering sunlight.
There were women around her, tall like tree trunks, their shadows long.
One of them—older, with silvery hair—was holding the afterbirth, which steamed gently. The old woman nibbled at it cautiously, and then, with a glance at Shadow, she ran away toward the smoking fire with her stolen treat.
The other women stared at Shadow's face. Their small, protruding noses wrinkled. Now that the greater pain was ebbing, Shadow became aware of an itching that had spread across her cheeks and forehead and nose; she scratched it absently.
A woman stood before her. She held the baby, her long fingers clamped around its waist. It had large pink ears, small, pursed lips, and wrinkled, bluish-black skin. Its head was swollen, like a pepper. It—he—opened his mouth and wailed.
He smelled strange.
The skinny woman thrust the baby at Shadow, letting him drop on her belly. Feebly the baby grasped at her fur, mouth opening and closing with a pop.
With hesitant hands, Shadow picked him up around the waist. He wriggled feebly. She turned him around so his face was toward her, and pressed his face against her chest. Soon his mouth had found her nipple, and she felt a warm white gush course through her body.
But the baby smelled wrong. She could hardly bear even to hold him.
The Running-folk let her stay the rest of the day, and through the night. But they gave her no more food. And when dawn came, they drove her away with stones and yells.
Her baby clamped to her chest, its big awkward head dangling, Shadow walked unsteadily across the savannah, toward the wooded crater wall.
## _R eid Malenfant_
Malenfant woke to the scent of bacon.
He surfaced slowly. The smell took him back to Emma and the home they had made in Clear Lake, near Houston, and even deeper back than that, to his parents, the sunlit mornings of his childhood.
But he wasn't at home, in Clear Lake or anywhere else.
When he opened his eyes he found walls of smoothed-over turf all around him, a roof of crudely cut planking, the whole covered in a patina of smoke and age. Light streamed in through unglazed windows, just holes cut in the sod covered by animal skin scraped thin. Under the smell of the bacon he could detect the cool green earthy scent of forest.
The day felt hot already. Thin air, Malenfant: hot days, cold nights, like living at altitude.
Nemoto's pallet was empty.
When he tried to sit up, pushing back the blankets of crudely woven fiber, his shoulder twinged sharply: injured, he was reminded, where a _Homo erectus_ had thrown a stone at him, prior to trying to eat him.
He swung his legs out of bed. He was in his underwear, including his socks, and his boots were set neatly behind the hut's small door. He could feel the ache of a faint hangover, and his mouth felt leathery. He remembered the beer he had consumed the night before, a rough, chewy ferment of some local vegetation, sluiced down from wooden cups.
The door opened, creaking on rope hinges. A woman walked in.
Malenfant snatched back the blankets, covering himself.
She was short, squat, dressed in a blouse and skirt dyed a bright, almost comical yellow. Her face protruded beneath a heavy brow, but her hair was tied back neatly and adorned with flowers.
She looked like a pro wrestler in drag. She curtsied neatly.
She was carrying Malenfant's coverall, which had been cleaned and patched at the shoulder. She put the coverall on his bed, and crossed to a small dresser, evidently homemade. There was a wooden bowl of dried flowers on top of the dresser. She scooped out the flowers and replaced them with a handful of pressed yellow blooms—marigolds, perhaps—that she drew from a pouch in her skirt. Her feet were bare, he saw, great spade-shaped toes protruding from under the skirt.
She curtsied again. "Breakfas', baas," she said, her voice a gruff rasp. She had not once met his eyes. She turned to go out the door.
"Wait," he said.
She stopped. He thought he saw apprehension in her stance, though she must have been twice his weight, and certainly had nothing to fear from him.
"What's your name?"
"Julia." It was difficult for her to make the "J" sound; it came out as a harsh squirt. _Choo-li-a_.
"Thanks for looking after me."
She curtsied once again and walked stolidly out of the room, her big feet padding on the wooden floor.
The settlement consisted of a dozen huts, of cut sod or stacked logs, with roofs of thick green blankets of turf. The huts were a uniform size and laid out like a miniature suburban street. The central roadway was crimson dust beaten flat by the passage of many feet, and lined with heavy rocks. Around each of the huts a small area was cordoned off by more lines of rocks. Some of the rocks were painted white. In the "gardens" plants grew, vegetables and flowers, in orderly rows.
Crude-looking carts were parked in the shadow of one wall, and other bits of equipment—what looked like spades, hoes, crossbows—were stacked in neat piles under bits of treated skin. There was even a neat, orderly latrine system: trenches topped by little cubicles and wooden seats.
The effect was oddly formal, like a barracks, a small piece of a peculiarly ordered civilization carved out of the jungle, which proliferated beyond the tall stockade that surrounded the huts. Last night McCann had been apologetic about the settlement's crudity, but with its vegetable-fiber clothing and carts and tools of wood and stone, it struck Malenfant as a remarkable effort by a group of stranded survivors to carve out of this unpromising jungle something of the civilization they had left behind.
But the huts' sod walls were eroded and heavily patched by mud. And several of the huts appeared abandoned, their walls in disrepair, their tiny gardens desiccated back to crimson dust.
There was nobody about—no humans, anyhow.
A man dressed in skins crossed the compound's little street, barefoot, passing from one hut to another. He was broad, stocky, like Julia. A Neandertal, perhaps.
In one corner of the compound two men worked at a pile of rocks, steadily smashing them one against the other, as if trying to reduce them to gravel. The men were naked, powerful. Malenfant could immediately see they were the _Homo erectus_ types. They were restrained by heavy ropes on their ankles, and they didn't seem aware of his presence. The display of their strength, unaccompanied by the control of minds, disturbed him.
But he could still smell bacon. Comparative anthropology could wait.
He followed his nose to a hut at the center of the compound. Within, a table had been set with wooden plates and cups and cutlery, and in a small kitchen area another Neandertal-type woman, older than Julia, was frying bacon on slabs of rock heated by a fire. In the circumstances, it seemed incredibly domesticated.
Nemoto was sitting at the table, chewing her way steadily through a slab of meat. She looked at him as he entered, and raised an eyebrow.
"... Malenfant. Good morning."
Malenfant turned at the voice, and his hand was grasped firmly.
Hugh McCann was wearing a suit, Malenfant was startled to see, with a collared shirt and even a tie. But the suit and shirt were threadbare, and Malenfant saw how McCann's belt dug into his belly.
McCann saw him looking. He said ruefully, "I never was much of a hand with the needle. And our bar-bar friends make fine cooks, but they don't have much instinct for tailoring, I fear."
Malenfant was fuddled by the scent of the food. "Bar-bars?"
"For _barbarians_ ," Nemoto said, her mouth full. "The Neandertals."
"They call themselves Hams," McCann said. "A Biblical reference, of course. But bar-bars they were to me as a boy, and bar-bars they will always remain, I fear." His accent was clearly British, but of a peculiarly strangulated type Malenfant hadn't encountered outside of World War II movies. And he gave Malenfant's name a strong French pronunciation. He took Malenfant's elbow and guided him toward the kitchen area. "What can we offer you? The bacon comes from the local breed of hog, and is fairly authentic, but the bird who laid those eggs was no barnyard chicken: rather some dreadful flightless thing like a bush turkey. Still, the eggs are pretty tasty." He flashed a smile at the Ham cook, showing decayed teeth.
First things first. Malenfant grabbed a plate and began to ladle it full of food. The wooden utensils were crude, but easy to use. He took his plate to the table, and sat with Nemoto, who was still eating silently. Malenfant sliced into his bacon. The well-cooked meat fell apart easily.
After a moment McCann joined them. "I expect last night is all a bit of a blur. You did rather go on a bust, Malenfant."
"Body fluid redistribution," Nemoto said dryly. "Low oxygen content. You just could not take it, Malenfant."
"I'll know better next time."
"Runners," Nemoto said.
"What?"
"The _Erectus/Ergaster_ breed. Mr. McCann calls them Runners—Running men, Running-folk."
"Quite a danger in the wild," McCann said around a mouthful of bacon. "That scrog of wood where we found you was hotching with them. But once broken they are harmless enough. And useful. A body strong enough for labor, hands deft enough to handle tools, and yet without the will or wit to oppose a man's commands—if backed up by a light touch of the _sjambok_ from time to time..."
Nemoto leaned forward. "Mr. McCann. You said that when you were a boy you called the Neandertals—that is, the Hams—bar-bars. So were there Hams in, umm, in the world you came from?"
McCann dug a fork into his scrambled egg, considering the question. He seemed more comfortable talking to Malenfant than to Nemoto, and he directed his remarks to him. "Look here," he said. "I don't know who you are or where you're from, not yet. But I'm going to be honest with you from the start. I don't mind telling you that yours are the first white faces we've seen since we've come here. Aside from those dreadful Zealot types, of course, but they're no help to us, and beyond the pale anyhow... Yes," he said. "Yes, there are Hams where I come from. There. That's a straight answer to a straight question, and I trust you'll treat me with the same courtesy."
"Where?" Nemoto pressed. "Where are your Hams? In Europe, Asia—"
"Yes. Well, they are now. But not by origin, of course. The Hams came originally from the New World."
Nemoto asked, "America?"
McCann frowned. "I don't know that name."
Malenfant eyed Nemoto. "What are you thinking?"
"An alternate Earth," Nemoto said simply.
Yes, he thought. McCann had come from an Earth, a different Earth, a world where Neandertals had survived to the present—a world where pre-European America had been in the occupation, not of a branch of _Homo sapiens_ , but of another species of humankind altogether, a different flesh... What an adventure that must have been, Malenfant thought, for a different Columbus.
Nemoto said softly, "I think we may be dealing with a whole sheaf of worlds here, Malenfant. And all linked by this peculiar wandering Red Moon."
McCann was listening intently. Malenfant saw how deeply cut were the lines in his face; he might have been fifty, but he looked older, careworn, intense, lit by a kind of desperation. He said, "You believe we come from different worlds."
"Different versions of Earth," Nemoto said.
He nodded. "And in this Earth of yours, there are no Hams?"
"No," Nemoto said steadily.
"Well, we have no Runners. The Runners may be native to this place, perhaps." He eyed them sharply. "And what about the others, the Elf-folk and the Nutcrackers..."
Malenfant said, "If you mean other breeds of hominids, or prehominids—no. Nothing between us and the chimps. The chimpanzees."
McCann's eyes opened wider. "How remarkable. How—lonely."
The Neandertal woman, with a bulky grace, came to the table and began to gather up their dishes.
They walked around the compound.
There was very little metal here: a few knives, bowls, shears. These tools, it seemed, had been cut from the wreck of the ship that had brought McCann and his colleagues here: like Nemoto and Malenfant, the English had got here under their own power. So the tools were irreplaceable and priceless—and they were a target for steady theft, by Hams within and without the compound. McCann said the Hams did not use the tools; they seemed to destroy them or bury them, removing this trace of novelty from the world.
There were many Hams, working as servants. And there appeared to be several of the so-called Runners, kept under control at all times, apparently domiciled outside the main stockade. He tried to put aside judgment. He was not the one who had battled to survive here for so long; and it was evident that this McCann and his companions came from a very different world from his.
And besides, McCann appeared to believe that he treated "his" hominids well.
They met one other of the English, a bloated-looking red-faced man with a Santa Claus beard and an immense pot belly that protruded from the grimy, much-patched remnant of a shirt. He was riding in a cart drawn by two of the Runners, harnessed with strips of leather like pack animals. Santa Claus glared at Malenfant and Nemoto as he passed them, and then went riding out of the stockade through gates smoothly pulled back by Hams.
"There goes Crawford in his Cape cart," whispered McCann conspiratorially. "Something of an oddball, between you and me. Well, we all are, I suppose, after all this time. I fear he's too much set in his ways to deal with you. Of course if he suspected you were French he'd shoot you where you stand!... Martyr to his lumbago, poor chap. And I fear he may have a touch of the black-water."
McCann talked quickly and fluently, as if he had been too long alone.
There had been twelve of them, it seemed—all men, all British, from an Empire that had thrived longer than in Malenfant's world. Their rocket ship had been driven by something called a Darwin engine.
McCann struggled to describe the history of his world, his nation. After bombarding them with a lot of detail, names of wars and kings and generals and politicians that meant nothing to Malenfant, he settled on a blunt summary.
"We are engaged in a sort of global war," McCann said. "That's been the shape of it for a couple of centuries now. Our forefathers struck out for new lands, in Asia and Africa and Australia—even the New World—as much out of rivalry as for expectations of gain."
But the ultimate "new land" had always hovered in the sky. Before the Red Moon had appeared in McCann's sky, a Moon had sailed there—not tiny, desiccated Luna, but a much fatter world, a world of water-carved canyons and aquifers and dust storms, a world that sounded oddly like Mars. Drawn by that Moon, the great nations of this other Earth had launched themselves into a space race as soon as the technology was available, decades before Malenfant's history had caught up.
Malenfant, battered by strangeness, found room for a twinge of nostalgia. He'd have exchanged McCann's fat Moon for Luna any time. If only a world like Mars had been found to orbit the Earth, instead of poor desiccated Luna—a world with ice and air, just waiting for an explorer's tread! With such a world as a lure, just three days away from Earth, how different history might have been. And how differently his own life, and Emma's, might have turned out.
"The lure of the Moon was everything, of course." McCann said. "From times before memory it has floated in the sky, fat and round and huge, with storms and ice caps and even, perhaps, traces of vegetation, visible with the naked eye. You could see it was another world in the sky, waiting for the tread of man, for the flag of empire, the ploughs of farmers... It was quite a chase. Got to stop the other chap getting there first, you see."
Malenfant was getting confused again. "Other chap? You mean the Americans?"
Nemoto said gently, "There are no Americans in his world, Malenfant."
"The French, of course," said McCann. "The blooming French!"
Colonies on this bounteous Moon had been founded in what sounded like the equivalent of the first half of the twentieth century. Since then wars had already been fought, wars on the Moon waged between spreading mini-empires of Brits and French and Germans.
But then, in McCann's universe, the Mars-Moon had disappeared, to be replaced by this peculiar, wandering Red Moon, with its own cargo of oceans and life. Once the world had gotten over its bewilderment—once the last hope of contacting the lost colonies on Mars-Moon was gone—a new race had begun to plant a flag in the Red Moon.
"... Or _Lemuria_ , as we call it," McCann said.
Nemoto said, "A lost continent beneath the Indian Ocean, once thought to have been the cradle of mankind."
McCann talked on: of how the dozen men had travelled here; of a disastrous landing that had wrecked their ship and killed three of them; of how they had sent heliograph and radio signals home and waited for rescue—and of how their Earth had flickered out of the sky, to be replaced by another, and another.
"A sheaf of worlds," murmured Nemoto, gazing at McCann.
When it was clear that no rescue was to come, some of the exploratory party had submitted to despair. One committed suicide. Another handed himself over to a party of Elf-folk for a hideous and protracted death.
The survivors had recruited local Hams, and used their muscles and Runner labor to construct this little township. They had found no others of their kind, save for the sinister-sounding Zealots, of whom McCann was reluctant to speak, who lived some distance from the compound.
It seemed that it had been the mysterious Zealots who had taught the indigenes their broken English—if inadvertently, through escaped slaves returning to their host populations. The Zealots had been here for centuries, McCann seemed to believe.
"Not much of a life," McCann said grimly. "No women, you see. Some of us sought relief with the Hams, even with Runners. But they aren't _women_. And there were certainly no children to follow." He smiled stoically. "Without women and children, you can't make a colony, can you? After a time you wonder why you bother to shave every day."
One by one the Englishmen had died, their neat little huts falling into disrepair.
McCann showed them a row of graves, outside the stockade gate, marked by bits of stone. The last to die had been a man called Jordan—"dead of paralytic shock," McCann said. McCann appeared especially moved to be at Jordan's graveside. Malenfant wondered if these withdrawn, lonely men, locked in civility and their memories of a forever lost home, had in the end sought consolation in each other.
But McCann, in a gruesome effort to play the good host, talked brightly of better times. "We had a life of sorts. We played cards—until they wore out—and we made chess sets, carving pieces from bits of balsa. We had no books, but we would spin each other yarns, recounting the contents of novels as best we remembered them. I dare say the shades of a few authors are restless at the liberties we took. Once or twice we even put on a play or two. Marlowe comedies mostly: _Much Ado About Nothing_ , that kind of thing. Just to amuse ourselves, of course.
"We used to play sports. Your average Ham can't kick a soccer ball to save his life, but he's a formidable rugby player. As for the Runners, they can't grasp the simplest principle of rules or sportsmanship. But, my, can they run! We would organize races. The record we got was under six seconds for the hundred-yard dash. _That_ fellow was rewarded with plenty of bananas and beer..."
McCann spoke of how the survivors, just four of them, had become withdrawn, even one from the other, as they waited gloomily for death. Crawford would disappear into the forest for days on end with squads of Hams, "fossicking around," as McCann put it. The others would rarely even leave their huts.
"And you?" Nemoto asked. "What is your eccentricity, Mr McCann?"
"A longing for company," he said immediately, smiling with self-deprecation. "That's always been my weakness, I'm afraid."
"Then it must have been hard for you here," Malenfant said.
"Indeed. But when my companions withdrew into themselves, I sought out the company of the lesser folk: the Hams, even the Runners at times. My companions took to calling me _Mowgli_. Perhaps you know the reference. I have attempted to civilize them, teach them skills—more advanced toolmaking, even reading. With little success, I am afraid. Your bar-bar is smarter than your Runner, and these presapients are smarter in turn than the pongid species, the Elves and Nutcrackers. Your bar-bar can be taught to use a new tool, you know—to _use_ it but never to make it. They can make things work but never understand _how_ they work, rather like human infants. And, like your Kaffir, your bar-bar can see the first stage of a thing, and maybe the second, but no more.
"And that, of course, is the difference between man and presapient. Wherever there are sub-men, who live only for the day and their own bellies, we must rule. But the work shapes one. The responsibility. It has made me pitiful and kindly, I would say, as I have learned something of their strange, twisted reasoning." He leaned toward them. "They have no chins, you see, none of them. And everybody knows that a weak chin generally denotes a weak race."
When evening came again, fires were built within the palm-thatched huts, and smoke rose through the roofs and the crude chimneys that pierced them. Malenfant saw a pair of bats, flapping uncertainly between the turbulent columns of smoke. They were big, as big as crows, with broad, rounded wings.
"Leaf-nose bats," Nemoto murmured.
"Don't tell me. Prehistoric bats."
Nemoto shrugged. "Perhaps. There are many bats here. They have occupied some of the niches never taken by the birds."
Malenfant watched the bats' slow, ungainly flapping. "They sure look unevolved."
"Ah, but they were the peak of aerial engineering when they hunted flies and mosquitoes over lakes full of dinosaurs, Malenfant. You should have a little more respect."
"I guess I should."
Nemoto whispered conspiratorially, "It all hangs together, Malenfant."
"What does?"
"McCann's account of his alternate Earth. A much larger Moon would raise immense tides. The oceans would not be navigable. McCann's America must once have been linked to Eurasia by land bridges, as ours was, for otherwise the Hams presumably couldn't have reached it. But when the land bridges were submerged, the Americas were effectively cut off—until iron-hulled ships and airplanes emerged, in the equivalent of our own twentieth century. Malenfant, it may have been easier to fly to the Moon than to reach America. Think of that."
"What does all this mean, Nemoto?"
"I am working on it," she said seriously. "Consider this, though. We are alone on our Earth, our closest relatives terribly distant. But McCann's world has a spectrum of hominid types—as it was on our own Earth, long ago. McCann's Earth may in some senses be more _typical_ than ours."
A party of Runners, supervised by a Ham, brought in a deer, slung between them, half-butchered.
"Look at that," muttered Nemoto. "I think that one is a mouse deer." It was small, the size of a dog, its coat yellow-brown spotted with white, and it had tusks in its upper jaw. "You see them in Africa. Actually it isn't really a deer at all. It is midway between pigs and deer, and more primitive than either. It climbs trees. It catches fish in the streams. Probably unchanged across thirty million years. Older than grass, Malenfant."
"And the other?"
This was a little larger than the mouse deer, with a black stripe down its back, and powerful hind legs: a creature evolved for the undergrowth, Malenfant thought.
"A duiker, I think," Nemoto said. "Another primitive form, the oldest of the antelopes. Sometimes hunts birds and feeds on carrion. Maybe here it eats bats. Everything is ancient here." Now she seemed agitated. "Perhaps these forms were brought here by the same mechanism that imported hominids. What do you think?"
"Take it easy."
Her small, thin face worked in the gathering gloom. "This is wrong, Malenfant."
"Wrong? What's wrong with it?"
"The ecology is—out of tune. Like a misfiring engine. It is a jumble of species and microecologies, a mixed-up place, fragments thrown together. Though many of the fragments are very ancient, there has been no time for these plants and animals to evolve together, to find an equilibrium. Periodically something disturbs this world, Malenfant, over and over, stirring it up."
Malenfant grunted. "Guess you can't go wandering across the reality lines without a little confusion."
But Nemoto would not take the matter lightly. "This is not right, Malenfant. All this _mixing_. There is a _reason_ the primitive hominids became extinct, a reason why the mouse deer's descendants evolved new forms. An ecology is like a machine, where all parts work together, interlocking. You see?"
Malenfant said, amused, "These deer and antelopes seem to have been prospering before they ran into some hunter's crossbow bolt."
"It shouldn't be this way, Malenfant. To meddle with ecologies, to short-circuit them, is irresponsible."
Malenfant shrugged. "Sure. And we cut down the forests to build shopping malls." He was feeling restless; maybe his first shock was wearing off. He'd had enough of McCann; he was eager to get out of here, get back to the lander—and progress in his primary mission, which was to find Emma.
But when he expressed this to Nemoto she laughed harshly. "Malenfant, we barely managed to survive our first few minutes after landing. Here we are safe. Have patience."
He seethed. But without her support, he didn't see what he could do about it.
## _M anekatopokanemahedo_
When she was Mapped to the Market—when the information that comprised her had been squeezed through cracks in the quantum foam that underlay all space and time—she was no longer, quite, herself, and that disturbed her greatly.
Manekato was used to Mapping. The Farm was large enough that walking, or transport by Workers, was not always rapid enough. But Mappings covering such a short distance were brief and isomorphic: she felt the same coming out of the destination station as entering it (just as, of course, principles of the identity of indiscernible objects predicted she should).
A Mapping spanning continents was altogether more challenging. To compensate for differences in latitude and altitude and seasons—early summer _there_ , falling into autumn _here_ —and to adjust for momentum differences—people on the far side of the spinning Earth were moving in the opposite direction to her—such a Mapping could be no more than homomorphic. What came out looked like her, felt like her. But it was not indiscernible from the original; it could not _be_ her.
Still, despite these philosophical drawbacks, the process was painless, and when she walked off the Mapping platform, her knuckles tentatively touching new ground, she found herself comfortable. The air was hot, humid but caused her no distress, and even its thinness at this higher altitude did not give her any discomfort.
And the air was still. _There was no Wind_. Thanks to the Air Wall wrapped around it, the Market was the only place on Earth from which the perpetual Wind was excluded. She had been prepared for this intellectually, of course. But to stand here in this pond of still air—not to feel the caressing shove of the Wind on her back—was utterly strange.
This crowded Mapping station was full of strangers. She peered around, feeling conspicuous, bewildered. Some of the people here were small, some tall, some squat, some thin; some were coated with hair that was red or black or brown, and some had no hair at all; some crawled close to the ground, and some almost walked upright, like their most distant ancestors, their hands barely brushing the ground. Manekato, who had spent her whole life on a Farm where everybody looked alike, tried to mask her shock and revulsion at so much _difference_.
She was met outside the station by a Worker, a runner from the Astrologers. She slid easily onto its broad back, wrapping her long arms around its chest, and allowed herself to be carried away.
Her first impression of the Market was of waste. The streets were broad, the buildings an inefficient variety of designs, and she could spot immediately places where heat would leak or dust gather, or where the layout must prevent optimally short journeys from being concluded.
All of this jarred with her instinct. The goal for every Farmer was to squeeze the maximum effectiveness and efficiency from every last atom—and beyond, to the infinitesimal. The mastery of matter at the subatomic level, resulting in such everyday wonders as Mapping and Workers, had brought that ultimate dream a little closer.
But, she reminded herself, this was the Market, not a Farm.
In the deepest past there had been a multitude of markets, where Farmers traded goods and information and wisdom. The transient population of the markets had always been predominantly male. Women were more tightly bound to the land, locked into the matriarchal Lineages that had owned the land since the times almost before history; men were itinerant, sent to other Farms for the purpose of trade and marriage.
But as technology had advanced and the Farms had become increasingly self-sufficient, the primary function of the markets had dwindled. One by one they had fallen into disuse. But the role of the markets as centers of innovation had been recognized—and, perhaps, their purpose in providing an alternate destiny for rootless men and boys. So some of the markets had been preserved.
At last only one Market remained: the grandest and most famous, perched here on the eroded peak of its equatorial mountain, supported now by tithes from Farms around the world. Here men, and a few women, dreamed their dreams of how differently things might be—and enough of those innovative dreams bore fruit that it was worth preserving.
It had been this way for two hundred thousand years.
The Worker carried her away from the Market's crowded center toward its fringe. The crowds thinned out, and Manekato felt a calming relief to be alone. Alongside an impossibly tall building the Worker paused and hunched down, letting her slip to the ground.
A door dilated in the side of the building. She glanced into the interior; it was filled with darkness.
Reluctant to enter immediately, she loped further along the gleaming, dust-free road. Not far beyond the building the ground fell away. She was approaching the rim of the summit plateau, worn smooth by the feet and hands of visitors. She leaned forward curiously. The mountain's shallow flanks fell away into thicker, murky air; far below she glimpsed green growing things.
And she saw the Air Wall.
It was like a bank of windblown cloud, moving swiftly, gray and boiling. But this cloud bank hung vertically from the sky, and the clouds streamed horizontally past her. Now that it was not masked by the buildings she could see how the great Wall curved around the mountaintop, enclosing it neatly. It stretched down like a curtain to the ground below, where dust storms perpetually beat against the struggling vegetation, and it stretched up toward the sky.
It was not easy for her to look up, for her back tilted forward, and her neck was thick, heavily muscled, adapted to fight the Wind. Besides, at home there was generally nothing to see but a lid of streaked, scudding cloud. But now she tipped back awkwardly, raising her chinless jaw.
It was like peering up into a tunnel, lined by scraps of hurrying cloud. And at the very end of the tunnel there was a patch of clear blue.
She had never before seen the sky beyond the clouds.
She shuddered. She hurried inside the building.
And there she met her brother.
## _R eid Malenfant_
While he waited for an opportunity to progress in his mission, Malenfant ate and drank as much as he could, and after the first day put his body through some gentle exercise. He stretched and pushed up and pounded around the red dust of the neat little stockade in his undershirt and shorts, while Ham servants watched with a kind of absent curiosity, and Runners hooted and shook their shackling ropes. The low gravity made him feel stronger, but conversely the reduced oxygen content of the low-pressure air weakened him. If he overexerted himself he would soon run out of air; his chest would ache, and, in the worst cases, black spots would gather around his vision.
But he would adapt. And for now, it did no harm to test his limits.
McCann took him for tourist-guide jaunts around the compound, and even beyond. He seemed childishly eager to show off what he and his companions had built here.
McCann said the English had tried to mine mudstone—a kind of natural brick—so as to build better houses. "We have the raw muscle, among the Runners and the Hams," McCann said. "That's fine for hauling, lifting, and dragging. But they can't be set to fine work, Malenfant; not without a man's constant supervision. You certainly can't send off a party even of the Hams to a mudstone seam and expect them to return with anything but a jumble of gouged-out, misshapen rocks—nothing like _bricks_ , you see—that's if they bring back anything at all."
There were a lot of pleasurable knickknacks to inspect, constructed over long hours by the ingenious hands of these bored Englishmen. Malenfant, a gadget fan, pored over locks, clocks, and slide rules, all made of wood.
McCann had even maintained a crude calendar system—though it was little more than marks on wood. "Like a rune staff," McCann said, grimacing. "How far we have fallen. But we haven't quite mastered the knack of papermaking, you see; needs must. And, besides, this wandering world has a damnably irregular sky. Even the stars swim about sometimes, you know. But we try to impose order. We do try."
Everything was made of wood, or stone, or bone, or material manufactured from vegetable products. You could make rope, for instance, from birch bark, pine roots, or willow. Ham women baked pine bread made from phloem, the soft white flesh just inside the tree's bark. You could drink the sap of birch trees, if you had to. And there were medicinal products: spruce resin to ease gut ache. And so on.
McCann said, "This benighted world is bereft of metals, you see—of sizable ore lodes, anyhow, so far as we could find. Of course the very dust is iron oxide—hematite, I think—but we have notably failed to establish a workable extraction regime... It was an early disappointment, and all the more severe for that. And we were reluctant to mine the only source of refined metals here—I mean our ship, of course. As long as we clung to hope that we might escape this jungle world, we were reluctant to turn our only vessel into pots and pans. All seems a little foolish now, doesn't it? And so ours is an economy of stone and wood. We have become like our woad-wearing forebears. Amusing, isn't it?"
They came to a hut where a Ham woman, somewhat bent, was ladling water from a wooden bucket at her feet. Malenfant, glimpsing machinery, poked his head inside the hut, and allowed his eyes to adjust to the shade.
A big wooden container sat on a stand above a smoldering fire. There was some kind of mash inside the container: The woman showed him, though she had to remove a lid sealed with some kind of wax to do it. Two narrow bamboo pipes led down from the container. Condenser pipes, Malenfant thought. The pipes finished in V-notches that tipped their contents neatly into gourds...
"It's a still," Malenfant breathed. "Holy shit. Hillbilly stuff. Just the way Jack Daniels started. God, I love this stuff."
McCann preened, inordinately proud; briefly Malenfant was taken back to his prelaunch inspections at Vandenberg and elsewhere.
Immediately outside the stockade the forest seemed sparse. The leaves were a pale green, lighter than usual, and lianas tangled everywhere, irregular. Though there were sudden patches of shade, much of the ground was open to the sun; there was no solid canopy here.
This area had been cleared, Malenfant realized—twenty, thirty years ago?—and then abandoned. And now, oblivious to the failed ambitions of the stranded English, the forest was claiming back the land. He gazed at the ground, and thought he discerned the straight line edges of forgotten fields, like Roman ruins.
But even out here there were signs of rudimentary industry. A charcoal pile had been constructed: just a heap of logs with earth piled over the top, steadily burning. And there was a tar pit, a hole in the ground filled with pine logs, buried under a layer of earth. The logs burned steadily, and crude wooden guttering brought out the tar.
They came to a stand of small oil-palm trees that clung to the banks of a stream. They were slim and upright with scruffy green fronds, holding onto the slope with prop-roots, like down-turned fingers curling out from the base of their pale gray trunks. Under the direction of one or two of the Hams, Runner workers gathered oil from the flesh of the nut and the kernel of the seeds, and sap from shallow cuts near the trees' bases.
"You cook with the oil, or you make soap with it," McCann said. "And if you were to hang a bucket under that cut in the trunk you'd be rewarded by ready-made palm wine, Malenfant. Nature is bountiful sometimes, even here. Though it takes human ingenuity to exploit it to the full, of course."
McCann even showed Malenfant the poignant ruin of a windmill. Crudely constructed, it was a box of wood already overgrown by vegetation and with daylight showing through cracks in its panels. Later McCann showed him elaborate drawings, crammed into the blank pages of yellowing log books. There had been ambitious schemes for different designs of mills—"magpie mills" with a tail to turn into the wind, and even a water mill—none of them realized. "We never had the labor, you see. Your Ham or your Runner is strong as an ox. But you can't teach him to build, or to maintain, anything more complex than a hand axe or a spear. He will go where you tell him, do what you tell him, but no more; he has no initiative or advanced skill, not a scrap. One had to oversee everything, every hand turned to the work. After a time—well, and with no hope for the future—one became rather disheartened."
McCann was obviously desperate for company, and it was hard to blame him. He challenged Malenfant to a game of chess—which Malenfant refused, never having grasped the game. Despite this McCann set out crudely carved wooden pieces, and moved them around the board in fast, well-practiced openings. "I played a lot with old Crawford before he lost his wits. I do miss the game. I even tried to teach the bar-bars to play!—but though they appear capable of remembering the moves of the pieces, not even the brightest of them, even Julia, could grasp its essence, the _purpose_. Still, I would have Julia or another sit where you are sitting, Malenfant, and serve as a sort of token companion as I played out solitary games..."
As he pushed the pieces around the board McCann bombarded Malenfant with anecdotes and memories, of his time here on the Red Moon and on his own lost version of Earth.
But the talk was unsatisfactory. They were exiles from different versions of parallel Earths. They could compare notes on geography and the broad sweep of history, but they had no _detail_ in common. None of the historical figures in their worlds seemed to map across to each other. Although McCann seemed to follow a variant of Christianity—something like Calvinism, so far as Malenfant could determine—his "Christ" was not Jesus, but a man called John; "Christian" translated, roughly, to "Johannen."
No doubt all this was fascinating as a study of historical inevitability. But it made for lousy small talk. McCann strove to mask his profound disappointment that Malenfant was not from the home where he had left a wife and child, a family from whom he had not heard since their world had disappeared from the sky.
Conversely Malenfant told McCann what he could of Emma, and asked if anyone like her had shown up, here on the Red Moon. But McCann seemed to know little of what went on beyond the limits of the stockade, and the scrap of Red Moon he and his colleagues controlled. Malenfant, frustrated, realized afresh he was going to have to find Emma alone.
McCann said now, "Solitary, seeking diversion, I discovered the intricate delights of the knight's tour." He swept the board empty of pieces, save for a solitary knight, which he made hop in its disturbingly asymmetrical fashion from square to square. "The knight must move from square to square over an empty board, touching all the cells, but each only once. An old schoolboy puzzle... I quickly discovered that a three-by-four board is the smallest on which such a tour can be made. I have discovered many tours on the standard chessboard, many of which have fascinating properties. A closed tour, for example, starts and ends at the same cell." The knight moved around the board with bewildering rapidity. "I do not know how many tours are possible. I suspect the number may be infinite." He became aware of Malenfant's uncomfortable silence.
Malenfant tried to soften his look—how sane would _you_ be after so many decades alone on Neandertal Planet, Malenfant?
Embarrassed, McCann swept the pieces into a wooden box. "Rather like our situation here, don't you think?" he said, forcing a smile. "We move from world to world with knight's hops, forward a bit and sideways. We must hope our tours are closed, too, eh?"
After the first night McCann gave the two of them separate huts. In this dwindling colony there was plenty of room.
Malenfant found it impossible to sleep. Lying in his battered sod hut, he gazed through his window as the night progressed.
He heard the calls of the predators as the last light faded. Then there was an utter stillness, as if the world was holding its breath—and then a breath of wind and a coolness that marked the approaching dawn.
Malenfant wasn't used to living so close to nature. He felt as if he was trapped within some vast machine.
His head rattled with one abortive scheme after another. He was a man who was used to taking control of a situation, of bullying his way through, of pushing until something gave. This wasn't his world, and he had arrived here woefully ill-equipped; he still couldn't see any way forward more promising than just pushing into the forest on foot, at random. He had to wait, to figure out the situation, to find an option with a reasonable chance of success. But still his enforced passivity was burning him up.
The door opened.
The Neandertal girl came into his hut. She was carrying a bowl of water that steamed softly, a fresh towel, a jug that might hold nettle tea.
He said softly, "Julia."
She stood still in the gray dawn light, the glow from the window picking out the powerful contours of her face. "Here, Baas."
"Do you know what's going on here?"
She waited.
He waved a hand. "All of this. The Red Moon. Different worlds."
"Ask Ol' Ones," she said softly.
"Who?"
"Th' Ol' Ones. As' them wha' for."
"The Old Ones? Where do they live?"
She shrugged, her shoulders moving volcanically. "In th' ol'est place."
He frowned. "What about you, Julia?"
"Baas?"
"What do you want?"
"Home," she said immediately.
"Home? Where is home?"
She pointed into the sky. "Gray Earth."
"Does Mr. McCann know you want to go home?"
She shrugged again. "Born here."
"What?"
She pointed to herself. "Born here. Mother. Moth' born here."
"Then this is your home, with Mr. McCann."
She shook her head, a very human gesture. She pointed again to the forest, and the sky.
Then she said, "You, Baas? What you wan'?"
He hesitated. "I came looking for my wife."
Her face remained expressionless. But she said, "Fam'ly."
"Yes. I guess so. Emma is my family. I came here looking for her."
"Lon' way."
"Yes. Yes, it was a long way. And I ain't there yet."
She walked toward him, rummaging in the pouch of her skirt.
"Thomas," she said.
"I know him. He found me."
"Took off of Runner in fores'." She held out something in the dark, something small and jewel-like that glittered in her palm.
He took it, held it up to the light of the window. It was a hand-lens, badly scuffed, snapped off at its mount. It was marked with the monogram of the South African air force.
_"Emma,"_ he breathed. He was electrified. So there were indeed things McCann didn't know, even about the Hams of his own household. "Julia, where—"
But she had gone.
## _M anekatopokanemahedo_
"I have three wives and six children. That is how it is done in my new home..." Babo was talking fast, nervously, and his knuckles rattled as he walked with her through the tall dark halls of the building. His body hair was plaited and colored in a fashion that repelled Mane's simple Poka tastes. "The Farm is fine, Mane, and bigger than that of the Poka Lineage, but its design is based on the triangle: plane-covering, of course, but cramped and cluttered compared to Poka's clean-lined hexagons."
"You always were an aesthete," she said dryly.
This whole building, she realized slowly, was a store of records piled up high from the lowest room to the highest. Physically, some of the records were stored in twinkling cubes that held bits of the quantum foam, minuscule wormholes frozen into patterns of meaning; and some were scraped onto parchment and animal skin.
"Some of these pieces are very ancient indeed," Babo said. "Dating back half a million years or more. And the Air Wall, you know, is a controlled storm. It is like a hurricane, but trapped in one place by subtle forces. It has raged here, impotent, for fifty thousand years—so that for all that time the Market has been in the eye of the storm—an eye that reveals the sky beyond the clouds, a sky opened for the study of the Astrologers..."
She stopped and glared at him. "Oh, Babo, I don't want to know about Air Walls or records! I never thought I would see you again—I didn't know you had become an Astrologer."
He sighed, ruminatively picking his nose. "I am no Astrologer. But the Astrologers sent for me. When I was younger I did spend some time here, working informally, before I reached the home of my wives. Many boys do, Mane. You matriarchs run the world, but there is much you do not know, even about those who sire your children!"
" _Why are you here_ , Babo?"
He wrapped his big hands over his head. "Because the Astrologers thought it would be _kinder_ that way. Kinder if your brother told you the news, rather than a stranger..."
"What news?"
He grabbed her hand, pulling her. "Come see the sky with me. Then I'll tell you everything."
Reluctantly, she followed.
The building was tall, and they had a long way to ascend. At first they used simple short-range isomorphic Mappers, but soon they came to more primitive parts of the building, and they had to climb, using rungs stapled to walls of crude bricks.
Babo led the way. "A remarkable thing," he called down to her. "We find climbing easy; our arms are strong, our feet well adapted to grasping. But it appears that our climbing ancestors evolved into creatures that, for a time, walked upright, on their hind feet. You can see certain features of the position of the pelvis—well. But we have given that up, too; now, once more, we walk on all fours, using our knuckles, clinging to the ground."
"If you tried to walk upright you would be knocked over by the Wind."
"Of course, of course—but then _why_ is it we carry traces of a bipedal ancestry? We are creatures of anomaly, Mane. We are not closely related to any of the animals on this Earth of ours—not one, not above a certain basic biochemical equivalence of course, without which we could not eat our food and would quickly starve to death. We can trace evolutionary relationships among all the world's creatures, one related to the other in a hierarchy of families and phyla— _except us_. We seem to be unique, as if we fell out of the sky. We have no evolutionary forebears, no bones in the ground that might mark the passing of those who came before us.
"Is it possible _we evolved somewhere else_?—a place where the Wind did not blow so strongly, where it was possible to walk upright?"
"What sort of place? And how could we have got here from there?"
"I don't know. Nobody knows. But the pattern of the bones, the biochemistry, is unmistakeable."
"Idle speculation, Babo, won't germinate a single seed."
"A Farmer's practical reply," he said sadly. "But we are surrounded by mystery, Manekato. The Astrologers hope that your mission will settle some of these fundamental quandaries. Oh, please keep climbing, dear Mane! We are soon there, and I will tell you everything."
With bad grace, clinging to the rungs with feet and hands, she continued her ascent.
They reached a platform, open to the sky. But there was no breeze, and the air felt as warm as it had inside the tower.
Babo walked around nervously, peering into the sky. "It is darkening already. Our days are short, because the planet spins quickly—did you ever reflect on that, Mane? It didn't have to be so. Earth could spin more slowly, and we would have leisurely days, and—oh, look!" He pointed with a long stabbing finger. "Look, a star!"
She peered up awkwardly. There was a single bright star, close to the zenith, set against the deepening blue of the sky.
"How strange," Babo breathed, "that before the first tentative Mappings no human eye saw a star."
Manekato grunted. "What of it? Stars are trivial. You don't need to _see_ them."
That was true, of course. Every child was expected to figure out the stars.
When Manekato was two years old she had been shut in a room with a number of other children, and a handful of artifacts: a grain of sand, a rock crystal, a bowl of water, a bellows, a leaf, other objects. And the children were told to deduce the nature of the universe from the contents of the room.
Of course the results of such trials varied—in fact the variations were often interesting, offering insights into scientific understanding, the nature of reality, the psychology of the developing mind. But most children, working by native logic, quickly converged on a universe of planets and stars and galaxies. Even though they had never seen a single star.
Stars were trivial mechanisms, after all, compared to the simplest bacterium.
"Ah, but the detail is everything," Babo said, "and that you can never predict, of course. That and the _beauty_. That was quite unexpected, to me. Oh, and one other thing. The _emptiness_ of the universe..."
Manekato's childhood cohort, like most others, had concluded—groping with an intuition of uniformity—that if _this_ world was inhabited, and the universe was _large_ —well, then, there must be many inhabited planets. She recalled what a great and unwelcome surprise it had been to learn that that was not true: that, as far as could be discerned, the universe was empty of the organization that would have marked the work of intelligence.
"It is a deep, ancient mystery," Babo said. "Why do we see no Farms in the sky? Of course we are a sedentary species, content to cultivate our Farms. But not every species need have the same imperatives as us. Imagine an _acquisitive_ species, that covets the territory of others."
She thought it through quickly. "That is outlandish and unlikely. Such a species would surely destroy itself in fratricidal battles, as the illogic of its nature worked itself out."
"Perhaps. But wouldn't we see the flaring of the wars, the mighty ruins they left behind? We should _see_ them, Mane."
She snapped, "Babo, get to the point."
He sighed and came to squat before her. Gently he groomed her, picking imaginary insects from her coat, as he had when they were children. "Mane, dear Mane, the Astrologers have read the stars..."
The word "astrology," in Manekato's ancient, rich language, derived from older roots meaning "the word of the stars." Here astrology had absorbed astronomy and physics and other disciplines; here astrology was no superstition, no foolishness, but one of the fundamental sciences. For if the universe was empty of mind save for humans, then the courses of the stars could have no meaning—save for their role in the affairs of humanity.
And now, Babo said, the Astrologers, peering into the sky and poring over records dozens of millennia deep, had discerned a looming threat.
## _J oshua_
Mary was in estrus. The scent of her seemed to fill the air of the hut, and the head of every man.
Joshua longed for the time of her blood to pass, and she and the other women could recede to the gray periphery of his awareness. For the deep ache aroused by Mary distracted him from the great conundrum which plagued him.
Over and over he thought of the great blue wings he had seen falling from the sky, bearing that fat black and white seed to its unknown fate in the forest at the top of the cliff. He had never seen such a thing before. _What was it?_
Joshua's was a world that did not countenance change. And yet, a stubborn awareness told him, there _was_ change. Once the people had lived on the Gray Earth. Now they lived here. So the past contained a change. And now the black and white seed had fallen from the sky, and whatever grew from it surely marked change to come in the future as well.
Change in the past, change in the future.
Joshua, helplessly conservative himself, had an instinctive grasp of parsimony: his world contained two extraordinary events—Gray Earth and sky seed—and surely they must be linked. But how? The elements of the conundrum revolved in his head.
Joshua had solved puzzles before.
Once, as a boy, he had found a place where Abel, his older brother, had knapped out a burin. It was just a patch of dune where stone flakes were scattered, in a rough triangle that showed where Abel had sat. Joshua had picked over the debris, curious. Later, in the hut, he had found the discarded burin itself. It was a fine piece of work, slender and sharp, and yet fitting easily into Joshua's small hand. And he remembered the spall outside.
He sat where his brother had sat—one leg outstretched, the other tucked underneath. He reached for bits of the spall, and tried to fit them back onto the finished tool. One after another he found flakes that nestled closely into the hollows and valleys of the tool, and then more flakes which clustered around them.
Soon there were more flakes than he could hold in his hands, so he put down his assemblage carefully, and climbed a little way up the cliff behind the hut. He found a young tree sprouting from a hollow, and bled it of sap. With the sticky stuff cradled in his hands he ran back to his workplace, and began to fix the flakes to the tool with dabs of the sap. The sap clung to his fingers, and soon the whole thing was a sticky mess. But he persisted, ignoring the sun that climbed steadily into the sky.
At last he had used up almost all the large flakes he could find on the ground, and there was nothing left there but a little dust. And he had almost reassembled the cobble from which the burin had been carved.
Shouting with excitement he ran into the hut, cradling his reconstruction. But he had received a baffled response. Abel had picked at the sticky assemblage of flakes, saying, "What, what?"
A cobble was a cobble, until it was turned into a tool, and then the cobble no longer existed. Just as Jacob had been a man until he died, and then there was only a mass of meat and bones, soon to be devoured by the worms. To turn a tool back into a cobble was almost as strange to the people as if Joshua had tried to turn Jacob's bones back into the man himself.
Eventually Abel crushed the little stone jigsaw. The gummy flakes stuck to his hand, and he brushed them off on the dusty ground, growling irritably.
But in some corner of his spacious cranium Joshua had never forgotten how he had solved the puzzle of the shattered cobble. Now, as he pondered the puzzle of the multiple earths and the falling seed, Joshua found that long-ago jigsaw cobble pricking his memory.
And when a second seed fell from the sky—another fat black and white bundle suspended under a blue canopy, landing where the first had lodged at the top of the cliffs—he knew that he could not rest until he had seen for himself what mighty tree might sprout from those strange seeds.
Joshua approached Abel and Saul and other men to accompany him on his jaunt up the cliff face. But there was no purpose to his mission—no game to be hunted, no useful rock, no foraging save for the huge enigmatic seeds which had slid silently over the surface of everybody else's mind.
And besides, everybody knew there was danger at the top of the cliff. The camp of the Zealots was there, in the center of a great clearing hacked crudely out of the forest. The Zealots were Skinny-folk. They were easily bested if you could ever get one engaged in close quarters. But the Zealots were cunning, and their heads were full of madness: They could baffle the most powerful of the Hams. They were best avoided.
Joshua tried to go alone. He set foot on the rough goat trail that led by gully and switchback turn up to that cliff-crest forest.
The trail was easy enough, but he soon turned back. The isolation worked on him, soon making him feel as if he didn't exist at all. The People of the Gray Earth needed nothing in life so much as each other.
But word of his project permeated the gossip-ridden hut. A few days later, to his surprise, he was approached by the young girl Mary, who asked him about the cliff, and the forest, and the strange sky seed.
And a day after that, to his greater surprise, she accompanied him on the trail.
She gossiped all the way to the top of the cliff. "... Ruth say Abel skinny as an En'lish. An' Ruth tell tha' to Miriam. An' Miriam tell Caleb, an' Caleb tell Abel. An' Abel throw rocks and skins all over th' hut. So Abel couple Miriam, and he tell Caleb about tha', and he tell Ruth. And Ruth say..."
Unlike himself she was no loner. She was immersed in her little society. By comparison it was as if he couldn't even see or hear the vibrant, engaged people she described.
All of which made it still stranger that she should choose to accompany him on this purposeless jaunt. But Mary was at a key moment in her life, and a certain wanderlust was in her blood right now. Soon she would have to leave the security of the hearths her mother built, and share her life with the men, and with the children who would follow. To cross from one side of a skin hut to the other was an immense journey for someone like Mary. And as nervous courage empowered her for that great adventure, she seemed ready, for the time being, to take on much more outlandish quests.
She was not in estrus, to Joshua's great relief. As he made his careful way up the cliff face he was pleased not to have the distraction of his own singing blood.
They reached the top of the cliff. Here they found a shrub laden with bright yellow fruit, and they sat side by side at the cliff's edge, plucking the fruit, their broad feet dangling in the air. They gazed out in silence toward the east, and the sea.
The sun was still rising, and its light glimmered from the sea's steel-gray, wrinkled hide. The distinct curve of the world was reflected in layers of scattered purple clouds which hovered over the sea. Joshua could see the grassy plain where he lived, sweeping towards the ocean, terminating in dune fields and pale sand. Near the squat brown shape of the hut itself, people moved to and fro, tiny and clear. He followed streams, shining lines of silver that led towards the sea.
A small group of antelopes picked their way through the morning grass. One of them looked up, as if staring directly at him.
Joshua felt himself dissolve, out from the center of his head, to the periphery of the world. There was no barrier around him, no layer of interpretation or analogy or nostalgia; for now he _was_ the plain and the sea and the clouds, and he was the slim doe that looked up at the cliff, just as he was the stocky, quiet man who gazed down from it. For a time he was immersed in the world's beauty in a way no human could have shared.
Then, by unspoken consent, Joshua and Mary folded their legs under them and stood. Side by side, they walked into the forest that crowded close to the cliff.
The green dark was a strong contrast to the bright sea vista. It was not a comfortable place to be.
Washed by the salty air off the sea, the forest was chill, thick with a clammy moisture that settled into Joshua's bones. And as they penetrated deeper the ground was covered in a tangled mass of roots, branches, leaves, and moss, so that in some places Joshua couldn't see the actual surface at all. He slipped, stumbled, and crashed over the undergrowth, making a huge amount of noise.
Mary started to shiver and complain, growing increasingly fearful. But Joshua pulled his skin wraps tighter around him and shoved his way deeper into the forest.
A shadow slid through the wood, just a little way ahead, utterly silent.
Joshua and Mary both froze. Joshua bunched his fists. Was it a Zealot?
The shadow slowed to a halt, and Joshua made out a squat, stocky body, with short legs and immensely long arms, the whole covered by a dark brown layer of hair. A hand reached out and grabbed a bamboo tree. The tree was pulled down until it cracked, and drawn toward a gaping mouth.
It was a Nutcracker-man. Joshua relaxed.
Mary stumbled closer to Joshua, making a cracking noise.
The Nutcracker-man turned his great head with its sculpted skull ridge and giant cheekbones. Perhaps he saw them; if he did he showed no concern. He pulled his bamboo toward his mouth and bit sideways at the trunk, seeking the pithy interior. As he chewed, the heavy muscles that worked his jaw expanded and contracted, making his entire head move.
Though slow and foolish and easily trapped, the Nutcrackers' muscles made them formidable opponents. But the Nutcrackers rarely ventured from their forests, and when they did they showed no instinct for aggression against the Hams. Likewise the Hams did not eat people. The two kinds of people had little in common and nothing to fight about, and simply avoided each other.
After a short time the Nutcracker-man finished his bamboo. He slid effortlessly away into the green, placing his hands and feet slowly and methodically, but he moved rapidly and almost noiselessly, soon outstripping any effort Joshua might have made to catch him.
Out of curiosity Joshua and Mary tried the bamboo. It took both of them to crack a trunk as thick as the one the Nutcracker had pulled over with one hand, and when he tried to bite into it Joshua's teeth slid off the trunk's glossy casing.
They moved deeper into the forest. The sun, showing in glittering fragments through the dense canopy, was now high. But Joshua caught occasional glimpses of the sea, and he kept it to his right, so that he knew he was working roughly the way the floating black and white seed had fallen. Mary kept close behind him. Her biceps showed, hard and massive, beneath the tight skins wrapped around her arms.
And now there was another shadow passing through the forest ahead. But this time there was much more noise. Maybe it was a bear, careless of who or what heard it. They both crouched down in a dense patch of tangled branches, and peered out fearfully.
The shadow was small, even slender.
It was just a man, and a feeble-looking man at that, with nothing like the bulk of a Ham, still less a Nutcracker. He was a Skinny: surely a Zealot. He wore skins wrapped closely around his limbs and torso, and he carried a length of bamboo tube. His face was covered by an ugly mass of black beard, and he was muttering to himself as he blundered noisily through the forest.
With some care he selected a broad-trunked tree. He sat down beneath it. He reached into his trousers to scratch his testicles, and emitted a long, luxurious fart. Then he raised the bamboo to his lips. To Joshua's astonishment, a foamy liquid gushed from the bamboo into the man's mouth. _"Up your arse, Praisegod Michael."_ He raised the flask, and drank again. Soon he began to wail. _"There is a lady, sweet and kind..."_
Mary clapped her hand over her mouth to keep from laughing. The Zealot was squealing like a sickly child.
Joshua was fascinated by the bamboo flask, by the way the murky liquid poured out into the man's mouth and down his bearded chin.
The Zealot finished off the contents of his flask. He settled farther back against his tree trunk, tucking his arms into his sleeves. He had a broad-rimmed hat on his head, and as he reclined it tipped down over his eyes, hiding his face. His mouth popped open, and soon rattling snores issued from it.
Joshua and Mary crept forward until they stood over the sleeping Zealot. Joshua bent to pick up the bamboo. He tipped it upside down. A little foamy fluid dripped onto his palm. He licked it curiously. The taste was sour, but seemed to fill his head with sharpness.
He inspected the bamboo more closely. Its end had been stopped by a plug of wood, and a loop of leather attached another plug that, with some experimentation, Joshua managed to fit into the open end of the tube, sealing it. Joshua's people carried their water in their hands, or sometimes plaited leaves or hollowed-out fruit. Though they would have been capable of it, it had never occurred to them to make anything like the Zealot's bamboo flask.
Mary, meanwhile, was crouching over the Zealot. She was studying his clothing. Joshua saw that it had been cut from finely-treated skin. The skin had been heavily modified, with whorls and zigzag lines and crosses scratched into it and colored with some white mineral. The edges of the various pieces of skin had been punctured. Then a length of vegetable twine had been pushed through the puncture holes, to hold the bits of skin together. Mary picked at the seams and hems with her blunt fingers; she had never seen anything like it.
Joshua found the patterns on the skins deeply disturbing. He had seen their like before, on other Zealot artifacts. To Joshua the patterns made by the markings were at the limit of his awareness, neither there nor not there, flickering like ghosts between the rooms of his mind.
Now Mary's searching fingers found something dangling around the man's neck on a piece of thread. It was a bit of bone, that was all, but it had been shaped, more finely than Abel's best tools.
Joshua studied the bone. Suddenly a man surged out of the carving: his face contorted, his hands outstretched, and his chest ripped open to reveal his heart.
Joshua screamed. He grabbed the bit of bone and yanked it so the thread around the Skinny's neck broke, and he hurled it away into the forest.
The Skinny woke with a gulping snore. He sat up abruptly, and his hat fell off his head. Seeing the two hulking Hams, he raised his hands to the sky and began to yell. _"Oh, Heaven help me! By God's wounds, help me!"_
Mary looked up into the sky, trying to see who he was speaking to. But of course there was nobody there. The Skinny-folk were immersed in madness: they would talk to the sky, the trees, the patterns on their clothes or ornaments, as if those things were people, but they were not.
So Mary sat on the Zealot's chest, crushing him to the ground; he gasped under her weight. "Stop talkin' sky! Stop!"
The bearded Zealot howled.
She slapped him across the face. The Zealot's head was jerked sideways, and he instantly became limp.
Mary backed away. "Dead?"
Joshua, reluctantly, bent closer. The Zealot had fouled himself, perhaps when Mary had leapt on him; a thin slime of filthy piss trickled from his trouser legs. But his chest rose and fell steadily. "No' dead."
Mary, her eyes wide under her lowering brow ridges, said, "Kill?"
Joshua grimaced. "Bad meat. Leave for th' bears."
"Yes," Mary said doubtfully. "Leave for th' bears."
They wiped their hands clean of the Zealot's filth on handfuls of leaves. Then they turned and pushed on, heading steadily north.
After a time, Joshua stepped cautiously into a clearing.
The trees here were battered and twisted. When he looked to the west, he saw how they had been smashed down and broken back to make a great gully through the forest.
And to the east, at the tip of this gully, was the seed from the sky.
He gazed at the blocky shape at the end of the huge trench, excitement warring with apprehension. It was a mound of black and white, half-concealed by smashed foliage. It was surrounded by bits of blue skin—or not skin; a bit of it fluttered against his leg, a membrane finer than any skin he had ever seen.
It was so strange, he could barely even make it out.
Mary, nervous, had stayed back in the fringe of the forest. " 'Ware," she said. "Zealots."
Joshua knew it was true. He could smell the smoke of their hearths, their burned meat. They were now very close to the Zealots' camp.
But the lure of the sky seed was irresistible. He began to work his way around the edge of the clearing, stepping over fallen tree trunks, shoving aside smashed branches, ready to duck back into the forest's green shadows.
The sky seed was big, bigger than any animal, perhaps as big as the hut where the people lived. He saw that the thing had fallen here after crashing through the trees, almost reaching the point where the forest gave out at the edge of the cliff itself.
But that was all the sense he could make of it.
He had no words to describe it, no experience against which to map it. Even the touch of it was unfamiliar: glossy black or white, the patches separated by clear straight lines, the soft surface neither hot nor cold, neither skin nor stone nor wood. It was difficult for him even to _see_ the thing. He would study some part of it—like the small neat puncture-holes on one part of its hide, surrounded by scorch-marks—but then his gaze would slide away from the strangeness, seeking some point of familiarity and finding none.
"Back," Mary hissed to Joshua.
He made out the telltale signs that Skinny-folk had been here: the narrow footmarks in the raw dirt, the remains of the burnt rolls of leaves they liked to carry in their mouths. The Zealots had indeed been here, too, inspecting the sky seed, just as he was.
But, despite the imminence of danger, he could not abandon this sky seed. It repelled him—yet it attracted him, like the carving on a Skinny-folk spear. Drawn close, driven away, he hovered.
He came to a sudden decision.
He bent and applied his shoulder to the blunt rear of the sky seed. It was lighter than it looked, and it ground forward through the dirt. But soon he was coming up against the resistance of the last battered trees at the cliff's edge.
"Joshua!" Mary hissed.
"Help push." And he applied himself again.
She tried to make him give up his self-appointed task, wheedling and plucking at his skins. But when she saw he wouldn't come away, she joined him at the back of the sky seed. She was not yet fully grown, but her strength was already immense, enough to drive the sky seed forward, crunching through the spindly cliff-edge trees.
With a screeching scrape, the sky seed pitched over the raw rock lip of the cliff and lurched out of sight. After a last tortured groan, silence fell.
## _M anekatopokanemahedo_
"Soon, something will appear in the sky," Babo said. "A satellite, like those of the outer planets. _Earth will have a Moon_ , for the first time in its history."
Manekato scratched her head. "How? By some gravitational deflection?"
"No. Like a Mapping, I think. But not a Mapping. The truth is nobody knows, Mane. But the Astrologers can see it is approaching, in the shivers of the starlight."
"It must be artificial, this moving of a Moon. A contrivance."
"Yes, of course. It is a deliberate act. But we do not know the agents or their motive."
Manekato thought through the implications. "There will be tides," she said. "Earthquakes. Great waves."
"Yes. And _that_ is the danger posed to our Farm, and some others."
Suddenly she was filled with hope. "Is that why I am here? Is it possible to avert this Moon—to save the Farm?"
"No," he said, sadly but firmly.
She pulled away from him. "You talked of my mission. What mission, if the Farm is doomed?"
"You must travel to the Moon," said Babo.
"Impossible," she snapped. "No Mapping has ever been attempted over such a distance."
"Nevertheless you must make it possible," Babo said. "You must use the resources of the Farm to achieve it."
"And if I reach the Moon?"
"Then you must find those responsible for sending this rogue here. You must make them remove it, and have them assure you it will not return." He forced a smile. "We are a species good at negotiation, Mane. The Lineages could not have survived otherwise. You are all but a matriarch, the matriarch of Poka Lineage. You will find a way. Go to the Moon, Mane—take this chance. I will be with you, if you wish. If you succeed, Poka will be granted new land. We have pledges..."
"And if I fail—or refuse?"
He stiffened. "Then our Lineage will die with us. Of course."
"Of course—"
There was a fizz of purple light, a stink of ozone. A Worker fell from the sky and landed in the center of the room. Semisentient, it raised a sketchy face and peered at them. Recognizing Manekato, it gave her the doleful news it had brought, its voice flat and unengaged.
Orphaned, brother and sister clung to each other as they wept.
## _R eid Malenfant_
After days of pressure from Malenfant, McCann agreed to lead them in an orderly expedition back to the crash site of the lander. Malenfant felt a vast relief, as if he was being let out of jail: at last, some progress.
First, McCann inspected them critically. "I'll have Julia fit you both with buckskin. One must go cannily. You'll stand out a mile in those sky-blue nursery rompers."
The buckskin gear turned out to be old and musty—presumably manufactured, with much labor, for deceased inhabitants of this place. And McCann loaned Malenfant and Nemoto calf-length leather boots, to keep out the snakes and the bugs. The boots were ill-fitting, and much worn. The gear was heavy, stiff and hot to wear, and its rough interior scratched Malenfant's skin. But it was substantial, feeling like a suit of armor, and was obscurely comforting.
McCann wore a suit of sewn skin and a Davy Crockett hat; he had a crossbow on his back, and a belt of flechettes over his shoulder. He looked capable, tough, and well-adapted.
Malenfant wrapped up his coverall and other bits of gear in a skin pack that he wore on his back. He insisted Nemoto do the same; he wanted to be sure they didn't have to return here if they got the chance to get away.
A party of six Hams was gathered in the courtyard. They were all squat, burly men. The Hams wore their peculiar wrappings of skin, tied in place by bits of thong or vegetable rope, not shaped or sewn. They carried weapons, spears, and clubs on loops of rope or tucked into their belts, and their broad elliptical heads were shaded by hats of woven grass.
One of them was Thomas, the man who had rescued Malenfant and Nemoto from the wild Runners in the first place.
Malenfant couldn't figure out why the Hams had gotten the lens to him (or come to that, how they knew he would be interested). Maybe they just like the story, Malenfant said, the guy who flies to another world in search of his wife. Just like the American taxpayer. Or maybe there are aspects of these quasi-people none of us will ever understand.
When Malenfant approached to thank him, Thomas shook his hand, an oddly delicate gesture he must have learned from the stranded English, taking care not to crush Malenfant's bones. But, when Malenfant questioned him away from the others, he would say nothing of where he had found Emma's lens.
Two Hams opened the gates of the stockade, and the little party formed up. McCann was to ride in a kind of litter—"What a Portugoose would call a _machila_ , I'm told." The litter, just a platform of wood, was to be borne by two Hams, and McCann had offered the same to Malenfant and Nemoto.
Malenfant had refused.
Nemoto had been skeptical. "You are sentimental, Malenfant. After a few hours you may long for a ride. And besides, the Hams are well capable of bearing our weight. They are treated well—"
"That's not the point."
"Survival is the point. What else?"
Anyhow, with the sun still climbing—with McCann's litter in the van, Malenfant and Nemoto walking in the center with Hams beside and behind them—the little party set off.
McCann said they would take a roundabout route to the lander. It would take longer, but would avoid the densest forest and so would be less problematic.
They walked through the forest. The air was laden with moisture and without a breath of wind. The sweat was soon dripping from Malenfant's scalp into his eyes, and his buckskin was clinging to his back as if glued there.
The Hams walked barefoot along a trail that was invisible to Malenfant, with their feet splayed at wide angles, making fast, short steps, almost delicate. Malenfant tried to keep up. But the brown sheets of dead leaves on top of wet mud made him slip, or he would walk into thorny lianas, or trip over the surface roots that splayed out from the boles of the largest trees. As the feet and legs of the Ham in front began to blur, he realized he was going to have to imitate the Ham's small movements, but he lost further ground as he tried to master the oddly precise mincing motions.
McCann walked alongside Malenfant, musing. "Hear how quiet it is. One does miss birdsong. Africa is full of birds, of course: parrots and plovers, kingfishers and skimmers. How sad a world without the song of birds, Malenfant."
Here was a canthium tree: a massive straight black trunk, branches spreading high above the palms. "Keep away from it," McCann said. "The flowers stink like corpses—to attract flies, you see, which carry its pollen. The presapients keep away from it. The trunk is covered in biting ants—" He froze, and held Malenfant's arm. "Look there. _An Elf_." He dropped to all fours and crawled forward, hiding behind a tree.
Malenfant followed suit. The two of them finished lying in cold mud, side by side, peering through a brush of greenery.
A man sat on a bough, a few feet off the ground—a dwarvish, naked, hairy man with a face like a chimp's, and no forehead to speak of. He had long legs like a human, long arms like an ape. He pulled twigs toward his face and bit off leaves, with thick, active lips. His face was black, his eyes brown, sheltered by a thick brow of bone. He moved slowly, thoughtfully.
A twig cracked.
The Elf stopped eating. He leaned forward, rocked from side to side to see better. He urinated, a stream of acrid piss that splashed to the floor not feet from Malenfant's face.
Then he turned away and called. _"Oo-hah!"_
Suddenly there were more of them, more Elves, shadowy figures with glinting eyes and empty hands. They had black faces and palms and soles. If they had crouched like chimpanzees it would have been okay, but they didn't; they stood eerily upright, as if their bodies had been distorted in some hideous lab. They were _wrong_ , and Malenfant shivered.
"There are ways to trap them," McCann whispered. "Though their more robust cousins the Nutcrackers provide better meat. You hunt with special spears, twelve feet long. Then you goad the Nutcracker man, until he charges onto your spear point..."
The first Elf man stood up straight on his bough. He opened his mouth wide, revealing pink gums and impressive canines, and let out a series of short, piercing barks. He slapped the tree trunk and rattled a branch.
The others joined in, whooping with rage. Their hair was suddenly erect, which made them look twice the size, and they stamped and shook branches in a frenzy. It was quite a display, Malenfant thought, a mass of noise and movement.
Then the man in the tree turned, bent over and let out an explosion of feces that showered over Malenfant and McCann.
Malenfant brushed gloopy shit off his head. "Jesus. What a situation."
McCann was laughing.
Now McCann's Hams stood up. They yelled and banged their spears together, or against logs and tree trunks.
The Elves turned and ran, melting into the green shadows as fast as they had appeared.
Malenfant was relieved when they broke out of the forest, just as McCann had promised, and he found himself walking through a more open country, a kind of parkland of grass and scattered clumps of trees.
Nemoto trudged sourly beside him, her small face hidden by a broad straw hat.
There were herbs in the grass, and when they were crushed by bare Neandertal feet they sent up a rich aroma. The sun was strong on Malenfant's face, and the blue Earth rode high in the sky. Malenfant felt lifted, exhilarated—even giddy, he thought, anoxic perhaps, and he made sure he kept his breathing deep and even, making the most of the thin air.
McCann noticed Malenfant's mood. With a touch of the stubby whip he called a _sjambok_ , he directed his Ham bearers to carry him closer to Malenfant. "Quite a day, isn't it, Malenfant? You know, I believe that with a knight's move of that mopani tree over _here_ one might take that kopje, with the thicket of wild banana, over _there_."
Malenfant forced a laugh. "Remember, I'm a checkers man."
McCann was clutching a battered Gladstone bag on his lap, from which he extracted water and ointments to dab on his face, neck, and wrists. He looked sideways at Malenfant, as if apologetically. "I fear I may have come across as something less than a man to you, on our first meetings."
"Not at all."
"It's just that one is so desperate for company. But you mustn't think that I am protesting my lot. I draw strength from the teachings of my father—I grew up in a kirk on the Scottish borders—which took a grip on my mind from early days. My father made me a fatalist in creed: Man is but a playing-piece in the hands of the Maker. Chess again, eh? And so it was foreordained that I should be brought to this distant shore. But I admit to a great deal of pleasure in my new home on a day like today. Much of it is familiar. In my time here I've spotted wildebeest, kudu, impala. There are few birds in flight, but you'll find flightless, clucking versions of quail, partridge, pheasant..."
"But it isn't your true home," Malenfant said gently. "Nor mine. It's not even from the right universe. Just as it isn't home for these Hams, is it?"
McCann eyed him sharply. "You've been talking to the fragrant Julia—their legend of the Gray Earth, the place in the sky from which they stumbled. Yes?" He laughed. "Well, it might even be true. Perhaps a party of bar-bars did fall through a shining portal, just as you say your wife did. But it was a blooming long time ago, Malenfant.
"Listen. Once upon a time old Crawford got it into his head that there might be something of value in the ground here—gold, diamonds, even hidden treasure of obscure origin, perhaps laid down by some race of supermen. And he went digging—especially in the hearths and caves of the bar-bars. He had to turf out a few of them to do that, for they will cling to their domiciles. He found no treasure. But what he did find was more bar-bars, or anyhow traces of them, their buried bones mixed in with those peculiar knobkerries and assegais they favor in the wild. There was layer upon layer of bone, said old Crawford, in every place he dug.
"Well, the meaning is obvious. These bar-bars have endured a long stretch on this exotic little world: They must surely have been here for hundreds of generations, thousands of years, or more. And in all that time they have clung to their dreams of home." He considered Malenfant. "You may think I am harsh with the bar-bars, Malenfant, or uncaring. I am not. Inferior they may be. But what memory lies buried in those deep skulls of theirs!—don't you think?"
The country began to rise. The little party grew strung out. The grass grew thinner, the underlying crimson soil more densely packed.
They reached the crest of a ridge and took a break. The ground was hard-packed here, covered thinly by bracken and little bushes like hazels. The party, drinking water from a pannikin handed around by a Ham, was surrounded by a thin, subsiding cloud of red dust.
Malenfant stepped forward. The ground fell away before him, and he saw that this ridge curved around, making a neat circle. It was a bowl of greenery. A few improbably tall trees sprouted, but much of the basin was covered by grass that was littered with color, the yellow and white of marigolds and lilies. Pools glistened on the uneven floor, ringed by lush primeval-looking ferns.
It was a crater, a classic impact formation a couple of miles across. Standing here, Malenfant heard distant calls and hoots. They were the cries of hominids, cousins to mankind, patrolling this forested crater. It was a startling, uplifting, utterly alien prospect.
McCann was standing beside him. "Here we stand, men born on different worlds, confronting a third. Do you know your Plutarch, Malenfant? 'Alexander wept when he heard from Anaxarchus that there was an infinite number of worlds... "Do you not think it lamentable that with such a vast multitude of worlds, we have not yet conquered one?" ' "He pointed with imperious confidence into the bowl of the crater. "There lies our _Redoubtable_ —or at least her corpse. Come, you men."
Brushing a walking-stick before him, he strode off down the flank of the crater. Malenfant and Nemoto, and the Hams with their litter, hurried to follow.
Malenfant came first on a rib of metal, heavily corroded, that arched into the air above him. Its smooth circular shape was a startling contrast to the fractal profusion of the greenery all around. He stepped under the rib, onto twisted and rusted metallic remnants that groaned under his weight. He found he was in a long cylindrical chamber, its walls extensively broken and corroded, open to the sky. When it was intact this tank must have been six or seven yards in diameter.
Thorn bushes pushed through the base of the cylinder, and creepers curled over its sides; above, a thick canopy turned the light dim, moist, and green. The ship had been a long time dead, and the vegetation had grown over and through it, concealing its remains.
McCann walked in alongside him, followed by Nemoto. The Hams lingered on the fringe of the deeper forest, leaning on the litter and sipping water. Thomas kept an eye on McCann, but his gaze slid over the lines of the ship, as if it was a thing of mists and shadows, not really there.
"This was the propellant tank," McCann said. He pointed with his stick. "You can see the bulkheads to either end, or what's left of 'em." McCann pushed on through mazes of piping and cables. Malenfant and Nemoto followed more cautiously, taking care of the sharp edges of twisted metal under their feet.
McCann's figure was stocky and competent, and swathed in his treated animal skins he looked somehow right against the background of the fallen, smashed-open ship; Malenfant wondered how often he visited this relic of home.
They passed through a ripped-open dome into another cylindrical tank. "Here we stored oxidants. Though of course much of the oxidant was drawn from the air."
"A ramjet," Malenfant said to Nemoto.
McCann came to a tangle of what looked like crude electrical equipment, valves and relays, so badly corroded it was an inseparable mass. "Control gear," he said. "For the pumps and valves and so forth." They passed through a more solid bulkhead, supported by heavy ribs, and arrived in what appeared to have been habitable quarters. There had been several decks, separated by two or three yards—but now tipped over, so the floors and ceilings had become walls. A fireman's pole ran along the length of this section, passing neatly through holes in the floors, horizontal now.
McCann pointed out highlights with his stick. "Stores." Malenfant saw the crumpled remnants of bulky machines, perhaps recycling and cleansing devices for air and water, and refrigerated stores for food, but damaged by fire and gutted; they lay in the dark of the rocket's hull like fetuses in unhatched dinosaur eggs. "Infirmary, galley, sleeping quarters and such." Little was left here save a bare frame that might have held bunk beds, a heavy table bolted to the tilted-over floor and fitted with leather restraints, perhaps intended for surgery, and the nubs of pipes and flues showed where galley equipment had been ripped out or salvaged.
"And the bridge." At the nub of the ship, this had been lined with polished oak panels, now scuffed, broken, and covered by lichen and moss. Brass portholes bore only fragments of the thick glass that had once lined them. There were heavy couch frames bolted to the floor, long since stripped of their soft coverings. Malenfant could make little of what must once have been instrument panels; now they were just rectangular hollows in the fascia, though he glimpsed tangles of wires behind.
McCann saw him looking. "Once we realized the old lady wasn't serviceable we stripped out what we could. We built a succession of radio transmitters and heliographs. We got replies, of course, as long as the Earth—I mean, _my_ Earth—still hovered in the sky. That, and promises of rescue, which assurances I have no doubt would have been fulfilled. We kept on trying even after Earth had gone, until the last generator seized up. Powered by a bicycling Runner, incidentally."
"I'm sorry," Malenfant said. "She must have been a beautiful ship."
"Oh, she was. Help me." Leaning on Malenfant's arm, he clambered stiffly up the hull wall, using gaping porthole sockets as hand- and footholds.
Malenfant followed him. Soon the two of them stood side by side on the outer hull of the habitable section, surrounded by gashes and treacherous-looking rents. But McCann was confident in his step.
From here Malenfant could make out the full sweep of the ship's length, a slim spear that must have been two hundred yards long. Its lovely back was broken; and green tendrils clutched at the ship, as if pulling it into the belly of the Moon that had killed it. But still a solitary fin poked out of the greenery, crumpled but defiant. The fin bore a faded roundel that reminded Malenfant of the logo of the Royal Air Force.
The Ham man, Thomas, walked beside the ship close to McCann, keeping his eyes on the Englishman.
"He is loyal," said Malenfant. "He looks out for you all the time."
"He knows I have done my best to improve the lot of his people."
Even if it didn't need improving, Malenfant thought. "But he seems to be having trouble looking at the rocket."
"The bar-bar mind is rigid, Malenfant. Conservative beyond imagining, they are utterly resistant to the new. At the beginning we had a devil of a battle to keep them from destroying our gear—even when tamed, a bar-bar still harbors destructive tendencies."
Malenfant recalled the fate of his shoulder camera. He said, "That almost seems superstitious."
"Oh, not that. There is no superstition among the bar-bars: There is no magic in their world, no sense of the numinous. To them the surface of the world is everything; they do not see hidden meanings, nor seek deeper explanations."
"They have no gods, then."
"Nor can they even conceive of the possibility." McCann smiled. "And what a loss that is. I am sure they are well spared propitiations to the savage and bloody gods of the jungle. But they cannot know the Mercy of the one true God. You understand, it is not merely that they do not know Him—they _cannot_. And without God, there is no order to their lives, no meaning—save what _we_ provide." He tapped Malenfant on the chest with the worn head of his walking stick. "I know you are uncomfortable with our relationship to these barbarians, Malenfant. I see it in your eyes. I've seen it in Africa, when men of conscience go among the Kaffirs there. But can't you see it is our duty to provide them with a Johannen way of life—even if they can't comprehend its meaning?—just as the philosophers and theologians have been proposing since the first steel clippers found these bar-bars' cousins running wild in the New World."
Malenfant studied Thomas's face, but could see no hint of reaction to McCann's sermonizing.
McCann began to talk briskly about the horsepower generated by the "Darwin engines" that had once powered the ship. "I know your little tub came gliding in like a bat. We applied a little more brute force. In the last stages of its descent the _Redoubtable_ was intended to land upright on Earth or Moon, standing on its rocket exhaust. And it should have taken off in the same manner."
"Direct ascent," Malenfant said. It was a mode that had been considered for Apollo's lunar landings, a whole ship traversing back and forth between Earth and Moon. But aside from the greater expense compared to the final Lunar Module design, landing such a giant ship with rockets would have posed stability problems, like an ICBM landing on its tail.
From McCann's descriptions, it sounded as if that had been the downfall of the _Redoubtable_.
"She was a veteran," McCann said softly. "She had done the Earth-Moon round trip a dozen times or more. But now we were dealing with a new Moon, you see. Well, we hastily modified her for her new mission. She landed on her fins well enough on the fields at Cosford, but this crater floor is no tarmacadam strip in Shropshire. She was top-heavy, and—" He fell silent, studying the ruined carcass of the ship. "I was navigator; I must share responsibility for the disaster that followed. Most of us got out, by the Mercy of God." He clapped Malenfant on the back, forcing a laugh. "And since then our lovely ship has been scavenged to make cooking pots."
"Erasmus Darwin," Nemoto called.
Malenfant looked down.
Nemoto was standing in the ruins of the habitable compartment, peering up at him. Her face was like a brown coin in the gloom. "The Darwin drive," she said. "Grandfather of Charles, who is probably the Darwin you're thinking of, Malenfant. In the 1770s he sketched a simple liquid-fuel rocket engine, along with a ramjet. In our world, the sketch languished unnoticed in his notebooks until the 1990s. But in Mr. McCann's world—"
McCann nodded. "The design was the seed around which a new generation of rockets and missiles grew. After the pioneering work of Congreve, the Brunels, father and son, became involved in the development of craft capable of carrying heavy loads into the atmosphere. The first dummy load was orbited around the Earth before the death of Victoria, Empress of the Moon, and the first manned flight beyond the atmosphere was launched from Ceylon in 1920... Ah, but none of this happened in your world, did it, Malenfant? It is a divergence of history. In your world Darwin was ignored or forgotten, his ideas no doubt rediscovered by some other, more vigorous nation."
"Something like that."
Nemoto moved on, working her way through the ship's gloomy interior.
McCann watched her, then leaned closer to Malenfant. "Always watching, thinking, _recording_ , your little Oriental friend—eh, Malenfant?"
"That's her way," Malenfant said cautiously. "And it's our mission. Part of it, anyhow."
"And quite the fount of knowledge about obscure British philosophers two centuries dead." McCann's eyes narrowed. "I have observed the gadget she carries."
Malenfant saw no point in lying. "It's called a softscreen."
"Its working is no doubt beyond my comprehension, but its purpose is clear enough. It is a repository of knowledge, from which Madam Nemoto sips as she requires. I am a man of this dismal jungle now, Malenfant, but you need not think me a fool."
"Take it easy, McCann."
McCann frowned, as if decoding the colloquialism. "Without my shelter you would both surely be 'taking it easy' beneath the crimson dust by now. Remember that." When Malenfant did not answer, McCann clapped him on the shoulder again. "Enough of one beached vessel; let us seek another. Come." McCann began to clamber down to the ground, into the helpful arms of the Ham who served him.
It took another two hours to reach the clearing dug out by the lander on its way down.
The lander was gone.
This was the place he remembered: the Gagarin avenue cut through the trees, the scattered bushes and branches—and even bits of blue parafoil, grimy, damp, still clinging to the damaged foliage. But the lander was gone.
McCann stalked over the grass, inspecting ripped-up bushes, scattered trees. "You're sure this is the place?"
"It can't be."
Nemoto approached him. "Malenfant, you are not a man who has trouble remembering where he parked the car."
Malenfant wanted to believe the lander was sitting someplace else, where it had fallen, as battered and crumpled and precious as when he and Nemoto had so foolishly become parted from it—a key part of the technological ladder that would take him, and Emma, home. But there could be no doubt.
" _We're stranded_ , Nemoto," he blurted. "As stranded as these damn English."
"Perhaps we always were," she said evenly.
He hitched his pack of tied-up skin, containing all his belongings, all that was left of Earth. "We're a pretty pathetic expeditionary force."
She shrugged. "We still have the most important tools: our minds, and our hands, and our knowledge." She eyed him. "What do you intend to do now?"
"Let's get out of here. We have to find the lander. There's nothing more we can achieve with these English. I hate to be a bad guest, but I'm not sure how well McCann will take our leaving."
"Not well, I fear," Nemoto said dryly. And she stepped back.
A hand clamped on Malenfant's arm. It was a Ham, not Thomas.
McCann came walking up, leaning on his stick, his broad face red and grim. "Thank you, Madam Nemoto," he said. "He has behaved just as you predicted."
Malenfant glared at Nemoto, disbelieving. "You betrayed me. You warned him I'd try something."
"You are very predictable, Malenfant." She sighed, impatient, her face expressionless. "You should not make the mistake of believing we share the same agenda. This new Moon, this Red Moon, is the greatest mystery in recorded history—a mystery that deepens with every day that passes, everything we learn. Unless we discover the truth behind it, we will have accomplished nothing."
"And you believe you can achieve that by staying here, with McCann?"
"We need a base, Malenfant. We need resources. We can't spend our whole lives looking over our shoulders for the next stone axe to fall, or grubbing around in the forest for food. These British have all that."
_"And what of Emma?"_
Nemoto said nothing, but McCann said smoothly, "Our scouts and hunters range far and wide, Malenfant. If she is here, we will find her for you."
If your Ham scouts tell you everything they see, Malenfant thought. He fingered the little lens in his pocket.
"Let's look at the matter in a sensible light," McCann said now. "I know you think little of me, Malenfant. But once again I assure you I am not a fool. I desire more than a chess partner; I desire escape from this place—what man wouldn't? Now you have fallen from the sky into my lap, and only a fool would let you go, for surely your _Americans_ will come looking for you from that blue Earth of yours. And when they do, they will find me."
"My world isn't your world," Malenfant snarled.
"But my world is lost," McCann said wistfully. "And I know you have an England. Perhaps I will find a place there." His face hardened, and Malenfant perceived a new toughness. This was, Malenfant remembered, a representative of a breed who had carved out a global empire—and on a much more hostile planet than Earth. "Providence has given me my chance and I must take it. I believe that in keeping you now, in following the promptings of my own infallible heart, I see the workings of Omnipotence. Is this moral arrogance? But without such beliefs man would never have left the trees and the caves, and remained like our presapient and pongid cousins." He glanced at Nemoto. "As for your companion's slight treachery—perhaps she is destined to betray you, over and over, on all Anaxarchus's infinity of worlds. What do you think?" And he brayed laughter.
The little column formed up for the homeward journey. The big Ham called Thomas took his place beside Malenfant.
And he winked broadly.
## _E mma Stoney_
A day after leaving the first troupe, Emma found another a group of Hams, women and a few infants foraging for berries and fruit. They had regarded her blankly, but then, seeing she was no threat—and, as not _one of them_ , of no conceivable interest—they had turned away and continued their gathering.
Emma waited patiently until they were done. Then she followed them back to their encampment.
She stayed there a couple of days, and then moved on, seeking another troupe.
And then on again.
Hams were basically the same, wherever she found them. Their tool-making, for instance. Though each group varied its kit a little according to circumstances, like the availability of different types of stone—and perhaps, she speculated, some slight cultural tradition—still, if something was not in their toolmaking repertoire, which was evidently very ancient and fixed, no Ham was interested.
They didn't even talk about their toolmaking, even while they jabbered endlessly about their intricate social lives. It was as if they were conscious while they were interacting with each other, but not while they were making tools, or even hunting.
After a time Emma began to get used to it. She reasoned that _she_ did many things she wasn't aware of, like breathing, and keeping her heart pumping. And she could think of times when she had performed quite complex tasks requiring skill, judgment, and the focus on a specific goal without knowing about it—such as driving all the way to work with her mind on some stunt of Malenfant's, only to "wake up" when she found herself in the car lot. Or she thought of her father, able to carve fine furniture from wood in his hobby workshop, but never able to tell her how it was done; all he could do was show her.
With the Hams, that circle of unawareness spread a little further, that was all. Or maybe it was just that you could get used to anything, given time.
Anyhow it didn't matter. She wasn't here to run experiments in hominid cognition. It was enough that she was able to use her fish and rabbits and other hunting produce as subtle bribes to gain favor—or at least as a hedge against exclusion.
Thus she worked her way through the forest, moving from one Ham group to the next, using them as stepping stones of comparative safety, one way or another travelling steadily east, day after day, seeking Malenfant.
But sometimes she glimpsed faces in the forest, just at the limit of her vision: hominid faces, watchful, like no species she had yet encountered. It seemed she had barely glimpsed the extent of her kin, on this strange world.
## _R eid Malenfant_
The details of the regime that would govern Malenfant's life coalesced with startling speed and efficiency—such speed, in fact, that Malenfant wondered who else McCann or the others had had cause to imprison.
Malenfant was free to come and go, within the stockade. But there was always a burly male Ham at his elbow, even sleeping outside his hut during the night.
He took to prowling around the perimeter fence. It was tall, and its ferocious spikes were daubed with a sticky, tarlike substance. For the first time it struck him that the fence was just as effective at keeping him in as keeping out the undesirables of the wilds beyond. And anyhow every time Malenfant tried to approach the fence too closely, he was immobilized by his Ham guard—as simple as that; one of those massive hands would clamp on his shoulder or elbow or even his head, exerting a strength he couldn't hope to match.
He tested his cage in other ways.
He spoke to Thomas, asking for his help. But Thomas would say nothing, giving no hint that he was prepared to follow up on that reassuring wink in the forest.
One night Malenfant tried climbing out of his hut's window. But though it was unglazed the window was small and high. By the time he had dropped clumsily to the ground his Ham keeper was standing over him, silhouetted by blue Earthlight, solid and silent as a rock.
He considered making other protests—going on a hunger strike, maybe. But he sensed McCann might simply let him starve; the steel he had glimpsed in the soul of this other-world Brit did not encourage him to seek weakness or pity. Alternatively McCann might have his Ham servants force-feed Malenfant, not a prospect he relished, since the Hams were muscled a little too heavily to be good nurses.
Anyhow he needed to build his strength for the days to come, and the search for Emma he confidently expected to be progressing sooner rather than later.
So, after a couple of days, Malenfant began to engage with McCann once more: eating with him, even walking around the compound, conversing. It was a peculiar arrangement, in which both of them clearly knew their relative positions of power and yet did not speak of it, as if they were engaged in some formal game.
Malenfant tried to find out as much as he could about this world from McCann. But the British had done little exploring more than a few day's travel away from their stockade. Their main business here had, after all, been ensuring their survival. And McCann's mind seemed peculiarly closed to Malenfant. The purpose of McCann's original mission had not been exploration, and still less science, but economic and political gain for his Empire; he was more like a prospector than a surveyor. But sometimes he spoke again of the deeper mission he felt he had taken on: to bring the word of his God, and his Christ-figure John, to the barbarian hominids of the Red Moon.
McCann was a man with a head full of agendas. It seemed to Malenfant he was barely able to see the Red Moon and its exotic inhabitants for what they were—just as the Hams had seemed unable to look directly at the wreckage of the _Redoubtable_.
Maybe every hominid species had such blind spots, mused Malenfant. He wondered what his own were.
For his part McCann pressed Malenfant about rescue.
Malenfant tried to describe the politics and economics of his home world. He knew it was extremely unlikely that the will to mount a further mission could be assembled on tide-ravaged Earth—even though the NASA support teams knew where the lander had come down, and had received those few minutes of footage to show he and Nemoto had survived, at least for a time.
McCann showed Malenfant the transceiver gear he and his companions had scavenged from the wreck of the _Redoubtable_. It was a formidable array of antique-type parts, huge glass valves and mica capacitors and big clattering relays. For years the British had nursed it, for instance keeping it continually powered to save the valves from the thermal shock of being switched on and off. But at last too many of the valves had failed, and other parts were corroded and damaged from prolonged exposure to the damp air. Malenfant tinkered with the gear, but he had less of an idea than McCann how to fix it.
In his own mind Malenfant's primary mission remained clear: to find Emma, and get the hell off this Moon. If he could help McCann on the way, fine; if Nemoto wanted to come home or stay here, it was up to her. But they were side issues. To Malenfant, only home and Emma mattered.
So they worked through their days. But as time passed it seemed to Malenfant that McCann grew steadily more anxious. Periodically he would peer up into the sky, as if seeking to reassure himself that Earth was still there.
And Malenfant barely saw Nemoto.
One morning, maybe a week into his captivity, he was woken as usual by Julia, with her wooden bowl of hot water and a fresh stone blade for him to shave. Dressed in her blouse and long skirt of sewn skin, with her muscled body moving powerfully, she looked absurd, like a chimpanzee in a child's dress.
She picked up his covered slop bucket, curtsied at him—"Baas"—and made to leave.
"Help me," Malenfant blurted.
She stopped by the door. Malenfant could see the shadow of a burly Ham male outside the door.
"Baas?"
"You know I'm being kept here against my will—umm, Boss McCann won't let me go. You helped me before. You gave me the lens—the clear stone. You know it came from Emma. I want to get out of here and find her, Julia. I don't want to hurt anybody, not Boss McCann, not anybody. I just want to get to Emma."
She shrugged, her mountainous shoulders rippling. "Breakfas'," she said.
Frustrated, he snapped, "Why do you stay here? Any one of you could take on McCann and his cronies. Even their crossbows couldn't hold you back if you put your mind to it."
She looked at him reproachfully. "Tired ol' men," she said, as if that was explanation enough. Then she turned and walked out, the slop pail carried effortlessly in one huge hand.
## _M anekatopokanemahedo_
The great Mapping, across a distance unprecedented in recorded history, could be regarded as a technological triumph. But to Manekato it had been like the working out of an intricate mathematical theorem, a theorem that proved the identity of certain points of space and time with certain other points. The fact that those other points were placed close to the surface of a world which had not even existed as the proof was developed scarcely added to the complexity of the procedure. And once the proof was established, the journey itself would be a mere corollary, of little interest save as an exercise for the young.
The proof had not been trivial, but it had not been overdemanding. Most adults, with a little effort, could have achieved the same result. Manekato had worked at the Mapping with part of her mind, with the rest consumed by her grief for her mother and her concerns over her own future.
On Mane's Earth, anybody could develop a space program in their spare time.
With her brother Babo and the woman who called herself Without-Name, Manekato stood on the crushed bones of her ancestors. The eternal Wind blasted over the rock, unnoticed. Above her hovered a great rippling lens of star-filled sky, as if a hole had been cut in the clouds: Thanks to simple Mapping techniques it was as if she were suspended in orbit, far above the clouds of Earth. But the three of them barely glanced up; it was a minor, uninteresting miracle.
This eroded volcanic core, once the heart of the Farm, was bare now. After her mother had died, Manekato had ordered the deletion of the great House. The walls of Adjusted Space had disappeared like a bursting bubble, as if fifty millennia of sturdy existence had been but a dream. Manekato had welcomed the simple geologic clarity of the mountain's eroded summit: she knew she could never live in the House, and it served no purpose save to preserve memories of unhappiness.
But she had retained the pit containing the ashes of her grandmothers, and to it she had added the last remains of Nekatopo.
Without-Name stalked around the perimeter of the ash pit, her knuckles pressing disrespectfully into the sealed-in dirt, leaving impressions of her hands and feet. A Worker followed this ill-mannered guest, restoring the pit's smoothness. "Destroy the pit," Without-Name told Manekato. "Fill it in. Delete it. It serves no purpose."
"The pit is the memory of my Lineage," said Manekato evenly.
Without-Name bared her teeth and growled. "This pit is not a memory. It is a hole filled with dust."
Babo protested, "The practice of adding oneself to the Farm's ground at the end of one's life is as old as our species. It derives from the sensible desire to use every resource to enrich the ground for one's descendants. Today the practice is symbolic, of course, but—"
"Symbolism. Pah! Symbolism is for fools."
Babo looked shocked.
If Without-Name enjoyed goading Manekato, she positively relished taunting Babo. "Only children chatter of an afterlife. We are nothing but transient dissipative structures. In your cherishing the bone dust of the dead you are seeking to deny the basic truth of existence: that when we die, we are gone."
Babo said defiantly, "I have visited the Rano Lineage and I saw the pit of your ancestors. You are a hypocrite. You say one thing and practice the other."
She raised herself to her hind feet and towered over him. She wore her body hair plucked clean in great patches over her body, and where hair remained it had been stiffened into great bristling spikes. It was a fashion from the other side of the world that made her seem oddly savage to Manekato. _"Not any more,"_ she hissed. "I salute death. I salute the cleansing it brings. There is only life—all that matters is the here and the now—and what can be achieved in the moment."
Manekato held back her emotions.
This Without-Name's preferred diminutive actually was—had been—Renemenagota. But she insisted she had abandoned her true name. "My land is to be destroyed," she had said. "And so is my Lineage. What purpose does a fossilized name serve?" Even the contradiction in her position—for _Without-Name_ was itself a name, of course, so that she was trapped in an oxymoron—seemed only to please her perversely. Manekato knew she must work with this woman, who was a refugee as she was, to study the rogue Moon and its fabricators; that had been the directive of the Astrologers. But Manekato felt that she had been the target of Without-Name's bitterness and discourtesy from the moment they had been thrown together...
There was a dazzling electric-blue flash, gone in an instant.
A shift in the Wind touched Manekato's face. She looked into the tunnel of stars.
"If you embrace experience," she said, "then you must embrace _that_."
Without-Name lifted her head awkwardly, and fell forward onto her knuckles.
Babo was already gazing at the sky, open-mouthed. Even the Workers were backing away, small visual sensors protruding from their hides, peering up at the dangerous sky.
Suddenly the Red Moon swam there, complete, huge.
## _R eid Malenfant_
Nemoto said in a monotone, "We are dealing with multiple universes. That much is clear. We have seen for ourselves multiple Moons. And we have hints of multiple Earths. The Earth of Hugh McCann is clearly quite different from our Earth—even if his history is interestingly convergent with ours. And the Hams talk of a Gray Earth, a third place where conditions may be different again..."
In the hut Malenfant had come to think of as the dining hall, Nemoto and Malenfant faced each other at either end of the long table. The table's wooden surface, polished to darkness by decades of use, was bare. An elderly Ham woman was preparing lunch.
It had taken days before Malenfant had been able to face Nemoto, such was his anger at her betrayal. But she was his only companion from home, and if he was ever going to get out of here he might need her help. As for Nemoto, it was as if the incident of the betrayal had simply been a step in some grand plan, which any rational person would accept as justified.
But she was changing, Malenfant saw: becoming more withdrawn, hollow-eyed—dangerously detached from the texture of the world around her, obsessed instead with huge ideas of origins and destinies.
So Malenfant listened coldly, as Nemoto described alternate realities.
"Malenfant, perhaps there are a cluster of alternate universes with identical histories up to the moment of some key event in the evolution of humanity—and differing after that only in the details of that event, and its consequences." Nemoto waved her hands vaguely, as if trying to indicate three-dimensional space around her. "Imagine the possible universes arrayed around us in a kind of probability space, Malenfant. Do you see that universes differing _only_ in the details of the evolution of mankind must somehow be _close_ to ours in that graph?"
"And you're saying this is what we're experiencing—a crossover between possible universes? Well, maybe. But it's just talk. What I don't see is how you can hop from one cosmos to the next."
Nemoto smiled coldly. "I do not know how that is possible, Malenfant. And what is more important is that I do not know _why_ anybody should wish to make it happen."
" _Why_... You think all of this is deliberate—somehow artificial?"
"Your Wheel in Africa looked artificial to me, Malenfant. Perhaps the Hams' Old Ones, if they exist, will be able to tell us what they intended."
"And you're going to ask them, I suppose."
"If they exist. If I can find them. What else is there to do? Malenfant, there is something else. I have raised with McCann the question of whether other life forms exist beyond the Earth _—his_ Earth, I mean. His scientists have looked for evidence, as ours have. They have found none. Philosophers there have propounded something similar to our Fermi Paradox to crystallize this observation."
"Why is this important?"
"I don't know yet. But it does appear odd that such a profound contradiction is to be found in both universes..."
Light flickered, startlingly blue, beyond the door frame. Malenfant gasped. The color had tugged at his heart—for it was the color of the flash from within the Wheel that had consumed Emma.
They hurried outside. There was something in the sky.
## _M anekatopokanemahedo_
In her first stunned glance Manekato made out a single vast continent, scorched red, and a blue-gray ocean from which the sun cast a single blunt highlight. The disk, almost full, was surrounded by a thin layer of blurred softness. An atmosphere, then. But no lights shone in the darkened, shadowed crescent.
The Wind buffeted Manekato, turbulent, suddenly uneven. Already it begins, she thought.
Small Workers, no larger than insects, hovered around Babo's head, defying the shifting breeze; she saw their light play over his face, dense with information. "Its gross parameters are as we anticipated," he said. "A Moon, a world, two-thirds of Earth's diameter, a quarter of its mass. It has an atmosphere—"
"It is not Farmed," Without-Name hissed. "Your jabber of numbers is meaningless, you fool. Look at it: _It is not Farmed_. This Moon is primordial."
Without-Name was right. Even without magnification Manekato could see great expanses where nothing lived: that ugly red scar of a continent, the naked oceans, those crude caps of ice. It was a world of waste, of unawakened resources.
Wild.
"Wild, yes," growled Without-Name. "Consider the comparison with our Earth. For two million years we have cherished every atom. We have carefully sustained the diversity of species. We have even sacrificed ourselves—billions of years of lost lives—refusing longevity in order to maintain the balance of the world."
Mane murmured, "An ecology consisting of a single species would not be sustainable."
Without-Name laughed. "You quote childish slogans. Think, Manekato! Our species has been shaped, even as we have shaped our world. But _nothing_ about that ugly Moon has been managed. We will have no place. We will have to fight to achieve our purposes, perhaps even to survive."
Mane was troubled by that perception, though she acknowledged it might contain a grain of truth.
"But," Babo said, an edge in his voice, "the Red Moon cannot be primordial—it must contain mind— _for it would not be here otherwise_."
Yes, Mane thought. Yes. And for that she was afraid of this monstrous Moon. It was a deep fear, of a type she had never suffered before, a fear suffused by a sense of powerlessness. She had to search deep into the recesses of her memory, poring through the most ancient roots of the million-year-old language with which all children were born, to find an ancient, obsolete word that suited what she felt:
_Superstition_.
Babo rattled more statistics of the Moon's composition, describing a ball of silicate rock and a small iron core. But as his courage grew his thinking seemed to clear. _"Earth,"_ he said. "That wandering Moon is made of the same material as Earth's outer layers. How can that be?"
The three of them began to talk rapidly, their minds developing and sharing hypotheses.
"Given the identity of substances this body cannot have formed elsewhere in the Solar System." "Could it have budded off of Earth while the planet was accreting from the primordial cloud of dust and ice?" "No, for then its proportions should resemble Earth's global composition, and this body shows a deficiency of iron and other heavy elements. It is more like a piece of the Earth's mantle, its outer layers, ripped up and wadded together and thrown into the sky." "Then an Earth must have formed, differentiated so the iron-rich rocks sank to the core, before the material to assemble this Moon was detached from the outer layers. But how would it happen?" "A vast volcanic event? But surely that would not be sufficiently violent—" "A collision. A rogue planetesimal, a giant, or even a planet. Such a collision might cause a splash of ejecta which could accrete into this Moon..."
Within seconds, then, they had unravelled the mystery of the Moon's origin, a deduction that had taken humans two centuries of geological science.
All around the Earth, other witnesses must be coming to a similar conclusion, and Manekato imagined a growing consensus of understanding whispering in Babo's ear.
"But," Manekato said, "if this Red Moon was born from Earth, it was not _our_ Earth."
"No," Babo said somberly. "For our Earth never suffered a catastrophic collision of that magnitude. We would see the results today, for example in the composition of the planet's core. And if our world had enjoyed the company of such a Moon everything would have been different in its evolution: Much of the primordial atmosphere would have been stripped off in the collision, leaving thinner air less rich in carbon dioxide; there would have been many subtle effects on tides and the world's spin..."
"On such a world," Manekato said, "one would not need a Mapping to see the stars. And in such a sky a Moon like this would ride. But such is not our world."
"Not our universe," said Without-Name bluntly. "Tell me then, Babo: what do your Astrologers have to say of a power which can Map a Moon, not merely from planet to planet, but _between universes_?"
"They have little to say," he said evenly. "That is why we must go there... There is something more." He uttered a soft command to his Workers.
A new Mapping was made, showing them a vision from a large Farm that straddled the equator of the planet.
A giant blue circle, hovering above the ground, was sweeping over the Farm's cultivated ground, upright and improbably tall. People stood and watched as it passed. Workers backed away before it. Children ran alongside it, laughing, levering themselves forward on their knuckles in their excitement.
And there were people falling out of the circle's empty disk.
No, not people, Manekato saw: _like_ people, naked hominids, some tall and hairless, some short and squat and covered in fine black hair. They flopped and gasped for breath like stranded fish, and their flimsy bodies were buffeted this way and that by the Wind.
"What does it mean, Babo?"
"One can predict the broad outline of events. But chaos is in the detail..." He waved his hand, banishing the image.
A gust of Wind howled across the bare, eroded plateau, powerful enough to make Manekato stagger.
Babo stepped forward. "It is time."
Manekato and Without-Name took his hands and each other's, so the three of them were locked together in a ring.
At the last moment Manekato asked, "Must it be so?"
Babo shrugged regretfully. "The predictions are exact, Mane. The focusing effect of the shoreline's shape here, the gradient of the ocean floor, the precise positioning of the new Moon in the sky: All of these have conspired to doom our Farm, and the Poka line with it."
Without-Name tipped back her head and laughed, the spikes that covered her body bristling and twisting. "And for all our vaunted power we can do nothing about it. This is a moment that separates past from future. It is a little death. My friends, welcome the cleansing!"
Manekato uttered a soft command.
The three of them rose into the air, through a body's height. The Mapping had begun.
_Mane..._
Surprised to hear her name called, Manekato looked down. One of the Workers, a battered old gadget from a long-forgotten crop, was peering up at her with a glinting lens. It was clinging to the ground with long stabilizing suckers, but the Wind battered at it, and its purple-black hide glistened with rain.
Memory stirred. There had been a Worker like this when she had clambered from her mother's womb, chattering excitedly, full of energy and curiosity. In those first days and weeks that Worker had fed her, instructed her, kept her from harm, and comforted her when she was afraid. She had not seen the old gadget for years, and had thought little of it. Could this be the same Worker? Why should it seek her now, as it was about to be destroyed?
A wall of rain swept over the mountaintop. The three of them were immediately soaked, and Manekato labored to breathe the harshly gusting air.
When the rain gust passed, the mountaintop had been swept bare; all the Workers were gone, surely destroyed. Manekato felt an odd, distracting pang—regret, perhaps?
But this was no time to dwell on the past; the nameless one was right about that.
The three of them ascended without effort.
She was still clothed in her body, her legs dangling, her hair soaked. But of course this body was a mere symmorph: differing from her original self in form, but representing the same idea. (And in fact, as she had been through hundreds of previous Mappings, that "original" body had itself been nothing but a symmorph, a copy of a copy reworked to suit temporary needs, though one tailored to remain close to her primary biological form as possible.)
But such a morphology was no longer appropriate. With a soundless word, she discarded the symmorph, and accepted another form.
Now she was smeared around Earth, immersing it in her awareness, as if it were a speck that floated in her eyeball.
The great Farms glittered over the planet: from pole to pole, around the equator, even on the floor and surface of the oceans, and in the clouds. It was as if the planet were encrusted with jewels of light and life and order. There were no barren red deserts, no frozen ice caps _here_.
But already, as the Red Moon began its subtle gravitational work, the first changes were visible. Huge ocean storms were unravelling the delicate ocean-floor and water-borne Farms. A vast line of earthquakes and ugly volcanism was unstitching an eastern continent. And, from an ocean which was sloshing like water in a disturbed bath, a train of immense tsunamis marched toward the land.
Soon the Poka Farm was covered—extinguished, scoured clear, even the bedrock shattered, the bone dust of her ancestors scattered and lost, beyond memory.
The jewel-like lights were failing, all over the world. There was nothing for her here.
She gazed at her destination, the new, wandering Moon.
## _R eid Malenfant_
Malenfant's world was stratified into layers of varying incomprehensibility.
At the base of it all was the stockade, the familiar sturdy fence and the huts of mud and wood: the physical infrastructure of the world, solid, imperturbable.
And then there were the people.
Hugh McCann was standing alone at the center of the colony's little street, hands dangling at his sides, gazing up at a corner of the sky. His mouth was open, and his cheeks glistening, as if he were weeping. Nemoto was shielding her eyes, so that she couldn't so much as glimpse the sky above.
He saw Julia and Thomas, close together near the gate. The Hams didn't seem disturbed by the fiery sky. They were stripping off their neat, sewn-together garments, revealing bodies that were ungainly slabs of corded muscle. They pulled on much cruder skin wraps, of the kind Malenfant had seen Thomas wear out in the bush, tying them up with thongs. More Hams were coming in through the open gate _(the gate is open, Malenfant!)_ , and they picked up the discarded English-type clothing and started to pull it on.
A shift change, he thought, wondering. As if the settlement was a factory maintained by a pool of labor beyond the stockade walls.
And in the sky...
You can't put off thinking about it any longer, Malenfant.
Start with the basics. There is the white sun, the yellow Earth _(yellow?)_. There are the clouds, stringy cirrus today, littered over the sky's dome. And beyond the clouds, in the spaces between sun and Earth—
What, Malenfant?
He saw bars, circles, lines, patterns that seemed to congeal and then disappear. If he stared fixedly at one point of the sky he would make out a fragment of texture, as if something were sliding by, something huge, beyond the roof of the world. But it never stayed stable in his vision—like an optical illusion, a form that oscillated between two interpretations, a bubble that flipped into a crater. And no matter how he tried he would find his eyes sliding away to the familiar, to the huts, the red dust of the ground.
"Why can't I see it?"
Nemoto kept her head down. "It's too far beyond your experience, Malenfant. Or above it. You think of your eyes as little cameras, your ears as microphones, giving you some objective impression of the true world. They are not. Everything you think you see is a kind of virtual-reality projection, based on sensory input, framed by prejudice about what the brain imagines _ought_ to be out there. Remember, we evolved as plain-dwelling hunter-gatherers, and our sensoriums are conditioned to the hundred-mile scale of Earth landscapes. Malenfant, you just aren't programmed to see—"
"The scaffolding in the sky."
"Whatever it is."
"Like the Hams. When we went to the wreck of the _Redoubtable_. It was as if they couldn't see it at all."
"Do you find the thought disturbing, Malenfant? To find you have the same limitations as Neandertals?"
"What's happening, Nemoto? What is coming down on us?"
"I could not begin even to guess."
McCann was standing alone, still weeping.
As Malenfant approached, McCann used his sleeve to wipe away the dampness on his cheeks, the dribble of mucus that had dangled from his nose. "Malenfant. You bear yourself well. The first Change I witnessed threw me into a cold grue of terror. But you have a stiff back; I could see that about you from the start."
"What are you talking about?"
"Can't you see?" And he stabbed a finger at the sky, at the Earth.
The new Earth.
The planet was a ball of yellow-white cloud, very bright. It was banded by watercolor streaks of varying hues. There were dark knots in the bands, perhaps giant storms. It reminded Malenfant of nothing so much as space-probe images of Jupiter or Saturn. It was a Banded Earth.
Deep unease settled into his gut. "What happened to the Earth?"
"Nothing, Malenfant," Nemoto said, her voice expressionless. "It's gone. Or rather, we have. The Red Moon has moved on to a fresh universe, another of the vast ensemble of possibilities—"
"And it has taken us with it," McCann said bitterly. "We have suffered another knight's move between possibilities. Now do you see why I weep? It is unmanly, perhaps—but now that the Red Moon has moved on from your world, any chance of rescue by your people is gone with it." He laughed, an ugly sound. "I have seen a whole succession of worlds skip through that dismal sky, Malenfant, each of them as bleak as the last—save only for yours, where I could see the glint of cities on the night side. And then your squat glider came floating down from the sky, and I allowed myself to hope, you see—a fool's mistake. But now hope is gone, and you are as stranded as I am—both of you—all of us in this Purgatory..."
Malenfant saw it in that instant; it was as if the world swivelled around him, taking on new, and unwelcome, configurations. The Red Moon had moved on. He was indeed stranded, beyond the reach of any help from those who knew him _—stranded in another universe_ , to which he had somehow been transported.
In a corner of his mind he wondered if poor impoverished Luna had been restored to the skies of Earth.
As the light show faded, the Hams—the "new shift"—were moving slowly around the stockade, picking up brooms and tools, heading for the huts. Beginning their work.
Malenfant said, "Why do they come here?"
McCann held up his hands, plucked at his threadbare jacket. "Look at me. I am old and fat and tired—and at that I am perhaps the best functioning of those who survived the crash of the _Redoubtable_. And now look at the bar-bars." He faced Malenfant. "You think I am some slavekeeper. How could I keep these people, if they did not wish to stay? Or—if I keep slaves, _where are the children?_ Where are the old, the lame?" He pointed beyond the gate. "There is a troupe of them out there. We keep up a certain trade, I suppose you'd call it. They sustain this little township with their labor, as you have seen. And in return, there are things we have which they covet: certain foodstuffs—and beer, Malenfant, your bar-bar gentleman likes his beer!"
Nemoto said levelly to Julia, "Why do you keep these English alive?"
Julia grinned, showing a row of tombstone teeth. "Tired ol' men," she said.
McCann eyed Malenfant ruefully. "Pity, you see; the pity of animals. They saw we had no women or children, that we were slowly dying. They regard us as pets, these Hams. _That_ is what we are reduced to."
"And all your talk of educating them in a Christian, umm, Johannen life—"
"A man does not welcome too much reality..."
That gate was still open. You're wasting time, Malenfant.
He found Julia. She was dressed in her native skins; no trace of her guise as a maid for the English remained. He pointed toward the open gate. He said, "Emma."
She nodded.
He went back to the others. "I'm out of here, McCann. Will you try to stop me?"
McCann laughed. "What difference does it make now? But what will you do?"
"What I came to do," Malenfant said bluntly.
"Ah—Emma. I wish I had the comfort of such a goal." McCann looked at Nemoto. "And you, Madam Nemoto? Will you stay with a beaten old man?"
Nemoto raised her face to the sky; flickering light reflected from her skin. "I will seek answers."
"Answers?" McCann snorted. "Of what use are answers? Can you eat answers, sleep under them, use them to ward off the Runners, the Elves?"
She shrugged. "I am not content to subsist, like you, like these Hams."
Malenfant felt reluctant to lose her, even though she had betrayed him. And besides, she was scarcely streetwise: he imagined her dreaming of sheaves of parallel universes as a shaped cobble stove in her skull... "Come with me."
She appraised him coolly. "We have always had different agendas, Malenfant."
McCann looked from one to the other. Impulsively he said, "I have been sedentary too long. Let me accompany you, Malenfant. I daresay I have a few tricks, born of long experience, which might yet save your hide."
Malenfant glanced at Julia, who had no reaction. "What about Crawford and the others?"
McCann clapped Thomas on his broad shoulder. "I see no reason why our friends should fail to look after three as well as they have looked after four."
Thomas nodded curtly.
Malenfant faced Nemoto. "I hope you find what you are looking for."
"I will see you again," she said.
"No," he said, flooded by a sudden certainty. "No, you won't. We'll never meet again."
She stared at him. Then she turned away.
## _M anekatopokanemahedo_
She was standing on a shining, smooth surface of Adjusted Space, bright yellow, softly warm under her bare feet. Babo and Without-Name still clung to her hands; she released them.
On the Red Moon, there was no Wind. She relished the luxury of not having to fight against the air's power, enjoying the ease with which she took each breath.
Around them were a dozen more people—more exiles from one ruined Farm or another, their symmorphs adorned with a startling variety of colors and stylings of skin and hair—and perhaps a hundred times as many Workers: Workers tall and slim, short and squat, Workers that flew and crawled and rolled and walked. As was customary, the people's new symmorphs were as close as possible in appearance to the shells they had abandoned on Earth.
The Mapping had taken account of the different physical conditions. Thus Manekato felt no discomfort as her lungs drank in the thin, oxygen-depleted air of this small world, and her new body would suffer no ill-effects from the relative lack of carbon dioxide. But she had taken care not to engineer out all of the Red Moon's experiential differences; for if she had there would scarcely be a purpose in coming here at all. Thus the air was cold and damp and laden with a thousand powerful, unfamiliar scents—and thus the lower gravity, just two-thirds of Earth's, tugged only feebly at her limbs.
Manekato loped through the crowd of gazing people and scuttling Workers. Her gait felt oddly clumsy in the low gravity, as if her muscles were suddenly overpowered. The yellow floor was perhaps a hundred paces across. It was a neatly circular disk of Adjusted Space, its smoothness comforting. She reached the rim of the disk. Tiny Workers streamed past her into the green world beyond, recording, interpreting, transmitting.
Beyond the platform was a wall of forest, concealing a dense green gloom. The trees were tall here: great spindly structures of wood, very different from the ground-hugging species of Wind-blasted Earth. Shadows flitted through that green darkness. She thought she saw eyes peering out at her, eyes like a mirror of her own.
Babo ran past her with a gurgled cry. He ran straight into the forest and clambered into the lowest branches of a tree, clumsily, but with enthusiasm and strength.
Manekato peered down. In the Moon's red dust grass grew, sprinkled with small flowers, white and yellow. She leaned forward, supporting her weight on one fist, and touched the grass. The blades were coarse, and other plants and moss crowded around, fighting over each scrap of soil. She saw leaves protruding from beneath the disk, crushed, bent back; some of the living things of this world had already died because of her presence.
The land here had never been Farmed: not once, not in all the billions of years this world had existed. Even this patch of grass-covered land, where billions of living things fought for life in every scrap, was disturbing, enthralling proof of that.
In front of the forest fringe she made out a small, brown-furred Worker—no, not a Worker, an _animal_ , its species probably unmodified by conscious design. It had a short, slim body, and four spindly legs; it bent a graceful neck, and a small mouth nibbled at the grass. It moved gracefully, but with a startling slowness, an unhurried languor that contrasted with the frantic scuttling of the people and the workers. By the look of the genitalia between its back legs its kind must reproduce in a mammalian fashion, rather than be nurtured directly from the ground...
_Nobody_ had nurtured this creature, she reminded herself; there had been no conscious process. It had been born in blood and pain and mucus, without the supervision of any human, and it found food to sustain its growth in this wild, unmanaged, undisciplined place.
On her world, there had been no parks or zoos for nine hundred millennia. Though the richness of the ecology was well understood and managed minutely—including the place of people within that ecology—there were no creatures save those which served a conscious purpose, no aspect of nature which was not thought through and controlled. Not so much as a stomach bacteria.
Manekato had known that this new Moon would be wild, but that its ecology would function nonetheless. But it was one thing to have a theoretical anticipation and another to be confronted with the fact. She felt as if she had entered the workings of some vast intricate machine, all the more remarkable for lacking a conscious designer or a controlling intelligence.
Now Babo came hurrying back from the forest. He clutched something in his arms that wriggled sluggishly.
Babo's legs were covered in scrapings of green moss, and his hair was dishevelled and dirty. But his eyes were bright, and he was breathing hard. "My arms are strong," he told his sister. "I can _climb_. It is as if this body of mine remembers its deepest past, many millions of years lost, even though the trees on Earth are mere wind-blown stubs compared to these mighty pillars..."
Without-Name asked, "What is it you carry?"
He held it out carefully. It had a slim body and a small head. Its legs were short and somewhat bowed, but Manekato could see immediately that this creature was designed—no, had _evolved_ —to walk bipedally. It was perhaps half of Babo's height, and much slimmer.
"It is a hominid," she said wonderingly.
"I found it in the tree," Babo said. "It is quite strong, but moves slowly. It was easy to catch."
Manekato reached to touch the creature's face.
The hominid whipped its head sideways and sank its teeth into Manekato's finger.
Manekato fell back with a small cry. Miniature Workers in her bloodstream caused the ripped flesh to close immediately.
_"Ha!"_ the creature yelled. _"Elf strong Elf good hurt stupid Ham hah!"_
This jabber meant nothing to Manekato.
Without-Name took the creature from an unresisting Babo. She held it up by its head. Dangling, the hominid hooted and thrashed, scratching at Manekato's arm with its legs and fists, but its motions were slow and feeble.
With a single, harsh motion Without-Name crushed the hominid's skull. The body shuddered once and was limp. Without-Name let the body fall to the ground, its head a bloody pulp. A Worker scuttled close and swept up the tiny corpse.
Babo looked at Without-Name, his face empty of expression. "Why did you do that?"
"There was no mind," said Without-Name. "There was no utility. Therefore there was no right to life. I have been dispossessed by this Moon. I will not rest until I have made the Moon my possession in turn."
Manekato suppressed her anger. "We did not come here to kill. We came to learn—to learn and negotiate."
Without-Name spat a gobbet of thick phlegm out onto the grass. "We all have our reasons to be here, Manekatopokanemahedo. You follow the foolish dreams of the Astrologers. _I_ am a Farmer."
"And," Manekato said slowly, "is that your ambition here? To subdue a new world, to turn it all into your dominion?"
"What higher ambition could there be?"
"But we have yet to find those who moved this world. _They_ were more powerful than these blades of grass, that wretched hominid. Remember that, Renemenagota, when you boast of what you will conquer."
Now Manekato saw that two burly Workers had brought another hominid for their inspection. It was taller, heavier than the last, but it was scrawny, filthy, hollow-eyed.
Again Without-Name picked up the specimen by its skull and lifted it easily off the ground. The creature cried and struggled, clearly in distress, but its movements were still more sluggish than the first's, and it made no attempt to injure Without-Name.
"Let it go," Manekato said evenly.
Without-Name studied her. "You are not of my Lineage. You do not have authority over me."
"Look at it, Renemenagota. _It is wearing clothes_."
Babo breathed deeply. "Do it," he said. "Or I will have the Workers stop you. _I_ have the authority for that, nameless one, thanks to the Astrologers you despise."
Without-Name growled her protest. But she released the hominid, which fell into a heap on the floor, and stalked away.
Manekato and Babo huddled over the hominid. It had curled into a fetal position; as gently as they could they turned it on its back and pried open its limbs.
"I think it is female," Babo said. "Its head is badly bruised, as is its neck, and it struggles to breathe. Without-Name has damaged it."
"Perhaps the Workers can repair it."
The hominid coughed and struggled to sit up. Babo helped it with a lift from a powerful hand.
_"My name,"_ the hominid said, _"is Nemoto."_
## _S hadow_
The antelope had gotten separated from its herd. It was running awkwardly, perhaps hampered by age or injury.
With fluid grace, the lion leapt onto the antelope's back, forcing it to the ground in a cloud of crimson dust. The antelope kicked and struggled, its back and haunches already horribly ripped. Then the lion inflicted a final, almost graceful bite to its throat. As its blood poured onto the dust of the savannah, Shadow saw surprise in the antelope's eyes.
More lions came loping up to feed.
Shadow remained huddled behind her rock—exposed on the open savannah, but downwind of the kill. She kept her baby quiet by cradling its big, deformed head tightly against her stomach.
The lions pushed their faces into the fallen antelope's carcass, digging into the entrails and the easily accessible meat of the fleshy areas. Soon their muzzles were crimson with blood, and their growls of contentment were loud. Shadow was overwhelmed by the iron stink of blood, and the sharp burning scent of the lion's fur—and by hunger; her mouth pooled with saliva.
Her face itched, and she scratched it.
At last the lions' purring growls receded.
Already more scavengers were approaching the carcass. Hyenas loped hungrily toward it in a jostling pack, and overhead the first bats were wheeling, huge carrion-eating bats, their wings black stripes against the sky.
And, from the crater's wooded rim, people emerged: Elf-folk like Shadow, men, women, and infants, melting out of the green shelter of the woods, their black pelts stark against the green and crimson of the plain. They eyed the carcass hungrily, and they carried sticks and cobbles.
But the hyenas were hungry, too, and in a moment they were on the antelope, burying their muzzles inside the great rips made by the lions' jaws, already fighting among themselves. Their lithe bodies clustered over the carcass, tails high in the air, from a distance like maggots working a wound.
The people moved in, yelling and waving their sticks and throwing their stones. Some of the dogs were hit by hurled cobbles. One man, a squat, manic creature with one eye closed by a huge scar, got close enough to pound one animal with a fat branch, causing the dog to yelp and stumble. But the dogs did not back away. A few of them tore themselves away from the meat long enough to rush at the hominids, barking and snapping, before hurling themselves back into the feast. Most simply ignored the people, gouging out as much meat as they could before being forced away by a dog bigger and stronger.
So it went, a web of complex but unconscious calculations: each hyena's dilemma over whether to attack the hominids, or whether to gamble that another dog would, leaving it free to take more meat; the hominids' estimation of the strength and determination of the hyenas versus their own hunger and the value of the meat.
This time, at least, the hyenas were too strong.
The Elf-folk troupe backed away sullenly. They found a place in the shade of the trees at the forest edge, staring with undisguised envy at the rich meat being devoured by the dogs.
At last the hyenas started to disperse. They had taken most of the meat, and the antelope was reduced to scattered bones and bits of flesh on a blood-stained patch of ground, as if it had exploded. Again the people came forward, and their stones and sticks drove away the last of the dogs.
There was little meat to be had. But there was still a rich resource here, which hominid tools could reach. The adults took the antelope's bones and, with brisk, skillful strikes of their shaped stones, they cracked them open. Soon many of the people were sucking marrow greedily. Children fought over scraps of flesh and cartilage.
Huge bats flapped down, their leathery wings black, vulturelike. They pecked at outlying bits of the carcass, bloodying their fur. The people tolerated them. But if the bats came too close they would be greeted by a stick wielded by a hooting hominid.
Shadow came out from behind her rock.
A child came up to her, curious, a bit of gristle dangling from her chin. But as Shadow neared, the child wrinkled her nose and stared hard at Shadow's face. Then she turned and ran for the security of her mother.
As Shadow approached the group, the people moved their children away from her, or growled, or even threw stones. But they did not try to drive her away.
Shadow saw a big older woman, the hair of her back oddly streaked with silver. This woman—Silverneck—was working assiduously at the remnant of a thigh bone. Shadow sat close to Silverneck, not asking for food, content not to be rejected.
The sun wheeled across the sky, and the people worked at the carcass.
At length Silverneck hurled away the last fragments of bone. She lay on her back, legs crossed, and crooked an arm behind her head. She belched, picked bits of marrow and bone from her teeth, and thrust a finger into one nostril with every sign of contentment.
Cautiously, her baby clinging to her back, Shadow crept closer. She started to groom Silverneck, picking gently through the hairs of Silverneck's shoulders. The older woman, reclining stiffly, submitted to this in silence, eyes closed as if asleep.
Shadow knew what she must do to win a place here. In her home forest she had watched women seeking favor with their seniors. Still cautious, Shadow moved toward Silverneck's waist and reached out to stroke the older woman's genitals, just as she had seen others do before.
A hand grasped her wrist, gentle but strong. Silverneck's face, worn almost bald by grooming, was a mass of wrinkles. And it showed disgust. She pulled her legs under her, and pushed Shadow away.
Shadow sat still, baffled, disturbed.
After a time Shadow again reached out to groom Silverneck. Again Silverneck submitted. This time Shadow did not try to cross the boundary to sexual contact, and Silverneck did not push her away.
As the shadows lengthened across the plain, the carrion-eating bats clustered closer around the remnants of the carcass. One by one the people started to drift back to the forest. The first roosting calls began to sound from the tree tops.
At last the old woman stretched and yawned loudly, bones popping. Then she got to her feet and ambled back toward the forest's edge.
Shadow sat where she was, waiting.
Silverneck looked back once, thoughtfully. Then she turned and moved on.
Shadow got to her feet, her baby clinging to her back. Hastily she rummaged through the carcass, but the marrow and meat had been chewed or sucked off every bone. Cramming bits of greasy skin into her mouth, she hurried after Silverneck into the forest.
## _M anekatopokanemahedo_
With a wave of his hand Babo conjured an image of the Red Moon—but it was not an image, rather a limited injective-recursive Mapping of the Moon into itself. The Moon turned for their benefit, a great hovering globe twice Babo's height. Manekato gazed at searing red desert-continent and steel ocean.
The little hominid who called herself Nemoto stood close to Manekato, her eyes wide, her smooth face bearing some unreadable expression.
"Your work is proceeding well," Manekato said to her brother.
"It is a routine application of familiar techniques; merely a question of gathering sufficient data... But already the key to this world's mysteries is clear."
"Ah." Manekato said somberly. She reached up and pointed at the huge volcano that dominated the western side of the rust-red continent. "You mean _that_."
"Yes, the volcanic anomaly," Babo said. "Which in turn must derive from some magmatic feature, a plume arising deep within the belly of this world."
_"You talk of the Bullseye?"_ Nemoto was watching them, straining to hear, turning her little head this way and that in order to position her small immobile ears.
Babo watched Nemoto uneasily. "Do you think she can follow us?"
"I have taught her a few words," said Manekato. "But our speech is too rapid for her to grasp; like all the creatures here on this oxygen-starved world, she is sluggish and slow-witted. I have had more success in decoding her own language. It is a little like the nonsense argots you used to make up for my amusement as a child, Babo."
Babo was still watching Nemoto. "She imitates your behavior well. Look how she gazes at the volcano! It's almost as if she can understand what she is seeing."
Manekato grunted. "Do not underestimate her, brother. I believe she is intelligent, to a degree. Consider the clothes she wears, her speech with its limited grammar, the tools she deploys—even her writing of symbols into her blocks of bound paper. Why, she claims to have come here, not through the blue portals, but in a spacecraft designed by others of her kind. And that she came to this Moon from _curiosity_. I found this as hard to believe as you, but she drew sketches which convinced me she is telling the truth."
"But even the making of clothes may be no more than the outcome of instinct, Mane," Babo said gently. "There is a kind of aquatic spider that makes diving-bells from its webbing, and nobody would argue that _it_ is intelligent. Perhaps some day we will discover a species, utterly without mind, which makes starships. Why not? And nor is symbol-making sufficient to demonstrate intelligence; there are social ants which—"
Manekato raised a hand to quiet him. "I am aware of the dangers of anthropomorphism. You think I have found a pet, here in this dismal place—that I am seeking intelligence where all I see is a reflection of my own self."
Babo rubbed her back affectionately. "Well, isn't that true?"
"Perhaps. But I strive to discount it. And meanwhile I have come to the belief that Nemoto and her kind may be—not merely intelligent—but _self-aware_."
Babo laughed. "Come now, Mane. Let us show her a mirror, and together we will watch her seek the hominid behind the glass."
"I already tried that test," Manekato said. "She was very insulted."
"If she is too proud to be tested, why does she follow you around?"
"For protection," Manekato said promptly. "You saw how Without-Name treated her when she first found her. Nemoto shows great fear of her."
Babo grunted. He crouched down before the hominid, Nemoto; his huge body was like a wall before her slim frame.
Nemoto returned his gaze calmly.
"... Intelligent, Mane? But the size of the cranium, the limited expanse of the frontal lobes— _the dullness of those eyes_. I do not get a sense of a person looking back out at me."
Manekato snapped, "And you can assess a creature's intelligence merely by looking at it?" She said, _"Nemoto."_
The hominid looked up at her.
_"You remember what I told you of the Mapping."_ Manekato strove to slow down her speech, and to pronounce each word of Nemoto's limited language clearly and distinctly.
Nemoto was frowning, concentrating hard. _"I remember. You defined a mathematical function to map the components of your body to material of the Moon."_ Her words, like her actions, were slow, drawn-out. _"The domain of this function was yourselves and your equipment, the range a subset of the Moon. When you had defined the Mapping..."_
"Yes?"
Nemoto struggled, but failed to find the words. _"I have much to learn."_
Babo grunted. "It is impressive that she knows there are limits to her knowledge. Perhaps that indicates some degree of self-awareness after all."
Manekato said, "Then I am winning the argument."
Babo grumbled good-naturedly. "Just remember we are here to study the Moon, and those who sent it spinning between the universes—not to converse with these brutish hominids, who were certainly not responsible."
Manekato studied Nemoto. The little creature was watching her with empty, serious eyes. _"Come,"_ said Manekato, and she held out her hand.
Nemoto took it with some reluctance.
Babo turned back to the refinement of his Mapping.
Manekato led Nemoto across the Mapped-in floor of the compound. They passed between structures that had been conjured out of Adjusted Space to shelter the people. Rounded yellow forms, to Mane's taste overornate, they made the compound look like a plate set before a giant, loaded with exotic shapes—and with insectlike humans, Workers, and hominids scuttling across it.
_"You must not let my brother upset you,"_ Manekato said evenly, striving to express herself correctly in the narrow confines of Nemoto's limited tongue.
_"He has no imagination,"_ said Nemoto.
Manekato barked laughter, and Nemoto flinched. _"I'll tell him you said that!... But he means you no harm."_
_"Unlike Without-Name, who does mean harm, and who has far too much imagination."_
_"That is insightful, and neatly phrased."_ She snapped her fingers and a Worker came scuttling. _"Well done, Nemoto. You deserve a banana."_
Nemoto regarded the yellow fruit proffered by the Worker with loathing.
Manekato shrugged. She popped the banana into her mouth and swallowed it whole, skin and all.
Nemoto said cautiously, _"I think your world has no Moon—none but this unwanted arrival."_
Manekato, interested, said, _"And what of it?"_
_"Our scientists have speculated how the destiny of my world might have differed if it had been born without a Moon."_
_"Really?"_ Manekato wondered briefly if "scientists" was correctly translated.
Nemoto took a deep breath. _"Our Moon was born in a giant impact, in the final stage of the violent formation of the Solar System. The effects on Earth were profound..."_
Manekato was fascinated by all this—not so much by the content, which seemed trivially obvious, but by the fact that Nemoto was able to spin together such a coherent statement at all—even if it was delivered in a maddeningly slow drawl. But Nemoto seemed desperate to retain Manekato's attention, to win her understanding—and perhaps her approval.
_"And what difference would all this make to the evolution of life?"_
Nemoto said, _"You come from a world that spins fast. There must be winds there—persistent, strong. Perhaps you were once bipeds, but now you walk on all fours; probably I could not stand upright on your world. Your trees must hugthe ground. And so on. Your air, derived from a primordial atmosphere never stripped off by impact, is thicker than mine, richer in carbon dioxide, probably richer in oxygen. You think fast, move fast, fuelled by the oxygen-rich air."_ She hesitated. _"And perhaps you die fast. Mane, I can expect to live for seventy years—years measured on your Earth, or mine. And you?"_
_"Twenty-five,"_ Manekato breathed. _"Or less."_ She was stunned by Nemoto's sudden acuity—but then the hominid had been observing her for days now, learning about Manekato as Manekato had learned about her; she had simply saved up her conclusions—as a good scientist should.
_"The evolution of life must have been quite different,"_ Nemoto said now. _"With lower tides your oceans must be less enriched of silt washed down from the continents. And there must be less global ocean movement. I would expect a significantly different biota_.
_"As for humans, I believe that our evolutionary paths diverged at the stage we call the 'Australopithecine,' Manekato. But the environment was different on our worlds, evoking a different adaptation. I would hazard that hunting is not a viable strategy for hominids on your world. Probably your short days were simply not long enough. You call yourself 'Farmers.' Perhaps your world encouraged the early development of agriculture."_
_" 'Australopithecines.' I don't know that word."_
_"The hominids called Nutcrackers and Elves here seem to be surviving specimens. From that root stock your kind took one path; mine took another."_
_"But, Nemoto—why do such divergent worlds have people at all? Why would hominid forms evolve on world after world—"_
_"Your kind did not originate on your Earth,"_ Nemoto said bluntly. _"Your scientists must have deduced that much."_
Manekato bristled. She tried to put aside her annoyance at being patronized by this monkey-thing. _"You are right. That much is evident. People share the same biochemical substrate as other living things, but are linked to no animal alive or of the past by any clear evolutionary path."_
_"But on my Earth there_ is _a clear evolutionary path to be traced from humans back into the past."_
_"So you are saying my line originated on your Earth? And how did my Australopithecine grandmothers get delivered to 'my Earth'?"_
Nemoto shrugged. _"Perhaps by this Red Moon, and its blue-ring scoops."_
It was a startling vision—especially coming from the mouth of this small-brained biped—but it had a certain cogency. Manekato was aware her mouth was dangling open; she shut it with a snap of her great teeth. _"Who would have devised such a mechanism? And why?"_
Nemoto's face pulled tight in the grimace Manekato had come to recognize as a smile. _"The Hams have a legend of the Old Ones, who built the world. I am hoping you will find them."_
Manekato glared at Nemoto: She was profoundly impressed by Nemoto's acuity, yet she was embarrassed at her own condescension toward the hominid. It was not a comfortable mixture. _"We will talk of this further."_
_"We must,"_ said Nemoto.
## _R eid Malenfant_
Malenfant counted them. Sixteen, seventeen, eighteen Runners: eighteen powerful, languid bodies relaxing on the barren ground. The band seemed to be settling here for the night. The three of them—Julia, Malenfant, Hugh McCann—hunkered down in the dirt. The grass beneath Malenfant's scuffed boots was sparse, and the Mars-red dust of the world showed through, crimson-bright where it caught the light of the setting sun.
This swathe of scrubby grassland was at the western border of the coastal forest strip NASA cartographers had christened the Beltway. Farther west of this point, beyond a range of eroded mountains, there was only the arid, baked interior of the great continent, hundreds of miles of red desert, an Australia in the sky. No doubt it was stocked with its own unique ecology exquisitely evolved to maximize the use of the available resources, Malenfant thought sourly, but it was an unremittingly hostile place for a middle-aged American—and of no interest to him whatsoever, unless it held Emma in its barren heart.
McCann moved closer to Malenfant, his buckskin clothes creaking softly. "How strange these pongids are," he said. "How very obviously ante-human. See the way they have made their crude camp. They have built a fire, you see, probably from a hot coal carried for tens of miles by some horny-handed wretch. They even have a rudimentary sense of the hearth and home: Look at that big buck voiding his bowels, off beyond the group—what an immense straining—everything these fellows do is mighty!
"But that is about the extent of their humanity. They have no tools, save the pebbles they pluck from the ground to be shaped; they carry nothing for sentiment—nothing at all, so their nakedness is deeper than ever yours or mine could be. And though they gather in little clusters, of mothers with infants, a few younger siblings, there is no community there.
"If you look into the eyes of a Runner, Malenfant, you see a bright primal presence, you see cleverness—but you do not see a _mind_. There is only the now, and that is all there will ever be. Whatever dim spark of awareness resides behind those deceptive eyes is trapped forever in a cage of inarticulacy... One must pity them, even as one admires them for their animal grace."
Malenfant grimaced. "Another lecture, Hugh?"
McCann sighed. "I have been effectively alone here too long, my reflections on the strange lost creatures who inhabit this place rattling around in my head. Would I were as conservative with my words as dear Julia, who, like the rest of her kind, speaks only when necessary."
Or maybe, Malenfant thought, she just hasn't got much to say to you, or me. He'd observed the Hams chattering among themselves, when they thought no human was watching them. For all his bush craft, McCann's understanding of the creatures around him was obviously shallow.
Without a word, Julia stood up and began to walk across the sparse scrub toward the Running-folk. McCann and Malenfant stayed crouched in the dirt.
The Runners turned to watch her approach. They were silent, still, like wary prey animals.
Julia got as far as the Runners' fire. She hunkered down there, making sure she didn't sit close to the meat. The Runners were still wary—one burly man bared his teeth at Julia, which she calmly ignored—but they didn't try to drive her away.
After a time an infant came up to her, bright eyes over a lithe little body. Julia reached out her massive hand, but its mother instantly snatched the child back.
Malenfant suppressed a sigh. Sometimes Julia would win the Runners' confidence quickly; other times it took longer. Tonight it looked as if Julia would have to spend the night in the Runners' rough camp before they could make any further progress.
As the days had worn on, Malenfant had lost count of the number of Runner groups they had tracked down. Julia was always given the lead, hoping to establish a basis of trust, and then Malenfant and McCann would follow up. Malenfant would produce his precious South African air force lens, his one indubitable trace of Emma, hoping for some spark of recognition in those bright animal eyes.
It hadn't worked so far, and Malenfant, despite his own grim determination, was gradually losing hope. But he didn't have any better ideas.
As Julia sat quietly with the Runners, the light leaked out of the sky. The predators began to call, their eerie howls carrying far on the still evening air.
Briskly, without speaking, Malenfant and McCann built a fire. They used dry grass for tinder, and had brought bundles of wood from the Beltway for fuel.
Malenfant's supper was a few mouthfuls of raw fish. The Runners used their fires primarily for warmth, not cooking. If McCann or Malenfant were to throw this tough, salty fish onto the fire, the smell of burned flesh would spook the Runners and quickly drive them away.
After that it was foot maintenance time. Malenfant eased off his boots and inspected the latest damage. There was a kind of flea that laid eggs under your toenail, and naturally it was Malenfant who was infected. When the critters started to grow in the soft flesh under there, feeding off his damn toe cheese, McCann said Julia would dig them out with her stone knives. Malenfant backed off from that, sterilized his pocket knife in the fire, and did it himself. But, Christ, it hurt, unreasonably so, and it made a bloody mess of his toes; for the next few days he had had a _lot_ of trouble walking.
When he was done with his feet, Malenfant started making pemmican. It was one of his long-term projects. You took congealed fat from cooked fish, and softened it in your hands. Then you used one of Julia's stone knives to grate the cooked flesh into powdery pieces and mixed it with the fat. You added some salt and berries and maybe a little grated nutmeg from McCann's pack, and then pulled the mess apart into lumps the size of a golf ball. You rolled the balls into cocktail-sausage shapes, and put them in the sun, to set hard.
He had already done the same with a haunch of antelope. It was simple stuff, dredged up from his memories of his astronaut survival training. But the treatment ought to make these bits of fish and meat last months.
McCann sat and watched him. He was nursing a wooden bowl filled with a tea made of crushed green needles from a spruce tree. Malenfant had been skeptical of what he saw as an English affectation, but the tea was oddly refreshing; Malenfant suspected the needles were full of vitamin C. But the tea was strongly flavored and full of sharp bits of needle (which he had learned to strain out through a sock).
McCann said, "Malenfant, you are a man of silence and unswerving intent. Your preparations are admirable and thorough. But to enter the desert is foolhardy, no matter how many pemmican cakes you make. Even if you could find your way through the mountains, there is only aridity beyond."
Malenfant growled, "We have this conversation roughly once a day, Hugh. We must have found all the Runner groups who work this area, and have come up blank. On the other hand, we know a lot of them work deeper into the desert." He squinted, peering into the harsh flat light of the arid western lands. "There could be dozens more tribes out there. We have to go find them."
McCann pulled a face and sipped his tea. "And seek out traces of your Emma."
Malenfant kept kneading his pemmican. "You've come this far, and I'm grateful. But if you don't want to follow me any further that's okay by me."
McCann smiled, tired. "I suppose I have attached myself to you—become a squire to your chessboard knight. On this desolate Red Moon we are all lost, you see, Malenfant—not just your Emma. And we all seek purpose."
Malenfant grunted, uncomfortable. "I'm grateful for your company. But why the hell you're doing it is your business, not mine. I never cared much for psychoanalysis."
McCann frowned at the term, but seemed to puzzle out its meaning. "You always look outward, don't you?—but perhaps it would serve you to look inward, from time to time."
"What is that supposed to mean?"
"For a man with such a powerful drive—a drive to a goal for which he is clearly prepared to give his life—you seem little interested in the origin of that drive." McCann raised a finger. "I predict you will puzzle it out in the end—though it may require you to find Emma herself before you do so."
They would take turns to stand watch: McCann first, then Malenfant.
Malenfant cleaned his teeth with a bit of twig. Then he settled down for his first sleep.
The nights here were always cold. Malenfant zipped up his jumpsuit, placed a bag of underwear under his hips to soften the hardness of the ground, and pulled a couple of layers of chute cloth over his body. He set his head on the pack in which he carried the remnant of his NASA coverall, his real-world underwear, and the rest of his few luxuries. Though he had gotten used to his suit of deerskin—it had softened with use, and after the first few days he suspected it stank more of him than its original owner—he clung to the few items he had salvaged from the ludicrous wreck of his mission as a kind of message to himself, a reminder that he hadn't been born in these circumstances, and maybe he wouldn't have to die in them either.
As usual he had trouble settling.
"I don't like to complain," he said at length.
"Of course not."
"This ground is like rock. I can't turn over without dislocating a hip."
"Then don't turn over."
So it went.
After three hours it was Malenfant's turn to stand watch. McCann shook Malenfant awake, pitching him into a cold, star-littered night. Malenfant shook out his blanket and went to take a leak. Sign of age, Malenfant.
Beyond the circle of light from their hearth, the desert was deep and dark, its emptiness broken only by the ragged glow of the Runners' fire.
Sometimes it scared him to think of what a wilderness it was that had claimed him. There were no cop cars cruising through that darkness, no watching choppers or surveillance satellites, nobody out there to help him—no law operating save the savagely impartial rule of nature.
And yet every day he was struck by the strange _orderliness_ of the place. Decaying animal corpses did not litter the ground, save for a handful of bleached bones here and there; it was rare to walk into so much as a heap of dung. There was death here, yes, there was blood and pain—but it was as if every creature, including the hominids, were a cog in some vaster machine, that served to sustain all their lives. And every creature, presumably unconsciously, accepted its place and the sacrifices that came with it.
All save one species of hominid, it seemed: _Homo sap_ himself, who was forever seeking to tear up the world around him.
The final time he woke that night, he found Julia looming over him. She was a vast silhouette whose disturbing scent of _other_ was enough to kick Malenfant's hind brain into wakefulness. He sat up, rubbing his eyes. His chute-silk blanket fell away, and all his warmth was lost to the cool, moist air. It was a little after dawn, and the world was drenched with a blue-gray light that turned the crimson sand purple.
The Runners had gone. He could just make them out, slim dark figures against the purple-gray desert, running easily and silently, far away into the desert.
He hadn't even gotten to show them his lens.
## _M anekatopokanemahedo_
There was a call from Babo, who was standing beneath his beautiful spinning globe. Manekato hurried to her brother, and Nemoto jogged after her.
The great rotating Moon-projection had been rendered semitransparent. And there was a hole in its very heart.
Something lurked there, blocky, enclosed—clearly artificial, very large. It was connected to the surface by a long, threadlike tube: not entirely straight, bending like a reed as it passed through the Moon's layers of core, thick mantle, and deep, hard lithosphere, so much thicker on this small cold world than the crustal layers of the Earth. The tube terminated in what looked like a small, compact crater, not far from the eastern shore of the world-spanning continent—not far from the location of the compound, in fact.
Manekato reached inside the Map. The misty layers of mantle and core resisted her gently, as if her fingers were pushing through some viscous liquid. She wrapped her fingers around the knot of machinery at the Map's center. It was dense and complex and well-anchored.
Nemoto watched her carefully.
"It is the world engine," said Babo.
Studying the globe as a whole, Manekato saw that the surface crater was diametrically opposite the summit of the great volcanic mountain, at the peak of the huge region of uplift that so distorted the figure of the world. Looking more closely she could see detail in the Map's misty outer layers: a disturbance in the core, a great plume in the deep-buried mantle, hot magmatic material working its way up through cracks in the mighty lithosphere towards that antipodal bulge.
"I cannot believe that such asymmetry is deliberate," Babo said.
"No," Manekato said. "The internal disturbances must be a result of the poor control of the Moon as it lurches from universe to universe. Perhaps the Moon is not meant to plummet about the cosmic manifold like this. The mechanism is poorly designed..."
"Or faulty. If it has been sweeping up hominids since early in our evolution, Mane, it must have been operating for millions of years."
"Perhaps even the great machines of the Old Ones are subject to failure."
"Quantum tunneling," said Babo. "That's how they do it. That's how this thing in the core sends this Moon from universe to universe."
Manekato said, "Tell me what you mean, brother."
"You understand the concept. An electron, say, does not have a precise position or velocity; rather it is embedded in a spreading cloud of probability. Given a measurement of its position, there is a small but finite chance that the electron will next be found—not close to the last position—but far away, outside any cage you care to throw around it—or at the heart of the sun—or in orbit around a distant star..."
"Yes, yes. _Or even another universe_. Is that your point?"
He scratched his head absently. "Well, we know that quantum tunneling can cause the nucleation of a new universe. The vacuum sustains a series of energy levels. A bubble of 'our' vacuum can tunnel to an otherwise empty space-time at a lower energy state, and there expand and become causally disconnected from our own..."
"We are talking of moving not an electron, but a world."
Babo shrugged. "I think we have the pieces of the puzzle now, at least; perhaps understanding will follow."
"In any case, our next object is clear," Manekato said. She pressed a finger into the crater at the top of the tube from the core; she could barely feel the feather-touch of its tiny rim. "We must go to this strange crater, learn all we can—and, perhaps, seek a way to direct the future course of this rogue Moon."
_"The manifold is a sheaf of possible universes,"_ Nemoto said.
Babo grimaced. "What did she say?"
Nemoto went on, _"I understand some of what you say. Perhaps the manifold universes were nucleated from a single primal universe by some such mechanism as quantum tunneling. Perhaps the nucleation of the universes was deliberate. Perhaps the Old Ones lived in the primal universe..."_
Babo bared his teeth at her, and Nemoto fell silent.
Manekato said dryly, "What's wrong?"
"She sees so much," Babo said. "Much further than I imagined. If she sees so much, will she not see that the achievements of the Old Ones are as far beyond us as..."
"As our Farms and our Maps are beyond her poor grasp?" She touched his shoulder, mock-grooming, seeking to calm him. "But would that be so bad? Would it hurt us to learn some of her humility?"
"I don't think she is so humble, Mane. Look at the defiance in that small face. It is unnatural. It is like being challenged by a Worker."
A cry rent the air.
Nemoto turned sharply. Manekato felt her ears swivel. It had been a cry of pain and despair—an animal's cry, but desolating nonetheless.
Nemoto began to run toward the place the cry had come from.
After a moment's hesitation, Manekato hurried after her pet.
_"Oh, let me up; I beg you, Madam Daemon, by the blood of Christ, let me up!"_
It was Without-Name, of course. She had caught another hominid. She had him sprawled on the smooth floor of the compound with her massive foot in the small of the back, so that he could do little but flop like a fish. He was wearing clothes of a cruder design than Nemoto's—scraps of skin sewn together with bits of hide, as if he had clambered inside the gruesome reconstruction of a dead animal. It seemed his capture had not been without incident. Blood leaked from a filthy wound on his forehead, and his right foot was dangling at an awkward angle, just a mass of blood, badly pulped. His blood and snot and sweat, even his urine, had spilled over the floor of Adjusted Space-time.
Others stood around the gruesome little tableau. Manekato was dismayed to see fascination on several faces, as if the blood-soaked allure of this world were seeping into more than one soul.
She rested a hand on Nemoto's shoulder. _"He is a member of your troupe? That is why you are distressed."_
_"No. I have never seen him before. And we don't have 'troupes.' But he is human, and he is suffering."_
Babo challenged Without-Name. "What new savagery is this, Renemenagota of Rano?"
"Am I the savage? Then what is this under my foot? We are not at home now, Manekato—we are not even on Earth. And if we wish to progress our inquiries we must abandon the techniques we would apply on the Earth."
"I don't understand."
"You gaze at a pretty Map while the real world is all around you—vibrant, primal." She slapped at the floor of Adjusted Space. "You even separate yourselves from the dirt. Have you stepped off this platform, Manekato, even once? I tell you, this is not a place for logic and Maps. It is a place of red and green, of life and blood and death—a place for the heart, not the head."
"And your heart tells you to torment this helpless wretch," Babo said.
"But not without a purpose," Without-Name said. "He comes from a troupe of hominids to the north of here. They live in crude shelters of wood and mud, and they call themselves _Zealots_. They are as intelligent as your pet, Manekato—but they are utterly insane, driven by dreams of a God they cannot see." She bellowed laughter, and applied more pressure with her heel to the Zealot's back; he groaned, his eyes rolling, as bones cracked. "These Zealots have been here for centuries. With their feeble eyes, their dim brains, they have seen this world which you are too frightened to touch. _They have seen the workings of the Old Ones_ , for they have been dragged from one cosmos to the next by their meddling. And they have formulated their own ambition in response: to spit in the face of the sky itself." She looked down at the sprawled, twitching hominid. "It is absurd. But in its way, it is magnificent. Hah! _These_ are the creatures of this world. I want to see what they see, know what they know. That way I will learn the truth about the Old Ones—and what must be done to defeat them."
Others growled assent behind her.
Manekato, deeply disturbed, stepped closer to Without-Name. "We did not come here to inflict pain."
"There is no pain here," Without-Name said easily. "For there is no sentience. You see only reflex, as a leaf follows the sunlight."
_"No."_ It was Nemoto. She stepped forward, evading the clutching hand of Manekato.
The nameless one gaped at her, briefly too startled to react.
_"I know that you understand me. I believe your species has superior cognition to my own. But nevertheless we have cognition. This man is aware of himself, of his pain. And he is terrified, for he is aware that you plan to kill him, Renemenagota."_
Without-Name reared up on her hind legs, and the man in the dust howled. _"You will not use my name."_
_"Let him go."_ Nemoto held out her arms, her hands empty.
The moment stretched. Without-Name towered over the slim form of the hominid.
Then Without-Name stepped off the fallen man and pushed him away with her foot. She dropped to her knuckles and laughed. "Your pet has an amusing defiance, Manekato. Nevertheless I tell you that these creatures of the Moon are the key to our strategy here. The key!" And she knuckle-walked away toward the forest, where she blended into the shadows of the trees.
Where she had shoved him, the fallen Zealot had left a trail of urine and blood. Workers hurried forward to tend him, and to clean the mess he had made.
Manekato approached the trembling hominid. _"Nemoto—I am sorry—"_
Nemoto shrugged off her touch. _"So you understand, at last. Let me reward you with a banana."_ And she stalked away, her anger visible in every step, every gesture.
## _R eid Malenfant_
"About the desert," McCann said. He took a half-burned twig and started to scrape at the red dust, sketching out a map. "Here is the Congo—I mean, the great river which rises in the foothills of the great volcano you call the Bullseye, the river that winds its way through the interior of the continent to debouche into the ocean beyond the forests. For much of its length the river's flow is confined to a series of ancient canyons, where the stream is fed by a series of underground tributaries. The north bank is very arid. But on its south bank _—here_ , for example—there are floodplains where the vegetation grows a little more thickly.
"Here is what I propose. We will cut across the plain, meeting the river valley at _this_ point, where there is a crossing place to the south bank, which is the greener. We will follow the river, heading steadily west, following it upstream as it works its way through the mountains, and using the vegetation and its inhabitants as our base resource. Thus we will seek out these shy Runner bands of yours. And if we fail to find your Emma before the character of the country changes—well, we will think of something else."
Malenfant felt tempted to argue with this strategy. But he had no better ideas of how to explore a continent-wide desert, in search of a single person. And there might be a logic to it: Whatever she was doing, whoever she was with, Emma surely couldn't be anywhere else but close to water.
The river, then. He nodded curtly. McCann grinned and scuffed over his map with the sole of his boot.
They heard a cry.
It was Julia. She was hunting a lame deer. She had stripped naked and was running flat out toward it; baffled by a rock outcropping, the animal turned the wrong way, and Julia fell on the animal's neck and wrestled it to the ground.
"Dinner is served," McCann said dryly.
"There must be an easier way to make a living," Malenfant said.
McCann shrugged. "You don't find much to admire about these non-human humans, do you, Malenfant? Don't you envy Julia her brutal strength, her immersion in the bloody moment, her uncomplicated heart?"
"No," Malenfant said quietly.
They entered the desert.
Malenfant sacrificed more parafoil silk to make a hat and a scarf for his neck, and he added a little silvered survival blanket to the top of his hat to deflect the sunlight. After the first couple of days his eyes hurt badly in the powerful light. In his pack was a small chemical-film camera; he broke this open with a rock, and tied the fogged film over his eyes with a length of chute cord.
McCann fared a little better. His ancient suit of skin, well-worn and much-used, had a hood he could pull over his head, and various ingenious flaps he could open to make the suit more or less porous.
Julia's squat bow-legged frame was made for short bursts of extreme energy, not for the steady slog of a desert hike. She struggled as her feet sank into the soft, stingingly hot sand. But she kept on, grinning, self-deprecating, her tongue lolling from her open mouth, her sparse hair plastered to the top of her head.
Anyhow it wasn't a desert, Malenfant supposed; not strictly. Life flourished, after a fashion. In the red dust shrubs and cacti battled for space with the ubiquitous stands of spiky spinifex grass. Lizards of species he couldn't identify scuttled after insects. He spotted a kind of mouse hopping by like a tiny kangaroo. He had no idea how such a creature could survive here; maybe it had some way of manufacturing its own water from the plants it chewed on.
Not a desert, then. Probably a climatologist would call it a temperate semidesert. But it was dry as toast, and hot enough for Malenfant.
It was a relief to them all when they reached the river.
Malenfant and Julia pulled off their clothes and ran with howls of relief into the sluggish water. McCann was a little more decorous, he stripped down to his trousers and paddled cautiously. Malenfant splashed silty-brown liquid into his face, and watched improbably large droplets hover around him; he felt as if his skin was sucking in the water directly through his pores.
Great islands floated past, natural rafts of reed and water hyacinth, emissaries from the continent's far interior, a startling procession of vegetation on its way to the sea. It was a reminder that this single mighty stream drained an area the size of India.
The river flowed sluggishly between yellow sandstone cliffs streaked with white and black. Here and there he saw sandbars strewn with black or brown boulders—mudstones and shales, said McCann, laid down in ancient swamps. The sedimentary strata here were all but horizontal, undisturbed: These were rocks that had remained stable for a great length of time, for a thousand million years and more. This Moon was a small, static world.
Life flourished close to the river. The bank was crowded with plants that craved the direct sunlight, bushes and lianas competing for space. Even behind them the first rank of trees was draped with lianas, ferns, and orchids, overshadowed only by the occasional climbing palm. Wispy manioc shrubs grew on the lower slopes. Speckled toads croaked all along the river bank, and fireflies the size of earwigs, each of them making a spark of green light, danced and darted in the tangled shadows of the trees.
A vast spiderweb stretched between two relatively bare tree trunks. It was heavy with moisture, and glistened silver-white, like strings of pearls. Looking closer, Malenfant saw that many spiders, maybe a hundred or more, inhabited the web. A social species of spider?
Objects hung from the higher branches of the palms, like pendulous fruit, leathery and dark brown, each maybe a foot long.
"They are bats," McCann murmured. "They have wing spans of a yard or more. Those are males. At night they call for the attention of females." He rammed his fingers into his nostrils, and cried, " _Kwok! Kwok!_ And the females fly up and down the line for hours, selecting the male who sings the most sweetly..."
After a time Julia clambered out of the water. She took a handful of palm oil from a wooden gourd in McCann's pack, and worked it into her skin, paying attention to every crease and the spaces between her fingers and toes. When she stood, her skin shone, lustrous. She was silent, beautiful.
McCann went fishing. He found a spot where the bank curved, cupping a still, shallow patch of water, thick with reeds. He took leaves from a pretty little bush with white flowers shaped like bells. He scattered the flowers in the river, over the still spot.
Above the shallower water, by the reed beds, dragonflies hovered and zigzagged, big scarlet creatures the size of small birds. Sometimes they dipped their abdomens into the river, breaking the sluggish, oily surface of the water. Perhaps they were laying eggs, Malenfant mused, wishing he knew more natural history; when you got down to it he knew very little about his own world, let alone this exotic new one.
To Malenfant's surprise, fish started coming to the surface in front of McCann, their fins breaking the oily meniscus, their mouths popping. Evidently they couldn't breathe. McCann, stocky, determined, splashed into the water and started grabbing the fish, holding their tails and slamming their heads against rocks on the bank.
Malenfant thought he saw something move through the water. He scrambled out fast.
It had been bigger than any fish, but not the distinctive shape of a croc or an alligator—something that must have been at least his size, and covered with sleek hair, like a seal. But neither of the others noticed anything, so he didn't mention it.
They spent a day at the side of the river, and replenished their stock of fish, then moved on, heading steadily west.
By noon the following day they had come to a place which showed signs of habitation. A small beach close to the river was littered with blackened scars, perhaps the marks of hearths, and neat rings of holes showed in the ground. When Malenfant walked his boots crunched over a litter of stone tools.
Julia cowered, her huge arms wrapped around her torso.
Malenfant asked, "What is it? A Runners' camp?"
McCann's face was grim. "Runners are not so permanent as this—nor do they make such structures. See these holes? They are for the wooden supports of tents and the like... But see the scattering of the fires, the heaps of discarded tools. Men do not conduct themselves so, Malenfant; we would build a single fire; we would take our tools with us. This is a Ham settlement—or was. And, look, the great thickness of the debris tells of a long occupation, which is of course typical of these dogged, infinitely patient Hams. But it was an occupation that was ended bloodily. Here, and here..." Stains on rocks, that might have been dried blood. "They are recent. It is the Zealots, Malenfant. We must be alert for their scouts."
Julia was clearly distressed here. They moved on quickly.
After that, another day's hike took them to the spot McCann had picked out as a possible crossing place. On the far side of the river, just as he had promised, the land was flatter and less rocky, and there was more life: a few shrubs, some straggling trees, even patches of green grass.
And, stretched between the banks, tied firmly to a rock on either side, there was a rope.
Malenfant and McCann inspected the rope dubiously. It seemed to be of vegetable fiber, woven tightly together into a thick cord.
McCann picked at the rope. "Look at this. I think this material has been worked by teeth."
"It isn't human, is it?"
McCann smiled. "Certainly this is not what our hands would make—but we have never observed the Hams or the Runners use ropes on such a scale, or to have the imaginative intellect to make a bridge—and still less the Elves or Nutcrackers." He looked around coolly. "Perhaps there are others here, other presapient types we have yet to encounter."
Malenfant grunted. "Well, whoever they are, I'm glad they came this way."
Malenfant crossed first. He went naked. He probed at the river bed with a wooden pole as he inched forward, and he dragged another rope, a length of chute cord, tied around his waist. The water never came higher than his ribs.
Once he was across, he and McCann started to transfer their packs of clothes and food. They used a carabiner clip from Malenfant's NASA jumpsuit to attach each pack to the ropes, then pulled at the chute cord to jiggle the packs across.
Julia came next. She entered the water with a dogged determination that overcame her obvious reluctance—which wasn't surprising, as her stocky frame was too densely packed for her to float; whatever else they were capable of, Neandertals couldn't swim. McCann fixed a loop of cord around her waist and clipped her to the chute line with the carabiner clip. Then he and Malenfant kept a tight hold of the chute line as she crossed—though whether they could have retrieved her great weight from the water if something had gone wrong, Malenfant wasn't sure.
It took no more than an hour for them all to get across. They spread out their gear to dry, and rested. Cleansed by the water, lying on warm rocks, Malenfant found he enjoyed the touch of the sun on his face, the arid breeze that blew off the desert.
Julia grunted, pointing at the river. There were creatures in the water.
They were sleek swimmers, their hair long and slicked down, their bodies streamlined. Their hands and feet were clearly webbed—but those hands had five fingers, and the small-brained heads had recognizable eyes and noses and mouths. They were churning in the water, clambering over each other like mackerel in a net. Oblivious of Malenfant and the others, they seemed to be lunging at the sky, their round eyes shining.
They were hominids.
"Swimmers," said McCann morosely. "Sometimes they'll steal fish off your line... The Hams have stories of how a Swimmer will aid you if you get yourself into trouble in the water, but I've never observed such a thing. And, do you know, they appear to sleep with only one eye shut at a time; perhaps they need to keep conscious enough to control their breathing..."
Malenfant imagined a troupe of Australopithecines, perhaps, scooped from some quasi-African plain a couple of million years ago, and dumped by the merciless working of the electric-blue portals on an isolated outcrop of rock on some watery Earth. Ninety-nine out of a hundred such colonies would surely have starved quickly—even if they hadn't drowned first. But a few survived, and learned to use the water, seeking fish and vegetation—and, in time, they left the land behind altogether...
And now here were their descendants, scooped up by another Wheel, stranded once again on the Red Moon.
Hominids like dolphins. How strange, Malenfant thought.
Something immense collided with the back of his head.
He was on the ground. He felt something pushing down on his back. A foot, maybe. One eye was pressed into the ground, but the other was exposed, and could see.
That fat new Earth still swam in the sky.
He heard a commotion. Maybe Julia was putting up a fight. A face—runtish, filthy—eclipsed the Banded Earth.
Once again the back of his head was struck, very hard, and he could think no more.
## _S hadow_
Shadow learned day by day how to live with these new people, here on the slope of the crater wall.
One morning she brought a bundle of ginger leaves she had collected from the forest. She approached the group of women that was, as usual, centered on Silverneck. She sat next to Silverneck, offering the leaves.
A woman called Hairless—left almost totally bald in her upper body by overgrooming—immediately grabbed all the leaves. She passed some to Silverneck and the others. When Shadow tried to get back some of her leaves, Hairless slapped her away.
So Shadow came up behind Hairless and began to groom her. Though Hairless flinched away at first, she submitted.
But now Hairless spotted the baby, clinging to Shadow's neck. She reached out and plucked the baby off Shadow, as if picking a fruit off a branch. Shadow did not resist. Hairless poked her finger in the baby's mouth and fingered his genitals. The baby squirmed, his huge head lolling.
While Hairless probed at her baby, Shadow stole back some leaves.
But Hairless developed a sudden disgust for the malformed infant. She thrust the child back at Shadow, jabbering.
Shadow retreated to the fringe of the group, chewing quietly on her prize.
Shadow was the lowest of the women here. She made her nests on the periphery of the group, and she kept as quiet as possible. Though she clung to Silverneck as much as she could, she was subject to abuse, violence, and theft of her food from men and women alike.
But this community was different from that of Termite and Big Boss. Here, sex was everything.
During some rough-and-tumble play between older infants, a chase and wrestle involved a boy taking the penis of another in his mouth. Soon the wrestling had dissolved into a bout of oral sex and other erotic games, after which the chasing began again.
One day two of the more powerful men came into conflict. One of them was Stripe, the dominant man, a tall, robust man with a stripe of gray hair down one side of his head. The other was One-eye, the shorter, more manic man who had taken it on himself to attack the pack of hyenas with a stick on the day Shadow had joined this new group. The fight, caused when One-eye didn't respond submissively enough to an early-morning show of power by Stripe, escalated from yelling and hair-bristling to a show of shoving and punching. At last one firm kick from Stripe put One-eye on his back.
The smaller man got up, confronting Stripe again. Both men's fur bristled, as if full of electricity—and both had erections. After another bout of shouting, they grew quieter, and One-eye, hesitantly, reached out and took Stripe's erection, rubbing it gently. After a time Stripe's bristling hair subsided, and he briskly cupped One-eye's scrotum.
The contact was quickly over. Neither man reached an orgasm, but orgasms were usually not the point.
Sex was everything. Couplings between men and women, and the older children, were frequent, both belly-to-back and belly-to-belly. Infants became excited during couplings, jumping over the adults involved and sometimes pressing their own genitals against the adults'. But contact between members of the same sex was common, too.
It was a lesson Shadow learned quickly. She learned how to avert a male fist by grasping a penis or scrotum, or taking it in her mouth, or allowing a brief copulation. She earned toleration by groups of women as they fed or groomed by rubbing breasts and genitals, or allowing herself to be touched in turn.
But still, things went badly for her, no matter how hard she worked. She was surrounded by hostility and disgust. The women would push her and her baby away, the men would hit her, and children would stare, wrinkle their noses at her, and throw stones or sticks.
There was something wrong, with herself and her baby. The wrongness began to be embedded in her, so that she accepted it as part of her life.
That was why she submitted to the attentions of One-eye without resisting.
Many of the men, at one time or another, initiated sexual contact with Shadow. She was young, and, save for the lingering _wrongness_ , healthy and attractive. But the contacts rarely led to ejaculation; the man, after being lost briefly in pleasure, would look at her, and his face would change, and he would push her away. After a time most of her contacts came from boys, eager to experiment with a mature woman, and men who for some reason were frustrated elsewhere; she learned to submit to their immature or angry fumblings, and the blows that came with them.
But One-eye was different. Of all the men, One-eye alone developed an obsession with Shadow.
At first his approaches to her were conventional. He would come to her with legs splayed and erection showing, sometimes shaking branches and leaves. She would submit, as she had learned to submit to any demand made of her, and he would take her into the shade of a tree.
But from the beginning his coupling was rough, leaving her breasts pinched and bitten, her thighs scratched and bruised.
After a time his demands became cruder. He would drop the formalities of the invitation and simply take her, wherever and whenever he felt like it—even if she was feeding, or suckling her child, or sleeping in her nest. He seemed to find her exciting and would quickly reach orgasm. But the speed of the couplings did not reduce their violence.
The other women rejected One-eye. If he approached them they would turn away, or run to the protection of the powerful women. His intent, manic strength repelled the women. And so he was forced to prey on the very old and young and weak, who were unable to defend themselves—them, and Shadow, for Shadow got no protection from the other women, not even Silverneck.
Bruised and bloodied, she submitted to his attentions, over and again, and the sex became harsher.
One day Shadow caught a glimpse of one reason why she continued to be shunned.
One-eye had used her particularly hard that day, and some old wounds had been opened by his roughness; she wanted to clear the dirt and blood from the injuries before they began to stink. Deep in the forest, high on the wall of the crater, she found a small, still pool. She leaned over the pool, reaching for the water.
A reflection peered back out at her.
She leapt back, jabbering in alarm. Her infant, feebly crawling in the leaves, fell on her belly and mewled.
Cautiously Shadow crept back to the pond. A face peered out at her, a face made grotesque with a bulbous nose and lumpy protrusions on its cheekbones and brow. The face was alarming and threatening—but of course it was her own face.
Screeching, she dug her fingernails into her face, the swellings there, and tried to rip it off, longing to throw it far away from her. But she succeeded only in making her face bleed, and great crimson drops splashed into the little pool that had betrayed her.
By now, Shadow had no memory of the infected stream from which she had drunk when she crossed the plain, and had no understanding of the fungus infection she had contracted.
She lay down in the leaves, thumb jammed in her mouth. Her child began to sneeze, loudly and liquidly.
Shadow uncurled. She rolled over and picked up her infant. She inspected the child's dribbling nose, then she plucked some leaves and wiped away the snot and dirt. Then she took the softly weeping child to her breast.
Far away she heard a hooting. It was the cry of One-eye, seeking to use her body once more. She curled tighter around her child.
The infant's cold grew steadily worse, developing into a fever that kept him awake during the night.
Shadow quickly grew exhausted, without energy enough even to feed herself, or keep herself properly clean. The swellings on her face now itched constantly. They hurt badly when struck. And they continued to grow, to the point where she could see the fleshy masses framing her eye sockets and cheekbones.
Even in the midst of all this, she was not spared One-eye's voracious demands.
She never resisted him. But out of his sight she would place her sickly infant down carefully on a bed of leaves or a nest of branches. If the coupling permitted it, she would look across that way, and even reach out to touch or stroke the child.
Eventually One-eye noticed this.
It enraged him. He was already lying on top of her. He pinched her chin in his right hand, making her face him, and he punched her hard on the lumps in her brow, making her scream. Then he grabbed her ankles and pushed them back toward her head, and entered her savagely.
When he was done he pushed her away and began to beat her, aiming precise blows at her belly and kidneys. When she curled in on herself he grabbed her arms and pulled her open, making her lie unprotected on her back, and rammed his fist over and over into her solar plexus.
The world dissolved into fragments, red as blood, white as bone.
When she came to she could barely move. Her belly and back were a mass of pain, and one eye was covered with a film of drying blood.
Silverneck had taken her baby. The older woman cradled him on her lap, and was even allowing him to suck on her cracked, dry nipples.
With a groan, Shadow let the world fall away again.
After a time, she was aware of a looming shape before her. Her child was sleeping uneasily at her breast. She cringed, trying to curl tighter.
But a gentle hand touched her shoulder, and pushed her gently back. It was Silverneck. She was carrying a pepper. Its stem had been pulled out, and it was full of water. Shadow drank greedily. But her lips were cracked and swollen, and she felt the water dribble down her chin.
It was dark before she found the strength to clamber a little way up into a tree, and construct a rough nest.
## _R eid Malenfant_
Malenfant was bent double. His arms were pinned behind his back. Something was jolting him, over and over. His head felt like it would explode. It was like the feeling you got after a few days on orbit, when your body fluid balance hadn't yet adjusted to microgravity, and blood pooled in your head.
But when he forced his eyes open—the light stabbed bright, making him squint—he saw, in glimpsed shards, a ground of rust-red dust, powerful bare legs pumping.
Not in orbit, it appears, Malenfant. He was being carried over somebody's shoulder, in a fireman's lift. But his head was upside down, and with every step his cheek crashed into the back of his carrier.
He threw up. It was a spasm of gut and throat; suddenly hot yellow-green fluid was spilling down the naked back before his eyes.
There was a loud hoot of protest. With a shrug he was thrown off the shoulder, as if he were as light as a feather, with a good two yards to fall to the ground.
The fall seemed long, slow-motion. He couldn't raise his bound arms to protect himself. He landed headfirst.
When he came to again his head ached even worse than before. He was lying on his side. All he could see was red dust, and a pair of grimy buckskin boots. His legs were free. But his arms, still pinned behind his back, felt like they were half-wrenched out of their sockets.
A buckskin boot dug into his stomach to tip him over, none too gently. He finished up on his back, as helpless as a landed fish. It felt as if his neck was in his own warm vomit.
Faces loomed over him. One pushed closer. It was a bearded man, aged perhaps forty; his face was round, greasy, suspicious.
Malenfant tried to speak. "Let me up," he gasped.
The man's eyes narrowed. "English? But no argot I ever heard. What are you, a Frenchie?" His accent was thick, the vowels twisted, almost incomprehensible.
Somebody said, "He's sick. Leave him. We ain't here for this."
Beyond the bearded man Malenfant saw McCann; he seemed composed, though his arms were bound. "Sprigge. In the bowels of Christ I beseech you. _He is an Englishman_."
The bearded man—Sprigge—glared at McCann. Then he turned back to Malenfant. "Get him up."
Ungentle hands dug into Malenfant's armpits and hauled him off the dirt. He managed to get his feet on the ground. But he couldn't keep his eyes targeted; they slid sideways in their sockets as if he were drunk, and when he was let go he fell back into the dirt.
His NASA boots were gone. His feet were bare, grimy and bleeding. They even took my socks, he thought. He wondered what had happened to his pack.
Sprigge stood over Malenfant again. "Get up or I leave you for the Elves."
Malenfant slumped forward. He managed to get up onto one knee, got one foot on the ground, and pushed himself up. This time he staggered, and his head still spun, but he stayed upright.
McCann said, "You can't expect the man to walk."
Sprigge nodded, and snapped a finger.
A huge Runner stepped up to Malenfant. He was naked, dust-encrusted—and his head was small, like a child's, though his face was weather-beaten and scarred. From the look of the dribble of vomit down his back, this had been Malenfant's reluctant mount.
The Runner kneeled in front of Malenfant, his hands making a stirrup. Malenfant stared stupidly.
McCann said, "Use him, Malenfant." Now Malenfant saw that McCann was sitting on the shoulders of another huge Runner, like a child riding on its father. The Runner's head was bowed, his eyes fixed on the dirt. McCann seemed relaxed, almost comfortable. "Follow my lead, Malenfant. One must keep up the front."
"... I."
Julia walked up to Malenfant. Her head was bowed, and her wraps of skin had been ripped away, leaving her naked. But her hands were unbound.
Sprigge touched his belt, where a whip was coiled.
Julia kept her gaze directed at the dirt, not looking the humans in the face. She said, "Carry Mal'fan'."
Sprigge barked a laugh. "So you use a Ham's quim, Sir Malenfant. Your punishment will sting if you let Praisegod Michael witness such iniquity." But he stepped back.
Julia slid her arms under Malenfant's body and lifted him effortlessly, like a child.
The party formed up and began to move off over the dirt.
The party was made up of perhaps a dozen Runners. Most were naked, though some wore loincloths. Some of them bore heavy packs, or loads on their heads and shoulders. Two of them were dragging the carcass of an immense bull antelope on a crude travois. The rest of the Runners had passengers: buckskin-clothed men sitting on their shoulders, stubby whips in their hands. All the Runners walked silently, just waiting for instruction. Several of them had scars striped on their shoulders and bellies.
There was one other hominid: a Ham, dressed in clothes as comparatively well-stitched as those of the humans. He carried a whip; perhaps he was a supervisor, a boss.
Malenfant saw that Julia's breasts were scratched, as if by fingernails, or teeth. "Did they hurt you?"
She did not answer.
McCann's Runner came trotting alongside. "She shouldn't speak to you, Malenfant," McCann said urgently. "It will be a whipping for her if she does, and perhaps for you. She knows how to behave with these types; you must learn, and fast. These brutes had a little gruesome fun with her, but yon Constable Sprigge stopped them. I sense there is a core of decency in that man, under the dirt and violence. Perhaps that will assist us as we deal with these Zealots..."
"Zealots," Malenfant growled.
McCann said grimly, "I did not expect to encounter them here. They are clearly expanding their area of operations—which is all the worse for us. Listen to me, Malenfant. Your romantic quest for Emma is going to have to wait. It's vital to keep up a front. All that keeps us from doing the carrying rather than being carried is that these fellows accept us as human beings. So you must act as if it is your privilege, no, your _right_ , to use the muscles of these poor creatures as if they belonged to you. And don't forget, you're English." He eyed Malenfant. "A colonial type like you might take it as a great indignity to have to impersonate a Britisher. But I believe any of these ruffians would run you through if they suspected you were a French or a Spaniard or a Portugoose..."
Malenfant said bitterly, "You know what? I miss America. In America you can travel more than a couple of miles without getting robbed, attacked, kidnapped, or trussed up."
"Chin up, sir. Chin up."
Malenfant's thinking dissolved. Lulled by the stink of the dust, his weakness, and Julia's steady warmth, he dozed.
Somewhere thunder cracked, and when he looked up he saw more fat clouds scudding across the sky.
Half a day after the capture of Malenfant and the others, the party reached the fringes of the Zealot empire.
They crossed a plain scattered with broken rock fragments. The rim of a broad young crater loomed over the horizon; perhaps they were in the crater's debris field. In any event it was slow, difficult going, as the Runners had to pick their way past huge sharp-edged boulders.
They came to a place where a thin, sluggish stream ran, and green growing things clustered close to its banks. The land had been cleared. Malenfant saw how the rocks had been piled up into waist-high dry stone walls, mile after mile of them. The rocks must have been broken up before they were moved, a hell of a labor—but then labor was cheap here.
In a field close to the river, a team of Runners was drawing a wooden plough. The four of them were bound together by a thick leather harness, and wooden yokes lay over their shoulders. The Runners were followed by a Ham, a stocky man who carried a long whip.
When Sprigge's party came alongside, the Ham overseer stared at Julia. Then he turned back to his charges and lashed them, a single stroke that cut across all four backs. The Runners, their faces empty, did not look up from the dirt they tilled.
"Good God," Malenfant said, disgusted.
"It would pay you not to blaspheme in this company," McCann said evenly. "And besides, is it any less cruel to use an ox or a horse for such a purpose?"
"Those draught animals aren't oxen, McCann. They are hominids."
"Hominids, but not people, Malenfant," McCann said sadly. "If they have no conception of pain—if even their Ham boss does not—then what harm is done?"
"You can't believe that's true."
McCann said stiffly, "I would sooner believe it than join those poor Runner gentlemen behind their plough."
They passed a small farmhouse, just a rough sod hut. In a yard of red mud, children were playing—they looked like human children, a boy and a girl. They gazed at the approaching party, then ran into the hut. A man emerged from the hut, stripped to the waist, bareheaded. He looked apprehensive.
From his Runner mount, Sprigge nodded to him. "No tithes to collect today, George."
'Aye, Master Sprigge.' The man George nodded back, cordially enough, but his eyes were wary, fixed on Sprigge as if he were a predator.
They moved on, following the river as it worked its way toward the Beltway forest. As the land became less arid, the cultivation spread away from the river bank. Soon Malenfant was surrounded by fields, toiling hominids, an occasional human. It might have been a scene from some vision of the old west, or maybe the European Middle Ages, if not for the humanoid forms of the beasts of burden here, the unmistakeably Neandertal features of their supervisors, and the unremitting crimson glower of the land itself.
But this was a genuine colony, he thought, a growing community, for all its ugliness—unlike the dying, etiolated English camp.
Rain began to fall. The rough path by the riverbank soon turned to mud, and the party trudged on in miserable silence. Malenfant tucked his head closer to Julia's chest. With remarkable kindness she leaned over him and sheltered him from the rain with her own bare back, and Malenfant could not find the strength to protest.
Again he dozed.
When he woke, he was dumped on his feet. They had reached the Zealot fortress, it seemed.
They were in a clearing, surrounded by dense wood; Malenfant hadn't even noticed they had come back to the forest. Ditches, ramparts, gates, and drawbridges stretched all the way around the township. Sharpened stakes were stuck in the sides of the ramparts, so that the compound bristled, like some great hedgehog of wood and mud.
A big gate was opened. They were pushed inside.
The encampment was a place of rambling muddy paths and ugly, low-tech buildings placed haphazardly. There was one central building that looked more sturdily built, mud brick on a wooden frame, like a chapel. Aside from that, the huts were so rough they seemed to have grown out of the debris that littered the muddy ground. They were built of stripped saplings and wattles, and laid over with palm fronds. Everything showed signs of much use and recycling; here was half of what looked like a dugout canoe, for example, serving as a chicken coop.
There were no straight lines anywhere, no squares or rectangles, no hard edges; everything was sloppy, all the lines blurred. It was as if the first arrivals here had just marked out trails where they wandered and put up their wattle-and-daub huts where they felt like. There was none of the regularity and discipline of the British compound: Malenfant sensed McCann's impatience at this disorderliness.
Malenfant's arms were untied. He could barely move them because of cramp, and he could feel where the cord had cut into his wrists.
With McCann, he was pushed into a dark, stinking sod hut. He couldn't see what had become of Julia. The hut was dark, the floor was just mud, uneven. A door of saplings bound together by liana twine blocked the door.
Malenfant limped to a dark corner and slumped there. The floor was greasy and black; when he lifted his hand a great slick sheen came away with it. The whole place stank like a toilet.
Termite passageways, like the stems of some dead plant, curled up the walls and disappeared into the wooden beams and the thatch. A gecko clambered across the ceiling, incurious.
He hadn't eaten or drunk anything since being hit over the head by the Zealots. He felt as if he had been systematically pummelled, all over his body, with a baseball bat. And here he was in some quasi-medieval prison block, lying in filth. The world he had come from—of NASA and Houston and Washington, of computers and phones and cars and planes—seemed utterly unreal, evanescent as the shining surface of a bubble, a dream.
What a mess, he thought.
McCann was waxing enthusiastic. "I see the pattern, Malenfant. The Hams and Runners surely do not have the wit to be rebellious or to long for escape; unlike human slaves it is unlikely they can conceive of _freedom_. Besides, if you get them young enough, you can quite easily break their spirits, as with a young horse. If each man controls, say, ten of the Ham bosses, and then each Ham in turn controls ten Runners, you have a formidable army of workers. And at the top of it you have this fellow Praisegod Michael of whom Sprigge has spoken, who creams off the tithes. It is like a vast, spreading, self-sustaining—"
"Prison camp," Malenfant said sourly.
"Oh, much more than that, Malenfant. Think how carefully the strata of this little society are defined. You have the humans, with of course their own ranks and order. Beneath them you have your Hams, who in turn lord it over the Runners. And since in this case each lower rank is clearly the intellectual inferior of that above, you have a social order that reflects the natural order. It is a hierarchy as stable as a cathedral."
Malenfant growled, "I thought you despised the Zealots. You wouldn't tell me a damn word about them."
"I think I am beginning to see I have underestimated them, Malenfant. Oh, this is a place of repellent squalor, of blood, and mud. It _is_ cruel, Malenfant. I don't deny it. But those subject to the greatest cruelty, as far as I can see, are those least capable of perceiving it. And as a social arrangement it is intricate and marvelous. One must admire efficiency when one finds it, whatever one's moral qualms."
He sounded brittle, almost feverish, Malenfant thought dully. This bizarre mood of his, his fan-worship of the Zealots, could probably evaporate as fast as it had come.
The hell with it. Malenfant closed his eyes.
But still, he saw Emma's face in his mind's eye, bright and clear, as if she stood before him. He probed a pocket on his sleeve. The spyglass lens still nestled there, hard and round under his fingers, comforting.
McCann went to a window—just a hole in the wall, unglazed. He called, "We need water and food. And tell him, Sprigge! Tell your Praisegod Michael we are Englishmen! It will go worse for you if you fail!"
McCann shook him awake. "We have an invitation to dinner, Malenfant! How jolly exciting."
A sullen Zealot had brought them a wooden pail of water. They both inspected this suspiciously; they were ferociously thirsty, but in the dim light diffusing from the window, the water looked cloudy.
McCann shrugged. "Needs must." He plunged his hands into the water and scooped up mouthfuls, which he gulped down.
Malenfant followed suit. The water tasted sour, but it had no odor.
When they were done they used the rest of the water to wash themselves. Malenfant cleaned dried blood and grit out of wounds on his bare feet, wrists, and neck.
McCann used the water to slick down his hair. He even produced a tie from one jacket pocket and knotted it around his neck. "Impression is everything," he said to Malenfant. "Outer form. Get that right and the rest follows. Eh?"
The door was pushed open, its leather hinges creaking. Sprigge walked in, looking as dusty as when they had all walked in from the plains. "You have your wish, gentlemen." He raised his fist. "But any defiance or dissimulation and you'll know my wrath."
McCann and Malenfant nodded silently.
They were led out of the hut, into a broad compound. It was raining, and the evening was drawing in. The ground was just red dirt, hard-packed by the passage of human feet. But it was heavily rain-soaked, and Malenfant felt the mud seep between his naked toes.
People moved between the huts, carrying food and tools or leading children by the hand. They seemed to be humans, but they were small, skinny, stunted folk, dressed in filthy skin rags. There were no lanterns, and the only light inside the huts came from fire hearths.
McCann murmured to him like a tour guide. "They do not approach us; the authority of this Praisegod Michael of theirs is binding. Look there. I think that hut yonder is a house of ill-fame."
"A what?... Oh. A brothel."
"Yes, but a brothel stocked with Runners—women and boys, so far as I can tell. There are contradictions here, Malenfant. We have a community run by this Praisegod fellow, seemingly on rigid religious lines. And yet here is a bordello operating openly."
The rain grew heavier. The Zealot compound was turning to a muddy swamp. The buildings seemed to slump in defeat, as if sliding back down into the earth from which they had been dragged. And the people, humans, Runners, and Hams alike were wan figures, all the same dun color, images of misery.
McCann stamped through puddles contemptuously. "These people don't know what they are doing," he barked. "We coped rather better. Culverts! Storm drains!" And with broad sweeps of his arms he sketched an ambitious drainage system.
They were brought to the compound's central structure, the solid-looking chapel. Well, maybe it really was a chapel; now Malenfant saw it had a narrow spire.
Sprigge led the two of them along a short, dark hallway. Grilles of tightly interwoven wooden laths were set in the floor. Malenfant glanced down. He thought he saw movement, eyes peering up at him. But the light was uncertain.
They arrived at a large, bright room. It had neat rectangular windows—unglazed, but covered with sheets of what looked like woven and scraped palm leaves, so that they admitted a cool yellow light. Lanterns burned on the walls, each just a stone bowl cupping oil within which a wick floated, burning smokily. At the head of the room was a stone fireplace, impressively constructed from heavy red blocks—perhaps ejecta from the crater field they had crossed. No fire burned beneath the blackened chimney stack, but there was a large, impressive crucifix set over the fireplace. At the other end of the room was a plain altar, set with goblets and plates, all of it carved from wood.
At the center of the room was a small, unevenly made, polished wooden table. A man sat behind the table, eating steadily. There were no plates; the man ate bits of fish and meat off what looked like slabs of thick bread.
The man wore a black robe that swept to the ground, with a napkin thrown over his shoulder. A band of silver-gray hair surrounded a crown that looked shaved, like a tonsure. His narrow face was disfigured by warts.
This was, presumably, Praisegod Michael. He ignored Malenfant and McCann.
Behind Praisegod two Ham women stood, backed up against the wall. They were both dressed in modest, all-covering dresses of soft leather, and they kept their eyes on the floor.
Sprigge nudged McCann, and indicated they should sit on the floor before the table. McCann complied readily enough. Malenfant followed his lead. Sprigge stepped back, and took a station at the corner of the room.
As Praisegod Michael ate, everybody in the room waited in silence.
Malenfant couldn't take his eyes off the food.
There was a puree of what looked like chicken mixed in with rice and some kind of nuts. An animal like a young piglet, roasted, had been carved and set before Michael, and he picked at its white flesh. Other side dishes included some kind of beans cooked in what smelled like meat stock, and mushrooms in a kind of cream, and a green salad. There was even wine—or anyhow it looked like wine, served in a delicately carved wooden goblet.
At length Praisegod Michael slowed down. More than half the piglet was left on its serving plate. Michael belched, and mopped his lip with a scrap of cloth.
Then he looked up, directly into Malenfant's eyes. Malenfant was jolted by the intensity of his gaze.
One of the Ham women behind him stepped forward. Malenfant was startled to recognize Julia. With heavy grace she took the unfinished dishes from Michael, and set them on the floor before McCann and Malenfant.
Malenfant reached straight for the pork, but McCann touched his arm.
McCann closed his eyes. "For this blessing, Lord, we thank You."
Michael watched coldly.
Now McCann began to eat, using his fingers to tear at the pork. Malenfant followed suit.
Michael spoke. "Your Ham girl is well-tempered," he said to Malenfant. His voice was deep, commanding, but his accent was powerfully strange.
Malenfant said, "She isn't _my_ anything."
McCann said quickly, "She has an even nature, and is wise for a Ham."
Michael's gaze swivelled to McCann. "I know of you, or at least men who speak like you. Once one was brought here."
McCann blanched. "Russell. Is he—"
"He died for his sins."
There was a long silence. McCann's eyes were closed, even as he chewed steadily on the meat. Then he said carefully, "There are only a handful of us—a handful, and Hams and Runners. We have no women, no children. We are weak old men," he said, looking directly at Michael. "We are no threat to your—umm, your expansion."
Michael got out of his chair. Tall, cadaverously thin, his arms clasped before his belly, he walked around the table and studied McCann and Malenfant. "My soldiers will spare them."
"They live in God," McCann said fervently.
Michael nodded. "Then let them die in God. But you talk of an _expansion_."
McCann said hastily, "I am sorry if—"
"Whenever anything in this world is exalted, or exalts itself, God will pull it down, for He alone will be exalted," said Praisegod Michael. His speech was rapid, his delivery flat. He laid his hand on Julia's flat brow; she did not react. "My language is not of kingdoms and kings, empires and emperors. No king I, but a Protector," he said.
McCann was nodding vigorously. "I see that. Yes, I see that. As men we are different—we come from different worlds—but differences between men are as nothing compared to the gulf between men and animals. There are few enough strong men scattered over this world, Praisegod Michael, to shoulder the responsibility."
Michael regarded him. "God hath poured this confused nation from vessel to vessel, until He poured it into my lap. Perhaps it is divine providence that brings you here."
McCann smiled. "Providence, by God's dispensation. Indeed."
Praisegod Michael turned to Malenfant. "And what of this one? His eye is defiant, his accent strange. What is your religion, man? Popish? Atheistical?"
McCann said quickly, "His faith is as strong as mine."
Michael smiled thinly. "Then perhaps he will have the courage to say it for himself." He seemed to come to a decision. "You are right. There are few enough decent men here. But can I trust you?... Tomorrow we hunt. Accompany me, and we will talk further." He knelt before his altar, his eyes closed.
Sprigge motioned Malenfant and McCann to follow him out of the room.
Back in their crude hut, McCann seemed excited. "He is English—that is clear enough—but I would say that his history must have split off from our own no later than our seventeenth century... Perhaps you number your dates differently. Well, it looks as if the Zealots have been here since then. But they seem to have made no significant progress, socially or mechanically, since those days..."
Malenfant said sourly, "What difference does it make?"
" _We understood each other_ , Malenfant. Don't you see? Myself and this Praisegod. His is a faith which has much in common with my own. He spoke of providences. Through providences, you see, God intervenes in the world, to make His will visible. And I have no doubt that Praisegod will count himself among the Elect—that is, those who are already destined to be saved—but he has surely been cast in a world of Reprobates, the already damned." He smiled, and his eyes glinted in the dark. "I understand him. I can do business with this man."
Malenfant frowned. "But his 'business' seems to be to enslave those he regards as lesser than him."
"Ah, but that's the delicious irony of it all, Malenfant—oh, but I forget, you slept and did not see—I spied a man coming out of the Runner bawdy-house, his trousers dangling around his knees. A more unspeakable wretch you never saw. But _I could make out clearly that he had a tail_. Malenfant, our grandiloquent Praisegod Michael, the savior of the world, has a monkey's tail!"
After a minute, Malenfant began to laugh. McCann joined in. Once they started, they couldn't stop.
## _J oshua_
Joshua and Mary, breathing hard, stepped gingerly over crushed branches and uprooted shrubs. They reached the edge of the cliff and peered down. The sky seed still lay where it had fallen when they had pushed it over the cliff: trapped well below the lip of the cliff, pinned by a ledge and a thick knot of shrubbery.
Joshua grinned. Every few days he had come clambering up the trail to this battered clearing, to see again what they had done to the sky seed.
The seed was safe here. The feeble muscles of the Zealots would never succeed in hauling this prize up from such a place—and the Nutcracker-folk, though good climbers, were surely too stupid even to envisage such a thing. Only the People of the Gray Earth, with their brains and powerful bodies, could retrieve the sky seed from where it rested, pinned against the cliff's gray breast—
Voices screamed, all around them.
They whirled, shocked.
There were only trees and bushes and leaves, some of them shaking violently, as if in a wind, though there was no wind.
From nowhere a spear flew. It lanced into Joshua's shoulder, neatly puncturing it through.
He was knocked back. He fell on the spear. It twisted, and there was savage pain.
And now something new descended over him, a thing of ropes and threads knotted together, that tangled up his legs and arms and head.
Leaves and twigs fell away, and suddenly there were people: men, all around them. They were Skinnies. They carried spears and knives that glinted. Still screaming, they threw themselves forward. It had all happened in a heartbeat, overwhelming, bewildering. The Zealots had just melted out of the trees: One instant they were not there, the next they were there, an overwhelming magic beyond Joshua's experience.
Their blows and kicks were feeble, but there were many of them, and they clung to Joshua's limbs while punching his stomach and chest and head. He heard Mary cry out, an angry, fearful roar.
"... Looks like Tobias was right. A fine old pair we trapped here!"
"Wrap up yon buck and give us a hand with the maid, will you? She's struggling like a bear..."
Joshua lay passively, defeated by shock as much as the spear, peering up at the indifferent sun. He saw that the men had got Mary on the ground, and had ripped open her skins.
"By the tears of the Lord—"
"Get her legs. Get her legs."
"The buck is for the minister. This one's for us, eh, lads?"
"Face like a bear but the tits of an angel. She's going to take a bit of stilling, though..."
Joshua came to himself. With a bellow he wrenched himself over, rolling onto his belly. Zealots, yelling, went flying. For a moment he was free of their weight and their blows. But the spear ground into the dirt, opening his wound wider, and he cried out.
But Joshua's struggle had distracted Mary's attackers, and she had got one arm loose. With a fist more massive than any Skinny's, she pounded at the temple of one of her assailants. Joshua heard the crunch of bone; a Zealot went down.
"God's wounds. Peter—Peter!"
"Get her, lads!"
Mary struggled to her feet, her ripped skins swinging, her small breasts glistening with blood. She had her back to the forest. The men, all save the fallen one, made a half-circle to face her, wielding their weapons. Their lust had been replaced by caution, Joshua saw, for even a half-mature Ham girl, if free, was more than a match for any one of the Skinnies.
But she could not defeat them all.
With a last, regretful glance at Joshua, she turned and crashed into the trees. Though she made an immense racket, she had soon disappeared out of sight, and Joshua knew that the Zealots could not follow her.
He let his head slump to the blood-soaked ground beneath his face.
A shadow crossed him. "This is for Peter."
A boot hurled at his face.
## _R eid Malenfant_
The morning after their capture, Malenfant and McCann found their door was not barred, no guard posted.
They crept out into light still tinged gray with dawn.
Already the business of the day was starting. Runners and Hams were working silently to sweep the ground clear of yesterday's debris, and to fill the water casks that sat outside each hut. It was strange to see specimens of _Homo neandertalensis_ and _erectus_ dressed in crudely-sewn parodies of clothing, their heads and bodies strikingly misshapen in the uncertain dawn light, coming and going as they pursued their chores. It was like a mockery of a human township.
Away from the Zealots, neither Hams nor Runners made any attempt to use human language; they simply got through their work with steady dullness, united in blank misery.
There was a specialized group of Runners who were used solely to carry passengers. Some of them wore primitive harnesses. But these unfortunates were stooped, with overdeveloped shoulders and necks, and what looked like permanent curves to their backs. Their shoulders and thighs bore bright red weals.
Malenfant said, "Look at those scars. These Zealot jockeys don't spare the whip."
McCann grunted, impatient. "Have you much experience in the husbandry of animals, Malenfant? None of them look terribly _old_ , do they?—I would wager that under excessive loading their bodies break down rather rapidly once the flush of youth is over.
"But the whip is surely necessary. In Africa I knew a man who tried to train elephants. You may know that while your Indian elephant has been tamed by the locals for centuries, your African runs wild. My acquaintance struggled to master his elephants, even though he imported experienced mahouts from India; freedom runs in the blood of those African tuskers, and they are far more intelligent than, say, a horse."
"Hence the whip."
"Yes. For it is only by severe and strict punishment that such intelligent beasts can be controlled. Even then, of course, you can never be sure; even in India the tamest-looking elephant with a grudge against his mahout may wait years, decades—but he will take his one chance and gore or trample his tormenter, careless of his fate.
"Now your Runner, who is after all a man, if a different stripe of man, is surely more intelligent than an elephant. Hence, as you say, the whip. And perhaps other practices have been developed. See there—that grizzled, rather bent old chap is tied up to the boy." The old man and the boy, sitting in the dirt, listless and naked, were attached by tight bonds around their ankles. "If you want to break an animal you will sometimes put him in with an older beast. The tamed creature may prove an example in the work to be done, and so forth. But in addition the young perceives there is no hope, you see, and quits his rebelliousness sooner."
Malenfant said, "I don't understand why these Runners don't just up and get out of here."
McCann pulled his walrus moustache. "These boys have probably been in captivity since they were very young—either born here, or wrest from their dead mothers' arms in the wild. They know nothing else; they cannot imagine freedom. And these wretches could not run off if you turned them free tomorrow. See how they limp—the scars on the backs of their ankles? Hamstrung. Perhaps that explains their demeanor of defeat. They are creatures evolved, surely, for one thing above all else—running—and if they cannot run anymore, they have no aspiration. Perhaps it is humane to excise the very possibility of escape; believe me, hope harms a creature far more than despair ever did..."
Praisegod Michael emerged from his chapel-like residence. His black robe flapped about his ankles, heavy, as he walked. He threw his arms wide, loudly sniffing the air. Then he fell to his knees, bowed his head, and began to pray.
Praisegod's hunting party formed up rapidly. There were to be five humans (or near-humans)—Praisegod, his man Sprigge and one other Zealot, and Malenfant and McCann—along with four Hams and ten Runner bearers.
One of the Hams was just a child, about the size of a human ten-year-old. This boy seemed dressed in clothing of a somewhat finer cut than most of the Zealots. Praisegod kept him close by, sometimes resting his hand on the boy's flattened skull, or cupping him under his chinless jaw. The boy submitted to this, and ran small errands for Praisegod.
Five of the Runners were to carry equipment—homemade spears and crossbows. The rest were there to carry the humans.
Malenfant's mount was to be one of the older, more broken-down specimens he had observed that morning. The hominid stood before him, as tall as Malenfant despite his stoop, his very human eyes empty of expression.
Malenfant flatly refused to climb aboard his shoulders.
McCann leaned toward him. "For God's sake, Malenfant," he hissed.
Praisegod Michael watched this with a thin amusement. "Do you imagine you spare this stooped one discomfort or indignity? There is no soul behind those deceptive eyes, sir, to experience such complicated passions. I trust your compassion will not pour away when your bare feet are bleeding and sore... But perhaps you are right; he is rather worn down." He nodded to Sprigge.
Sprigge tapped the old Runner's elbow, and he obediently knelt on the ground. Sprigge stepped behind him and drew a knife from his belt—metal, very old, sharpened and polished until the blade was a thin, fragile remnant.
"Shit." Malenfant lunged forward, but McCann grabbed his arm.
Distracted by the commotion, the Runner saw the knife. His battered face twisted in animal rage. He started to rise, perhaps for the first time in his life defying those who used him.
But Sprigge wrestled him to the ground and knelt on his back. He sliced the knife through the old Runner's throat. Blood spurted, a brighter red shining in the crimson dirt. Still the Runner fought; he didn't stop struggling until his head had been all but sawn off his body.
McCann released Malenfant. "The rogue elephant and the mahout, Malenfant," he whispered grimly. "And if you defy, you will only make matters worse for the creatures here."
"Thank you, sir," Praisegod said to Malenfant, his look calculating, mocking. "You perceived a lack which I have been remiss in correcting. Well, it is done, and the sun is already high. Come now." And he slapped the face of his own mount, who trotted away to the west, away from the rising sun.
The others hastily mounted, and the hunting party proceeded at a steady jog after Praisegod, the Runners' bare feet thumping into the earth, the Hams following the graceful Runners as best they could with their awkward, bow-legged style.
They reached the fringe of the forest, and moved out onto the plain.
The forest floor hadn't been so bad for Malenfant's bare feet, save for bites, for which he'd no doubt suffer later. But after a half-mile of desert his feet were aching and bloody. And as the miles wore away he began to dig deep into his already shallow reserves of energy. Malenfant knew they had had no choice but to go along with Praisegod Michael's invitation to join his hunt, which was obviously some kind of bullshit character test. He tried to see it as an opportunity. But there was nowhere to run, nowhere to hide.
He found his thoughts dissolving, his purpose reducing merely to a determination to keep one foot moving in front of the others, to show no weakness.
The weather fell apart. A lid of boiling cloud settled over the sky, making the small world seem flat and enclosed, washing the colors out of everything. And then the rain came, a ferocious storm that stippled the crimson sand with miniature craters. Much of the water drained quickly into the dry soil, but soon rivulets were running over the ground, and the sand turned into clinging mud.
Praisegod called a halt. The humans dismounted. Malenfant rested, hands on his knees, breathing deep of the thin air.
Under the brisk supervision of the Hams, the Runners unloaded sheets of sewn-together leather. They quickly put together a kind of teepee.
The Zealots, with McCann and Malenfant, huddled in the teepee. Inside there was a stink of old leather and damp bodies and clothing. The other hominids were excluded—all save Praisegod's Ham boy, who snuggled close to the Zealot; Praisegod stroked his cheek with in-turned fingers. The other Hams had a few sections of skin that they held up over themselves, to keep the rain off their heads.
As for the Runners, they had no shelter at all. They huddled together under a rain so thick it turned the air gray, their knees tucked into their chests, naked, visibly shivering.
McCann saw Malenfant watching the Runners. "You should not concern yourself," he said. "In the wild they have no conception of shelter. If it rains they get wet; if they catch a chill they die. Nothing in their present circumstances changes that."
Praisegod had been reading passages in a book, a clumsy thing of scraped-leather pages, presumably a Bible or a prayer book. He leaned forward, as if trying to find a more comfortable position for the comical, stubby tail he must have curled up under his robe. "I suspect you fear the rain, Malenfant."
Malenfant frowned. "Ah, bullshit. All this turbulent weather has got to be a result of that new Earth in the sky. It's a bigger world: You're going to get tides, quakes, atmospheric disruptions—"
"Your language is a jabber. Perhaps you believe the rain will wash away this puny world, and you along with it. Well, it will not; for if this island resisted the very Flood itself, a little local rain will not harm it now."
"Ah." McCann was smiling. Malenfant could tell what he was thinking. _This is what this guy believes. Don't say anything to contradict him_. McCann said, "We are on an island, an island that survived the Flood. Yes, of course." He glanced out at the huddled Runners. "And that explains _them_."
Praisegod said, "They are less than men yet more than the animals. What can they be but _Homo diluvii testi_ —witnesses of the Flood? This island was spared the rising waters; and so were its inhabitants, who must have crowded here with the ignorant instincts of any animal."
"Then," said McCann carefully, "we are privileged to glimpse the antediluvian order of things."
"Privileged or damned," Sprigge muttered, staring at the Neandertal boy on Praisegod's lap. "This place is an abomination."
"Not an abomination," snapped Praisegod. "It is like a strange reflected Creation. Man was born to look up at the orders of beings above him, the angels, prophets, saints, and apostles, who serve the Holy Trinity. Here, we look _down_ , down on these creatures with men's hands and faces and even tongues, but creatures without mind or soul, who sprawl in the mud."
They talked further, an incoherent conversation of disconnected fragments, peppered by misunderstanding, suffused by mistrust. But Malenfant slowly learned something of Praisegod Michael.
The Zealot township had been a godless place when Praisegod was a child, given to anarchy and lawlessness, weakened by the endless green lure of the forest. But—so Michael was told by his parents—God was involved in every detail of life. God watched the daily deeds of men and punished their sins, and the Elect—those who obeyed God's law—would be saved. Praisegod learned this in prayer and torment, in misery and distrust, at the hands of what sounded to Malenfant like abusive parents.
And then they abandoned him, just melted away into the bush, leaving the child to the tender mercies of the townspeople.
Life had been very hard for the young Praisegod, it seemed. But eventually he had rediscovered the religion inside himself. He drew strength from this inner core. And when the growing, toughening Praisegod had come to see that he himself was one of the Elect, his duty had become clear: to devote himself to God's fight and the establishment of His kingdom on this fragmentary world.
He had pursued that goal from then on with an ever-burning zeal and an unswerving fixity of purpose that had turned this gaunt, lisping, wart-ridden preacher into something like a man of true destiny.
But there was a cost, of course.
To the Zealots, it seemed to Malenfant, the other hominids, the presapients, barely even existed. They had no language, no clothing, no religion, and therefore they had absolutely no rights under God or man. They were animals, no more than that, regardless of the curiosity of their gaze, the pain in their cries, their misery in enslavement: simply a resource for exploitation.
Malenfant leaned forward. "I'm curious. What do you want, Praisegod Michael? What do you want to achieve among all these animals?"
Michael's eyes were bright. "I seek only to emulate Ramose, who led his nation out of Egypt to the land of Canaan..." Malenfant soon realized that this "Ramose" was a kind of analogue of Moses from his own timeline, like the John who had replaced Christ in McCann's history. "I believe I have seen the providence of God, for surely it is by His dispensation I have been given my place here. And I have no choice but to follow that providence."
McCann seemed to be growing agitated. "But one must search for the truth of providences, Praisegod Michael. One must be wary of the exaltation of the self."
Michael just laughed. "You have not lived in this land long. You will learn that it is only _I_ who stands between these mindless apes and chaos itself." His hands, apparently without conscious volition, stroked the Neandertal boy's broad chest. He glanced out of the teepee's flap door; the rain had slackened. "Come. Time enough for theology later. For now there is a hunt to be made, bellies to be filled." And he led the way out of the teepee.
"The man is too much," McCann said, glowering at Praisegod's back. "He takes divinity on himself. He is close to blasphemy. He likens himself to Bay—that is, his own twisted version of Bay." Malenfant guessed that Bay was another of Moses' parallel-historical pseudonyms. "Malenfant, the man is a self-aggrandizing monster. He must be stopped. Otherwise, what will come to pass, as Praisegod's blasphemous hordes swarm like locusts over this wretched Moon?"
Malenfant shrugged. For all McCann's talk of Praisegod's ambitions, he found it hard to take seriously anybody who lived in a mud hut. "He's vicious. But he's a shithead. Anyhow I thought you were going to do business with him."
McCann glared at him, angry, frustrated. And Malenfant saw that McCann's mood had switched, just as he had feared. It was as if a veneer had been stripped away.
Malenfant felt only dismay. He just wanted to get out of here; if McCann went off the rails, he had no idea how he was going to handle the situation.
Now there was a commotion up ahead. Sprigge had reached the huddle of Hams. Two of them were standing unsteadily, while the third sprawled in the mud. Sprigge began to beat the Hams vigorously.
"It is the wine," Praisegod remarked. "They steal it from us and hide it in their clothing. Though their bellies are large, their brains are small, and they cannot take it as men can."
The Runners watched apathetically as the Hams were chastised.
The sky cleared rapidly. Through high thin clouds the sunlight returned. The red dust began to steam under their feet, making the air humid.
A little after noon, they reached the fringe of a belt of dense forest. They made a rough camp in the shade of the wood, spreading out their clothes and goods to dry. The Runners were tied up by their necks or ankles to tree trunks, but were able to forage for food among the roots of the trees.
McCann nodded. "Efficient. It saves their carrying their own provision. And while their fingers are nimble with food, their minds are too empty to puzzle out knots."
Sprigge was to lead a hunting party into the forest. He would take four Runners, and—as a punishment—all three Hams, who seemed to have crashed into catastrophic hangovers. Both McCann and Malenfant were invited to join them; McCann agreed to go, but Malenfant refused.
Praisegod settled down on a sheet of leather. The other Zealot, a squat, silent man, dug foodstuffs from out of the Runners' packs and laid them out. Praisegod nibbled on nuts, fruit, and dried meat; he pressed tidbits into the mouth of his Ham boy, fingering the child's lips each time.
Malenfant sat in the dirt, waiting for a turn at the food. The silent Zealot sat alone some distance away, chewing on something that looked like beef jerky; he watched Malenfant warily.
Praisegod said, "So you declined to join the hunt, Sir Malenfant." He smiled coldly. "You are not a hunter, then—not a woodsman or a man of the heath either, I would say. What, then? A scholar?"
"A sailor, I guess."
"A sailor." Praisegod chewed thoughtfully. "In my father's day some effort was made to escape this antediluvian island. Men took to the desert, which stretches west of this place. And they built boats and took to the sea, which stretches away to the east. Most did not come back, from either longitude. Those who did reported only emptiness—deserts of sand or water, the land populated by lowly forms. Of course you and your friend have yet to confess what marvelous ship, or providential accident, brought _you_ here."
"So that you can use it to get out of here," Malenfant said cautiously. "Is that what you want?"
Praisegod said, "I do not long for escape. I know what _you_ want, Reid Malenfant, for I have discussed your state of mind with your wiser companion. You seek your wife. You have wagered your life, in fact, on finding her. It is a goal with some nobility, but a goal of the body, not the soul."
Malenfant smiled coldly. "It's all I have."
The hunting party returned.
Two of the Runners carried limp, hairy bodies, slung over their shoulders. They looked to Malenfant like the chimplike Elf-folk. One was an adult, but the other was an infant, just a scrap of brown-black fur. The other two Runners bore a net slung on a horizontal pole. A third Elf squirmed within the net, frightened, angry, jabbering, a bundle of muscle and fur and long, humanlike limbs. Malenfant could see heavy, milk-laden breasts.
Praisegod got up to greet the party, an expression of anticipation on his cadaverous face. His Ham boy clung to Praisegod's robe and stayed behind him, evidently frightened of the Elf's jabber. Under Sprigge's sharp commands, two of the Runners and the Hams set to constructing a large fire, with a spit set over it.
McCann approached Malenfant, his hands scratched by branches and brambles, his face red with exertion. His mood seemed to have swung again. "Quite an adventure, Malenfant! You should have seen it. The Runners are remarkable. They crept like shadows through that forest, closing on those helpless pongids like Death himself. They caught these three, and though the Elves fought, our fellows would have despatched them all in seconds if not for Sprigge's command..."
The Hams had wrestled the live Elf to the ground, and were cautiously lifting away the net. The Elf squirmed and spat—and Malenfant thought she looked longingly at the corpse of the infant, piled carelessly on top of the adult's body. Perhaps she was the child's mother.
Praisegod walked around the little campsite until he had found a fist-sized rock. He turned to Malenfant, holding out the rock. "Sir, you omitted the hunt. Will you share in the kill?"
Malenfant folded his arms.
"No?" Praisegod motioned to Sprigge.
Now, at a sharp command from Sprigge, a Runner approached, bearing a fire-hardened spear. With a single powerful gesture he skewered the Elf, ramming the pole into her body through her anus, pushing until its tip emerged bloody from her mouth.
This time it was Malenfant who had to restrain McCann.
The Elf was still alive when the Hams lifted the pole onto the spit frames—Malenfant heard her body rip as it slumped around its impaling pole—and, he thought, she was still alive, if barely, when a burly Runner went to work on her skull, curling back the flesh and cracking the skull as if it were the shell of a boiled egg.
Praisegod studied Malenfant. "Perhaps it would have been merciful to kill it first. Or perhaps not; this creature cannot comprehend its fate in any case. It is the brains, you see; freshness is all for that particular delicacy."
McCann broke away from Malenfant. He strode toward Praisegod Michael, his fists bunched, his face purple. "Now I know what you are, Praisegod. No Bay, no Ramose! 'Him the Almighty Power / Hurl'd headlong flaming from th' Ethereal Sky / With hideous ruin and combustion down / To bottomless perdition.' You are no man of God. This is Hell, and you are its Satan!"
Sprigge slammed his fist into the back of McCann's head, and the Englishman went sprawling.
Praisegod Michael seemed unperturbed. "Blasphemy and anarchy, sir. Flogging, branding and tongue-boring will be your fate. That is God's law, as I have interpreted it."
McCann tried to rise. But Sprigge kicked his backside, knocking him flat again. Two of the Runners ripped McCann's jacket from his back, exposing an expanse of pasty skin, and Sprigge loosened his whip.
Malenfant watched this, his own fists bunched.
Don't do it, Malenfant. This isn't your argument; it's not even your damn world. Think of Emma. She is all that matters.
But as Sprigge raised his arm for the first lash, Malenfant hit him in the mouth, hard enough to knock him flat.
He didn't remember much after that.
## _S hadow_
For days after her latest beating at One-eye's hands, Shadow had stayed in her nest. There was a little fruit here, and dew to be sucked from the leaves. She found something like contentment, simply to be left in peace.
But the child developed rashes on his belly and inner thighs, and Shadow herself lost a lot of hair around her groin. Her hair, and the child's, were matted with urine and feces. In her illness she had failed to clean the child, or herself when the child fouled her.
She clambered down from the tree and set the child on the ground. When Shadow propped him up the child was actually able to sit up by himself—wobbling, his legs tangled, that great strange head bobbing like a heavy fruit, but sitting up nevertheless. She bathed him gently, with cool clean water from a stream. The coolness made the rash subside. The child's infection was subsiding, too, and his nose was almost free of snot.
The child clapped his little hands together, looked at them as if he had never seen them before, and gazed up at his mother with wide eyes.
Shadow embraced him, suddenly overwhelmed by her feelings, warm and deep red and powerful.
And a great mass caromed into her back, knocking her flat.
Her child was screaming. She forced herself to her knees and turned her head.
One-eye had the infant. He was sitting on the ground, holding the baby by his waist. The child's heavy head lolled to and fro. One-eye was flanked by two younger men, who watched him intently. One-eye flicked the side of the child's skull with a bloody finger, making the head roll further.
Shadow got to her feet. Her back was a mass of bruises. She walked forward unsteadily, and with every step pain lanced. She stood before One-eye and held her hands out for her child.
One-eye clutched the child closer to his chest, not roughly, and the child scrabbled at his fur, seeking to cling on. The other men watched Shadow with a cold calculation.
Shadow stood there, bewildered, hot, exhausted, aching. She didn't know what One-eye wanted. She sat on the ground and lay back, opening her legs for him.
One-eye grinned. He held the child before him. And he bit into the front of its head. The child shuddered once, then was limp.
Shadow's world dissolved into crimson rage. She was aware of the child's body being hurled into the air, blood still streaming from the wound in his head, as limp as a chewed leaf. She lunged at One-eye, screaming in his face, clawing and biting. One-eye was knocked flat on the ground, and he raised his hands before his bloody face to ward off her blows.
Then the other men got hold of her shoulders and dragged her away. She kicked and fought, but she was weakened by her long deprivation, her beatings, and her illness; she was no match for two burly men. At last they took her by an arm and a leg. They swung her in the air and hurled her against a broad tree trunk.
The men were still there, One-eye and the others, sitting in a tight circle on the ground. They were working at something. She heard the rip of flesh, smelled the stink of blood. She tried to rise, but could not, and she fell back into darkness.
The next time she woke she was alone. The light was gone, and only pale yellow earthlight, filtered through the forest canopy, littered the ground.
She crawled to where the men had been sitting.
She picked up one small arm. A strip of gristle at the shoulder showed where it had been twisted from the torso. The hand was still in place, perfectly formed, clenched into a tiny fist.
She was high in a tree, in a roughly prepared nest. She didn't remember getting there. It was day, the sun high and hot.
She remembered her baby. She remembered the tiny hand.
By the time she clambered down from the tree, her determination was as pure as fast-running water.
## _E mma Stoney_
Emma trudged wearily over the soft sand of the ocean shore. The ocean itself was a sheet of steel, visibly curving at the horizon, and big low-gravity waves washed across it languidly.
This strip of yellow-white beach lay between the ocean and a stretch of low dunes. Further inland she saw a grassy plain, a blanket of green that rippled as the wind touched it, studded here and there by knots of trees. A herd of grazing animals moved slowly across the plain, their collective motion flowing, almost liquid; they looked like huge wild horses. The stretch of savannah ended in a cliff of some dark volcanic rock, and a dense forest spilled over the lip of the cliff, a thick green-black. It was a scene of life, of geological and biological harmony, characterized by the scale and slow pace of this world. In any other context it might have been beautiful.
But Emma walked warily, the rags of her flight suit flapping around her, her loose pack strapped to her back with bits of vegetable rope, a wooden spear in one hand and a basalt axe in the other. Beautiful or not, this was a world full of dangerous predators—not least, the humans.
And then she saw a flash of blue fabric, high on the cliff.
She walked up the beach toward the cliff, trying to ignore the hammering of her heart.
Every day her mood swung between elation and feverish hope, to bitterness that bordered on despair. One day at a time, Emma. Think like a Ham. Take it one day at a time.
But now she could see the lander itself. She broke into a run, staring, wishing her eyes had a zoom feature.
It was unmistakeably NASA technology, like a stubby scale-model Space Shuttle, with black and white protective tiles. It was surrounded by shreds of its blue parafoil. But it was stuck in a clump of trees, halfway down the cliff; it looked like some fat moth clinging to the rock.
"Nice landing, Malenfant," she murmured.
Disturbingly, she saw no sign that anybody had done anything constructive up there. There were no ropes leading up or down the cliff, no stars and stripes waving, no SOS sign carved into the foliage.
Maybe the crew hadn't survived the crash.
She put that thought aside. They could have gotten out, before the lander had plummeted over the cliff, even ejected on the way down. There were many possibilities. At the very least, there should be stuff she could use—tools, a first aid kit, maybe even a radio.
Messages from home.
What was for sure was that she was going to have to get up that cliff to find out. And she wasn't going to make it up there alone.
There was an encampment of Hams, a squat hut of skin weighted down with stone, almost directly under the blue flash. She could see them moving around before the hut, slabs of muscle wrapped in crudely-cut skins.
That was how she was going to get up that damn cliff.
She forced herself to slow. One step at a time, Emma; you know the protocol. It was going to be hard to be patient, to engage a new group of Hams once again. But that was what she was going to have to do.
She dropped her pack at the edge of the sea, and splashed her face with salt water. Then she walked up and down the beach, picking out bits of scattered driftwood. She found a long, straight branch, and selected a handful of thorny sticks. She took her favored hand axe and, with a skill born of long hours of practice and many cut fingers, she made notches in one end of the stick, wide enough to fit the thorny twigs. Then she took a bit of rawhide string from her pack, and wrapped it around the stick, lashing the barbs in place.
Thus, one harpoon.
She slipped off her boots and socks and coverall and waded into the shallows, harpoon raised.
Fishing had become her speciality. It didn't seem to have occurred to any of the Ham communities here to figure out how to catch fish, either in the ocean or in freshwater streams. Fish meat, exotic but appealing, made a good bribe.
There was a ripple at her feet, a roughly diamond shape that emerged briefly from the sand. She stabbed down hard, feeling the crunch of breaking wood.
She found she had speared a skate, a big brown fleshy square of a fish, maybe two feet across. Skate buried themselves in the mud, coming up at night to hunt shellfish. Her catch was wriggling violently, and it was all she could do to hold onto the harpoon. With a grunting effort she heaved the skate over her head and out onto the sand, where it flopped, slowly dying. One bit of lingering squeamishness was a reluctance to kill anything; acknowledging the hypocrisy, she let her victims die instead.
She splashed out of the water. Briskly she inspected her harpoon, considering whether it was worth keeping; she had learned to conserve her energy and time, never throwing away anything that might be used again. But the barbs were broken. She stripped off the hide string and stuffed it back into her pack, and let the bits of the harpoon fall, abandoning this thing she had made that would have been beyond her imagining a few months ago, forgetting it as carelessly as any every-day-a-new-day Ham craftsman.
With her hand axe she skinned and gutted the fish. You had to avoid the guts, and the skin could be coated by toxic mucus or dangerous spines: tricks she remembered from her childhood camping-in-the-woods days.
Then she pulled on her coverall and boots, picked up the skate meat and her pack, and walked steadily up the beach toward the Ham encampment.
These Hams accepted her silent presence in the corner of their hut, as readily as every other group she had encountered. They predictably turned away from her first offer of skate meat. But she continued to bring home gifts from the sea, until they had, one by one, experimentally, begun to taste the pale, sharp flesh.
So she settled into her corner of the communal hut, wrapping herself each night in grimy bits of parachute canvas, watching the Hams, waiting for some opportunity to find a way up the cliff to the lander.
She learned their names—Abel and Ruth and Saul and Mary—odd quasi-Biblical names, presumably bequeathed to them, like their fractured English, by some ancient contact with humans, Zealots or other "Skinny-folk." She tried to follow their complex social interactions, much of it centering on speculative gossip about the vigorous child-woman Mary.
They were typical Hams. Come to that, _all_ Hams were typical Hams.
Their English was broken—mispronounced, with missing or softened _G_ and _K_ and _th_ sounds and vowels that blurred to sameness. The language had tenses—past, future—and there were even conditionals, used for instance by gossiping women as they speculated what would follow if Mary gave herself to Saul, or if she fell for Abraham's clumsy wooing first. But their language was elemental, with a simple vocabulary focusing on each other, their bodies, the hut.
As for Mary herself, she was clearly at the center of a storm of hormonal change, relishing and fearing all the attention she got at the same time. But she never teased, Emma observed, never led any of the men on. Deceit seemed utterly unknown to these people. They were clever in many ways, but whatever they used those big brains for it wasn't for lying to each other, as humans did.
All this dubious anthropological speculation served to occupy her mind. But it was all spectacularly useless when it came to bringing her closer to her central goal of reaching the big black and white moth suspended on the cliff over their heads, in which none of the Hams showed the slightest interest.
## _M anekatopokanemahedo_
Manekato pushed into the forest. The foliage was dense, dark green, damp, cold, and it seemed to clutch at her face and limbs. The shadows stretched deep all around her, concealing subtle, elusive forms, as if wild creatures were Mapping themselves into and out of existence all around her.
Briefly she considered going back to the compound and seeking a new symmorph—perhaps with better dark-adapting vision. But as she worked deeper into the wood her body moved increasingly easily, her feet and hands clutching at branches and roots, and a clear sense of direction worked with her powerful hearing to guide each footfall. She dismissed her fears; she even felt a certain deep exhilaration. We came from the forest, she thought, and it is to the forest that I now return.
She was seeking Without-Name, who had left the encampment of exiles.
Even before her final departure Without-Name had taken to spending increasingly long times away from the compound. After her challenge by Nemoto over the captured Zealot, she had not brought back further "specimens," but at times Manekato thought she had glimpsed blood on her dirt-matted fur, and even on her lips.
To her surprise the little hominid Nemoto had expressed sympathy with Without-Name. _"Without-Name is out of control. But she is right. You are too slow, too cerebral, Mane. Perhaps your minds have grown overornate, and are strangled by their own complexity. It is time to confront the Old Ones, not to theorize over them..."_
It had been deeply shocking for Manekato to hear such critical sentiments expressed by a mere lower hominid.
Still, Without-Name had become an increasing distraction, a wild blood-stained rogue planet crashing through the orderly solar system of purpose and knowledge acquisition that Manekato had sought to establish. Babo and others had expressed relief when Without-Name had finally failed to return from one of her ambiguous jaunts. But Manekato had sensed that Without-Name would cause them all severe and unwelcome problems yet.
Finally Manekato had been disturbed by a cacophony of cries, coming from deep in the nearby belt of forest. Something there had died, in great pain and anguish; and Manekato had had a powerful intuition that it was time for her to seek out Without-Name and meet her on her own terms.
And so here she was, just another hominid picking her way through the forest.
She emerged from the bank of trees. Beyond a stretch of rock-strewn ground, a low cliff rose: broken and eroded, perhaps limestone, pocked with hollows and low caves, overgrown with moss and struggling trees. Somewhere water trickled.
The sky was clouded over. The place was claustrophobic, enclosing. She could smell blood, and dread gathered in her heart.
A hominid walked out of one of the caves. To judge by the sewn skins he wore, he was a Zealot, like the specimen Without-Name had brought back to the camp. He carried a crossbow, and his tunic and leggings were splashed with dirt and blood. He saw Manekato, standing alone at the edge of the forest. His eyes widened. He dropped his bow and ran back into the cave. _"Daemons! Strange Daemons!"_
Manekato gathered her courage. She stepped forward, crossing the rock-strewn floor.
She paused in the cave's entrance, giving her eyes time to adapt to this deeper dark. The cave's roof was a layer of rock just above her head. It was worn smooth, as if by the touching of many fingers; perhaps this place had been inhabited for many generations. The cave stank of hominid, of crudely prepared food, of stale urine and feces and sweat—and of blood.
A shadow moved before her. As it approached the light, it coalesced into the form of Without-Name. Her fur was splashed with blood, and a gouge had been cut into her arm.
"I suppose I have been expecting you," she growled. "Are you aware what a target you provide, silhouetted against the light? We have not fought a war for a million years, Manekato; we have lost our instincts for survival."
"What have you done, Renemenagota?" Manekato reached out and touched the wound in the other's arm. It was a deep slice over the bicep, still leaking blood—it had not even been cleaned. "I see your victims did not submit quietly."
Without-Name barked laughter. "It was glorious. Come."
She turned and led the way deeper into the cave, and Manekato followed reluctantly.
At the back of the cave a lamp of what looked like burning animal fat flickered in a hollow on one wall; the rock above was streaked with black grease. By its light Manekato saw she was walking over scorched patches of dirt—hearths, perhaps, all cold and disused. Bits of stone and bone and wood were scattered everywhere. At the rear of the cave, animal skins had been stretched over rough frames of wood.
There were hominids here. They were Zealots, dressed in their characteristic garb of crudely sewn skin. When Manekato knuckle-walked towards them they yelled and grabbed their weapons.
Without-Name held up her hands. _"She is weak. She will not harm you."_
The Zealots hurried out of her way, jabbering their alarm to each other.
Beyond the Zealots there was a mound of slumped forms.
They were hominids, all dead. They were the powerful squat creatures Nemoto called Hams. They had been slaughtered by crossbow bolts and spear thrusts. They had not died easily: ripped throats and gouged eyes and severed limbs testified to that, as did the injuries nursed by the Zealots. Blood soaked through the grisly heap, and spilled guts glistened on the floor beneath.
Without-Name's eyes glittered. "You cannot engage these fellows hand-to-hand; the power of their stocky bodies is simply too great. But they work strictly short range. And so they fell to our bows and throwing spears as they tried to close with us, one after the other. Once they were down it was a case of moving in to finish them off. But they fought on even with their bellies torn open, their throats cut. Well, this was their home for uncounted generations—you can see that—they were fighting as we would for our Farms..."
Manekato discerned a smaller bundle, laid on top of the heap of corpses. It was an infant, its age impossible to tell, one leg bent back at an impossible angle. "Did this little one give you a good spectacle, Renemenagota?"
Without-Name shrugged. "The Zealots took most of the smaller infants back to their stockade. You can't tame an adult Ham, you see; you have to get them young to break them. This one wouldn't leave its mother's side. The efforts to remove it resulted in a snapped leg." She grinned, her teeth showing bright in the gloom. "Praisegod Michael was here. Their leader, you see; the leader of the Zealots. He uttered words over the corpses, blessing them, commending their souls to the afterlife he believes awaits us—or rather awaits _his_ sort of hominid; he isn't so sure about the rest of us. Michael said his prayers over this little creature and then cut its throat. A delicious contradiction, don't you think?
"You should see the ambition that burns in Michael's eyes! He dreams of cleansing his world of such _creatures of the Devil_ as this—what an ambition!—but he has lacked the understanding to make it so. He was wary of me when I approached him—no, contemptuous, because for him I am less than human. But I forced him to listen to me. I made him see that by taking his captives and training them properly, he increases his resources, you see, which he can deploy for further conquest; once initiated, it is a simple exponential growth."
"You spoke to this monster—you are _working_ with him?" Manekato said tightly, "Whoever this _Praisegod_ is, his reasons for wishing to destroy the Hams and the others have surely more to do with the flaws in his own heart than any ideological justification."
Without-Name grabbed her arm and held it tight; Manekato felt moisture, blood and sweat, soaking into her fur. "Of course Praisegod Michael is mad. But it is a glorious madness."
Manekato pried her arm away from Without-Name's grip. Regretfully she said, "Glorious or not, I have to stop you."
Without-Name laughed. "You do not have the imagination or the courage for that, Manekato."
The Zealots were returning to the pile of Ham corpses. They were cutting away ears and hands, perhaps as trophies. But their movements were characteristically sluggish, like pale worms moving in the dark.
## _J oshua_
Joshua lay on the filth-crusted floor of his cell.
He was left alone for days. It was worse than any beating. There was nobody to look at him.
The People of the Gray Earth were never alone by choice. They spent their entire lives in their tight-knit communities, surrounded day and night by the same faces, change coming only through the slow tide of birth and death. Some women spent their entire lives within a hundred paces of where they were born. Even parties of hunters who ranged farther in search of big game would not mix with other groups of hominids, even other Hams; strangers were like faces in a dream, remote, not real.
He tried to picture the hut, the people coming and going about their business. He tried to recall the faces of Abel and Saul and Mary and Ruth and the others. The life of the people was going on, even though he was not there to be looked at—just as it had continued after the death of Jacob, the endless round of days and nights, of eating and sleeping and fornicating, of birth and love and death.
Jacob was dead. Was Joshua dead?
Away from others, Joshua was not even fully conscious. As the light came and went, he felt himself crumble. He was the walls, the filthy floor, the patch of daylight in the roof.
... And yet he was not alone, for there were people in the walls.
Faint marks had been scratched there, perhaps by fingernails, or with bits of stone. Some of them were so ancient they were crusted with dirt, and could only be detected by the touch of his fingertips. Perhaps they were made by Skinny or Nutcracker-man or Elf or Runner. But not by Ham, for no Ham made marks like these.
Scratches on the wall. Patterns that pulled at his consciousness. Boxes and circles and lines that longed to speak to him.
He was in a cave. But it was not a cave, for its walls were made of rocks piled one on top of the other. Sometimes the people would build walls, lines of rubble loosely piled, to help keep out the small animals that foraged at night. Joshua knew what a wall was. But _these_ walls went up, high above Joshua's head, too high for him to reach.
And there was a roof made of rocks, too, suspended over his head. On first waking here, he had cringed, thinking a sky full of rocks was descending on him. But the roof did not fall. He learned to uncurl, even to stand—though each time he woke from sleep he forgot about the roof, and whimpered in terror and curled in a corner of the cell.
The only light here came from a hole in the roof. He saw the days come and go through that hole, night succeeding day. He would lie on his back staring at the little circle of light. But when it rained, the water would pour through the hole, and he would huddle in a corner, shivering.
Sometimes a face would appear in the hole, the face of a Skinny. Stuff would be thrown down at him. Sometimes it would be food that he would scrabble to collect from the floor. The food was poor, scraps of cut-up vegetable or fruit peel or bits of gristle, some of it already chewed, sour with the saliva of Skinnies. But he devoured it all, for he was constantly hungry.
Sometimes they would hurl down water at him, usually brackish and stinking, enough to drench him. It would drain away out of a hole in the center of the blackened, worn floor, taking much of his own shit and piss with it. When the water came he would stand with his mouth and hands open, catching as much as he could. And when it had finished he would scrape at the filth-blackened floor with his fingers, collecting as much of the water as he could, even lick the floor with his tongue.
But sometimes all the Skinnies would throw down was their own thin shit, or they would piss in the hole, trying to hit him as he scurried from side to side.
His memories of how he had come here were blurred.
He remembered the clearing. After Mary had escaped he had been picked up by many Skinnies, all grunting with the effort. With every jolt his shoulder had blazed with pain. They had thrown him onto a platform made of strips of cut-up wood. And then the platform had been dragged away, along broad trails burned into the woods.
He remembered entering the stockade. It was a great wall of sharpened tree trunks driven into the ground, many times higher than Joshua could have reached. Inside there were huts of sod and wood, dark hovels whose stink had struck him as he was dragged past. There were many animals, goats and rabbits and ducks. There were many, many Skinnies, with grimy skin and brown teeth.
And there were Hams. They dragged at ropes and pushed bits of wood and dug at the ground. Joshua had hooted to the Hams, seeking help. Though the Hams were few, they could surely overpower these Skinny folk easily. But they had not responded, not even looked up, and he had been silenced by a slamming blow to his mouth.
They had removed his skins, and he was naked. And he had been thrown into this darkened cell.
The punishment had started immediately.
There had been Skinnies around him. Some of them were grinning. One of them carried a stick whose tip glowed bright red. Joshua stared at the glowing stick; it was one of the most beautiful colors he had ever seen. For one brief instant he left his aching body, and was the fiery glow.
But then the Skinnies shoved him on his back, trapping his limbs. The man with the glowing stick held it before Joshua's face—he could feel heat, like a fire—the man rammed it into the wound in his shoulder.
Only fragments after that, dark red fragments soaked with pain. Fragments, fading into dark.
But Joshua welcomed the presence of those who beat him. For at least, then, he was not alone.
One day he saw faces in the scratches on the wall. Faces that peered out at him, the faces of Skinnies.
No, not faces: one face, over and over.
The face of a man, thin, bearded, a circle over his head. The man looked at him, but did not look at him. Sometimes Joshua yelled at him, punched the face. But the wall would return, scraping his knuckles, and the man, not replying, would disappear into his web of scratches.
Joshua was dead. He was in a hole in the ground, like Jacob. But there were no worms here. There were only the faces, looking at him, not looking at him.
He screamed. He cowered in the corner, as he did when his captors pissed on him.
That was how the Skinnies found him one day, when they burst into his cell with their clubs and rocks and whips. They mocked him, kicking at his back and kidneys, and they pulled him out of the corner and stretched him.
A leering face hovered over him. "We'll break you yet, boy, while there's still some work left in that hulking body of yours." He arched his back, trying to see the man in the wall.
There was laughter. "He's looking for Jesus."
Running footsteps. A boot launched at his face. He felt a tooth smash at the back of his mouth.
"Help!" he cried. "Help me, Chee-sus!"
The jailers staggered back, open-mouthed, staring.
A day and a night. His tooth was a pit of pain.
Skinnies were in the cell. Joshua scuttled to his corner, expecting the usual blows.
But a net was thrown over him. He did not resist. His hands and arms and feet and legs were tightly bound, and then his legs folded behind his back and tied up to his waist.
Wrapped in the net, he was dragged out of his cell.
Outside was a long, narrow cavern. There was no daylight, but fires burned in pits on the wall. He saw only the floor and walls, the lumping shadows of his jailers as they dragged him, letting his bruised limbs and head rattle on the floor.
They paused, and there was a clanking, clattering noise. Joshua lifted his head dully.
He was facing an open cell. A man sat in the cell, a Skinny. But this was a Skinny like none Joshua had ever seen. He had no hair on the top of his head, none at all, although stubble clustered on his cheek. And his clothing, though filthy, blood-stained and torn, was not like the skin the Zealots wore. It was blue: a blue membrane, like the wings of the sky seed.
Joshua, electrified, gasped with recognition.
The man was looking at him. "My name is Reid Malenfant," he said gently. "If you get out of here, remember that. Malenfant."
Joshua worked his mouth; it was crusted with blood and his lips were cracked. "Mal'fan'."
Malenfant nodded. "Good luck to you, friend."
And then the door was slammed.
## _S hadow_
She stayed away from the others. She slept in nests at the periphery of the crater-wall forest, and fed from trees and shrubs far from the movements of the rest of the group.
She searched for cobbles in streams and on the exposed, eroded crater walls.
She had not grown old enough to acquire more than the most basic tool-making skills. So it took her many tries, chipping at cobbles with stones and bits of bone, before she had manufactured something that felt right. It was a lens-shaped cobble, with one crudely sharpened edge, that fit neatly in her hand.
Through these days her determination burned, clear and unwavering.
Burned until she was ready.
## _J oshua_
Joshua was in a new place. The walls were white, like snow. The floor shone, smooth as a bamboo trunk.
Joshua stood naked at the cell's center. Heavy ropes bound his hands before him and his feet, and the ropes were fixed to a great bar dug out of the rock floor beneath him. There were big holes in the walls covered by palm fronds, and through them Joshua could see daylight. He sniffed deeply, but his cavernous nostrils were clogged with snot and blood.
There were people in the walls.
The marks on these walls were not mere scratches. They were vivid images in bright bloodred and night black, and in them Joshua saw the thin, bearded man. The man was much clearer here than in the deep cell—so clear he never went away—and there were many of him, shining brightly, even one version of him fixed to a tree trunk and bleeding.
Joshua cowered.
"Well might you avert your eyes from the Lord's countenance."
Joshua turned. A man had spoken. A Skinny. He was taller than Joshua, his hair gray, and his black clothing swept to the ground. His black robes were skin, finely worked, black like charcoal from a hearth.
Joshua cringed. But no blow came. There was only a hand on his forehead, light, almost curious, exploring his brow ridges.
"Well might you hide your face for shame of what you are. And yet you called out for the Lord's help—so the brutes assigned to break you assured me... Stand up, boy." Joshua received a hard toe cap to the side of his leg. "Up, Ham."
Reluctantly Joshua stood.
The man had a sharp nose, and warts on his face, and eyes such a pale blue they made Joshua think of the sky. He walked around Joshua, and touched his chest and back. His hands were very soft. "I did ask for you to be cleaned up," he said absently. "Well. You may call me Praisegod Michael. Do you understand? I am Praisegod Michael. _Praisegod_."
"Prai'go'."
"Praisegod Michael, yes." Praisegod peered into his eyes. "What brows, what a countenance... And you, do you give yourself a name?" When Joshua didn't reply, Praisegod pointed to his own chest. "Praisegod Michael." And he pointed to Joshua.
Joshua spoke his name. When he moved his mouth his smashed tooth hurt; he could feel pulp leak into his mouth.
Praisegod laughed. " _Joshua_. My fathers named your fathers, when they found themselves sharing this Purgatorial place with you... And now you pass on the names one to the other, down through the generations, like heirlooms in the hands of apes. Very well, Joshua. And _what_ are you?"
The man's thin face, with its flat brow and high, bulging forehead, terrified Joshua. He had no idea what Praisegod wanted.
Praisegod produced a short, thick whip. With practiced motions he lashed at Joshua's shoulder. The pain was great, for that was the site of Joshua's spear wound. But the skin was not broken.
"If you do not answer, you will be treated so," Praisegod said evenly. "But let me answer for you. You have a man's name, but you are not a man. _You are a Ham_. That is another name my father gave yours, and it is appropriate. Do you know who Ham was?"
A failure to reply brought a fresh lash of the whip.
"Ham, father of Canaan, son of Noah. He failed to respect his father. 'Cursed be Canaan; a servant of servants shall he be unto his brethren.' Genesis 9:25. A servant of servants, yes; that is your place, boy. But then you know nothing of Noah, do you? You are an animal—a magnificent one, perhaps, and yet an animal even so. From your misshapen head to your splayed feet you signify antediluvian stock—if not pre-Adamite, indeed." Praisegod seemed to be growing angry. Joshua watched him dully. "The world was cleansed of your kind by the waters of the Flood. But you survive beyond your time in this dismal pit. And now you call on the Lord Himself—"
Another lash to the shoulders, and Joshua flinched. Then a blow to the back of the legs forced Joshua to his knees.
Praisegod Michael grabbed Joshua by his hair, making him raise his head. "Look on His merciful face. What can _you_ know of His benison? Do you know what my fathers suffered to bring the Word to this world? When they fell here, they had nothing: nothing but the clothes they wore. _They were set upon by beasts like yourself;_ they starved; they fell prey to diseases. And yet they survived, and built this community, all by the strength of their hands, and their faith.
"And in all this they remembered the Word. They had no Book with them, not a single copy. But they remembered. They would sit around their fires and recite the verses, one after another, seeking to recall it for their children, for they knew they had no way home.
"And _that_ is how the Word of the Lord came to this pit. And now you, an animal of the field, with your thunderbolts of stone, you presume to call on Him for help?"
Joshua folded over himself, letting the whip fall. He felt his flesh break, and the whip dug deeper into the wounds it had made.
## _S hadow_
The fungal growth now framed her vision, black as night.
When she heard the roosting calls of the people, she slid through the trees. The people nested, silhouetted high against a cloud-laced earth-blue sky. She recognized One-eye by the grunting snores he made, the stink of a body she had come to know too well.
She slipped up the trunk of the tree, her long hands and feet gripping. With scarcely a rustle, she clung to branches above One-eye's rough nest.
He lay on his back, hands wide, legs splayed, one foot dangling over the edge of his nest. His mouth was open, and a thin stream of drool slid down his chin. He had an erection, dark in the earthlight.
She clung to the branches with her feet and legs, and hung upside down over him. She took his penis in her mouth and sucked it gently, rubbing the shaft with her lips. In his sleep, he moaned.
Then she bit down, as savagely as she could.
He screamed and thrashed. She could hear answering hoots from surrounding nests.
She flung herself down on him. His eyes were wide and staring, and she thought she could smell blood on his breath. He was stronger than she was, but he was already in intense agony, and she had the advantage of surprise. He pushed feebly at her face with one hand. She grabbed the hand, pulled a finger into her mouth, and nipped off a joint with a single savage bite. He howled again, and she spat the bloody joint into his open mouth, making him gag.
Then she raised her shaped cobble and slammed it against the side of his head.
## _J oshua_
A day and a night, here in this white place, without food or water.
Men scrubbed him roughly. They mopped away the blood and shit from the floor.
Praisegod was prone to swings of mood, which Joshua neither understood nor could predict. Sometimes there was coldness, cruelty, beatings. But sometimes Praisegod would gaze at him with bright eyes, and run his hands over his battered body, as a mother might stroke a child. Joshua quickly learned to dread such moments, for they always finished in the most savage beatings of all.
And yet he longed for Praisegod Michael to stay, rather than leave him alone.
He lay on his side, staring at the marks on the walls—not the face of _Chee-sus_ , but strange angular lines, the loops and whorls. Bewildered by pain and exhaustion, he stared and stared, trying to lose himself in the lines, trying to see the faces there.
"What is it you see, boy? Can you read? Can you read the Lord's words? Do you hear what they tell you?" Showing his sporadic, chilling tenderness, Praisegod Michael was kneeling on the floor, with Joshua's head on his lap.
His mouth dry, his tongue thick, Joshua whispered, "People."
"People?" Praisegod Michael stared at the marks. " _These_ are words, and _these_ are pictures. The words speak to us... Ah, _but they do not_ , do they? Marks on the wall do not speak. They are _symbols_ , of the sounds we make when we speak, which are themselves symbols of the thoughts we concoct... Is that what you mean?" His hands explored Joshua's body with a rough eagerness. "What lies inside that cavernous head of yours? The words you utter are themselves symbolic—but your kind have no books, no art. Is that why you cannot understand? Would you like me to tell you what those letters say to me?" He pointed at the wall. " 'After this I looked, and there before me was a door standing open in Heaven.' Revelation 4,1."
"Heav'n," Joshua mumbled.
"The sky, child, where we will pass when we die."
Joshua twisted his head to see Praisegod's face. "Dead."
"No." Praisegod was almost crooning, and he rocked Joshua back and forth. "No, you poor innocent. You are alive. And when you die, you will be alive again in Christ—if His mercy extends to your kind..."
"Dead," said Joshua. "Dead. Gone. Like Jacob."
"Dead but not gone! The corpse in the ground is the seed that is planted in the earth. So we will all bloom in the spring of the Lord. 'And I saw the dead, great and small, standing before the throne, and books were opened.' But I am talking in symbols again, ain't I? A man is not a seed. But a man is _like_ a seed."
Suddenly he pushed Joshua away. The Ham's head clattered on the floor, jarring his aching tooth.
"You can know nothing of what I speak, for your head is empty of symbols... Ah, but what if my religion is nothing but symbols—is that what you are thinking?—the symbol of the seed, the Mother and Child—a dream concocted by words rattling in my empty head?" Now Joshua felt kicks, hard, frantic, aimed at his back and buttocks. "O you witness to the Flood, O you underman! See how you have planted doubts in my mind! How clever you are, how cunning! You and that Daemon of the forest, Renemenagota, she of the ape build and mocking, wise eyes... The Daemons make me promises. They can take my vision and make it real, make this antediluvian island a godly place. So they say. So _she_ says. Ah, but in her dark eyes I sense mockery, Joshua! Do you know her? _Did she send you?_... How you madden me! Are you agents of Satan, sent to confuse me with your whispers of God's work?"
But now Praisegod leaned over Joshua again and grabbed his face. Joshua saw how his eyes were red and brimming with tears, his face swollen as if by weeping. " _Can sin exist here?_ The brutes who serve me have their Runner women, their whores with the bodies of angels and the heads of apes. I, I am not of that kind... But now, here! Here!" He grabbed Joshua's bound hands and pushed them into his crotch; Joshua could feel a skinny erection. "You are destroying me!"
And the beatings went on.
Joshua lay on the floor, his own blood sticky under his face. Pieces moved around in his head, just as they had before: when he saw the sky seed fall from the sky, when he put together the cobble from the bits of shattered stone.
The kind Skinny's face peered through a cloud of pain and black-edged exhaustion.
He whispered, " 'Fore me was door standin' open Heaven."
Praisegod Michael was here. Panting, he gazed into Joshua's eyes. "What did you say?"
But Joshua was, for now, immersed in his own head, where the pieces were orbiting one another, the flakes sticking to the core of the cobble one by one. The Gray Earth. The seed that fell from the air. The door in the sky.
Joshua was, in his way, a genius. Certainly none of his kind had experienced such a revelation before.
"Heav'n," he said at last.
Praisegod Michael pushed his ear close to his mouth to hear.
"Heav'n is th' Gray Earth. Th' seed. Th' seed takes th' people. Th' people pass through th' door. Door to heaven. To Gray Earth."
"By God's eyes." Praisegod Michael stumbled back. "Is it possible you _believe_?"
Joshua tried to raise his head. "Believe," he said, for he did, suddenly, deeply and truly. "Th' door in th' sky. Th' Gray Earth."
Praisegod Michael stalked around the cell, muttering. "I have never heard an ape-thing like yourself utter such words. Is it possible you have _faith_? And if so, must you therefore have a _soul_?" Again he stroked the heavy ridges over Joshua's eyes, and he pressed his gaunt body close to the Ham's. "You intrigue me. You madden me. I love you. I despise you." He leaned closer to the Ham, and kissed him full on the lips. Joshua tasted sourness, a rank staleness.
_"Graah_ — _"_ Praisegod rolled away, lying sprawled on the floor, and vomited, so that thin bile spread across the shining floor.
Then he stood, trembling, striving for composure. "I would kill you. But if you have the soul of a _man_ —I will not risk damnation for you—if you have not damned me already!" He smiled, suddenly cold, still. "I will send you out. You will spread the Word to your kind. You will be a Saul of the apes." He raised his pale eyes to the light from the window. "A mission, yes, with you as my acolyte _—you_ , a pre-Adamite man-ape."
Joshua stared at him, understanding nothing, thinking of a door in the sky.
But now Praisegod stood over him again, and again he spoke tenderly. "I will help you." He reached into his clothing and produced a knife. It was not of stone; it glittered like ice, though Joshua could see how worn and scuffed it was. "No beast should speak the Word of God. Here." He put his fingers inside Joshua's mouth. The fingers tasted of burning. He pushed down, until Joshua's mighty jaw dropped.
Then, without warning, he grabbed Joshua's tongue and dragged it out of his mouth. Joshua felt the slash, a stab of pain.
Blood sprayed over Praisegod Michael.
## _S hadow_
The next morning the women surrounded Silverneck, as usual. With their infants clambering over them, they munched on figs.
With a crash, One-eye fell from his tree. His hands and feet left a smear of blood where they touched bark or leaves, for several of his fingers and toes had been nipped off. White bone showed in a huge deep wound on the side of his head. And his penis was almost severed, dangling by a thread of skin. His fur was matted by blood and piss and panic shit.
The women stared.
He looked about vaguely, as if blinded, and he mewled like an infant. Then he stumbled away, alone, into the deeper forest.
Shadow walked out of the tree cover.
Silverneck moved aside for her. One of the younger women growled, but Shadow punched her in the side of the head, so hard she was knocked sideways. Shadow sat with the group, and clawed figs into her mouth. But nobody looked at her, nobody groomed her, and even the children avoided her.
That night, when the roosting calls went out, One-eye did not return.
## _R eid Malenfant_
Malenfant was kept chained up in a dark, filthy cell. It was just a brick-lined pit, its damp mud floor lined with packed-down filth. The only light came from a grilled window high in the ceiling. The door was heavy with a massive wooden bolt on the outside.
He reached out to touch the walls. The bricks were rotten. Maybe he could dig out handholds and climb up to that window.
And then what? What then, after you climb out into the middle of Praisegod's courtyard?
You are not dealing with rational people, Malenfant.
It was true Praisegod had built a place of relative order here. But this was an island of rigidity in a world of fluidity and madness, a world where mind itself was at a premium, a world where the very stars regularly swum around the sky, for all Praisegod's zeal and discipline—just as, Malenfant suspected, Praisegod's own inner core of horror constantly threatened to break through his surface of control.
There was nothing he could do, nothing to occupy his mind.
Sometimes the most courageous thing was doing nothing. _Do-nothing heroics_ : Was that a phrase from Conrad? If there was really, truly no way you could change your situation, the last thing you wanted to do was to pour so much energy into fighting your fear that you burned yourself up before the chance came for a break.
As he sat in the dark and the filth, utterly alone, Malenfant wondered how long his own do-nothing heroics would sustain him.
At last he was brought before Praisegod Michael.
At Praisegod's chapel-residence Malenfant was kept waiting, standing before Praisegod's empty desk bound hand and foot, for maybe an hour.
Finally Praisegod walked in, slowly, contemplative, his Ham boy at his side. Praisegod didn't look at Malenfant. He sat at his desk, and a Ham girl brought in a tray of chopped fish set on slabs of hard, dark bread, with a bowl of what looked like mustard and a wooden goblet of wine. Praisegod ate a little of the fish, dipping it in the mustard, and then he passed the rest to the Ham boy, who sat on the floor and ate ravenously.
Praisegod's manner seemed distracted to Malenfant, almost confused. He said rapidly, "I have been forced to punish Sir McCann. You see why—you witnessed his blasphemous disrespect. His soul is hard, set in a mold of iniquity. But you—you are different. You seek the woman you love; you are moved by a chivalrous zeal. In you I see a soul that could be turned to higher goals."
"Don't count on it," Malenfant said.
Praisegod's eyes narrowed. "You should not presume on God's grace."
"This place has nothing to do with God," Malenfant said evenly, staring hard at Praisegod. "You play with human lives, but you don't even see that much, do you? Praisegod, this place—this Moon—is an artifact. Not made by God. _Humans_. Men, Praisegod. Men as different from you or me as we are different from the Elves, maybe, but men nevertheless. They are moving this whole damn Moon from one reality strand to the next, from Earth to Earth. And everything you see here, the mixing up of uncounted possibilities is because of that moving. Because of _people_. Do you get it? God has nothing to do with it."
Praisegod closed his eyes. "This is a time of confusion. Of change... I think you may yet serve my purpose, and therefore God's. But I must shape you, like clay on the wheel. But there is much bile in you that must be driven out." He nodded to Sprigge. "A hundred stripes to start with."
Malenfant was dragged out of the room. "You're a savage, Praisegod. And you run a jerkwater dump. If this is some holy crusade, why do you allow your men to run a forced brothel?"
But Praisegod wasn't listening. He had turned to his Ham boy, and stroked his misshapen head.
Malenfant was taken to a room further down the dismal corridor.
He found himself stretched out over an open wooden frame, set at forty-five degrees above the horizontal. His feet were bound to the base of the frame. Sprigge wrapped rope around his wrists and pulled Malenfant's arms above his head until his joints ached.
Sprigge looked Malenfant in the eye. "I have to make it hard," he said. "It'll be the worse for me if I spare you."
"Just do your job," Malenfant said sourly.
"I know Praisegod well enough. That fat Englishman just riled him. He thinks you might be useful to him. But you must play a canny game. If you go badly with him, he'll ill use you, Malenfant. I've seen that before, too. He has a lot of devices more clever than my old whip, I'll tell you. He has gadgets that crush your thumbs or fingers until they are as flat as a gutted fish. Or he will put a leg-clamp on you, a thing he'll use on recalcitrant Runner folk, and every day we have to turn it a little tighter, until the bones are crushed and the very marrow is leaking into your boots."
Malenfant tried to lift his head. "I don't have any boots."
"Boots will be provided."
A joke? He could dimly make out Sprigge's face, and it bore an expression of something like compassion—compassion, under a layer of dirt and weathered scars and tangled beard, the mask of a hard life. "Why do you follow him, Sprigge? He's a madman."
Sprigge tested the bonds and stepped back. "Sometimes the lads go off into the bush. They think life is easier there, that they can have their pick of the bush women, not like the bleeding whores they keep here. Well, the bush folk kill them, if the animals or the bugs don't first. As simple as that. Without Praisegod we'd all be prey, see. He organizes us, Sir Malenfant. We're housed and we're fed and nobody harms us. And now that he's taken up with the Daemons—well, he has big ideas. You have to admire a man for that."
Malenfant thought, What the hell is a Daemon? He felt his jacket being pulled off his back. The air was damp and cold.
"Now, a hundred stripes is a feeler, Sir Malenfant. I know how you'll bear it. But you'll live; remember that." He stepped away, into the dark.
Malenfant heard running footsteps.
And then he heard the lash of the whip, an instant before the pain shot through his nervous system. It was like a burn, a sudden, savage burn. He felt blood trickling over his sides and falling to the floor, and he understood why the frame under him had to be open.
More of Sprigge's "stripes" rained down, and the pain cascaded. There seemed to be no cut-off in Malenfant's head, each stroke seemingly doubling the agony that went before, a strange calculus of suffering.
He didn't try to keep from crying out.
Maybe he lost consciousness before the hundred was done.
At last he was hit by a rush of water—it felt ice-cold—and then more pain reached him, sinking into every gash on his back, like cold fire.
Sprigge appeared before him. "The salty back," he said, cutting Malenfant's wrists free. "It'll help you heal."
Malenfant fell to the floor, which stank of his own blood, like the iron scent of the crimson dust of this rusted Red Moon.
A heavy form moved around him in the dark. He cowered, expecting more punishment.
But there was a hand on his brow, water at his lips. He could smell the dense scent of a Ham—perhaps it was Julia. The Ham helped him lie flat on his belly, with his ripped jacket under his face. His back was bathed—the wounds stung with every drop—and then something soft and light was laid over his back, leaves that rustled.
The square window in the ceiling above showed diffuse gray-blue. It was evening, or very early morning.
He was left alone after that, and he slept, falling into a deeper slumber.
When he woke again that square of sky was bright blue. By its light he saw that the leaves on his back were from a banana tree. His pain seemed soothed.
"... Malenfant. Malenfant, are you there?"
The voice was just a whisper, coming from the direction of the door.
Malenfant got his hands under his chest, pushed himself up to a crawling position. He felt the leaves fall away from his back. His bare chest was sticky with his own dried blood, and with every move he felt scabs crack, wounds ache.
He crawled to the wall by the door, kneeling there in the mud and blood.
"McCann?"
"Malenfant! By God it's good to hear the voice of a civilized man. Have they hurt you?"
Malenfant grimaced. "A 'feeler,' Sprigge called it."
"It could get worse, Malenfant."
"I know that."
McCann's voice sounded odd—thick, indistinct, as if he were talking around something in his mouth. _Flogging_ , _branding, tongue-boring_ , Malenfant recalled. The penalty for blasphemy.
"What have they done to you, Hugh?"
"My punishment was enthusiastically delivered," McCann lisped. "One must admire their godly zeal... And the beatings are not the half of it. Malenfant, he has me laboring in the fields: pulling ploughs, along with the Runner slaves. It is not the physical trial—I can barely add an ounce to the mighty power of my Runner companions—but the indignity, you see. Praisegod has made me one with the sub-men, and his brutish serfs mock me as I toil."
"You can stand a little mockery."
"Would that were true! Praisegod understands how to hurt beyond the crude infliction of blows and cuts and burns; and the shame of this casting-down has hurt me grievously—and _he_ knows it. But his punishment will not last long, Malenfant. I am not so young nor as fit as I was; soon, I think, I will evade Praisegod's monstrous clutches once and for all... But it need not be so for you. Malenfant, I think Praisegod has some sympathy for you—or purpose, at least. Tell him whatever it is you think he wants to hear. That way you will be spared his wrath."
Malenfant said softly, "You were the one who said you could do business with him."
"Do as I say, not as I do," McCann hissed. "It is my faith, Malenfant, my faith. Praisegod arouses in me a righteous rage which I cannot contain, whatever the cost to myself. But he is an intelligent man, a cunning man. I suspect his grasp of his ugly crew here was slipping. I have heard the men mutter. They tell fortunes, you know, with cowrie shells—much handled, shining like old ivory... Superstition! A fatal flaw for a regime whose legitimacy comes entirely from religion. He was on his uppers, Malenfant, until quite recently. But now his inchoate ambitions have found a new clarity, a _plausibility_. He has found new allies: these Daemons, whoever or whatever they are. He has suddenly become a much more credible, and dangerous, figure... If I had half a brain I would stay in his fold.
"But you are different, Malenfant. Without faith—a paradoxically enviable condition!—you have no moral foundation to inhibit you; you must lie and cheat and steal; you must kowtow to Praisegod; you must do everything you can, everything you _must_ , to survive."
"I'll try," Malenfant gasped.
"Will you, my friend? Will you truly? There is a darkness in you, Malenfant. I saw it from the beginning. You may choose, without knowing it, to use Praisegod as the final instrument of your own destruction."
"What the hell are you talking about?"
"You must look into your heart, Malenfant. Think about the logic of your life... The day advances. Soon I will be called to my work in the fields, and I must sleep if I can."
"Take care of yourself, Hugh."
"Yes... God be with you, my friend."
That night Malenfant called McCann's name. The only reply was a kind of gasping, inarticulate, and a moist slithering.
The night after that Malenfant called for McCann, over and over, but there was no reply.
## _E mma Stoney_
She had first become aware of Joshua as an absence. There was a spare place at the hearths of Ruth and others, portions of meat left set aside by the hunters. It was a pattern she had noticed before when somebody had recently died; the Hams clearly remembered their dead, and they made these subtle tributes of absence—halfway to a ritual, she supposed.
Then, one day, Joshua came back.
Within a couple of days it was clear Joshua was not like the other Hams.
He was perhaps twenty-five years old, as much as she was any judge of the ages of these people. His body bore the marks of savage beatings, and his tongue seemed to be damaged, making his speech even more impenetrable than the rest.
No Hams lived alone. But Joshua lived alone, in his cave beyond the communal space around the hut. Hams did not go naked—but Joshua did, wearing not so much as a scrap of skin to cover his filth-encrusted genitals. Hams cut their hair and, crudely, shaved their beards with stone knives. Joshua did not, and his hair was a mane of black streaked with gray, his beard long but rather comically wispy under that huge jaw. Hams joined in the activities of the community, making tools, gathering and preparing food, repairing clothes and the hut. Joshua did none of this.
Hams did not make markings, or symbols of any kind—in fact they showed loathing of such things. Joshua covered the walls of his cave with markings made by stone scrapers and bits of bone. They might have been faces; he sketched rough ovals and rectangles, crisscrossed by interior lines—noses, mouths?—over and over. The marks were crude scratches, as if made by a small child—but still, they were more than she had ever observed any other Ham to make.
The other Hams tolerated him. In fact, since he did no gathering or hunting, by providing him with food they were keeping him alive, as she had seen other groups sustain badly injured, sickly, or elderly individuals. Perhaps they thought he was ill, beyond his body's slowly healing wounds.
Certainly, by the standards of his kind, he was surely insane. Studying this Ham hermit from afar, Emma concluded that whatever his story, she had best avoid him.
But when Joshua spotted her, the matter was taken out of her hands.
She was walking up the beach from the sea. Her catch of fish had been good that day, and she had used a scrap of blue chute cloth from her pack to carry it all.
Joshua was sitting outside his cave, muttering to himself. When he saw her blue cloth, he got to his feet, hooted loudly, and came running.
Other Hams, close to the hut, watched dully.
Joshua capered before her, muttering, his accent thicker than any she had heard before. He was gaunt, and his back was still red with half-healed welts. But he might have been three times her weight.
Emma reached for the stone knife she kept tucked in her belt. "Keep back, now."
He grabbed the blue cloth, spilling the fish on the sand. He sniffed the cloth with his giant, snot-crusted nostrils, and wiped it over his face. "This," he shouted. _"This!"_
She frowned. "What is it? What are you trying to tell me?"
"Th' door in the sky," he said. "Th' door in Heaven. Th' wings of th' seed." His voice was horribly indistinct—and when he opened his mouth to yell these things at her, she saw a great notch had been cut out of his tongue.
She should get out of here, flee to the sanctuary of the hut, get away from his deranged grasp. But she stayed. For no other Ham had used phrases like "the door in the sky."
She asked cautiously, "What door?"
"Th' sky seed. Th' Gray Earth. Th' seed fell th' sky."
She understood it in a flash. She whirled and pointed to the lander, stranded on the cliff face. "Is that what you're talking about? The lander—the thing that fell from the sky?" She grabbed back the bit of cloth. "Under a parachute. A blue chute, wings, like this."
For answer he bellowed, "Sky seed!" And he turned away and ran full tilt toward the foot of the cliff, beneath the lander.
Emma watched him go, her heart thumping.
She could stay here her whole life and never persuade the Hams to help her get to the lander. Maybe it took an insane Ham even to conceive of such a project. A Ham like Joshua.
Now or never, Emma.
She grabbed her pack and ran after Joshua.
There was a trail, of sorts, that led from the beach to the top of the cliff. At least Joshua showed her the way; she couldn't have managed at all otherwise. But it was a trail for Hams—or maybe goats—certainly not for humans. The scrambling and climbing was a major challenge for Emma, never superfit, never any kind of climber. Nevertheless, by sheer force of will, she kept up.
At the top of the cliff she fell back, exhausted, her heart pumping and her lungs scratching for air. It was like her first few days after the portal, when she had struggled to acclimatize to this strange mountaintop world.
Joshua immediately plunged into the cliff-top forest. Emma forced herself to her feet and followed.
Joshua crashed through the dense forest by main force, pushing aside branches, saplings, and even some mature trees. He seemed careless of the noise he made and the trail he left behind—again unlike most Hams, who took care to pass silently through the dangerous twilight of the forest.
At last they pushed into a clearing. Here the trees had been battered flat, she saw, and bits of blue cloth clung to scattered branches. Her heart thumped harder. Joshua ran across the clearing to the far side, where a last line of trees had been broken down, exposing blue-gray sky. She followed him.
She found herself at the lip of the cliff, looking down on a trail of scraped rock and bits of cloth and chute cord. And there, really not so far beneath the lip of the cliff, like a fat bug trapped in some huge spiderweb, lay the lander.
Joshua squatted on his haunches and pointed down at the lander. "Sky seed," he said excitedly. "Sky seed!"
She gazed hungrily down at the lander: crumpled, battered, stained and weathered, but intact. She saw no sign that anybody had climbed out of it since its plummet down the cliff.
From here the lander looked very small. Specifically, she couldn't see any sign of an engine pack, no way the thing could get itself off the ground and back to Earth.
She sat back, forcing herself to think. Sitting here with a Neandertal the internal politics of America seemed a remote abstraction—but still she couldn't believe that the US government would sanction any kind of one-way mission, even for someone as persuasive as Reid Malenfant. But that meant—she thought, her brain working feverishly—that the engine had to be somewhere else.
She grabbed Joshua's arms, and immediately regretted it; his skin was covered in filth and scabs. He flinched back from her touch, as if she intended to hurt him. She let go, and held her empty hands up before him. "I'm sorry... Listen to me. _There must be another lander_. I mean, another sky seed. A second one." But Hams did not count. She held her hands up to mime two landers coming down from the west, one after the other. But Hams did not use symbols.
She pointed, bluntly. "Sky seed. Down there. Sky seed." She pointed into the forest, at random. "Over there."
He frowned. He pointed west, deeper into the forest. "Ov' there."
She took a deep breath. _I knew it_.
But now Joshua was jabbering, pointing at the lander and the sky. "Sky seed. Praisegod. 'There 'fore me was door standin' open Heav'n.' Sky seed in Heav'n. People of th' Gray Earth. People of Heav'n." And on and on, a long, complex, baffling diatribe.
She peered into his ridged eye sockets, struggling to understand what was going through that mind—so alien from hers, and damaged, too.
Bit by bit she got it.
Joshua had seen the lander come down from the sky. He had seen the second lander, too. She knew that Hams believed their people came from a place in the sky, which they called the Gray Earth. Joshua, alternately, called it Heaven. As best she could make out he wanted to use the lander to take his people home, to Gray Earth, to Heaven.
"Was it the Zealots who taught you about Heaven? Did the Zealots hurt you? Did this Praisegod hurt you?"
'Prai'go' Michael," he mumbled. "Mal'fan."
Suddenly she couldn't breathe. She grabbed his shoulders, mindless of the filth, resisting his flinching. _"What did you say?"_
"Mal'fan'. Zealots. Mal'fan'."
The Zealots had Malenfant. _Malenfant was here_.
She sat back on her haunches, breathing in gasps. "Do you know where Malenfant is being held?—no, you can't tell me that. But you could show me." She studied Joshua, who gazed back at her. "Listen to me. There is something you want. There is something I want. This is what we will do. You take me to Malenfant... If you do this, I will give you the lander. It will take you home, to Heaven, to the Gray Earth."
It took a long time to make him understand all of this. It might have been the first time in the history of these Hams, she thought, that anybody had tried to strike a bargain.
And, as she had absolutely no intention of using the lander for anything else but getting herself and Malenfant out of here, it might have been the first time anybody had told a Ham a lie.
## _R eid Malenfant_
Uncounted days after his whipping, Malenfant was again dragged before Praisegod Michael.
Malenfant stood as straight as he could, his arms tied behind his back, a new skin jacket over his upper body. He seethed with resentment at his own pain and humiliation, anger at what he suspected had become of McCann, and a kind of self-righteous disgust at Praisegod.
Get a hold of yourself, Malenfant. Do business, remember.
"What now, Praisegod? Another beating?"
Praisegod walked around Malenfant. Malenfant saw how his right leg spasmed, as if he wished to flee; he seemed unusually agitated. Praisegod Michael was a man of depths, all of them murky.
Praisegod's Ham boy sat on the edge of the desk, staring at Malenfant.
"I do not wish to punish you, Sir Malenfant. I can tell you have twice the mentation of Sprigge, here. I would rather obtain your support."
"You know nothing about me."
Praisegod said, "Where we came from does not matter, Malenfant. For we cannot escape this place; men have spent their lives to prove that. And as your friend McCann understood, what unites men, in this world of animals, is greater than that which separates us. All that matters is that we are here, now, and we must make the best of it. Though it has the face of a work of Satan, this island is a world made by God—of course it is; to argue otherwise would be to support the heresy of Manichaeus. Therefore it is perfectible, and therefore there is good work to be done here by righteous men... There is much to be done here."
Malenfant eyed him. Praisegod was a shithead, yes. He wasn't about to conquer the Red Moon. But a shithead like this could cause a lot of suffering to a lot of people, and near-people. "Perfectible? Right. I know your kind. You intend to build an empire, Praisegod. A _perfect_ empire, soaked in blood."
"What is blood?" Praisegod said easily. "If men stand against us, they will be as stubble before our swords. And as for the rest, to spill the blood of an animal is not a sin, Malenfant. Indeed, given that these soulless apes show a mockery of man's features, I am convinced that to cleanse the worlds of their obscene forms is a duty."
"So you will use the Hams and the Runners as a resource to build your empire on this Moon. And when the hominids' usefulness has passed, you will exterminate them."
Praisegod's predator's eyes gleamed. "It is time for your answer, Malenfant."
Malenfant closed his eyes.
_Stay alive_ , Malenfant. That's all that matters. The creatures on this Red Moon mean nothing to you. A little while ago you didn't even know they existed. (But some of them have helped me, even saved my life...) And they are not even human. (But they are differently human...) This Praisegod may be difficult, but he is powerful. If you can work with him he may even help you achieve your goal—which is, was, and always will be to find Emma. (But he's a psychopathic monster...)
He imagined he heard Emma's mocking voice.
_You can't do it, can you? You never were too good at politics, were you, Malenfant?—even in NASA—anyplace where the ancient primate strategies of knowing when to fight and when to groom, when to dominate and when to submit, were essential. Ah, but this is about more than politics, isn't it, Malenfant?Are you growing a conscience? You, who lied his way to Washington and back to get his BDB off the ground, who used up people and spat them out on the way to achieving what you wanted? Now you stand here on this jungle Moon and you can't swallow a few preachy platitudes to save your own worthless hide?_
Or, he thought, maybe McCann was right about me. So was my mother-in-law, come to that. Maybe all I ever wanted to do was crash and burn.
Praisegod's foot was tapping out its nervous drumbeat. The Ham boy, seeming to sense the tension between the two men, slid off the desk and crawled behind Praisegod's chair.
Malenfant took a breath. He said, "Why are you really so dead set against the hominids?" He glanced at the Neandertal boy; one eye and a thatch of ragged dark hair protruded from behind the chair leg. "Does this boy warm your bed, Praisegod Michael? Is that why you have to destroy him?"
Malenfant saw white all the way around Praisegod's pupils, and a dribble of blood and snot was leaking from his nose. The man stood before Malenfant, close enough to smell the fishy stink of his breath. He whispered, "This time the whips will fillet the flesh off you, until the men will be flogging your neck and the soles of your feet. And I, I will prevail, in the light of His countenance."
Malenfant had time for an instant of satisfaction. Got through to you, you bastard. Then he was clubbed to his knees.
## _E mma Stoney_
She spent days in the cliff-top forest, spying, scouting.
This patch of forest was damp and thin. There were extensive clearings where old trees had fallen to the ground in chaotic tangles of branches. Paths wound among the trees, marked out through rotting leaves, fungus-ridden trunks, brambles, and crushed saplings. Many of these paths were made no doubt by animals, or perhaps hominids, the Nutcracker-folk or the Elf-folk. But some of them were, unmistakeably, the work of humans; straight, sometimes rutted by wheels.
And the human paths converged on a township, a brooding, massive structure at the heart of the forest. It was the fortress of the Zealots.
The great gate of the compound would open a couple of times a day to let out or admit parties, apparently for hunting and provisioning. The open gates, swinging on massive hinges of rope, revealed a shabby cluster of huts and fire-pits within. The Zealot foragers, always men, always dressed in drab green-stained skins, were armed with pikes and bows and arrows. They stayed alert as they made their way along the paths they had worn between the trees.
The returning parties would call out informal _halloos_ to let those inside know they wanted in. Nobody seemed to feel the need for passwords or other identifiers. But the gate openings were brief, and the forest beyond was always carefully watched by armed men. The foragers would return with sacks full of the forest's fruits, or with bats or animals, commonly small hogs, or even grain and root vegetables brought in from the hinterland that must stretch beyond the forest.
But they would also bring home Elves, even the occasional Nutcracker, suspended limply from poles, heads lolling. The Zealots had no taboo, it seemed, over consuming the flesh of their apparent near-relatives—which she heard them call, in their thick, strangulated accent, _bush meat_. The hunters seemed to prize the hands and ears of infant Elves, which they would hack off and wear around their necks as gruesome trophies.
Also, less frequently, they brought home captured Runners. The Runners were always returned alive. The men and boys were evidently beaten into submission, their backs bearing the scars of whips and their faces misshapen from blows; they trudged through the forests with ropes around their necks and wrists, and with their long legs hobbled so they had to shuffle. She supposed the male Runners were brought back to the stockade as slave labor. Their strong, supple bodies and clever hands well qualified them for the role.
Perhaps some of the captured women and girls were used that way, too, but Emma suspected they had a darker fate in store. They were returned to the township with bite marks and scratches on their breasts and blood running down their legs. Some of the boys seemed to have been similarly abused. Evidently the hunters took the breaking-in of a new captive as a perk of the job. Emma had no way of knowing how many of these victims had fought too hard, and ended their lives in the forest in uncomprehending misery beneath the grunting bodies of the Zealots.
She was relieved her instinct had always been to keep out of sight of these people. She didn't quite know what reaction they would have to finding a human woman alone in the forest, but she didn't feel inclined to take a chance on their charity.
At last her spying paid off. She overheard a group of hunters, as they lazed in the shade of a fig tree, feeding themselves on its plump fruit and talking loosely. Their gossip was of a major expedition—it almost sounded military—to take on a new group the Zealots called the _Daemons_. The Zealots sounded alternately apprehensive and excited about the coming conflict; there was much speculation about the quality of the women among the Daemons.
Emma knew nothing about these Daemons, and couldn't care less. But if a large number of the township's able bodies was going to be taken away, she sensed a window of opportunity.
She sat in the cave before Joshua, holding his massive head with both her hands on his filthy cheeks, making him face her. "Hunting Praisegod Michael. Tomorrow. Hunting Praisegod. Do you understand?"
"Hunt Prai'go'," he said at last, thickly, his damaged tongue protruding. "Tomorr'."
"Yes. Tomorrow. Wait until tomorrow. All right?"
He gazed back at her, his eyes containing an eerie sharpness that none of his people seemed to share. Perhaps there was madness there—but even so, it was a much more human gaze than any she had encountered since losing Sally and Maxie. But there was absolutely no guile in those eyes, none at all, and no element of calculation or planning.
She released him.
He picked up a rock he had been knapping, and resumed working on it, steady, patient. She sat down in the corner of the cave, her legs drawn up to her chest, arms wrapped around her knees, watching him. The blue-gray glow of the sky, leaching of light, reflected in his eyes as he worked; often, like most Ham knappers, he didn't even look at the stone he was working.
Tomorrow, this child-man would have to take part in a concerted assault.
Not for the first time Emma wondered what the hell she was doing here. _How have I come so far? I'm an accountant, for God's sake..._
She had spent the days waiting for the Zealots' expedition trying to raise a fighting force from among the Hams. But she had quickly learned that it was impossible to turn these huge, powerful, oddly gentle creatures into anything resembling soldiers—not in a short time, probably not if she kept at it forever. She had hit at last on the notion of making the assault a hunt, the one activity where the Hams did appear to show something resembling guile.
But even now she didn't know how many of them she could count on. She, and Joshua, had managed to enthuse a few of the younger men to join the battle. But when she approached them the next day even the most ardent would-be warriors would have forgotten all about the project.
Another problem was that the Hams' _only_ notion of actual combat was hand-to-hand: just yesterday she had seen three of the men wrestle an overgrown buck antelope to the ground with their bare hands. It was a strategy that had worked for them so far, evidently, or the cold hand of natural selection would long ago have eliminated them—even if they paid the price in severe injuries and shortened life spans. But it wasn't a strategy that would work well in a war, even against the disorganized and weakened rabble she hoped the Zealots would prove to be.
In the end, she realized, the Hams would fight (or not) according to their instinct and impulse, and they would fight the way they always had, come what may. She would just have to accept that, and deal with the consequences.
Joshua turned the rock over in his hands, running his scarred fingertips over the planes he had exposed, gazing intently at it. Unlike her, he wasn't fretting about tomorrow. She sensed a stillness about his mind, as if it were a clear pool, clear right to the bottom, and in its depths all she could see was the rock. It was as if Joshua and the rock blurred together, becoming a single entity, as if his self-awareness were dimming, as if he were more aware of the microstructure of the rock even than of himself.
With her head echoing as ever with hopes and fears and schemes, Emma couldn't begin to imagine how that might feel. But she knew she envied him. Since starting to live with the Hams she had often wished she could simply switch off the clamor in her head, the way they seemed to.
Now Joshua lifted his worn bone hammer—the only possession he cherished—and, with the precision of a surgeon, tapped the rock. A flake fell away. It was a scraper, she saw, an almost perfect oval.
He lifted his head and grinned at her, his scarred tongue protruding.
The Zealots' attacking army had drawn up in rough order outside the stockade, armed with their crossbows and knives and pikes. There looked to be fifty men and boys, and they had been followed by about as many Runner bearers, all of them limping, their arms full of bundles of weapons and provisions.
Emma watched the soldiers prepare, curious. The pikemen, in addition to their immensely long pikes, had leather armor: breastplates and backplates, what they called gorgets to protect their throats, and helmets that they called pots. They carried provisions in leather packs they called snapsacks. There was even a cavalry, of sorts; but the soldiers rode the shoulders of men, of Runners. They were marshalled by an insane-looking cleric type, in a long robe of charcoal-blackened skin—and by a hominid, a vast, hulking gorillalike creature with rapid, jerky movements and swivelling ears. Was it a Daemon? At least eight feet tall, it looked smart, purposeful; Emma hadn't seen its like before.
Not your problem, Emma.
The army, its preparations nearly done, sang hymns and psalms. Then a man they called Constable Sprigge stood on a rock before them, and began to pray. "Lord, you know how busy I must be this day. If I forget Thee, do not Thou forget me..." Emma found the wry soldiers' prayer oddly moving.
And with that the army marched off through the forest. The Zealot fortress was as weakened as it would ever be.
She crouched by the stockade gate, her heart beating like a hammer drill, clutching the shortest, sharpest thrusting spear she could find. She surveyed her own motley army. In the end, only the big man, Abel—Joshua's brother—and the oddly adventurous girl Mary had elected to join her and Joshua on this expedition. Three Hams counted physically for a lot more than twice as many Zealots. And she was planning nothing more than a smash-and-grab raid, a commando operation, a mission with a single goal. But still, there were only four of them—three child-people and herself, and she was certainly no soldier.
She was frightened for the Hams, already guilty for the harm they would surely suffer today—and, of course, profoundly frightened for herself, middle-aged accountant turned soldier. But this was the only way she could see to get to Malenfant. And getting to him was the only way she was ever going to get out of this dismal, bizarre place—if he really was here, if he was still alive, if she hadn't somehow misunderstood Joshua, fooled by his damaged tongue and her own aching heart. And so she put aside her fears and doubts and guilt, for there was no choice.
She kept her Hams quiet until she was sure the ragged Zealot army was out of hearing.
## _M anekatopokanemahedo_
The compound was calm, quiet, orderly. Workers trundled to and fro over the bright yellow floor of Adjusted Space, pursuing their unending chores.
But not a person moved. They stood or sat or lay in a variety of poses, like statues, or corpses, arrayed beneath the huge turning Map of the world. The core activity here was internal, as each person contemplated the vast conundrum of the Red Moon.
After two million years of continuous civilization, nobody rushed.
But to Manekato, after her vivid experiences in the forest, it was like being in a mausoleum. She found a place of shade and threw herself to the ground. A Worker came over and offered her therapeutic grooming, but Manekato waved it away.
Nemoto came to her. She carried her block of paper, much scribbled on. She sat on the floor, cross-legged, and regarded Manekato gravely. _"Renemenagota of Rano represents a great danger."_
Manekato snapped her teeth angrily. _"What do you know of the hearts of people? You are not even a person. You are like a Worker..."_
But Nemoto showed no distress. _"Person or not, I may perceive certain truths more clearly than you. I see, for instance, that you are troubled on a deep level. You are human, but you are still animal, too, Manekato. And your animal side is repelled by the cold efficiency of this place you have built, and is drawn to the dark mysteries of the forest. Perhaps my lesser kind have a better understanding of the shadows of our hearts."_ But there was defiance in her pronunciation of that word _lesser_.
Manekato felt shamed. Hadn't she just taken out her own distress and confusion on a weaker creature—this Nemoto—just as Without-Name had punished the hominids she had captured? She propped herself up on her elbows. _"What is it you want?"_
_"I have a hypothesis,"_ said the little hominid.
Manekato sighed. More of Nemoto's theories: partial, immature, expressed badly and at the pace of a creeping glacier—and yet suffused by an earnest need to be understood, listened to, approved. She nodded, a gesture she had learned from Nemoto herself.
Nemoto began to spread pages of her paper block over the floor. The paper bore columns labeled _Earth, Banded Earth, Gray Earth (Hams)_ , and so on, though some columns were headed by nothing but query marks. And the paper was covered with a tangle of lines and arrows that linked the columns one to the other.
_"I have elaborated my views,"_ Nemoto said. _"I have come to believe that this Red Moon has played a key role in human evolution. Consider. How do new species arise, of hominids or any organism? Isolation is the key. If mutations arise in a large and freely mixing population, any new characteristic is diluted and will disappear within a few generations. But when a segment of the population becomes isolated from the rest, dilution through interbreeding is prevented. Then, when a new characteristic appears within the group—and provided it is beneficial to the survival of the group and the individuals within it—it will be reinforced. Thus the isolated group may, quite rapidly, diverge from the base population_.
_"And when those barriers to isolation are removed, the new species finds itself in competition with its predecessors. If it is better adapted to the prevailing conditions, it will survive by outcompeting the parent stock. If not, it declines_.
_"When our scientists believed there was only one Earth, they suspected the evolution of humanity had been the consequence of a number of speciation steps. The apelike bipedal Australopithecines gave rise to tool users, who in turn produced erect hairless creatures capable of walking on the open plain, who in turn gave rise to various species of_ Homo sapiens _—the family that includes myself. It is believed that at some points in history there were many hominid species, all derived from the base Australopithecine stock, living together on the Earth. But my kind_ —Homo sapiens sapiens— _proved the fittest of them all. By outcompetition, the variant species were removed_.
_"Presumably, each speciation episode was instigated by the isolation of a group of the parent stock. We had generally assumed that the key isolating events were caused by climate changes: rising or falling sea levels, the birth or death of forests, the coming and going of glaciation. It was a plausible picture. Before we knew of the Red Moon."_
_"And now your radical hypothesis—"_
Nemoto tapped her papers. _"What if the vagaries of the Red Moon were involved in all this? Look here. This central column sketches the history of the Earth."_
"Your _Earth_."
Nemoto smiled, her small naked face pinched. _"Assume that the base Australopithecine stock evolved on Earth. Imagine that the Red Moon with its blue Wheel portals scooped up handfuls of undifferentiated Australopithecines and, perhaps some generations later, deposited them on a variety of subtly different Earths."_
_"It is hard to imagine a more complete isolation."_
_"Yes. And the environments in which they were placed might have had no resemblance to those from which they were taken. In that case our Australopithecineswould have had to adapt or die. Perhaps one group was stranded on a world of savannah and open desert—"_
_"Ah. You are suggesting that the hairless, long-legged Runners might have evolved on such a world."_
"Homo erectus— _yes. Other worlds produced different results. And later, the Red Moon returned and swept up samples of those new populations, and handed them on to other Earths—or perhaps returned them to where they had come from, to compete with the parent stock, successfully or otherwise_.
_"My species shares a comparatively recent common ancestor with creatures like the Hams—which are of the type we call Neandertals, I think. Perhaps a group of that ancestral stock was taken to the world the Hams call the Gray Earth, where they evolved the robust form we see now. And, later, a sample of Hams was returned to the Earth. Later still, groups of_ Homo sapiens sapiens _—that is, my kind—were swept here from the Earths of the groups called the English and the Zealots, and no doubt others."_ She gazed at her diagrams. _"Perhaps even my own kind evolved on some other Earth, and were brought back by the Moon in some ancient accident."_
Manekato picked her nose thoughtfully. _"Very well. And my Earth—which you have labeled 'Banded Earth'?"_
Somewhat hesitantly, Nemoto said, _"It seems that your Earth may have been seeded by Australopithecine stock from_ my _Earth. You seem to have much in common, morphologically, with the robust variant of australopithecines to be seen in the forests here, called Nutcrackers."_
Manekato lay back and sighed, her mind racing pleasurably. _"You fear you have offended me by delegating my world to a mere offshoot. You have not. And your scheme is consistent with the somewhat mysterious appearance of my forebears on Earth—my Earth."_ She glanced at Nemoto's sketches. _"It is a promising suggestion. This strange Moon might prove to be the crucible of our evolution: Certainly it is unlikely that hominid forms could have evolved independently on so many diverse Earths. But such is the depth of time involved, and such is the complexity of the mixing achieved by our wandering Moon, the full picture is surely more complicated than your sketch—and it is hard to believe that_ your _Earth just happens to be the primary home of the lineage... And how is it that so many of these other Earths share, not just hominid cousins, but a shared history, even shared languages? Your own divergence from the Zealot type must be quite ancient—their peculiar tails attest to that—and yet your history evidently shares much in common with them."_
Nemoto frowned, her small face comically serious. _"That is a difficulty. Perhaps there is such a thing as historical convergence. Or perhaps thewandering of the Moon has induced mixing even in historical times. Cultural, linguistic transmission—"_
It was a simplistic suggestion, but Manekato did not want to discourage her. _"Perhaps. But the truth may be more subtle. Perhaps the manifold of universes is larger than you suppose. If it were arbitrarily large, then there would be an arbitrarily close match to any given universe."_
Nemoto puzzled through that. _"Just as I would find my identical twin, in a large enough population of people."_
_"That's the idea. The closer the match you seek, the more unlikely it would be, and the larger the population, of, umm, candidate twins you would need to search."_
_"But the degree of convergence between, say, the Zealot universe and my own—language, culture, even historical figures—is so unlikely that the manifold of possibilities would have to be very large indeed."_
_"Infinite,"_ said Mane gently. _"We must consider the possibility that the manifold of universes through which we wander is in fact infinite."_
Nemoto considered that for a while. Then she said, _"But no matter how large the manifold, I still have to understand_ why _this apparatus of a reality-wandering Moon should have been devised in the first place—and who by."_
Manekato studied Nemoto, wishing she could read the hominid's small face better. _"Why show me your schema now?"_
_"Because,"_ Nemoto said, _"I believe all of this, this grand evolutionary saga, is now under threat."_
Manekato frowned. _"Because of the failure of the world engines?"_
_"No,"_ Nemoto said. _"Because of you. And Renemenagota of Rano."_
A shadow fell over Manekato's face. "Your monkey may be right, Mane. You should listen to it."
It was Without-Name. She stepped forward, carelessly scattering Nemoto's spidery diagrams.
## _E mma Stoney_
Emma lifted her head. _"Hall-oo! Hall-oo!"_ Her call, though pitched higher than that of the men who mostly ventured outside the stockade, was, she was sure, a pretty accurate imitation of the soft cries of returning hunters.
Within a couple of minutes she heard an answering grunt, and the rattle of heavy wooden bolts being slid back.
All or nothing, she thought. Malenfant—or death.
When the heavy gate started to creak open, she yelled and threw herself at it. Her flimsy mass made no difference. But the Hams immediately copied her, making a sound like a car ramming a tree. The splintering gate was smashed back, and she heard a howl of pain.
The Hams surged forward. There were people in the compound, women and children. As three immense Hams came roaring in among them, they ran screaming.
Emma glanced around quickly. She saw a litter of crude adobe huts, one substantial chapel-like building at the centre, a floor of dust stamped flat by feet and stained with dung and waste. She smelled shit, stale piss.
Now the door to one of the buildings flew open. Men boiled out, pulling on clothing. Inside the building's smoky darkness Emma glimpsed naked Runner women, some of them wearing mockeries of dresses, others on beds and tables, on their backs or their bellies, legs splayed, scarred ankles strapped down.
Grabbing pikes and clubs and bows, the men ran at Abel, howling. With a cry of pleasure Abel joined with them. He brushed aside their clubs as if they were twigs wielded by children. He got two of the Zealots by the neck, lifted them clean off the ground, and slammed their heads together, making a sound like eggs cracking.
But now the bowmen had raised their weapons and let fly. Emma, despising herself, huddled behind Abel's broad back. She heard the grisly impact of arrows in Abel's chest. He fell to his knees, and blood spewed from his mouth.
The archers were struggling to reload. Mary hurled herself at them, fists flailing.
Emma grabbed Joshua's arm. "Malenfant! Quickly, Joshua. Malenfant—where?"
For answer he ran toward the chapel-like central building. Emma touched Abel's back apologetically, and ran after Joshua toward the chapel. She seethed with rage and adrenaline and fear. This had better be worth the price we're paying, Malenfant.
## _M anekatopokanemahedo_
Manekato stood quickly. Nemoto hurried behind her, sheltering behind her bulk. Babo came running to join them, his legs and arms levering him rapidly over the floor of Adjusted Space. Other people gathered in a loose circle around this central confrontation, watching nervously. Workers scuttled back and forth, seeking tasks, trying to discern the needs of the people, ignored.
For the first time it struck Manekato just how physically big Without-Name was—towering over a lesser hominid like Nemoto, but larger than Manekato, too, larger than any of the other people on this expedition. Physical size did not matter at home, on civilized Earth. But on this savage Moon, strength and brute cunning were key survival factors; and Without-Name seemed to relish her unrestrained power.
And now Manekato noticed a new hominid following in Without-Name's wake. It was a male, taller than Nemoto, rake-thin, and he was dressed in a tight robe of animal skin stained black, perhaps by charcoal. He drew a Ham boy after him. The boy was dressed in elaborate clothing, and he had a collar around his neck, connected to a lead in the tall hominid's hand.
Babo said tightly, "And is this your Praisegod Michael, Renemenagota of Rano?"
Without-Name raised one hand.
Crossbow bolts thudded into Babo's belly and chest and upper arms. He cried out softly, dull surprise on his face. He crumpled forward and fell on the bolts, making them twist, and his cries deepened. A Worker rushed to tend Babo's wounds, but Without-Name kicked it away.
Manekato, stunned, saw that the circular platform was surrounded by hominids—Zealots, in their sewn skins. Some of them, bizarrely, were riding on the shoulders of Running-folk. They seemed afraid, but they held up their crossbows and spears with defiance.
Praisegod Michael passed his hands over Babo's shuddering form, making a cross in the air. " 'Behold, Esau my brother is a hairy man, and I am a smooth man...' "
Manekato found words. "Renemenagota—what are you doing?"
"Providing you with a purpose."
"Your army of hominids would be no match for the power we could deploy," Manekato whispered.
"Of course not— _if_ you choose to deploy it," Without-Name said mockingly. "But you won't, will you? Meanwhile these hominids believe they are soldiers of God. They have only their simple handmade weapons, but their heads are on fire. And so their crossbow bolts will best all your learning and technology. And under my guidance, they will sweep the world."
Now Nemoto stepped out from behind Manekato. Without-Name eyed the little hominid with undisguised loathing.
But Praisegod Michael faced her, apparently unsurprised to find her here. _"You are the one called Nemoto. Malenfant told me I would find you here."_
_"I know your kind,"_ Nemoto said. She turned to Manekato. _"You must stop this, here and now. You have not seen such things before, Manekato. With Renemenagota's organizational skill, Michael and his fellows will march on, overwhelming others with their savagery and determination, armed with an unwavering faith that will lead them to their deaths if necessary. Those they do not destroy will be forcibly converted to the creed. By the second generation the conquered will regard themselves as soldiers of the conquering army. We are limited creatures, Manekato, and we do not have the strength of mind to fight off a contagion of seductive but lethal ideas. You must stop this for the slaughter that will follow if you don't."_
Babo twisted on the ground, his hands clamped to his stomach, his face a rictus of pain. "Yes," he hissed. "Exponential growth, Mane. They will conquer, acquire resources to fuel further expansion, thus acquiring still more, and all driven by a dazzling virus of the mind."
Manekato said, "It is—unbelievable."
Nemoto faced her. _"Manekato, you must save us from ourselves—and save this machine-world from the deadly manipulation of Renemenagota."_
Without-Name stood before her, her immense biceps bunched, gazing into her eyes, so close Manekato could smell blood on her breath. "Perhaps this ape-thing is right, Manekato. Will you take its advice?—Ah, but then you would have to become like me, wouldn't you, and how you dread that! You must destroy me—but you cannot, can you, Mane?"
Babo, on the floor, groaned and raised one bloody arm. "But I can, Renemenagota of Rano."
A sudden wind, hot and dense, billowed before Manekato's face.
People staggered back, crying out. Nemoto took hold of Babo's arm, anchoring herself against the gusts.
A tube of whirling air formed over the platform. It was the end of a winding column that stretched down from the sky, silvery-gray, suddenly tightly defined. It was a controlled whirlwind, like that which had stormed around the Market for two hundred thousand years.
And in the heart of the column of tortured air was Renemenagota. She raised her fists, briefly bipedal like those who she had sought to lead. But she could land no blows on the twisting air, and it paid no heed to her screamed defiance.
In a brief blur of brown and black, she was gone.
The whirlwind shrivelled, shrinking back up into the lid of cloud that had covered the sky. A cloud of crimson dust came drifting down on the platform.
Mane, stunned, bewildered, looked around. Nemoto still clung to the fallen Babo. Of the ring of armed Zealots there was no sign.
Praisegod had been bowled over. He lay on his back on the platform, his black clothing scattered around him. His eyes flickered, cunning, calculating, the eyes of a trapped animal seeking a way out.
But his pet Ham boy stood over him.
Praisegod lifted his hand to the boy, asking for help, forcing a smile.
The boy bunched his fist and rammed it into Praisegod's chest, through clothing, skin, an arch of ribs.
Praisegod shuddered and flopped like a landed fish. The Ham's squat face was expressionless as he rummaged in that bloody cavern. Then the Ham boy grimaced, and the muscles of his arms contracted.
Praisegod's head arched back, and his voice was a rasp. _"Why have you forsaken me?"_
Then, his heart crushed, he was still.
## _E mma Stoney_
There was a lot of shouting going on. Mary was running around the compound, busily engaging her foe. Though Abel had fallen, Mary was moving too quickly for the archers to get an accurate sight on her, and every time she got close enough she was slamming heads, breaking arms, and generally kicking ass with a joyous vigor.
The chapel, built of mud brick around a sturdy wooden frame, was as substantial as it looked. Emma ducked into the building and slammed the door, and ran a heavy wooden bolt into a notch.
Within seconds fists were hammering on the door.
"Quickly," she said to Joshua. "Malenfant. Where?"
But Joshua did not reply, and when she turned, she saw that he was facing a crucifix, gazing at the gentle, anguished face of a Messiah. Joshua cringed, but was unable to look away.
The yelling at the door was growing intense, and the first hints of organized battering were detectable. Emma couldn't wait any longer. She cast around the little chapel, shoving aside furniture and a small, ornately carved wooden altar.
And she found a hatchway.
The hatch opened on a small, dark shaft, fitted with stubby wooden rungs. Emma clambered down hastily, to find herself in a short corridor. A single wicker torch burned fitfully in a holder. She grabbed it and hurried along the corridor.
The corridor led to two wooden doors. One door was swinging open, and Emma recoiled. The cell within was just a pit, with a filth-crusted floor and blackened, scratched walls; it stank of blood and vomit and urine.
The other door was shut. Emma hammered on it. "Malenfant! Are you there?" The wood was so filthy her hands came away smeared with deep black.
No reply.
Struggling to hold up the torch, she made out a thick bolt, just wood, a smaller copy of the one on the compound gate. She hesitated for a heartbeat, her hand on the bolt.
She reminded herself that she actually had no idea what lay on the other side of this door. But you've come this far, Emma.
She pulled back the bolt, dragged open the door. She held the torch in front of her protectively.
There were two people here. One was sitting on the floor, hands crossed over her chest for protection—her, for it was a woman, in a long dress that looked finely made. But despite the dress and the tied-back hair, that protruding face and the ridged eyes marked her out as a Ham.
The other was a man. He was wearing a blue coverall, and he was curled up in the dirt, folded on himself.
Emma hurried to him. Gently she lifted aside his arm, to reveal his face. "Do you know me? Do you know where you are? Oh, Malenfant..."
He opened his eyes, and his face worked. "Welcome to hell," he whispered.
The Ham woman slipped her arms under Malenfant and cradled him, with remarkable tenderness. She said her name was Julia; her English, though slurred by the deficiencies of the Ham palate, was well-modulated and clear.
With Malenfant limp but seemingly light as a baby in Julia's arms, they clambered out of the pit and back into the chapel.
Still the Zealots battered at the door. Joshua remained in his apelike crouch, his head buried in his big arms. He was whimpering, as if horrified by what he had done.
Gently Emma pulled his arm away from his face. His cheeks were smeared with tears. "No time," she said. "Mary. Skinnies hurt Mary. Joshua help."
It took an agonizing minute of repetition, with the hammering on the door turning into a splintering, before he responded.
He got to his feet with a roar. He ran to the door, dragged it open, and with a sweep of his massive arm he knocked aside the scrambling crowd of Zealot men. He forced his way outside, calling for Mary.
Julia followed, carrying Malenfant. Emma stayed close by her side, cradling Malenfant's lolling head.
# _PART FOUR_
# **World Engine**
#
## _R eid Malenfant_
"You always were a heathen bastard, Malenfant. No wonder Praisegod had it in for you. I remember the trouble we had when we chose a church. Even though it was a time when overt religiosity was a career asset if you wanted to be part of the public face of NASA."
"I did like that chapel at Ellington. Kind of austere, for a Catholic chapel. Not too many bleeding guys on the wall. And I liked the priest. Monica Chaum, you could go bowling with."
"Well, I liked the chapel, too, Malenfant. I found it comforting. A place to get away from the squawk boxes and the rest, when you were in orbit."
" _On_ orbit. You never told me that."
"There are lots of things you don't know about me, Malenfant. I remember one Christmas Eve when you were up there, doing whatever you did. _Christmas Eve_ , and I was alone. I was sick of it all, Malenfant. I wanted to go to church, but I didn't want people gawping. So I asked Monica if she would open up the church for me. Well, she dug out the organist, and she went through the church lighting all the candles, just as they would be lit for the Midnight Mass that night, and the organist played the program planned for the service. When I walked in and saw it was all there just for me—well, it was one of the most beautiful sights I ever saw."
"I remember that Christmas. I asked Monica to get you a gift. It was a dress. I picked it out."
"Oh, Malenfant. It was at least five sizes too big. Monica had to apologize; _she_ knew. No wonder you can't figure out the Fermi Paradox, Malenfant, if you don't know your own wife's dress size... I never liked being alone, you know."
"Nobody does. I guess that's why we're here, why we swung down from the damn trees. Every one of us is looking for somebody..."
"Stop it. Even now, you'd rather talk about issues, about human destiny and the rest of the garbage, anything but us. Anything but _me_. When you're gone I'll be alone here, Malenfant—truly alone, more alone than any person I can think of—to all intents and purposes the only one of my kind, on the whole Moon, in this whole _universe_... It's unimaginable. I'm an accountant, Malenfant. It's not supposed to be like this. Not for me. And it's all your fault. Do you want to know what I'm afraid of—really afraid of?"
"Tell me."
"Chronic reactive depression. You ever heard of that? I looked it up once. You can die of loneliness, Malenfant. Four months, that's all it takes. You don't have to be a failure. Just—outcast."
"I'm sorry."
"Bullshit."
## _S hadow_
There was little food to be had on the plain. The Elf-folk had carried some food from their crater-wall forest, figs and bananas and apples. But now the sun was setting, the footsteps made by the people in the bare patches of dust were little pools of shadow, and most of the food was gone. Plaintively, as they trooped after Shadow across the dusty grass, many of them looked back to the forest they had left.
They came to the site of an old kill. The bones were so scattered and worn by the teeth of successive predators and scavengers that it was impossible to tell what animal it might once have been.
Nevertheless Shadow stopped here. She sat amid the bones and, with a grunt, passed water into the dirt. The fungal growth on her face was a thick mask over her brow and cheeks and nose, making her look alien, ferocious, and some of the more livid scars on her body seemed to glow as bright red as the dust at her feet.
The others followed her lead: first Stripe, the strongest of the men, then Silverneck and the women who followed her. Infants clambered down to the dusty ground and plucked yellow grass blades, stuffing them into their mouths with rust-red fingers.
The adults huddled together uneasily. On this vast tabletop of a landscape the Elf-folk were a dark knot, easily visible, horribly vulnerable. Nevertheless Shadow seemed content to stay here, and so stay they must.
None of the people sat close to Shadow.
Some of them made small offerings to her: a fig, an apple they had carried in their hands. Soon a small pile of food built up. Without acknowledging the people, Shadow reached down and took pieces of the food.
The sun sank further, its edge dipping below rounded hills. A nervy young man, Shiver, emitted a hesitant, hooting roosting call. But there were no trees here to make nests, and the gentle, eerie sound only made the people huddle still closer.
Silverneck sat on the fringe of the group. She picked up a bone from the litter around her. It was a section of a skull. The face was almost intact: she pushed her fingers into eye sockets, nostrils. This might have been a person, an Elf, a Ham, a Nutcracker, a Runner. She ran her finger along it, picking out scrapes and notches, made by teeth or, perhaps, tools. She was almost naked of fur now, so frantically had she been groomed by the other women in these days of turmoil and doubt. Her remaining hairs clung in patches to her blue-black skin and stuck out from her body; the low reddening sunlight made her hair glow, as if she were surrounded by a soft cloud.
Shiver was sitting close to a woman, Palm, barely out of her adolescence. She in turn was resting against her mother's stolid back. Shiver was eating an apple, slowly, his eyes fixed on Palm. His erection was obvious. Shiver started flicking bits of the apple at Palm; the half-chewed fragments landed at her feet, or on her lap.
Without looking at Shiver, Palm picked up the morsels and popped them in her mouth. Gradually, in silence, all but imperceptibly, Shiver moved closer to the girl, his erection dangling before him.
With a sigh, Palm folded back from her mother and lay on the ground, legs separated, her arms stretched above her head. Shiver slid over her and entered her, all in one liquid, silent movement. With a few thrusts he reached orgasm, and withdrew smoothly. Seconds later he and Palm were sitting side by side as if nothing had happened.
Stripe, the boss man, absently grooming Silverneck, had noticed none of this challenge to his status.
Shadow had watched it all. But she cared nothing for such reproductive play. Shadow's dominance had nothing to do with the community's traditional bonds, sex, and children.
After the death of One-eye she had soon become the strongest of the women. And the men—even mighty Stripe—had learned to submit to her power. Though many of them outsized her, her naked, unbridled aggression gave her an edge in most contests. Many of the men and boys cradled hands and feet missing fingers or toes, nipped away by Shadow as an indelible mark of their defeat.
And now she had led them all far from home, far from the trees and shrubs and streams and clearings they knew, across this crimson plain—for a purpose only Shadow, in the deepest recesses of her mind, understood.
A small boy approached Shadow. He had his eyes fixed on the pile of fruit before her. His mother, Hairless, growled warningly, but he feigned not to hear. The boy grabbed his infant sister, and, pulling a twisted, funny face, began to wrestle with her. She joined in, chortling. Soon he was on top of her, making playful pelvic thrusts, and then she rolled on top of him. But every roll took them closer to Shadow's food pile.
As soon as the boy was close enough, his hand whipped out to grab a fig. He tucked it in his mouth, immediately abandoning his play, and walked back toward his mother.
One of the women laughed at his clever deceit.
A sharpened cobble hissed through the air. It caught the boy at the top of his spine, laying open the flesh. He howled and went down. Hairless hurried forward and grabbed him. He curled up in her lap, screaming with pain, as she tended the wound.
Stripe picked up the bloody cobble, wiped it on the grass, and passed it back to Shadow.
The group sat in silence, save for the screams of the boy, which took a long time to subside.
The sun slid beneath the horizon. Light bled from the sky.
The people huddled in a close circle. The adults had their backs to the dark, with the children and infants at the center of the circle. Without fire, without weapons that could strike at a distance save a handful of stones, these hominids were defenseless against the creatures that prowled the savannah night.
Nobody but the infants would sleep tonight. But they feared Shadow more than they feared the dark.
When the dawn came, they found that the boy who had stolen Shadow's fig had gone. As the group moved on, Hairless, his mother, was inconsolable. She had to be half-carried by her sisters and mother, until the memory had started to fade.
At last they reached the cover of trees. This was a forest that lapped at the foot of a tall mountain range; bare rock shone high above. With relief, they slipped into the trees' shadows. Some submitted to ancient green impulses and clambered high into the trees to make nests, even though the day was not yet half over.
But Shiver, clambering high, found a nest already made. He broke it apart, hooting loudly, his fur standing on end.
Then others joined in the noise, for they began to find discarded fruit peel, and even an abandoned termite-fishing stick. They sniffed and licked these remnants; they were fresh. Others had been here, and recently.
And then, as they spread deeper through the new forest, seeking shoots and fruit-bearing shrubs and trees, a child yelled. The adults came crashing through the undergrowth to see, their hair bristling.
A small girl was standing at the edge of a clearing where a great tree had fallen; its carcass lay on the ground, surrounded by crushed bushes. The girl was facing a child a little older than she was. It was another girl, standing unsteadily, gazing back nervously.
It was in fact Tumble, Shadow's small sister. But Shadow did not recognize her. And Tumble, even if she had remembered Shadow, would not have known this scarred creature with her grotesque fungal mask.
Shadow had come home: transformed, unrecognizable, infused with a new and deadly purpose.
It was no coincidence that the encounter had taken place so quickly. As the forest remnants had continued to shrink back, the Nutcracker-men, living in the green heart of the forest, had managed to hold their territory against the incursions of hungry Elf-folk. So the Elves had been restricted to the shrinking forest fringe, patrolling ever closer to its border with the mountains or the plain.
The little girl stepped forward, and tentatively touched Tumble's face. Tumble nipped her finger playfully. In a moment they were rolling in the dry leaves, wrestling. When the little girl reached for Tumble's genitals, Tumble shrank back, but then she submitted, curiously, to the gentle touch. Then they chased each other over the fallen tree trunk, and started to play together with the fallen leaves. They pushed them into great piles, and rolled in the leaves, throwing handfuls over their heads and rubbing them against their faces.
Now, on the far side of the little clearing, silent shadows flitted through the trees. They were adults, some carrying infants. Led by Stripe and Silverneck, the people stepped forward into the clearing. A loose circle of watchful adults surrounded the playing children.
Only Shadow stayed in the dark green shade.
Silverneck walked forward. She was met by a large, calm woman. She was Termite, Shadow's mother. Cautiously, eyes locked, the women began to groom, plucking at each other's hair. More children joined in the play on the forest floor.
The men were more tentative. They eyed each other warily and made subdued displays, showing bristling hair and waving erections.
Suddenly Shiver ran forward toward the other men. He yelled, stamped and slapped at the ground and drummed with his flat hands on a tree trunk, uttering loud, fierce calls. Then he retreated quickly to the safety of his own group.
He was imitated by a burly man from the other group. This was Little Boss. His display of strength was vivid. He hurled rocks on the ground, making them shatter, and pulled branches this way and that. Never as dominant since the death of his mentor, Big Boss, he was still a massive, powerful presence. The invading men retreated subtly, raising their fists and hooting. But Little Boss, too, drew back to his friends.
So it went on, with the children playing, the women grooming or making tentative sexual contact, and a display of noisy aggression by the men. But not a single punch or kick was landed, or stone thrown in earnest.
Now one small, muscular man broke out of the group and approached the woman Hairless. He was Squat, another of Shadow's original group. He seemed fascinated by Hairless's baldness, and he stroked her bare blue-black skin. She responded, cupping his scrotum in her hand.
Within a few minutes they had coupled, belly to belly.
After that the groups separated, the men issuing a few last threats to each other, the women apologetically abandoning their grooming. Mothers had to pry their children away from their fascinating new playmates.
Shadow watched all this. And when her old family group dispersed into the trees, she followed.
## _M anekatopokanemahedo_
The delegation of angry and fearful citizens was led by a stocky, sullen woman called Hahatomane, of the Nema Lineage.
They met at the center of the platform of Adjusted Space. Manekato waited patiently, resting easily on her knuckles, with Babo and Nemoto to either side of her. Hahatomane stood facing her, with her followers in a rough triangle behind her, and attended by Workers that crawled or hovered.
"What is it you want to talk about, Hahatomane of Nema?"
"That should be obvious," Hahatomane said. She glanced into the sky, where the rising Earth was a fat banded ball, almost full. "Renemenagota of Rano is already dead. Many others of us have suffered unspeakable deprivations. This is a foolish quest, devised by foolish Astrologers, which will not help germinate a single seed. We have done what we can. We should leave Workers here to complete the rest, and return to Earth before more of us lose our lives or our sanity."
Babo stepped forward. Though the medical Workers had striven to heal his injuries, the Zealots' crossbow bolts had been laced with an exotic poison of vegetable oils and fish extracts, and he suffered internal agonies that caused a heavy limp. "But you have no place on Earth, Hahatomane. Your Farm is destroyed by the tides and quakes, and the Nema Lineage is extinguished."
Hahatomane kept her gaze locked on his sister. "You do us a dishonor by keeping a man and your ugly _hominid_ by your side, Manekato of Poka," she said. "I do not hear the words of this one."
"Then you should," Manekato said quietly. "For we are all hominids. We are all people, in fact, of one flavor or another."
Hahatomane bared her teeth, an unconscious but primal gesture. "We do not recognize you as any form of leader, Manekato."
"Fine. If you wish to leave, do so."
"And you—"
"I intend to stay on this Moon until I have unraveled the mystery of its design."
Hahatomane growled. "Then none of us can leave."
Everybody understood that this was true. If this expedition were a success its members would be honored, even allowed to carve out new Farms. But if Hahatomane were to split the group, those who abandoned the project could expect nothing but contempt. This was the true source of Manekato's power, and Hahatomane knew it.
Hahatomane's shoulders hunched, as if she longed to launch herself at Manekato's throat—and perhaps it would be healthier if she did, Mane thought. Hahatomane said, "You drag us all into your folly, Manekato of Poka. I for one will be happy to witness your inevitable disillusion."
"No doubt on that day you will remind me of this conversation," Manekato said.
Hahatomane snorted her frustration and turned away. Her followers scattered, bemused and disappointed, and Workers scuttled after them, bleating plaintively.
Manekato sat on the yellow floor. Now that the confrontation was over she felt the strength drain out of her. Babo absently groomed her, picking nonexistent insects from the heavy fur on her back. Nemoto sat cross-legged. She had a large bunch of young, bright yellow bananas, and she passed the fruit to Manekato and Babo.
"You did well," Babo said; then, glancing at Nemoto, he repeated the remark in her tongue, slowing his speech to suit her sluggish oxygen-starved pace of thinking.
Manekato grunted, and spoke in Nemoto's language. _"But I would rather not endure such encounters. We faced off like two groups of Elf-creatures, in their matches of shouting and wrestling. Hahatomane's group even surrounded themselves with Workers to make themselves look larger and stronger, just as male Elves will make their hair bristle in their aggressive displays."_
Nemoto laughed softly. _"We are all hominids here, all primates."_
Babo said, _"But it is cruel to be reminded of it so bluntly. Perhaps there is something in the bloody air of this place which has infected us."_
_"That is foolish and unscientific,"_ Manekato said. _"Even Earth is no paradise of disembodied intelligence and pure reason."_ She glanced at the banded planet that shone brightly in the sky. _"Think about it. Why have we clung to our scraps of land for so many thousands of generations?"_
Babo looked offended. _"To cultivate every atom, the final goal of Farming, is to pay the deepest homage to the world which bore us—"_
_"That's just rationalization, brother. We cling to our land because it is an imperative that comes to us from the deepest past, from the time before we had minds. We cling to our land for the same reasons that Nutcrackers cling to their tree nests—because that is what we do; it is in our genes, our blood. And what of the exclusion we suffered when we lost our Farms?_ Why _must it be so? What is that but savage cruelty—what is that but sublimated aggression, even murder? No, brother. This Moon has not polluted our souls; we brought the blood and the lust with us."_
_"You should not be so harsh on yourselves,"_ Nemoto said.
Even now Manekato felt a frisson of annoyance that this small-brained hominid was trying to comfort her.
But Babo said, _"She's right. Isn't it possible to celebrate what we have achieved, despite our limitations? Can we not see how we have risen above our biological constraints?"_
Manekato said, _"That is true of your kind, Nemoto. You spoke of the contagions of madness that sweep your people. And yet those grand obsessions havedriven your kind to a certain greatness: a deep scientific description of the universe, an exploration of your world and others, even a type of art... Achievements that press against the boundaries of your capabilities. We, by comparison, have done little to transcend our biology—have done little for the past two million years, in fact, but squat on our Farms. Two million years of complacency."_
_"Again that is harsh,"_ Nemoto said. _"Two million years of peace, given the savagery in your breast, is not a small achievement. We must all strive to embrace the context provided by this place—perhaps that is one of its purposes."_
_"Yes,"_ said Babo. _"There are many ways to be a hominid. The Red Moon is teaching us that."_
_"And,"_ said Nemoto, _"we must anticipate meeting the Old Ones, who may be superior to us all. Then we will see how long a shadow we cast in their mighty light."_
Babo said, _"But are you content with such abstractions, Nemoto? Don't you long for home, too?"_
Nemoto shrugged. _"My home is gone. One day there were eight billion people in the sky; the next they had all vanished. The shock continues to work through my psychology. I don't welcome exploring the scar."_
The three of them sat in their small ring, soberly eating the sweet young bananas, while Workers politely scuttled to and fro, removing the discarded skins.
## _R eid Malenfant_
Much of the time he slept, drifting through uneasy, green-tinged dreams of the kind that had plagued him since the day he had come to this unnatural Moon. And then the dreams would merge into a fragmented wakefulness, fringed by blood and pain, with such soft transitions he couldn't have said where dream finished and reality began.
He was lying on his side—he could tell that much—with his arms and legs splayed out in front of him, like a GI Joe fallen off the shelf. He didn't even know where he was. He was surrounded by wood and earth. Some shelter, he supposed, something constructed by hands and eyes and brains, human or otherwise.
It was all very remote, as if he were looking down a long tunnel lined with brown and green and bloodred.
He supposed he was dying. Well, there wasn't a damn thing he could do about it, and he had no desire to fight it.
But if he could feel little with his busted-up body—taste nothing of the glop that was ladled into his mouth, barely sense the warm palm oil that was rubbed into his limbs—there was one thing he could still feel, one anguished pinpoint that pushed into him whenever he made out Emma's face.
Regret.
"Regret what, Malenfant?"
"Regret I'm going to die not knowing _why_."
"You're dying because some psychopathic religious nut had you beaten to death. That's why."
"But _why the Red Moon_? Why the Fermi Paradox—"
"Malenfant, for Christ's sake, is this the time or the place for—"
'emma, give me a break. This is my deathbed. What other time and place is there? That damn Paradox baffled me my whole life. I thought the showing up of this Red Moon, for sure the strangest event in human history since Joshua made the sun stand still in the sky, had to have something to do with that flaw in the universe. I guess I _hoped_ it did. But..."
"But what?"
"It didn't work out that way. Emma, it just got more mysterious. Nemoto saw that immediately. Not only did we suddenly find that we inhabit just one of a whole bunch of universes, there are no signs of extraterrestrial intelligence in the other universes either. Not a trace. It's Fermi writ large—as if there is something wrong not just with this universe, but all our cosmic neighbors..."
"Malenfant, none of this matters. Not any more."
"But it does. Emma, find the advanced guys. The ones with the light shows in the sky. That's what you've got to do. Ask _them_ what the hell is going on here. Maybe they caused it. All this, the multiple realities, the wandering Moon. Maybe they even _caused_ Fermi, in some way. That's what you must do, after..."
"After you're gone? Poor Malenfant. I know what's really bothering you. It's not that the question is unanswered. It's the idea that you won't be around when the answer comes. You always did think you were the center of everything, Malenfant. You can't stand to think that the universe will go on without you."
"Doesn't everybody feel that way?"
"Actually, no, not everybody, Malenfant. And you know what? The universe _will_ go on. You don't have to save it. It doesn't need you to keep space expanding or the stars shining. We'll keep on finding out new stuff, visiting new places, finding new answers, even when you aren't around to make it happen."
"Some bedside manner, babe."
"Come on, Malenfant. We are what we are, you and I. I can't imagine us changing now."
"I guess."
## _S hadow_
She slid through the forest, stepping on roots and rocks to avoid dead leaves and undergrowth, silent save for the brush of her fur on the leaves. Her hair was fully erect, and her fungal mask seemed to glow with purpose and power.
There were three men with her. They were tense, fearful.
Shadow turned back to the men and grinned fiercely, knowing how her teeth shone white under the hairless protuberance over her brow and cheeks. They grinned back, and they punched and slapped each other, seeking courage. The smallest and youngest, Shiver, absently sucked the forefinger of his right hand; it was a stump, the first two joints nipped off by Shadow.
Shadow moved forward once more, and the men followed.
She froze. She had heard the soft whimper of an infant—and there, again.
She roared and charged forward, crushing through low shrubbery.
A woman and child were in the low branches of a tree. They had been eating fruit; the forest floor beneath the tree was littered with bits of yellow skin. The woman was called Smile. She was in fact a sister of Termite's, an aunt of Shadow. Shadow did not know this—nor would it have made any difference if she had known.
Smile tumbled out of her tree. She landed with a roll on the forest floor, got to her feet and turned to flee. But her child, less than three years old, was still in the tree. He clung to a branch, screaming. So Smile ran back, scrambled up the tree, collected the child, and dropped back to the ground. But she had lost her advantage; now the attackers were on her.
Shadow grabbed her by the shoulders and pulled her to the ground. Shiver joined in, kicking and stamping. Stripe grabbed the infant from his mother's arms. He held the child by his feet and flailed him this way and that, slamming him against a tree trunk. The child was soon limp, and Stripe hurled him away, sending the little body spinning into a clump of undergrowth.
With grim determination, Smile fought against the odds. She twisted and bit Shiver hard on the shoulder. He howled. She managed to ram his body into Shadow and the others, momentarily reducing them to a tangle of flailing limbs.
That was enough of a break for Smile to get away. She scrambled into a fig tree. Stripe followed her. But Smile clambered around the branches, evading him, screaming. Now Shadow clambered up the tree, more stiffly than Stripe, for her lifetime of injuries and beatings had left their mark.
But as she approached, Smile made an almighty leap. She crashed into the branches of another tree, and tumbled to the ground. In an instant she was on her feet. She ran to the foliage where her child had fallen, picked up the limp body, and ran into the deeper woods.
Shiver pursued, but she was soon out of his reach. He ran back and forth across the bloodied forest floor, howling and throwing rocks and kicking at the trees, ridding himself of his desperate aggression.
Shadow fell on Stripe. She jabbered at him, and hailed blows on his head and shoulders. He huddled over, long arms protecting his head and chest.
For now Smile had been spared. But it was only the beginning.
Shadow's next target was Little Boss. She took six men with her, armed with sticks and rocks, and patrolled the forest until she found him.
Little Boss was alone, drinking from a small stream. Beside him was a pile of cobbles, suitable for making sharp new tools. When he heard Shadow's party approach, he stood straight, hair immediately erect, and snarled defiance. By this time, the newcomers' murderous aggression was well known among Little Boss's group. But when he saw how many men had come with Shadow, Little Boss turned to run.
He was built for power, not speed.
Shiver was the first to catch him, seizing his legs and throwing him to the ground. Shadow pinned him down, sitting on his head and holding his shoulders. The other men fell on Little Boss, attacking with a savagery only impeded by the fact that they got in each other's way.
At last Shadow and the men backed off. Charged with energy, fists clenched, mouths and stone tools stained by blood, the men ran to and fro, howling and pounding their weapons against tree trunks and rocks.
Little Boss remained motionless for a time. Then, uttering faint screams, he sat up. He had great gashes on his face, legs, and back. He could not move one leg. The ground where he had lain was stained by blood and panic shit. He looked back at his assailants, who were capering and howling their rage. He opened his mouth, as if to cry defiance. But a great bubble of bloody mucus formed there, and his voice was a strangle. When the bubble broke, Little Boss fell back, rigid as a falling tree.
Shadow fell on the body immediately. She pulled it by its ankles out into the clearing, sat on its chest, and immediately began to slice away its flesh with a new stone cobble.
With degrees of reluctance or enthusiasm, the others joined her. Soon they were all feeding.
The miniature war was brief but savage.
Shadow's only tactic was to isolate her targets and destroy them. But it was a tactic beyond the grasp of her opponents, and it worked over and over. The women, especially if burdened by infants, were easy prey. The men were picked off one by one, always by overwhelming force.
And as Shadow's group fed day after day on fresh meat, they grew stronger, and hungrier.
It finished as Shadow watched her acolytes fall on the body of her mother. In her last moments, before they opened her chest, Termite reached out a bloody hand to Shadow, who stayed unmoved.
And then Shadow went alone into the forest to hunt down the last free man, her brother, Claw. When Shadow returned to her warmongering group, the object she clutched in her hand was his heart.
But when the opponents were annihilated, the group, filled with a rage for blood and murder, anxious for more meat, began to fall on each other.
## _R eid Malenfant_
He remembered how his father, on learning of his inoperable tumor, had suddenly rediscovered the Episcopalian faith of his youth. Somehow that had hurt Malenfant—as if his father, in those last months, had chosen to draw away from him. But he hadn't been about to deny his dad the comfort he sought.
It had always seemed to him that religion was a kind of bargain. You gave over your whole life, a portion of your income and half your intellect, in return for a freedom from the fear of death. Maybe it wasn't such a bad bargain at that.
But look at the Hams: Julia and the rest, these Moon-bound Neandertals, as rational and smart as any human being, just as aware of the human tragedy of death and pain and loss—and yet, it seemed, quite without the consolation of religion. But they seemed able to cope with the dreadful truth of life without hiding from it.
Well, maybe they were tougher than humans.
And what about you, Malenfant, now the black meteor is approaching at last? Don't you need comfort—forgiveness—the prospect of continued existence beyond the grave of crimson dust that will soon welcome your bones?
Too late for me now, he thought. But it doesn't seem to trouble me. Maybe I'm more like a damn Neandertal than a human.
Or maybe Emma was right: that nothing mattered so much to him about where he was going, compared to what he was escaping from.
Julia was here, her concerned, Moonlike face swimming in the gloom before his eyes. He wondered absently if it was night or day.
After a time, Emma was here. She frowned, wiped at his mouth with a scrap of leaf, and tried to give him water.
"Things to tell you."
"You need to save your strength for drinking. Eating. All that good stuff."
"No time."
"If you're going to start lecturing me about Fermi again—"
"I did my best, Emma."
"I know you did."
"I came all the way to this damn Moon to find you. I went to the White House. I built a rocket ship."
"That always was the kind of stuff you were good at, Malenfant."
"Looking out for you?"
"No," she said sadly. "The grand gesture."
"I found you. But I can't do anything for you."
She looked at him, her eyes blank, oddly narrowed. "But was that ever the idea?"
"What else?"
"You're a complicated man, Reid Malenfant. Your motives aren't simple."
"Your mother thinks I've been trying to kill you for years."
"Oh, it's not that, Malenfant. It's not me you're trying to destroy. _It's you_. It's just that I'm sometimes in the way..."
He frowned, deeply disturbed, remembering fragments of conversations with McCann, Nemoto. "What are you talking about?"
"What about Praisegod Michael?"
"He was a psychopath. I had to—"
"You had to _what_? Malenfant, it wasn't your fight. What does Praisegod Michael matter to you, or me? If you really had been devoted to the cause of getting to me, you'd have said anything he wanted to hear, to keep your skin intact. But not you. You walked into his guns, Malenfant. Deliberately. And you must have known you couldn't win. On some level you _wanted_ him to do this to you."
"I was looking for you," he said stubbornly. "That's why I came to the Moon."
"I'm sorry, Malenfant. I see what I see."
He licked his lips with a tongue that felt like a piece of wood.
"Tell me this," she said now. "When we were in that damn T-38 over Africa, when the Wheel appeared in the sky—"
"Yeah."
_"You could have turned away."_
He closed his eyes. He thought back to those moments, the glittering sky-bright seconds of the crash, when he and Emma had been suspended in the deep African light, before the enigmatic alien artifact.
... Yes. He remembered how the aerosurfaces had bit, just for a second. He had felt the stick respond. He knew he could turn the nose of the plane away from the Wheel. It was a chance. He didn't take it.
"Yes," he rasped. "And then—"
And then there had been that instant of _exuberance_ —the sense of relief, of freedom, as the T-38 hurtled at the Wheel, as he felt the little jet slide out of his control, as the great blue circle had rushed toward him, and he had reached the point where he could do no more.
"How did you know? The slaved instruments—"
"I didn't need to watch instruments, Malenfant. I know you. It's just—the way you are, the kind of person you are. You could no more help it than you could stop breathing, or keep from farting in your sleep."
"I do that?"
"I never knew when would be a good time to tell you."
"You picked a doozy."
"Poor Malenfant. The universe never has made much sense to you, has it?—not from the grandness of the Fermi Paradox, not yourself, on down to your relationship with your first grade teacher."
"She really was an asshole."
"I've always known all about you, what you are, what you could not help but become. Right from the beginning, I've known. And I went along with you anyway. What does that say about me?... Maybe we're alike, you and I." She reached up and passed her hands over his eyes. "Sleep now."
But sleep eluded him, though regret lingered.
"Listen, Malenfant. I've decided. You're right. I'm going to go on, to track down the Daemons— _Homo superior_ , whatever they are. Every time this damn Moon shifts, people suffer and die, right here on the Moon, and on all the Earths. What gives those guys the right to screw up so many lives—so many billions of lives?"
"And you intend to stop them."
"Malenfant, I don't know what I intend. I haven't had a plan since the day I fell through that blue Wheel and found myself here, covered in shit. I'll do what you always did. I'll improvise."
"Take care."
"Because you won't be around to look out for me? Malenfant, if it escaped your notice, _I_ rescued _you_. All _you_ did was lose your spacecraft, your sole companion, and all your gear, and get yourself thrown in jail. Twice."
"Anger can make you feel good."
"... Yes. Maybe that's what I need. An enemy. Somebody to be mad at. Other than you, that is."
"Why here?"
"What?"
"Why is it finishing like this, here, now, so far from home?"
"You always did ask big questions, Malenfant. Big, unanswerable questions. Why are there no aliens? Why is there something, rather than nothing?"
"I mean it. Why did I have to run into a petty thug like Praisegod? Why couldn't it have been more—"
"More meaningful? But it is meaningful, Malenfant. There's a logic. And it has nothing to do with the Red Moon or the Fermi Paradox, or any of that. It's _you_ , Malenfant. It's _us_. Your whole life has a logic leading up to this place and time. It just had to be this way."
"The universe is irrelevant. That's what you're saying."
"I guess so... But there are other universes. We know that now. We've seen them. Are there other destinies for us, Malenfant?... _Malenfant!_ "
The tunnel was long now, and filling with an oily darkness. Her face was like a distant beacon, a point of light like a star in a telescope, and he struggled to see her. There was a dim awareness of hands working his body, hands pounding at his chest, heavy hands, not human.
The light went out, the last light.
Soft lips brushed his brow, gentle as a butterfly's wings, yet the most vivid event in all the collapsing universe.
Enough, he thought, gratefully, fearfully.
## _M anekatopokanemahedo_
It was time for the Mapping to the crater that promised to reveal the secrets of the world engine.
The people stood in a rough circle at the center of the platform. The yellow floor was bare again, the temporary structures it had borne unraveled, space-time allowed to heal. The great turning Map of the Red Moon had been folded away also, having served its purpose. There was nothing left but the platform, and its cargo of people.
Beyond there was only the unmanaged forest, where, perhaps, curious eyes gazed out at the creatures they had learned to call Daemons.
Manekato sought out Nemoto. The little hominid stood alone, ignored by the rest. She wore her much-repaired blue coverall, and over her shoulder she bore the bag of parachute fabric that contained her few artifacts.
Manekato knew that it would serve no purpose to tell Nemoto that possessions were meaningless, for anything desired could be reproduced at will, over and over, Mapped out of the raw stuff of the universe itself. In this, oddly, Manekato's kind had much in common with the more primitive hominids here. The Hams and Runners would manufacture tools for a single use and then discard them, without sentiment or longing. Perhaps Manekato shared with them some deep sense of the unstinting bounty of the universe—there would always be another rock to make a hand axe—an intuition which Nemoto, caught between the two, coming from a culture of acquisition and limits, could never share.
Manekato sighed, aware of the drift of her thinking. As always, just as Without-Name had complained, too many philosophical ruminations!—Enough, Mane. It is time to act.
She took Nemoto's hand; it lay against her own, tiny and white and fragile. _"Are you ready?"_
Nemoto forced a smile. _"I have been fired across space by a barely-controlled explosion devised by primitives. By comparison you are masters of space and time. I should feel confident in your hands."_
_"But you don't."_
_"But I don't."_
Manekato said gently, _"A Mapping is only a matter of logic. You are a creature of logic, Nemoto; I admire that in you. And in the working out of logic, there is nothing to fear."_
_"Yes,"_ Nemoto said softly. But her hand tightened in Manekato's.
In due course, the Mapping was expressed.
Hand in hand, the people and their Workers—and one frightened _Homo sapiens_ —drifted upward from the platform. The great shield of Adjusted Space folded away beneath them, leaving a disk of light-starved, barren, crushed land. But Manekato knew that the denuded patch would soon be colonized by the vigorous life forms here, and she felt no guilt.
Then the Mapping's deep logic worked into her bones, and she was smeared over the sky.
She hung among the stars, suspended in a primal triumvirate of bodies: Earth, sun and Moon, the only bodies in all the universe that showed as more than a point of light to a naked human eye. But this was not Nemoto's Earth, or her sun; and it was nobody's Moon. How strange, she thought.
She had no body, and yet she was aware of Nemoto's hand in her own.
_"Nemoto?"_
_"... How can I hear you?"_
_"It doesn't matter. Can you see the Red Moon?"_
_"I see it all at once!—but that is impossible. Oh, Mane..."_
_"Try not to understand. Let the logic guide you."_
_"But it is a world. It is magnificent,"_ Nemoto said. _"It seems absurd, grandiose, to suppose that this is a mere cog in some vast machine."_
It took Manekato a moment to secure the translation of "cog." _"Look at the stars, Nemoto."_
_"I can't see them. The sun dazzles me."_
_"You can see them if you choose,"_ Manekato said gently.
_"... Yes,"_ Nemoto said at length. _"Yes, I see them. How wonderful."_
_"Are they the same stars as shine on your Earth?"_
_"I think so. And they are just as silent. Are we alone in all the human universes, Manekato?"_
_"Perhaps."_ She glared at the unchanging stars. _"But if we are alone, the stars have no purpose save what they can offer humanity. My people have sat on their Farms for two million years,"_ Manekato said, _"a vast desert of time we could have spent cultivating the sky. Long enough, Nemoto. When this is over—Ah. I think—"_
And then the Mapping was done.
The platform coalesced, as space-time adjusted itself for the convenience of the expedition. People moved here and there, speaking softly, trailed by Workers. Few of them showed much interest in their new environs; already the first shelters were coalescing, sprouting from the platform like great flat fungi.
Once again Manekato found herself injected into a new part of the Red Moon.
This place was bright, more open than the forest location. And she could smell ocean salt in the air. To the east, the way the gentle, salt-laden breeze came, the land rose, becoming greener, until it reached a crest that was crowned by a line of trees. As she studied the ridge of rock, she saw how it curved away from her. It was the rim of a crater. To the west was a broad plain of rock and crimson dust, all but barren. In the far distance, beyond a rippling curtain of heat haze, hominids ran across the plain. They moved silently and without scent, like ghosts.
Nemoto had slumped to the ground. She peered into her bag, rummaging through its contents, as if unable to believe that a Mapping could be completed without losing some key piece of her battered and improvised equipment.
Babo came to Manekato. "Interesting. She behaves like an infant after her first Mapping. But then we arrive in the world _knowing_ that reality has certain properties. Deep in our hind brains, the parts we share with these sub-human hominids and even more ancient lines, we store the deep intuition that a thing is either _here_ or _there_ , that it either exists or it does not—it cannot spontaneously leap between the two states. And Mapping violates all that. Perhaps we should admire Nemoto for keeping her sanity."
"Yes." Manekato rubbed his head fondly. "For now our companions are all too busy rebuilding their houses to have much to complain about. Shall we investigate what we have come so far to see?"
He raised his hand, preparing to execute another short-range Mapping.
She grabbed his arm. "No. Renemenagota was a monster. But I have come to believe that some of her intuition was sound." Deliberately she walked forward, knuckles and feet working confidently, until she had stepped off the platform and onto the raw native ground. She scraped at the dirt, and clouds of crimson dust drifted into the air. Soon her feet and lower legs were stained a pale pink.
Babo grinned, showing white teeth. "You're right, Mane. We are creatures designed for walking. Let us walk." He jumped off the platform, landing with hands and feet flat, evoking more billows of dust.
Side by side they loped away from the compound and began to scale the wall of the crater.
## _S hadow_
The Nutcracker-woman was eating her way through a pile of figs. A child played at her feet, rolling and scrabbling in dead leaves. The woman was about the same height as one of the Elf-folk, and she was covered in similar black-brown hair. But her belly seemed swollen compared to an Elf's—it housed a large stomach capable of fermenting her low-quality feed—and her head was a sculpture of bone, with a great crested ridge over the top of her skull, and immense cheekbones to which powerful muscles were anchored.
A rock hurtled out of the surrounding foliage. It slammed into the trunk of the fig with a rich hollow noise, then fell to the earth.
The Nutcracker-woman screeched and scrambled back. She stared at the fallen stone. At last, cautiously, she poked it with one finger, as if it were a living thing, a bat that had stunned itself on the tree. But the stone lay still, unresponsive.
And now a stick came spinning from another part of the foliage.
The Nutcracker-woman got to her feet, gathered up her infant, and looked about suspiciously, sniffing the air with her broad, dirty nostrils. She took a step away from the fig tree.
Shadow struck.
## _M anekatopokanemahedo_
The ground rose steadily.
Manekato could feel a layer of hard, compact rock beneath a thin skim of dust. Green things grew here, grass and shrubs and even a few low trees, but they struggled to find purchase. It was dry; there was no sign of the springs that sometimes could be observed bubbling from the shattered walls of craters. And, though the rise of the slope was steady, it was not becoming noticeably steeper.
The morphology of this formation was like no other impact crater or volcanic caldera she had encountered. The rim of a crater this size should be more sharply defined: a circular ridge, perhaps eroded into hillocks, with a splash plain of rubble and ejecta beyond. There was none of that here; the "crater" was just an upraised blister erupting from an empty plain.
She glanced at Babo. She saw his mouth was working as he studied the rock, the vegetation, the dust, thinking, analyzing.
Babo saw her looking, and grinned. "I know what you're thinking," he said. " _Artificial_. But then, we know this Red Moon is a thing of artifice, and we suspect this crater may be the key to its secrets. Why should we expect anything but artifice here, of all places?"
The climb had already been long, and Manekato halted and rested her weight on her clenched knuckles. Babo raised a handful of crimson dust and let it drift off in the air; she could smell its rich iron tang, and some of it stuck to the sweat-soaked palm of his hand.
She glanced to the west, over the landscape from which they had climbed. The Adjusted-Space platform nestled at the foot of this slope, a bright splash, oddly ugly. Beyond it a plain of crimson dust stretched away, its color remarkably bright, marked by the pale green of vegetation clumps. The horizon of this small world curved noticeably, a smeared band of muddy gray. The sky was a dome littered by high clouds, and to the west she saw the dingy stain of volcanic dust streaking the air.
It was not a spectacular view, but something in its sweep tugged at her imagination. If she were anywhere on her Earth she would see the work of people, and it had never before struck her quite how claustrophobic that could be. _This_ was an empty, unmade land.
Babo pointed. "Look. Down there."
She saw that near the foot of the crater wall a group of hominids were working their way through the sparse coating of vegetation towards a fig tree. She thought they were Elves, the small, gracile creatures Nemoto called _Australopithecines_. They moved with stealth, and they approached the tree from several directions, surrounding it.
"I think they are hunting something," Babo said. "... Ah. Look, there. Under the tree. It is another hominid."
Manekato saw it now: a burly black-furred form, with a bony, crested skull and distended belly, this was the alternate variant of Australopithecines called a Nutcracker. This hominid had swollen, milk-laden breasts: a female. An infant huddled close to this mother.
The Elves crept closer.
Manekato murmured, "Must this world see more sentience dissipated needlessly?"
"It is not our affair, Mane," Babo said gently. "They are only animals."
"No," she said softly.
## _S hadow_
The Elf-folk charged into the clearing.
The Nutcracker-woman squealed, dropped her child, and scrambled up the fig tree for safety. The child tried to climb after her, but her hands and feet were small and poor at grasping, and she fell back again.
Shadow was the first to grab the infant.
Shiver had the temerity to attempt to snatch a limb of the infant for himself; they might have torn it apart between them. But Shadow pulled the infant to her chest, in a parody of parental protectiveness, and bared her teeth at Shiver.
The Nutcracker-folk mother dropped out of her tree, screaming her rage, mouth open to show rows of flat teeth. Nutcracker-folk were powerfully built, and were formidable opponents at close quarters. She charged at Shadow.
But Stripe lunged forward. His big bulk, flying through the air, knocked her flat. But the Nutcracker-woman wrapped her big arms around Stripe's torso and began to squeeze. Bones cracked, and he howled.
Now more of the men threw themselves at the Nutcracker-woman. Shadow saw that some of them had erections. This was the first time they had hunted one of the Nutcracker-folk. The men had grown accustomed to using the Elf-women of the forest before killing them. Perhaps this Nutcracker-woman, when subdued, would provide similar pleasure.
Shadow took the Nutcracker infant by her scrawny neck and held her up. Her short legs dangled, and huge eyes in a small pink face gazed at Shadow. But she could never be mistaken for the child of an Elf; the exotic bony ridges of her skull saw to that.
Shadow opened her mouth, and placed the child's forehead between her lips.
## _M anekatopokanemahedo_
As the Nutcracker mother fought for her life, as the wild-looking Elf woman, battered and scarred, lifted the helpless infant by its neck, Manekato raised her head and roared in anguish.
## _S hadow_
... And there was a flash of bright white light, and searing pain filled her head.
When Shadow could see again, the men were lying on the ground, some clutching their eyes, as dazzled and shocked as she was.
Of the Nutcracker mother and child there was no sign.
The men sat up. Stripe looked at Shadow. There was no prey, no meat. Stripe bared his teeth and growled at her.
## _M anekatopokanemahedo_
Babo touched Manekato's shoulder. "You should not have done that," he said regretfully.
"The Nutcracker woman _knew_ , Babo. She knew the pain she would endure if she lost her infant. Perhaps the child itself knew."
"Mane—"
"No more," she said. "No more suffering of creatures who understand that they suffer. Let that be the future of this place."
One by one the scattered Elves were clambering to their feet. Still rubbing their eyes, they stumbled back toward the plain—all but one, the woman who had captured the infant. She stood as tall as she could on the rocky slope, gazing up in suspicion. Manekato and Babo were well sheltered by the trees here, and the creature could surely suspect no causal connection between Manekato and her own defeat anyhow. But nevertheless the Elf howled, baring broken teeth to show pink gums, and she hurled a rock as far as she could up the slope.
Then she turned and loped away, limping, her muscles working savagely even as she walked.
Manekato shuddered, wondering what, in this creature's short and broken life, could have caused such anguish and anger.
Babo sat on his haunches. "An Air Wall," Babo said. "We will erect an Air Wall to exclude unwelcome hominids, and other intruders. We will move the platform inside the cordon."
"Yes..."
"No more blood and pain, Mane."
They turned, and began to clamber farther up the crater wall.
It was not long before they had reached the summit of the crater rim wall—and found themselves facing a broad plateau. A thin breeze blew, enough to cool Manekato's face, and to ruffle her fur. The rock here was crimson red, like a basalt or perhaps a very compact and ancient sandstone. It was bare of vegetation and very smooth, as if machined, and covered by a hard glaze that glistened in the sun's weak light. There was little dust here, only a few pieces of scattered rock debris.
It was as if the crater had been filled in. "I don't remember this from the Mapped image," Babo said, disturbed.
Manekato dug her fingers into the fur on his neck. "Evidently we have limits."
"But it means we don't know what we will find, from now on."
"Isn't that a good thing? Isn't that why we came? Come, brother, let us walk, and let us remember our humility."
They walked forward, for perhaps a mile. And then they came to a circular pit, geometrically perfect. It was only yards across. Light leaked out of it, trapped by dust motes, a shaft that reached dimly to the sky.
Manekato's imagination quailed. She reached for Babo's hand, reluctantly reminded of how she had guided Nemoto through the strangeness of the Mapping.
Babo grinned at his sister. "This is strange and frightening—perhaps it is our turn to be humbled now—but I am sure we will find nothing that will not yield to the orderly application of science."
"Your faith is touching," she said dryly.
He laughed.
"But it is not time to approach it yet," she said.
"No. We must study it."
"Not just that." They regarded each other, sharing a deep instinctive wisdom. "This is not for us alone, but for all hominids."
"Yes," he said. "But how long must we wait?"
"I think we will know..."
There was a blue flash, painfully bright, that seemed to fill Mane's head; it reminded her uncomfortably of the punishment she had imposed on the Elf-folk.
She raised her head. "... Ah. Look, Babo."
In the sky swam a new world. It looked like a vast ball of steel. Its atmosphere seemed clear, save for streaks and whorls of cloud. But beneath the cloud there was no land: not a scrap of it, no continents or islands, nothing but an ocean that gleamed gray, stretching unbroken from pole to pole. There weren't even any polar caps to speak of: just crude, broken scatterings of pack ice, clinging to this big world's axes. The only feature away from the poles was a glowing ring of bloodred, a vast undersea volcano, perhaps. And here and there she saw more soot-black streaks of dust or smoke, disfiguring the world ocean; drowned or not, this was a geologically active world.
It was a startling, terrifying sight—Manekato's hind brain knew from five million years of observation that things in the sky weren't supposed to change suddenly, arbitrarily—and she tried not to cower.
"It is a new Earth," Babo said thinly. "So we have completed a transition, riding this rogue Red Moon. How interesting."
"Yes." She clutched her brother's hands. Despite his cool words, he was trembling. "And now we are truly of this world, Babo."
It was true. For Banded Earth, Manekato's Earth, had gone.
## _E mma Stoney_
With Joshua, Mary, and Julia, Emma walked south, towards the place where—as the Hams put it—the wind touched the ground.
Emma was pretty much toughened up by now. So long as she avoided leg ulcers, getting tangled up in lianas or bramble, and the snakes and the multitude of insects that seemed to target any bare flesh like heat-seeking missiles, she was able to maintain a steady plod, covering miles and miles each day, across desert or semi-scrub or savannah or even through denser forest.
The Hams had more trouble. Their sheer strength vastly exceeded her own, but long-distance walking was alien to their physiques. They looked awkward as they barrelled along, and after a couple of days she could see how they suffered aches in the hips and knees of their bow legs, and the low arches of their great flat feet. Also, she suspected, such sedentary creatures as these must suffer a deeper disturbance as they dragged themselves across the landscape, far from any settled community. But, though they moaned wordlessly and rubbed at the offending parts of their anatomies, they never complained, not to her or each other.
The days were long and hot, and the nights, spent under the crudest of lean-tos, cold and cruelly uncomfortable. The Hams seemed capable of sleeping wherever they lay down, their great muscled bodies tensed and hard even in their sleep, like marble sculptures. But Emma had to work hard to get settled, with bits of parachute silk wrapped around her, and socks and vests bundled into a ball under her head.
Much of this stuff was Malenfant's.
She had forced herself to take everything from him that might prove useful, even the little lens that had found its way from her hands to his. It wasn't sentiment—sentiment would have driven her to bury the stuff with him—but a question of seeking advantages that might prolong her own survival. Not that there was much left, even though Malenfant had come to this Red Moon as part of a purposeful expedition, unlike her own helpless tumble through the Wheel. Idiot, Malenfant.
Anyhow, each night she immersed her face in the ragged bits of Malenfant's clothing, seeking the last traces of his scent.
Day after day, they walked. The Hams never wavered in their course, each clumsy step directed by a wordless navigation.
It occurred to Emma to wonder how people who moved house less often than empires rose and fell on Earth were able to find their way across such challenging distances. She tried to discuss this with Julia. But Julia was unforthcoming. She shrugged her mighty shoulders. "Lon' time. People come, people go. This way, tha'. See?"
No, Emma didn't see. But maybe it was something to do with their long Neandertal timescales—far longer than any human.
The Hams, squatting in their caves and huts, made nothing like the seasonal or annual congregations associated with human communities. But there had to be occasional contacts even so, for example when outlying hunting parties crossed each other's paths, or maybe when a group was forced to move by some natural disaster, a cave flood or a land slip.
And such was the static nature of the Ham world that even very occasional contacts—not even once a generation—would suffice to keep you up to date. Once you knew that Uncle Fred and Aunt Wilma lived in those limestone caves two days' hike west of here, you could be absolutely sure that they would always be there. And so, over generations, bit by bit, from one small clue after another, the Hams and their forefathers built up a kind of map of the world around them. The Ham world was a place of geological solidity, the locations of their communities as anchored as the positions of mountains and rocks and streams, shifting only with the slow adjustments of climate.
It was an oddly comforting worldview, filled with a certain calm and order: where nothing ever changed much, but where each person had her own place in the sun, along with every rock and stream. But it wasn't a human worldview. _People rooted like trees_... Though she struggled to understand, it was beyond her imagination.
And of course she might be quite wrong. Maybe the Hams worked on infra-sound like the elephants, or on telepathy, or astral projection. She didn't know, and as Julia was unable to answer questions Emma was barely able to frame, she guessed she was never going to know.
And anyhow, after the first few days' walk, the direction they were all travelling became obvious even to her. Far to the south a column of darkness reached up to the sky: not quite straight, with a sinuous, almost graceful curve. It was a permanent storm, tamed, presumably, by some advanced technology she couldn't even guess at.
It was, of course, the fortress of _Homo superior_ , whoever and whatever they were.
The Hams plodded on, apparently unaffected by this vision. But when the twister's howling began to be audible, banishing the deep silences of the night, Emma found it hard to keep up her courage.
The weeping came to her in the night.
Or in the morning when she woke, sometimes from dreams in which she fled to an alternate universe where she still had him with her.
Or, unexpectedly, during the day as they walked or rested, as something—the slither of a reptile, the chirp of an insect, the way the sunlight fell on a leaf—reminded her unaccountably of him.
She knew she was grieving. She had seen it in others; she knew the symptoms. It wasn't so much that she was managing to function despite her grief; rather, she thought, this unlikely project to go challenge _Homo superior_ was something to occupy the surface of her mind, while the darker currents mixed and merged beneath. Therapy, self-prescribed.
The Hams seemed to understand grief. So they should, she thought bleakly; their lives were harder than any human's she had known, brief lives immersed in loss and pain. But they did not try to soothe her or, God forbid, cheer her up.
_There is no consolation_ , they seemed to be telling her. The Hams had no illusion of afterlife or redemption or hope. It was as if they were vastly mature, ancient, calm, compared to self-deluding mayfly humans, and they seemed to give her something of their great stolid strength.
And so she endured, day by day, step by step, approaching the base of that snake of twisting air.
It didn't surprise Emma at all when the Hams, with the accuracy of expert map readers, walked out of the desert and straight into an inhabited community.
It was a system of caves, carved in what looked like limestone, in the eroded rim wall of what appeared to be a broad crater. The upper slopes were coated thinly by tough grass or heather, but the sheltered lower valleys were wooded. And the crater was at the very bottom of that huge captive twister, which howled continually, as if seeking to be free.
As she approached she made out the bulky forms of Hams, wrapped in their typical skin sheets, coming and going from scattered cave mouths that spread high up the hillsides.
Emma could see the advantages of the site. The cave mouths were mostly north-facing, which would maximize the sunlight they captured and shelter them from the prevailing winds. She suspected the elevated position of the caves was a plus, too. Maybe the migration paths of herd animals came this way. Hams preferred not to have to go too far to find their food; sitting in their caves, gazing out over the broken landscape around the crater, all they would have to do was wait for their food supply to come their way.
... But that wind snake curled into the air above their heads, strange, inexplicable, filling the air with its noise—even if it didn't disturb so much as a dust grain. You'd think it would bother the Hams. She saw no sign that it did.
Emma and her companions walked to the foot of the crater wall, and began to clamber up. The adults glanced down at their approach, but turned away, incurious.
The first person who showed any interest in them was a child: stark naked, a greasy bundle of muscle and fat no more than three years old, with one finger lodged in his cavernous nostril. This little boy stared relentlessly at Emma and followed her, but at a safe distance of a yard or so; if she tried to get closer he backed away rapidly until his buffer of safety was restored. Ham children were much more like human children than their adult counterparts. But Ham kids grew fast; soon they lost the open wonder of youth, and settled into the comfortable, stultifying conservatism of adulthood.
She stepped into the mouth of the largest cave. The noise of the whirlwind was diminished. The sun was bright behind them, and Emma, dazzled, peered into the gloom.
The walls were softened and eroded, as if streaked with butter. There was a powerful stink of meat, coming from haunches and skins stacked at the back of the cave. The place was not designed for the convenience of people, she saw; the roof was so low in places that the Hams had to duck to pass, and crude lumps of rock stuck out awkwardly from the walls and floor. She recognized the usual pattern of Ham occupation: a floor strewn thick with trampled-down debris, an irregular patchwork of hearths. The roof was coated with soot from innumerable fires, and the walls at head height and below were worn away and blackened by the touch of bodies, generation after generation of them. This place had been lived in a _long_ time.
Emma found a piece of wall that seemed unoccupied. She dumped her pack and sat down in the dirt.
A woman approached the travellers. Bent, her hair streaked with white, a tracery of scars covering her bare arms, she looked around eighty, but was probably no older than thirty-five or forty. She began to jabber in a guttural language Emma did not understand, with no discernible traces of English or any other human language. Julia seemed uncertain how to reply, but Mary and Joshua answered confidently. Neither party seemed ill at ease or even surprised to see the other.
Julia came to Emma.
Emma said, "So can we stay?"
Julia nodded, a _Homo sap_ gesture Emma knew she affected for her benefit. "Stay."
With relief Emma leaned back against the creamy, cool wall of the cave. She opened her pack and dug out her parachute silk blanket and a bundle of underwear to use as a pillow. The ground here, just crimson dust, much trodden and no doubt stuffed with the bones of Ham grandmothers, was soft by comparison with what she had become used to; soon she felt herself sliding toward sleep.
But she could hear the howl of that tame whirlwind, relentless, unnatural, profoundly disturbing.
She spent a full day doing nothing but letting her body recover, letting her head become used to the sights and sounds and smells of this new place.
Right outside the cave entrance, a stream of clear water worked its way through rocky crevices towards the impact-broken plain below. Its course was heavily eroded, so that it cascaded between lichen-crusted, round-bottomed pools. The people used the pools for washing and preparing food, though they drank from the higher, cleaner streams.
Emma waited until she wasn't in anybody's way. Then she drank her fill of the stream, and washed out her underwear, and spread it out to dry over the sunlit rocks.
As she tended her blistered feet and ulcerated legs, and made small repairs to her boots and underwear, she watched the hominids around her.
Her Ham companions seemed to settle in quickly, according to their nature. Mary, strong and powerful, spent happy hours wrestling with the younger men, besting them more often than not. By the end of the day she was hardening spear points in a hearth, apparently preparing for a hunt.
Julia seemed to make friends with a group of women and children who spent much of their time clustered around one hearth—she blended in so well, in fact, that Emma soon had trouble distinguishing her from her companions, as if she had been here all her life.
Joshua, a loner in his own community, was a loner here. He settled into a small, solitary cave, and Emma saw little of him. But the Hams here seemed to tolerate his eccentricities, as had his own people.
As for Emma, she was largely ignored, much as she been with her other communities of Hams. Unable to shake off a feeling of sufferance—after all, how would a Neandertal stray be treated if she wandered into a human community?—she did her best to keep out of everybody's way.
There was one old man who seemed to take a liking to her, however— _old_ , meaning maybe ten years younger than she was. He was badly disfigured by a swathe of scar tissue that lapped up from where his right ear should have been to the crown of his head. She didn't have a word in common with this guy, and she couldn't ask him about his injury. But this wounded, smiling man seemed vaguely curious about her: curious enough, anyhow, to offer her meat. The meat was a prime cut, apparently from the shoulder of some animal—an antelope, maybe, but it could have been a rhino for all she knew. It was a groaning bloody slab two fingers thick and twice the size of a dinner plate. Her benefactor watched with absent interest as she rigged up a frame of sticks to cook it over the nearest fire.
It seemed he had no English name. She took to thinking of him as Scarhead.
The meat was frankly delicious, though she longed for green vegetables, gravy, and a mellow Bordeaux to go with it.
The Hams worked hard, of course. But it struck her how _happy_ they all seemed—or if not that, content. Evidently the game was bountiful here, the living easy; all these guys had to do was sit around and wait for the meat to come wandering past, season after season. They even had fresh running water, day and night, right outside the cave. She remembered fantasies as a child of finding Candyland, where all the trees were chocolate and the streams lemonade, where you didn't have to work for anything, where you could take as much as you liked, just by reaching out. Was the way these people lived so different from that?
But what would humans do, she mused, if they stumbled on a situation like this?
Well, they wouldn't be satisfied with the generosity of Candyland. They'd breed until the caves were overflowing. The hunters would start ranging farther until all the animals in the area were eaten or driven away. Then agriculture would start, with everybody forced to bend their bodies to back-breaking toil, day after day. As the population exploded the forests would be cut back, the animals decimated.
Then would come the famines and the wars.
So much for Candyland. Maybe these Hams weren't just as smart as humans, she mused; maybe they were actually smarter.
On the third day she walked out of the caves, alone, and set off up the eroded hillside.
The rocks were broken and worn, and cut deeply by gullies in some of which water still flowed. She found that the easiest way to make progress was to lower herself into one of the gullies and clamber up its smooth, sloping sides, taking care not to slip on moss or lichen, until the channel petered out and she had to transfer to another.
Though she was soon panting hard and sweating into her coverall, she could feel her heart and lungs pump, the muscles of her newly powerful legs tingling. You're in the best shape you've been in for years, girl.
The noise of the tame whirlwind howled even louder. She resolutely ignored it.
Just below the summit she sat on a patch of bare rock, gathering her breath, getting the hassles of the climb out of her head. The eroded hillside, deeply punctured by its limestone gullies and caves, swept away beneath her. The sun was still low; it was maybe ten in the morning local time.
She stood and turned away from the plain. She walked up the last few paces to the crater's summit plateau, and faced the wind.
It was a wall of churning air: a cylinder, laden with dust, that must have been a couple of miles wide. It looked flat on her puny human scale, like the wall of a vast building. But it snaked into the sky, diminishing as her gaze followed it, and at its highest extremity it curled in the air, threadlike. The whole thing was streaked horizontally, like the clouds of Jupiter, by billows of crimson dust. The flow of the air seemed smooth, though here and there she saw bits of rock and vegetation, even a few snapped-off trees. But the rock at the wind's shimmering foot was worn bare.
The violence, the energy, was startling; it was like a waterfall, a rocket launch. A deep part of her mind couldn't accept that it was _controlled_ by anything: the animal in her, conditioned by a million years of experience, knew that this lethal expression of nature's power was unpredictable, beyond propitiation.
Nevertheless she walked forward. After a few paces, she felt the first breath of wind, and a speckle of dust on her cheek.
When she got to within maybe a hundred paces of that dense wall of dust the air grew turbulent. She staggered but kept on, leaning into the wind to keep to a rough straight line, and the dust bit harder, stinging her mouth and eyes.
She shielded her eyes. Only maybe fifty paces to the dust. Forty-nine, forty-eight... The air was a powerful physical presence, battering at her torso and face, whipping her hair, snatching the breath from her lungs.
And now she was inside the dust, suddenly, as if walking into a sandstorm. The dust was a thick glowing cloud around her, obscuring the sky, the rock, even the twister itself; and when she looked downwind she saw how she cast a kind of shadow in the streaming particles.
Afresh surge hit her, unexpectedly violent. She fell sideways, rolled a couple of times, and hit her head on a rock.
She lay there for a moment. Then she got to all fours on the worn-bare rock and tried crawling.
She fell again, rolled back, tried again. Her hands and the skin of her cheeks were streaked with tiny cuts, where sharp bits of rock had bitten into her. Still she kept trying.
Lacking a plan B, she tried again the next day.
And the next.
She tried wrapping herself in her parachute silk, to keep out the dust and bits of rock. She just got blown away faster. So she tied the silk tightly around herself, an outer-body garment with slits for her hands, a mask over her face. She managed to get further into that central wall of dust, maybe ten paces deep, before the sheer strength of the wind stopped her progress.
She tried crawling in, all the way. That didn't work.
The Hams watched all this, bemused.
She considered schemes with ropes and pitons and rock hammers, where she would make a kind of ladder that she could "climb," across the face of the barren windswept rock, all the way to the center. But she had no rope or pitons or rock hammers, and couldn't come up with any way of making them.
She explored the cave system, but found no way through that way.
And if she couldn't go under the twister wall, she surely couldn't go over it; it looked to her as if that tunnel of tortured air stretched all the way out of the atmosphere. (She did toy with insane schemes of retrieving Malenfant's lander and firing it up into some kind of Alan Shepard suborbital trajectory that would take her up and over the wall of air, and re-enter right into the eye of the storm. But—despite her various rash promises to Joshua to pilot him and the lander all the way to his mythical Gray Earth—she didn't know how to fly the lander, still less how to rig it for such a flight, still less how to land it.)
On the tenth day of trying, as she lay clinging to the rock, sucking air from dust through a sheet of muslin, somebody walked past her.
Mouth gaping, bits of chute silk flapping around her, she watched as a Ham man and child walked hand in hand into the teeth of the storm, blurring. Granted the Hams were stronger than she was—both of them probably, even the boy—but they weren't _that_ strong. They weren't even leaning into the damn wind.
Then she noticed, just before they disappeared into gray-red dust, that their skin wraps were hanging loose around them. The churning air wasn't touching them.
She spent more days watching.
The Hams had always used the other side of the crater as part of their domain for hunting and gathering. They had trails leading that way, so ancient they were actually worn into the rock. When a Ham walked such a trail, heading for the crater's interior, she just carried on through the wall of wind and dust.
The Hams weren't the only ones.
A flock of bats flapped clumsily into the crimson mist one day, their fragile wings unaffected by the tearing air. She spotted a young deer, apparently lost, that stumbled out of the dust, gazed around with wide eyes at the world beyond, then bolted back into the wind storm. Even other hominids could make it through: notably Runners, and one Nutcracker she spotted.
But not herself—and, for some reason, not the chimplike Elves, an association she found insulting.
She tried to interrogate the Hams. "Julia, how come you can get through the wind and I can't?"
An intense frown creased that powerful face. "Hams live here." She waved her arm. " _Still_ live here."
"All right. But why am I kept out?"
A shrug.
"What is it I'm not allowed to see? Is there some kind of installation in there, a base? Are the Hams allowed to go up to it? Do you have any, umm, trade with whoever built it?"
None of this meant much to Julia. "Funny stuff." She waved her fingers before her face. "Hard to see."
Emma sighed. So the Hams might be wandering around or through some kind of fabulous _Homo superior_ base without even looking at it, interested only in their perennial pursuits, perhaps not even _capable_ of seeing it from out of their bony cages of conservatism.
And that, presumably, was why the Daemons let the Hams wander at will past their meteorological moat. The Hams would restrict themselves, going where they had always gone inside the crater, doing what they had always done, taking not a step beyond their self-imposed boundaries; they would not interfere with whatever projects and designs the Daemons were developing in there. Whereas noisy, curious, destructive _Homo sap_ types like herself would not rest until they had barged their way into the Daemons' shining city.
Breaking this demeaning exclusion became an obsession with her.
She focused on the Hams. She kept trying their trails. She carried Ham tools and weapons as if intent on some Ham-type gathering and hunting. She tried walking in with a party of Hams, her slim form tucked into a line of their great hulking bodies. But the wind seemed to whip _through_ their immense muscular forms, to grab at her and push her aside.
She pushed the deception further. She purloined some skins and wrapped herself up like a Ham. Slouching, bending her legs, she practiced the Hams' powerful, clumsy gait. She let her hair grow ragged and filthy, and even smeared clay on her face, letting it dry in a hopeful imitation of a Ham's bulky facial morphology, the high cheekbones and the bony crest over the eyes.
Then, joining another foraging party, she slouched toward the wind, her gait rolling, keeping her distinctive _Homo sap_ chin tucked into her chest.
The wind wasn't fooled.
Furious, she stamped back to the caves, and sought out Joshua.
"You have to help me."
Joshua stared at her. He was ragged, filthy, sitting in a debris-strewn cave that managed to be remarkably ill-appointed, even by the Paleolithic standards of this Red Moon.
"Wha' for?"
She sighed, forgiving him his squalor, and kneeled in the dirt before him. _"I want to know,"_ she said. "I want to know what they are doing in there—and who _they_ are. If they are responsible for dragging this Moon around the realities—I mean, for changing the sky—I want to know why they are doing it. And to make them understand the damage they are causing, the suffering. Do you see?"
He frowned at her. "Deal," he said simply.
"Yes," she said wearily. "Yes, we had a deal. We still have a deal. You help me, and I'll try to help you get to the Gray Earth. Just as I promised." God forgive me for lying, she thought.
But his eyes narrowed, almost calculating. "Fin' a way."
"Yes, I'll find a way. We'll go back to the lander and—"
His massive hand shot out and grabbed her wrist. The grip was painful, but she knew that he was using only a fraction of his strength, that if he chose he could probably crush her bone.
"No lies."
He means it, she thought. He knows my kind too well. "Okay. No lies. I'll find a way. Get me through the wind wall and I'll work on it, I'll find a way. I promise, Joshua. Please, my arm..."
He squeezed harder—just a little—but it was like a vice closing over her flesh. Then he released her. He sat back, baring his teeth in a wide grin. "How?"
"How can I get through the wind wall? I've been thinking about that. Whatever controls the wind is too smart to be fooled by appearance. It's not enough that I look like a Ham. But maybe if I can learn to _think_ like a Ham..."
Scarhead dragged a couple of haunches of meat from the back of the cave. For one brief moment the old guy looked the image of the cartoon caveman. He threw the meat down on the trampled ground, then went back into the cave to fetch tools.
Emma had once more donned her best-effort Neandertal disguise. She got to the ground gingerly, conscious of the need to keep her face rigid so as not to crack her mask of clay.
As usual, nobody showed the slightest interest in her—by now, not even the children.
The meat was, gruesomely, a couple of legs, intact from hoof to shoulder, perhaps from a horse. The limbs were already skinned, fresh, bloody, steaming slightly. Flies buzzed languidly around the exposed flesh.
Scarhead returned. He threw his handfuls of tools on the ground and sat cross-legged. He grinned, and the low morning sun made his scar tissue glisten.
She inspected the tools with absent interest. There were limestone pebbles gathered from the beds of rivers, used as chopping tools, and dark basalt blocks shaped into bi-faced hand axes and cleavers. These were working tools, each of them heavily worn and blood-splashed.
Before she left the Earth she'd known nothing of technology like this, and if she had been confronted with this collection of pebbles and rocks she would have dismissed them as nothing but random debris. Now she knew differently. Tools like these, or the still more primitive artifacts of the Runners, had kept her alive for months.
Scarhead held out a hand axe to her.
She took the rock, feeling its rough texture. She turned it over in her hands, testing its weight, feeling how it fit perfectly into her small human hand—for, of course, Scarhead had chosen it to suit her grip.
Now Scarhead held up a fresh lump of obsidian, hammers of bone and rock. He said bluntly, "Copy." He grabbed one of the horse legs, and began to saw at the joint between the scapula and humerus, between shoulder and leg. His stone blade rasped as he cut through tough tendons and ligaments.
She tried. Just manhandling the heavy limb proved a challenge to her; the joints were gruesomely stiff, the meat slippery and cold in her hands.
She sighed. "Could I see the vegetarian menu?"
Scarhead just stared at her.
No smart-ass _H sap_ jokes, Emma; today you're a Neandertal, remember?
She kept trying. She worked the knife into the meat until she had exposed the tendons beneath the shoulder. The meat, cold and slippery against her legs, was purple-red and marbled with fat; it was coldly dead, and yet so obviously, recently attached to something alive.
Turning the stone tool in her hand, she sought to find the sharpest edge. She managed to insert her blade into the joint and sawed at the tough ligaments, scraping them until they gave, like tough bits of rope.
Scarhead grunted.
Surprised, she raised her hand. The tool's edge had cut into her flesh, causing long straight-line gashes that neatly paralleled the lifeline on her palm. She hadn't even felt the cuts happen—but then the blade on a stone knife could be sharper than a metal scalpel; it could slide right into you and you'd never know it. She saw belatedly that Scarhead's working hand was wrapped in a hunk of thick, toughened animal skin, and a kind of apron was draped over his lap.
... And now the pain hit, sharp and deep like a series of paper cuts, and she yowled. She went to a stream to drench her cut palm in cold water until the slow bleeding had stopped.
Scarhead waited patiently for her, no expression she could read on his broad, battered face.
You aren't doing too well here, Emma.
She tried again. She spread a skin apron over her lap, and improvised a protective binding for her hand from a bit of tough leather. Then she resumed her work at the ligaments and tendons.
Think about the work, Emma. Think about the feel of the stone, listen to the rasp of the tendons, smell the coagulated blood; feel the sun on your head, listen to the steady breathing of Scarhead...
She reached bone. Her axe scraped against the hard surface, almost jarring from her hand. She pulled the axe back and turned it over, exposing fresh edge, and began to dig deeper into the joint, seeking more tendon to cut.
A last tough bit of gristle gave way, and the leg disarticulated.
She stared, oddly fascinated, at the bone joints. Even Malenfant, who had never shown the slightest interest in biology, might have been interested at this bit of natural engineering, if he had gotten to take it apart in his own hands.
And she was still analyzing. _Wrong_.
She glanced up at Scarhead. Not watching her, apparently immersed in the work, he had begun to fillet the meat from the shoulder joint he was holding. Emulating his actions, she did the same. She dug her blade into the gap between meat and bone, cutting the muscle that was attached to the bone surface. She soon found the easiest way was to prop the scapula on the ground between her legs, and pull at the muscle with one hand to expose the joint, which she cut with the other hand. She got into a rhythm of turning the axe in her hand, to keep exposing fresh edge.
She tried not to think about anything—not Earth, Malenfant, the wind wall, the destiny of mankind, her own fate—nothing but the feel of the sun, the meat in her hand, the scrape of stone on bone.
For brief moments, as the hypnotic rhythms of the butchery tugged at her mind, she got it.
It was as if _she_ was no longer the little viewpoint camera stuck behind her eyes; it was as if her consciousness had dispersed, so that _she_ was her working hands, or spread even further to her tool, the flesh and bone she worked, and the trails and bits of forest and scrub and the crater walls and the migrating herds and all the other details of this scrap of the world, a scrap inhabited by the Hams, unchanging, for generation upon generation upon generation.
... Her hands had finished the butchery. On one side of her, a flensed shoulder-bone; on the other, a neat stack of filleted meat.
She looked into cavernous eyes, feeling the sun's heat, feeling the pleasurable ache of her arms and hands. She forgot the name she had given him, forgot her own name, forgot herself in his deep stare.
Shadows beside her. It was Joshua, and Julia... No, no names; these people simply were who they were, everybody in their world knew them, without the need for labels. She took their hands and let herself be raised to her feet.
The Hams led her up the hillside, away from the caves, toward the place where the unnatural wind moaned.
It was not like a dream; it was too detailed for that. She felt the sharpness of every grain of red dust under her feet, the lick of the air on her cheeks, the salty prickle of sweat on her brow and neck, the sharp, almost pleasant ache of her cut palm. It was as if a veil had been removed from her eyes, stops from her ears and nose, so that the colors were vivid and alive—red earth, green vegetation, blue sky—and the sounds were clear, grainy, loud, their footsteps crunching into the earth, the hiss of wind over the scrubby grass that clung to these upper slopes. It was like being a child again, she thought, a child on a crisp summer's Saturday morning, when the day was too long for its end to be imagined, the world too absorbing to be analyzed.
Was _this_ how it was to be a Neandertal? If so, how—enviable.
They had reached the crest of the crater-rim hill. They began to walk forward, in a line, hand in hand.
That wall of air spread across the land before her, a cylinder so wide it looked flat. She felt a lick of wind, touching her cheek, disturbing her hair, the first prickle of dust on her skin. She dropped her head, concealing her _Homo sap_ protruding chin, and walked steadily on. She concentrated on the sun, the texture of the ground, the bloody iron scent of the dusty air.
Anything but the wind.
They went into the dust. She walked steadily, between her Ham friends, immersed in crimson light. She was ten paces inside the dust. Then fifteen, past her previous record. Twenty, twenty-one, twenty-two...
Maybe it was the counting. Hams did not count.
The wind hit her like a train.
Her hands were wrenched from the Hams' grip. She was lifted up off the ground, flipped on her back, and slammed down again.
The light dimmed to a dull Venusian red. Suddenly she couldn't see Julia or Joshua, nothing but a horizontal hail of dust particles and bits of rock, looming out of infinity as if she was looking into a tunnel. If she turned her head into the wind she could hardly breathe.
Another gust—she was rolled over—she scrabbled at the ground. And then she was lifted up, up into the air, limbs flailing, like a cow caught by a Midwest tornado. She was immersed in a shell of whirling dust; she couldn't see ground or sky, couldn't tell how far away the ground was, couldn't even tell which way up she was. But she could tell she was falling.
She screamed, but her cry was snatched away. _"Malenfant!"_
She was on her back. She could feel that much. But there was no wind: no hot buffeting gusts at her face, no sting of grit on her exposed skin. Nothing but a remote howl.
She opened her eyes.
She was looking up into a dark tunnel, like gazing up from the depths of a well, towards a circle of cloud-scattered blue sky. The light was odd, grayish-red, as if shadowed. Was she back in the caves? She tried to sit up. Pain lanced through her back and stomach.
A face loomed above her, silhouetted by the patch of bright sky, back-lit by diffuse gray light. "Take it easy. We don't think any bones are broken. But you are cut and bruised and badly winded. You may be concussed." The face was thin, capped by a splash of untidy black hair. Emma stared at an oddly jutting chin, weak cheekbones, an absurd bubble skull with loose scraps of hair. It was a woman's face.
It came into focus. A _human_ woman.
The woman frowned. "Do you understand me?"
When she tried to speak Emma found her mouth full of dust. She coughed, spat, and tried again. "Yes."
"You must be Emma Malenfant."
"Stoney," Emma corrected automatically. "As if it makes a difference now." She saw the woman was wearing a faded blue coverall, scuffed and much-repaired, with a NASA meatball logo on her chest. "You're Nemoto. Malenfant's companion."
Nemoto regarded her gravely, and with a start Emma recognized for the first time the Oriental cast of her features. A lesson, she thought wryly. Compared to the distance between humans and other hominids, the gap between our races really is so small as to be unnoticeable.
"... Malenfant is dead," she said hesitantly. "I'm sorry."
She thought she saw hope die, just a little, in Nemoto's blank, narrowing eyes.
"I don't know how well you knew him. I—"
"We have much to discuss, Emma Stoney."
"Yes. Yes, we do."
Nemoto slid an arm under Emma's back and helped Emma sit up. Everything worked, more or less. But her belly and back felt like one immense bruise, and she was having trouble breathing.
She was sitting on crimson dirt. A few paces away from her, waiting patiently, she saw Joshua and Julia. She grinned at them, and Julia gave her an oddly human wave back.
Beyond them was strangeness.
A yellow floor sprawled over the ground—seamless and smooth, obviously artificial. There were buildings on this floor, rounded structures the same color and apparently made of the same material, as if they had grown seamlessly from out of the floor, as if the whole thing was a sculpture of half-melted cheddar cheese.
Hominids were moving among the structures. They walked on feet and knuckles, big and bulky, too remote for her to make out details. Like gorillas, she thought, like the creature she had seen leaving the Zealot stockade with the ragtag army. Could they be Daemons?
She looked over her shoulder. She saw that wall of wind, streaked with dirt and ripped-up vegetation. But now she could see how it curved inward, _around_ her—confining her here, not excluding her. And when she looked up it stretched into the sky, making a twisting, slowly writhing tunnel.
She was inside the twister.
"Ha!" she said, and she punched the air. "Fooled 'em, by God."
Nemoto was frowning. There was an edge about her, a tension that seemed wound tight. "It was not like that. You did not 'fool' anybody. The Daemons watched your approach. They watched as you plastered clay on your face and butchered your meat—"
"How did they watch me?"
Nemoto waved at the air. "They can see what they like, go wherever they want to, at a gesture. They call it Mapping."
"I don't understand."
Nemoto leaned down, thrusting her face at Emma, anger sparking. "Your efforts to deceive them were comical. Embarrassing. They could not have succeeded. _It was me_ , Emma Stoney. I was the one who practiced deceit in the end; I convinced them to admit you. I tried to spin your absurd stunt into an act of true cognition. I told them that deceit is a sign of a certain level of intelligence. But I said you were aware of the shallowness of your deceit. You _intended_ to demonstrate an ability to bluff and counterbluff, thus showing multiple levels of cognition which—"
Emma raised a hand. "I think I get it." Holding Nemoto's hand, she pulled herself to her feet. "I wish I could say I was so smart. Intentionally, anyhow. Umm, I guess it's appropriate to thank you."
She heard heavy footsteps. She turned.
One of the gorilla-things was coming toward her. It—no, _she_ , she had breasts—she walked using her knuckles. But she moved fast, more than a walk: it was a knuckle-sprint, a knuckle-gallop, startlingly fast for such a huge animal.
The creature must have been eight feet tall. The ground seemed to shake.
Emma felt Nemoto's hand slide into hers. "Show no fear. Her name is Manekato, or Mane. She will not harm you."
The Daemon stood before Emma. She straightened up, her massive black-haired bulk towering, and her hands descended on Emma's shoulders, powerful, heavy, humanlike. Emma felt overwhelmed by weight, solidity, the powerful rank stench of chest hair. She raised her hands and pressed against that black chest, pushing with all her strength against the surging muscle. Effortlessly, it seemed, the Daemon pressed closer, bringing her shining black face close to Emma's. The mouth opened, and Emma glimpsed a pink cavern and tongue, two huge spikelike upper canines, and smelled a breath sweet as milk.
Two ears swivelled towards Emma, like little radar dishes.
Then the Daemon backed off, dropping to rest her weight on her knuckles once more. She growled and hooted.
Nemoto was smiling thinly. "That was English. You will get used to her pronunciation. Mane asks, _What is it you want?_ "
"Tell her I want—"
"Tell her yourself, Emma Stoney."
Emma faced Manekato, gazed into deep brown gorilla eyes. "I've come here looking for answers." She waved a hand. "Don't you see the damage you cause?"
Mane frowned, a distinctly puzzled expression, and she peered at Nemoto, as if seeking clarification there. Just as with the Hams, Emma had the distinct and uncomfortable feeling that she wasn't even asking the right questions.
Again Nemoto had to translate for Emma. _"You think we made this. The engine that moved the world. Child, the Old Ones are far above us—so far they are as distant from me as from you. Do you not understand that?"_
Emma shuddered. But she said belligerently, "I just want to know what is going on."
This time, Emma made out Mane's guttural words for herself. "We hoped you could tell us."
That first night, Emma stayed in the shelter the Daemons had given Nemoto—despite Nemoto's obvious reluctance to share. A second bed was "grown" inside the little shelter's main room for Emma, fully equipped with mattress, pillow, and sheets; the gorilla-thing called Mane apologized to Emma for the crowding, but promised a place of her own by the next night.
Unlike the rounded, quasi-organic feel of the other structures on the disk floor, Nemoto's residence was a boxy design with rectangular doors and windows, giving it a very human feel. But, like the other structures, it seemed to have grown from the smooth, oddly warm, bright yellow substrate. It was as if the whole place were a seamless chunk of pepper-yellow plastic that had popped out of some vast mold.
But the Daemons had provided for Nemoto well. She had a bed with a soft mattress and sheets of some smooth fabric. She was given fruit and meat to eat; she even had a box the size of a microwave oven, with pretty much the same function. There were spigots for hot and cold water, a bathroom with a toilet that flushed.
Holiday Inn it wasn't, but it was close enough, Emma thought, in the circumstances. Nemoto said the flush toilet, for instance, had taken a couple of prototypes to get right.
None of this had anything to do with the way the Daemons lived their lives. They seemed to have no desire for privacy when defecating or urinating, for instance; they just let go wherever they happened to be, making sure they didn't splash the food. The magic floor absorbed the waste, no doubt recycling it for some useful purpose, and would even dispel odors. The Daemons, though, were understanding, or at least tolerant, of Nemoto's biological and cultural hang-ups.
Anyhow it suited Emma fine.
There were sanitary towels. Emma fell on these and stole as many as she could carry away.
There was coffee (or a facsimile).
There was a shower.
She luxuriated in her first hot wash for months, using soap and shampoo that didn't smell as if it had come oozing straight out of the bark of a tree. At first the water just ran black-red at her feet, as if every pore on her body was laden with crimson dirt. By the time she had washed out her hair two, three times, it began to _feel_ like her hair again. She cleaned out the black grime from beneath her fingernails. She looked around for a razor, but could find none; so she used one of her stone blades, purloined from a Neandertal community many miles away, to work at her armpits.
Towelling herself dry, Emma stood by the window of Nemoto's little chalet, peering out at the Daemons' encampment.
Feeling oddly like a primatologist in a hide, she watched little knots of the huge gorillalike creatures knuckle-walking to and fro. _H. superior_ or not, they all looked alike, for God's sake. And little cartoon robots buzzed everywhere, rolling, hopping, and flying. She had to remind herself that these really were creatures capable of flying between worlds, of putting on a light show in the sky to shame the aurora borealis, of _growing_ a city in the jungle.
But as she watched, one of the "gorillas" flickered out of existence, reappearing a few minutes later on the other side of the compound.
At that moment Emma knew, deep in her gut, that there was indeed nothing primitive about these shambling, knuckle-walking, hairy slabs of muscle, despite her _Homo sap_ prejudices.
And it made it still more terrifying that it was not the Daemons who were responsible for moving the Moon, but another order of creatures beyond even them. She felt that she was at the bottom of a hierarchy of power and knowledge, unimaginably tall.
She hit her first soft pillow in months. Emma spent twelve hours in deep, dreamless sleep.
When she hauled herself out of bed the next day, Nemoto made her brunch (French toast, by God). But Nemoto was largely silent, volunteering little of her experiences here.
Emma, in turn, resented this silence. After all Nemoto had spent a good deal of time with Malenfant—spent most of his last few months alive with him, in fact, when Emma had been about as far from him as she could be. But Emma wasn't about to beg for scraps of information about her own damn husband.
I am not, Emma thought, going to get along easily with this woman.
Manekato came visiting. She crouched to get her eight-feet-tall bulk inside Nemoto's shelter, then sat squat on the floor, a gorilla in a too-small cage. Her accent was thick, her voice a Barry White growl. But when she spoke slowly, Emma found she understood her.
Manekato said, "You have talked. Nemoto has shared with you what she has learned."
Nemoto and Emma shared a glance.
Emma said, "Actually, no."
Mane slapped her huge thigh, apparently in frustration. "You are the same species! You are alone here, far from home! Why can you not cooperate?"
Nemoto said easily, "You are showing your prejudice, Manekato. You must see us as individuals. We are the same species, but that does not determine our goals—any more than you and Renemenagota had identical motivations."
The name meant nothing to Emma.
Mane turned to Emma, her huge head swivelling. "Very well. Em-ma? Why have you come here?"
Emma thought about that. "I want to go home."
Manekato said, "I regret that is not within my gift. _I_ cannot go home."
Emma closed her eyes for a moment, letting her last sliver of hope disappear. She should have expected this, of course. If it were possible to reach Earth, Nemoto would surely have been sent there by now.
She opened her eyes and met Mane's gaze. "Then I want to go to the center."
"The center?"
"The place where everything happens."
Nemoto grinned. "She wants to see the world engine."
Mane asked, "Why?"
Emma felt angry. _Who are you to ask? It isn't yours, any more than it is human_... "Because I've come this far. Because I've kept myself alive on this damn Moon that took my husband's life, and I want to know what the hell it is all for."
"What difference would _knowing_ make?"
"It just would," Emma snapped. "And I resent your cross-questioning."
Mane paused. Then she said gently, "Em-ma, how did you come here?"
"It was an accident. I, umm, fell through a portal. A Wheel, a blue circle."
"Yes. We know of such devices. But your mate, Mal-en-fant, came here purposefully, with Nemoto."
"He came to rescue me."
"How is it Mal-en-fant had the technology to travel to the Red Moon? Did he invent it from scratch?"
Emma glanced at Nemoto, who showed no reaction. Mane was asking her questions to which Nemoto must have already given answers; perhaps this was some obscure test.
"No," Emma said. "We had travelled to our own Moon—umm, a lifeless world—long before the Red Moon showed up. The technical base was there."
"Why did you go to this Moon? For science, for learning?"
"For politics," Nemoto said sourly. "For irrational purposes. For typical _Homo sapiens_ reasons."
"It wasn't just that," Emma said, frowning. "You don't live with an astronaut your whole life without figuring out some of the bigger picture. Manekato, we went to the Moon because we are a species that explores. We go places even when there is no immediate purpose. 'Why choose this as our goal? Why climb the highest mountain? Why... fly the Atlantic? We choose to go to the Moon... because that goal will serve to organize and measure the best of our abilities and skills...' "
Nemoto laughed. "President Kennedy's 1961 speech. It is a long time since I heard those words."
"Malenfant was fond of quoting it."
"So," Mane said, "you intended to live on your Moon, to colonize it."
"Ultimately, I guess, yeah."
"And then?"
"And then the other planets," Emma said vaguely. "Mars, the asteroids, the moons of Jupiter."
"And then?"
"And then the stars, I guess. Alpha Centauri... You'd have been better asking Malenfant." She studied Manekato, trying to read the expressions that passed over that broad, blue-black face. "Every intelligent species must have the same kind of goals. Expansion, colonization. Mustn't they? Especially every intelligent variety of hominid."
Nemoto was shaking her head. "Not so, it seems."
Emma was growing irritated again; she wasn't enjoying being treated as the dope of the class. "Why are _you_ here, Manekato?"
"Like you," Mane said evenly, "when this Red Moon appeared in our skies—and it disrupted our world as much as it did yours—we asked the question _why_."
Emma leaned forward. "But _why you_ , Mane, rather than somebody else?"
Mane frowned. "I came because I had no home."
It turned out that Mane's home, which she referred to as a _Farm_ , had been wiped off the face of her Earth by Red Moon tides.
"She came here because she was forced," Nemoto said.
"You could have rebuilt someplace else."
"There is nowhere else," Mane said. She pulled at an ear that was all but buried in thick black fur. "It was the end of my Lineage. A Lineage that stretched back through a hundred thousand generations." She sighed, and began to scratch at the other ear.
Emma sat, stunned. A hundred thousand generations? If each generation was, say, twenty years at minimum—why, that added up to _two million years_.
Nemoto said, "Emma, these people are _not like us_. They are much more like the Hams. They sit on those Farms of theirs, forever and a day. They do not covet what their neighbors possess. There is no robbery, no territorial or economic expansion, no nation, no war."
"And if you lose your Farm—"
"If you lose your Farm, you die. Or anyhow your Lineage does."
"That's terrible," Emma said to Mane. "What do they do? Sterilize you? Take your children?"
But it seemed that once again she had asked the wrong question. Mane asked blankly, _"They?"_
"Nobody has to enforce it," Nemoto said. "It just happens. The families let themselves die out. It is seen as a price worth paying for ecological stability. Emma, the Daemons have _evolved_ this way, shaped by their cultural imperatives. Two million years, remember."
Emma shook her head, uncomfortable under Mane's steady gaze. She felt defiant. "Humans wouldn't live like that. We wouldn't accept it."
Mane kept pulling her ear. "What would you do?"
Emma shrugged. "The family would go on. The _Mayflower_ syndrome. We'd carve a place out of the wilderness—"
"But there is no wilderness," Mane said. "Even without war, even if you found a space not already cultivated, you would be forced to occupy a region, delineated in space, time, and energy flow, already exploited by another portion of the ecology."
It took some time for Emma to figure that out. "Yes," she said. "There is bound to be some environmental impact. But—"
"Other species would find reduced living space. Diversity would fall. And so it would go on. Soon the world would be covered from pole to pole by humans, fighting over the diminishing resources." Mane nodded. "Such was the ambition of Praisegod Michael. At least you are consistent."
"The Daemons limit their numbers," said Nemoto. "They don't overrun their Earth. By respecting the stability of the ecosystem that provides for them they have survived for millions of years. They even accept their short life spans, though it would be trivial for them to do something about that."
"A brief life burns brightly," said Manekato.
Emma shook her head. "I still say humans couldn't live like that."
Nemoto said slyly, "The Hams do. And they are _almost_ human."
"Are you saying we should live like Neandertals, in caves, wearing skins, wrestling buffalo, watching our children die young?"
Mane said, "Are the Hams suffering?"
No, Emma thought. Actually they are happy. But her pride was hurt; she stayed silent.
Mane leaned forward, and Emma could smell her milk-sweet breath. "The lion takes only the last deer in the herd. She does not dream of having so many cubs that the plains would be full of nothing but lions. There are simple laws. Most species figure them out; you are the exception. An ecology of a single species is not viable. A diverse, stable world would provide for you."
Candyland, Emma said.
"We have a story," Mane said. "A mother was dying. She called her daughter. She said, 'This is the most beautiful Farm in the world.' And so it was. The mother said, 'When I die, you will be free to act. Do with it what you will.' The daughter pondered these words.
"And when the mother died, the daughter took a torch and set fire to her Farm—every bit of it, the buildings and crops and creatures.
"When asked why she had done this—for of course, without a Farm, her Lineage would be extinguished—the daughter said, 'One night of glory is better than a thousand years of toil.' " The big Daemon actually shuddered as she finished her tale.
"We have a similar legend," Emma said. "There was a warrior, called Achilles. The gods gave him a choice: a brief life of glory, or a long, uneventful life in obscurity. Achilles chose the glory." She looked up at Mane. "In my culture, that story is regarded as uplifting."
Mane turned her tremendous head. "The tale I told you is, umm, a scary story. Intended to frighten the children into proper behavior."
Nemoto said grimly, "But we will go on anyhow. To the planets, the stars. If we get the chance; if we survive the human-induced extinction event that is unfolding on our Earth. Because we don't have a choice." She eyed Manekato bleakly. "Sure, our strategy is flawed. But it has a deadly internal logic. We're stuck on this road we have chosen. We have to keep expanding, or we'll die anyhow."
"There is that," Mane said gently. She stood, and with startling clumsiness rammed her head against the low roof of the chalet. "You wish to see the engine of the world. So do I, Em-ma. We will go together."
Nemoto nodded warily. "How? Will you Map us?"
Manekato laid a hand on Emma's scalp. It was heavy, gentle, the pads of flesh on the palm soft. "We have found we cannot Map there. But it would not be appropriate anyway. We are all hominids together, here on this Red Moon. Let us do what hominids do. We will walk to our destiny."
Four of them would be travelling together: Emma, Nemoto, Manekato—and Julia, the Ham. As Emma was preparing to leave, Julia had walked out of nowhere, with every sign of staying at Emma's side until they reached whatever there was to find, at the center of this wind-wrapped crater.
Manekato loomed over the three of them, the massive muscles of her shoulders as big as Emma's skull. "Now we go, we four, to discover the secret of the universe." She threw back her mighty head and laughed, a roar that rattled off the smooth-walled structures of the compound. And, without hesitation, she walked off the yellow platform floor, heading for the interior of the crater, and the forest that lay there.
The little column turned single file and spread out. The going was easy over the dust-strewn rock, and Emma, hardened by her weeks of living rough, found it easy to keep up with Manekato's knuckle-gallop. But when she looked back she saw that Nemoto was laboring, lagging behind Emma by a hundred yards. Julia walked at her side, stolid, slow, patient, her own awkward gait endearingly clumsy.
Emma waited until Nemoto caught up. Nemoto did not look her in the eye; she plodded on, her gait showing a trace of a limp. Emma clapped her on the shoulder. "I guess the human species isn't going to conquer the stars if we can't even walk a couple of miles, Nemoto."
"I am not as acclimatized as you," Nemoto said.
"Despite all that astronaut training you must have had. Whereas _I_ was just thrown here on my ass from out of the blue sky—"
"Punish me if you like. Your misfortunes are not my fault."
"Right. You came here to rescue me. Or was it just to give me somebody even worse off than I am?"
Julia moved between them. "No' worry, Emma. I help."
Emma grinned. "Just throw her over your shoulder if she gives you any trouble. Nemoto—even if they can't Map there, I don't understand why the Daemons haven't been to this center before."
"They have been studying it. They can be remarkably patient. And—"
"Yes?"
"I think they have been waiting for us."
Emma observed, "Nobody's carrying anything."
Julia shrugged. "Fores' has food. Fores' has water."
"You see?" Nemoto glared. "These _others_ do not think as we do. Julia _knows_ that the land will provide everything she needs: food, water, even raw materials for tools. It is a different set of assumptions, Emma Stoney. Just as Manekato said. _They_ see the universe as essentially bountiful, a generous mother land. We see the universe as an enemy nation, to be occupied and mastered."
"So we're inferior in every way," Emma grumbled, resentful.
"Not that," Nemoto said. "But we are different. The Daemons' intellectual capacity is obvious—the rapidity of their comprehension, the richness and precision of their thinking. But they come from a world where hunters, indeed predators of any kind, cannot prosper. Even their games are cooperative, all concerned with building things."
"What about religion? What do they believe?"
Nemoto shrugged. "If they have a religion it is buried well, in their minds and their culture. They need not worship sublimated mothers or seeds as we do, because they control nature—at least, below the Red Moon. And without the metaphor of the seed, of renewal, they have no urge to believe in a life beyond the grave."
"Like the Hams."
"Yes. The Hams, Neandertals, have much more in common with the Daemons than we do. And remember this, Emma Stoney. Mane's people regard us as less intelligent than them. Save for academic interest or sentimentality, they have no more interest in _talking_ to us than you would have in chatting to a Colobus monkey. This is the framework within which we must operate, no matter how hurtful to your _Homo sapiens_ ego."
They reached a patch of forest. Manekato plunged into it, seeking fruit. The others followed more slowly.
Keeping Manekato's broad back in sight, Emma stepped cautiously over a muddy, leaf-strewn ground. Roots snaked everywhere, as if put there to trip her. In some places the trees towered high. She could see the canopy, where the thick branches of each tree spread out, making an almost horizontal roof of greenery. The trunks themselves were dense with life, with lianas that looped and sagged, and ferns and orchids sprouting like underarm hair from every crevice and fork. Though it was humid and still, the moist air felt almost cool on her cheeks, as if this were fall. There was a mild, pervading stench of decaying vegetation.
A shadow flitted between the tree trunks, a round, uncertain form dimly glimpsed among the shadowy verticals.
Emma stopped dead, heart hammering.
Manekato was a massive, reassuring form at her side. "It is a Nutcracker. A vegetarian hominid which—"
"I know about Nutcrackers."
Manekato peered curiously into her face. "I sense fear."
Emma found her breath was shallow; she tried to control it. "Does that surprise you?"
"You are already far from home. Without prior preparation, without aid, you have survived in this place for many weeks. What more is there for you to fear now?"
"Humans aren't creatures of the forest, like the Elves or the Nutcrackers. We are creatures of the open. Like the Runners."
"Ah." Apologetically Manekato reached for her and, with thick, gentle, leather-skinned fingers, she probed at Emma's shoulders, elbows, hips. "It is true. You are designed for steady walking, for running, over long distances. You sweat—unlike me—so that you can control your heat loss efficiently in the open sunlight. Yes, your link with the forest is lost deep in the past. And so you see it, not as a place of bounty and safety, but of threat."
"We have tales. Fictions. Many of them are scary. They involve dense forests, being lost in the woods."
Manekato showed ferocious teeth. "And if an Elf were able, it would frighten its companions with tales of being trapped in the open, with no forest cover in sight, at sunset, as the predators begin to feed... But that hominid appeared to be fleeing. Little threatens the Nutcrackers, here in their forest domain; they are strong and smart. Curious." Mane loped forward, more slowly than before, her massive form moving with barely a rustle through the crowded foliage. Emma followed in her tracks.
Then Mane slowed, peering down at something on the ground.
Emma heard the buzzing of flies. Then came the stench, the rotting-meat stench: sanitized out of the world she had come from, a smell she would not get used to no matter how long she lasted on this strange, mixed-up Moon.
The smell of death.
It looked like a chimp that had been hit by a truck. Its hairy skin was broken by wounds and lesions, and a watery fluid leaked from a gaping mouth and empty eye-sockets. Maggots squirmed in the lesions, giving the corpse a semblance of life. The body seemed to be deliquescing, in fact, its flesh and bones dissolving right out from within its skin and pouring into the ground.
There was an infant sitting on the ground beside the adult, presumably its mother, a round bundle of misery.
"Now we know why that Nutcracker was fleeing," Emma said.
Nemoto, panting hard, joined Emma. "I have seen this before. Do not touch anything."
"What is it?"
"Something like the Ebola virus, I think. It starts with a headache, a fever. As your cells fill with the replicating virus your immune system collapses. Your skin turns to pulp; you hemorrhage; your gut fills with blood; blood leaks from your eyes, mouth, nose, ears, anus. When you die your body turns to slime. If somebody picks up the corpse, they contract it, too, and die in turn. There is no vaccine or cure. I guess that is why the others of this one's troupe have abandoned it, and its child."
"I have made this one safe," Mane murmured. "There is no infection here." Emma hadn't seen her do anything.
The baby raised its head and studied Emma. The little Nutcracker, surely no more than a year old, was surrounded by scrapings of thin white infant scut.
Emma said to Mane, "It's safe to pick it up?"
"Yes."
Emma pulled a piece of cloth over her mouth and nose and stepped forward, toward the infant. The infant cowered back, but it was weak and hungry and scared, and let Emma tuck her hands under its armpits.
She lifted it easily, though it was heavier than she had thought, a boulder of hair and bone. "Well, it's a girl; I can tell that much." The infant had brown-black eyes, creamy white at the edges. Her skin beneath the hair was black, and wrinkles ran across her brow, between her eyes and over her stubby ape nose, giving her a troubled expression. Her mouth was open, and was a startlingly bright pink inside. The hair on her body was thick and coarse, but on her head, over that improbable crest of bone, the hair was sparser.
Emma held the baby against her chest. The little body was very warm. The sad, small black face tucked into a fold of Emma's coverall, and Emma bent to kiss the bony crest on the top of her head. She smelled leaves.
Then the infant hugged her tight with legs and arms, tensed, and defecated in a stream that spilled down Emma's trouser legs.
Julia made claw hands. "Leopards. Hyenas. Chomp baby Nu'cracker."
"Right," Emma said. "Smart baby. You only take a dump when you're mother is holding you."
Nemoto was watching her. "Emma Stoney, I hope you're not considering bringing that infant with you."
Emma hadn't thought that far ahead. "Why not?"
"Because you do not know how to look after it."
"Her. I don't know how to look after _her_."
"You know nothing of the ecology of these creatures. You are sentimental."
"She is right," Mane said mournfully. The big Daemon loomed over the little tableau, like an adult standing over a child with her doll. "This infant has been abandoned by its kind. It will shortly die, of starvation, predation, disease. Death is commonplace for all hominid species, Emma. Among the Nutcrackers, in fact, the men compete for access to groups of women and children. And sometimes if one man displaces another, he will destroy the children of his defeated opponent."
"All very evolutionarily sound," Emma said coldly. "But I'm keeping her."
She felt a massive hand on her back: Julia's. "Lonely," said the Ham.
"Yes. Yes, I'm lonely, Julia. I lost my husband, my world, my life. For all your kindness, of course I'm lonely."
"All," Julia said softly. "Lonely."
Nemoto prowled about the little clearing, agitated, avoiding the corpse. "We are the lonely hominids. On Earth it is thirty-five thousand years since we last encountered another hominid species. Maybe it was our relentless expansion that drove the last of the Neandertals to extinction; maybe it was our fault—but whatever the cause it was surely the last contact. And when we look out into the sky, we see nothing but emptiness. An empty world in an empty universe. No wonder we have been at war with our planet since before records began. Earth had betrayed us, orphaned us: what else was there to do? Yes, we are lonely, all of us. Lonely and frightened. But do you really think making a pet of an orphaned Australopithecine is going to make any difference?"
Emma felt Mane's heavy, gentle hand touch the top of her head, distant, comforting.
They approached the center.
People moved over the rocky ground. They were Daemons, little clusters of them walking to and fro, bearing incomprehensible pieces of equipment, occasionally flickering into and out of existence in that baffling, utterly disturbing way of theirs.
Beyond the Daemons, Emma thought she could see light shining up from the ground, caught by swirling dust motes. She shivered.
Nemoto was silent, tense.
They reached the center of the clearing. Emma stepped forward gingerly.
There was a hole in the ground, a few yards wide, like a well. Light shone from it, up into the dusty air, like an inverted sunbeam.
Emma felt cold with awe.
She sat on the grass with the Nutcracker infant and reached for a flask of milk from her pack. She opened up the yellow plastic-feel flask, exposing a nipple, and tipped it toward the infant's head, making soothing noises. The infant grabbed the yellow flask with hands and feet, and she began to suck at the nipple, very hard. Milk splashed into her mouth and over her face, and over Emma.
Emma wiped milk from her lap and eyes. "I should do this with an apron."
"You shouldn't do it at all," Nemoto said sourly. "You should give her back to her kind."
"Nutcrackers don't adopt orphans. You know that."
Mane stood over them like a block of granite. "We could make the infant acceptable to a troupe of its kind."
Emma scowled. "How?"
Nemoto said, "Emma, if they can travel between worlds just by thinking about it, the Daemons can surely fool some half-evolved ape."
Mane reproached her, "Nutcrackers are fully evolved. Just differently evolved."
The infant finished the milk, or at any rate lost patience with the bottle. She threw it over her head. Then she touched the milk that had pooled on Emma's chin, and opened her mouth to make fast, rasping cries. "Hah hah hah!"
"She's laughing at me," Emma said.
"I am not surprised," Nemoto said.
"I'll find some running water and wash us both up."
Julia, watching, grinned. "Nutcracker don' wash!"
Nemoto grimaced. "This is not a toy, still less a human child! Soon you will be stinking as badly as her! Emma, give up this sentimentality. Give her back to her own kind." She seemed obsessed with the issue of the infant.
Emma looked up at Manekato, and she looked into her own heart. "Not yet," she said.
There was a moment of stillness. In this open space the sun was warm on her face, invigorating, its light making the dusty air shine. The infant Nutcracker gurgled and plucked at Emma's sleeve. Manekato walked to the lip of the tunnel. She stood silently, on crimson earth, peering into the well in the Moon, its diffuse light picking out the folds in her blue-black skin. Emma wondered what she was thinking, what the tunnel was saying to her.
Mane turned. "It is time." She held out her hands.
Yes, Emma thought. Somehow she knew it, too. She stood up, brushing dust off her coveralls. The Nutcracker child clambered up into her arms. She settled her distorted head against Emma's chest and promptly fell asleep.
Nemoto stood reluctantly. Emma could see she was trembling, utterly afraid. Mane took Emma's hand, and Nemoto's, and Julia took Nemoto's other hand. Cradling the infant, Emma walked up to the lip of the well.
The shaft at her feet was a cylinder, walled by what looked like sparkling glass, a wall that receded downward to infinity. Lights had been buried in the walls every few yards, so the shaft was brilliantly lit, like a passageway in a shopping mall, the multiple reflections glimmering from the glass walls. Conduits snaked along the tunnel, their purpose unclear. The shaft was vertical, perfectly symmetrical, and there was no mist or dust, nothing to obscure her view.
Momentarily dizzy, Emma stepped back, anchored herself again on the surface of the Red Moon.
Nemoto said, "What is this?"
Mane said evenly, "It is a tunnel in the Moon."
"But what is it for?"
"We don't know."
Emma said, "How deep is it?"
"We don't know that either," Manekato said. "We have tried sending—" she hesitated "— _radio signals_ and other emissions into the well. No echo has returned."
"But," said Emma, "it can't be longer than the width of the Moon. Even if it came out the other side... It can't be longer than that."
"We don't know," Mane said. _"We did not put it here."_
Nemoto said tightly, "What do we have to do?"
Mane regarded her with her large eyes, pupils black, the whites flecked with yellow. "I think you know."
Yes, Emma knew—though she didn't understand how she knew. A prickly wave of vertigo swept over her. Malenfant, she thought desperately, you should be here to see this. You would _love_ it. But _me_...
There was no more time, no time for thinking, for doubt. Without a word, the five of them stepped off the lip of the tunnel, into the air.
For a moment they floated there in space, bathed in the light from the heart of the world, like cartoon characters for whom the laws of physics are momentarily suspended.
And then they began to sink, gently.
There was nothing beneath her feet. The air was full of light.
Slow as a snowflake, tugged by a force that felt like gravity—and yet it could not be gravity—Emma fell toward the heart of the Moon. There was no noise save the rustle of clothing, their soft breathing, no smell save the lingering iron-and-blood stink of the crimson dust of the Red Moon.
She could tell she was falling. Lines in the wall, like depth markers, were already rising up past her, mapping her acceleration. But it was as if she were suspended here, in the glowing air; she had no sense of speed, no vertigo from the depths beneath her.
She could hear her own heart pound.
Nemoto was laughing, manic.
Emma held the black bundle of fur closer to her chest, drawing comfort from the Nutcracker's solid animal warmth. "I don't know what the hell is so funny."
Nemoto's face was twisted, a mask of fear and denial. "We are not in the hands of some omnipotent, infallible god. This is no more than a gadget, Emma. More ancient than our species, more ancient than worlds perhaps, very advanced—but very old, and cranky, and probably failing as well. And we are relying on it for our lives. _That_ is what strikes me as funny."
Their speed picked up quickly.
In seconds, it seemed, they had already passed through the fine layers of the Red Moon's outer geology. Now they sailed past giant chunks of rock that crowded against the glassy, transparent tunnel walls like the corpses of buried animals.
"The megaregolith," Nemoto murmured. "In the later stages of its formation this little world must have been just as bombarded as our own Luna. Under the surface geology, the craters and cracks, this is what you get. Pulverization, shattered rock, mile upon mile of it. We are already far beyond the reach of any human mining, Emma. We are truly sinking deep into the carcass of this world."
Mane regarded her, curious, judgmental. "You are analytical. You like to find names for what you see."
"It helps me cope," Nemoto said tightly.
The material beyond the walls turned smooth and gray. This must be bedrock, Emma thought, buried beyond even the probing and pulverizing of the great primordial impactors. Unlike Earth, on this small world there had been no tectonic churning, no cycling of rocks from surface to interior; these rock layers had probably lain here undisturbed since the formation of the Red Moon.
Already they must be miles deep.
Despite the gathering warmth of the tunnel, despite her own acceleration, she had a sense of cold, of age and stillness.
She had no real sense of how long she had been falling—it might have been seconds, or minutes—perhaps time flowed as deceptively here as space, as gravity. But she was reluctant to glance at a watch, or even look up to the receding disk of daylight above. She was not like Nemoto, determinedly labeling everything; rather she felt superstitious, as if she might break the spell that held her in the air if she questioned these miracles too hard.
They dropped through a surprisingly sharp transition into a new realm, where the rock beyond the walls glowed of its own internal light. It was a dull gray-red, like a cooling lava on Earth.
"The mantle," Nemoto whispered. "Basalt. Neither solid nor liquid, a state that you don't find on the surface of a planet, rock so soft it pulls like taffy."
Soon the rock brightened to a cherry-pink, rushing upward past them. It was like dropping through some immense glass tube full of fluorescing gas. Gazing at that shining pink-hot rock just yards away, Emma felt heat, but that was surely an illusion.
The baby Nutcracker stirred, eyes closed, wiping her broad nose on Emma's chest.
Falling, falling. Thick conduits surrounded them now, crowding the tunnel, flipping from bracket to bracket. She wondered what their purpose was; neither Nemoto nor Mane offered an opinion.
For the first time she felt a lurch, like an elevator slowing. Looking down along the forest of conduits, she could see that they were approaching a terminus, a platform of some dull, opaque material that plugged the tunnel.
She asked, "Where are we?"
Mane said, "Thousands of miles deep. Some two-thirds of the way to the center of the Moon."
They slowed, drifting to a crawl maybe a yard above the platform. Emma landed on her feet, still clutching the infant—an easy landing, even if it had reminded her of her involuntary skydive.
Now she glanced at the watch Nemoto had loaned her. The fall had taken twenty minutes.
The smooth surface was neither hot nor cold, a subdued white, stretching seamlessly from one side of the shaft to the other.
Emma put down the infant Nutcracker. With a happy grunt the infant urinated, a thin stream that pooled on the gleaming floor.
In this place of shining geometric perfection, all the hominids looked misshapen, out of place: Julia with her heavy-browed skull, the Daemon with her looming gorilla body, her fast, jerky motions and her eerily swivelling ears, and Nemoto and Emma, the proud ambassadors of _Homo sapiens_ , huddled close together in their dusty, much-patched coveralls. We are barely evolved, Emma thought—even Mane—unformed compared to the chill, effortless perfection of this place.
"... Noise," Julia said. She turned her great head, peering around. "Noise. Lights."
Nemoto scowled, peering around, up into the tunnel that receded into infinity over their heads. "I cannot hear anything."
"There is much information here," Mane said gently. She had closed her eyes. "You must—let it in."
"I don't know how," Nemoto said miserably.
Emma glanced down at the infant Nutcracker. She was crawling on legs and knuckles and peering into the floor, as if it were the surface of a pond. Emma, stiffly, got to her knees beside the child. She stared at the floor, looked where the infant looked.
There was a flash of blue light, an instant of searing pain.
The floor had turned to glass. With the Nutcracker, she was kneeling on nothingness. She gasped, pressed her hands against the hard surface. No, not glass: there was no reflection, nothing but the warm feel of the floor under her hands and knees.
And below her, a huge chamber loomed.
She felt Nemoto's hand on her shoulder, gripping tight, as if for comfort.
Emma said, "Can you see it?"
"Yes, I see it."
Emma glimpsed a far wall. It was covered with lights, like stars. But these stars marked out a regular pattern of equilateral triangles. Artificial, then. She looked from side to side, trying to make out the curve of that remote wall. But it was too far away for her to make out its shape, too far beyond her puny sense of scale.
"It's a hole," she said. "A chamber at the heart of the Moon."
"It is whatever it seems to be."
"The chamber looks flattened. Like a pancake."
"No," Nemoto murmured. "It is probably spherical. You have the eyes of a plains ape, Emma Stoney. Evolved for distances of a few hundred miles, no more. Even the sky looks like a flat lid to you. Humans aren't evolved to comprehend spaces like this—a cave thousands of miles across, a cave big enough to store a world."
"Those lights are regular. Like fake stars on a movie set."
"Perhaps they are the mouths of tunnels, like this one."
"Leading to more holes on the surface?"
"Or leading somewhere else." Nemoto's voice was quavering. "I don't know, Emma. I understand none of this."
But you understand more than me, Emma thought. Which is, perhaps, why you are more frightened.
There was motion in the heart of the chamber. Blueness. Vast wheels turning. A churning, regular, like a huge machine.
The Nutcracker child gurgled, her eyes shining. She seemed enchanted by the turning wheels, as if the whole display, surely a thousand miles across, were no more than a nursery mobile.
"Blue rings," Nemoto breathed.
Emma squinted, wishing her eyes would dark-adapt faster. "Like the Wheel, the portal I fell through to come here."
Nemoto said, "This technology has a unifying, if unimaginative, aesthetic."
"It is the world engine," Mane said simply.
Emma saw the turning wheels reflected in Mane's broad, glistening eyes. "What is a _world engine_?"
"Can you not see? Look deeper."
"... Ah," Nemoto said.
At the heart of the turning rings, there was a world.
It was like Earth, but it was not Earth. Turning slowly in the light of an off-stage sun, it was wrapped in a blanket of thick, ragged cloud. Emma glimpsed land that was riven by bright-glowing cracks and the pinpricks of volcanoes. Plumes of black smoke and dust streaked the air, and lightning cracked between fat purple clouds.
"Not a trace of ocean," Nemoto murmured. "Too hot and dry for that."
"Do you think it is Earth?—or any of the Earths?"
"If it is, it is a young Earth, an Earth still pouring out the heat of its formation..."
"The sky," Mane said, her voice quavering, "is full of rock."
Emma glanced up.
... And for an instant she saw what the Daemon saw: a different point of view, as if she were standing on that burnt, barren land, on bare rock so hot it glowed, close to a river of some sticky, coagulating lava. She looked up through rents in fat, scudding clouds—into a sky that was covered by a lid of rock, an inverted landscape of mountains and valleys and craters.
She gasped, and the vision faded.
Emma saw again the hot young world, and another beside it now, a Moonlike world, evidently cooler than Earth, but large, surely larger than Mars, say. The two planets sat side by side, like an orange and an apple in a still life.
But they were approaching each other.
"I think we are watching the Big Whack," Nemoto murmured. "The immense collision that devastated young Earth, but created the Earth-Moon system..."
The planets touched, almost gently, like kissing. But where they touched a ring of fire formed, shattering the surface of both worlds, a spreading splash of destruction into which the smaller body seemed to implode, like a fruit being drained of its flesh.
"The collision took about ten minutes," Nemoto said softly. "The approach speed was tens of thousands of miles per hour. But a collision between such large bodies, even at such speeds, would look like slow motion."
A vast fount of material, glowing liquid rock, gushed into space from the impact point. Emma glimpsed the impacting planetesimal's gray curve, a last fragment of geometric purity, lost in the storm of fire. A great circular wave of fire spread out around the Earth from the impact point.
A ring of glowing light began to coalesce in Earth orbit. As it cooled it solidified into a swarm of miniature bodies. And then spiral arms formed in the glowing moonlet cloud. It was a remarkable, beautiful sight.
"This is how the Moon was born," Nemoto said. "The largest of those moonlets won out. The growing Moon swept up the remnant particles, and under the influence of tidal forces rapidly receded from Earth. Earth itself, meanwhile, was afflicted by huge rock tides, savage rains as the ocean vapor fell back from space. It took millions of years before the rocks had cooled enough for liquid water to gather once more."
"You know a lot about this stuff, Nemoto."
Nemoto turned, her face underlit by the glow of Earth's violent formation. "A few months ago a new Moon appeared in Earth's sky. I wanted to know how the old one had got there. I thought it might be relevant."
Emma glanced at Mane. The Daemon stood with her knuckles resting lightly on invisibility. Her eyes were closed, her face blank. Julia's eyes were closed, too.
"What do they see?" she whispered to Nemoto. "What do they hear?"
"Perhaps more than this show-and-tell diorama. Manekato said this place, this tunnel in the Moon, was _information-rich_. Julia is as smart as we are, but different. Manekato is smarter still. I don't know what they can apprehend, how far they can see beyond what we see."
"... Hey. What happened to the Earth?"
The glowing, devastated planet had blown apart. Fragments of its image had scattered to corners of the chamber—where the fragments coalesced into new Earths, new Moons, a whole family of them. They hung around the chamber like Christmas tree ornaments, glowing blue or red or yellow, each lit by the light of its own out-of-view sun.
_Other Earths:_
Emma saw a fat, solitary world, banded with yellow cloud.
Here was another cloud-striped world, but the clouds swirled around a point on its equator—no, it was a world tipped over so that its axis pointed to its sun, like Uranus (or was it Neptune?).
Here was an Earth like Venus, with a great shroud of thick clouds that glowed yellow-white, nowhere broken.
Here was a world with a fat, cloud-shrouded Moon that seemed to loom very close. This Earth was streaked by volcanic clouds. It lacked ice caps, and its unrecognizable continents were pierced by shining threads that must have been immense rivers. This world must be battered by the great tides of air, water, and rock raised by that too-close companion.
Most of the Earths seemed about the size of Earth—of _the_ Earth, Emma's Earth. But some were smaller—wizened worlds that reminded her of Mars, with huge continents of glowering red rock and brooding weather systems squatting over their poles. And some of the Earths were larger. These monster planets were characteristically wreathed in thick, muddy atmospheres and drowned in oceans, water that stretched from pole to pole, with a few eroded islands protruding above the surface, rooted on some deep-buried crust.
The Moons varied, too. There seemed to be a spectrum of possible Moons. The smallest were bare gray rock like Luna, those somewhat larger cratered deserts of crimson rock more or less like Mars. Some were almost Earthlike, showing thick air and ice and the glint of ocean—like the Red Moon itself. There were even Earths with pairs of Moons, Emma saw, or triplets. One ice-bound Earth was surrounded, not by a Moon, but by a glowing ring system like Saturn's.
Emma looked, without success, for a blue Earth with a single, gray, modest Moon.
"The Big Whack collision shaped Earth and Moon," Nemoto murmured. "Everything about Earth and Moon—their axial tilt, composition, atmosphere, length of day, even Earth's orbit around the sun—was determined by the impact. But it might have turned out differently. Small, chance changes in the geometry of the collision would have made a large difference in the outcome. Lots of possible realities, budding off from that key, apocalyptic moment."
"What are we looking at here? Computer simulations?"
"Or windows into other possible realities. It is a glimpse of the vast graph of probability and possibility, of alternates that cluster around the chaotic impact event." Nemoto seemed coldly excited. "This is the key, Emma Stoney. The Big Whack was the pivotal event whose subtly different outcomes produced the wide range of Earths we have encountered..."
Emma barely understood what she was saying.
Julia grunted. "Gray Earth," she said. She was pointing to the tipped-over, Uranuslike Earth.
Emma said, "Where you came from."
"Home," Julia said simply.
Nemoto said, "I recognize _that_ one." She pointed to the fat, solitary Earth, banded by Jupiterlike clouds. "A Moonless Earth, an Earth where the great impact did not happen at all. It may be the Earth they call the Banded Earth, which seems to be the origin of these Daemons."
Mane laid gentle, patronizing hands on their scalps. "Analyze, analyze. Your minds are very busy. You must watch, listen."
"Ooh." It was the Nutcracker infant. She was crawling over the invisible floor, chortling at the light show.
Emma glanced down. The various Earths had vanished, to be replaced by a floor of swirling, curdled light.
It was a galaxy.
"Oh, my," she muttered. "What now?"
The galaxy was a disk of stars, flatter than she might have expected, in proportion to its width no thicker than a few sheets of paper. She thought she could see strata in that disk, layers of structure, a central sheet of swarming blue stars and dust lanes sandwiched between dimmer, older stars. The core, bulging out of the plane of the disk like an egg yolk, was a compact mass of yellowish light; but it was not spherical, rather markedly elliptical. The spiral arms were fragmented. They were a delicate blue laced with ruby-red nebulae and the blue-white blaze of individual stars—a granularity of light—and with dark lanes traced between the arms. She saw scattered flashes of light, blisters of gas. Perhaps those were supernova explosions, creating bubbles of hot plasma hundreds of light years across.
But the familiar disk—shining core, spiral arms—was actually embedded in a broader, spherical mass of dim red stars. The crimson fireflies were gathered in great clusters, each of which must contain millions of stars.
The five of them stood over this vast image—if it _was_ an image—Daemon and Ham and humans and Nutcracker baby, squat, ungainly, primitive forms.
"So, a galaxy," said Emma. " _Our_ Galaxy?"
"I think so," Nemoto said. "It matches radio maps I have seen." She pointed, tracing patterns. "Look. That must be the Sagittarius Arm. The other big structure is called the Outer Arm."
The two major arms, emerging from the elliptical core, defined the Galaxy, each of them wrapping right around the core before dispersing at the rim into a mist of shining stars and glowing nebulae and brooding black clouds. The other "arms" were really just scraps, Emma saw—the Galaxy's spiral structure was a lot messier than she had expected—but still, she thought, the sun is in one of those scattered fragments.
The Galaxy image began to rotate, slowly.
Emma could see the stars swarming, following individual orbits around the Galaxy core, like a school of sparkling fish. And the spiral arms were evolving, too, ridges of light sparking with young stars, churning their way through the disk of the Galaxy. But the arms were just waves of compression, she saw, like the bunching of traffic jams, with individual stars swimming through the regions of high density.
"A galactic day," Nemoto breathed. "It takes two hundred million years to complete a turn."
Oh, Malenfant, Emma thought again, you should be here to see this. Not me—not _me_.
Nemoto said, "But _whose_ Galaxy is it?"
"That is a good question," Mane said. "It is our Galaxy—that is, it belongs to all of us. The Galactic background is common to the reality threads bound by the Earth-Moon impact probability sheaf—"
"Whoa," Emma said. "Nemoto, can you translate?"
Nemoto frowned. "Think of the Galaxy, a second before the Earth-Moon impact. All those stars have nothing whatsoever to do with the Big Whack, and will not be affected by it. The Galaxy will turn, whether the Moon exists or not, whether humans evolve or not..."
Mane said, "Our Galaxy looks the same as _yours_. And it is unmodified."
Emma snapped, "What does _that_ mean?"
Nemoto said, "That there is no sign of life, Emma."
"But we're looking at a whole damn _galaxy_. From this perspective the sun is a dot of light. The place could be swarming with creatures like humans, and you wouldn't see it."
Nemoto shook her head. "The Fermi Paradox. In our universe, and Mane's, there has been time for a thousand empires to sweep over the face of the Galaxy. Some of the signs of their passing ought to be very visible."
"Like what?"
"Like they might tamper with the evolution of the stars. Or they might mine the black hole at the Galaxy's core for its energy. Or they might wrap up the Galactic disk in a shell to trap all its radiant energy. Emma, there are many possibilities. It is very likely that we would see _something_ even when we peer at a Galaxy from without like this."
"But we don't."
"But we don't. Humanity seems to be alone in our universe, Emma; Earth is the only place where mind arose." Nemoto confronted Mane. "And _your_ universe is empty, too. As was Hugh McCann's. Perhaps that is true in all the universes in this reality sheaf."
Emma murmured, "The Fermi paradox."
Nemoto seemed surprised she knew the name.
"Something is happening to the Galaxy," Mane said.
They clustered close to watch.
The Galaxy was spinning fast now. All over the disk the stars were flaring, dying. Some of them, turning to red embers, began to drift away from the main body of the disk.
Emma picked up the Nutcracker infant and clutched her to her chest. "It is—shrivelling," she said.
"We are seeing vast swathes of time," Nemoto said sombrely. "This is the future, Emma."
" _The future?_ How is that possible?"
Suddenly the stars died. All of them went out, it seemed, all at once.
The Galaxy seemed to implode, becoming much dimmer.
At first Emma could make out only a diffuse red wash of light. Perhaps there was a slightly brighter central patch, surrounded by a blood-colored river, studded here and there by dim yellow sparkles. That great central complex was embedded in a diffuse cloud; she thought she could see ribbons, streamers in the cloud, as if material were being dragged into that pink maw at the center.
Further out still, the core and its orbiting cloud seemed to be set in a ragged disk, a thing of tatters and streamers of gas. Emma could make out no structure in the disk, no trace of spiral arms, no lanes of light and darkness. But there were blisters, knots of greater or lesser density, like supernova blisters, and there was that chain of brighter light points studded at regular intervals around the disk. Filaments seemed to reach in from the brighter points toward the bloated central mass.
Emma said, "What happened to all the stars?"
"They died," Nemoto said bluntly. "They grew old and died, and there wasn't enough material left to make any more. And then, _this_." Nemoto pointed. "The wreck of the Galaxy. Some of the dying stars have evaporated out of the Galaxy. The rest are collapsing into black holes—those blisters you see in the disk. That central mass is the giant black hole at the core."
" _When_ is this?"
Nemoto hesitated, thinking, and when she spoke again, she sounded awed. "Umm, perhaps a hundred thousand billion years into the future—compared to the universe's present age _five thousand times_ older."
The numbers seemed monstrous to Emma. "So this is the end of life."
"Oh, no." Mane replied. She pointed to the clusters of brighter light around the rim of the galactic corpse. " _These_ seem to be normal stars: small, uniform, but still glowing in the visible spectrum."
"How is that possible?"
"Those stars can't be natural," Nemoto said. She turned to Emma, her eyes shining. "You see? Somebody must be gathering the remnant interstellar gases, forming artificial birthing clouds... Somebody is farming the Galaxy, even so far in the future. Isn't it wonderful?"
" _Wonderful?_ The wreck of the Galaxy?"
"Not that," Nemoto said. "The existence of life. They still need stars and planets, and warmth and light. But their worlds must be huddled close to these small, old stars—probably gravitationally locked, keeping one face in the light, one in the dark... I think this is, umm, a biography," Nemoto said. "This whole vast show. The story of a race. _They_ are trying to tell us what became of them."
"A very human impulse," said Mane.
Emma shrugged. "But why should they care what we think?"
Nemoto said, "Perhaps they were our descendants..."
Mane said nothing, her eyes wide as she peered at the crimson image, and Emma wondered what strange news from the future was pouring into her head.
And now the Galaxy image whirled again, evolving, changing, dimming.
Emma hugged the baby hominid and closed her eyes.
## _M anekatopokanemahedo_
This is how it is, how it was, how it came to be.
It began in the afterglow of the Big Bang, that brief age when stars still burned.
Humans arose on an Earth. Emma, perhaps it was your Earth. Soon they were alone, and forever after.
Humans spread over their world. They spread in waves across the universe, sprawling and brawling and breeding and dying and evolving. There were wars, there was love, there was life and death. Minds flowed together in great rivers of consciousness, or shattered in sparkling droplets. There was immortality to be had, of a sort, a continuity of identity through copying and confluence across billions upon billions of years.
Everywhere they found life: crude replicators, of carbon or silicon or metal, churning meaninglessly in the dark.
Nowhere did they find mind—save what they brought with them or created—no _other_ against which human advancement could be tested.
They were forever alone.
With time, the stars died like candles. But humans fed on bloated gravitational fat, and achieved a power undreamed of in earlier ages. It is impossible to understand what minds of that age were like, minds of time's far downstream. They did not seek to acquire, to breed, or even to learn. They needed nothing. They had nothing in common with their ancestors of the afterglow.
Nothing but the will to survive. And even that was to be denied them by time.
The universe aged: indifferent, harsh, hostile, and ultimately lethal.
There was despair and loneliness.
There was an age of war, an obliteration of trillion-year memories, a bonfire of identity. There was an age of suicide, as even the finest chose self-destruction against further purposeless time and struggle.
The great rivers of mind guttered and dried.
But some persisted: just a tributary, the stubborn, still unwilling to yield to the darkness, to accept the increasing confines of a universe growing inexorably old.
And, at last, they realized that something was wrong. _It wasn't supposed to have been like this_.
Burning the last of the universe's resources, the final downstreamers—lonely, dogged, all but insane—reached to the deepest past...
## _E mma Stoney_
Nemoto was muttering, perhaps to Emma or Manekato, or perhaps to herself, as she impatiently swept lianas and thorn tangles out of her path. "Evolution has turned out to be a lot more complicated than we ever imagined, of course. Well, everything is more complicated now, in this manifold of realities. Even though Darwin's basic intuition was surely right..." And so on.
Carrying the sleeping Nutcracker infant, Emma walked through the forest. Ahead she could see the broad back of Manekato.
Emma let Nemoto talk.
"... Even before this Red Moon showed up in our skies we had developed major elaborations to the basic Darwinian model. Darwin's 'tree of life' is no simple tree, it turns out, no simple hierarchy of ancestral species. It is a tangle—"
"Like this damn jungle," Emma said, trying to turn the monologue into a conversation. "Lianas and vines cutting across everywhere. If it were just the trees it would be easy."
"A crisscross transfer of genetic information, this way and that. And now we have this Red Moon wandering between alternate Earths, the Wheels returning to different Africas over and over, scooping up species here and depositing them there, making an altogether untidy mess of the descent of mankind—and of other species; no wonder this world is full of what Malenfant called 'living fossils.' Surely without the Red Moon we would never have evolved, we _Homo sapiens sapiens. Homo erectus_ was a successful species, lasting millions of years, covering the Earth. We did not _need_ to become so smart..."
It had been some days since their jaunt into the tunnel in the Moon. Nemoto had spent the time with Manekato and other Daemons, struggling to interpret the experience. For her part, Emma had barely been able to function once those visions of the aging Galaxy had started to blizzard over her—even though it had been, apparently, just a fraction of the information available in that deep chamber, for those minds capable of reading it.
But she remembered the last glimpse of all.
_... It was dark. There were no dead stars, no rogue planets. Matter itself had long evaporated, burned up by proton decay, leaving nothing but a thin smoke of neutrinos drifting out at lightspeed_.
_But even now there was something rather than nothing_.
_The creatures of this age drifted like clouds, immense, slow, coded in immense wispy atoms. Free energy was dwindling to zero, time stretching to infinity. It took these cloud-beings longer to complete a single thought than it once took species to rise and fall on Earth..._
That ultimate, dismal vision was slow to dispel, like three-in-the-morning fears of her own death. She knew she didn't have the mental toughness to confront all this, special effects or not. Unlike Nemoto, perhaps.
Or perhaps not. To Nemoto, the whole thing seemed to have been more like a traumatic shock than an imparting of information. She had come out of the experience needing human company, in her reticent way, and needing to talk. But when she talked it was about Charles Darwin and the Red Moon, or even Malenfant and the politics of NASA, anything but the central issue of the Old Ones.
Emma concentrated on the leafy smell of the child, the crackle of dead leaves, the prickle of sunlight on her neck, even the itch of the ulcers on her legs. _This_ was reality, of life and breath and senses.
Manekato had stopped, abruptly. Nemoto fell silent. They were in a small scrap of clearing, by the side of the lichen-covered corpse of a huge fallen tree. Manekato lifted herself up on her hind legs, sniffed the air and swivelled her ears, and belched with satisfaction. "Here," she said. "The Nutcrackers will come." With a massive thump she sat on the ground, and began exploring the bushes around her for berries.
Emma gratefully put down the infant Nutcracker and sat beside her. The leaves were slippery and damp; the morning was not long advanced. She considered giving the infant some more milk, but the child had already discovered Manekato's fruit, and was clambering up the Daemon's impassive back.
Nemoto sat beside Emma. Her posture was stiff, her arms wrapped around her chest, her right heel drumming on the ground. Emma laid one hand on Nemoto's knee. Gradually the drumming stopped.
And, suddenly, Nemoto began to talk.
"They made the manifold."
"Who did?"
"The Old Ones. They constructed a manifold of universes—an infinite number of universes. They _made_ it all." Nemoto shook her head. "Even framing the thought, conceiving of such ambition, is overwhelming. But they did it."
Manekato was watching them, her large eyes thoughtful.
Emma said carefully, "How did they do this, Nemoto?"
"The branching of universes, deep into the hyperpast," Manekato murmured.
Emma shook her head, irritated. "What does that mean?"
Nemoto said, "Universes are born. They die. We know two ways a universe can be born. The most primitive cosmos can give birth to another through a Big Crunch, the mirror-image of a Big Bang suffered by a collapsing universe at the end of its history. Or else a new universe can be budded from the singularity at the heart of a black hole. Black holes are the key, Emma, you see. A universe which cannot make black holes can have only one daughter, produced by a Crunch. But a universe which is complex enough to make black holes, like ours, can have many daughters, baby universes connected to the mother by space-time umbilicals through the singularities."
"And so when the Old Ones tinkered with the machinery—"
"We don't know how they did it. But they changed the rules," Nemoto said.
Emma said hesitantly, "So they found a way to create a lot more universes."
Manekato said, "We believe the Old Ones created, not just a multiplicity of daughter universes, but _an infinite number_." The bulky Daemon studied Emma's face, seeking understanding.
"Infinity is significant, you see," Nemoto said, too rapidly. "There is, umm, a qualitative difference between a mere large number, however large, and infinity. In the infinite manifold, in that infinite ensemble, _all_ logically possible universes must exist. And therefore _all_ logically possible destinies must unfold. Everything that is possible _will_ happen, somewhere out there. They created a grand stage, you see, Emma: a stage for endless possibilities of life and mind."
"Why did they do this?"
"Because they were lonely. The Old Ones were the first sentient species in their universe. They survived their crises of immaturity. And they went on, to walk on the planets, to touch the stars. But everywhere they went—though perhaps they found life—they found no sign of mind, save for themselves."
"And then the stars went out."
"And the stars went out. There are ways to survive the darkness, Emma. You can mine energy from the gravity wells of black holes, for instance... But as the universe expanded relentlessly, and the available energy dwindled, the iron logic of entropy held sway. Existence became harsh, straitened, in an energy-starved universe that was like a prison. Some of the Old Ones looked back over their lonely destiny, which had turned into nothing but a long, desolating struggle to survive, and—well, some of them rebelled."
The infant crawled over Manekato's stolid head and down her chest, clutching great handfuls of hair. Then she curled up in the Daemon's lap, defecated efficiently, and quickly fell asleep. Emma suppressed a pang of jealousy that it was not _her_ lap.
"So they rebelled. How?"
Nemoto sighed. "It's all to do with quantum mechanics, Emma."
"I was afraid it might be."
Manekato said, "Each quantum event emerges into reality as the result of a feedback loop between past and future. Handshakes across time. The story of the universe is like a tapestry, stitched together by uncountable trillions of such tiny handshakes. If you create an artificial timelike loop to some point in space-time within the negative light cone of the present—"
"Whoa. In English."
Manekato looked puzzled.
Nemoto said, "If you were to go back in time and try to change the past, you would damage the universe, erasing a whole series of consequential events. Yes? So the universe starts over, from the first point where the forbidden loop would have begun to exist. As the effects of your change propagate through space and time, the universe knits itself into a new form, transaction by transaction, handshake by handshake. The wounded universe heals itself with a new set of handshakes, working forward in time, until it is complete and self-consistent once more."
Emma tried to think that through. "What you're telling me is that changing history is possible."
"Oh, yes," said Nemoto. "The Old Ones must have come to believe _they had lived through the wrong history_. So they reached back, to the deepest past, and made the change—and the manifold was born."
Emma thought she understood. So this had been the purpose the Old Ones had found. Not a saga of meaningless survival in a dismal future of decay and shadows. The Old Ones had reached back, back in time, back to the deepest past, and put it right, by creating infinite possibilities for life, for mind.
She said carefully, "I always wondered if life had any meaning. Now I know. The purpose of the first intelligence of all was to reshape the universe, in order to create a storm of mind."
"Yes," Manekato said. "That is a partial understanding, but—yes."
"Whew," Emma said.
Nemoto seemed to be shivering, exhausted. "I feel as if I have been gazing through a pinhole at the sun; I have stared so long that I have burned a hole in my retina. And yet there is still so much more to see."
"You have done well," Manekato said gently.
Nemoto snapped, "Do I get another banana?"
"We must all do the best we can." Manekato's massive hand absently stroked the Nutcracker; the child purred like a cat.
"But," Emma said, "the Old Ones must have wiped out their own history in the process. Didn't they? They created a time paradox. Everybody knows about time paradoxes. If you kill your grandmother, the universe repairs itself so you never existed..."
"Perhaps not," Manekato murmured. "It seems that conscious minds may, in some form, survive the transition."
"Do not ask how," Nemoto said dryly. "Suffice it to say that the Old Ones seem to have been able to look on their handiwork, and see that it was good... mostly."
"Mostly?"
Nemoto said, "We think that we, unwilling passengers on this Red Moon, are, umm, exploring a corner of the manifold, of that infinite ensemble of universes the Old Ones created. Remember the Big Whack. Remember how we glimpsed many possible outcomes, many possible Earths and Moons, depending on the details of the impact."
"It is clear," Manekato said, "that within the manifold there must be a sheaf of universes, closely related, all of them deriving from that primal Earth-shaping event and its different outcomes."
Nemoto said, "Many Earths. Many realities."
"And in some of those realities," Manekato said, "what you call the Fermi Paradox was resolved a different way."
"You mean, alien intelligences arose."
"Yes." Nemoto rubbed her nose and glanced uneasily at the sky. "But in every one of those alien-inhabited realities, _humans got wiped out_ —or never evolved in the first place. Every single time."
"How come?"
Nemoto shrugged. "Lots of possible ways. Interstellar colonists from ancient cultures overwhelmed Earth before life got beyond the single-cell stage. Humankind was destroyed by a swarm of killer robots. Whatever. The Old Ones seem to have selected a bundle of universes—all of them deriving from the Big Whack—in which there was _no_ life beyond the Earth. And they sent this Moon spinning between those empty realities, from one to the other—"
"So that explains Fermi," Emma said.
"Yes," said Nemoto. "We see no aliens _because we have been inserted into an empty universe_. Or universes. For our safety. To allow us to flourish."
"But why the Red Moon, why link the realities?"
"To express humanity," Manekato said simply. "There are many different ways to be a hominid, Em-ma. We conjecture the Old Ones sought to explore those different ways: to promote evolutionary pulses, to preserve differing forms, to make room for different types of human consciousness."
Emma frowned. "You make us sound like pets. Toys."
Manekato growled; Emma wondered if that was a laugh. "Perhaps. Or it may be that we have yet to glimpse the true purpose of this wandering world."
Emma said, "But I still don't get it. Why would these superbeing Old Ones care so much about humanity?"
Nemoto frowned. "You haven't understood anything, Emma. _They were us_. They were our descendants, our future. _Homo sapiens sapiens_ , Emma. And their universe-spanning story is our own lost future history. _We_ built the manifold. _We_ —our children—are the Old Ones."
Emma was stunned. Somehow it was harder to take, to accept that these universe-making meddlers might have been—not godlike, unimaginable aliens—but the descendants of humans like herself. What hubris, she thought.
Nemoto said now, "That was the purpose, the design of the Red Moon. But now the machinery is failing."
"It is?"
"The sudden, frequent and irregular jumps. The instabilities, the tides, the volcanism. It shouldn't be happening that way."
Emma turned back to Manekato. "Let me get this straight. The Red Moon has been the driver of human evolution. But now it is breaking down. So what happens next?"
"We will be on our own," said Nemoto. She raised her thin hands, turned them over, spread the fingers. "Our evolutionary destiny, in hominid hands. Does that frighten you?"
"It frightens _me_ ," Manekato said softly.
For a moment they sat silently. Emma was aware of the dampness of the breeze, the harsh breathing of the big Daemon. On impulse she put her hand on Manekato's arm. Her fur was thick and dense, and her skin hot—hotter than a human's, perhaps a result of her faster metabolism.
"... Wait," Manekato said softly, peering into the trees.
Shadows moved there: shadows of bulky, powerful forms. They paused, listening. There were at least three adults, possibly more. Emma could make out the characteristic prow-shaped silhouettes of their skulls.
The Nutcracker infant roused from her sleep. Bleary-eyed, she peered into the trees and yowled softly.
The shadows moved closer, sliding past the trees, at last resolving into recognizable fragments: curling fingers, watchful eyes, the unmistakeable morphology of hominids. One of them, perhaps a woman, extended a hand.
The infant clambered off Manekato's lap and stood facing the Nutcracker woman, nervous, uncertain.
The Nutcracker woman took a single step into the clearing, her eyes fixed on the infant. The child whimpered, and took a hesitant step forward.
Nemoto hissed to Emma, "Listen to me. I have a further theory. The Old Ones did not disappear into some theoretical universe-spanning abstraction. _They are still here_. Wouldn't they want to be immersed in the world they made, to eat its fruit, to drink its water? Maybe they have become these Nutcrackers, the most content, pacific, unthreatened, _mindless_ of all the hominid species. They shed everything they knew to live the way hominids are supposed to, the way we never learned, or forgot. What do you think?"
The infant glanced back at Emma, knowing. Then, with a liquid motion, the Nutcracker woman scooped up the child and melted into green shadows.
Back in the Daemons' yellow-plastic compound, Emma luxuriated in a hot shower, a toweling robe, and a breakfast of citrus fruit.
Luxuriate, yes. Because you know you aren't going to enjoy this much longer, are you, Emma? And maybe you'll never live like this again, not ever, not for the rest of your life.
You will miss the coffee, though.
She dressed and emerged from her little chalet. The sky was littered with cloud, the breeze capricious and laden with moisture. Storm coming.
She saw Nemoto arguing with Manekato. Nemoto looked, in fact, as if she still wasn't getting a great deal of sleep; black smudges made neat hyperbolae around her eyes. By contrast, Manekato was leaning easily on her knuckles, her swivelling ears facing Nemoto, her great black-haired body a calming slab of stillness. And Julia, the Ham girl, was standing close by, listening gravely.
When Emma approached, Mane turned to her, smooth and massive as a swivelling gun-turret. "Good morning, Em-ma."
"And to you. Nemoto, you look like shit."
Nemoto glowered at her.
"What's the hot topic?"
"Future plans." Nemoto's foot was characteristically tapping the plastic-feel floor like a trapped animal, about the nearest she got to expressing a true emotion.
"Gray Earth," Julia said.
"... Oh. The deal we made."
"The deal _you_ made," Nemoto said. "Over and over again. You said you would take the Hams back to their home world, if they helped you."
"I know what I said."
"Well, now it is payback time."
Emma sighed. She stepped forward and took Julia's great hands; her own fingers, even hardened by weeks of rough living, were pale white streaks compared to Julia's muscular digits. "Julia, I meant what I said. If I could find a way I would get you people home." She waved toward the latest Earth in the sky, a peculiarly shrunken world with a second Moon orbiting close to it. "But you can see the situation for yourself. Your world is gone. It's lost. You see—"
Nemoto said, "Emma, you have made enough mistakes already. It would pay you, pay us both, not to patronize this woman."
Emma said, "I'm sorry." So I am, she thought. But I made a promise I couldn't keep, and I knew it when I made it, and now I just have to get out of this situation as gracefully as I can. That's life. "The point is the Gray Earth _isn't_ coming back. Not in any predictable way." She looked up at Mane. "Is it?"
The great Daemon rubbed her face. "We are studying the world engine. It is ancient and faulty." She grunted. "Like a bad-tempered old hominid, it needs love and attention."
Emma frowned. "But you think you might get it to work again?"
Mane patted Emma's head. "Nemoto frequently accuses me of underestimating you. I am guilty. But you are symmetrically guilty of overestimating me. We cannot repair the world engine. We cannot understand its workings. Perhaps in a thousand years of study... For now we can barely _see_ it."
Nemoto shuddered. "We are all on very low rungs of a very tall ladder."
But Mane said, "There is no ladder. We are all different. Difference is to be cherished."
"And that's what we humans must learn," Emma said.
"You will not learn it," Manekato said cheerfully, "for you will not survive long enough." She sighed, a noise like a steam train in a tunnel. "However, to return to the point, we believe we may be able to direct the wandering of the Red Moon, to a limited extent. Prior to shutting down the world engine altogether."
"Gray Earth come," Julia said again, and her face relaxed from its mock-human smile into the gentle, beatific expression Emma had come to associate with happiness.
Emma held her breath. "And Earth," she said. "My Earth; our Earth. Can you reach that, too?"
"The Daemons can make _one_ directed transition," said Nemoto gravely. "And they are going to use it to take us to the universe of the Gray Earth."
"Because of me?"
"Because of you."
Emma studied Nemoto. "I sense you're pissed at me," she said dryly.
Nemoto glowered. "Emma, _these are not humans_. They don't lie, the Hams and the Daemons. It's all part of the rule set with which they have managed to achieve such longevity as species. A bargain, once struck, is absolutely rigid."
"But what's the big deal? Even if the Daemons manage to bring us back to the Gray Earth universe, they can just send the Hams home. As many as want to go. They can just _Map_ them there."
Nemoto shook her head. "You aren't thinking right. The deal was with us, not the Daemons. We have to get them home. Whichever way we can."
"The lander?"
Nemoto just glared. Then she walked away, muttering, scheming, her whole body tense, her gait rigid, like a machine.
# _PART FIVE_
# **Manifold**
#
## _E mma Stoney_
Hello, Malenfant. I want to tell you I'm all right.
I know that's not what you'd want to hear. The notion that I'm alive, I'm prospering without _you_ , is anathema. Right?
But then you probably aren't listening at all.
You never did listen to me. If you had you wouldn't have screwed up our entire relationship, from beginning to end. You really are an asshole, Malenfant. You were so busy saving the world, saving _me_ , you never thought about yourself. Or me.
But I miss you even so.
I guess you know I'm alone here. Even Nemoto has gone, off to a different fate, in some corner of the manifold or other...
## _M ary_
There were more yesterdays than tomorrows. Her future lay in the black cold ground, where so many had gone before her: Ruth, Joshua, even one of her own children.
And there came a day when they put old Saul in the ground, and Mary found herself the last to remember the old place, the Red Moon where she had been born.
It didn't matter. There was only today.
Nemoto was not so content, of course.
Even in the deepest times of the Long Night, Nemoto would bustle about the cave, agitated, endlessly making her incomprehensible objects. Few watched her come and go. To the younger folk, Nemoto had been here all their lives, not really a person, and so of no significance.
But Mary remembered the Red Moon, and how its lands had run with Skinnies like Nemoto. Mary understood. Nemoto had brought them here, home to the Gray Earth. Now it was Nemoto who was stranded far from her home.
And so Mary made space for Nemoto. She would protect Nemoto when she fell ill, or injured herself. She would even give her meat to eat, softening the deep-frozen meat with her own strong jaws, chewing it as she would to feed a child.
But one day, Nemoto spat out her mouthful of meat on the floor of the cave. She raged and shouted in her jabbering Skinny tongue, and pulled on her furs and gathered her tools, and stamped out of the cave.
She returned staggering and laughing, and she carried a bundle under her arms. It was a bat, dormant, still plump with its winter fat, its leathery wings folded over. Nemoto jabbered about how she would eat well of fresh meat.
Nemoto consumed her bat, giving warm titbits to the children. But when she offered them the bloated, pink-gray internal organs of the bat, mothers pulled the children away.
After that, Nemoto would never be healthy again.
There was a time of twilights, blue-purple shading to pink. And then, at last, the edge of the sun was visible over the horizon: just a splinter of it, but it was the first time it had shown at all for sixty-eight days. There was already a little meltwater to be had. And the first hibernating animals—birds and a few large rats—were beginning to stir, sluggish and vulnerable to hunting in their torpor.
The people capered and threw off their furs.
Nemoto was growing more ill. She suffered severe bouts of diarrhea and vomiting. She lost weight. And her skin grew flaky and sore.
Mary tried to treat the diarrhea. She brought salt water, brine from the ocean diluted by meltwater. But she did not know how to treat the poisoning which was working its way through Nemoto's system.
The days lengthened rapidly. The ice on the lakes and rivers melted, causing splintering crashes all over the landscape, like a long, drawn-out explosion. In this brief temperate interval between deadly cold and unbearable heat, life swarmed. The people gathered the fruit and shoots that seemed to burst out of the ground. They hunted the small animals and birds that emerged from their hibernations.
And soon a distant thunder boomed across the land. It was the sound of hoofed feet, the first of the migrant herds. The men and women gathered their weapons, and headed toward the sea.
It turned out to be a herd of giant antelopes: long-legged, the bucks sporting huge unwieldy antlers. The animals were slim and streamlined, and the muscles of their legs and haunches were huge and taut. And they ran like the wind. Since most of this tipped-up world was, at any given moment, either freezing or baking through its long seasons, migrant animals were forced to travel across thousands of miles, spanning continents in their search for food, water, and temperate climes.
But predators came, too, streamlined hyenas and cats, stalking the vast herds. Those predators included the people, who inhabited a neck of land between two continents, a funnel down which the migrant herds were forced to swarm.
The antelope herd was huge. But it passed so rapidly that it was gone in a couple of days, a great river of flesh that had run its course.
The people ate their meat and sucked rich marrow, and waited for their next provision to come to them, delivered up by the tides of the world.
The air grew hotter yet. Soon the fast-growing grass and herbs were dying back, and the migrant animals and birds had fled, seeking the temperate climes.
The season's last rain fell. Mary closed her eyes and raised her open mouth to the sky, for she knew it would be a long time before she felt rain on her face again.
The ground became a plain of baked and cracked mud.
The people retreated to their cave. Just as its thick rock walls had sheltered them from the most ferocious cold of the winter, so now the walls gave them coolness.
Nemoto's relentless illness drove her to her pallet, where she lay with a strip of skin tied across her eyes.
At length there came a day when the sun failed even to brush the horizon at its lowest point. For sixty-eight days it would not rise or set, but would simply complete endless, meaningless circles in the sky, circles that would gradually grow smaller and more elevated.
The Long Day had begun.
Nemoto said she would not go into the ground until she saw a night again. But Nemoto's skin continued to flake away, as the bat she had woken took its gruesome revenge.
There came a day when the sun rolled along the horizon, its light shimmering through the trees which flourished there.
Mary carried Nemoto to the mouth of the cave—she was light, like a thing of twigs and dried leaves.
Nemoto screwed up her face. "I do not like the light," she said, her voice a husk. "I can bear the dark. But not the light. I long for tomorrow. For tomorrow I will understand a little more. Do you follow me? I have always wanted to _understand_. Why I am here. Why there is something, rather than nothing. Why the sky is silent."
"Lon' for tomorrow," Mary echoed, seeking to comfort her.
"Yes. But _you_ care nothing for tomorrow, or yesterday. Here especially, with your Long Day and your Long Night, as if a whole year is made of a single great day."
Overhead, a single bright star appeared, the first star since the spring.
Nemoto gasped. She was trying to raise her arm, perhaps to point, but could not. "You have a different pole star here. It is somewhere in Leo, near the sky's equator. Your world is tipped over, you see, like Uranus, like a top lying on its side; that is how the impact shaped it. And so for six months, when your pole points at the sun, you have endless light; and for six months endless dark... Do you follow me? No, I am sure you do not."
She coughed, and seemed to sink deeper into the skins. "All my life I have sought to understand. I believe I would have pursued the same course, whichever of our splintered worlds I had been born into. And yet, and yet—" She arched her back. "And yet I die alone."
Mary took her hand. It was as delicate as a bundle of dried twigs. "Not alone."
Nemoto tried to squeeze Mary's hand; it was the gentlest of touches.
And the sun, as if apologetically, slid beneath the horizon. A crimson sunset towered into the sky.
Mary placed her in the ground, the ground of this Gray Earth.
The memory of Nemoto faded, as memories did. But sometimes, sparked by a scent, or the salty breeze that blew off the sea, Mary would think of Nemoto, who had not died alone.
## _E mma Stoney_
Alone.
Yes, Malenfant, I'm alone. I know I have company—various specimens of _Homo superior_ , who you never got to meet, and the Hams, including your Julia, who didn't get to ride back to the Gray Earth. But I'm alone even so. I'm a pet of the Daemons. They are—kindly. So are the Hams. I feel like I'm drowning in chocolate.
I've decided to leave. I'm going upriver, into the heart of the continent. I'm intending to hook up with another band of Runners. I did that before; I can do it again. They range far into the continent's interior, the desert. They know how to find water, how to eat, how to survive out there. If anybody knows a way across the red center it will be the Runners.
I want to see the Bullseye up close, that big volcanic blister. Although maybe it won't be so spectacular. Like you used to say about Olympus Mons on Mars: too big for the human eye to take in, right? Well, those mile-deep rift canyons around its base look like they'd be worth a snapshot.
But I want to go on beyond that.
Maybe I can get past the Bullseye, all the way to the other side of the continent. There is another Beltway over there, Malenfant, another strip of greenery on the western edge of the continent. Nemoto told me you didn't see any dwellings or structures, from Earth or when you orbited the Moon. But maybe there are people there even so, in the western Beltway.
Maybe they are like me. Maybe they are like the Hams, or the Daemons, or maybe another form we haven't dreamed of before. Nobody seems to know. Not the Daemons, not even the Hams.
I can hear your voice. I know what you're saying. I know it's dangerous. Doubly so for a person alone. But I'm going anyhow. I'm tougher than I used to be, Malenfant.
I'll tell you what I'd like to find, in the other Beltway, or someplace else. The place where humans evolved.
We know the Hams were shaped by conditions on the Gray Earth. We think the Daemons are descended from a bunch of Australopithecines that wandered over to the Banded Earth millions of years ago. And so on.
Well, presumably humans came from a group of Runners, similarly isolated. Maybe there were several of Nemoto's "speciations": one to produce some archaic form, a common ancestor of humans and Neandertals—Hams—and then others to produce the Hams, and us. And maybe others. Other cousins.
I think I'd like to find that place. To meet the others.
Nobody knows everything there is to know about this Red Moon. It's a big place. It's full of people.
Full of stories.
## _M anekatopokanemahedo_
Babo shrugged massively, as Manekato groomed him. "It may yet be possible to use the world engine, if only in a limited way..."
"To do what?"
"We can explore the manifold. We can Map to other realities. Other possibilities. You don't have to send a whole Moon to do that."
Mane pondered. "But what is there to look for?"
"In fact there is a valid goal," Babo said carefully.
The Astrologers, he told Manekato, believed that the universe—any given universe—was a fundamentally comprehensible system. If a system was comprehensible, then an entity must exist that could comprehend it. Therefore an entity must exist that could comprehend the entire universe, arbitrarily well—or rather _She_ must exist, as Babo put it.
"The God of the Manifold," Manekato said dryly.
The catch was that there was a manifold of possible universes, of which this was only one. So She may not exist in this universe.
Anyhow, it—She—was to be the ultimate goal of the Daemons' quest.
"Of course," Babo said, "She may actually be an expression of the manifold itself—or perhaps the manifold itself, the greater structure of reality strands, _is itself_ self-referential, in some sense conscious. Or perhaps the manifold is itself merely one thread in a greater tapestry—"
"A manifold of manifolds."
"And perhaps there is a further recursion of structure, no end to the hierarchies of life and mind, which—"
Mane held up her hands. "If we find Her: what will we ask Her?"
Babo picked his nose thoughtfully. "I asked Em-ma that. She said, 'Ask Her if She knows what the hell is going on.' "
Mane touched her brother's head. "Then that is what we will ask. Come, brother; we have much to do."
Hand in hand, the two of them loped toward the forest, seeking shade and food.
## _S hadow_
Shadow found a scrap of meat.
It was on the ground, under a fig leaf, where she had been looking for fruit. It was just a scrap, half-chewed, not much more than a bit of gristle. Shadow scrabbled it up off the floor. Her fingers were stiff now, her vision poor, and she had trouble making her hands do what she wanted them to do.
She sat on the ground and chewed the gristle, sucking away the dust and the tang of somebody else's saliva. The meat was well-chewed. There was barely any flavor; any blood; she couldn't even tell what animal it had come from. But it was tough, and the way it scraped between her teeth made her ache with hunger. She swallowed it only when she had reduced it to a shred of fiber, too ragged to hold or gnaw.
She had not eaten meat for a long, long time: not yesterday, not the days she remembered in vivid, nonchronological, blood-soaked glimpses, not as far back as she could remember.
... She became aware of their scent first. The scent of fur, musk, blood. Then their shadows.
All around her.
They had come on her silently. But they had been coming for her, one way or another, since the day when she had failed to kill the Nutcracker infant, in that blinding flash of light. She tried to run, willing all her strength into legs, which had once been so strong. But her life had been very hard, and she was slow.
Young hands grabbed her legs. She fell facefirst into the dirt.
She twisted, trying to get on her back. But those strong hands kept a grip of her ankle. Her grimace of hatred and defiance turned to a yell of pain, as bones snapped.
They fell on her. Both her legs were held. Somebody sat on her head, and dark stinking fur pushed into her mouth and nose and eyes. She flailed and got one blow on hard flesh. But then her arms were pinned down. She couldn't see who they were.
The blows began to fall. Kicks, stamps, jumps, punches. Bodies hurling themselves onto her. She glimpsed others running around the main group of assailants, landing kicks and blows when the chance came. It was a bedlam, of screams, pain, motion. Still she couldn't make out their faces.
Thumbs pressing into her eyes. Strong hands working at one of her arms, twisting. Bright red pain in her shoulder and elbow, the crunch of ligament and bone.
Termite!... But her mother was long dead, of course.
The pain lessened. With relief, she fell into darkness.
## _E mma Stoney_
You know, I think I always knew we couldn't manage to live together. But I think I always dreamed we would get to die together.
But it's been quite a ride. I wouldn't have missed it for the world, Malenfant. For _all_ the worlds.
Of course there is another possibility. Maybe I should go with the Daemons, off into the manifold. If this really is a manifold of infinite universes, anything is possible. No, strike that—anything that _can_ happen _will_ happen, someplace.
And so there must be one reality where you're waiting for me. There _must_ be. A whole universe, just for us. Kind of romantic, don't you think?
I'm still blown away by what I've learned of the Old Ones.
The Old Ones created infinite possibility—infinite opportunities for life, for mind. What higher mission could there be? And what really overwhelms me is that they may have been _us_. Or at least humans from some variant of our future history. _Us_ : We did this. Think of that.
You'd have loved it, Malenfant. But of course, maybe you already know all about it.
To redesign an infinite ensemble of universes: what terrible responsibility, what arrogance... Maybe they really were us. It sounds just the kind of thing your average _Homo sap_ would do for a dare.
An _H. sap_ like Reid Malenfant.
Is it all your fault? Malenfant, _what did you do_ , out there in the forest of realities?
Time to go. Good-bye, Malenfant, good-bye.
## _M axie_
The people walk across the grass. Maxie's legs are walking. He is following Fire.
The sky is blue. The grass is sparse, yellow. The ground is red under the grass. The people are slim black forms scattered on red-green.
The people call to each other.
"Berry? Sky! Berry!"
"Sky, Sky, here!"
The sun is high. There are only people on the grass. The cats sleep when the sun is high. The hyenas sleep. The Nutcracker men and the Elf men sleep in their trees. Everybody sleeps except the Running-folk. Maxie knows this without thinking.
There is a blue light, low in the sky.
Maxie looks at the blue light. The blue light is new. The blue light is still. It watches him. It is a bat. Or an eye.
Maxie grins. He cares nothing for the blue light.
He walks on, across the hot crimson dust.
|
{
"redpajama_set_name": "RedPajamaBook"
}
| 1,915
|
\section{Introduction}
Research in formal security aims to provide rigorous definitions for different notions of security as well as methods to analyse a given system with regard to the security goals. Restricting the information that may be available to a user of the system (often called an agent) is an important topic in security. Noninterference~\cite{GogMes,goguen84} is a notion that formalizes this. Noninterference uses a security policy that specifies, for each pair of agents, whether information is allowed to flow from one agent to the other. To capture different aspects of information flow, a wide range of definitions of noninterference has been proposed, see, e.g.,~\cite{DBLP:conf/csfw/YoungB94,DBLP:conf/csfw/Millen90,DBLP:conf/esorics/Oheimb04,DBLP:conf/sp/WittboldJ90}.
In this paper, we study systems where in different parts different policies apply. This is motivated by the fact that different security requirements may be desired in different situations, for instance, a user may want to forbid interference between his web browser and an instant messenger program while visiting banking sites but when reading a news page, the user may find interaction between these programs useful.
As an illustrating example, consider the system depicted in Fig.~\ref{fig:admin changes policy}, where three agents are involved: an administrator $A$ and two users $H$ and $L$. The rounded boxes represent system states, the arrows represent transitions. The labels of the states indicate what agent $L$ observes in the respective state; the labels of the arrows denote the action, either action $a$ performed by $A$ or action $h$ performed by $H$, inducing the respective transition.
Every action can be performed in every state; if it does not change the state (i.\,e., if it induces a loop), the corresponding transition is omitted in the picture.
The lower part of the system constitutes a secure subsystem with respect to the bottom policy: when agent $H$ performs the action $h$ in the initial state, the observation of agent $L$ changes from $0$ to $1$, but this is allowed according to the policy, as agent $H$ may interfere with agent $L$---there is an edge from $H$ to $L$.
Similarly, the upper part of the system constitutes a secure subsystem with respect to the top policy: interference between $H$ and $L$ is not allowed---no edge from $H$ to $L$---and, in fact, there is no such interference, because $L$'s observation does not change when $h$ performs an action.
\begin{wrapfigure}[11]{r}{6.2cm}
\scalebox{0.75}{
\begin{tikzpicture}[tikzglobal]
\begin{scope}[transnodedistance]
\node[transsysstate,initial] (q0) {${\tt obs}_L\colon 0$} ;
\node[transsysstate,above=of q0] (q2) {${\tt obs}_L\colon 0$} ;
\node[transsysstate,right=of q0] (q1) {${\tt obs}_L\colon 1$} ;
\node[transsysstate,above=of q1] (q3) {${\tt obs}_L\colon 0$} ;
\end{scope}
\path (q0) edge node {$a$} (q2) ;
\path (q0) edge node {$h$} (q1) ;
\path (q2) edge node {$h$} (q3) ;
\newcommand{\agents}{
\node[agent] (A) [left] {$A$};
\node[agent] (H) [right of=A] {$H$};
\node[agent] (L) [below of=H] {$L$};
}
\node[localpolicy] (q4) [right=of q1] {
\agents
\\
};
\path[policyedge] (H) edge (L);
\node[localpolicy] (q5) [right=of q3] {
\agents
\\
};
\begin{pgfonlayer}{grayboxes}
\node[graybox, fit= (q0) (q1) (q4)] (sub0) {};
\node[graybox, fit= (q2) (q3) (q5)] (sub1) {};
\end{pgfonlayer}
\end{tikzpicture}
}
\vspace*{-6mm}
\label{fig:admin changes policy}
\caption{System with local policies}
\end{wrapfigure}
However, the entire system is clearly insecure: agent $A$ must not interfere with anyone---there is no edge starting from $A$ in either policy---but when $L$ observes ``$1$'' in the lower right state, $L$ can conclude that $A$ did \emph{not} perform the $a$ action depicted.
Note that interference between $H$ and $L$ is allowed, unless $A$ performs action $a$.
But $L$ must not get to know whether $a$ was performed. To achieve this, interference between $H$ and $L$ must never be allowed.
Otherwise, as we have just argued,
$L$ can---by observing $H$'s actions---conclude that in the current part of the system, interference between $H$ and $L$ is still legal and thus
$A$ did not perform $a$.
In other words, in the policy of the lower part,
the edge connecting $H$ and $L$ can never be ``used'' for an actual information flow.
We call such edges \emph{useless}.---Useless edges are a key issue arising in systems with local policies.
\paragraph{Our results.} We develop a theory of noninterference with local policies which takes the aforementioned issues into account.
Our contributions are as follows:
\begin{enumerate}
\item We provide new and natural definitions for noninterference with local policies, both for the transitive~\cite{GogMes,goguen84} (agent $L$ may only be influenced by agent $H$ if there is an edge from $H$ to $L$ in the policy) and for the intransitive setting~\cite{HY87} (interference between $H$ and $L$ via ``intermediate steps'' is also allowed).
\item We show that policies can always be rewritten into a normal form which does not contain any ``useless'' edges (see above).
\item We provide characterizations of our definitions based on unwindings, which demonstrate the robustness of our definitions and from which we derive efficient verification algorithms.
\item We provide results on the complexity of verifying noninterference. In the transitive setting, noninterference can be verified in nondeterministic logarithmic space (\complexityclassname{NL}). In the intransitive setting, the problem is \complexityclassname{NP}-complete, but fixed-parameter tractable with respect to the number of agents.
\end{enumerate}
Our results show significant differences between the transitive and the intransitive setting. In the transitive setting, one can, without loss of generality, always assume a policy is what we call uniform, which means that each agent may ``know'' (in a precise epistemic sense) the set of agents that currently may interfere with him. Assuming uniformity greatly simplifies the study of noninterference with local policies in the transitive setting. Moreover, transitive noninterference with local policies can be characterized by a simple unwinding, which yields very efficient algorithms.
In the intransitive setting, the situation is more complicated. Policies cannot be assumed to be uniform, verification is \complexityclassname{NP}-complete, and, consequently, we only give an unwinding condition that requires computing exponentially many relations. However, for \emph{uniform} policies, the situation is very similar to the transitive setting: we obtain simple unwindings and efficient algorithms.
As a consequence of our results for uniform policies, we obtain an unwinding characterization of IP-security~\cite{HY87} (which uses a single policy for the entire system). Prior to our results, only an unwinding characterization that was \emph{sound}, but not \emph{complete} for IP-security was known~\cite{rushby92}. Our new unwinding characterization immediately implies that IP-security can be verified in nondeterministic logarithmic space, which improves the polynomial-time result obtained in~\cite{emsw11}.
\textit{Related Work}. Our intransitive security definitions generalize IP-security~\cite{HY87} mentioned above. The issues raised against IP-security in~\cite{meyden2007} are orthogonal to the issues arising from local policies. We therefore study local policies in the framework of IP-security, which is technically simpler than, e.g., TA-security as defined in~\cite{meyden2007}.
Several extensions of intransitive noninterference have been discussed, for instance, in~\cite{RoscoeG99,DBLP:journals/jcs/MyersSZ06}. In~\cite{Leslie-DYNAMICNONINTERFERENCE-SSE-2006}, a definition of intransitive noninterference with local policies is given, however, the definition in~\cite{Leslie-DYNAMICNONINTERFERENCE-SSE-2006} does not take into account the aforementioned effects, and that work does not provide complete unwinding characterizations nor complexity results.
\section{State-based Systems with Local Policies}
\label{sect:preliminaries}
We work with the standard state-observed system model, that is, a system is a deterministic finite-state automaton where each action belongs to a dedicated agent and each agent has an observation in each state. More formally, a \emph{system} is a tuple $M=(S,s_0,\ensuremath{A},\mathtt{step},\mathtt{obs},\mathtt{dom})$, where $S$ is a finite set of \emph{states}, $s_0\in S$ is the \emph{initial state}, $A$ is a finite set of \emph{actions}, $\mathtt{step}\colon S\times\ensuremath{A}\rightarrow S$ is a \emph{transition function}, $\mathtt{obs}\colon S\times D\rightarrow O$ is an \emph{observation function}, where $O$ is an arbitrary set of observations, and $\mathtt{dom}\colon\ensuremath{A}\rightarrow D$ associates with each action an agent, where $D$ is an arbitrary finite set of agents (or security domains).
For a state $s$ and an agent $u$, we write ${\tt obs}_u(s)$ instead of ${\tt obs}(s,u)$. For a sequence $\alpha\in\ensuremath{A}^*$ of actions and a state $s\in S$, we denote by $s\cdot\alpha$ the state obtained when performing $\alpha$ starting in $s$, i.e., $s\cdot\epsilon=s$ and $s\cdot\alpha a=\mathtt{step}(s\cdot\alpha,a)$.
A \emph{local policy} is a reflexive relation
${\dintrel{}}\subseteq D\times D$.
To keep our notation simple, we do not define subsystems nor policies for subsystems explicitly.
Instead, we assign a local policy to every state and denote the policy in state $s$ by $\dintrel{s}$.
We call the collection of all local policies $\ensuremath{(\dintrel s)_{s\in S}}\xspace$ the \emph{policy} of the system.
If $(u,v) \in {\dintrel{s}}$ for some $u, v \in D$, $s \in \States$, we
say $u \dintrel{s} v$ is an \emph{edge} in $\ensuremath{(\dintrel s)_{s\in S}}\xspace$.
A system has a \emph{global policy} if all local policies $\dintrel{s}$ are the
same in all states, i.e., if $u\dintrel sv$ does not depend on $s$. In this case, we denote the single policy by $\rightarrowtail$ and only write $u \rightarrowtail v$.
We define the set $\infagents us$ as the set of agents that \emph{may interfere} with $u$ in $s$, i.e., the set $\set{v\ \vert\ v\dintrel su}$.
In the following, we fix an arbitrary system $M$ and a policy $\ensuremath{(\dintrel s)_{s\in S}}\xspace$.
In our examples, we often identify a state with an action sequence leading to it from the initial state $s_0$, that is, we write $\alpha$ for $s_0 \cdot \alpha$, which is well-defined, because we consider deterministic systems. For example, in the system from Fig.~\ref{fig:admin changes policy}, we denote the initial state by $\epsilon$ and the upper right state by $ah$. In each state, we write the local policy in that state as a graph. In the system from Fig.~\ref{fig:admin changes policy}, we have $H\dintrel{\epsilon}L$, but $H\not\dintrel{a}L$. In general, we only specify the agents' observations as far as relevant for the example, which usually is only the observation of the agent $L$. We adapt the notation from Fig.~\ref{fig:admin changes policy} to our definition of local policies, which assigns a local policy to every state: we depict the graph of the local policy inside the rounded box for the state, see Fig.~\ref{fig:dpsecure_system}.
\section{The Transitive Setting}
\label{sect:transitive}
In this section, we define noninterference for systems with local policies in the transitive setting, give several characterizations, introduce the notion of useless edge, and discuss it.
The basic idea of our security definition is that an occurrence of an action which, according to a local policy, should not be observable by an agent $u$ must not have any influence on $u$'s future observations.
\begin{definition}[t-security\xspace]
\label{def:local-dpurge-secure}
The system $M$ is t-secure\xspace iff for all $u \in D$, $s \in \States$, $a \in A$ and $\alpha \in A^*$ the following implication holds:
\begin{equation*}
\text{If } \dom(a) \not \dintrel{s} u, \text{ then }
{\tt obs}_u(s \cdot \alpha) = {\tt obs}_u(s \cdot a \alpha) \enspace.
\end{equation*}
\end{definition}
\setlength\intextsep{0pt}
\begin{wrapfigure}[9]{r}{0pt}
\scalebox{0.75}{
\begin{tikzpicture}[tikzglobal]
\newcommand{\agents}{
\node[agent] (A) [left] {$A$};
\node[agent] (B) [right of=A] {$B$};
\node[agent] (L) [below of=A] {$L$};
}
\node[initial,systemstate] (q0) {
\agents
\\
\hline
${\tt obs}_L \colon 0$ \\
};
\path (A) edge node {} (L)
(B) edge node {} (L);
\node[systemstate] (q1) [right=of q0] {
\agents
\\
\hline
${\tt obs}_L \colon 1$ \\
};
\path (B) edge node {} (L);
\node[systemstate] (q2) [below=of q1] {
\agents
\\
\hline
${\tt obs}_L \colon 2$ \\
};
\path (A) edge node {} (L);
\path (q0) edge node {$b$} (q1)
edge node {$a$} (q2)
(q1) edge [bend left=10] node {$b$} (q2)
(q2) edge [bend left=10] node {$a$} (q1);
\end{tikzpicture}
}
\caption{A t-secure\xspace system}
\label{fig:dpsecure_system}
\end{wrapfigure}
Fig.~\ref{fig:dpsecure_system} shows a t-secure\xspace system. In contrast, the system in Fig.~\ref{fig:admin changes policy} is not t-secure\xspace, since $A \not \dintrel{\epsilon} L$, but ${\tt obs}_L(ah) \neq {\tt obs}_L(h)$.
\subsection{Characterizations of t-Security\xspace}
In Theorem~\ref{thm:dp_characterizations}, we give two characterizations of t-security\xspace, underlining that our definition is quite robust. The first characterization is based on an operator which removes all actions that must not be observed. It is essentially the definition from Goguen and Meseguer~\cite{GogMes,goguen84} of the ${\tt purge}$ operator generalized to systems with local policies.
\begin{definition}[purge for local policies]
\label{sec:dynamic-transitive-purge}
For all $u \in D$ and $s \in \States$ let $ {\tt purge}(\epsilon, u,
s) = \epsilon$ and for all $a \in A$ and $\alpha \in
A^*$ let
\begin{align*}
{\tt purge}(a \alpha, u, s) & =
\begin{cases}
a \ {\tt purge}(\alpha, u, s \cdot a) &\text{if } \dom(a) \dintrel{s} u
\\
{\tt purge}(\alpha, u, s) & \text{otherwise} \enspace.
\end{cases}
\end{align*}
\end{definition}
The other characterization is in terms of unwindings, which we define for local policies in the following, generalizing the definition of Haigh and Young~\cite{HY87}.
\begin{definition}[transitive unwinding with local policies]
A \emph{transitive unwinding} for $M$ with a policy $\ensuremath{(\dintrel s)_{s\in S}}\xspace$ is a family of equivalence relations $(\sim_u)_{u\inD}$ such that for every agent $u \in D$, all states $s, t \in \States$ and
all $a \in A$, the following holds:
\begin{itemize}
\item If $\dom(a) \not\dintrel{s} u$, then $s \sim_u s\cdot a$. \hfill \textnormal{ \textnormal{(LR$_\dpu$)}\xspace---local respect}
\item If $s \sim_u t$, then $s\cdot a \sim_u t \cdot a$. \hfill \textnormal{\textnormal{(SC$_\dpu$)}\xspace---step consistency}
\item If $s \sim_u t$, then ${\tt obs}_u(s) = {\tt obs}_u(t)$. \hfill \textnormal{\textnormal{(OC$_\dpu$)}\xspace---output consistency}
\end{itemize}
\end{definition}
Our characterizations of t-security\xspace are spelled out in the following theorem.
\begin{theorem}[characterizations of t-security\xspace]
\label{thm:dp_characterizations}
The following are equivalent:
\begin{enumerate}
\item The system $M$ is t-secure\xspace.
\label{thm:dp_characterization_def}
\item For all $u \in D$, $s \in \States$, and $\alpha, \beta \in A^*$ with ${\tt purge}(\alpha, u, s) = {\tt purge}(\beta, u, s)$, we have ${\tt obs}_u(s \cdot \alpha) = {\tt obs}_u(s \cdot \beta)$.
\label{thm:dp_characterization_purge}
\item There exists a transitive unwinding for $M$ with the policy $\ensuremath{(\dintrel s)_{s\in S}}\xspace$.
\label{thm:dp_characterization_unwind}
\end{enumerate}
\end{theorem}
Unwinding relations yield efficient verification procedure. For verifying t-security, it is sufficient to compute for every $u \in D$ the smallest equivalence relation satisfying \textnormal{(LR$_\dpu$)}\xspace and \textnormal{(SC$_\dpu$)}\xspace and check that the function ${\tt obs}_u$ is constant on every equivalence class. This can be done with nearly the same algorithm as is used for global policies, described in~\cite{emsw11}. The above theorem directly implies that t-security\xspace can be verified in nondeterministic logarithmic space.
\subsection{Useless Edges}
An ``allowed'' interference $v\dintrel su$ may contradict a ``forbidden'' interference $v\not\dintrel{s'}u$ in a state $s'$ that should be indistinguishable to $s$ for $u$.
In this case, the edge $v\dintrel su$ is useless. What this means is that an edge $v\dintrel su$ in the policy may be deceiving and should not be interpreted as ``it is allowed that $v$ interferes with $u$'', rather, it should be interpreted as ``it is not explicitly forbidden that $v$ interferes with $u$''. To formalize this, we introduce the following notion:
\begin{definition}[t-similarity\xspace]
States $s$, $s'$ are \emph{t-similar\xspace} for an agent $u \in D$, denoted $s \approx_u s'$, if there exist $t \in \States$, $a \in A$, and $\alpha \in A^*$ such that $\dom(a) \not\dintrel{t} u$, $s = t \cdot a \alpha$, and $s' = t\cdot \alpha$.
\end{definition}
Observe that t-similarity\xspace is identical with the smallest equivalence relation satisfying \textnormal{(LR$_\dpu$)}\xspace and \textnormal{(SC$_\dpu$)}\xspace. Also observe that the system $M$ is t-secure\xspace if and only if for every agent $u$, if $s\approx_u s'$, then ${\tt obs}_u(s)={\tt obs}_u(s')$.
The notion of t-similarity\xspace allows us to formalize the notion of a useless edge:
\begin{definition}[useless edge]
An edge $v\dintrel su$ is \emph{useless} if there is a state~$s'$ with $s\approx_u s'$ and $v\not\dintrel{s'}u$.
\end{definition}
For example, consider again the system in Fig.~\ref{fig:admin changes policy}. Here, the local policy in the initial state allows information flow from $H$ to $L$. However, if $L$ is allowed to observe $H$'s action in the initial state, then $L$ would know that the system is in the initial state, and would also know that $A$ has not performed an action. This is an information flow from $A$ to $L$, which is prohibited by the policy.
Useless edges can be removed without any harm:
\begin{theorem}[removal of useless edges]
\label{theorem:uniformpolicies}
Let $\ensuremath{(\dintrel{s}')_{s\in S}}\xspace$ be defined by
\begin{align*}
{\dintrel s'} = {\dintrel s} \setminus \{v \dintrel s u \mid v
\dintrel s u \text{ is useless}\} \qquad \text{for all $s \in S$.}
\end{align*}
Then $M$ is t-secure\xspace w.\,r.\,t.\xspace $\ensuremath{(\dintrel s)_{s\in S}}\xspace$ iff $M$ is t-secure\xspace w.\,r.\,t.\xspace $\ensuremath{(\dintrel{s}')_{s\in S}}\xspace$.
\end{theorem}
The policy $\ensuremath{(\dintrel{s}')_{s\in S}}\xspace$ in Theorem~\ref{theorem:uniformpolicies} has no useless edges, hence every edge in one of its local policies represents an \emph{allowed} information flow---no edge contradicts an edge in another local policy. Another interpretation is that any information flow that is \emph{forbidden} is \emph{directly} forbidden via the absence of the corresponding edge. In that sense, the policy is closed under logical deduction.
We call a policy $\ensuremath{(\dintrel s)_{s\in S}}\xspace$ \emph{uniform} if $\infagents u{s}=\infagents u{s'}$ holds for all states $s$ and $s'$ with $s\approx_u s'$. In other words, in states that $u$ should not be able to distinguish, the exact same set of agents may interfere with $u$. Hence $u$ may ``know'' the set of agents that currently may interfere with him. Note that a policy is uniform if and only if it does not contain useless edges. (This is not true in the intransitive setting, hence the seemingly complicated definition of uniformity.) Uniform policies have several interesting properties, for example, with a uniform policy the function ${\tt purge}$ behaves very similarly to the setting with a global policy: it suffices to verify action sequences that start in the initial state of the system and ${\tt purge}$ satisfies a natural associativity condition on a uniform policy.
\section{The Intransitive Setting}
\label{sect:intransitive case}
In this section, we consider the intransitive setting, where,
whenever an agent performs an action, this event may transmit information about
the actions the agent has performed himself as well as information about
actions by other agents that was previously transmitted to him. The
definition follows a similar pattern as that of t-security\xspace:
if performing an action sequence $a\alpha$ starting in a state $s$ should
not transmit the action $a$ (possibly via several intermediate steps)
to the agent $u$, then $u$ should be unable to deduce from his
observations whether $a$ was performed.
To formalize this,
we use Leslie's extension~\cite{Leslie-DYNAMICNONINTERFERENCE-SSE-2006} of Rushby's definition~\cite{rushby92} of ${\texttt{sources}}$.
\begin{definition}[sources]
For an agent $u$ let
$\dsrc{\epsilon}us =\set{u}$
and for
$a \in A$, $\alpha \in A^*$, if
$\dom(a) \dintrel{s} v$ for some $v \in \dsrc{\alpha}{u}{s\cdot
a}$, then let
$\dsrc{a\alpha}us = \dsrc{\alpha}{u}{s\cdot a} \cup \set{dom(a)}$, and else let $\dsrc{a\alpha}us = \dsrc{\alpha}{u}{s\cdot a}$.
\end{definition}
The set $\dsrc{a\alpha}us$ contains the agents that ``may know'' whether the action~$a$ has been performed in state $s$ after the run $a\alpha$ is performed: initially, this is only the set of agents $v$ with $\dom(a)\dintrel sv$. The knowledge may be spread by every action performed by an agent ``in the know:'' if an action $b$ is performed in a later state $t$, and $\dom(b)$ already may know that the action $a$ was performed, then all agents $v$ with $\dom(b)\dintrel{t}v$ may obtain this information when $b$ is performed.
Following the discussion above, we obtain a natural definition of security:
\begin{definition}[i-security\xspace]
The system $M$ is i-secure\xspace iff for all $s \in \States$, $a\inA$, and $\alpha\inA^*$, the following implication holds.
\begin{align*}
\text{If $\dom(a)\notin\dsrc{a\alpha}{u}{s}$, then ${\tt obs}_u(s\cdot
a\alpha)={\tt obs}_u(s\cdot\alpha)$.}
\end{align*}
\end{definition}
The definition formalizes the above: if, on the path $a\alpha$, the action $a$ is not transmitted to $u$,
then $u$'s observation must not depend on whether $a$ was performed; the runs $a\alpha$ and $\alpha$ must be indistinguishable for $u$.
Consider the example in
Fig.~\ref{fig:admin changes policy}.
The system remains insecure in the intransitive setting:
as $A$ must not interfere with any agent in any state,
we have $\dom(a)\notin\dsrc{ah}L{\epsilon}$, where again, according to our convention, $\epsilon$ denotes the initial state.
So, the system is insecure, since ${\tt obs}_L(ah)\neq{\tt obs}_L(h)$.
\subsection{Characterizations and Complexity of i-Security\xspace}\label{sect:intransitive unwinding exponential}\label{sect:dipsecty characterizations}
We now establish two characterizations of intransitive noninterference with local policies and study the complexity of verifying i-security\xspace. Our characterizations are analogous to the ones obtained for the transitive setting in Theorem~\ref{thm:dp_characterizations}. The first one is based on a purge function, the second one uses an unwinding condition. This demonstrates the robustness of our definition and strengthens our belief that i-security\xspace is indeed a natural notion.
We first extend Rushby's definition of ${\tt ipurge}$ to systems with local policies.
\begin{definition}[intransitive purge for local policies]
For all $u \in D$ and all $s \in \States$, let $\dipurge{\epsilon}us=\epsilon$
and, for all
$a \in A$ and $\alpha \in A^*$, let
\begin{align*}
\dipurge{a\alpha}{u}{s} & =
\begin{cases}
a \ \dipurge{\alpha}u{s\cdot a} & \mathtext{ if }\dom(a)\in\dsrc{a\alpha}{u}{s}, \\
\dipurge{\alpha}u{s} & \mathtext{ otherwise}.
\end{cases}
\end{align*}
\end{definition}
The crucial point is that in the case where $a$ must remain hidden from
agent~$u$, we define $\dipurge{a\alpha}us$ as $\dipurge\alpha us$ instead
of the possibly more intuitive choice $\dipurge\alpha u{s\cdot a}$, on which
the security definition in~\cite{Leslie-DYNAMICNONINTERFERENCE-SSE-2006} is
based.
We briefly explain the reasoning behind this choice. To this end, let $\ensuremath{{\tt ipurge}}'$ denote the alternative definition of \ensuremath{{\tt ipurge}}\xspace outlined above. Consider the sequence $ah$, performed from the initial state in the system in Fig.~\ref{fig:admin changes policy}. Clearly, the action~$a$ is purged from the trace, thus the result of $\ensuremath{{\tt ipurge}}'$ is the same as applying $\ensuremath{{\tt ipurge}}'$ to the sequence $h$ starting in the upper left state. However, in this state, the action $h$ is invisible for $L$, hence $\ensuremath{{\tt ipurge}}'$ removes it, and thus purging $ah$ results in the empty sequence. On the other hand, if we consider the sequence $h$ also starting in the initial state, then $h$ is not removed by $\ensuremath{{\tt ipurge}}'$, since $H$ may interfere with $L$. Hence $ah$ and $h$ do not lead to the same purged trace---a security definition based on $\ensuremath{{\tt ipurge}}'$ does not require $ah$ and $h$ to lead to states with the same observation. Therefore, the system is considered secure in the $\ensuremath{{\tt ipurge}}'$-based security definition from~\cite{Leslie-DYNAMICNONINTERFERENCE-SSE-2006}. However, a natural definition must require $ah$ and $h$ to lead to the same observation for agent $L$, as the action $a$ must always be hidden from $L$.
We next define unwindings for i-security\xspace and then give a characterization of i-security\xspace based on them.
\begin{definition}[intransitive unwinding]
An \emph{intransitive unwinding} for the system $M$ with a policy $\ensuremath{(\dintrel s)_{s\in S}}\xspace$ is a family of relations $(\direl{D'})_{D'\subseteq D}$ such that $\direl{D'}\subseteq \States \times \States$ and for all $D'\subseteq D$, all $s, t \in \States$ and all $a \in A$, the following hold:
\begin{itemize}
\item $s\direl{\set{u\in D\ \vert\ \dom(a)\not\dintrel{s} u}} s\cdot a$.
\hfill \textnormal{(LR$_\dip$)}\xspace
\item If $s\direl{D''} t$, then $s\cdot
b\direl{D''} t\cdot b$, where $D''= D'$ if
$\dom(b)\inD'$,
\\
and
else $D''= D'\cap\set{u\ \vert\ \dom(b)\not\dintrel{s} u}$.
\hfill \textnormal{(SC$_\dip$)}\xspace
\item If $s\direl{D'} t$ and $u\inD'$, then ${\tt obs}_u(s)={\tt obs}_u(t)$,
\hfill \textnormal{(OC$_\dip$)}\xspace
\end{itemize}
\end{definition}
Intuitively, $s\direl{D'} t$ expresses that there is a common reason for
all agents in~$D'$ to have the same observations in $s$ as in $t$,
i.e., if there is a state $\tilde s$, an action~$a$ and a sequence
$\alpha$ such that $s= \tilde s\cdot a\alpha$, $t=\tilde s
\cdot\alpha$, and $\dom(a)\notin\dsrc{a\alpha}u{\tilde s}$ for
\emph{all} agents $u\inD'$.
\begin{theorem}[characterization of i-security\xspace]
\label{theorem:intransitive unwinding and ipurge characterization}
The following are equivalent:
\begin{enumerate}
\item The system $M$ is i-secure\xspace.
\item For all agents $u$, all states $s$, and all action sequences
$\alpha$ and $\beta$ with
\\
$\dipurge\alpha us=\dipurge\beta us$, we
have ${\tt obs}_u(s\cdot\alpha)={\tt obs}_u(s\cdot\beta)$.
\item There exists an intransitive unwinding for $M$ and $\ensuremath{(\dintrel s)_{s\in S}}\xspace$.
\end{enumerate}
\end{theorem}
In contrast to the transitive setting, the unwinding characterization of
i-security\xspace does not lead to a polynomial-time algorithm to verify
security of a system, because the number of relations needed to consider is exponential in the number of agents in the system. Unless $\complexityclassname{P}=\complexityclassname{NP}$, we cannot do significantly better, because the verification problem is \complexityclassname{NP}-complete; our unwinding characterization, however, yields an FPT-algorithm.
\begin{theorem}[complexity of i-security\xspace]
\label{theorem:intransitive case np complete}
Deciding whether a given system is i-secure\xspace with respect to a policy is \complexityclassname{NP}-complete and fixed-parameter tractable with the number of agents as parameter.
\end{theorem}
\subsection{Intransitively Useless Edges}
\begin{wrapfigure}[11]{r}{0pt}
\scalebox{0.75}{
\begin{tikzpicture}[tikzglobal]
\newcommand{\agentshdl}{
\node[agent] (h) [left] {$H$};
\node[agent] (d) [right=2.5mm of h] {$D$};
\node[agent] (l) [below=2.5mm of d] {$L$};
}
\newcommand{\agentshd}{
\node[agent] (h) {$H$};
\node[agent] (d) [below=2.5mm of h] {$D$};
}
\newcommand{\agentsdl}{
\node[agent] (d) {$D$};
\node[agent] (l) [below=2.5mm of d] {$L$};
}
\node[initial,systemstate] (q0) {
\agentshd
\\
\hline
${\tt obs}_L \colon 0$ \\
};
\path[policy] (h) edge (d);
\node[systemstate] (q1) [right= 10mm of q0] {
\agentshdl
\\ \hline
${\tt obs}_L \colon 0$ \\
};
\path[policy] (h) edge (l);
\path[policy] (d) edge (l);
\node[systemstate] (q3) [right=10mm of q1] {
${\tt obs}_L \colon 1$ \\
};
\node[systemstate] (q2) [below=5mm of q1] {
\agentsdl \\
\hline
${\tt obs}_L \colon 0$ \\
};
\path[policy] (d) edge (l);
\node[systemstate] (q4) [right=10mm of q2] {
${\tt obs}_L \colon 2$ \\
};
\node[systemstate] (q5) [above=of q3] {
${\tt obs}_L \colon 0$ \\ };
\node[systemstate] (q6) [left=20mm of q5] {
${\tt obs}_L \colon 0$ \\ };
\path (q0) edge node {$h_1$} (q1);
\path (q0) edge [bend right] node {$h_2$} (q2);
\path (q1) edge node {$d$} (q3);
\path (q2) edge node {$d$} (q4);
\path (q1) edge node {$h_1$} (q5);
\path (q1) edge node {$h_2$} (q6);
\end{tikzpicture}}
\caption{Intransitively useless edge}\label{fig:redundant edge}
\end{wrapfigure}
In our discussion of t-security\xspace we observed that local policies may contain edges that can never be used. This issue also occurs in the intransitive setting, but the situation is more involved. In the transitive setting, it is sufficient to ``remove any incoming edge for $u$ that $u$ must not know about'' (see Theorem~\ref{theorem:uniformpolicies}). In the intransitive setting it is not: when the system in Fig.~\ref{fig:redundant edge} is in state $h_1$, then agent $L$ must not know that the edge $D\dintrel{} L$ is present, since states $\epsilon$ and $h_1$ should be indistinguishable for $L$, but clearly, the edge cannot be removed without affecting security. However, useless edges still exist in the intransitive setting, even in the system from Figure~\ref{fig:redundant edge}, as we will show below.
To formally define useless edges, we adapt t-similarity\xspace to the intransitive setting in the natural way.
\begin{definition}[i-similarity\xspace]
For an agent $u$, let $\approx^i_u$ be the smallest equivalence
relation on the states of $M$ such that for all $s \in \States$, $a
\in A$, $\alpha\in A^*$, if $\dom(a)\notin\dsrc{a\alpha}us$, then $s\cdot a\alpha\approx^i_us\cdot\alpha$. We call states $s$ and $s'$ with $s\approx^i_u s'$ \emph{i-similar\xspace for $u$}.
\end{definition}
Using this, we can now define intransitively useless edges:
\begin{definition}[intransitively useless edge]
Let $e$ be an edge in a local policy of \ensuremath{(\dintrel s)_{s\in S}}\xspace and let $({\hat \rightarrowtail}_s)_{s\in S}$ be the policy obtained from \ensuremath{(\dintrel s)_{s\in S}}\xspace by removing~$e$. Let $\approx^i_u$ and ${\hat \approx}^i_u$ be the respective i-similarity relations. Then $e$ is \emph{intransitively useless} if $s\approx^i_u s'$ if and only if $s {\hat \approx}^i_u s'$ for all states $s$ and $s'$ and all agents $u$.
\end{definition}
An edge is intransitively useless if removing it does not forbid any information flow that was previously allowed. In particular, such an edge itself cannot be used directly.
Whether an edge is useless does not depend on the observation function of the system, but only on the policy and the transition function, whereas a definition of security compares observations in different states.
If the policy does not contain any intransitively useless edges, then there is no edge in any of its local policies that is contradicted by other aspects of the policy. In other words, the set of information flows \emph{forbidden} by such a policy is closed under logical deduction---every edge that can be shown to represent a forbidden information flow is absent in the policy.
Fig.~\ref{fig:redundant edge} shows a secure system with an intransitively useless edge. The system is secure (agent $L$ knows whether in the initial state, $h_1$ or $h_2$ was performed, as soon as this information is transmitted by agent $D$). The edge $H\dintrel{h_1}L$ is intransitively useless, as explained in what follows.
The edge allows $L$ to distinguish between the states $h_1, h_1h_1, h_1h_2$. However, one can verify that $h_2h_1\approx^i_L h_1$, $h_2h_1h_1\approx^i_L h_2h_1$, $h_2h_1h_1\approx^i_L h_1h_1$, $h_2h_1h_2\approx^i_L h_2h_1$, and $h_2h_1h_2\approx^i_L h_1h_2$ all hold.
Symmetry and transitivity of $\approx^i_L$ imply that all the three states $h_1,h_1h_1,h_1h_2$ are $\approx^i_L$-equivalent. Hence the edge $H\dintrel{h_1}L$ is indeed intransitively useless (and the system would be insecure if $h_1$, $h_1h_1$, and $h_1h_2$ would not have the same observations).
Intransitively useless edges can be removed without affecting security:
\begin{theorem}[removal of intransitively useless edges]
\label{theorem:redundant edges}
Let $\ensuremath{(\dintrel{s}')_{s\in S}}\xspace$ be obtained from $\ensuremath{(\dintrel s)_{s\in S}}\xspace$ by removing a set of edges which are intransitively useless. Then $M$ is
i-secure\xspace
with respect to $\ensuremath{(\dintrel s)_{s\in S}}\xspace$ if and only if $M$ is
i-secure\xspace
with respect to $\ensuremath{(\dintrel{s}')_{s\in S}}\xspace$.
\end{theorem}
This theorem implies that for every policy $\ensuremath{(\dintrel s)_{s\in S}}\xspace$, a policy $\ensuremath{(\dintrel{s}')_{s\in S}}\xspace$ without intransitively useless edges that is equivalent to $\ensuremath{(\dintrel s)_{s\in S}}\xspace$ can be obtained from $\ensuremath{(\dintrel s)_{s\in S}}\xspace$ by removing all intransitively useless edges.
\subsection{Sound Unwindings and Uniform Intransitive Policies}
The exponential size unwinding of i-security\xspace given in Section~\ref{sect:intransitive unwinding exponential} does not yield a polynomial-time algorithm for security
verification. Since the problem
is \complexityclassname{NP}-complete, such an algorithm---and hence an unwinding that is both small and easy to compute---does not exist, unless $\complexityclassname{P}=\complexityclassname{NP}$. In this section, we define unwinding conditions that lead to a polynomial-size unwinding and are \emph{sound} for
i-security\xspace, and are \emph{sound and complete} for
i-secure\xspace in the case of uniform policies. Uniform policies are (as in the transitive case) policies in which every agent ``may know'' the set of agents who may currently interfere with him, that is, if an agent $u$ must not distinguish two states by the security definition, then the set of agents that may interfere with $u$ must be identical in these two states.
Formally, we define this property as follows.
\begin{definition}[intransitive uniform]
A policy $\ensuremath{(\dintrel s)_{s\in S}}\xspace$ is \emph{intransitively uniform}, if for all agents $u$ and states $s$, $s'$ with $s\approx^i_u s'$, we have that $\infagents{u}{s}=\infagents{u}{s'}$.
\end{definition}
Note that this definition is very similar to the uniformity condition for the transitive setting, but while in the transitive setting, uniform policies and policies without useless edges coincide, this is not true for intransitive noninterference (in fact, neither implication holds).
Uniformity, on an abstract level, is a natural requirement and often met in concrete systems, since an agent usually knows the sources of information available to him. In the uniform setting, many of the subtle issues with local policies do not occur anymore; as an example, i-security\xspace\ and the security definition from~\cite{Leslie-DYNAMICNONINTERFERENCE-SSE-2006} coincide for uniform policies. Uniformity also has nice algorithmic properties, as both, checking whether a system has a uniform policy and checking whether a system with a uniform policy satisfies i-security\xspace, can be performed in polynomial time. This follows from the characterizations of i-security in terms of the unwindings we define next.
\begin{definition}[uniform intransitive unwinding]
A \emph{uniform intransitive unwinding} for $M$ with a policy \ensuremath{(\dintrel s)_{s\in S}}\xspace\ is a family of equivalence relations $\sim^{\tilde s,v}_u$ for each
choice of states $\tilde s$ and agents $v$ and $u$,
such that for all $s, t \in \States$, and all $a \in A$, the following holds:
\begin{itemize}
\item If $s\sim^{\tilde s,v}_ut$, then ${\tt obs}_u(s)={\tt obs}_u(t)$.
\hfill \textnormal{(OC$_{\dip}^{\uniform}$)}\xspace
\item If $s\sim^{\tilde s,v}_ut$, then $\infagents{u}{s}=\infagents{u}t$.
\hfill \textnormal{(PC$_\dip^\uniform$)}\xspace
\item If $s\sim^{\tilde s,v}_ut$ and $a\in
A$ with $v\not\dintrel{\tilde s}\dom(a)$, then $s\cdot
a\sim^{\tilde s,v}_ut\cdot a$.
\hfill \textnormal{(SC$_\dip^\uniform$)}\xspace
\item If $\dom(a)\not\dintrel{\tilde s}u$,
then $\tilde s\sim^{\tilde s,\dom(a)}_u \tilde s\cdot a$.
\hfill \textnormal{(LR$_\dip^\uniform$)}\xspace
\end{itemize}
\end{definition}
In the following theorem intransitive uniformity and i-security\xspace (for uniform policies) are characterized by almost exactly the same unwinding. The only difference is that for uniformity we require policy consistency \textnormal{(PC$_\dip^\uniform$)}\xspace, since we are concerned with having the same \emph{local policies} in certain states, while for security, we require \textnormal{(OC$_{\dip}^{\uniform}$)}\xspace, since we are interested in \emph{observations}.
\pagebreak
\begin{theorem}[uniform unwinding characterizations]
\label{theorem:polynomial unwinding characterization of intransitive uniformity and security}
\begin{enumerate}
\item The policy $\ensuremath{(\dintrel s)_{s\in S}}\xspace$ is intransitively uniform if and only if there is a uniform intransitive unwinding for $M$ and $\ensuremath{(\dintrel s)_{s\in S}}\xspace$ that satisfies \textnormal{(PC$_\dip^\uniform$)}\xspace, \textnormal{(SC$_\dip^\uniform$)}\xspace, and \textnormal{(LR$_\dip^\uniform$)}\xspace.
\item If $\ensuremath{(\dintrel s)_{s\in S}}\xspace$ is intransitively uniform, then $M$ is i-secure\xspace if and only if there is a uniform intransitive unwinding that satisfies \textnormal{(OC$_{\dip}^{\uniform}$)}\xspace, \textnormal{(SC$_\dip^\uniform$)}\xspace and \textnormal{(LR$_\dip^\uniform$)}\xspace.
\end{enumerate}
\end{theorem}
In particular, if an unwinding satisfying all four conditions exists, then a system is secure.
Due to Theorem~\ref{theorem:intransitive case np complete}, we cannot hope that the above unwindings completely characterize i-security\xspace, and indeed the system in Fig.~\ref{fig:redundant edge} is
i-secure\xspace but not intransitively uniform.
However, for uniform policies, Theorem~\ref{theorem:polynomial unwinding characterization of intransitive uniformity and security} immediately yields efficient algorithms to verify the respective conditions via a standard dynamic programming approach:
\begin{corollary}[uniform unwinding verification]
\begin{enumerate}
\item Verifying whether a policy is intransitively uniform can be performed in nondeterministic logarithmic space.
\item For systems with intransitively uniform policies, verifying whether a system is
i-secure\xspace
can be performed in nondeterministic logarithmic space.
\end{enumerate}
\end{corollary}
The above shows that the complexity of intransitive noninterference with local policies comes from the \emph{combination} of local policies that do not allow agents to ``see'' their allowed sources of information with an intransitive security definition. In the transitive setting, this interplay does not arise, since there a system always can allow agents to ``see'' their incoming edges (see Theorem~\ref{theorem:uniformpolicies}).
\subsection{Unwinding for IP-Security\xspace}
In the setting with a global policy, i-security\xspace
is equivalent to IP-security as defined in~\cite{HY87}. For
IP-security, Rushby gave unwinding conditions that are sufficient, but
not necessary.
This left open the question whether there is an unwinding condition that \emph{exactly} characterizes IP-security, which we can now answer positively as follows. Clearly, a policy that assigns the same local policy to every state is intransitively uniform. Hence our results immediately yield a characterization of IP-security with the above unwinding conditions, and from these, an algorithm verifying IP-security in nondeterministic logarithmic space can be obtained in the straight-forward manner.
\begin{corollary}[unwinding for IP-security\xspace]
\begin{enumerate}
\item A system is IP-secure if and only if it has an intransitive unwinding satisfying \textnormal{(OC$_{\dip}^{\uniform}$)}\xspace, \textnormal{(SC$_\dip^\uniform$)}\xspace, and \textnormal{(LR$_\dip^\uniform$)}\xspace.
\item IP-security can be verified in nondeterministic logarithmic space.
\end{enumerate}
\end{corollary}
\section{Conclusion}
We have shown that noninterference with local policies is considerably different from noninterference with a global policy: an allowed interference in one state may contradict a forbidden interference in another state. Our new definitions address this issue. Our purge- and unwinding-based characterizations show that our definitions are natural, and directly lead to our complexity results.
We have studied generalizations of Rusby's IP-security~\cite{rushby92}. An interesting question is to study van der Meyden's TA-security~\cite{meyden2007} in a setting with local policies. Preliminary results indicate that such a generalization needs to use a very different approach from the one used in this paper.
\bibliographystyle{alpha}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 500
|
Include Medications and Medical Supplies: If you take medicine or use a medical treatment on a daily basis, be sure you have what you need on hand to make it on your own for at least a week and keep a copy of your prescriptions as well as dosage or treatment information. If it is not possible to have a week-long supply of medicines and supplies, keep as much as possible on hand and talk to your pharmacist or doctor about what else you should do to prepare. If you undergo routine treatments administered by a clinic or hospital or if you receive regular services such as home health care, treatment or transportation, talk to your service provider about their emergency plans. Work with them to identify back-up service providers within your area and other areas you might evacuate to.
Include Emergency Documents: Include copies of important documents in your emergency supply kits such as family records, medical records, wills, deeds, social security number, charge and bank accounts information and tax records. It is best to keep these documents in a waterproof container. If there is any information related to operating equipment or lifesaving devices that you rely on, include those in your emergency kit as well. If you have a communication disability, make sure your emergency information list notes the best way to communicate with you. Also be sure you have cash or travelers checks in your kits in case you need to purchase supplies.
Consider two kits: In one, put everything you will need to stay where you are and make it on your own. The other should be a lightweight, smaller version you can take with you if you have to get away.
keep your emergency supplies. If you use a wheelchair or other medical equipment, show friends how to use these devices so they can move you if necessary and teach them how to use any lifesaving equipment or administer medicine in case of an emergency. Practice your plan with those who have agreed to be part of your personal support network.
Some of the things you can do to prepare for the unexpected, such as assembling an emergency supply kit and making an emergency plan are the same regardless of the type of emergency. However, it's important to stay informed about what might happen and know what types of emergencies are likely to affect your region. Be prepared to adapt this information to your personal circumstances and make every effort to follow instructions received from authorities on the scene. Above all, stay calm, be patient and think before you act.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,983
|
L'Aéroport de Broye-lès-Pesmes est une base aérienne désaffectée (depuis ) situé au nord de la commune française de Broye-Aubigney-Montseugny, dans le département de la Haute-Saône en Franche-Comté.
Depuis 1959, c'était un aérodrome de dispersion de l'OTAN. Il a notamment été utilisé par les appareils de la 2e escadre de chasse de la Base aérienne 102 de Dijon.
Il héberge depuis 2005 le site d'émission du Radar GRAVES (Grand Réseau Adapté à la Veille Spatiale), système de détection de satellites évoluant en orbite terrestre basse.
Notes et références
Broye-lès-Pesmes
Broyes
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,065
|
\section{Introduction}
In recent years, there has been growing interest in electron-based dissociation (ExD) – primarily electron capture (ECD) \cite{zubarev1998electron} and electron transfer dissociation (ETD)\cite{Syka2004PeptideAndProtein} in protein mass spectrometry.
These fragmentation methods allow the cleavage of the backbone of a protein or peptide without significantly disrupting other bonds (even preserving noncovalent interactions) and as such, much effort has gone into the use of ExD methods for top-down sequencing, as well as the study of labile post-translational modifications and even binding sites of non-covalent ligands\cite{garcia2007characterization,haakansson2001electron,ayaz2009vivo,ge2009top,tsybin2011structural,fornelli2012analysis,cournoyer2005deamidation,li2010glutamine,xie2006top,jackson2009use,yin2010elucidating,goth2016gas}.
Additionally, considerable efforts have been made to determine preferential reaction pathways and cleavage sites in ExD of known precursors, to obtain insight into gas-phase protein/peptide conformation \cite{breuker2002detailed,oh2002secondary,skinner2012ubiquitin,skinner2013charge,zhang2011native,zhang2013native,zhang2014exploring,lermyte2014etd,lermyte2015electron,zhang2016native,lermyte2017conformational} as well as to investigate the reaction mechanism \cite{turecek2003n,turecek2003mechanism,chung2010backbone}.
Ideally, reaction products are not only identified, but also quantified in these efforts. Because of the information-rich nature of top-down ExD spectra, data processing is usually performed with the help of specialized software.
The first, and arguably most critical step in this data processing is usually spectral deisotopisation, i.e. reducing the multitude of signals observed in the m/z dimension due to various charge states and isotopologues to a minimal set of components and abundances.
Most of the readily available software tools for this – e.g. {\tt THRASH}\cite{horn2000automated}, {\tt MASH}\cite{guner2014mash,cai2016mash}, {\tt DeconMSn}\cite{mayampurath2008deconmsn}, {\tt Decon2LS}\cite{jaitly2009decon2ls} – utilize an averagine-scaling approach\cite{senko1995determination} to determine charge states, monoisotopic masses, and ion intensities.
As this requires resolution of the (aggregated) isotope peaks, these tools are mostly used to process FTICR or Orbitrap data, particularly as they can natively process Bruker and/or Thermo data files (in fact, a modified {\tt THRASH} algorithm, called {\tt SNAP}, is built into the Bruker DataAnalysis software).
Observed isotope clusters are often composed of multiple overlapping isotope distributions (envelopes), each generated by ions whose chemical formulas differ by one (or a few) hydrogen atoms.
These shifts (by an integer number of hydrogen masses) are commonly observed in ExD spectra and provide information on reaction pathways\cite{lermyte2017conformational,o2006long,tsybin2007ion}.
As such, it is desirable to preserve the information contained in observed isotope distributions during and after the deconvolution procedure.
Thus, there is a need for software tools which are able to process high-resolution tandem MS data from a variety of instruments, utilize the high-resolution information (e.g. properly assign highly resolved peaks) to perform thorough data analysis, and provide the user with information regarding preferred cleavage sites and relative probabilities of competing reaction pathways.
Ideally, this should not require the user to possess extensive expertise regarding statistics and/or gas-phase ion/ion chemistry.
Recently, we have demonstrated the use of an in-house developed software for deconvoluting complex isotope clusters occurring in top-down ETD spectra acquired on a Waters Synapt G2 Q-IM-TOF instrument\cite{lermyte2015understanding}.
Furthermore, we have shown how this allows us to infer branching ratios and how this correlates to collision cross-sections and gas-phase conformations of ubiquitin\cite{lermyte2017conformational}.
Here, we present in detail the above computational workflow, together with extensions that shed further light onto the electron transfer driven reactions.
The Python implementation of that workflow, called {\tt MassTodonPy}, is made publicly available for download via Python Package Index.
\noindent\textbf{Paper organization.}
In the rest of the article we describe the stages of the proposed workflow: (1) the preprocessing of the spectrum, (2) the generation of potentially observable chemical formulas, (3) the deconvolution of spectra, which involves the estimation of the intensities of the potential products of the considered set of reactions, (4) the pairing of fragment ions, resulting in estimates of the probabilities of the considered reactions and fragmentations.
The workflow was tested \textit{in silico} and on around 200 mass spectra.
Finally, we mention some possible extensions to the workflow.
\begin{table*}[t]
\centering
\begin{tabular}{rlcl}
\textbf{PTR} &\ce{[M + nH]^{n+} + A^{.-}} &\ce{->}& \ce{[M + (n-1) H]^{(n-1)+} + AH} \\
\textbf{ETnoD} &\ce{[M + nH]^{n+} + A^{.-}} &\ce{->}& \ce{[M + nH]^{(n-1)+.} + A} \\
\textbf{ETD} &\ce{[M + nH]^{n+} + A^{.-}} &\ce{->}& \ce{[C + xH]^{x+} + [Z + (n - x)H]^{(n-x-1)+.} + A}\\
\textbf{HTR} &\ce{[C + xH]^{x+}} &\ce{->}& \ce{[C + (x - 1)H]^{x+}}\\
&\ce{[Z + (n - x)H]^{(n-x-1)+}} &\ce{->}& \ce{[Z + (n - x + 1)H]^{(n-x-1)+}}
\end{tabular}
\caption{Considered chemical reactions. \ce{M} stands for either a precursor ion or a fragment ion. The HTR reaction can happen only after ETD and consists in the transfer of a hydrogen atom from the $c$ to the $z$ fragment.}\label{chemical_reactions}
\end{table*}
\section{Materials and methods}
\noindent\textbf{Data Preprocessing.}
We assume that the input spectrum was already calibrated.
The spectrum should not be centroided, as {\tt MassTodon} does its own centroiding, as described later in the peak picking section.
To attenuate the possibility of fitting to noise peaks, some parts of the mass spectrum need to be trimmed out.
We offer two simple ways to do it.
The first way focuses on the intensity of individual peaks and amounts to trimming out peaks with intensity below a user-provided threshold.
The second way retains only the heighest peaks whose joint intensity cover the user specified percentage of the total intensity in the spectrum.
To make that idea more clear, consider a spectrum comprised of three peaks with intensities equal to 1000, 990, and 10.
Also, set the joint threshold at 99\%.
The intensity of the first peak amounts to $\frac{1000}{1000+990+10} = 50\%$ of the entire intensity in the spectrum.
The intensity of the first two peaks amounts to $\frac{1000+990}{1000+990+10} = 99.5\%$ of the overal intensity. It is the smallest set of heighest peak that jointly surpass the required threshold of $99\%$ and so only these peaks are left, and the third one is trimmed out.
Observe that the same effect would be achieved if trimming out peaks with intensity above 990.
For each run of the second trimming spectrum we calculate that implicit cut-off and store it for the inspection of the user.
Finally, the mass to charge ratios are rounded to better match the theoretical spectra, as described later on.
\noindent\textbf{Generating chemical formulas.}
{\tt MassTodon} exhausetively finds the formulas of all molecular species that might be present in the set of considered reactions.
The theoretical envelopes of these molecules are then fitted to the spectral data in a later stage.
The presented workflow considers a set of known chemical reactions triggered by the electron transfer, c.f. Table~\ref{chemical_reactions}.
The Proton Transfer Reaction (PTR) and the non-dissociative Electron Transfer Dissociation (ETnoD) do not result in any fragments; they affect the charge state and the mass of the cation alone.
The Electron Transfer Dissociation (ETD), potentially followed by the transfer of a hydrogen (HTR), result in $c$ and $z$ fragments\citep{RoepstorffScheme}.
We assume that PTR and ETnoD may occur multiple times on the same ions, including the $c$ and $z$ fragments.
We assume that fragments cannot further fragment, as the inner fragments are scarcely ever observed experimentally in ETD.
The number of fragments depends on the charge of the precursor filtered during $\text{MS}_1$, denoted $Q$, its amino acid sequence and the existing modifications.
We neglect the ordering of reactions within one pathway.
Thus, the product of the PTR reaction followed by the ETnoD reaction is the same as the product of the ETnoD reaction followed by the PTR reaction.
In general, reaction pathways leading to the same product are indiscernible until the last stage of the algorithm.
Every molecular species is described by its elemental composition and charge $q$.
Each reaction consumes one charge.
During PTR, the radical passes from anion to cation reducing its charge without significantly changing its mass (we neglect the mass of the electron).
This motivates the introduction of an additional quantity, the \textit{quenched charge} $g$, that describes the number of extra hydrogen masses with respect to precursor's hydrogen content, see \citet{lermyte2015understanding}.
An increase in $g$ corresponds to an increase in one atomic mass unit and does not change the charge state.
To exemplify the above concept, consider triply charged Substance P, \ce{\text{\tt RPKPQQFFGLM}^{3+}}.
The mass of its monoisotopic isotopologue equals $1347.712$[u], when rounded to the third decimal place.
Add the mass of two protons and one quenched charge and divide it by the two present charges to get $\frac{1347.712 + 3 \times 1.008}{2} = 675.368$[Da].
Thus, the regions of the mass spectrum close to that value can contain ions belonging to that molecular species.
Consider the case of filtering only triply charged precursors during the MS1, i.e. selecting ions with m/z around $450.245$[Da].
It is then possible to state that these ions must have underwent exactly one ETnoD.
This is because ETnoD would reduce their charge by one, \ce{3^+ -> 2^+}, and increases the number of quenched charges by one, see Table~\ref{chemical_reactions}.
Further on we show how to infer the number of reactions both from precursor ions and fragments.
While studying the above example, it is important to notice that other sources of ions can explain the same peak.
In particular, consider the second most probable isotopologue of the precusor ion that underwent the PTR reaction.
One of the \ce{^{12}C} carbon atoms in this isotopologues is exchanged for a heavier isotopic variant, \ce{^{13}C}.
These ions are only slightly less likely to be found in the sample than the monoisotopic ions: on average in $29.6\%$ of cases for this ion source versus $43.1\%$ for the monoisotopic peak.
Their mass is $1348.716$[u]. When equiped with two charges, their m/z equals $\frac{1348.716 + 2 \times 1.008}{2} = 675.366$.
Most instruments would not resolve the $0.002$[Da] difference between the two molecular species.
However, confusing the two ions sources leads to a poor estimate of the relative relative extent of PTR versus ETnoD.
Based on one peak alone it is impossible to correctly identify the relative proportions of different molecular species.
However, in most cases it is possible to differentiate between various molecular species by looking at their isotopic distributions as a whole.
This opens the possibility to evaluate how much of observed intensity can be attributed to particular ion sources.
Further on we show how this can be achieved.
Observe that quenched charge may also be used to record information on a hydrogen transfered during HTR.
This is convenient, as there is nor real difference between a quenched charge and a \textit{regular} hydrogen atom within one molecule.
Consider then a precursor that undergoes a direct HTR reaction: the c fragment must then have a quenched charge equal to -1, which we consider a valid possibility.
In that case alone does this quantity assume a negative value.
During the fragmentation, the remaining charge and quenched charge (if positive) are distributed among the fragments.
One might expect the charge state of smaller fragments to be limited, due to Coulomb repulsion.
For this reason, \textsc{MassTodon} omits formulas with too many charges per a given number of amino acids, the default being set to 5.
In case of Substance P, this means that we could not observe a $c_{3}$ fragment with two charges, but we could observe a $c_5$ fragment.
The charge distance parameter can be adjusted by the user.
If one considered only the PTR and ETnoD reactions, the precursor molecule could result exactly in $\frac{Q(Q+1)}{2}$ different molecular species.
Each product can be further fragmented into pairs of different $c$ and $z$ fragments.
The number of such pairs is $K$ - the number of amino acids in the provided sequence, minus the number of prolines, that cannot be fragmented easily by electron transfer due to their ring structure.
Then, each fragment can again undergo several PTR and ETnoD reactions.
The number of all fragments is thus of the order of $\mathcal{O}(K Q^4)$.
\begin{figure*}[t]
\begin{subfigure}[b]{0.7\textwidth}
\centering
\includegraphics[width=\linewidth]{images/deconvolution_graph6}
\caption{$\mathcal{C}$, a connected component of $\mathcal{G}$}\label{fig::deconvolution_principles}
\end{subfigure}
\begin{subfigure}[b]{0.29\textwidth}
\begin{subfigure}[b]{\textwidth}
\includegraphics[width=\textwidth]{images/cc}
\caption{Graph form of $\mathcal{C}$}
\label{fig::graph form of G}
\end{subfigure}
~
\begin{subfigure}[b]{\textwidth}
\centering
\includegraphics[width=.6\textwidth]{images/one_envelope_little_support}
\caption{A molecule with with scarse experimental support}
\label{fig::poor support}
\end{subfigure}
\end{subfigure}
\caption{A connected component $\mathcal{C}$ of the \textit{deconvolution graph} $\mathcal{G}$.
Experimental peaks are shown in pink.
Among the nodes of $\mathcal{G}$ we find the molecules $M$, their isotopologues $I$, and experimental groups $G$.
The probability $p$ of meeting $I$ among the $M$ ions decorates the edge between $I$ and $M$.
Edges between $I$ and $G$ are not plotted for clarity in (a); we do mark however their corresponding flow variables, $x$.
They denote the amount of experimental intensity attributed to a given isotopologue.
The aim of the deconvolution is to establish total intensities of $\text{M}_1$ and $\text{M}_2$, denoted respectively as $\alpha_A$ and $\alpha_B$. In (b) we show $\mathcal{C}$ as a graph.
The experimental peaks (in pink) are depicted only for clarity of the representation and are not actually in $\mathcal{G}$.
In (c) we show a molecule $M$ with scarse experimental support.}
\end{figure*}
\noindent\textbf{Generating the isotopic distributions.}
The isotopic distribution of a given molecular species models the expected signal one could register in the mass spectrometer.
In terms of the presented workflow, it offers a way to relate the intensities of peaks in the mass spectrum with different m/z ratios, whenever they follow a predicted pattern.
Each reaction product is described by its elemental composition, charge $q$, and quenched charge $g$.
This information is sufficient to generate the theoretic isotopic distibution using any isotopic calculator.
To perform calculations here, we use the \textsc{IsoSpec} algorithm \cite{lacki2017isospec}.
Given the elemental composition, \textsc{IsoSpec} produces a series of infinitely resolved isotopologues, representable as tuples (mass, probability).
To avoid the combinatorial explosion in their number\cite{Valkenborg2012Isotopic}, \textsc{IsoSpec} reports only the smallest possible set of peaks, such that their cumulative probability does not fall under some user specified threshold, e.g. $99.9\%$.
The masses of the envelopes are adjusted according to formula $\frac{m+q+g}{q}$ to obtain valid mass over charge ratios.
To model low resolution spectra, one does not need infinitely resolved theoretical envelopes.
Whenever small differences between the m/z ratios cannot be discerned, one can safely aggregate peaks with similar m/z ratios.
In our workflow, we ask the user to provide a measure of the instrument's resolution in terms of one parameter alone -- the peak's m/z tolerance $tol$.
Experimental peaks are deemed to potentially originate from a molecule $M$ if their m/z ratios are within the $tol$ distance from a theoretical isotopologue $I$ of that molecule.
This is shown in Figure~\ref{fig::deconvolution_principles}.
By default, we assume that differences between m/z ratios an order of magnitude smaller than $tol$ cannot be discerned.
This implies a finite granularity of the spectrum: if $tol$ amounted to 0.05 [u], then the smallest difference between peaks would be that of 0.001 [u].
To obtain such spectrum, peaks with the same first three significant digits are aggregated, i.e. they are represented by one peak with the same rounded m/z and intensity equal to the total intensity of these peaks.
In general, given tolerance $tol$, we round the spectrum to the significant digit given by $\lceil - log_{10}(tol)\rceil$ and then aggregate it.
By convention, we still call the so obtained cluster of isotopologues an isotopologue.
The same operation is performed on the experimental m/z ratios.
This step reduces the size of the deconvolution problem and speeds up the peak picking and the deconvolution.
That said, one should not provide an underestimate of $tol$ to speed up computations.
This is because highly resolved spectra offer the possibility to discern between isotopologues of different molecular species and to better identify their joint intensities.
\noindent\textbf{Peak picking.}
The aim of the peak picking is to assign peaks in the mass spectrum to the potential molecular species.
This is done by comparing the m/z ratios of the experimental peaks with these of the peaks in the theoretical isotopic envelopes, as described in the previous section and visualized in Figure~\ref{fig::deconvolution_principles}.
Figure~\ref{fig::deconvolution_principles} also shows that finding potential explanations for a given experimental peaks corresponds to finding all intervals of the form $[\frac{m}{z} - tol, \frac{m}{z} + tol]$ to which its m/z value belongs.
To find these intervals effectively, we make use of the interval trees data structure \citep{cormen2009leiserson}.
Different intervals might overlap, as is the case for isotopologues $I_{A1}$ and $I_{B0}$ in Figure~\ref{fig::deconvolution_principles}.
The intersections of these intervals partition the m/z axis into regions that can be traced back to originate from different sets of molecules and regions that cannot be explained by any of the products of the considered reactions.
Experimental peaks inside such intersections (there might be more then one) form experimental groupings $G$.
The total intensity within one such groupings is stored and denoted by $G_\text{\tt intensity}$.
After these operations, the experimental peaks do not play any more role in calculations and can be deleted.
Considered together, molecules $M$, their isotopologues $I$, and the experimental groupings $G$ form nodes of the \textit{deconvolution graph}, $\mathcal{G}$, as shown in Figures~\ref{fig::deconvolution_principles} and \ref{fig::graph form of G}.
In $\mathcal{G}$, molecule nodes $M$ are naturally joined with their isotopologue nodes $I$, that are in their turn joined with experimental groupings $G$ they could explain.
Graph $\mathcal{G}$ is usually composed of several connected components, like the one presented in Figure~\ref{fig::deconvolution_principles}.
While picking the peaks, one can easily spot molecules $M$ with poor experimental support, as shown in Figure~\ref{fig::poor support}.
More precisely, if the sum of probabilities of isotopologues of $M$ connected to some $G$ does not exceed some percentual threshold $P$ (by default, $70\%$), then we can discard it.
This additional preprocessing eliminates substances that alone could not explain more than the $P$ percent of the total experimental intensity within the considered subproblem, and thus makes part of the overall variable selection procedure we consider.
Each connected component of $\mathcal{G}$ gives rise to some deconvolution problem, as several molecules might compete for the explanation of the given range of the mass spectrum.
These problems might be solved independently and simultaneously rather than sequentially.
{\tt MassTodonPy} offers both ways of performing these calculations.
\noindent\textbf{Deconvolution.}
The problem of deconvolving the intensities within one connected component of graph $\mathcal{G}$ is reminiscent of linear regression.
Indeed, the goal is to express the observed signal as a weighted sum of the isotopic envelopes.
Denoted the weight by $\alpha$, as in Figure~\ref{fig::deconvolution_principles}.
It can be interpreted as the total intensity of a given molecular species in the mass spectrum.
In particular, $\alpha$ cannot be negative.
In advance, one does not know how to redistribute the intensity of $I$ among the neighboring experimental groupings $G$.
This motivates the introduction of the \textit{flows} between $G$ and $I$, denoted by $x_G^I$,
For instance, in Figure~\ref{fig::deconvolution_principles} isotopologue $I_{B0}$ is linked with experimental intentensities $G_2$ and $G_3$.
It absorbs $x_{G_2}^{I_{B0}}$ of the intensity of $G_2$, and $x_{G_3}^{I_{B0}}$ of the intensity of $G_3$.
$I_{B0}$ should contribute $x_{G_2}^{I_{B0}} + x_{G_3}^{I_{B0}}$ to $M_B$.
On the other hand, this should be equal to a fraction $p_{B_0}$ of the total intensity of $M_B$, denoted by $\alpha_B$.
In other words, $p_{B_0} \alpha_B = x_{G_2}^{I_{B0}} + x_{G_3}^{I_{B0}}$.
Similarly, in general the intensities of isotopologues $I$ and molecules $M$ are related via a set of linear restrictions $\alpha_M p_M^I = \sum_{G: G \leftrightarrow I} x^I_G$, where under the sum we iterate over all experimental groups $G$ that neighbor isotopologue $I$.
It is sensible to choose molecular intensities $\alpha$ and isotopologue intensities $x$ to assure a minimial divergence between the observed group intensities $G_\text{\tt intensity}$ and the total outflows of intensity from these nodes towards the isotopologue nodes.
The overall deconvolution problem can thus be formalized as
\begin{align*}
\min_{x,\alpha} \sum_{G} \big(G_\text{intensity} - \sum_{I: G\leftrightarrow I} x^I_G \big)^2\quad\text{so that}\\
\alpha_M p_M^I = \sum_{G: G \leftrightarrow I} x^I_G, \quad x^I_G \geq 0
\end{align*}
To minimize the risk of numerical instability and perform model selection one can include in the cost function additional penalty terms\cite{james2013introduction},
{\footnotesize\begin{align*}
L_1^x \sum_{G\leftrightarrow I} x^I_G + L_1^\alpha \sum_M \alpha_M + L_2^x \sum_{G\leftrightarrow I} (x^I_G)^2 + L_2^\alpha \sum_M \alpha_M^2.
\end{align*}}
By default, we set $L_1^\alpha, L_2^\alpha, L_1^x,$ and $L_2^x$ to $0.001$.
The penalty terms after $L_1^\alpha$ and $L_1^x$ should round small estimates to zero, as in the lasso model selection approach\cite{james2013introduction}.
The above problem can be efficiently solved with quadratic programming.
MassTodon relies on the {\tt CVXOPT} Python module\cite{andersen2013cvxopt} that
solves quadratic programmes with a path following algorithm.
After each deconvolution, we calculate and report various error statistics.
These include the sum of the absolute values of the errors, the sum of overestimated values, and the sum of the underestimated values.
The above quantities are also divided by the total ion current or the total intensity within the tolerance regions of any of the theoretically molecular species.
The cost function is minimized simultaneously in $x$s and $\alpha$s.
Only $\alpha$s are analyzed in the next, final stage of the algorithm.
\noindent\textbf{Pairing of the observed ions.}
Up to this step, the algorithm obtained estimates of intensities of each considered product molecule, uniquely defined by its type (precursor, $c$ or $z$ fragment), charge $q$, quenched charge $g$.
Previously\cite{lermyte2017conformational}, we described a method for the retrieval of information on the branching ratios, i.e. the probabilities of ETnoD and PTR, entirely based upon estimates of the intensities of the non-fragmented ions.
Given a non-fragmented molecular species with charge $q$ and quenched charge $g$, one can retrieve the numbers of the PTR and ETnoD reactions by solving
\begin{align}
q &= Q - N_\text{PTR} - N_\text{ETnoD},\label{eq::q} \\
g &= N_\text{ETnoD},\label{eq::g}
\end{align}
for $N_\text{PTR}$ and $N_\text{ETnoD}$.
Eq.~\eqref{eq::q} states that each reaction reduces the observed charge by one.
Eq.~\eqref{eq::g} traces the origin of all quenched charges on the precursor molecules solely to the ETnoD reaction.
The estimate of the probability of ETnoD then equals
\begin{equation*}
\hat{p} = \frac{ \sum_i N^i_\text{ETnoD}I_i }{ \sum_i (N^i_\text{ETnoD} + N^i_\text{PTR}) I_i }.
\end{equation*}
The index $i$ iterates over different observed precursors.
The nominator counts ions that underwent ETnoD, $I_i$ is the estimated intensity of the precursor with charges $(q_i, g_i)$.
The denominator additionally contains the count of ions undergoing PTR.
The above estimator relies on the presumed proportionality of the signal intensity to the actual number of molecules.
With the above method one cannot retrieve the probabilities of fragmentations.
This is because counts of reactions are not directly accessible and only estimates of the overall intensity of $c$ and $z$ fragments are at hand.
To unveil the number of fragmentation events, one has to pair back the matching $c$ and $z$ fragments.
There exists a whole range of possible pairing strategies.
The two extremes are: (1) ions come from entirely separate groups of precursors, and (2) the observed fragments are generated by a minimal number of precursors.
For instance, in Figure~\ref{fig::pairing problem} we show a situation where 5 $c$ and 3 $z$ matching fragments were observed (filled circles).
In principle, one could say, that at the beginning of the experiment there were together 8 precursor molecules, but after the fragmentation one of each fragments always lost all of its charge.
This is the \textit{lavish} interpretation, as shown in Figure~\ref{fig::lavish pairing}.
If the ions could not be observed only because of loosing the entire charge, then this scenario would require a lot of reaction events to explain the outcome.
Figure~\ref{fig::parsimonious pairing} depicts the other possibility.
Here, a maximal pairing is performed, and only two $c$ fragments have to be paired with $z$ fragments with a depleted charge (dashed circles).
This approach is much more \textit{parsimonious} in terms of reactions needed to explain the experimental results.
\begin{figure}[t]
\begin{subfigure}[b]{0.55\linewidth}
\includegraphics[width=\textwidth]{images/pairing_lavish}
\caption{Lavish}
\label{fig::lavish pairing}
\end{subfigure}
\begin{subfigure}[b]{0.4\linewidth}
\centering
\includegraphics[width=.7\textwidth]{images/pairing_parsimonious}
\caption{Parsimonious}
\label{fig::parsimonious pairing}
\end{subfigure}
\caption{Two interpretations of observing 5 $c$ and 3 $z$ matching fragments: lavish (a) and parsimonious (b). Nodes with dashed edges symbolize cations that never reach the detector. (a) maximizes the number of missing cations needed to explain the spectrum, while (b) minimizes that number.
}\label{fig::pairing problem}
\end{figure}
Irrespectful of the above strategies, only matching ions should be paired, i.e. a $c_k$ fragment should be matched only with a $z_{K-k}$ fragment, where $K$ is the total number of amino acids in a given sequence.
Moreover, pairing should include natural restrictions on the charge states $(q_c,q_z)$ and quenched charges $(g_c,g_z)$ of both fragments.
The \textit{basic algorithm} we propose to solve the pairing problem disregards quenched charges $g_c$ and $g_z$: intensities of fragments $c_k$, $z_{K-k}$ fragments with appropriate charge are summed.
Also, we entirely neglect the presence of HTR in the whole analysis, as it renders the whole procedure too complex: all fragments that could have been taken either for ETD or HTR products are considered to be purely ETD products.
We then construct the \textit{pairing graph}, see Figure~\ref{fig::pairing graph}.
The nodes correspond to different observed molecular species and store information on their total estimated intensity.
Special dummy nodes are added to denote the matching cofragments that had lost all their charge.
Our approach assumes that the only way ions can end up being undetected is solely through the total loss of charge.
Edges are drawn between $c$ and $z$ nodes with complimentary sequences if their total charge plus one (the fragmentation producing fragments takes away one charged) does not exceed that of the precursor chosen in MS1 stage of the experiment, $q_c + q_z + 1 \leq Q$.
\begin{figure}[t]
\includegraphics[width=.9\linewidth]{images/table_algos}
\caption{Summary of the proposed pairing algorithms.}\label{fig::algorithms}
\end{figure}
\begin{figure*}[t]
\begin{subfigure}[b]{0.49\linewidth}
\centering
\includegraphics[width=.7\linewidth]{images/pairing_graph}
\caption{Pairing Graph}
\label{fig::pairing graph}
\end{subfigure}
\begin{subfigure}[b]{0.49\linewidth}
\centering
\includegraphics[width=\linewidth]{images/max_flow_problem}
\caption{Max Flow Problem}
\label{fig::max flow problem}
\end{subfigure}
\caption{ A \textit{pairing graph} (a) and it rrepresentation as a \textit{max flow} optimization problem (b). Nodes with dashed edges correspond to ions that lost their charge; other nodes correspond to observed fragments. In (a), ion charge are shown as red plus signs. Gray edges mark possible pairings. Red dashed line between $c_3$ and $z_2$ marks an impossible pairing: if combined, both fragments must have originated from a 6+ precursor, which was not possible. The task is to redistribute the intensity in nodes along the edges. This comes at a cost $Q - q_c - q_z$. To turn (a) into (b), one has to: (1) remove unobserved ion nodes (2) direct remaining edges from $c$ to $z$ (3) add sink S and terminal T (4) add edges directed from S to $c$ nodes and from $z$ nodes to T and add capacities equal to observed ion intensities (5) add edges from S to $z$ fragments and edges from $c$ fragments to T: these correspond to pairings with unobserved ions. This representation is possible for \textit{basic} and \textit{intermediate} pairing algorithms.
}\label{fig::pairing to max flow}
\end{figure*}
The pairing of fragments correspond to the redistribution of the estimated intensities $I$ in the nodes along the edges of the pairing graph.
Assigning intensity to an edge diminishes the intensities in both end nodes by the same amount.
All intensity must be assigned to some edges.
Assigning intensity comes at a cost reflecting the number of reactions the pair of ions underwent during the whole experiment.
In the basic approach, fragments with charges $(q_c, q_z)$ together underwent $Q - 1 - q_c - q_z$ reactions.
The optimization task we are about to set up lets us forget the extra fragmentation count, fixing these costs at $Q - q_c - q_z$, equal to the total number of ETnoD and PTR reactions that both fragments underwent, $N_\text{ETnoD}^{cz} + N_\text{PTR}^{cz}$.
Note that this equation holds also for pairings involving cofragments that entirely disappeared due to the loss of all charge.
The pairing problem turns into an optimization problem where one wants to minimize the total number of reactions that could have produced the observed $c$ and $z$ fragments.
More specifically, we face a constrained linear optimization task:
\begin{align}
\min_{ I_{cz}:\,c \in \mathcal{A}_C,\,z \in \mathcal{A}_Z }
\sum_{ \mbox{\tiny$\begin{matrix}
c \in \mathcal{A}_C \\
z \in \mathcal{A}_Z
\end{matrix}$} } (N^{cz}_\text{ETnoD} + N^{cz}_\text{PTR}) I_{cz}\label{eq::cost of pairing} \\
\forall_{c\in\mathcal{O}_C}\,\, I_c = \sum_{z\in\mathcal{A_Z}} I_{cz},\,\,\forall_{z\in\mathcal{O}_Z}\,\, I_z = \sum_{c\in \mathcal{A_C}} I_{cz}. \label{eq::equality constraints}
\end{align}
Above, $\mathcal{O}_C$ and $\mathcal{O}_Z$ denote sets of observed $c$ and $z$ nodes, and $\mathcal{A}_C$ and $\mathcal{A}_Z$ additionally contain the unobserved cofragments.
The above simplifies to a \textit{max flow} problem: subtract flows between observed fragments from both sides of equalities in \eqref{eq::equality constraints} and what results are the expressions for flows between observed fragments and their unobserved cofragments.
Plugging these into Eq.~\eqref{eq::cost of pairing} and some simple algebra results in
\begin{align*}
\max_{ I_{cz}:\,c \in \mathcal{A}_C,\,z \in \mathcal{A}_Z }
\sum_{\mbox{\tiny$\begin{matrix}
c \in \mathcal{O}_C \\
z \in \mathcal{O}_Z
\end{matrix}$}} I_{cz} \quad\text{s.t.}\\
\forall_{c\in\mathcal{O}_C} I_c \geq \sum_{z\in\mathcal{O}_Z} I_{cz},\quad \forall_{z\in\mathcal{O}_Z} I_z \geq \sum_{c\in\mathcal{O}_C} I_{cz}.
\end{align*}
Of course, all flows $I_{cz}$ are non-negative.
To solve the max flow problem we use the Edmonds-Karp algorithm\cite{edmonds1972theoretical} as implemented in the {\tt NetworkX} Python module\cite{hagberg-2008-exploring}.
The solution to the above problem provides us with estimates of the total intensities of ions undergoing a specific type of fragmentation.
In particular, this lets us estimate the probabilities of fragmentation along the protein.
It also lets us estimate the probability with which the precursor will fragment.
However, this setting does not offer any possibility to estimate the number of ETnoD and PTR reactions from fragments.
These might become important in case of experiments where bigger and more charged substances are studied, or when much of the precursor ions reacted away, mostly through fragmentation.
To provide a solution to the above problems, we have developed another algorithm -- the \textit{intermediate} approach.
In this approach we do not aggregate the intensities of observed ions with different quenched charges.
As a result, the \textit{pairing graph} contains more nodes, both observed and dummy ones.
Have we followed the previous approach, then each observed fragment could match several unobserved cofragments, all amounting to the same overall number of reactions but differing in specific numbers of ETnoD and PTR among them.
Unfortunately, the existence of many unobservable cofragments would prevent us from reducing the problem to a \textit{max flow} optimization, making it impossible to derive equations for all flows between observed and unobserved fragments.
To solve this problem, we reduce the number of potential dummy nodes and combine them together.
The edges between existing fragments now convey information necessary to tell how many PTR and ETnoD reactions happened on both fragments throughout their history, including the period before any fragmentation ooccurred.
Similarly to equations \eqref{eq::q} and \eqref{eq::g}, the numbers of PTR and ETnoD reactions on a given pair of fragments characterized by charges $(q_c, q_z)$ and quenched charges $(g_c, g_z)$ follow equations
\begin{align}
N_\text{PTR} &= Q - 1 - q_c - q_z - g_c - g_z\notag\\
N_\text{ETnoD} &= q_c + q_z.\notag
\end{align}
Note that due to aggregation, the same cannot be said about edges between the observed and unobserved ions.
Otherwise said, if a mass spectrum does not contain \textit{pairable} fragments, then the only source of information on the numbers of ETnoD and PTR reactions can be obtained solely from the precursor products.
Finally, we investigated a third solution to the \textit{pairing problem}, the \textit{advanced} approach.
It includes the introduction of additional penalty terms to the cost function,
\begin{equation*}
\lambda_1 \sum_{
\mbox{\tiny$\begin{matrix}
c \in \mathcal{A}_C \\
z \in \mathcal{A}_Z
\end{matrix}$}} I_{cz} +
\lambda_2 \sum_{
\mbox{\tiny$\begin{matrix}
c \in \mathcal{A}_C \\
z \in \mathcal{A}_Z
\end{matrix}$}} I_{cz}^2.
\end{equation*}
Above, $\lambda_1$ corresponds to a lasso-type penalty and $\lambda_2$ - a ridge penalty.
This approach was investigated mainly for its ability to automatically round the estimates of small flows to zero.
The above problem cannot be cast into the \textit{max flow} setting because of the quadratic terms in the cost function.
For this reason, we use yet again the general purpose {\tt CVXOPT} solver.
\begin{algorithm}\scriptsize
\caption{\textit{In silico} spectra generator}\label{alg::simulation}
\begin{algorithmic}
\State
\INPUT
\State A list $\mathcal{I}$ comprising $N$ precursor ions with a given charge $Q$ and sequence $F$.
\State Probabilities of reactions $p_\text{PTR}, p_\text{ETnoD}, p_\text{ETD}$.
\State Overall intensity $I$ of the process.
\State Standard deviation of mass inaccuracy $\sigma$.
\OUTPUT
\State A mass spectrum.
\State
\State Draw the placements of charges $q$ along the fasta sequence.
\State Set experiment time to zero, $T = 0$.
\While{ T < 1 }
\State Increase T by a random time interval sampled from
\State \quad the exponential distribution with intensity $I\sum_i N_i q_i^2$.
\State Extract ion $M$ from $\mathcal{I}$ with probability prop. to $N_i q_i^2$.
\State Draw R from PTR, ETnoD, and PTR,
\State \quad with probabilities $p_\text{PTR}, p_\text{ETnoD}, p_\text{ETD}$.
\If{ R = ETD }
\If{ fragmentation occurred twice }
\State Discard ion M.
\Else
\State Draw the fragmentation spot.
\State Add fragments with $q>0$ to $\mathcal{I}$.
\EndIf
\Else
\State Reduce charge by one.
\State Adjust the quenched charge.
\State Add $M$ to list $\mathcal{I}$.
\EndIf
\EndWhile
\ForAll{ $M$ in $\mathcal{I}$ }
\State Randomly choose the isotopic variant of $M$.
\State Blur its mass with gaussian noise.
\EndFor
\State bin the spectrum
\end{algorithmic}
\end{algorithm}
\section{Results and Discussion}
\noindent\textbf{In Silico results.} In order to test the entire workflow, we conducted \textit{in silico} experiments.
A chemical process was simulated using a tailored Gillespie algorithm\cite{gillespie1977exact}, as described in Algorithm~\ref{alg::simulation}.
Briefly, the process generates a random series of three chemical reactions (PTR, ETnoD, and ETD; HTR is neglected) occurring in particular moments of time. The length of time intervals between reaction events is random and depends upon the number of charged ions at particular charge state, following \citet{McLuckey1999-su}.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{images/deconvolution_errors}
\caption{Error rates of the deconvolution procedure on \textit{in silico} data for different numbers of initial precursor ions (N = 1~000, 10~000, 100~000) and under different amounts of mass inaccuracy $sigma$ (on $x$ axis). The tolerance interval in \textsc{MassTodon} was set to $0.05$ [Da]. To measure error we sum the absolute differences of peak heights and normalize the result to the number of the precursor ions (the result does not need to sum to 100\%).
}
\label{fig::in silico errors}
\end{figure}
{\tt MassTodon} was tested in various conditions: we checked all the combinations of settings of different initial numbers of precursors, $N = 1000, 10000$, or $100000$ ions, initial precursor charges $Q = 3, 6, 9,$ and $12$, three levels of the standard deviation of mass accuracy $\sigma$, and $12$ different sets of probabilities of reactions.
Strongest correlation with deconvolution error was noticed for spectra with low ion content and large mass inaccuracy, see Figure \ref{fig::in silico errors}.
The algorithm works best when there is enough ions to form a well sampled isotopic distribution (in case of our simulations -- 100~000 ions).
In case of high-resolution mass data, when thousands of isotopologue peaks are present in the mass spectrum, it might be thus advisable to provide MassTodon with a spectrum binning results of several runs of the instrument.
It is also vital not to underestimate the size of the tolerance interval.
Of course, the above remarks are intrinsic to any peak assigning procedure that uses peak intensities, rather than relying solely on their mass over charge ratios.
While running simulation descibed by Algorithm~\ref{alg::simulation} we store the numbers of each molecule $M$ drawn in the process. We have compared these numbers with the estimates of MassTodon to check the quality of the applied deconvolution procedures.
Figure~\ref{fig::in silico errors} reports the obtained error rates.
Interestingly, the number of ions in the sample proves is of limited importance if one is interested in the estimation of the probabilities of ETnoD and PTR reactions, as shown in Figure~\ref{fig::signed prob error estimate}.
We note, that the parsimonious approach we have taken on average only slightly overestimates values of the true parameters, showing a preference towards the PTR reaction.
Note also, that the \textit{basic} approach to the pairing problem seems to offer estimates with the smallest variance.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{images/signed_prob_error_plot}
\caption{The distribution of distance between the estimates $(\hat{p}_\text{ETnoD}, \hat{p}_\text{PTR})$ and the true values $(p_\text{ETnoD}, p_\text{PTR})$ for different approaches we take, measured by the euclidean distance normalized to the maximal distance $\sqrt{2}$. Estimates in the blue regions favor PTR, while those in the yellow - ETnoD. The distributions are conditional on the number of initial precursor ions (N = 1~000, 10~000, 100~000) and different level of mass inaccuracy $sigma$ (on $x$ axis).}
\label{fig::signed prob error estimate}
\end{figure}
\noindent\textbf{Experimental results.}
Mass spectra have been acquired for purified Substance P and ubiquitin as described in detail in the previous publications\cite{lermyte2015understanding,lermyte2015characterization}.
The outcomes of \textsc{MassTodon} can be used to compare more easily mass spectra gathered under different instrumental settings.
Figures~\ref{fig::substance P wall} and \ref{fig::ubi wall} explore the differences and similarities of the information conveyed in different mass spectra, including their percentual content of products of all studied reactions, the probabilities of fragmentation, and intensities and probabilities of the ETnoD and PTR reactions.
\textsc{MassTodon} provides point estimates of the above parameters.
Given that the analysis of one spectrum is reasonably fast (check Figure~\ref{fig::runtime}) we decided to rely on bootstrap procedures\cite{efron1994introduction,wasserman2013all} to estimate the standard deviations of the above parameters.
In particular, each mass spectrum was randomly reshuffled multiple times.
We assume that each bootstrap spectrum to be composed out of $N$ ions.
The m/z ratios of these ions were then independently drawn among the original ratios, with probabilities equal to the heights of the corresponding peaks, normalized to the total ion current.
The number of observed molecules in the spectrum $N$ is not truly known in advance.
In our simulations, we assumed that the whole spectrum consist of around 100~000 molecules.
We draw 250 random spectra for each real one and run \textsc{MassTodon} on each one of them.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{images/runtime_plot}
\caption{{\tt MassTodonPy} runtime distribution. The analysis contains all the stages of the algorithm, including running all three \textit{pairing algorithms}. The 3+ precursors correspond to substance P spectra; other results are obtained for ubiquitin. Usually, it takes more time to process a spectrum randomly reshuffled by bootstrap than the original version. Runtimes were obtained using the sequential version of the algorithm, which solves the \textit{deconvolution problems} one after another. It is possible to reduce this time for larger problems using the multiprocessing option.}
\label{fig::runtime}
\end{figure}
Figure \ref{fig::fitting errors on subP} shows the overall fitting quality in case of the substance P spectra. On average, the products of the considered reactions cannot explain on average between 30\% to 40\% of the mass spectrum.
Shifting our attention only to those regions of the mass spectrum fall within the range of any potential product, the error estimates drops in a range between 10 to 20\%.
Note that for spectra gather at wave height fixed at 150 and wave velocity between 700 to 1500 the errors grow significantly.
Figure~\ref{fig::fragmentations} presents the estimates of probabilities of fragmentation for substance P. Interestingly, the probabilities are almost constant across different experimental settings.
They are also almost uniformly distributed along the possible fragmentation sites (proline not being one of them).
This is what would be expected of a small molecule, like substance P, with a trivial tertiary structure.
Again, significant departures from this pattern emerge in the same region of wave velocity.
Figure~\ref{fig::etnod ptr intensities} seems to shed some light on the nature of these anomalies.
It presents the estimates of the intensity of ions that underwent ETnoD and PTR, which is a proxy for the number of these events to happen on the molecules of substance P within the sample.
In particular, it can be noted that the range of wave velocity between 700 to 1500 contains a particularly small amount of ions that could have been prescribed to ETnoD or PTR.
By comparison, all estimates where these intensities were above 40~000 show a much smaller amount of variance.
Note also, that Figure~\ref{fig::etnod ptr intensities} suggests that the relative ratios of ETnoD and PTR remain stable under most experimental settings, with the exception of small wave velocities.
These ratios can be interpreted as relative probabilities of the ETnoD and PTR reactions, conditional on one of the reaction happening.
Interestingly, a similar pattern reemerges in case of mass spectra of ubiquitin, as shown in Figure~\ref{fig::ubi wall}.
In case of the mass spectra where the filtered precursor molecule was bearing 6 charges, the ETnoD vastly dominates over PTR.
In one of our previous papers\cite{lermyte2015understanding} we show, that this might be related to the insufficient capability of only 6 protons to induce enough denaturation of the protein within the instrument.
In other words, the fragmentation cannot happen because the two fragments wrap around each other, giving rise to a higher percentage of the ETnoD products.
\begin{figure*}[t]
\begin{subfigure}[b]{\linewidth}
\begin{subfigure}[b]{0.33\linewidth}
\centering
\caption{Mismatch and Fitting Errors}\label{fig::fitting errors on subP}\label{fig::fit errors}
\includegraphics[width=\textwidth]{images/fit_error}
\end{subfigure}
\begin{subfigure}[b]{0.66\linewidth}
\centering
\caption{Probabilities of Fragmentation}\label{fig::fragmentations}
\includegraphics[width=\textwidth]{images/Fragmentation_plot}
\end{subfigure}
\end{subfigure}
\begin{subfigure}[b]{\linewidth}
\includegraphics[width=\textwidth]{images/substance_P_intensities_of_etnod_ptr}
\caption{Intensities of ETnoD and PTR}\label{fig::etnod ptr intensities}
\end{subfigure}
\caption{
Selected results of the \textsc{MassTodon} as run on substance P spectra. The instrumental settings were obtained for two two strips of settings in the two dimensional space comprising wave height and velocity. Results in (a) and (b) show bootstrap estimates (250 repetitions). Results in (c) contain additionally lines linking together the estimates obtained for the actual mass spectra. Figure (a) shows estimates of the mismatch error and the fitting error. Both are calculated using the normalized $l_1$ distance, $E(p,q) = \frac{ \sum_k |p_k - q_k| }{ \sum_k p_k + \sum_k q_k }$, where $p$ and $q$ are maps with keys $k$ (different m/z ranges) and values $p_k$ and $q_k$ (i.e. real intensities and their estimates). In case of the mismatch error, we compare in this way the estimated spectrum versus whole experimental mass spectrum, which includes peaks that are not among the studied reaction products. The fit error restricts this comparison to the regions of the mass spectrum that actually could be explained by some theoretical product of some reaction. Figure (b) shows estimates of the probabilities of fragmentation along the backbone of substance P, whose amino sequence is RPKPQQFFGLM. Fragmentation on prolines (P) is deemed highly unlikely due to the ring structure of this amino acid. The vertical orange dashed lines correspond to probability equal to $1/9$, which would be attained assuming a fully uniform probability of fragmentation. Figure (c) shows the estimates of the intensity of the ETnoD and PTR reactions. Values of intensities in the $y$ axis have been transformed by a square root scaling in order to expose the behaviour of the lower estimates.
}\label{fig::substance P wall}
\end{figure*}
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{images/ubi_plot}
\caption{Estimates of the probabilities of ETnoD and PTR conditional on one of these events happening. Red dots correspond to estimates performed on real data. The black box plots, mostly extremely narrow, correspond to 250 sample bootstrap estimates. Precursor charge $Q$ is shown in top-left parts of the panels. Each panel is subdivided into subpanels corresponding to different experimental settings. \textit{Attention:} left panels correspond to different levels of preactivations. For $Q=9$ the energy of preactivation was set to 15, while for $Q = 6$ to 20. The $x$ axis shows the retention time RT, while the $y$ axis shows the percentual content of the ETnoD and PTR reactions.
For the spectrum gathered at $Q=9$ and RT = 20, without pre-activation and without the supplementary activation, there were no ions found that could undergo ETnoD or PTR in the real spectrum under the given threshold on the intensity (results contain the $95\%$ of the highest peak in that spectrum), so the red dot is missing.
During the bootstrap procedure, small percentages of peaks apparently corresponding to PTR or ETnoD products appeared above that threshold, leading to the
}\label{fig::ubi wall}
\end{figure*}
\section{Conclusion}
As high-performance mass spectrometers and the use of ExD methods become more prevalent, there will be an increasing demand for software methods to assist in processing the resulting, considerable amounts of data.
Here, we have presented a user-friendly software package to analyze high-resolution ETD data, deconvolute isotope distributions, and infer information about various competing reaction pathways occurring under ETD conditions.
Future work will focus on casting the entire framework into a Bayesian setting, in order to provide the user with better understanding of the uncertainties of the estimates and potential correlations of results.
In particular, the user might be interested to what degree some parts of the mass spectrum could be alternatively explained by other substances.
Obtaining such information could be done by looking at the joint distribution of the counts of molecules that compete for the explanation of a given part of the spectrum.
Moreover, it would be interesting to free the user from the need to specify the tolerance parameter.
This should be obtained automatically and potentially vary in different ranges of the m/z half line.
The implementation of the {\tt MassTodon} algorithm is freely available for downloads from the Python Package Index.
Installation instructions and documentation can be found at \href{http://masstodonpy.readthedocs.io}{\tt readthedocs}.
Source code is available for download from \href{https://matteolacki.github.io/MassTodonPy/}{\tt github}.
The software is distributed under the terms of the GNU GPL V3 public license.
\noindent\textbf{Acknowledgements.}
We would like to thank Michał Aleksander Ciach for his help in implementing the \textit{in silico} simulator.
Finally, we would like to thank dr Piotr Dittwald for his continuous support.
This work is supported by Polish NCN grants 2014/12/W/ST5/00592, 2015/17/N/ST6/03565 and partially by the Flemish SBO grant InSPECtor, 120025, IWT.
{\scriptsize
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,402
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.